In this paper, we investigate discrete regularity estimates for a broad class of temporal numerical schemes for parabolic stochastic evolution equations. We provide a characterization of discrete stochastic maximal ell ^p-regularity in terms of its continuous counterpart, thereby establishing a unified framework that yields numerous new discrete regularity results. Moreover, as a consequence of the continuous-time theory, we establish several important properties of discrete stochastic maximal regularity such as extrapolation in the exponent p and with respect to a power weight. Furthermore, employing the H^infty -functional calculus, we derive a powerful discrete maximal estimate in the trace space norm D_A(1-frac{1}{p},p) for p in [2,infty ).

Abstract We build a solvability theory of elliptic boundary-value problems in normed Sobolev spaces of generalized smoothness for any integrability exponent $$p>1$$ p > 1 . The smoothness is given by a number parameter and a supplementary function parameter that varies slowly at infinity. These spaces are obtained by a combination of the methods of the complex interpolation with number parameter between Banach spaces and the quadratic interpolation with function parameter between Hilbert spaces applied to classical Sobolev spaces. We show that the spaces under study admit localization near a smooth boundary and describe their trace spaces in terms of Besov spaces with the same supplementary function parameter. We prove that a general differential elliptic problem induces Fredholm bounded operators on appropriate pairs of the spaces under study. We also find exact sufficient conditions for solutions of the problem to have a prescribed generalized or classical smoothness on a given set and establish corresponding a priori estimates of the solution. These results are specified for parameter-elliptic problems.

As flip chip packaging technologies advance, silicon-level failure highlights the importance for robust design evaluation methodologies early in the development cycle. To support this need, we have developed a multi-scale finite element analysis (FEA) framework that includes both 3D and 2D multi-scale simulations, to capture stress distributions at critical failure locations. These models demonstrated comparable stress trends, validating their effectiveness in identifying high-risk regions. To further investigate micro-crack propagation risks at submicron scales, a 2D fracture mechanics model is introduced. This model enables detailed analysis of how key design parameters, such as aluminum pad (AP) spacing and BEOL metal trace spacing, influence failure susceptibility. Notably, these parameters exhibit opposing effects on two different failure modes, underscoring the importance of trade-off analysis. This paper presents a simulation-driven framework for assessing chip-level failure risks during the early stages of silicon-level design. Rather than prescribing specific design solutions, this study emphasizes the use of developed modeling framework to guide early-stage design exploration on silicon-level failure risk. The modeling framework provides a foundation for integrating modeling-driven insights into silicon-level design and process development, ensuring robust chip design to avoid failures downstream in packages.

Abstract We study the space of traces associated with arbitrary full free products of unital, separable $C^*$ -algebras. We show that, unless certain basic obstructions (which we fully characterize) occur, the space of traces always results in the same object: the Poulsen simplex, that is, the unique infinite-dimensional metrizable Choquet simplex whose extreme points are dense. Moreover, we show that whenever such a trace space is the Poulsen simplex, the extreme points are dense in the Wasserstein topology. Concretely for the case of groups, we find that, unless the trivial character is isolated in the space of characters, the space of traces of any free product of non-trivial countable groups is the Poulsen simplex. Our main technical contribution is a new perturbation result for pairs of von Neumann subalgebras $(M_{1},M_{2})$ of a tracial von Neumann algebra M , providing necessary conditions under which $M_{1}$ and a small unitary perturbation of $M_{2}$ generate a II $_{1}$ factor.

This paper designs a W-band planar structure double-beam magnetron injection gun based on the adiabatic compression theory and the momentum conservation law for a planar gyrotron. Using CST software for simulation and optimization, an obliquely placed cathode is introduced to correct the electron beam deflection when an axisymmetric magnetic field is used as usual. The design parameters of the electron gun are an anode voltage of 60 kV and a beam current of 5 A, with a transverse-to-axial velocity ratio of 1.15. The transverse velocity spread is 7.25%, and the longitudinal one is 10.12%. The electron gun was then fabricated and experimentally tested. The experimental trace spacing and deflection angle are consistent with the simulation. Furthermore, the electron data produced with the ParticleTracking module are used in the particle-in-cell (PIC) calculations. The PIC results exhibit the output power can achieve 62.5 kW, which is 85% of the theoretical prediction (73.8 kW).

To address the demand for high-precision exploration of natural gas hydrates in the northern South China Sea, this paper presents a novel high-precision small-scale 3D seismic exploration technology. The research team independently developed a seismic acquisition system, incorporating innovative designs such as a narrow trace spacing of 3.125 m and a short streamer length of 150 m. By integrating advanced processing techniques, including pre-stack noise suppression, spectral broadening, and refined velocity analysis, the system significantly enhances the precision and spatial resolution of shallow seismic data. During field trials in the Qiongdongnan basin, the system successfully acquired 3D seismic data over an area of 50 km2, enabling fine-scale imaging of sub-seabed strata within the upper 300 m. This represents a notable improvement in resolution compared to conventional 3D seismic technologies. When benchmarked against international counterparts such as P-cable, our system demonstrates distinct advantages in terms of exploration depth (reaching 1800 m) and dominant frequency range (spanning 10~390 Hz). The research findings provide a reliable technical approach for the detailed characterization of natural gas hydrates and the inversion of reservoir parameters, thereby holding significant practical value for advancing the industrial development of natural gas hydrates in China’s offshore areas.

Abstract. This study revisits seismic reflection data from the Central Scandinavian Caledonides, initially acquired during campaigns in the late 1980s and early 1990s. Previous analyses faced challenges in merging and imaging due to varying trace spacing and data gaps, particularly in the western parts. To address these limitations, we spatially resampled the data to a consistent trace spacing, carefully merged segments, and migrated the entire merged section. This approach resulted in a revised seismic profile, with notable changes in the western section where the image reveals key differences compared to earlier interpretations. The updated profile indicates near-continuous reflections across merged segments, resolving issues of abrupt breaks present in some prior publications. Enhanced imaging in the western section unveils new structural details, including collapsed diffractions and shorter reflective segments, offset from one another. These reflecting segments in the Skardöra antiform are interpreted as representing dolerite sills that were once continuous over a larger area but have been offset by normal faulting. This reinterpretation suggests a significantly thinner upper allochthon in the west than in previous interpretations. These results emphasize the importance of careful data integration and migration for seismic interpretation, shedding new light on the structural complexity of the western Scandinavian Caledonides. The study contributes to refining geological models and advancing understanding of the region's tectonic history.

Abstract The uniform tracial completion of a ‐algebra with compact trace space is obtained by completing the unit ball with respect to the uniform 2‐seminorm . The trace problem asks whether every trace on the uniform tracial completion is the ‐continuous extension of a trace on . We answer this question positively in the case of ‐algebras that tensorially absorb the Jiang–Su algebra, such as those studied in the Elliott classification programme.

Surface-related multiple suppression is a critical step in seismic data processing, while traditional adaptive matching subtraction methods often distort primaries, resulting in either the leakage of primaries or the residue of surface-related multiples. To address these challenges, we propose a field-parameter-guided semi-supervised learning (FPSSL) method to more effectively eliminate surface-related multiples. Field parameters refer to the time–space coordinate information derived from the seismic acquisition system, including offsets, trace spaces, and sampling intervals. These parameters reveal the relative positional relationships of seismic data in the time–space domain. The FPSSL framework comprises a supervised network module (SNM) and an unsupervised network module (USNM). The input and output data of the SNM are a small sample of full wavefield data and the weights of a polynomial function, respectively. A linear weighted sum method is employed to represent the SNM outputs (weights), the full wavefield data, and field parameters as a polynomial function of the primaries, which is matched with adaptive subtraction label data. The trained SNM generates preliminary estimates of the primaries and multiples with improved lateral continuity from full wavefield data, both of which are used as inputs to the USNM. The USNM is essentially an optimization operator that refines the underlying nonlinear mapping relationship between primaries and full wavefield data using the local wavefield feature loss function, thereby obtaining more accurate prediction results with respect to primaries. Examples from synthetic data and real marine data demonstrate that the FPSSL method surpasses the traditional L1-norm adaptive subtraction method in suppressing multiples, significantly reducing the leakage of primaries and the residuals of surface-related multiples in the estimated demultiple results. The effectiveness and efficiency of our proposed method are verified through two sets of synthetic data and one marine data example.

The high input/output demands of memory packages require precise trace width and spacing, posing challenges for contemporary package design. Substrate copper trace cracks are a major reliability issue during temperature cycling tests (TCTs). This study offers a detailed analysis of copper trace crack mechanisms through experimental observations, material characterization, and numerical simulations. Common failure modes of trace cracks are identified from experimental data, pinpointing initiation sites and propagation paths. Young's modulus of copper foil samples is assessed using four testing methods, revealing consistent trends across samples from different substrate suppliers. Sample A with higher E/H values tested via nanoindentation correlated with lower failure rates in the experiment. Stress-strain testing on copper foil was successfully performed at the lower TCT temperature limit of -65 °C, providing vital input for finite element (FE) models. The simulations show strong alignment with trace crack locations under different failure modes. The impact of copper trace width and material properties is illustrated in numerical models by comparing variations in plastic strain responses, which show differences of up to 40% and 30%, respectively. The simulation design of the experiments (DOE) indicates that high-strength solder resist (SR) can significantly enhance temperature cycling performance by reducing SR and copper trace stress and strain by up to 75%. The accumulation of plastic strain in copper traces is predicted to increase up to four times when SR breaks at the crack location, underscoring the importance of SR in copper trace reliability.

We report a surprising and deep structural property of boundary integral operators occurring in first-kind boundary integral equations associated with Hodge–Dirac and Hodge–Laplace operators for de Rham Hilbert complexes on a bounded domain \Omega in a Riemannian manifold. We show that, as regards their induced bilinear forms, those boundary integral operators are Hodge–Dirac and Hodge–Laplace operators in the weak sense, this time set in a trace de Rham Hilbert complex on the boundary \partial\Omega whose underlying spaces of differential forms are equipped with non-local inner products defined through layer potentials. On the way to this main result we conduct a thorough analysis of layer potentials in operator-induced trace spaces and derive representation formulas.

Stress solution of static linear elasticity with mixed boundary conditions via adjoint linear operators

Abstract Over the past years, rapid progress has been made on soft‐matter electronics for wearable and implantable devices, for bioelectronics and optogenetics. Liquid Metal (LM) based electronics are especially popular, due to their long‐term durability, when subject to repetitive strain cycles. However, one major limitation has been the need for tethering bioelectronics circuits to external power, or the use of rigid bulky batteries. This has motivated a growing interest in wireless energy transfer, which demands circuit miniaturization. However, miniaturization of LM circuits is challenging due to low LM‐substrate adhesion, LM smearing, and challenges on microchip‐interfacing. In this article, these challenges are addressed by high‐resolution laser‐assisted micropatterning of biphasic LM composites and vapor‐assisted LM microchip soldering. Through the development of a search algorithm for optimization of the biphasic ink coil performance, micro coils with trace spacing of 50 µm are designed and implemented that can harvest a significant amount of energy (178 mW cm−2) through near field inductive coupling. Miniaturized soft‐matter circuits with integrated SMD chips such as NFC chips, capacitors, and LEDs that are implemented in a few minutes through laser patterning, and vapor‐assisted soldering. In the context of optogenetics, where lightweight, miniaturized systems are needed to provide optical stimulation, soft coils stand out in terms of their improved conformability and flexibility. Thus, this article explores the applications of soft coils in wearable and implantable devices, with a specific focus on their use in optogenetics.

We show that the space of traces of free products of the form C(X1)⁎C(X2), where X1 and X2 are compact metrizable spaces without isolated points, is a Poulsen simplex, i.e., every trace is a pointwise limit of extreme traces. In particular, the space of traces of the free group Fd on 2≤d≤∞ generators is a Poulsen simplex, and we demonstrate that this is no longer true for many virtually free groups. Using a similar strategy, we show that the space of traces of the free product of matrix algebras Mn(C)⁎Mn(C) is a Poulsen simplex as well, answering a question of Musat and Rørdam for n≥4. Similar results are shown for certain faces of the simplices above, such as the face of finite-dimensional traces or amenable traces.

Glass has excellent electrical and mechanical properties for advanced electronic packaging, and even photonics applications. With a large bulk resistance, low moisture sensitivity, and dimensional stability, glass wafers have properties that allow them to be inserted in many of the processes typically used for silicon, and also allow advantaged performance in some applications, especially those involving high frequency. Thin glass is required to keep latency low and form-factor contained. In this presentation, we will describe the capability to produce void-free, hermetic, copper-filled vias, ranging in aspect ratio from 2:1 to 8:1. We also describe progress in multilayer metallization of thin glass interposers. Five redistribution layers (3 on one side, and 2 on the other) are explored in order to accommodate a relatively high density of interconnects. Daisy-chain structures demonstrate continuity as a function of RDL trace line and space. We report bow measurements on thin glass, exploring the effects of film stress of multiple RDL layers both in a configuration supported by a temporary handle, and free-standing.

In the decomposition of gauge-theory amplitudes into kinematic and color factors, the color factors (at a given loop order L) span a proper subspace of the extended trace space (which consists of single and multiple traces of generators of the gauge group, graded by powers of N). Using an iterative process, we systematically construct the L-loop color space of four-point amplitudes of fields in the adjoint representation of SU(N), SO(N), or Sp(N). We define the null space as the orthogonal complement of the color space. For SU(N), we confirm the existence of four independent null vectors (for L ≥ 2) and for SO(N) and Sp(N), we establish the existence of seventeen independent null vectors (for L ≥ 5). Each null vector corresponds to a group-theory constraint on the color-ordered amplitudes of the gauge theory.

Abstract On the reference tetrahedron $$K$$ K , we construct, for each $$k \in {\mathbb {N}}_0$$ k ∈ N 0 , a right inverse for the trace operator $$u \mapsto (u, \partial _{\textbf{n}} u, \ldots , \partial _{\textbf{n}}^k u)|_{\partial K}$$ u ↦ ( u , ∂ n u , … , ∂ n k u ) | ∂ K . The operator is stable as a mapping from the trace space of $$W^{s, p}(K)$$ W s , p ( K ) to $$W^{s, p}(K)$$ W s , p ( K ) for all $$p \in (1, \infty )$$ p ∈ ( 1 , ∞ ) and $$s \in (k+1/p, \infty )$$ s ∈ ( k + 1 / p , ∞ ) . Moreover, if the data is the trace of a polynomial of degree $$N \in {\mathbb {N}}_0$$ N ∈ N 0 , then the resulting lifting is a polynomial of degree N. One consequence of the analysis is a novel characterization for the range of the trace operator.

In October 2021, I interviewed three Chinese professional dancers who had completed undergraduate dance degrees in China and pursued their MFA in Dance degrees in the US. Their bi-cultural experiences and unique training narratives motivated me to re-search the Chinese dancing bodies beyond the borders. In this article, I draw on Robin Nelson’s methodologies of Practice/Performance as Research and Diana Taylor’s Performance Studies theory of the archive to examine the performativity of my interviewees’ dance experiences and my own educational experiences in both the US and in China. I explore how Chinese dancers, with their distinctive, yellow-skinned appearance, embody the national spirit and cultural traditions of China as they navigate various dance practices across the diverse landscape of the United States. Through their daily movements, these dancers inhabit a space of in-betweenness, leaving indelible but untraceable traces of their experiences on their bodies. My analysis delves into how these dancers, through their encounters in the field of dance and performance in the US, reinterpret their cultural memories, social identities, and socialist ideologies through choreography. Drawing from my own cross-border experiences between the US and China, I approach the examination of their performances through interviews, textual analysis, and movement interpretations.

As the need for chiplet integration becomes more prevalent in industry, the current standard of silicon interposers is not sustainable for broad adoption. Among the top technologies competing in this race are embedded bridge chip structures, chips-last fan-out, and chips first-fan out. Deca’s M-SeriesTM, a fully organic chips-first planar structure, promises a new way to achieve equivalent density to benchmark high density interconnects without the need for the complex structures and processes. Deca Technologies, Siemens EDA, and ASE formed a collaboration to design, verify, build, and perform physical analysis on an M-Series chiplet test vehicle. For this proof of concept, a 10-die chiplet package was designed using Siemens EDA’s Xpedition high density advanced packaging (HDAP) technologies. Siemens EDA’s Calibre was then used to perform verification and signoff. With design, verification and signoff complete, follow up plans are to build a test vehicle with ASE for physical characterization. Deca’s technology has surpassed other fan-out technologies in the chiplet integration race with Adaptive Patterning achieving significantly higher density for both vias and traces while enabling designs with the tightest device bond pad pitch. The planar surface of M-Series enables trace width and spacing of 2 microns and Adaptive Patterning makes possible a smaller copper stud size (competitors add an additional copper layer to increase the size) yielding a 20 micron copper stud pitch. In addition to an 80% reduction in the area required for a set number of copper studs, this technology also provides the highest integrity electrical connections with superior via contact areas. When compared to competing technologies, a 500% increase in via contact area can be achieved for the same bond pad pitch. This test vehicle design proves that Adaptive Patterning provides a compelling solution which surpasses competitors in the chiplet integration race.

Abstract We introduce the notion of tracial amenability for actions of discrete groups on unital, tracial C$$^*$$ ∗ -algebras, as a weakening of amenability where all the relevant approximations are done in the uniform trace norm. We characterize tracial amenability with various equivalent conditions, including topological amenability of the induced action on the trace space. Our main result concerns the structure of crossed products: for groups containing the free group $$F_2$$ F 2 , we show that outer, tracially amenable actions on simple, unital, $$\mathcal {Z}$$ Z -stable C$$^*$$ ∗ -algebras always have purely infinite crossed products. Finally, we give concrete examples of tracially amenable actions of free groups on simple, unital AF-algebras.