Papers

Classical nonabelian Hodge theory identifies Dolbeault and de Rham moduli spaces by providing a real-analytic isomorphism. In this paper, motivated by the Kapustin–Witten theory, we study this correspondence in the more general framework of perfect complexes on proper varieties, paying special attention to the surface case. We establish a Lagrangian correspondence which relates the shifted symplectic geometries by Pantev–Toën–Vaquié–Vezzosi (PTVV) between the derived stacks of flat and Higgs perfect complexes. We investigate the existence of derived twistor structures of hyperkähler type on the moduli stack of perfect complexes endowed with $\lambda$-connections by Deligne–Hitchin–Simpson. We establish a version of the AKSZ/PTVV transgression, Lagrangian intersection, and (hyperkähler) symplectic reduction theorems in this context. Moreover, we prove that the derived Riemann–Hilbert correspondence of Porta and Holstein–Porta, which states an equivalence of derived analytic stacks of perfect complexes on $X_{\mathrm{Betti}}$ and $X_{\mathrm{DR}}$, is compatible with the natural shifted–symplectic structures. We then study the relation between the shifted (pre-)twistor structures and the shifted symplectic forms on the fibers, and prove that the analytic Deligne–Hitchin–Simpson moduli stack on a smooth projective variety $X$ has a canonical $2(1-\dim X)$ shifted pretwistor structure over $\mathbb{P}^1_{\mathbb{C}}$, a result which has been anticipated for some time. In particular, the moduli stack of solutions to the Kapustin–Witten equations modulo gauge equivalence on a smooth proper complex algebraic surface exibits a $(-2)$-shifted (pre)twistor structure as a family over $\mathbb{P}^1_{\mathbb{C}}$.

We describe the Poisson bracket of a Lagrangian field theory expressed in the framework of $L_\infty$ algebras. If the symplectic structure on phase space is defined following recent work, we show that the Poisson bracket can be computed with the Peierls formula, confirming the well-known result of the covariant phase space formalism. Of particular interest is the Poisson bracket in nonlocal theories. In $p$-adic string theory we find that a straightforward construction of the Poisson bracket is obstructed by higher derivative instabilities. We also discuss the inverse relation between the Poisson bracket and symplectic structure in the language of homological algebra, extending some ideas in the mathematical physics literature.

We consider infinite-type IETs arising from elementary examples of finite-area translation surfaces of infinite genus such as the Baker's surface. We call such IETs tail-reversing and we show that for any tail-reversing permutation the subset of the simplex of lengths $\Delta$ for which the corresponding infinite-type IET is uniquely ergodic contains a dense $G_{\delta}$ set with respect to the $\ell^1$-topology. To this end, we generalize Rauzy-Veech induction to a large class of infinite-type IETs, where we prove a minimality criterion as a generalization of Keane's criterion in the finite setting. We then restrict ourselves to tail-reversing IETs and obtain our genericity result through a combinatorial analysis of their infinite-type Rauzy diagrams. Moreover, we derive an explicit condition for a tail-reversing IET to be uniquely ergodic by studying the diameter of its induction matrices.

Contrastive embedding models trained with scale-invariant losses are typically paired with distance metrics like cosine similarity, effectively ignoring embedding magnitudes. However, surprisingly, empirical studies reveal that despite this, these "discarded" norms seem to correlate with semantic properties such as concept specificity, token frequency, and human uncertainty. In this work, we provide a formal theoretical framework explaining this phenomenon. By analyzing the optimization dynamics, we derive an analytic formula demonstrating that embedding length naturally encodes this information as a byproduct of the training process. We also show how this gives rise to signals that can serve as "free" calibration tools in specific models and retrieval tasks, providing a grounded explanation for a previously heuristic observation.

We investigate the problem of remote estimation (at a monitor) of a discrete-time joint Markov process with individual components which can be observed with dedicated sensors. At a given time slot, the monitor has the option of staying idle or sending a pull request to one of the sensors to obtain a partial state value, while the sensors are assumed to have heterogeneous sampling costs. Our goal is to develop a monitor pull policy, i.e., determining when and towards which sensor to send a pull request, in order to minimize a weighted sum of average age of incorrect information (AoII), or in short age, and sampling costs. As the communication model, we assume an erasure channel with a fixed one-slot delay from each sensor to the monitor. In this setting, the monitor does not perfectly know either the state of the process or the age, at any given time. We first obtain a sufficient statistic, namely belief, representing the joint distribution of the age and the current state of the observed process, by using the history of all pull requests and observations. Then, we formulate the optimization problem as a continuous state-space Markov decision process (MDP), namely belief-MDP, for the solution of which we propose two model predictive control (MPC) methods, namely MPC without terminal costs (MPC-WTC), and reinforcement learning MPC (RL-MPC). The effectiveness of the proposed methods is validated by numerical examples.

We present a novel {\delta}f particle-in-cell (PIC) method for the kinetic simulation of electrostatic plasmas in which the bulk density, acting as a control variate, is evolved using symplectic neural networks (SympNets). The SympNets are used as an approximation of the backward flow and trained using the particle trajectories. We introduce a periodic variant of the SympNet architecture that encodes the spatial periodicity of the problem into the network itself. We validate the approach with numerical results in 1D1V and 3D3V for the Vlasov-Poisson system.

The problem of the mean-square optimal estimation of the linear functionals which depend on the unknown values of a multidimensional homogeneous random field from observations of the field with noise is considered. The minimax (robust) method of estimation is applied in the case where the spectral densities of the fields are not known exactly while some sets of admissible spectral densities are given. Formulas that determine the least favourable spectral densities and the minimax spectral characteristics are derived for some special sets of admissible densities.

Let $q$ be an integer and let $f$ be a normalised Hecke eigenform of integral weight for the full modular group. Let $L(s,\mathrm{sym}^j f)$ denote the $j$-th symmetric power $L$-function associated to $f$, and let $\lambda_{\mathrm{sym}^j f}(n)$ denote its $n$-th coefficient. We study the behaviour of the partial sum of $\lambda_{\mathrm{sym}^j f}(n)$, and of its second moment, taken over those sums of $m$ squares that are congruent to $1$ modulo $q$. As an application, we investigate the shifted convolution sum of $\lambda_{\mathrm{sym}^j f}(n)$ against a $k$-full kernel function, for any $k \geq 2$. We also study the number of sign changes of $\lambda_{\mathrm{sym}^j f}(n)$ twisted with a $k$-full kernel function, again over sums of $m$ squares. Throughout, $m$ is even with $m \in \{2,4,6,8,10,12\}$.

We introduce and study peeling and wrapping operations for families of compact convex sets. The two peeling procedures considered in the paper are the $m$-point peeling, obtained by intersecting the convex hulls remaining after all possible deletions of $m$ members of the family, and the recursive convex hull peeling, obtained by repeatedly removing the contributing sets, that is, those members whose deletion strictly changes the convex hull. Using polarity, we also introduce the dual wrapping operations for intersections of convex sets. The deterministic part of the paper develops the geometric framework needed for these constructions. In particular, we study contributing sets under general position assumptions, explain the role of compactness of convex hulls of subfamilies, and prove continuity results for both peeling procedures with respect to a suitable vague convergence of locally finite point measures on the space of compact convex sets. The probabilistic part applies this framework to $K$-hulls generated by random samples from a convex body $K$. Assuming that $K$ is strictly convex and regular, we prove that the m-point and recursive peelings of the polar bodies associated with the random $K$-hulls converge in distribution to the corresponding peelings of the limiting Poisson object. By polarity, this also yields distributional convergence of the associated wrapping operations for the rescaled random sets themselves.

We extend the Sternberg–Weinstein coupling construction to the Jacobi geometry setting. Starting from a Jacobi Hamiltonian $G$-space and a principal bundle equipped with a connection whose curvature satisfies some nondegeneracy condition, we show that the associated bundle naturally carries a Jacobi structure compatible with the canonical ones on the fibers. This construction provides a unified framework encompassing the symplectic, locally conformal symplectic, and contact cases. It reveals new coupling phenomena related to the presence of the Reeb vector field.

We show that, when $d=o(n)$, every $d$-regular $n$-vertex graph contains a spanning subgraph whose degree distribution is nearly uniform, i.e., for each $0\leq i\leq d$, there are $(1+o(1))n/(d+1)$ vertices with degree $i$. This proves a conjecture of Alon and Wei on irregular subgraphs and strengthens a previous result of Fox, Luo and Pham.

Let $X$ be a Picard-rank-one del Pezzo manifold of dimension $n\geq 4$ over an algebraically closed field of characteristic zero. Okamura proved that the unpointed Kontsevich spaces $\overline{M}_{0,0}(X,d)$ are irreducible of the expected dimension for every $d\geq 1$. We refine this result by studying pointed evaluation fibers. First, we prove that for every $d\geq 1$, the one-pointed evaluation morphism $\overline{M}_{0,1}(X,d)\to X$ has geometrically irreducible generic fiber. Second, in the very ample cases $H^n=3,4,5$, we prove that for every $d\geq 2$, the two-pointed evaluation morphism $\overline{M}_{0,2}(X,d)\to X\times X$ has geometrically irreducible generic fiber.

Let $f$ be a normalised Hecke eigenform of even integral weight for the full modular group $\mathrm{SL}(2,\mathbb{Z})$, let $L(s,\mathrm{sym}^{j}f)$ be the $j$th symmetric power $L$-function attached to $f$, and let $\lambda_{\mathrm{sym}^{j}f}(n)$ denote its $n$th Dirichlet coefficient. For each even integer $m$ with $2 \le m \le 12$, we establish upper bounds for the partial sums of $\lambda_{\mathrm{sym}^{j}f}(n)$ and asymptotic formulas for those of $\lambda_{\mathrm{sym}^{j}f}^{2}(n)$ taken over integers represented as a sum of $m$ squares. As an application, we obtain lower bounds for the number of sign changes of $\lambda_{\mathrm{sym}^{j}f}(n)$ along these sums of $m$ squares.

We show that the Hankel flat extension formulation of the cactus algorithm is equivalent to a completion problem for multiplication tensors of Artinian Gorenstein algebras. The unknown Hankel moments are canonically identified with the undetermined tensor coefficients, and under this identification the symbolic multiplication matrices and their commutation equations coincide. This shows that the usual degree extension formulation is a coordinate realization of a variable extension problem with marked generators. We further use Borel-fixed and squat staircases to reduce the family of candidate basis shapes in the resulting algorithm.

Semantic communication (SemCom) aims to preserve semantic meaning and task-oriented information beyond conventional message recovery over wireless channels. The adoption of SemCom in shared-access wireless networks introduces new vulnerabilities for multi-user semantic inference. This paper considers a SemCom system for two transmitters communicating with a common receiver over a multiple access channel. Each transmitter maps source information into latent semantic representations, while the receiver jointly reconstructs and classifies the semantic information for both transmitters. A selective over-the-air backdoor (Trojan) attack is presented in which an adversary transmits a low-power trigger waveform over the air and injects it into the shared received signal during training. By transmitting the trigger again during testing, this stealthy, low-power attack selectively manipulates the semantic inference for one transmitter while minimally affecting the inference of the other transmitter. To mitigate this vulnerability, a trigger-aware defense mechanism is developed to preserve correct semantic labels under trigger-contaminated wireless observations. The results demonstrate both the vulnerability of shared-access SemCom systems to selective over-the-air backdoor attacks and the effectiveness of trigger-aware robust training for semantic protection.

Naslund and Sawin used the slice-rank method for diagonal tensors to prove that $$|\mathcal{F}|=O\!\left(n^{1/2}\left(\frac{3}{2^{2/3}}\right)^n\right)$$ for any sunflower-free family $\mathcal{F}\subseteq 2^{[n]}$. We prove a lemma similar to the slice-rank lemma for the newly defined $i$-triangular tensors, and use it to achieve a polynomial-factor improvement of the bound of Naslund and Sawin by proving that $$|\mathcal{F}|=O\!\left(n^{1/6}\left(\frac{3}{2^{2/3}}\right)^n\right)$$ for any sunflower-free family $\mathcal{F}\subseteq 2^{[n]}$.

A vertex $v$ of an oriented graph $D$ is a Seymour vertex if the number of outneighbors of $v$ is less than or equal to the number of second outneighbors of $v$ (vertices at directed distance exactly $2$ from $v$). Seymour's Second Neighborhood Conjecture claims that every oriented graph contains a Seymour vertex. We propose the following stronger conjecture (Conjecture 1.4): given any oriented graphs $D$, there exists a minimum outdegree vertex $s$, and a Seymour vertex $v^*$ such that the directed distance from $s$ to $v^*$ is at most $\lfloor\frac{\delta^+(D)}{3} \rfloor$, where $\delta^+(D)$ denotes the minimum outdegree of $D$. The main contribution of this article is to show that Conjecture 1.4 is true for all oriented graphs having minimum outdegree at most $7$. This, in particular, improves the results of Kaneko and Locke (2001) which (implicitly) proved Conjecture 1.4 for all oriented graphs having minimum outdegree at most $6$. In particular, this yields Seymour's Second Neighborhood Conjecture for this class of oriented graphs. We also construct a family of digraphs attaining the proposed distance bound, showing that it cannot be improved in general. Finally, by adapting existing arguments, we show that if Seymour's Second Neighborhood Conjecture is true then given any oriented graph $D$ and any vertex $v$, it is always possible to find a Seymour vertex $v^*$ such that the directed distance from $v$ to $v^*$ is at most the outdegree of $v$.

Existing semantic communications have exhibited satisfactory performance in many tasks, but secure image transmission remains insufficiently explored. We propose a novel secure image semantic communication (SISC) framework over multiple-input multiple-output (MIMO) fading channels. To ensure high-quality image reconstruction for the legitimate semantic user (SU) and simultaneously interfere with the eavesdropper (Eve), we design a semantic noise generation (SNG) network. This network generates a beneficial semantic noise map based on both the source features and the SU channel state information (CSI). An efficient channel estimation enhanced network is incorporated to obtain the accurate CSI and enhance the system performance. Furthermore, to improve the secure image reconstruction quality, we develop an efficient transceiver beamformer optimization algorithm, where the formulated problem is solved using the constrained stochastic successive convex approximation method. In the proposed SISC framework, semantic noise generation and beamforming optimization work together to ensure secure and high-quality image transmission. Numerical results demonstrate that the proposed semantic noise aided transmission scheme effectively protects image information from leakage to Eve while maintaining high-fidelity image reconstruction at SU.

Let $K_1,\ldots,K_k\subset\mathbb R^n$ be origin-symmetric measurable sets of finite volume such that \[ \sum_{1\le i<j\le k}\langle x_i,x_j\rangle\le \binom{k}{2}, \qquad \forall\,x_i\in K_i, x_j\in K_j. \] We prove the sharp many-body Blaschke–Santaló type inequality \[ \prod_{i=1}^k |K_i|\le |B^n|^k \] proposed by Kalantzopoulos and Saroglou, and characterize all equality cases. The proof combines multi-marginal optimal transport with a pseudo-Euclidean volume estimate. Using the geometric–functional equivalence of Kalantzopoulos and Saroglou, we also establish the functional version inequality proposed by Kolesnikov and Werner.

Many modern generative modeling methods, including diffusion models, normalizing flows, and flow matching, estimate transport maps or plans between distributions without explicitly targeting an optimal transport (OT) map. In applications like generative modeling, the transport cost itself is irrelevant, and this makes it natural to target maps which are more tractable from either a statistical or computational standpoint. In this short note, we formalize the task of estimating any valid transport map in a rigorous minimax framework. One consequence of this framing is that it yields sample complexity lower bounds for any method whose learned object is evaluated as a transport map or plan, including flow matching and diffusion-based generative models, in settings where direct analysis would be challenging due to the analytic complexity of the methods and their target maps. We observe that, under standard, though strong, stability assumptions from the OT literature, estimating any valid transport map is statistically as hard as estimating the OT map. We complement these results with some examples showing that when these stability assumptions fail, alternative transport maps can be learned substantially more accurately than the OT map. Our minimax framing provides a rigorous foundation for understanding the statistical limits of modern transport-based generative methods and clarifies when targeting sub-optimal maps can provide real statistical advantages.

Let $ G $ be a compact metrizable Abelian group, $ L^{1}(G) $ its group algebra and $ M(G) $ its measure algebra. For each proper subset $ E $ of the dual group $ \hat{G} $, let $ L^{1}_{E}(G)=\{f\in L^{1}(G):\hat{f}=0 \text{ on } \hat{G}\setminus E \}$ and $M_{E}=\{\mu\in M(G):\hat{\mu}=0 \text{ on }\hat{G}\setminus E\} $. If $ M_{E}(G)=L^{1}_{E}(G) $ then the set $ E $ is said to be a Riesz sets. If $ M_{E}(G)*M_{E}(G)\subseteq L_{E}^{1}(G) $ then $ E $ is said to be a small-2 set. The main results of this paper are the following: 1. Every small-2 set is a Riesz set. 2. The ideal $ L^{1}_{E}(G) $ is Arens regular iff $ E $ is a Riesz set. Let $ A=L_{E}(G) $ and equip $ A^{**} $ with the first Arens product. 3. The centre of $ A^{**} $ is $ Z(A^{**})=A+N(A^{**}) $, where $ N(A^{**})=\{r\in A^{**}:rA^{**}=\{0\}\} $. These results settle three long-standing open problems in this area.

Complex unmanned systems comprising satellites, unmanned aerial vehicles (UAVs), unmanned ground vehicles (UGVs), and quadruped robots are increasingly deployed to perform large-scale sensing and autonomous operations. We propose a world model-empowered sensing, communication, decision (SCD) integration framework for complex unmanned communication networks. The proposed architecture establishes a closed-loop system where a unified world model jointly optimizes time-sensitive sensing, wireless communication, and intelligent decision-making. To regulate sensing freshness and reduce redundant data generation, we propose a time-sensitive age of information (AoI)-driven sensing mechanism that dynamically schedules sensing updates based on task urgency and predictive uncertainty. Furthermore, a predictive world model is developed to jointly represent environmental dynamics, wireless channel evolution, and agent mobility within a hybrid deterministic-stochastic latent space. This enables proactive communication scheduling and decision evaluation via latent rollout. To support large-scale heterogeneous coordination, a multi-granularity knowledge graph is further designed to organize cross-population relationships among satellites, UAVs, UGVs, and ground agents. Numerical results demonstrate that the proposed SCD framework outperforms conventional systems, highlighting the significant potential of world models for supporting unmanned systems.

We prove that every class in the narrow ray class group modulo an integral ideal $\mathfrak q$ of a fixed number field is represented by a product of three prime ideals of norm at most $ ( N\mathfrak q)^{\max(1,3\alpha,4\alpha_0)+\kappa} $ for any $\kappa>0$, where $\alpha$ is the exponent in short character sum bounds for general non-principal ray class characters and $\alpha_0$ comes from a bounded-order subconvexity input for Hecke $L$-functions. Wu's subconvexity bound gives the admissible choice $\alpha=\alpha_0=103/256$, hence the explicit bound $(N\mathfrak q)^{103/64+\kappa}$. This improves the previous $O_K((N\mathfrak q)^3)$-scale bound of Deshouillers, Gun, Ramaré, and Sivaraman. We also prove that a positive proportion of ray classes are represented by products of two prime ideals. The proof extends the multiplicative dense-model and transference framework of Matomäki–Teräväinen to narrow ray class groups.

We characterize weakly regularized continual classification in homogeneous models as sequential projections onto task margin sets. This result generalizes prior analyses restricted to either stationary (single-task) deep models or continual linear models. We show that global convergence generally fails, even for simple models linear in data but nonlinear in parameters. Nevertheless, by leveraging results from nonconvex projection theory, we identify regularity properties of homogeneous deep networks that guarantee local linear convergence under random and cyclic task sequences. Finally, we extend our analysis to continual regression, unifying the framework for homogeneous models.

We generalize to certain families of even Artin groups several classical results on right-angled Artin groups. In particular, we compute the cohomology ring, describe the pro-$p$ completion, and determine the $p$-Zassenhaus restricted Lie algebra in the FC case. As a by-product, we prove a rigidity result that implies that if two even Artin groups of FC type are isomorphic, then for every prime $p$, the $p$-parts of their defining graphs are isomorphic.

For $r \geq 2$ and graphs $H_1, \ldots, H_r, G$, we say that $G$ is $(H_1, \ldots, H_r)$ vertex-Ramsey, or $(H_1, \ldots, H_r)_v$-Ramsey, if whenever we colour the vertices of $G$ with colours from the set $[r]=\{1,2, \ldots, r\}$ there exists $j \in [r]$ such that some copy of $H_j$ in $G$ is monochromatic in colour $j$. Given any fixed collection of graphs $H_1, \ldots, H_r$, Luczak, Ruciński and Voigt and Kreuter determined in the 1990s the threshold edge probability $p$ at which the binomial random graph $G(n,p)$ becomes $(H_1, \ldots, H_r)_v$-Ramsey. More recently, Das, Morris and Treglown investigated the vertex-Ramsey property in the randomly perturbed setting. When $r=2$ they determined the number of random edges one must add to a dense graph to ensure that with probability $1-o(1)$ the resulting graph is $(H_1, H_2)_v$-Ramsey whenever one of $H_1$ or $H_2$ is a clique. They posed the problem of extending their results to all pairs of graphs $(H_1, H_2)$. In this paper we resolve a more general form of their problem and determine for any $r\geq 2$ and $r$-tuple of graphs $(H_1, \ldots, H_r)$ the number of random edges one must add to a dense graph to ensure that with probability $1-o(1)$ the resulting graph is $(H_1, \ldots, H_r)_v$-Ramsey.

A complete classification of weak$^*$~continuous, measure-valued valuations is established on star bodies in $\R^n$. Consequences are an integral representation of rotation equivariant, measure-valued valuations and a characterization of dual area measures.

We show convergence of solutions to a Navier–Stokes/Allen–Cahn system as the interfacial thickness $\varepsilon>0$ tends to zero for well-prepared initial data as long as the limit system possesses a sufficiently smooth solution. The limit system consists of a two-phase Navier–Stokes system separated by a sharp interface in the presence of surface tension coupled to a convective mean curvature flow equation. In comparison to previous results we obtain improved convergence estimates for higher-order norms. These enable us to prove convergence in the case of three space dimensions and non-constant viscosity, which was unknown before. The convergence results relies crucially on uniform higher-order estimates for the associated linearized Navier–Stokes/Allen–Cahn system in suitably weighted $L^2$-Sobolev spaces. Here a novel problem-adapted weight proportional to the sum of $\varepsilon$ and the distance to the sharp interface of the limit, which gives improved and sharp estimates, is an important new ingredient. This approach can be potentially adapted to other sharp interface limits as well.

The continuous integral R2 indicator is a Pareto-compliant refinement of the classical finite-weight-vector R2 indicator, used in performance assessment, bounded archiving for a-posteriori multi-objective optimization, and skyline selection in databases. This work introduces a bidirectional perspective mapping between continuous integral R2 computation and integration over unions of anchored axis-aligned boxes. After translating the ideal point of a minimization problem to the origin, approximation points become strictly positive loss vectors, and the subgraph of the lower weighted Tchebycheff envelope over the weight simplex maps to the complement of an anchored-box union in reciprocal objective space. The Jacobian gives an absolute R2 formula as a weighted complement volume with density $(x_1+\cdots+x_N)^{-(N+1)}$, while differences of R2 values become finite weighted hypervolume differences. Hence, hypervolume algorithms that emit box decompositions can be reused by replacing ordinary box volumes with closed-form weighted box integrals. For $N$ objectives, this gives an output-sensitive overhead $O(2^N M)$ for an $M$-box decomposition, or $O(M)$ for fixed $N$. Using existing box-decomposition approaches, the integral R2 can be computed in $O(n \log n)$ for $N=2,3$, in $O(n^2)$ for $N=4$, and in $O\left(n^{\lfloor (N-1)/2\rfloor+1}\right)$ for $N\geq4$, with $n$ denoting the size of the approximation set. On the lower-bound side, exact value computation has an $\Omega(n\log n)$ lower bound in the algebraic decision-tree model already in two objectives, this bound lifts to every fixed $N\geq2$, and exact computation is $\#P$-hard when $N$ is part of the input. Together, the proposed perspective mapping provides a powerful tool for transferring algorithmic and structural results between anchored-box union and hypervolume theory and integral R2 computation.

We study a system formed by $M$ particles moving in 3 dimensions and interacting with two heat reservoirs, each with $N\gg M$ particles. The system and the reservoirs interact via random collisions and thus evolve via a Kac-type master equation. The initial state of the reservoirs is given by two non-centered Maxwellian distributions; they have temperature $T_+$ and $T_-$ and have average velocity $\vec p_+$ and $\vec p_-$, respectively. We prove that, for times shorter than $\sqrt{N}/M$, the interaction with the two reservoirs is well-approximated by the interaction with two shearing dynamic Maxwellian thermostats (i.e. heat reservoirs with $N=\infty$). As a byproduct of our analysis, we obtain a uniform in time approximation when $T_+=T_-$ and $\vec p_+=\vec p_-$.