We study the rapid stabilization of general linear systems, when the differential operator $\mathcal{A}$ has a Riesz basis of eigenvectors. We find simple sufficient conditions for the rapid stabilization and the construction of a relatively explicit feedback operator. We use an $F$-equivalence approach \textcolor{black}{relying on Fredholm transformation} to show a stronger result: under these sufficient conditions the system is equivalent to a simple exponentially stable system, with arbitrarily large decay rate. In particular, our conditions improve the existing conditions of rapid stabilization for non-parabolic operators such as skew-adjoint systems.

Configuring stream processing systems for efficient performance, especially in cloud-native deployments, is a challenging and largely manual task. We present an experiment-driven approach for automated configuration optimization that combines three phases: Latin Hypercube Sampling for initial exploration, Simulated Annealing for guided stochastic search, and Hill Climbing for local refinement. The workflow is integrated with the cloud-native Theodolite benchmarking framework, enabling automated experiment orchestration on Kubernetes and early termination of underperforming configurations. In an experimental evaluation with Kafka Streams and a Kubernetes-based cloud testbed, our approach identifies configurations that improve throughput by up to 23% over the default. The results indicate that Latin Hypercube Sampling with early termination and Simulated Annealing are particularly effective in navigating the configuration space, whereas additional fine-tuning via Hill Climbing yields limited benefits.

The contribution of the weak interaction to the hyperfine splitting of the ground state of muonium is investigated. The amplitudes of the one- and two-quantum exchange determined by the Z and W bosons are calculated. One-loop corrections in the photon and Z boson propagators and their contribution to the hyperfine structure of the spectrum are obtained.

We provide sufficient conditions for the existence of a trace on the essential $\mathrm{C}^\ast$-algebra of a (not necessarily Hausdorff) étale groupoid $G$ which extends an invariant measure $μ$ on the unit space of $G$. In particular, it suffices for the isotropy groups of $G$ to be amenable, or for $G$ to be essentially free with respect to $μ$. We also show that $G$ is essentially free with respect to an invariant measure $μ$ if and only if $μ$ extends to a unique trace on the full $\mathrm{C}^\ast$-algebra of $G$. We work in the generality of possibly infinite measures and, accordingly, possibly unbounded traces. Moreover, whenever possible, we state our results for twisted groupoids. As an application, we show that gauge-invariant algebras of finite-state self-similar groups admit a unique tracial state.

A type assignment system for lambda-calculus enjoys the principal typing property if every typable term M has a special typing, called principal, from which all typings for M can be obtained via suitable operations. The existence of principal typings in various intersection type disciplines has long been established using both semantical and syntactical approaches. Historically, on the syntactical side, proofs of this property and the description of type inference (semi-)algorithms computing principal typings have been complicated by many subtle technicalities; the present work aims at providing a more accessible formulation. To this end, we revisit some classical notions, and identify three elementary operations (substitution, expansion, erasure) that allow to build any type derivation in a system characterizing head normalization. We then use such operations in the design of an inference semi-algorithm that computes the principal typing of all and only the strongly normalizing terms, thus contributing to a modern perspective on results originally proven more than 40 years ago

Terms in the lambda-calculus can be represented as planar trees decorated with symbols for abstraction and application, and having variables as leaves. In this paper, we concentrate on the branches of such trees, rather than on the trees themselves. We reformulate several well-known notions of beta-reduction in this view. In a natural manner, this reconsideration eventually leads to a new form of beta-reduction, being expanding in the sense that the reduction of term t1 to term t2 entails that the tree of t1 is a subtree of the tree of t2.

We continue the investigation into the computational status of the existence of moduli of regularity (and their use for rates of convergence) in the sense of Kohlenbach, Lopez and Nicolae (2019), carried out w.r.t. classical reverse mathematics and Weihrauch degrees in a previous paper and determine the amount of LEM involved. We also show that the existence of a modulus of regularity always yields an algorithm for the computation of a zero in the case of continuous real-valued functions F on a compact metric space K (in F equipped with a modulus of uniform continuity and K given in standard representation) whenever such a zero exists. If K is a compact subset of a uniformly convex Banach space X and the zero set of F is convex one can compute even the zero of minimal norm. A modulus of regularity can also be used to compute the left-most infinite path of an infinite 0/1-tree. We also show that there is no proof-theoretically tame nonstandard uniformity principle which would make it possible to replace in the regularity assumption compactness by metric boundedness and still guarantee classically correct bounds.

Formal reasoning about inductively defined relations and structures is widely recognized not only for its mathematical interest but also for its importance in computer science, and has applications in verifying properties of programs and algorithms. Recently, several proof systems of inductively defined predicates based on sequent calculus including the cyclic proof system CLKID-omega and the infinite-descent proof system LKID-omega have attracted much attention. Although the relation among their provabilities has been clarified so far, the logical complexity of these systems has not been much studied. The infinite-descent proof system LKID-omega is an infinite proof system for inductive definitions and allows infinite paths in proof figures. It serves as a basis for the cyclic proof system. This paper shows that the logical complexity of the provability in LKID-omega is (Pi-1-1)-complete. To show this, first it is shown that the validity for inductive definitions in standard models is equivalent to the validity for inductive definitions in standard term models. Next, using this equivalence, this paper extends the truth predicate of omega-languages, as given in Girard's textbook, to inductive definitions by employing arithmetical coding of inductive definitions. This shows that the validity of inductive definitions in standard models is a (Pi-1-1) relation. Then, using the completeness of LKID-omega for standard models, it is shown that the logical complexity of the provability in LKID-omega is (Pi-1-1)-complete.

In the pure Calculus of Constructions (CC) one can define data types and function over these, and there is a powerful higher order logic to reason over these functions and data types. This is due to the combination of impredicativity and dependent types, and most of these features can already be observed in polymorphic (second order) dependent type theory $λ$P2. The impredicative encoding of data types (in $λ$P2 or CC) is powerful but not fully satisfactory: for example, the induction principle is not provable. As a matter of fact, it can be shown that induction is not provable for whatever possible representation of data types. In a recent paper, Awodey, Frey and Speight show that in an extension of $λ$P2 with Sigma-types, identity types with uniqueness of identity proofs and function extensionality, it is possible to define data types for which the induction principle is provable. More recently it has been shown that in this extension of $λ$P2, also quotient types can be defined with the proper induction principle, and, using quotient types, coinductive types can be defined with the proper coinduction principle. This leaves various questions open: Are quotient types with induction principle not definable in the original $λ$P2? And how about coinductive types, is it impossible to get a strong coinduction principle in $λ$P2? Looking at it from the other side: which of the extensions used are really needed to make induction and coinduction work? In this paper, we contribute partial answers to these questions: parametric quotient types are not definable in $λ$P2 and the well-known definable stream type does not have a coinduction principle. For the latter question we show that, if we just extend $λ$P2 with Sigma-types and identity types with uniqueness of identity proofs, we still cannot prove an induction principle for the natural numbers. So function extensionality is crucial in making induction provable. We show these results by studying models of $λ$P2 where the types representing these principles are empty, so these models act as counter models to the derivability of the principles.

In this paper we: (1) propose Lightweight C (LC), namely a core calculus that formalizes a proper subset of the ANSI C without preprocessor directives; (2) define Colored LC (CLC), namely LC endowed with ANSI C preprocessor directives; and (3) define a type system for CLC that guarantees that all programs to be generated by the C preprocessor are well-typed C programs. We believe that the simple formalization provided by CLC could be useful also for teaching purposes. Stefano Berardi spent most of his academic career at the Department of Computer Science of the University of Turin, where he conducts outstanding research on the logical foundations of computer science and on type-based program analyses. Over the years, he taught many courses, from BSc courses on programming with C to PhD courses on program analysis. Therefore, this paper fully falls within Stefano Berardi's research and teaching activities.

We present and explain two unpublished remarks of Stefano Berardi connected to game semantics.

Foundations of computer science are a key area in theoretical research, one to which Stefano has made significant contributions, particularly from a logical and proof-theoretic perspective. Recently, we have been involved, with him, in teaching an introductory course on this topic, guided by the idea that understanding and writing ordinary, discursive proofs is a valuable skill for future programmers. This shared experience has inspired the pedagogical approach at the basis of this paper. Behind specific foundational topics in computer science lie core techniques that are best taught through examples. However, standard textbooks often do not place enough emphasis on these ubiquitous techniques and frequently lack examples that are directly relevant to informatics. We believe that highlighting fundamental techniques, rather than focusing solely on specific foundational topics, would offer significant pedagogical benefits for an introductory course. In this paper we propose transitive closure of relations as a case study supporting our approach. While all proofs are elementary, we claim that this is a suitable topic for putting to work paradigmatic notions -- intrinsically tied to computational thinking -- that can serve as structural anchors for a course in the foundations of computer science. In particular, we highlight the techniques employed in proofs, that constitute a comprehensive summary of those that are normally taught in an introductory logic course, and the abstract structures, that allow to connect transitive closure with Kleene star (via quantales) and closure operators (on complete lattices). We then outline a series of further examples that may be used, as in our case study on stars, as a hands-on approach to basic analytic skills to be learned in a course on the foundations of computing. -- To Stefano Berardi on the occasion of his birthday.

We present generalized algebraic theories corresponding to slightly modified versions of two of the type theories in our paper Type Theory with Explicit Universe Polymorphism. We first present a generalized algebraic theory for categories with families with extra structure corresponding to Martin-Lof type theory with an external tower of universes. We then present a generalized algebraic theory for level-indexed categories with families with extra structure corresponding to Martin-Lof type theory with explicit universe polymorphism: a theory with universe level judgments, internally indexed universes, and level-indexed products. In this way we get abstract characterizations of the two theories as initial models of their respective generalized algebraic theories. We thus abstract from details of the grammar and inference rules of the type theories and highlight their high-level structure. More broadly, the present work can be viewed as a case study of a uniform approach to categorical logic based on generalized algebraic theories and categories with families. We also discuss the relevance to Voevodsky's initiality conjecture project.

This work introduces a novel framework of uniform realizability that unifies and generalizes various realizability interpretations of logic, particularly focussing on the treatment of atomic formulas and quantifiers. Traditional realizability interpretations (such as Kleene's number realizability) require explicit witnesses for existential quantifiers. In contrast, newer approaches, such as in the first author's uniform Heyting arithmetic, Herbrand realizability of non-standard arithmetic, or in the "classical" realizability of arithmetic, (some) quantifiers, are treated uniformly. The proposed notion of uniform realizability abstracts these differences, parametrising the interpretation by a given treatment of atomic formulas, accounting for both classical and modern variants. The approach is illustrated using several realizability interpretations of Heyting arithmetic.

Structural recursion is a common technique used by programmers in modern languages and is taught to introductory computer science students. But what about its dual, structural corecursion? Structural corecursion is an elegant technique, supported in languages like Haskell and proof assistants such as Rocq or Agda. It enables the design of compositional algorithms by decoupling the generation and consumption of potentially infinite or large data collections. Despite these strengths, structural corecursion is generally considered more advanced than structural recursion and is primarily studied in the context of pure functional programming. Our aim is to illustrate the expressive power of different notions of structural corecursion in the presence of classical reasoning. More specifically, we study coiteration and corecursion combined with the classical callcc operator, which provides a computational interpretation of classical reasoning. This combination enables interesting stream-processing algorithms. As an application, we present a corecursive, control-based proof of the Infinite Pigeonhole Principle and compare it with the continuation-passing proof of Escardó and Oliva in Agda. To further demonstrate the power of mixing corecursion and control, we give an implementation of the Axiom of Countable Choice. In contrast to the usual continuation-passing implementations of this axiom, which rely on general recursion whose termination is established externally, our approach justifies termination by coiteration alone.

Finitary/static semantics in the form of intersection type assignments have become a paradigm for analysing the fine structure of all sorts of lambda-models. The key step is the construction of a filter model isomorphic to a given lambda-model. A property of great interest of filter lambda-models is sensibility, i.e. the interpretation of all unsolvable terms is the least element. The flexibility of intersection type assignments derives from their parametrisation on intersection type theories. We construe intersection type theories as special meet-semilattices and show that appropriate morphisms, in the opposite category of meet-semilattices, preserve sensibility of the induced lambda-models. Interestingly the set of saturated sets together with the set of lambda-terms is such a meet-semilattice, thus showing that arguments based on Tait-Girards's computability amount to the construction of a morphism. We characterise two classes of intersection type theories which induce sensible filter models. The first is non-effective while the second is effective and it amounts to the generalisation of Mendler's criterion to intersection types and head normalising terms. The complete characterisation of sensible filter models however still escapes.

Dynamic structural equation models (DSEMs) combine time-series modeling of within-person processes with hierarchical modeling of between-person differences and differences between timepoints, and have become very popular for the analysis of intensive longitudinal data in the social sciences. An important computational bottleneck has, however, still not been resolved: whenever the underlying process is assumed to be latent and measured by one or more indicators per timepoint, currently published algorithms rely on inefficient brute-force Markov chain Monte Carlo sampling which scales poorly as the number of timepoints and participants increases and results in highly correlated samples. The main result of this paper shows that the within-level part of any DSEM can be reformulated as a linear Gaussian state space model. Consequently, the latent states can be analytically marginalized using a Kalman filter, allowing for highly efficient estimation via Hamiltonian Monte Carlo. This makes estimation of DSEMs computationally tractable for much larger datasets -- both in terms of timepoints and participants -- than what has been previously possible. We demonstrate the proposed algorithm in several simulation experiments, showing it can be orders of magnitude more efficient than standard Metropolis-within-Gibbs approaches.

A new phenomenological dark energy model, originally associated to the large-scale structure formation and considered as a solution to the fine-tuning and coincidence problems related to the cosmological constant, was analyzed within the framework of General Relativity in a Friedman-Robertson-Walker spacetime and its model parameters were estimated using cosmic chronometers and recent DESI data. It turns out that the proposed model can serve as an alternative evolving dark energy model with a novel equation of state function, apart from other popular propositions in the literature. Due to the form of this phenomenological energy density ansatz, which starts to rise with the nonlinear structure growth in the universe and falls with the domination of cosmic voids, we prefer to call it structure-induced dark energy. Observational constraints show that it is not only a suitable solution for the fundamental problems such as coincidence or fine-tuning problems, it gives flexibility, when considering the cosmic tensions and presents a new perspective on the evolving dark energy models.

Spatial autocorrelation in regression models can lead to downward biased standard errors and thus incorrect inference. The most common correction in applied economics is the spatial heteroskedasticity and autocorrelation consistent (HAC) standard error estimator introduced by Conley (1999). A critical input is the kernel bandwidth: the distance within which residuals are allowed to be correlated. However, this is still an unresolved problem and there is no formal guidance in the literature. In this paper, I first document that the relationship between the bandwidth and the magnitude of spatial HAC standard errors is inverse-U shaped. This implies that both too narrow and too wide bandwidths lead to underestimated standard errors, contradicting the conventional wisdom that wider bandwidths yield more conservative inference. I then propose a simple, non-parametric, data-driven bandwidth selector based on the empirical covariogram of regression residuals. In extensive Monte Carlo experiments calibrated to empirically relevant spatial correlation structures across the contiguous United States, I show that the proposed method controls the false positive rate at or near the nominal 5% level across a wide range of spatial correlation intensities and sample configurations. I compare six kernel functions and find that the Bartlett and Epanechnikov kernels deliver the best size control. An empirical application using U.S. county-level data illustrates the practical relevance of the method. The R package SpatialInference implements the proposed bandwidth selection method.

Sagittarius A* (Sgr A*) is a compact radio source at the Galactic center. Observations have confirmed that its mass is approximately (4.1)*10$^{6}$ M$_{\odot}$, and Sgr A* is generally believed to be powered by gas accretion onto a supermassive black hole. Multifrequency radio observations of the pulsar J1745-2900, about 0.12 pc away from Sgr A*, reveal an unusually large Faraday rotation. Combined with X-ray observations, this indicates that there is a strong magnetic field (greater than 8 mG) leading to a low $β$ plasma at large scales.We show that the gas starts to be captured by the black hole below tens of thousands of the Schwarzschild radii $r_S$, where the gas pressure starts to dominate. Assuming that the accretion rate along magnetic fields at large scales decreases with the distance to the black hole following a power law, it is shown that, with an accretion disk below tens of $r_S$, as revealed with the EHT observations, there should be a supersonic wind above such a small accretion disk, and the accretion flow may be convection-dominated from tens of $r_S$ to tens of thousands of $r_S$. Detailed modeling is warranted.

The Robinson Splitting Theorem states that a c.e. degree $\mathbf{b}$ splits over any low c.e. degree $\mathbf{c}<\mathbf{b}$. We prove that a weaker version of this theorem holds in models of $\mathrm{P}^-+\mathrm{I}Σ_1$, with lowness replaced by superlowness.

While Transformers have achieved remarkable success in LLMs through superior scalability, their application in industrial-scale ranking models remains nascent, hindered by the challenges of high feature sparsity and low label density. In this paper, we propose SORT (Systematically Optimized Ranking Transformer), a scalable model designed to bridge the gap between Transformers and industrial-scale ranking models. We address the high feature sparsity and low label density challenges through a series of optimizations, including request-centric sample organization, local attention, query pruning and generative pre-training. Furthermore, we introduce a suite of refinements to the tokenization, multi-head attention (MHA), and feed-forward network (FFN) modules, which collectively stabilize the training process and enlarge the model capacity. To maximize hardware efficiency, we optimize our training system to elevate the model FLOPs utilization (MFU) to 22%. Extensive experiments demonstrate that SORT outperforms strong baselines and exhibits excellent scalability across data size, model size and sequence length, while remaining flexible at integrating diverse features. Finally, online A/B testing in large-scale e-commerce scenarios confirms that SORT achieves significant gains in key business metrics, including orders (+6.35%), buyers (+5.97%) and GMV (+5.47%), while simultaneously halving latency (-44.67%) and doubling throughput (+121.33%).

Bounded continuous data on the unit interval frequently arise in applied fields and often exhibit a non-negligible proportion of observations at the boundaries. Inflated regression models address this feature by combining a continuous distribution on the unit interval with a discrete component to account for zero- and/or one-inflation. In this paper, we propose a class of Bayesian structured additive quantile regression models for inflated bounded continuous data that accommodates zero- and/or one-inflation. The proposed approach enables direct modeling of both the conditional quantiles of the continuous component and the probabilities of observing zeros and/or ones, with structured additive predictors incorporated in both parts, including nonlinear effects, spatial effects, random effects, and varying-coefficient terms. Posterior inference is carried out using Markov chain Monte Carlo algorithms implemented through the software Liesel, a probabilistic programming framework for semiparametric regression. The practical performance of the proposed models is illustrated through simulation studies and two real-data applications: one analyzing the proportion of traffic-related fatalities across Brazilian municipal districts, and another evaluating speech intelligibility in cochlear implant recipients under different experimental conditions.

An anomalous back discharge movement phenomenon is induced by a set of dielectric barrier corona discharges (DBCD) at unipolar half-sine voltage waveforms, where the back discharge has a time delay that relates to the applied voltage level. An ionic wind model is employed to analyze the physical behavior. Theoretical explanation and quantitative analysis are presented in this study based on abundant experimental results of 5 typical insulating materials and a FEP insulating cable. A numerical model is derived, which indicates that the back discharge can be activated under a relatively low potential voltage level in this study. The results highlight that the back discharge movement phenomenon behaves distinctly under half-sine voltage with negative polarity, yielding a significantly different partial discharge (PD) pattern with positive polarity. Besides, PD amplitude dependent on dielectric thickness is demonstrated by plotting in phase resolved partial discharge (PRPD) pattern. Furthermore, comparative experiments are conducted with respect to the variation of air gap length and dielectric geometry, manifesting different influences on PD amplitude.

We present a timing analysis of the high-cadence optical light curve of the high-redshift flat-spectrum radio quasar PKS\,0805$-$07 obtained during \textit{TESS} Sector~34 (MJD $59230.90$--$59239.90$). We search for short-timescale quasi-periodic oscillations (QPOs) using complementary time-series techniques, including the Lomb--Scargle periodogram (LSP) and the weighted wavelet $Z$-transform (WWZ), and evaluate their significance against red-noise variability using Monte Carlo simulations. The LSP reveals a dominant modulation at $f \approx 0.597\,\mathrm{d^{-1}}$ ($P \approx 1.7\,\mathrm{d}$) exceeding the $99.99\%$ confidence level, while the WWZ independently recovers a consistent timescale at the $\sim 99.9\%$ level and shows that the signal is temporally localized rather than persistent across the full light curve. The modulation spans $\sim 5$ coherent cycles, indicating a transient quasi-periodic feature. We discuss possible physical interpretations of the detected modulation. In a disk-based scenario, orbital motion of a hotspot near the innermost stable circular orbit implies a black hole mass of $M_{\rm BH}\sim7.2\times10^{8}\,M_\odot$, consistent with typical FSRQ values. Alternatively, magnetohydrodynamic kink instabilities in the relativistic jet can naturally produce day-scale variability for standard blazar parameters and account for the transient character of the signal. We conclude that the observed modulation is consistent with a compact, short-lived structure embedded within stochastic jet variability.

The graded Lie algebra associated with the Nottingham group over a field of prime characteristic serves as a fundamental example of Nottingham algebras, a class of infinite-dimensional, positively graded thin algebras. This paper completes the classification of Nottingham algebras initiated in earlier papers, proving both existence and uniqueness results that determine all such algebras up to isomorphism.

The integration of artificial intelligence as a design tool for metasurfaces, and the implementation of a deep-learning model pose a challenge in the development of an automated solution due to high resources requirements. The presented work introduces a network-layer solution to configure such environment for end user objectives, and for an underlying physical-layer technology. An architecture is developed to design an anomalous reflector by employing the Redhat Openshift AI (RHOAI) technology to support an automated machine learning operations (MLOps) framework in smart radio environments. This entails the design of lossless impenetrable metasurfaces characterized by a scalar surface impedance for an optimal anomalous reflection, achieved by optimizing the number of the Floquet modes through the utilization of a local power conservation constraint qualified as a fitness function. The metasurfaces design process is implemented by using a conditional generative adversarial network (cGAN). An extended cGAN with a surrogate model assists in a high-quality freeform metasurfaces design, where it introduces a swift simulation tool for the metasurfaces design process and analysis of the far-field model. The paper focuses on the challenges of building such a system, and potential abstraction layers. The training accuracy value of the proposed model demonstrates the feasibility and benefits of deploying in containerized environment of Red Hat Openshift in comparison with other deployments of ResNet-50 reported in literature.

Electronic phases in quantum materials are often governed by nanoscale inhomogeneity, where local order develops within spatially confined regions or puddles. A prominent example is an incommensurate charge-density-wave (I-CDW) that comprises locally commensurate domains. In 2$H$-NbSe$_2$, such an I-CDW state persists alongside lattice anharmonicity and superconductivity, raising fundamental questions about the dynamical stabilization of CDW order in puddles. Here, we probe the puddle-dynamics in 2$H$-NbSe$_2$. Raman scattering reveals a strong Fano-coupling between the interlayer shear vibration and the CDW amplitude mode. Time-resolved reflectivity measurement shows a low-frequency ~0.15 THz coherent overdamped oscillation onsetting within the CDW regime at ~17 K, pointing towards a so far unexplored transition. This we identify as a Fano-coupled phonon-CDW hybrid emerging from the collective dynamics of CDW puddles. These dynamics highlight how lattice pinning and electronic correlations in layered materials affect the CDW order, which is crucial for the design of novel Van der Waals devices.

Radiative-conductive systems are intrinsically nonlinear due to the quartic temperature dependence of thermal radiation. Under fixed total heating power, convexity arguments imply that nonuniform temperature distributions radiate more efficiently and therefore exhibit a lower mean temperature than their isothermal counterparts. However, this conclusion remains qualitative, and an explicit quantitative relation between temperature heterogeneity and mean temperature reduction has been lacking. Here we derive a variance-based analytical expression linking the area-averaged temperature to the corresponding isothermal equilibrium temperature in a nonuniformly heated radiative--conductive system. By integrating the governing equation and performing a systematic second-order expansion about the ambient temperature, we show that the decrease of the mean temperature relative to the isothermal equilibrium value is linearly proportional to the temperature variance, with a proportionality coefficient set solely by the ambient temperature. This result transforms the convexity-based inequality into a quantitative statistical relation within the perturbative regime and provides a physically transparent framework for describing nonlinear radiative averaging in thermally heterogeneous systems.

This paper establishes strong and weak convergence rates for slow-fast systems driven by $α$-stable processes with jump coefficients. Unlike existing studies on multiscale systems driven by additive Lévy white noise, our model incorporates multiplicative noise, which brings essential challenges in deriving the exponential ergodicity for the frozen process, particularly gradient estimates. We derive exponential ergodicity in two different ways: the coupling method and the spatial periodic method; then the gradient estimate is developed by heat kernel asymptotic expansion. Moreover, under sufficient Hölder regularity of the time-dependent coefficients of the slow process, we can yield an optimal strong convergence rate of order $1-\frac{1}{α_{2}}$ and a weak convergence rate of order 1. Furthermore, explicit formulas for the tangent map between tangent spaces of $S^{d-1}$ as well as its Jacobian determinant are obtained, where the map is induced by a nonlinear immersion.