Trace Space Research Articles

All-loop group-theory constraints for four-point amplitudes of SU(N), SO(N), and Sp(N) gauge theories

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.

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

We revisit stress problems in linear elasticity to provide a perspective from the geometrical and functional-analytic points of view. For the static stress problem of linear elasticity with mixed boundary conditions we present the associated pair of unbounded adjoint operators. Such a pair is explicitly written for the first time, despite the abundance of the literature on the topic. We use it to find the stress solution as an intersection of the (affinely translated) fundamental subspaces of the adjoint operators. In particular, we treat the equilibrium equation in the operator form, which involves the spaces of traces on a part of the boundary, known as the Lions-Magenes spaces. Our analysis of the pair of adjoint operators for the problem with mixed boundary conditions relies on the properties of the analogous pair of operators for the problem with the displacement boundary conditions, which we also include in the paper.

Inter-Viewing Socialist Dancing Bodies

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.

Tracially amenable actions and purely infinite crossed products

AbstractWe 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.

Investigation of REV scale and anisotropy for 2D permeable fracture Networks: The role of geological entropy

To investigate the intrinsic association between geometrical properties and permeability of anisotropic fractured media, a methodology of geological entropy is applied to anisotropic fracture networks for the evaluation of seepage fields. By integrating the synthetic effect of fracture geometrical parameters (e.g., trace length, spacing, dip angle, and aperture), the global connectivity of fracture networks is quantified by the entropy scale (Hs), while the directional entropy scale (Hs,α) is used to characterize the local geometrical anisotropy in arbitrary orientations. A series of hydraulic representative elementary volumes (REVs) of fracture patterns with various geometrical properties are acquired based on continuum analysis. The permeability in the continuum scale is obtained from steady-state flow simulations. Sensitivity analyses are performed to understand the geometrical dependence of permeability parameters regarding entropy indices. It is found that a systematic increase in trace length, spacing, and dip angle is correlated with an increase, decrease, and equilibrium in entropy, respectively, which corresponds to a change in hydraulic REV sizes that can be expressed as a power-law function related to the Hs. Moreover, Hs,α can well explain the permeability anisotropy resulting from the spatial order of fracture networks, and a unified mathematical relationship for the effective permeability coefficients in arbitrary orientations versus the Hs,α is proposed, independent of variations in fracture patterns and scales. The results indicate that geological entropy, related to the geometrical properties and spatial distribution of fractures, is appropriate to characterize the permeability of anisotropic systems.

Trace Operator on von Koch’s Snowflake

AbstractWe study properties of the boundary trace operator on the Sobolev space $$W^1_1(\Omega )$$ W 1 1 ( Ω ) . Using the density result by Koskela and Zhang (Arch. Ration. Mech. Anal. 222(1), 1-14 2016), we define a surjective operator $$Tr: W^1_1(\Omega _K)\rightarrow X(\Omega _K)$$ T r : W 1 1 ( Ω K ) → X ( Ω K ) , where $$\Omega _K$$ Ω K is von Koch’s snowflake and $$X(\Omega _K)$$ X ( Ω K ) is a trace space with the quotient norm. Since $$\Omega _K$$ Ω K is a uniform domain whose boundary is Ahlfors-regular with an exponent strictly bigger than one, it was shown by L. Malý (2017) that there exists a right inverse to Tr, i.e. a linear operator $$S: X(\Omega _K) \rightarrow W^1_1(\Omega _K)$$ S : X ( Ω K ) → W 1 1 ( Ω K ) such that $$Tr \circ S= Id_{X(\Omega _K)}$$ T r ∘ S = I d X ( Ω K ) . In this paper we provide a different, purely combinatorial proof based on geometrical structure of von Koch’s snowflake. Moreover we identify the isomorphism class of the trace space as $$\ell _1$$ ℓ 1 . As an additional consequence of our approach we obtain a simple proof of the Peetre’s theorem (Special Issue 2, 277-282 1979) about non-existence of the right inverse for domain $$\Omega $$ Ω with regular boundary, which explains Banach space geometry cause for this phenomenon.

Dynamical comparison and [formula omitted]-stability for crossed products of simple C⁎-algebras

We establish Z-stability for crossed products of outer actions of amenable groups on Z-stable C⁎-algebras under a mild technical assumption which we call McDuff property with respect to invariant traces. We obtain such result using a weak form of dynamical comparison, which we verify in great generality. We complement our results by proving that McDuffness with respect to invariant traces is automatic in many cases of interest. This is the case, for instance, for every action of an amenable group G on a classifiable C⁎-algebra A whose trace space T(A) is a Bauer simplex with finite dimensional boundary ∂eT(A), and such that the induced action G↷∂eT(A) is free. If G=Zd and the action G↷∂eT(A) is free and minimal, then we obtain McDuffness with respect to invariant traces, and thus Z-stability of the corresponding crossed product, also in case ∂eT(A) has infinite covering dimension.

Robust nonlocal trace spaces and Neumann problems

We prove trace and extension results for fractional Sobolev spaces of order s∈(0,1). These spaces are used in the study of nonlocal Dirichlet and Neumann problems on bounded domains. The results are robust in the sense that the continuity of the trace and extension operators is uniform as s approaches 1 and our trace spaces converge to H1/2(∂Ω). We apply these results in order to study the convergence of solutions of nonlocal Neumann problems as the integro-differential operators localize to a symmetric, second order operator in divergence form.

ON THE CORRECT STATEMENTS OF BOUNDARY VALUE PROBLEMS FOR PDE'S IN BANACH-SOBOLEV SPACES

Abstract. In this report we consider the Banach Functional Space    X , defined on measure space   m ;  , where n R   is bounded domain, m is the Lebesgue measure in n R . It is defined the Banach Sobolev Space    m X W , generated by norm of    X . Regarding these spaces the elliptic equation is considered and the concepts of classical, local, weak and strong solutions are given. One method for defining the trace, trace operator and trace space generated by space    1 X W are introduced. These concepts allow us to consider the boundary value problems for differential equations regarding the spaces    m X W . Finally, it is considered the Dirichlet problem for polyharmonic equations in Sobolev spaces, generated by norm of symmetric spaces and the correct solvability of the corresponding problem is proved.

A unified hp-HDG framework for Friedrichs' PDE systems

This work proposes a unified hp-adaptivity framework for hybridized discontinuous Galerkin (HDG) method for a large class of partial differential equations (PDEs) of Friedrichs' type. In particular, we present unified hp-HDG formulations for abstract one-field and two-field structures and prove their well-posedness. In order to handle non-conforming interfaces we simply take advantage of HDG built-in mortar structures. With split-type mortars and the approximation space of trace, a numerical flux can be derived via Godunov approach and be naturally employed without any additional treatment. As a consequence, the proposed formulations are parameter-free. We perform several numerical experiments for time-independent and linear PDEs including elliptic, hyperbolic, and mixed-type to verify the proposed unified hp-formulations and demonstrate the effectiveness of hp-adaptation. Two adaptivity criteria are considered: one is based on a simple and fast error indicator, while the other is rigorous but more expensive using an adjoint-based error estimate. The numerical results show that these two approaches are comparable in terms of convergence rate even for problems with strong gradients, discontinuities, and singularities.

Dynamic positioning method and precision analysis of marine seismic vertical cables

AbstractSimilar to conventional ocean bottom seismometers and ocean bottom cables, the attitude and position of vertical cables deployed on the seafloor should be calibrated. However, due to the effect of ocean currents, the cable attitude is not stationary. Therefore, a new method for the dynamic repositioning of vertical cables is required. The dynamic repositioning of the hydrophones on vertical cables plays an important role in accurately describing the structure of the stratum and establishing an accurate velocity model. To realize the dynamic positioning of marine vertical cables, the shot lines are divided according to time, the first break time objective equation is established and the receiver position in each time period is obtained using the inversion methods outlined in this work. By analysing the trace spacing, first break time, shot point coordinate accuracy, temporal resolution, shot line spacing and other factors after repositioning, it is found that the first break time pickup accuracy and shot line spacing are the main factors that affect the repositioning accuracy. Furthermore, the resolution is limited by the timing of a single shot line. Additionally, the trace spacing after repositioning is consistent with the actual situation, which verifies the correctness and feasibility of this method.

An Adaptive and Quasi-periodic HDG Method for Maxwell’s Equations in Heterogeneous Media

With the aim to continue developing a hybridizable discontinuous Galerkin (HDG) method for problems arisen from photovoltaic cells modeling, in this manuscript we consider the time harmonic Maxwell’s equations in an inhomogeneous bounded bi-periodic domain with quasi-periodic conditions on part of the boundary. We propose an HDG scheme where quasi-periodic boundary conditions are imposed on the numerical trace space. Under regularity assumptions and a proper choice of the stabilization parameter, we prove that the approximations of the electric and magnetic fields converge, in the L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{L}^2$$\\end{document}-norm, to the exact solution with order hk+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$h^{k+1}$$\\end{document} and hk+1/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$h^{k+1/2}$$\\end{document}, resp., where h is the meshsize and k the polynomial degree of the discrete spaces. Although, numerical evidence suggests optimal order of convergence for both variables. An a posteriori error estimator for an energy norm is also proposed. We show that it is reliable and locally efficient under certain conditions. Numerical examples are provided to illustrate the performance of the quasi-periodic HDG method and the adaptive scheme based on the proposed error indicator.

ProtFe: Low-Cost Secure Power Side-Channel Protection for General and Custom FeFET-Based Memories

Ferroelectric Field Effect Transistors (FeFETs) have spurred increasing interest in both memories and computing applications, thanks to their CMOS compatibility, low-power operation, and high scalability. However, new security threats to the FeFET-based memories also arise. A major threat is the power analysis side-channel attack (P-SCA), which exploits the power traces of the memory access to obtain data information. There have been several effective efforts on resistive nonvolatile memories (NVMs), but they fail to meet the requirements for secure FeFET-based memories due to the different capacitive FeFETs load. Directly applying these existing countermeasures to the P-SCA protection for FeFETs induces huge challenges, especially for the balance between power side-channel resistance and corresponding overheads. To address this issue, we leverage the unique features of FeFETs and propose ProtFe , namely the protection methods for FeFET-based memories, including the pipelined multi-step write strategy ( PiMWrite ) and the split array design ( SpA ). PiMWrite is proposed for general FeFET-based memories, and inserts specially designed intermediate states to mitigate information leakage with pipelined steps to reduce overheads. SpA is proposed for custom FeFET-based memories, and simultaneously writes two split portions of the array with shared minimized peripherals to go beyond the balance between security and overheads. Simulation results show that PiMWrite expands the search space of a single power trace to 21× and involves nearly zero hardware penalties. SpA presents 33× search space improvement with negligible latency, 0.6% area, and only 7.1% energy overhead. ProtFe achieves improved balance between security and overheads, compared with the state-of-the-art works.

Semi‐automatic identification of discontinuity parameters in rock masses based on Unmanned Aerial Vehicle photography

Efficient and accurate extraction of discontinuity geometric parameters is significant for evaluating the stability and seepage characteristics of a rock mass. Traditional means of rock discontinuity measurement are susceptible to the influence of terrain and have issues such as high operational workload and low efficiency. An approach was proposed to comprehensively and efficiently obtain the geometric parameter information of the discontinuity, using an Unmanned Aerial Vehicle (UAV), which consists of the following five procedures: (1) measurement of a rock slope with UAV, (2) creation of three‐dimensional (3D) point cloud model for the slope, (3) voxel filtering of point cloud data, (4) planar segmentation based on modified region growing (MRG) algorithm, and (5) acquisition of parameter set of the discontinuity (orientation, trace length, and spacing). A new code, FacetDetect, has been developed based on this method to identify discontinuity from a 3D point cloud. Meanwhile, the trace length and spacing are calculated by the code written in Python. A comparison with the three‐point method reveals that most errors are less than 3° for dip angles and dip directions. These deviations are reasonable and confirm the reliability of the method. Overall, this method is a valuable reference for automatic discontinuity interpretation and related fields.

Evaluation of Dimensional Characteristics of Conductive Pattern Elements of 3D-Printed Circuit Boards

Introduction. A review of foreign publications on 3D printing methods shows the possibility of their application for the manufacture of printed circuit boards (PCBs), while demonstrating the growth of interest in this field. One of the available methods is fused deposition modeling, which can be used to form substrates with channels for traces. At present, the minimum channel width comprises 100 µm, which corresponds to the 5th class of PCB accuracy. However, there is no data on how to obtain such dimensions or on the magnitude of their deviation from the given nominal values.Aim. To determine the influence of such factors as nozzle diameter and the number of contours on the dimensional characteristics of PCBs (linear dimensions; trace channel width; trace spacing; via diameters).Materials and methods. To obtain linear models, a full factorial experiment was carried out. The influence of the following materials was evaluated – polylactide (PLA), acrylonitrile butadiene styrene (ABS) and glass-filled ABS; slicers – Ultimaker Cura and PrusaSlicer; 3D printers – Picaso 3D Designer Classic 2016 and Hercules Strong 2017; brass nozzle with diameters ∅0.2 and ∅0.4 mm. The possibility of using brass nozzles coated with chemical nickel for printing with glass-filled filaments was considered. The study was conducted using the facilities of the "Center for Additive Technologies" of Bauman Moscow State Technical University.Results. The possibility of manufacturing PCBs up to the 2nd and 3rd accuracy classes using nozzles with a diameter of ∅0.4 and ∅0.2 mm, respectively, was shown. It was found that galvanic deposition of chemical nickel on a ∅0.2 mm brass nozzle makes it possible to increase the PCB wearability when printing with glass-filled filaments. Recommendations are proposed for selecting an optimal combination of nozzle diameter and the number of contours.Conclusion. The conducted analysis of mathematical models shows the significance of such parameters as nozzle diameter and the number of contours. The results can be used when creating new PCB layouts

Boundary value problems with rough boundary data

We consider linear boundary value problems for higher-order parameter-elliptic equations, where the boundary data do not belong to the classical trace spaces. We employ a class of Sobolev spaces of mixed smoothness that admits a generalized boundary trace with values in Besov spaces of negative order. We prove unique solvability for rough boundary data in the half-space and in sufficiently smooth domains. As an application, we show that the operator related to the linearized Cahn–Hilliard equation with dynamic boundary conditions generates a holomorphic semigroup in Lp(R+n)×Lp(Rn−1).

On research of dispersion characteristics of multi-component surface waves from traffic-induced seismic ambient noise

As one of the commonly used sources in passive surface wave exploration, traffic-induced seismic ambient noise is widely used in urban underground velocity structure detection at present. However, characteristics of the dispersion curve extracted from such kind of ambient noise lack systematical research, especially for the multi-component surface wave. Due to the complex character of the real-world underneath structures, dispersion characteristics concluded from real-world traffic-induced seismic ambient noise that is hard to verify. The numerical simulation technique is a good choice for systematical research of characteristics of dispersion curves extracted from multi-component traffic-induced seismic ambient noise. Most of the numerical examples that appeared in the present literature, however, are based on the dispersion-curve-based numerical simulation technique. Hence, most of these can only simulate the vertical component. In this paper, the two-dimensional finite difference method is used to simulate multi-component traffic seismic ambient noise. The characteristics of original traffic noise are analyzed, and the effect of trace intervals on surface wave imaging of different types of traffic-induced noise is studied. Rayleigh wave dispersion energy image is extracted by the passive surface-wave frequency wavenumber transformed method. The differences in Rayleigh-wave dispersion images in traffic noise with different components are analyzed and compared. The results show that Rayleigh wave dispersion energy images appearing different characteristics in different components. The surface wave dispersion information presented by different types of traffic noise is frequency-dependent. The trace spacing also strongly affect the surface-wave dispersion energy images extracted from different types of traffic-induced noise. The numerical results are significant for guiding the traffic-induced seismic noise data acquisition and processing.

Development of novel low-mass module concepts based on MALTA monolithic pixel sensors

The MALTA CMOS monolithic silicon pixel sensors has been developed in the Tower 180 nm CMOS imaging process. It includes an asynchronous readout scheme and complies with the ATLAS inner tracker requirements for the HL-LHC. Several 4-chip MALTA modules have been built using Al wedge wire bonding to demonstrate the direct transfer of data from chip-to-chip and to read out the data of the entire module via one chip only. Novel technologies such as Anisotropic Conductive Films (ACF) and nanowires have been investigated to build a compact module. A lightweight flex with 17 μm trace spacing has been designed, allowing compact packaging with a direct attachment of the chip connection pads to the flex using these interconnection technologies. This contribution shows the current state of our work towards a flexible, low material, dense and reliable packaging and modularization of pixel detectors.

Force Computation for Dielectrics Using Shape Calculus

Abstract We are concerned with the numerical computation of electrostatic forces/torques in only piece-wise homogeneous materials using the boundary element method (BEM). Conventional force formulas based on the Maxwell stress tensor yield functionals that fail to be continuous on natural trace spaces. Thus their use in conjunction with BEM incurs slow convergence and low accuracy. We employ the remedy discovered in [P. Panchal and R. Hiptmair, Electrostatic force computation with boundary element methods, SMAI J. Comput. Math. 8 (2022), 49–74]. Motivated by the virtual work principle which is interpreted using techniques of shape calculus, and using the adjoint method from shape optimization, we derive stable interface-based force functionals suitable for use with BEM. This is done in the framework of single-trace direct boundary integral equations for second-order transmission problems. Numerical tests confirm the fast asymptotic convergence and superior accuracy of the new formulas for the computation of total forces and torques.

Traces for Hilbert complexes

We study a new notion of trace operators and trace spaces for abstract Hilbert complexes. We introduce trace spaces as quotient spaces/annihilators. We characterize the kernels and images of the related trace operators and discuss duality relationships between trace spaces. We elaborate that many properties of the classical boundary traces associated with the Euclidean de Rham complex on bounded Lipschitz domains are rooted in the general structure of Hilbert complexes. We arrive at abstract trace Hilbert complexes that can be formulated using quotient spaces/annihilators. We show that, if a Hilbert complex admits stable “regular decompositions” with compact lifting operators, then the associated trace Hilbert complex is Fredholm. Incarnations of abstract concepts and results in the concrete case of the de Rham complex in three-dimensional Euclidean space will be discussed throughout.