Papers

We present and analyze an iterative refinement (IR) framework for improving the precision dependence of algorithms that combine a quantum linear system algorithm (QLSA) with quantum state tomography. Existing QLSAs achieve polylogarithmic dependence on the inverse error tolerance to prepare a quantum state encoding the solution, but extracting a classical description of such a state via tomography typically introduces a polynomial dependence on the target precision. To retain polylogarithmic dependence in the inverse error tolerance throughout the entire process, we develop an IR scheme that solves a sequence of related linear systems to progressively refine the solution while requiring only fixed-precision quantum subroutines, finally obtaining a classical description of a high-precision solution. We analyze the proposed framework under three input models: quantum-read/classical-write RAM (QRAM), linear combinations of unitaries (LCU), and sparse-access oracles. We evaluate the proposed scheme through numerical experiments on both quantum simulators and real quantum hardware. The results demonstrate that iterative refinement efficiently improves the precision of the solution and exhibits robustness to hardware noise.

We prove upper bounds for certain number field counting functions using Serre's modularity conjecture (now a theorem of Khare–Wintenberger). These results are comparable to sharp upper bounds for the number of abelian extensions with fixed or bounded discriminant, with Serre's modularity conjecture playing the role of class field theory.

We study the spatial $(n+1)$-body problem formed by one heavy central mass together with $n$ equal masses placed on a single orbit of a polyhedral rotation group $H\in\{T,O,I\}$, so that $n=|H|\in\{12,24,60\}$. Imposing the symmetry $q_{L}=L\,q_{I}$ for $L\in H$ reduces the problem to a single $2\pi$-periodic reference curve, with reduced action $A_{H}=A_{0}+\varepsilon A_{1}$, in which $\varepsilon$ is the inverse central mass and $A_{0}$ is the Kepler action. At $\varepsilon=0$ the critical set contains, as one connected component, the five-dimensional manifold of Kepler ellipses of minimal period $2\pi$, which we prove to be a nondegenerate critical manifold. A Lyapunov–Schmidt reduction along this manifold turns the continuation problem into the search for nondegenerate critical points of an explicit function $\Phi(e,\psi)$ of the eccentricity $e$ and the spatial orientation $\psi$, a nondegeneracy we verify by a computer-assisted proof. We thereby obtain, for each of the three groups, families of periodic solutions of the $(n+1)$-body problem with $n+1\in\{13,25,61\}$ bodies, bifurcating from Kepler ellipses and carrying the full tetrahedral, octahedral, or icosahedral symmetry.

Directly self-starting implicit methods are attractive for transient analysis because they avoid auxiliary starting procedures while retaining the original first- or second-order governing equations. However, most existing formulations usually fix the last sub-step at the end of each time interval, which restricts the attainable order. This study develops a generalized $s$-sub-step implicit framework by releasing this constraint and treating all sub-step locations as design variables. The resulting methods admit a unified Runge–Kutta representation for both first- and second-order transient systems and preserve identical effective matrices over all sub-steps. Accuracy conditions are derived by simultaneously matching the numerical amplification factor and load operator, thereby accounting for both homogeneous and forced responses. For $s=1,~\cdots,~6$, two complementary families are obtained: $s$th-order members with user-controllable high-frequency numerical dissipation and adjustable sub-step locations, and $(s+1)$th-order members obtained by selecting the sub-step locations, with fixed dissipation. The latter reach up to seventh-order accuracy without increasing the number of sub-steps, although some high-order members are $A(\alpha)$-stable with stability angles extremely close to $90^\circ$. Analytical amplitude and phase errors further reveal parity-dependent superconvergence in undamped systems, and appropriate parameter selections can substantially increase either phase or amplitude accuracy beyond the formal order. Numerical benchmarks confirm the predicted convergence orders and the controllable suppression of spurious high-frequency responses.

We prove that the curvature equation $\sum_{j=1}^n \alpha_j E_j(\tau_{\mathcal{M}})=\sum_{j=1}^n \alpha_j E_j(\tau_{\mathcal{N}})$ yields uniqueness up to translation for any two closed $C^2_+$ hypersurfaces $\mathcal{M}, \mathcal{N}\hookrightarrow\mathbb{R}^{n+1}$ whenever $(\alpha_1,\ldots,\alpha_n)\in\mathbb{R}_{\geq 0}^n\setminus\{0\}$ is log-concave and has no internal zeros. This gives an affirmative answer to a uniqueness question posed by Firey in the $C^2_+$ class.

The Boros-Moll sequences $\{d_i(m)\}_{0\leq i\leq m}$ are defined as $$d_i(m)=2^{-2m}\sum_{k=i}^m 2^k \binom{2m-2k}{m-k}\binom{m+k}{k}\binom{k}{i}.$$ Consider the ratio sequence $$u_i(m)=\frac{d_{i-1}(m)d_{i+1}(m)}{d_i(m)^2}.$$ Chen and Gu conjectured that $\{u_i(m)\}_{2\leq i \leq m-2}$ is both reverse ultra log-concave and log-concave. In this paper, we prove the reverse ultra log-concavity conjecture using bounds of Chen-Gu and Zhao, and prove the log-concavity conjecture asymptotically by showing that $\{u_i(m)\}_{2\leq i \leq m-2}$ is strictly log-concave for all sufficiently large $m$. The key ingredient in the latter result is a recurrence of Kauers and Paule, which we interpret as a nonlinear discrete dynamical system through a backward map. We construct an approximation to the ratio sequence using its stable limiting fixed point and combine localization and contraction arguments with finite-difference estimates and separate interior and edge analyses to obtain the desired strict log-concavity.

Exact algorithms for solving mixed-integer bilevel linear programs (MIBLPs) typically rely on sequences of lower and upper bounds that converge to the optimal value. These procedures are commonly initialized using the single-level relaxation (SLR), obtained by omitting the follower's optimality condition and solving the resulting single-level optimization problem. In this paper, we investigate whether, for broad classes of MIBLPs, the resulting standard bounds admit uniform improvements that can be computed within the same computational complexity regime. For pure continuous bilevel linear programs, we show that, unless $P = NP$, neither the SLR-based lower bound nor its associated upper bound can be uniformly improved in polynomial time, even for the class of min-max problems. We then extend this analysis to the class of pure integer min-max bilevel linear programs under the assumption that the polynomial hierarchy does not collapse. First, we show that the continuous relaxation of the SLR admits no uniform polynomial-time computable improvement. We then prove that neither the SLR itself nor its associated upper bound admits a uniform improvement by a polynomial-time algorithm with access to a mixed-integer linear programming (MILP) oracle. Importantly, this rules out uniform improvements by iterative MILP-based approaches, including cutting-plane-based and decomposition algorithms. Overall, our results demonstrate that the SLR-based bounds are, in a complexity-theoretic sense, unimprovable systematically within their natural computational regimes.

We prove that a very general quartic sixfold admits no decomposition of the diagonal, and in particular is not stably rational. The proof uses specialization to an explicit quartic sixfold birational to a quadric bundle of the type considered by Colliot-Thélène and Ojanguren. We also prove stable irrationality for very general six-dimensional intersections of four quadrics.

Low-voltage (LV) distribution feeders increasingly require internal state information, yet real-time telemetry is often limited to feeder-head active and reactive power, while downstream demand and production are described only by static contract metadata. This paper proposes a physics-based AC circuit-embedding reconstruction framework for this extreme-sparsity regime. Feeder-head measurements, matched metadata, and class-level profiles first determine bus-level load and production priors. These priors parameterize auxiliary constant-power elements coupled to the physical feeder through impedance branches, and the feeder state is reconstructed by solving the resulting nonlinear AC circuit equilibrium. The resulting AC equilibrium enforces the assigned virtual-element powers, while weighted least squares (WLS) reconciles measurements and pseudo-measurements through weighted residual fitting. The method is evaluated on a real 459-bus Portuguese LV feeder model using an in-sample-calibrated synthetic benchmark of 672 fifteen-minute snapshots. The proposed method converged for all 672 snapshots and achieved voltage RMSE, MAE, and maximum absolute error of 0.0033, 0.0014, and 0.0337 p.u., respectively, outperforming both static-prior WLS baselines. Under the same constructed priors, matched-prior WLS achieved a lower RMSE of 0.0029 p.u., while the proposed circuit embedding reconstructs the feeder state by solving the nonlinear AC equilibrium, bypassing WLS residual fitting entirely and requiring no specification or tuning of pseudo-measurement variances.

Let $\mathbb{F}_{q^n}$ be a finite field where $q=p^l$ ($p$ is a prime) and a Dickson nearfield $R$ associated to the field. In this note we investigate the basic properties of the generalized distributive set $D(a,b)$ where $a,b\in R$. The set is well-known when $n=1, 2$. For $n>2$, we characterize when $D(a,b)$ is a subfield of $\mathbb{F}_{q^n}$. Furthermore we find the unique largest subfield of $\mathbb{F}_{q^n}$ contained in $D(a,b)$. We also provide a condition to guarantee that $D(a,b)$ is not a subfield of $\mathbb{F}_{q^n}$. We end the note by posing the question of when $D(a,b)$, as a space over $\mathbb{F}_p$, has a primitive normal basis.

In this paper, a rarefaction wave under space-periodic perturbation for the generalized Korteweg-de Vries-Burgers equation is considered. It is shown that if the initial perturbation around the rarefaction wave is suitably small, then the solution of the system tends to the rarefaction wave. The stability of solution under periodic perturbation is an interesting and important problem since the perturbation keeps oscillating at the far fields. The key of proof is to construct a suitable ansatz carrying the same oscillation as the solution. Then we can find cancellations between solutions and ansatz such that the perturbation belongs to some Sobolev space. The nonlinear stability can be obtained by the energy method.

This work investigates an inverse problem for a general class of linear reaction-diffusion systems incorporating multiple delayed contributions, namely retarded diffusion, retarded time derivatives, and retarded source terms. The objective is to simultaneously recover the unknown time lag $\tau>0$ and spatially heterogeneous coefficients from boundary flux measurements alone. The recovery strategy exploits the singular temporal behavior generated by an incompatibility between the prescribed initial history and the boundary data. In contrast to prior inverse problems for delay equations, which assume $\tau$ known and are confined to ODE or abstract settings, our approach operates in a parabolic PDE framework with spatially varying coefficients. Once $\tau$ is identified, we establish a Lipschitz stability estimate via a Carleman inequality, in the case of time-independent coefficients $p=p(x)$, $q=q(x)$. This is the first result to simultaneously recover an unknown delay and spatially dependent coefficients in a delayed parabolic PDE from boundary data.

We study first-order differential equations $f'=R(e^z,f)$, where $R\in\C(t,w)$. We prove that a meromorphic solution on the whole complex plane is algebraic over $\C(e^z)$ unless $R$ is a polynomial of degree at most two in its second variable. Such an algebraic solution necessarily has the form $S(e^{z/q})$, with $S$ rational and $q$ a positive integer, giving an affirmative answer to Question~6.2 in Gundersen's collection. The geometric input is Guillot's theorem on single-valued trajectories of meromorphic vector fields on surfaces. The additional argument compares the fibration provided by that theorem with the original exponential coordinate. A ramification calculation forces every finite nonzero branch value $a$ to satisfy $Da=a$, which excludes such a value and reduces the comparison to a two-point cover. Divisor divisibility and relative algebraic closedness then recover a Riccati equation over the original coefficient field. No growth hypothesis is imposed. We also identify the algebraic degree with the least integer deck period and describe the growth and value fibres of the resulting solutions.

Using set-theoretic methods, we prove that every $\mathrm{AW}^*$-algebra is normal, resolving a question of Wright that has stood open for 46 years. This was previously known in the case of $\mathrm{AW}^*$-factors, by work of Saitô and Wright. We also show that if there is a model $V$ of $\mathrm{ZFC}$ with an $\mathrm{AW}^*$-algebra that fails to be monotone complete, then in some forcing extension of $V$ there is an $\mathrm{AW}^*$-factor that fails to be monotone complete, implying that any $\mathrm{ZFC}$ proof that all $\mathrm{AW}^*$-factors are monotone complete yields a $\mathrm{ZFC}$ proof that all $\mathrm{AW}^*$-algebras are monotone complete. Both results build on transfer principles for Boolean-valued factor representations originally developed by Ozawa. This work was assisted by the Danus LLM orchestration system.

We study Ringel's brick chain filtrations over finite-dimensional algebras: filtrations whose factors are filtered by copies of individual bricks, ordered so that morphisms from earlier bricks to later ones vanish. First, using submodule varieties, we prove that the largest submodule of a fixed module belonging to a torsion class takes only finitely many values as the class varies, answering Pavón's question. We deduce finiteness of brick chain filtrations and the bound $2^{d^2}$ for modules of dimension $d$. For $\tau$-tilting finite algebras, we bound their number by the multinomial coefficient determined by simple composition multiplicities. Finally, we construct families of bricks over the three-arrow Kronecker algebra whose filtration counts grow exponentially in the square of composition length. We prove this growth using a Littlewood–Richardson formula for submodule counts and the hook-length formula. In particular, the counts eventually exceed the factorial of composition length. These results answer Ringel's two questions.

Let $\Omega$ be the Lorentz future cone in $\mathbb{R}^{d+1}$ with respect to the Lorentz product and let $T^M$ be the $M$-fold product of the future tube $T=\mathbb{R}^{d+1}+i\Omega$. The Lorentz group $\mathrm{SO}_0(1,d)$ acts diagonally on $T^M$, and its complexification $\mathrm{SO}(1,d)^\mathbb{C}$ acts on $\mathbb{C}^{(d+1)\times M}$. We prove that the domain $G^\mathbb{C}\cdot T^M$ is a Stein manifold for any connected unipotent subgroup $G$ of $\mathrm{SO}_0(1,d)$.

For $-1<\alpha\leq1/2$, let $J_\alpha$ be the Bessel function of the first kind and let $j_{\alpha,2}$ be its second positive zero. Define $\beta(\alpha)<\alpha+1$ by $$ \int_0^{j_{\alpha,2}}u^{-\beta(\alpha)}J_\alpha(u)\,du=0. $$ We prove that $\beta''(\alpha)>0$ for $-1<\alpha\leq1/2$, including the left second derivative at $\alpha=1/2$. The continuous extension $\beta(-1)=0$ is strictly convex on $[-1,1/2]$, strengthening Askey's 1993 convexity conjecture. The analytic argument uses a positive expansion of a primitive and a vanishing weighted sum. Increasing ratios of consecutive weights give a negative covariance term, while one comparison controls slope and curvature. The remaining algebraic step proves eight rational inequalities by finite exact polynomial calculations, reproduced in an appendix.

We determine the sharp measure contraction exponents of two families of Grushin-type metric measure spaces. The radial Grushin space $\mathbb{G}^{n+m}$ is $\mathbb{R}^{n}\times\mathbb{R}^{m}$, equipped with Lebesgue measure and generated by $X_i=\partial_{x_i}$ and $Y_j=|x|\partial_{y_j}$, for $1\leq i\leq n$ and $1\leq j\leq m$. We prove that $\mathbb{G}^{n+m}$ satisfies $\operatorname{MCP}(K,N)$ if and only if $N\geq n+4m$ and $K\leq 0$. We also show that, for $\alpha\geq1$, the $\alpha$-Grushin plane generated by $X=\partial_x$ and $Y_\alpha=|x|^\alpha\partial_y$ satisfies $\operatorname{MCP}(K,N)$ if and only if $K\leq0$ and $N\geq N_\alpha$, where \[ N_\alpha := 1+\max_{L>1} \frac{(2\alpha+1)L}{(L-1)^{2\alpha+1}+1}. \] This resolves the conjecture posed in arXiv:2010.16350 and, for integer $\alpha\geq2$, provides the first examples of real-analytic sub-Riemannian structures with noninteger curvature exponent. Both results recover the known curvature exponent $5$ of the classical Grushin plane, corresponding respectively to $n=m=1$ and $\alpha=1$.

We give a class of local diffeomorphisms which are homotopic to toral hyperbolic endomorphisms, but which are not topologically semi-conjugate, within the homotopic class of the identity, to any Anosov local diffeomorphism. In particular, we show that for a non-invertible partially hyperbolic $C^1$-smooth local diffeomorphism with expanding directions on the $2$-torus, if it is semi-conjugate to an Anosov local diffeomorphism, then it is also an Anosov local diffeomorphism.

A Hawkes process is a simple point process whose intensity depends on its history; the resulting dynamics are generally non-Markovian. We establish a sample-path moderate deviation principle for a nonlinear Hawkes process in the full moderate regime. Since a Poisson cluster representation is unavailable for nonlinear Hawkes processes, we use the past configuration as a Markov state and construct a potential, or Poisson corrector, for the centered stochastic intensity. A monotone Poisson coupling shows that the add-one increment of the corrector is uniformly bounded. The centered counting process is consequently the sum of a martingale with bounded jumps and an exponentially negligible boundary term. Exponential stabilization of the predictable quadratic variation follows from the process-level large deviation principle for nonlinear Hawkes processes. The martingale moderate deviation theorem then yields the result for every scale between the central-limit and large-deviation scales. The same construction gives a response formula for the asymptotic variance and, in particular, verifies that the variance dominates the stationary mean intensity in the self-exciting case.

We introduce Selector-Anchored Diffusion with Local Innovation (D+LI), a distributed vector-assembly primitive that constructs complete local replicas from independently owned authoritative blocks. Orthogonal selectors preserve block identity, local innovation anchors each owner's contribution, and Laplacian diffusion propagates these contributions across the network. We show that pure averaging with raw additive selector sources cannot isolate this replica. The D+LI error operator decomposes coordinatewise as the identity minus an owner-grounded Laplacian, yielding global contraction with a factor independent of the total and block dimensions. For time-varying outer targets, we establish a finite-T input-to-state stability (ISS) bound and a self-consistent estimator-controller gain condition for Lipschitz coordination maps, including non-potential ones. For semi-algebraic potential problems, we prove Kurdyka-Lojasiewicz (KL) finite-length convergence of the coupled decision-assembly iteration with a fixed finite communication budget per outer iteration under explicit timescale-separation conditions. Three numerical experiments demonstrate contraction tightness and finite-horizon tracking, verify an admissible KL timescale, and illustrate discrete and non-potential outer maps.

We show that on any closed orientable real-analytic Riemannian surface, any smoothly immersed closed immortal curve-shortening flow converges to a unique non-constant closed geodesic.

We study a nonlocal 2-Hessian equation, given by an infimum of linear deformations of the fractional Laplacian, that is, $\inf_{A\in {A}_2}\Delta^s (u\circ A)(A^{-1}x)$. We characterize the class of coefficient matrices ${A}_2$, which determines the behavior of the operator, and provide a detailed geometric description of the possible degeneracies. Our main theorem shows that the nonlocal 2-Hessian equation remains uniformly elliptic for strictly positive right-hand sides, which leads to regularity estimates. The results hold under weaker hypotheses than those previously considered in the literature, and for the full range $s\in(0,1)$. In particular, no convexity on the solutions is required. Our hypotheses can be interpreted as nonlocal counterparts of the local notions of semiconcavity and 2-convexity. Moreover, all the results are stable as $s\to1$, recovering the local case. The geometric methods developed here are new, even in the local setting, and may be relevant to a broader class of nonlocal fully nonlinear equations and curvature-type problems.

We study Stein neighborhood bases for unions of closed complex and totally real submanifolds, with particular attention to a linear configuration with noncompact intersection. We prove that the set $X=\{w=0\}\cup\{\Im z=\Im w=0\}\subset\C^2$ admits a Stein neighborhood basis. The proof starts from the explicit plurisubharmonic defining function $\rho(z,w)=(\Im w)^2+\sinh^2(\Im z)|w|^2$ and obtains the necessary non-uniform control at infinity by adding a locally uniformly convergent series of nonnegative plurisubharmonic terms constructed by Oka–Weil approximation. We also give an example of a closed complex submanifold and a closed totally real submanifold in $\C^2$ with compact intersection whose union has no Stein neighborhood basis, showing that the unrestricted union problem has a negative answer.

Let $\kk$ be a field of characteristic $p$ and $R=\kk[x_1,\dots,x_n]$. Let $D$ denote the ring of $\kk$-linear differential operators on $R$. Let $\varphi:R\to R$ be a $\kk$-algebra endomorphism whose Jacobian is a unit in $R$. We show that there is a unique $\kk$-algebra endomorphism $\Phi:D\to D$ such that $\Phi|_R=\varphi$ and that the restriction of scalar via $\Phi$ preserves holonomicity. We also construct an example of a $\kk$-algebra endomorphism of $D$ that does not preserve holonomicity.

Computing the closest point on a clothoid is a recurring task in geometric design, road and railway alignment, and path planning. The efficient algorithm of Frego and Bertolazzi addresses this problem, but its candidate-selection analysis assumes at most one local minimum per search interval. We exhibit admissible configurations with two local minima, raising the question of whether the existing strategy accounts for every possible minimum. Using the geometry of the clothoid evolute, we prove that, for any query point and any proper no-inflection planar clothoid segment with tangent-angle variation at most $2\pi$, the squared-distance function has at most three stationary points; if all three are local extrema, their order is min-max-min. This establishes the completeness of the original candidate-selection logic beyond the one-minimum premise. It also shows that no interior search is needed when neither endpoint derivative test is active, allowing unnecessary midpoint searches to be omitted while retaining numerical fallback. Numerical experiments demonstrate reductions in iteration count and evaluation time.

We study high-dimensional conditioned dynamics obtained from an arbitrary number of independent copies of a mixing Markov chain. The dynamics is conditioned to avoid a family of holes whose stationary measure vanishes as the dimension grows, and we ask whether the resulting process admits a quasi-stationary measure close to the stationary product measure. Our problem is motivated by conditioning on holes with complicated geometry in high dimension, such as sets arising naturally from large deviations of suitable observables. Our main tool is the ANOVA decomposition, which separates functions according to their dependence on different subsets of coordinates. This allows us to exploit the product structure of the dynamics and obtain contraction estimates that are uniform in the dimension. Combined with a Keller Liverani perturbation argument, these estimates yield the existence of quasi-stationary densities converging to the stationary density as the size of the hole vanishes. Under stronger one-step mixing and regularization assumptions, we obtain sharper convergence in a Sobolev norm, with a square-root dependence on the measure of the hole. The improvement relies on the increasingly strong contraction of higher-order ANOVA components, thereby overcoming the usual loss of control with dimension. Finally, we apply our results to additive-noise Markov chains on the circle with smooth, uniformly positive transition densities

We study the Hopf decomposition of subgroup actions of the Teichmüller modular group on the Thurston boundary with respect to the Thurston measure class. We identify the conservative part with the big horospherical limit set modulo null sets. For a basepoint with trivial stabilizer in the subgroup, the ideal boundary of the associated Dirichlet polyhedron is wandering, and its subgroup translates cover the dissipative part modulo null sets. The dissipative part also agrees with the set of Dirichlet points modulo null sets. The description using Dirichlet polyhedra relies on a separation theorem for extremal length: every level set of an extremal length ratio at distinct points of Teichmüller space has measure zero. Kaimanovich's Radon–Nikodym criterion characterizes the two parts by the divergence and convergence, respectively, of a series of extremal length ratios. For the Torelli group of a closed surface of genus at least two, we use radial limits of the period map to prove that its conical limit set has measure zero. Together with the conservativity established by Choi, Gekhtman, Yang, and Zheng, our geometric characterization implies that its big horospherical limit set has full measure and that the ideal boundary of every Dirichlet polyhedron has measure zero.

We rigorously justify the zero–dispersion limit of linear water waves over a sharp bottom step, recovering the shallow-water equations as a transmission problem, and describe the localized dispersive effects generated near the discontinuity. To obtain estimates uniform in the shallow-water parameter, we study an abstract functional framework for the linear water waves equations in two-dimensional fluid domains with a Lipschitz curved bottom. The framework is based on fractional powers of the Dirichlet–Neumann operator, and we characterize low-regularity energy spaces using adapted Rellich identities. At higher regularity, the associated velocity potential develops corner singularities. For the step geometry, we use an explicit conformal mapping to obtain a version of Grisvard's shift theorem, where we derive an explicit formula for the singular part of the potential. The dependence on the horizontal variable reveals a narrow transition scale generated by the step. The convergence toward the shallow-water transmission problem holds away from this transition region, while a localized dispersive correction remains near the step. We illustrate this discrepancy numerically and identify the resulting step-induced dispersive boundary layer. Building on these observations, we consider the strongly dispersive regime and derive a consistent model that explicitly describes this boundary layer for a small step.

We develop a thermodynamically consistent diffuse-interface model for multicomponent membranes. The proposed model is derived from a coupled free energy functional that incorporates protein-dependent bending elasticity, diffuse surface tension, a volume penalty, and a membrane-associated Ohta–Kawasaki energy. Applying the Onsager variational principle, we derive a coupled $L^2$ gradient flow system and its energy dissipation law. We then construct a stabilized alternating ETD1 scheme and a stabilized alternating ETDRK2 scheme with Strang-type composition (alternating Strang-ETDRK2). To the best of our knowledge, the proposed alternating Strang-ETDRK2 scheme has not previously been developed and analyzed for coupled phase field systems. We further prove the discrete energy dissipation for both schemes under some regularity assumptions on the numerical solutions. Numerical experiments in two and three dimensions validate the discrete energy dissipation law, and present the protein segregation and membrane deformation produced by the proposed model.