Trace Map Research Articles

Quantum traces and embeddings of stated skein algebras into quantum tori

The stated skein algebra of a punctured bordered surface (or equivalently, a marked surface) is a generalization of the well-known Kauffman bracket skein algebra of unmarked surfaces and can be considered as an extension of the quantum special linear group \({\mathcal {O}}_{q^2}(SL_2)\) from a bigon to general surfaces. We show that the stated skein algebra of a punctured bordered surface with non-empty boundary can be embedded into quantum tori in two different ways. The first embedding can be considered as a quantization of the map expressing the trace of a closed curve in terms of the shear coordinates of the enhanced Teichmüller space, and is a lift of Bonahon-Wong’s quantum trace map. The second embedding can be considered as a quantization of the map expressing the trace of a closed curve in terms of the lambda length coordinates of the decorated Teichmüller space, and is an extension of Muller’s quantum trace map. We explain the relation between the two quantum trace maps. We also show that the quantum cluster algebra of Muller is equal to a reduced version of the stated skein algebra. As applications we show that the stated skein algebra is an orderly finitely generated Noetherian domain and calculate its Gelfand-Kirillov dimension.

Harnessing cortical plasticity via gabapentinoid administration promotes recovery after stroke.

Stroke causes devastating sensory-motor deficits and long-term disability due to disruption of descending motor pathways. Restoration of these functions enables independent living and therefore represents a high priority for those afflicted by stroke. Here, we report that daily administration of gabapentin, a clinically approved drug already used to treat various neurological disorders, promotes structural and functional plasticity of the corticospinal pathway after photothrombotic cortical stroke in adult mice. We found that gabapentin administration had no effects on vascular occlusion, haemodynamic changes nor survival of corticospinal neurons within the ipsilateral sensory-motor cortex in the acute stages of stroke. Instead, using a combination of tract tracing, electrical stimulation and functional connectivity mapping, we demonstrated that corticospinal axons originating from the contralateral side of the brain in mice administered gabapentin extend numerous collaterals, form new synaptic contacts and better integrate within spinal circuits that control forelimb muscles. Not only does gabapentin daily administration promote neuroplasticity, but it also dampens maladaptive plasticity by reducing the excitability of spinal motor circuitry. In turn, mice administered gabapentin starting 1 h or 1 day after stroke recovered skilled upper extremity function. Functional recovery persists even after stopping the treatment at 6 weeks following a stroke. Finally, chemogenetic silencing of cortical projections originating from the contralateral side of the brain transiently abrogated recovery in mice administered gabapentin, further supporting the conclusion that gabapentin-dependent reorganization of spared cortical pathways drives functional recovery after stroke. These observations highlight the strong potential for repurposing gabapentinoids as a promising treatment strategy for stroke repair.

The quantum trace as a quantum non-abelianization map

We prove that the balanced Chekhov–Fock algebra of a punctured triangulated surface is isomorphic to a skein algebra which is a deformation of the algebra of regular functions of some abelian character variety. We first deduce from this observation a classification of the irreducible representations of the balanced Chekhov–Fock algebra at odd roots of unity, which generalizes to open surfaces the classification of Bonahon, Liu and Wong. We re-interpret Bonahon and Wong’s quantum trace map as a non-commutative deformation of some regular morphism between this abelian character variety and the [Formula: see text]-character variety. This algebraic morphism shares many resemblances with the non-abelianization map of Gaiotto, Moore, Hollands and Neitzke. When the punctured surface is closed, we prove that this algebraic non-abelianization map induces a birational morphism between a smooth torus and the relative [Formula: see text] character variety.

A Study on the Method of Calculating Rainfall Thresholds Based on a Multimethod Approach to Provide Heavy Rain Disaster Impact Information

Various disasters have recently occurred in Korea because of weather changes caused by climate change, and the scale of the damage is also increasing. In terms of the weather phenomena and their socioeconomic impacts, weather-related disasters cause the most damage. Therefore, this study suggests a method of calculating rainfall thresholds to predict the impact of heavy rains. To calculate rainfall thresholds on the basis of a multimethod approach, this study divided Korea into a 1 km × 1 km grid and further split it into four categories: standard watershed, urban, river, and inundation trace maps. This study verified the results of the rainfall thresholds in the standard basin and city center by referring to the relevant data on the rainy season in summer 2020 and the Typhoon Haishen in September. Verification results showed that the flooding that occurred in the grid actually caused the urban flooding in Busan in July 2020. Moreover, the calculation on the grid that caused flooding in Gangneung, which was hit by Typhoon Haishen in September, reflected the actual damage. When used in impact forecasting, the proposed method of calculating rainfall thresholds and predicting rainfall information is expected to be able to provide data on the impact of heavy rains and facilitate rainfall forecasting.

Longitudinal manganese-enhanced magnetic resonance imaging of neural projections and activity.

Manganese-enhanced magnetic resonance imaging (MEMRI) holds exceptional promise for preclinical studies of brain-wide physiology in awake-behaving animals. The objectives of this review are to update the current information regarding MEMRI and to inform new investigators as to its potential. Mn(II) is a powerful contrast agent for two main reasons: (1) high signal intensity at low doses; and (2) biological interactions, such as projection tracing and neural activity mapping via entry into electrically active neurons in the living brain. High-spin Mn(II) reduces the relaxation time of water protons: at Mn(II) concentrations typically encountered in MEMRI, robust hyperintensity is obtained without adverse effects. By selectively entering neurons through voltage-gated calcium channels, Mn(II) highlights active neurons. Safe doses may be repeated over weeks to allow for longitudinal imaging of brain-wide dynamics in the same individual across time. When delivered by stereotactic intracerebral injection, Mn(II) enters active neurons at the injection site and then travels inside axons for long distances, tracing neuronal projection anatomy. Rates of axonal transport within the brain were measured for the first time in "time-lapse" MEMRI. When delivered systemically, Mn(II) enters active neurons throughout the brain via voltage-sensitive calcium channels and clears slowly. Thus behavior can be monitored during Mn(II) uptake and hyperintense signals due to Mn(II) uptake captured retrospectively, allowing pairing of behavior with neural activity maps for the first time. Here we review critical information gained from MEMRI projection mapping about human neuropsychological disorders. We then discuss results from neural activity mapping from systemic Mn(II) imaged longitudinally that have illuminated development of the tonotopic map in the inferior colliculus as well as brain-wide responses to acute threat and how it evolves over time. MEMRI posed specific challenges for image data analysis that have recently been transcended. We predict a bright future for longitudinal MEMRI in pursuit of solutions to the brain-behavior mystery.

Dendritic Domain-Specific Sampling of Long-Range Axons Shapes Feedforward and Feedback Connectivity of L5 Neurons.

Feedforward and feedback pathways interact in specific dendritic domains to enable cognitive functions such as predictive processing and learning. Based on axonal projections, hierarchically lower areas are thought to form synapses primarily on dendrites in middle cortical layers, while higher-order areas are posited to target dendrites in layer 1 and in deep layers. However, the extent to which functional synapses form in regions of axo-dendritic overlap has not been extensively studied. Here, we use viral tracing in the secondary visual cortex of male mice to map brain-wide inputs to thick-tufted layer 5 pyramidal neurons. Furthermore, we provide a comprehensive map of input locations through subcellular optogenetic circuit mapping. We show that input pathways target distinct dendritic domains with far greater specificity than appears from their axonal branching, often deviating substantially from the canonical patterns. Common assumptions regarding the dendrite-level interaction of feedforward and feedback inputs may thus need revisiting.

Dye tracing and concentration mapping in coastal waters using unmanned aerial vehicles

Coastal water flows facilitate important nutrient exchanges between mangroves, seagrasses and coral reefs. However, due to the complex nature of tidal interactions, their spatiotemporal development can be difficult to trace via traditional field instrumentations. Unmanned aerial vehicles (UAVs) serve as ideal platforms from which to capture such dynamic responses. Here, we provide a UAV-based approach for tracing coastal water flows using object-based detection of dye plume extent coupled with a regression approach for mapping dye concentration. From hovering UAV images and nine subsequent flight surveys covering the duration of an ebbing tide in the Red Sea, our results show that dye plume extent can be mapped with low omission and commission errors when assessed against manual delineations. Our results also demonstrated that the interaction term of two UAV-derived indices may be employed to accurately map dye concentration (coefficient of determination = 0.96, root mean square error = 7.78 ppb), providing insights into vertical and horizontal transportation and dilution of materials in the water column. We showcase the capabilities of high-frequency UAV-derived data and demonstrate how field-based dye concentration measurements can be integrated with UAV data for future studies of coastal water flow dynamics.

The Chebyshev–Frobenius homomorphism for stated skein modules of 3-manifolds

We study the stated skein modules of marked \(3-\)manifolds. We generalize the splitting homomorphism for stated skein algebras of surfaces to a splitting homomorphism for stated skein modules of \(3-\)manifolds. We show that there exists a Chebyshev–Frobenius homomorphism for the stated skein modules of 3-manifolds which extends the Chebyshev homomorphism of the skein algebras of unmarked surfaces originally constructed by Bonahon and Wong. Additionally, we show that the Chebyshev–Frobenius map commutes with the splitting homomorphism. This is then used to show that in the case of the stated skein algebra of a surface, the Chebyshev–Frobenius map is the unique extension of the dual Frobenius map (in the sense of Lusztig) of \({\mathcal {O}}_{q^2}(SL(2))\) through the triangular decomposition afforded by an ideal triangulation of the surface. In particular, this gives a skein theoretic construction of the Hopf dual of Lusztig’s Frobenius homomorphism. A second conceptual framework is given, which shows that the Chebyshev–Frobenius homomorphism for the stated skein algebra of a surface is the unique restriction of the Frobenius homomorphism of quantum tori through the quantum trace map.

Silver-mean canonical quasicrystalline-generated phononic waveguides

We investigate propagation of harmonic axial waves in a class of periodic two-phase phononic rods whose elementary cells are designed adopting the quasicrystalline silver mean Fibonacci substitution rule. The stop-/pass-band spectra of this family are studied with the aid of a trace-map formalism which provides a geometrical interpretation of the recursive rule governing traces of the relevant transmission matrices: the traces of two consecutive elementary cells can be represented as a point on a surface defined by an invariant function of the circular frequency, and the recursivity implies the description of an orbit on the surface. We show that, for a sub-class of silver mean-generated waveguides, the orbits predicted by the trace map at specific frequencies are periodic. The configurations for which this occurs, called canonical, are also associated with periodic stop-/pass-band diagrams along the frequency domain. Several types of periodic orbits exist and each corresponds to a self-similar portion of the dynamic spectra whose scaling law can be studied by linearising the trace map in the neighbourhood of the orbit. The obtained results provide both a new piece of theory to better understand the behaviour of classical two-phase composite periodic waveguides and an important advancement towards design and realisation of phononic quasicrystalline-based metamaterials.

The moduli of sections has a canonical obstruction theory

Abstract We give a detailed proof that locally Noetherian moduli stacks of sections carry canonical obstruction theories. As part of the argument, we construct a dualising sheaf and trace map, in the lisse-étale topology, for families of tame twisted curves when the base stack is locally Noetherian.

QDC-2D: A Semi-Automatic Tool for 2D Analysis of Discontinuities for Rock Mass Characterization

Quantitative characterization of discontinuities is fundamental to define the mechanical behavior of discontinuous rock masses. Several techniques for the semi-automatic and automatic extraction of discontinuities and their properties from raw or processed point clouds have been introduced in the literature to overcome the limits of conventional field surveys and improve data accuracy. However, most of these techniques do not allow characterizing flat or subvertical outcrops because planar surfaces are difficult to detect within point clouds in these circumstances, with the drawback of undersampling the data and providing inappropriate results. In this case, 2D analysis on the fracture traces are more appropriate. Nevertheless, to our knowledge, few methods to perform quantitative analyses on discontinuities from orthorectified photos are publicly available and do not provide a complete characterization. We implemented scanline and window sampling methods in a digital environment to characterize rock masses affected by discontinuities perpendicular to the bedding from trace maps, thus exploiting the potentiality of remote sensing techniques for subvertical and low-relief outcrops. The routine, named QDC-2D (Quantitative Discontinuity Characterization, 2D) was compiled in MATLAB by testing a synthetic dataset and a real case study, from which a high-resolution orthophoto was obtained by means of Structure from Motion technique. Starting from a trace map, the routine semi-automatically classifies the discontinuity sets and calculates their mean spacing, frequency, trace length, and persistence. The fracture network is characterized by means of trace length, intensity, and density estimators. The block volume and shape are also estimated by adding information on the third dimension. The results of the 2D analysis agree with the input used to produce the synthetic dataset and with the data collected in the field by means of conventional geostructural and geomechanical techniques, ensuring the procedure’s reliability. The outcomes of the analysis were implemented in a Discrete Fracture Network model to evaluate their applicability for geomechanical modeling.

COVID-19 Monitoring and Response for Military Bases in Singapore—Perspectives and Lessons From January to June 2020

ABSTRACTIntroductionMilitary forces around the world face an increased risk of the spread of communicable diseases, due to the close living quarters and congregated nature of the military camps. The Singapore Armed Forces (SAF) implemented a multi-pronged surveillance and containment strategy to reduce the risk of a coronavirus disease 2019 (COVID-19) outbreak within the local military camps. This paper details the epidemiological investigations of the COVID-19 cases in the SAF and highlights the strategies and public health measures undertaken, aligned with the national COVID-19 control strategy, to reduce the risk of COVID-19 transmission in the military camps.Materials and MethodsMedical data of our military personnel who were infected with COVID-19 during the first 180 days of the pandemic were extracted from the military electronic health records. Contact tracing and activity mapping results were obtained from unit-level epidemiological data. A review of the organization’s response plans, instructions, and orders was conducted to collate the measures implemented across the same time period.ResultsPrompt contact tracing and activity mapping was done for each of the 24 SAF military personnel diagnosed with COVID-19 between February 2020 and June 2020, with possible links among the cases identified and investigated.ConclusionA combination of strategies in the formulation of public health measures based on key principles of early warning and surveillance, prompt diagnosis, and early containment were successful in preventing the formation of COVID-19 clusters within the SAF. This will provide a framework for the management of future pandemics within the military setting, driven by strong governance and leadership, to meet the military’s need to maintain operational readiness in a safe manner.

Virtual outcrop geology comes of age: The application of consumer-grade virtual reality hardware and software to digital outcrop data analysis

In this work, the application of consumer-grade virtual reality (VR) hardware and software to the analysis of digital outcrop data is explored. Here, we utilize a widely available VR hardware platform (HTC Vive) and VR based freeform 3D visual arts software package (Google Tilt Brush), to digitize fault traces from photo-textured digital outcrop models of exposures of the Penrith Sandstone Formation (Lacy's Caves), in northwest England. Using MATLAB routines provided herein, triangular meshes output from Tilt Brush are used as the basis for 3D fracture trace map extraction, which in turn, are used to generate fracture properties (trace length, orientation, areal fracture intensity). We compare the results of this analysis to two equivalent datasets obtained from the Lacy's Caves model using digital outcrop analysis deployed via a conventional flat panel display: namely (1) a 3D trace map extracted using optical ray tracing from manually interpreted calibrated images and (2) 3D traces fitted directly the Lacy's Caves textured mesh using manual polyline interpretation within an established digital outcrop analysis software platform (OpenPlot).Fault statistics obtained using VR based analysis are broadly equivalent to those acquired from 3D trace maps extracted using the flat panel display deployed analyses presented herein. In this case study, it was found that VR based digital outcrop analysis provided faster data acquisition than the comparative pixel-based approach, which requires linkage and merging of traces mapped from multiple contiguous images. Manual raster analysis and optical ray tracing did however provide 3D trace maps with significantly higher areal fault intensity, with VR analysis incurring censoring of finer fault traces, due to the limited resolution of the outcrop model textured mesh. Whilst data acquisition times and resultant fault intensities proved similar between the VR and OpenPlot workflows, it was noted anecdotally, that the VR analysis holds some advantages for the operator when interpreting models exhibiting complex geometries, such as mine workings and caves systems, with the clip point implemented within the viewport of conventional digital outcrop analysis software tools obstructing the user from obtaining an optimum view of the outcrop surface.VR based digital outcrop analysis techniques, such as those presented here, provide an immersive analytical environment to the operator. This allows users to fuse powerful 3D visualizations of photo-realistic outcrop models with geological interpretation and data collection, fulfilling the early promise of ‘virtual outcrops’ as an analytical medium that can emulate traditional fieldwork. It is hoped that this study and its associated code library will facilitate the evaluation of emerging VR technologies for digital outcrop applications, by provided access to VR analytical tools for non-specialists in virtual reality systems. Finally, prospects for the use of VR technology within the field of digital outcrop geology, as well as within the wider geosciences, are also discussed.

Intrusion-related orogenic gold deposit in the East Kunlun belt, NW China: A multiproxy investigation

The Wulonggou gold district is one of the most important gold producers of the East Kunlun metallogenic belt in China. Due to its complex geological background and limited investigations, the age of mineralization and ore-forming process remain unclear. Here, we focus on the Yanjingou gold deposit which is the largest and most typical deposit in the Wulonggou gold district, with a view to investigate the metallogenic mechanism and timing of ore formation with a view to evaluate the ore-forming processes and sources through a comprehensive study of the ore mineralogy, monazite and zircon U-Pb dating, in situ trace element analysis and mapping, and in situ S isotopes. Our results show that the magmatism and mineralization were coeval, during 239 Ma and 244 Ma in the Middle Triassic. Four stages of mineralization are identified with distinct textures, mineral assemblages, trace element characteristics and S isotopes. Stage 1 is represented by euhedral pyrite bearing quartz veins, which originated from deep metamorphic fluid, and was rarely affected by magmatic hydrothermal fluids. Stage 2 is characterized by the occurrence of abundant oriented arsenopyrite, which originated from the mixture of deep metamorphic fluid and magmatic hydrothermal fluid. Stage 3 is characterized by the occurrence of polymetallic sulfide of copper, with two generation of pyrite (stage 3 Py1 and Py2). Stage 3 Py1 formed from magmatic hydrothermal fluids whereas Py2 originated from deep metamorphic fluid and was also slightly influenced by magmatic hydrothermal fluids. Stage 4 is represented by late calcite and pyrite combination, which originated from another stage of deep metamorphic fluid. Stage2 and 3 are the main gold mineralization stages, and Au has a close relationship with As, and an obvious inverse correlation with Co and Ni. Combined with the trace element characteristics, we infer that the denudation degree of the Yanjingou gold deposit is very shallow with good prospects for exploration. Through multi-proxy investigation of the different stages and types of pyrite, we postulate that the Yanjingou gold deposit is an orogenic gold deposit related to intrusive rocks formed during Middle Triassic compressional transformation.

Exact Mobility Edges in 1D Mosaic Lattices Inlaid with Slowly Varying Potentials

AbstractA family of 1D mosaic models inlaid with a slowly varying potential is proposed. Combining the asymptotic heuristic argument with the theory of trace map of transfer matrix, mobility edges (MEs), and pseudo‐MEs (PMEs) in their energy spectra are solved semi‐analytically, where ME separates extended states from weakly localized ones and PME separates weakly localized states from strongly localized ones. The nature of eigenstates in extended, critical, weakly localized, pseudo‐critical, and strongly localized is diagnosed by the local density of states, the Lyapunov exponent, the localization tensor, and fractal dimension. Numerical calculation results are in excellent quantitative agreement with theoretical predictions.

Corrigendum: Lineage Tracing and Fate Mapping in Xenopus Embryos.

Corrigendum: Lineage Tracing and Fate Mapping in Xenopus Embryos.

Warpage Simulation Study by Trace Mapping Method for FCCSP with ETS Substrate

Abstract ETS (Embedded trace substrate) has become as the mainstream substrate for FCCSP since it has fine trace, better trace dimension control and low cost advantages which compared to normal substrate. But it usually encountered more serious warpage issue for bare substrate and complete package which may influence D/B (die bonding) and SMT yield rate due to its coreless characteristic. Especially for ETS substrate with special trace pattern design (ex. larger Copper area), bare substrate may appear peculiar warpage contour and led to serious non-wetting issue at specific location during D/B process. Thus, if it can predict warpage value and contour accurately for bare substrate and package is an important topic. In this paper, a FCCSP package with ETS substrate was chosen to study trace impact. Bare substrate and package warpage simulation models w/ and w/o considering trace pattern by trace mapping method were performed and compared to shadow moiré results. Analysis results showed that simulation w/ considering trace pattern could get more accurate warpage value and more similar warpage contour for bare substrate and package warpage.

Finite étale extensions of Tate rings and decompletion of perfectoid algebras

In this paper, we examine the behavior of ideal-adic separatedness and completeness under certain ring extensions using trace map. Then we prove that adic completeness of a base ring is hereditary to its ring extension under reasonable conditions. We aim to give many results on ascent and descent of certain ring theoretic properties under completion. As an application, we give conceptual details to the proof of the almost purity theorem for Witt-perfect rings by Davis and Kedlaya. Witt-perfect rings have the advantage that one does not need to assume that the rings are complete and separated.

On the number of zeros to the equation f(x1)+... + f(xn)=a over finite fields

Let p be a prime, k a positive integer and let Fq be the finite field of q=pk elements. Let f(x) be a polynomial over Fq and a∈Fq. We denote by Ns(f,a) the number of zeros of f(x1)+⋯+f(xs)=a. In this paper, we show that∑s=1∞Ns(f,0)xs=x1−qx−xMf′(x)qMf(x), where Mf′(x) stands for the derivative of Mf(x) andMf(x):=∏m∈Fq⁎Sf,m≠0(x−1Sf,m) with Sf,m:=∑x∈FqζpTr(mf(x)), ζp being the p-th primitive unit root and Tr being the trace map from Fq to Fp. This extends Richman's theorem which treats the case of f(x) being a monomial. Moreover, we show that the generating series ∑s=1∞Ns(f,a)xs is a rational function in x and also present its explicit expression in terms of the first 2d+1 initial values N1(f,a),...,N2d+1(f,a), where d is a positive integer no more than q−1. From this result, the theorems of Chowla-Cowles-Cowles and of Myerson can be derived.

Theoretical and Experimental Investigation of Warpage Evolution of Flip Chip Package on Packaging during Fabrication.

This study attempts to investigate the warpage behavior of a flip chip package-on-package (FCPoP) assembly during fabrication process. A process simulation framework that integrates thermal and mechanical finite element analysis (FEA), effective modeling and ANSYS element death-birth technique is introduced for effectively predicting the process-induced warpage. The mechanical FEA takes into account the viscoelastic behavior and cure shrinkage of the epoxy molding compound. In order to enhance the computational and modeling efficiency and retain the prediction accuracy at the same time, this study proposes a novel effective approach that combines the trace mapping method, rule of mixture and FEA to estimate the effective orthotropic elastic properties of the coreless substrate and core interposer. The study begins with experimental measurement of the temperature-dependent elastic and viscoelastic properties of the components in the assembly, followed by the prediction of the effective elastic properties of the orthotropic interposer and substrate. The predicted effective results are compared against the results of the ROM/analytical estimate and the FEA-based effective approach. Moreover, the warpages obtained from the proposed process simulation framework are validated by the in-line measurement data, and good agreement is presented. Finally, key factors that may influence process-induced warpage are examined via parametric analysis.