Tile Surface Research Articles

Surface defect segmentation of magnetic tiles based on cross self-attention module

The detection of magnetic tile quality is an essential link before the assembly of permanent magnet motor. In order to meet the high standard of magnetic tile surface defect detection and realize the rapid and automatic segmentation of magnetic tile defects, a magnetic tile surface defect segmentation algorithm based on cross self-attention model (CSAM) is proposed. It adopts high-low level semantic feature fusion method to build the dependency relationship between the deep and shallow features. Multiple auxiliary loss functions are used to constrain the network and reduce the noise in the deep features. In addition, an image enhancement method is also designed to solve the problem of insufficient annotated data. The experimental results show that the network can achieve 79.6% mIoU and 98.5% PA, which can meet the high standard requirements of magnetic tile manufacturing.

Assessment of Materials on Covid-19 Spread Prevention in Public Buildings

This research is an attempt to suppress the spread of COVID-19 in public buildings through the application of materials that may prevent the spreading of SARS-CoV-2 while at the same time improving audial comfort in such buildings. Study on the capability of materials to break further spreading of the virus applied by measuring the presence of SARS-CoV-2 RNA on the material based on exposure time (0, 3, 6, and 9 minutes). This RNA tracing procedure is repeated on the surface of aluminum, tiles, carbon steel, glass, gypsum, copper, and painted cement on the wall. The results showed that the CT value (cycle threshold) of all materials was approximately 25, which means that they still have the potential to spread the virus. Slightly different results occur on the gypsum surface where the viral envelope (gene E) is not detected with a CT value of 36, meaning no chance of the virus spreading. The same materials will undergo measurement of sound absorptive characteristics to assess their effect on the audial comfort in the building. The results of the study are expected to be a reference before their application as interior materials in public buildings.

Preparation of Spinel-Type Black Pigments Using Microwave-Assisted Calcination of Stainless Steel Dust: The Effect of Manganese Molar Content

Stainless steel dust is rich in valuable metal elements including Fe, Cr, Ni and Mn, which can be utilized to prepare Fe–Cr–Ni–Mn series black pigments. Meanwhile, manganese can absorb the majority of the visible light wavelength range, which improves the color rendering performance of Fe–Cr–Ni–Mn series black pigments. However, the coloring mechanism of manganese in the above black pigments is not clear. Therefore, the effect of manganese oxide content on the preparation of spinel-type black pigments from microwave-assisted calcination of stainless steel dust was studied in this work. The results show that with the increase in MnO content in the raw mixture, the crystal plane spacing of black pigments increases from 0.2525 nm to 0.2535 nm, the grain size grows from 61.4619 nm to 79.7171 nm, and the lattice constant grows from 0.8377 to 0.8406 nm. Moreover, the band gap is decreased from 1.483 eV to 1.244 eV, the absorbance increases significantly and has a consistent absorbance in the visible range, and the L*, a* and b* values reduce from 41.8, 0.6, 1.6 to 32.0, 1.0, 0.8, respectively. MnO can react with the spinel in stainless steel dust, forming Mn3O4, MnCr2O4 and Ni (Fe,Cr)O4 in the system, with a regular polyhedral structure. The prepared pigments have excellent thermal stability at 1100 °C and good compatibility with transparent glazes, which can be adhered to the surface of ceramic tiles after calcination to demonstrate better compatibility as the content of MnO increases.

Balanced multi-scale target score network for ceramic tile surface defect detection

Ceramic tiles, as a prevalent building material, exhibit a wide variety of types and high demand. Traditional manual inspection methods relying on human visual observation suffer from low efficiency and unreliable accuracy. Current automated detection methods mostly rely on traditional image processing techniques for feature extraction, followed by machine learning-based classification. However, faced with the diversity of tile types and defect categories, fine-tuning and deployment processes require significant human and material resources, while detection efficiency remains limited. In this study, we first construct a high-resolution dataset for studying surface defects in ceramic tiles (CT surface defects dataset), encompassing multiple batches and various patterns of tiles. Subsequently, data analysis is conducted to address the scale and quantity differences in defect distribution. We propose an improved approach by introducing a content-aware feature recombination method and a dynamic attention mechanism to enhance the classical single-stage object detection algorithm YOLOv5. These enhancements aim to reduce information loss in features and enhance the expression of multi-scale features. Furthermore, we design a loss function that mitigates score differences for multi-scale defects. The proposed approach mitigates the discrepancy in contribution among different scale targets caused by imbalanced quantities. It effectively prevents the model from excessively favoring a specific scale target during the learning process. Experimental results demonstrate the superior accuracy and efficiency of our detection method.Compared to the baseline network YOLOv5, our approach achieved improvements of 4.9% in AP (Average Precision), 6% in APs (small-scale objects), and 8% in APl (large-scale objects). Furthermore, we achieved a 3.9% improvement in detecting white point defects, which are most affected by small-scale objects, and a 4.1% improvement in detecting discolored spot defect, which are most affected by class imbalance.

MCAW-YOLO: An Efficient Detection Model for Ceramic Tile Surface Defects

Traditional manual visual detection methods are inefficient, subjective, and costly, making them prone to false and missed detections. Deep-learning-based defect detection identifies the types of defects and pinpoints their locations. By employing this approach, we could enhance the production workflow, boost production efficiency, minimize company expenses, and lessen the workload on workers. In this paper, we propose a lightweight tile-defect detection algorithm that strikes a balance between model parameters and accuracy. Firstly, we introduced the mobile-friendly vision transformer into the backbone network to capture global and local information. This allowed the model to comprehend the image content better and enhance defect feature extraction. Secondly, we designed a lightweight feature fusion network. This design amplified the network’s detection capability for defects of different scales and mitigated the blurriness and redundancy in the feature maps while reducing the model’s parameter count. We then devised a convolution module incorporating the normalization-based attention module, to direct the model’s focus toward defect features. This reduced background noise and filtered out features irrelevant to defects. Finally, we utilized a bounding box regression loss with a dynamic focusing mechanism. This approach facilitated the prediction of more precise object bounding boxes, thereby improving the model’s convergence rate and detection precision. Experimental results demonstrated that the improved algorithm achieved a mean average precision of 71.9%, marking a 3.1% improvement over the original algorithm. Furthermore, there was a reduction of 26.2% in the model’s parameters and a 20.9% decrease in the number of calculations.

Environmentally friendly tea saponin foam detergents costabilized by GO-OH/SiO2 nanoparticles for removing radioactive surface contaminants

Currently, bare silica nanoparticles with low surface activity and weak stabilizing foam ability have restricted applicability in treating surface radioactive pollutants. To address potential issues with SiO2 as a foam stabilizer, a highly efficient foam stabilizer was fabricated by compounding silica with hydroxylated graphene oxide (GO-OH), which was prepared by hydrothermal modification of graphene oxide (GO). Then, an environmentally friendly foam detergent with high stability and high decontamination efficiency was developed using the natural surfactant tea saponin (TAS) as a foaming agent and a composite of silica and GO-OH (GO-OH/SiO2) as a foam stabilizer, which is named tea saponin foam detergent. For different types of surfactants, the structural and morphological characteristics of GO-OH, such as hydrophilic and surface oxygen groups, as well as the synergistic influence of GO-OH with SiO2 on foamability, foam stability, and detergency of the foam detergent, were studied. As a consequence, the TAS foam detergent has good stability with a drain half-life of 68 min. By hydrothermal modification of GO, the contact angle of water on GO-OH in air dropped to 63.2°, which was lower than that (73.0°) on GO. Furthermore, the hydrophilicity of GO-OH increased, and more oxygen-containing groups on GO-OH enhanced its absorption and enrichment of the radionuclide uranium and improved the decontamination rate of the TAS foam detergent. Moreover, when GO-OH was combined with SiO2, the dispersion and stability of SiO2 in the foam detergent improved, and the foam stability of the detergent subsequently further improved. Adding 2 wt% SiO2 and 0.1 wt% GO-OH to the detergent enhanced the foam drainage half-life of the detergent from 40 min to 68 min and improved the stability of the detergent by 70%. In addition, the synergistic action of GO-OH/SiO2 with TAS surfactant increased the liquid film thickness of the foam and the airflow resistance to support the weight-bearing capacity of the foam skeleton. As a result, it slowed the rate of foam disproportionation, reduced the bubble size, increased the bubble number, and finally improved the foam stability of the detergent. For simulated radioactive uranium pollutants on the surfaces of glass, stainless steel, and tiles, the foam detergent with the synergistic action of GO-OH/SiO2 and TAS surfactant had decontamination rates of 91.7%, 91.16% and 91.76%, respectively, which were 14.96%, 15.67% and 19.39% higher than that of TAS foam detergent without GO-OH/SiO2. The synergistic enhancement of GO-OH and SiO2 on foam stability and its adsorption and combination action with uranyl ions would be the main reasons to improve the detergency of the TAS foam detergent. Therefore, TAS foam detergent stabilized synergistically by GO-OH/SiO2 with high stability and efficient decontamination will be widely applied in nuclear emergencies and decommissioning decontamination of nuclear facilities.

Research on tile surface defect detection by improved YOLOv5

Traditional methods for detecting surface defects on ceramic tiles result in misdetection and missed detection, which makes it difficult to guarantee product stability and consistency within the same batch. Therefore, this article proposes an improved YOLOv5 algorithm for detecting surface defects on ceramic tiles. Firstly, the Res2Net module is combined with self-attention to fully utilize local and global information and improve the feature extraction effect of defects. Secondly, the GS-BiFPN neck network is designed to enhance the fusion capability of shallow detail and deep semantic information and alleviate ambiguity and redundancy on the feature map. Then, a lightweight attention module is introduced to improve the detection capability of difficult-to-recognize defects and anti-background interference. Finally, the SIoU loss function improves the model’s convergence speed and accuracy. Experimental results demonstrate that the improved algorithm’s mean average precision (mAP) reaches 73.3%, 6.3% higher than the baseline model. Even when compared with YOLOv7-tiny, the mAP of the improved algorithm has increased by 8.7%. Additionally, the detection speed of the model can reach 92 frames per second, which can meet the requirements of ceramic tile surface defect detection in industrial scenarios.

Effective photocatalytic sterilization based on composites of Ag/InVO4/BiOBr: Factors, mechanism and application

We hypothesized that photocatalysts with a low band gap could be useful in the sterilization of ceramic tiles in the natural environments of toilets using natural light in those settings. Certain photocatalysts can produce reactive oxygen species (ROS) under light illumination, which in turn are bactericidal. The properties of the BiOBr-containing photocatalysts were tuned by creating junctions and heterostructures with Ag and InVO4 and studied with respect to their bactericidal effect in dispersion. The bactericidal mechanism was studied through experiments in which active species were captured and via electron paramagnetic resonance (EPR) spectroscopy. At an optimal dosage of 0.5 g/L, the Ag/InVO4/BiOBr composite had a sterilization efficacy of 99.9999 % in 30 min under visible light illumination of 1000 W. It retained a sterilization efficacy of 99.999 % after four cycles. Anions such as Cl−, SO42−, and NO3− were shown to have no negative impact on sterilization efficacy. It was shown that the holes in the composite photocatalyst and hydroxyl radicals (·OH) were mechanistically critical for the sterilization. The photocatalysts were also studied in the field in the natural environment of a restroom, where they were loaded on ceramic tiles. Samples were collected from the surface of the ceramic tiles and analyzed for bacterial cultures and microbial diversity. The results were compared in the scope of the sterilization ability of various agents at the microbial level. The ceramic tiles loaded with Ag/InVO4/BiOBr showed the least amount of bacteria on their surfaces, and the microbial community richness was also the lowest.

OpenFOAM modeling of beryllium melt motion and splashing from first wall in ITER

Beryllium (Be) is a material which will be used as a plasma facing component in ITER due to its unique properties of high thermal conductivity, low density, and high strength. However, under extreme conditions of high temperature and pressure, Be can melt at the surface of tiles and molten droplets can be ejected into the reactor leading to disruption of fusion plasma. The pressure, mass density, velocity of Be vapor, and variations of temperature at the melt layer interface can influence the splashing of Be melt. The Computational Fluid Dynamics (CFD) model based on the OpenFOAM toolbox, a free open source CFD software package, was developed to treat the coupled flow of liquid Be metal and its vapor. The vapor-melt interface is modeled using the volume of fluid (VOF) approach implemented in the interCondensatingEvaporatingFoam solver that solves the continuity, momentum, heat conduction, and VOF equations. This CFD model is capable to predict the hydrodynamic effects of Be vapor on the melt layer motion, splashing, non-linear growth of melt waves, and ejection of molten droplets. The modeling accounts for the effects of thermal, viscous, gravitational, and surface tension forces at the vapor-melt interface. In this research, we used the interCondensatingEvaporatingFoam solver to simulate the effects of Be phase change and the development of melt motion with the droplets ejected from the surface. The CFD model accounts for inter-phase change between Be liquid and Be vapor. The evaporation model was validated against the Stefan phase-change problems. The influence of heat and mass transfer across the vapor-melt interface on melt layer stability is also investigated. The results provide an understanding of how the rate of phase change affects the development of melt structures and waves at the vapor-melt interface.

Assessing children’s potential exposures to harmful metals in tire crumb rubber by accelerated photodegradation weathering

Whether a tire crumb rubber (TCR) playground would expose children to potentially harmful chemicals such as heavy metals is an open question. The released metals available for pickup on the surface of TCR tiles was studied by accelerated 2-year aging of the TCRs in the NIST-SPHERE (National Institute of Standards and Technology Simulated Photodegradation via High Energy Radiant Exposure). The dermal contact was mimicked by a method of composite surface wiping from US Environmental Protection Agency throughout the weathering process. The surface release of ten most concerned harmful metals (Be, Cr, Cu, As, Se, Cd, Sb, Ba, Tl, Pb) was monitored through the course of aging. The cumulative release of Cu, As, Tl, and Sb reached potentially harmful levels at various times within 3 years, although only Cr was found at a harmful level on the surface of the tiles. Taking the cleansing effect of precipitation or periodic cleansing with rain into account, TCR playgrounds may still be safe for use.

Tile Surface Defect Detection Based on Improved Faster R-CNN

With the increasing population, there is a high demand for housing, commercial buildings, and lifestyle facilities. The development of modern manufacturing technology has led to higher quality control requirements for tiles. This article presents a ceramic tile defect detection method based on Faster R-CNN, primarily focused on detecting surface defects or cracks on tiles. The method replaces VGG16 with the ResNet-101 network as a new backbone and utilizes depth-wise separable convolution to address efficiency issues resulting from the backbone replacement. Finally, Soft-NMS is employed to optimize regression boxes and prevent missed detections. Experimental results demonstrate that the improved algorithm achieves a mAP of 70.4%, a 13.2% enhancement over the original algorithm, highlighting the effectiveness and feasibility of the proposed approach.

Non-destructive detection for mosaic ceramic surface defects based on convolutional neural networks

Abstract Mosaic ceramic art pattern with noble, elegant features, it is a unique form of art creation in ancient Greece and the ancient Rome period has been loved by artists and created a lot of classic large-scale exterior mosaic ceramic art works. Small size square mosaic ceramic as the basic raw material for the creation of large exterior mosaic art, it directly affects the quality of the work created by the artist, so these ceramic mosaic ceramic materials need to undergo rigorous inspection to meet the needs of the artist’s high-quality art creation. However, small size multi-color square mosaic ceramics are different from ordinary large target ceramics, they have the characteristics of small size and easy reflection, currently mainly using manual inspection, the existing automatic inspection methods have the problem of low efficiency and accuracy, cannot meet the needs of artists for the quantity and quality of mosaic ceramics. To solve these problems, this paper proposes a new convolutional network-based fast nondestructive testing method for detecting square mosaic tiles. The detection method is based on the convolutional neural network YOLOv5s model, and by introducing the AF-FPN module and the data enhancement module, the method further improves the recognition performance of the model relative to the original YOLOv5s model and achieves the fast detection of surface defects on square mosaic ceramics. The experimental results show that the detection method for small size multicolor square mosaic ceramic tile surface minor defects detection rate of up to 94 % or more, a single square mosaic ceramic detection time of 0.41 s. The method takes into account the detection accuracy and speed, can be fast and accurate screening of high-quality, defect-free small size multicolor square mosaic ceramic, to meet the artist’s requirements for high-quality mosaic ceramic raw materials Quality and quantity requirements, to ensure the quality of the creation of mosaic art patterns, to better show the charm of the mosaic art patterns role. At the same time, the method can not only be applied to the detection of mosaic ceramics, the method can also be applied to have a similar small volume, easy to reflect the characteristics of small target object defect detection.

A fluorescent enhancement method for footwear impressions in ice melt products

There are limited options reported for the detection of footwear impressions in ice melt products. Enhancement of footwear impressions in ice melt products using a fluorescent metal-organic complex was identified in this work. The deprotonation of 8-hydroxyquinoline with a solution of strong base led to the formation of a complex between calcium and magnesium ions commonly found in winter ice melt products. Three ice melt products containing 0.01–99% calcium were compared for enhancement with 8-hydroxyquinoline reagent. With 254 nm light, footwear impressions enhanced with this method fluoresced on a semi-porous tile surface. This application of 8-hydroxyquinoline is a simple, efficient method to enhance footwear impressions, and offers a stable alternative reagent compared to the acetone solution used in previous work.

Agricultural land management alters the biogeochemical cycling capacity of aquatic and sediment environments

Nutrient loading from agricultural landscapes continues to pose a threat to downstream aquatic systems, contributing to degraded water quality and eutrophication. Nutrient loading to adjacent aquatic environments is influenced by the type of applied fertilizer (chemical and manure), field management, and seasonal weather patterns. The risk these nutrients pose to downstream systems may be mitigated by the capacity of the microbial community in receiving water catchments to process those nutrients. This study evaluated the interactive effects of agricultural practices and seasonal weather patterns on nutrient cycling communities in catchment ponds within a long-term Agriculture and Agri-Food Canada research trial, located in the Lake Erie watershed in Ontario, Canada. Microbial nitrogen (N) and P (phosphorus)-cycling functions in water and sediment environments, receiving direct tile drainage and surface runoff flow, were quantified with respect to nutrient loads from liquid cow manure, solid cow manure, inorganic phosphorus, and a control fertilizer treatment, over a full year. High bacterial nitrification potential in the water column coincided with nitrogen loading, which was driven by manure/fertilizer application, tillage practices, and precipitation. The highest overall phosphorus concentrations occurred in the liquid cow manure treatment, but the availability of phosphorus in all treatments was strongly associated with microbial phosphorus mobilization capacity in both aquatic and sediment environments. Seasonality also influenced the aquatic environment, where N and P- cycling potential significantly increased during the winter months. Microbial functional potential in the water column was strongly influenced by the external environment, including season and precipitation, while the functional potential in the sediment was more resistant to environmental fluctuations, and was primarily influenced by long-term management fertilizer/manure practices. Overall, this study provides evidence that agricultural management alters the capacity of the microbial communities in adjacent aquatic environments to process nutrients.

Chlorinated paraffins in multimedia during residential interior finishing: Occurrences, behavior, and health risk

Though with bioaccumulation and toxicity, chlorinated paraffins (CPs) are still high produced and widely utilized in various daily necessities for extender plasticization and flame retardation. CPs can be released during the reprocessing processes of finishing materials and distributed in multi-environmental media. Herein, concentrations and compositions of CPs in four representative media including interior finishing materials, PM10, total suspended particulate (TSP), and dust samples collected from eight interior finishing stages were studied. Unexpectedly, CP concentrations in ceramic tiles was found to be high with a mean value of 7.02 × 103 μg g−1, which could be attributed to the presence of CPs in the protective wax coated on ceramic tiles surfaces. Furthermore, the pollution characteristics of short-chain and medium-chain CPs (SCCPs and MCCPs) in those samples were inconsistent. According to the investigation regarding Kdust−TSP and KPM10-TSP, the occurrence and distribution of CPs in indoor atmospheric particles (PM10 and TSP) and dust were highly affected by reprocessing processes (cutting, hot melting, etc.) compared to that in the finishing materials. Moreover, dermal contact was the primary pathway of CP exposure for the occupational population (interior construction workers) for most interior finishing stages, and the interior finishing process is the prime CP exposure period for the occupational groups. As suggested by our assessment, though hardly posing an immediate health risk, CPs exposure still presents unneglected adverse health effects, which calls for adequate personal protections during interior finishing, especially in developing countries.

Damages of TZM as plasma facing material under transient heat load

Titanium-Zirconium-Molybdenum (TZM) is applied as the first wall material of EAST due to its excellent performance. To investigate the service behavior of TZM under transient heat fluxes, the postmortem inspection of in-situ damaged TZM tiles by plasma disruption heat load in EAST was analyzed. Meanwhile, the EMS-60 electron gun has been also used to test the transient heat load behaviors of TZM. There are some similar characteristics between the in-situ melting TZM and the ex-situ melting TZM tiles, but there are also some obvious differences. Specifically, in some melting zone, the surface morphology of partial in-situ TZM melting region is like that of ex-situ TZM melting areas, and their convex surface is relatively smooth, while the concave surface is relatively rough, and the solidification layer undergoes recrystallization. In other melting areas, there are many small pits on the surface of the TZM tile that was melted in-situ, but there are no obvious pits on the surface of the ex-situ TZM sample, and part of the surface of the TZM tile that was melted in-situ is like a big pit, and the melted TZM droplets in this melting area tend to splash towards the periphery. The capillary waves around the castle structure in the ex-situ test samples are the result of wetting effect of liquid metal and capillary force, and the holes in the resolidified layer is caused by the overheating boiling of TZM liquid. On the other hand, the number and propagation direction of cracks in the in-situ TZM tile are more complex than those in the ex-situ TZM sample. Firstly, the number of cracks on the in-situ TZM tile is large and dense, while the number of cracks on the surface of the ex-situ melting sample is small and scattered. The crack width of the in-situ tile is greater than that of the ex-situ tile. Secondly, the crack extension direction on the in-situ melted TZM is diverse, while the extension direction of the surface crack on the ex-situ TZM sample is relatively simple, and there is only one kind of crack on the tile surface. Such results provide reference for understanding the thermal behavior of TZM under transient heat flux in fusion devices.

A non-contact inspection method of tile debonding using tuned acoustic wave and laser doppler vibrometer

The debonding of tiles from walls is currently a critical issue in modern cities. This study develops a non-contact acoustic approach integrating an effective numerical model for accurate inspection of tile debonding from a wall, with the aid of a directional sound source and a laser doppler vibrometer (LDV). Typical engineering tile debonding cases are characterised and analysed using the non-contact inspection method. Numerical modelling based on full-field modelling is performed to simulate and predict the inspection process. In both experiments and simulations, the acoustic wave with a sweeping frequency band is generated by the sound source and directed to the tiles. Activated vibration signals are captured at multiple points on the tile surface and then processed, and an out-of-surface velocity map of the tile is plotted to identify the debonding area, where good agreements between the numerical and experimental results can be seen. The results show that different shapes of the debonding areas under the tiles can be well detected, proving the effectiveness of the method.

Technological and economical investigation of glaze preparation using dry stirred media mill

Ceramic tiles are the most common building material for floor and wall coverings in many countries. Glazed tiles are produced from mixture of frits and some additional raw materials applied on the surface of green tiles and subjected to a firing process. A new method of processing of glaze that is dry stirred media mill was investigated in a pilot grinding plant. The produced glaze particle size, shape and surface area are measured. The comparison was made with the product of conventional wet discontinue ball mills using the same wall tile glaze recipes. The results indicated that dry stirred media mill can provide product that have finer particle size distribution, more stable product compared to the conventional wet ball milling. The glaze thermal expansion and optical properties such as colour (L, a and b parameters) of the produced glazes were measured and comparison was also made in details. Finally, the microstructural characteristics of the produced glazes were determined using scanning electron microscopy (SEM). The results have shown that dry stirred media mill enhances glaze properties and process economy considerably.

Numerical assessment of thermal performance of W/Cu monoblock during steady-state operation and plasma vertical displacement events under ITER-like conditions

This paper employs a two-dimensional conduction model to assess the spatiotemporally thermal performance of ITER W/Cu monoblock under ITER-like conditions. The model considers the heat conduction processes including the evaporation, radiation and melting of both W armor and Cu heat sink materials under a heat load either during steady-state discharge or during plasma vertical displacement event (VDE). Moreover, the model employs coupling cooling water condition instead of the fixed cooling temperature at bottom in authors’ previous work. For the steady-state case, the heat flux distribution on the target tile, the input data to the model, was specifically evaluated by a particle-in-cell code to take into account its geometrical effect. For the VDE case, the heat flux distribution was simply assumed to be uniform on the tile surface. The simulated results obtained from the heat conduction model indicate that cooling water conditions have important influence on the thermal performance of W/Cu monoblock, especially for Cu heat sink materials during steady state. In the case of VDE, the baffle tile surface would undergo intense vaporization and melting, while whether Cu heat sink materials melted was determined by the combination of the exposed heat load and its duration. Based on typical VDE heat loads, detailed analysis is presented herein.PACS: 52.55.Fa, 52.55.Rk, 52.40.Hf, 52.65.-y

New Kronecker–Weyl type equidistribution results and Diophantine approximation

An interesting result of Veech more than 50 years ago is a parity, or mod 2, version of the Kronecker–Weyl equidistribution theorem concerning the irrational rotation sequence {qalpha }, q=0,1,2,3,ldots If alpha is badly approximable and bin (0,1) satisfies bne {malpha } for any min {mathbb {Z}}, then the parity of cardinalities of the sets {1leqslant qleqslant N,{:},{qalpha }in [0,b)} as Nrightarrow infty is evenly distributed. We first answer a question of Veech and establish a stronger form of the mod n analog of his result. Furthermore, for irrational alpha and b={malpha } for some min {mathbb {N}}, we give a simple yet precise characterization of those cases that give rise to even distribution. We also obtain time-quantitative description of some very striking violations of uniformity—this part is particularly number theoretic in nature, and involves Ostrowski representations of positive integers and alpha -expansions of real numbers. The Veech discrete 2-circle problem can also be visualized as a problem that concerns 1-direction geodesic flow on a surface obtained by modifying the surface comprising two side-by-side squares by the inclusion of symmetric barriers and gates on the vertical edges, with appropriate modification of the vertical edge identifications. We establish a far-reaching generalization of this case to ones that concern 1-direction geodesic flow on surfaces obtained by modifying a finite square tiled translation surface in analogous but not necessarily symmetric ways.