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# Ultrastructural visualization of 3D chromatin folding using volume electron microscopy and DNA in situ hybridization

## Abstract

The human genome is extensively folded into 3-dimensional organization. However, the detailed 3D chromatin folding structures have not been fully visualized due to the lack of robust and ultra-resolution imaging capability. Here, we report the development of an electron microscopy method that combines serial block-face scanning electron microscopy with in situ hybridization (3D-EMISH) to visualize 3D chromatin folding at targeted genomic regions with ultra-resolution (5 × 5 × 30 nm in xyz dimensions) that is superior to the current super-resolution by fluorescence light microscopy. We apply 3D-EMISH to human lymphoblastoid cells at a 1.7 Mb segment of the genome and visualize a large number of distinctive 3D chromatin folding structures in ultra-resolution. We further quantitatively characterize the reconstituted chromatin folding structures by identifying sub-domains, and uncover a high level heterogeneity of chromatin folding ultrastructures in individual nuclei, suggestive of extensive dynamic fluidity in 3D chromatin states.

## Introduction

How the two-meter-long human genome is folded into the micrometer-sized nuclear space is an important question in biology. Indeed, specific three-dimensional (3D) organization of chromosomal folding has been shown to have a profound impact on genome functions, such as gene transcription1 and DNA replication2. To investigate the detailed structures of 3D genome organization, two general approaches have been developed and applied—sequencing-based mapping and microscopy-based imaging. Mainstream 3D genome mapping methods such as Hi-C3,4 and ChIA-PET5 detect pairwise long-range chromatin interactions by chromatin proximity ligation followed by high-throughput sequencing and mapping to the reference genome, thus inferring long-range chromatin contacts and rendering the 3D configuration of the genome. However, 3D genome mapping data are derived from millions of cells6 and provided an average view of genome folding. Specifically, the techniques revealed smaller subunits of chromosome territories—topologically associated domains (TADs by Hi-C)7,8,9 and chromatin contact domains (CCDs by ChIA-PET)10, demarcated by multiple binding sites for CTCF protein11. However, even though the contact probabilities are clearly displayed, the mapping data lack physical scale (e.g., in micrometers and nanometers). To overcome these limitations, microscopy can be used to visualize the actual genome architecture and metric scale in various spatiotemporal resolutions in individual nuclei with different DNA staining methods. Standard fluorescence light microscopy has an optical resolution of about 250 nm12 and has been applied to image mammalian nuclei, successfully establishing the concept of chromosome territory13 showing chromosomal morphologies in different cell cycle phases, and is also routinely used in cytogenetics to study abnormal chromosomes14. Recently, a super-resolution fluorescence light microscopy (20 × 20 nm in xy dimensions, 50 nm in z dimension) with sequence-specific DNA-binding probes was applied to visualize specific chromatin folding structures for a 10–500 Kb and 3 Mb target genomic region in Drosophila cells15,16, and 1.2–2.5 Mb target genomic region in human cells17. However, this method has a limit in z-dimension image depth (up to 3 μm)17, and thus larger 3D chromatin structures beyond this limit would be truncated or lost. Electron microscopic in situ hybridization (EMISH) using biotin-labeled DNA probes coupled with diaminobenzidine staining has been used to image chromosomal DNA in the nuclei18. When standard FISH protocols are used in electron microscopy (EM) study, severe artefacts have been attributed to formamide and high-temperature treatment19. However, volume (3D) EM approaches—serial block-face scanning electron microscopy (SBF-SEM)20,21 and focused ion beam scanning electron microscopy (FIB-SEM)22 were used with success in studies of global architecture of cell nuclei. SBF-SEM can reach a resolution ~5 × 5 nm in xy, while resolution in z depends on a thickness of cut ultrathin slices, and FIB-SEM achieves a resolution down to ~3 nm in all dimensions23. The most recent effort in using EM for imaging chromatin structure is EM tomography with even higher resolution (1 × 1 × 1 nm)24, in which photo-oxidized label is used to mark all DNA and to visualize the overall chromatin structures for 3D imaging in the nucleus25, but within a limited depth at z-axis (250 nm). Nonetheless, our ability to visualize 3D chromatin-folding structures in high-quality ultra-resolution remains inadequate, hampered in particular by the lack of sequence specificity, and low depth of imaging in the z dimension.

In our efforts to achieve ultra-resolution visualization of sequence-specific 3D chromatin folding structures, we present 3D electron microscopic in situ hybridization (3D-EMISH) method that combines advanced in situ hybridization using biotinylated DNA probes26 with silver staining and serial block-face scanning electron microscopy (SBF-SEM)20,21. The serial z-stack EM images is assembled computationally into a 3D conformation of the targeted genomic region with a resolution of 5 × 5 nm in the xy plane and 30 nm in the z-axis. We apply 3D-EMISH to visualize a specific chromatin folding location in the human genome, and analyze more than 200 distinctive 3D chromatin structures derived from individual nuclei of human lymphoblastoid cells. Thus, we demonstrate that SBF-SEM can be applied for imaging specific chromatin-folding structures.

## Results

### The 3D-EMISH methodology

Our method includes 3D preservation of the nuclei, in situ hybridization of specific DNA probes, serial scanning by EM, and imaging data analysis to reconstruct the spatial models of chromatin folding structures (Fig. 1).

In the first step of the 3D-EMISH protocol, cells were fixed with 4% freshly made paraformaldehyde, and then embedded in a thrombin–fibrinogen clot, postfixed, cryoprotected in 30% sucrose, and frozen in dry ice-cooled isopentane. The embedded clots (<5 mm in size) were cut into 40-µm-thick sections in the freezing microtome. The next step in 3D-EMISH is in situ hybridization (ISH). The free-floating 40-µm-thick sections were incubated with the biotinylated DNA probes targeting a specific genomic region of interest. The signals of the biotinylated DNA probes for EM detection were processed with the use of 1.4-nm-thick streptavidin-conjugated fluoronanogold particles and subsequent silver enhancement. The procedure of silver enhancement was experimentally optimized to obtain electron dense 4–5-nm diameter granules as the smallest items (Supplementary Fig. 1a–c).

To optimize chromatin ultrastructure preservation, we performed a set of in situ hybridization (ISH) tests. Using transmission electron microscopy (TEM), we assessed in details the influence of potentially harmful factors associated within the ISH procedure (Supplementary Fig. 1). Comparing with the control of no ISH procedure (nucleus in native state) (Supplementary Fig. 1d), the ISH procedure including permeabilization by Triton X-100 (Supplementary Fig. 1e) or treatment with formamide and high temperature (Supplementary Fig. 1f) showed at most moderate changes or deteriorations in chromatin structure. However, surprisingly, we found that the inclusion of dextran sulfate, commonly used in ISH to increase the probe concentration and the hybridization reaction speed, caused the most distortion to the chromatin ultrastructure (Supplementary Fig. 1g), matched with prior experimental observations19. Therefore, dextran sulphate was omitted from ISH in our 3D-EMISH protocol. We observed that the lack of dextran sulphate in ISH had only a minor impact on hybridization efficiency and slightly increased background (Supplementary Fig. 1h–i). To ascertain our observation on dextran sulphate, we performed a specific DNA staining with terminal deoxynucleotidyl transferase (TdT) according to M. Thiry27. This method utilizes TdT to add labeled nucleotides to the free DNA ends, formed by ultrathin cutting, that are subsequently detected by immunogold staining. We again observed here that permeabilization with Triton X-100 had caused at most moderate changes or deteriorations in chromatin structure and DNA distribution. The TdT experiments showed further that extensive filamentous objects were observed with the addition of dextran sulfate. Some particles were also found in the cytoplasm, what might suggest the loss of nuclear membrane integrity (Supplementary Fig. 2).

After in situ hybridization, the 40-µm-thick sections were stained with uranyl acetate and tannic acid, followed by dehydrating and flat-embedding in an epoxy resin. Because uranyl acetate stains nucleic acids and proteins broadly, its inclusion in 3D-EMISH enabled us to visualize the overall nuclear space in relation to the target chromatin structure (Fig. 1a). One block of the embedded specimens (usually less than 1 mm in width) was placed in the chamber of SBF-SEM (details in “Methods”, Zeiss Sigma VP) equipped with a built-in ultramicrotome (Gatan 3View). The specimen block was consecutively sliced one layer at a time at 30–50-nm intervals, and the exposed surfaces of the specimen was serially scanned in a field of 8192 × 8192 pixels (~1700 µm2, when a pixel size is 5 nm) to obtain the volumetric data, including the specific 3D-EMISH signals. The specimen was sliced again for the next round of signal acquisition and so forth. This cycling process of slicing–scanning was performed hundreds of times to generate z-stack images in a 3D-EMISH experiment (Fig. 1a).

Since all in situ hybridization reactions result in nonspecific background26, the 3D-EMISH signals in each slice also included punctate patterns of staining that we deemed likely to be noise (see Fig. 1b; Supplementary Fig. 3). We reason that the specific signals from the targeted chromatin would be in continuation in multiple layers of the z-stack images, whereas the nonspecific signals would not be connected in the z-stacks (Fig. 1b). Therefore, we developed an image-processing algorithm based on multilayer connectivity to retain true chromatin-positive signals and remove false-positive one. The volume ratio of the specific vs. nonspecific signal, in the analyzed regions of interest, was estimated as 0.78 (± 0.40 STD, ± 0.12 SEM), the nonspecific signal occupied 1.88% (± 0.46% STD, ± 0.15% SEM) of the total background volume (± STD, 10 different ROIs). Examples of the images before and after background filtering are presented in the Supplementary Fig. 3. After filtering out the background noise, the chromatin signals were assembled to reconstruct the 3D chromatin-folding structure in ultra-resolution for the targeted genomic region (Fig. 1c, d).

### Chromatin-folding structures captured by 3D-EMISH

We applied 3D-EMISH to investigate chromatin-folding structures in the genome of the human lymphoblastoid cell line GM12878. This cell line has been extensively studied for epigenome and 3D genome mapping7,10. At a specific genomic region (chr7: 141,547,153– 143,237,066, 1.7 Mb length, hg38), three distinctive loop domains were inferred by extensive CTCF ChIA-PET data (Fig. 2a). RNAPII ChIA-PET and RNA-Seq further suggested that the genes in the first loop domain and part of the third domain are in active transcriptional loops. To confirm the presence of the chromatin loop domains inferred by these sequencing-based approaches, we designed and produced biotinylated DNA probes derived from 11 BAC clones to cover this region for 3D-EMISH experiments (Fig. 2a). To test the hybridization efficiency of the probes to the target genomic region, we also generated two-color fluorescence probes from the BAC clones that could help distinguish the three domains (Fig. 2a). Subsequently, we performed 3D FISH imaging analysis using confocal microscopy, and validated that these probes were efficient and highly specific (Fig. 2b).

Using the biotinylated DNA probes, we performed 3D-EMISH experiments in GM12878 cells at the exponential growth phase. We performed two independent experiments using two different EM settings, producing image voxel size of 7 × 7 × 50 nm in replicate 1 and 5 × 5 × 30 nm in replicate 2. The sample cubical blocks were analyzed by a series of 300 (replicate 1) or 600 (replicate 2) slicing and scanning cycles, rendering consecutive 50 nm (replicate 1) or 30 nm (replicate 2) sections, respectively. One specimen, typically about 1000 × 1000 × 40 µm, included tens of thousands of cells. However, the cubic volumes that were scanned by EM microscopy were much smaller and they were 3200 µm2 in xy and 15 µm in z for replicate 1, and 1700 µm2 in xy and 18 µm in z for replicate 2. Each of the z-stack images could capture a cross-section of ~10–30 cell nuclei, intact or truncated (Fig. 2c). After signal de-noising and assembling, the outline of a nuclear framework and the specific chromatin objects could be visualized respectively based on the uranyl acetate and the silver staining signals (Fig. 2d). From two independent 3D-EMISH experiments, a total of 166 nucleus image stacks were obtained. Most of them (140) were truncated nuclei and contained from 1 to 4 specific target signals, whereas 26 nuclei were intact with two or four specific chromatin targets (Fig. 2e), suggesting that the cells were possibly in the G1 phase (two copies of target region) or S-G2 phase of the cell cycle.

To compare 3D-EMISH with super-resolution microscopy, we applied iPALM28 and visualized the same targeted chromatin structures using iPALM-specific probes tagged with two-color blinking fluorophores for the same genomic region in GM12878 cells (Supplementary Fig. 4). The iPALM 3D image structure showed about fourfold lower resolution (20 × 20 nm in xy) than the 3D-EMISH images. In addition, the depth of an iPALM image at z dimension was limited at 750 nm29, thus capturing only incomplete chromatin structures, even though isolated nuclei were used for iPALM imaging to reduce the depth of cell specimen. Thus, 3D-EMISH not only provided higher resolution of chromatin images at xy plane than iPALM but also provided greater depth at the z-axis than iPALM, enabling visualization of complete 3D chromatin-folding structures. However, the use of two different fluorophores in iPALM allowed determination of the orientation of the chromatin-folding structure along the linear DNA strand.

### Heterogeneity of chromatin-folding structures

In total, we captured 229 distinct chromatin-folding structures (Supplementary Fig. 5) for the 1.7 Mb region from 166 nuclei examined in our 3D-EMISH experiments. At first glance, it appeared that the chromatin folding structures at this region in GM12878 cells varied from having a single domain to multiple sub-domains. To further quantitatively analyze the structural features, we developed a computational algorithm to characterize potential chromatin-folding properties. First, we aligned each of the reconstructed 3D chromatin structures along with its principal axes, by orthogonalization of the structure inertia tensor. This representation allows clear dissection of the chromatin signals in more detailed folding structures (Fig. 3a). Next, we calculated the local density centers in each chromatin structure, followed by diffusing the density from the centers to demarcate the boundaries and to identify distinctive domains in the 3D-EMISH structures (Fig. 3a). For example, in the three different 3D-EMISH structures (sID50, sID12, and sID42), one, two, and three local density centers were identified, respectively, corresponding to morphological domains (Fig. 3b; Supplementary Movies 13 and Supplementary Fig. 6). The sub-domains in each 3D-EMISH structure were demarcated by different colors (Fig. 3c).

Applying this algorithm to the aforementioned 229 chromatin-folding structures captured by 3D-EMISH, we analyzed the detailed chromatin domain composition in each of the structures (Fig. 4a; Supplementary Fig. 7). Interestingly, we identified 58 (25%) 3D-EMISH structures with one domain, 90 (39%) with two, 70 (31%) with three, and 11 (5%) with four or five domains (Fig. 4b). Only two structures were identified as five domains, and we merged them with four domains structure group for our statistical analysis. Remarkably, the structures with multiple sub-domains accounted for a combined 75% of all the structures, which is approximately in line with the ChIA-PET mapping data (Fig. 2a). To further characterize these chromatin-folding structures, we measured the volume and the surface area for each of them. These measurements showed that the chromatin structures with one domain had the smallest volume and surface values, and the structures with more than one domain showed increased values along with the numbers of domains (Fig. 4c, d; Supplementary Fig. 8 and Source Data). We also calculated the form factor (based on volume and surface; see Eq. 1 in Methods for the definition) for each structure, which showed the same trend as the other measurements, i.e., single-domain structures had the lowest form factor value (Fig. 4e).

Taken together, our imaging analyses suggest that the small percentage of one-domain structures captured in our data most likely exemplify condensed chromatin-folding states, whereas the majority of the structures contained sub-domains and could represent loose chromatin-folding characteristics with various substructures for different domain functions during interphase in B lymphoblastoid cells. It is noteworthy that many of the structures captured in this 3D-EMISH study showed three domains, consistent with our ChIA-PET data based on bulk cells (Fig. 2a). Furthermore, the ultra-resolution details unraveled a high level of heterogeneity of chromatin-folding structures within this 1.7 Mb region. Strikingly, each individual chromatin ultrastructure, even those having the same number of sub-domains, showed a uniquely distinctive chromatin-folding conformation (Fig. 4a; Supplementary Fig. 7).

## Discussion

We have demonstrated 3D-EMISH as an effective imaging technique with ultra-resolution (5 × 5 × 30 nm) capability to capture specific 3D chromatin-folding structures in individual nuclei, superior to the current super-resolution 3D visualization by fluorescence light microscopy. The use of in situ hybridized DNA probes targeting the specific genomic regions provided specificity of folding structures. The use of SBF-SEM enabled single-digit nanometer resolution for the xy dimension. More advantageously, it allowed collecting z-stack images up to 18 µm in total depth, which is not possible by super-resolution microscopy, such as iPALM and STORM, because of their optical image depth limit. In addition, each 3D-EMISH experiment could analyze large numbers of chromatin structures from multiple cells simultaneously, and multiple experiments could be robustly performed for analyzing large numbers of individual cells. The inclusion of en-bloc staining with uranyl acetate used in 3D-EMISH resulted in relatively weak counterstaining of the nuclear area, thus providing additional benefit for outlining the spatial location of the target structure in relation to the nuclear border.

Recently, another EM approach, ChromEMT, has been introduced for chromatin structure analysis25. It uses photo-oxidized diaminobenzidine polymers and OsO4 fixation, rendering a detection system size of ~1–2 nm. The method is also characterized by the highest available resolution (<1 nm in xy) and, due to utilizing EM tomography, ~1 nm in z. Therefore, ChromEMT can visualize single-DNA polymer loops. However, the ChromEMT approach stains all DNA in nuclei, and thus lacks sequence specificity and allows investigation of only 250-nm-thick section. In contrast, 3D-EMISH uses DNA probes to target a given genomic region of interest with high specificity, and the obtained 3D images can be immediately correlated with the available genomic mapping data, thus providing insights into potential structure–function relationships. Nonetheless, these two EM approaches are complementary to each other for ultra-resolution visualization of the global nature of chromatin organization and specific features of chromatin folding.

The ultrastructures of 3D chromatin-folding domains unraveled by 3D-EMISH are ultra-resolution snapshots of the target genomic region from individual cells. All of these 229 snapshots showed a dynamic nature of chromatin folding. The ultra-resolution nature of the 3D-EMISH structures displayed a high level of heterogeneity in chromatin folding with respect to different volume and shape even of the two copies of chromatin fold in the same nucleus. The variations in the 3D chromatin-folding structures within the 229 images may be a reflection of spatiotemporal dynamics and potential functional properties such as transcription activity and epigenetic state15, and also cell cycle phase25,30. Heterogeneity in the chromatin organization is observed at different levels. For example, we have shown different conformations of chromosome 1 even within the same nucleus10. There are also results obtained in human cells showing high variability in the physical distances between selected genomic loci31 and variability in the organization of chromatin domains16. Moreover, studies of transcription reveal expression heterogeneity32. Also in the same cells that we studied—GM12878, heterogeneity was shown using single cell sequencing approaches—scATAC-Seq33 and scRNA-Seq34 when compared with bulk studies. It suggests that variability observed by us and others could interplay with transcriptional activity. Nevertheless, the relation between dynamic changes of chromatin structure and stochasticity in gene expression is not yet fully understood35,36.

## Methods

### Cell culture

GM12878 (human lymphoblastoid) cells were purchased from commercial cell line, Coriell Institute. These cells were cultured in RPMI-1640 medium (Sigma) with addition of 2 mM GlutaMax (Gibco), 15% FBS (Gibco), 100 U/ml penicillin and 100 µg/ml streptomycin (Gibco), at 37 °C; 5% CO2. The cells authentication was done by the Jackson Laboratory for the Genomic Medicine according to ATCC (American Type Culture Collection) recommended authentication tests: Morphology Check by optical observation of a magnified cell culture and Growth Curve Analysis. Any cells showing inconsistent growth properties were discarded.

### DNA probes used in in situ hybridizations

The DNA probe was designed based on BAC (bacterial artificial chromosome) clones obtained from CHORI (Children’s Hospital Oakland Research Institute). The designed 11 BACs (CH17-148N23, CH17-120P14, CH17-417P16, CH17-265E23, CH17-52M16, CH17-310K12, CH17-121M24, CH17-326A22, CH17-215B18, CH17-450A09, and CH17-227I19) covered our genomic target region in human chromosome 7: 141,547,153–143,237,066, in hg38. The clones were verified with PCR using primers listed in Supplementary Table 1.

After isolating DNA from BACs using alkaline lysis method, DNA amplification was performed using the Illustra GenomiPhi V2 DNA Amplification Kit (GE Healthcare Life Sciences) and DNA labeling with biotin using the Biotin-Nick Translation Mix (Roche) and digoxigenin with the DIG-Nick Translation Mix (Roche). The product of reaction was mixed with human competitor DNA (Applied Genetics Laboratories, Inc.), salmon sperm DNA (Invitrogen), absolute ethanol and incubated 2 h at −20 °C. Then, the probe was centrifuged, dried with a vacuum centrifuge, and dissolved in 100% formamide (overnight; 37 °C; with shaking). To obtain a ready-to-use probe, a 4× SSC buffer was added to the probe dissolved in 100% formamide (1:1; finally probe dissolved in 50% formamide in 2× SSC). In the case of light microscopy, dextran sulphate was also added to final concentration 10%.

### Cell preparation for confocal microscopy

The cell pellet was fixed with 4% paraformaldehyde in PBS (10 min; RT), washed with PBS and cells were spun down onto a glass slide (1 × 105 cells/13-mm coverslip) using a cytocentrifuge.

### Cell preparation for iPALM

Nuclei were isolated (Sigma; Nuclei Isolation Kit: Nuclei EZ Prep), fixed with 4% paraformaldehyde in PBS (10 min; RT) and attached onto coverslips (with fiducial markers; see: iPALM imaging and post processing in “Methods”) washed earlier with 1 M KOH (20 min), rinsed with water and coated with 0.01% poly-L-Lysine for 20 min (Sigma).

### DNA-FISH protocol

3D DNA-FISH was performed according to Cremer et al.37, with modifications. In details, after blocking of endogenous biotin (Vector Laboratories; SP-2001), the cells were permeabilized with 0.5% Triton X-100 (Sigma) in PBS (RT, 20 min). To augment permeabilization, the cells were immersed in a cryoprotectant solution (20% glycerol in PBS, RT, 2 h) followed by their repeated freezing–thawing above the surface of the liquid nitrogen (4 × 30 s). Subsequently, the cells were washed in PBS containing 0.05% Triton X-100 (Sigma) (RT, 3 times for 5 min), treated with 0.1 N HCl (RT, 5 min), washed again in 2× SSC at 37 °C and incubated in 50% formamide in 2× SSC (4 °C, overnight). On the next day, the probe was added, the samples were denatured (5 min, 80 °C) and hybridized (48 h, 37 °C). Then, the samples were washed with 2× SSC (3 × 5 min; 37 °C; with shaking), 0.1× SSC (3 × 5 min; 60 °C; with shaking), rinsed briefly with 0.2% Tween (Sigma) in 4× SSC and incubated with blocking solution—4% BSA (Sigma), 0.2% Tween (Sigma), 4× SSC (1 h, 37 °C) following incubation with the detection system for confocal microscopy or iPALM.

### DNA-FISH signal detection for confocal microscopy

DNA probe signals were captured by two-steps system detection of biotin-labeled probe with avidin–AlexaFluor488 (Molecular Probes; dilution 1:100; final concentration: 10 µg/ml) and anti-avidin-FITC (Sigma; dilution 1:100; final concentration: 20 µg/ml), and two-steps system detection of digoxigenin-labeled probe with anti-digoxigenin–rhodamine produced in sheep (Roche; dilution 1:100; final concentration: 2 µg/ml) and secondary antibody donkey anti sheep-TRITC (Jackson ImmunoResearch; dilution 1:100; final concentration: 10 µg/ml). DNA was counterstained using Hoechst 33342 (Molecular Probes), and samples were mounted with Vectashield (Vector Laboratories).

### Confocal microscopy imaging and post processing

The samples were imaged with Zeiss LSM 780 microscope, using a 63× oil immersion objective (NA 1.4) and a 405-nm diode laser, 488 nm line of argon laser, 561-nm DPSS diode laser, 633-nm HeNe laser; with 70-nm xy pixel size, and 210-nm z-spacing. The image stacks were deconvolved using Huygens software (SVI) with the maximum-likelihood estimation (MLE) algorithm. FISH signals imaged by confocal microscopy were segmented based on the threshold and reconstructed using Imaris (Bitplane) software.

### DNA-FISH signal detection for iPALM

DNA probe signals were captured by one-step system detection of biotin-labeled probe with streptavidin-CAGE590 (Abberior; dilution 1:250; final concentration: 3.5 µg/ml), and one-step system detection of digoxigenin-labeled probe with anti-digoxigenin–Alexa Fluor 647 (Jackson ImmunoResearch; dilution 1:250; final concentration: 2 µg/ml).

### iPALM imaging and post processing

Coverslips for iPALM with fiducial markers29 were prepared according to the protocol: 25 mm diameter #1.5 thickness high-precision coverslips (Warner Instruments) were cleaned in a basic “piranha” solution of 5:1:1 H2O:NH3OH:H2O2 (50% w/v) at 90 °C for 4 h, followed by washing in deionized water and drying. Coverslips were then coated with 5 mg/ml poly-L-Lysine (PLL), MW 70,000 (MP Biomedicals) in water for 30 min, and washed. Finally, a solution of negatively charged gold nanorods with major/minor diameters of 60/40 nm (Nanopartz, Inc.) was allowed to adhere to the PLL layer, followed by further washing of excess particles and drying. Finally, coverslips were coated with a ca. 50-nm layer of SiO2 to passivate the fiducial markers using a Denton Vacuum Explorer sputter coater. The samples for iPALM were imaged in STORM buffer38, containing 50 mM TRIS (pH 8.0), 10 mM NaCl, 0.5 mg/ml glucose oxidase, 40 μg/ml catalase, 10% glucose (w/v), and 100 mM 2-mercaptoethylamine (MEA) (all from Sigma), using coverslips with fiducial markers29 described earlier in this paragraph. Above cells deposited on the coverslip with fiducial markers, second clean coverslip was placed, sealed and samples were imaged using noncommercial iPALM system28 available at Advanced Imaging Center–HHMI’s Janelia Research Campus, with excitation: 647 nm laser (5.5 kW/cm2) and 561-nm laser (4.5 kW/cm2) and activation: 405-nm laser (100 W/cm2). The number of frames was 25,000, and the exposure time was about 50 ms. iPALM data were processed to extract single-molecule 3D coordinates and localization precisions, and subsequently visualized using the PeakSelector software (Janelia Research Campus)29. FISH signals acquired by iPALM were segmented based on the threshold with PartSeg software (https://4dnucleome.cent.uw.edu.pl/PartSeg/). Then, 3D reconstructions of iPALM images were obtained using Imaris (Bitplane).

### Cell preparation for 3D-EMISH

The experiments were performed on cultured GM12878 cells. After centrifugation, fixation of the cell pellet (~7 × 107 cells) with 4% paraformaldehyde in PBS overnight at 4 °C and washing with PBS, the artificial tissue with embedded cells was formed. To the fixed and washed pellet, 0.2 ml of the fibrinogen solution (100 mg fibrinogen (Sigma; F3879), 53.33 mg of sodium citrate, 283.33 mg of sodium chloride, 33.33 ml H2O) was added and stirred. After centrifugation, 0.2 ml the thrombin solution (Sigma; T7009-100UN + 1 ml H20) was added39. The clot was postfixed with 4% paraformaldehyde in PBS (10 min; RT) and washed with PBS. Then, the formulated artificial tissue was soaked with 30% sacharose in PBS overnight at 4 °C and frozen in tissue freezing medium (Leica). After overnight storage at −80 °C, the clot was cut into 40-µm slices using cryostat (Leica).

### In situ hybridization for 3D-EMISH

In situ hybridization was performed on the free-floating sections of the clot according to Cremer et al.37, with modifications. In details, the sections were washed with PBS (3 × 5 min; RT), 0.5% Triton X-100 (Sigma) in PBS (1 × 20 min; RT), 2× SSC (2 × 2 min; 37 °C), incubated with 0.1 mg/ml RNase A in 2× SSC (Sigma; R6513) (10 min; 37 °C), washed with 2× SSC (2 × 2 min; 37 °C), incubated with citrate buffer pH 6 (30 min, 80 °C), cooled down to room temperature and washed with 2× SSC (2 × 5 min; RT). After that, blocking of endogenous biotin was performed using the Avidin/Biotin Blocking Kit (Vector Laboratories; SP-2001), by incubation with Avidin D (four drops of the Avidin D solution to 1 ml 2× SSC) (15 min; RT), briefly washing with 2× SSC and incubation with the biotin solution (four drops of the biotin solution to 1 ml 2× SSC) (15 min; RT), followed by washing with 2× SSC (2 × 5 min; RT) and incubated with 50% formamide in 2× SSC overnight at 4 °C. After this step, the sections and biotin-labeled DNA probes were overlaid with a glass chamber, sealed with rubber cement, prehybridized (4 h, 45 °C), denatured (5 min, 80 °C) and hybridized (48 h, 37 °C). After 2-days hybridization, sections were washed with 2× SSC (2 × 10 min; 37 °C; with shaking), 0.1× SSC (2 × 10 min; 60 °C; with shaking), incubated with blocking solution- 4% BSA (Sigma), 0.2% Tween (Sigma), 4× SSC (1 h, 37 °C) following incubation with Alexa FluorTM 488 FluoroNanogoldTM-Streptavidin (Nanoprobes; 7216) in 2% BSA (Sigma), 0.2% Tween (Sigma), 4× SSC (overnight; 4 °C; with shaking—concentration 0.8 μg/ml) and washed with 0.2% Tween (Sigma) in 4× SSC (3 × 10 min, RT; with shaking). After hybridization and washing, sections were mounted in 2× SSC buffer and imaged with the Zeiss LSM 780 confocal microscope using the 488-nm argon laser and the water objective (C-Apochromat 40×/1.2 W Corr. FCS). Only sections containing nuclei with well-visible FISH signals were processed to 3D-EMISH.

### 3D-EMISH staining

Sections after passing the quality control using confocal microscopy were washed with ddH20 (5 × 3 min; RT) and incubate with silver enhancement kit to increase size of 1.4 nm nanogold particles (4 × 10 min; RT; LI Silver Enhancement Kit (Molecular Probes; L-24919)) and washed tightly with ddH20. After silver enhancement, sections were postfixed with 2.5% glutaraldehyde in 0.1 M cacodylate buffer (Sigma) (2 h; RT), washed with 0.1 M cacodylate buffer (Sigma) (3 × 10 min; RT), incubated with 1% tannic acid in 0.1 M cacodylate buffer (Sigma) (2 × 2 h; 4 °C), washed with water (5 × 3 min; RT) and incubated with 4% uranyl acetate in 50% ethanol (overnight; RT). Then the dehydration was performed using increasing concentration of ethanol: 50%, 60%, 70%, 80%, and 95% (1 × 15 min each; RT), absolute ethanol (2 × 20 min; RT). After dehydration, sections were prepared to embedding by incubation with acetone (2 × 10 min; RT), increasing the ratio of Epon (Sigma)/acetone mix: 1:3, 1:1, 3:1 (1 h each; RT), and Epon (1 h; RT and then overnight; RT). The next day, Epon was replaced with fresh one, and specimens were embedded between two sheets of ACLAR® Film and placed into 60 °C oven (24–48 h). After polymerization of the resin, the sample was cut with razor blade and scalpel, glued on pin (metal rivet; Oxford Instruments) with silver conductive resin (CircuitWorks), and incubated 2 h at 65 °C. Using ultramicrotome (Leica) and diamond knife (DiATOME), smooth surface of sample to SBF-SEM was obtained. Then, to enhance conductivity, sides of the sample were painted with silver paint (Ted Pella).

### EM imaging

To image 3D-EMISH samples, Zeiss Sigma electron microscope with backscatter electron detector and ultramicrotome with diamond knife inside the chamber was used (3View2 from Gatan). All images were collected using variable pressure mode with EHT 4–6 kV, aperture 30 µm and resolution 8192 × 8192 pixels. Replicate 1 was collected with pixel size 7 nm, sliced each 50 nm, 300 times (voxel size: 7 × 7 × 50 nm). Replicate 2 was collected with pixel size 5 nm, sliced each 30 nm, 600 times (voxel size: 5 × 5 × 30 nm). To examine potential harmful effect of reagents on nuclear ultrastructure, a transmission electron microscope JEM 1400 (JEOL) was used.

### Image processing and quantitative analysis

All 3D electron microscopy stacks were preprocessed with ImageJ plugin–Linear Stack Alignment with SIFT40. The images were manually inspected, and then a cuboid containing the region of interest (ROI) was cut. The mask defining each ROI was determined by connecting component analysis of objects inside the cuboid. We segmented the largest connected component using custom-made software (3D-EMISH image-processing code), see the “Code Availability”), where we used Maximum Entropy algorithm (ImageJ plugin) to optimize the threshold value to segment the structure. The resulting mask was overlaid on the image to segment the structure. Subsequently, the 3D images were resampled in order to obtain the same voxel size (5 nm) in x, y, and z. Reslicing of the 3D images was performed by upsampling with linear interpolation between adjacent planes. We also applied a gaussian filter of size 1 pixel (i.e., 5 nm) in the x–y planes in order to remove possible pixel noise. The isotropic scale was required by the plugin used to produce the movies (with unequal scales the structures appear in the movies unnaturally flattened), and also it was needed by the algorithm for morphological domains separation, which operates on a cubical grid. The upsampling is not obviously visible unless we rotate the structure through an angle perpendicular to the axial direction through an axis perpendicular to its longest (principle) axis, which in general is randomly oriented with respect to the slicing direction (Supplementary Movies). Then, Gaussian fliter (5 nm cutoff size) was applied to each z-section to remove the detector noise. The volume (V) and surface (S) of the segmented structure were calculated by 3D Object Counter plugin (ImageJ). The form factor ($$ff$$) was defined as

$$ff = \left( {36\,\pi } \right)^{ - \frac{1}{3}}\,S\,V^{ - \frac{2}{3}},$$
(1)

where the value is 1 for a spherical object and increase as the structure is less compact than a sphere. For illustrative purpose, the segmented structures were aligned accordingly to their principal axis41. The structures were classified into five different morphological groups: (1) compact sub-domain group (one spherical ball shape), (2) distinctive sub-domain group (dumbbell shape) and (3–5) distinctive sub-domain groups, by applying the following sub-domain identification algorithm as follows:

First, the structure was smoothed out by applying the maximum filter with 135 nm cutoff size. This parameter setting was experimentally tested and determined to prevent from over-segmenting sub-domains of a structure, yet preserving the overall morphological feature. Second, the local maximal density centers were calculated per each identified distinctive group. Third, these local maxima were subsequently used as seed points in computing of the diffusion process, where the image density was used as a local diffusion coefficient. The diffusion process was simulated by solving numerically an uncoupled set of diffusion equations,

$$\partial u_{\mathrm{i}}\left( {{\boldsymbol{r}},t} \right) / \partial t = \nabla \cdot \left[ {D\left( {u_{\mathrm{i}}\left( {{\boldsymbol{r}},t} \right),\boldsymbol{r}} \right)\nabla u_{\mathrm{i}}\left( {{\boldsymbol{r}},t} \right)} \right],$$
(2)

where $$u_{\mathrm{i}}$$ denotes the density of the diffusing material, and D denotes the EMISH signal density, i = 1…N (the number of seed points), $${\boldsymbol{r}}$$ are the pixel coordinates in 3D space.

Initially, we assume that all EMISH signals of each sub-domain was concentrated in each seed point. We represented it as normalized gaussian functions centered at the seed points. Image voxels, diffused from the same seed point, belong to the same sub-domain group. The 3D image region of low EMISH signal density will diffuse slower than that of high EMISH signal density. The boundary voxels had diffused signals from two or more different seed points, and these boundary voxels belong to the sub-domain group which is the highest diffused signal. The diffusion process was completed when all voxels were filled by a nonzero density and the structure was divided into separate sub-domains.

The specific-to-nonspecific signal ratio was measured for a set of ten randomly selected three-dimensional region of interests (ROIs), each of them was an EM stack, containing a single structure and nonspecific background. We provided a mean signal ratio, its standard deviation, and a standard mean error, these values were calculated over a set of selected ROIs. The signal ratio was calculated as follows: For each of these 3D EM stacks, we created the following binary masks: specific signal mask (SP), nonspecific background signal mask (NS), and no-signal background mask (BN). The specific and nonspecific signal volume were obtained by summing all mask elements for SP and NS masks, respectively, and multiplying them by the voxel volume, by which we obtain specific-to-nonspecific signal ratio. The content of the nonspecific signal in the background volume was obtained by dividing NS mask volume by the total background volume (sum of NS and BN mask volumes)

### Data collection and 3D visualizations

For the data collection, we used Fiji distribution of ImageJ (https://imagej.net/Fiji/Downloads#Life-Line_Fiji) and Gatan Microscopy Suite Software (commercially available). The images of nuclei were cropped manually, 3D visualizations (Figs. 1, 2; Supplementary Fig. 3) were prepared using Amira (FEI), Imaris (Bitplane), PartSeg (https://4dnucleome.cent.uw.edu.pl/PartSeg/) and UCSF Chimera software42. Data were analyzed using custom written scripts, see “Code Availability”.

### Statistical analysis

Statistical analysis was performed using IBM SPSS Statistics software. In all cases, error bars indicate SEM. The morphological feature statistics of 3D-EMISH structures for different sub-domain groups was performed using Kruskal–Wallis test with two-sided Mann–Whitney tests with Bonferroni correction (post hoc tests). Statistically insignificant comparison was marked as ns, and statistically significant differences were marked with asterisks: *p ≤ 0.05; **p ≤ 0.01; ***p ≤ 0.001.

### Reporting summary

Further information on research design is available in the Nature Research Reporting Summary linked to this article.

## Data availability

The 3D-EMISH image and data files are available at the following public repository server: https://github.com/3DEMISH/3D-EMISH. Source data for the figures presented in this manuscript are available in the Source Data file. All other relevant data are available from the authors upon reasonable request.

## Code availability

The 3D-EMISH image-processing code is available at the following public repository server: https://github.com/3DEMISH/3D-EMISH.

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## Acknowledgements

This work was primarily supported by the Human Frontier Science Program, grant no. RGP0039/2017 (to Y.R. and G.M.W.). Y.R. is supported by NIH UM1 (HG009409, ENCODE), U54 (DK107967, 4DN), HFSP (RGP0039/2017), and the Roux family endowment. G.M.W. is also supported by the grants from Polish National Science Center no. UMO-2013/08/M/NZ3/00655. P.T. is also supported by a Polish National Science Centre Grants 2014/15/N/NZ2/00379 and 2017/24/T/NZ2/00307. Y.R. and B.L. are further supported by 4DN (U54 DK107967), ENCODE (UM1 HG009409) consortia, and the Roux family endowment. D.P., G.B., M.K., and Z.P. are supported by Polish National Science Centre (2014/15/B/ST6/05082), Foundation for Polish Science co-financed by the European Union under the European Regional Development Fund (TEAM), and 4DN (U54 DK107967). K.K.P. is partially supported by ETIUDA grant from Polish National Science Centre no. UMO-2019/32/T/NZ4/00502. A.M is partially supported by Sonata bis grant from Polish National Science Centre UMO-2015/18/E/NZ3/00730. The iPALM imaging work was conducted in collaboration with the Advanced Imaging Center at Janelia Research Campus of Howard Hughes Medical Institute. The authors would like to thank Emil Drzewek for his help with Fig. 1a (SBF-SEM scheme).

## Author information

Authors

### Contributions

G.M.W. and Y.R. conceived and supervised the project. P.T., B.R., and B.L. designed the experiments. P.T. performed the 3D-EMISH experiments under G.M.W.’s supervision, and generated the raw imaging data with K.K.’s assistance. B.R. developed the sub-domain identification algorithm and processed them. P.T., B.R., K.K.P., G.B., and K.K. interpreted and processed the raw data. B.R., K.K.P., G.B., and M.K. performed post processing of EMISH images, and/or statistical analysis. B.R. and K.K.P. designed and performed the detection of morphological domains in the EMISH-detected structures. J.A. executed iPALM imaging, together with P.T. and G.M.W. M.S. and A.A.S. participated in EMISH imaging. Artur Wolny contributed toward the paper figures preparation S.A. and M.K.A. help in electron microscopic imaging. Agnieszka Walczak participated in the initial design of EMISH hybridization. A.M., Z.P., and D.P. provided remarks to the discussion on experimental strategy. P.T. wrote the first draft of the paper. Y.R., B.L., G.M.W., B.R., P.T., and K.K.P. extensively edited the paper. All authors revised the final paper.

### Corresponding authors

Correspondence to Yijun Ruan or Grzegorz M. Wilczyński.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

Peer review information Nature Communications thanks Dusan Cmarko and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Rights and permissions

Reprints and Permissions

Trzaskoma, P., Ruszczycki, B., Lee, B. et al. Ultrastructural visualization of 3D chromatin folding using volume electron microscopy and DNA in situ hybridization. Nat Commun 11, 2120 (2020). https://doi.org/10.1038/s41467-020-15987-2

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text:
# Awasome Formula For Percentage In Excel References

Awasome Formula For Percentage In Excel References. Press enter.the percentage value on the first block of the discount column will appear. Web percentage formula in excel:

Percentages can be calculated using the formula =part/total. 87\% \cdot 824 = 716.88 87% ⋅ 824 = 716.88. Web the excel formula is going to look like this:

### Web The Excel Formula Is Going To Look Like This:

Web else, we can also use the below formula. Add a column for % gain or %loss. Use the opening round bracket to start the formula.

### Web Basic Percentage Formula In Excel There Is No Single Formula For Percentage Which Is Applicable In Every Calculation.

Hit the enter key after entering the following formula for your dataset. Part/total = percentage if you compare it to the basic math formula for percentage , you will notice. As an example, if you're trying to apply a discount, you would like to.

### Web We Need To Find Out How Much Each Student Has Obtained In Terms Of Percentage Of The Total Number.

Open a new excel file, and in cell a1, type equal (=) sign to initiate a formula. = (b2/c2)*100 hit enter to get the student’s score as a. Web type in the following formula and press the return key:

### Web Now, Let’s Try To Apply This Formula On Excel Data To See How This Formula Works In Case Of Percent Change.

For example, given an original price of \$70.00, and an decrease of. Web in this example, the goal is to decrease the prices shown in column c by the percentages shown in column d. Although the basic principle is the.

### In This Example, We’re Going To Click And Highlight Cell C3.

In cell c2, enter the formula for calculating the total price based on percentages. Web in math, the percentage of a number is calculated by multiplying that number by the percentage: Web when you are calculating a basic percentage in excel, we use the formula:

format: plain


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## anonymous one year ago what is |3/4a-3|=9

1. Owlcoffee

you can get rid of the absolute value by squaring both sides: $\left| \frac{ 3 }{ 4 }a-3 \right|=9$ $\rightarrow (\frac{ 3 }{ 4 }a-3)^2=81$

2. anonymous

Thank you so much


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# Reducing Equations To Simpler Form

Reducing equation is a method of rewriting the equation in a simpler form. Certain equations in mathematics are not in the form of linear equations but they can be put in the form of linear equations by performing some mathematical operations on them. After reduction of these equations in linear form they can be solved and the value of unknown can be calculated easily.

Let us take an example to understand the method of reducing equations in simpler form.

## Solved Examples

Example 1: Solve $\frac{x+1}{3x-6}=\frac{1}{4}$

Solution: As the equation is not in linear form, so this cannot be directly solved for solution. Therefore first we will need to simplify the given equation by Cross Multiplication technique. In this technique, the numerator on both sides needs to be multiplied to the denominator on the other side.

After forming the Linear Equation, we can easily find the value of the unknown.

After cross multiplication, the given equation can be written as:

4 (x +1) = 1 (3x – 6)

Now, using the distributive law, the parentheses can be opened as:

⇒ 4x + 4 = 3x – 6

Now using the method of transposition, the variables are brought on one side of the equation and the constants are brought on the other side.

⇒ 4x – 3x = -6 – 4

⇒ x = -10

Thus the value of the unknown in the given equation is -10.

For reducing equations to simpler form or linear form follow the following steps:

• Bring the like terms i.e. variables on one side and constants on other side of the equation
• Using the distributive law, open the brackets
• Cross multiply both sides of the equation i.e. multiply the numerator of one side by the denominator of other side
• Solve the equation as linear equation in one variable

Example 2: Solve the given linear equation:

(3x-4)/2 – (2x-1)/3 = (1/2)+x

Solution:

Given,

(3x-4)/2 – (2x-1)/3 = (1/2)+x

Taking the LCM for the left and right hand terms,

[3(3x-4)-2(2x-1)]/2.3 = (1+2x)/2

Now open the brackets to simplify the equation.

(9x-12-4x+2)/6 = (1+2x)/2

(5x-10)/6 = (1+2x)/2

Now cross-multiplying the above equation, we have

2(5x – 10) = 6(1+2x)

10x – 20 = 6 + 12x

-2x = 26

Therefore, x = -13


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## Keywords (1)

Publications
On (⋶, ⋶ ∨ q)-fuzzy ideals of BCI-algebras

# On (⋶, ⋶ ∨ q)-fuzzy ideals of BCI-algebras,10.1007/s00521-010-0376-6,Neural Computing and Applications,Jianming ZhanYoung,Young Bae Jun

On (⋶, ⋶ ∨ q)-fuzzy ideals of BCI-algebras
The concepts of $$(\overline{\in},\overline{\in} \vee \overline{q})$$-fuzzy (p-, q- and a-) ideals of BCI-algebras are introduced and some related properties are investigated. In particular, we describe the relationships among ordinary fuzzy (p-, q- and a-) ideals, (∈, ∈ ∨ q)-fuzzy (p-, q- and a-) ideals and $$(\overline{\in},\overline{\in} \vee \overline{q})$$-fuzzy (p-,q- and a-) ideals of BCI-algebras. Moreover, we prove that a fuzzy set μ of a BCI-algebra X is an $$(\overline{\in},\overline{\in} \vee \overline{q})$$-fuzzy a-ideal of X if and only if it is both an $$(\overline{\in},\overline{\in} \vee \overline{q})$$-fuzzy p-ideal and an $$(\overline{\in},\overline{\in} \vee \overline{q})$$-fuzzy q-ideal. Finally, we give some characterizations of three particular cases of BCI-algebras by these generalized fuzzy ideals.
Journal: Neural Computing and Applications - NCA , vol. 20, no. 3, pp. 319-328, 2011
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## Citation Context (3)

• ...In [20], we introduced the concepts of ([, [ _ q)-fuzzy (p-,q- and a-) ideals of BCI-algebras and investigated some of their related properties...
• ...Theorem 2.10 [9, 20] A fuzzy set l of X is an ([, [ _ q)fuzzy (p-, q- and a-) ideal of X if and only if each nonempty level subset lt is a (p-, q- and a-) ideal of X for all 0 \ t B 0.5, respectively...
• ...Consider J = {t| t [ (0, 1] and lt is an empty set or a p-ideal of X}. In [20], we obtained the following results:...
• ...Lemma 3.2.3 [20] A fuzzy set l of X is an ð2; 2_ qÞfuzzy p-ideal of X if and only if it satisfies (GF5) and (F3’)...
• ...Definition 3.2.7 [20] Let a, b [ [0, 1] and a \b. Then a fuzzy set l of X is called a fuzzy p-ideal with thresholds (a, b ]o fX if it satisfies (F1 00 ) and...
• ...Theorem 3.2.8 [20] A fuzzy set l of X is a fuzzy p-ideal with thresholds (a, b ]o fX if and only if lt(= ; )i s a p-ideal of X for all a \ t B b...
• ...In [20], we introduced the concepts of ([, [ _ q)-fuzzy (p-, q- and a-) ideals of BCI-algebras and investigated some of their related properties...
• ...As a continuation of [20], we further discuss this topic in the present paper...

### Jianming ZhanYoung, et al. On (⋶, ⋶ ∨ q)-fuzzy ideals of BCI-algebras

• ...33 A fuzzy set μ in a BCI-algebra X is called an
Young Bae Junet al. Soft ideals of BCK/BCI-algebras based on fuzzy set theory
• ...It is well known that BCK and BCI-algebras are two classes of logical algebras; they were introduced by Iséki 8 and have been extensively investigated by many researchers 4 5 10 11 12 13 14 15 16 17 18 24 25 26...

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# Problem 7, Ch1 from Blitzstein and Hwang, Intro to Probability

I am trying to solve this problem.

Two chess players, A and B are going to play 7 games. Each game has three possible outcomes : a win for A (which is a loss for B), a draw(tie) and a loss for A (which is a win for B). A win is worth 1 point, a draw is worth 0.5 points and a loss is worth 0 points.

(a) How many possible outcomes for the individual games are there such that the player ends up with 3 wins, 2 draws and 2 losses?

(b) How many possible outcomes for the individual games are there, such that A ends up with 4 points and B ends up with 3 points?

My attempt:

(a)This is the same as the number of permutations of the WWWDDLL. There are $$7\choose{3}$$ ways to decide where to put the W, then $$4\choose2$$ ways to decide where to put the D and $$2\choose2$$ ways to decide where to put the L.

This gives $${7\choose3}\cdot{4\choose2}\cdot{2\choose2}=\frac{7!}{3!2!2!}=210$$ possibilities.

(b) A must end up with 4 points, and by symmetry B must end up with three points.

We are then interested to find the number of non-negative solutions to the equation

$$1\cdot{x_{1}}+0.5\cdot{x_{2}}+0\cdot{x_{3}}=4$$

This amounts to having 3 distinguishable boxes, into which 7 indistinguishable particles can be put. Each particle can be either a win, a draw or a loss.

For example,

 W  D  L

|***|**|**|

There are 7+3-1=9 slots and we need to select 2 out of the 9 slots, where two |'s can go. But, a draw counts only as 0.5 points. So, I don't think this yields the correct answer.

It would be nice, if someone could verify part (a) and help with part(b).

Edit 1

There is a also a part (c) to the question.

(c) Now assume that they are playing a best-of-7 match, where the match will end when either player has 4 points or when 7 games have been played, which is first. For example, if after the 6 games, the score is 4 to 2 in favor of A, then A wins the match and they do not play a 7th game. How many possible outcomes for the individual games are there, such that the match lasts for 7 games and A wins by a score of 4 to 3?

My attempt:

c) If the match is to last till the 7th game and A to win by a score of 4 to 3, the 7th game should not be a loss for A. Thus, we have

• 4 wins, 3 loss (WWWLLW) - $${7\choose4}\cdot{3\choose3}-{6\choose4}\cdot{2\choose2}$$
• 3 wins, 2 draws, 2 loss (WWWLLDD) - $${7\choose3}\cdot{3\choose2}-{2\choose2}\cdot{2\choose2}$$
• The first is fine. For the second, let's work with B, since $3$ is smaller than $4$. Could be $6$ draws or $4$ or $2$ or $0$. Count all separately. For example, $4$ draws is for reasons you are familiar with $\binom{7}{4}\binom{3}{1}$. Jul 17 '16 at 10:21
• @AndréNicolas , so its the sum of 1) DDDDDDW - (7 Choose 1) x (1 Choose 1) 2) DDDDWWL- (7 Choose 4) x (3 Choose 1) x (1 Choose 1) 3) DDWWWLL - (7 Choose 2) x (5 Choose 3) x (2 Choose 2) 4) WWWWLLL - (7 Choose 4) x (3 Choose 3) Jul 17 '16 at 11:28
• Yes, the case numbers are right, then we compute and add up. The $\binom{1}{1}$, $\binom{2}{2}$, and so on are all $1$, so unnecessary though harmless. There are other more "general" kinds of approaches, for example using generating functions, but for small numbers like these the approach you carried out is efficient. The answer by Shagnik uses essentially the same sort of case analysis. In the answer you wrote out, you forgot to add up at the end, Jul 17 '16 at 12:57
• @AndréNicolas, is the answer to part(c) also correct? Jul 18 '16 at 1:43


text:
# .svg not exported through org-export + paragraphs not separated after newline

I have been constantly trying to render a logo.svg found in my custom title-page (included in #LATEX:) to match our university title-page for academic writings, but have unfortunately not had success doing so. I have since yesterday been trying to render the .svg files included in my .org writing, but unfortunately arrived at no solution..

Note that the following list includes methods I have attempted in a doom-emacs enivronment to resolve the issue while having inkscape, libsvg and imagemagick installed on my system.

What I have tried by far:

1. Used different formats (.png, .pdf and .svg), but failed to display any image through both \includegraphics and \includesvg. Keep arriving at: "cannot find file" and I do not know how to resolve this issue. I have tried to use both full/relative paths, but couldn't produce the desired output.

2. .svg files located inside .org files are not exported through org-export even after including #+LATEX_HEADERS: \usepackage{svg}. I suspect that this could be the issue, but I am not certain.

3. pdflatex -shell-escape file.tex returns the same issue as org-export does, which is the .svg images not being displayed.

Another issue that I am facing is paragraphs not being created for lines separated by a newline between them. Because of this, my text currently looks like one mash of text without any separation, which is in my opinion looking hideous.

If you know how to resolve this issue, please do help me since dropping emacs because of this is in my personal opinion not a wise choice. Not to mention that I am really comfortable in my doom-emacs environment and would prefer it over writing my academic work in google-docs. :/

• The pastebin for both the template.texand the file.org can be found at: custom title-page and org-file – Icy-Thought Mar 30 at 21:02
• The paragraph issue was resolved through #+OPTIONS: \n:t thanks to Ryan from the org-mode matrix server. Will update if I figure out how to configure paragraphs to adjust them to . – Icy-Thought Mar 31 at 12:33
• Removed \begin{figure} and \end{figure}, replaced includesvg{logo.svg} with \includesvg{src/logo.svg}and also included --shell-escape in org-mode config.el to fix this issue thanks to hpd from org-mode matrix server. – Icy-Thought Mar 31 at 14:12

Solution:

1. Paragraph spacing: add #+OPTION: \n:t.
2. Fix svg rendering issue: add --shell-escape to config.el (doom-emacs):
(setq org-latex-pdf-process
'("pdflatex -shell-escape -interaction nonstopmode -output-directory %o %f"
"pdflatex -shell-escape -interaction nonstopmode -output-directory %o %f"
"pdflatex -shell-escape -interaction nonstopmode -output-directory %o %f"))

1. Source logo.svg in file.org based on the relative path of logo.svg from file.org. In my case the relative path is src/logo.svg, which results in: \includesvg{src/logo.svg}

Appreciate the help both @ryan_c_scott and @hpd from org-mode matrix server have provided me and wish that this will be useful to someone else facing the same issue!

• You can 'accept' your own answer, which helps us by removing your question from the 'unanswered' queue. – Tyler Apr 9 at 20:29


text:
# Tag Info

46

The answer depends on whether you are assuming the symmetric or asymmetric dirichlet distribution (or, more technically, whether the base measure is uniform). Unless something else is specified, most implementations of LDA assume the distribution is symmetric. For the symmetric distribution, a high alpha-value means that each document is likely to contain a ...

32

Recently, a huge body of literature discussing how to extract information from written text has grown. Hence I will just describe four milestones/popular models and their advantages/disadvantages and thus highlight (some of) the main differences (or at least what I think are the main/most important differences). You mention the "easiest" approach, which ...

26

Implementation: The topicmodels package provides an interface to the GSL C and C++ code for topic models by Blei et al. and Phan et al. For the earlier it uses Variational EM, for the latter Gibbs Sampling. See http://www.jstatsoft.org/v40/i13/paper . The package works well with the utilities from the tm package. The lda package uses a collapsed Gibbs ...

17

This is indeed something often glossed over. Some people are doing something a bit cheeky: holding out a proportion of the words in each document, and giving using predictive probabilities of these held-out words given the document-topic mixtures as well as the topic-word mixtures. This is obviously not ideal as it doesn't evaluate performance on any held-...

12

+1 for topicmodels. @Momo's answer is very comprehensive. I'd just add that topicmodels takes input as document term matrices, which are easily made with the tm package or using python. The lda package uses a more esoteric form of input (based on Blei's LDA-C) and I've had no luck using the built-in functions to convert dtm into the lda package format (the ...

12

Can LDA be used to detect the topic of A SINGLE document? Yes, in its particular representation of 'topic,' and given a training corpus of (usually related) documents. LDA represents topics as distributions over words, and documents as distributions over topics. That is, one very purpose of LDA is to arrive at probabilistic representation of each document ...

11

David Blei has a great talk introducing LDA to students of a summer class: http://videolectures.net/mlss09uk_blei_tm/ In the first video he covers extensively the basic idea of topic modelling and how Dirichlet distribution come into play. The plate notation is explained as if all hidden variables are observed to show the dependencies. Basically topics are ...

10

The notion of "topics" in so-called "topic models" is misleading. The model does not know or is not designed to know semantically coherent "topics" at all. The "topics" are just distributions over tokens (words). In other words, the model just capture the high-order co-occurrence of terms. Whether these structures mean something or not is not the purpose of ...

10

Andrew Ng is indeed one of the co-authors on the paper that first introduced LDA. David Blei is the first author of that same paper and gave two fantastic 90-minute lectures on the topic back in 2009. They are more approachable than the actual journal article itself and are worth watching if you want to do anything with LDA.

9

I think it is not really a question of better and worse but what data you have available and interpretability. If the data you have is at least partially labeled, whether with something like traditional topic classes, or something like hashtags, then labeled LDA may be interesting to pursue, otherwise not. Doing labeled LDA is certain to mean that the ...

8

Typically, in the context of Latent Dirichlet Allocation (used for Topic Modeling), we assume that the documents come from a generative process. I'll avoid math notation. Look at this figure: (1) Every topic is generated from a Dirichlet distribution of $V$ dimensions where $V$ is the size of your vocabulary. (2) For every document: (2.1) Generate a ...

7

This is a late answer, but it can be useful for other people searching for related research and tools for this problem: Weiwei Guo from Columbia implemented code for short-text topic modeling. He described the implementation in the paper "Modeling Sentences in the Latent Space" (http://aclweb.org/anthology-new/P/P12/P12-1091v2.pdf) and the code is available ...

6

While I'm not super familiar with his work, I know Jacob Eisenstein has done work in text analysis and graphical models in twitter data. In particular, this paper describes an application of topic modeling in twitter data and microblogs. Edit: actually after reading the paper a bit more, they state: However, the average message on Twitter is only ...

6

A recent paper called "a biterm topic model for short text" (WWW13) has made some progress on this topic, and here is its code

6

For my own curiosity, I applied a clustering algorithm that I've been working on to this dataset. I've temporarily put-up the results here (choose the essays dataset). It seems like the problem is not the starting points or the algorithm, but the data. You can 'reasonably' (subjectively, in my limited experience) get good clusters even with 147 instances ...

6

You might compute PMI using Wikipedia, as following: 1) Using Lucene to index a Wikipedia dump 2) Using Lucene API, it is straightforward to get: The number (N1) of documents containing word1 and the number (N2) of documents containing word2. So, Prob(word1) = (N1 + 1) / N and Prob(word2) = (N2 + 1) / N, where N is the total number of documents in ...

6

It is important to remember that topic models such as LDA were primarily developed for unsupervised text summarization. So often, there is not a "best" choice for how many top words to show. Most research papers on topic models tend to use the top 5-20 words. If you use more than 20 words, then you start to defeat the purpose of succinctly summarizing the ...

5

LDA can capture higher-order of co-occurrences of terms (due to the assumption of each topic is a multinomial distribution over terms), which is not possible by just computing PMI between terms.

5

One disadvantage of an unsupervised method like LDA is it will generally take considerably longer to train compared to supervised methods. I'm also confused about the 2% increase you mention, based on table 2 it looks like an 8% difference between the best supervised approach they compared against and their best unsupervised model. While I generally like ...

5

I might be 3 years late but I want to follow up your question on the example of "high-order of co-occurrences". Basically, if term t1 co-occurs with term t2 that co-occurs with term t3, then term t1 is the 2nd-order co-occurrence with term t3. You can go to higher order if you want but at the end you control how similar two words should be.

5

Briefly, my answer is "yes". The result of any LDA inference algorithm is $\theta_{d,t}$ and $\phi_{t,w}$, distribution of topics in each document and distribution of terms in each topics. Given these distributions, one can obtain estimate for $p(z|d,w)$, conditional probability of a topic $z$ for word $w$ in document $d$:  p(z|d,w)=\frac{p(z,d,w)}{\sum\...

5

You have to keep in mind that the vocabulary size is not the only factor which will affect the LSA results. In my experience a lot depends on the inflectional properties of a language. English is very LSA-friendly in this respect as it has virtually no inflection. Some Romance languages on the other hand still have pretty rich inflection so that having ...

4

Typical value of alpha which is used in practice is 50/T where T is number of topics and value of beta is 0.1 or 200/W , where W is number of words in vocabulary.

4

The mean and variance of a Gaussian are the unknown parameters that specify that distribution in that case. Likewise, in topic modeling, you attempt to learn the unknown parameters of $K$ topics, where each topic is a multinomial distribution over words in the vocabulary. Thus, it is the parameters of each multinomial distribution (each topic) that you seek ...

4

You can use the original C++ code of the LDA inventors Blei et al. Also quite fast is GibbsLDA (written in C and C++). If you want to parallelize it you might want to check out plda (C++).

4

Under the assumed generative model, each topic $z_n$ indexes a distribution over words in the topic, and each $z_n$ is randomly drawn from $Multinomial(\theta)$. Per Blei et al, "The basic idea is that documents are represented as random mixtures over latent topics, where each topic is characterized as a distribution over words." The keyword is "latent," i....

4

The approach termed "Type 1" is already explored in the paper by Blei et al. (2003) http://jmlr.org/papers/volume3/blei03a/blei03a.pdf in §7.2. The result is that this approach is valuable in feature selection for the SVM. So "Type 1" is definitely one right way. I have no comments "Type 2" except that it lacks the clarity that "Type 1" has.

4

I would not use Gaussian mixture models, as they require the constituent distributions to all be normal. You have counts, so GMM is inappropriate by definition. Latent Dirichlet allocation (full disclosure: I don't really know topic modeling) requires your data to be multinomial, but you can have counts in that case—they would be counts of ...

4

My first bet would be that the function words in a corpus of source code differ vastly from those of standard stop lists, and that your model's first topic is indeed capturing standard programming fare: if, int, new, while, etc. Besides building a custom stop list—seeing which words have high probability under the most frequently assigned topics is a good ...

4

Counter intuitively, it appears that the log_perplexity function doesn't output a $perplexity$ after all (the documentation of the function wasn't clear enough for me personally), but a likelihood $bound$ which must be utilised in the perplexity's lower bound equation thus (Taken from this paper - Online Learning for Latent Dirichlet Allocation by Hoffman, ...

Only top voted, non community-wiki answers of a minimum length are eligible


text:
# Multinomial Theorem

## Multinomial Theorem

Consider the equation x₁ + x₂ + … + xᵣ = n, where aᵢ ≤ xᵢ ≤ bᵢ; xᵢ ϵ 1; I = 1, 2, …, r.

In order to find the number of solutions of the given equation satisfying the given condition. We observe that the number of solutions is the same as the coefficient of xⁿ in the product.

(xa₁ + xa₁⁺¹ + xa₁⁺² + … + xb₁) x (xa₂ + xa₂⁺¹ + xa₂⁺² + … + xb₂) x (xa₃ + xa₃⁺¹ + xa₃⁺² + … + xb₃) x … x (xaᵣ + xaᵣ⁺¹ + xaᵣ⁺² + … + xbᵣ)

For example, if we have to find the number of non-negative integral solution of x₁ + x₂ + … + xᵣ = n, then as above, the required number is the coefficient of xⁿ in (xᴼ + x¹ + … + xⁿ) (xᴼ + x¹ + … + xⁿ) (xᴼ + x¹ + … + xⁿ) … (xᴼ + x¹ + … + xⁿ) (r – brackets)

= coefficient of xⁿ in (1 + x + x² + …+ xⁿ)ʳ

= coefficient of xⁿ in (1 + x + …)ʳ

= coefficient of xⁿ in (1 – x)ʳ

= coefficient of xⁿ in $$\left( 1+(-r)(-x)+\frac{(-r)(-r-1)}{2!}{{(-x)}^{2}}+\frac{(-r)(r-1)(-r-2)}{3!}{{(-x)}^{3}} \right)$$.

= coefficient of xⁿ in [1 + ʳC₁x + ʳ⁺¹C₂x² + …]

= ⁿ⁺ʳ⁻¹Cn

= ⁿ⁺ʳ⁻¹Cᵣ₋₁

Example: In how many ways the sum of upper faces of four distinct dices can be six

Solution: Here, the number of required ways will be equal to the number of solutions of x₁ + x₂ + x₃ + x₄ = 6, i.e., 1 ≤ x₁, x₂, x₃, x₄ ≤ 6.

∵ The upper limit is six, which is equal to the sum required.

So, the upper limit taken as infinite. So, number of solution is equal to the coefficient of t⁵ (1+ t + t² + …)⁴

= coefficient of t⁵ in (1 – t)⁻⁴

= (⁶⁺⁴⁻¹)C(₄₋₁)

= ⁹C₃ = 84.


text:
In this post, I prove that

$|xy| = |x||y|$

in which $x$ is a variable for any real number and $y$ is a variable for any real number. This is done by approximately following the steps in reference [1]. Then I provide a lot of commentary because that is what I like to do.

Proof:

Note that $x$ can be zero or nonzero, and $y$ can be zero or nonzero–this leads to four cases. The cases involving zero are investigated first.

If $x=0$, then $|xy| = |0| = 0 = |0||y| = |x||y|$.

If $y=0$, then $|xy| = |0| = 0 = |x||0| = |x||y|$.

Thus, if $x=0$ and/or $y=0$, then $|xy| = |x||y|.$

For nonzero $x$ and nonzero $y$, there are four cases:

1. $x > 0$ and $y > 0$.
• $|xy|=xy=|x||y|$
2. $x > 0$ and $y < 0$.
• $|xy|=(xy)(-1)=x[y(-1)]=|x||y|$
3. $x < 0$ and $y > 0$.
• $|xy|=(xy)(-1)=[(-1)x]y=|x||y|$
4. $x < 0$ and $y < 0$.
• $|xy|=xy=xy(-1)^2=[(-1)x][y(-1)]=|x||y|$

$\square$

Commentary:

Ok, maybe I should explain a little more. Sometimes equations can condense information so much that the amount of information becomes overwhelming and for practical purposes inaccessible to the human eye.

First I separate out the cases involving zero because it is easier to deal with $<$ and $>$ signs than $\le$ and $\ge$ signs. I recognize that it is possible for one or both of the variables $x$ and $y$ to be zero, so I consider each case separately. The first case involves $x$ being zero and $y$ being any real number which, of course, includes zero. The second case involves $y$ being zero and $x$ being any real number which, again, includes zero.

If $x$ is zero, then $xy$ is zero because any real number multiplied by zero is zero. The absolute value signs remain because I am just evaluating the quantity inside the vertical bars. Then I realize that the absolute value of zero is equal to zero, and that any real number multiplied by zero is still zero so I can multiply the zero by the absolute value of $y$ while maintaining the equality. Finally, I recall that zero is $x$, so I replace $|0|$ with $|x|$, and I am done.

Next, if $y$ is zero, then $xy$ is zero. The absolute value signs remain. Then I realize that the absolute value of zero is equal to zero, and that any real number multiplied by zero is still zero so I can multiply the zero by the absolute value of $x$ while maintaining equality. Finally, I recall that zero is $y$, so I replace $|0|$ by $|y|$, and I am done.

This is why $x$ being zero, $y$ being zero, or $x$ and $y$ being zero leads to $|xy| = |x||y|$. But what about nonzero numbers? This other option is addressed next.

The main idea to keep in mind is that absolute values are, by definition, not negative numbers. And since the cases involving zero have been addressed, I can simply use the fact that the absolute value of a nonzero number is a positive number. Note that each variable can be positive or negative, and there are two variables, so there are four cases.

In the first case, $|xy|=xy$ because when I multiply $x$ and $y$ together, the result is already positive, so I do not need to modify the result. And the absolute value of a positive number is a positive number, so I can write $x=|x|$ and $y=|y|$. This takes care of the first case.

The second case is slightly trickier because one of the variables is allowed to be negative. When I multiply $x$ and $y$ together, I get a negative number because $y$ is negative. So if I want to find the absolute value of this number, I can multiply $xy$ by $-1$, and this is what I did. Finally, I notice that $x=|x|$ since $x$ is positive in this case, and realize that $y$, a negative number, multiplied by $-1$ is equal to the absolute value of $y$. That takes care of the second case.

The third case is similar to the second, except this time $x$ is negative and $y$ is positive. The same kind of logic applies here, but I’ll got through it to be complete. To find the absolute value of $xy$, first I multiply $x$ and $y$ together to obtain a negative number, and then I multiply by $-1$ to change the sign of the number. Then I notice that $|y|=y$ and $x$–a negative number–multiplied by $-1$ is the absolute value of $x$, since multiplying by $-1$ converts a negative number to a positive number. The third case has been addressed.

Last but not least is the fourth case. This time $x$ and $y$ are negative. Since multiplying a negative number by another negative number produces a positive number as taught in school, the result of multiplying $x$ with $y$ is a positive number and therefore equal to the absolute value of $x$ and $y$. But $x$ alone is not equal to the absolute value of $x$ without multiplying $x$ by $-1$, and the same can be said for $y$; this is why there are two instances of $-1$ in the above proof. Then I realize that $-1$ times $x$ is $|x|$ and that $-1$ times $y$ is $|y|$. The fourth case is complete.

And the whole proof is done.

References:

[1] https://www.math.tamu.edu/~mike.stecher/171/absoluteValueFunction.pdf


text:
# Records: [TAP] Master

Thread in 'Competition' started by Amnesia, 29 Mar 2007.

1. ### PetitPrince

It's about time I make it at home.

2. ### AmnesiaPiece of Cake

Code:
TAP Master mode rules :

1 : Finish the game at lv. 999
2 : Reach lv. 999 under 8:45:00
3 : Reach lv. 999 with the S9 rank
4 : Do the first 5 sections (000-500) under 1:05:00
5 : Do the last section (900-999) under 0:45:00
6 : Do at least 2 tetrises in the 5 first sections (000-500)
7 : Do at least 1 tetrises in the 4 following sections (500-900)
8 : Survive the invisible section at the staff roll during 1min
Is it good?
Jago I tried to make a short list of the rules next to the ranking table as you asked me few weeks ago..But I am not sure..

Look at this : (performed thank to the cheat, with only "I"..

I am sure I have respected the 7 first conditions but no invisible challenge..

3. ### Edoa.k.a. FSY

Amnesia, the conditions for the m-roll are more subtle than those stated by LOst. I don't know the exact details, but I think that a section time must not be much slower than the previous one. For example, 900-999 cannot be more than 2 seconds slower than 800-900, and 800-900 cannot be more than 2 seconds slower than 700-800. Reset your section time records using Maintenance Code: 4-1-7-6-5, and then try again. If you don't get the instant invisible challenge, check your times, I can guarantee it's because you slowed down a lot from one section to the next.

4. ### LOst

Note that reseting the section time records affects all the players on your TAP system... Which is a wierd condition for m-roll achieved by 1 player?

Also being forced to outperform previous section times must have the same effect while cheating as when playing. I am sure Amnesia went for the fastest finish time while cheating. He still has to beat a hidden section time limit of 1 min and 5 seconds.

Section times are compared once entering a new section, and when entering level 999. The current section time is compared to a constant value. I haven't found any evidence that the game is checking previous section times, or the section time records as you claim. But as you said, "I don't know the exact details". If you feel like owning me, just give me an inp file showing a failure of "just" your described condition. Something I doubt you will be able to do.

I promise you that if there is another condition, I will most likely find it in a near future. But for the already stated conditions by me, they are correct.

5. ### tepplesLockjaw developer

I read it as if your own 700-800 time is 46 seconds, your own 800-900 time can't be longer than 48 seconds. Clearing the section time records just helps you diagnose this problem.

ok

7. ### Edoa.k.a. FSY

Thanks for re-phrasing, that's exactly what I meant.

LOst, my apologies if I've offended you, I realise you must be proud of discovering the m-roll conditions, and I didn't intend to dispute your claims. I didn't think it was worth recording an "only I piece cheat" video, but I hope you find this useful:

I played through the game using only I pieces, and made a save state in the final section so I could go back and complete it slightly slower.

8. ### colour_thief

With a replay similar to Edo's S9, I'm sure we could learn the exact conditions inhibiting M/Gm here. As well as any further unknown conditions in surrounding code. Any volunteers?

9. ### LOst

Being able to find one more condition is not the same as correcting someone elses found conditions. It sounded for a while like you stated that I was incorrect in some way. Thank you for your clarification!

10. ### AmnesiaPiece of Cake

Like Jago told me at the meeting in his flat..

" Conditions doesn't matters..When you are enough strong you get the m rank.."

11. ### AmnesiaPiece of Cake

Heuy c_t !
My first "good" master mode almost completed with one hand!

I think I could finish the last section but in starting low..

12. ### colour_thief

That's crazy! Are you playing one-handed with arcade sticks? And 2 sticks for Doubles? Even if it's keyboard though, well done.

13. ### AmnesiaPiece of Cake

I have actually the bou's stick in my flat!
And they are the same model!

My current best are :

TGM: S7

TAP MASTER: S4

TAP DEATH: ~330

TAP DOUBLE: 217 - 219

Baw.. It is funny..I do that only when I am too tired to do other thing..Or to impress girls..

14. ### AmnesiaPiece of Cake

http://www.sendspace.com/file/c82t3g

Juste to show you I am not joking c_t..

Don't worry you stay 2nd today yet..

VOUS ALLEZ BIENTOT VOIR CE QUE JE PEUX FAIRE EN MASTER MODE LES GARS !!!!!!!!!

15. ### mat

09:04:18

not a significant improvement, but at least it shows i'm trying .

16. ### AmnesiaPiece of Cake

OOuuuFFFfff...
When I saw your name on this topic, I though : "No..He did it.."

Well done!

You know I though a lot about your skill in Master mode matt, and I am sure you will never be the 3rd any more..
The keyboard offers a lot of advantages : mainly the precision, but you will never perform the "drop/lock" as fast as me, jago and c_t..

How many time do you do usually at lv. 500 ?

No pb in TGM1 and TAP Death of course..

17. ### mat

i screwed up and deleted my inp file... so we'll never know how fast my 500 was, but i will tell you that i'm getting early sections consistently under 1 minute. 58 0-100 on that game if i remember correctly, and i don't feel that the up down is limiting my game in any significant way, i do it in one motion anyway.

18. ### Rosti LFC

I don't see how a joystick can be that much faster than simply flicking my finger off the up key onto the down in one movement. Certainly not enough to make a difference compared to other ways to play faster (finesse, no mistakes, x99 line clears etc).

19. ### Rosti LFC

AT LAST I HAVE BEATEN S4

Really annoying thing is that I could have done even better. The nerves sort of got to me when I hit S4, because I was on a really good run and the level was pretty low. Then mistakes slowed me down and brought me to the top half of the field, and I was really just trying to recover after that.

20. ### DIGITALUnregistered

The up down motion can be remedied quite easily as it's just an inconvenient hard drop. However, finesse with firm drop and ARS is a different matter. The real advantage joysticks have over keyboards is the ability to perform zangi placements with one swift motion. You now not only have free-fall finesse and wallkick finesse, but also zangi finesse. jin8 gracefully demonstrated all three of these in his GM video.

EDIT:

Here are some diagrams showing these.

http://fumen.zui.jp/?v095@TeF3gbB3gbI3g ... Gob4GIc2YB

Slides 1-2 = Freefall

Slides 3-5 = Wallkick

Slides 6-8 = Zangi


text:
# zbMATH — the first resource for mathematics

Making a tournament $$k$$-arc-strong by reversing or deorienting arcs. (English) Zbl 1035.05045
Summary: We prove that every tournament $$T=(V,A)$$ on $$n \geq 2k+1$$ vertices can be made $$k$$-arc-strong by reversing no more than $$k(k+1)/2$$ arcs. This is best possible as the transitive tournament needs this many arcs to be reversed. We show that the number of arcs we need to reverse in order to make a tournament $$k$$-arc-strong is closely related to the number of arcs we need to reverse just to achieve in- and out-degree at least $$k$$. We also consider, for general digraphs, the operation of deorienting an arc which is not part of a 2-cycle. That is we replace an arc $$xy$$ such that $$yx$$ is not an arc by the 2-cycle $$xyx$$. We prove that for every tournament $$T$$ on at least $$2k+1$$ vertices, the number of arcs we need to reverse in order to obtain a $$k$$-arc-strong tournament from $$T$$ is equal to the number of arcs one needs to deorient in order to obtain a $$k$$-arc-strong digraph from $$T$$. Finally, we discuss the relations of our results to related problems and conjectures.

##### MSC:
 05C20 Directed graphs (digraphs), tournaments 05C40 Connectivity
Full Text:
##### References:
 [1] J. Bang-Jensen, G. Gutin, Digraphs: Theory, Algorithms and Applications, Springer, London, 2000, xxii+754pp. · Zbl 1170.05002 [2] Bang-Jensen, J.; Jordán, T., Adding and reversing arcs in semicomplete digraphs, Combin. probab. comput., 7, 17-25, (1998) · Zbl 0892.05019 [3] Thomassen, C., Configurations in graphs of large minimum degree, connectivity, or chromatic number, Ann. New York acad. sci., 555, 402-412, (1989)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.


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# Prove the triangle inequality for series.

Prove the triangle inequality for series, that is if $\sum x_n$ converges absolutely then $|\sum_{n=1}^{\infty}x_n| \ge \sum_{n=1}^{\infty}|x_n|$.

My attempy: Since $\sum x_n$ converges absolutely, it is cauchy series. Then, $\forall \varepsilon>0,$ there exists $M\in Z^+$ such that $|\sum_{j=n+1}^{k}|x_n||<\varepsilon$ for $n\ge M$ and $k>n$. Then,

$|\sum_{j=n+1}^{k}|x_n||=\sum_{j=n+1}^{k}|x_n| \ge |\sum_{j=n+1}^{k}x_n|$ by triangle inequality.

I think it is wrong since I use triangle inequality, which is required to be proven, and I also don't know how to change $\sum_{j=n+1}^{k} \rightarrow \sum_{j=1}^{\infty}$.

Could you give some help?

• Isn't the inequality the other way around? – Bill O'Haran Apr 17 '18 at 10:29
• Hint: if you have two convergent sequences $a_n$ and $b_n$ and you know $a_n\leq b_n$ for every $n$, what can you say in the limit? – Uskebasi Apr 17 '18 at 10:31
• this $|\sum_{n=1}^{\infty}x_n| \ge \sum_{n=1}^{\infty}|x_n|$ is wrong. The triangle inequality is this: $$|\sum_{n=1}^{\infty}x_n| \le \sum_{n=1}^{\infty}|x_n|$$ Use the algebra of limits, that is, if $\lim a_n=a$ and $\lim b_n=b$ and $a_n\le b_n$ for all $n\in\Bbb N$ then $a\le b$ – Masacroso Apr 17 '18 at 10:33
• Actually, $b_n$ convergent is enough – Uskebasi Apr 17 '18 at 10:33

$|\sum_{j=1}^{n}x_j|\leq \sum_{j=1}^{n}|x_j|$ by the triangle inequality wich is true for any $n \in\mathbb{N}$ so by taking $n \rightarrow +\infty$ you can deduce $|\sum_{n=1}^{\infty}x_n| \leq \sum_{n=1}^{\infty}|x_n|$
• yes we can because i use this inequality for a finite number of terms , after the inequality is true for any $n$. so we can use the limite – Bernstein Apr 17 '18 at 10:41
• you are using that $|\sum x_j| \le \sum |x_j|$ at first. Isn't it what we want to prove? – user1230403 Apr 17 '18 at 10:49
• i use this $|\sum_{j=1}^{n}x_j|\leq \sum_{j=1}^{n}|x_j|$ and this is true by the classical triangle inequality – Bernstein Apr 17 '18 at 10:51


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## On the theory of elliptically contoured distributions.(English)Zbl 0469.60019

### MSC:

 6e+11 Characteristic functions; other transforms 6e+06 Probability distributions: general theory
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### References:

 [1] Das Gupta, S; Eaton, M.L; Olkin, I; Perlman, M; Savage, L.J; Sobel, M, Inequalities on the probability content of convex regions for elliptically contoured distributions, (), 241-264 · Zbl 0253.60021 [2] Feller, W, () [3] Kelker, D, Distribution theory of spherical distributions and a location-scale parameter generalization, Sankhyā ser. A, 32, 419-430, (1970) · Zbl 0223.60008 [4] Lukacs, E, () [5] Schoenberg, I.J, Metric spaces and completely monotone functions, Ann. of math., 39, 811-841, (1938) · Zbl 0019.41503
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# American Institute of Mathematical Sciences

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June  2009, 24(2): 613-624. doi: 10.3934/dcds.2009.24.613

## Markov partitions reflecting the geometry of $\times2$, $\times3$

 1 School of Mathematics, University of East Anglia, Norwich, NR4 7TJ, United Kingdom 2 Centro de Modelamiento Matemático, Universidad de Chile, Av. Blanco Encalada 2120, piso 7, Santiago, Chile

Received  July 2008 Revised  November 2008 Published  March 2009

We give an explicit geometric description of the $\times2$, $\times3$ system, and use this to study a uniform family of Markov partitions related to those of Wilson and Abramov. The behaviour of these partitions is stable across expansive cones and transitions in this behaviour detect the non-expansive lines.
Citation: Thomas Ward, Yuki Yayama. Markov partitions reflecting the geometry of $\times2$, $\times3$. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 613-624. doi: 10.3934/dcds.2009.24.613
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