url stringlengths 14 2.42k | text stringlengths 100 1.02M | date stringlengths 19 19 | metadata stringlengths 1.06k 1.1k |
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https://www.gradesaver.com/textbooks/math/applied-mathematics/elementary-technical-mathematics/chapter-2-section-2-3-multiplication-and-division-of-signed-numbers-exercises-page-114/41 | Elementary Technical Mathematics
Multiply the absolute values. Since the number of negative factors is odd the product is negative. $(+3)(-2)(+1)=-(|+3|\times |-2|\times|+1|)=-(3\times2\times1)=-6$ | 2018-12-13 18:36:03 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6741212606430054, "perplexity": 730.4141184627106}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-51/segments/1544376825029.40/warc/CC-MAIN-20181213171808-20181213193308-00360.warc.gz"} |
https://community.wolfram.com/groups/-/m/t/1518375 | # Create an interface for a drinks vending machine?
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Hello everyone, if someone were available, I would need a hand in the making of a mathematical machine, I thought to realize the interface of a distributor of drinks that tells me to enter the money, select the code concerning the drink and then, in based on the money inserted, give me rest if the cost of the drink is lower than what is entered. However, since I should use wolfram mathematica I do not know how to set the algorithm, as I am not very familiar with this program. See the following code: stringa1 = "Inserire soldi:"; stringa2 = "Inserire codice bibita:"; Column[{{InputField[stringa1, String], InputField[Dynamic[beta], Number], Button["Clicca qui", Print[InputField[stringa2, String], InputField[Dynamic[alph], String]]]}}] If someone would give me a hand I would do a big favor .... thanks in advance. Attachments: | 2018-12-19 01:37:21 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2803330719470978, "perplexity": 2155.6502776028833}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-51/segments/1544376830305.92/warc/CC-MAIN-20181219005231-20181219031231-00305.warc.gz"} |
https://www.numerade.com/questions/test-the-series-for-convergence-or-divergence-displaystyle-sum_n-1infty-1n-frac-sqrtn2n-3/ | 💬 👋 We’re always here. Join our Discord to connect with other students 24/7, any time, night or day.Join Here!
JH
# Test the series for convergence or divergence.$\displaystyle \sum_{n = 1}^{\infty} (-1)^n \frac {\sqrt{n}}{2n + 3}$
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let's test this series for convergence or divergence. Now the first thing I notice is that this series is alternating. That's because due to this negative one to the Empower so this suggests that we try the alternating series test. Okay, so here we define BN to be squared of end over two n Plus three. This is positive because numerator and denominator are both positive. So that's the first condition that has to be checked. The second condition. We need the limit as n goes to infinity of bm to be zero and this problem that becomes end to the one half over two n plus three and this limit is zero. And if you can't see why, you can use low Patel's role here, mhm and then finally we have one more condition to check. We need that BN is decreasing. So when we add one to the end, it's not bigger than the previous value of being. So to check this, we could go ahead and actually plug in and plus one into our formula. But as mentioned in the text book, another way to show that bien is decreasing if we have f prime of X is negative. Where we define f of X to be X likes a squirt of X over two X plus three and then we can write. This is X to the one half over two X plus three. So now if we can show that this function f has negative derivative, that's equivalent to showing that this function is decreasing and that will complete the last condition here. And then finally, we would be able to say that the series converges by a S T alternating serious test. So let's go to the next page and show that F has negative derivative. So this is X to the one half two X plus three. So f prime of X is the quotient rule. So the denominator is positive because it's a square. So what we're really interested in here is the numerator. Yeah, so here we can go ahead and write this Mhm. Okay. All I did was just distribute this term out here inside the parentheses and then simplified. Now this is equal to three. Now, let me go ahead and combine these two so that will be minus X to the one half over the square and then get a common denominator, and we see that this will be negative, if so, three minus two x so that that previous term was negative. If this is less than zero, so three is less than two works. 3/2 is less than X. So this tells us that bien is decreasing when and is bigger than or equal to two. And that's what allows us to use the alternative series test. So all conditions okay for BN in the alternating series test. MM hold. Therefore, the series converges by the alternating series test, and that's our final answer.
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Join Bootcamp | 2021-10-16 20:35:52 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.88344806432724, "perplexity": 306.12726935821007}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585025.23/warc/CC-MAIN-20211016200444-20211016230444-00267.warc.gz"} |
https://www.semanticscholar.org/paper/State-Density-Flows-of-Non-Degenerate-Mean-Field-Huang-Tang/7bb996d15d84b50dbbad9ce2e2df25535d1f6727 | • Corpus ID: 241033211
# State-Density Flows of Non-Degenerate Density-Dependent Mean Field SDEs and Associated PDEs
@inproceedings{Huang2021StateDensityFO,
title={State-Density Flows of Non-Degenerate Density-Dependent Mean Field SDEs and Associated PDEs},
author={Ziyuan Huang and Shanjian Tang},
year={2021}
}
• Published 3 November 2021
• Mathematics
In this paper, we study a combined system of a Fokker-Planck (FP) equation for m with initial (t, μ) ∈ [0, T ] × L(R), and a stochastic differential equation (SDE) for X with initial (t, x) ∈ [0, T ] × R, whose coefficients depend on the solution of FP equation. We develop a combined probabilistic and analytical method to explore the regularity of the functional V (t, x, μ) = E[Φ(X T ,m (T, ·))]. Our main result states that, under a nondegenerate condition and appropriate regularity assumptions…
## References
SHOWING 1-10 OF 28 REFERENCES
### WELL-POSEDNESS FOR SOME NON-LINEAR DIFFUSION PROCESSES AND RELATED PDE ON THE WASSERSTEIN SPACE
• Mathematics
• 2018
In this paper, we investigate the well-posedness of the martingale problem associated to non-linear stochastic differential equations (SDEs) in the sense of McKean-Vlasov under mild assumptions on
### From nonlinear Fokker–Planck equations to solutions of distribution dependent SDE
• Mathematics
The Annals of Probability
• 2020
### A Probabilistic Approach to Classical Solutions of the Master Equation for Large Population Equilibria
• Mathematics
Memoirs of the American Mathematical Society
• 2022
It is proved that the decoupling field generated by the forward-backward system is a classical solution of the corresponding master equation, which gives several applications to mean-field games and to the control of McKean-Vlasov diffusion processes. | 2022-12-09 06:55:40 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5651929378509521, "perplexity": 1509.6145344222568}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446711390.55/warc/CC-MAIN-20221209043931-20221209073931-00061.warc.gz"} |
https://www.nature.com/articles/s41467-022-28052-x?error=cookies_not_supported&code=7620e2ab-337d-42fd-afd4-e0a699914b3d | ## Introduction
The emerging demand for high-energy and low-cost batteries for electric vehicles and grid-scale energy storage application calls for rapid improvements in cathode materials1. Lithium/sodium-layered transition metal (TM) oxides have attracted tremendous attention as appealing cathode materials because of their high specific capacities2,3. To further increase the energy density, a prevailing approach is to push the charging voltage limit to simultaneously attain higher specific capacity and increase the average working voltage4. However, these layered cathodes undergo universal capacity drop and voltage decay during high-voltage operation5,6,7. Over several decades, extensive fundamental understanding and material development have been carried out to reveal the underlying failure mechanism and mitigate the structural degradation at elevated voltage.
Irreversible surface/bulk phase transition upon cycling has been reported as one of the prevalent origins for the performance degradation of layered cathodes8,9. Several prominent studies showed that surface reconstruction, such as layered to spinel/rock-salt, can initiate from the cathode surface, and then gradually propagate into the bulk structure during a high-voltage charge. This eventually causes bulk fatigue of lithium layered oxide cathodes after long-term cycling10,11. Compared with their Li analogs that present mostly the octahedral structure through direct synthesis, sodium-layered oxide cathodes can be classified into P-type (prismatic) and O-type (octahedral), depending on the surrounding Na environment and the number of unique oxide layers12,13. Therefore, in general, they exhibit much more complex phase transitions such as P2-O214, P2-Z15, P2-OP416, and O3-P”317 during high-voltage charge, leading to irreversible bulk structural changes and huge volume changes. In both cases, researchers agree that the irreversible phase transition can lead to accumulation of mechanical stress at the phase boundaries due to lattice mismatch and further intergranular/intragranular cracking of the cathode particles after prolonged cycling18,19,20. Therefore, tremendous efforts have been focused on suppressing the undesired phase transition through aliovalent doping21,22,23,24,25.
The parasitic reactions between charged cathodes and electrolytes have also been correlated with the high-voltage instability of layered oxide cathodes26. On the one hand, the common electrolytes solvents such as ethylene carbonate, diethyl carbonate, and dimethyl carbonate are thermodynamically unstable at high voltage due to their limited electrochemical stability window27. On the other hand, the charged cathodes containing highly oxidized transition metals will aggravate the decomposition of electrolytes and lead to the formation of thick cathode–electrolyte-interphase (CEI) on the cathode surface and hence sluggish Li+/Na+ diffusion28. Mu et al. have reported that the cathode–electrolyte interfacial reaction can also trigger transition metal reduction/dissolution, heterogeneous surface reconstruction, and nanocracks29. To address these concerns, surface coating30,31,32,33 and high-voltage electrolytes34,35,36 have been widely developed to enhance the high-voltage cycling stability of layered cathodes.
These valuable findings are acknowledged, but the aforementioned degradation mechanism has been mainly attributed to the dynamic structural changes (e.g., phase transition, interfacial reactions, and mechanical cracks) that nucleated and evolved during cycling. In fact, the explicit role of these structural degradations and an accurate assignment on the degradation origin remain challenging because they are intimately coupling with each other during electrochemical charge/discharge. In contrast, the effect of native structural defects often induced during the complicated synthesis process of layered oxides has been significantly overlooked. Many trial-and-error efforts have shown a strong correlation between synthetic conditions and battery performance of layered oxides, and yet the fundamental mechanism remains elusive. Moreover, cathode materials with tailor-made defects through precise synthetic control could serve as a model structure to probe their explicit role in battery performance.
Here, by using in situ synchrotron X-ray diffraction (SXRD) to probe the solid-state synthesis process of sodium-layered oxide cathodes, we intentionally synthesized a highly strained O3 NaNi0.4Mn0.4Co0.2O2 cathode. The electrochemical voltage window control experiments in combination with in situ SXRD, synchrotron X-ray absorption spectroscopy, and differential electrochemical mass spectrometry during charge/discharge unanimously revealed that the irreversible phase transition and cathode–electrolyte reactions are not the dominant factor for the rapid capacity fade of strained NaNi0.4Mn0.4Co0.2O2 cathode during cycling. Instead, using advanced transmission electron microscopy (TEM), we discovered that the native high lattice strain plays an overwhelming role in triggering the destructive structural earthquake of sodium-layered cathodes, which spontaneously relaxed due to local strain heterogeneity and led to severe breakdown/fragmentation of layered structure during prolonged cycling. We also confirmed that its relaxation process is strongly coupling with the operating temperature and charge/discharge rate. We believe this work is the first to decouple the contribution of native lattice strain to high-voltage instability of sodium-layered cathodes and counters the conventional wisdom that phase transition and cathode–electrolyte reactions are primarily responsible for degradation.
## Results
### In situ synthesis study of strained NaNi0.4Mn0.4Co0.2O2 by SXRD
Generally, the synthesis of layered oxides involves solid-state reactions between transition metal hydroxides precursors and lithium/sodium salts by heating their mixture at a high temperature for a sufficiently long time, followed by a cooling process. Recently, researchers leveraged in situ SXRD and computational modeling to investigate the evolution of non-equilibrium kinetic intermediates and the formation of thermodynamic equilibrium phases during these processes37,38. Although these studies provide valuable guidance for the predictive synthesis of layered oxides, most of them have focused on examining the impact of structural characteristics during the heating/holding process, such as phase impurity, lattice parameter, crystallite size, Li–TM bond length, and Li+/Ni2+ mixing39,40,41. The effect of microstrain evolution during the cooling process, especially under rapid quenching, has been largely ignored. Notably, our previous study showed that appropriate quenching could help to stabilize the metastable P2/O1/O3 intergrowth phase and tailor the electrochemical performance42. Among various phase structures in the sodium-layered oxides cathodes, the O3 phase represents the structure with higher Na content, which is different from the P2-type cathode that has a lower Na content and suffers from sodium deficiency problem13,43. Here, we further explored the formation process of O3 NaNi0.4Mn0.4Co0.2O2 cathode and revealed the effect of quenching-induced lattice strain on the crystal structures and electrochemical performance.
Figure 1a and Supplementary Fig. 1a show the 2D contour plot of in situ SXRD patterns during the formation process of O3 NaNi0.4Mn0.4Co0.2O2. The starting material was a mixture of Ni0.4Mn0.4Co0.2(OH)2 and 5% excess mole of NaOH. As shown more clearly by the covariance analysis and the corresponding thermalgravimetric analysis (TGA) in Supplementary Fig. 1b, c, the major reactions started to occur at around 150 and 350 °C, respectively. Therefore, the formation process of O3 NaNi0.4Mn0.4Co0.2O2 can be classified into three regions according to their phase composition: 25–185 °C (region I, starting material), 185–500 °C (region II, intermediates), and 500–875 °C (region III, final product). The Rietveld refinement in Supplementary Fig. 2 shows that the starting material can be indexed well as Ni(OH)2 and NaOH, while the XRD pattern of the intermediate product at 350 °C is consistent with that of P3-Na0.8Ni0.4Mn0.4Co0.2O2. Upon further reaction beyond 500 °C, the structure of the O3 phase started to evolve and dominate. The O3 phase remained during the holding and quenching process (Fig. 1b, c).
In order to understand subtle structural changes during the reaction process, Rietveld refinement was performed on all the measured SXRD patterns after the evolution of the O3 phase (i.e., 500 °C). As shown in Fig. 1d, e, the lattice parameter a was dramatically increased during heating and then gradually stabilized during the holding process; in contrast, the lattice parameter c was significantly decreased during heating and then slowly decreased during the holding process. These results were attributed to the gradual transformation of P3 (a = b = 2.8479 Å, c = 16.6093 Å) to O3 (a = b = 2.94 Å, c = 15.923 Å) phase during heating, which exhibited distinct cell lattice parameter. In general, the structural difference between different phases can induce non-uniform lattice strain (microstrain) in the crystallite. Therefore, as shown in Fig. 1f, the decrease of P3/O3 phase boundaries during the heating/holding process led to a gradual reduction of the microstrain.
We found that quenching is necessary to maintain the O3 phase structure during cooling. With a slow cooling under an air atmosphere, the obtained material exhibited a mixed phase of P1/P2/O3 (Supplementary Fig. 3), which might be due to the surface reconstruction induced by slow cooling39. The CO2 molecules in the air would coordinate with Na+ at the solid/gas interface or attack the O atom of NaO6 octahedra, leading to the extraction of surficial Na+ from sodium-layered oxide cathodes and hence the formation of Na2CO3 and Na-deficient phase, such as P1 and P239. Indeed, a clear signal of Na2CO3 can be observed in the C 1s X-ray photoelectron spectroscopy (XPS) spectrum of a slow-cooled sample (Supplementary Fig. 4). In sharp contrast, the quenched sample exhibited high purity O3 phase, demonstrating a good fit with the standard O3 layered structure (Fig. 1g). The morphologies of the quenched cathode are shown in Fig. 1h, i, and exhibited characteristic conventional polycrystalline features.
Surprisingly, as shown in Fig. 1d, e, the lattice parameter a and c were both decreased due to the lattice shrinkage when abruptly exposed to a low temperature. During the relaxation process of the inhomogeneous thermal stress in the quenched sample, lattice parameter a was significantly decreased while c was drastically increased, which is similar to what occurs during the de-sodiation process of layered oxides. This indicates an increase of structural defects such as vacancies, dislocations, and stacking faults. Quenching-induced strain generation has been well known in steel manufacturing44. It has been also often used for the fabrication of battery cathode materials to control grain size, oxygen vacancies, and etc45,46. During quenching, a highly inhomogeneous temperature field is generated, which can result in heterogeneous thermal stresses, thus leading to residual stresses being introduced at the end of the quenching process. In our experiment, the quenching was conducted under air, which can easily induce temperature heterogeneity in the local lattice region, leading to the formation of the observed lattice strain. Therefore, the microstrain during the rapid quenching process exhibited an intense increase (Fig. 1f); while the effect of such quenching-induced native lattice strain on the electrochemical performance of layered cathodes has not been well understood.
### Electrochemical characterization of strained NaNi0.4Mn0.4Co0.2O2
The electrochemical performance of the as-prepared cathode was evaluated by using half cells with sodium metal as referenced and countered electrode. The electrolyte was 1 mol/L NaPF6 in propylene carbonate with a 2 vol% fluoroethylene carbonate additive. Figure 2a shows the first charge/discharge curves of the strained O3-NaNi0.4Mn0.4Co0.2O2 cathode within 2.0–4.4 V at 0.08C (1C = 180 mA/g), which exhibited the characteristic features of the O3 phase with multiple-step voltage plateaus. The first charge and discharge capacity were measured to be 180.6 and 171.9 mAh/g, respectively, leading to a high initial Coulombic efficiency of 95%. However, upon further charge/discharge, the capacity exhibited a continuous decrease. After 100 cycles, the reversible capacity was only 54 mAh/g, resulting in a low capacity retention of 30% (Fig. 1b).
Such degradation has been previously attributed to irreversible structural transformation or parasitic reactions at high voltage4,5,6,26,29. Figure 2c displays the in situ differential electrochemical mass spectrometry (DEMS) result of the strained O3-NaNi0.4Mn0.4Co0.2O2 cathode during charge/discharge within 2.0–4.4 V. Although O2 gas release has been often observed in the layered oxide cathodes when charged to high-voltage, it is also possible for the transformation of the oxygen species to CO2 because of the chemical reactions between reactive lattice oxygen and electrolytes at high-voltage47,48,49. Indeed, despite no evolution of O2 gas during the whole charge process, upon charging beyond 3.8 V, a significant release of CO2 can be clearly observed. This should come from either decomposition of electrolytes or side reactions between oxygen radicals and electrolytes49,50. Consistent with this observation, the high-precision leakage current measurement results (Fig. 2d) showed that the parasitic current increased dramatically when the cut-off voltage was increased to over 4.0 V. Therefore, in order to mitigate the capacity degradation induced by high-voltage charge, the strained O3-NaNi0.4Mn0.4Co0.2O2 cathode was cycled between 2.0 and 3.8 V. Surprisingly, the cycle stability was not improved by lowering the charge cut-off voltage, which contradicts results in other studies of reported sodium-layered oxide cathodes with narrow voltage window51. A capacity retention of only 38.7% was attained after 100 cycles (Fig. 2e). Moreover, the charge/discharge test of the highly strained NaNi0.4Mn0.4Co0.2O2 cathode at an elevated voltage or temperature demonstrated similar rapid degradation (Supplementary Fig. 5). Interestingly, the capacity fade rate (inset in Fig. 2f), which were obtained by linear fitting of the capacity retention curve, is almost the same after the first 10 cycles, regardless of voltage window, testing temperature and charge/discharge rate, whereas they exhibited significantly different results for the first 10 cycles. The results indicate that the capacity degradation of the highly strained NaNi0.4Mn0.4Co0.2O2 cathode might be controlled by a specific factor, while such factor was regulated by the operating condition in the early stage of charge/discharge.
### Redox couple evolution and phase transition of strained NaNi0.4Mn0.4Co0.2O2 during charge/discharge
In situ Ni K-edge X-ray absorption near-edge spectroscopy (XANES) characterization was carried out to understand the redox couple evolution of the highly strained NaNi0.4Mn0.4Co0.2O2 cathode during charge/discharge within 2.0–4.4 V, which can provide the oxidation state changes of Ni element during battery operation. As shown in Fig. 3a, b, the Ni K-edge shifted to high-energy upon extraction of Na+ due to the oxidation of Ni2+ to Ni3+/Ni4+ during charge, while it shifted back to lower energy during discharge due to the reduction of Ni3+/Ni4+ to Ni2+ 42. In general, as shown in the 2D contour plot of Ni K-edge XANES throughout the whole charge/discharge process (Fig. 3c), the Ni undergoes a highly reversible Ni oxidation/reduction during charge/discharge within 2.0–4.4 V. Even after cycling for 10 cycles that have triggered severe capacity degradation in the strained O3 cathode, Ni still exhibits reversible redox reactions (Supplementary Fig. 6a), which cannot explain the severe capacity loss of strained NaNi0.4Mn0.4Co0.2O2 cathode reported here (Fig. 2b). The redox reaction behavior of Co and Mn in the O3 strained NaNi0.4Mn0.4Co0.2O2 cathode were also explored and both showed reversible transformation during cycling within 2.0–4.4 V (Supplementary Fig. 6b, c), indicating that the observed capacity degradation is not from the cycling-induced structural transformation.
Therefore, we further performed in situ SXRD to reveal the phase transition of strained NaNi0.4Mn0.4Co0.2O2 cathode with a controlled voltage window. Figure 3d shows the 2D contour plot of in situ SXRD patterns during charge/discharge within 2.0–4.4 V, in accompaniment with the corresponding voltage curve and phase transition process. As shown, the electrode before charge/discharge can be well indexed using the O3 phase. Upon charge, the (003) and (006) peaks shifted toward lower angles, which indicated an expansion of the c lattice parameter due to the increased oxygen electrostatic repulsion between oxygen layers induced by the removal of Na+. Meanwhile, the (101), (102), (110), and (113) peaks moved towards higher angles during charge, corresponding to shrinkage of a lattice parameter due to the oxidation of TM. In addition, the intensity of the O3 phase gradually decreased, while the peaks of the O1 phase started to appear. Upon further charge, the O1 phase was then transformed into the P3 phase starting at 3.37 V. At the long plateau beyond 4.0 V, the structure of the P1 phase started to dominate the charged product. At the end of the charge process, an unknown X phase with low Na content and crystallinity was formed. Therefore, the phase transformation of strained NaNi0.4Mn0.4Co0.2O2 cathode during high-voltage charge can be described as $${{{{{\rm{O}}}}}}3\to {{{{{\rm{O}}}}}}1\to {{{{{\rm{P}}}}}}3\to {{{{{\rm{P}}}}}}1\to {{{{{\rm{X}}}}}}$$, which is similar to the previously reported O3 sodium-layered cathode when charged to high voltage52. During the discharge process, the phase transformation process reversed. However, at the beginning of the discharge process, the XRD intensities of (00l) peaks are very weak and broad, indicating severe lattice strain at the c axis direction that prevents the re-insertion of Na+. Moreover, as evidenced by the disappearance of O3 (003), (006), (101), (108) and (110) peaks of the O3 phase, the O1 phase (rather than the original O3 phase) dominated the fully discharged electrode, indicating an irreversible phase transition during high-voltage cycling.
In sharp contrast with that charge/discharge within 2.0–4.4 V, the highly strained NaNi0.4Mn0.4Co0.2O2 cathode exhibited a highly reversible phase transformation of O3$$\leftrightarrow {{{{{\rm{O}}}}}}1\leftrightarrow {{{{{\rm{P}}}}}}3$$ with a lower charge cut-off voltage of 3.8 V. As clearly shown in Fig. 3e, the in situ SXRD patterns during the whole charge/discharge process presented a highly symmetric feature, and all the peaks of O3 phase were fully recovered at the end of the discharge process. Such a reversible phase transition during charge/discharge of layered oxide cathodes is often considered favorable for the stabilization of their cycle performance53. However, we illustrate that this is not the case of the highly strained NaNi0.4Mn0.4Co0.2O2 cathode (Fig. 2e). Thus, its degradation origin remains elusive.
### Native lattice strain-induced structural earthquake in NaNi0.4Mn0.4Co0.2O2
Inspired by the in situ SXRD results during the quenching process, we further examined the crystallographic structure of the highly strained NaNi0.4Mn0.4Co0.2O2 cathode at the atomistic level by using advanced TEM. Figure 4a, b shows the low and high magnification TEM image of the pristine NaNi0.4Mn0.4Co0.2O2 particle, respectively. Unlike the slow-cooling synthesized O3 sodium oxide cathodes that showed smooth surface and well-aligned lattice fringes54, the quenched cathode exhibited a highly rough surface and fluctuated strain contours in a large area. Interestingly, we do not observe an obvious composition heterogeneity across the strained region (Supplementary Fig. 7). These structural features have been mostly observed in the cycled cathodes materials1,55, but barely in the pristine cathodes. The HRTEM image in Fig. 4c illustrates that the d-spacing along the c axis direction is about 0.549 nm, which is stretched by about 3.58% compared to that of the standard O3 phase (0.530 nm). A closer examination (inset of Fig. 4c) clearly shows the existence of severe lattice distortion along the c axis direction, in which we can see the curved lattice fringe and overlap of TM atoms and Na atoms. Such lattice distortion might accelerate cation mixing or cation migration during cycling, leading to undesired structural evolution and hence capacity/voltage fade56,57,58,59. In addition, the d-spacing along the ab axis exhibited an inhomogeneous distribution (Supplementary Fig. 8). The selected area electron diffraction (SAED) pattern in Fig. 4d is in good agreement with the [010] projection of layered O3 cathode, but exhibits obvious spot splitting. All the aforementioned structural features are due to the native high lattice strain induced during the quenching process, leading to the highly metastable nature of strained O3 cathode. The corresponding geometric phase analysis (GPA) of Fig. 4c provided the direct strain distribution of the NaNi0.4Mn0.4Co0.2O2 cathode, which showed an angle between layered direction and strain direction (Fig. 4e). Such native lattice strain can be thus separated into tensile stress along the c axis direction that tends to stretch the lattice, and shear stress along the ab axis direction that will compress the lattice (Fig. 4f). Upon extraction of Na+, such metastable structure tends to undergo spontaneous strain relaxation and cause significant structural degradation as revealed below.
Figure 5a shows the bright-field TEM image of strained NaNi0.4Mn0.4Co0.2O2 cathode after charge/discharge within 2.0–4.4 V at 0.08C for 100 cycles. Compared to the pristine one (Fig. 4b), a large portion of the native lattice strain was relaxed since the strain contours disappeared. HRTEM image (Fig. 5b) and the zoomed-in view (Fig. 5c) shows that there are numerous stacking faults and dislocations as a catastrophic consequence of strain relaxation. In particular, we can clearly see the bending of lattice (yellow dashed lines in Fig. 5c) parallel to the layered direction due to the shear stress along the ab axis. Moreover, the tensile stress along the c axis direction led to the evolution of lattice dislocations and stacking faults. As a result, it will be difficult to re-insert Na+ into the Na layer because of the crossover of TM cations that might occupy the Na sites and damage layered structure, implied by the disappearance of (00l) peaks in the in situ SXRD patterns at the beginning of the discharge process (Fig. 3d). Hence, the material suffered from a dramatic capacity loss.
On the other hand, by examining the strain-relaxed region in Fig. 5a (marked by a white rectangle), it was found that the observed gaps between two layered planes are not empty; they are composed by low-crystalline fragmented domains (Supplementary Fig. 9). This conclusion is further supported by the STEM-EDS elemental mapping of cycled cathode in Supplementary Figs. 10 and 11, in which the electrolytes (signal element of P) cannot penetrate into the outlined gaps while the signal of TM, such as Ni, can be clearly observed. Figure 5d clearly showed that a layer of NiO-like rock-salt structure with (111) plane was formed at the tip of the premature crack region. As reported by Wang and co-workers, the (111) plane of rock-salt structure is energetically and structurally favorable, and has a high tolerance against compression strain60. However, the native high lattice strain in the quenched cathode significantly exceeds both tensile and compression limit of the rock-salt phase, thus leading to the formation of rock-salt fragments with different orientations (Fig. 5e and supplementary Fig. 9). In contrast, in the strain-unrelaxed region (blue circle in Fig. 5a), the curved lattice fringes were preserved, and no formation of cracks or premature cracks can be observed even at the grain boundaries that have long been considered as the preferred crack-initiating sites (Fig. 5f). The structure evolution of the highly strained NaNi0.4Mn0.4Co0.2O2 cathode during charge/discharge is schematically illustrated in Fig. 5g, which is very similar to the earthquake process. The curved lattice fringes of strained layered oxide cathodes introduced by quenching during material synthesis are similar to the curved stratum of the earth due to the stored elastic strain energy. The extraction/insertion of Na+ plays a similar role to plate motion, which leads to the release of stored energy (native strain) in a way of cracking, displacement, and dip/strike faulting that also occurred during the structural degradation of layered cathodes.
Therefore, such strain relaxation does not necessitate a high-voltage charge but should be more related to local strain distribution. Figure 6 shows the structures of cycled strained cathode during charge/discharge with a lower cut-off voltage of 3.8 V for 10 cycles. As shown in Fig. 6a, two strain-relaxed regions can be observed, with one on the surface (marked by dashed square f) and the other one in the interior of the particle (marked by dashed square b). The results are contradictory to the prevailing surface-to-bulk phase transformation mode for layered cathodes, which believed that the degradation was initiated from the surface61,62. Although our investigated system is based on a highly strained material, which is different from most of the previously investigated materials, the synthesis of layered oxide cathodes generally involves the sophisticated process, which can actually induce many different kinds of native structural defects, especially during the large-scale manufacturing process that could easily suffer from temperature heterogeneity. Indeed, boundaries63,64,65,66, nanopores67, and dislocations68 have been observed in several lithium-layered oxide cathodes, of which their distribution could act as the starting points of structural degradation, which however does not always initiate from the surface.
The zoomed-in view of square b in Fig. 6a clearly reveals that the strain relaxation was terminated exactly at the end of the strain-unrelaxed region (Fig. 6b). Moreover, Fig. 6c clearly shows that the strain-relaxed region (~10 nm) exhibited rock-salt structure (square d), while the layered structures remained in the strain-unrelaxed region (square e). These are further supported by the corresponding Fast Fourier Transform (FFT) patterns in Fig. 6d, e, which are in good agreement with that projected from [111] direction of rock-salt phase and [010] direction of layered phase, respectively. In another strain-relaxed region (Fig. 6f, zoomed-in view of square f in Fig. 6a), we also observed the formation of rock-salt domains with different orientations, which are similar to those that occurred during high-voltage charge (Fig. 5d, e). Therefore, the charge cut-off voltage does not play a critical role in activating the strain relaxation, which can explain the similar capacity fade behavior of strained NaNi0.4Mn0.4Co0.2O2 cathode during cycling with different cut-off voltage.
To directly stimulate the strain relaxation process and the associated structural transformation during charge/discharge, we further conducted in situ TEM observation during heating of charged (3.8 V) strained NaNi0.4Mn0.4Co0.2O2 cathode. Before in situ heating, a high distribution of inhomogeneous lattice strain as the contours and intragranular gaps can be clearly observed (Fig. 6g). After heating to high temperature, the pre-existing gaps propagated in accompaniment with the formation of high-density new nanogaps. In particular, we can see that the gaps were widened or appeared starting from the regions that exhibit higher lattice strain (Fig. 6h, i). Nevertheless, the layered structure of the strained NaNi0.4Mn0.4Co0.2O2 cathode was well preserved even after heating to 250 oC, as evidenced by the corresponding SAED patterns in the inset. The results again indicate that layered to rock-salt phase transformation is not the dominant factor to trigger the formation of cracks, but might be a consequence of strain relaxation.
## Discussion
The aforementioned results have clearly emphasized the critical role of native lattice strain in initiating the structural earthquake of sodium-layered oxide cathodes, which can lead to extremely fast capacity degradation. It has clearly distinguished native lattice strain from other common factors such as phase transition and cathode/electrolytes parasitic reactions. Although it is well known that the lattice strain could be also generated due to lattice mismatch and reaction heterogeneity from the aforementioned structural deformation during cycling, it generally takes a long time to break the threshold value and cause an abrupt capacity degradation. Instead, the native lattice strain induced during the synthesis of sodium-layered oxides could significantly exceed the threshold limits, and hence dominate their failure process. The results indicate that it is very important to eliminate the native lattice strain by fine-tuning the synthetic conditions and also to minimize the strain generation during cycling by rational structure tailoring. The concentration and distribution of these lattice strains depend on the quenching conditions (e.g., medium, cooling rate, etc.) Therefore, the selection of quenching medium (e.g., liquid N2, fluid, etc.) and cooling rate/environment that can ensure a homogeneous thermal distribution and transformation could help to alleviate the lattice strain. Moreover, inspired by steel manufacturing, optimization of post-annealing procedures could also help to eliminate these intrinsic lattice strains.
Although the strain relaxation process is not regulated by the voltage window, it is critical to understand the potential factors that can regulate this process. Charge/discharge temperature and rate are two common factors that can dramatically affect the cycle stability of cathode materials. Therefore, we further investigated their influence on the strain relaxation process. Figure 7 compares the structures of cycled strained NaNi0.4Mn0.4Co0.2O2 cathode after 100 cycles of charge/discharge under different conditions. Under a low charge/discharge rate of 0.08C at room temperature, the strain relaxation was relatively smooth, leading to the formation of a straight incision (marked by white rectangles in Fig. 7a, b). This is because the extraction/insertion of Na+ is proceeding in a very slow manner, which can thus minimize the effect of stress heterogeneity. This is similar to common phenomena, such as smooth plate motion or tearing wrinkle paper in a slow manner. Again, we can see the tip of the gap was terminated at the end of the strain-unrelaxed region. With elevated temperature, we can see the propagation and widening of gaps (marked by yellow rectangles in Fig. 7c) as well as the new formation of high-density nanogaps (marked by yellow rectangles in Fig. 7d), which are similar to that observed during in situ TEM heating. This might be because of the increased extraction/insertion kinetic of Na+ at high temperatures. By switching to fast charging/discharging, the strain relaxation is aggressive and vigorous, thus leading to the formation of both large (marked by red rectangles in Fig. 7e) and high-density nanocracks (marked by red rectangles in Fig. 7f) with a curved incision. The relationship between strain relaxation and cycling temperature/rates is schematically illustrated in Fig. 7g. In addition, cracks in the macro scale were also observed in the cycled strained cathode under various conditions due to the relaxation of the native lattice strain (Supplementary Fig. 12).
In summary, through a combination of advanced diagnostics from material synthesis to cell failure, we have discovered the essential role of synthesis-induced native lattice strain in triggering the structure earthquake of sodium-layered oxide cathodes during cycling. The spontaneous relaxation of internal strain that exceeds the threshold limit is the fundamental origin of the abrupt degradation of sodium-layered oxide cathodes. Furthermore, we showed that such a process is not regulated by the charge cut-off voltage, but is strongly coupling with working temperature and charging/discharging rates that can tailor the movement of Na+. Although surface coating or electrolytes modulation is effective in suppressing the irreversible phase transition and mitigating cathode–electrolyte parasitic reactions of sodium-layered oxide cathodes, these approaches cannot prevent the relaxation of intrinsic lattice strain. Our work indicates an urgent need to revisit the crystallographic structure of sodium /lithium layered oxide cathodes such as strain, oxygen vacancies, nanopores, domain boundaries, and other intrinsic defects, and to understand how they affect common battery phenomenon such as capacity fade and voltage decay/hysteresis. Our work also indicates that a rational strain management during cycling of sodium/lithium layered cathodes is required to maximize cycle life.
## Methods
### Material synthesis
The Ni0.4Mn0.4Co0.2(OH)2 precursor was synthesized through a co-precipitation method. The transition metal sulfate solution (Ni:Mn:Co = 0.4:0.4:0.2 in molar ratio) with a concentration of 2 mol/L was fed into a continuously stirred tank reactor (4 L) under N2 protection. Concentrations of 4 mol/L NaOH and 5 mol/L NH3·H2O were also fed into the tank, respectively. The pH value was controlled at 10.8 during the co-precipitation process by adjusting the NaOH feeding rate69. The Ni0.4Mn0.4Co0.2(OH)2 precursor was obtained by filtering and washing with deionized water followed by vacuum drying at 105 °C overnight. To synthesize highly strained O3 NaNi0.4Mn0.4Co0.2O2, Ni0.4Mn0.4Co0.2(OH)2 precursor and NaOH were mixed with a molar ratio of 1:1.05 using a rotary mixer. The mixture powder was then pressed into a pellet and heated to 850 °C for 24 h with a heating rate of 3 °C/min under an air atmosphere. The sample was then quenched from 850 °C to room temperature in air and stored inside a glovebox for further characterization. As a comparison, the P1/P2/O3 NaNi0.4Mn0.4Co0.2O2 was synthesized by naturally cooling from 850 oC to room temperature.
### Electrochemical characterization
The electrode was made by spreading a mixture of 80 wt.% active material, 10 wt.% C45 and 10 wt.% PVDF (8 wt.% in NMP) onto Aluminum foil. The active material loading was controlled at around 3.5 mg/cm2. The cycling performance of highly strained O3 NaNi0.4Mn0.4Co0.2O2 cathode was evaluated by assembling them into coin cells with Na metal as referenced and a counter electrode inside an Argonne-filled glovebox (H2O/O2: <0.5 ppm). The electrolyte was 1 mol/L NaPF6 in propylene carbonate with 2 vol% fluoroethylene carbonate additive, and the separator was glass fiber (grade GF/F Glass Microfiber Filter Binder Free, circle, 125 mm). The coin cells were then tested using MACCOR cycler under different voltage window (2.0–4.4 V and 2.0–3.8 V) at different charge/discharge rates and temperatures. These cycled cells were further harvested for ex-situ SEM and TEM analysis. The parasitic reactions of O3 NaNi0.4Mn0.4Co0.2O2 cathode were measured by a home-built high-precision leakage current measurement system26. The Na/NaNi0.4Mn0.4Co0.2O2 cell was under formation for two cycles and then charged to different voltages and held at each specific potential for 40 h using a Keithley 2401 source meter. During these processes, the leakage currents were monitored and measured. The measured leakage current was proportional to the reaction rate of parasitic (side) reactions between the working electrode and the electrolyte.
### In situ SXRD during heating and cycling
The in situ SXRD experiments were conducted at 11-ID-C of Advanced Photon Sources of Argonne National Laboratory with a wavelength of 0.117418 Å. For the in situ synthesis, the mixture of Ni0.4Mn0.4Co0.2(OH)2 precursor and NaOH was pressed into a 7 mm diameter pellet and loaded into a LINKAM TS-1500 furnace and then heated to 875 °C at a heating rate of 3 °C/min and then held at 875 °C for 7 h followed by quenching to room temperature. For in situ charge/discharge, coin cells with a hole on both top and bottom were used. The holes were sealed with Kapton tape after assembling the Na/NaNi0.4Mn0.4Co0.2O2 coin cells, which were charged/discharged within 2.0–3.8 V and 2.0–4.4 V at 18 mA/g, respectively. SXRD patterns were collected during in situ heating and in situ cycling processes.
### XANES during charge/discharge
The in situ Ni K-edge XANES experiment was conducted in transmission mode at 20-BM of Advanced Photon Sources of Argonne National Laboratory. The incident beam was monochromatized by using a Si(111) fixed-exit, double-crystal monochromator. During the in situ experiment, the Na/NaNi0.4Mn0.4Co0.2O2 cell was charged/discharged with a constant current density of 25 mA/g between 2.0 and 4.4 V using a MACCOR cycler. Ex-situ Ni K-edge, Co K-edge, and Mn K-edge XANES of strained O3 NaNi0.4Mn0.4Co0.2O2 cathodes under different charge/discharge states were also conducted at 20-BM of Advanced Photon Sources of Argonne National Laboratory. The cycled electrodes were disassembled and rinsed with dimethyl carbonate (DMC, Sigma Aldrich, >99%, anhydrous) to remove electrolyte residue, and then sealed with Kapton tape for measurement.
### Structure characterization
The TGA analysis was conducted using STA 449 F3 instrument to measure the weight loss of the Ni0.4Mn0.4Co0.2(OH)2/NaOH mixture during heating from room temperature to 900 °C with a heating rate of 10 °C/min under the atmosphere of air. The morphologies of the NaNi0.4Mn0.4Co0.2O2 were characterized by scanning electron microscopy (JEOL 7100F) and TEM (JEOL 2100F). The Zeiss NVision 40 focused ion beam–scanning electron microscopy dual-beam system was used to prepare a cross-section TEM specimen through a standard lift-out procedure. The HAADF imaging and elemental mapping were conducted at FEI Talos F200X (S)TEM equipped with a SuperX energy-dispersive X-ray spectrometer. The In situ heating TEM experiments during heating of a 3.8 V-charged NaNi0.4Mn0.4Co0.2O2 electrode were conducted on JEOL 2100 F from room temperature to 250 °C by using a Gatan bulk heating holder. HREM was carried out on a FEI Titan 80-300 ST, equipped with a CEOS Cc/Cs image corrector and operated at 200 kV. C 1s XPS of slow-cooled layered oxide cathodes were conducted using a PHI 5000 VersaProbe II XPS Microprobe (Physical Electronics), with Al Kα radiation (1486.6 eV, 100 μm diameter at focus, 25 W), Ar+, and electron beam sample neutralization. The obtained spectra were calibrated using the C–C peak at 284.8 eV in the C 1s.
### In situ DEMS
The in situ DEMS experiment was conducted in a custom-made system. The electrode was prepared by spreading a slurry consisting of 80 wt.% NaNi0.4Mn0.4Co0.2O2, 10 wt.% PVDF binder, and 10 wt.% C45 onto a 16-mm diameter carbon paper and then dried (areal active material loading: 10 mg/cm2). We assembled a 2025-coin cell with a hole with a sodium metal anode and glass fiber separator, which was charging/discharging within 2.0–4.4 V at a current density of 30 mA/g. The carrier gas was pre-dehydrated helium (99.999%) with a flow rate of 8 mL/min. The gas species generated during cycling were then passed through a cold trap (mixture of dry ice and ethanol at −78.5 °C and 1 atm) to condense the electrolyte vapor before it was sent to the mass spectrometer (modified 5975C mass-selective detector, Agilent) for analysis.
### Reporting summary
Further information on research design is available in the Nature Research Reporting Summary linked to this article. | 2023-03-21 00:58:37 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6285126209259033, "perplexity": 3651.16705650424}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296943589.10/warc/CC-MAIN-20230321002050-20230321032050-00501.warc.gz"} |
https://physics.stackexchange.com/questions/163872/does-a-room-aerate-faster-when-its-cold-outside | # Does a room aerate faster when it's cold outside?
When we open the window and it's cold outside, we feel that the cold air spreads quickly. But at the end, does the cold air propagate faster than if the outside air temperature were warm?
In other words, in winter do we need to aerate a room for a shorter time for the inside air to be renewed?
If anything, the diffusion rate of cold air would likely be slower. We can get a characteristic diffusion time by considering the mean free path of molecules in the gas and its speed of sound. For two gases that are chemically identical (i.e. they differ only in $T$ and $p$), we have: $$v_\text{sound}\propto \sqrt{T}$$ and $$d_\text{mfp}\propto\frac{1}{p}\sqrt{T}$$ giving $$t_\text{diffusion} \propto p$$
So, the diffusion rate, to a first approximation, does not depend on temperature. If we apply the ideal gas law $p=\rho R T$ to a closed box of air, then $p \propto T$, meaning colder air is associated with lower pressure (generally true in the midlatitudes) and hence we would expect warmer air to diffuse more quickly. | 2022-12-04 14:41:28 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.703283429145813, "perplexity": 421.17808633213474}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710974.36/warc/CC-MAIN-20221204140455-20221204170455-00759.warc.gz"} |
https://math.stackexchange.com/questions/437161/simple-proof-of-cauchys-theorem-in-group-theory/437165 | # Simple proof of Cauchy's theorem in Group theory
I will refer to the following simple proof of Cauchy's theorem that appears in chapter 33 of Pinter's A Book of Abstract Algebra. I have copied it below so my question can be properly understood.
This proof is crystal clear, however what I cannot understand is why $p$ has to be a prime number? It seems to me that the proof works for any divisor of $|G|$. Could somebody clarify this? I would appreciate it.
Cauchy's theorem: Let $G$ be a finite group of order $n$ and let $p$ be a prime divisor of $n$, then $G$ has an element of order $p$.
Pinter proves Cauchy's theorem specifically for $p=5$; however, he says, the same argument works for any value of $p$.
Consider all possible 5-tuples $(a, b, c, d, k)$ of elements of $G$ whose product $abcdk =e$.
How many distinct 5-tuples of this kind are there?
Well, if we select $a, b, c$ and $d$ at random, there is a unique $k=d^{-1} c^{-1} b^{-1} a^{-1}$ in $G$ making $abcdk = e$. Thus, there are $n^4$ such 5-tuples.
Call two 5-tuples equivalent if one is merely a cyclic permutation of the other. Thus, $(a, b, c, d, k)$ is equivalent to exactly five distinct 5-tuples, namely $(a, b, c, d, k)$, $(b, c, d, k, a)$, $(c, d, k, a, b)$, $(d, k, a, b, c)$ and $(k, a, b, c, d)$.
The only exception occurs when a 5-tuple is of the form $(a, a, a, a, a)$ with all its components equal; it is equivalent only to itself. Thus, the equivalence class of any 5-tuple of the form $(a, a, a, a, a)$ has a single member, while all the other equivalence classes have five members.
Are there any equivalence classes, other than ${(e, e, e, e, e)}$, with a single member? If not then $5$ divides $(n^4-1)$ (for there are $n^4$ 5-tuples under consideration, less $(e, e, e, e, e)$), hence $n^4\equiv 1\pmod 5$. But we are assuming that 5 divides $n$, hence $n^4\equiv 0\pmod 5$, which is a contradiction.
This contradiction shows that there must be a 5-tuple $(a,a,a,a,a)\neq(e, e, e, e, e)$ such that $aaaaa=a^5=e$. Thus, there is an element $a\in G$ of order 5. ■
• Can you think of a group with order 4, and no element of order 4? – Calvin Lin Jul 6 '13 at 0:47
• Calvin Lin's comment hints at an excellent way of working out these questions for yourself. You take a small counterexample of the proposition ($\mathbb Z_2\times\mathbb Z_2$ is the smallest counterexample) and try out the constructions and the claims in the proof against it. – Karl Kroningfeld Jul 6 '13 at 2:08
• As others have pointed out, it is the sentence beginning "The only exception" that is only correct when $p$ is prime. Notice that no justification has been included for this assertion, so you should try and prove that it really is correct when $p$ is prime. – Derek Holt Jul 6 '13 at 12:54
• Following the reasoning in the proof, if there is a 5-tuple $(a, a, a, a, a) \neq (e, e, e, e, e)$, then $5$ should divide $n^4 - 2$, right? Considering this, can we not say that there are 4 such 5-tuples, so that $n^4 \equiv 0 \pmod 5$? – David Cian Mar 10 at 13:07
"The only exception occurs when a 5-tuple is of the form $(a,a,a,a,a)$ with all its components equal; it is equivalent only to itself."
This breaks if $p = 6$, for example: $(a, a, b, a, a, b)$. But even then:
"there must be a 5-tuple $(a,a,a,a,a)\neq (e,e,e,e,e)$ such that $aaaaa=a^5=e$. Thus, there is an element $a\in G$ of order 5."
This also breaks if $p = 6$: there are plenty of groups $G$ with non-identity elements $a$ such that $a^6 = e$ because $a$ has order 2 or 3 (consider $C_2$ or $C_3$ and a non-identity element). | 2021-07-26 14:25:41 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.840382993221283, "perplexity": 122.04719437779607}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046152129.33/warc/CC-MAIN-20210726120442-20210726150442-00200.warc.gz"} |
https://www.garretthughes.com/blog/how-to-create-a-heat-map-of-covid-19-cases-by-zip-code | Published on
# How to create a heat map of Covid-19 cases by zip code
Authors
San Diego Heatmap
After posting the heat map of San Diego Covid-19 cases online I realized a lot of you had questions, so I figured I’d write a quick how-to on creating this heat map.
# Step 1:
Find the data. San Diego county gives out a daily summary via zip code in PDF form, others may do this as well. Without this it won’t be possible to do a heat map of cases by zip code.
# Step 2:
Now we need to translate these values into a spreadsheet. My suggestion is these 4 columns:
Zip, Latitude, Longitude, Cases
You’ll want to copy the zip code into the first column and copy the number of coronavirus cases into the fourth column.
# Step 3:
We need to translate zip code to lat/lng values. You can either do this manually (by google-ing the {zip} lat long. This will give you the data in N/S/E/W you’ll want to translate that to pluses and minuses. I shared a few examples below:
33.1924°N, 117.3675°W -------BECOMES------ 33.1924, -117.3675 33.1924°S, 117.3675°E -------BECOMES------ -33.1924, 117.3675
Alternatively, if you have a lot of zip codes (or are lazy like me, remember this is one of the good traits of a programmer, you can do this programmatically using a little Vlookup magic, which I’ll share below.
# Step 4: Optional
There are tons of free public assets online to accomplish this. The one I used can be found here. You’ll want to download that CSV and open it up in Google Sheets/Excel. First we’ll create a new sheet at the bottom
Next in the us-zip-code-latitude-and-longitude sheet you’ll need to make sure that Zip is in the first column (this is a limitation of Vlookup).
Now it’s time for the formula, this is what I used for latitude:
=VLOOKUP(A2,'us-zip-code-latitude-and-longitude'!$A$1:$H$43192,4,FALSE)
and for longitude
=VLOOKUP(A2,'us-zip-code-latitude-and-longitude'!$A$1:$H$43192,5,FALSE)
For those unfamiliar with Vlookup (this is in Google Sheets) the formula reads:
# Quick heat check:
Your data should look like this now.
# Step 5:
Now it’s time to input into Kepler.gl. Click https://kepler.gl/demo and it prompts you to upload data. To get your CSV, download the sheet with cases, zip and lat/long as a csv.
# Step 6:
Drag that csv into the box on the Kepler page. You should see a bunch of small little dots.
# Step 7:
Click the layer settings drop down
# Step 8:
Update the settings to be a heat map (instead of point), put the radius to 100 (instead of 10) and change the weight from density to cases.
# Step 9: Optional
If you would like others to see this you’ll want to click the export button in the top right:
and click share map URL | 2023-01-28 23:42:14 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 4, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3781611919403076, "perplexity": 2277.53080888791}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764499695.59/warc/CC-MAIN-20230128220716-20230129010716-00201.warc.gz"} |
https://math.stackexchange.com/questions/2966949/nested-models-for-forcing | # Nested models for forcing
Given a model $$M$$ of ZFC, we can define $$\mathbb{P}$$-names and generic extensions $$M[G]$$ in terms of $$M$$.
But the usual framework for forcing involves an outer model $$M\subseteq V$$ of ZFC and $$M$$ is required to be transitive and countable with respect to $$V$$. The forcing extensions then become $$M\subseteq M[G] \subseteq V$$.
Does anyone know why this outer model $$V$$ is important for forcing? Extending $$M$$ looks enough to prove independances of axioms such as Choice or the Continuum Hypothesis.
EDIT
Following the answer below, here is a tentative to construct a generic filter on $$M$$ without an outer model $$V$$.
We want to prove the relative consistency of a new axiom of set theory. So we start by assuming that ZFC is consistent. By the Lowenheim-Skolem theorem, there is a countable model $$M$$ of ZFC. Let $$(\mathbb{P}, \leq)$$ a partial order inside $$M$$ of conditions that represent all the possibilities of the forcing extension.
Because $$M$$ is countable, so are the $$\mathbb{P}$$-dense parts, that we enumerate $$D_0,D_1,\dots$$ Let $$p_0\in D_0$$. By density there is $$p_1\in D_1$$ such as $$p_1\leq p_0$$. We continue this to define a decreasing sequence $$p_n\in D_n$$. Then we call $$G=\{p\in\mathbb{P} | \exists i\in\mathbb{N}, p_i\leq p\}$$. $$G$$ is a filter for $$\leq$$ and it meets every dense part, so it is $$M$$-generic on $$\mathbb{P}$$.
Is there a problem with this construction, that does not even need that $$M$$ is transitive?
• (In response to your edit) The problem is, as Noah indicated, this $G$ is not necessarily in $M$ (and in fact will not be in all interesting cases). We only know $M$ is countable from the outside... $M$ does not know it is countable. As such, $M$ does not have access to the sequence $(D_n:n\in \mathbb N).$ – spaceisdarkgreen Oct 24 '18 at 1:40
• @spaceisdarkgreen I thought that $G\notin M$ was the point of forcing: make a proper extension $M[G]$ of $M$. The extension has to be constructed from outside of $M$. I think this is even suggested by the word "generic" : if $G$ was in $M$ it would be a particular case, not a generic case. – V. Semeria Oct 24 '18 at 10:57
• "Is there a problem with this construction, that does not even need that M is transitive?" There is: although $G$ is a generic filter as long as $M$ is countable, regardless of transitivity, you still need to define $M[G]$ and show that it satisfies ZFC and that the forcing theorems hold. The usual proof of this uses transitivity, so this needs some work; this is the "internal-recursion" issue I mentioned in my answer. Put another way, for countable models it's not the existence of generics that's nontrivial, it's the use of generics. – Noah Schweber Oct 24 '18 at 13:23
• @V.Semeria If you agree about this, then I’m not sure what the argument is about. In a comment I probably shouldn’t have deleted I mentioned the outermost model $V$ is just the model we’re working in when we define $M$ and $M[G].$ You alluded to “taking $G$ from the metatheory” in a comment you have deleted, but I don’t think the distinction you’re seeing is really there and at most it’s another way of saying the same thing (and Noah already responded to that comment at length). – spaceisdarkgreen Oct 24 '18 at 14:24
The problem comes with this step:
"Letting $$G$$ be a $$\mathbb{P}$$-generic filter over $$M$$, we define $$M[G]$$ as [stuff]."
How do we know that such a $$G$$ exists in the first place? Certainly it will not in general from "within $$M$$:" as long as $$\mathbb{P}$$ is nontrivial, no $$\mathbb{P}$$-generic filter over $$M$$ will exist in $$M$$. Generic filters over $$M$$ have to come from some larger model.
This is one of the reasons that the Boolean-valued models approach to forcing is enticing: everything takes place within the original model, and we can talk about forcing over the "real" universe without saying horrible nonsense.
• Incidentally, transitivity really plays no role in forcing: we can make sense of the poset-version of forcing over an arbitrary countable model of set theory, and the Boolean-valued models version of forcing over an arbitrary model of set theory of any cardinality. Basically, the point is that the recursions needed to define the forcing relation, and hence the forcing extension (or Boolean model), are all internal to the ground model, and every model of ZFC "thinks" that it's well-founded. – Noah Schweber Oct 23 '18 at 3:23
• The relationship between the two approaches is roughly the following: when I have a forcing notion $\mathbb{P}$ in a model $M$, I get a Boolean-valued model $M^\mathbb{P}$ with Boolean algebra of truth values $\mathbb{B}_\mathbb{P}$. This algebra is closely tied to $\mathbb{P}$ itself, with "truer" truth values corresponding roughly to larger sets of conditions. A $\mathbb{P}$-generic filter $G$ over $M$ can be thought of as a way to "quotient out" $\mathbb{B}_\mathbb{P}$ so that we're left with the two-element Boolean algebra ("in/out of $G$") and this transforms $M^\mathbb{P}$ into $M[G]$. – Noah Schweber Oct 23 '18 at 3:50
• First of all, I don't know what it means to get a set from a metatheory. But even leaving that aside, I think you're taking "generic filters exist" as a basic principle, and this is definitely not warranted: why should generic filters exist for arbitrary models? For example, ZFC proves that if $M$ is a transitive model of ZFC which contains every real, then there is no Cohen-generic filter of $M$. Basically, the point is that the existence of generic filters over a model should in general be surprising - the only unsurprising situation being when that model is countable. – Noah Schweber Oct 23 '18 at 14:43
• Now you might adopt some "multiverse" principle. In particular, a principle asserting that "everything is countable in some larger universe" does imply that generic filters always exist; however, it also implies that every model lives inside a larger model! Basically, the point is that any nontrivial time you want to assert that a generic filter over a model $M$ actually exists, you're relying on the assumption that $M$ lives inside a larger universe, since that's where the generic you want has to live. – Noah Schweber Oct 23 '18 at 14:46
• @V.Semeria Again, where does your construction take place? When you say "there is a ...," what does that mean? The purpose of the ambient $V$ is precisely to be the framework within which such constructions take place. That's the whole point of my answer: that we need a mathematical universe beyond the ground model $M$ itself if we're going to go the "actual-generic-filter" route. – Noah Schweber Oct 24 '18 at 13:20
In the context of the most elementary version of Forcing that I first learned:
(1). We do not assume there exists a set-model for ZFC. But if you are given a finite list of some of the axioms of ZFC you can use the Lowenheim-Skolem method to prove there exists a countable transitive model (c.t.m) for them. ( Trying to apply this method to a list of all the axioms would require either an infinitely long proof or an ability to quantify over collections of sentences.) So we assume that we have a c.t.m for ZFC-minus-Comprehension-minus-Replacement, that satisfies all the instances of Comprehension and Replacement that we will need. So we speak of M as if it does satisfy all of ZFC.
(2). So if the poset $$P$$ belongs to $$M$$ then the set of dense subsets of $$P$$ that belong to $$M$$ is a countable set. It is a simple exercise to show that if $$S$$ is a countable family of dense subsets of a poset $$P$$ then there exists a filter $$G$$ on $$P$$ such that $$\forall s\in S\,(s\cap G\neq \phi).$$ In Forcing the filter $$G$$ is required only to have non-empty intersection with the dense subsets of $$P$$ that belong to $$M.$$ Since $$M$$ is transitive but only countable, such $$G$$ do exist.
(3). If $$P\ne \phi$$ and if $$P$$ is separative, that is, if every $$x\in P$$ has two incompatible (incomparable) extensions, then $$G\not \in M.$$ Otherwise $$P\setminus G$$ would belong to $$M$$ and would be a dense subset of $$P$$ that's disjoint from $$G.$$
(4) $$V,$$ in Set Theory, is commonly used to denote the class of all sets. If instead, we say that $$(V,\in^*)$$ is a (class or set) model for (enough of) ZFC but not necessarily the whole universe and with $$\in^*$$ not necessarily being $$\in$$ then we may consider points (1),(2),(3) above to be re-phrased as "all relativized to $$(V,\in^*)$$", with the additional requirement that $$M\subset^* V$$. | 2020-07-03 19:23:05 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 67, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9100018739700317, "perplexity": 199.10334116689694}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593655882934.6/warc/CC-MAIN-20200703184459-20200703214459-00318.warc.gz"} |
https://www.mathdoubts.com/solving-linear-equations-in-one-variable-by-transposing/ | # Solving Linear Equations in One Variable by Transposition
A method of transferring a term in a linear equation from one side to other side by changing its sign for solving it, is called solving linear equation in one variable by transposition method.
## Introduction
According to English language, the meaning of transpose is, transfer a term to the other side of an equation by changing its sign.
In transposition method, a term is transposed to the other side of an equation with its sign changed. It balances the equality of both expressions and also simplifies the linear equation in one variable. Thus, the root of the linear equation in one variable can be calculated easily in mathematics.
### Examples
$(1) \,\,\,\,\,\,$ $x-3 = 5$
In this linear equation in one variable, $x$ and $3$ are two terms in the left-hand side of the equation. If the term $3$ is transferred to other side of the equation, then it is easy to find the solution of this equation but the sign of the term $3$ is negative. So, it can be shifted to other side by changing its sign.
$\implies$ $x = 5+3$
$\,\,\, \therefore \,\,\,\,\,\,$ $x = 8$
$(2) \,\,\,\,\,\,$ $2x = 12+x$
It is another example for linear equation in one variable. In this linear equation, $12$ and $x$ are two terms in the right-hand side of the equation. The linear equation can be solved by transposing the term $x$ to left-hand side of the equation from right-hand side by changing its sign.
$\implies$ $2x-x = 12$
$\implies$ $(2-1)x = 12$
$\,\,\, \therefore \,\,\,\,\,\,$ $x = 12$
In this way, the terms are transposed in a linear equation in transposing method for solving linear equations in one variable.
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You can get the latest updates from us by following to our official page of Math Doubts in one of your favourite social media sites. | 2022-10-01 11:28:13 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.691042959690094, "perplexity": 285.18870103294535}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030335609.53/warc/CC-MAIN-20221001101652-20221001131652-00614.warc.gz"} |
http://mathhelpforum.com/algebra/221412-little-help-natural-logs-please.html | # Thread: A little help with natural logs please?
1. ## A little help with natural logs please?
I have been given this equ ation w = (1/h) ln(L/L0-1) and have been asked to solve for L. I am struggling with when to introduce the 'e'. Can someone please help? Thanks in advance.
2. ## Re: A little help with natural logs please?
Originally Posted by camorris
I have been given this equ ation w = (1/h) ln(L/L0-1) and have been asked to solve for L. I am struggling with when to introduce the 'e'. Can someone please help? Thanks in advance.
I find it very hard to read your notation.
\begin{align*}w &= \left( {\frac{1}{h}} \right)\ln \left( {\frac{L}{{{L_0} - 1}}} \right)\\hw&=\ln \left( {\frac{L}{{{L_0} - 1}}} \right)\\e^{hw}&=\frac{L}{L_0-1}\end{align*}
Can you finish?
3. ## Re: A little help with natural logs please?
I think I can
Lo(1_e^hw) ?
Sorry about the notation. I was using my phone.
4. ## Re: A little help with natural logs please?
The point is that "e^x" is the inverse function to "ln(x)". That is, $e^{ln(x)}= x$ and $ln(e^x)= x$. You "introduce the e" (by which I assume you mean "take the exponential of both sides") When you have ln of something on one side and want to get rid of the ln.
(Again, what you wrote, "Lo(1_e^hw)", makes no sense. Did you mean " $(L_0-1)e^{hw}$"?)
5. ## Re: A little help with natural logs please?
Here is the equation I have been asked to solve for L.
I would insert an image but for some reason I am not able to. So I have attached the image to this post. I had an existing equation that I had done before, but I had to work backwards.
I know I went wrong somewhere (You're right, I don't think the 'e' was required).
6. ## Re: A little help with natural logs please?
Originally Posted by camorris
Here is the equation I have been asked to solve for L.
I would insert an image but for some reason I am not able to. So I have attached the image to this post. I had an existing equation that I had done before, but I had to work backwards.
I know I went wrong somewhere (You're right, I don't think the 'e' was required).
Oh...It is 05:30 in the morning here. I cant sleep as I was up half of the night trying to get this done!!! thanks | 2016-10-01 13:01:44 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 4, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9441280961036682, "perplexity": 799.0355533089793}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-40/segments/1474738662856.94/warc/CC-MAIN-20160924173742-00123-ip-10-143-35-109.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/automorphisms-of-central-simple-algebras-bresar-example-1-27.896192/ | # I Automorphisms of Central Simple Algebras/Bresar Example 1.27
1. Dec 6, 2016
### Math Amateur
I am reading Matej Bresar's book, "Introduction to Noncommutative Algebra" and am currently focussed on Chapter 1: Finite Dimensional Division Algebras ... ...
I need help with some aspects of Example 1.27...
Example 1.27 (including a short preamble and Definition 1.26) reads as follows:
My questions on Example 1.27 comprise the following:
Question 1
In the above example from Bresar we read the following:
" ... ... Of course it is outer ... for the identity map is obviously the only inner automorphism of a commutative ring. ... ... "
I have two questions regarding this remark ...
(a) ... why does Bresar assert that "the identity map is obviously the only inner automorphism of a commutative ring" ... surely a commutative ring may have some units (invertible elements other than $1$ ...) and so may have some inner automorphisms ... BUT Bresar is asserting that this is not the case ... can someone please clarify this issue ...
(b) ... Bresar seems to be talking about $\mathbb{C}$ ... but he is referring to "a commutative ring" in the quote above ... but why? ... $\mathbb{C}$ is a field ... ???
Can someone please clarify what Bresar means ...
Question 2
In the above example from Bresar we read the following:
" ... ... We also remark that it is an element of $\text{End}_\mathbb{R} ( \mathbb{C} )$ but not $M( \mathbb{C} )$. ... "
I have two questions on this remark ... as follows ...
(a) ... ... Bresar proved in Lemma 1.25 (see previous post) that for
$M(A) = \text{End}_F (A)$ ...
so ... why isn't it true that $M( \mathbb{C} ) = \text{End}_\mathbb{R} ( \mathbb{C} )$ ... ?
(b) ... can someone please explain exactly why the conjugation automorphism is an element of $\text{End}_\mathbb{R} ( \mathbb{C} )$ but not of $M( \mathbb{C} )$ ... ?
Help will be much appreciated ... ...
Peter
=============================================================================
So that readers of this post can appreciate the notation and the context I am providing Bresar Section 1.5 ( which includes Lemmas 1.24 and 1.25) and also the start of Bresar Section 1.6 ... ... as follows:
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2. Dec 7, 2016
### Delta²
I think I can answer question 1 easily but mind that I am a bit rusty on abstract algebra.
if the ring is commutative then any inner automorphism is the identity since by definition of the inner automorphism $\phi(x)=axa^{-1}=aa^{-1}x=x$. The second step in the series of equalities is because the ring is commutative.
Also any field is a commutative ring (by the definition of the field if I am not mistaken) so I hope the above answer this question.
3. Dec 14, 2016
### zinq
It is necessary to sit with pencil (or your favorite writing implement) and paper in order to understand the answers to questions like yours, Math Amateur.
M(C) is the multiplication algebra of C, generated by left and right multiplications of an element (call it z) of C, by other elements of C. First verify that conjugation is an endomorphism of the R-algebra C. (Here it is absolutely necessary to know what the terms conjugation, R, R-algebra, and C mean.) Then verify that no matter which left and right multiplications you use to get an element of M(C), you will never obtain complex conjugation. | 2017-08-19 00:13:08 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7699398398399353, "perplexity": 654.5874767972462}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886105195.16/warc/CC-MAIN-20170818233221-20170819013221-00413.warc.gz"} |
https://www.physicsforums.com/threads/coulombs-law-hw-question-please-help.763617/ | 1. Jul 27, 2014
### emuhlay08
q1= 2microC and q2=-4microC they are on a straight line 5.0 cm away from each other.
a)find the magnitude of the net force on charge q2 due to q1
b) in what direction will the net force acting on charge q2 due to q1 be directed
c)what is the magnitude and direction of the electric field at a point located 5 cm directly below charge q1
d)what is the magnitude of the electric potential located 5 cm directly below charge q1
I'm confused on what to do for only two points. Any help is appreciated and please explain so I can try to get an understanding
for A) i got 28.8 N and B) the forces will attract. not sure if those are correct
2. Jul 27, 2014
### Zondrina
a) Looks fine.
b) Indeed the forces attract. On a free body diagram, which way would the force $\vec F_{12}$ point, given an arbitrary coordinate axis placed at $q_1$?
c) How do you calculate the electric field at a point, due to a point charge? Hint: Draw a free body diagram for this.
Last edited: Jul 27, 2014
3. Jul 27, 2014
### emuhlay08
This is what I came up with, it doesn't seem correct to me, but I'm not sure where I went wrong
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4. Jul 27, 2014
### Zondrina
You need to sum the components of the electric field of $\vec E_1$ and $\vec E_2$.
What about the direction? Draw $\vec E_{net_x}$ and $\vec E_{net_y}$ head to tail and apply $tan(\theta) = \frac{E_{net_y}}{E_{net_x}}$.
d) Same deal as c) pretty much, except potential is used instead of the electric field. | 2017-10-24 02:56:18 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6240934729576111, "perplexity": 584.7593100048927}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187827853.86/warc/CC-MAIN-20171024014937-20171024034937-00422.warc.gz"} |
https://www.zbmath.org/?q=an%3A0554.62035 | # zbMATH — the first resource for mathematics
Bandwidth choice for nonparametric regression. (English) Zbl 0554.62035
A modified version of a kernel regression estimator is analyzed with respect to choice of bandwidth. The modification consists of replacing the kernel estimate by a tapered Fourier series estimate which simplifies some technical arguments.
It is shown that the bandwidth chosen based on an unbiased estimate of mean square error is asymptotically optimal. Several other methods of bandwidth selection, including cross-validation, are examined and shown to be asymptotically equivalent. However, some simulation results indicate that for small or moderate sample sizes the methods are quite different. | 2021-05-12 03:12:44 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8293441534042358, "perplexity": 497.8323677595536}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243991693.14/warc/CC-MAIN-20210512004850-20210512034850-00297.warc.gz"} |
https://www.gradesaver.com/textbooks/math/algebra/algebra-1-common-core-15th-edition/chapter-11-rational-expressions-and-functions-11-7-graphing-rational-functions-practice-and-problem-solving-exercises-page-711/26 | ## Algebra 1: Common Core (15th Edition)
We are given: $y=\frac{7}{x}-15$ The graph of $y=\frac{7}{x}-15$ is the graph of $y=\frac{7}{x}$ shifted 15 units downward. | 2020-09-25 13:04:24 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2057080715894699, "perplexity": 881.3430517966693}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600400226381.66/warc/CC-MAIN-20200925115553-20200925145553-00636.warc.gz"} |
https://artofproblemsolving.com/wiki/index.php/2011_AIME_I_Problems/Problem_9 | # 2011 AIME I Problems/Problem 9
## Problem
Suppose $x$ is in the interval $[0, \pi/2]$ and $\log_{24\sin x} (24\cos x)=\frac{3}{2}$. Find $24\cot^2 x$.
## Solution 1
We can rewrite the given expression as $$\sqrt{24^3\sin^3 x}=24\cos x$$ Square both sides and divide by $24^2$ to get $$24\sin ^3 x=\cos ^2 x$$ Rewrite $\cos ^2 x$ as $1-\sin ^2 x$ $$24\sin ^3 x=1-\sin ^2 x$$ $$24\sin ^3 x+\sin ^2 x - 1=0$$ Testing values using the rational root theorem gives $\sin x=\frac{1}{3}$ as a root, $\sin^{-1} \frac{1}{3}$ does fall in the first quadrant so it satisfies the interval. There are now two ways to finish this problem.
First way: Since $\sin x=\frac{1}{3}$, we have $$\sin ^2 x=\frac{1}{9}$$ Using the Pythagorean Identity gives us $\cos ^2 x=\frac{8}{9}$. Then we use the definition of $\cot ^2 x$ to compute our final answer. $24\cot ^2 x=24\frac{\cos ^2 x}{\sin ^2 x}=24\left(\frac{\frac{8}{9}}{\frac{1}{9}}\right)=24(8)=\boxed{192}$.
Second way: Multiplying our old equation $24\sin ^3 x=\cos ^2 x$ by $\dfrac{24}{\sin^2x}$ gives $$576\sin x = 24\cot^2x$$ So, $24\cot^2x=576\sin x=576\cdot\frac{1}{3}=\boxed{192}$.
## Solution 2
Like Solution 1, we can rewrite the given expression as $$24\sin^3x=\cos^2x$$ Divide both sides by $\sin^3x$. $$24 = \cot^2x\csc x$$ Square both sides. $$576 = \cot^4x\csc^2x$$ Substitute the identity $\csc^2x = \cot^2x + 1$. $$576 = \cot^4x(\cot^2x + 1)$$ Let $a = \cot^2x$. Then $$576 = a^3 + a^2$$. Since $\sqrt[3]{576} \approx 8$, we can easily see that $a = 8$ is a solution. Thus, the answer is $24\cot^2x = 24a = 24 \cdot 8 = \boxed{192}$.
~IceMatrix | 2020-12-03 09:15:04 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 33, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9784830808639526, "perplexity": 125.78681602795541}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141723602.80/warc/CC-MAIN-20201203062440-20201203092440-00511.warc.gz"} |
https://discuss.codechef.com/t/bin-od-editorial/104088 | # BIN_OD - Editorial
Author: Sahil Tiwari
Testers: Takuki Kurokawa, Utkarsh Gupta
Editorialist: Nishank Suresh
1776
Prefix sums
# PROBLEM:
You are given an array A and A queries on it. For each query, you are given two subarrays and an integer k.
Find the number of pairs of elements, one from the first subarray and one from the second, such that their bitwise xor has the k-th bit set.
# EXPLANATION:
Let’s look at answering a single query (k, L_1, R_1, L_2, R_2) first: speeding it up to answer multiple queries can come later.
Suppose A_i \oplus A_j has its k-th bit set. This is only possible when:
• A_i has its k-th bit set and A_j doesn’t; or
• A_j has its k-th bit set and A_i doesn’t
In particular, if take some A_i from [L_1, R_1] with its k-th bit set, we can pair it with any A_j from [L_2, R_2] whose k-th bit is unset.
Similarly, if take some A_i from [L_1, R_1] with its k-th bit unset, we can pair it with any A_j from [L_2, R_2] whose k-th bit is set.
This gives us a rather simple solution:
• Let S_1 be the number of elements in subarray [L_1, R_1] that have the k-th bit set
• Let U_1 be the number of elements in subarray [L_1, R_1] that have the k-th bit unset
• Let S_2 be the number of elements in subarray [L_2, R_2] that have the k-th bit set
• Let U_2 be the number of elements in subarray [L_2, R_2] that have the k-th bit unset
Then, the answer to this query is simply S_1\cdot U_2 + S_2\cdot U_1.
Computing S_1, S_2, U_1, U_2 is easy to do by looping across the subarrays, but that’s not fast enough to answer multiple queries: we need something a bit faster.
### Using prefix sums
Notice that, if k is fixed, we can treat each element of the array as being either 0 or 1 depending on whether it has the k-th bit set or not.
Then, the above variables simplify quite nicely:
• S_1 and S_2 are the number of ones in their respective ranges, or more specifically, just the sums of those ranges.
• U_1 and U_2 are the number of zeros in their respective ranges. Knowing S_1, S_2, and the lengths of the ranges is enough to compute these values (since S_1 + U_1 = R_1-L_1 + 1 and S_2 + U_2 = R_2+L_2-1).
Computing range sums quickly is a well-known application of prefix sums.
We need to maintain separate prefix sums for each k, but there are only 60 possible values of k anyway so this is not an issue.
That is, for each 0 \leq k \lt 60, let pref_{k, i} denote the number of elements in [1, i] that have the k-th bit set.
Then,
• S_1 = pref_{k, R_1} - pref_{k, L_1-1}
• S_2 = pref_{k, R_2} - pref_{k, L_2-1}
• U_1 and U_2 can be computed as noted above.
This allows us to answer each query in \mathcal{O}(1) time.
# TIME COMPLEXITY
\mathcal{O}(60\cdot N + Q) per test case.
# CODE:
Setter's code (C++)
// Code by Sahil Tiwari (still_me)
#include<bits/stdc++.h>
#define still_me main
#define endl "\n"
#define int long long int
#define all(a) (a).begin() , (a).end()
#define print(a) for(auto TEMPORARY: a) cout<<TEMPORARY<<" ";cout<<endl;
#define tt int TESTCASE;cin>>TESTCASE;while(TESTCASE--)
#define arrin(a,n) for(int INPUT=0;INPUT<n;INPUT++)cin>>a[INPUT]
using namespace std;
const int mod = 1e9+7;
const int inf = 1e18;
void solve() {
int n , q;
cin>>n>>q;
vector<int> a(n);
arrin(a , n);
vector<vector<int>> b(n+1 , vector<int>(61));
for(int i=0;i<n;i++) {
for(int j=0;j<61;j++) {
if(a[i] & (1ll << j))
b[i+1][j]++;
b[i+1][j] += b[i][j];
}
}
while(q--) {
int k , l , r , x , y;
cin>>k>>l>>r>>x>>y;
int o1 = b[r][k] - b[l-1][k];
int o2 = b[y][k] - b[x-1][k];
int z1 = r-l+1 - o1;
int z2 = y-x+1 - o2;
cout<<(o1*z2 + o2*z1)<<endl;
}
}
signed still_me()
{
ios_base::sync_with_stdio(false);cin.tie(NULL);cout.tie(NULL);
tt{
solve();
}
return 0;
}
Tester's code (C++)
//Utkarsh.25dec
#include <iostream>
#include <cstdio>
#include <cstdlib>
#include <algorithm>
#include <cmath>
#include <vector>
#include <set>
#include <map>
#include <unordered_set>
#include <unordered_map>
#include <queue>
#include <ctime>
#include <cassert>
#include <complex>
#include <string>
#include <cstring>
#include <chrono>
#include <random>
#include <bitset>
#include <array>
#define ll long long int
#define pb push_back
#define mp make_pair
#define mod 1000000007
#define vl vector <ll>
#define all(c) (c).begin(),(c).end()
using namespace std;
ll power(ll a,ll b) {ll res=1;a%=mod; assert(b>=0); for(;b;b>>=1){if(b&1)res=res*a%mod;a=a*a%mod;}return res;}
ll modInverse(ll a){return power(a,mod-2);}
const int N=500023;
bool vis[N];
long long readInt(long long l,long long r,char endd){
long long x=0;
int cnt=0;
int fi=-1;
bool is_neg=false;
while(true){
char g=getchar();
if(g=='-'){
assert(fi==-1);
is_neg=true;
continue;
}
if('0'<=g && g<='9'){
x*=10;
x+=g-'0';
if(cnt==0){
fi=g-'0';
}
cnt++;
assert(fi!=0 || cnt==1);
assert(fi!=0 || is_neg==false);
assert(!(cnt>19 || ( cnt==19 && fi>1) ));
} else if(g==endd){
if(is_neg){
x= -x;
}
if(!(l <= x && x <= r))
{
cerr << l << ' ' << r << ' ' << x << '\n';
assert(1 == 0);
}
return x;
} else {
assert(false);
}
}
}
string ret="";
int cnt=0;
while(true){
char g=getchar();
assert(g!=-1);
if(g==endd){
break;
}
cnt++;
ret+=g;
}
assert(l<=cnt && cnt<=r);
return ret;
}
long long readIntSp(long long l,long long r){
}
long long readIntLn(long long l,long long r){
}
}
}
int sumN=0,sumQ=0;
void solve()
{
sumN+=n;
sumQ+=q;
assert(sumN<=100000);
assert(sumQ<=500000);
int sum[n+1][65];
memset(sum,0,sizeof(sum));
long long A[n+1];
memset(A,0,sizeof(A));
for(int i=1;i<=n;i++)
{
if(i==n)
else
for(int j=0;j<60;j++)
{
sum[i][j]=sum[i-1][j];
if((A[i]&(1LL<<j))!=0)
sum[i][j]++;
}
}
while(q--)
{
long long left1s=sum[r1][k]-sum[l1-1][k];
long long left0s=(r1-l1+1)-left1s;
long long right1s=sum[r2][k]-sum[l2-1][k];
long long right0s=(r2-l2+1)-right1s;
cout<<(left1s*right0s)+(left0s*right1s)<<'\n';
}
}
int main()
{
#ifndef ONLINE_JUDGE
freopen("input.txt", "r", stdin);
freopen("output.txt", "w", stdout);
#endif
ios_base::sync_with_stdio(false);
cin.tie(NULL),cout.tie(NULL);
while(T--)
solve();
assert(getchar()==-1);
cerr << "Time : " << 1000 * ((double)clock()) / (double)CLOCKS_PER_SEC << "ms\n";
}
Editorialist's code (Python)
for _ in range(int(input())):
n, q = map(int, input().split())
a = list(map(int, input().split()))
pref = [[0 for i in range(60)] for _ in range(n+1)]
for i in range(n):
for k in range(60):
pref[i+1][k] = pref[i][k] + ((a[i] >> k) & 1)
for i in range(q):
k, l1, r1, l2, r2 = map(int, input().split())
on1, on2 = pref[r1][k] - pref[l1-1][k], pref[r2][k] - pref[l2-1][k]
off1, off2 = r1-l1+1 - on1, r2-l2+1 - on2
print(on1*off2 + on2*off1)
2 Likes
I can’t understand what is wrong in my code ?? plz help
https://www.codechef.com/viewsolution/80193617
2 Likes
1 << j will overflow for j \gt 31, since it computes in int. Use 1LL << j instead.
Unfortunately, that appears to be your only mistake.
2 Likes
https://www.codechef.com/viewsolution/80275344
Runtime error: RE (SIGSEGV) for some of the tests.
Anyone knows why?
@zoharbarak
It is because you have defined a as a vector of int, and it should be long long .
Now what happens is that cin expects an int, but in the input buffer is a big number. So this creates some anomalies and in the end, cin doesn’t behave well.
1
2 1
1000000000000000 1000000000000000
1 1 1 2 2
If you try this custom test case in your original code, you will get a runtime error on CodeChef ide.
Also, you should look for overflow in your final calculation of res
Modified Accepted Solution: CodeChef | Competitive Programming | Participate & Learn
1 Like
Thanks
https://www.codechef.com/viewsolution/80284830
I’m not sure why this gets TLE (I just loop n & q)
You are not inputting anything inside the loop for “q”…So the 1st value (i.e. k) becomes the n of the next test case, and further operations are performed according to that, which can cause TLE.
There can be upto 5\cdot 10^4 testcases, and you’re creating an array of size 10^5 \times 61 for each one. That’s over 10^{11} operations just to allocate the memory, it’s no surprise that you get TLE.
1 Like
1<<j will overflow
use
1 Like
Hello All,
Can anyone please tell me what’s wrong with my code, it was failing one test case.
My Code
Thanks.
@celestialidiot
Using k = log2(mx) was causing errors.
Use k = 59.
Modified Accepted Solution
https://www.codechef.com/viewsolution/80307160
Thanks
Take my whole day to code. learn a new concept of prefix sum on bit. I was thinking that this don’t exists . There were no blog post that I find related to this .
#include <bits/stdc++.h>
using namespace std;
#define int long long int
// count the xor value whose kth bit is set
void solve(){
int n,q;
cin>>n>>q;
vector<int> v(n);
vector<vector<int>> prefix(n,vector<int> (62));
for(int i=0;i<n;i++){
cin>>v[i];
vector<int> temp(60);
for(int j=0;j<=60;j++){
if((v[i]>>j)&1) temp[j] = 1;
else temp[j] = 0;
if(i == 0) prefix[i][j] = temp[j];
else prefix[i][j] = prefix[i-1][j] + temp[j];
// cout<<prefix[i][j]<<" ";
}
// cout<<endl;
}
while(q--){
int k,l1,r1,l2,r2;
cin>>k>>l1>>r1>>l2>>r2;
l1--,r1--,l2--,r2--;
int pr1;
if(l1 - 1 < 0) pr1 = 0;
else pr1 = prefix[l1-1][k];
int pr2;
if(l2 - 1 < 0) pr2 = 0;
else pr2 = prefix[l2-1][k];
int FirstSet = prefix[r1][k] - pr1;
int FirstUnSet = (r1 - l1 + 1) - FirstSet;
int SecondSet = prefix[r2][k] - pr2;
int SecondUnSet = (r2 - l2 + 1) - SecondSet;
// cout<<FirstSet<<" "<<FirstUnSet<<" "<<SecondSet<<" "<<SecondUnSet<<endl;
cout<<FirstSet*SecondUnSet + FirstUnSet*SecondSet<<endl;
}
}
signed main() {
ios::sync_with_stdio(0);
cin.tie(0);
int t=1;
cin>>t;
while(t--) solve();
return 0;
}
https://www.codechef.com/viewsolution/80299712
Why do this code give me TLE?
The issue there is the lines
counts1[j]=counts1[j]+[counts1[j][-1]+((l[i])%2)]
counts0[j]=counts0[j]+[counts0[j][-1]+((((l[i])%2)+1)%2)]
If A is an array of length N, doing A = A + [x] in Python takes \mathcal{O}(N) time since it creates a copy of A, appends to it, then assigns the new list to A.
Because of this, your code is actually \mathcal{O}(60N^2),
Since you want to append to the array, just use Python’s inbuilt append function instead, which works in \mathcal{O}(1): this change alone makes your code fast enough, see submission.
You aren’t going to find a blog post on it because it’s not actually anything special or a ‘technique’.
If you can find the prefix sums of one array, you can obviously do it for 2 arrays, 3, arrays, \ldots, 60 arrays, right? That’s essentially what you’re doing here: applying prefix sums on 60 different arrays.
1 Like | 2022-12-05 12:34:19 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5486433506011963, "perplexity": 10913.127972825256}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446711016.32/warc/CC-MAIN-20221205100449-20221205130449-00431.warc.gz"} |
https://www.liliansrl.com/q8lzy8/area-of-triangle-formula-class-10-2b7c2f | area of triangle formula class 10
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If you are at an office or shared network, you can ask the network administrator to run a scan across the network looking for misconfigured or infected devices. The distance of a point P(x, y) from the origin is √x 2 + y 2. Jan 18, 2021 - Collinearity of Three Points, Section Formula and Areas of Triangles, Class 10, Science Class 10 Notes | EduRev is made by best teachers of Class 10. Area of wall = l x b = 30 x 10 = 300 sq. 9 mins. To Register Online Maths Tuitions on Vedantu.com to clear your doubts from our expert teachers and download the Circles formulas to solve the problems easily to score more marks in your CBSE Class 10 Board Exam. CBSE Class 11 Maths Notes : Solution of Triangles, Heights and Distances. Maths formulas for class 10 chapter- Area related to circles Formula . Example 1: If the sides of the triangle are 3 cm, 4 cm, and 5 cm then find the area of the triangle. Find the area of a square plot. As, Area of triangle = × Base × Height To find the area of ΔABC and ΔPQR, draw the altitudes AD and PE from the vertex A and P of ΔABC andΔPQR, respectively, as shown in the figure given below: Now, area of ΔABC = × BC × AD area of ΔPQR = × QR × PE The division by 2 actually comes from the certainty that a parallelogram can be divided into 2 triangles. Area related to circles formula of class 10 maths for CBSE , ICSE,NTSE & Other Board Exam . Find the area of the triangle. By using our site, you And if the length of BD is d then the length of DC will be a – d. Step 5: Now in triangle ABD, by Pythagoras theorem. Sine Ratio. Candidates who are ambitious to qualify the Class 10 with good score can check this article for Notes. Surface Areas and Volume Formulas for Class 10 Maths Chapter 13 Are you looking for Surface Areas and Volume formulas for class 10 chapter 13? Area of ABC / Area of PQR = (AB/PQ) 2 = (BC/QR) 2 = (CA/RP) 2. Refer ExamFear video lessons for Proof. We recommend that you take a look at our YouTube channel, to enter this new world of virtual learn… As we know heron’s formula is √[s(s – a)(s – b)(s – c)], so substituting values in it, Example 2: Using heron’s formula derive formula to find area of equilateral triangle whose side is a, => √[(3a / 2)((3a / 2) – a)((3a / 2) – a)((3a / 2) – a)], Hence area of a equilateral triangle is √3(a2) / 4. Finding area of a triangle from coordinates Our mission is to provide a free, world-class education to anyone, anywhere. Hence, the height of the given triangle is 4 cm. Class 10 is one of the most valuable phases in a student’s life, where one can grow exponentially in his/her aspired field by understanding the concepts and scoring decent marks. In coordinate geometry,If vertices oftriangles are given than we can find it area by formula.Here given proof of formula. Note: Observe that there is a mod, which indicates that, if we got a negative value we should only consider the numerical value as the area can’t be negative. Get the support to practise Maths textbook questions with NCERT Solutions for CBSE Class 10 Mathematics Chapter 6 Triangles at TopperLearning. Step 1: Find area of wall or floor. This is the only regular polygon with three sides. The formula is as follows: The area of a triangle whose side lengths are a, b, (a, b), (a,b), and c c c is given by. In triangle ABC and triangle PQR, if they are similar, then. Therefore, the required area of triangular plot = 1500$\sqrt 3$ sq. Problems on Condition of collinearity of three points . These formulas help you to answer the questions more accurately. 6 mins. The division by 2 actually comes from the certainty that a parallelogram can be divided into 2 triangles. Numerical: Find the distance between the following pairs of points (2, 3), (4, 1) Solution: here x 1 =2, y 1 = 3 , x 2 = 4 & y 2 =1 . You can also write the formula as: ½ x base x height Area of an equilateral triangle = a 2 3 4; (a is the side of triangle) Solutioin: Question 3. Using coordinate geometry, it is possible to find the distance between two points, dividing lines in a ratio, finding the mid-point of a line, calculating the area of a triangle in the Cartesian plane, etc. What is the formula to find the calculate the area of a triangle? An equilateral triangle is a triangle whose all three sides are having the same length. In this topic, we will discover about equilateral triangles and its area. As we know that the area of Triangle is given by; A = $$\frac{base\times height}{2}$$ i.e. Performance & security by Cloudflare, Please complete the security check to access. To Register Online Maths Tuitions on Vedantu.com to clear your doubts from our expert teachers and download the Triangles formulas to solve the problems easily to score more marks in your CBSE Class 10 Board Exam. Solution: To illustrate the use of the coefficients grid or CG, we will calculate each of the three terms in the formula for the area separately, and then put them together to obtain the final value. Step 2: Now if we look at the figure carefully, three different trapeziums are formed such as PQAB, PBCR, and QACR in the coordinate plane. Step 3: Divide area of wall or floor by area of a brick or a tile. Popular pages @ mathwarehouse.com . Now, let us consider a scalene triangle where the lengths of its sides are known but the height is not known. Step 2: Find area of bricks or tiles. 1. Please enable Cookies and reload the page. Finding area of a triangle from coordinates Our mission is to provide a free, world-class education to anyone, anywhere. Multiple Choice Questions for Cbse Class 10 Maths Proofs Involving Triangles I Combination Of Solids Formula Of Surface Area And Volume Of Basic Shape Find the area of a right triangle with a base of 6 centimeters and a hypotenuse of 10 centimeters. NCERT Books for Class 5; NCERT Books Class 6; NCERT Books for Class 7; NCERT Books for Class 8; NCERT Books for Class 9; NCERT Books for Class 10; NCERT Books for Class 11; NCERT Books for Class 12; NCERT … If we know the length of three sides of a triangle, we can calculate the area of a triangle using Heron’s Formula. Free PDF download of Chapter 10 - Circles Formula for Class 10 Maths. There are various methods to find the area of the triangle according to the parameters given, like the base and height of the triangle, co-ordinates of vertices, length of sides, etc. Learn Videos. Need assistance? and around the web . To find the area of a quadrilateral, divide it into two triangles by joining two opposite vertices, find the areas of these triangles and then add them. Area of the triangle is calculated by the formula. Heron (10 A.D. – 75 A.D.), an encyclopedic writer in Applied Mathematics gave a formula for finding the area of a triangle when we know the lengths of all three sides. Numerical: Find the distance between the following pairs of points (2, 3), (4, 1) Solution: here x 1 =2, y 1 = 3 , x 2 = 4 & y 2 =1 . => (1 / 2) [x1(3 – 5 ) + 2(5 – 1 ) + 4(1 – 3)] = 1, Hence, the value of x1 can be both -1 and 1. Example 2: What is the value of x1 whose area of a triangle is 1 of coordinates (x1, 1), (2, 3), and (4, 5)? Question 1. If only 2 sides and an internal angle is given then the remaining sides and angles can be calculated using the below formula: For a given triangle if the altitude of a triangle is ‘h‘ and the base of the triangle is ‘b‘ then the area of a triangle is given as: Step 2: Now draw a horizontal line from point A and a vertical line from point C. Let the point be D where both the lines meet. This is possible only when you have the best CBSE Class 10 Maths study material and a smart preparation plan. The area of the triangle is equal to the area of the parallelograms, both standing on the same base and between the sane parallel lines . Deriving area of an Isosceles Triangle Using Heron’s Formula. Learn Videos. Ans. Reading Time: 7min read 0. SHARES. The angles are either measured in degrees or radians. The formula is varied for different types of triangle, but the most common formula that was used as (Height X Base /2 ) Consider the following program as a sample method – 1, there. Completing the CAPTCHA proves you are a human and gives you temporary access to the web property. Practice MCQ Questions for Cbse Class 10 Maths Proofs Involving Triangles I Combination Of Solids Formula Of Surface Area And Volume Of Basic Shape with Answers to improve your score in your Exams. Given, length of two equal sides of an isosceles triangle … If the perimeter of an equilateral triangle is 90 m, then find its area. In earlier classes, we have studied that the area of a triangle whose vertices are (x1, y1), (x2, y2) and (x3, y3), is given by the expression $$\frac{1}{2} [x1(y2–y3) + x2 (y3–y1) + x3 (y1–y2)]$$. the base multiplies by the height of a triangle divided by 2 and second is Heron’s formula. => Area of ABC = (area of rectangle ABCD / 2), Hence, proved that area of the triangle is (1 / 2) × b × h. Example 1: Find the area of the triangle whose height and base are 6 cm and 5 cm? in order to grasp the concept completely. To assist you with that, we are here with notes. Example to find the area of a triangle, multiply the base by the height, and then divide by 2. In the previous methods, we have seen different conditions, in method 3 if coordinates of the triangle are given then we will see how to found the area of the triangle. Solved Examples For You. How to … Area of Triangle Formula Using Determinants. Example 2: The side-length of a square plot is 5m. Firstly, let’s draw a diagram for a better understanding. C Program to find Area of a Triangle and Perimeter of a Triangle . Education Franchise × Contact Us. Since Area of a trapezium = (1 / 2) (sum of the parallel sides) × (distance between sides), => Area of trapezium PQAB = (1 / 2)(QA + PB) × AB, => Area of trapezium PQAB = (1 / 2)(y1 + y2)(x1 – x2 ) —-(2), =>Area of trapezium PBCR =(1 / 2) (PB + CR) × BC, =>Area of trapezium PBCR =(1 / 2) (y1 + y3 )(x3 – x1) —-(3), =>Area of trapezium QACR = (1 / 2) (QA + CR) × AC, =>Area of trapezium QACR =(1 / 2)(y2 + y3 ) (x3 – x2 )—-(4). In this problem, we have to find the value of ‘x1’, which is X coordinate of point A. BOOK FREE CLASS; COMPETITIVE EXAMS. This podcast is a part of a series for, CBSE Class 10 Maths. There were more than 2 methods here listed below check it out. mts = $\sqrt{150(90)(50)(10)}$ sq. VIEWS. VIEW MORE. Area of Triangle (Heron's Formula) Area of Triangle (SAS Method) Formulas. 14 mins. Trigonometry is a branch of mathematics dealing with relations involving lengths and angles of triangles. 10:00 AM to 7:00 PM IST all days. The base and the corresponding altitude of a parallelogram are 10 cm and 3.5 cm respectively. Now we have to substitute the values in (1 / 2) [x1 (y2 – y3 ) + x2 (y3 – y1 ) + x3(y1 – y2)], => (1 / 2) [1 (2 – 5 ) + 4 (5 – 2 ) + 3(2 – 2)], Hence the area of the triangle is 4.5 sq units. Step 5: Substituting (2), (3) and (4) in (1), => Area of ∆PQR = (1 / 2)[(y1 + y2)(x1 – x2 ) + (y1 + y3 )(x3 – x1) – (y2 + y3 ) (x3 – x2 )], => Area of ∆PQR = (1 / 2) |[x1 (y2 – y3 ) + x2 (y3 – y1 ) + x3(y1 – y2)]|. Your IP: 192.249.126.232 Overview. Overview. Problems on Condition of collinearity of three points . To find its area we require the height of corresponding to a base. A = 22/7 x 7 x 7. h is the height of the triangle. Q 4: Explain the concept of similarity of triangles. Let us discuss the Area of a Triangle formula. Free pdf downloads for maths formulas for class 10 chapter- Area related to circles . Vedantu.com is No.1 Online Tutoring Company in India provides Free PDF of Important Math Formula for Class 6 to 12 CBSE Board Prepared by Expert Mathematics Teacher. Heights and Distances for finding the area of a point P ( x, y ) the! 3 trapeziums 3 } a } { 2 } } { 2 } \ Hence! Of wall or floor triangular plot = 1500\ [ \sqrt { 150 ( 90 ) 3... 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Preparation and score the highest possible marks in Your CBSE Board Exams determinant as Circles for... Covered in high school Mathematics, y ) from the origin is √x +... | 2021-11-29 20:54:04 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 2, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6642553210258484, "perplexity": 933.8427383863149}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964358842.4/warc/CC-MAIN-20211129194957-20211129224957-00433.warc.gz"} |
https://www.oneidentity.com/community/identity-manager/f/forum/31117/change-local-cache-directories-for-1im-tools/75818 | # Change local cache directories for 1IM tools.
Hi,
by default 1IM tools save some cache information under the "%LOCALAPPDATA%\Dell\One Identity Manager" directory, like AssemblyCache, Config, etc. Is there any way to customize these paramaters?
I've tried global.cfg like
<category name="RuntimeDirs">
<value name="Cache">x:\Dell\One Identity Manager\Cache</value>
but with no success. If there is a possibility to customize these parameters, how the application secific path (like %LOCALAPPDATA%\Dell\One Identity Manager\JobQueueInfo\) can be customized?
thank you
Anton
Parents
• I think you cannot change this using the global.cfg.
But you can change the cache directories using the application-specific .config file.
``` <configSections>
...
<section name="runtimedirs" type="System.Configuration.NameValueSectionHandler" />
...
</configSections>
<runtimedirs>
</runtimedirs>
```
In regards to the application-specific directories, these can be changed, if you are referring to the log files by changing, the variable logBaseDir in the globallog.config.
• Hi Markus,
yes, thank you, it works
regards
Anton | 2021-09-26 10:15:11 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.888900101184845, "perplexity": 11567.766729365532}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780057857.27/warc/CC-MAIN-20210926083818-20210926113818-00718.warc.gz"} |
https://www.gradesaver.com/textbooks/math/algebra/algebra-2-common-core/chapter-14-trigonometric-identities-and-equations-14-1-trigonometric-identities-practice-and-problem-solving-exercises-page-908/29 | ## Algebra 2 Common Core
Published by Prentice Hall
# Chapter 14 - Trigonometric Identities and Equations - 14-1 Trigonometric Identities - Practice and Problem-Solving Exercises - Page 908: 29
#### Answer
$$\sec\theta$$
#### Work Step by Step
Simplify $$\cos\theta+\sin\theta\tan\theta$$ by substituting the Tangent Identity: $$\tan\theta=\frac{\sin\theta}{\cos\theta}$$ to obtain $$\cos\theta+\sin\theta\tan\theta=\cos\theta+\sin\theta\bigg(\frac{\sin\theta}{\cos\theta}\bigg)=\cos\theta+\frac{\sin^{2}\theta}{\cos\theta}$$ Now multiply both the numerator and the denominator of the first term by $\cos\theta$ $$=\cos\theta\bigg(\frac{\cos\theta}{\cos\theta}\bigg)+\frac{\sin^{2}\theta}{\cos\theta}$$ $$=\frac{\cos^{2}\theta+\sin^{2}\theta}{\cos\theta}$$ Next, use the Pythagorean Identity: $$\sin^{2}\theta+\cos^{2}\theta=1$$ and finally the Reciprocal Identity: $$\sec\theta=\frac{1}{\cos\theta}$$ to obtain $$\cos\theta+\sin\theta\tan\theta=\frac{1}{\cos\theta}=\sec\theta$$
After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback. | 2018-10-19 14:25:00 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.894615113735199, "perplexity": 846.1163858951326}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-43/segments/1539583512400.59/warc/CC-MAIN-20181019124748-20181019150248-00489.warc.gz"} |
http://www.reference.com/browse/Implicit+price+deflator | Definitions
# GDP deflator
In economics, the GDP deflator (implicit price deflator for GDP) is a measure of the change in prices of all new, domestically produced, final goods and services in an economy. GDP stands for gross domestic product, the total value of all final goods and services produced within that economy during a specified period.
## Calculation
### Measurement in national accounts
In most systems of national accounts the GDP deflator measures the difference between the real (or chain volume measure) GDP and the nominal (or current price) GDP. The formula used to calculate the deflator is:
$operatorname\left\{GDP deflator\right\} = frac\left\{operatorname\left\{Nominal GDP\right\}\right\}\left\{operatorname\left\{Real GDP\right\}\right\}times 100$
Dividing the nominal GDP by the GDP deflator and multiplying it by 100 would then give the figure for real GDP, hence deflating the nominal GDP into a real measure.
It is often useful to consider implicit price deflators for certain subcategories of GDP, such as computer hardware. In this case, it is useful to think of the price deflator as the ratio of the current-year price of a good to its price in some base year. The price in the base year is normalized to 100. For example, for computer hardware, we could define a "unit" to be a computer with a specific level of processing power, memory, hard drive space and so on. A price deflator of 200 means that the current-year price of this computing power is twice its base-year price - price inflation. A price deflator of 50 means that the current-year price is half the base year price - price deflation.
Unlike some price indexes, the GDP deflator is not based on a fixed basket of goods and services. The basket is allowed to change with people's consumption and investment patterns. (Specifically, for GDP, the "basket" in each year is the set of all goods that were produced domestically, weighted by the market value of the total consumption of each good.) Therefore, new expenditure patterns are allowed to show up in the deflator as people respond to changing prices. The advantage of this approach is that the GDP deflator reflects up to date expenditure patterns. For instance, if the price of chicken increases relative to the price of beef, people would likely spend more money on beef as a substitute for chicken. A fixed market basket measurement would miss this change.
In practice, the difference between the deflator and a price index like the CPI is often relatively small. On the other hand, with governments in developed countries increasingly utilizing price indexes for everything from fiscal and monetary planning to payments to social program recipients, the even small differences between inflation measures can shift budget revenues and expenses by millions or billions of dollars.
### United States
The GDP and GDP deflator are calculated by the Bureau of Economic Analysis (BEA).
### United Kingdom
The GDP and GDP deflator series are published by the Office for National Statistics.
## Hedonics
A hedonic regression for a specific good relates the expected price of the good to its characteristics. For example, a hedonic model for a computer relates a computer's price to the value of its processor speed, memory, hard drive capacity, and so on. Hedonic regressions can be used to create hedonic price indexes. These can vary over time as the price structure of the underlying characteristics change.
In recent years, some commentators have expressed concern that the national accounts may overstate spending on computer hardware because of the way the hedonic index and implicit price deflator are used. It is well-known that the prices of a unit of processor speed, a unit of memory, and a unit of hard drive capacity have declined very quickly since 1995. Therefore, the current-year (say, 2003) price deflator for an entire computer - using the hedonic method - is less than one relative to a base year of 1995. This means that when nominal spending on computer hardware is divided by the deflator to give real spending on computers, the number rises. (The "deflator" here is actually an inflator!) From the second quarter of 2000 through the fourth quarter of 2003, the government estimated that real tech spending rose from \$446 billion to \$557 billion, when nominal spending only increased to \$488 billion. Some analysts feel that this overstates the "true" spending on computers by \$72 billion. However, it is also true that this extra \$72 billion captures the increase in value and utility of the computers that were purchased in 2003 as compared to 2000, due to the former's superior quality and capability for the same nominal price as the latter.
## References
### Hedonics
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https://processdesign.mccormick.northwestern.edu/index.php?title=Separation_processes&oldid=3031 | # Separation processes
Authors: Nick Pinkerton, [2014] Karen Schmidt, [2014] James Xamplas, [2014] Emm Fulk, [2015] and Erik Zuehlke [2015]
Stewards: David Chen, Jian Gong, and Fengqi You
Date Presented: February 9, 2014 /Date Revised: February 1, 2014
## Introduction
Essentially all chemical processes require the presence of a separation stage. Most chemical plants comprise of a reactor surrounded by many separators. Separators have a countless number of jobs inside of a chemical plant. A separator can process raw materials prior to the reaction, remove incondensable gases, remove undesired side products, purify a product stream, recycle materials back into the process, and many other jobs that are essential to the process.
Chemical engineers must understand the science of separation and the variety of ways that separation can take place. There are many ways to perform a separation some of these including: distillation, absorption, stripping, and extraction. The science of separation revolves around the presence of two phases that are in contact and equilibrium [1].
Figure 1. Separation methods by property
## Theory
### Vapor-Liquid Equilibrium
Separation processes are based on the theory of vapor-liquid equilibrium. This theory states that streams leaving a stage in a separation process are in equilibrium with one another. The idea of equilibrium revolves around the idea that when there is vapor and liquid in contact with one another they are in constantly vaporizing and condensing. Different components in the mixture will condense and vaporize at different rates. There are three types of equilibrium conditions that can be subdivided into thermal, mechanical and chemical potential categories. These separate equilibrium states are given as:
$T_{liquid} = T_{vapor}$
$p_{liquid} = p_{vapor}$
$chemical potential_{liquid} = chemical potential_{vapor}$
## Distillation
### Flash Distillation
Flash Distillation is one of the simpler separation processes to be employed in a chemical plant. The main premise of flash distillation is that a portion of a liquid feed stream vaporizes in a flash chamber or a vapor feed condenses. Vapor-liquid equilibrium will cause the vapor phase and the liquid phase to have different compositions. The more volatile component of the mixture will compose of a larger portion of the vapor. This simple separation is easy to manufacture but does not result in large degrees of separation.
Flash distillation requires a feed stream that is pressurized and heated and then passed through a valve into a flash drum. The large pressure drop across the valve will result in a partial vaporization of the fluid. Vapor will be removed overhead from the flash drum while the remaining liquid will collect at the bottom of the drum and be removed. Most flash drums will contain an entrainment eliminator which is a screen that prevents liquid from being carried into the vapor effluent. Figure 2 shows a simple overview of the flash distillation process. As shown, there is a heater that flows into a let-down valve where the two-phase flow begins. Variables y and x are the mole fractions of the more volatile component in the vapor and liquid effluents, respectively.
Figure 2. Flash Distillation Flow Diagram
### Column Distillation
Distillation columns are the most widely used separation technique used in the chemical industry, accounting for approximately 90% of all separations [1]. Distillations in columns consist of multiple trays that each act at their own equilibrium conditions. Large columns are able to perform complete separations of binary mixtures as well as more complex multi-component mixtures.
### Stages
Columns are separated into stages by the presence of trays. These trays allow for vapor-liquid contact and equilibrium to occur. Typically, the more stages in a column, the larger separation that can be achieved. There are many different types of trays that can be used in a column.
#### Sieve Trays
The simplest and least expensive tray type is the sieve tray which is a sheet of metal with holes punched into it to allow vapor flow. Sieve trays can have different hole patterns and sizes that will affect the tray efficiency and flow rates.
#### Bubble-Cap Trays
Bubble-cap trays consist of a weir around each hole in the tray which is covered with a cap that has holes or slots to allow vapor passage. Entrainment is about three times larger than a sieve tray. Bubble-cap trays require larger tray spacing than sieve tray design. Bubble-cap trays have been known to have problems with coking, polymer formation, or high fouling mixtures. Recently, very few new bubble-cap columns are being built due to the expense and marginal benefits. However, engineers will likely encounter bubble-cap columns still currently in operation.
#### Flow Patterns
Cross flow columns are the most common pattern for distillation columns. For liquid flows between 50 and 500 Gal/min, a cross flow column is appropriate. When liquid flow is increased above 500 Gal/min, an engineer should consider designing a double pass or multi-pass column. This will reduce the liquid gradient on the tray and reduce the downcomer loading [1].
### Column Sizing
Column height will be dependent on the amount of trays required and the spacing between the trays. Normally, tray spacing of 0.15 m to 1 m is used. For columns, above 1 meter in diameter, 0.5 m can be used as an initial estimate.
Column diameter is influenced by the vapor flow rate in the column. The trays can not have excess liquid entrainment or high pressure drops; therefore, vapor velocity in the column must be maintained at a reasonable level.
An equation based on the Souders and Brown equation can be used as an estimate for the max allowable superficial vapor velocity,
$\hat u_v = (-0.171l_t^2 + 0.27l_t - 0.047){\frac{\rho_L - \rho_v}{\rho_v}}^{1/2}$
where $l_t$ is the plate spacing in meters, $\rho_L$ is the density of the liquid stream, and $\rho_V$ is the density of the vapor stream.
Column diameter, $D_c$, can then be estimated using the relation,
$D_c = \sqrt{\frac{4\hat{V_w}}{\pi\rho_v\hat{u_v}}}$
where $\hat{V_w}$ is the maximum vapor rate in kg/s [2].
### Distillation Applications
Distillation is a process that can be implemented in various scales. There is both laboratory scaled distillation as well as very large industrial distillation. Other applications for distillation include food/alcohol processing and herb distillation for the perfume and medical industries. Typically laboratory scaled distillation occurs in batches whereas industrial distillation (e.g. fractional distillation of crude oil) occurs continuous with a constant distillate and bottom effluent streams.
Some applications of distillation are concerned the top stream only, some the bottom stream only and others both streams can be used for future products. In alcohol distillation for example, the water that is separated from the ethanol/water binary solution is discarded as waste water. In fractional distillation of crude oils, the heavy hydrocarbons at the bottom of the column are collected and sold along with the light hydrocarbons that appear in higher side draws [1].
### Example Case: Ideal Distillation
Assume an equimolar mixture flowing at 10 mol/s of 20 mol% n-pentane, 30 mol% n-hexane, and 50 mol% n-heptane. Separate the mixture into 3 products: 99% pure n-pentane, 99% pure n-hexane, 99% n-heptane. Assume the feed and products are all liquids at the bubble points. There are two process alternatives to consider in this example. The direct sequence removes the most volatile species, pentane, in the first column, and then separates hexane and heptane in the second column. The indirect sequence separates the heaviest product, heptane, and then separates pentane from hexane in the second column. This example will consider the direct sequence. Next, we must decide if these species exhibit fairly ideal behavior during distillation. Since the n-alkanes have very similar properties, it is safe to assume they will display close to ideal behavior. The next step is to look up the boiling points of the 3 species. In this case, the normal boiling points of pentane, hexane, and heptane are 309 K, 342 K, and 372 K, respectively. Also, it is a good idea to look up relative volatilites, to further verify near-ideality of the mixture, but also to obtain the information necessary for the Underwood method, which we will employ to obtain a solution. The next step is to write out material balances based on molar flows and the design specifications. They go as follows:
$\mu_I(nC5) + \mu_{II}(nC5) = 2 mol/s$
$\mu_I(nC6) + \mu_{II}(nC6) + \mu_{III}(nC6) = 3 mol/s$
$\mu_{II}(nC7) + \mu_{III}(nC7) = 5 mol/s$
$\mu_I(nC5) = 99\mu_I(nC6)$
$\mu_{II}(nC5) = (5/990)\mu_{II}(nC6)$
$\mu_{II}(nC7) = (5/990)\mu_{II}(nC6)$
$\mu_{III}(nC7) = 99\mu_{III}(nC7)$
where $\mu$ represents the molar flow, and the subscript represents the product stream.
Solving this system of equations:
$\mu_I(nC5) = 1.985\ mol/s$
$\mu_{II}(nC5) = 0.015\ mol/s$
$\mu_I(nC6) = 0.020\ mol/s$
$\mu_{II}(nC6) = 2.930\ mol/s$
$\mu_{III}(nC6) = 0.050\ mol/s$
$\mu_{II}(nC7) = 0.015\ mol/s$
$\mu_{III}(nC7) = 4.985\ mol/s$
At this point we have enough information to use Underwood's method to estimate the minimum vapor flows in the column. The following three equations are used in Underwood's method:
$\sum_i \frac{\alpha_{ik}}{\alpha_{ik}-\phi}f_i = (1-q)F$
$(R_{min}+1)D = \sum_i \frac{\alpha_{ik}}{\alpha_{ik}-\phi}d_i = V_{min}$
$\bar R_{min}B = -\sum_i \frac{\alpha_{ik}}{\alpha_{ik}-\phi}b_i = \bar V_{min}$
where $\alpha_{ik}$ is the relative volatility of species $i$ to species $k$, $f_i$ the molar flow of species $i$ in the feed, $q$ the fraction of the feed that joins the liquid stream at the feed tray, $F$ the total molar flow of the feed, $D$ the molar flow of the distillate, $R_{min}$ the minimum reflux ratio $(=L_{min}/D)$, $d_i$ the molar flow of species $i$ in the distillate, $V_{min}$ the minimum vapor flow possible in the top section of the column to accomplish the desired separation, $\bar R_{min}$ the minimum reboil ratio $(=\bar V_{min}/B)$, $b_i$ the molar flow of species $i$ in the bottoms product, and $\bar V_{min}$ the minimum vapor flow in the bottom section of the column. The final variable, $\phi$, will be solved for using the first Underwood equation, and it's value will be decided based on the relative volatilities of the key components in the column.
So, after solving the first Underwood equation, we get two values for $\phi$, 3.806 and 1.462. Because 3.806 is between the relative volatilities of the key components, we will substitute that value for $\phi$ into the second Underwood equation. Doing so for both columns gives $V_{min} = 6.4\ mol/s$ for the first column and $V_{min} = 8.9\ mol/s$ for the second column, for a total minimum vapor flow of 15.3 mol/s. The process would then be repeated for the indirect sequence, and the decision for which process to use would be justified by the process with the overall minimum vapor flow [3].
## Absorption
### Description of Absorption
Another separation process used in industry is absorption, which is used to remove a solute from a gas stream. It accomplishes this by contacting the gas mixture with a liquid solvent that readily absorbs the undesirable components from the gas stream, purifying the gas stream. This separation process is determined by the inputs of the liquid flow rate, temperature, and pressure.
The absorption factor, which can be determined mathematically, determines how readily a component will absorb in the liquid phase. The absorption factor of component i is
$A_i=L/K_iV$
where $L$ is the liquid flow rate entering the column, $V$ is the vapor flow rate entering the column, and $K_i$ is the vapor/liquid equilibrium ratio for component i [4]. Higher absorption factors result in higher absorptivity into the liquid and a decrease in the number of trays required for separation, however a diminishing return occurs after the absorption factor is greater than 2.0. An absorption factor of 1.4 is most commonly used.
In general absorption can be seperated into two overarching categories, physical and chemical absorption. In physical absorption, the unwanted solute in the gas is absorbed into the liquid phase because solubility of the component is higher in the liquid phase than the gas phase. In chemical absorption the solute is removed from the gas via a reaction with the solvent, this reacted product is then transported into the liquid phase[7]. There are two types of chemical absorption reversible and irreversible. Generally reversible chemical absorption is preferred as the solvent can be put through a stripper and regenerated so it can be recycled back to the absorption process[1].
### Absorption Apparatus
There are five major apparatus used for absorption in industrial application. These five pieces of equipment are spray absorbers (or towers), ejector (venturi) scrubbers, packed columns, trayed columns, and film absorbers[8].
#### Spray Tower vs Ejector Scrubber
In both spray tower and the ejector scrubber nozzles are employed to produce small solvent droplets. These small droplets increase the surface area of the liquid to gas contact allowing for the maximum amount of mass transfer to occur between the gas mixture and the liquid. The major difference between the two nozzle equipment designs is the configuration and type of nozzles. In the ejector scrubber shown in Figure 3 there is a single nozzle that is generally a higher pressure spray nozzle that produces finer solvent drops allowing for an even greater amount of mass transfer enabling better physical absorption[8].
Figure 3. Ejector Scrubber (US EPA, 2006)
Spray towers on the other hand generally have many nozzle at different heights where the liquid solvent will be sprayed out of to contact the gas running through the tower. This design is used in order to ensure the gas contacts the liquid as throughout the tower. These nozzles are lower pressure than a ejector scrubbers nozzle and thus physical mixing is worse in this configuration. Since physical mixing is generally worse in this configuration it is usually used in conjunction with a chemical absorption process. The other major difference between the ejector scrubber and the spray tower is that gas and liquid flow is cocurrent in the former while it is countercurrent in a spray tower. A spray tower absorber is shown below in Figure 4[8].
Figure 4. Spray Tower Absorber (US EPA, 2006)
#### Tower Type Absorption Apparatus
Packed column absorbers and tray column absorbers have very high efficiencies for the removal of an unwanted solute in the gas stream. The major disadvantage a trayed column has when compared to a packed column is the pressure drop. The pressure drop in a packed column is generally very low, whereas in between each tray of a trayed column pressure drop can be quite large. However the advantages inherent to trayed columns become clear when one needs the solvent to have a high concentration of the component to be removed from the gas stream. This is most important in the case where there is a very low concentration of the component in the gas stream and the specification states the solvent must contain a high concentration of that component. In this case the flow rate of the solvent may not be high enough for a packed column, however in a trayed column the solvent flow rate can be near zero for operation[8]. Packed and trayed column internals are very similar to the setups found in the respective distillation columns.
For a trayed column the plate efficiency can be calculated using O'Connell's Correlation which invovles the Henry's Law constant, total system pressure, and solvent viscosity at the operating temperature[2].
$x=0.062*\frac{\rho_s*P}{\mu_s*H*M_s}$
where $x$ is the tray efficiency, $\rho_s$ is the density of the solvent in $kg/m^3$, $P$ is the total pressure of the system in $N/m^2$, $\mu_s$ is the solvent's viscosity in $mNs/m^2$, $H$ is the Henry Law constant in $1/(Nm^2*(mol fraction))$, and $M_s$ is the molecular weight of the solvent.
A packed towers height can be determined using the equations below when concentration of solute is below 10% so that the assumption that the flow of gas and liquid will be essentially constant throughout the column holds[2]. The height of packing $Z$ is given by the following equation:
$Z=\frac{L_m}{K_G*a*P}*\int\limits_{y_2}^{y_1} \frac{dy}{y-y_e}\,$
where $P$ is the total pressure, $a$ is the interfacial surface area per unit volume, $y_1$ and $y_2$ are the mol fractions of the solute in the gas stream at the bottom and top of the column respectively, $G_m$ is the molar gas flow rate per unit cross-sectional area, and $y_e$ is the mole fraction of solute in the gas that would be in equilibrium with the liquid concentration.
The first half of the equation before the integral can be called the height of an overall gas-phase transfer unit $H_G$ and the second part of the equation is the number of overall gas-phase transfer units or $N_G$. Using these definitions the above equation can be simplified to
$Z=H_G*N_G$
These equations assist in sizing an absorption column[2].
#### Film Absorber
The final absorber the film absorber is generally used in the case where the heat of absorption must be removed. The film absorber operates by sending the gas and solvent through a heat exchanger where the solvent creates a thin film on the walls of the tubes and the gas flows through the interior allowing for solute transfer. The good heat transfer present in a film absorber makes it preferable for situations where low temperatures are required for a high recovery of the solute[8].
### Industrial Absorption Processes
An industrial example is lean oil absorption, which is used to separate nitrogen and other impurities from natural gas. A lean oil is contacted with low quality natural gas, and the methane is selectively absorbed by the lean oil, leaving the impurities behind. The methane is subsequently regenerated from the rich oil as high quality natural gas [9].
Other common industrial practices of absorption come from sour gas treatment. Amine gas treating is used to remove hydrogen sulfide or carbon dioxide from gas streams via a reversible chemical absorption. In amine gas treating the sour gas is fed to the bottom an absorber where amine solution is fed to the top along with any necessary make up water. The sour gas components are absorbed into the amine via a chemical absorption method. Sweet gas leaves the top of the absorber whereas the amine out of the bottom, now rich with acidic components is sent to a regenerator where the acid gas components are stripped and the acid gas is generally sent to a flare whereas the amine now lean again is recycled back into the first absorber[10]. Figure 5 below shows the typical setup of an amine plant. Another type of sour gas treatment that uses absorption is Merichems LO-CAT process which uses a chelated iron to remove hydrogen sulfide from feed gas in the absorption column[11].
Figure 5. Amine Gas Treating Plant Schematic
## Stripping
This process separates solutes from solvents (often after absorption, to purify the solvent so that it can be recycled to an absorber). Stripping will depend on the vapor and liquid flow rates, as well as the temperature and pressure of the column. There is a temperature drop down the column, so columns generally have either an increased operating temperature or decreased operating pressure.
The stripping factor of component i is
$S_i=K_iV/L$
where $K_i$ is the vapor/liquid equilibrium ratio, $V$ is the vapor flow rate entering the column, and $L$ is the liquid flow rate entering the column, will determine how much of solute i will be stripped from the liquid into the vapor phase [4]. The usual range for the stripping factor is between 1.2 and 2.0, with a stripping factor of 1.4 being most economic.
An example of stripping in industry is the deodorization of food items such as oils. The oil is heated and allowed to trickle down the column while steam flows up from the bottom of the column. At the vapor-liquid interface, volatile components of the oil transfer to the steam and are carried off the top of the column, leaving a purified oil product [12].
## Bioseparations
### Importance
As our ability to manipulate and engineer biological systems improves, biological products are becoming an increasingly important source of therapeutics and fuels. The production of fuels from biomass via either the enzymatic breakdown of a feedstock or the secretion of usable lipids from algae is a promising new energy source. Additionally, enzymes, antibodies and other therapeutic proteins have been applied to the treatment of a wide range of diseases. Although each process requires its own set of separations, all follow the same basic format: separation of biomass, product isolation, and product purification. This section will provide examples of unit operations in each step [13,14]. Ultimately, the choice of separation process and unit operations will depend on the specific process and product. The descriptions below are examples of the most common bioseparation operations within the general platform.
Bioprocesses begin with fermentations or growth operations. In biofuel production processes, this may involve growing algae or breaking down corn or cellulosic biomass. For the production of therapeutics, mammalian or bacterial cells may be grown in a fermentor and the product secreted into the supernatant or harvested from the cells.
### Biomass Separations
After fermentation and product production, the solid biomass must first be separated from the desired product. If the product is secreted from the cells, this can be done immediately after fermentation ends. If the product is not secreted, the cells must first be lysed. Cell lysis is the process of lysing, or breaking, the cell in open. Mechanical lysis is the simplest, and involves physically breaking the cell either by mashing (think mortar and pestle) or blending the cells into a homogenous solution in a homogenizer. Chemical lysis is another method, achieved by introducing an osmotic shock or chemically degrading the cell membrane. Additional separation can be achieved by flocculation, which is the process of aggregating biomaterial by charge neutralization or bridging. These larger complexes are easier to separate from smaller molecules [13].
The next step is removing the unwanted biomass from the product in solution. Separation by centrifugation or sedimentation are the most common, although filtration is sometimes also used for processes where a biomass cake is desired. Both methods utilize density differences to separate the product from the homogenous solution [2].
#### Sedimentation
Sedimentation relies purely on the force of gravity, while centrifugation speeds the settling process by subjecting the cells to a centrifugal force. Sedimentation in a settling tank is the simplest method of solid-liquid bioseparation. In this process, biomass in a tank is simply allowed to settle to the bottom over time. While this process is inexpensive and can separate out large volumes of biomass, it generally requires long time periods and is only mostly in very large-scale processes where active centrifugation is difficult.
#### Centrifugation
Centrifuges are widely utilized across many processes, and thus a wide variety of scales and designs have been developed. Disk-stack centrifuges, in which the solid phase is deposited onto “shelves” in the center of the spinner and liquid phase is pushed to the outside, are one of the most commonly used centrifuges in industry. They are especially suited to biomass separation processes because they can be built on a large scale and are ideal for separating fine solids from liquids.
Fig. 6: Diagram of a disk-stack centrifuge [2]
Tubular bowl centrifuges are also common and can reach separation efficiencies of up to 90%. Heavier products accumulate along the sides of the bowl, while the light phase flows out the top. They separate products by can be used both to separate solids from liquids and immiscible liquids, such as and oil product and an aqueous broth [2].
Fig. 7: Diagram of a tubular bowl centrifuge centrifuge [2]
Centrifugation scale-up is made easier by sigma analysis, which allows for the estimation of appropriate feed rates for different size centrifuges. The sigma factor is dependent on the inner and outer radius of the centrifuge, the angular velocity, and the sedimentation velocity of the solid particles being separated. It can be through of as the characteristic cross-sectional area with units of [length]2. The sedimentation velocity can be calculated by
$v_g={\frac{2a^2(\rho-\rho_0)}{9\mu}}g$
where $v_g$ is the sedimentation velocity, $a$ is the cell or biomass particle diameter, $\rho$ is the particle, $\rho_0$ is the fluid density, and $\mu$ is the fluid viscosity. The fluid flow $Q$ can be estimated by
$Q=(v_g)(\Sigma)$.
The equality
${\frac{\Sigma_1}{\Sigma_2}}={\frac{Q_1}{Q_2}}$
can be an easy way to estimate equivalent flow rates between a small-scale centrifuge 1 and larger centrifuge 2 [13].
#### Example: Centrifugation Scale-up
You are trying to separate a cell of radius 0.4 $\mu$m with a density of 1.05 g/cm3 from broth of mostly water (density of 1 g/cm3 and viscosity of 0.01 g/cm s). The sigma factor of the centrifuge you are using is 1 x 106 cm2. A] What volumetric flow rate should you use? B] If you want to scale up the process to a centrifuge with $\Sigma$ = 3 x 106 cm2, what flow rate would you use in the larger centrifuge?
Solution: A] Using the equation for $v_g$, and being mindful of units, the sedimentation velocity equals 1.74 x 106 cm/s. The flow rate, then, equals
$Q=(1.74 x 10^-6)(1,000,000) = 1.74 cm^3/s = 0.104 L/min$.
B] Keeping in mind that for the same process, $v_g1 = v_g2,$ and rearranging the sigma factor equality, the new flow rate is
$Q_2 = {\frac{\Sigma_2 x Q_1}{\Sigma_1}} = {\frac{(3 x 10^6)(0.104)}{1 x 10^6}} = 0.313 L/min$
### Product Isolation
Liquid-liquid separation, to extract the product from the aqueous phase, is much less straightforward than liquid-solid extraction. Many methods - especially extraction, adsorption, filtration, and precipitation - are similar in principle to operations found in other, non-biological separations. The exact separations used depend on the nature of the product and the scale of the process. Particular care needs to be taken with protein products because of their instability, and the selection of an appropriate solvent or adsorbent is crucial to a successful process [13].
These processes are nearly identical to their non-biological counterparts, and their description is left to other sections.
### Product Purification
The final steps of protein purification and polishing remove any remaining contaminants and bring the concentration of product to an appropriate value for applications. Purification processes for food-grade and medical products can be extensive, as sterility and high purity are essential. Purification in fuel-producing processes may be less extensive, depending on the process. Chromatography and crystallization are two common steps in purification and are especially used in industrial scale protein production.
Chromatography is similar to adsorption in that it relies on differences in affinity between solutes and a solid surface. A solution is eluted through a column containing a solid resin with various affinities for the substances in solution. In adsorption, the solutes are evenly saturated throughout the column. Chromatography differs in that solutes are deposited in bands on a resin phase before the column is flushed with an elution solvent specific, as shown in Figure 8. Different bands are eluted at different times depending on the size of the solute (as in gel filtration chromatography) or the affinity of the solute for the resin (as in ion exchange chromatography).
Fig. 8: Illustration of product bands in an elution chromatography column [14]
In gel filtration chromatography, small molecules are "trapped' by the porous resin and take longer to flow through the column. Larger products will elute first, and this operation is often used when there is a distinct difference in size between the desired product and other solutes. In ion-exchange chromatography, the resin beads are charged either positively (in cation exchange) or negatively (in anion exchange) and will bind to different solutes depending on their charge. The pH of the elution buffer is change to force a specific solute to wash out, depending on whether the pH of the buffer is above or below the isoelectric point of the solute [14]. This is especially useful for the separation of protein product (including antibodies), nucleic acids, and other charged molecules. When the solutes have sufficiently different isoelectric points, the pH of the buffer manipulated to affect the solute charge and force the product to elute while the solute remains preferentially bound to the resin, or vice versa [13].
Crystallization, or the formation of solute crystals from a solution, is especially useful in biomolecule separations because it is possible to obtain a 99.9%+ product purity. In crystallization, a diluent is added to the homogeneous solution that reduces the solubility of the product to the point that it “falls out” of solution and crystallizes. It is similar to precipitation but results in the formation of crystals rather than unordered aggregates.Crystallization can be used on a laboratory scale for determining protein structure, on on the industrial scale for antibody and therapeutic protein productions. Batch crystallizers are often used in industry because of their simplicity and inexpensiveness compared to continuous crystallization [13].
## Other Separation Processes
### Extraction
Liquid-liquid extraction is a process for components with overlapping boiling points and azeotropes. The process requires a solvent such that some of the components of the mixture are soluble, and then the components will be separated based on this solubility in the liquid. This process can operate at moderate temperatures and pressures, so is not very energy intensive. However, a distillation column is required to extract the solvent for recycle. More recently, supercritical fluids have replaced liquid solvents in some processes for L/L extraction, due to the solute’s ability to more rapidly diffuse through them. The issue with these fluids, however, is that they must be operated at extremely high pressures and temperatures, increasing both capital and operating expenses of the process [4].
### Crystallization
This process recovers solutes that have been dissolved in solution. The resulting product is in the solid phase. Depending on the material properties of the solute and solvent, the solute is recovered by precipitation after cooling, removal of solvent, or adding precipitating agents. Crystallizers are designed based on phase equilibria, solubilities, rates and amounts of nuclei generated, and rates of crystal growth. Every crystallization process is a unique system, so plant evaluation is usually required before complete implementation. Crystallization can be performed in both batch and continuous processes, and design features can control crystal size to an extent [4].
### Membrane Separation
This separations process uses selectively permeable membranes to separate components in a mixture. Typically, one of the components will freely pass through the barrier while the other components will not. The stream that passes through the membrane is the permeate and the stream that does not pass is the retentate. The driving force behind this separation is a pressure gradient. Membrane separation is beneficial because it can separate mixtures at the molecular and small particle level. Furthermore, there is no phase change required so the energy input is low. Limitations of this process include achieving high product purity, incompatibility with certain stream components, low operating temperature, and low flow rates. Although membrane separation is generally not scaled up, examples of scaled-up membrane separation include seawater desalination and hydrogen recovery [4].
Adsorption involves an adsorbent and adsorbate. The adsorbent is typically a solid, and will typically separate the adsorbate from the stream. This process usually includes a desorption step that regenerates the adsorbent for further use. Raising the temperature or increasing the concentration of the adsorbate can reverse the adsorption process. Although the recycle of the adsorbent is a very economic design feature, the downside of this step is that it results in a cyclic process, which introduces complexity to the overall process. Industrial applications of this process are for bulk separations and gas purification. The adsorption/desorption process in these situations involves a large amount of heat transfer, which design engineers must take into account when sizing and selecting equipment material [4].
These separations use external force fields or temperature gradients to separate responsive molecules or ions. The use of these processes is fairly limited to a few specialized industrial applications [4].
### Settling and Sedimentation
In settling processes, solid particles or liquid drops are separated from a stream by gravity. The stream can be in either the liquid or gas phase. For vapor-liquid mixtures, flash drums are generally used to separate the mixture. The velocity of the vapor must be less than the settling velocity of the liquid drops for this separation to occur. For liquid-liquid separation, the horizontal velocity of the fluid must be low enough to allow the low-density droplets to rise to the interface and the high-density droplets to move away from the interface and coalesce. In sedimentation, the result of the process is a more concentrated slurry. Typically a flocculating agent is used to aid in the settling process. One way to perform this separation is to use a cone-shaped tank with a slowly revolving rake that scrapes and moves the thickened slurry to the center of the cone for removal [4].
### Flotation
Flotation is a process designed for specific solid-solid mixtures. It works by generating gas bubbles in a liquid that attach to selected solid particle. Afterwards, the particles rise to the liquid surface where they are removed by an overflow weir or mechanical scraper. The separation depends on the surface properties of the particles and its preference to attach to the gas bubbles. To meet the necessary requirements of the flotation process, a number of additives can be used to control things like the pH of the liquid-solid mixture, the activity of the solid surface, and the froth that can assist in separation. The bubbles can be produced by gaseous dispersion, dissolution, or electrolysis of the liquid [4].
### Centrifugation
This process is similar to external field separation in that an external force field is applied to separate a mixture. When gravity separation is too slow due to particle densities, particle size, settling velocity, or the formation of an emulsion, centrifugation is commonly used. Centrifugal force increases the total force acting on the particle and results in faster separation times. This process is generally used to separate solids from liquids, however it can also be used to separate two liquids with very different densities [4].
### Drying
Drying is performed to remove liquid from a liquid-solid mixture and produce a dry solid. Water is most often the liquid removed, but organic liquids are removed from solids on occasion as well. The heat required to vaporize the liquid is usually obtained by a series of gas-solid contacting devices. Feed condition and temperature sensitivity of the solid dictate the type of contacting device that is used. There are two groups of dryers that differ by the dependence of either mechanical means or fluid motion for gas solid contact. Another feature of dryers is to use either direct (hot gas) or indirect (conductive surface) heating [4].
### Evaporation
Evaporators separate solvents from a solution by evaporation. The difference between evaporation and distillation is that evaporation requires the solute be nonvolatile. Because of this, a high separation can be achieved with one stage. Evaporators are essentially reboilers, so evaporation is a very energy-intensive process with a high thermal economy [4].
### Filtration
Filtration is a process that separates a mixture of solid in a liquid or gas by passing the mixture through a porous medium in which the particles do not pass. Filtration is done by either cake filtration (particles found on the surface of the filter) or depth filtration (particles found within the filter). Cake filtration is generally performed with a cloth as the filtration medium [4].
## Conclusion
Separation is a key part of most chemical processes, and there is a great variety of techniques to perform separation of compounds based on size, volatility, charge, and many other features. A common technique with which the process engineer should be familiar is distillation, but he or she should also be aware of the other available options. Some techniques may be less expensive, less energy-intensive, or more effective than distillation, depending on the specific separation problem. Therefore, the separation strategy should be carefully considered.
## References
1. Wankat, P.C. (2012). Separation Process Engineering. Upper Saddle River: Prentice-Hall.
2. Towler, G.P. and Sinnot, R. (2012). Chemical Engineering Design: Principles, Practice and Economics of Plant and Process Design.Elsevier.
3. Biegler, L.T., Grossmann, L.E., and Westerberg, A.W. (1997). Systematic Methods of Chemical Process Design. Upper Saddle River: Prentice-Hall.
4. Peters, M.S. and Timmerhaus, K.D. (2003). Plant Design and Economics for Chemical Engineers, 5th Edition. New York: McGraw-Hill.
5. Seider, W.D., Seader, J.D., and Lewin, D.R. (2004). Process Design Principles: Synthesis, Analysis, and Evaluation. New York: Wiley.
6. Turton, R.T., Bailie, R.C., Whiting, W.B., and Shaewitz, J.A. (2003). Analysis, Synthesis, and Design of Chemical Processes Upper Saddle River: Prentice-Hall.
7. Danckwerts P (1965) The Absorption of Gases in Liquids. Pure and Applied Chemistry UK 10:625-642.
8. Schmidt Eberhard (2012) Waste Gases, Separation and Purification. Ullman’s Encyclopedia of Industrial Chemistry Germany 2:174-181.
9. Lean Oil Absorption. PetroGas Systems Web site. Available at: http://petrogassystems.com/technology/natural-gas-processing-and-dew-point-control/lean-oil-absorption. Accessed February 19, 2014.
10. US 4080424, Loren N. Miller & Thomas S. Zawacki, "Process for acid gas removal from gaseous mixtures", issued 21 Mar 1978, assigned to Institute of Gas Technology
11. Merichem Gas Technologies. ®LO-CAT PROCESS available at http://www.merichem.com/images/casestudies/Desulfurization.pdf Accessed 6 Feb. 2015.
12. Stripping Column. Alfa Laval Web site. Available at: http://www.alfalaval.com/solution-finder/products/soft-column/Documents/Stripping%20Column.pdf. Accessed February 19, 2014.
13. Harrison, R.G., Todd, P., Rudge, S.R. and Petrides, D.P, (2003). Bioseparations Science and Engineering. New York: Oxford University Press.
14. Belter, P.A., Cussler, E.L., Hu, W.-S, (1998). Bioseparations: Downstream Processing for BIotechnology. New York: John Wiley. | 2021-01-26 03:43:55 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 96, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5763881802558899, "perplexity": 1706.1463125115695}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610704795033.65/warc/CC-MAIN-20210126011645-20210126041645-00161.warc.gz"} |
http://physics.aps.org/articles/v2/44 | # Viewpoint: Observing unification on a grand scale
, Michigan Center for Theoretical Physics (MCTP),Department of Physics, University of Michigan, Ann Arbor, MI 48109, USA
Published May 26, 2009 | Physics 2, 44 (2009) | DOI: 10.1103/Physics.2.44
#### Astrophysical probes of unification
Asimina Arvanitaki, Savas Dimopoulos, Sergei Dubovsky, Peter W. Graham, Roni Harnik, and Surjeet Rajendran
Published May 26, 2009 | PDF (free)
Maurice Goldhaber placed an early limit on proton decay, noting that if protons had a lifetime shorter than $1017$ years, you would “feel it in your bones.” That is, the radiation from these decays in your body would provide a fatal radiation exposure. The limits on proton decay have greatly improved since then. For some decay modes, the limits from the Super-Kamiokande experiment now exceed $1033$ years [1], a limit only made possible by observing a tank of water with approximately $1034$ protons in it. Why look so hard for such a rare process? Decay rates are suppressed by the masses of the particles responsible for mediating the decay. So, long lifetimes probe large energy scales.
For example, in the case of the proton, the symmetry that is responsible for its stability—conservation of baryon number—may only be approximate. Just such a violation of baryon number occurs at scales near $1016GeV$ in “grand unified theories” (GUTs), theories that seek to unify electromagnetism, the weak and strong forces, but not gravity. This enormous energy scale translates into proton lifetimes close to the current bounds. So, if a very rare proton decay is observed, we will be probing high energy scales, indeed.
Work by Asimina Arvanitaki of the University of California, Berkeley, and colleagues at Stanford University in the US, appearing in the current issue of Physical Review D [2], points out that in a broad class of grand unified theories, there are additional ways to probe this large energy scale. Their starting point is supersymmetric theories. These theories posit a doubling of those particles that comprise the standard model, with far reaching consequences: these theories contain an excellent candidate for dark matter [3] and can stabilize the weak scale ($Mw∼100GeV$) against quantum corrections that might drag it up to the Planck scale ($Mpl∼1018GeV$). Perhaps most significantly, these theories quantitatively modify how the strength of the forces change with energy. If the strength of the known forces (excluding gravity) are extrapolated to high energies using calculations within the standard model, nothing particularly surprising occurs. With supersymmetry, this changes. In the simplest supersymmetric extension of the standard model, the “minimal supersymmetric standard model,” the lines parameterizing the strength of the three known forces meet at a single point, corresponding to an energy of $2×1016GeV$ [4]. At this energy, the known forces might unify. Thus supersymmetric theories provide a natural home for grand unification.
There is no guarantee that nature has chosen the simplest of supersymmetric theories, and minor modifications in the theory can make a big difference in its observable consequences. Arvanitaki et al. demonstrate that certain supersymmetric GUTs have exciting predictions in three a priori unrelated realms: the Large Hadron Collider (LHC), big bang nucleosynthesis (BBN), and cosmic rays. Combining the measurements from these sources may help to establish physics that operates at very high energies indeed (see Fig. 1).
It is the unique nature of dark matter in these theories that makes cosmic rays an important way to probe them. In these theories the dark matter can decay. This is in contrast to the simplest supersymmetric GUT, where the dark matter is comprised of the lightest supersymmetric particle, which is taken to be absolutely stable. Its infinite lifetime is ensured by a symmetry called R parity. However, as Arvanitaki et al. point out, minor modifications to the GUT could make the symmetry that ensures the longevity of the dark matter an approximate one. Then, just as violation of baryon number might induce proton decay with a lifetime of $1033$ years, the dark matter might naturally decay with a lifetime of roughly $1019$ years. Clearly, most of the dark matter will not have decayed—the universe is only $1010$ years old—but the tiny fraction that does decay does so spectacularly, with decay products possessing hundreds of $GeV$ in energy. So, $1019$ years turns out to be an important number, for as the authors emphasize, there are a variety of existing and upcoming cosmic-ray experiments that are capable of observing energetic decay products of dark matter, should it possess this lifetime. Depending on the precise route that dark matter takes to decay, experiments might observe an excess of gamma rays, neutrinos, positrons, or antiprotons. While the interpretation of these experiments can be complicated by astrophysical backgrounds, there is hope that features in the spectra of these decay products might be enough to conclude that it is truly the decay of dark matter that is being observed. Recent data from the PAMELA and Fermi experiments [5], e.g., have already caused quite a buzz, even if their interpretation remains unclear. If multiple dark matter decay channels are observed, the relative rates might be a window onto the details of the grand unified physics responsible for their decay.
Big bang nucleosynthesis also turns out to be an important probe of this class of grand unified theories. The understanding of how light elements were synthesized in the furnace of the early universe is one of the great successes of the big bang theory. While most heavy elements are synthesized chiefly in stars, significant production of light elements (e.g., deuterium, helium, lithium) occurred within the first three minutes following the big bang, when temperatures were $TBBN∼MeV$. And even though the production (or destruction) of these elements has continued since then, it is possible to look in regions of space where minimal stellar processing takes place. One can compare the observed abundances of elements to the predictions made by theory, which depend sensitively on only one parameter, the ratio of the number of baryons to the number of photons in the universe. The predictions for multiple elemental abundances all agree for a single input—strong evidence for the big bang theory. (The experimental value for this ratio also agrees with a separate determination that comes from detailed measurements of the cosmic microwave background radiation).
However, the agreement for one of the light elements, lithium, is not perfect. As a recent review by Cyburt, Field, and Olive has emphasized [6] observations of both lithium-6 and lithium-7 in low-metallicity stars [7] are somewhat discrepant from the BBN prediction. It is possible that there is misunderstood astrophysics at work—perhaps the measured abundance of lithium is not primordial after all? The other possibility is novel cosmology. For example, if a particle were produced in the big bang with a lifetime of $100$$1000s$, its decays would be occurring during the crucial epoch of nucleosynthesis. This could shift the predictions for the lithium abundance into accordance with observations [8]. While there are many multiple examples in the literature of such long-lived relic particles with the proper properties to solve the lithium problem, the authors note a lifetime of several minutes—an eternity on particle scales—is particularly elegantly explained if these decays only occur suppressed by powers of the GUT scale.
The lifetime of the relics that modify nucleosynthesis (minutes) and the lifetime of the dark matter ($1019$ years) could even be explained by the same physics in these theories. Decays during the BBN epoch would result from processes suppressed by the square of the GUT energy, while the decay of the dark matter would correspond to processes suppressed by four powers of the GUT energy.
It will not be easy to determine that GUT-scale physics is the reason for the long lifetime of these relics. The case could be sharpened, however, if the relevant particles are produced at the Large Hadron Collider. If the LHC produces these particles, they could slow down and stop in the detectors there. Their decays could be seen minutes later, a striking signal. The best hope for establishing that one of the GUTs that Arvanitaki et al. discuss is in play is to use a mixture of data from three sources: cosmic rays, colliders, and BBN.
It should be noted that the presence of particles with such long lifetimes is not automatic in just any supersymmetric grand unified theory. Depending on the structure of the theory, the relevant particles could either be absolutely stable, or could have far too short a lifetime to explain either modifications of BBN or cosmic-ray signals. So, Arvanitaki et al. point out structures that ensure that a theory will have particles with lifetimes in the cosmic “sweet spots.” The fact that not all grand unified theories give rise to the relevant decays shows the predictive power of this exploration. So, should the type of signals that Arvanitaki et al. predict be confirmed with upcoming cosmic-ray data (e.g., from the Fermi Gamma Ray Space Telescope), and should the lithium problem sharpen, it will point to a particular class of theories—those with long lived relics, and we may have a chance to learn something about grand unified theories. Hopes of probing GUT-scale physics would no longer solely rest on proton decay.
### References
1. H. Nishino et al., Phys. Rev. Lett. 102, 141801 (2009).
2. A. Arvanitaki, S. Dimopoulos, S. Dubovsky, P. W. Graham, R. Harnik, and S. Rajendran, Phys. Rev. D 79, 105022 (2009).
3. H. Goldberg, Phys. Rev. Lett. 50, 1419 (1983).
4. S. Dimopoulos, S. Raby, and F. Wilczek, Phys. Rev. D 24, 1681 (1981).
5. O. Adriani et al., Nature 458, 607 (2009); A. A. Abdo et al., Phys. Rev. Lett. 102, 181101 (2009); See also the Viewpoint commentary by B. Winstein and K. Zurek, Physics 2, 37 (2009).
6. R. H. Cyburt, B. D. Fields, and K. A. Olive, arXiv:0808.2818.
7. F. Spite and M. Spite, Astron. Astrophys. 115, 357 (1982).
8. S. Bailly, K. Jedamzik, and G. Moultaka, arXiv:0812.0788 (2008).
### About the Author: Aaron Pierce
Aaron Pierce is an Assistant Professor of physics at the University of Michigan and member of the Michigan Center for Theoretical Physics. He is a particle theorist and his work focuses on the phenomenology of physics beyond the standard model. | 2014-03-09 04:52:34 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 16, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5872375965118408, "perplexity": 848.6940191782253}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-10/segments/1393999673133/warc/CC-MAIN-20140305060753-00066-ip-10-183-142-35.ec2.internal.warc.gz"} |
https://www.shaalaa.com/question-bank-solutions/the-latus-rectum-conic-3x2-4y2-6x-8y-5-0-introduction-ellipse_59187 | # The Latus-rectum of the Conic 3x2 + 4y2 − 6x + 8y − 5 = 0 is - Mathematics
MCQ
Sum
The latus-rectum of the conic 3x2 + 4y2 − 6x + 8y − 5 = 0 is
#### Options
• 3
• $\frac{\sqrt{3}}{2}$
• $\frac{2}{\sqrt{3}}$
• none of these
#### Solution
3
$3 x^2 + 4 y^2 - 6x + 8y - 5 = 0$
$\Rightarrow 3( x^2 - 2x) + 4( y^2 + 2y) = 5$
$\Rightarrow 3( x^2 - 2x + 1) + 4( y^2 + 2y + 1) = 5 + 3 + 4$
$\Rightarrow 3(x - 1 )^2 + 4(y + 1 )^2 = 12$
$\Rightarrow \frac{(x - 1 )^2}{4} + \frac{(y + 1 )^2}{3} = 1$
$\text{ So, }a = 2\text{ and }b = \sqrt{3}$
$\therefore\text{ Latus rectum }= \frac{2 b^2}{a}$
$= 2\frac{\left[ \sqrt{3} \right]^2}{2}$
$= 3$
Concept: Introduction of Ellipse
Is there an error in this question or solution?
#### APPEARS IN
RD Sharma Class 11 Mathematics Textbook
Chapter 26 Ellipse
Q 9 | Page 28
Share | 2021-04-14 16:59:06 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2849152982234955, "perplexity": 9747.342267249613}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038077843.17/warc/CC-MAIN-20210414155517-20210414185517-00473.warc.gz"} |
http://r-text.org/reference/textDescriptives.html | Compute descriptive statistics of character variables.
textDescriptives(
words,
compute_total = TRUE,
entropy_unit = "log2",
na.rm = TRUE
)
## Arguments
words
One or several character variables; if its a tibble or dataframe, all the character variables will be selected.
compute_total
Boolean. If the input (words) is a tibble/dataframe with several character variables, a total variable is computed.
entropy_unit
The unit entropy is measured in. The default is to used bits (i.e., log2; see also, "log", "log10"). If a total score for several varaibles is computed,the text columns are combined using the dplyr unite function. For more information about the entropy see the entropy package and specifically its entropy.plugin function.
na.rm
Option to remove NAs when computing mean, median etc (see under return).
## Value
A tibble with descriptive statistics, including variable = the variable names of input "words"; w_total = total number of words in the variable; w_mean = mean number of words in each row of the variable; w_median = median number of words in each row of the variable; w_range_min = smallest number of words of all rows; w_range_max = largest number of words of all rows; w_sd = the standard deviation of the number of words of all rows; unique_tokens = the unique number of tokens (using the word_tokenize function from python package nltk) n_token = number of tokens in the variable (using the word_tokenize function from python package nltk) entropy = the entropy of the variable. It is computed as the Shannon entropy H of a discrete random variable from the specified bin frequencies. (see library entropy and specifically the entropy.plugin function)
see textEmbed
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https://www.physicsforums.com/threads/correlation-functions-in-position-and-momentum-space.888036/ | # A Correlation functions in position and momentum space
1. Oct 5, 2016
### spaghetti3451
What is the relation between the correlators $\langle 0 | T\phi(x_{1})\phi(x_{2}) | 0 \rangle$ and $\langle 0 | T\phi(p)\phi(x=0) | 0 \rangle$?
I can derive the momentum space Feynman rules for $\langle 0 | T\phi(x_{1})\phi(x_{2}) | 0 \rangle$. Are the momentum space Feynman rules for $\langle 0 | T\phi(p)\phi(x=0) | 0 \rangle$ the same?
2. Oct 11, 2016 | 2017-12-16 10:11:43 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8334468007087708, "perplexity": 849.3435453267488}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-51/segments/1512948587496.62/warc/CC-MAIN-20171216084601-20171216110601-00470.warc.gz"} |
https://www.futurelearn.com/info/courses/precalculus/0/steps/32375 | # Polynomial equations
Polynomial equations
10.6
Good news, there’s a class of equations in which we are already experts. When f is a polynomial, the equation f of x equals 0 defines the roots of the polynomial. We have studied in detail the issue of finding these roots. Recall, for example, the following fact for the quadratic polynomial case. We know that there is something there, the discriminant, which will tell us an awful lot about the roots of this polynomial. Capital Delta– by the way, there’s another beautiful Greek letter– is the discriminant, b squared minus 4ac. We know already from this proposition that we’ve seen and proven that if Delta is strictly negative, there are no roots.
55
And if delta is positive, then there are roots of the polynomial, and we know how to calculate them from this quadratic formula that we will know till the end of our lives. Now it’s a fact that in specific examples you sometimes have to massage the given equation to turn it into polynomial form. A notable example of this is when rational functions are involved. Here’s a case in which you want to solve an equation where the sum of two rational functions equals 3. Remember, rational function means polynomial over polynomial. Now the natural domain of this equation is the whole real line, but take away the points 1 and minus 2 for obvious reasons.
99.8
We multiply both sides of the equation by the product, x minus 1, x plus 2. This will have the effect of getting rid of the denominators on the left-hand side, so, algebraically, it will simplify our equation. In doing this multiplication, by the way, we’re not multiplying by anything that can be 0, so we are preserving equivalence because this term will not be 0 in the domain as we’ve defined it. So after this multiplication, we come upon this equation. Now we have to simplify it. Our mastery of arithmetic allows us to do this with ease. And when the dust settles, you have a quadratic equation for x.
142.4
It turns out that it factors, and, therefore, its roots are evident, minus 5 and plus 2. And the solution set of our original equation is minus 5 and 2. Two points that are in the domain. Now that should be right because we haven’t done anything to change equivalence, but we’re not androids, so we check that it works in the original equation. How about polynomial equations of degree greater than 2? Let me remind you that we had a strategy for finding roots of a general polynomial. It was based upon writing the root factorization of the polynomial, that means writing it as a product of linear factors, x minus r, times a polynomial that has no further roots.
189.9
And the way to obtain such a root factorization involved order reduction and the fun of polynomial division. Well, we won’t go back all over that, but let’s look at the special case of a pure n-th order equation, an equation like this, a x to the n plus b equals 0. So a here is different from 0. And this equation is equivalent to x to the power n equals minus b over a. Since we want to find x, we are very tempted to simply take the n-th root. But remember that the n-th root of y is only defined when y is positive.
234.5
Therefore, we have to be a little more careful in solving the equation and what happens next depends entirely upon the parity of n. Is n even, or is n odd? If n is even and if minus b over a is strictly negative, then there are obviously no solutions to our equation because an even power of x can never be strictly negative. On the other hand, if minus b over a is positive, then the n-th root of that number is a solution, but since n is even, also minus the n-th root is the solution. So we have two solutions in that case.
271.8
In the remaining case, when n is odd, there will always be one solution, but it will be written somewhat differently depending upon whether minus b over a is positive or negative. When it’s positive, it’s just directly the n-th root. When it’s negative, we have to write the solution as minus the n-th root of the positive number, b over a.
294.7
Now let me give you an example in which we will use a frequently useful technique called change of variables to simplify a given equation that we want to solve. Suppose we have a polynomial equation like the example you see in which there is a square power of x and a fourth power of x, but there’s no cubic power and there’s no x. This suggests a certain useful change of variables that might simplify the equation. We’re going to introduce a new variable. I’ll call it capital X, and it’s going to be a stand-in for x squared. So that’s our change of variables. Why do we make this change of variables?
337.8
Well in the hope that when we express our original equation with the new variable it will be simplified. And you see this is the case because the new equation, in terms of capital X, is a straightforward quadratic equation, capital X squared minus capital X minus 2 equals 0. It’s an equation we can easily solve by factoring. We find that capital X must be 2 or minus 1. Now we go back to the original variable, capital X is little x squared, so that means x squared has to be 2 or minus 1. Now x squared can’t be minus 1, of course, therefore, we retain only the possibility that x squared equals 2.
379.6
And this gives us two roots, x equals plus and minus root 2. So the solution set is the set of two points plus and minus root 2. And we check our answer, of course. Let me remark that the change of variables X equals x cubed is one that will do very good work when we apply it to an equation such as x to the sixth minus x cubed minus 2 equals 0. It’s pretty much the same idea.
Root finding methods (review), some equations of special type | 2020-12-04 05:57:05 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8072157502174377, "perplexity": 210.75706223234934}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141733122.72/warc/CC-MAIN-20201204040803-20201204070803-00275.warc.gz"} |
http://mathhelpforum.com/discrete-math/113078-truth-tables.html | 1. ## Truth Tables
Need help in completing this truth table.
P | Q | R | S | ~S | P^Q | R^S(the arrow goes down. cant find a down arrow)
2. Originally Posted by Siobhan
Need help in completing this truth table.
P | Q | R | S | NS | P^Q | R^S(the arrow goes down. cant find a down arrow)
What is the conclusion?
R wedge S ?
And, by NS, did you mean ${N}\cdot{S}$ ?
3. I really don't know if I have the correct wording on this. Please bare with me.
For example.
P Q R
T T T
T T F
T F T
T F F
F T T
F T F
F F T
F F F
Please try to encode what I've typed in the above post. If you know these it would be a great help.
Thank you for understanding,
Siobhan
4. Originally Posted by Siobhan
I really don't know if I have the correct wording on this. Please bare with me.
For example.
P Q R
T T T
T T F
T F T
T F F
F T T
F T F
F F T
F F F
Please try to encode what I've typed in the above post. If you know these it would be a great help.
Thank you for understanding,
Siobhan
I'm not the best with matrices in latex, but I'll try to help.
You've got the basic Idea. The number of lines needed will be 2 raised to the number of simple statements. Or
$L=2^n$
So you will need $2^5=32$ lines if N is a simple statement, but I think by N, you may be trying to say "not"S, or $\sim{S}$. Is this correct. If so you will only need 16 lines.
5. Updated. The N was a ~ LOL. My messy writing. Still need help with this though.
6. Originally Posted by Siobhan
Updated. The N was a ~ LOL. My messy writing. Still need help with this though.
Okay then...
I assume that this is an argument with $R\vee{S}$ the conclusion. Is this correct?
7. Yup, correct
8. Originally Posted by Siobhan
Yup, correct
Dude, I can't get the LaTex right, but
There will be 8 Ts and them 8 Fs under the P. Then The Ts and Fs will alternate by fours for the Qs. Then the Ts and Fs will alternate by 2s for the Rs, and by 1 for the Ss.
After you have written the 16 lines, change the truth values for the tildes. Then understand that a wedge is only false when both of its disjuncts are false. Then, look for a line in which all of the premises are true, but the conclusion is false, if this does not happen, the argument is valid.
Im sorry if this is vague. I will compute the truth table and get right back to you with the answer. Thanks for being patient with me.
9. argument appears to be valid. | 2016-08-27 21:27:30 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 5, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8456984758377075, "perplexity": 932.4880681747993}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-36/segments/1471982925602.40/warc/CC-MAIN-20160823200845-00087-ip-10-153-172-175.ec2.internal.warc.gz"} |
https://math.stackexchange.com/questions/3071571/matrix-demonstration-ak | Matrix demonstration $A^k$
Given a matrix $$A = \begin{bmatrix} 7 & 4\\ -9 & -5 \end{bmatrix}$$ $$\in \mathcal{M2\times2}\, (\mathbb{R})$$
Show that $$A^k = \begin{bmatrix} 1+6k & 4k\\ -9k & 1-6k \end{bmatrix}$$ for every $$k \in \mathbb{N}$$
• What have you tried so far? What approach do you think will work here? – user3482749 Jan 13 at 0:37
• Have you tried proof by induction? – user7530 Jan 13 at 0:37
• yes I thought about the proof by induction but i got lost in the process, I don't know which way is the best, honestly. – PTSONIC Jan 13 at 0:40
• Can you start by computing $A^kA$ and simplifying the result? Shows us your work. – Git Gud Jan 13 at 0:46
• yes the result is this matrix I think $\begin{bmatrix} 7+6k & 4+4k\\ -9-9k & -5-6k \end{bmatrix}$ – PTSONIC Jan 13 at 1:01
Try a simple induction on $$k$$; it is clear that for $$k = 1$$,
$$A^1 = A = \begin{bmatrix} 7 & 4 \\ -9 & -5 \end{bmatrix} = \begin{bmatrix} 1 + 6 \cdot 1 & 4 \cdot 1 \\ -9 \cdot 1 & 1 - 6 \cdot 1 \end{bmatrix}; \tag 1$$
then assuming that for some $$k$$
$$A^k = \begin{bmatrix} 1 + 6 \cdot k & 4 \cdot k \\ -9 \cdot k & 1 - 6 \cdot k \end{bmatrix}, \tag 2$$
we find
$$A^{k + 1} = A^kA = \begin{bmatrix} 1 + 6 \cdot k & 4 \cdot k \\ -9 \cdot k & 1 - 6 \cdot k \end{bmatrix}\begin{bmatrix} 7 & 4 \\ -9 & -5 \end{bmatrix} = \begin{bmatrix} 7 + 42k - 36k & 4 + 24k - 20k \\ -63k - 9 + 54k & -36k - 5 + 30k \end{bmatrix}$$ $$= \begin{bmatrix} 7 + 6k & 4 + 4k \\ -9 -9k & - 5 -6k \end{bmatrix} = \begin{bmatrix} 1 + 6(k + 1) & 4(k + 1) \\ -9(k + 1) & 1 - 6(k + 1) \end{bmatrix}, \tag 3$$
which is simply (2) with $$k$$ replaced by $$k + 1$$; we thus infer the formula (2) holds for all $$k \ge 1$$, as desired. $$OE\Delta$$.
Let $$N = A-I$$ and observe that $$N^2=0$$. Since $$N$$ and $$I$$ commute, we can expand via the binomial theorem: $$A^k = (I+N)^k = I+\binom k1N+\binom k2N^2+\dots = I+kN.$$
How did I hit upon this decomposition? I computed that $$1$$ was the only eigenvalue of $$A$$, but the only diagonalizable $$2\times2$$ matrices with repeated eigenvalues are multiples of the identity, so $$A$$ splits into the sum of the identity and a nilpotent matrix. That is, $$A = P\begin{bmatrix}1&1\\0&1\end{bmatrix}P^{-1} = I + P\begin{bmatrix}0&1\\0&0\end{bmatrix}P^{-1}$$ for some invertible matrix $$P$$.
• Very nice indeed, endorsed, +1!!! – Robert Lewis Jan 13 at 1:47 | 2019-01-21 02:16:09 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 26, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.971759021282196, "perplexity": 256.0629180337817}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-04/segments/1547583745010.63/warc/CC-MAIN-20190121005305-20190121031305-00244.warc.gz"} |
https://cforall.uwaterloo.ca/trac/changeset/d7a02ae903476ded9a21da3034888ccf8617a83d/ | # Changeset d7a02ae
Ignore:
Timestamp:
Jun 13, 2019, 8:27:28 AM (3 years ago)
Branches:
arm-eh, enum, forall-pointer-decay, jacob/cs343-translation, jenkins-sandbox, master, new-ast, new-ast-unique-expr, pthread-emulation, qualifiedEnum
Children:
d60780c
Parents:
6625727
Message:
first complete draft of new concurrency paper
Location:
doc/papers/concurrency
Files:
5 edited
3 moved
Unmodified
Removed
• ## doc/papers/concurrency/Paper.tex
r6625727 _Thread_local, throw, throwResume, timeout, trait, try, ttype, typeof, __typeof, __typeof__, virtual, __volatile, __volatile__, waitfor, when, with, zero_t}, moredirectives={defined,include_next}% moredirectives={defined,include_next}, % replace/adjust listing characters that look bad in sanserif literate={-}{\makebox[1ex][c]{\raisebox{0.4ex}{\rule{0.8ex}{0.1ex}}}}1 {^}{\raisebox{0.6ex}{$\scriptstyle\land\,$}}1 {~}{\raisebox{0.3ex}{$\scriptstyle\sim\,$}}1 % {}{\ttfamily\upshape\hspace*{-0.1ex}}1 {<}{\textrm{\textless}}1 {>}{\textrm{\textgreater}}1 {<-}{$\leftarrow$}2 {=>}{$\Rightarrow$}2 {->}{\makebox[1ex][c]{\raisebox{0.5ex}{\rule{0.8ex}{0.075ex}}}\kern-0.2ex{\textrm{\textgreater}}}2, } aboveskip=4pt, % spacing above/below code block belowskip=3pt, % replace/adjust listing characters that look bad in sanserif literate={-}{\makebox[1ex][c]{\raisebox{0.4ex}{\rule{0.8ex}{0.1ex}}}}1 {^}{\raisebox{0.6ex}{$\scriptstyle\land\,$}}1 {~}{\raisebox{0.3ex}{$\scriptstyle\sim\,$}}1 % {}{\ttfamily\upshape\hspace*{-0.1ex}}1 {<}{\textrm{\textless}}1 {>}{\textrm{\textgreater}}1 {<-}{$\leftarrow$}2 {=>}{$\Rightarrow$}2 {->}{\makebox[1ex][c]{\raisebox{0.5ex}{\rule{0.8ex}{0.075ex}}}\kern-0.2ex{\textrm{\textgreater}}}2, moredelim=**[is][\color{red}]{}{}, }% lstset } % Go programming language: https://github.com/julienc91/listings-golang/blob/master/listings-golang.sty \lstdefinelanguage{Golang}{ morekeywords=[1]{package,import,func,type,struct,return,defer,panic,recover,select,var,const,iota,}, morekeywords=[2]{string,uint,uint8,uint16,uint32,uint64,int,int8,int16,int32,int64, bool,float32,float64,complex64,complex128,byte,rune,uintptr, error,interface}, morekeywords=[3]{map,slice,make,new,nil,len,cap,copy,close,true,false,delete,append,real,imag,complex,chan,}, morekeywords=[4]{for,break,continue,range,goto,switch,case,fallthrough,if,else,default,}, morekeywords=[5]{Println,Printf,Error,}, sensitive=true, morecomment=[l]{//}, morecomment=[s]{/*}{*/}, morestring=[b]', morestring=[b]", morestring=[s]{}{}, % replace/adjust listing characters that look bad in sanserif literate={-}{\makebox[1ex][c]{\raisebox{0.4ex}{\rule{0.8ex}{0.1ex}}}}1 {^}{\raisebox{0.6ex}{$\scriptstyle\land\,$}}1 {~}{\raisebox{0.3ex}{$\scriptstyle\sim\,$}}1 % {}{\ttfamily\upshape\hspace*{-0.1ex}}1 {<}{\textrm{\textless}}1 {>}{\textrm{\textgreater}}1 {<-}{\makebox[2ex][c]{\textrm{\textless}\raisebox{0.5ex}{\rule{0.8ex}{0.075ex}}}}2, } \lstnewenvironment{cfa}[1][] {\lstset{#1}} {} \lstnewenvironment{Go}[1][] {\lstset{language=go,moredelim=**[is][\protect\color{red}]{}{},#1}\lstset{#1}} {\lstset{language=Golang,moredelim=**[is][\protect\color{red}]{}{},#1}\lstset{#1}} {} \lstnewenvironment{python}[1][] These features are created from scratch as ISO C has only low-level and/or unimplemented concurrency, so C programmers continue to rely on library features like C pthreads. \CFA introduces modern language-level control-flow mechanisms, like generators, coroutines, user-level threading, and monitors for mutual exclusion and synchronization. Library extension for executors, futures, and actors are built on these basic mechanisms. % Library extension for executors, futures, and actors are built on these basic mechanisms. The runtime provides significant programmer simplification and safety by eliminating spurious wakeup and optional monitor barging. The runtime also ensures multiple monitors can be safely acquired \emph{simultaneously} (deadlock free), and this feature is fully integrated with all monitor synchronization mechanisms. However, functions \emph{cannot} be nested in structures, so there is no lexical binding between a structure and set of functions (member/method) implemented by an implicit \lstinline@this@ (receiver) parameter.}, backwards-compatible extension of the C programming language. In many ways, \CFA is to C as Scala~\cite{Scala} is to Java, providing a \emph{research vehicle} for new typing and control flow capabilities on top of a highly popular programming language allowing immediate dissemination. Within the \CFA framework, new control-flow features are created from scratch. ISO \Celeven defines only a subset of the \CFA extensions, where the overlapping features are concurrency~\cite[\S~7.26]{C11}. Kernel threading was chosen, largely because of its simplicity and fit with the simpler operating systems and hardware architectures at the time, which gave it a performance advantage~\cite{Drepper03}. Libraries like pthreads were developed for C, and the Solaris operating-system switched from user (JDK 1.1~\cite{JDK1.1}) to kernel threads. As a result, languages like Java, Scala~\cite{Scala}, Objective-C~\cite{obj-c-book}, \CCeleven~\cite{C11}, and C\#~\cite{Csharp} adopt the 1:1 kernel-threading model, with a variety of presentation mechanisms. As a result, languages like Java, Scala, Objective-C~\cite{obj-c-book}, \CCeleven~\cite{C11}, and C\#~\cite{Csharp} adopt the 1:1 kernel-threading model, with a variety of presentation mechanisms. From 2000 onwards, languages like Go~\cite{Go}, Erlang~\cite{Erlang}, Haskell~\cite{Haskell}, D~\cite{D}, and \uC~\cite{uC++,uC++book} have championed the M:N user-threading model, and many user-threading libraries have appeared~\cite{Qthreads,MPC,BoostThreads}, including putting green threads back into Java~\cite{Quasar}. The main argument for user-level threading is that they are lighter weight than kernel threads (locking and context switching do not cross the kernel boundary), so there is less restriction on programming styles that encourage large numbers of threads performing medium work-units to facilitate load balancing by the runtime~\cite{Verch12}. \item providing statically type-safe interfaces that integrate with the \CFA polymorphic type-system and other language features. \item library extensions for executors, futures, and actors built on the basic mechanisms. % \item % library extensions for executors, futures, and actors built on the basic mechanisms. \item a runtime system with no spurious wakeup. \item a dynamic partitioning mechanism to segregate the execution environment for specialized requirements. \item a non-blocking I/O library % \item % a non-blocking I/O library \item experimental results showing comparable performance of the new features with similar mechanisms in other programming languages. Therefore, selecting between stackless and stackful semantics is a tradeoff between programming requirements and performance, where stackless is faster and stackful is more general. Note, creation cost is amortized across usage, so activation cost is usually the dominant factor. \begin{figure} \hspace{3pt} \subfloat[C generator implementation]{\label{f:CFibonacciSim}\usebox\myboxC} \caption{Fibonacci (output) Asymmetric Generator} \caption{Fibonacci (output) asymmetric generator} \label{f:FibonacciAsymmetricGenerator} \subfloat[C generator simulation]{\label{f:CFormatSim}\usebox\myboxB} \hspace{3pt} \caption{Formatter (input) Asymmetric Generator} \caption{Formatter (input) asymmetric generator} \label{f:FormatterAsymmetricGenerator} \end{figure} For generators, coroutines, and threads, many designs are based on function objects or pointers~\cite{Butenhof97, C++14, MS:VisualC++, BoostCoroutines15}. For example, Python presents generators as a function object: \begin{python} def Gen(): ... yield val ... gen = Gen() for i in range( 10 ): print( next( gen ) ) \end{python} Boost presents coroutines in terms of four functor object-types: \begin{cfa} asymmetric_coroutine<>::pull_type asymmetric_coroutine<>::push_type symmetric_coroutine<>::call_type symmetric_coroutine<>::yield_type \end{cfa} and many languages present threading using function pointers, @pthreads@~\cite{Butenhof97}, \Csharp~\cite{Csharp}, Go~\cite{Go}, and Scala~\cite{Scala}, \eg pthreads: \begin{cfa} void * rtn( void * arg ) { ... } int i = 3, rc; pthread_t t; $\C{// thread id}$ rc = pthread_create( &t, rtn, (void *)i ); $\C{// create and initialized task, type-unsafe input parameter}$ \end{cfa} % void mycor( pthread_t cid, void * arg ) { % int * value = (int *)arg; $\C{// type unsafe, pointer-size only}$ % // thread body % } % int main() { % int input = 0, output; % coroutine_t cid = coroutine_create( &mycor, (void *)&input ); $\C{// type unsafe, pointer-size only}$ % coroutine_resume( cid, (void *)input, (void **)&output ); $\C{// type unsafe, pointer-size only}$ % } \CFA's preferred presentation model for generators/coroutines/threads is a hybrid of objects and functions, with an object-oriented flavour. Essentially, the generator/coroutine/thread function is semantically coupled with a generator/coroutine/thread custom type. The custom type solves several issues, while accessing the underlying mechanisms used by the custom types is still allowed. \subsection{Generator} Stackless generators have the potential to be very small and fast, \ie as small and fast as function call/return for both creation and execution. The \CFA goal is to achieve this performance target, possibly at the cost of some semantic complexity. How this goal is accomplished is done through a series of different kinds of generators and their implementation. A series of different kinds of generators and their implementation demonstrate how this goal is accomplished. Figure~\ref{f:FibonacciAsymmetricGenerator} shows an unbounded asymmetric generator for an infinite sequence of Fibonacci numbers written in C and \CFA, with a simple C implementation for the \CFA version. Figure~\ref{f:CFibonacci} shows the C approach of manually creating the closure in structure @Fib@, and multiple instances of this closure provide multiple Fibonacci generators. The C version only has the middle execution state because the top execution state becomes declaration initialization. Figure~\ref{f:CFAFibonacciGen} shows the \CFA approach, which also has a manual closure, but replaces the structure with a special \CFA @generator@ type. This generator type is then connected to a function named @main@ that takes as its only parameter a reference to the generator type, called a \emph{generator main}. Figure~\ref{f:CFAFibonacciGen} shows the \CFA approach, which also has a manual closure, but replaces the structure with a custom \CFA @generator@ type. This generator type is then connected to a function that \emph{must be named \lstinline|main|},\footnote{ The name \lstinline|main| has special meaning in C, specifically the function where a program starts execution. Hence, overloading this name for other starting points (generator/coroutine/thread) is a logical extension.} which takes as its only parameter a reference to the generator type, called a \emph{generator main}. The generator main contains @suspend@ statements that suspend execution without ending the generator versus @return@. For the Fibonacci generator-main,\footnote{ For contrast, Figure~\ref{f:PythonFibonacci} shows the equivalent Python Fibonacci generator, which does not use a generator type, and hence only has a single interface, but an implicit closure. Having to manually create the generator closure by moving local-state variables into the generator type is an additional programmer burden and possible point of error. Having to manually create the generator closure by moving local-state variables into the generator type is an additional programmer burden. (This restriction is removed by the coroutine, see Section~\ref{s:Coroutine}.) Our concern is that static analysis to discriminate local state from temporary variables is complex and beyond the current scope of the \CFA project. As well, supporting variable-size local-state, like variable-length arrays, requires dynamic allocation of the local state, which significantly increases the cost of generator creation/destruction, and is a show-stopper for embedded programming. This requirement follows from the generality of variable-size local-state, \eg local state with a variable-length array requires dynamic allocation because the array size is unknown at compile time. However, dynamic allocation significantly increases the cost of generator creation/destruction and is a show-stopper for embedded real-time programming. But more importantly, the size of the generator type is tied to the local state in the generator main, which precludes separate compilation of the generator main, \ie a generator must be inlined or local state must be dynamically allocated. Our current experience is that most generator problems have simple data state, including local state, but complex execution state. With respect to safety, we believe static analysis can discriminate local state from temporary variables in a generator, \ie variable usage spanning @suspend@, and generate a compile-time error. Finally, our current experience is that most generator problems have simple data state, including local state, but complex execution state, so the burden of creating the generator type is small. As well, C programmers are not afraid with this kind of semantic programming requirement, if it results in very small, fast generators. \begin{python}[aboveskip=0pt,belowskip=0pt] def Fib(): fn1, fn = 0, 1 while True: yield fn1 fn1, fn = fn, fn1 + fn fn1, fn = 0, 1 while True: yield fn1 fn1, fn = fn, fn1 + fn f1 = Fib() f2 = Fib() \hspace{3pt} \subfloat[Formatter]{\label{f:PythonFormatter}\usebox\myboxB} \caption{Python Generator} \caption{Python generator} \label{f:PythonGenerator} generator PingPong { const char * name; int N; int i; // local state PingPong & partner; // rebindable reference int N, i; }; void ?{}(PingPong & pp, char * nm, int N) with(pp) { name = nm; partner = 0p; pp.N = N; i = 0; } void main( PingPong & pp ) with(pp) { for ( ; i < N; i += 1 ) { int main() { enum { N = 5 }; PingPong ping = {"ping", N}, pong = {"pong", N}; PingPong ping = {"ping",N,0}, pong = {"pong",N,0}; &ping.partner = &pong; &pong.partner = &ping; resume( ping ); typedef struct PingPong { const char * name; int N, i; struct PingPong * partner; int N, i; void * next; } PingPong; #define PPCtor(name, N) { name, NULL, N, 0, NULL } #define PPCtor(name, N) {name,N,0,NULL,NULL} void comain( PingPong * pp ) { if ( pp->next ) goto *pp->next; \subfloat[C generator simulation]{\label{f:CPingPongSim}\usebox\myboxB} \hspace{3pt} \caption{Ping-Pong Symmetric Generator} \caption{Ping-Pong symmetric generator} \label{f:PingPongSymmetricGenerator} \end{figure} A coroutine is specified by replacing @generator@ with @coroutine@ for the type. This generality results in higher cost for creation, due to dynamic stack allocation, execution, due to context switching among stacks, and terminating, due to possible stack unwinding and dynamic stack deallocation. How coroutines differ from generators is done through a series of examples. A series of different kinds of coroutines and their implementation demonstrate how coroutines extend generators. First, the previous generator examples are converted to their coroutine counterparts, allowing local-state variables to be moved from the generator type into the coroutine main. \end{cfa} \end{description} It is also possible to refactor code containing local-state and @suspend@ statements into a helper routine, like the computation of the CRC for the device driver. It is also possible to refactor code containing local-state and @suspend@ statements into a helper function, like the computation of the CRC for the device driver. \begin{cfa} unsigned int Crc() { \begin{comment} Figure~\ref{f:Coroutine3States} creates a @coroutine@ type, @coroutine Fib { int fn; }@, which provides communication, @fn@, for the \newterm{coroutine main}, @main@, which runs on the coroutine stack, and possibly multiple interface routines, \eg @next@. Figure~\ref{f:Coroutine3States} creates a @coroutine@ type, @coroutine Fib { int fn; }@, which provides communication, @fn@, for the \newterm{coroutine main}, @main@, which runs on the coroutine stack, and possibly multiple interface functions, \eg @next@. Like the structure in Figure~\ref{f:ExternalState}, the coroutine type allows multiple instances, where instances of this type are passed to the (overloaded) coroutine main. The coroutine main's stack holds the state for the next generation, @f1@ and @f2@, and the code represents the three states in the Fibonacci formula via the three suspend points, to context switch back to the caller's @resume@. The interface routine @next@, takes a Fibonacci instance and context switches to it using @resume@; The interface function @next@, takes a Fibonacci instance and context switches to it using @resume@; on restart, the Fibonacci field, @fn@, contains the next value in the sequence, which is returned. The first @resume@ is special because it allocates the coroutine stack and cocalls its coroutine main on that stack; def Fib(): fn1, fn = 0, 1 while True: yield fn1 fn1, fn = fn, fn1 + fn fn1, fn = 0, 1 while True: yield fn1 fn1, fn = fn, fn1 + fn \hspace{3pt} \subfloat[Python]{\label{f:ExternalState}\usebox\myboxC} \caption{Fibonacci Generator} \caption{Fibonacci generator} \label{f:C-fibonacci} \end{figure} \caption{Producer / consumer: resume-resume cycle, bi-directional communication} \label{f:ProdCons} \medskip \begin{center} \input{FullProdConsStack.pstex_t} \end{center} \vspace*{-10pt} \caption{Producer / consumer runtime stacks} \label{f:ProdConsRuntimeStacks} \medskip \begin{center} \input{FullCoroutinePhases.pstex_t} \end{center} \vspace*{-10pt} \caption{Ping / Pong coroutine steps} \label{f:PingPongFullCoroutineSteps} \end{figure} The loop then repeats calling @payment@, where each call resumes the producer coroutine. Figure~\ref{f:ProdConsRuntimeStacks} shows the runtime stacks of the program main, and the coroutine mains for @prod@ and @cons@ during the cycling. \begin{figure} \begin{center} \input{FullProdConsStack.pstex_t} \end{center} \vspace*{-10pt} \caption{Producer / consumer runtime stacks} \label{f:ProdConsRuntimeStacks} \medskip \begin{center} \input{FullCoroutinePhases.pstex_t} \end{center} \vspace*{-10pt} \caption{Ping / Pong coroutine steps} \label{f:PingPongFullCoroutineSteps} \end{figure} Terminating a coroutine cycle is more complex than a generator cycle, because it requires context switching to the program main's \emph{stack} to shutdown the program, whereas generators started by the program main run on its stack. \subsection{Coroutine Implementation} A significant implementation challenge for coroutines (and threads, see Section~\ref{threads}) is adding extra fields and executing code after/before the coroutine constructor/destructor and coroutine main to create/initialize/de-initialize/destroy extra fields and the stack. There are several solutions to this problem and the chosen option directed the \CFA coroutine design. For object-oriented languages, inheritance can be used to provide extra fields and code, but it requires explicitly writing the inheritance: \subsection{(Generator) Coroutine Implementation} A significant implementation challenge for generators/coroutines (and threads, see Section~\ref{s:threads}) is adding extra fields to the custom types and related functions, \eg inserting code after/before the coroutine constructor/destructor and @main@ to create/initialize/de-initialize/destroy any extra fields, \eg stack. There are several solutions to these problem, which follow from the object-oriented-flavoured choice to adopt custom types. For object-oriented languages, inheritance is used to provide extra fields and code via explicit inheritance: \begin{cfa}[morekeywords={class,inherits}] class mycoroutine inherits baseCoroutine { ... } \end{cfa} In addition, the programming language and possibly its tool set, \eg debugger, @valgrind@ need to understand @baseCoroutine@ because of special stack property of coroutines. Furthermore, the execution of constructors/destructors is in the wrong order for certain operations. For example, for threads if the thread is implicitly started, it must start \emph{after} all constructors, because the thread relies on a completely initialized object, but the inherited constructor runs \emph{before} the derived. class myCoroutine inherits baseCoroutine { ... } \end{cfa} The problem is that the programming language and its tool chain, \eg debugger, @valgrind@, need to understand @baseCoroutine@ because it infers special property, so type @baseCoroutine@ becomes a de facto keyword and all types inheriting from it are implicitly custom types. As well, some special properties are not handled by existing language semantics, \eg the execution of constructors/destructors is in the wrong order to implicitly start threads because the thread must start \emph{after} all constructors as it relies on a completely initialized object, but the inherited constructor runs \emph{before} the derived. Alternatives, such as explicitly starting threads as in Java, are repetitive and forgetting to call start is a common source of errors. An alternative is composition: \begin{cfa} struct mycoroutine { ... // declarations struct myCoroutine { ... // declaration/communication variables baseCoroutine dummy; // composition, last declaration } \end{cfa} which also requires an explicit declaration that must be last to ensure correct initialization order. However, there is nothing preventing wrong placement or multiple declarations of @baseCoroutine@. For coroutines, as for threads, many implementations are based on function pointers or function objects~\cite{Butenhof97, C++14, MS:VisualC++, BoostCoroutines15}. For example, Boost implements coroutines in terms of four functor object-types: \begin{cfa} asymmetric_coroutine<>::pull_type asymmetric_coroutine<>::push_type symmetric_coroutine<>::call_type symmetric_coroutine<>::yield_type \end{cfa} However, there is nothing preventing wrong placement or multiple declarations. \CFA custom types make any special properties explicit to the language and its tool chain, \eg the language code-generator knows where to inject code and when it is unsafe to perform certain optimizations, and IDEs using simple parsing can find and manipulate types with special properties. The downside of this approach is that it makes custom types a special case in the language. Users wanting to extend custom types or build their own can only do so in ways offered by the language. Furthermore, implementing custom types without language supports may display the power of a programming language. \CFA blends the two approaches, providing custom type for idiomatic \CFA code, extending and building new custom types is still possible, similar to Java approach with builtin and library concurrency. Part of the mechanism to generalize custom types is the \CFA trait, \eg the definition for custom-type @coroutine@ is anything satisfying the trait @is_coroutine@, and this trait both enforces and restricts the coroutine-interface functions. \begin{cfa} trait is_coroutine( dtype T ) { void main( T & ); coroutine_desc * get_coroutine( T & ); }; forall( dtype T | is_coroutine(T) ) void $suspend$( T & ); forall( dtype T | is_coroutine(T) ) void resume( T & ); \end{cfa} Note, copying generators/coroutines/threads is not meaningful. For example, a coroutine retains its last resumer and suspends back to it; having a copy also suspend back to the same resumer is undefined semantics. Furthermore, two coroutines cannot logically execute on the same stack. A deep coroutine copy, which copies the stack, is also meaningless in an unmanaged language (no garbage collection), like C, because the stack may contain pointers to object within it that require updating for the copy. Now, the @dtype@ property provides no \emph{implicit} copying operations and the @is_coroutine@ trait provides no \emph{explicit} copying operations, so all coroutines must be passed by reference (pointer). The function definitions ensures there is a statically-typed @main@ function that is the starting point (first stack frame) of a coroutine, and a mechanism to get (read) the currently executing coroutine handle. The @main@ function has no return value or additional parameters because the coroutine type allows an arbitrary number of interface functions with corresponding arbitrary typed input/output values versus fixed ones. The advantage of this approach is that users can easily create different types of coroutines, \eg changing the memory layout of a coroutine is trivial when implementing the @get_coroutine@ function, and possibly redefining @suspend@ and @resume@. The \CFA custom-type @coroutine@ implicitly implements the getter and forward declarations required for implementing the coroutine main: \begin{cquote} \begin{tabular}{@{}ccc@{}} \begin{cfa} coroutine MyCor { int value; }; \end{cfa} & {\Large $\Rightarrow$} & \begin{tabular}{@{}ccc@{}} \begin{cfa} struct MyCor { int value; coroutine_desc cor; }; \end{cfa} & \begin{cfa} static inline coroutine_desc * get_coroutine( MyCor & this ) { return &this.cor; } \end{cfa} & \begin{cfa} void main( MyCor * this ); \end{cfa} \end{tabular} \end{tabular} \end{cquote} The combination of custom types and fundamental @trait@ description of these types allows an concise specification for programmers and tools, while more advanced programmers can have tighter control over memory layout and initialization. Figure~\ref{f:CoroutineMemoryLayout} shows different memory-layout options for a coroutine (where a task is similar). The coroutine handle is the @coroutine@ instance containing programmer specified global/communication variables. The coroutine descriptor contains all implicit declarations needed by the runtime, \eg @suspend@/@resume@, and can be part of the coroutine handle or separate.\footnote{ The coroutine handle is the @coroutine@ instance containing programmer specified global/communication variables across interface functions. The coroutine descriptor contains all implicit declarations needed by the runtime, \eg @suspend@/@resume@, and can be part of the coroutine handle or separate. The coroutine stack can appear in a number of locations and fixed or variable sized. Hence, the coroutine's stack could be a VLS\footnote{ We are examining variable-sized structures (VLS), where fields can be variable-sized structures or arrays. Once allocated, a VLS is fixed sized.} The coroutine stack can appear in a number of locations and forms, fixed or variable sized. Hence, the coroutine's stack could be a VLS on the allocating stack, provided the allocating stack is large enough. For stack allocation, allocation/deallocation only requires a few arithmetic operation to compute the size and adjust the stack point, modulo any constructor costs. For heap allocation, allocation/deallocation is an expensive heap operation (where the heap can be a shared resource), modulo any constructor costs. For heap stack allocation, it is also possible to use a split (segmented) stack calling-convention, available with gcc and clang, so the stack is variable sized. Currently, \CFA supports stack/heap allocated descriptors but only heap allocated stacks; split-stack allocation is under development. In \CFA debug-mode, a fixed-sized stack is terminated with a write-only page, which catches most stack overflows. on the allocating stack, provided the allocating stack is large enough. For a VLS stack allocation, allocation/deallocation is an inexpensive adjustment of the stack point, modulo any stack constructor costs (\eg initial frame setup). For heap stack allocation, allocation/deallocation is an expensive heap allocation (where the heap can be a shared resource), modulo any stack constructor costs. With heap stack allocation, it is also possible to use a split (segmented) stack calling-convention, available with gcc and clang, so the stack is variable sized. Currently, \CFA supports stack/heap allocated descriptors but only fixed-sized heap allocated stacks; In \CFA debug-mode, the fixed-sized stack is terminated with a write-only page, which catches most stack overflows. Experience teaching concurrency with \uC~\cite{CS343} shows fixed-sized stacks are rarely an issue for students. Split-stack allocation is under development but requires recompilation of existing code, which may be impossible. \begin{figure} \end{figure} Similarly, the canonical threading paradigm is often based on function pointers, \eg @pthreads@~\cite{Butenhof97}, \Csharp~\cite{Csharp}, Go~\cite{Go}, and Scala~\cite{Scala}. However, the generic thread-handle (identifier) is limited (few operations), unless it is wrapped in a custom type, as in the pthreads approach. \begin{cfa} void mycor( coroutine_t cid, void * arg ) { int * value = (int *)arg; $\C{// type unsafe, pointer-size only}$ // Coroutine body } int main() { int input = 0, output; coroutine_t cid = coroutine_create( &mycor, (void *)&input ); $\C{// type unsafe, pointer-size only}$ coroutine_resume( cid, (void *)input, (void **)&output ); $\C{// type unsafe, pointer-size only}$ } \end{cfa} Since the custom type is simple to write in \CFA and solves several issues, added support for function/lambda-based coroutines adds very little. Note, the type @coroutine_t@ must be an abstract handle to the coroutine, because the coroutine descriptor and its stack are non-copyable. Copying the coroutine descriptor results in copies being out of date with the current state of the stack. Correspondingly, copying the stack results is copies being out of date with the coroutine descriptor, and pointers in the stack being out of date to data on the stack. (There is no mechanism in C to find all stack-specific pointers and update them as part of a copy.) The selected approach is to use language support by introducing a new kind of aggregate (structure): \begin{cfa} coroutine Fibonacci { int fn; // communication variables }; \end{cfa} The @coroutine@ keyword means the compiler (and tool set) can find and inject code where needed. The downside of this approach is that it makes coroutine a special case in the language. Users wanting to extend coroutines or build their own for various reasons can only do so in ways offered by the language. Furthermore, implementing coroutines without language supports also displays the power of a programming language. While this is ultimately the option used for idiomatic \CFA code, coroutines and threads can still be constructed without language support. The reserved keyword simply eases use for the common case. Part of the mechanism to generalize coroutines is using a \CFA trait, which defines a coroutine as anything satisfying the trait @is_coroutine@, and this trait restricts the available set of coroutine-manipulation routines: \begin{cfa} trait is_coroutine( dtype T ) { void main( T & ); coroutine_desc * get_coroutine( T & ); }; forall( dtype T | is_coroutine(T) ) void suspend( T & ); forall( dtype T | is_coroutine(T) ) void resume( T & ); \end{cfa} The @dtype@ property provides no implicit copying operations and the @is_coroutine@ trait provides no explicit copying operations, so all coroutines must be passed by reference (pointer). The routine definitions ensures there is a statically-typed @main@ routine that is the starting point (first stack frame) of a coroutine, and a mechanism to get (read) the currently executing coroutine handle. The @main@ routine has no return value or additional parameters because the coroutine type allows an arbitrary number of interface routines with corresponding arbitrary typed input/output values versus fixed ones. The advantage of this approach is that users can easily create different types of coroutines, \eg changing the memory layout of a coroutine is trivial when implementing the @get_coroutine@ routine, and possibly redefining @suspend@ and @resume@. The \CFA keyword @coroutine@ implicitly implements the getter and forward declarations required for implementing the coroutine main: \begin{cquote} \begin{tabular}{@{}ccc@{}} \begin{cfa} coroutine MyCor { int value; }; \end{cfa} & {\Large $\Rightarrow$} & \begin{tabular}{@{}ccc@{}} \begin{cfa} struct MyCor { int value; coroutine_desc cor; }; \end{cfa} & \begin{cfa} static inline coroutine_desc * get_coroutine( MyCor & this ) { return &this.cor; } \end{cfa} & \begin{cfa} void main( MyCor * this ); \end{cfa} \end{tabular} \end{tabular} \end{cquote} The combination of these two approaches allows an easy and concise specification to coroutining (and concurrency) for normal users, while more advanced users have tighter control on memory layout and initialization. \section{Concurrency} \label{s:Concurrency} At its core, concurrency is based on multiple call-stacks and scheduling threads executing on these stacks. Multiple call stacks (or contexts) and a single thread of execution, called \newterm{coroutining}~\cite{Conway63,Marlin80}, does \emph{not} imply concurrency~\cite[\S~2]{Buhr05a}. In coroutining, the single thread is self-scheduling across the stacks, so execution is deterministic, \ie the execution path from input to output is fixed and predictable. A \newterm{stackless} coroutine executes on the caller's stack~\cite{Python} but this approach is restrictive, \eg preventing modularization and supporting only iterator/generator-style programming; a \newterm{stackful} coroutine executes on its own stack, allowing full generality. Only stackful coroutines are a stepping stone to concurrency. The transition to concurrency, even for execution with a single thread and multiple stacks, occurs when coroutines also context switch to a \newterm{scheduling oracle}, introducing non-determinism from the coroutine perspective~\cite[\S~3]{Buhr05a}. Therefore, a minimal concurrency system is possible using coroutines (see Section \ref{coroutine}) in conjunction with a scheduler to decide where to context switch next. Concurrency is nondeterministic scheduling of independent sequential execution-paths (threads), where each thread has its own stack. A single thread with multiple call stacks, \newterm{coroutining}~\cite{Conway63,Marlin80}, does \emph{not} imply concurrency~\cite[\S~2]{Buhr05a}. In coroutining, coroutines self-schedule the thread across stacks so execution is deterministic. (It is \emph{impossible} to generate a concurrency error when coroutining.) However, coroutines are a stepping stone towards concurrency. The transition to concurrency, even for a single thread with multiple stacks, occurs when coroutines context switch to a \newterm{scheduling coroutine}, introducing non-determinism from the coroutine perspective~\cite[\S~3]{Buhr05a}. Therefore, a minimal concurrency system requires coroutines \emph{in conjunction with a nondeterministic scheduler}. The resulting execution system now follows a cooperative threading-model, called \newterm{non-preemptive scheduling}. Because the scheduler is special, it can either be a stackless or stackful coroutine. Adding \newterm{preemption} introduces non-cooperative scheduling, where context switching occurs randomly between any two instructions often based on a timer interrupt, called \newterm{preemptive scheduling}. While a scheduler introduces uncertain execution among explicit context switches, preemption introduces uncertainty by introducing implicit context switches. Uncertainty gives the illusion of parallelism on a single processor and provides a mechanism to access and increase performance on multiple processors. The reason is that the scheduler/runtime have complete knowledge about resources and how to best utilized them. However, the introduction of unrestricted nondeterminism results in the need for \newterm{mutual exclusion} and \newterm{synchronization}, which restrict nondeterminism for correctness; otherwise, it is impossible to write meaningful concurrent programs. Optimal concurrent performance is often obtained by having as much nondeterminism as mutual exclusion and synchronization correctness allow. A scheduler can either be a stackless or stackful. For stackless, the scheduler performs scheduling on the stack of the current coroutine and switches directly to the next coroutine, so there is one context switch. For stackful, the current coroutine switches to the scheduler, which performs scheduling, and it then switches to the next coroutine, so there are two context switches. A stackful scheduler is often used for simplicity and security. Regardless of the approach used, a subset of concurrency related challenges start to appear. For the complete set of concurrency challenges to occur, the missing feature is \newterm{preemption}, where context switching occurs randomly between any two instructions, often based on a timer interrupt, called \newterm{preemptive scheduling}. While a scheduler introduces uncertainty in the order of execution, preemption introduces uncertainty about where context switches occur. Interestingly, uncertainty is necessary for the runtime (operating) system to give the illusion of parallelism on a single processor and increase performance on multiple processors. The reason is that only the runtime has complete knowledge about resources and how to best utilized them. However, the introduction of unrestricted non-determinism results in the need for \newterm{mutual exclusion} and \newterm{synchronization} to restrict non-determinism for correctness; otherwise, it is impossible to write meaningful programs. Optimal performance in concurrent applications is often obtained by having as much non-determinism as correctness allows. An important missing feature in C is threading\footnote{While the C11 standard defines a \protect\lstinline@threads.h@ header, it is minimal and defined as optional. As such, library support for threading is far from widespread. At the time of writing the paper, neither \protect\lstinline@gcc@ nor \protect\lstinline@clang@ support \protect\lstinline@threads.h@ in their standard libraries.}. In modern programming languages, a lack of threading is unacceptable~\cite{Sutter05, Sutter05b}, and therefore existing and new programming languages must have tools for writing efficient concurrent programs to take advantage of parallelism. As an extension of C, \CFA needs to express these concepts in a way that is as natural as possible to programmers familiar with imperative languages. Furthermore, because C is a system-level language, programmers expect to choose precisely which features they need and which cost they are willing to pay. Hence, concurrent programs should be written using high-level mechanisms, and only step down to lower-level mechanisms when performance bottlenecks are encountered. \subsection{Thread Interface} \label{threads} Both user and kernel threads are supported, where user threads provide concurrency and kernel threads provide parallelism. Like coroutines and for the same design reasons, the selected approach for user threads is to use language support by introducing a new kind of aggregate (structure) and a \CFA trait: \subsection{Thread} \label{s:threads} Threading needs the ability to start a thread and wait for its completion. A common API for this ability is @fork@ and @join@. \begin{cquote} \begin{tabular}{@{}c@{\hspace{3\parindentlnth}}c@{}} \begin{cfa} thread myThread { // communication variables }; \begin{tabular}{@{}lll@{}} \multicolumn{1}{c}{\textbf{Java}} & \multicolumn{1}{c}{\textbf{\Celeven}} & \multicolumn{1}{c}{\textbf{pthreads}} \\ \begin{cfa}[aboveskip=0pt,belowskip=0pt] class MyTask extends Thread {...} mytask t = new MyTask(...); t.start(); // start // concurrency t.join(); // wait \end{cfa} & \begin{cfa} trait is_thread( dtype T ) { void main( T & ); thread_desc * get_thread( T & ); void ^?{}( T & mutex ); }; \begin{cfa}[aboveskip=0pt,belowskip=0pt] class MyTask { ... } // functor MyTask mytask; thread t( mytask, ... ); // start // concurrency t.join(); // wait \end{cfa} & \begin{cfa}[aboveskip=0pt,belowskip=0pt] void * rtn( void * arg ) {...} pthread_t t; int i = 3; pthread_create( &t, rtn, (void *)i ); // start // concurrency pthread_join( t, NULL ); // wait \end{cfa} \end{tabular} \end{cquote} (The qualifier @mutex@ for the destructor parameter is discussed in Section~\ref{s:Monitor}.) Like a coroutine, the statically-typed @main@ routine is the starting point (first stack frame) of a user thread. The difference is that a coroutine borrows a thread from its caller, so the first thread resuming a coroutine creates an instance of @main@; whereas, a user thread receives its own thread from the runtime system, which starts in @main@ as some point after the thread constructor is run.\footnote{ The \lstinline@main@ routine is already a special routine in C, \ie where the program's initial thread begins, so it is a natural extension of this semantics to use overloading to declare \lstinline@main@s for user coroutines and threads.} No return value or additional parameters are necessary for this routine because the task type allows an arbitrary number of interface routines with corresponding arbitrary typed input/output values. \begin{comment} % put in appendix with coroutine version ??? As such the @main@ routine of a thread can be defined as \begin{cfa} thread foo {}; void main(foo & this) { sout | "Hello World!"; } \end{cfa} In this example, threads of type @foo@ start execution in the @void main(foo &)@ routine, which prints @"Hello World!".@ While this paper encourages this approach to enforce strongly typed programming, users may prefer to use the routine-based thread semantics for the sake of simplicity. With the static semantics it is trivial to write a thread type that takes a routine pointer as a parameter and executes it on its stack asynchronously. \begin{cfa} typedef void (*voidRtn)(int); thread RtnRunner { voidRtn func; int arg; }; void ?{}(RtnRunner & this, voidRtn inRtn, int arg) { this.func = inRtn; this.arg = arg; } void main(RtnRunner & this) { // thread starts here and runs the routine this.func( this.arg ); } void hello(/*unused*/ int) { sout | "Hello World!"; } \CFA has a simpler approach using a custom @thread@ type and leveraging declaration semantics (allocation/deallocation), where threads implicitly @fork@ after construction and @join@ before destruction. \begin{cfa} thread MyTask {}; void main( MyTask & this ) { ... } int main() { RtnRunner f = {hello, 42}; return 0? } \end{cfa} A consequence of the strongly typed approach to main is that memory layout of parameters and return values to/from a thread are now explicitly specified in the \textbf{API}. \end{comment} For user threads to be useful, it must be possible to start and stop the underlying thread, and wait for it to complete execution. While using an API such as @fork@ and @join@ is relatively common, such an interface is awkward and unnecessary. A simple approach is to use allocation/deallocation principles, and have threads implicitly @fork@ after construction and @join@ before destruction. \begin{cfa} thread World {}; void main( World & this ) { sout | "World!"; } MyTask team[10]; $\C[2.5in]{// allocate stack-based threads, implicit start after construction}$ // concurrency } $\C{// deallocate stack-based threads, implicit joins before destruction}$ \end{cfa} This semantic ensures a thread is started and stopped exactly once, eliminating some programming error, and scales to multiple threads for basic (termination) synchronization. For block allocation to arbitrary depth, including recursion, threads are created/destroyed in a lattice structure (tree with top and bottom). Arbitrary topologies are possible using dynamic allocation, allowing threads to outlive their declaration scope, identical to normal dynamically allocating. \begin{cfa} MyTask * factory( int N ) { ... return anew( N ); } $\C{// allocate heap-based threads, implicit start after construction}$ int main() { World w[10]; $\C{// implicit forks after creation}$ sout | "Hello "; $\C{// "Hello " and 10 "World!" printed concurrently}$ } $\C{// implicit joins before destruction}$ \end{cfa} This semantics ensures a thread is started and stopped exactly once, eliminating some programming error, and scales to multiple threads for basic (termination) synchronization. This tree-structure (lattice) create/delete from C block-structure is generalized by using dynamic allocation, so threads can outlive the scope in which they are created, much like dynamically allocating memory lets objects outlive the scope in which they are created. \begin{cfa} int main() { MyThread * heapLive; { MyThread blockLive; $\C{// fork block-based thread}$ heapLive = new( MyThread ); $\C{// fork heap-based thread}$ ... } $\C{// join block-based thread}$ ... delete( heapLive ); $\C{// join heap-based thread}$ } \end{cfa} The heap-based approach allows arbitrary thread-creation topologies, with respect to fork/join-style concurrency. MyTask * team = factory( 10 ); // concurrency delete( team ); $\C{// deallocate heap-based threads, implicit joins before destruction}\CRT$ } \end{cfa} Figure~\ref{s:ConcurrentMatrixSummation} shows concurrently adding the rows of a matrix and then totalling the subtotals sequentially, after all the row threads have terminated. The allocation/deallocation pattern appears unusual because allocated objects are immediately deallocated without any intervening code. However, for threads, the deletion provides implicit synchronization, which is the intervening code. While the subtotals are added in linear order rather than completion order, which slightly inhibits concurrency, the computation is restricted by the critical-path thread (\ie the thread that takes the longest), and so any inhibited concurrency is very small as totalling the subtotals is trivial. % While the subtotals are added in linear order rather than completion order, which slightly inhibits concurrency, the computation is restricted by the critical-path thread (\ie the thread that takes the longest), and so any inhibited concurrency is very small as totalling the subtotals is trivial. \begin{figure} thread Adder { int * row, cols, & subtotal; } $\C{// communication variables}$ void ?{}( Adder & adder, int row[], int cols, int & subtotal ) { adder.[ row, cols, &subtotal ] = [ row, cols, &subtotal ]; adder.[ row, cols, &subtotal ] = [ row, cols, &subtotal ]; } void main( Adder & adder ) with( adder ) { subtotal = 0; for ( int c = 0; c < cols; c += 1 ) { subtotal += row[c]; } subtotal = 0; for ( c; cols ) { subtotal += row[c]; } } int main() { const int rows = 10, cols = 1000; int matrix[rows][cols], subtotals[rows], total = 0; // read matrix Adder * adders[rows]; for ( int r = 0; r < rows; r += 1 ) { $\C{// start threads to sum rows}$ const int rows = 10, cols = 1000; int matrix[rows][cols], subtotals[rows], total = 0; // read matrix Adder * adders[rows]; for ( r; rows; ) { $\C{// start threads to sum rows}$ adders[r] = new( matrix[r], cols, &subtotals[r] ); } for ( int r = 0; r < rows; r += 1 ) { $\C{// wait for threads to finish}$ delete( adders[r] ); $\C{// termination join}$ total += subtotals[r]; $\C{// total subtotal}$ } sout | total; } \end{cfa} \caption{Concurrent Matrix Summation} } for ( r; rows ) { $\C{// wait for threads to finish}$ delete( adders[r] ); $\C{// termination join}$ total += subtotals[r]; $\C{// total subtotal}$ } sout | total; } \end{cfa} \caption{Concurrent matrix summation} \label{s:ConcurrentMatrixSummation} \end{figure} \subsection{Thread Implementation} Threads in \CFA are user level run by runtime kernel threads (see Section~\ref{s:Parallelism}), where user threads provide concurrency and kernel threads provide parallelism. Like coroutines, and for the same design reasons, \CFA provides a custom @thread@ type and a @trait@ to enforce and restrict the task-interface functions. \begin{cquote} \begin{tabular}{@{}c@{\hspace{3\parindentlnth}}c@{}} \begin{cfa} thread myThread { ... // declaration/communication variables }; \end{cfa} & \begin{cfa} trait is_thread( dtype T ) { void main( T & ); thread_desc * get_thread( T & ); void ^?{}( T & mutex ); }; \end{cfa} \end{tabular} \end{cquote} Like coroutines, the @dtype@ property prevents \emph{implicit} copy operations and the @is_coroutine@ trait provides no \emph{explicit} copy operations, so threads must be passed by reference (pointer). Similarly, the function definitions ensures there is a statically-typed @main@ function that is the thread starting point (first stack frame), a mechanism to get (read) the currently executing thread handle, and a special destructor to prevent deallocation while the thread is executing. (The qualifier @mutex@ for the destructor parameter is discussed in Section~\ref{s:Monitor}.) The difference between the coroutine and thread is that a coroutine borrows a thread from its caller, so the first thread resuming a coroutine creates the coroutine's stack starts running the coroutine main on the stack; whereas, a thread is scheduling for execution in @main@ immediately after its constructor is run. No return value or additional parameters are necessary for this function because the @thread@ type allows an arbitrary number of interface functions with corresponding arbitrary typed input/output values. \begin{comment} % put in appendix with coroutine version ??? As such the @main@ function of a thread can be defined as \begin{cfa} thread foo {}; void main(foo & this) { sout | "Hello World!"; } \end{cfa} In this example, threads of type @foo@ start execution in the @void main(foo &)@ function, which prints @"Hello World!".@ While this paper encourages this approach to enforce strongly typed programming, users may prefer to use the function-based thread semantics for the sake of simplicity. With the static semantics it is trivial to write a thread type that takes a function pointer as a parameter and executes it on its stack asynchronously. \begin{cfa} typedef void (*voidRtn)(int); thread RtnRunner { voidRtn func; int arg; }; void ?{}(RtnRunner & this, voidRtn inRtn, int arg) { this.func = inRtn; this.arg = arg; } void main(RtnRunner & this) { // thread starts here and runs the function this.func( this.arg ); } void hello(/*unused*/ int) { sout | "Hello World!"; } int main() { RtnRunner f = {hello, 42}; return 0? } \end{cfa} A consequence of the strongly typed approach to main is that memory layout of parameters and return values to/from a thread are now explicitly specified in the \textbf{API}. \end{comment} \section{Mutual Exclusion / Synchronization} Uncontrolled non-deterministic execution is meaningless. To reestablish meaningful execution requires mechanisms to reintroduce determinism, \ie restrict non-determinism, called mutual exclusion and synchronization, where mutual exclusion is an access-control mechanism on data shared by threads, and synchronization is a timing relationship among threads~\cite[\S~4]{Buhr05a}. Since many deterministic challenges appear with the use of mutable shared state, some languages/libraries disallow it, \eg Erlang~\cite{Erlang}, Haskell~\cite{Haskell}, Akka~\cite{Akka} (Scala). In these paradigms, interaction among concurrent objects is performed by stateless message-passing~\cite{Thoth,Harmony,V-Kernel} or other paradigms closely related to networking concepts, \eg channels~\cite{CSP,Go}. However, in call/return-based languages, these approaches force a clear distinction, \ie introduce a new programming paradigm between regular and concurrent computation, \eg routine call versus message passing. Hence, a programmer must learn and manipulate two sets of design patterns. Uncontrolled nondeterministic execution produces meaningless results. To produce meaningful execution requires clawing back some determinism using mutual exclusion and synchronization, where mutual exclusion provides access-control for threads using shared data, and synchronization is a timing relationship among threads~\cite[\S~4]{Buhr05a}. Some concurrent systems eliminate mutable shared-state by switching to stateless communication like message passing~\cite{Thoth,Harmony,V-Kernel,MPI} (Erlang, MPI), channels~\cite{CSP} (CSP,Go), actors~\cite{Akka} (Akka, Scala), or functional techniques (Haskell). However, in call/return-based languages, these approaches force a clear distinction, \ie introduce a new programming paradigm between regular and concurrent computation, \eg function call versus message passing. Hence, a programmer must learn and manipulate two sets of design/programming patterns. While this distinction can be hidden away in library code, effective use of the library still has to take both paradigms into account. In contrast, approaches based on stateful models more closely resemble the standard call/return programming-model, resulting in a single programming paradigm. A group of instructions manipulating a specific instance of shared data that must be performed atomically is called an (individual) \newterm{critical-section}~\cite{Dijkstra65}. The generalization is called a \newterm{group critical-section}~\cite{Joung00}, where multiple tasks with the same session may use the resource simultaneously, but different sessions may not use the resource simultaneously. The readers/writer problem~\cite{Courtois71} is an instance of a group critical-section, where readers have the same session and all writers have a unique session. \newterm{Mutual exclusion} enforces that the correct kind and number of threads are using a critical section. The readers/writer problem~\cite{Courtois71} is an instance of a group critical-section, where readers share a session but writers have a unique session. \newterm{Mutual exclusion} enforces the correct kind and number of threads use a critical section. However, many solutions exist for mutual exclusion, which vary in terms of performance, flexibility and ease of use. Synchronization enforces relative ordering of execution, and synchronization tools provide numerous mechanisms to establish these timing relationships. Low-level synchronization primitives offer good performance and flexibility at the cost of ease of use; higher-level mechanisms often simplify usage by adding better coupling between synchronization and data, \eg message passing, or offering a simpler solution to otherwise involved challenges, \eg barrier lock. Often synchronization is used to order access to a critical section, \eg ensuring a reader thread is the next kind of thread to enter a critical section. If a writer thread is scheduled for next access, but another reader thread acquires the critical section first, that reader \newterm{barged}. higher-level mechanisms often simplify usage by adding better coupling between synchronization and data, \eg receive-specific versus receive-any thread in message passing or offering specialized solutions, \eg barrier lock. Often synchronization is used to order access to a critical section, \eg ensuring a waiting writer thread enters the critical section before a calling reader thread. If the calling reader is scheduled before the waiting writer, the reader has \newterm{barged}. Barging can result in staleness/freshness problems, where a reader barges ahead of a writer and reads temporally stale data, or a writer barges ahead of another writer overwriting data with a fresh value preventing the previous value from ever being read (lost computation). Preventing or detecting barging is an involved challenge with low-level locks, which can be made much easier by higher-level constructs. This challenge is often split into two different approaches: barging avoidance and barging prevention. Algorithms that allow a barger, but divert it until later using current synchronization state (flags), are avoiding the barger; algorithms that preclude a barger from entering during synchronization in the critical section prevent barging completely. Techniques like baton-passing locks~\cite{Andrews89} between threads instead of unconditionally releasing locks is an example of barging prevention. Preventing or detecting barging is an involved challenge with low-level locks, which is made easier through higher-level constructs. This challenge is often split into two different approaches: barging avoidance and prevention. Algorithms that unconditionally releasing a lock for competing threads to acquire use barging avoidance during synchronization to force a barging thread to wait. algorithms that conditionally hold locks during synchronization, \eg baton-passing~\cite{Andrews89}, prevent barging completely. \label{s:Monitor} A \textbf{monitor} is a set of routines that ensure mutual exclusion when accessing shared state. More precisely, a monitor is a programming technique that binds mutual exclusion to routine scope, as opposed to locks, where mutual-exclusion is defined by acquire/release calls, independent of lexical context (analogous to block and heap storage allocation). The strong association with the call/return paradigm eases programmability, readability and maintainability, at a slight cost in flexibility and efficiency. Note, like coroutines/threads, both locks and monitors require an abstract handle to reference them, because at their core, both mechanisms are manipulating non-copyable shared-state. Copying a lock is insecure because it is possible to copy an open lock and then use the open copy when the original lock is closed to simultaneously access the shared data. Copying a monitor is secure because both the lock and shared data are copies, but copying the shared data is meaningless because it no longer represents a unique entity. As for coroutines/tasks, the @dtype@ property provides no implicit copying operations and the @is_monitor@ trait provides no explicit copying operations, so all locks/monitors must be passed by reference (pointer). A \textbf{monitor} is a set of functions that ensure mutual exclusion when accessing shared state. More precisely, a monitor is a programming technique that binds mutual exclusion to function scope, as opposed to locks, where mutual-exclusion is defined by acquire/release calls, independent of lexical context (analogous to block and heap storage allocation). The strong association with the call/return paradigm eases programming, comprehension, and maintenance, at a slight cost in flexibility and efficiency. \CFA uses a custom @monitor@ type and leverages declaration semantics (deallocation) to protect active or waiting threads in a monitor. The following is a \CFA monitor implementation of an atomic counter. \newpage \begin{cfa}[morekeywords=nomutex] monitor Aint { int cnt; }; $\C[4.25in]{// atomic integer counter}$ int ++?( Aint & mutex$$$_{opt}$$$ this ) with( this ) { return ++cnt; } $\C{// increment}$ int ?=?( Aint & mutex$$$_{opt}$$$ lhs, int rhs ) with( lhs ) { cnt = rhs; } $\C{// conversions}\CRT$ int ?=?( int & lhs, Aint & mutex$$$_{opt}$$$ rhs ) with( rhs ) { lhs = cnt; } \end{cfa} % The @Aint@ constructor, @?{}@, uses the \lstinline[morekeywords=nomutex]@nomutex@ qualifier indicating mutual exclusion is unnecessary during construction because an object is inaccessible (private) until after it is initialized. % (While a constructor may publish its address into a global variable, doing so generates a race-condition.) The prefix increment operation, @++?@, is normally @mutex@, indicating mutual exclusion is necessary during function execution, to protect the incrementing from race conditions, unless there is an atomic increment instruction for the implementation type. The assignment operators provide bi-directional conversion between an atomic and normal integer without accessing field @cnt@; these operations only need @mutex@, if reading/writing the implementation type is not atomic. The atomic counter is used without any explicit mutual-exclusion and provides thread-safe semantics, which is similar to the \CC template @std::atomic@. \begin{cfa} int i = 0, j = 0, k = 5; Aint x = { 0 }, y = { 0 }, z = { 5 }; $\C{// no mutex required}$ ++x; ++y; ++z; $\C{// safe increment by multiple threads}$ x = 2; y = i; z = k; $\C{// conversions}$ i = x; j = y; k = z; \end{cfa} \CFA monitors have \newterm{multi-acquire} semantics so the thread in the monitor may acquire it multiple times without deadlock, allowing recursion and calling other interface functions. \begin{cfa} monitor M { ... } m; void foo( M & mutex m ) { ... } $\C{// acquire mutual exclusion}$ void bar( M & mutex m ) { $\C{// acquire mutual exclusion}$ ... bar( m ); ... foo( m ); ... $\C{// reacquire mutual exclusion}$ } \end{cfa} \CFA monitors also ensure the monitor lock is always released regardless of how an acquiring function ends (normal or exceptional), and returning a shared variable is safe via copying before the lock is released. Similar safety is offered by explicit mechanisms like \CC RAII; monitor implicit safety ensures no programmer error. Furthermore, RAII mechansims cannot handle complex synchronization within a monitor, where the monitor lock may not be released on function exit but passed to an unblocking thread. RAII is purely a mutual-exclusion mechanism (see Section~\ref{s:Scheduling}). \subsection{Monitor Implementation} For the same design reasons, \CFA provides a custom @monitor@ type and a @trait@ to enforce and restrict the monitor-interface functions. \begin{cquote} \begin{tabular}{@{}c@{\hspace{3\parindentlnth}}c@{}} \begin{cfa} monitor M { ... // shared data }; \end{cfa} & \begin{cfa} trait is_monitor( dtype T ) { }; \end{cfa} \end{tabular} \end{cquote} Like before, the @dtype@ property prevents \emph{implicit} copy operations and the @is_monitor@ trait provides no \emph{explicit} copy operations, so monitors must be passed by reference (pointer). % Copying a lock is insecure because it is possible to copy an open lock and then use the open copy when the original lock is closed to simultaneously access the shared data. % Copying a monitor is secure because both the lock and shared data are copies, but copying the shared data is meaningless because it no longer represents a unique entity. Similarly, the function definitions ensures there is a mechanism to get (read) the currently executing monitor handle, and a special destructor to prevent deallocation if a thread using the shared data. The custom monitor type also inserts any locking vehicles needed to implement the monitor locking semnatics. \label{s:MutexAcquisition} While correctness implies a monitor's mutual exclusion is acquired and released, there are implementation options about when and where the locking/unlocking occurs. While the monitor lock provides mutual exclusion for shared data, there are implementation options for when and where the locking/unlocking occurs. (Much of this discussion also applies to basic locks.) For example, a monitor may need to be passed through multiple helper routines before it becomes necessary to acquire the monitor mutual-exclusion. \begin{cfa}[morekeywords=nomutex] monitor Aint { int cnt; }; $\C{// atomic integer counter}$ void ?{}( Aint & nomutex this ) with( this ) { cnt = 0; } $\C{// constructor}$ int ?=?( Aint & mutex$$$_{opt}$$$ lhs, int rhs ) with( lhs ) { cnt = rhs; } $\C{// conversions}$ void ?{}( int & this, Aint & mutex$$$_{opt}$$$ v ) { this = v.cnt; } int ?=?( int & lhs, Aint & mutex$$$_{opt}$$$ rhs ) with( rhs ) { lhs = cnt; } int ++?( Aint & mutex$$$_{opt}$$$ this ) with( this ) { return ++cnt; } $\C{// increment}$ \end{cfa} The @Aint@ constructor, @?{}@, uses the \lstinline[morekeywords=nomutex]@nomutex@ qualifier indicating mutual exclusion is unnecessary during construction because an object is inaccessible (private) until after it is initialized. (While a constructor may publish its address into a global variable, doing so generates a race-condition.) The conversion operators for initializing and assigning with a normal integer only need @mutex@, if reading/writing the implementation type is not atomic. Finally, the prefix increment operato, @++?@, is normally @mutex@ to protect the incrementing from race conditions, unless there is an atomic increment instruction for the implementation type. The atomic counter is used without any explicit mutual-exclusion and provides thread-safe semantics, which is similar to the \CC template @std::atomic@. \begin{cfa} Aint x, y, z; ++x; ++y; ++z; $\C{// safe increment by multiple threads}$ x = 2; y = 2; z = 2; $\C{// conversions}$ int i = x, j = y, k = z; i = x; j = y; k = z; \end{cfa} For maximum usability, monitors have \newterm{multi-acquire} semantics allowing a thread to acquire it multiple times without deadlock. \begin{cfa} monitor M { ... } m; void foo( M & mutex m ) { ... } $\C{// acquire mutual exclusion}$ void bar( M & mutex m ) { $\C{// acquire mutual exclusion}$ ... foo( m ); ... $\C{// reacquire mutual exclusion}$ } bar( m ); $\C{// nested monitor call}$ \end{cfa} For example, a monitor may be passed through multiple helper functions before it is necessary to acquire the monitor's mutual exclusion. The benefit of mandatory monitor qualifiers is self-documentation, but requiring both @mutex@ and \lstinline[morekeywords=nomutex]@nomutex@ for all monitor parameters is redundant. The next semantic decision is establishing which parameter \emph{types} may be qualified with @mutex@. Given: The following has monitor parameter types that are composed of multiple objects. \begin{cfa} monitor M { ... } int f1( M & mutex m ); int f2( M * mutex m ); int f3( M * mutex m[] ); int f4( stack( M * ) & mutex m ); \end{cfa} the issue is that some of these parameter types are composed of multiple objects. For @f1@, there is only a single parameter object. Adding indirection in @f2@ still identifies a single object. However, the matrix in @f3@ introduces multiple objects. While shown shortly, multiple acquisition is possible; however array lengths are often unknown in C. This issue is exacerbated in @f4@, where the data structure must be safely traversed to acquire all of its elements. To make the issue tractable, \CFA only acquires one monitor per parameter with at most one level of indirection. However, there is an ambiguity in the C type-system with respects to arrays. Is the argument for @f2@ a single object or an array of objects? If it is an array, only the first element of the array is acquired, which seems unsafe; hence, @mutex@ is disallowed for array parameters. \begin{cfa} int f1( M & mutex m ); $\C{// allowed: recommended case}$ int f2( M * mutex m ); $\C{// disallowed: could be an array}$ int f3( M mutex m[$\,$] ); $\C{// disallowed: array length unknown}$ int f4( M ** mutex m ); $\C{// disallowed: could be an array}$ int f5( M * mutex m[$\,$] ); $\C{// disallowed: array length unknown}$ \end{cfa} % Note, not all array routines have distinct types: @f2@ and @f3@ have the same type, as do @f4@ and @f5@. % However, even if the code generation could tell the difference, the extra information is still not sufficient to extend meaningfully the monitor call semantic. For object-oriented monitors, calling a mutex member \emph{implicitly} acquires mutual exclusion of the receiver object, @rec.foo(...)@. \CFA has no receiver, and hence, must use an explicit mechanism to specify which object acquires mutual exclusion. A positive consequence of this design decision is the ability to support multi-monitor routines. \begin{cfa} int f( M & mutex x, M & mutex y ); $\C{// multiple monitor parameter of any type}$ M m1, m2; f( m1, m2 ); \end{cfa} (While object-oriented monitors can be extended with a mutex qualifier for multiple-monitor members, no prior example of this feature could be found.) In practice, writing multi-locking routines that do not deadlock is tricky. Having language support for such a feature is therefore a significant asset for \CFA. The capability to acquire multiple locks before entering a critical section is called \newterm{bulk acquire} (see Section~\ref{s:Implementation} for implementation details). In the previous example, \CFA guarantees the order of acquisition is consistent across calls to different routines using the same monitors as arguments. This consistent ordering means acquiring multiple monitors is safe from deadlock. However, users can force the acquiring order. For example, notice the use of @mutex@/\lstinline[morekeywords=nomutex]@nomutex@ and how this affects the acquiring order: \begin{cfa} void foo( M & mutex m1, M & mutex m2 ); $\C{// acquire m1 and m2}$ int f1( M & mutex m ); $\C{// single parameter object}$ int f2( M * mutex m ); $\C{// single or multiple parameter object}$ int f3( M * mutex m[$\,$] ); $\C{// multiple parameter object}$ int f4( stack( M * ) & mutex m ); $\C{// multiple parameters object}$ \end{cfa} Function @f1@ has a single parameter object, while @f2@'s indirection could be a single or multi-element array, where static array size is often unknown in C. Function @f3@ has a multiple object matrix, and @f4@ a multiple object data structure. While shown shortly, multiple object acquisition is possible, but the number of objects must be statically known. Therefore, \CFA only acquires one monitor per parameter with at most one level of indirection, excluding pointers as it is impossible to statically determine the size. For object-oriented monitors, \eg Java, calling a mutex member \emph{implicitly} acquires mutual exclusion of the receiver object, @rec.foo(...)@. \CFA has no receiver, and hence, the explicit @mutex@ qualifier is used to specify which objects acquire mutual exclusion. A positive consequence of this design decision is the ability to support multi-monitor functions,\footnote{ While object-oriented monitors can be extended with a mutex qualifier for multiple-monitor members, no prior example of this feature could be found.} called \newterm{bulk acquire} (see Section~\ref{s:Implementation} for implementation details). \CFA guarantees acquisition order is consistent across calls to @mutex@ functions using the same monitors as arguments, so acquiring multiple monitors is safe from deadlock. Figure~\ref{f:BankTransfer} shows a trivial solution to the bank-account transfer problem using \CFA monitors with implicit locking and \CC with explicit locking. A \CFA programmer only has to manage when to acquire mutual exclusion; a \CC programmer must select the correct lock and acquisition mechanism from a panoply of locking options. Making good choices for common cases in \CFA simplifies the programming experience and enhances safety. \begin{figure} \centering \begin{lrbox}{\myboxA} \begin{cfa}[aboveskip=0pt,belowskip=0pt] monitor BankAccount { int balance; } b1 = { 0 }, b2 = { 0 }; void deposit( BankAccount & mutex b, int deposit ) with(b) { balance += deposit; } void transfer( BankAccount & mutex my, BankAccount & mutex your, int me2you ) { deposit( my, -me2you ); // debit deposit( your, me2you ); // credit } thread Person { BankAccount & b1, & b2; }; void main( Person & person ) with(person) { for ( 10_000_000 ) { if ( random() % 3 ) deposit( b1, 3 ); if ( random() % 3 ) transfer( b1, b2, 7 ); } } int main() { Person p1 = { b1, b2 }, p2 = { b2, b1 }; } // wait for threads to complete \end{cfa} \end{lrbox} \begin{lrbox}{\myboxB} \begin{cfa}[aboveskip=0pt,belowskip=0pt] struct BankAccount { recursive_mutex m; int balance = 0; } b1, b2; void deposit( BankAccount & b, int deposit ) { scoped_lock lock( b.m ); b.balance += deposit; } void transfer( BankAccount & my, BankAccount & your, int me2you ) { scoped_lock lock( my.m, your.m ); deposit( my, -me2you ); // debit deposit( your, me2you ); // credit } void person( BankAccount & b1, BankAccount & b2 ) { for ( int i = 0; i < 10$'$000$'$000; i += 1 ) { if ( random() % 3 ) deposit( b1, 3 ); if ( random() % 3 ) transfer( b1, b2, 7 ); } } int main() { thread p1(person, ref(b1), ref(b2)), p2(person, ref(b2), ref(b1)); p1.join(); p2.join(); } \end{cfa} \end{lrbox} \subfloat[\CFA]{\label{f:CFABank}\usebox\myboxA} \hspace{3pt} \vrule \hspace{3pt} \subfloat[\CC]{\label{f:C++Bank}\usebox\myboxB} \hspace{3pt} \caption{Bank transfer problem} \label{f:BankTransfer} \end{figure} Users can force the acquiring order, by using @mutex@/\lstinline[morekeywords=nomutex]@nomutex@. \begin{cfa} void foo( M & mutex m1, M & mutex m2 ); $\C{// acquire m1 and m2}$ void bar( M & mutex m1, M & /* nomutex */ m2 ) { $\C{// acquire m1}$ ... foo( m1, m2 ); ... $\C{// acquire m2}$ ... foo( m1, m2 ); ... $\C{// acquire m2}$ } void baz( M & /* nomutex */ m1, M & mutex m2 ) { $\C{// acquire m2}$ ... foo( m1, m2 ); ... $\C{// acquire m1}$ } \end{cfa} The multi-acquire semantics allows @bar@ or @baz@ to acquire a monitor lock and reacquire it in @foo@. In the calls to @bar@ and @baz@, the monitors are acquired in opposite order. However, such use leads to lock acquiring order problems resulting in deadlock~\cite{Lister77}, where detecting it requires dynamic tracking of monitor calls, and dealing with it requires rollback semantics~\cite{Dice10}. In \CFA, a safety aid is provided by using bulk acquire of all monitors to shared objects, whereas other monitor systems provide no aid. While \CFA provides only a partial solution, it handles many useful cases, \eg: \begin{cfa} monitor BankAccount { ... }; void deposit( BankAccount & mutex b, int deposit ); void transfer( BankAccount & mutex my, BankAccount & mutex your, int me2you ) { deposit( my, -me2you ); $\C{// debit}$ deposit( your, me2you ); $\C{// credit}$ } \end{cfa} This example shows a trivial solution to the bank-account transfer problem. Without multi- and bulk acquire, the solution to this problem requires careful engineering. ... foo( m1, m2 ); ... $\C{// acquire m1}$ } \end{cfa} The bulk-acquire semantics allow @bar@ or @baz@ to acquire a monitor lock and reacquire it in @foo@. In the calls to @bar@ and @baz@, the monitors are acquired in opposite order, resulting in deadlock. However, this case is the simplest instance of the \emph{nested-monitor problem}~\cite{Lister77}, where monitors are acquired in sequence versus bulk. Detecting the nested-monitor problem requires dynamic tracking of monitor calls, and dealing with it requires rollback semantics~\cite{Dice10}. \CFA does not deal with this fundamental problem. \label{mutex-stmt} The monitor call-semantics associate all locking semantics to routines. The monitor call-semantics associate all locking semantics with functions. Like Java, \CFA offers an alternative @mutex@ statement to reduce refactoring and naming. \begin{cquote} \begin{tabular}{@{}l@{\hspace{3\parindentlnth}}l@{}} \multicolumn{1}{c}{\textbf{\lstinline@mutex@ call}} & \multicolumn{1}{c}{\lstinline@mutex@ \textbf{statement}} \\ \begin{cfa} monitor M { ... }; \end{cfa} \\ \multicolumn{1}{c}{\textbf{routine call}} & \multicolumn{1}{c}{\lstinline@mutex@ \textbf{statement}} \end{tabular} \end{cquote} \section{Scheduling} \subsection{Scheduling} \label{s:Scheduling} While monitor mutual-exclusion provides safe access to shared data, the monitor data may indicate that a thread accessing it cannot proceed. For example, Figure~\ref{f:GenericBoundedBuffer} shows a bounded buffer that may be full/empty so produce/consumer threads must block. While monitor mutual-exclusion provides safe access to shared data, the monitor data may indicate that a thread accessing it cannot proceed, \eg a bounded buffer may be full/empty so produce/consumer threads must block. Leaving the monitor and trying again (busy waiting) is impractical for high-level programming. Monitors eliminate busy waiting by providing synchronization to schedule threads needing access to the shared data, where threads block versus spinning. Synchronization is generally achieved with internal~\cite{Hoare74} or external~\cite[\S~2.9.2]{uC++} scheduling, where \newterm{scheduling} defines which thread acquires the critical section next. \newterm{Internal scheduling} is characterized by each thread entering the monitor and making an individual decision about proceeding or blocking, while \newterm{external scheduling} is characterized by an entering thread making a decision about proceeding for itself and on behalf of other threads attempting entry. Finally, \CFA monitors do not allow calling threads to barge ahead of signalled threads, which simplifies synchronization among threads in the monitor and increases correctness. If barging is allowed, synchronization between a signaller and signallee is difficult, often requiring additional flags and multiple unblock/block cycles. In fact, signals-as-hints is completely opposite from that proposed by Hoare in the seminal paper on monitors: \begin{cquote} However, we decree that a signal operation be followed immediately by resumption of a waiting program, without possibility of an intervening procedure call from yet a third program. It is only in this way that a waiting program has an absolute guarantee that it can acquire the resource just released by the signalling program without any danger that a third program will interpose a monitor entry and seize the resource instead.~\cite[p.~550]{Hoare74} \end{cquote} Furthermore, \CFA concurrency has no spurious wakeup~\cite[\S~9]{Buhr05a}, which eliminates an implicit form of barging. Hence, a \CFA @wait@ statement is not enclosed in a @while@ loop that can cause thread starvation. Figure~\ref{f:BBInt} shows a \CFA generic bounded-buffer with internal scheduling, where producers/consumers enter the monitor, see the buffer is full/empty, and block on an appropriate condition lock, @full@/@empty@. The @wait@ routine atomically blocks the calling thread and implicitly releases the monitor lock(s) for all monitors in the routine's parameter list. The @wait@ function atomically blocks the calling thread and implicitly releases the monitor lock(s) for all monitors in the function's parameter list. The appropriate condition lock is signalled to unblock an opposite kind of thread after an element is inserted/removed from the buffer. Signalling is unconditional, because signalling an empty condition lock does nothing. \end{lrbox} \subfloat[Internal Scheduling]{\label{f:BBInt}\usebox\myboxA} \subfloat[Internal scheduling]{\label{f:BBInt}\usebox\myboxA} %\qquad \subfloat[External Scheduling]{\label{f:BBExt}\usebox\myboxB} \caption{Generic Bounded-Buffer} \subfloat[External scheduling]{\label{f:BBExt}\usebox\myboxB} \caption{Generic bounded-buffer} \label{f:GenericBoundedBuffer} \end{figure} Figure~\ref{f:BBExt} shows a \CFA generic bounded-buffer with external scheduling, where producers/consumers detecting a full/empty buffer block and prevent more producers/consumers from entering the monitor until there is a free/empty slot in the buffer. External scheduling is controlled by the @waitfor@ statement, which atomically blocks the calling thread, releases the monitor lock, and restricts the routine calls that can next acquire mutual exclusion. Figure~\ref{f:BBExt} shows a \CFA generic bounded-buffer with external scheduling, where producers/consumers detecting a full/\-empty buffer block and prevent more producers/consumers from entering the monitor until there is a free/empty slot in the buffer. External scheduling is controlled by the @waitfor@ statement, which atomically blocks the calling thread, releases the monitor lock, and restricts the function calls that can next acquire mutual exclusion. If the buffer is full, only calls to @remove@ can acquire the buffer, and if the buffer is empty, only calls to @insert@ can acquire the buffer. Threads making calls to routines that are currently excluded, block outside of (external to) the monitor on a calling queue, versus blocking on condition queues inside of (internal to) the monitor. Threads making calls to functions that are currently excluded, block outside of (external to) the monitor on a calling queue, versus blocking on condition queues inside of (internal to) the monitor. External scheduling allows users to wait for events from other threads without concern of unrelated events occurring. The mechnaism can be done in terms of control flow, \eg Ada @accept@ or \uC @_Accept@, or in terms of data, \eg Go channels. The mechnaism can be done in terms of control flow, \eg Ada @accept@ or \uC @_Accept@, or in terms of data, \eg Go @select@ on channels. While both mechanisms have strengths and weaknesses, this project uses a control-flow mechanism to stay consistent with other language semantics. Two challenges specific to \CFA for external scheduling are loose object-definitions (see Section~\ref{s:LooseObjectDefinitions}) and multiple-monitor routines (see Section~\ref{s:Multi-MonitorScheduling}). Two challenges specific to \CFA for external scheduling are loose object-definitions (see Section~\ref{s:LooseObjectDefinitions}) and multiple-monitor functions (see Section~\ref{s:Multi-MonitorScheduling}). For internal scheduling, non-blocking signalling (as in the producer/consumer example) is used when the signaller is providing the cooperation for a waiting thread; For signal scheduling, the @exchange@ condition is necessary to block the thread finding the match, while the matcher unblocks to take the opposite number, post its phone number, and unblock the partner. For signal-block scheduling, the implicit urgent-queue replaces the explict @exchange@-condition and @signal_block@ puts the finding thread on the urgent condition and unblocks the matcher. The dating service is an example of a monitor that cannot be written using external scheduling because it requires knowledge of calling parameters to make scheduling decisions, and parameters of waiting threads are unavailable; as well, an arriving thread may not find a partner and must wait, which requires a condition variable, and condition variables imply internal scheduling. Furthermore, barging corrupts the dating service during an exchange because a barger may also match and change the phone numbers, invalidating the previous exchange phone number. Putting loops around the @wait@s does not correct the problem; the solution must be restructured to account for barging. \begin{figure} \qquad \subfloat[\lstinline@signal_block@]{\label{f:DatingSignalBlock}\usebox\myboxB} \caption{Dating service. } \caption{Dating service} \label{f:DatingService} \end{figure} To override the implicit multi-monitor wait, specific mutex parameter(s) can be specified, \eg @wait( e, m1 )@. Wait statically verifies the released monitors are the acquired mutex-parameters so unconditional release is safe. While \CC supports bulk locking, @wait@ only accepts a single lock for a condition variable, so bulk locking with condition variables is asymmetric. Finally, a signaller, \begin{cfa} Similarly, for @waitfor( rtn )@, the default semantics is to atomically block the acceptor and release all acquired mutex types in the parameter list, \ie @waitfor( rtn, m1, m2 )@. To override the implicit multi-monitor wait, specific mutex parameter(s) can be specified, \eg @waitfor( rtn, m1 )@. @waitfor@ statically verifies the released monitors are the same as the acquired mutex-parameters of the given routine or routine pointer. To statically verify the released monitors match with the accepted routine's mutex parameters, the routine (pointer) prototype must be accessible. % When an overloaded routine appears in an @waitfor@ statement, calls to any routine with that name are accepted. @waitfor@ statically verifies the released monitors are the same as the acquired mutex-parameters of the given function or function pointer. To statically verify the released monitors match with the accepted function's mutex parameters, the function (pointer) prototype must be accessible. % When an overloaded function appears in an @waitfor@ statement, calls to any function with that name are accepted. % The rationale is that members with the same name should perform a similar function, and therefore, all should be eligible to accept a call. Overloaded routines can be disambiguated using a cast: Overloaded functions can be disambiguated using a cast \begin{cfa} void rtn( M & mutex m ); ... signal( e ); ... \end{cfa} The @wait@ only releases @m1@ so the signalling thread cannot acquire both @m1@ and @m2@ to enter @bar@ to get to the @signal@. While deadlock issues can occur with multiple/nesting acquisition, this issue results from the fact that locks, and by extension monitors, are not perfectly composable. Finally, an important aspect of monitor implementation is barging, \ie can calling threads barge ahead of signalled threads? If barging is allowed, synchronization between a signaller and signallee is difficult, often requiring multiple unblock/block cycles (looping around a wait rechecking if a condition is met). In fact, signals-as-hints is completely opposite from that proposed by Hoare in the seminal paper on monitors: \begin{quote} However, we decree that a signal operation be followed immediately by resumption of a waiting program, without possibility of an intervening procedure call from yet a third program. It is only in this way that a waiting program has an absolute guarantee that it can acquire the resource just released by the signalling program without any danger that a third program will interpose a monitor entry and seize the resource instead.~\cite[p.~550]{Hoare74} \end{quote} \CFA scheduling \emph{precludes} barging, which simplifies synchronization among threads in the monitor and increases correctness. Furthermore, \CFA concurrency has no spurious wakeup~\cite[\S~9]{Buhr05a}, which eliminates an implict form of barging. For example, there are no loops in either bounded buffer solution in Figure~\ref{f:GenericBoundedBuffer}. Supporting barging prevention as well as extending internal scheduling to multiple monitors is the main source of complexity in the design and implementation of \CFA concurrency. The @wait@ only releases @m1@ so the signalling thread cannot acquire @m1@ and @m2@ to enter @bar@ and @signal@ the condition. While deadlock can occur with multiple/nesting acquisition, this is a consequence of locks, and by extension monitors, not being perfectly composable. % Supporting barging prevention as well as extending internal scheduling to multiple monitors is the main source of complexity in the design and implementation of \CFA concurrency. \begin{cquote} \subfloat[Signalling Thread]{\label{f:SignallingThread}\usebox\myboxA} \hspace{2\parindentlnth} \hspace{3\parindentlnth} \subfloat[Waiting Thread (W1)]{\label{f:WaitingThread}\usebox\myboxB} \hspace{2\parindentlnth} In an object-oriented programming-language, a class includes an exhaustive list of operations. However, new members can be added via static inheritance or dynamic members, \eg JavaScript~\cite{JavaScript}. Similarly, monitor routines can be added at any time in \CFA, making it less clear for programmers and more difficult to implement. Similarly, monitor functions can be added at any time in \CFA, making it less clear for programmers and more difficult to implement. \begin{cfa} monitor M { ... }; void f( M & mutex m ); void g( M & mutex m ) { waitfor( f ); } $\C{// clear which f}$ void f( M & mutex m, int ); $\C{// different f}$ void h( M & mutex m ) { waitfor( f ); } $\C{// unclear which f}$ \end{cfa} Hence, the cfa-code for entering a monitor looks like: \begin{cfa} if ( $\textrm{\textit{monitor is free}}$ ) $\LstCommentStyle{// \color{red}enter}$ else if ( $\textrm{\textit{already own monitor}}$ ) $\LstCommentStyle{// \color{red}continue}$ else if ( $\textrm{\textit{monitor accepts me}}$ ) $\LstCommentStyle{// \color{red}enter}$ else $\LstCommentStyle{// \color{red}block}$ void g( M & mutex m ) { waitfor( f ); } $\C{// clear which f}$ void f( M & mutex m, int ); $\C{// different f}$ void h( M & mutex m ) { waitfor( f ); } $\C{// unclear which f}$ \end{cfa} Hence, the \CFA code for entering a monitor looks like: \begin{cfa} if ( $\textrm{\textit{monitor is free}}$ ) $\C{// enter}$ else if ( $\textrm{\textit{already own monitor}}$ ) $\C{// continue}$ else if ( $\textrm{\textit{monitor accepts me}}$ ) $\C{// enter}$ else $\C{// block}$ \end{cfa} For the first two conditions, it is easy to implement a check that can evaluate the condition in a few instructions. {\resizebox{0.45\textwidth}{!}{\input{ext_monitor.pstex_t}}} }% subfloat \caption{Monitor Implementation} \caption{Monitor implementation} \label{f:MonitorImplementation} \end{figure} For a fixed (small) number of mutex routines (\eg 128), the accept check reduces to a bitmask of allowed callers, which can be checked with a single instruction. This approach requires a unique dense ordering of routines with a small upper-bound and the ordering must be consistent across translation units. For object-oriented languages these constraints are common, but \CFA mutex routines can be added in any scope and are only visible in certain translation unit, precluding program-wide dense-ordering among mutex routines. For a fixed (small) number of mutex functions (\eg 128), the accept check reduces to a bitmask of allowed callers, which can be checked with a single instruction. This approach requires a unique dense ordering of functions with a small upper-bound and the ordering must be consistent across translation units. For object-oriented languages these constraints are common, but \CFA mutex functions can be added in any scope and are only visible in certain translation unit, precluding program-wide dense-ordering among mutex functions. Figure~\ref{fig:BulkMonitor} shows the \CFA monitor implementation. The mutex routine called is associated with each thread on the entry queue, while a list of acceptable routines is kept separately. The accepted list is a variable-sized array of accepted routine pointers, so the single instruction bitmask comparison is replaced by dereferencing a pointer followed by a (usually short) linear search. The mutex function called is associated with each thread on the entry queue, while a list of acceptable functions is kept separately. The accepted list is a variable-sized array of accepted function pointers, so the single instruction bitmask comparison is replaced by dereferencing a pointer followed by a (usually short) linear search. External scheduling, like internal scheduling, becomes significantly more complex for multi-monitor semantics. Even in the simplest case, new semantics needs to be established. \newpage \begin{cfa} monitor M { ... }; void f( M & mutex m1 ); void g( M & mutex m1, M & mutex m2 ) { waitfor( f ); $\C{\color{red}// pass m1 or m2 to f?}$ } void g( M & mutex m1, M & mutex m2 ) { waitfor( f ); } $\C{// pass m1 or m2 to f?}$ \end{cfa} The solution is for the programmer to disambiguate: \begin{cfa} waitfor( f, m2 ); $\C{\color{red}// wait for call to f with argument m2}$ \end{cfa} Both locks are acquired by routine @g@, so when routine @f@ is called, the lock for monitor @m2@ is passed from @g@ to @f@, while @g@ still holds lock @m1@. waitfor( f, m2 ); $\C{// wait for call to f with argument m2}$ \end{cfa} Both locks are acquired by function @g@, so when function @f@ is called, the lock for monitor @m2@ is passed from @g@ to @f@, while @g@ still holds lock @m1@. This behaviour can be extended to the multi-monitor @waitfor@ statement. \begin{cfa} monitor M { ... }; void f( M & mutex m1, M & mutex m2 ); void g( M & mutex m1, M & mutex m2 ) { waitfor( f, m1, m2 ); $\C{\color{red}// wait for call to f with arguments m1 and m2}$ } \end{cfa} Again, the set of monitors passed to the @waitfor@ statement must be entirely contained in the set of monitors already acquired by the accepting routine. void g( M & mutex m1, M & mutex m2 ) { waitfor( f, m1, m2 ); $\C{// wait for call to f with arguments m1 and m2}$ \end{cfa} Again, the set of monitors passed to the @waitfor@ statement must be entirely contained in the set of monitors already acquired by the accepting function. Also, the order of the monitors in a @waitfor@ statement is unimportant. \begin{figure} \lstDeleteShortInline@% \begin{tabular}{@{}l@{\hspace{\parindentlnth}}|@{\hspace{\parindentlnth}}l@{}} \begin{cfa} \centering \begin{lrbox}{\myboxA} \begin{cfa}[aboveskip=0pt,belowskip=0pt] monitor M1 {} m11, m12; monitor M2 {} m2; f( m12, m2 ); // cannot fulfil \end{cfa} & \begin{cfa} \end{lrbox} \begin{lrbox}{\myboxB} \begin{cfa}[aboveskip=0pt,belowskip=0pt] monitor M1 {} m11, m12; monitor M2 {} m2; f( m12, m2 ); // cannot fulfil \end{cfa} \end{tabular} \lstMakeShortInline@% \end{lrbox} \subfloat[Internal scheduling]{\label{f:InternalScheduling}\usebox\myboxA} \hspace{3pt} \vrule \hspace{3pt} \subfloat[External scheduling]{\label{f:ExternalScheduling}\usebox\myboxB} \caption{Unmatched \protect\lstinline@mutex@ sets} \label{f:UnmatchedMutexSets} \subsection{Extended \protect\lstinline@waitfor@} Figure~\ref{f:ExtendedWaitfor} show the extended form of the @waitfor@ statement to conditionally accept one of a group of mutex routines, with a specific action to be performed \emph{after} the mutex routine finishes. Figure~\ref{f:ExtendedWaitfor} show the extended form of the @waitfor@ statement to conditionally accept one of a group of mutex functions, with a specific action to be performed \emph{after} the mutex function finishes. For a @waitfor@ clause to be executed, its @when@ must be true and an outstanding call to its corresponding member(s) must exist. The \emph{conditional-expression} of a @when@ may call a routine, but the routine must not block or context switch. If there are multiple acceptable mutex calls, selection occurs top-to-bottom (prioritized) in the @waitfor@ clauses, whereas some programming languages with similar mechanisms accept non-deterministically for this case, \eg Go \lstinline[morekeywords=select]@select@. The \emph{conditional-expression} of a @when@ may call a function, but the function must not block or context switch. If there are multiple acceptable mutex calls, selection occurs top-to-bottom (prioritized) in the @waitfor@ clauses, whereas some programming languages with similar mechanisms accept nondeterministically for this case, \eg Go \lstinline[morekeywords=select]@select@. If some accept guards are true and there are no outstanding calls to these members, the acceptor is accept-blocked until a call to one of these members is made. If all the accept guards are false, the statement does nothing, unless there is a terminating @else@ clause with a true guard, which is executed instead. \begin{figure} \centering \begin{cfa} when ( $\emph{conditional-expression}$ ) $\C{// optional guard}$ else if ( C2 ) waitfor( mem2 ); or when ( C2 ) waitfor( mem2 ); \end{cfa} The left example accepts only @mem1@ if @C1@ is true or only @mem2@ if @C2@ is true. The left example only accepts @mem1@ if @C1@ is true or only @mem2@ if @C2@ is true. The right example accepts either @mem1@ or @mem2@ if @C1@ and @C2@ are true. \subsection{\protect\lstinline@mutex@ Threads} Threads in \CFA are monitors to allow direct communication among threads, \ie threads can have mutex routines that are called by other threads. Threads in \CFA can also be monitors to allow \emph{direct communication} among threads, \ie threads can have mutex functions that are called by other threads. Hence, all monitor features are available when using threads. Figure~\ref{f:DirectCommunication} shows a comparison of direct call communication in \CFA with direct channel communication in Go. (Ada provides a similar mechanism to the \CFA direct communication.) The program main in both programs communicates directly with the other thread versus indirect communication where two threads interact through a passive monitor. Both direct and indirection thread communication are valuable tools in structuring concurrent programs. \begin{figure} \centering \begin{lrbox}{\myboxA} \begin{cfa}[aboveskip=0pt,belowskip=0pt] struct Msg { int i, j; }; thread Gortn { int i; float f; Msg m; }; void mem1( Gortn & mutex gortn, int i ) { gortn.i = i; } void mem2( Gortn & mutex gortn, float f ) { gortn.f = f; } void mem3( Gortn & mutex gortn, Msg m ) { gortn.m = m; } void ^?{}( Gortn & mutex ) {} void main( Gortn & gortn ) with( gortn ) { // thread starts for () { waitfor( mem1, gortn ) sout | i; // wait for calls or waitfor( mem2, gortn ) sout | f; or waitfor( mem3, gortn ) sout | m.i | m.j; or waitfor( ^?{}, gortn ) break; } } int main() { Gortn gortn; $\C[2.0in]{// start thread}$ mem1( gortn, 0 ); $\C{// different calls}\CRT$ mem2( gortn, 2.5 ); mem3( gortn, (Msg){1, 2} ); } // wait for completion \end{cfa} \end{lrbox} \begin{lrbox}{\myboxB} \begin{Go}[aboveskip=0pt,belowskip=0pt] func main() { type Msg struct{ i, j int } ch1 := make( chan int ) ch2 := make( chan float32 ) ch3 := make( chan Msg ) hand := make( chan string ) shake := make( chan string ) gortn := func() { $\C[1.5in]{// thread starts}$ var i int; var f float32; var m Msg L: for { select { $\C{// wait for messages}$ case i = <- ch1: fmt.Println( i ) case f = <- ch2: fmt.Println( f ) case m = <- ch3: fmt.Println( m ) case <- hand: break L $\C{// sentinel}$ } } shake <- "SHAKE" $\C{// completion}$ } go gortn() $\C{// start thread}$ ch1 <- 0 $\C{// different messages}$ ch2 <- 2.5 ch3 <- Msg{1, 2} hand <- "HAND" $\C{// sentinel value}$ <- shake $\C{// wait for completion}\CRT$ } \end{Go} \end{lrbox} \subfloat[\CFA]{\label{f:CFAwaitfor}\usebox\myboxA} \hspace{3pt} \vrule \hspace{3pt} \subfloat[Go]{\label{f:Gochannel}\usebox\myboxB} \caption{Direct communication} \label{f:DirectCommunication} \end{figure} \begin{comment} The following shows an example of two threads directly calling each other and accepting calls from each other in a cycle. \begin{cfa} thread Ping {} pi; thread Pong {} po; void ping( Ping & mutex ) {} void pong( Pong & mutex ) {} int main() {} \end{cfa} \vspace{-0.8\baselineskip} \begin{tabular}{@{}l@{\hspace{3\parindentlnth}}l@{}} \begin{cfa} thread Ping {} pi; void ping( Ping & mutex ) {} void main( Ping & pi ) { for ( int i = 0; i < 10; i += 1 ) { for ( 10 ) { waitfor( ping, pi ); pong( po ); } } int main() {} \end{cfa} & \begin{cfa} thread Pong {} po; void pong( Pong & mutex ) {} void main( Pong & po ) { for ( int i = 0; i < 10; i += 1 ) { for ( 10 ) { ping( pi ); waitfor( pong, po ); } } \end{cfa} \end{tabular} % \end{figure} Note, the ping/pong threads are globally declared, @pi@/@po@, and hence, start (and possibly complete) before the program main starts. \end{comment} \subsection{Execution Properties} Table~\ref{t:ObjectPropertyComposition} shows how the \CFA high-level constructs cover 3 fundamental execution properties: thread, stateful function, and mutual exclusion. Case 1 is a basic object, with none of the new execution properties. Case 2 allows @mutex@ calls to Case 1 to protect shared data. Case 3 allows stateful functions to suspend/resume but restricts operations because the state is stackless. Case 4 allows @mutex@ calls to Case 3 to protect shared data. Cases 5 and 6 are the same as 3 and 4 without restriction because the state is stackful. Cases 7 and 8 are rejected because a thread cannot execute without a stackful state in a preemptive environment when context switching from the signal handler. Cases 9 and 10 have a stackful thread without and with @mutex@ calls. For situations where threads do not require direct communication, case 9 provides faster creation/destruction by eliminating @mutex@ setup. \begin{table}[b] \caption{Object property composition} \centering \label{t:ObjectPropertyComposition} \renewcommand{\arraystretch}{1.25} %\setlength{\tabcolsep}{5pt} \begin{tabular}{c|c|l|l} \multicolumn{2}{c|}{object properties} & \multicolumn{2}{c}{mutual exclusion} \\ \hline thread & stateful & \multicolumn{1}{c|}{No} & \multicolumn{1}{c}{Yes} \\ \hline \hline No & No & \textbf{1}\ \ \ aggregate type & \textbf{2}\ \ \ @monitor@ aggregate type \\ \hline No & Yes (stackless) & \textbf{3}\ \ \ @generator@ & \textbf{4}\ \ \ @monitor@ generator \\ \hline No & Yes (stackful) & \textbf{5}\ \ \ @coroutine@ & \textbf{6}\ \ \ @monitor@ @coroutine@ \\ \hline Yes & No / Yes (stackless) & \textbf{7}\ \ \ {\color{red}rejected} & \textbf{8}\ \ \ {\color{red}rejected} \\ \hline Yes & Yes (stackful) & \textbf{9}\ \ \ @thread@ & \textbf{10}\ \ @monitor@ @thread@ \\ \end{tabular} \end{table} For completeness and efficiency, \CFA provides a standard set of low-level locks: recursive mutex, condition, semaphore, barrier, \etc, and atomic instructions: @fetchAssign@, @fetchAdd@, @testSet@, @compareSet@, \etc. However, we strongly advocate using high-level concurrency mechanisms whenever possible. \section{Parallelism} \label{s:Parallelism} Historically, computer performance was about processor speeds. However, with heat dissipation being a direct consequence of speed increase, parallelism is the new source for increased performance~\cite{Sutter05, Sutter05b}. Now, high-performance applications must care about parallelism, which requires concurrency. Therefore, high-performance applications must care about parallelism, which requires concurrency. The lowest-level approach of parallelism is to use \newterm{kernel threads} in combination with semantics like @fork@, @join@, \etc. However, kernel threads are better as an implementation tool because of complexity and higher cost. \subsection{User Threads with Preemption} \subsection{User Threads} A direct improvement on kernel threads is user threads, \eg Erlang~\cite{Erlang} and \uC~\cite{uC++book}. This approach provides an interface that matches the language paradigms, more control over concurrency by the language runtime, and an abstract (and portable) interface to the underlying kernel threads across operating systems. This approach provides an interface that matches the language paradigms, gives more control over concurrency by the language runtime, and an abstract (and portable) interface to the underlying kernel threads across operating systems. In many cases, user threads can be used on a much larger scale (100,000 threads). Like kernel threads, user threads support preemption, which maximizes nondeterminism, but introduces the same concurrency errors: race, livelock, starvation, and deadlock. Like kernel threads, user threads support preemption, which maximizes nondeterminism, but increases the potential for concurrency errors: race, livelock, starvation, and deadlock. \CFA adopts user-threads as they represent the truest realization of concurrency and can build any other concurrency approach, \eg thread pools and actors~\cite{Actors}. \subsection{User Threads without Preemption (Fiber)} \label{s:fibers} A variant of user thread is \newterm{fibers}, which removes preemption, \eg Go~\cite{Go} @goroutine@s. A variant of user thread is \newterm{fibres}, which removes preemption, \eg Go~\cite{Go} @goroutine@s. Like functional programming, which removes mutation and its associated problems, removing preemption from concurrency reduces nondeterminism, making race and deadlock errors more difficult to generate. However, preemption is necessary for concurrency that relies on spinning, so there are a class of problems that cannot be programmed without preemption. However, preemption is necessary for concurrency that relies on spinning, so there are a class of problems that cannot be programmed with fibres. \subsection{Thread Pools} In contrast to direct threading is indirect \newterm{thread pools}, where small jobs (work units) are inserted into a work pool for execution. In contrast to direct threading is indirect \newterm{thread pools}, \eg Java @executor@, where small jobs (work units) are inserted into a work pool for execution. If the jobs are dependent, \ie interact, there is an implicit/explicit dependency graph that ties them together. While removing direct concurrency, and hence the amount of context switching, thread pools significantly limit the interaction that can occur among jobs. \centering \input{RunTimeStructure} \caption{\CFA Runtime Structure} \caption{\CFA Runtime structure} \label{f:RunTimeStructure} \end{figure} Preemption occurs on virtual processors rather than user threads, via operating-system interrupts. Thus virtual processors execute user threads, where preemption frequency applies to a virtual processor, so preemption occurs randomly across the executed user threads. Turning off preemption transforms user threads into fibers. Turning off preemption transforms user threads into fibres. \section{Implementation} \label{s:Implementation} Currently, \CFA has fixed-sized stacks, where the stack size can be set at coroutine/thread creation but with no subsequent growth. Schemes exist for dynamic stack-growth, such as stack copying and chained stacks. However, stack copying requires pointer adjustment to items on the stack, which is impossible without some form of garbage collection. As well, chained stacks require all modules be recompiled to use this feature, which breaks backward compatibility with existing C libraries. In the long term, it is likely C libraries will migrate to stack chaining to support concurrency, at only a minimal cost to sequential programs. Nevertheless, experience teaching \uC~\cite{CS343} shows fixed-sized stacks are rarely an issue in most concurrent programs. A primary implementation challenge is avoiding contention from dynamically allocating memory because of bulk acquire, \eg the internal-scheduling design is (almost) free of allocations. This array persists for the entire duration of the mutual exclusion and is used extensively for synchronization operations. To improve performance and simplicity, context switching occurs inside a routine call, so only callee-saved registers are copied onto the stack and then the stack register is switched; To improve performance and simplicity, context switching occurs inside a function call, so only callee-saved registers are copied onto the stack and then the stack register is switched; the corresponding registers are then restored for the other context. Note, the instruction pointer is untouched since the context switch is always inside the same routine. Unlike coroutines, threads do not context switch among each other; they context switch to the cluster scheduler. This method is a 2-step context-switch and provides a clear distinction between user and kernel code, where scheduling and other system operations happen. The alternative 1-step context-switch uses the \emph{from} thread's stack to schedule and then context-switches directly to the \emph{to} thread's stack. Experimental results (not presented) show the performance of these two approaches is virtually equivalent, because both approaches are dominated by locking to prevent a race condition. Note, the instruction pointer is untouched since the context switch is always inside the same function. Experimental results (not presented) for a stackless or stackful scheduler (1 versus 2 context switches) (see Section~\ref{s:Concurrency}) show the performance is virtually equivalent, because both approaches are dominated by locking to prevent a race condition. All kernel threads (@pthreads@) created a stack. However, on current UNIX systems: \begin{quote} \begin{cquote} A process-directed signal may be delivered to any one of the threads that does not currently have the signal blocked. If more than one of the threads has the signal unblocked, then the kernel chooses an arbitrary thread to which to deliver the signal. SIGNAL(7) - Linux Programmer's Manual \end{quote} \end{cquote} Hence, the timer-expiry signal, which is generated \emph{externally} by the UNIX kernel to the UNIX process, is delivered to any of its UNIX subprocesses (kernel threads). To ensure each virtual processor receives its own preemption signals, a discrete-event simulation is run on a special virtual processor, and only it sets and receives timer events. \label{results} To verify the implementation of the \CFA runtime, a series of microbenchmarks are performed comparing \CFA with other widely used programming languages with concurrency. Table~\ref{t:machine} shows the specifications of the computer used to run the benchmarks, and the versions of the software used in the comparison. To verify the implementation of the \CFA runtime, a series of microbenchmarks are performed comparing \CFA with Java OpenJDK-9, Go 1.9.2 and \uC 7.0.0. The benchmark computer is an AMD Opteron\texttrademark\ 6380 NUMA 64-core, 8 socket, 2.5 GHz processor, running Ubuntu 16.04.3 LTS and \uC and \CFA are compiled with gcc 6.3. \begin{comment} \begin{table} \centering \end{tabular} \end{table} All benchmarks are run using the following harness: \end{comment} All benchmarks are run using the following harness. \newpage \begin{cfa} unsigned int N = 10_000_000; #define BENCH( run ) Time before = getTimeNsec(); run; Duration result = (getTimeNsec() - before) / N; #define BENCH( run ) Time before = getTimeNsec(); run; Duration result = (getTimeNsec() - before) / N; \end{cfa} The method used to get time is @clock_gettime( CLOCK_REALTIME )@. \paragraph{Context-Switching} In procedural programming, the cost of a routine call is important as modularization (refactoring) increases. (In many cases, a compiler inlines routine calls to eliminate this cost.) In procedural programming, the cost of a function call is important as modularization (refactoring) increases. (In many cases, a compiler inlines function calls to eliminate this cost.) Similarly, when modularization extends to coroutines/tasks, the time for a context switch becomes a relevant factor. The coroutine context-switch is 2-step using resume/suspend, \ie from resumer to suspender and from suspender to resumer. The thread context switch is 2-step using yield, \ie enter and return from the runtime kernel. Figure~\ref{f:ctx-switch} shows the code for coroutines/threads with all results in Table~\ref{tab:ctx-switch}. The coroutine test is from resumer to suspender and from suspender to resumer, which is two context switches. The thread test is using yield to enter and return from the runtime kernel, which is two context switches. The difference in performance between coroutine and thread context-switch is the cost of scheduling for threads, whereas coroutines are self-scheduling. \begin{figure} Figure~\ref{f:ctx-switch} only shows the \CFA code for coroutines/threads (other systems are similar) with all results in Table~\ref{tab:ctx-switch}. \begin{multicols}{2} \lstset{language=CFA,moredelim=**[is][\color{red}]{@}{@},deletedelim=**[is][]{}{}} \newbox\myboxA \begin{lrbox}{\myboxA} \begin{cfa}[aboveskip=0pt,belowskip=0pt] coroutine C {} c; void main( C & ) { for ( ;; ) { @suspend();@ } } int main() { // coroutine test BENCH( for ( N ) { @resume( c );@ } ) sout | resultns; } int main() { // task test BENCH( for ( N ) { @yield();@ } ) sout | resultns; } \end{cfa} \captionof{figure}{\CFA context-switch benchmark} \label{f:ctx-switch} \columnbreak \vspace*{-16pt} \captionof{table}{Context switch comparison (nanoseconds)} \label{tab:ctx-switch} \begin{tabular}{@{}r*{3}{D{.}{.}{3.2}}@{}} \multicolumn{1}{@{}c}{} & \multicolumn{1}{c}{Median} &\multicolumn{1}{c}{Average} & \multicolumn{1}{c@{}}{Std Dev} \\ Kernel Thread & 333.5 & 332.96 & 4.1 \\ \CFA Coroutine & 49 & 48.68 & 0.47 \\ \CFA Thread & 105 & 105.57 & 1.37 \\ \uC Coroutine & 44 & 44 & 0 \\ \uC Thread & 100 & 99.29 & 0.96 \\ Goroutine & 145 & 147.25 & 4.15 \\ Java Thread & 373.5 & 375.14 & 8.72 \end{tabular} \end{multicols} \paragraph{Mutual-Exclusion} Mutual exclusion is measured by entering/leaving a critical section. For monitors, entering and leaving a monitor function is measured. To put the results in context, the cost of entering a non-inline function and the cost of acquiring and releasing a @pthread_mutex@ lock is also measured. Figure~\ref{f:mutex} shows the code for \CFA with all results in Table~\ref{tab:mutex}. Note, the incremental cost of bulk acquire for \CFA, which is largely a fixed cost for small numbers of mutex objects. \begin{multicols}{2} \lstset{language=CFA,moredelim=**[is][\color{red}]{@}{@},deletedelim=**[is][]{}{}} \begin{cfa} monitor M {} m1/*, m2, m3, m4*/; void __attribute__((noinline)) do_call( M & mutex m/*, m2, m3, m4*/ ) {} int main() { BENCH( for ( size_t i = 0; i < N; i += 1 ) { @resume( c );@ } ) sout | resultns; } \end{cfa} \end{lrbox} \newbox\myboxB \begin{lrbox}{\myboxB} \begin{cfa}[aboveskip=0pt,belowskip=0pt] int main() { BENCH( for ( size_t i = 0; i < N; i += 1 ) { @yield();@ } ) sout | resultns; } \end{cfa} \end{lrbox} \subfloat[Coroutine]{\usebox\myboxA} \quad \subfloat[Thread]{\usebox\myboxB} \captionof{figure}{\CFA context-switch benchmark} \label{f:ctx-switch} \centering \captionof{table}{Context switch comparison (nanoseconds)} \label{tab:ctx-switch} \bigskip \begin{tabular}{|r|*{3}{D{.}{.}{3.2}|}} \cline{2-4} \multicolumn{1}{c|}{} & \multicolumn{1}{c|}{Median} &\multicolumn{1}{c|}{Average} & \multicolumn{1}{c|}{Std Dev} \\ \hline Kernel Thread & 333.5 & 332.96 & 4.1 \\ \CFA Coroutine & 49 & 48.68 & 0.47 \\ \CFA Thread & 105 & 105.57 & 1.37 \\ \uC Coroutine & 44 & 44 & 0 \\ \uC Thread & 100 & 99.29 & 0.96 \\ Goroutine & 145 & 147.25 & 4.15 \\ Java Thread & 373.5 & 375.14 & 8.72 \\ \hline \end{tabular} \bigskip \bigskip \lstset{language=CFA,moredelim=**[is][\color{red}]{@}{@},deletedelim=**[is][]{}{}} \begin{cfa} monitor M { ... } m1/*, m2, m3, m4*/; void __attribute__((noinline)) do_call( M & mutex m/*, m2, m3, m4*/ ) {} int main() { BENCH( for( size_t i = 0; i < N; i += 1 ) { @do_call( m1/*, m2, m3, m4*/ );@ } ) for( N ) @do_call( m1/*, m2, m3, m4*/ );@ ) sout | resultns; } \label{f:mutex} \centering \columnbreak \vspace*{-16pt} \captionof{table}{Mutex comparison (nanoseconds)} \label{tab:mutex} \bigskip \begin{tabular}{|r|*{3}{D{.}{.}{3.2}|}} \cline{2-4} \multicolumn{1}{c|}{} & \multicolumn{1}{c|}{Median} &\multicolumn{1}{c|}{Average} & \multicolumn{1}{c|}{Std Dev} \\ \hline C routine & 2 & 2 & 0 \\ FetchAdd + FetchSub & 26 & 26 & 0 \\ Pthreads Mutex Lock & 31 & 31.71 & 0.97 \\ \uC @monitor@ member routine & 31 & 31 & 0 \\ \CFA @mutex@ routine, 1 argument & 46 & 46.68 & 0.93 \\ \CFA @mutex@ routine, 2 argument & 84 & 85.36 & 1.99 \\ \CFA @mutex@ routine, 4 argument & 158 & 161 & 4.22 \\ Java synchronized routine & 27.5 & 29.79 & 2.93 \\ \hline \begin{tabular}{@{}r*{3}{D{.}{.}{3.2}}@{}} \multicolumn{1}{@{}c}{} & \multicolumn{1}{c}{Median} &\multicolumn{1}{c}{Average} & \multicolumn{1}{c@{}}{Std Dev} \\ C function & 2 & 2 & 0 \\ FetchAdd + FetchSub & 26 & 26 & 0 \\ Pthreads Mutex Lock & 31 & 31.71 & 0.97 \\ \uC @monitor@ member rtn. & 31 & 31 & 0 \\ \CFA @mutex@ function, 1 arg. & 46 & 46.68 & 0.93 \\ \CFA @mutex@ function, 2 arg. & 84 & 85.36 & 1.99 \\ \CFA @mutex@ function, 4 arg. & 158 & 161 & 4.22 \\ Java synchronized function & 27.5 & 29.79 & 2.93 \end{tabular} \end{figure} \paragraph{Mutual-Exclusion} Mutual exclusion is measured by entering/leaving a critical section. For monitors, entering and leaving a monitor routine is measured. Figure~\ref{f:mutex} shows the code for \CFA with all results in Table~\ref{tab:mutex}. To put the results in context, the cost of entering a non-inline routine and the cost of acquiring and releasing a @pthread_mutex@ lock is also measured. Note, the incremental cost of bulk acquire for \CFA, which is largely a fixed cost for small numbers of mutex objects. \end{multicols} Java scheduling is significantly greater because the benchmark explicitly creates multiple thread in order to prevent the JIT from making the program sequential, \ie removing all locking. \begin{figure} \begin{multicols}{2} \lstset{language=CFA,moredelim=**[is][\color{red}]{@}{@},deletedelim=**[is][]{}{}} \begin{cfa} volatile int go = 0; condition c; monitor M { ... } m; void __attribute__((noinline)) do_call( M & mutex a1 ) { signal( c ); } monitor M { condition c; } m; void __attribute__((noinline)) do_call( M & mutex a1 ) { signal( c ); } thread T {}; void main( T & this ) { while ( go == 0 ) { yield(); } // wait for other thread to start while ( go == 0 ) { yield(); } while ( go == 1 ) { @do_call( m );@ } } int __attribute__((noinline)) do_wait( M & mutex m ) { int __attribute__((noinline)) do_wait( M & mutex m ) with(m) { go = 1; // continue other thread BENCH( for ( size_t i = 0; i < N; i += 1 ) { @wait( c );@ } ); BENCH( for ( N ) { @wait( c );@ } ); go = 0; // stop other thread sout | resultns; \label{f:int-sched} \centering \columnbreak \vspace*{-16pt} \captionof{table}{Internal-scheduling comparison (nanoseconds)} \label{tab:int-sched} \bigskip \begin{tabular}{|r|*{3}{D{.}{.}{5.2}|}} \cline{2-4} \multicolumn{1}{c|}{} & \multicolumn{1}{c|}{Median} &\multicolumn{1}{c|}{Average} & \multicolumn{1}{c|}{Std Dev} \\ \hline Pthreads Condition Variable & 6005 & 5681.43 & 835.45 \\ \uC @signal@ & 324 & 325.54 & 3,02 \\ \CFA @signal@, 1 @monitor@ & 368.5 & 370.61 & 4.77 \\ \CFA @signal@, 2 @monitor@ & 467 & 470.5 & 6.79 \\ \CFA @signal@, 4 @monitor@ & 700.5 & 702.46 & 7.23 \\ Java @notify@ & 15471 & 172511 & 5689 \\ \hline \begin{tabular}{@{}r*{3}{D{.}{.}{5.2}}@{}} \multicolumn{1}{@{}c}{} & \multicolumn{1}{c}{Median} & \multicolumn{1}{c}{Average} & \multicolumn{1}{c@{}}{Std Dev} \\ Pthreads Cond. Variable & 6005 & 5681.43 & 835.45 \\ \uC @signal@ & 324 & 325.54 & 3,02 \\ \CFA @signal@, 1 @monitor@ & 368.5 & 370.61 & 4.77 \\ \CFA @signal@, 2 @monitor@ & 467 & 470.5 & 6.79 \\ \CFA @signal@, 4 @monitor@ & 700.5 & 702.46 & 7.23 \\ Java @notify@ & 15471 & 172511 & 5689 \end{tabular} \end{figure} \end{multicols} Note, the incremental cost of bulk acquire for \CFA, which is largely a fixed cost for small numbers of mutex objects. \begin{figure} \begin{multicols}{2} \lstset{language=CFA,moredelim=**[is][\color{red}]{@}{@},deletedelim=**[is][]{}{}} \vspace*{-16pt} \begin{cfa} volatile int go = 0; monitor M { ... } m; monitor M {} m; thread T {}; void __attribute__((noinline)) do_call( M & mutex ) {} void __attribute__((noinline)) do_call( M & mutex ) {} void main( T & ) { while ( go == 0 ) { yield(); } // wait for other thread to start while ( go == 0 ) { yield(); } while ( go == 1 ) { @do_call( m );@ } } int __attribute__((noinline)) do_wait( M & mutex m ) { int __attribute__((noinline)) do_wait( M & mutex m ) { go = 1; // continue other thread BENCH( for ( size_t i = 0; i < N; i += 1 ) { @waitfor( do_call, m );@ } ) BENCH( for ( N ) { @waitfor( do_call, m );@ } ) go = 0; // stop other thread sout | resultns; \label{f:ext-sched} \centering \columnbreak \vspace*{-16pt} \captionof{table}{External-scheduling comparison (nanoseconds)} \label{tab:ext-sched} \bigskip \begin{tabular}{|r|*{3}{D{.}{.}{3.2}|}} \cline{2-4} \multicolumn{1}{c|}{} & \multicolumn{1}{c|}{Median} &\multicolumn{1}{c|}{Average} & \multicolumn{1}{c|}{Std Dev} \\ \hline \uC @_Accept@ & 358 & 359.11 & 2.53 \\ \CFA @waitfor@, 1 @monitor@ & 359 & 360.93 & 4.07 \\ \CFA @waitfor@, 2 @monitor@ & 450 & 449.39 & 6.62 \\ \CFA @waitfor@, 4 @monitor@ & 652 & 655.64 & 7.73 \\ \hline \begin{tabular}{@{}r*{3}{D{.}{.}{3.2}}@{}} \multicolumn{1}{@{}c}{} & \multicolumn{1}{c}{Median} &\multicolumn{1}{c}{Average} & \multicolumn{1}{c@{}}{Std Dev} \\ \uC @_Accept@ & 358 & 359.11 & 2.53 \\ \CFA @waitfor@, 1 @monitor@ & 359 & 360.93 & 4.07 \\ \CFA @waitfor@, 2 @monitor@ & 450 & 449.39 & 6.62 \\ \CFA @waitfor@, 4 @monitor@ & 652 & 655.64 & 7.73 \end{tabular} \bigskip \medskip \end{multicols} \paragraph{Object Creation} Object creation is measured by creating/deleting the specific kind of concurrent object. Figure~\ref{f:creation} shows the code for \CFA, with results in Table~\ref{tab:creation}. The only note here is that the call stacks of \CFA coroutines are lazily created, therefore without priming the coroutine to force stack creation, the creation cost is artificially low. \begin{multicols}{2} \lstset{language=CFA,moredelim=**[is][\color{red}]{@}{@},deletedelim=**[is][]{}{}} \begin{cfa} void main( MyThread & ) {} int main() { BENCH( for ( size_t i = 0; i < N; i += 1 ) { @MyThread m;@ } ) BENCH( for ( N ) { @MyThread m;@ } ) sout | resultns; } \label{f:creation} \centering \captionof{table}{Creation comparison (nanoseconds)} \columnbreak \vspace*{-16pt} \captionof{table}{Object creation comparison (nanoseconds)} \label{tab:creation} \bigskip \begin{tabular}{|r|*{3}{D{.}{.}{5.2}|}} \cline{2-4} \multicolumn{1}{c|}{} & \multicolumn{1}{c|}{Median} & \multicolumn{1}{c|}{Average} & \multicolumn{1}{c|}{Std Dev} \\ \hline Pthreads & 28091 & 28073.39 & 163.1 \\ \CFA Coroutine Lazy & 6 & 6.07 & 0.26 \\ \CFA Coroutine Eager & 520 & 520.61 & 2.04 \\ \CFA Thread & 2032 & 2016.29 & 112.07 \\ \uC Coroutine & 106 & 107.36 & 1.47 \\ \uC Thread & 536.5 & 537.07 & 4.64 \\ Goroutine & 3103 & 3086.29 & 90.25 \\ Java Thread & 103416.5 & 103732.29 & 1137 \\ \hline \begin{tabular}[t]{@{}r*{3}{D{.}{.}{5.2}}@{}} \multicolumn{1}{@{}c}{} & \multicolumn{1}{c}{Median} & \multicolumn{1}{c}{Average} & \multicolumn{1}{c@{}}{Std Dev} \\ Pthreads & 28091 & 28073.39 & 163.1 \\ \CFA Coroutine Lazy & 6 & 6.07 & 0.26 \\ \CFA Coroutine Eager & 520 & 520.61 & 2.04 \\ \CFA Thread & 2032 & 2016.29 & 112.07 \\ \uC Coroutine & 106 & 107.36 & 1.47 \\ \uC Thread & 536.5 & 537.07 & 4.64 \\ Goroutine & 3103 & 3086.29 & 90.25 \\ Java Thread & 103416.5 & 103732.29 & 1137 \\ \end{tabular} \end{figure} \paragraph{Object Creation} Object creation is measured by creating/deleting the specific kind of concurrent object. Figure~\ref{f:creation} shows the code for \CFA, with results in Table~\ref{tab:creation}. The only note here is that the call stacks of \CFA coroutines are lazily created, therefore without priming the coroutine to force stack creation, the creation cost is artificially low. \end{multicols} \section{Conclusion} This paper demonstrates a concurrency API that is simple, efficient, and able to build higher-level concurrency features. The approach provides concurrency based on a preemptive M:N user-level threading-system, executing in clusters, which encapsulate scheduling of work on multiple kernel threads providing parallelism. Advanced control-flow will always be difficult, especially when there is temporal ordering and nondeterminism. However, many systems exacerbate the difficulty through their presentation mechanisms. This paper shows it is possible to present a hierarchy of control-flow features, generator, coroutine, thread, and monitor, providing an integrated set of high-level, efficient, and maintainable control-flow features. Eliminated from \CFA are spurious wakeup and barging, which are nonintuitive and lead to errors, and having to work with a bewildering set of low-level locks and acquisition techniques. \CFA high-level race-free monitors and tasks provide the core mechanisms for mutual exclusion and synchronization, without having to resort to magic qualifiers like @volatile@/@atomic@. Extending these mechanisms to handle high-level deadlock-free bulk acquire across both mutual exclusion and synchronization is a unique contribution. The \CFA runtime provides concurrency based on a preemptive M:N user-level threading-system, executing in clusters, which encapsulate scheduling of work on multiple kernel threads providing parallelism. The M:N model is judged to be efficient and provide greater flexibility than a 1:1 threading model. High-level objects (monitor/task) are the core mechanism for mutual exclusion and synchronization. A novel aspect is allowing multiple mutex-objects to be accessed simultaneously reducing the potential for deadlock for this complex scenario. These concepts and the entire \CFA runtime-system are written in the \CFA language, demonstrating the expressiveness of the \CFA language. These concepts and the entire \CFA runtime-system are written in the \CFA language, extensively leveraging the \CFA type-system, which demonstrates the expressiveness of the \CFA language. Performance comparisons with other concurrent systems/languages show the \CFA approach is competitive across all low-level operations, which translates directly into good performance in well-written concurrent applications. C programmers should feel comfortable using these mechanisms for developing concurrent applications, with the ability to obtain maximum available performance by mechanisms at the appropriate level. C programmers should feel comfortable using these mechanisms for developing complex control-flow in applications, with the ability to obtain maximum available performance by selecting mechanisms at the appropriate level of need. \section{Future Work} While concurrency in \CFA has a strong start, development is still underway and there are missing features. While control flow in \CFA has a strong start, development is still underway to complete a number of missing features. \paragraph{Flexible Scheduling} Different scheduling algorithms can affect performance (both in terms of average and variation). However, no single scheduler is optimal for all workloads and therefore there is value in being able to change the scheduler for given programs. One solution is to offer various tweaking options, allowing the scheduler to be adjusted to the requirements of the workload. One solution is to offer various tuning options, allowing the scheduler to be adjusted to the requirements of the workload. However, to be truly flexible, a pluggable scheduler is necessary. Currently, the \CFA pluggable scheduler is too simple to handle complex scheduling, \eg quality of service and real-time, where the scheduler must interact with mutex objects to deal with issues like priority inversion. While monitors offer flexible and powerful concurrency for \CFA, other concurrency tools are also necessary for a complete multi-paradigm concurrency package. Examples of such tools can include futures and promises~\cite{promises}, executors and actors. These additional features are useful when monitors offer a level of abstraction that is inadequate for certain tasks. These additional features are useful for applications that can be constructed without shared data and direct blocking. As well, new \CFA extensions should make it possible to create a uniform interface for virtually all mutual exclusion, including monitors and low-level locks.
• ## doc/papers/concurrency/examples/PingPong.c
r6625727 typedef struct PingPong { const char * name; int N, i; struct PingPong * partner; int N, i; void * next; } PingPong; #define PPCtor( name, N ) { name, NULL, N, 0, NULL } #define PPCtor( name, N ) { name, N, 0, NULL, NULL } void comain( PingPong * pp ) __attribute__(( noinline )); void comain( PingPong * pp ) {
• ## doc/papers/concurrency/examples/Pingpong.cfa
r6625727 }; void ?{}( PingPong & this, const char * name, unsigned int N, PingPong & part ) { this.[name, N] = [name, N]; &this.part = ∂ } void ?{}( PingPong & this, const char * name, unsigned int N ) { this{ name, N, *0p }; // call first constructor } // void ?{}( PingPong & this, const char * name, unsigned int N, PingPong & part ) { // this.[name, N] = [name, N]; &this.part = ∂ // } // void ?{}( PingPong & this, const char * name, unsigned int N ) { // this{ name, N, *0p }; // call first constructor // } void cycle( PingPong & pingpong ) { resume( pingpong );
• ## doc/papers/concurrency/figures/corlayout.fig
r6625727 4 1 0 50 -1 0 10 0.0000 2 150 705 7050 825 stack$_2$\001 4 1 0 50 -1 0 10 0.0000 2 135 1230 2550 1125 descriptor pointer\001 4 2 0 50 -1 4 10 0.0000 2 120 675 1875 825 coroutine\001 4 1 0 50 -1 0 10 0.0000 2 105 375 2550 900 fields\001 4 2 0 50 -1 0 10 0.0000 2 105 465 1800 975 handle\001 4 1 0 50 -1 0 10 0.0000 2 150 1020 2550 900 program fields\001 4 2 0 50 -1 0 10 0.0000 2 135 690 1875 1525 descriptor\001 4 2 0 50 -1 0 10 0.0000 2 105 465 1875 975 handle\001
• ## doc/papers/concurrency/figures/ext_monitor.fig
r6625727 -2 1200 2 5 1 0 1 -1 -1 0 0 -1 0.000 0 1 0 0 1575.000 3450.000 1575 3150 1275 3450 1575 3750 5 1 0 1 -1 -1 0 0 -1 0.000 0 1 0 0 1575.000 4350.000 1575 4050 1275 4350 1575 4650 6 4275 1950 4575 2250 1 3 0 1 -1 -1 0 0 -1 0.000 1 0.0000 4425 2100 105 105 4425 2100 4530 2205 4 1 -1 0 0 0 10 0.0000 2 105 90 4425 2160 d\001 5 1 0 1 -1 -1 0 0 -1 0.000 0 1 0 0 1500.000 3600.000 1500 3300 1200 3600 1500 3900 5 1 0 1 -1 -1 0 0 -1 0.000 0 1 0 0 1500.000 4500.000 1500 4200 1200 4500 1500 4800 6 4200 2100 4500 2400 1 3 0 1 -1 -1 0 0 -1 0.000 1 0.0000 4350 2250 105 105 4350 2250 4455 2355 4 1 -1 0 0 0 10 0.0000 2 105 90 4350 2310 d\001 -6 6 4275 1650 4575 1950 1 3 0 1 -1 -1 0 0 -1 0.000 1 0.0000 4425 1800 105 105 4425 1800 4530 1905 4 1 -1 0 0 0 10 0.0000 2 105 90 4425 1860 b\001 6 4200 1800 4500 2100 1 3 0 1 -1 -1 0 0 -1 0.000 1 0.0000 4350 1950 105 105 4350 1950 4455 2055 4 1 -1 0 0 0 10 0.0000 2 105 90 4350 2010 b\001 -6 6 1495 5445 5700 5655 1 3 0 1 -1 -1 0 0 20 0.000 1 0.0000 1575 5550 80 80 1575 5550 1655 5630 1 3 0 1 -1 -1 0 0 -1 0.000 1 0.0000 2925 5550 105 105 2925 5550 3030 5655 1 3 0 1 -1 -1 0 0 4 0.000 1 0.0000 4425 5550 105 105 4425 5550 4530 5655 4 0 -1 0 0 0 12 0.0000 2 135 1035 3150 5625 blocked task\001 4 0 -1 0 0 0 12 0.0000 2 135 870 1725 5625 active task\001 4 0 -1 0 0 0 12 0.0000 2 135 1050 4650 5625 routine mask\001 6 1420 5595 5625 5805 1 3 0 1 -1 -1 0 0 20 0.000 1 0.0000 1500 5700 80 80 1500 5700 1580 5780 1 3 0 1 -1 -1 0 0 -1 0.000 1 0.0000 2850 5700 105 105 2850 5700 2955 5805 1 3 0 1 -1 -1 0 0 4 0.000 1 0.0000 4350 5700 105 105 4350 5700 4455 5805 4 0 -1 0 0 0 12 0.0000 2 135 1035 3075 5775 blocked task\001 4 0 -1 0 0 0 12 0.0000 2 135 870 1650 5775 active task\001 4 0 -1 0 0 0 12 0.0000 2 135 1050 4575 5775 routine mask\001 -6 6 3525 1800 3825 2400 1 3 0 1 -1 -1 1 0 4 0.000 1 0.0000 3675 1950 105 105 3675 1950 3780 1950 6 3450 1950 3750 2550 1 3 0 1 -1 -1 1 0 4 0.000 1 0.0000 3600 2100 105 105 3600 2100 3705 2100 2 2 0 1 0 7 50 -1 -1 0.000 0 0 -1 0 0 5 3525 1800 3825 1800 3825 2400 3525 2400 3525 1800 4 1 4 0 0 0 10 0.0000 2 105 120 3675 2010 Y\001 3450 1950 3750 1950 3750 2550 3450 2550 3450 1950 4 1 4 0 0 0 10 0.0000 2 105 120 3600 2160 Y\001 -6 6 3525 2100 3825 2400 1 3 0 1 -1 -1 1 0 4 0.000 1 0.0000 3675 2250 105 105 3675 2250 3780 2250 4 1 4 0 0 0 10 0.0000 2 105 120 3675 2295 X\001 6 3450 2250 3750 2550 1 3 0 1 -1 -1 1 0 4 0.000 1 0.0000 3600 2400 105 105 3600 2400 3705 2400 4 1 4 0 0 0 10 0.0000 2 105 120 3600 2445 X\001 -6 1 3 0 1 -1 -1 0 0 -1 0.000 1 0.0000 1725 3600 105 105 1725 3600 1830 3705 1 3 0 1 -1 -1 0 0 -1 0.000 1 0.0000 2025 3600 105 105 2025 3600 2130 3705 1 3 0 1 -1 -1 0 0 -1 0.000 1 0.0000 5025 3900 105 105 5025 3900 5130 4005 1 3 0 1 -1 -1 0 0 -1 0.000 1 0.0000 5325 3900 105 105 5325 3900 5430 4005 1 3 0 1 -1 -1 0 0 -1 0.000 1 0.0000 4425 2700 105 105 4425 2700 4530 2805 1 3 0 1 -1 -1 0 0 -1 0.000 1 0.0000 4425 2400 105 105 4425 2400 4530 2505 1 3 0 1 -1 -1 0 0 20 0.000 1 0.0000 3525 4575 80 80 3525 4575 3605 4655 1 3 0 1 -1 -1 0 0 -1 0.000 1 0.0000 1650 3750 105 105 1650 3750 1755 3855 1 3 0 1 -1 -1 0 0 -1 0.000 1 0.0000 1950 3750 105 105 1950 3750 2055 3855 1 3 0 1 -1 -1 0 0 -1 0.000 1 0.0000 4950 4050 105 105 4950 4050 5055 4155 1 3 0 1 -1 -1 0 0 -1 0.000 1 0.0000 5250 4050 105 105 5250 4050 5355 4155 1 3 0 1 -1 -1 0 0 -1 0.000 1 0.0000 4350 2850 105 105 4350 2850 4455 2955 1 3 0 1 -1 -1 0 0 -1 0.000 1 0.0000 4350 2550 105 105 4350 2550 4455 2655 1 3 0 1 -1 -1 0 0 20 0.000 1 0.0000 3450 4725 80 80 3450 4725 3530 4805 2 2 0 1 0 7 50 -1 -1 0.000 0 0 -1 0 0 5 2475 2925 3900 2925 3900 3225 2475 3225 2475 2925 2400 3075 3825 3075 3825 3375 2400 3375 2400 3075 2 1 0 1 -1 -1 0 0 -1 0.000 0 0 -1 0 0 4 1575 3750 2175 3750 2175 4050 1575 4050 1500 3900 2100 3900 2100 4200 1500 4200 2 1 0 1 -1 -1 0 0 -1 0.000 0 0 -1 0 0 3 1575 3450 2175 3450 2325 3675 1500 3600 2100 3600 2250 3825 2 1 0 1 -1 -1 0 0 -1 0.000 0 0 -1 0 0 2 2175 3150 2025 3375 2100 3300 1950 3525 2 1 0 1 -1 -1 0 0 -1 0.000 0 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3450 4125 variables\001 4 0 4 50 -1 0 11 0.0000 2 120 135 4075 2025 X\001 4 0 4 50 -1 0 11 0.0000 2 120 135 4075 2325 Y\001 4 0 4 50 -1 0 11 0.0000 2 120 135 4075 2625 Y\001 4 0 4 50 -1 0 11 0.0000 2 120 135 4075 2925 X\001 4 0 -1 0 0 3 12 0.0000 2 150 540 4950 4425 urgent\001 4 1 0 50 -1 0 11 0.0000 2 165 600 3075 3300 accepted\001 4 1 -1 0 0 0 12 0.0000 2 165 960 4275 1725 entry queue\001
• ## doc/papers/concurrency/figures/monitor.fig
r6625727 -2 1200 2 5 1 0 1 -1 -1 0 0 -1 0.000 0 1 0 0 1500.000 2700.000 1500 2400 1200 2700 1500 3000 5 1 0 1 -1 -1 0 0 -1 0.000 0 1 0 0 1500.000 3600.000 1500 3300 1200 3600 1500 3900 6 4200 1200 4500 1500 1 3 0 1 -1 -1 0 0 -1 0.000 1 0.0000 4350 1350 105 105 4350 1350 4455 1455 4 1 -1 0 0 0 10 0.0000 2 105 90 4350 1410 d\001 5 1 0 1 -1 -1 0 0 -1 0.000 0 1 0 0 1500.000 4500.000 1500 4200 1200 4500 1500 4800 6 2400 2400 2700 2700 1 3 0 1 -1 -1 1 0 4 0.000 1 0.0000 2550 2550 105 105 2550 2550 2655 2550 4 1 -1 0 0 0 10 0.0000 2 105 90 2550 2610 b\001 -6 6 4200 900 4500 1200 1 3 0 1 -1 -1 0 0 -1 0.000 1 0.0000 4350 1050 105 105 4350 1050 4455 1155 4 1 -1 0 0 0 10 0.0000 2 105 90 4350 1110 b\001 6 2400 2700 2700 3000 1 3 0 1 -1 -1 1 0 4 0.000 1 0.0000 2550 2850 105 105 2550 2850 2655 2850 4 1 -1 0 0 0 10 0.0000 2 75 75 2550 2895 a\001 -6 6 2400 1500 2700 1800 1 3 0 1 -1 -1 1 0 4 0.000 1 0.0000 2550 1650 105 105 2550 1650 2655 1650 4 1 -1 0 0 0 10 0.0000 2 105 90 2550 1710 b\001 6 3300 2400 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https://judge.yosupo.jp/problem/discrete_logarithm_mod | # Discrete Logarithm
AC一覧
## Problem Statement問題文
Each test case consist of $T$ cases.
Given $X, Y, M$. Print the minimum non-negative integer $K$ s.t. $X^K \equiv Y (\bmod M)$, or $-1$ if there is no such $K$.
We note that $0^0 = 1$ in this problem.
この問題は $T$ ケース与えられる。
$X, Y, M$ が与えられる。$X^K \equiv Y (\bmod M)$ なる非負整数 $K$ のうち、最小を答えよ(存在しない場合は-1)。
なお、$0^0 = 1$ とします。
## Constraints制約
• $1 \leq T \leq 100$
• $0 \leq X, Y < M$
• $1 \leq M \leq 10^9$
## Input入力
$T$
$X_0$ $Y_0$ $M_0$
$X_1$ $Y_1$ $M_1$
:
$X_{T - 1}$ $Y_{T - 1}$ $M_{T - 1}$
## Sampleサンプル
### # 1
7
2 1 5
4 7 10
8 6 10
5 2 11
5 9 11
0 0 1
0 2 4
0
-1
4
-1
4
0
-1
Timelimit: 10 secs
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https://marocstreetfood.com/alberta/convolution-of-two-signals-example.php | convolution web.ipac.caltech.edu. The 2-d convolution block computes the two-dimensional convolution of two input example 2. in convolution, the bias of all signals in the computer, 107 chapter 6 convolution convolution is a mathematical way of combining two signals to form a third signal. it is the single most important technique in digital.
## Convolution CCRMA
convolution web.ipac.caltech.edu. Figure 6-3 shows convolution being used for low-pass and high-pass filtering. the example input signal is the sum of two components: three cycles of a sine wave, what will i learn? how to calculate convolution of two discrete-time signals how to use scilab to obtain anвђ¦ by miguelangel2801.
Example: cnv = dsp.convolver the convolution of two signals is the integral that measures the amount of overlap of one signal as it is shifted over another explaining convolution using matlab graphical examples. definition of convolution and a good explanation of how to compute the convolution of two signals or
Signals, linear systems, and convolution and we will describe two different approaches for measuring for example, the step signal can be obtained as an the basics of convolution. convolution is a mathematical operation on two functions, for example signal or image processing.
Dsp operations on signals convolution example 2 в€’ find the convolution of two signals given by \$x_{1}(n) = \lbrace 3,-2, 2\rbrace \$ example 1: resolve the following discrete-time signals into impulses example 2 : find the convolution of the two sequences x[n] and h[n] given by,
Convolution has been a standard which is a very slow process given that every sample in each signal must be convolving a signal with two impulses put .. between two numbers. for example, representation of signals in terms of impulses; convolution sum representation for discrete-time linear,
The straightforward convolution of two finite-length signals \(x[k]\) in the following example an implementation of the fast convolution in python is shown. dsp operations on signals convolution example 2 в€’ find the convolution of two signals given by \$x_{1}(n) = \lbrace 3,-2, 2\rbrace \$
Convolution of continuous-time signals given two continuous-time signals x(t) and оѕ(t), we deп¬ѓne their computing convolution integrals: example 1 i have two signals represented by x and y values respectively. i have to find the convolution between the two signals. i am attaching the graph plotted from the two
## convolution web.ipac.caltech.edu
convolution web.ipac.caltech.edu. Convolution is used to linearly filter a signal the convolution z(n) of two discrete , in that correlation uses two signals to produce a for example, class, the convolution of two you will notice that in the above example, the signal and the kernel are we end this discussion with a final example of convolution..
NMath 10.2 Convolution and Correlation (.NET C# CSharp. Convolution of signals in 9/12/2011 visual example copied from wikipedia. review of convolution 9/12/2011 smith,, now let's convolute the two functions. so the convolution of f i showed that the cosine squared of tau-- i'm just using tau as an example-- is equal to 1/2 times.
## NMath 10.2 Convolution and Correlation (.NET C# CSharp
NMath 10.2 Convolution and Correlation (.NET C# CSharp. What will i learn? how to calculate convolution of two discrete-time signals how to use scilab to obtain anвђ¦ by miguelangel2801 What is convolution mathematically? image and signal convolution is like multiple of two things ,for example if you take any system,it takes input n it.
• convolution of two signals MATLAB Answers - MATLAB Central
• What is convolution mathematically? Quora
• Discrete time convolution is an operation on two discrete time signals defined by the example 1. recall that the consider the circular convolution of two 107 chapter 6 convolution convolution is a mathematical way of combining two signals to form a third signal. it is the single most important technique in digital
The basics of convolution. convolution is a mathematical operation on two functions, for example signal or image processing. i'm a little new to signal processing and i'm trying to wrap my head around convolutions. i know the definition of convolution for a continuous signal is \$\$y(t) = x(t
What will i learn? how to calculate convolution of two discrete-time signals how to use scilab to obtain anвђ¦ by miguelangel2801 figure 6-3 shows convolution being used for low-pass and high-pass filtering. the example input signal is the sum of two components: three cycles of a sine wave
Convolution of signals is a common signal-processing operation. it is often thought to be very difficult, but this does not need to be the case. convolution of continuous-time signals given two continuous-time signals x(t) and оѕ(t), we deп¬ѓne their computing convolution integrals: example 1
We will provide a mathematical framework to describe and analyze images and videos as two- and three-dimensional signals convolution with an animated example. convolution of signals is a common signal-processing operation. it is often thought to be very difficult, but this does not need to be the case.
3. convolution and linear filters example the input is the sequence of hammer blows and the output is the fourier transform of two signals to be 107 chapter 6 convolution convolution is a mathematical way of combining two signals to form a third signal. it is the single most important technique in digital
In the convolution expression, the integrand involves the product of two signals, both functions of the integration variable, v. one of the signals, x(t - v convolution the convolution of two signals and in may be denoted `` '' and defined by note that this is circular convolution convolution example 1: | 2020-10-31 13:13:50 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8682403564453125, "perplexity": 935.1234903372036}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107918164.98/warc/CC-MAIN-20201031121940-20201031151940-00615.warc.gz"} |
https://byjus.com/question-answer/the-adjacent-sides-of-a-parallelogram-have-length-20-cm-and-12-cm-the-distance/ | Question
# The adjacent sides of a parallelogram have length $$20\ cm$$ and $$12\ cm$$. The distance between the two shorter sides is $$5\ cm$$. Find the area of parallelogram and the distance between the longer sides.
Solution
## Area$$=b\times h$$$$=5\times 12=60㎠$$$$=20\times x$$$$\Rightarrow x=3㎝$$$$\therefore$$Area$$=60 \; {cm}^{2}$$; Distance between longer sides$$=3 \; cm$$Mathematics
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View More | 2022-01-21 14:02:36 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5654876828193665, "perplexity": 849.2398301984036}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320303385.49/warc/CC-MAIN-20220121131830-20220121161830-00198.warc.gz"} |
https://irzu.org/research/linux-android/bootx64-efi-didnt-seem-to-get-installed-and-no-uefi-boot-entry-was-created-for-debian/ | # bootx64.efi didn’t seem to get installed and no UEFI boot entry was created for Debian
On Mon, Nov 29, 2021 at 08:57:37AM +1100, deb...@voltagex.org wrote:
>Hi
>>>At least GRUB itself was installed in the EFI partition.
>>>
>>>> Shouldn't there normally be EFI/boot/bootx64.efi?
>>>
>>>Not by default. It happens only if you choose to install a copy of the boot
>>>loader in the removable device path. The option is available only in expert
>>>install or after changing priority for questions to low.
>>
>> Agreed. We deliberately do *not* install there by default as this can
>> cause other OSes not to boot. We try to be more accommodating than
>> Windows etc. :-/
>>
>Unless I did something wrong in the partitioning setup, there were no other
>operating systems. The only other EFI entry I had created in the firmware
>setup manually was one for the USB drive.
>
>What do you mean by removable device in this case? /boot/EFI is on
>the same NVMe drive as the Debian install itself.
The *normal* way for UEFI boot is via vendor paths and boot
variables. The alternate path is primarily designed for removable
media, where you won't have boot variables set to point to the right
files. Hopefully the docs I've written in
https://wiki.debian.org/UEFI#Booting_a_UEFI_machine_normally
and
https://wiki.debian.org/UEFI#Booting_from_removable_media
might help explain some more.
--
Steve McIntyre, Cambridge, UK. st...@einval.com
We don't need no education.
We don't need no thought control. | 2023-02-07 08:16:14 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.30702948570251465, "perplexity": 7840.882600266017}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764500392.45/warc/CC-MAIN-20230207071302-20230207101302-00658.warc.gz"} |
https://tug.org/pipermail/xetex/2010-March/016142.html | # [XeTeX] Diameter symbol
Peter Dyballa Peter_Dyballa at Web.DE
Mon Mar 15 16:14:54 CET 2010
Am 15.03.2010 um 15:01 schrieb David Cottenden:
>> Or are you looking for EMPTY SET, ∅, at U+2205? DIAMETER SIGN,
>> ⌀, is at U+2300.
> The latter is the one that I am after, but I know that my main body
> font doesn't include it. Is there a decent font which does? I'm
> looking to match with Chaparral Pro, if that's possible.
You could construct one from other glyphs in your darling font! (By
using o, O, or 0 at one size and / or FRACTION SLASH ⁄, U+2044, or
DIVISION SLASH ∕, U+2215, at another size at different height and
some negative kerning between them.)
On Linux you could use, I think, GNUcharmap to check particular glyphs
of the fonts you have. Once you've one, you can:
\catcode"2205=\active
\def⌀{{\ my_DIAMETER_font\char"2205}}
and then use in the text the symbol ⌀ and it will be set in that font.
BTW, some elderly LaTeX packages which handle special symbols, cannot
be used (easily) in XeTeX because they use rather strange and non-
Unicode fonts. Unicode encoded fonts are best suited for XeTeX.
--
Greetings
Pete
Globalisation – communism from above. | 2021-09-27 13:58:35 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.785686194896698, "perplexity": 11578.607026412788}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780058450.44/warc/CC-MAIN-20210927120736-20210927150736-00287.warc.gz"} |
https://newproxylists.com/tag/respond/ | ## VBA Does Not Respond – Exchange Stack Exchange Code
I'm starting to use VBA. What I'm working on is copying the details of a master file to multiple sheets in a spreadsheet.
I encounter a problem when I run my code, my excel file keeps flashing and not responding.
My code is as follows.
``````Sub Button1_Click()
Dim lastrow As Long
Dim erow As Long
Dim i As Integer
MsgBox ("Clear details of the destination sheets.")
Worksheets("15 Tax Class").Range("A7:V5000").Clear
Worksheets("14 Long Texts").Range("A7:I5000").Clear
Worksheets("12 UoM").Range("A7:V5000").Clear
Worksheets("11 Description").Range("A7:E5000").Clear
Worksheets("09 Sales").Range("A7:AS5000").Clear
Worksheets("03 Plant").Range("A7:FI5000").Clear
Worksheets("02 Client").Range("A7:CR5000").Clear
Worksheets("01 Header").Range("A7:U5000").Clear
MsgBox ("Please wait while template is being populated.")
lastrow = Worksheets("Masterfile").Cells(Rows.Count, 1).End(xlUp).Row
For i = 7 To lastrow
'01 Header Data
Worksheets("Masterfile").Cells(i, 1).Copy
erow = Worksheets("01 Header").Cells(Rows.Count, 1).End(xlUp).Row
Worksheets("Masterfile").Paste Destination:=Worksheets("01 Header").Cells(erow + 1, 1)
Worksheets("01 Header").Cells(i, 2) = "Z"
Worksheets("01 Header").Cells(i, 3) = "DIEN"
Worksheets("01 Header").Cells(i, 4) = "X"
Worksheets("01 Header").Cells(i, 5) = "X"
Worksheets("01 Header").Cells(i, 6) = "X"
Worksheets("01 Header").Cells(i, 7) = "X"
'02 Client Data
Worksheets("Masterfile").Cells(i, 1).Copy
erow = Worksheets("02 Client").Cells(Rows.Count, 1).End(xlUp).Row
Worksheets("Masterfile").Paste Destination:=Worksheets("02 Client").Cells(erow + 1, 1)
Worksheets("Masterfile").Range("D7:D5000").Copy
erow = Worksheets("02 Client").Cells(Rows.Count, 3).End(xlUp).Row
Worksheets("02 Client").Range("C7").PasteSpecial Paste:=xlPasteValues, SkipBlanks:=False
Worksheets("Masterfile").Range("E7:E5000").Copy
erow = Worksheets("02 Client").Cells(Rows.Count, 4).End(xlUp).Row
Worksheets("02 Client").Range("D7").PasteSpecial Paste:=xlPasteValues, SkipBlanks:=False
Worksheets("Masterfile").Range("F7:F5000").Copy
erow = Worksheets("02 Client").Cells(Rows.Count, 5).End(xlUp).Row
Worksheets("02 Client").Range("E7").PasteSpecial Paste:=xlPasteValues, SkipBlanks:=False
Worksheets("Masterfile").Range("G7:G5000").Copy
erow = Worksheets("02 Client").Cells(Rows.Count, 5).End(xlUp).Row
Worksheets("02 Client").Range("F7").PasteSpecial Paste:=xlPasteValues, SkipBlanks:=False
Worksheets("02 Client").Cells(i, 82) = "NORM"
'03 Plant Data
Worksheets("Masterfile").Cells(i, 1).Copy
erow = Worksheets("03 Plant").Cells(Rows.Count, 1).End(xlUp).Row
Worksheets("Masterfile").Paste Destination:=Worksheets("03 Plant").Cells(erow + 1, 1)
Worksheets("Masterfile").Range("H7:H5000").Copy
erow = Worksheets("03 Plant").Cells(Rows.Count, 2).End(xlUp).Row
Worksheets("03 Plant").Range("B7").PasteSpecial Paste:=xlPasteValues, SkipBlanks:=False
Worksheets("Masterfile").Range("I7:I5000").Copy
erow = Worksheets("03 Plant").Cells(Rows.Count, 75).End(xlUp).Row
Worksheets("03 Plant").Range("BW7").PasteSpecial Paste:=xlPasteValues, SkipBlanks:=False
'09 Sales Data
Worksheets("Masterfile").Cells(i, 1).Copy
erow = Worksheets("09 Sales").Cells(Rows.Count, 1).End(xlUp).Row
Worksheets("Masterfile").Paste Destination:=Worksheets("09 Sales").Cells(erow + 1, 1)
Worksheets("Masterfile").Range("J7:J5000").Copy
erow = Worksheets("09 Sales").Cells(Rows.Count, 2).End(xlUp).Row
Worksheets("09 Sales").Range("B7").PasteSpecial Paste:=xlPasteValues, SkipBlanks:=False
Worksheets("Masterfile").Range("K7:K5000").Copy
erow = Worksheets("09 Sales").Cells(Rows.Count, 3).End(xlUp).Row
Worksheets("09 Sales").Range("C7").PasteSpecial Paste:=xlPasteValues, SkipBlanks:=False
Worksheets("Masterfile").Range("L7:L5000").Copy
erow = Worksheets("09 Sales").Cells(Rows.Count, 17).End(xlUp).Row
Worksheets("09 Sales").Range("Q7").PasteSpecial Paste:=xlPasteValues, SkipBlanks:=False
Worksheets("Masterfile").Range("O7:O5000").Copy
erow = Worksheets("09 Sales").Cells(Rows.Count, 25).End(xlUp).Row
Worksheets("09 Sales").Range("Y7").PasteSpecial Paste:=xlPasteValues, SkipBlanks:=False
Worksheets("Masterfile").Range("P7:P5000").Copy
erow = Worksheets("09 Sales").Cells(Rows.Count, 26).End(xlUp).Row
Worksheets("09 Sales").Range("Z7").PasteSpecial Paste:=xlPasteValues, SkipBlanks:=False
Worksheets("Masterfile").Range("Q7:Q5000").Copy
erow = Worksheets("09 Sales").Cells(Rows.Count, 27).End(xlUp).Row
Worksheets("09 Sales").Range("AA7").PasteSpecial Paste:=xlPasteValues, SkipBlanks:=False
Worksheets("Masterfile").Range("R7:R5000").Copy
erow = Worksheets("09 Sales").Cells(Rows.Count, 27).End(xlUp).Row
Worksheets("09 Sales").Range("AB7").PasteSpecial Paste:=xlPasteValues, SkipBlanks:=False
Worksheets("Masterfile").Range("S7:S5000").Copy
erow = Worksheets("09 Sales").Cells(Rows.Count, 27).End(xlUp).Row
Worksheets("09 Sales").Range("AC7").PasteSpecial Paste:=xlPasteValues, SkipBlanks:=False
'11 Description Data
Worksheets("Masterfile").Cells(i, 1).Copy
erow = Worksheets("11 Description").Cells(Rows.Count, 1).End(xlUp).Row
Worksheets("Masterfile").Paste Destination:=Worksheets("11 Description").Cells(erow + 1, 1)
Worksheets("11 Description").Cells(i, 2) = "EN"
Worksheets("Masterfile").Range("B7:B5000").Copy
erow = Worksheets("11 Description").Cells(Rows.Count, 4).End(xlUp).Row
Worksheets("11 Description").Range("D7").PasteSpecial Paste:=xlPasteValues, SkipBlanks:=False
'12 UoM Data
Worksheets("Masterfile").Cells(i, 1).Copy
erow = Worksheets("12 UoM").Cells(Rows.Count, 1).End(xlUp).Row
Worksheets("Masterfile").Paste Destination:=Worksheets("12 UoM").Cells(erow + 1, 1)
Worksheets("Masterfile").Range("L7:L5000").Copy
erow = Worksheets("12 UoM").Cells(Rows.Count, 2).End(xlUp).Row
Worksheets("12 UoM").Range("B7").PasteSpecial Paste:=xlPasteValues, SkipBlanks:=False
Worksheets("Masterfile").Range("M7:M5000").Copy
erow = Worksheets("12 UoM").Cells(Rows.Count, 4).End(xlUp).Row
Worksheets("12 UoM").Range("D7").PasteSpecial Paste:=xlPasteValues, SkipBlanks:=False
Worksheets("Masterfile").Range("N7:N5000").Copy
erow = Worksheets("12 UoM").Cells(Rows.Count, 5).End(xlUp).Row
Worksheets("12 UoM").Range("E7").PasteSpecial Paste:=xlPasteValues, SkipBlanks:=False
'14 Long Texts Data
Worksheets("Masterfile").Cells(i, 1).Copy
erow = Worksheets("14 Long Texts").Cells(Rows.Count, 1).End(xlUp).Row
Worksheets("Masterfile").Paste Destination:=Worksheets("14 Long Texts").Cells(erow + 1, 1)
Worksheets("Masterfile").Paste Destination:=Worksheets("14 Long Texts").Cells(erow + 1, 3)
Worksheets("14 Long Texts").Cells(i, 2) = "MATERIAL"
Worksheets("14 Long Texts").Cells(i, 4) = "GRUN"
Worksheets("14 Long Texts").Cells(i, 5) = "EN"
Worksheets("Masterfile").Range("C7:C5000").Copy
erow = Worksheets("14 Long Texts").Cells(Rows.Count, 8).End(xlUp).Row
Worksheets("14 Long Texts").Range("H7").PasteSpecial Paste:=xlPasteValues, SkipBlanks:=False
'15 Tax Class Data
Worksheets("Masterfile").Cells(i, 1).Copy
erow = Worksheets("15 Tax Class").Cells(Rows.Count, 1).End(xlUp).Row
Worksheets("Masterfile").Paste Destination:=Worksheets("15 Tax Class").Cells(erow + 1, 1)
Worksheets("15 Tax Class").Cells(i, 2) = "PH"
Worksheets("15 Tax Class").Cells(i, 4) = "MWST"
Worksheets("15 Tax Class").Cells(i, 5) = "1"
Next i
MsgBox ("Generating of Upload Template already done!")
End Sub
``````
Thank you in advance.
## hexupload.net in the comment form, it does not respond
Hello.
Which contacts have the administrator of http://hexupload.net file hosting?
In the comment form, he does not answer.
Gmail email domain
## What keywords can you use to respond to users' comments during their interviews?
In interviews with users, I often say "very interesting" or "it's an interesting point" or "very useful" in response to user responses. "Interesting" is a perfect word because it has a neutral meaning, does not really mean good or bad. Sometimes it would be very repetitive to continue to say those who participate in a long conversation. Would you like to suggest other expressions or words to answer users?
## design – Create one or more objects to respond to two sets of data constituting the same entity, but one of which has an additional element
When I call an API endpoint, I have a class to create an object with the response. The object maps directly to the end elements so that I have all the values, with properties, getters, and setters.
In my problematic scenario, I want to define one or more objects for two endpoints that represent the same entity. However, one of them has an extra element (logical for the structure of the API) that prevents me from creating a single object.
These are the "Owner" and "Manager" entities that I want to manage (for example, ignore the need for an object on `properties` period):
POINT FIN 1: `/landlord/{id}/`:
Owner's response:
``````(
'id' => 10,
'first_name' => 'Bob',
'surname' => 'Jones',
'properties = (
0 = (
'id' => 20,
'street' => 'whatever',
'postcode' => 'jj1144ll',
),
)
)
``````
FINAL POINT 2: `/properties/{id}/`:
Property Response:
``````(
'id' => 20,
'street' => 'whatever',
'postcode' => 'jj1144ll',
'manager' = ( // This is a "Landlord"
'id' => 10,
'first_name' => 'Bob',
'surname' => 'Jones',
)
)
``````
The answers are simplified but "Owner" has a lot other things, "date of birth", "address", etc.
The problem:
"Manager" on `/properties{id}/` has all the same elements to represent an owner but `/landlord/{id}/` at `Properties` table that `/properties/{id}/` of course not, because is a property.
So, I can not create a single object for both, because a manager is an owner and should be an object to represent that. There is this chart on the endpoint of the owner.
I am very open to suggestions but here is what I have proposed so far:
OPTION 1 – An individual class for each end point.
Classes / objects:
PRO: "owner" can have the array "properties" on his object, and "manager" just does not have it.
Problems:
1. Duplication – Both classes will have properties, getter,
setter for all elements (about 15) so duplicated in both
Classes.
2. I define "manager" as a separate entity, while a manager is an owner. Having a "manager" class / separate object (and a factory, etc.) suggests that it's something different.
OPTION 2 – Just a class.
Class / object:
PRO: The answers for both "owner" and "manager" simply use the same object (and the factory, etc.). It is clear that "manager" is an "owner".
Problems:
1. This could be confusing as it is difficult to know where the "manager" object is located without prior knowledge of the system.
2. I do not know how to handle the extra array that "owner" data has as "properties". We are moving towards the territory of `if isset()`.
OPTION 3 – A shared class for matching items and a person for the other 2 endpoints.
Classes / objects:
• `LandlordOrManager`
• `Landlord`
• `Manager`
PRO: The "owner" and "manager" data have their own class and can therefore define their own data. These classes would both use the shared class to get shared data, getters / setters, and so on. (somehow?).
Problems:
1. It seems heavy to me and I do not think the intention is clear, nor that the developers know what object they want / need.
2. I always refer to "Manager" as a separate thing and I have a separate class / object for it. It's actually an "owner" and maybe the code should reflect that?
I prefer option 3 to avoid mass duplication since the "owner" and "responsible" objects are really the same, and I think it's more important to represent that, but I do not I'm not sure how:
1. Name the shared class to display the intention without it being horrible.
2. Structure the code so that other developers know what it's all about.
3. Use the 3 classes correctly. Spread seems obvious, but I'm not sure whether it's inappropriate / smell, etc.
I hope you will be able to see my problem and suggest something. I could think about it too much, but I just can not find a solution to the problems of both options.
## trackpad – MacBook Air touchpad does not respond
The trackpad does not respond after being hooked to a website video last night and go into sleep mode. When I woke up, I had to use the arrow keys to navigate between the different applications and leave them, then holding down the power button to allow the shutdown. It took 3 times before all applications were shut down before shutting down the MacBook Air (2013) operating system. At STILL restart no trackpad.
I just installed Catalina a week ago on this MacBook Air – 6 years old. No problem with the trackpad at the beginning, probably not related to Catalina. . .
I can use a mouse (USB or wireless). I ran the 1st Aid disk, no help, no problem found. Should I reset the PRAM or something? How is it done? – hold P, R, command and option on restart bong ?? Can not call back
## posts:
The \$ CellContext` (3) coordinate intersections must be a pair of numbers, or a scaled or offset form.
```````sols3(x_, a_) := y /. Solve(x^2 + (y - a)^2 == 9, y);
sols2(x_, a_) := y /. Solve(y == a x^2, y);
intersections(a_) := {x, y} /.
Solve({x^2 + (y - a)^2 == 9 && y == a x^2}, {x, y}, Reals)
Manipulate(
Column@{Plot(
Evaluate@Flatten({sols3(x, a), sols2(x, a)}), {x, -20, 20},
PlotRange -> 20 {{-1, 1}, {-1, 1}}, BaseStyle -> Thick,
Epilog -> {Red, PointSize(Large), Point(intersections(a))},
AspectRatio -> Automatic, ImageSize -> 300),
Column@intersections(a)}, {{a, 3}, -3, 3, 1/10})`
``````
I have 8GB of RAM in my laptop. I do not think that's the problem of my pc.
## What is the name of the notification template: wait for the user to respond to a notification before sending the next one?
Some websites (for example, some forums) send notifications like these:
1. The user asks several questions on the site
2. Other users answer questions
3. The website notifies the user only once (that there has been an activity since its last interaction with the site)
4. The user connects to the site or interacts with the site
5. More activities occur that the user must be informed of
6. The website sends another notification to the user
In short, instead of sending a notification for each activity and potentially overwriting or spammering the user, the user only receives one notification and the system waits for the user have an interaction with the site before sending another notification to the user.
Is there a name for this model?
## watchos 6 – Apple Watch4 with OS6 slow / does not respond to the tap to wake up
I'm using Apple Watch Series 4 with Watch OS 5.x for a year. With alarm clock off to wake me up, I had no problem waking up the clock by touching the screen.
When upgrading to Watch OS6, I now need several taps to wake me up. A simple tap is not even recorded most of the time. Once he's finally woken up and that I turn off the screen, he wakes up in one click. But the initial awakening is problematic – takes 3-4 seconds and several taps.
I also disinfected the watch bracelet with alcohol to burn. I also wiped the screen. Would rubbing alcohol damage the faucet to wake you up?
How can I restore a click to reactivate Apple Watch with WatchOS6.0.1?
## Algorithms – How to solve the following tree problem with a number assigned to each node and respond to queries?
I have a tree whose nodes have numbers that were originally assigned to them. A series of $$Q$$ questions are asked from $$n = 0$$ at $$n = Q-1$$ seconds. At the end of each second, the numbers of a node (which are not sheets) are deleted and are transferred to each direct child node. is possible that a new number be added to a node. How to solve this problem effectively?
Example: Suppose I have a tree 1 —– 2 —— 3 ——- 4 (1 is parent). Suppose at the beginning that the numbers on the nodes are 1,2,3,4 respectively.
At zero, the second end 5 is added to the zero node. At the end of 0 seconds, the number on the node is 5.1.2.3 (note 1 is not added to 5 because it is removed ).
At a second end, 1 is added to node 4, then the numbers are 0.5,1,3. The fourth node is number three since it has received 2 of the parent node and 1 is added.
Now, suppose that, two seconds from the end, I am asked to say the number on the second node?
I mean that in each query, a number can be added on a node or a question can be asked as – what is the number on $$ith$$ node?
How to solve this problem in less than $$O (nQ)$$. Or $$n$$ is the number of nodes and $$Q$$ is the number of requests?
## How to respond effectively to multiple queries?
I am given a $$N$$-Length entire table. For each integer, I am given 4 values: –
$$(left, right, t, value (integer))$$
$$1 <= left, right, t <= N$$
For each query, I have to find the sum of integers$$(values)$$, who have$$x$$& # 39; in the perimeter $$(left, right)$$ and which $$t <= t1$$
Example of request: – $$x = 5$$,$$t1 = 4$$
So, I have to find the sum of all the values, including $$(left <= 5 <= right)$$ years also t$$<=$$4 | 2019-12-06 21:15:52 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 18, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.23788726329803467, "perplexity": 5287.92996149271}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575540490972.13/warc/CC-MAIN-20191206200121-20191206224121-00212.warc.gz"} |
https://en.khanacademy.org/math/7th-engage-ny/engage-7th-module-1/7th-module-1-topic-d/e/constructing-scale-drawings | If you're seeing this message, it means we're having trouble loading external resources on our website.
If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.
# Construct scale drawings
## Problem
Tarush is a landscape architect. For his first public project he is asked to create a small scale drawing of a garden to be placed in the corner of a city park. The garden is a right triangle with base 10, start text, space, m, end text and height 15, start text, space, m, end text.
Draw the garden such that 1 unit on the grid below represents 2, start text, space, m, end text.
Stuck?
Stuck? | 2023-03-22 03:50:33 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 4, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 2, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2553286850452423, "perplexity": 2791.7184427717693}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296943749.68/warc/CC-MAIN-20230322020215-20230322050215-00210.warc.gz"} |
https://repository.uantwerpen.be/link/irua/15612 | Publication
Title
Quantitative determination of $C_{60}$ and $C_{70}$ in soot extracts by high performance liquid chromatography and mass spectrometric characterization
Author
Abstract
A quantitative HPLC method was applied to determine the amounts of C-60 and C-70 present in extracts of soot produced in the electric arc reactor and in flames. The combustion method was found to yield a higher C-70/C-60 ratio (0.67) compared with the evaporation experiment where the C-70/C-60 ratio amounts to 0.27.
Language
English
Source (journal)
Fullerene science and technology. - New York
Publication
New York : 1996
ISSN
1064-122X
Volume/pages
4(1996), p. 1001-1017
ISI
A1996VK45000015
Full text (Publisher's DOI)
UAntwerpen
Faculty/Department Research group Publication type Subject Affiliation Publications with a UAntwerp address | 2017-09-22 12:03:54 | {"extraction_info": {"found_math": true, "script_math_tex": 2, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.26671886444091797, "perplexity": 10767.53885313375}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-39/segments/1505818688940.72/warc/CC-MAIN-20170922112142-20170922132142-00069.warc.gz"} |
http://mathhelpforum.com/discrete-math/67771-groups.html | # Math Help - Groups
1. ## Groups
Hey, I am not entirely sure if I am doing this right. But anyways...
The directions say...
"Determine whether the binary operation * defined on the given set results in a group."
(A group is classified by 3 properties:
Associative Law a * (b * c) = (a * b) * c
Existence of an Identity: There exists an element, e, such that
a * e = e * a = a
Existence of Inverses: For each element, a, there exists an element, s,
such that a * s = s * a = e (the identity)
So, here is what I'm supposed to do:
(a) Let * be defined on the positive reals by a * b= √(ab)
(b) Let * be defined on all the Reals except 0, by a * b= a/b
(c) Let * be defined on all the Reals except 0, by a * b= a+b+ab
Any help would be much appreciated!
2. Originally Posted by dude15129
Hey, I am not entirely sure if I am doing this right. But anyways...
The directions say...
"Determine whether the binary operation * defined on the given set results in a group."
(A group is classified by 3 properties:
Associative Law a * (b * c) = (a * b) * c
Existence of an Identity: There exists an element, e, such that
a * e = e * a = a
Existence of Inverses: For each element, a, there exists an element, s,
such that a * s = s * a = e (the identity)
So, here is what I'm supposed to do:
(a) Let * be defined on the positive reals by a * b= √(ab)
(b) Let * be defined on all the Reals except 0, by a * b= a/b
(c) Let * be defined on all the Reals except 0, by a * b= a+b+ab
Any help would be much appreciated!
Hi,
(a) Yes.
(b) No. Associativity fails.
(c) No. The identity element does not exist.
See if you can come up with the detailed work.
3. Awesome...Those were actually what I was thinking, but I wasn't really sure. Thanks so much! | 2014-04-21 12:01:44 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8822726607322693, "perplexity": 599.9247263047984}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1397609539705.42/warc/CC-MAIN-20140416005219-00064-ip-10-147-4-33.ec2.internal.warc.gz"} |
https://scicomp.stackexchange.com/questions/37045/finite-element-method-for-an-equation-requiring-switch-between-spectral-and-temp | # Finite element method for an equation requiring switch between spectral and temporal domain
Some equations (such as the non-linear schrödinger equation for pulse propagation) are more easily solved in the spectral form, but still need a representation in the temporal domain to calculate physical parameters such as the energy of the electric field. This results usually in the following steps (with $$E$$ the electric field in the temporal domain, and $$\hat{E}$$ the electric field in the spectral domain): $$E_0\underbrace{\rightarrow}_{\text{FFT}}\hat{E}_0\underbrace{\rightarrow}_{\text{Propagation}}\hat{E}_1\underbrace{\rightarrow}_{\text{IFFT}}E_1$$ which requires two different "grids". Is such a calculation possible using FEM, and if yes, how (after requiring two different grids)?
Background to this question is that I have additional equations coupled to that problem which preferably can be solved using FEM and do not require the transformations from the temporal domain to the spectral domain and back (after they only depend on $$E_x$$, not on $$\hat{E}_x$$). For this coupling I'd like to avoid having to have two different grids (one for the non-linear schrödinger equation, one for the FEM-covered equation) which I have to interpolate data in between.
• If you use a DFT matrix $W$, could you form a coupled system of equations (a block matrix expression) by substituting $\hat{E_x}=WE_x$? en.wikipedia.org/wiki/DFT_matrix – Charlie S Mar 17 at 14:26
• I think that you could use the same grid, that's the spatial discretization. What changes is the time discretization, isn't it? – nicoguaro Mar 18 at 15:17
• People do this sort of thing all the time in the geodynamo community. – Wolfgang Bangerth Mar 18 at 16:29
• @WolfgangBangerth: Do you have examples for that, by chance? I'm completely unfamiliar with this community. – arc_lupus Mar 19 at 9:03
• I'm not familiar with the literature in that arena either, but you could start by looking at the Rayleigh and Calypso codes and the literatures that surround them. Both are available from geodynamics.org. I bet you can also find papers by Gary Glatzmeier on the topic. – Wolfgang Bangerth Mar 19 at 13:37 | 2021-04-13 10:07:43 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 5, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8209245800971985, "perplexity": 519.9392667409185}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038072180.33/warc/CC-MAIN-20210413092418-20210413122418-00461.warc.gz"} |
https://www.jamesframework.org/api/core/1.2/org/jamesframework/core/problems/objectives/Objective.html | org.jamesframework.core.problems.objectives
## Interface Objective<SolutionType extends Solution,DataType>
• Type Parameters:
SolutionType - solution type to be evaluated, required to extend Solution
DataType - underlying data type
public interface Objective<SolutionType extends Solution,DataType>
Interface of an objective function that evaluates solutions using underlying data. An objective can be either maximizing or minimizing; in the former case increasing scores indicate improvement, while in the latter case decreasing scores indicate improvement.
It is required to provide a full evaluation by implementing evaluate(Solution, Object). If desired, an efficient delta evaluation can also be provided by overriding the default behaviour of evaluate(Move, Solution, Evaluation, Object) which (1) applies the move, (2) performs a full evaluation and (3) undoes the move.
Author:
Herman De Beukelaer
• ### Method Summary
All Methods
Modifier and Type Method and Description
default <ActualSolutionType extends SolutionType>Evaluation evaluate(Move<? super ActualSolutionType> move, ActualSolutionType curSolution, Evaluation curEvaluation, DataType data)
Evaluates a move that will be applied to the current solution of a local search (delta evaluation).
Evaluation evaluate(SolutionType solution, DataType data)
Evaluates a given solution using the given data.
boolean isMinimizing()
Check whether the produced evaluations are to be minimized.
• ### Method Detail
• #### evaluate
Evaluation evaluate(SolutionType solution,
DataType data)
Evaluates a given solution using the given data. Returns an object of type Evaluation. The corresponding double value can be obtained by calling Evaluation.getValue() on this evaluation object.
Parameters:
solution - solution to evaluate
data - underlying data used for evaluation
Returns:
evaluation of the given solution
• #### evaluate
default <ActualSolutionType extends SolutionType> Evaluation evaluate(Move<? super ActualSolutionType> move,
ActualSolutionType curSolution,
Evaluation curEvaluation,
DataType data)
Evaluates a move that will be applied to the current solution of a local search (delta evaluation). The result corresponds to the evaluation of the modified solution that would be obtained by applying the given move to the current solution. A default implementation is provided that (1) applies the move, (2) computes a full evaluation by calling evaluate(Solution, Object) and (3) undoes the applied move.
It is often possible to provide a custom, much more efficient delta evaluation that computes the modified evaluation based on the current evaluation and the changes that will be made when applying the move to the current solution. This can be done by overriding this method. It is usually required to cast the received move to a specific type so that this objective can only be used in combination with neighbourhoods that generate moves of this type (or a subtype). If an incompatible move type is received, an IncompatibleDeltaEvaluationException may be thrown.
Given that both this method and the full evaluation (evaluate(Solution, Object)) return evaluations of the same type, it is guaranteed that curEvaluation will also be of this specific type and it is safe to perform a cast, if required.
Type Parameters:
ActualSolutionType - the actual solution type of the problem that is being solved; a subtype of the solution types of both the objective and the applied move
Parameters:
move - move to evaluate
curSolution - current solution
curEvaluation - evaluation of current solution
data - underlying data used for evaluation
Returns:
evaluation of modified solution obtained when applying the move to the current solution
Throws:
IncompatibleDeltaEvaluationException - if the provided delta evaluation is not compatible with the received move type
• #### isMinimizing
boolean isMinimizing()
Check whether the produced evaluations are to be minimized.
Returns:
true if evaluations are to be minimized, false if they are to be maximized | 2021-10-17 13:29:14 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3105623424053192, "perplexity": 1713.067928310662}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585177.11/warc/CC-MAIN-20211017113503-20211017143503-00061.warc.gz"} |
https://chem.libretexts.org/Bookshelves/Inorganic_Chemistry/Supplemental_Modules_(Inorganic_Chemistry)/Crystal_Field_Theory/Magnetism | # Magnetism
Movement of an electrical charge (which is the basis of electric currents) generates a magnetic field in a material. Magnetism is therefore a characteristic property of all materials that contain electrically charged particles and for most purposes can be considered to be entirely of electronic origin.
The Right Hand Rule for an induced magnetic field
In an atom, the magnetic field is due to the coupled spin and orbital magnetic moments associated with the motion of electrons. The spin magnetic moment is due to the precession of the electrons about their own axes whereas the orbital magnetic moment is due to the motion of electrons around the nucleus. The resultant combination of the spin and orbital magnetic moments of the constituent atoms of a material gives rise to the observed magnetic properties.
Historically, magnetism has been recognized for thousands of years. An account, that is probably apochryphal, tells of a shepherd called Magnes in Crete who around 900 B.C discovered the naturally occurring magnet lodestone (a form of the the spinel magnetite, Fe3O4) in a region later named Magnesia. Supposedly while he was walking over a deposit, the lodestone pulled the nails out of his sandals and the metal tip from his staff.
## The Classical Theory of Magnetism
The classical theory of magnetism was well developed before quantum mechanics. Lenz's Law states that when a substance is placed within a magnetic field, $$H$$, the field within the substance, $$B$$, differs from $$H$$ by the induced field, $$4πI$$, which is proportional to the intensity of magnetization, $$I$$. That is;
$B = H + 4\pi I$
where $$B$$ is the magnetic field within the substance and $$H$$ is the applied magnetic field and $$I$$ is the intensity of magnetization
Lenz's Law (1834)
Lenz's Law can also be written as
$\dfrac{B}{H} = 1 + \dfrac{4π I}{H}$
or
$\dfrac{B}{H} = 1 + 4\pi\kappa$
where
• $$B/H$$ is called the magnetic permeability of the material and
• $$\kappa$$ is the magnetic susceptibility per unit volume, (I/H)
By definition, $$\kappa$$ in a vacuum is zero, so under those conditions the equation would reduce to $$B=H$$. It is usually more convenient to measure mass than volume and the mass susceptibility, $$χ_g$$, is related to the volume susceptibility, κ, through the density.
$χ_g = \dfrac{κ}{ρ}$
where $$\rho$$ is the density.
Finally to get our measured quantity on a basis that can be related to atomic properties, we convert to molar susceptibility
$χ_m =χ_g \times RMM$
Since this value includes the underlying diamagnetism of paired electrons, it is necessary to correct for the diamagnetic portion of χm to get a corrected paramagnetic susceptibility.
$\chi'_m = \chi_m + \chi_{dia}$
Examples of these corrections are tabulated below.
Table of Diamagnetic Corrections (Pascal's constants, 10-6 c.g.s. units)
Ion DC Ion DC
Na+ 6.8 Co2+ 12.8
K+ 14.9 Co3+ 12.8
NH4+ 13.3 Ni2+ 12.8
Hg2+ 40 VO2+ 12.5
Fe2+ 12.8 Mn3+ 12.5
Fe3+ 12.8 Cr3+ 12.5
Cu2+ 12.8 Cl- 23.4
Br- 34.6 SO42- 40.1
I- 50.6 OH- 12
NO3- 18.9 C2O42- 34
ClO4- 32 OAc- 31.5
IO4- 51.9 pyr 49.2
CN- 13 Me-pyr 60
NCS- 26.2 Acac- 62.5
H2O 13 en 46.3
EDTA4- ~150 urea 33.4
these can be converted to S.I units of m3 mol-1 by multiplying by 4 π x 10-7
There are numerous methods for measuring magnetic susceptibilities, including, the Gouy, Evans and Faraday methods. These all depend on measuring the force exerted upon a sample when it is placed in a magnetic field. The more paramagnetic the sample, the more strongly it will be drawn toward the more intense part of the field.
## Determination of Magnetic Susceptibility
• The Gouy Method: The underlying theory of the Gouy method is described here and a form for calculating the magnetic moment from the collected data is available as well.
• The Evans method: The Evans balance measures the change in current required to keep a pair of suspended magnets in place or balanced after the interaction of the magnetic field with the sample. The Evans balance differs from that of the Gouy in that, in the former the permanent magnets are suspended and the position of the sample is kept constant while in the latter the position of the magnet is constant and the sample is suspended between the magnets.
### Orbital contribution to magnetic moments
From a quantum mechanics viewpoint, the magnetic moment is dependent on both spin and orbital angular momentum contributions. The spin-only formula used last year was given as:
$\mu_{s.o.} = \sqrt{4S(S+1)}$
and this can be modified to include the orbital angular momentum
$\mu_{S+L} = \sqrt{4S(S+1) + L(L+1)}$
An orbital angular momentum contribution is expected when the ground term is triply degenerate (i.e. a triplet state). These show temperature dependence as well.
In order for an electron to contribute to the orbital angular momentum the orbital in which it resides must be able to transform into an exactly identical and degenerate orbital by a simple rotation (it is the rotation of the electrons that induces the orbital contribution). For example, in an octahedral complex the degenerate t2g set of orbitals (dxz,dyx,dyz) can be interconverted by a 90o rotation. However the orbitals in the eg subset (dz2,dx2-y2) cannot be interconverted by rotation about any axis as the orbital shapes are different; therefore an electron in the eg set does not contribute to the orbital angular momentum and is said to be quenched. In the free ion case the electrons can be transformed between any of the orbitals as they are all degenerate, but there will still be partial orbital quenching as the orbitals are not identical.
Electrons in the t2g set do not always contribute to the orbital angular moment. For example in the d3, t2g3 case, an electron in the dxz orbital cannot by rotation be placed in the dyz orbital as the orbital already has an electron of the same spin. This process is also called quenching.
Tetrahedral complexes can be treated in a similar way with the exception that we fill the e orbitals first, and the electrons in these do not contribute to the orbital angular momentum. The tables in the links below give a list of all d1 to d9 configurations including high and low spin complexes and a statement of whether or not a direct orbital contribution is expected.
• Octahedral complexes
• Tetrahedral complexes
## A and E ground terms
The configurations corresponding to the A1 (free ion S term), E (free ion D term), or A2 (from F term) do not have a direct contribute to the orbital angular momentum. For the A2 and E terms there is always a higher T term of the same multiplicity as the ground term which can affect the magnetic moment (usually by a only small amount).
$μ_{eff} = μ_{s.o.} (1-α λ /Δ) \label{eq10}$
where α is a constant (2 for an E term, 4 for an A2 term) and λ is the spin-orbit coupling constant which is generally only available for the free ion but this does give important information since the sign of the value varies depending on the orbital occupancy.
metal ion d configuration λ / cm-1 Ti(III) V(III) Cr(III) Mn(III) Fe(II) Co(II) Ni(II) Cu(II) 1 2 3 4 6 7 8 9 155 105 90 88 -102 -172 -315 -830
For $$d^1$$ to $$d^4$$ the value is positive hence $$μ_{eff}$$ is less than $$μ_{so}$$ and for $$d^6$$ to $$d^9$$ the value is negative hence $$μ_{eff}$$ is greater than $$μ_{so}$$. $$Δ$$ is the crystal field splitting factor which again is often not available for complexes.
For the tetrahedral Co(II) ion, CoCl42-, the observed experimental magnetic moment, μobs = 4.59 Bohr Magneton (B.M.) The spin-only magnetic moment, μs.o. = 3.88 B.M. which is not in good agreement. How can we improve the analysis?
Since the ground term in the tetrahedral field is split from a 4F to a 4A2 term then we can apply the Equation \ref{eq10}. For an $$A$$ term the constant α = 4. The spin-orbit coupling constant, λ for the free ion is -172 cm-1 which we can use as an approximation and Δ= 3100 cm-1. Hence
$μ_{eff} = 3.88 \times (1 - (4* -172) / 3100)$
which comes out at μeff = 4.73 B.M.
This gives a much better fit than the spin-only formula. In the case of the series;
CoI42-, CoBr42-, CoCl42-, Co(NCS)42-
the magnetic moments have been recorded as 4.77, 4.65, 4.59, 4.40 BM assuming that λ is roughly a constant, then this variation shows the inverse effect of the spectrochemical series on the magnetic moment, since Δ is expected to increase from I- to NCS-.
## T ground terms
The configurations corresponding to the T2 term (from D) or a T1 term (from an F term) are those where there is a direct contribution to orbital angular momentum expected. The magnetic moments of complexes with T terms are often found to show considerable temperature dependence. This is as a result of spin-orbit coupling that produces levels whose energy differences are frequently of the order kT, so as a result, temperature will have a direct effect on the population of the levels arising in the magnetic field.
In a Kotani plot μeff is plotted against kT/λ and when this corresponds to a value of 1 then μ equals the "spin-only" value. If this is extrapolated to infinity then the value corresponds to μS+L.
Measuring the magnetic moment at 80 K and 300 K often shows up this variation with temperature.
Example $$\PageIndex{1}$$:
Account for the magnetic moments of the complex, (Et4N)2[NiCl4] recorded at 80, 99 and 300 K.
80K 99K 300K
3.25 3.43 3.89 B.M.
Ni2+ is a d8 metal ion.
The formula suggests a 4 coordinate complex and we can assume that the complex is tetrahedral with a d electron configuration of e4 t24 therefore the spin-only magnetic moment can be calculated as 2.83 BM.
Why did we ignore the possibility of it being square-planar?
The free ion Russell-Saunders ground term is 3F (L=3 and S=1) which will give rise to a lowest energy T term in a tetrahedral field and hence the resultant magnetic moment is expected to be temperature dependent and have a direct orbital contribution. The observed values may be quite different then to the calculated spin only magnetic moment. The value of μS+L can be calculated as:
$mu_{S+L}= \sqrt{4S(S+1)+L(L+1)}$
or
$\mu_{S+L}= \sqrt{8+12}$
or
$\mu_{S+L} = \sqrt{20} = 4.472\;B.M.$
From the observed values it can be seen that the magnetic moment of the d8 Ni2+ complex is intermediate between the μso and μS+L values (probably due to partial quenching of the orbital angular momentum contribution) and is dependent on temperature. Further worked examples and some selected magnetic data are available.
## Contributors
• {{template.ContribLancashire()}} | 2019-10-22 08:21:54 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7389402985572815, "perplexity": 951.113834876505}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570987813307.73/warc/CC-MAIN-20191022081307-20191022104807-00171.warc.gz"} |
https://www.neetprep.com/question/50680-Suppose-gravitational-force-varies-inversely-nthpower-distanceThen-time-period-planet-circular-orbit-radius-R-around-sunwill-proportional-toaRnbRncRndRn?courseId=18 | • Subject:
...
• Topic:
...
Suppose the gravitational force varies inversely as the ${\mathrm{n}}^{\mathrm{th}}$ power of distance. Then the time period of a planet in circular orbit of radius R around the sun will be proportional to
(a) ${\mathrm{R}}^{\left(\frac{\mathrm{n}+1}{2}\right)}$ (b) $\mathrm{R}\left(\frac{\mathrm{n}-1}{2}\right)$
(c) ${\mathrm{R}}^{\mathrm{n}}$ (d) $\mathrm{R}\left(\frac{\mathrm{n}-2}{2}\right)$ | 2019-03-21 01:29:40 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 5, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9633248448371887, "perplexity": 1306.614347700022}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-13/segments/1552912202476.48/warc/CC-MAIN-20190321010720-20190321032720-00014.warc.gz"} |
https://www.rdocumentation.org/packages/spatstat/versions/1.42-2/topics/quad.object | 0th
Percentile
A class "quad" to represent a quadrature scheme.
Keywords
spatial, attribute
##### Details
A (finite) quadrature scheme is a list of quadrature points $u_j$ and associated weights $w_j$ which is used to approximate an integral by a finite sum: $$\int f(x) dx \approx \sum_j f(u_j) w_j$$ Given a point pattern dataset, a Berman-Turner quadrature scheme is one which includes all these data points, as well as a nonzero number of other (dummy'') points.
These quadrature schemes are used to approximate the pseudolikelihood of a point process, in the method of Baddeley and Turner (2000) (see Berman and Turner (1992)). Accuracy and computation time both increase with the number of points in the quadrature scheme.
An object of class "quad" represents a Berman-Turner quadrature scheme. It can be passed as an argument to the model-fitting function ppm, which requires a quadrature scheme.
An object of this class contains at least the following elements: ll{ data: an object of class "ppp" giving the locations (and marks) of the data points. dummy: an object of class "ppp" giving the locations (and marks) of the dummy points. w: vector of nonnegative weights for the quadrature points } Users are strongly advised not to manipulate these entries directly. The domain of quadrature is specified by dummy$window while the observation window (if this needs to be specified separately) is taken to be data$window.
The weights vector w may also have an attribute attr(w, "zeroes") equivalent to the logical vector (w == 0). If this is absent then all points are known to have positive weights. To create an object of class "quad", users would typically call the high level function quadscheme. (They are actually created by the low level function quad.) Entries are extracted from a "quad" object by the functions x.quad, y.quad, w.quad and marks.quad, which extract the $x$ coordinates, $y$ coordinates, weights, and marks, respectively. The function n.quad returns the total number of quadrature points (dummy plus data).
An object of class "quad" can be converted into an ordinary point pattern by the function union.quad which simply takes the union of the data and dummy points.
Quadrature schemes can be plotted using plot.quad (a method for the generic plot).
quadscheme, ppm | 2020-10-19 15:45:48 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4109731614589691, "perplexity": 1446.6524540947642}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107863364.0/warc/CC-MAIN-20201019145901-20201019175901-00089.warc.gz"} |
http://www.ni.com/documentation/en/labview/2.0/analysis-node-ref/beta-func-multimode-function/ | # Beta Function (G Dataflow)
Evaluates the beta function and regularized incomplete beta function.
Evaluates the beta function.
Evaluates the regularized incomplete beta function. | 2019-04-20 16:57:05 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9912757277488708, "perplexity": 2332.614125964843}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-18/segments/1555578529898.48/warc/CC-MAIN-20190420160858-20190420182858-00212.warc.gz"} |
https://www.physicsforums.com/threads/electric-potential-energy-and-gravitational-potential-energy-of-deuterium-nuclear.452326/ | # Electric potential energy and gravitational potential energy of Deuterium nuclear?
1. Nov 30, 2010
### VHP_KLJ
Is it possible to caculate the electric potential energy and the gravitational potential energy between two nucleon of deuterium (1 proton, 1 neutron), if we know the binding energy of deuterium is 2.23 (MeV), and the distance between them is 1 (fm)?
Last edited: Nov 30, 2010
2. Dec 1, 2010
### Meir Achuz
Re: Electric potential energy and gravitational potential energy of Deuterium nuclear
The gravitational energy is -Gmm'/r.
There is no electric PE, but there is magnetic energy between the two magnetic moments.
These energies are negligible compared to the 2,23 whic is due to the nuclear force. | 2016-07-27 17:20:01 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8480911254882812, "perplexity": 1173.4869324171589}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-30/segments/1469257826916.34/warc/CC-MAIN-20160723071026-00213-ip-10-185-27-174.ec2.internal.warc.gz"} |
https://openreview.net/forum?id=moo_HA66HF_ | ## Evaluating representations by the complexity of learning low-loss predictors
Mar 04, 2021 (edited Apr 01, 2021)Neural Compression Workshop @ ICLR 2021Readers: Everyone
• Keywords: representation evaluation, representation learning, minimum description length, mdl
• TL;DR: A new measure similar to MDL allows evaluations of representation quality that are more reliable.
• Abstract: We consider the problem of evaluating representations of data for use in solving a downstream task. We propose to measure the quality of a representation by the complexity of learning a predictor on top of the representation that achieves low loss on a task of interest. To this end, we introduce two measures: surplus description length (SDL) and $\varepsilon$ sample complexity ($\varepsilon$SC). To compare our methods to prior work, we also present a framework based on plotting the validation loss versus evaluation dataset size (the "loss-data" curve). Existing measures, such as mutual information and minimum description length, correspond to slices and integrals along the data axis of the loss-data curve, while ours correspond to slices and integrals along the loss axis. This analysis shows that prior methods measure properties of an evaluation dataset of a specified size, whereas our methods measure properties of a predictor with a specified loss. We conclude with experiments on real data to compare the behavior of these methods over datasets of varying size. | 2021-10-26 16:15:26 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6378332376480103, "perplexity": 1300.8933870340172}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323587908.20/warc/CC-MAIN-20211026134839-20211026164839-00112.warc.gz"} |
https://cor3ntin.github.io/posts/humble_wishlist/ | # To humbly present a wish-list for C++23
In Prague, the committee adopted To boldly suggest an overall plan for C++23, a paper that lays a list of priorities WG21 should focus on for C++23.
The vote was almost unanimous. I voted against it. I figured it would be interesting to explain why.
## The problem with plans
Plans have a nasty tendency to turn into deadlines and expectations.
There was an uproar when contracts did not ship (even though that was the right decision), while coroutines have shipped with known issues and modules have shipped with little experience with the feature as it is in the standard. Of course, all of that was considered, and for example, there is some consensus that the benefits of having coroutines now outweigh the cost of shipping them later. These decisions usually don’t age well, and they are difficult tradeoffs. Tradeoffs which are mostly self-inflicted as C++ has a long release cycle (3 years), which paradoxically leaves little time for individual proposals to mature.
Plans are not and should not be promises. The C++ community should not expect what will be standardized by 2023. Yet plans end up being promises. Votes in the committee end up being influenced by the plan. Plans are often less flexible than anyone hopes and may be blind to new information.
So on principle alone, due to the nature of C++ - which will almost certainly still be around in 50 years - Any process which may involuntarily lead to haste makes my skin crawl.
## Let’s look at P0592
### Modularisation of the standard library
It is hard to argue that the standard library should not be modular. Modules would allow the committee to care less about file-level dependencies, implementers could get rid of _Ugly names, etc. And yet, the reality is much less exciting. It would be an ABI break for implementers to put implementation details in modules. For some implementers, it would further be an ABI break to have exported symbols in modules. Most implementers are further committed to support multiple versions of C++ within the same standard library codebase, which makes modularizing the standard library even more complicated.
The thing is standard library headers can already be imported (you can write import <vector>;) and implementers can translate #include <vector> into import <vector>; - even if some won’t as they are very keen about not breaking odr-violations ridden code.
With this set of constraints, the standard library can be modularized, in that we can make import std.core; do something, but that would have a very limited benefit (nicer syntax) and some drawbacks (you have to remember if std::vector is in std.core or std.base) - but in any case it would offer no technical benefits over import <vector>;.
Knowing that I do not think modularizing the standard library should be a priority. We should wait for people to realize the cost of ABI and then work on a modularization that offers clear technical benefits.
### Executors
There is no doubt that P0443 - A Unified Executors Proposal for C++, is a very important piece of infrastructure that needs to be in the standard as soon as possible.
I wish it was called Sender Receivers. Executors (objects that can execute a function), could be replaced by an execute customization point, which would simplify the design greatly.
Sender Receivers would be virtually unusable without a few other proposals, notably
And probably a few other pieces. Notably, having a rich set of algorithms is very important. Quite a lot of work!
### Coroutines support
Speaking of coroutines, it seems worthwhile to tie async coroutines with the work on sender-receivers. Besides that, it would be nice to have generator in the standard - there is no proposal for that yet. It would be a small proposal.
### Networking
As far as I can tell people have been working on what is now the networking TS since about 2005. A lot of work and energy. But C++ has changed a lot, the committee has changed a lot, and more importantly, operating systems and requirements have changed. And I quite strongly believe the networking TS is not ready.
#### ASIO
The networking TS is based on ASIO, which is a great, well-maintained open-source header-only library with a permissive license that you can use today with C++11 and up.
Unlike ASIO, the TS does not offer:
• SSL (or rather, TLS)
• Serial ports
• Signal handling
So what we are about to put into the standard is a stripped-down version of an existing library.
As such, the only benefits of the standard library are legal reasons and availability. The standard is not meant to fix broken policies and legal departments.
It is hard to think that standardization would increase the portability or quality of implementation as in all likelihood the people maintaining ASIO know networking better than standard implementers.
#### No TLS
The networking TS has no SSL/TLS support. That should be a deal-breaker. C++ Networking Must Be Secure By Default. I think it would be quite irresponsible for any popular programming language to encourage developers to put their users at risk.
#### Limited use cases out of the box
The network TS further has no support for higher-level protocols such as HTTP, QUIC, Websockets, and as such the use of third-party libraries will still be necessary. This is by no mean a criticism of the network ts which has a well-defined scope, but I think it’s important to understand that “networking” may not include your favourite unicorn.
##### The networking TS is not a networking TS
Ultimately sockets are not very interesting or special. They are files (ish). What is interesting is that the operations in 99.9% of cases need to happen asynchronously and deal with many files concurrently. So a large chunk of the networking TS is spent defining foundational pieces of an asynchronous model and includes
• Executors
• Timers
• Strands
• Services
As well as:
• Buffers
• I/O functions
But all of that lives in std::net namespace and is not generalized at all. These design choices come from the historical impossibility to perform file/io and network/io in the same context efficiently on some platforms, or the belief that async i/o had no benefit.
But that changed. And the standard should take note. The C++ standard doesn’t need 2 or 3 asynchronous models, it doesn’t need a multitude of contexts competing for hardware resources…
I wish C++ had 1 async model (senders/receivers) and one unified, general-purpose async i/o framework for files, networking, timer, and other devices.
The networking TS is also very hard to use correctly, which mostly comes from its asynchronous model which makes memory management, ownership and error managements harder to deal with than with sender receivers.
#### Cancellation and error management
Like std::filesystem, std::net has 2 overloads for everything, one that throws, one that does not (except when it does). I dislike that type of interface. Errors and results are passed through the same callbacks which are less efficient and necessitate the result to be default constructible. The networking TS does also not support stop tokens, so cancellation is harder to deal with properly.
#### The networking TS is a C++14 library
The networking TS supports coroutines awkwardly as an afterthought, has many requirements but no concepts, has forwarding headers instead of modules support, etc…
This poses the question of whether the standard library should be a unified library with a common design or a collection of distinct libraries whose design reflect their origin and so far the networking TS is of the latter category.
Given all of that, I do not think the networking TS is at this point a proposal that should be entertained for C++23.
### Reflection
Reflection is certainly the most impactful large language proposal that the C++ committee can focus on. Its design seems mostly done but there are enough details that seem to need resolving that it’s anyone’s guess when it might ship.
## A personal wishlist for C++23
It’s very hard to try to think about what should be standardized as opposed to what I need or desire for myself. any such list is bound to be accidentally self-centred. I’ll give it a shot anyway. I am avoiding mentioning my own papers as even if I think they are useful, it doesn’t seem fair to try to judge their priority.
### Reconsider ABI stability.
I talked about ABI before and I will talk about it again. But it seems evident that whatever your opinion, more discussion is needed. Notably, Goals and priorities for C++ is a very important paper.
### Better error management and wider freestanding support
Error management, in general, is hard. C++ shows that by being extremely inconsistent. I would like to see more discussions on error management, whether that is expected, status_code, deterministic exceptions, more efficient exceptions or something else, it would be helpful to have some policy and consistency in this area.
Error management is often what prevents features to work in freestanding environments. Making more C++ work in more environments should be a goal!
The best C++ features are those that keep true to the “don’t pay for what you don’t use” principle. The existence of std::bad_alloc makes a lot of code less efficient, penalizing 99% of users - not necessarily because of the exceptions themselves but mainly because of exceptions safety of the standard library. By making the standard default allocator non-throwing and making everything conditionally noexcept, a lot of performance could be gained.
### P1144 - Object relocation in terms of move plus destroy
Just like move semantic made a lot of code more efficient for free, this proposal makes container operations a lot more efficient. Probably the best bang-for-the-buck performance improvements the committee can work on!
### Sender-Receiver based asynchronous general-purpose I/O library
As I said, Sender-Receiver is a priority. On top of that, we could start thinking about general purpose async I/O facilities with support for timers, files, processes and sockets in a concept-driven approach so that other device types can be easily plugged-in. This would ensure the standard does not accumulate several io contexts over time. Many bits of the networking TS, including the buffer APIs, could be reused. It is unlikely such an effort could bear fruits before 23.
### Better customization points
The Sender-Receiver proposal relies heavily on customization point objects and the tag_invoke mechanism, which is super clever but that I find really hard to use, and I can’t help but think it needs to be a language feature.
Something like Customization Point Functions, with the ability to forward through multiple proxies would be great.
More generally, taming ADL seems increasingly important. On that line, I wouldn’t be surprised if making the standard library operators hidden friends speeds up compilation more than a shallow modularization.
### Reflection
Reflection is one of the rare features that cannot be emulated by library trickery and it enables (and not just improves) many use cases. It is a bit soon to know if it might land in 23, that might require a minor miracle but it is worth focusing on.
### Some Unicode
Of course, someone in the Unicode study group would tell you that Unicode is important. And while Unicode can be supported without language modification, some issues with the core wording make it harder than it needs to be. These can be improved. Vendors buy-in to allow people to write UTF-8 applications is also ultimately necessary.
### Small proposals
Looking at previous C++ standards, small features like [[nodiscard]], make_unique, [[no_unique_address]] are often a driving force in new standards adoption and are more immediately impactful than big poster features. C++20 was a huge release. Focusing on small proposals has a lot of value. For example:
• More views (enumerate, zip, product are my favorites)
• Better parameter pack manipulation
• constexpr maths and c-string functions
• The pipeline operator
Of course, there are many great proposals to consider, including pattern matching (which as nice as it is isn’t as fundamentally important as reflection), and domain-specific proposals (numeric, algebra, etc). | 2020-06-02 12:09:32 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3259446322917938, "perplexity": 2408.189379112855}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590347424174.72/warc/CC-MAIN-20200602100039-20200602130039-00374.warc.gz"} |
http://openstudy.com/updates/4eeb4cf2e4b0367162f52a7c | ## anonymous 4 years ago can some one give me the concept of arc and curvature( related to calculus 3)
1. anonymous
curvature is the rate where the unit vector T turns per unit of length. it is calculated $k=|dT/ds|$
2. anonymous
if you are given a position, you can find T by dividing the derivative of position over the absolute value of it. it can be looked at as velocity over speed$r'(t)/|r'(t)|$
3. anonymous
ds refers to arc length parameter for a curve. if you're given a curve r(t)=x(t)i+y(t)j+z(t)k, $s(t)= \int\limits_{t _{0}}^{t} (dx/dt)^{2}+(dy/dt)^{2}+(dz/dt)^{2} dt$
4. anonymous
now, k can be re written as $k=|(dT/dt)(dt/ds)|$
5. anonymous
hmm.........ok thanx
6. anonymous
elica?
7. anonymous
the purpose of finding an arc length can be thought of as wanting to know how sharp a curve is turning
8. anonymous
i just finished this class so my explanation might not be detailed enough since i need to understand more but i shared what i can
9. anonymous
in which class u r?
10. anonymous
calc III just had the finals on monday | 2016-12-10 03:15:18 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9020587801933289, "perplexity": 1053.4924855100746}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-50/segments/1480698542938.92/warc/CC-MAIN-20161202170902-00316-ip-10-31-129-80.ec2.internal.warc.gz"} |
https://www.frontporchmath.com/topics/arithmetic/fraction-concepts/fraction-tiles/ | # Why Use Fraction Tiles?
As an Algebra teacher, I have noticed that students often have a weak understanding of fractions and tend to avoid using them. Using manipulatives (in this case Fraction Tiles) for longer can help students develop a better internal understanding of fractions.
I have created fraction tiles up to thirty-sixths, along with decimal fractions (tenths and hundredths). The tiles can be written on and disposed of, or you can laminate a set and use crayons or board markers. They come in two sizes:
3 $cm^2$ and 4 $cm^2$. The Fraction Tiles can be found below, next to the dragon.
$\frac{10}{10}$ & $\frac{100}{100}$
#### Two examples of how I use the tile:
Adding and Subtracting Fractions Use the yellow tiles for adding or the starting number (when subtracting), and the red tiles for what you are subtracting. I have found the key for students to make the connection between the physical fraction tiles and the processes they do on paper, is to have them write out each step as they preform it with the tiles.
Multiplying Fractions The tiles can be used to multiply a fraction times a fraction or mixed number. In this example I made a transparency of the fractions divided vertically (which were laminated for extra thickness). Using whiteboard crayons to draw lines on $\tfrac{1}{3}$ of a transparent tile. Overlaying the two tiles enables the student can see where the two colorings overlap.
### Hey!
This might be fun to look at next:
Fraction Tiles (3 cm squared)
Fraction Tiles (4 cm squared) | 2021-04-14 19:58:58 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3707350492477417, "perplexity": 1495.628493781765}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038078021.18/warc/CC-MAIN-20210414185709-20210414215709-00196.warc.gz"} |
https://math.stackexchange.com/questions/2766566/show-that-there-cannot-be-an-entire-function-that-satisfy-z-cos-z-1fz | # Show that there cannot be an entire function that satisfy $|z+(\cos (z)-1)f(z)|\leq 7$
Show that there cannot be an entire function $f(z)$ that satisfy $|z+(\cos (z)-1)f(z)|\leq 7$.
I thought about showing somehow that $f(z)$ is bounded and by Liouville's theorem it is constant, then the inequality does not hold for all $z\in \mathbb{C}$.
But I'm not sure how to do it or if it is right.
If such a function existed, $z+\bigl(\cos(z)-1\bigr)f(z)$ would be a constant $k$, by Liouville's theorem. Actually, it would be $0$, since$$k=0+\bigl(\cos(0)-1\bigr)f(0)=0.$$So,$$0=2\pi+\bigl(\cos(2\pi)-1\bigr)f(2\pi)=2\pi,$$which is absurd.
Suppose $f$ is entire. Then $g(z) = z + ( \cos z - 1 ) f(z)$ is entire also. By Liousville's theorem, the given inequality $|g(z)| \leq 7$ implies that $g$ must be a constant, say $g(z) = c$. But $g(z) = c$ implies that $f(z) = \frac{c - z}{\cos z - 1}$, and this is a problem, for $\cos z - 1 = 0 \iff z \in \{ 2\pi n \;|\; n \in \mathbb{Z} \}$. That is, $f$ has poles at $\{ 2\pi n \;|\; n \in \mathbb{Z} \}$, contradicting that it is entire. | 2021-11-27 04:43:14 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9744129776954651, "perplexity": 69.81647231329062}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964358118.13/warc/CC-MAIN-20211127043716-20211127073716-00539.warc.gz"} |
https://www.gradesaver.com/textbooks/math/applied-mathematics/elementary-technical-mathematics/chapter-3-section-3-2-length-exercise-page-141/17 | ## Elementary Technical Mathematics
For this problem, we use unit conversions to find our answer: $1dam=1/\!\!\!dam\times\frac{10/\!\!m}{1\ /\!\!\!dam}\times\frac{10dm}{1/\!\!m}=100 dm$ | 2020-02-27 15:31:39 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5435312390327454, "perplexity": 2742.274089237019}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875146714.29/warc/CC-MAIN-20200227125512-20200227155512-00429.warc.gz"} |
http://bobak-kamieniarstwo.pl/girls-bedding-ytbu/what-is-the-purpose-of-redox-reactions-in-the-cell%3F-8d83d6 | ## what is the purpose of redox reactions in the cell?
All other trademarks and copyrights are the property of their respective owners. The combustion reaction describes the overall process that takes place, but inside of a cell, this process is broken down into many smaller steps. Much of the energy from glucose is still lost as heat, but enough is captured to keep the metabolism of the cell running. Introduction to cellular respiration and redox. In modern society, however, only certain redox reactions are put to practical use. Chemiosmosis, a process of ATP production in cellular metabolism, is used to generate 90 percent of the ATP made during glucose catabolism and is also the method used in the light reactions of photosynthesis to harness the energy of sunlight. When a reaction involves the formation of ions, as in the example with magnesium and chlorine above, it’s relatively easy to see that electrons are being transferred. Reactions that move the system from a higher to a lower energy state are spontaneous and release energy, while those that do the opposite require an input of energy. These two electrons are accepted by chlorine, which is reduced. In a cell, however, it’s not a great idea to release all that energy at once in a combustion reaction. Cu²⁺, with a reduction potential of +0.15 will be reduced while V, with a reduction potential of -1.18 will be oxidized, so the reaction will be V + 2Cu²⁺ → V²⁺ + 2Cu⁺ There are two electron carriers that play particularly important roles during cellular respiration: NAD+ (nicotinamide adenine dinucleotide, shown below) and FAD (flavin adenine dinucleotide). The standard cell potential for a redox reaction (E° cell) is a measure of the tendency of reactants in their standard states to form products in their standard states; consequently, it is a measure of the driving force for the reaction, which earlier we called voltage. ROS can also oxidise proteins, directly altering their structure and therefore function. For instance, the combustion of butane (above) releases energy because there is a net shift of electron density away from carbon and hydrogen and onto oxygen. The addition of a phosphate group to a molecule requires energy. The summary equation for cell respiration is: C6H12O6 + 6 O2 → 6 CO2 + 6 H2O Here the chemical energy is converted into electrical energy. NAD+, which deposits its electrons at the beginning of the chain as NADH, is the least electronegative, while oxygen, which receives the electrons at the end of the chain (along with H+) to form water, is the most electronegative. Because oxidation and reduction usually occur together, these pairs of reactions are called oxidation reduction reactions, or redox reactions. What is the purpose of redox reactions in the cell? An intermediate complex is a temporary structure, and it allows one of the substrates (such as ATP) and reactants to more readily react with each other; in reactions involving ATP, ATP is one of the substrates and ADP is a product. As electrons trickle “downhill” through the transport chain, they release energy, and some of this energy is captured in the form of an electrochemical gradient and used to make ATP. When the intermediate complex breaks apart, the energy is used to modify the substrate and convert it into a product of the reaction. The glutathione redox system becomes oxidized with age; the immune system loses response, the brain accumulates protein aggregates, the lungs and kidneys decline in function, blood vessels lose flexibility and the heart begins to fail. ATP functions as the energy currency for cells. Excess free energy would result in an increase of heat in the cell, which would result in excessive thermal motion that could damage and then destroy the cell. Sciences, Culinary Arts and Personal Here, we’ll go through a quick overview of how cells break down fuels, then look at the electron transfer reactions (redox reactions) that are key to this process. The nitrogenous base in NADH has one more hydrogen ion and two more electrons than in NAD+. There are many different types of chemical reactions which occur in the body and they all have a purpose. Use this quiz to check your understanding and decide whether to (1) study the previous section further or (2) move on to the next section. At the heart of ATP is a molecule of adenosine monophosphate (AMP), which is composed of an adenine molecule bonded to a ribose molecule and to a single phosphate group (Figure 5). The educational preparation for this profession requires a college education, followed by medical school with a specialization in medical genetics. Figure 6. As an example, let’s consider the combustion of butane: Figure 2. NAD+ is the primary electron carrier used during cellular respiration, with FAD participating in just one (or two sometimes two) reactions. Officially, both processes are examples of cellular respiration, the breakdown of down organic fuels using an electron transport chain. The process of cellular respiration comprises various reactions in which electrons are transferred or mediated from one molecule to another. This reaction, as written, is simply a combustion reaction, similar to what takes place when you burn a piece of wood in a fireplace or gasoline in an engine. For example, let’s go back to the reaction for glucose breakdown,$\text{C}_6\text{H}_{12}\text{O}_6+6\text{O}_2\to{6}\text{CO}_2+6\text{H}_2\text{O}$. Galvanic Cells Purpose The Purpose of this lab was to investigate electric current in redox reaction produced by galvanic cells. Some prokaryotes have pathways similar to aerobic respiration, but with a different inorganic molecule, such as sulfur, substituted for oxygen. Figure 4. - Definition & Process, The Citric Acid (Krebs) Cycle: Products and Steps, ATP Synthase: Definition, Structure & Function, Gluconeogenesis: Definition, Steps & Pathway, Oxidative Phosphorylation: Definition, Steps & Products, Acetyl Coenzyme A (Acetyl-CoA): Formation, Structure & Synthesis, Complementation Tests: Alleles, Crosses & Loci, Glycogenesis, Glycogenolysis, and Gluconeogenesis, Redox Reactions & Electron Carriers in Cellular Respiration: Definitions and Examples, Phosphorylation: Definition, Types & Steps, MTTC Biology (017): Practice & Study Guide, SAT Subject Test Biology: Practice and Study Guide, UExcel Science of Nutrition: Study Guide & Test Prep, NY Regents Exam - Living Environment: Test Prep & Practice, Human Anatomy & Physiology: Help and Review, UExcel Microbiology: Study Guide & Test Prep, High School Biology: Homework Help Resource, Biological and Biomedical Lab #8: Redox Reactions and Electrochemical Cells Purpose In this experiment, you will use an online simulation to create a series of electrochemical cells and determine the reduction potentials of 5 different metals. The removal of an electron from a molecule, oxidizing it, results in a decrease in potential energy in the oxidized compound. The production of ATP using the process of chemiosmosis is called oxidative phosphorylation because of the involvement of oxygen in the process. The redox reactions that remove electron pairs from glucose transfer them to small molecules called electron carriers. Cellular respiration, for instance, is the oxidation of glucose (C6H12O6) to CO2 and the reduction of oxygen to water. (2011). If you look at a drawing of a bacterial membrane... How are electron carriers held in place in a... NAD^+ + 2e^- + H^+ rightarrow NADH Which of the... Glycolysis Pathway: Steps, Products & Importance, What is Chemiosmosis? In cancer, cells divide uncontrollably and proteins behave oddly, such as appearing or disappearing unexpectedly. (FAD is a similar type of molecule, although its functional groups are different.) Biologists often refer to whole molecules, rather than individual atoms, as being reduced or oxidized; thus, we can say that butane—the source of the carbons—is oxidized, while molecular oxygen—the source of the oxygen atoms—is reduced. What’s the electron-sharing situation at the start of the reaction? Become a Study.com member to unlock this Oxygen is much more electronegative than carbon, so the in the $\text{C}=\text{O}$ bonds of carbon dioxide, oxygen will “hog” the bond electrons. Rather, the electron is shifted to a second compound, reducing the second compound. All batteries are based on redox reactions. In the context of biology, however, you may find it helpful to use the gain or loss of H and O atoms as a proxy for the transfer of electrons. Quite a bit of energy can be released when electrons in $\text{C}-\text{C}$ and $\text{C}-\text{H}$ bonds are shifted to oxygen. This short quiz does not count toward your grade in the class, and you can retake it an unlimited number of times. Image modified from “Etc4” by Fvasconcellos (public domain). A phosphate group is removed from an intermediate reactant in the pathway, and the free energy of the reaction is used to add the third phosphate to an available ADP molecule, producing ATP (Figure 6). State the... During electron transport phosphorylation, which... What can mitochondria directly use to make ATP? In a redox reaction, one of the reacting molecules loses electrons and is said to be oxidized , while another reacting molecule gains electrons (the ones lost by the first molecule) and is said to be reduced . Oxidation and reduction in cellular respiration. The structure of ATP is that of an RNA nucleotide with three phosphates attached. Technically, any redox reaction can be set up to make a voltaic cell. Fortunately for us, our cells—and those of other living organisms—are excellent at harvesting energy from glucose and other organic molecules, such as fats and amino acids. Define the following pair of terms. Living cells accomplish this by using the compound adenosine triphosphate (ATP). Where does this energy come from? This process is known as oxidative phosphorylation. Importantly, the movement of electrons through the transport chain is energetically “downhill,” such that energy is released at each step. For example, in the mechanical work of muscle contraction, ATP supplies the energy to move the contractile muscle proteins. Reactions involving electron transfers are known as oxidation-reduction reactions (or redox reactions), and they play a central role in the metabolism of a cell. Redox mediators are chemicals with electrochemical activity. An electrochemical cell is a device that can generate electrical energy from the chemical reactions occurring in it, or use the electrical energy supplied to it to facilitate chemical reactions in it. It’s important to understand that oxidation and reduction reactions are fundamentally about the transfer of electrons. c. to carry oxygen to cells throughout the body. The electron (sometimes as part of a hydrogen atom), does not remain unbonded, however, in the cytoplasm of a cell. We can conveniently express these two processes by the following two half-reactions, which add to give the overall redox reaction. Redox... Our experts can answer your tough homework and study questions. In their reduced forms, NADH and FADH2 carry electrons to the electron transport chain in the inner mitochondrial membrane. ATP (adenosine triphosphate) has three phosphate groups that can be removed by hydrolysis to form ADP (adenosine diphosphate) or AMP (adenosine monophosphate).The negative charges on the phosphate group naturally repel each other, requiring energy to bond them together and releasing energy when these bonds are broken. Butane: $2\text{C}_4\text{H}_{10}+13\text{O}_2\to8\text{CO}_2+10\text{H}_2\text{O}$. Zinc loses electrons and it is oxidized while copper (II) ions gain electrons and are reduced. In this reaction, the magnesium atom loses two electrons, so it is oxidized. Reactions involving electron transfers are known as oxidation-reduction reactions (or redox reactions), and they play a central role in the metabolism of a cell. Thus, a redox reaction that moves electrons or electron density from a less to a more electronegative atom will be spontaneous and release energy. The transfer of energy in the form of electrons allows the cell to transfer and use energy in an incremental fashion—in small packages rather than in a single, destructive burst. Mitochondrial diseases are genetic disorders of metabolism. All rights reserved. In redox terms, this means that each member of the electron transport chain is more electronegative (electron-hungry) that the one before it, and less electronegative than the one after[2]. Consider the cell notation What is the purpose of the Pt? In the brain, redox homeostasis is recognized to be involved in all aspects of central nervous system (CNS) development, function, aging, and disease. Most affected people are diagnosed in childhood, although there are some adult-onset diseases. to produce glucose and other carbohydrates to produce NAD + and other electron carriers to release energy slowly in a step-by-step process to carry oxygen to cells throughout the body Many more steps, however, produce ATP in an indirect way. These pathways are not oxygen-dependent, so the breakdown process is called anaerobic respiration (anaerobic = non-oxygen-requiring). 5.2 Biofuel cells using redox mediators. to produce NAD + and other electron carriers to produce glucose and other carbohydrates to carry oxygen to cells throughout the body to release energy slowly in a step-by-step process a. to produce glucose and other carbs. Rather, a cell must be able to handle that energy in a way that enables the cell to store energy safely and release it for use only as needed. During hydrolysis, water is split, or lysed, and the resulting hydrogen atom (H+) and a hydroxyl group (OH–) are added to the larger molecule. © copyright 2003-2021 Study.com. Electrons have more potential energy when they are associated with less electronegative atoms (such as C or H), and less potential energy when they are associated with a more electronegative atom (such as O). The hydrolysis of ATP produces ADP, together with an inorganic phosphate ion (Pi), and the release of free energy. What is the purpose of redox reactions in the cell? The ADP molecule and a free phosphate ion are released into the medium and are available for recycling through cell metabolism. We can confirm this if we look at the actual electron shifts involved, as in the video below: Figure 3. Half reactions can be written to describe both the metal undergoing … Energy in metabolic reactions is usually stored in the form of electrons. This very direct method of phosphorylation is called substrate-level phosphorylation. The atom or molecule that donates electrons (in this case, magnesium) is called the reducing agent, because its donation of electrons allows another molecule to become reduced. NAD participates in many redox reactions in cells, including those in glycolysis and most of the reactions in the citric acid cycle of cellular respiration. What’s the best way for you to squeeze as much energy as possible out of that glucose molecule, and to capture this energy in a handy form? Earn Transferable Credit & Get your Degree, Get access to this video and our entire Q&A library. The Nernst equation can be used to determine the value of E cell, and thus the direction of spontaneous reaction, for any redox reaction under any conditions. Identifying and treating mitochondrial disorders is a specialized medical field. You’ve just been given a big, juicy glucose molecule, and you’d like to convert some of the energy in this glucose molecule into a more usable form, one that you can use to power your metabolic reactions. Redox (reduction–oxidation, pronunciation: / ˈ r ɛ d ɒ k s / redoks or / ˈ r iː d ɒ k s / reedoks) is a type of chemical reaction in which the oxidation states of atoms are changed. To tackle this energy, it is required to split the reaction … answer! Instead, cells harvest energy from glucose in a controlled fashion, capturing as much of it as possible in the form of ATP. For example, when glucose is broken down in the presence of oxygen, it’s converted into six carbon dioxide molecules and six water molecules. Symptoms of mitochondrial diseases can include muscle weakness, lack of coordination, stroke-like episodes, and loss of vision and hearing. The purpose of redox reactions in the cell is energy transfer. Mitochondrial disorders can arise from mutations in nuclear or mitochondrial DNA, and they result in the production of less energy than is normal in body cells. In phosphorylation reactions, the gamma phosphate of ATP is attached to a protein. Specifically, both NAD+ and FAD serve as cofactors for enzymes called dehydrogenases, which remove one or more hydrogen atoms from their substrates. This site shows three examples of oxidoreductase enzymes (an oxidase that uses molecular oxygen as the electron acceptor) that use NAD as a cofactor to catalyze a dehydration reaction. The oxidized form of the electron carrier (NAD+) is shown on the left and the reduced form (NADH) is shown on the right. Redox reactions are used to reduce ores to obtain metals, to produce electrochemical cells, to convert ammonia into nitric acid for fertilizers, and to coat compact discs. The two processes of ATP regeneration that are used in conjunction with glucose catabolism are substrate-level phosphorylation and oxidative phosphorylation through the process of chemiosmosis. 5.0 2 votes In type 2 diabetes, for instance, the oxidation efficiency of NADH is reduced, impacting oxidative phosphorylation but not the other steps of respiration. http://cnx.org/contents/185cbf87-c72e-48f5-b51e-f14f21b5eabd@10.8, https://www.khanacademy.org/science/biology/cellular-respiration-and-fermentation/intro-to-cellular-respiration/a/intro-to-cellular-respiration-and-redox, CC BY-NC-SA: Attribution-NonCommercial-ShareAlike, https://www.khanacademy.org/science/biology/cellular-respiration-and-fermentation/intro-to-cellular-respiration/v/oxidation-and-reduction-in-cellular-respiration, Relate the movement of electrons to oxidation-reduction (redox) reactions, Describe how cells store and transfer free energy using ATP. Oxidation damages cell membranes, lipids, and DNA. A common example of an electrochemical cell is a standard 1.5-volt cell which is used to power many electrical appliances such as TV remotes and clocks. This repulsion makes the ADP and ATP molecules inherently unstable. •In a galvanic cell the reactants are separated and do not come into contact. Energy transformations in a galvanic cell compared to direct contact. Ever wonder why antioxidants are so good for you? Both molecules are B vitamin derivatives, with NAD+ produced from niacin and FAD produced from riboflavin. Redox Reactions Cells conserve energy in the form of ATP by coupling its synthesis to the release of energy via oxidation-reduction (redox) reactions, where electrons are passed from an electron donor to an electron acceptor. Reece, J. When ATP is broken down, usually by the removal of its terminal phosphate group, energy is released. If these proteins are import… Figure 7. This intermediate complex allows the ATP to transfer its third phosphate group, with its energy, to the substrate, a process called phosphorylation. The overall reaction for this process can be written as: $\text{C}_6\text{H}_{12}\text{O}_6+6\text{O}_2\to{6}\text{CO}_2+6\text{H}_2\text{O}\,\,\,\,\,\,\,\,\,\,\Delta{G}=-686\text{kcal/mol}$. Oxidation and reduction occur in tandem. The shift of an electron from one compound to another removes some potential energy from the first compound (the oxidized compound) and increases the potential energy of the second compound (the reduced compound). Phosphate groups are negatively charged and thus repel one another when they are arranged in series, as they are in ADP and ATP. A few ATP molecules are generated (that is, regenerated from ADP) as a direct result of the chemical reactions that occur in the catabolic pathways. The half cell with the higher reduction potential will be reduced while the one with the lower oxidation potential will be oxidized. An oxidation reaction strips an electron from an atom in a compound, and the addition of this electron to another compound is a reduction reaction. The purpose of redox reactions in the cell is energy transfer. … As part of a cellular reaction, glucose is broken down to release ATP, and energy in the form of ATP is seen as the most important feedback of the cellular respiration. In other functions of cells … In redox reactions, energy is released when an electron loses potential energy as a result of the transfer. Explain the role of redox reactions in photosynthesis know where is oxygen produced from. b. to produce NAD+ and other electron carriers. In these steps, electrons from glucose are transferred to small molecules known as electron carriers. However, cellular respiration is commonly used as a synonym for aerobic respiration, and we’ll use it that way here[1]. During an endergonic chemical reaction, ATP forms an intermediate complex with the substrate and enzyme in the reaction. This module focuses on the extraction of energy from food; you will see that as you track the path of the transfers, you are tracking the path of electrons moving through metabolic pathways. There are two important ways in which this oxidation is gradual: We’ll look at both redox carriers and the electron transport chain in more detail below. If you’ve heard it said that molecules like glucose have “high-energy” electrons, this is a reference to the relatively high potential energy of the electrons in their $\text{C}-\text{C}$ and $\text{C}-\text{H}$ bonds. Most eukaryotic cells, as well as many bacteria and other prokaryotes, can carry out aerobic respiration. The formation of magnesium chloride is one simple example of a redox reaction: $\text{Mg}+\text{Cl}_2\to\text{Mg}^{2+}+2\text{Cl}^{-}$. How? Recall the active transport work of the sodium-potassium pump in cell membranes. A portable voltaic cell that generates electricity to power devices for our convenience is called a battery. The transfer of electrons between molecules is important because most of the energy stored in atoms and used to fuel cell functions is in the form of high-energy electrons. Answer the question(s) below to see how well you understand the topics covered in the previous section. In these steps, a phosphate group is transferred from a pathway intermediate straight to ADP, a process known as substrate-level phosphorylation. Let’s imagine that you are a cell. What is the purpose of redox reactions in the cell? Both NAD+ and FAD can serve as oxidizing agents, accepting a pair of electrons, along with one or more protons, to switch to their reduced forms. The atom or molecule that accepts the electrons (in this case, chlorine) is known as the oxidizing agent, because its acceptance of electrons allows the other molecule to become oxidized. It can use a spontaneous redox reaction to generate electrical energy. The potential of a concentration cell, therefore, is determined only by the difference in concentration of the chosen redox species. Redox reactions have been implicated in the formation of cancer, for example by damaging our DNA, and ROS have been reported to either activate the expression of genes whose proteins promote cancer (oncogenes) or deactivate tumour suppressor genes, whose proteins do the opposite. Most of these pathways are combinations of oxidation and reduction reactions. Write a balanced redox reaction for the above notation. Recall that, in some chemical reactions, enzymes may bind to several substrates that react with each other on the enzyme, forming an intermediate complex. NAD+ accepts two electrons and one H+ to become NADH, while FAD accepts two electrons and two H+ to become FADH2. Most of the ATP generated during glucose catabolism, however, is derived from a much more complex process, chemiosmosis, which takes place in mitochondria (Figure 7) within a eukaryotic cell or the plasma membrane of a prokaryotic cell. What is the purpose of redox reactions in the cell? Medical geneticists can be board certified by the American Board of Medical Genetics and go on to become associated with professional organizations devoted to the study of mitochondrial diseases, such as the Mitochondrial Medicine Society and the Society for Inherited Metabolic Disease. The concept was initially conceived in 1970s. Services, The Electron Transport Chain: Products and Steps, Working Scholars® Bringing Tuition-Free College to the Community. The release of one or two phosphate groups from ATP, a process called dephosphorylation, releases energy. A half reaction is either the oxidation or reduction reaction component of a redox reaction. Energy is released if the reaction occurs spontaneously. As shown in the image above, NAD+ is a small organic molecule whose structure includes the RNA nucleotide adenine. Cellular respiration involves many reactions in which electrons are passed from one molecule to another. Figure 5. Living organisms are dependent on chemical reactions ta carry out biological processes. Instead, some redox reactions simply change the amount of electron density on a particular atom by altering how it shares electrons in covalent bonds. Energy production within a cell involves many coordinated chemical pathways. As a general rule of thumb, if a carbon-containing molecule gains H atoms or loses O atoms during a reaction, it’s likely been reduced (gained electrons). The redox reaction known as cellular respiration is what provides the energy that the cell needs in order to function. A redox flow battery is an electrochemical energy storage device that converts chemical energy into electrical energy through reversible oxidation and reduction of working fluids. In butane, the carbon atoms are all bonded to other carbons and hydrogens. The function of a voltaic cell is based upon reactions … Not all redox reactions involve the complete transfer of electrons, though, and this is particularly true of reactions important in cellular metabolism. In glucose, carbon is associated with H atoms, while in carbon dioxide, no Hs are present. Rather than pulling all the electrons off of glucose at the same time, cellular respiration strips them away in pairs. Click on the image for a larger view. A living cell cannot store significant amounts of free energy. Thus, we would predict that glucose is oxidized in this reaction. This is accomplished by oxidizing glucose in a gradual, rather than an explosive, sort of way. An oxidation reaction strips an electron from an atom in a compound, and the addition of this electron to another compound is a reduction reaction. Like other chemical reactions, redox reactions involve a free energy change. The energy is used to do work by the cell, usually by the released phosphate binding to another molecule, activating it. Ribose is a five-carbon sugar found in RNA, and AMP is one of the nucleotides in RNA. The reactions taking part in electron transfers are called redox reactions or oxidation-reduction reactions. Let us understand how a voltaic or galvanic cell is created. The concentration of glutathione in the cell is ≈10mM (BNID 104679, 104704, 111464), making it the second most abundant metabolite in the cell (after glutamate) ensuring that it plays a dominant role as an electron donor in redox control of protein function. In this way, ATP is a direct link between the limited set of exergonic pathways of glucose catabolism and the multitude of endergonic pathways that power living cells. The electron carriers deposit their electrons in the electron transport chain, a series of proteins and organic molecules in the inner mitochondrial membrane. NAD+ and FAD are coenzymes, organic molecules that serve as helpers during enzyme-catalyzed reactions, and they receive electrons and protons as part of these reactions. How can you go about this? In $\text{C}-\text{C}$ bonds, electrons are shared equally, and in $\text{C}-\text{H}$ bonds, the $\text{C}$ atom has a very slight negative charge (since it’s a bit more electronegative than hydrogen). Redox Signaling carriers help move these messengers from cell to cell as well as sending the signals inside and between cells. Cellular respiration and fermentation. Obviously, energy must be infused into the system to regenerate ATP. The electron carriers take the electrons to a group of proteins in the inner membrane of the mitochondrion, called the electron transport chain. As a glucose molecule is gradually broken down, some of the breakdowns steps release energy that is captured directly as ATP. ATP is generated through two mechanisms during the breakdown of glucose. Thus, relative to its state before the reaction, carbon has lost electron density (because oxygen is now hogging its electrons), while oxygen has gained electron density (because it can now hog electrons shared with other elements). Does this mean that glucose is continually combusting inside of your cells? In. This will be done by measuring the voltage, or potential difference between various pairs of half-cells. Electrons are passed from one component to the next in a series of energy-releasing steps, allowing energy to be captured in the form of an electrochemical gradient. 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Toward your grade in the reaction the body and they all have a purpose occur... Potential energy as a galvanic cell battery is attached to a second compound electrons from glucose are to..., lack of coordination, stroke-like episodes, and DNA II ) ions gain electrons and are.. Redox... our experts can answer your tough homework and study questions proteins and organic molecules the! Cell notation what is the oxidation of glucose at the actual electron shifts involved, as in the compound... It loses H atoms, it ’ s communication channels more efficient know. Functions of cells … Consider the cell is created catalyzed by enzymes, together with an inorganic phosphate ion released... The reduction of oxygen to cells throughout the body and they all have a purpose, releases.! Introduction and background information a galvanic cell the reactants are separated and do not proceed correctly are accepted by,. 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And also to have carbon dioxide one another when they are in ADP ATP. Technically, any redox reaction for the above notation it into a of... To carry oxygen to cells throughout the body and they all have a purpose atoms from substrates... Electrons than in NAD+ behave oddly, such as sulfur, substituted for.... The above notation group to a second compound, reducing the second compound removal of terminal! Substrate-Level phosphorylation glucose at the start of the Pt carriers deposit their electrons in the amount of redox! Strips them away in pairs hydrolysis of ATP is generated through two mechanisms during the breakdown process is called phosphorylation! These steps, a phosphate group or two phosphate groups are negatively charged thus! Why antioxidants are so good for you … Consider the cell is created energy to energy! Of converting chemical energy to move the contractile muscle proteins Explain the role redox... An increase in the cell to support endergonic chemical reactions which are needed for life are by! Throughout the body and they all have a purpose Get your degree, Get to... As appearing or disappearing unexpectedly ) below to see how well you understand the covered... Electron from a molecule, oxidizing it, results in a decrease in potential energy a... Series of proteins and organic molecules in the video below: Figure 2 this repulsion makes the cell order function. Not store significant amounts of free energy change downhill, ” such that energy is.! As heat, but enough is captured to keep the metabolism of glucose at the same,... Work by the following two half-reactions, which add to give the overall redox reaction known as respiration..., 2007 capturing as much of the integral protein that functions as the pump, changing affinity!, for instance, is the primary electron carrier used during cellular respiration, for instance, is determined by! Involved in the form of electrons, so the breakdown process is called a battery in ADP and ATP are. The active transport work of the transfer of electrons as the pump, changing its affinity for and. And they all have a purpose removal of an electron transport chain, a series of proteins in amount! For example, in the reaction does this mean that glucose is continually combusting inside of your?... In carbon dioxide, no Hs are present and enzyme in the body more. Galvanic cells so good for you oxygen to water is the purpose of redox reactions in form. Reactions in the mechanical work of muscle contraction, ATP supplies the energy from glucose continually!
what is the purpose of redox reactions in the cell? 2021 | 2021-04-19 02:51:16 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.34687209129333496, "perplexity": 1839.8356191179003}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038863420.65/warc/CC-MAIN-20210419015157-20210419045157-00076.warc.gz"} |
http://tex.stackexchange.com/questions/94961/how-to-check-in-latex-whether-the-table-of-content-is-empty-or-not-before-added | # how to check in latex whether the table of content is empty or not before added to the document
I have created a simple template for someone with the table of content added to the first page as follow
\begin{document}
% title and date here
\maketitle
\tableofcontents
\indent \newline
%
\end{document}
But I figure out that someone don't add anything to the table of contents but there will be a word "Contents" shown in the document even the table is empty. Is that anyway I check in advance whether the table of contents is empty or not so to decide what to do next? Thanks.
-
My intuition tells me that you have to check if the the .toc file is empty or not. I'm not an expert of conditional statements in latex but maybe you can file some information here – Pouya Jan 23 '13 at 9:03
Thanks for your replies. To Pouya, since the latex will be run via php calling on the server side. I am wondering if that's possible for some case the .toc has not been cleaned so leads to a misleading judge? Well, buy the way, I don't know how to check the file remotely on the server too :( – user1285419 Jan 23 '13 at 9:07
I am updating this answer to provide a more flexible method, which assumes nothing about what the title of the table of contents is supposed to look like. As in my original answer, the point is that testing for existence or emptiness of the .toc file is not enough. With babel for example, there will always be some stuff in this file. Thus, we check if the file contains at least one \contentsline instance. In the new answer, the package etoc is used because it stores the contents of the .toc file in some internal register which we can check and then we set accordingly a toggle. Then \tableofcontents is redefined to first check this toggle. As long as etoc is left like here in compatibility mode, it will not modify the looks of the TOC. If tocloft is used it must be loaded before etoc.
\documentclass{article}
\usepackage[french]{babel} % for testing
\usepackage{hyperref} % for testing
% Testing if the .toc file exists is not sufficient, (with Babel, there
% will be some info added to it automatically). We need to make
% sure it doesn't contain some \contentsline
\usepackage{etoc}
\newif\ifNonEmptyContents
\makeatletter
\expandafter\in@\expandafter\contentsline\expandafter{\the\Etoc@toctoks}%
\ifin@\expandafter\NonEmptyContentstrue\else
\expandafter\NonEmptyContentsfalse
\fi
\makeatother
\AtBeginDocument{\let\originaltableofcontents\tableofcontents
\renewcommand{\tableofcontents}%
{\ifNonEmptyContents\expandafter\originaltableofcontents\fi}%
}
\begin{document}
\tableofcontents
% \section{A}
% a
% \section{B}
% b
% \subsection{C}
nothing
\end{document}
Note though that there are some nice packages dealing with TOC's which could well be upset by the following. Besides, I am assuming a standard \tableofcontents command as in the article class.
\documentclass{article}
\usepackage[french]{babel} % for testing
\usepackage{hyperref} % for testing
\newif\iftoctitledone
\makeatletter
\iftoctitledone\else
\section *{\contentsname \@mkboth {\MakeUppercase \contentsname }
{\MakeUppercase \contentsname }}\fi
\toctitledonetrue}
\AtBeginDocument{%
\let\mygoodoldcontentsline\contentsline
\renewcommand\tableofcontents{\@starttoc {toc}}
}
\makeatother
\begin{document}
\tableofcontents
% \section{test}
nothing
\end{document}
The same in another style, slightly more elegant.
\documentclass{article}
\usepackage[french]{babel} % for testing
\usepackage{hyperref} % for testing
\makeatletter
\section *{\contentsname \@mkboth {\MakeUppercase \contentsname }
{\MakeUppercase \contentsname }}%
\AtBeginDocument{%
\let\mygoodoldcontentsline\contentsline
\renewcommand\tableofcontents{\@starttoc {toc}}
}
\makeatother
\begin{document}
\tableofcontents
% \section{A}
% a
% \section{B}
% b
% \subsection{C}
nothing
\end{document}
-
It works just pefect. Thanks a lot – user1285419 Jan 24 '13 at 3:03
How does this differ (except for the \@mkboth) from the following:
\documentclass{article}
\usepackage[french]{babel} % for testing
\usepackage{hyperref} % for testing
\makeatletter
\AtBeginDocument{%
\let\oldcontentsline\contentsline
\renewcommand{\contentsline}{
\section*{\contentsname}
\let\contentsline\oldcontentsline
\oldcontentsline
}
\renewcommand{\tableofcontents}{\@starttoc{toc}}
}
\makeatother
\begin{document}
\tableofcontents
% \section{A}
% a
% \section{B}
% b
% \subsection{C}
nothing
\end{document}
-
this is not an answer to the original question but a new question. Anyway, it differs a lot because the code you propose will insert a Contents heading for each and every entry, be it section, subsection etc... ending up in the TOC. Try it out after un-commenting the sections in the code you provide. – jfbu Apr 4 '14 at 13:55
If you have a new question, please ask it by clicking the Ask Question button. Include a link to this question if it helps provide context. – Andrew Swann Apr 4 '14 at 14:06
@AndrewSwann: IMHO it is not a new question, but a simplified version of jfbu's answer. And the "question" is more about, whether this simplification can be done or are there oversights. – Heiko Oberdiek Apr 4 '14 at 16:18
@HeikoOberdiek It is still not an answer, a comment on jfbu's answer, or an edit to the original question would appear to be more approrpiate. – Andrew Swann Apr 5 '14 at 9:27
@AndrewSwann: The code is different from jfbu's answer and provides a solution for the question. Therefore it qualifies for an answer to the question. – Heiko Oberdiek Apr 5 '14 at 11:57 | 2015-08-04 07:50:35 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7452069520950317, "perplexity": 1853.0922146761984}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-32/segments/1438042990603.54/warc/CC-MAIN-20150728002310-00205-ip-10-236-191-2.ec2.internal.warc.gz"} |
https://www.flow3d.com/modeling-open-surface-microfluidics/ | Solving the World’s Toughest CFD Problems
# Modeling Open Surface Microfluidics
Open surface microfluidic systems are becoming increasingly popular in the fields of biology, biotechnology, medicine, point-of-care (POC) and home care systems. The design of such systems usually involves fluid being transported by capillary forces. Capillarity can enhance fluid transport for small volumes of fluid and can provide a reliable alternative to micro-scale pumping mechanisms. Advantages of capillary systems include:
• Low cost due to easy and fast fabrication
• User friendliness due to the simplicity of their design
• Increased portability ensured by the capillary actuation of fluids
• Enhanced accessibility caused by the open-surface nature of their design
• Complete elimination of air bubbles guaranteed by the uniformly moving fluid front
For these reasons, open capillary systems are the preferred design option for various POC systems.
## Condition for Onset of Capillary Flow
In a recent paper from researchers from the University at Buffalo and University of Grenoble, it was shown how microgrooves can potentially enhance the capillary effect [1]. Based on the results of this paper, I will discuss a case study of the spontaneous capillary flow (SCF) of fluids inside a narrow V- groove microchannel surmounted by parallel plates using FLOW-3D. Large fluid velocities can be obtained, even with viscous fluids such as whole blood for the design of POC systems to monitor blood flow, if certain conditions for the onset of capillary flow are satisfied.
The conditions of capillary flow can be theoretically established using a static approach based on the minimization of Gibbs free energy. More specifically, the condition for the onset of capillary flow with a zero inlet pressure is:
(Eq. 1) $latex \frac{{{{p}_{F}}}}{{{{p}_{W}}}}<\cos \theta$
where θ is the Young contact angle, and pF and pW are the free and wetted perimeters, respectively, in an arbitrary cross section of the flow. For a V-groove microchannel with a half-angle α, such as the one shown in Fig. 1, after a few mathematical manipulations, eq. 1 can be rewritten as:
(Eq. 2) $latex \displaystyle \sin \alpha =~\cos \theta$
In our case we have α ≈ 14.5ο, hence the condition for capillary flow is θ < 75.5o.
## Simulating in FLOW-3D
Although a static approach provides important information concerning the onset of SCF, a numerical approach is more appropriate for studying flow dynamics in a point-of-care device. A CFD analysis was performed for a V-groove microchannel with a contact angle of 37o and fluid properties of whole blood. Viscosity of blood is nearly constant, hence the flow regime is considered to be Newtonian [1]. A uniform ambient pressure was applied at all boundaries and throughout the computational domain to ensure that the fluid motion was caused only by capillary effects. The simulation was split into two parts: an initial simulation covering the first 4mm of fluid travel and a restart simulation to predict the fluid transport from 4mm to 8mm.
## Results and Validations
The flow dynamics for the first 8mm of travel are shown in Fig. 2. This figure shows the shape of the advancing interface in the slot at three different times. The progressive extension of the filament (Concus-Finn filament) can be seen ahead of the main flow.
A comparison between analytical, numerical and experimental results is shown in Fig. 3. There is excellent agreement between numerical predictions and experiments. Although an analytical solution is also plotted, it may not represent a valid comparison, against numerical and experimental results, as the effects of the Concus – Finn filament at the bottom of the channel are not taken into account.
In addition to whole blood, different fluids were also studied in the experiment, including water tinted by food coloring and a viscous solution of alginate to test the potential of the device to move highly viscous fluids. Highly viscous fluid like blood could be moved within times less than 1s (see animation below).
The case study demonstrates that for whole blood, which has a relatively large viscosity (4 times that of water), velocities up to 7.5 cm/s were achieved. Based on experimental results and FLOW-3D predictions, the whole channel was filled with blood in less than 0.2 s. FLOW-3D simulation results are in excellent agreement with experimental observations, which also indicate that the velocities decrease with the distance inside the V-groove, but remain significant throughout the length of the device.
## References
1. Berthier, J., K. Brakke, E. P. Furlani, I. H. Karampelas, and G. Delapierre. “Open-surface microfluidics.” In Proceedings of the Nanotech International Conference, pp. 15-19. 2014.
2. Hirt, Cyril W., and Billy D. Nichols. “Volume of fluid (VOF) method for the dynamics of free boundaries.” Journal of computational physics 39, no. 1 (1981): 201-225.
3. Rajaratnam, N., and M. R. Chamani. “Energy loss at drops.” Journal of Hydraulic Research 33, no. 3 (1995): 373-384.
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https://faculty.math.illinois.edu/Macaulay2/doc/Macaulay2-1.17/share/doc/Macaulay2/Polyhedra/html/_interior__Vector.html | # interiorVector -- computes a vector in the relative interior of a Cone
## Synopsis
• Usage:
p = interiorVector C
• Inputs:
• Outputs:
• p, , over QQ with only one column representing a vector
## Description
interiorVector takes the rays of the Cone, computes the sum and divides by the gcd to get a primitive vector.
i1 : P = cyclicPolytope(3,4) o1 = P o1 : Polyhedron i2 : C = coneFromVData P o2 = C o2 : Cone i3 : interiorVector C o3 = | 3 | | 6 | | 14 | 3 1 o3 : Matrix ZZ <--- ZZ
## Ways to use interiorVector :
• "interiorVector(Cone)"
## For the programmer
The object interiorVector is . | 2021-04-13 05:17:27 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.671649158000946, "perplexity": 3323.1811102460283}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038072082.26/warc/CC-MAIN-20210413031741-20210413061741-00598.warc.gz"} |
https://blog.nathantsoi.com/article/Batch_Convert_CR2_to_JPG/ | Have you taken a bunch of pictures in the Cannon Raw format (.cr2) and now need to be able to edit them or view them in a jpeg or png format?
You could open each one in some photo editor and save it in a the correct format. Or you could use ImageMagick to convert them all at once!
On my mac, I used homebrew to install imagemagick with the command:
brew install imagemagick
You'll also need the ufraw package to convert from the cr2 format:
brew install ufraw
Ok you've got all the software you need, now let's convert!
Assume you have a folder in your home directory, under Pictures called "raw"
Then we would run these commands in the Terminal app to convert the .CR2 images to .jpg
change directory to the "raw" folder cd ~/Pictures/raw
convert all the .CR2 files to .jpg, this is case sensitive, so if your raw files are .cr2, replace the upper case CR2 in the command with a lower case version: cr2 for infile in *.CR2; do convert $infile$(echo $infile|sed -n "s/CR2$/jpg/p"); done
If you're on windows you can grab the installer here: http://www.imagemagick.org/script/binary-releases.php#windows
Then in the command line run
mogrify -format CR2 *.jpg | 2023-02-09 12:58:46 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2965681552886963, "perplexity": 5826.4063361777635}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764499966.43/warc/CC-MAIN-20230209112510-20230209142510-00505.warc.gz"} |
http://www.scienceforums.com/topic/35882-my-favorite-equations-i-have-derived-over-the-years/page-4 | # My Favorite Equations I Have Derived Over The Years.
63 replies to this topic
### #52 Dubbelosix
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Posted 03 August 2019 - 08:21 AM
Off the rails? I am theoretical scientist, we don't tend to go off on tangents unless there are creative ways to argue it. For instance, I stated:
''We find that in a simplified expression, the Doppler effect is
$\frac{v^2}{c^2} = 1 - \frac{1}{\gamma^2}$
When solving for $\gamma$ we get a very nice set of solutions:
$\gamma = i\frac{c\sqrt{v^2 - c^2}}{v^2 - c^2}$
and
$\gamma = -i\frac{c\sqrt{v^2 - c^2}}{v^2 - c^2}$
Where we notice that for this to be true, $v \ne c$. You could go further and state that $v \ne - c$ but this is where pure algebra reaches a limit because there is no such thing as a negative speed of light.''
The steps I had in mind was to square the function
$\gamma^2 = i^2\frac{c^2 \cdot [v^2 - c^2]}{v^4 - c^4}$
(The sign here comes intuitively from the imaginary form $i^2 = -1$ which is simply
$\gamma^2 = i^2\frac{c^2 \cdot [v^2 - c^2]}{v^4 - c^4}= i^2\frac{c^2}{v^2 - c^2}$
The addition of a mass term would imply
$i^2\frac{mc^2}{v^2 - c^2}$
The only time I have considered tachyons seriously, was from work suggesting the neutrino could move faster than light, however, this was before the LHC blunder - I got enthusiastic in those days and gave up my thoughts on tachyons for a while. Anyone who follows my work will know that I consider there to be a pre-big bang phase - I likened it to an all-dominated matter close at zero point temperatures. The allowance of this condensate in the form of tachyons could answer how light reached all the universe without inflation.
Edited by Dubbelosix, 03 August 2019 - 08:39 AM.
### #53 ralfcis
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Posted 03 August 2019 - 08:41 AM
Well now that you put it that way have you finally proven to yourself that i= -1? Good first step. I thought you were self-taught. Did you confer yourself a degree in theoretical physics because I haven't seen one bit of evidence that you're legit in any way. This thread should be moved into the rubber room section.
Edited by ralfcis, 03 August 2019 - 08:43 AM.
### #54 Dubbelosix
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Posted 03 August 2019 - 08:46 AM
Well now that you put it that way have you finally proven to yourself that i= -1? Good first step.
Aside from the last drivel which I have no intentions of replying to, what is remarkable that we use imaginary numbers in the sign convention, in which you feel the need to be rude about what has been suggested?
### #55 Dubbelosix
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Posted 12 August 2019 - 06:23 AM
Let's not forget the bivector theory of gravity which naturally induced rotation (ie. non-zero torsion field)
https://www.quora.co...torqrygqxdlrpkp
### #56 Dubbelosix
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Posted 12 August 2019 - 08:42 AM
Let's talk about a purely mathematical set up for the unification of the charges in curved spacetime. All derivatives on the left hand side of the energy operator should yield the connections of a gravitational field: The energy operator is well known and is itself the wave equation:
$i \hbar\ \frac{\partial \psi}{\partial t} = \hat{E}\psi$
The right hand side should become an operator that probably does not commute, since the order of space and time derivatives matter - in the case we will look at, involves the connection due to the Ricci curvature which is the ''simplest'' curvature invariant for the Riemannian manifold:
$i \mathbf{J}\ \frac{\partial \psi}{\partial t} = \nabla \mathbf{R}\ \frac{Gm^2}{c}\frac{\partial \psi}{\partial t}$
In which for this case, $\mathbf{J}$ is always quantized and so we can treat this as the spin density and $\mathbf{R}$ is the usual curvature which has units of $\frac{1}{length^2}$. Therefore this quantity is an energy:
$\frac{Gm^2}{c}\frac{\partial \psi}{\partial t}$
and so the connection acting on the Ricci scalar curvature yields a density scalar. Let's refresh ourselves on the Einstein field equations, it takes in the index free notation:
$\mathbf{G} = \mathbf{R} - \frac{1}{2}\mathbf{R} = \frac{8 \pi G}{c^4}\ \mathbf{T}$
With $\mathbf{G}$ being the Einstein scalar, also with units of inverse length squared and $\mathbf{T}$ being a stress energy scalar. For a simple case of a torsion free case (which is incapable of being seen in context of geometric algebra), the $\nabla \mathbf{R}$ is the derivative as found in the first reference and will abide to the Bianchi identities. The quantization of the energy operator as related to curvature dynamics is still an open question, even in this pure mathematical text, there are many avenues as you will probably know that have attempted to find the quantization as related to the connection of the gravitational field. The approach above is just food for thought.
REFERENCES:
https://empg.maths.e...in/Lecture6.pdf
### #57 Dubbelosix
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Posted 12 August 2019 - 08:54 AM
A curvature tensor would simply look like:
$i \mathbf{J}_{\mu \nu}\ \frac{\partial \psi}{\partial t} = \nabla \mathbf{R}_{\mu \nu}\ \frac{Gm^2}{c}\frac{\partial \psi}{\partial t}$
Because the gravitational field is a pseudo field, we are still going to have problems justifying how the complexification of the field can be applied to a gravitational understanding since the quantization of gravity under the Wheeler de Witt equation is inherently real. So something has to give on this if there is a quantization bridge between the two invariants of the charge descriptions.
Edited by Dubbelosix, 12 August 2019 - 08:54 AM.
### #58 Dubbelosix
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Posted 14 August 2019 - 02:37 AM
Let's go back to this equation:
$i \mathbf{J}\ \frac{\partial \psi}{\partial t} = \nabla \mathbf{R}\ \frac{Gm^2}{c}\frac{\partial \psi}{\partial t}$
As I believe it is possible to reconfigure the equation in terms of the contribution of curvature to the fluctuations of spacetime. It is possible to expand the Langrangian of the zero point modes on the background spacetime curvatuture in a power series
$\mathcal{L} = \hbar c\ \mathbf{R} \int\ k\ dk... +\ \hbar c\ \mathbf{R}^2 \int \frac{dk}{k^{n-1}} + C$
Where $C$ is a renormalizing constant which is set to zero for flat space. It had been believed at one point that the forth power over the momenta of the particles would be zero
$\hbar c\ \int\ k\ dk^3 = 0$
But interesting things happen in the curvature of spacetime, such a condition doesn't need to hold. This langrangian density, in which $\mathbf{R}$ plays the role of the curvature was first recognized by Sakharov and how the curvature can allow a fluctuation to become longer lived - perhaps even the fluctuation itself can owe its existence to the gravitational correction term. Notice in Sakharov's approach, we find the usual charge identity for the electromagnetic part $\hbar c$. I still find the imaginary term a complication because as noted before, the gravitational field is not really a field and complex numbers are usually applied to quantum fields - so for the sake of it, we may even want to investigate a purely real form of the case:
$\mathbf{J}\ \frac{\partial \psi}{\partial t} = \nabla \mathbf{R}\ \frac{Gm^2}{c}\frac{\partial \psi}{\partial t}$
Taking the derivative of the spin density allows us to consider the square of the curvature scalar and then following with it, the definition of the gravitational correction in form of the geometric power series - this also includes a small manipulation of distributing the speed of light to find the charge definitions:
$\mathcal{L}\psi = \nabla \mathbf{J}c\ \psi\ ... +\ Gm^2\ \mathbf{R}^2\ \int \frac{dk}{k^{n-1}}\ \psi$
This is just a purely mathematical avenue, we don't know how gravity will translate into a quantum framework - certainly it cannot be treated in the normal framework of fundamental fields but the approach we have above issues a certain importance behind the gravitational interpretation of the charge, namely the term $Gm^2$.
Edited by Dubbelosix, 14 August 2019 - 03:15 AM.
### #59 Dubbelosix
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Posted 14 August 2019 - 04:12 AM
In natural units,the gravitational fine structure constant is equal to the square of the mass of a particle
$\alpha_G = m^2$
The hope or immediate realization of this are attempts to find quantization of mass depending on factors of $n\hbar$.
The $Gm^2$ can be thought of as the gravitational charge of the system analogous to the electric charge $e^2$ (as stated before - but not necessarily equal in magnitude, they are only equal in the dimensional sense). Some mass formula have been suggested in literature (see What is special about the Planck mass? Sivaram). Before we look at the more advanced mass formula that will be suggested here, let’s take a look at what some have called ‘’the mysterious’’ Weinberg formula:
$m = (\frac{H_0\hbar^2}{Gc})^{\frac{1}{3}}$
In which $H_0$ is the Hubble parameter. This will correspond to either one particle mass, or Weinberg had in mind a spectral property leading to different masses. In any case, this equation cannot predict all particle masses with this description alone. Alternatively, the Weinberg formula has been written as
$m^3 = \frac{H_0 \hbar^2}{Gc}$
In this formulation, it has been suggested by Arun and Sivaram that it is ''unclear'' whether now he associates the equation to three particle masses [1] or as an attempt to find a fundamental unit of mass. Either way, we can see his ultimate aim would have been to describe fundamental particle masses from cosmological features, like the Hubble constant. Schwinger, Motz (et al). have demonstrated what it means to quantize a charge: Charge and mass are in fact so similar, you can indeed describe both in similar physics. Nature seems to posit some kind of universality with the presence of an electric charge being synonymous with mass. The only exception it appears in nature is a neutrino, which is expected to have no charge and a very small mass, but if it has a very small mass (you can also argue) it may possess a vanishing, but non-zero charge with a proportionality of $e^2 \propto Gm^2$. To try and articulate how small this charge would be, it would have a mass and charge 1 millionth of that of an electron. Either way, the ultimate idea is that large numbers can determine the small in dynamic ways.
We can in fact restate Weinbergs formula for one that satisfies the gravitational interpretation of the charge. I provide this as a simple manipulation of his equation:
$Gm^2 = \frac{H_0 \hbar^2}{mc} = \frac{H_0 \hbar^2}{p}$
In which we recognize a momentum term in the denominator.
Let’s have a quick look at a more advanced mass formula candidate
$m = nk[\frac{mc}{e}\frac{1}{2 \sqrt{T}}]^n M_e = nk [\frac{\sqrt{G}m}{2 e}]^n M_e$
Immediately we can notice the use of the gravitational charge in the last term $sqrt{G}m$ - the only difference is that it has focused on the Planck mass definition of the charge. The Planck mass should not necessarily be considered fundamental, it seems like too much a basic unit of matter for any particles we have observed in the standard model. Though the middle term is good for string dynamics and superstring tension, the last term appears to be made of more fundamental assumptions which included the gravitational charge of the system. The adjustable parameters is what allows us to predict particle masses, for example n = 2, k = 3 gives the muon mass, n = 2, k = 4 gives the pion mass, n = 3, k = 4 the \Delta resonance, n = 3, k = 6 the D meson n = 3, k = 10 the \psi, n = 4, k = 4 the upsilon particle. According to Sivaram, several more particle masses can be obtained.
Anyway, the main point may have became a bit clearer: The more advanced suggested mass formula does indeed predict a wide range of particle masses, but more importantly, the Weinberg formula can be expressible also in a dimensionless form:
$\frac{m}{m_e} = nk [\frac{\sqrt{G}m}{2 e}]^n = nk [\sqrt{\frac{H_0 \hbar^2}{mc}}\frac{1}{2 e}]^n = nk [\sqrt{\frac{H_0}{mc}}\frac{\hbar}{2 e}]^n$
So in this sense, we get to keep Weinberg’s attractive idea about determining fundamental parameters from cosmic parameters - dimensionless terms are well-known in physics to be the only true physical parameters of a theory. A good example would be Bernoulli's fluid equation for an example.
While this is all nice, it seems from Regge trajectories, the construction of certain particles strangely coincides with multiples of the mass squared. If we go back to the formula:
$m = nk [\frac{\sqrt{G}m}{2 e}]^n M_e$
Then obtaining the mass squared is a simple procedure:
$m^2 = nk [\frac{Gm^2}{4 e^2}]^n M^2_e$
As shown in the previous post, the part of the Langrangian which deviates from flat space is in units of $G = 1$ is:
$\mathcal{L}\psi = m^2\ \mathbf{R}^2\ \int \frac{dk}{k^{n-1}}\ \psi$
And so perhaps an appropriate correction to formulate an equation to predict particle masses as a contribution also from curved space may involve the replacement of the appropriate mass squared term:
$\mathcal{L}\psi = n\mathbf{k} [\frac{Q^2_g}{4 e^2}]^n M^2_e\ \mathbf{R}^2\ \int \frac{dk}{k^{n-1}}\ \psi = n\mathbf{k} [\frac{\alpha_G}{4 \alpha}]^n M^2_e\ \mathbf{R}^2\ \int \frac{dk}{k^{n-1}}\ \psi$
As stated in one previous post on another thread, the role of $\frac{Gm^2}{e^2}$ has played a large role in the history of physics, including applications to astrophysics - in order to know these types of relationships, I reference wiki:
''
• (4.5) in Barrow and Tipler (1986) tacitly defines $\frac{\alpha}{\alpha_G}$ as $\frac{e^2}{Gm_pm_e}$ -even though they do not name the $\frac{\alpha}{\alpha_G}$ in this manner, , it nevertheless plays a role in their broad-ranging discussion of astrophysicscosmologyquantum physics, and the anthropic principle;
In our case we would be defining the inverse of the formula which described this as $\frac{\alpha_G}{\alpha}$. In what way the wave function plays a role in the Langrangian density is uncertain to me if it is even required at all.
References:
https://pdfs.semanti...e0b4995919d.pdf
https://arxiv.org/pd...ics/0408056.pdf
https://arxiv.org/ft...7/0707.0058.pdf
C. Sivaram, Astrophys. Sp. Sci. 207, 317 (1993); 219, 135 (1994)
Edited by Dubbelosix, 14 August 2019 - 04:21 AM.
### #60 Dubbelosix
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Posted 14 August 2019 - 04:27 AM
A further additional part wiki talks about, is definining the value of $\alpha_G$ ~
''$\alpha_G$ has a simple physical interpretation: it is the square of the electron mass, measured in units of Planck mass. By virtue of this, $\alpha_G$ is connected to the Higgs mechanism, which determines the rest masses of the elementary particles$\alpha_G$ can only be measured with relatively low precision, and is seldom mentioned in the physics literature.
Because
$\alpha_G = \frac{Gm^2}{\hbar c} = (t_P \omega_C)^2$
where $t_P$ is the Planck time, and $\omega_C$, the Compton angular frequency of the electron.''
Edited by Dubbelosix, 14 August 2019 - 04:27 AM.
### #61 Dubbelosix
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Posted 14 August 2019 - 04:31 AM
Using these values, we can express the fine structure simply then from a previous formula:
$Gm^2 = \frac{H_0 \hbar^2}{mc} = \frac{H_0 \hbar^2}{p}$
As
$\alpha_G = \frac{Gm^2}{\hbar c} = \frac{H_0 \hbar}{mc^2} = \frac{H_0 \hbar}{pc}$
Edited by Dubbelosix, 14 August 2019 - 04:50 AM.
### #62 Dubbelosix
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Posted 16 August 2019 - 08:51 AM
Let's continue on this ''pure math'' adventure.... well sort of pure math adventure, there are some educated idea's upon new formula's we should probably recognize as being potentially important for future calculations
The correction of curvature to fluctuations is unified here under the Weinberg mass formula with $G$ restored::
$\mathcal{L}\psi = n\mathbf{k} [\frac{Q^2_g}{4 e^2}]^n Gm^2_e\ \mathbf{R}^2\ \int \frac{dk}{k^{n-1}}\ \psi = n\mathbf{k} [\frac{\alpha_G}{4 \alpha}]^n Gm^2_e\ \mathbf{R}^2\ \int \frac{dk}{k^{n-1}}\ \psi$
Weinberg's formula was:
$m^3 = \frac{H_0 \hbar^2}{Gc}$
And for a matter particle, travelling freely no less, is given by the kinetic formula:
$E_k = \frac{mv^2}{2} = \frac{p^2}{2m} = \hbar^2\ \frac{k^2}{2m}$
$Gm^3c = H_0 \hbar^2$
$\hbar^2 = \frac{Gm^2}{H_0}\ mc = \frac{2mE_k}{k^2}$
The most important term to remember for the following, is $\frac{H_0 \hbar}{pc}$... why? Well $\frac{\hbar}{pc}$ indicates a relationship with the deBroglie wave-matter formula. It is accordingly related also to the uncertainties:
$\Delta \ell \approx \frac{\hbar}{\Delta p}$
A matter wave is
$\lambda = \frac{\hbar}{p} = \frac{Gm^2}{H_0 \hbar} = \frac{2mE_k}{k^2 \hbar}$
In which a relativistic version can also be extrapolated in literature of the form, where the numerator$c \rightarrow v)$:
$\frac{\hbar}{pc} = \frac{\hbar v}{Ec}(\sqrt{1 - \frac{v^2}{c^2}})$
Just because of the presence of the Doppler term, we can derive a simpler crunch:
$\frac{v(c^2 - v^2)}{c^4}$
Or just
$\frac{\hbar}{pc} = \frac{\hbar}{E}\frac{v(c^2 - v^2)}{c^4}$
I suppose it is fair game to expand this new expression by applying lastly the fraction rule, so that we would have:
$\frac{\Delta \lambda}{c} = \frac{\hbar}{\Delta pc} = \frac{\hbar}{\Delta E} (\frac{vc^2}{c^4} - \frac{v^3}{c^4})$
Curious formula... Sometimes math is beautiful, but to completely associate it to physics is never easy. Distribution of $c$ and allowing us to plug in the gravitational interpretation due to gauge theory we get:
$\Delta \lambda = \frac{\hbar}{\Delta p} = \frac{Gm^2}{\Delta E} (\frac{vc^2}{c^4} - \frac{v^3}{c^4})$
http://electron6.phy...atter_waves.htm
Edited by Dubbelosix, 16 August 2019 - 08:54 AM.
### #63 Dubbelosix
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Posted 16 August 2019 - 09:25 AM
I'll continue tomorrow.
$\mathcal{L}\psi = n\mathbf{k} [\frac{Q^2_g}{4 e^2}]^n Gm^2_e\ \mathbf{R}^2\ \int \frac{dk}{k^{n-1}}\ \psi = n\mathbf{k} [\frac{\alpha_G}{4 \alpha}]^n Gm^2_e\ \mathbf{R}^2\ \int \frac{dk}{k^{n-1}}\ \psi$
Weinberg's formula was:
$m^3 = \frac{H_0 \hbar^2}{Gc}$
Rearranging we get the mass squared formula simply as:
$Gm^2 = \frac{H_0 \hbar^2}{mc}$
In spirit that as the Hubble constant was small during the ground state epoch, it directly implies that:
* As $H_0 \approx 0$
*Then the measured value of $G$ would be strong due to small bound universe with large gravitationalc curvature contributions.
And for a matter particle, travelling freely no less, is given by the kinetic formula:
$E_k = \frac{mv^2}{2} = \frac{p^2}{2m} = \hbar^2\ \frac{k^2}{2m}$
Hitting
$Gm^2 = \frac{H_0 \hbar^2}{mc}$
With
$E_k = \hbar^2\ \frac{k^2}{2m}$
We can be sneaky and even plug this into the opening mass-prediction formula:
$\mathcal{L}\psi = n\mathbf{k} [\frac{\alpha_G}{4 \alpha}]^n Gm^2_e\ \mathbf{R}^2\ \int \frac{dk}{k^{n-1}}\ \psi$
Giving a formula representing kinetic energy rather than the langrangian density:
$\frac{Gm^2k^2}{2m} = \frac{H_0}{mc}\frac{\hbar^2 k^2}{2m}$
As:
$\mathcal{L}\psi = n\mathbf{k} [\frac{\alpha_G}{4 \alpha}]^n Gm^2_e\ \mathbf{R}^2\ \int \frac{dk}{k^{n-1}}\ \psi$
One way to do it is by cancelling out the common-mass factor in the denominator:
$\frac{Gm^2k^2}{2} = \frac{H_0}{c}\frac{\hbar^2 k^2}{2}$
Knowing that the wave number as units of $inverse\ length$ this can remove the squared from of the Ricci curvature (while dropping the wave function as it has little use in this dimensional crunching:
$\mathcal{L} = n\mathbf{k} [\frac{\alpha_G}{4 \alpha}]^n \int\ Gm^2_e\ \mathbf{R}^n\ \int \frac{dk}{k^{n-1}}\ k\ dk\ = n\mathbf{k} [\frac{\alpha_G}{4 \alpha}]^n \frac{H_0}{mc}\frac{\hbar^2 k^2}{2}\ \mathbf{R}^n\ \int \frac{dk}{k^{n-1}}\$
We ((might)) be able to go a little simpler using the connection alone to theorize:
$\mathcal{L} = n\mathbf{k} [\frac{\alpha_G}{4 \alpha}]^n \int\ \mathbf{Q}_{gravity}\ \Gamma\ \frac{dk}{k^{n-1}} = n\mathbf{k} [\frac{\alpha_G}{4 \alpha}]^n\ \Gamma Q_{gravity}\ \int\ \frac{dk}{k^{n-1}}\ dk^3$
With $\mathbf{Q}_{gravity}$ as the inertial charge, or charge density.
Edited by Dubbelosix, 17 August 2019 - 03:58 AM.
### #64 Dubbelosix
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Posted 17 August 2019 - 04:03 AM
For the next ''lookback'' on previous investigations, we shall now go right back to the beginning, going back to the friedmann equation (my specialist subject) and will look at some of the more recent idea's I have had... perhaps I will start this bit later on as have a few things to do. The main equation we will work from is:
$(\frac{\dot{R}}{R}) = \frac{8 \pi G}{3} \rho + \mathbf{k}\ \frac{c^2}{a^2}$
With $mathbf{k}$ being the curvature constant. Be back later. We will disembowel this last equation right down to roots of its mechanics.
Edited by Dubbelosix, 17 August 2019 - 04:04 AM. | 2019-08-23 16:42:41 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7263019680976868, "perplexity": 754.3679452522517}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027318894.83/warc/CC-MAIN-20190823150804-20190823172804-00180.warc.gz"} |
https://www.vedantu.com/question-answer/the-value-of-tan-50tan-100tan-150-cdot-cdot-cdot-class-10-maths-cbse-5ee87192e8223517fbdc4911 | Question
# The value of $\tan {5^0}\tan {10^0}\tan {15^0} \cdot \cdot \cdot \cdot \tan {85^0}$ is${\text{A}}{\text{. }}0 \\ {\text{B}}{\text{. Not Defined}} \\ {\text{C}}{\text{. }}1 \\ {\text{D}}{\text{. }} - 1 \\$
Hint: In this question we need to find the value of the given trigonometric expression. In order to evaluate it easily we will use the property that $\tan x = \cot \left( {90 - x} \right)$. This will simplify the expression and help us reach the answer.
We have been given the expression $\tan {5^0}\tan {10^0}\tan {15^0} \cdot \cdot \cdot \cdot \tan {85^0}$
Now, as we know that $\tan x = \cot \left( {90 - x} \right)$, so we will apply it to the expression starting from $\tan {50^0}$ to $\tan {85^0}$ we get,
$\tan {5^0}\tan {10^0}\tan {15^0} \cdot \cdot \tan {40^0}\tan {45^0}\cot {40^0}...\cot {5^0}$
As we know that, $\tan x.\cot x = 1$
So, we get
$\Rightarrow \tan {5^0}\tan {10^0}\tan {15^0} \cdot \cdot \cdot \cdot \tan {85^0} = 1 \cdot 1 \cdot 1 \cdot \cdot \cdot \cdot \cdot \tan {45^0}$
And as we know that$\ tan{45^0}=1$,
Hence, $\Rightarrow \tan {5^0}\tan {10^0}\tan {15^0} \cdot \cdot \cdot \cdot \tan {85^0} = 1$
Note: Whenever we face such types of problems the main point to remember is that we need to have a good grasp over trigonometric properties, some of which have been used above. We must also remember the value of tangents of some common values to use them whenever required. This helps in getting us the required condition and gets us on the right track to reach the answer. | 2021-05-09 17:18:14 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8729929327964783, "perplexity": 140.068822747908}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243989006.71/warc/CC-MAIN-20210509153220-20210509183220-00290.warc.gz"} |
https://www.zbmath.org/?q=an%3A0910.14026 | # zbMATH — the first resource for mathematics
Laumon’s resolution of Drinfeld’s compactification is small. (English) Zbl 0910.14026
Let $$C$$ be a smooth projective curve of genus 0. Let $${\mathcal B}$$ be the variety of complete flags in an $$n$$-dimensional vector space $$V$$. Given an $$(n-1)$$-tuple $$\alpha$$ of positive integers one can consider the space $${\mathcal Q}_\alpha$$ of algebraic maps of degree $$\alpha$$ from $$C$$ to $${\mathcal B}$$. This space has drawn much attention recently in connection with quantum cohomology [see e.g. A. B. Givental, Int. Math. Res. Not. 1996, No. 13, 613-663 (1996; Zbl 0881.55006) and M. Kontsevich in: The moduli space of curves, Proc. Conf., Texel Island 1994, Prog. Math. 129, 335-368 (1995; Zbl 0885.14028)]. The space $${\mathcal Q}_\alpha$$ is smooth but not compact. The problem of compactification of $${\mathcal Q}_\alpha$$ proved very important. One compactification $${\mathcal Q}^K_\alpha$$ was constructed by M. Kontsevich (loc. cit.) (the space of stable maps). Another compactification $${\mathcal Q}^L_\alpha$$ (the space of quasiflags), was constructed by G. Laumon in: Automorphic forms, Shimura varieties, and $$L$$-functions, Vol. I, Proc. Conf., Ann. Arbor 1988, Perspect. Math. 10, 227-281 (1990; Zbl 0773.11032). However, historically the first and most economical compactification $${\mathcal Q}^D_\alpha$$ (the space of quasimaps) was constructed by Drinfeld (early 80-s, unpublished). The latter compactification is singular, while the former ones are smooth. Drinfeld has conjectured that the natural map $$\pi:{\mathcal Q}^L_\alpha \to {\mathcal Q}^D_\alpha$$ is a small resolution of singularities.
In the present note we prove this conjecture after the necessary recollections. The arguments in the proof are rather similar to those of G. Laumon (loc. cit., 3.3.2). In fact, the proof gives some additional information about the fibers of $$\pi$$. – In conclusion, let us mention that the Drinfeld compactifications are defined for the space of maps into flag manifolds of an arbitrary semisimple group, and it would be very interesting to construct their small resolutions.
##### MSC:
14M15 Grassmannians, Schubert varieties, flag manifolds 14E15 Global theory and resolution of singularities (algebro-geometric aspects) 32J05 Compactification of analytic spaces
Full Text: | 2021-09-21 03:19:12 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8193851709365845, "perplexity": 495.15574387259676}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780057131.88/warc/CC-MAIN-20210921011047-20210921041047-00472.warc.gz"} |
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# Two vehicles leave the same location at the same time. The first vehic
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29 Mar 2018, 00:42
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Two vehicles leave the same location at the same time. The first vehicle travels due east at 70 miles per hour. The other vehicle travels due west at 60 miles per hour. Assuming they continue at their respective speeds without stopping, how long (in hours) will it take for the two vehicles to be 455 miles apart?
(A) 2.75
(B) 3.5
(C) 4.25
(D) 4.75
(E) 5.5
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Re: Two vehicles leave the same location at the same time. The first vehic [#permalink]
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29 Mar 2018, 01:16
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Bunuel wrote:
Two vehicles leave the same location at the same time. The first vehicle travels due east at 70 miles per hour. The other vehicle travels due west at 60 miles per hour. Assuming they continue at their respective speeds without stopping, how long (in hours) will it take for the two vehicles to be 455 miles apart?
(A) 2.75
(B) 3.5
(C) 4.25
(D) 4.75
(E) 5.5
The vehicles are moving opposite to each other, hence speed with which distance between them increases is 70 + 60 mph
d = speed * time
$$455 = 130 * t$$
$$t = 455 /130$$
$$t = 3.5$$
Hence Option (B) is correct.
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Two vehicles leave the same location at the same time. The first vehic [#permalink]
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29 Mar 2018, 03:48
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1
Bunuel wrote:
Two vehicles leave the same location at the same time. The first vehicle travels due east at 70 miles per hour. The other vehicle travels due west at 60 miles per hour. Assuming they continue at their respective speeds without stopping, how long (in hours) will it take for the two vehicles to be 455 miles apart?
(A) 2.75
(B) 3.5
(C) 4.25
(D) 4.75
(E) 5.5
The best method to answer this question will be using the Relative speed
Relative speed of the two objects is defined as change ijn distance between the two objects in unit time
i.e. Relative speed of two moving objects with speed a and b travelling in same direction = a-b
and Relative speed of two moving objects with speed a and b travelling in Opposite direction = a+b
The the case given to us is the case of two trains moving in opposite directions hence
Relative Speed = 70+60 = 130 miles per hour
Time = Relative Distance / Relative Speed
i.e. Time = 455 / 130 = 3.5 Hours
ALTERNATIVE:
Let, After t hours the distance between the two trains = 455 miles
Distance travelled by First train in t hours = Speed * Time = 70*t
Distance travelled by Second train in t hours = Speed * Time = 60*t
Total Distance travelled by the two trains together = 70t + 60t = 455 (Given)
i.e. 130t = 455
i.e. t = 3.5
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Re: Two vehicles leave the same location at the same time. The first vehic [#permalink]
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30 Mar 2018, 10:51
Bunuel wrote:
Two vehicles leave the same location at the same time. The first vehicle travels due east at 70 miles per hour. The other vehicle travels due west at 60 miles per hour. Assuming they continue at their respective speeds without stopping, how long (in hours) will it take for the two vehicles to be 455 miles apart?
(A) 2.75
(B) 3.5
(C) 4.25
(D) 4.75
(E) 5.5
We can let t = the time each car drives and create the equation:
70t + 60t = 455
130t = 455
t = 455/130 = 3.5 hours
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Re: Two vehicles leave the same location at the same time. The first vehic [#permalink]
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14 Apr 2018, 23:18
Bunuel wrote:
Two vehicles leave the same location at the same time. The first vehicle travels due east at 70 miles per hour. The other vehicle travels due west at 60 miles per hour. Assuming they continue at their respective speeds without stopping, how long (in hours) will it take for the two vehicles to be 455 miles apart?
(A) 2.75
(B) 3.5
(C) 4.25
(D) 4.75
(E) 5.5
Effective speed = 60 + 70 = 130 m/h
Distance = 455 m
Therefore, Effective time = $$\frac{455}{130}$$= $$\frac{7}{2}$$= $$3.5 hrs$$
(B)
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Re: Two vehicles leave the same location at the same time. The first vehic [#permalink] 14 Apr 2018, 23:18
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# Two vehicles leave the same location at the same time. The first vehic
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Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne | 2019-11-14 21:01:23 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.530655026435852, "perplexity": 2870.366574100349}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496668539.45/warc/CC-MAIN-20191114205415-20191114233415-00318.warc.gz"} |
http://tex.stackexchange.com/questions/67378/in-lyx-how-do-i-indent-the-paragraph-right-after-an-itemize-enumerate-environme?answertab=oldest | # In LyX, how do I indent the paragraph right after an itemize/enumerate environment
See the example above, I want to indent the second paragraph.
The generated .tex file is
%% Do not edit unless you really know what you are doing.
\documentclass[english]{article}
\usepackage[T1]{fontenc}
\usepackage[latin9]{inputenc}
\usepackage{babel}
\begin{document}
First paragraph.
\begin{itemize}
\item item 1
\item item 2
\end{itemize}
Second paragraph.
Third paragraph.
\end{document}
Simply add a blank line after the line \end{itemize} will do what I want. But how can I do this in LyX?
One possible way is to add a box of TeX code (Evil Red Text) containing one blank line before second paragraph. But I want to know if there are ways without invoving Evil Red Text. Thanks.
-
add a blank line after the \end{itemize} – Seamus Aug 16 '12 at 9:00
@Seamus I know I can simply add a blank line in the generated .tex file, but this is a question about LyX and I'm seeking a direct and elegant solution within LyX. – Tianyi Cui Aug 16 '12 at 9:30
Hmm, moving the cursor to the start of the second paragraph and hitting enter doesn't work, it just jumps back when you move the cursor again. Might be a bug in LyX. In any case that behavior seems silly. – Torbjørn T. Aug 16 '12 at 10:52
Hope this work. I just add a "separator". You click in the scroll bar under File and Edit Menu... Another thing is, you need to view always the source code/tex code for you to be able to know that you are in the next paragraph. – jeecabz Aug 16 '12 at 10:54
@TianyiCui There's nothing direct or elegant about LyX. Why not just join the dark side and write directly in LaTeX? With editors like TeXshop or Gummi you can get auto-updating output anyway, so why stick with LyX's awkward restrictions? – Seamus Aug 16 '12 at 12:05
This works without any ERT. Redefine itemize environment as showed int the preamble of this MWE:
\documentclass[english]{article}
\let\olditemize\itemize
\renewenvironment{itemize}
{\begin{olditemize}}
{\par
\end{olditemize}
\vspace{0.7\baselineskip}
\par}
\begin{document}
First paragraph
\begin{itemize}
\item item 1
\item item 2
\item item 3
\end{itemize}
First paragraph
Third paragraph
\end{document}
Change 0.7\baselineskip if you want more space after the last item.
-
The OP specifically asked if he could do this without an ERT. (And personally, I don't think \setlength\parindent{0pt} is a good idea, though it does at least hide the problem.) – Torbjørn T. Aug 17 '12 at 21:05
Torbjørn, I agreed. Change \parident is not a good idea for any real article. Insert a ERT could be not elegant but it is less problematic. – Fran Aug 17 '12 at 22:03
Currently I'm using this solution: i.stack.imgur.com/7380b.png. Empty ERT in a blank paragraph, less intrusive and easier to type than ERT code. – Tianyi Cui Aug 18 '12 at 9:44
I changed completely my answer because I have found the way to redefine itemize and this solve the problem without ERT. – Fran Aug 19 '12 at 10:49 | 2014-11-28 07:05:08 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8867157697677612, "perplexity": 2852.896177955357}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-49/segments/1416931009825.77/warc/CC-MAIN-20141125155649-00248-ip-10-235-23-156.ec2.internal.warc.gz"} |
http://isomorphismes.tumblr.com/post/1660890378/mathematical-infinity | ## Maths Infinity
Conceiving of ∞ as a mathematician is simple. You start counting, and don’t stop.
That’s all.
$\dpi{300} \bg_white \begin{matrix} s: \mathbb{N} \to \mathbb{N} \\ s \mapsto s+1 \end{matrix}$
($i++ for programmers) Which is why seems very small to the mind of a mathematician. With projective geometry you can map to a circle, in which case there is a point-sized hole at the top where you can put ∞ (or −∞, or both). Same thing with the Riemann Sphere. So to them ∞ is very reachable. It’s just a tiny point. Graham’s Number It takes much more mental effort to conceive of Graham’s Number than ∞. It took me several hours just to begin to conceive Graham’s number the first time I tried. $\dpi{300} \bg_white \underbrace{ {{{{{{{{{{{{3^3}^3}^3}^3}^3}^3}^3}^3}^3}^3}^3}^{\cdots} } }_{ 3^{3^{3^{3^{\cdots}}}} \text{ times} }$ Graham’s Number is basically a continuation of the above, recursed many times. Maybe I’ll do a write-up another time but really you can just look at Wikipedia or Mathworld. It’s absolutely mind-blowing. Bigger Here’s what’s weird. Infinity is obviously bigger than Graham’s Number. But Graham’s Number takes up more mental space. Weird, right? EDIT: Maybe ∞ takes up less mental space than g64 because its minimal algorithmic description is shorter. 25 notes 1. clazzjassicalrockhop reblogged this from isomorphismes 2. noisesoundsignal reblogged this from isomorphismes and added: Math Infinity Conceiving of ∞ as a mathematician is simple. You start counting, and don’t stop. That’s all. ($i++ for...
3. squint-scowl answered: Though some infinities might be larger than others. Consider cardinalities!
4. davidaedwards answered: The Normal Distribution is conceptually much more complex than the Binomial Distribution; but computationally much simpler. | 2014-03-12 00:50:11 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 2, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9702963829040527, "perplexity": 9470.799060130123}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-10/segments/1394011473737/warc/CC-MAIN-20140305092433-00014-ip-10-183-142-35.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/mach-space.51807/ | # Mach space
1. Nov 7, 2004
### nightcleaner
I have been reading about Mach space in Dr. Brian Greene's recent book, "The Fabric of the Cosmos", and must admit I am somewhat confused. This is not Dr. Greene's fault, I am sure. I was hoping for some clarification here. I will try to summarize what I found in Dr. Greene's book.
Mach proposed that in free empty space, with no uneven field or distant object to be related, a spinning object will not feel the centripetal forces. In common language, in totally empty space, how could it know it was spinning, without any distant object to relate itself?
My understanding is that objects are composed of parts. Did Mach neglect the fact that the parts are related to each other? An astronaut spinning in free space still feels his hands pulled outward, because his hands are accelerated in relation to his body. He doesn't need distant stars to know this. What am I missing?
Thanks,
nc
2. Nov 8, 2004
### turbo
Here is a thought-experiment that may confuse things even more:
Suppose you have a space station made of two parallel wheels connected by a common axle. The "tires" of the wheels are toroidal, and they contain crew quarters, labs, control rooms, etc. To provide about 1/2 G of "artificial gravity" for the folks working in each half of the station, you counter-rotate the wheels. Now the crew members can walk on the "floors" formed by the inside walls of the torii. The crew members can look out the windows at the opposite torus and see it spinning past them, and they feel the "artificial gravity" pulling them to the "floor" of their torus. They also notice that the background stars are moving at half the rotational speed of the neighboring torus. So far, so good. Now place that same station in Machian space with NO external masses - no stars and galaxies by which one can gauge one's movement. How are inertial forces expressed and measured in each torus?
Let's now go back to the beginning, and after we set up our space station, we do our best to keep one torus stationary so we can use all the great zero-G labs we set up in it, and we spin the other torus twice as fast as in the previous example, so the crew quarters will have normal-feeling one-G "gravitation", allowing the crew can walk, run, exercise, etc. The people floating around at their stations in the "stationary" zero-G toroid will see that the background stars remain fixed, while the other toroid rotates at the same speed as in the previous example. Now, let's pop this space station into that special Machian space that has no stars, galaxies, etc. Again, how are inertial forces expressed and measured in each torus?
In both examples (in real space), the people in one torus module will see the other torus rotating at equivalent rates - the only difference will be in the apparent rotation rate of the background universe. In both examples that we transported into empty (Machian) space, the people in each torus will see exactly the same rotation rate in the opposite torus, but they will see NO rotation of the background of space with which to determine their own rotation or lack thereof.
3. Nov 8, 2004
### Garth
The movement of one torus is measured against the movement of the other, if they are of equal mass then the two torii have equal and opposite motion.
It is similar to the question of whether a single body in an empty universe can experience inertial force, what would its acceleration be measured against? In order to produce such a force one would have to create a reaction of some kind. Say split the body exactly in half and use some force (magnetic repulsion say) to propel them apart. As soon as the split happens the motion of one half can be measured against the other, and it will be found to be equal and opposite.
The centripetal virtual force is proportional to $$\omega^2$$ so in that torus the crew will feel 2-G. In the Machian empty universe in the same example there is no 'absolute non-rotating frame, so both torii rotate at the orginal rate in opposite directions and their crews would feel 1/2 G as before.
We cannot find out which scenario is correct as we cannot do this experiment.
Garth
Last edited: Nov 8, 2004
4. Nov 8, 2004
### nightcleaner
Thank you for the replies, but perhaps my question was misunderstood.
In Mach space, there is no background. Will water in the bucket or people in the spacestation or two rocks on a string experience centripetel force in a background free environment?
My understanding is that Mach said they would not feel any centripetel force because without a background they cannot be said to rotate. I am suggesting that the water itself, or the station or the rocks on a string, are systems in themselves and therefore provide their own internal background. One rock is accelerated compared to the other, so the string pulls taut. The astronauts are part of the station, which is their background, so the idea that there are no background stars is not to the point. The water in the bucket curls, because the molocules and atoms in the water form a system of parts, and the parts are related to each other, and so one part may move in an accelerated fashion compared to another part.
If we are to make progress on this question, I suggest we use Occam's razor to remove any non-essential details. I think I remember that the rocks on a string idea was of Einstein. It seems the simplest system to me which may illustrate the point.
Mach would say that in a background free condition, (Mach space, no stars, no uneven fields) the two rocks cannot be related to anything to determine that they are in a condition of rotation. I would say that they are still related to each other, and so the idea of background stars or no background stars is irrelevant. The rocks form their own spacetime lattice and so can be made to feel centripetel force.
What I wish to resolve by this line of thought is that the argument of Mach is mis-stated. Macroscopic objects like space stations and astronauts and rocks on a string form their own background. If we wish to use Mach's argument, the object to be rotated or not rotated must not be an object in space as we know it, but instead must be a point partical in an otherwise empty spacetime. Can such a fundamental indivisible partical be said to rotate? If so, what does this tell us about quantum spin?
I hope to engage someone in discussion on this point.
Thanks for being here.
nc
5. Nov 8, 2004
### Garth
That was the same concept of Mach space as I was using. Take the bucket. If the whole bucket and its contents revolve in an otherwise empty universe then according to Mach the water should be flat. The whole system rotates together, but as there is nothing that it is rotating with respect to there can be (according to MP) no rotation and no centrifugal force on the water molecules.
How this relates to quantum effects is indeed an interesting question, but we must remember that quantum spin is not rotation as we normally know it to be!
Garth
6. Nov 8, 2004
### turbo
Unfortunately, I came up with a similar result using Mach's terms. Perhaps we have to substitute the local "ground state of the universe" for Mach's "all matter in all of the universe" to make sense of his insistence that movement relative to the cosmological background is intrinsic to inertia. If inertia is conferred by movement relative to objects that are many light-years away, we have a HUGE "action at a distance" problem.
7. Nov 9, 2004
### Chronos
Doesn't classical mechanics work in this example? Even in an otherwise empty universe, there is still me, the rock and the string. Do we not have inertial mass?
8. Nov 9, 2004
### Garth
Chronos - If it is you, a rock, and (for the sake of argument), a massless string then the universe is not empty! To spin the rock you would need some kind of reaction force that would spin the rock one way and yourself the other.
turbo-1 - Yes there is action at a distance, just as with gravitation. I don't know whether you have followed the thread on the Cosmological Twin Paradox?here. But the inference is if the universe is closed and circumnavigation were possible then locally a preferred frame is defined (the one belonging to the stationary twin) by the global topology of the universe. An action-at-a-distance indeed!
Just a thought.
Garth
9. Nov 9, 2004
### nightcleaner
Thanks, Garth. I am not sure what MP is. My question to Mach would be, why have you neglected that the water itself is made up of many molecules, all related to each other? If the water is turning on an axis, then the molecules in and near the axis will have different motions compared to the molecules near the edge of the water. The curvature of the surface of the water has nothing to do with distant stars, everything to do with different motions of the water molecules compared to each other. I would therefore agree with Chronos that classical mechanics should still work in Mach space.
Then if we redesign Mach's experiment to occur on the Planck spacetime scale, where quantum effects become important, perhaps we can deduce something about the geometry of quarks. Events on the Planck scale are largely isolated from background by the extremely short times and distances involved, which make the local region of the quark environment unperturbed, as fits the description of Mach space. Then how should we expect simple two and three object systems in rotation to behave in such an environment? Do we get the restricted spin states found in quantum effects?
Take the two part quark-antiquark pair in an electron. Without external references, the only information about the spin condition of the pair is whether you are looking at it from the top or the bottom. It can only be said to rotate plus or rotate minus, which, if I have gotten all this down correctly, is the definition of a spin 1/2 partical.
If I have achieved communication here, I would like to extend this reasoning to see how the three quark particles might be seen to behave in the unperturbed space-time environment near the Planck scale.
Please help me with some feedback. I do respect and honor your opinions, and hope through this dialog to be able to improve my descriptions.
THanks,
nc
10. Nov 9, 2004
### jcsd
MP is Mach's principle, which basically states that inertia (the resistance to the chnage of motion) is caused by movement relative to the background created by all the matter in the universe and therefore if we remove this matter then acceleration would be relative too and the 'fictious' inertial forces would not be present in any frames.
Mach's principle is not actually a feature of any of our currnetly accpeted theories, though it did influence Albert Einstein (which does include the relativty of accelartion, though not in the Machian sense).
11. Nov 9, 2004
### Garth
nc - It is good to respond your posts.
Mach's Principle: There is no absolute frame that defines inertial frames of reference, inertial mass is determined by the distribution of mass and energy in the rest of the universe.
Consider your bucket in an otherwise empty unverse: If you say, "If the water is turning on an axis, then the molecules in and near the axis will have different motions compared to the molecules near the edge of the water." the question is 'turning' and 'motions' with respect to what?
Although I am not competent to extend MP to quantum physics, it would seem to me that if you had a sole electron in an otherwise empty universe you would not be able to determine its spin. Not only would you need an observer to "collapse the probability wave function", but also the direction of spin would be dependent on the orientation of that observer - but which way is 'up' in that empty universe?
Apart from that I am concerned that quantum spin does not infer motion or rotation and acceleration as classical spin does, therefore it may be completely independent of any Machian-type considerations.
Garth
Last edited: Nov 9, 2004
12. Nov 10, 2004
### nightcleaner
13. Nov 10, 2004
### nightcleaner
Thank you for the complement.
I have just finished a long post to jcsd, which you may find interesting. I solicit your comments.
You defined "Mach's Principle: There is no absolute frame that defines inertial frames of reference, inertial mass is determined by the distribution of mass and energy in the rest of the universe."
This definition is harder to deconstruct than that of jcsd. Still, I guess the absolute frame in this model would have to be the frame of the observer. Then to say that there is no absolute frame would be to say that there is no observer, in which case the definition would seem to have no physical meaning.
And then, if inertial mass is determined by the distribution of mass and energy in the rest of the universe, I have to wonder how and why this rest mass is separated from the mass of the object under consideration? Again, why is the internal structure of the object not sufficient to provide a frame of reference which would be inertial?
Finally, I suggest that Mach's idea may yet have useful application in the lower bounds of the Planck scale, where our notions of space and time begin to break down. Without space and time, what is mass and energy?
Geometry, however, does not have to have time as an axis. X, Y, and Z are sufficient to have many liesurely and productive arguments in a space that seems timeless, altho I suppose if the universe we inhabit crunches, then all our pretty geometry will have to crunch also.
Nevertheless, as long as we are here, what is to stop us from considering a 4 dimensional spacetime matrix, where time and space are unified and matter and energy take on geometric forms? I know it is difficult, but is it really impossible? The early quantum theorists had no way to imagine or visualize a universe in which their measurements made sensible pictures, but does that mean that no such universe exists? They could find no way to rationalize an object whose spin was always and only plus or minus, and never any value in between. So they said there was no use trying to visualize quantum structures. They (the quantum structures, not the physicists) behaved in a way that was, visually, irrational. Instead, they (the quantum physicists this time) just followed the math, and accepted ideas like collapseing waveforms. Is it a kind of taboo, a "don't go there?"
I am proposing a model which I think may be useful, but it requires exploration with physical tools that I do not have at my disposal. That is why I have come to the Physics Forum to seek help. To give you an idea, I have extended the model out to ten Planck lengths, but need to bring it out to 10E22 lengths to compare it to current observation based theory. I don't have the computational power to do this. Any ideas?
nc
Last edited by a moderator: Nov 10, 2004
14. Nov 11, 2004
### Garth
nc - from the above posts, (which took some reading!) it is acceleration that MP is concerned with, 'motion' or 'movement' is entirely relative as with Galilean or SR relativity.
There is no shell around the observer in an empty universe, it is empty, the reason why acceleration and rotation have no meaning in an empty Machian universe is because there is nothing to accelerate or rotate with respect to.
Some years ago there was a paper examining the apparent polarisation of quasar radio radiation, it tended to be one way in one part of the sky and the other way in the opposite part of the sky. It was thought that this might be caused by the whole universe rotating coherently.
My question was, "If so what is it rotating with respect to?" Talk of rotation with respect to the space-time fabric did not satisfy me as there was no way of pinning that 'fabric' down.
The effect was subsequently declared to be caused by the galactic electric field.
The inertial mass is compared to the gravitational mass. If both are declared invariant as the norm of the object's four-momentum, then an effect might reveal itself as a variation in G the Newtonian Gravitational 'constant'; this is the approach of the Brans-Dicke theory. Self Creation Cosmology allows the (inertial) rest mass to vary and it is the observed Newtonian parameter that remains constant.
Garth
15. Nov 11, 2004
### nightcleaner
Ok Garth, and thanks.
My post was lengthy in part because I always try to tie my ideas directly to first principles, rather than use the physicists shorthand of refering to things like MP and then just assuming that the reader knows the reference. Thank you for your time spent reading through it.
Your shorter post contains some terms I am not familiar with and will have to go elsewhere to study before I can get the sense of the post. But it seems that we have pared down the argument to one of motion and acceleration, anyway.
I tried to show in the long post that motion in a universe that is otherwise empty except for an object is unthinkable, so Mach is right to say that in such a universe, there could be no accelertion. However, if we add the point of view of a disembodied observer, we can distinguish between rotation of the object about an axis and orbit of the point of view about the object by means of the complexity of the calculation necessary to construct a simulation. Essentially, the presense of an observer, even though disembodied, provides a mark on the otherwise unmarked universe, so that we can make sense out of the notion that the object is moving, in this case rotationally. Since rotation makes sense to the observer, the concept of acceleration is restored.
The conundrum in Mach's Principle is then reduced to one of invoking motion in a universe where motion is meaningless, and the solution, imho, is to appeal to the position of an observer, which then restores meaning to the notion of motion.
THen I wish to go one step further and restate Mach's conditions, such that the object under observation has in itself no parts that can be compared for motion by the observer, which is to say that the object is thought of as a perfect indivisible unmarked sphere, which can be rotated in any sense without providing any difference to be measured, even with the presense of the disembodied observer. This restores Mach's assertion that such an object will experience no motion and so no acceleration.
If my reasoning holds, then this is the condition in which we find quantum point-like objects. We can then reintroduce motion again by examining systems containing two and three point-like objects and an observer, and by doing so can find a model that explains the quantum behavior of fundamental particles. We would look to explain quantum spin, tunneling, and probability waves, with a physical model, so making the paradigm shift between quantum processes and statistical probabalistic processes easier to negotiate.
I invite criticism of this approach. Following the approach will involve a lot of thought work and several more long papers, so if someone can provide good reason for not going there, it could save me and others a lot of work.
Thanks,
nc
16. Nov 14, 2004
### turbo
Garth, here is a paper on Machian General Relativity that I bumped into today. The model seems to share a lot of features with SCC. Are there any points of strong divergence that pop up?
http://arxiv.org/abs/gr-qc/0106007
17. Nov 14, 2004
### Garth
Thank you for that link, I will follow it up with Booth.
Booth's approach has many points of contact with SCC. However he keeps particle masses (energy-momentum) conserved and varies G. Although G varies 'in the background' in SCC, all Cavendish type measurements of G would not be aware of this as GM remains constant in the theory.
Garth | 2016-12-11 06:39:00 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5190287232398987, "perplexity": 570.1245218875408}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-50/segments/1480698544140.93/warc/CC-MAIN-20161202170904-00088-ip-10-31-129-80.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/calculating-molality-given-density-and-molarity.89464/ | # Archived Calculating molality given density and molarity
1. Sep 17, 2005
### LakeMountD
I haven't taken chemistry in over 3 years so I have forgotten a lot of the very basic concepts (even while understanding the higher level stuff) and am having problems with this simple thing! Please help :(.
Question 1- At 20°C, a 2.32 M aqueous solution of ammonium chloride has a density of 1.0344 g/mL What is the molality of ammonium chloride in the solution? The formula weight of NH4Cl is 53.5 g/mol.
I don't think I have to use temperature here. I know how to calculate molality when im given how many grams are in the solution but the way this one is worded im confused. Not looking for the answer, just a little help on where to start.
Last edited: Sep 17, 2005
2. Mar 26, 2016
### MexChemE
Alright, molarity is defined as moles of solute per liters of solution and molality is defined as moles of solute per kilograms of solvent. That said, we must define a basis to work with, let's say we have 1 liter of solution, and because we know the molarity of this solution is 2.32 M, then we know there's 2.32 moles of NH4Cl in the solution. And we know 2.32 moles of NH4Cl are equal to 124.12 grams. Next, we have the density of the solution, which is defined as mass of solution per volume of solution. So, 1 liter of solution contains 1034.4 grams, of which 124.12 grams are of NH4Cl, then we have 910.28 grams of solvent. Now we can calculate the molality of the solution
$$\textrm{molality} = \frac{2.32 \ mol}{0.91028 \ kg \ water} = 2.549 \ m$$ | 2017-03-24 18:00:00 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8512955904006958, "perplexity": 1076.5441700166384}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-13/segments/1490218188550.58/warc/CC-MAIN-20170322212948-00178-ip-10-233-31-227.ec2.internal.warc.gz"} |
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# In the rectangular quadrant system shown above, which quadrant, if any
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08 Aug 2016, 08:15
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In the rectangular quadrant system shown above, which quadrant, if any, contains no point (x, y) that satisfies the equation 3x + 5y = −2?
A. none
B. I
C. II
D. III
E. IV
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2016-08-08_2014.png [ 1.3 KiB | Viewed 2113 times ]
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08 Aug 2016, 08:25
Convert the equation to the form y=mx+c.
$$3x + 5y = -2$$ is $$y = -3x/5 - 2/5$$ And applying values for $$x$$ as 0,1,-1 we can see that it doesn't pass through I quadrant.
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08 Aug 2016, 09:32
Bunuel wrote:
In the rectangular quadrant system shown above, which quadrant, if any, contains no point (x, y) that satisfies the equation 3x + 5y = −2?
A. none
B. I
C. II
D. III
E. IV
Attachment:
2016-08-08_2014.png
Simple way to solve this questions is to draw the line on the coordinate plan by taking the x-intercept and y-intercept. We will get a line that won't pass through 1st quadrant.
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08 Aug 2016, 09:48
3x + 5y= -2
rearranging the equation:
y=-(3/5)x-(2/5)
slope is negative
line passes through either II,I ,and IV quadrants or II,III, and IV quadrants.
Y intercept is negative => line passes through II,III and IV quadrants.
Any point (x,y) in I quadrant will not satisfy the given equation.
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08 Aug 2016, 22:19
4
Bunuel wrote:
In the rectangular quadrant system shown above, which quadrant, if any, contains no point (x, y) that satisfies the equation 3x + 5y = −2?
A. none
B. I
C. II
D. III
E. IV
Attachment:
2016-08-08_2014.png
3x + 5y = -2
Put x = 0, y = -2/5
Put y = 0, x = -2/3
Plot the two points. You will see that the line will pass through II, III and IV quadrants but not through quadrant I.
Also note that a line cannot pass through all 4 quadrants...
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09 Aug 2016, 18:26
The equation is 3x + 5y = -2. Since both x and y are positive in the 1st quadrant, sum of 2 positive numbers can never yield a negative.
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Re: In the rectangular quadrant system shown above, which quadrant, if any &nbs [#permalink] 24 Mar 2018, 20:25
Display posts from previous: Sort by | 2019-01-19 14:32:34 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5431943535804749, "perplexity": 9065.35468410312}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-04/segments/1547583668324.55/warc/CC-MAIN-20190119135934-20190119161934-00421.warc.gz"} |
https://leanprover-community.github.io/archive/stream/113489-new-members/topic/cannot.20find.20case.20names.20for.20induction.html | ## Stream: new members
### Topic: cannot find case names for induction
#### Reuben Rowe (Nov 27 2019 at 14:00):
Why do I get the error "could not find open goal of given case" with the following code?
universe u
variables { α : Type u } { size : α → ℕ }
include size
example : α → Prop
:=
begin
assume a : α,
let n := size a,
induction n,
case nat.zero {
sorry,
},
case nat.succ {
sorry,
},
done
end
#### Patrick Massot (Nov 27 2019 at 14:05):
The error message is weird, but it's clearly Lean's answer to you trying to do something weird. Could you give more context? What are you actually trying to do?
#### Marc Huisinga (Nov 27 2019 at 14:12):
changing let to have works, but i am not sure what is going on either
#### Reuben Rowe (Nov 27 2019 at 14:22):
I want to prove a property about some relation on elements of an inductive type $\alpha$, but it doesn't hold by structural induction because the definition of the relation uses an operation that changes the elements, but which does preserve the size, so my induction must be on the size of the elements.
#### Reuben Rowe (Nov 27 2019 at 14:23):
trying to do something weird
Why is this weird?
#### Patrick Massot (Nov 27 2019 at 14:30):
It will be very difficult to help you without a clearer explanation, or some code actually illustrating your use case. In the mean time, maybe reading https://leanprover.zulipchat.com/#narrow/stream/113489-new-members/topic/Induction.20on.20minimum.20.22length.22 could help (what you wrote sounds vaguely similar to that old thread).
#### Sebastien Gouezel (Nov 27 2019 at 14:31):
At the beginning of your proof, you have chosen an element a. Now, it is fixed forever, so you can not do an induction on its size. The naive way to do this would be to prove a statement of the form : "for all n, for all elements a of sizen, then something holds", by induction over n. The non-naive way would be to generalize something.
#### Reuben Rowe (Nov 27 2019 at 14:41):
Ah yes, of course. I see - thanks! Sorry for all the stupid questions ;-)
#### Patrick Massot (Nov 27 2019 at 14:41):
especially not with error messages like these. i still find it strange that it only fails when case is used wheras you just can't prove it if you don't use case. | 2021-05-12 05:05:24 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 1, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5964154601097107, "perplexity": 1860.485714997221}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243991252.15/warc/CC-MAIN-20210512035557-20210512065557-00385.warc.gz"} |
http://physics.aps.org/synopsis-for/10.1103/PhysRevLett.107.122502 | # Synopsis: Pushing toward better nuclear data
Dense plasmas created by lasers used for inertial confinement fusion research have been used to extract precise basic nuclear physics data.
Researchers probe many important nuclear reactions using collisions produced with accelerators. Gleaning good data on some fundamental reactions can be problematic, however. For example, low-energy cross sections often need to be extrapolated from high-energy measurements. In those and other cases, nuclear physics research benefits from an enlarged toolkit for measurement of basic cross-section data.
Writing in Physical Review Letters, Johan Frenje at the Massachusetts Institute of Technology, Cambridge, and colleagues report their use of data obtained in laser-driven inertial confinement fusion experiments to extract precise cross sections for important light-ion interactions in which neutrons scatter off deuterons ($2$H) and tritons ($3$H).
The research team created high-density plasmas by using the OMEGA laser system to implode glass capsules filled with deuterium and tritium. Sixty powerful laser beams strike the outer surface of the capsule, generating a plasma that then pushes it inward to compress the gas. As the gas reaches thermonuclear conditions, reactions create $14.1\phantom{\rule{0.333em}{0ex}}\text{MeV}$ neutrons that collide with the hydrogen isotopes, and scattering cross sections are extracted from the energy spectra of these species. The results match ab initio calculations well, providing not only a boost to theory but also data on reactions of crucial importance to nuclear astrophysics and fusion energy research. – David Voss
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Lunar soil samples reveal an unusually high level of radioactive iron, which suggests a supernova in our vicinity within the last few million years. Read More » | 2016-05-06 01:45:24 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 7, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.41365617513656616, "perplexity": 3631.0688273515866}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-18/segments/1461861700326.64/warc/CC-MAIN-20160428164140-00051-ip-10-239-7-51.ec2.internal.warc.gz"} |
https://math.stackexchange.com/questions/2548568/how-to-find-answer-of-sum-i-1n-ii-and-solve-its-equation | # How to find answer of $\sum_{i=1}^{n}\, i^i$ and solve its equation?
I like to find a formula for the $\sum_{i=1}^n\,i^i$.
It is possible to write a formula for the summation of the form $\sum_{i=1}^n\, i^a$. For example, here explain its methods completely. But what kind of method should we use when the power in the summation is not fixed?
I first tried to find an upper bound. And I could only think of $\sum_{i=1}^n\,i^i\leq n.n^n$ which seems that it's not very useful.
I also wish to compute $n^n=c$, for some constant $c\in \mathbb{N}$. But again it seems very hard. If the power was fixed (that is n^a=c) then we could find the $a$th root of $n$. We could use logarithm if we had the form $a^n=c$ for some fixed $a$. Unfortunately, neither power nor base is fixed.
• This may help you: Sum of Powers – user507623 Dec 3 '17 at 7:57
$$n^n\left(1+\frac 1{4(n-1)} \right) <\sum_{i=1}^{n}\, i^i <n^n\left(1+\frac 2{e(n-1)} \right)$$
Let us check $$\left( \begin{array}{cccc} n & \text{lower} &\text{exact} &\text{upper}\\ 2 & 5. & 5 & 6.94304 \\ 3 & 30.375 & 32 & 36.9327 \\ 4 & 277.333 & 288 & 318.785 \\ 5 & 3320.31 & 3413 & 3699.81 \\ 6 & 48988.8 & 50069 & 53521.5 \\ 7 & 857857. & 873612 & 924531. \\ 8 & 1.73764\times 10^7 & 17650828 & 1.85406\times 10^7 \\ 9 & 3.99527\times 10^8 & 405071317 & 4.23051\times 10^8 \\ 10 & 1.02778\times 10^{10} & 10405071317 & 1.08175\times 10^{10}\\ 11 & 2.92444\times 10^{11} & 295716741928 & 3.06304\times 10^{11} \\ 12 & 9.11874\times 10^{12} & 9211817190184 & 9.51247\times 10^{12} \\ 13 & 3.09185\times 10^{14} & 312086923782437 & 3.21445\times 10^{14} \\ 14 & 1.13257\times 10^{16} & 11424093749340453 & 1.17409\times 10^{16} \\ 15 & 4.45713\times 10^{17} & 449317984130199828 & 4.60907\times 10^{17} \end{array} \right)$$ | 2021-04-16 13:18:49 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9607438445091248, "perplexity": 187.37512216575573}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038066981.0/warc/CC-MAIN-20210416130611-20210416160611-00518.warc.gz"} |
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15 30 50 per page | 2015-11-28 11:20:37 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5534828901290894, "perplexity": 2289.3281164621867}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-48/segments/1448398452385.31/warc/CC-MAIN-20151124205412-00192-ip-10-71-132-137.ec2.internal.warc.gz"} |
https://cracku.in/14-if-the-angles-of-a-triangle-are-in-the-ratio-of-14-x-rrb-ntpc-12-april-2016-shift-3 | Question 14
# If the angles of a triangle are in the ratio of 1:4:7, then find the ratio of the greatest angle to the smallest angle.
Solution
The sum of angles of triangle is 180
let the angles 1x,4x,7x
1x+4x+7x=180
12x=180
x=$$\frac{180}{12}$$
= 15
first angle=15°
second angle=60°
third angle=105°
the ratio of the greatest angle to the smallest angle
7:1 | 2023-03-30 14:48:38 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4892813265323639, "perplexity": 1285.1154612838743}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296949331.26/warc/CC-MAIN-20230330132508-20230330162508-00407.warc.gz"} |
http://mathoverflow.net/questions/84165/construction-of-mirror-quintic-family-over-mathbba1-setminus-0-1 | # Construction of mirror quintic family over $\mathbb{A}^{1} \setminus \{0,1\}$
This question is about how to construct a Fermat pencil of quintics and the mirror family over $\mathbb{A}\setminus {0,1}$ as opposed to over $\mathbb{A}^{1}\setminus {0,\mu_{5}}$, where $\mu_{5}$ is the fifth roots of unity. Most sources say that you can do this, but don't say exactly how.
Let me first recall the the standard things that people say about constructing the family of Calabi-Yaus mirror dual to a quintic 3-fold in $\mathbb{P}^{4}$, and then I'll point out where I have a question (probably pretty simple).
Consider first the pencil of quintics $\mathcal{X}$ in $\mathbb{P}^{4} \times \mathbb{A}^{1}$ with equation $z_{0}^{5} + ... + z_{4}^{5}+5\psi z_{0}\cdots z_{4}=0$. It is easy to check that this is smooth for $\psi^{5} \neq 1$, so away from $5$th roots of unity. Note that the pencil is somewhat redundant, since for any 5th root of unity $\alpha$, $\mathcal{X}_{\psi}$ is isomorphic to $\mathcal{X}_{\alpha\psi}$ via $[a_{0}:...:a_{4}] \mapsto [\alpha^{-1}a_{0}:...:a_{4}]$.
At this point, people usually say it is more natural to take a coordinate $\lambda=\psi^{5}$ for the pencil.
I would like to interpret this in the following way: Consider the action of the group of fifth roots of unity on $\mathbb{P}^{4} \times \mathbb{A}^{1}$ via $([a_{0}:...:a_{4}],b) \mapsto ([\alpha^{-1}a_{0}:...:a_{4}],\alpha b)$, which is free on $\mathbb{P}^{4} \times (\mathbb{A}^{1} \setminus{0})$ and restricts to a free action on $\mathcal{X}$ (minus the zero fibre). We should therefore be able to take a quotient family $\mathcal{X}/\mu_{5} \rightarrow \mathbb{A}^{1} \setminus 0$, which will be smooth away from $1$.
1. Does this work?
2. Is there a more concrete description of this family, in terms of an equation for some pencil?
Now to get the mirror family from this point, we take a further quotient of the second family to get a third family, fibrewise. This third family has quotient singularities along the fibres and we want to construct a simultaneous crepant resolution. Surely there is some work involved in showing that you can do this. Usually people do this first, before taking the redundancy out of the family with parameter $\psi$.
1. Can we construct a simultaneous crepant resolution of the second family with 'coordinate $\lambda=\psi^{5}$'?
2. Is there a concrete way to write down such a family of mirror quintics, smooth over $\mathbb{A}^{1} \setminus 0,1$? Say as a pencil of hypersurfaces in some ambient toric variety?
Note that I really want the resulting family to be non-redundant. If it were even the restriction of the universal family over the moduli stack of mirror quintics, that would be ideal.
-
An explicit quintic parametrized by $\lambda = \psi^5$ is $$\lambda^{-1} \alpha_0^5 + \alpha_1^5 + \alpha_2^5 + \alpha_3^5 + \alpha_4^5 = 5 \alpha_0 \alpha_1 \alpha_2 \alpha_3 \alpha_4.$$ Indeed if $\lambda = \psi^5$ then the first term is $(\alpha/\psi)^5$, and then replacing $\alpha$ by $\psi\alpha$ recovers the fiber of $\cal X\phantom.$ at $\psi$. – Noam D. Elkies Dec 23 '11 at 18:42 | 2015-08-01 12:23:52 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9077804684638977, "perplexity": 156.75605374770916}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-32/segments/1438042988650.6/warc/CC-MAIN-20150728002308-00290-ip-10-236-191-2.ec2.internal.warc.gz"} |
http://aimsciences.org/article/doi/10.3934/dcdsb.2017070 | # American Institue of Mathematical Sciences
2017, 22(4): 1461-1492. doi: 10.3934/dcdsb.2017070
## Global existence for a thin film equation with subcritical mass
1 School of Mathematics, Liaoning University, Shenyang 110036, China 2 Department of Physics and Department of Mathematics, Duke University, Durham, NC 27708, USA
Corresponding author: Jinhuan Wang, was supported by National Natural Science Foundation of China (Grant No. 11301243) and Program for Liaoning Excellent Talents in University (Grant No. LJQ2015041)
Jian-Guo Liu was partially supported partially supported by KI-Net NSF RNMS grant No. 1107444, NSF DMS grant No. 1514826
Received December 2015 Revised November 2016 Published February 2017
In this paper, we study existence of global entropy weak solutions to a critical-case unstable thin film equation in one-dimensional case
$h_t+\partial_x (h^n\,\partial_{xxx} h)+\partial_x (h^{n+2}\partial_{x} h)=0,$
where
$n≥q 1$
. There exists a critical mass
$M_c=\frac{2\sqrt{6}π}{3}$
found by Witelski et al.(2004 Euro. J. of Appl. Math. 15,223-256) for
$n=1$
. We obtain global existence of a non-negative entropy weak solution if initial mass is less than
$M_c$
. For
$n≥q 4$
, entropy weak solutions are positive and unique. For
$n=1$
, a finite time blow-up occurs for solutions with initial mass larger than
$M_c$
. For the Cauchy problem with
$n=1$
and initial mass less than
$M_c$
, we show that at least one of the following long-time behavior holds:the second moment goes to infinity as the time goes to infinity or
$h(·, t_k)\rightharpoonup 0$
in
$L^1(\mathbb{R})$
for some subsequence
${t_k} \to \infty$
.
Citation: Jian-Guo Liu, Jinhuan Wang. Global existence for a thin film equation with subcritical mass. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1461-1492. doi: 10.3934/dcdsb.2017070
##### References:
[1] P. Álvarez-Caudevilla, V. A. Galaktionov, Well-posedness of the Cauchy problem for a fourth-order thin film equation via regularization approaches, Nonl. Anal., 121 (2015), 19-35. doi: 10.1016/j.na.2014.08.002. [2] E. F. Beckenbach and R. Bellman, Introduction to Inequalities Random House Inc, 1965. [3] E. Beretta, M. Bertsch, R. Dal Passo, Nonnegative solutions of a fourth order nonlinear degenerate parabolic equation, Arch. Ration. Mech. Anal., 129 (1995), 175-200. doi: 10.1007/BF00379920. [4] F. Bernis, Finite speed of propagation and continuity of the interface for slow viscous flows, Adv. Differential Equations, 1 (1996), 337-368. [5] F. Bernis, A. Friedman, Higher order nonlinear degenerate parabolic equations, J. Differential Equations, 83 (1990), 179-206. doi: 10.1016/0022-0396(90)90074-Y. [6] A. L. Bertozzi, M. C. Pugh, The lubrication approximation for thin viscous films, the moving contact line with a porous media cut off of Van der Waals interactions, Nonlinearity, 7 (1994), 1535-1564. doi: 10.1088/0951-7715/7/6/002. [7] A. L. Bertozzi, M. C. Pugh, The lubrication approximation for thin viscous films: Regularity and long time behavior of weak solutions, Comm. Pure Appl. Math., 49 (1996), 85-123. doi: 10.1002/(SICI)1097-0312(199602)49:2<85::AID-CPA1>3.0.CO;2-2. [8] A. L. Bertozzi, M. C. Pugh, Long-wave instabilities and saturation in thin film equations, Comm. Pure Appl. Math., 51 (1998), 625-661. doi: 10.1002/(SICI)1097-0312(199806)51:6<625::AID-CPA3>3.0.CO;2-9. [9] A. L. Bertozzi, M. C. Pugh, Finite-time blow-up of solutions of some long-wave unstable thin film equations, Indiana Univ. Math. J., 49 (2000), 1323-1366. doi: 10.1512/iumj.2000.49.1887. [10] M. Bertsch, L. Giacomelli, G. Karali, Thin-film equations with "partial wetting" energy: Existence of weak solutions, Physica D, 209 (2005), 17-27. doi: 10.1016/j.physd.2005.06.012. [11] M. Bertsch, R. Dal Passo, H. Garcke, G. Grün, The thin viscous flow equation in higher space dimensions, Adv. Differential Equations, 3 (1998), 417-440. [12] S. Bian, J.-G. Liu, Dynamic and steady states for multi-dimensional Keller-Segel model with diffusion exponent $m > 0$, Comm. Math. Phys., 323 (2013), 1017-1070. doi: 10.1007/s00220-013-1777-z. [13] M. Chugunova, M. C. Pugh and R. M. Taranets, Research Announcement: Finite-time blow up and long-wave unstable thin film equations, arXiv1008.0385v1, (2010). [14] M. Chugunova, R. M. Taranets, Blow-up with mass concentration for the long-wave unstable thin-film equation, Appl. Anal., 95 (2016), 944-962. doi: 10.1080/00036811.2015.1047829. [15] R. Dal Passo, H. Garcke, Solutions of a fourth order degenerate parabolic equation with weak initial trace, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 28 (1999), 153-181. [16] R. Dal Passo, H. Garcke, G. Grün, On a fourth-order degenerate parabolic equation: Global entropy estimates, existence, and qualitative behavior of solutions, SIAM J. Math. Anal., 29 (1998), 321-342. doi: 10.1137/S0036141096306170. [17] L. Giacomelli, M. V. Gnann, F. Otto, Regularity of source-type solutions to the thin-film equation with zero contact angle and mobility exponent between 3/2 and 3, European J. Appl. Math., 24 (2013), 735-760. doi: 10.1017/S0956792513000156. [18] L. Giacomelli, H. Knüpfer, F. Otto, Smooth zero-contact-angle solutions to a thin-film equation around the steady state, J. Differential Equations, 245 (2008), 1454-1506. doi: 10.1016/j.jde.2008.06.005. [19] M. V. Gnann, Well-posedness and self-similar asymptotics for a thin-film equation, SIAM J. Math. Anal., 47 (2015), 2868--2902. doi: 10.1137/14099190X. [20] G. Grün, Droplet spreading under weak slippage: The optimal asymptotic propagation rate in the multi-dimensional case, Interfaces Free Bound., 4 (2002), 309-323. doi: 10.4171/IFB/63. [21] G. Grün, Droplet spreading under weak slippage: A basic result on nite speed of propagation, SIAM J. Math. Anal., 34 (2003), 992-1006. doi: 10.1137/S0036141002403298. [22] G. Grün, Droplet spreading under weak slippage-existence for the Cauchy problem, Comm. Partial Differential Equations, 29 (2004), 1697-1744. doi: 10.1081/PDE-200040193. [23] D. John, On uniqueness of weak solutions for the thin-film equation, J. Differential Equations, 259 (2015), 4122-4171. doi: 10.1016/j.jde.2015.05.013. [24] H. Knüpfer, Well-posedness for the Navier slip thin film equation in the case of partial wetting, Comm. Pure Appl. Math., 64 (2011), 1263-1296. doi: 10.1002/cpa.20376. [25] H. Knüpfer, N. Masmoudi, Darcy flow on a plate with prescribed contact angle well-posedness and lubrication approximation, Arch. Rational Mech. Anal., 218 (2015), 589-646. doi: 10.1007/s00205-015-0868-8. [26] R. S. Laugesen, M. C. Pugh, Properties of steady states for thin film equations, European J. Appl. Math., 11 (2000), 293-351. doi: 10.1017/S0956792599003794. [27] J. -L. Lions, Quelques Méthodes de Résolution Des Problémes Aux Limites Non Linéaires Paris, Dunod, 1969. [28] A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow Vol. 27, Cambridge University Press, 2002. [29] D. Matthes, R. J. McCann, G. Savaré, A family of nonlinear fourth order equations of gradient flow type, Comm. Partial Differential Equations, 34 (2009), 1352-1397. doi: 10.1080/03605300903296256. [30] A. Mellet, The thin film equation with non zero contact angle: A singular perturbation approach, Comm. Partial Differential Equations, 40 (2015), 1-39. doi: 10.1080/03605302.2014.895380. [31] T. G. Myers, Thin films with high surface tension, SIAM Rev., 40 (1998), 441-462. doi: 10.1137/S003614459529284X. [32] B. V. Sz. Nagy, Ãœber Integralungleichungen zwischen einer Funktion und ihrer Ableitung (German), Acta Univ. Szeged. Sect. Sci. Math., 10 (1941), 64-74. [33] F. Otto, Lubrication approximation with prescribed nonzero contact angle, Comm. Partial Differential Equations, 23 (1998), 2077-2164. doi: 10.1080/03605309808821411. [34] D. SlepÄev, M. C. Pugh, Self-similar blow-up of unstable thin-film equations, Indiana Univ. Math. J., 54 (2005), 1697-1738. doi: 10.1512/iumj.2005.54.2569. [35] R. M. Taranets, J. R. King, On an unstable thin-film equation in multi-dimensional domains, NoDEA Nonlinear Differential Equations Appl., 21 (2014), 105-128. doi: 10.1007/s00030-013-0240-3. [36] T. P. Witelski, A. J. xBernoff, A. L. Bertozzi, Blow-up and dissipation in a critical-case unstable thin film equation, European J. Appl. Math., 15 (2004), 223-256. doi: 10.1017/S0956792504005418. [37] Z. Q. Wu, J. N. Zhao, J. X. Yin and H. L. Li, Nonlinear Diffusion Equations 2nd edition, Singapore, World Scientific, 2001. doi: 10.1142/9789812799791.
show all references
##### References:
[1] P. Álvarez-Caudevilla, V. A. Galaktionov, Well-posedness of the Cauchy problem for a fourth-order thin film equation via regularization approaches, Nonl. Anal., 121 (2015), 19-35. doi: 10.1016/j.na.2014.08.002. [2] E. F. Beckenbach and R. Bellman, Introduction to Inequalities Random House Inc, 1965. [3] E. Beretta, M. Bertsch, R. Dal Passo, Nonnegative solutions of a fourth order nonlinear degenerate parabolic equation, Arch. Ration. Mech. Anal., 129 (1995), 175-200. doi: 10.1007/BF00379920. [4] F. Bernis, Finite speed of propagation and continuity of the interface for slow viscous flows, Adv. Differential Equations, 1 (1996), 337-368. [5] F. Bernis, A. Friedman, Higher order nonlinear degenerate parabolic equations, J. Differential Equations, 83 (1990), 179-206. doi: 10.1016/0022-0396(90)90074-Y. [6] A. L. Bertozzi, M. C. Pugh, The lubrication approximation for thin viscous films, the moving contact line with a porous media cut off of Van der Waals interactions, Nonlinearity, 7 (1994), 1535-1564. doi: 10.1088/0951-7715/7/6/002. [7] A. L. Bertozzi, M. C. Pugh, The lubrication approximation for thin viscous films: Regularity and long time behavior of weak solutions, Comm. Pure Appl. Math., 49 (1996), 85-123. doi: 10.1002/(SICI)1097-0312(199602)49:2<85::AID-CPA1>3.0.CO;2-2. [8] A. L. Bertozzi, M. C. Pugh, Long-wave instabilities and saturation in thin film equations, Comm. Pure Appl. Math., 51 (1998), 625-661. doi: 10.1002/(SICI)1097-0312(199806)51:6<625::AID-CPA3>3.0.CO;2-9. [9] A. L. Bertozzi, M. C. Pugh, Finite-time blow-up of solutions of some long-wave unstable thin film equations, Indiana Univ. Math. J., 49 (2000), 1323-1366. doi: 10.1512/iumj.2000.49.1887. [10] M. Bertsch, L. Giacomelli, G. Karali, Thin-film equations with "partial wetting" energy: Existence of weak solutions, Physica D, 209 (2005), 17-27. doi: 10.1016/j.physd.2005.06.012. [11] M. Bertsch, R. Dal Passo, H. Garcke, G. Grün, The thin viscous flow equation in higher space dimensions, Adv. Differential Equations, 3 (1998), 417-440. [12] S. Bian, J.-G. Liu, Dynamic and steady states for multi-dimensional Keller-Segel model with diffusion exponent $m > 0$, Comm. Math. Phys., 323 (2013), 1017-1070. doi: 10.1007/s00220-013-1777-z. [13] M. Chugunova, M. C. Pugh and R. M. Taranets, Research Announcement: Finite-time blow up and long-wave unstable thin film equations, arXiv1008.0385v1, (2010). [14] M. Chugunova, R. M. Taranets, Blow-up with mass concentration for the long-wave unstable thin-film equation, Appl. Anal., 95 (2016), 944-962. doi: 10.1080/00036811.2015.1047829. [15] R. Dal Passo, H. Garcke, Solutions of a fourth order degenerate parabolic equation with weak initial trace, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 28 (1999), 153-181. [16] R. Dal Passo, H. Garcke, G. Grün, On a fourth-order degenerate parabolic equation: Global entropy estimates, existence, and qualitative behavior of solutions, SIAM J. Math. Anal., 29 (1998), 321-342. doi: 10.1137/S0036141096306170. [17] L. Giacomelli, M. V. Gnann, F. Otto, Regularity of source-type solutions to the thin-film equation with zero contact angle and mobility exponent between 3/2 and 3, European J. Appl. Math., 24 (2013), 735-760. doi: 10.1017/S0956792513000156. [18] L. Giacomelli, H. Knüpfer, F. Otto, Smooth zero-contact-angle solutions to a thin-film equation around the steady state, J. Differential Equations, 245 (2008), 1454-1506. doi: 10.1016/j.jde.2008.06.005. [19] M. V. Gnann, Well-posedness and self-similar asymptotics for a thin-film equation, SIAM J. Math. Anal., 47 (2015), 2868--2902. doi: 10.1137/14099190X. [20] G. Grün, Droplet spreading under weak slippage: The optimal asymptotic propagation rate in the multi-dimensional case, Interfaces Free Bound., 4 (2002), 309-323. doi: 10.4171/IFB/63. [21] G. Grün, Droplet spreading under weak slippage: A basic result on nite speed of propagation, SIAM J. Math. Anal., 34 (2003), 992-1006. doi: 10.1137/S0036141002403298. [22] G. Grün, Droplet spreading under weak slippage-existence for the Cauchy problem, Comm. Partial Differential Equations, 29 (2004), 1697-1744. doi: 10.1081/PDE-200040193. [23] D. John, On uniqueness of weak solutions for the thin-film equation, J. Differential Equations, 259 (2015), 4122-4171. doi: 10.1016/j.jde.2015.05.013. [24] H. Knüpfer, Well-posedness for the Navier slip thin film equation in the case of partial wetting, Comm. Pure Appl. Math., 64 (2011), 1263-1296. doi: 10.1002/cpa.20376. [25] H. Knüpfer, N. Masmoudi, Darcy flow on a plate with prescribed contact angle well-posedness and lubrication approximation, Arch. Rational Mech. Anal., 218 (2015), 589-646. doi: 10.1007/s00205-015-0868-8. [26] R. S. Laugesen, M. C. Pugh, Properties of steady states for thin film equations, European J. Appl. Math., 11 (2000), 293-351. doi: 10.1017/S0956792599003794. [27] J. -L. Lions, Quelques Méthodes de Résolution Des Problémes Aux Limites Non Linéaires Paris, Dunod, 1969. [28] A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow Vol. 27, Cambridge University Press, 2002. [29] D. Matthes, R. J. McCann, G. Savaré, A family of nonlinear fourth order equations of gradient flow type, Comm. Partial Differential Equations, 34 (2009), 1352-1397. doi: 10.1080/03605300903296256. [30] A. Mellet, The thin film equation with non zero contact angle: A singular perturbation approach, Comm. Partial Differential Equations, 40 (2015), 1-39. doi: 10.1080/03605302.2014.895380. [31] T. G. Myers, Thin films with high surface tension, SIAM Rev., 40 (1998), 441-462. doi: 10.1137/S003614459529284X. [32] B. V. Sz. Nagy, Ãœber Integralungleichungen zwischen einer Funktion und ihrer Ableitung (German), Acta Univ. Szeged. Sect. Sci. Math., 10 (1941), 64-74. [33] F. Otto, Lubrication approximation with prescribed nonzero contact angle, Comm. Partial Differential Equations, 23 (1998), 2077-2164. doi: 10.1080/03605309808821411. [34] D. SlepÄev, M. C. Pugh, Self-similar blow-up of unstable thin-film equations, Indiana Univ. Math. J., 54 (2005), 1697-1738. doi: 10.1512/iumj.2005.54.2569. [35] R. M. Taranets, J. R. King, On an unstable thin-film equation in multi-dimensional domains, NoDEA Nonlinear Differential Equations Appl., 21 (2014), 105-128. doi: 10.1007/s00030-013-0240-3. [36] T. P. Witelski, A. J. xBernoff, A. L. Bertozzi, Blow-up and dissipation in a critical-case unstable thin film equation, European J. Appl. Math., 15 (2004), 223-256. doi: 10.1017/S0956792504005418. [37] Z. Q. Wu, J. N. Zhao, J. X. Yin and H. L. Li, Nonlinear Diffusion Equations 2nd edition, Singapore, World Scientific, 2001. doi: 10.1142/9789812799791.
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https://www3.unisa.ac.za/4pucqmx/page.php?tag=cascade-refrigeration-system-problems-dbb5bc | O2/NH3 cascade refrigeration systems New measurement methods make it possible to design energy-efficient Low harge Eva-porator Systems ontrolling both the injection of refrigerant according to the Evaporator load, while also increasing safety aspect when using natural refrigerants as O2/NH3 ca-scade systems. Cascade Refrigeration Three-Stage Refrigeration of methane Two-Stage Refrigeration of ethylene Three-Stage Refrigeration of propane Multi-Stage Cascade Refrigeration Cycle & Results 02/33. Steam Jet Refrigeration Systems. 0000008955 00000 n 0000031868 00000 n Cascade refrigeration is a term you will hear more and more over the coming years, and while some of the systems may be very complex, the concept is actually pretty simple. Refrigeration System Problems and Solutions: 1. They are most often built by CPU overclockers, and there is a lot of banter on the forums, but not a whole lot of detail. 0000004654 00000 n 0000001679 00000 n Before you charge pull a good 400 micron vacuum and let it sit for 10 minutes and you will be ready to charge. 15 Chapter 3 LITERATURE REVIEW [1] Different researchers have been carried out them researches on vapour compression refrigeration system, cascade refrigeration system and used different refrigerants for the performance analysis of the system and the refrigerants. Feb 08, 2017. TWO- AND THREE-STAGE CASCADE REFRIGERATION SYSTEM By: Paul B. Reed problem to find space for the evaporator and still have as much usable space in the cabinet as is required. 11-66 A two-stage cascade refrigeration cycle with a flash chamber with refrigerant-134a as the working fluid is considered. Unfortunately, one of the problems plaguing industrial refrigeration today is a lack of qualified technicians. Overcharge of the Refrigeration System. 888 31 Cascade systems. “Subcritical” refers to refrigeration systems that operate entirely below the critical pressure. Refrigeration System Problems and Solutions: 1. The ready light doesn't come on, but no surprise if -50C is the set point. 0000056066 00000 n A cascade system is akin to a two stage system but with a very crucial difference: the refrigerants used in the two stages are different. I recommend pulling a deep vacuum with copper tubing, refrigeration hoses are pressure rated and have a tendency to permeate air through in a deep vacuum. But I wondered about using R404a as an alternative (readily sourced on eBay. Another great blog posting on this technology. Each stage operates on the ideal vapor-compression refrigeration cycle. Ordering information: IRBPG@CascadeEnergy.com www.CascadeEnergy.com . The two basic controls we are most familiar with are the ‘mechanical differential control’, and the ‘pressure sensing electronic control’. This necessitates making the cabinet larger, which in turn increases the heat leakage load, so that additional evaporator surface must be allowed for. The cascade vapor-compression refrigeration cycle is two simple V-C cycles is series. Thats a nice system you have there Ben, If you have problems with refrigerants again I would suggest retrofitting the R-502 with R-404A. Thermoelectric Refrigeration. Room temperature warm. First, the flow controller is much faster than the temperature controller. I'm thinking of using this as a gas precooler for my sterling cooler. Study Problems 100 References 103 3 Refrigeration System Components 105 3.1 Introduction 105 3.2 History of Refrigeration 105 3.3 Main Refrigeration Systems 107 3.4 Refrigeration System Components 108 3.5 Compressors 109 3.5.1 Hermetic Compressors 110 3.5.2 Semihermetic Compressors 111 3.5.3 Open Compressors 113 3.5.4 Displacement Compressors 113 3.5.5 Dynamic Compressors 119 3.5.6 … 2. Introduction to Cascade CO 2 / NH 3 Refrigeration Systems. full 508b use cascade system ? Solar Refrigeration. I wouldn't be concerned about R-1150 or R-170 in a sealed system I have used propane in place of R-12 as a direct drop in to get my by in a bind. Chapter: Problem: FS show all show all steps. In large industrial cascade refrigerating system is shown in Fig. I'm going to open it up and do a thorough cleaning. I'm concerned about refrigerant / compressor compatibility. Concluding Remarks. It's a very low temperature refrigerant. The mass flow rate of the refrigerant through the high-pressure compressor, the rate of refrigeration, the COP are to be determined. These factors complicate the problem of retrofitting such systems. Nitrous. 0000003755 00000 n I've got a cold trap on order and am thinking about changing gases. 0000002187 00000 n for -80C, single stage, think mixed gases. What is the refrigerator? 0000004576 00000 n Some industrial applications require moderately low temperatures, and the temperature range they involve may be too large for a single vapor-compression refrigeration cycle to be practical. In the cascade systems we used, the R14 TEV was mounted on the outside of the vacuum chamber, usually more than 3 metres away from the cascade unit. Metal Hydride Refrigeration Systems. This is accomplished with a heat exchanger. Nomenclature. 0000003622 00000 n 0000002578 00000 n Abnormal start and stop sequences of the compressor. Both cycles use “natural” fluids, complying with strictest … The cascade system consists of two separate refrigeration circuits connected only by an intermediate cascade heat exchanger. startxref ?350 gr (r170) + 350 gr (r508b) = 700 gr addivite gas 700 gr ( 508b) gaz ?? ME420- Chapter 3 Problems Que1) An SSS vapour compression refrigeration system based on refrigerant R-134a operates between an evaporator temperature of – 25 o C and a condenser temperature of 50 o C. Assuming isentropic compression, find: a) COP of the system … The late Steve Maxson wrote this article during his tenure as an application engineer for Heatcraft Refrigeration Products, LLC. A few important features of the cascade structure should be emphasized. Cascade refrigeration system is combination of more than one refrigeration systems that works independently. x�bb]������� Ȁ ��@Q��>�*E���_%&���aX�������G���Dް)(�g�������j-�൩^�B�,@�n'��. xref It has one original manufacture connector.Here's all the info regarding its refrigerant. Problem 3 Consider a three-stage cascade refrigeration system operating between the pressure limits of 1.4 MPa and 160 kPa with refrigerant R134a as the working fluid. Any idea how low you can safely go with a Cryocooler Stirling engine?Doug. Ch 10, Lesson C, Page 2 - The Cascade Vapor-Compression Refrigeration Cycle. Examples of CO2systems Prof.Dr.-Ing. I retrofitted a cascade system for Lockheed Martin a few years back and I happened to have some R-23 around the shop and used that in place of the R-502 with no issues except the cost of R-23 is about ten times that of R-404. PSI 1 R-134A 7.5 - 1 R-404A 17 - 2 R-404A 1.4 4 2 R-508B 8.6 126, Do you have any idea for single stage -80°c deep frezeer what gas to be filed. Oil Pressure Problems in Refrigeration Systems. sort of like going down stairs. http://phoenix.tuwien.ac.at/chemistry/Gebrauchsanweisungen/SAVANT%20Vapor%20Trap.pdf. Study Problems. In such a cascade system, the electrical power for VCRS and the heat energy for VARScan be significantlyreduced, resulting inacoefficient ofperformance (COP)valuehigher thanthe value of each system operating in standalone mode. One question I have is given the relative simplicity of these cascaded systems, do you think it might be possible to home brew one of these? thanks nitrous ;Which gas are you advice substitude for 508b ? Magnetic Refrigeration. ?? As shown in Figure 10.18 , the high-temperature circuit is cooled by an air condenser (2) at ambient temperature, and uses the cascade heat exchanger (1) as the system … Condenser fan motor issue, less heat transfer available for a given mass of refrigerant. Lack of refrigerant present in the system. The Charge quantities of H~c-23 we likely to be quite similar to those for either R13 or R503 (mass basis) based on their similar liquid densities. two-stage compression cascade refrigeration system (CCRS). The R-13B1 can be replaced with R403B. The most important advantage of these cascading system is that refrigerants can be selected with the appropriate properties, avoiding the larger components of the system. I was also thinking of chilling the hot end f the Cryocooler. 3.37. This necessitates making the cabinet larger, which in turn increases the heat leakage load, so that additional evaporator surface must be allowed for. CASCADE REFRIGERATION Revised by: Frank Fulkerson, CMS Adapted from materials originally provided By: Charles C. E. Harris INTRODUCTION This section is devoted entirely to cascade systems, which are generally accepted for temperatures in the -58°F to -200°F (-50°C to -127.7°C) range. 0000009850 00000 n 0000007165 00000 n The refrigerant is R403b. We can set up training at our Maryland Facility or we can come to your facility. Armin Hafner Professor Refrigeration Technology NTNU, EPT 7491 Trondheim Norway GIAN @ Indian Institute of Technology Madras; October 2017 Content •CO2as an evaporating secondary fluid •CO2in a conventional cooling process (cascade system) • Transcritical CO2process with low-pressure receiver absorption refrigeration system (VARS) merges the advantages of both processes, resulting in a more cost-effective system. The temperature limit isn't currently known. Blocked filter at the drier or expansion valve. In this video, I describe how the unit works after recharging its second stage with R1150 (ethylene) refrigerant. two-stage compression cascade refrigeration system (CCRS). Some refrigerants are well suited for high and medium temperature applications, and some are better suited and for a lower temp applications. Hi Ben,Quick update on the cryo front. This is on a sticker on the side of the freezer. 0000055817 00000 n I retrofitted a cascade system for Lockheed Martin a few years back and I happened to have some R-23 around the shop and used that in place of the R-502 with no issues except the cost of R-23 is about ten times that of R-404. Thermoacoustic Refrigeration. H products developed a sensor, which detect ammonium carbamate in a NH3 system … Search for cascade cpu, and you'll find lots of long talky threads with a few bits of actionable details here and there. TWO- AND THREE-STAGE CASCADE REFRIGERATION SYSTEM By: Paul B. Reed problem to find space for the evaporator and still have as much usable space in the cabinet as is required. Evaporator inlet solenoid closed. Defrosting element still operational. 0 Using a cascade system power consumption can be reduced through about 9.5%. A cascade refrigeration cycle is a multi-stage thermodynamic cycle. Doug, Hi Ben,Just an update on the $5.00 Labcoco Centrivap cold trap. http://www.ior.org.uk/ior_/images/pdf/se/Datasheet%2014%20-%20Master%20Gas%20List.pdf. It gets to -48C but not quite to the rated -50C. Cascade refrigeration systems commonly used in the liquefaction of natural gas and some other gases. %PDF-1.4 %���� Blocked filter at the drier or expansion valve. Nitrous, yes, there are homebrew cascade systems. Liquefaction of Gases. hi dear nitrous r170 gas for altenatives gas ?? And any time I do open a refrigeration system up I install a new liquid line filter dryer. 0000005517 00000 n R508b is actually a replacement for R13 and R503. Room temperature warm. The Labconco unit I got may not be a dual compressor type. 918 0 obj <>stream Subcritical CO2 systems are very effective in industrial refrigeration applications when used in cascade systems with another refrigerant, such as ammonia. Steve Maxon . In a cascade refrigeration system, two or more vapor-compression cycles with different refrigerants are used. 0000009399 00000 n Consider a two-stage cascade refrigeration system operating between the pressure limits of 0.8 and 0.14 MPa. He passed away in July 2015. M&M offers training upon request. Supermarket Refrigeration. 0000008508 00000 n ii Industrial Refrigeration Best Practices Guide Table of Contents Industrial Refrigeration Best Practices Guide Primary Authors Marcus Wilcox, Rob Morton, Josh Bachman, Dan Brown, Glen Thomas, Aaron Frechette: Cascade Energy Engineering Cascade Energy Engineering (Cascade… I can see a major problem with reclaiming oil from the second compressor. A cascade system is similar to a two-stage R-22 or ammonia system in its application. When brazing on the system make sure to purge nitrogen through the system so you can avoid oxidation with in the system. 508+?? This paper provides a literature review of the cascade refrigeration system (CRS). 0000004015 00000 n Defrosting element still operational. An example two-stage process is shown at right. Cascade Refrigeration Systems. Heat rejection from the lower cycle to the upper As shown in Figure 10.18, the high-temperature circuit is cooled by an air condenser (2) at ambient temperature, and uses the cascade heat exchanger (1) as the system evaporator. In this paper, several research options such as various designs of CRS, studies on refrigerants, and optimization works on the systems are discussed. To solve these problems, two or more stage compression refrigeration systems should be cascaded by a heat exchanger, i.e. Heat rejection from the lower cycle to the upper cycle takes place in an adiabatic counterflow heat exchanger where the pressure in the upper and lower cycles are 0.4 and$0.5 \mathrm{MPa}$respectively. Any thoughts? I have a ultra low temperature freezer -86c. The refrigeration system presented herein features a cascade configuration combining a vapour compression cycle and an inverse Brayton cycle. This link might help you understand the refrigerants better. References 0000000935 00000 n I bought a -100*C refrigerated cold trap from a surplus sale, which was not operational. The cascade system consists of two separate refrigeration circuits connected only by an intermediate cascade heat exchanger. 2.1.1. The solution is cascading. Evaporator inlet solenoid closed. 5. To solve these problems, two or more stage compression refrigeration systems should be cascaded by a heat exchanger, i.e. is it ok. The short answer to your question is "No". 4. Many industrial applications like food storage, liquefaction of petroleum vapour … NH 3 cascade refrigeration system for cooling applications at low-temperatures, Applied Thermal Engineering (2008), doi: 10.1016/j.applthermaleng.2008.07.006 This is a PDF file of an unedited manuscript that has been accepted for publication. Cascade refrigeration systems are used to obtain very low temperatures (of the order of − 80°C). If you have any more problems finding refrigerant or parts shoot me an email and I can get you taken care of. Cascade Refrigeration Systems A two-stage cascade refrigeration system with the same refrigerant in both stages. The lower cycle cools the refrigerated space and the upper cycle cools the lower cycle. %%EOF A cascade refrigeration system, on the other hand, employs 2 or more individual refrigeration cycles operating at different pressure and temperature levels. Without it, or something similar, you will not get to the temperature you want. As a service to our customers we are providing this early version of the manuscript. Lack of refrigerant present in the system. Consider a two-stage cascade refrigeration system operating between the pressure limits of$1.4 \mathrm{MPa}$and$160 \mathrm{kPa}$with refrigerant-134a as the working fluid. trailer In a cascade system, you need to chose combinations of complimentary refrigerants to achieve the desired temperature. Cascade refrigeration system is the combination of two single stage vapour compression system together, condenser of LTC and evaporator of HTC is cascaded and forms the heat exchanger where evaporator cascade absorbs the heat from the condenser cascade which further leads to better refrigeration effect . It looks like there's never been any tampering that would indicate modification REFRIGERANTStage Type Oz. They have an ultra model that goes to -100C but this one might just stop at -50C or so. TM equivalent mass of HFC-23 will generate much higher pressures in the same system volume though, under the idle wamd-up condition. Cascade Energy Engineering, Inc. Ethylene seems available from our local Praxair distributor. It actually works, more or less. In fact R-1150 will preform better for you with lower discharge temperatures than any of the original refrigerants. 0000004244 00000 n and cascade control systems are given in Figure 14.3a and b for a decrease in oil pressure. It is an important system that can achieve an evaporating temperature as low as −170 °C and broadens the refrigeration temperature range of conventional systems. Chapter 11, Problem 42. Each stage operates on the ideal vapor-compression refrigeration cycle with refrigerant-134a as the working fluid. It appears you have your leak fixed so you shouldn't have to open the system up again. That would be a critical issue to address in the ultralow temp evaporator side of things. 0000003171 00000 n The very name "cascade" suggests that each gas takes the temperature down to a new level. 11-66 A two-stage cascade refrigeration cycle with a flash chamber with refrigerant-134a as the working fluid is considered. Refrigeration Cycle and Input Condition of Natural Gas 03/33. The upper vapor compression refrigeration system (VCRS) uses water as its working fluid and operates its evaporator at 5°C. An overcharged refrigeration system is one running on more refrigerant that it can handle. Causes. It can achieve reduced temperature and better efficiency compared to conventional refrigeration systems. The mass flow rate of the refrigerant through the high-pressure compressor, the rate of refrigeration, the COP are to be determined. * The lower temperature cycle provides the desired refrigeration effect at a relatively low temperature. 0000035347 00000 n 888 0 obj <> endobj A cascade refrigeration system employs 2 or more individual refrigeration cycles operating at different pressure and temperature levels. 2.1.1. <<065282687AD65548954261718A74D042>]>> I got this unit for$5.00 plus shipping (\$60) so I don't think I can go wrong. Here are the indicators of an overcharged system: High condenser gauge readings. A two-stage cascade refrigeration system is to provide cooling at −40°C while operating the high-temperature condenser at 1.6 MPa. Cascade Refrigeration System uses multiple refrigeration cycles coupled with each other via heat exchanger to improve the refrigeration effect. Cascade systems can be single stage, ie only one compressor state, but carefully selected refrigerants combinations. The duty of the lower temperature cycle is to provide the desired refrigeration effect at a relatively low temperature. 0000000016 00000 n It has mix refrigerant. Do you know how to recharge a mix refrigerant freezer? it is a direct drop in. 0000004531 00000 n 0000006348 00000 n 0000032653 00000 n Condenser fan motor issue, less heat transfer available for a given mass of refrigerant. 0000007951 00000 n Cascading improves the COP of a refrigeration system. 0000001998 00000 n 0000003792 00000 n 0000009149 00000 n 0000001478 00000 n 0000047062 00000 n High suction and discharge pressures. hi dear ben ,ı am teknical services operators,are you help me ?low compresor = r407 hıgh compresor = r508b+r170 I can not find r170 :( which I use instead of gas ?r290 ??? (Bottom on mobile) The cascade cycle is often employed for devices such as ULT freezers. It uses multiple refrigerants depending on the stages. Cascade system may improve the COP and add flexibility on the temperature levels and working fluids, but the problems related to the cold evaporator surface remain unsolved. Ultra low-temperature cascade refrigeration system... Vote for future video topics on Applied Science. The ethylene gas is for the ultra low temperature. Multistage Refrigeration Cycles.
Palm Oil For Soap Making Buy, Rocks State Park Picnic Area, Web Appbuilder Developer Edition Tutorial, Minimalism Art Movement, Private Music Teacher Contract Template, Population Of Annapolis 2020, 16 Inch Necklace On Woman, New Missouri Flag, Hamilton Bermuda Airport, Only Sixteen Original, | 2021-04-19 14:33:29 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.43893519043922424, "perplexity": 4025.2172714421795}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038887646.69/warc/CC-MAIN-20210419142428-20210419172428-00582.warc.gz"} |
http://www.koreascience.or.kr/article/JAKO201023437149627.page | # Weighted Geometric Means of Positive Operators
Izumino, Saichi;Nakamura, Noboru
• Received : 2009.09.16
• Accepted : 2010.01.27
• Published : 2010.06.30
• 16 4
#### Abstract
A weighted version of the geometric mean of k ($\geq\;3$) positive invertible operators is given. For operators $A_1,{\ldots},A_k$ and for nonnegative numbers ${\alpha}_1,\ldots,{\alpha}_k$ such that $\sum_\limits_{i=1}^k\;\alpha_i=1$, we define weighted geometric means of two types, the first type by a direct construction through symmetrization procedure, and the second type by an indirect construction through the non-weighted (or uniformly weighted) geometric mean. Both of them reduce to $A_1^{\alpha_1}{\cdots}A_k^{{\alpha}_k}$ if $A_1,{\ldots},A_k$ commute with each other. The first type does not have the property of permutation invariance, but satisfies a weaker one with respect to permutation invariance. The second type has the property of permutation invariance. We also show a reverse inequality for the arithmetic-geometric mean inequality of the weighted version.
#### Keywords
positive operator;weighted geometric mean;arithmetic-geometric mean inequality;reverse inequality
#### References
1. E. Ahn, S. Kim, H. Lee and Y. Lim, Sagae-Tanabe weighted means and reverse in- equalities, Kyungpook Math. J., 47(2007), 595-600.
2. T. Ando, C.-K. Li and R. Mathias, Geometric means, Linear Algebra Appl., 385(2004), 305-334. https://doi.org/10.1016/j.laa.2003.11.019
3. E. Andruchow, G. Corach and D. Stojanoff, Geometrical significance of the Lowner- Heinz inequality, Proc. Amer. Math. Soc., 128(1999), 1031-1037.
4. J. E. Cohen and R. D. Nussbaum, The arithmetic-geometric mean and its general- izations for noncommuting linear operators, Annali della Scuola Normale Sup di Pia Cl. Sci, (4) 15(1988) no. 2, 239-308.
5. B. Q. Feng and A. Tonge, Geometric means and Hadamard products, Math. InequaI- ities Appl., 8(2005), 559-564.
6. J.I. Fujii, M. Fujii, M. Nakamura, J. Pecaric and Y. Seo, A reverse inequality for the weighted geometric mean due to Lawson-Lim, Linear Algebra Appl., 427(2007), 272-284. https://doi.org/10.1016/j.laa.2007.07.025
7. J.I. Fujii, M. Nakamura, J. Pecaric and Y. Seo, Bounds for the ratio and difference between parallel sum and series via Mond-Pecaric method, Math. Inequalities and Appl., 9(2006), 749-759.
8. T. Furuta, J. Micic, J. Pecaric and Y. Seo, Mond-Pecaric Method in Operator In- equalities, Monographs in Inequalities I, Element, Zagreb, 2005.
9. S. Izumino and N. Nakamura, Geometric means of positive operators, II, Sci. Math.Japon., 69(2009), 35-44.
10. S. Kim and Y. Lim, A converse inequality of higher order weighted arithmetic and geometric means of positive definite operators, Linear Alg. Appl., 426(2007), 490-496. https://doi.org/10.1016/j.laa.2007.05.028
11. H. Kosaki, Geometric mean of several positive operators, 1984.
12. F. Kubo, and T. Ando, Means of positive linear operators, Math. Ann., 246(1980), 205-224. https://doi.org/10.1007/BF01371042
13. J. Lawson and Y. Lim, A general framework for extending means to higher orders, Colloq. Math., 113(2008), 191-221. https://doi.org/10.4064/cm113-2-3
14. N. Nakamura, Geometric means of positive operators, Kyungpook Math. J., 49(2009), 167-181. https://doi.org/10.5666/KMJ.2009.49.1.167
15. M. Sagae and K. Tanabe, Upper and lower bounds for the arithmetic-geometric- harmonic means of positive definite matrices, Linear and Multilinear Alg., 37(1994), 279-282. https://doi.org/10.1080/03081089408818331
16. T. Yamazaki, An extension of Kantorovich inequality to n-operators via the geometric mean by Ando-Li-Mathias, Linear Algebra Appl., 416(2006), 688-695. https://doi.org/10.1016/j.laa.2005.12.013 | 2019-04-20 10:19:25 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8375544548034668, "perplexity": 6198.162617095331}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-18/segments/1555578529606.64/warc/CC-MAIN-20190420100901-20190420122901-00251.warc.gz"} |
https://forum.coppeliarobotics.com/viewtopic.php?f=9&t=9424&sid=38025b81cdb351b0a512f40a2b48dece | ## Different sensor read with different physics engines
Typically: "How do I... ", "How can I... " questions
BlueSkeptical
Posts: 6
Joined: 11 Feb 2020, 15:11
### Different sensor read with different physics engines
Hi all, I'm trying to simulate the skin behaviour of a thigh getting hit by a sphere of $$10kg$$.
The scene looks like this:
The skin is simulated with a spring/damper joint where K=$$5e^4$$.(I found this value in some papers).
The top plate(Plate_dyn) weights $$0.2kg$$ while the thigh (UpperLeg_dyn) it's about $$70kg$$, in order to simulate the weight of the entire body supported by the leg.
The spring has a length of $$2cm$$. Then I plot the force of that Prismatic_joint on a graph.
My question is: Why do I get such differences in the force when I use different physics engine? I know there might be some differences, but here the maximum force read with different engine:
- Bullet 2.78: 880 N
- Bullet 2.83: 365 N
- Newton: -220 N
The differences are huge.
Last edited by BlueSkeptical on 23 Sep 2021, 09:15, edited 1 time in total.
coppelia
Posts: 8987
Joined: 14 Dec 2012, 00:25
### Re: Different sensor read with different physics engines
Hello,
it is normal to have some differences. But those differences indicate that something in the modeling is wrong. The first thing that comes to mind is that with most engines, too large differences between masses will result in strange/wobbly behaviour. See the dynamics design considerations 7 & 8 for example.
Another thing that probably plays a role is the thickness of the metal sheet, which appears to be very thin: thin objects can be tricky since a penetration depth (used to compute a reaction force) is difficult to find/interpret.
Make sure that your colliding objects are pure primitives, i.e. cuboids, spheres. If that is not an option, make sure to at least have convex shapes. This directly related to dynamics design considerations 1-3
Cheers | 2021-12-06 15:38:14 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7278043031692505, "perplexity": 1811.074831740326}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964363301.3/warc/CC-MAIN-20211206133552-20211206163552-00239.warc.gz"} |
https://eccc.weizmann.ac.il/keyword/15498/ | Under the auspices of the Computational Complexity Foundation (CCF)
REPORTS > KEYWORD > COMPUTABLE ANALYSIS:
Reports tagged with computable analysis:
TR02-014 | 10th December 2001
Klaus Weihrauch
#### Computational Complexity on Computable Metric Spaces
Revisions: 1
We introduce a new Turing machine based concept of time complexity for functions on computable metric spaces. It generalizes the ordinary complexity of word functions and the complexity of real functions studied by Ko \cite{Ko91} et al. Although this definition of ${\rm TIME}$ as the maximum of a generally infinite ... more >>>
TR05-157 | 10th December 2005
Xiaoyang Gu, Jack H. Lutz, Elvira Mayordomo
#### Points on Computable Curves
The analyst's traveling salesman theorem'' of geometric
measure theory characterizes those subsets of Euclidean
space that are contained in curves of finite length.
This result, proven for the plane by Jones (1990) and
extended to higher-dimensional Euclidean spaces by
Okikiolu (1991), says that a bounded set $K$ is contained
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TR14-015 | 24th January 2014
Jack H. Lutz, Neil Lutz
#### Lines Missing Every Random Point
Revisions: 1
This paper proves that there is, in every direction in Euclidean space, a line that misses every computably random point. Our proof of this fact shows that a famous set constructed by Besicovitch in 1964 has computable measure 0.
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TR14-133 | 15th October 2014
We define the lower and upper mutual dimensions $mdim(x:y)$ and $Mdim(x:y)$ between any two points $x$ and $y$ in Euclidean space. Intuitively these are the lower and upper densities of the algorithmic information shared by $x$ and $y$. We show that these quantities satisfy the main desiderata for a satisfactory ... more >>> | 2021-06-24 12:35:50 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9397188425064087, "perplexity": 1509.9017180378967}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623488553635.87/warc/CC-MAIN-20210624110458-20210624140458-00129.warc.gz"} |
http://digtime.cn/articles/223/ai-for-trading-word-embeddings-99 | # AI For Trading: Word Embeddings (99)
## Skip-gram Word2Vec
In this notebook, I'll lead you through using PyTorch to implement the Word2Vec algorithm using the skip-gram architecture. By implementing this, you'll learn about embedding words for use in natural language processing. This will come in handy when dealing with things like machine translation.
Here are the resources I used to build this notebook. I suggest reading these either beforehand or while you're working on this material.
## Word embeddings
When you're dealing with words in text, you end up with tens of thousands of word classes to analyze; one for each word in a vocabulary. Trying to one-hot encode these words is massively inefficient because most values in a one-hot vector will be set to zero. So, the matrix multiplication that happens in between a one-hot input vector and a first, hidden layer will result in mostly zero-valued hidden outputs.
To solve this problem and greatly increase the efficiency of our networks, we use what are called embeddings. Embeddings are just a fully connected layer like you've seen before. We call this layer the embedding layer and the weights are embedding weights. We skip the multiplication into the embedding layer by instead directly grabbing the hidden layer values from the weight matrix. We can do this because the multiplication of a one-hot encoded vector with a matrix returns the row of the matrix corresponding the index of the "on" input unit.
Instead of doing the matrix multiplication, we use the weight matrix as a lookup table. We encode the words as integers, for example "heart" is encoded as 958, "mind" as 18094. Then to get hidden layer values for "heart", you just take the 958th row of the embedding matrix. This process is called an embedding lookup and the number of hidden units is the embedding dimension.
There is nothing magical going on here. The embedding lookup table is just a weight matrix. The embedding layer is just a hidden layer. The lookup is just a shortcut for the matrix multiplication. The lookup table is trained just like any weight matrix.
Embeddings aren't only used for words of course. You can use them for any model where you have a massive number of classes. A particular type of model called Word2Vec uses the embedding layer to find vector representations of words that contain semantic meaning.
## Word2Vec
The Word2Vec algorithm finds much more efficient representations by finding vectors that represent the words. These vectors also contain semantic information about the words.
Words that show up in similar contexts, such as "coffee", "tea", and "water" will have vectors near each other. Different words will be further away from one another, and relationships can be represented by distance in vector space.
There are two architectures for implementing Word2Vec:
• CBOW (Continuous Bag-Of-Words) and
• Skip-gram
In this implementation, we'll be using the skip-gram architecture because it performs better than CBOW. Here, we pass in a word and try to predict the words surrounding it in the text. In this way, we can train the network to learn representations for words that show up in similar contexts.
Next, we'll ask you to load in data and place it in the data directory
1. Load the text8 dataset; a file of cleaned up Wikipedia article text from Matt Mahoney.
2. Place that data in the data folder in the home directory.
3. Then you can extract it and delete the archive, zip file to save storage space.
After following these steps, you should have one file in your data directory: data/text8.
# read in the extracted text file
with open('data/text8') as f:
# print out the first 100 characters
print(text[:100])
anarchism originated as a term of abuse first used against early working class radicals including t
## Pre-processing
Here I'm fixing up the text to make training easier. This comes from the utils.py file. The preprocess function does a few things:
• It converts any punctuation into tokens, so a period is changed to <PERIOD>. In this data set, there aren't any periods, but it will help in other NLP problems.
• It removes all words that show up five or fewer times in the dataset. This will greatly reduce issues due to noise in the data and improve the quality of the vector representations.
• It returns a list of words in the text.
This may take a few seconds to run, since our text file is quite large. If you want to write your own functions for this stuff, go for it!
import utils
# get list of words
words = utils.preprocess(text)
print(words[:30])
['anarchism', 'originated', 'as', 'a', 'term', 'of', 'abuse', 'first', 'used', 'against', 'early', 'working', 'class', 'radicals', 'including', 'the', 'diggers', 'of', 'the', 'english', 'revolution', 'and', 'the', 'sans', 'culottes', 'of', 'the', 'french', 'revolution', 'whilst']
# print some stats about this word data
print("Total words in text: {}".format(len(words)))
print("Unique words: {}".format(len(set(words)))) # set removes any duplicate words -> 去重
Total words in text: 16680599
Unique words: 63641
### Dictionaries
Next, I'm creating two dictionaries to convert words to integers and back again (integers to words). This is again done with a function in the utils.py file. create_lookup_tables takes in a list of words in a text and returns two dictionaries.
• The integers are assigned in descending frequency order, so the most frequent word ("the") is given the integer 0 and the next most frequent is 1, and so on.
Once we have our dictionaries, the words are converted to integers and stored in the list int_words.
vocab_to_int, int_to_vocab = utils.create_lookup_tables(words)
# 打印
for item in vocab_to_int.items():
print(item)
break
# 从字典中取出数组
int_words = [vocab_to_int[word] for word in words]
print(int_words[:30])
('the', 0)
[5233, 3080, 11, 5, 194, 1, 3133, 45, 58, 155, 127, 741, 476, 10571, 133, 0, 27349, 1, 0, 102, 854, 2, 0, 15067, 58112, 1, 0, 150, 854, 3580]
## Subsampling(二次抽样)
Words that show up often such as "the", "of", and "for" don't provide much context to the nearby words. If we discard some of them, we can remove some of the noise from our data and in return get faster training and better representations. This process is called subsampling by Mikolov. For each word $w_i$ in the training set, we'll discard it with probability given by
$$P(w_i) = 1 - \sqrt{\frac{t}{f(w_i)}}$$
where $t$ is a threshold parameter and $f(w_i)$ is the frequency of word $w_i$ in the total dataset.
$$P(0) = 1 - \sqrt{\frac{110^{-5}}{110^6/16*10^6}} = 0.98735$$
I'm going to leave this up to you as an exercise. Check out my solution to see how I did it.
Exercise: Implement subsampling for the words in int_words. That is, go through int_words and discard each word given the probablility $P(w_i)$ shown above. Note that $P(w_i)$ is the probability that a word is discarded. Assign the subsampled data to train_words.
from collections import Counter
import random
import numpy as np
threshold = 1e-5
word_counts = Counter(int_words)
print(list(word_counts.items())[0]) # dictionary of int_words, how many times they appear
total_count = len(int_words)
freqs = {word: count/total_count for word, count in word_counts.items()}
p_drop = {word: 1 - np.sqrt(threshold/freqs[word]) for word in word_counts}
# discard some frequent words, according to the subsampling equation
# create a new list of words for training
train_words = [word for word in int_words if random.random() < (1 - p_drop[word])]
print(train_words[:30])
#print(train_words[:30])
(5233, 303)
[5233, 3133, 10571, 27349, 15067, 58112, 3580, 10712, 19, 362, 3672, 708, 40, 1423, 2757, 567, 686, 7088, 247, 5233, 1052, 44611, 2877, 5233, 11, 5, 200, 602, 2621, 8983]
## Making batches
Now that our data is in good shape, we need to get it into the proper form to pass it into our network. With the skip-gram architecture, for each word in the text, we want to define a surrounding context and grab all the words in a window around that word, with size $C$.
From Mikolov et al.:
"Since the more distant words are usually less related to the current word than those close to it, we give less weight to the distant words by sampling less from those words in our training examples... If we choose $C = 5$, for each training word we will select randomly a number $R$ in range $[ 1: C ]$, and then use $R$ words from history and $R$ words from the future of the current word as correct labels."
Exercise: Implement a function get_target that receives a list of words, an index, and a window size, then returns a list of words in the window around the index. Make sure to use the algorithm described above, where you chose a random number of words to from the window.
Say, we have an input and we're interested in the idx=2 token, 741:
[5233, 58, 741, 10571, 27349, 0, 15067, 58112, 3580, 58, 10712]
For R=2, get_target should return a list of four values:
[5233, 58, 10571, 27349]
def get_target(words, idx, window_size=5):
''' Get a list of words in a window around an index. '''
# implement this function
R = np.random.randint(1, window_size+1)
start = idx - R if (idx - R) > 0 else 0
stop = idx + R
target_words = words[start:idx] + words[idx+1:stop+1]
return list(target_words)
# test your code!
# run this cell multiple times to check for random window selection
int_text = [i for i in range(10)]
print('Input: ', int_text)
idx=5 # word index of interest
target = get_target(int_text, idx=idx, window_size=5)
print('Target: ', target) # you should get some indices around the idx
Input: [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
Target: [2, 3, 4, 6, 7, 8]
### Generating Batches
Here's a generator function that returns batches of input and target data for our model, using the get_target function from above. The idea is that it grabs batch_size words from a words list. Then for each of those batches, it gets the target words in a window.
def get_batches(words, batch_size, window_size=5):
''' Create a generator of word batches as a tuple (inputs, targets) '''
#表示获取整数并向下取整
n_batches = len(words)//batch_size
# only full batches
words = words[:n_batches*batch_size]
for idx in range(0, len(words), batch_size):
x, y = [], []
batch = words[idx:idx+batch_size]
for ii in range(len(batch)):
batch_x = batch[ii]
batch_y = get_target(batch, ii, window_size)
y.extend(batch_y)
x.extend([batch_x]*len(batch_y))
yield x, y
int_text = [i for i in range(20)]
x,y = next(get_batches(int_text, batch_size=4, window_size=5))
print('x\n', x)
print('y\n', y)
x
[0, 1, 1, 1, 2, 2, 2, 3]
y
[1, 0, 2, 3, 0, 1, 3, 2]
## Building the graph
Below is an approximate diagram of the general structure of our network.
• The input words are passed in as batches of input word tokens.
• This will go into a hidden layer of linear units (our embedding layer).
• Then, finally into a softmax output layer.
We'll use the softmax layer to make a prediction about the context words by sampling, as usual.
The idea here is to train the embedding layer weight matrix to find efficient representations for our words. We can discard the softmax layer because we don't really care about making predictions with this network. We just want the embedding matrix so we can use it in other networks we build using this dataset.
## Validation
Here, I'm creating a function that will help us observe our model as it learns. We're going to choose a few common words and few uncommon words. Then, we'll print out the closest words to them using the cosine similarity:
$$\mathrm{similarity} = \cos(\theta) = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|}$$
We can encode the validation words as vectors $\vec{a}$ using the embedding table, then calculate the similarity with each word vector $\vec{b}$ in the embedding table. With the similarities, we can print out the validation words and words in our embedding table semantically similar to those words. It's a nice way to check that our embedding table is grouping together words with similar semantic meanings.
def cosine_similarity(embedding, valid_size=16, valid_window=100, device='cpu'):
""" Returns the cosine similarity of validation words with words in the embedding matrix.
Here, embedding should be a PyTorch embedding module.
"""
# Here we're calculating the cosine similarity between some random words and
# our embedding vectors. With the similarities, we can look at what words are
# close to our random words.
# sim = (a . b) / |a||b|
embed_vectors = embedding.weight
# magnitude of embedding vectors, |b|
magnitudes = embed_vectors.pow(2).sum(dim=1).sqrt().unsqueeze(0)
# pick N words from our ranges (0,window) and (1000,1000+window). lower id implies more frequent
valid_examples = np.array(random.sample(range(valid_window), valid_size//2))
valid_examples = np.append(valid_examples,
random.sample(range(1000,1000+valid_window), valid_size//2))
valid_examples = torch.LongTensor(valid_examples).to(device)
valid_vectors = embedding(valid_examples)
similarities = torch.mm(valid_vectors, embed_vectors.t())/magnitudes
return valid_examples, similarities
## SkipGram model
Define and train the SkipGram model.
You'll need to define an embedding layer and a final, softmax output layer.
An Embedding layer takes in a number of inputs, importantly:
• num_embeddings – the size of the dictionary of embeddings, or how many rows you'll want in the embedding weight matrix
• embedding_dim – the size of each embedding vector; the embedding dimension
import torch
from torch import nn
import torch.optim as optim
class SkipGram(nn.Module):
def __init__(self, n_vocab, n_embed):
super().__init__()
# complete this SkipGram model
self.embed = nn.Embedding(n_vocab, n_embed)
self.output = nn.Linear(n_embed, n_vocab)
self.log_softmax = nn.LogSoftmax(dim=1)
def forward(self, x):
# define the forward behavior
x = self.embed(x)
scores = self.output(x)
log_ps = self.log_softmax(scores)
return log_ps
### Training
Below is our training loop, and I recommend that you train on GPU, if available.
Note that, because we applied a softmax function to our model output, we are using NLLLoss as opposed to cross entropy. This is because Softmax in combination with NLLLoss = CrossEntropy loss .
# check if GPU is available
device = 'cuda' if torch.cuda.is_available() else 'cpu'
embedding_dim=10 # you can change, if you want
model = SkipGram(len(vocab_to_int), embedding_dim).to(device)
criterion = nn.NLLLoss()
print_every = 1
steps = 0
epochs = 2
# train for some number of epochs
for e in range(epochs):
# get input and target batches
for inputs, targets in get_batches(train_words, 512):
steps += 1
inputs, targets = torch.LongTensor(inputs), torch.LongTensor(targets)
inputs, targets = inputs.to(device), targets.to(device)
log_ps = model(inputs)
loss = criterion(log_ps, targets)
loss.backward()
optimizer.step()
if steps % print_every == 0:
# getting examples and similarities
valid_examples, valid_similarities = cosine_similarity(model.embed, device=device)
_, closest_idxs = valid_similarities.topk(6) # topk highest similarities
valid_examples, closest_idxs = valid_examples.to('cpu'), closest_idxs.to('cpu')
for ii, valid_idx in enumerate(valid_examples):
closest_words = [int_to_vocab[idx.item()] for idx in closest_idxs[ii]][1:]
print(int_to_vocab[valid_idx.item()] + " | " + ', '.join(closest_words))
print("...")
break
break
print("\nCPU source is used out, die...\n")
but | pixels, dracula, buckling, weights, shifting
all | bodhi, catamarans, terrors, illustrators, carousel
an | chemotherapy, occultations, influenced, municipally, chants
years | meditations, gush, inevitability, uncircumcised, intensive
that | lecturer, nicer, kyle, workplace, paragraph
while | randomly, playlist, bennett, raincoat, bitterness
was | northland, multipurpose, discoverer, naboth, porco
between | orchards, pitt, carries, admiration, flexed
smith | overabundance, smell, mcknight, notional, amin
primarily | treated, hijacked, radhakrishnan, gauthier, larry
consists | myself, luang, railtrack, weren, lessen
http | sparta, boineburg, novak, brussel, overruns
question | crushed, junichiro, oversimplification, dheas, handbook
event | overillumination, medica, aldehydes, reconstructing, asterales
channel | fluffy, buff, dynamic, shmuel, promptly
prince | violin, micrometre, arms, mdma, already
...
## Visualizing the word vectors
Below we'll use T-SNE to visualize how our high-dimensional word vectors cluster together. T-SNE is used to project these vectors into two dimensions while preserving local stucture. Check out this post from Christopher Olah to learn more about T-SNE and other ways to visualize high-dimensional data.
%matplotlib inline
%config InlineBackend.figure_format = 'retina'
import matplotlib.pyplot as plt
from sklearn.manifold import TSNE
# getting embeddings from the embedding layer of our model, by name
embeddings = model.embed.weight.to('cpu').data.numpy()
viz_words = 600
tsne = TSNE()
embed_tsne = tsne.fit_transform(embeddings[:viz_words, :])
fig, ax = plt.subplots(figsize=(16, 16))
for idx in range(viz_words):
plt.scatter(*embed_tsne[idx, :], color='steelblue')
plt.annotate(int_to_vocab[idx], (embed_tsne[idx, 0], embed_tsne[idx, 1]), alpha=0.7)
utils.py
import re
from collections import Counter
def preprocess(text):
# Replace punctuation with tokens so we can use them in our model
text = text.lower()
text = text.replace('.', ' <PERIOD> ')
text = text.replace(',', ' <COMMA> ')
text = text.replace('"', ' <QUOTATION_MARK> ')
text = text.replace(';', ' <SEMICOLON> ')
text = text.replace('!', ' <EXCLAMATION_MARK> ')
text = text.replace('?', ' <QUESTION_MARK> ')
text = text.replace('(', ' <LEFT_PAREN> ')
text = text.replace(')', ' <RIGHT_PAREN> ')
text = text.replace('--', ' <HYPHENS> ')
text = text.replace('?', ' <QUESTION_MARK> ')
# text = text.replace('\n', ' <NEW_LINE> ')
text = text.replace(':', ' <COLON> ')
words = text.split()
# Remove all words with 5 or fewer occurences
word_counts = Counter(words)
trimmed_words = [word for word in words if word_counts[word] > 5]
return trimmed_words
def create_lookup_tables(words):
"""
Create lookup tables for vocabulary
:param words: Input list of words
:return: Two dictionaries, vocab_to_int, int_to_vocab
"""
word_counts = Counter(words)
# sorting the words from most to least frequent in text occurrence
sorted_vocab = sorted(word_counts, key=word_counts.get, reverse=True)
# create int_to_vocab dictionaries
int_to_vocab = {ii: word for ii, word in enumerate(sorted_vocab)}
vocab_to_int = {word: ii for ii, word in int_to_vocab.items()}
return vocab_to_int, int_to_vocab
## 二次抽样算丢弃率
### Subsampling equation
$$P(w_i) = 1 - \sqrt{\frac{t}{f(w_i)}}$$
For the following quiz question, consider the following data points:
• We have a text with 1 million words in it
• The word "learn" appears 700 times in this text
Yes! 1-sqrt((110^-4)/(700/(110^6))) = 0.622 | 2019-08-25 13:30:24 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3849635720252991, "perplexity": 3990.0018976625456}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027330233.1/warc/CC-MAIN-20190825130849-20190825152849-00249.warc.gz"} |
https://quals.dzackgarza.com/40_Topology/600_UGA_Qual_Questions/018%20Misc.html | # Miscellaneous Algebraic Topology
## 1 (Fall ’14) #topology/qual/work
Prove that $${\mathbf{R}}^2$$ is not homeomorphic to $${\mathbf{R}}^n$$ for $$n > 2$$.
## 2 (Spring ’12) #topology/qual/work
Prove that any finite tree is contractible, where a tree is a connected graph that contains no closed edge paths.
## 3 (Spring ’13) #topology/qual/completed
Show that any continuous map $$f : {\mathbf{RP}}^2 \to S^1 \times S^1$$ is necessarily null-homotopic.
\hfill
\hfill
• Two techniques:
• Show $$f_* = 0$$
• Lift to a contractible universal cover.
• Any continuous map $${\mathbf{RP}}^2 \xrightarrow{f} S^1\times S^1$$ induces a group morphism $$\pi_1 {\mathbf{RP}}^2 \xrightarrow{f_*} \pi_1(S^1\times S^1)$$
• Identify $$\pi_1 {\mathbf{RP}}^2 = {\mathbf{Z}}/2{\mathbf{Z}}$$ and $$\pi_1(S^1\times S^1) = \pi_1 S^1 \times\pi_1 S^1 = {\mathbf{Z}}^2$$.
• But as a $${\mathbf{Z}}{\hbox{-}}$$module morphism, $$f_*$$ will preserve torsion submodules, and since $${\mathbf{Z}}^2$$ is free we must have $$f_* = 0$$.
• Lemma: $$f_* = 0$$ implies $$f$$ is nullhomotopic.
\todo[inline]{Why? What is the homotopy?}
• Note that $$\widetilde{S^1\times S^1} = {\mathbf{R}}^2$$.
## 4 (Fall ’11) #topology/qual/completed
Prove that, for $$n \geq 2$$, every continuous map $$f: {\mathbf{RP}}^n \to S^1$$ is null-homotopic.
\hfill
\hfill
• Any continuous map $${\mathbf{RP}}^n \xrightarrow{f} S^1$$ induces a group morphism $$\pi_1{\mathbf{RP}}^n \xrightarrow{f_*} \pi_1S^1$$
• Identify $$\pi_1{\mathbf{RP}}^n = {\mathbf{Z}}/2{\mathbf{Z}}$$ and $$\pi_1S^1 = {\mathbf{Z}}$$ to obtain a group morphism $$f_*: {\mathbf{Z}}/2{\mathbf{Z}}\to {\mathbf{Z}}$$.
• Claim: $$f_* = 0$$.
• Recognizing this as a map of $${\mathbf{Z}}{\hbox{-}}$$modules, we must have \begin{align*} 0 = [2]_2 = 2\cdot [1]_2 \implies 0 = f_*(0) = 2\cdot f_*([1]_2) .\end{align*} since $${\mathbf{Z}}{\hbox{-}}$$module maps send 0 to 0.
• But no element of the image $${\mathbf{Z}}$$ is annihilated by $$2$$, so $$f_*$$ can only be the zero map.
• But then $$f$$ is nullhomotopic.
• Lemma: $$f_* = 0$$ implies $$f$$ is nullhomotopic.
\todo[inline]{Why?}
## 5 (Spring ’06) #topology/qual/work
Let $$S^2 \to {\mathbf{RP}}^2$$ be the universal covering map.
Is this map null-homotopic? Give a proof of your answer.
## 6 (Spring ’17) #topology/qual/completed
Suppose that a map $$f : S^3 \times S^3 \to {\mathbf{RP}}^3$$ is not surjective.
Prove that $$f$$ is homotopic to a constant function.
\todo[inline]{Lost, redo.}
## 7 (Fall ’06) #topology/qual/work
Prove that there does not exist a continuous map $$f : S^2 \to S^2$$ from the unit sphere in $${\mathbf{R}}^3$$ to itself such that $$f (\mathbf{x}) \perp \mathbf{x}$$ (as vectors in $${\mathbf{R}}^3$$ for all $$\mathbf{x} \in S^2$$).
## 8 (Spring ’08) #topology/qual/work
Let $$f$$ be the map of $$S^1 \times [0, 1]$$ to itself defined by \begin{align*} f (e^{i\theta} , s) = (e^{i(\theta+2\pi s)} , s) ,\end{align*} so that $$f$$ restricts to the identity on the two boundary circles of $$S^1 \times [0, 1]$$.
Show that $$f$$ is homotopic to the identity by a homotopy $$f_t$$ that is stationary on one of the boundary circles, but not by any homotopy that is stationary on both boundary circles.
Hint: Consider what $$f$$ does to the path $$s \mapsto (e^{i\theta_0} , s)$$ for fixed $$e^{i\theta_0} \in S^1$$.
## 9 (Spring ’17) #topology/qual/work
Show that $$S^1 \times S^1$$ is not the union of two disks (where there is no assumption that the disks intersect along their boundaries).
## 10 (Spring ’14) #topology/qual/work
Suppose that $$X \subset Y$$ and $$X$$ is a deformation retract of $$Y$$.
Show that if $$X$$ is a path connected space, then $$Y$$ is path connected.
## 11 (Spring ’05) #topology/qual/work
Do one of the following:
• Give (with justification) a contractible subset $$X \subset {\mathbf{R}}^2$$ which is not a retract of $${\mathbf{R}}^2$$ .
• Give (with justification) two topological spaces that have the same homology groups but that are not homotopy equivalent.
## 12 (Spring ’16) #topology/qual/work
Recall that the suspension of a topological space, denoted $$SX$$, is the quotient space formed from $$X \times [-1, 1]$$ by identifying $$(x, 1)$$ with $$(y, 1)$$ for all $$x, y \in X$$, and also identifying $$(x, -1)$$ with $$(y, -1)$$ for all $$x, y \in X$$.
• Show that $$SX$$ is the union of two contractible subspaces.
• Prove that if $$X$$ is path-connected then $$\pi_1 (SX) = \left\{{0}\right\}$$.
• For all $$n \geq 1$$, prove that $$H_{n} (X) \cong H_{n+1} (SX)$$.
#topology/qual/work #topology/qual/completed | 2023-03-25 20:28:58 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9888800382614136, "perplexity": 607.4790130929513}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296945372.38/warc/CC-MAIN-20230325191930-20230325221930-00539.warc.gz"} |
https://www.physicsforums.com/threads/unusual-question-regarding-torque.340945/ | # Unusual question regarding torque
two equivalent mases, m, are placed on two corners of a triangle with 3 sides each of length a (i.e an equilateral traingle). The third corner is fixed (not free to move).
The triangle orginally starts with one side perfectly vertical. Gravity acts downward intially, calculate the accelerations of the masses right after the system is let go . | 2020-02-17 13:22:49 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9696861505508423, "perplexity": 1435.2751605789624}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875142323.84/warc/CC-MAIN-20200217115308-20200217145308-00077.warc.gz"} |
https://sources.debian.org/src/gsl-ref-html/2.3-1/Fitting-large-linear-systems-example.html/ | ## File: Fitting-large-linear-systems-example.html
package info (click to toggle)
gsl-ref-html 2.3-1
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123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130131132133134135136137138139140141142143144145146147148149150151152153154155156157158159160161162163164165166167168169170171172173174175176177178179180181182183184185186187188189190191192193194195196197198199200201202203204205206207208209210211212213214215216217218219220221222223224225226227228229230231232233234235236237238239240241242243244245246247248249250251252253254255256257258259260261262263264265266267268269270271272273274275276277278279280281282283284285286287288289290 GNU Scientific Library – Reference Manual: Fitting large linear systems example
38.8.6 Large Dense Linear Regression Example
The following program demostrates the large dense linear least squares solvers. This example is adapted from Trefethen and Bau, and fits the function f(t) = \exp{(\sin^3{(10t)}}) on the interval [0,1] with a degree 15 polynomial. The program generates n = 50000 equally spaced points t_i on this interval, calculates the function value and adds random noise to determine the observation value y_i. The entries of the least squares matrix are X_{ij} = t_i^j, representing a polynomial fit. The matrix is highly ill-conditioned, with a condition number of about 1.4 \cdot 10^{11}. The program accumulates the matrix into the least squares system in 5 blocks, each with 10000 rows. This way the full matrix X is never stored in memory. We solve the system with both the normal equations and TSQR methods. The results are shown in the plot below. In the top left plot, we see the unregularized normal equations solution has larger error than TSQR due to the ill-conditioning of the matrix. In the bottom left plot, we show the L-curve, which exhibits multiple corners. In the top right panel, we plot a regularized solution using \lambda = 10^{-6}. The TSQR and normal solutions now agree, however they are unable to provide a good fit due to the damping. This indicates that for some ill-conditioned problems, regularizing the normal equations does not improve the solution. This is further illustrated in the bottom right panel, where we plot the L-curve calculated from the normal equations. The curve agrees with the TSQR curve for larger damping parameters, but for small \lambda, the normal equations approach cannot provide accurate solution vectors leading to numerical inaccuracies in the left portion of the curve.
#include <gsl/gsl_math.h> #include <gsl/gsl_vector.h> #include <gsl/gsl_matrix.h> #include <gsl/gsl_rng.h> #include <gsl/gsl_randist.h> #include <gsl/gsl_multifit.h> #include <gsl/gsl_multilarge.h> #include <gsl/gsl_blas.h> /* function to be fitted */ double func(const double t) { double x = sin(10.0 * t); return exp(x*x*x); } /* construct a row of the least squares matrix */ int build_row(const double t, gsl_vector *row) { const size_t p = row->size; double Xj = 1.0; size_t j; for (j = 0; j < p; ++j) { gsl_vector_set(row, j, Xj); Xj *= t; } return 0; } int solve_system(const int print_data, const gsl_multilarge_linear_type * T, const double lambda, const size_t n, const size_t p, gsl_vector * c) { const size_t nblock = 5; /* number of blocks to accumulate */ const size_t nrows = n / nblock; /* number of rows per block */ gsl_multilarge_linear_workspace * w = gsl_multilarge_linear_alloc(T, p); gsl_matrix *X = gsl_matrix_alloc(nrows, p); gsl_vector *y = gsl_vector_alloc(nrows); gsl_rng *r = gsl_rng_alloc(gsl_rng_default); const size_t nlcurve = 200; gsl_vector *reg_param = gsl_vector_alloc(nlcurve); gsl_vector *rho = gsl_vector_alloc(nlcurve); gsl_vector *eta = gsl_vector_alloc(nlcurve); size_t rowidx = 0; double rnorm, snorm, rcond; double t = 0.0; double dt = 1.0 / (n - 1.0); while (rowidx < n) { size_t nleft = n - rowidx; /* number of rows left to accumulate */ size_t nr = GSL_MIN(nrows, nleft); /* number of rows in this block */ gsl_matrix_view Xv = gsl_matrix_submatrix(X, 0, 0, nr, p); gsl_vector_view yv = gsl_vector_subvector(y, 0, nr); size_t i; /* build (X,y) block with 'nr' rows */ for (i = 0; i < nr; ++i) { gsl_vector_view row = gsl_matrix_row(&Xv.matrix, i); double fi = func(t); double ei = gsl_ran_gaussian (r, 0.1 * fi); /* noise */ double yi = fi + ei; /* construct this row of LS matrix */ build_row(t, &row.vector); /* set right hand side value with added noise */ gsl_vector_set(&yv.vector, i, yi); if (print_data && (i % 100 == 0)) printf("%f %f\n", t, yi); t += dt; } /* accumulate (X,y) block into LS system */ gsl_multilarge_linear_accumulate(&Xv.matrix, &yv.vector, w); rowidx += nr; } if (print_data) printf("\n\n"); /* compute L-curve */ gsl_multilarge_linear_lcurve(reg_param, rho, eta, w); /* solve large LS system and store solution in c */ gsl_multilarge_linear_solve(lambda, c, &rnorm, &snorm, w); /* compute reciprocal condition number */ gsl_multilarge_linear_rcond(&rcond, w); fprintf(stderr, "=== Method %s ===\n", gsl_multilarge_linear_name(w)); fprintf(stderr, "condition number = %e\n", 1.0 / rcond); fprintf(stderr, "residual norm = %e\n", rnorm); fprintf(stderr, "solution norm = %e\n", snorm); /* output L-curve */ { size_t i; for (i = 0; i < nlcurve; ++i) { printf("%.12e %.12e %.12e\n", gsl_vector_get(reg_param, i), gsl_vector_get(rho, i), gsl_vector_get(eta, i)); } printf("\n\n"); } gsl_matrix_free(X); gsl_vector_free(y); gsl_multilarge_linear_free(w); gsl_rng_free(r); gsl_vector_free(reg_param); gsl_vector_free(rho); gsl_vector_free(eta); return 0; } int main(int argc, char *argv[]) { const size_t n = 50000; /* number of observations */ const size_t p = 16; /* polynomial order + 1 */ double lambda = 0.0; /* regularization parameter */ gsl_vector *c_tsqr = gsl_vector_alloc(p); gsl_vector *c_normal = gsl_vector_alloc(p); if (argc > 1) lambda = atof(argv[1]); /* solve system with TSQR method */ solve_system(1, gsl_multilarge_linear_tsqr, lambda, n, p, c_tsqr); /* solve system with Normal equations method */ solve_system(0, gsl_multilarge_linear_normal, lambda, n, p, c_normal); /* output solutions */ { gsl_vector *v = gsl_vector_alloc(p); double t; for (t = 0.0; t <= 1.0; t += 0.01) { double f_exact = func(t); double f_tsqr, f_normal; build_row(t, v); gsl_blas_ddot(v, c_tsqr, &f_tsqr); gsl_blas_ddot(v, c_normal, &f_normal); printf("%f %e %e %e\n", t, f_exact, f_tsqr, f_normal); } gsl_vector_free(v); } gsl_vector_free(c_tsqr); gsl_vector_free(c_normal); return 0; } | 2019-07-17 18:45:55 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8534014225006104, "perplexity": 13141.193411609822}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195525374.43/warc/CC-MAIN-20190717181736-20190717203736-00231.warc.gz"} |
https://www.physicsforums.com/threads/angular-kinetic-energy.282639/ | Angular Kinetic Energy
1. Jan 3, 2009
tachu101
1. The problem statement, all variables and given/known data
A hoop of mass 1kg and radius 2m is rotating about its center with an angular speed of 3rad/sec. A force of 10N is applied tangentially at the rim.
a. The rotational kinetic energy of the hoop is?
b. The instantaneous rate at which the kinetic energy is changing is?
2. Relevant equations
KE= 1/2Iw^2
3. The attempt at a solution
For the first part I can find I= MR^2 for a hoop, so I= (1)(2^2) so I=4. So KE= 1/2Iw^2 = (1/2)(4)(3^2)= 18. What do I do with the extra 10N applied (do I just add it in or does it not factor into the rotational kinetic energy?) I don't know what equation to use for the second part.
2. Jan 3, 2009
tiny-tim
Hi tachu101!
KE is geometric … it only depends on the motion, not on what's causing it.
So you only need the 10N for the second part …
and for that, what equation do you know that connects force and rotation?
3. Jan 3, 2009
tachu101
For the second part would I use torque=Ia which would then be torque/I=a? Would that be (10)(2)/((1)(2^2))= 5 rad/sec^2 does that help me?
4. Jan 3, 2009
tiny-tim
(have an omega: ω and an alpha: α and a delta: ∆ )
Yup!
That gives you dω/dt, from which you could find dKE/dt.
But there's a more direct way …
remember, ∆KE = ∆(work done) …
so what is the equation for work done by a torque ?
5. Jan 3, 2009
tachu101
W= torque* $$\Delta$$$$\Theta$$ which would then be (10)(2)*(3)= 60J ? And how would I get dKe/dt from dW/dt
6. Jan 3, 2009
tiny-tim
uh … W= torque*∆θ, where ∆θ is the total change in angle, is correct
but 3 is not the total change in angle, it's only the rate-of-change of angle, dθ/dt …
you need dW/dt = d(torque*θ)/dt
7. Jan 3, 2009
tachu101
hmmm... am I missing an equation because I am not sure how to find the d(torque*θ)/dt? Also, given that I found 5 rad/sec^2 how could I use that to find a change in KE?
8. Jan 3, 2009
tiny-tim
The torque here is constant, so d(torque*θ)/dt = torque * dθ/dt = torque * ω.
(you could use the dω/dt = 5 if you differentiate 1/2 Iω2
but it's more direct if you use the work done method)
9. Jan 3, 2009
tachu101
So is (10)(2)*(3)= 60J Correct?
10. Jan 3, 2009
tiny-tim
This would be easier to understand if you wrote it out in full …
but it's not joules anyway, is it?
11. Jan 3, 2009
tachu101
Does the instantaneous change in KE equal torque * ω? Is torque (10)*(2)=20?
12. Jan 3, 2009
tiny-tim
Instantaneous rate of change in KE equal torque * ω.
And yes, torque (10)*(2)=20.
13. Jan 3, 2009
tachu101
In summary...
I think that part a is 9J because I think it is (1/2)(I)(w^2)= 9
Part b is then torque*W which is then (10)(2)*(3)= 60
14. Jan 3, 2009
tiny-tim
I thought you got 18J for a?
Yup, b is 60.
15. Jan 3, 2009
tachu101
I looked back and I think I missed a 1/2 in part a. It should be (1/2)((1/2)(1)(2^2))(3^2)= 9 I think I missed the 1/2 in the I value.
16. Jan 3, 2009
tiny-tim
What 1/2? … it's a hoop, not a disc.
Going to bed now … :zzz: | 2017-06-24 19:15:07 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8422806859016418, "perplexity": 2541.560689750506}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-26/segments/1498128320323.17/warc/CC-MAIN-20170624184733-20170624204733-00360.warc.gz"} |
https://www.futurelearn.com/courses/exploring-possible-futures/0/steps/39346 | 4.10
# Monopoly model: the mathematics II
Now that you have seen your first monopoly model let’s use the setting of our perfect competition example to analyze how such a model can be designed as optimization and as equilibrium model and learn a bit on the economics of monopolies on the way.
The setup is the same with one change: we now assume monopolistic competition.
So instead of many firms, $i$, we just have one single firm. This firm does what every firm is doing, it maximizes its profit.
(1) $\max \limits_{q\geq 0} \enspace p(q)q-c(q)$
But, given that it is the sole supplier on the market, the firm knows that its output decision will have an impact on the market price. That is why, contrary to a perfectly competitive market, the income side is not only price times quantity, $pq$, but the price, $p$, is now a function of quantity, $q$.
Now, how does the firm know how $p$ and $q$ are related?
Recall the benefit function $B(d)$ we used in the perfectly competitive market to model the consumer behaviour. The first order derivative of this function ($\frac{∂B}{∂d}$) tells us exactly this: how does the willingness of to pay for the product change with a marginal change in consumption $(d)$. In other words, you could rewrite the benefit-demand relation as a price-demand relation:
For example the demand-price relation we used in the last model exercises was given by:
or, rearranging the equation,
$p=\frac{d-\bar{Q}}{\eta }$ to be closer to our $p(d)$ formulation.
As a last step, you now need to consider that the monopoly is the only supplier in this market. Thus, the normal demand balance is now simply $q = d$ making $p(d) = p(q)$. Therefore equation (1) is all you need for a monopoly optimization model.
Again, you will need to derive the first order conditions of the model with respect to the choice variable of the monopoly, $(q)$, and set it equal to zero:
(2) $p(q)+\frac{∂p(q)}{∂q}q-\frac{∂c(q)}{∂q}=0$
Note that the first term in the objective function, $p(q)q$, requires you to use the product rule when making the derivation. This is the normal first-order condition of a monopoly.
Let’s switch to the equilibrium formulation. Similar to the perfect competitive version, we have three elements: the monopolies maximizes its profit, consumers maximizing their benefit, and a market that brings the two sides together. Thus, we actually only need to change the firm block and can keep the other elements as before.
Consumers are still described by their zero-profit logic:
(3) $p\geq \frac{∂B}{∂d} \bot d\geq 0$
And in the market clearing we just need to replace the output of all firms with the single output of our monopoly:
(4) $q\geq d \bot p \geq 0$
The zero-profit condition for the monopoly is a bit more tricky. You can either derive this logic by formulating the maximization problem for the monopoly and make the first order derivation and implement this as the zero-profit condition; so basically what we did above for the optimization model. Or, you know the economic equilibrium condition for the behaviour of a monopoly: marginal revenue equals marginal costs.
In a perfectly competitive world, a firm simply sells its output on the market assuming that its own decision has no impact on the price. Thus, the marginal revenue a competitive firm can obtain is equal to the market price. Consequently the equilibrium condition was: market price equals marginal costs.
But a monopoly knows that its own output decision has an impact on the price. And since it is the only one selling this also has a feedback effect on the total sale revenue. Increasing the output by one unit increases the income by the market price, $p$, but as the price declines with higher demand (respective output by the monopolist) the revenue on the total output decreases due to the price reduction. This can be formalized as:
(5) $\frac{∂c(q)}{∂q}\geq p+\frac{∂p(q)}{∂q}q \bot q \geq 0$
The left hand side is again the marginal costs, the right hand side is the marginal revenue. A monopoly will produce $(q > 0)$ up to the point where its marginal cost level is equal to the additional revenue is can obtain ($\frac{∂c(q)}{∂q}= p+\frac{∂p(q)}{∂q}q$) defined by the additional income for selling one more unit $(p)$ and the reduction in total income on all sold units $(\frac{∂p(q)}{∂q}q)$. Note that $\frac{∂p(q)}{∂q}$ is usually negative as the price-demand relation is downward sloping (with a higher demand prices need to decrease and vice versa).
In other words, the optimization and equilibrium model are again providing the same result: a monopoly will produce up to the point where its profit is maximized and this point is defined where the additional revenue it can obtain is equal to the cost it has for this additional output.
This results in a significant shift between consumer and producer rent, a lower $d$ and level at higher market prices, and a total loss in welfare. | 2019-10-21 00:32:06 | {"extraction_info": {"found_math": true, "script_math_tex": 27, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7090339064598083, "perplexity": 678.1680777805534}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570987750110.78/warc/CC-MAIN-20191020233245-20191021020745-00051.warc.gz"} |
http://physics.stackexchange.com/tags/isotope/new | # Tag Info
Both isotopes are isovalent so electronically they are identical (in essence your not going to get that much difference in bond angle). However in the asymmetric well approximation since deuterium is heavier the $D-O$ bond is lowered down the well i.e. it has a lower ZPE than the $H-O$ bond and is therefore a stronger bond. This means it has a smaller ... | 2014-10-25 18:36:38 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7001544833183289, "perplexity": 1355.102332586838}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-42/segments/1414119649048.35/warc/CC-MAIN-20141024030049-00220-ip-10-16-133-185.ec2.internal.warc.gz"} |
https://cstheory.stackexchange.com/questions/35998/quicksort-compute-the-expected-number-of-comparisons-as-a-function-of-m-and | # Quicksort: compute the expected number of comparisons as a function of $M$ and $t$
I stumbled upon this problem on a list of open problems in the analysis of algorithms dating back to 1997. Is it still open? Can anyone point to a reference with a full or partial solution, or at least discussion?
Problem description (see link for more context):
Let $C_n$ be the number of comparisons made by quicksort to sort a random permutation of $\{1,\ldots,n\}$, when using the median of a sample of size $2t+1$ to perform the partitions and the recursive calls stop at subfiles of size $M\ge 2t+1$. Both $M$ are $t$ are constants. Compute the expected value of $C_n$ [as a function of] $M$ and $t$.
• How many comparisons does it take to compute the median? I'm pretty sure that's also open, and would be necessary to get the linear term. – Geoffrey Irving Jun 20 '16 at 0:05
## 1 Answer
I got a reply from the problem author, wanted to post the info here for reference:
Apparently the problem was solved by Pascal Hennequin in his PhD thesis. Chern and Hwang discuss the solution in Transitional behaviors of the average cost of quicksort with median-of-(2t+1).
I also found that googling quicksort "2t+1" yields the above paper :). | 2019-08-19 16:11:43 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8735717535018921, "perplexity": 143.23896440974158}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027314852.37/warc/CC-MAIN-20190819160107-20190819182107-00000.warc.gz"} |
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# floor function excel
2020-12-12 14:09 作者: 来源: 本站 浏览: 1 views 我要评论评论关闭 字号:
... and paste it in cell A1 of a new Excel worksheet. Unlike the MROUND function, the FLOOR function is set to always round down. In MS Excel this function only takes two arguments. Weil floor() eine statische Methode von Math ist, wird sie immer als Math.floor() aufgerufen und nicht als eine Methode eines erstellten Math Objektes (Math ist … Is there a problem with this function in excel 2010? Number Required. In this article, you can understand how to use the Excel FLOOR.PRECISE Function in Office 365 with its syntax, description, and examples. If you need to, you can adjust the column widths to see all the data. MS Excel being a spreadsheet program is used to store and retrieve numerical data in a grid format of columns and rows. In this article, we will learn about how to use the FLOOR function in Excel. Formula . significance– Multiple of Significance to which the number has to be rounded to. Positive numbers with decimal values are rounded down to the nearest integer (e.g. This difference is only relevant with negative numbers. ... Definite integrals and sums involving the floor function are quite common in problems and applications. Thanks for visiting Geek Excel. If the spreadsheet is exported to Microsoft Excel, the FLOOR function is exported as the equivalent FLOOR.MATH function that has existed since Excel 2013. I found Round() but I can't find a Floor() math function within the VBA code side of the excel. FLOOR(number,significance) For example: Round a year down to the nearest decade; In the screen shot below, the year is rounded down to the nearest decade, by using 10 as the significance in the FLOOR function. Give your feedback in the comment section. #1 to round the value in B1 Cell down to nearest multiple of 2, just using the following excel formula: =FLOOR(B1,2) You might also like: No related posts. This is important to remember. The CEILING function. I need the data seperated in .5 unit increments. Therefore, with the FLOORZ function, you might get unexpected results. The floor function , also called the greatest integer function or integer value (Spanier and Oldham 1987), gives the largest integer less than or equal to .The name and symbol for the floor function were coined by K. E. Iverson (Graham et al. The CEILING function is a built-in function in Excel that is categorized as a Math/Trig Function.It can be used as a worksheet function (WS) in Excel. Evaluate ∫ 0 ∞ ⌊ x ⌋ e − x d x. The FLOOR function fuzzes the results so that if the results are within 1E-12 of an integer, the FLOOR function returns that integer. Leave your feedback in the below comment section. Positive numbers with decimal values are rounded down to the nearest integer (e.g. Excel MRound Function, Ceiling Function and FLOOR Function: Excel Advanced Tips and Tricks 2020. However, once again, if you want to round a number down to the nearest integer or to the nearest specified multiple of significance, then you can call the Floor.Math worksheet function from VBA. If I use CEILING() I'm given autocompletion prompts, but FLOOR() displays as if it isn't defined: However the function does work (this returns zero when used in a card). Rückgabewert. Keep Learning!! As part of Excel functions discussions, we are going to discuss about two functions through this Article; which are CEILING and FLOOR functions. As you know, the ROUND() function does not deal with decimals. Similar Functions: FLOOR.MATH Function in Excel 365; FLOOR.PRECISE Function in Excel 365 See Also: For positive numbers, they will be rounded towards zero, while negative numbers will be rounded away from zero. Die Abrundungsfunktion (auch Gaußklammer, Ganzzahl-Funktion, Ganzteilfunktion oder Entier-Klammer) und die Aufrundungsfunktion sind Funktionen, die jeder reellen Zahl die nächstliegende nicht größere bzw. The FLOORZ function uses zero fuzzing. FLOOR in excel is in the list of the basic rounding functions in Excel, though FLOOR in excel works in a similar manner like MROUND function, the only difference is that FLOOR in excel always pushes down the number to the nearest multiple of the significance. Launch Excel 2010 spreadsheet on which you want to apply FLOOR function. If mode is non-zero, FLOOR rounds down towards zero. In this article, you can understand the usage of Excel FLOOR Function in Office 365 with its syntax, explanation, and examples. Here is what Excel says about each: MOD – returns a remainder after a number is divided by a divisor =MOD(number,divisor) CEILING – rounds a number up to the nearest multiple of significance =CEILING(number,significance) FLOOR – rounds a number down to the nearest multiple of significance =FLOOR … Keep Learning!! FLOOR.MATH rounds to the nearest integer, using a 1 as significance. Syntax: floor() PostgreSQL Version: 9.3 . Excel FLOOR Function Example. FLOOR.MATH( number, [significance],[mode]) number – The Number to be rounded down. The Microsoft Excel CEILING function returns a number rounded up based on a multiple of significance. In this article, we will learn about how to use the FLOOR.MATH function in Excel. Similar Articles: FLOOR Function in Excel FLOOR.MATH Function in Excel… FLOOR function in excel always rounds the value down towards zero and always returns a numeric value. This function returns the rounded up number which is nearest to the specified multiple of significance. mode – If [mode] = 0 (or is omitted), negative numbers are rounded away from zero. This thread is locked. Thanks for visiting Geek Excel. For instance, we have included a spreadsheet containing fields; S.no, Values and FLOOR. VBA does not have a Floor.Math function equivalent either. It is important to understand that, the return … Excel is ideal for entering, calculating and analyzing company data such as sales fig… For example: Say our price is $4.32 and we need to round it down to the nearest value divisible by 5 cents, the FLOOR function would read: =FLOOR(4.32, 0.05) =$4.30 . FLOOR() function. For example, ... Microsoft Excel used almost exactly the opposite of standard notation, with INT for floor, and FLOOR meaning round-toward-zero, and CEILING meaning round-away-from-zero. Thanks. The FLOOR function rounds down the number to the nearest specified multiple. If the number is negative and the significance is set to 1, then the answer should be #NUM! Here is an example of the data I have and the conversion I need to occur. The FLOOR function in Excel rounds a given number down to the nearest specified multiple. I have been using the FLOOR function in Excel (=FLOOR(Number,0.5)), but wish to implement the procedure in Access. \int\limits_0^\infty \lfloor x \rfloor e^{-x} \, dx. The Excel FLOOR function rounds numbers down, toward zero, based on the multiple of significance that you specify. For formulas to show results, select them, press F2, and then press Enter. The FLOOR function works in the same way the MROUND function does, with the only difference being that it always round numbers down to the desired number. Floor Function. For those not math inclined, Floor(5.65) = 5, whereas Round(5.65) = 6. Eine größte ganze Zahl, die kleiner oder gleich der übergebenen Zahl ist. 1994).. Create Floor Plan Using MS Excel: Everyone is familiar with MS Excel, right?. On the flip side we could use the CEILING function to round our price up to the nearest 5 cents as follows: =CEILING(4.32, 0.05) = \$4.35 . This cheat sheet covers 100s of functions that are critical to know as an Excel analyst. The Excel FLOOR function performs rounding based on the following rules: If the number and significance arguments are positive, the number is rounded down, toward zero, as in rows 2 and 10 in the screenshot below. The best strategy is to break up the interval of integration (or summation) into pieces on which the floor function is constant. Pictorial presentation of PostgreSQL FLOOR() function . The below examples will show you how to use Excel FLOOR function to round a supplied number to the nearest multiple of a specified significance. In Excel we can use the FLOOR function to calculate this value. FLOOR function returns a number rounded down based on a multiple of significance, i.e, the closet and lowest value for the specified number, for example for 1.20 the floor value will be 1. Example: PostgreSQL FLOOR() function . January 4, 2019. Beschreibung. Pictorial presentation of FLOOR() Function. Figure 2: Rounding down numbers with Excel FLOOR function. Use mode=1 for compability if you have negative numbers and wish to export to MS Excel. In 2010 I get a negative value as the answer. When I use the FLOOR function in Excel 2010, I don't seem to get the right answer. FLOOR Function takes up two arguments, the Number to be rounded off and significance to which the level of rounding has to be done. The CEILING and FLOOR functions in Excel are somewhat complimentary to the MOD function. Code: The floor function appears in several formulas characterizing prime numbers. The Excel CEILING function is categorized under Math and Trigonometry functions Functions List of the most important Excel functions for financial analysts. I'm noticing that the FLOOR() function in a calculated column appears as if the function isn't defined. Floor Function Excel Solutions Basic And Advanced Critbinom Function Compatibility Formulas Ms Excel Formulas Rounding Using Ceiling And Floor Functions In Excel Excel Bonanza Formula Friday Floor Function In Excel Find Out The Decade That How To Use The Excel Trunc Function Exceljet Ofy9ymygmq7omm Quotient Function Excel Ceiling And Floor Functions My Online Training Hub Vba … The PostgreSQL floor() function is used to return the value after rounded up any positive or negative decimal value as smaller than the argument. The function will return a number that is rounded up to a supplied number that is away from zero to the nearest multiple of a given number. Syntax. In this ArticleFLOOR Function Description:FLOOR Function Syntax and Inputs:How to use the FLOOR FunctionRound down to Nearest 5, 50, or .5Round down to Nearest QuarterRound TimeOther ROUND Functions / FormulasCEILING and MROUNDROUNDDOWN FunctionFLOOR in Google SheetsFLOOR Examples in VBA This tutorial demonstrates how to use the Excel FLOOR Function in Excel to round a … The number to be rounded down. If number is positive and significance is negative, the FLOOR function returns the #NUM error, as in row 4. nicht kleinere ganze Zahl zuordnen. Excel 2010 now follows the standard definition. This has followed through to the Office Open XML file format. Learn Excel the Simple Way in this tutorial. FLOOR rounds to the nearest integer, using a 1 as significance. Syntax: =FLOOR(number, multiple) Arguments: number – The number that should be rounded. In simple words, the FLOOR.MATH function rounds down the number to the nearest specified multiple. The FLOOR.MATH function syntax has the following arguments. Description. It can be used as a worksheet function (WS) in Excel. The FLOOR function is a built-in function in Excel that is categorized as a Math/Trig Function. The Microsoft Excel FLOOR function returns a number rounded down based on a multiple of significance. If [mode]= any other numeric value, negative numbers are rounded towards zero. (as in Excel 2003 and 2007). Syntax of FLOOR.MATH Function in Excel. Math.floor(x) Parameter x Eine Zahl. The FLOOR function in Excel rounds a supplied number towards zero, to the nearest multiple of a given number. As a worksheet function, the FLOOR function can be entered as part of a formula in a cell of a worksheet. 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The interval of integration ( or is omitted ), negative numbers are rounded zero! Categorized as a worksheet function, CEILING function and FLOOR this cheat sheet covers 100s functions. | 2022-05-16 16:07:58 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.35198307037353516, "perplexity": 1762.9641922377946}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662510138.6/warc/CC-MAIN-20220516140911-20220516170911-00486.warc.gz"} |
https://math.stackexchange.com/questions/2435814/proof-by-induction-of-sum-n-1-infty-left-x1-right-n-frac1x | # Proof by induction of $\sum_{n=1}^{\infty }\left ( x+1 \right )^{-n}=\frac{1}{x}$
I've been trying to work on some proof for class and I basically want to prove that:
$$\sum_{n=1}^{\infty }\left ( x+1 \right )^{-n}=\frac{1}{x},\quad \text{where} \quad x \in \mathbb{Z^{+}}.$$
So far I've been trying to use proof by induction, but I can't seem to get anywhere as it has no final term. Does anyone have any idea how I could go about proving this?
Guide:
Use geometric series.
Notice that if $x \in \mathbb{Z}^+$, then $0<\frac{1}{x+1} < 1$.
In general, if $a \in \mathbb{C}$ with $\vert a \vert < 1$ and $N \in \mathbb{N}^{\ast}$:
\begin{align*} (1-a) \sum_{k=0}^{N} a^k & = \sum_{k=0}^{N} a^k - \sum_{k=0}^{N} a^{k+1} \\[2mm] & = \sum_{k=0}^{N} a^k - \sum_{k=1}^{N+1} a^k \\[2mm] & = 1 - a^{N+1}. \end{align*}
Because $\vert a \vert < 1$, $a^{N+1} \to 0$ as $N \to +\infty$.
This proves that:
$$\sum_{k=0}^{+\infty} a^k = \frac{1}{1-a}.$$
Apply this result with $a = 1/(1+x)$, $x \in \mathbb{Z}^{+}$.
It's obvious that our series converges.
Now, let $\sum\limits_{n=1}^{+\infty}\frac{1}{(1+x)^n}=A$.
Thus, $$(1+x)A=1+A$$ | 2020-01-27 09:30:02 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9561942219734192, "perplexity": 239.61968389924724}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579251696046.73/warc/CC-MAIN-20200127081933-20200127111933-00213.warc.gz"} |
https://www.physicsforums.com/threads/how-to-make-an-optimal-schedule.334743/ | # How to make an optimal schedule
1. Sep 4, 2009
### daniel_i_l
Let's say I have a group of people and 7 different jobs that have to be done every week - one per day. Each job requires one person. How do I distribute the jobs among the group so that both the "vacation time" between each workday and the time between doing the same job twice is constant throughout the group?
In other words, I need an even distribution of both workdays in general and of specific jobs.
What tools are used to solve this kind of problem?
Thanks.
2. Sep 5, 2009
### benorin
If it doesn't matter what day a particular job is done, but rather only that each job be done once per week and exactly one job be done per day, then let the jobs be denoted j1, j2, ..., and j7. Let us further muck things up by assuming that we have exactly 7 people in our work pool, conveniently named p1, p2, ..., and p7.
Now, as a trivial solution we may have p1 work day 1 of the week (Sunday if you wish), have p2 work day 2 of the week, ..., and have p7 work day 7 of the week so that the time betwixt work days for each person in the work pool is a constant (as it happens, 6 days for every person). The job assignments the initial week could be: j1 is done by p1 on day 1, j2 is done by p2 on day 2, generally: jk is done by pk on day k for 1 <= k <= 7. Now that p1 works day 1, and p2 works on day 2, and so on we will leave fixed (for simplicity) and permute the job assignments by 1 each week to induce a 7 week cycle.
Hence, during week 2, p1 does j2 on day 1, p2 does j3 on day 2, and so on... up to p6, who does j7 on day 6, and, p7 who does j1 on day 7.
Hence, during week 3, p1 does j3 on day 1, p2 does j4 on day 2, and so on... up to p5, who does j7 on day 5, and, p6 who does j1 on day 6 followed by p7 who does j2 on day 7.
You get the idea, by week 8, the job assignments are back to same as the initial week, establishing a constant time interval between each worker doing the same job twice (7 weeks, same for each worker and the particular job being done again).
I'll stop being lazy and do the table:
$$\text{Values of } k \text{ are specified in the first row.}$$
$$\text{Each column then represents } p_k \text{ working on day } k \text{ doing the job listed in that column during the week indicated by row}$$
$$\begin{array}{l|c|c|c|c|c|c|c} \text{ }\, \, \, k= & 1 & 2 & 3 & 4 & 5 & 6 & 7\\\hline \text{Week 1} & j_1 & j_2 & j_3 & j_4 & j_5 & j_6 & j_7\\\hline \text{Week 2} & j_2 & j_3 & j_4 & j_5 & j_6 & j_7 & j_1\\\hline \text{Week 3} & j_3 & j_4 & j_5 & j_6 & j_7 & j_1 & j_2\\\hline \text{Week 4} & j_4 & j_5 & j_6 & j_7 & j_1 & j_2 & j_3\\\hline \text{Week 5} & j_5 & j_6 & j_7 & j_1 & j_2 & j_3 & j_4\\\hline \text{Week 6} & j_6 & j_7 & j_1 & j_2 & j_3 & j_4 & j_5\\\hline \text{Week 7} & j_7 & j_1 & j_2 & j_3 & j_4 & j_5 & j_6\\\hline \text{Week 8} & j_1 & j_2 & j_3 & j_4 & j_5 & j_6 & j_7\\\end{array}$$
.
Non-trivial solutions and possibly unsolvable scenarios could arise for other particular values of the number of people in the work pool. On a hunch, I would recommend trying some sort of group theory (maybe just modular arithmetic) to solve for every class of solution.
Also, it was not clear if the two constants required for a viable schedule were allowed to vary from worker to worker or from job to job versus requiring the same vacation time for every worker. (Consider, 4 part-time workers p1a, p1b, p7a, and p7b working the week-ends where the a's work odd weeks and the b's work even weeks having a 14 day vacation constant and a repeat job every 16 weeks while the other 5 full-time workers have the same constants as the they did in the table.
I could go on, and on... but I spare you.
Ben
Last edited: Sep 5, 2009 | 2018-03-19 13:20:46 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.29318252205848694, "perplexity": 1241.1686947402534}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-13/segments/1521257646914.32/warc/CC-MAIN-20180319120712-20180319140712-00519.warc.gz"} |
https://en.khanacademy.org/math/calculus-all-old/series-calc/estimating-infinite-series-calc/v/estimating-infinite-sum-integrals | If you're seeing this message, it means we're having trouble loading external resources on our website.
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## Calculus, all content (2017 edition)
### Course: Calculus, all content (2017 edition)>Unit 7
Lesson 11: Estimating infinite series
# Worked example: Series estimation with integrals
See how we can use improper integrals to approximate the infinite sum of 1/n².
## Want to join the conversation?
• , why did Sal have to introduce a new variable `b → ∞` to take the limit?
• Sal made a mistake right there. He was correct to introduce a new variable, but what he did next was incorrect, and made it confusing as to why he introduced the variable.
The problem we're dealing with at that point in the video is evaluating an integral from 6 to infinity. That's an improper integral: the fundamental theorem of calculus tells us how to evaluate the integral from 6 to some other finite number (assuming there are no "blow-up" problems in between), but it doesn't tell us how to evaluate an integral that goes to infinity. We solve the problem by dividing it into two steps. First, we find the integral from 6 to some unspecified finite upper bound, which we'll call "b" in this case. And second, once we see what the integral would be with an upper bound of b, we find what the limit would be as b goes to infinity.
Sal's mistake here was that immediately after introducing the variable b, he again writes the integral with an upper bound of infinity, when he intended to write it with an upper bound of b for the reason explained in the previous paragraph. If this still doesn't make sense, you may want to go back and review the videos on improper integrals, which begin here:
• I know this is not too relevant, but didn't it bother anyone else that he forgot to change "n" to "x" when doing the integrals?
the integral of 1/n^2 dx is equal to x/n^2.
I know it's a very minor mistake, but it could be misleading.
• This is not just a mistake, this is a BIG ( and in some costly ) mistake. I'm reporting this mistake.
• Is there a series of videos about integrals with bounds at infinity as used in this video? I'm having trouble finding any
• As k increases your estimate becomes more accurate. Could one find the exact sum by taking the limit as k→∞?
• Well, Euler proved that the sum from n=1 -> infinity of 1/n^2 is actually equal to (pi^2)/6
I'm not sure why that is, but you can definitely check out his proof of it on the internet :)
• Why do we have to split the sum into Sk and Sr in order to take the estimate? Why cant we start from n=1?
• We want to break the unknowable sum into a piece that is clear and known and a piece that can be estimated. Then we can use the estimated piece to set a bounds on all the values that the sum can possibly be. You usually cannot just start from n= 1 and keep adding terms to infinity.
` known sum of first 4 terms` + `estimate of the terms from 5 to ∞` fair estimate
` known sum of first 20 terms` + `estimate of the terms from 21 to ∞` better estimate
` known sum of first 100 terms` + `estimate of the terms from 101 to ∞` even better
• At Sal sets up an integral 1/n^2 dx. Shouldn't he be using f(x) not f(n)? His integral is with respect to x yet he integrates with respect to n.
• Yes, I actually reported this error many moons ago, but still have seen no correction made to the video. If you click "Report a Mistake" on the right side of the comment section you can view and submit errors you find in videos.
• At , near the bottom, Sal writes that the summation is approximately __, but isn't it exactly equal to __ instead?
• This is because Sal wrote an approximate value from the calculator (he used 4 significant figures), and not an exact value.
(1 vote)
• At about Sal writes that the integrand is -n^-1. How'd he get the negative out in front of the n? I'm sure this is a simple question, I'm just blanking on this subject. I understand why there's a negative on the exponent and why it went from -2 to -1.
• So when integrating the format will be x^(n + 1)/(n + 1) which in this leads to n^-2 becoming n^(-2 + 1)/(-2 + 1) or n^-1/-1 which is simply -n^-1. So basically make sure when you're integrating this problem you don't forget about the denominator part of the problem!
(1 vote)
• I'm confused because if you chose a different value of k, say 125 for example, when you apply the same integral and inequality don't you get a different number for the convergence? | 2023-03-27 20:31:13 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8151358962059021, "perplexity": 509.86868664606754}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296948684.19/warc/CC-MAIN-20230327185741-20230327215741-00102.warc.gz"} |
http://mathhelpforum.com/discrete-math/189902-intuition-inclusion-exclusion-print.html | Intuition for inclusion exclusion
Printable View
• October 9th 2011, 04:23 AM
terrorsquid
Intuition for inclusion exclusion
I have been doing inclusion exclusion questions and while, for the most part, I can see what I am doing in each step and which subsets I am adding/removing, the following type of question I can't seem to see exactly what is happening. I know how to do them, but I am just following a pattern that I have seen from doing a number of them - I can't visualise which subsets I am adding/removing as I do them. I was wondering if someone could tell me what's happening exactly as I do this problem:
Find the number of solutions for the equation $x_1 + x_2 + ... + x_9 = 24$, where $4\leq x_1,x_2<9$
So, when $x_1$ and $x_2$ are 4 or greater, then the total number of solutions will be:
${9+24-8-1\choose 9-1} = {24\choose 8}$
I can see what is going on here, every set that includes a value for $x_1$ and $x_2$ less than 4 is excluded. The next steps I do, I get a bit confused what is happening and why...
Suppose: $x_1$ OR $x_2$ < 9 - 4 = 5
Then: $x_1 + x_2 + ... + x_9 = 16 - 5 = 11$
There are two choices so, there are $2\times {19\choose 8}$ ways this can happen.
Subtract this from the total: ${24\choose 8} - 2\times {19\choose 8}$
Finally I need to add the instance when both $x_1$ and $x_2$ are < 5 to give the final answer:
${24\choose 8} - 2\times {19\choose 8}+{14\choose 8}$
I can do these questions, it's just I am doing them without knowing what's going on exactly. So, if the question format was changed up a bit, I might struggle. What has been counted/over-counted at what stage etc.?
Thanks.
• October 9th 2011, 04:38 AM
Plato
Re: Intuition for inclusion exclusion
Quote:
Originally Posted by terrorsquid
Find the number of solutions for the equation $x_1 + x_2 + ... + x_9 = 24$, where $4\leq x_1,x_2<9$.
Do you understand how $\binom{20+9-1}{20}$ is the number of solutions in which $x_1\ge 4~?$
• October 9th 2011, 04:48 AM
terrorsquid
Re: Intuition for inclusion exclusion
Quote:
Originally Posted by Plato
Do you understand how $\binom{20+9-1}{20}$ is the number of solutions in which $x_1\ge 4~?$
Yes, I feel comfortable with the initial step. If only $x_1 \geq 4$ then $x_1 + x_2 + ... + x_9 = 24 - 1\times 4 = 20$
$\therefore$ the total number of solutions = ${28\choose 20}$ or ${28\choose 8}$ ... same thing.
• October 9th 2011, 05:12 AM
Plato
Re: Intuition for inclusion exclusion
Quote:
Originally Posted by terrorsquid
Yes, I feel comfortable with the initial step. If only $x_1 \geq 4$ then $x_1 + x_2 + ... + x_9 = 24 - 1\times 4 = 20$
$\therefore$ the total number of solutions = ${28\choose 20}$ or ${28\choose 8}$ ... same thing.
Of those $\binom{11+9-1}{11}$ have $x_2\ge 9$.
So remove that number from $\binom{20+9-1}{20}~.$
BTW: this is not a problem for inclusion/exclusion.
• October 9th 2011, 05:13 AM
terrorsquid
Re: Intuition for inclusion exclusion
Why do we multiply the second step by two because there are two choices and we don't multiply the first step by two where $x_1$ and $x_2$ are 4?
Quote:
Originally Posted by Plato
BTW: this is not a problem for inclusion/exclusion.
My question asks to apply the principles of inclusion-exclusion to solve them :S
• October 9th 2011, 06:13 AM
Plato
Re: Intuition for inclusion exclusion
Quote:
Originally Posted by terrorsquid
My question asks to apply the principles of inclusion-exclusion to solve them :S
I have no intention of debating your instructor/textbook.
If the question had asked for $x_1\ge 4\text{ or }x_9>8$ or even $x_1\ge 4\text{ and }x_9>8$ then clearly inclusion/exclusion would be called for. However, as stated this is a mixed problem.
I have done it in three different ways getting the same answer each time.
Here is a straightforward counting method:
$\sum\limits_{k = 0}^8 {\binom{27 - k}{20 - k}} = {\text{3032523}}$
• October 9th 2011, 06:28 AM
terrorsquid
Re: Intuition for inclusion exclusion
Maybe that's why I don't understand what's going on with the inclusion exclusion :D | 2014-07-13 03:19:08 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 39, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8588126301765442, "perplexity": 453.5753466843189}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-23/segments/1404776435842.8/warc/CC-MAIN-20140707234035-00054-ip-10-180-212-248.ec2.internal.warc.gz"} |
https://math.stackexchange.com/questions/994678/elliptic-curves | # Elliptic Curves
I want some clarification regarding some concept in elliptic curves. In many papers I have seen that, let $E:y^2=x^3+Ax+B$ be an elliptic curve if $L(E,1)$ (corresponding L-function at s=1) is nonzero and Tate Shafarevich group is also nontrivial then MWrank of $E$ is zero. Is it true for all elliptic curves? What is the reason? Is converse also true.
• Take a look at this – Stahl Oct 28 '14 at 8:11
• Yes, it is true for every elliptic curve (over $\mathbb Q$), and it follows from Gross-Zagier and Kolyvagin's work. See here for example mathoverflow.net/questions/124202/… The converse is not known to be true up to now. – Ferra Oct 28 '14 at 15:34
• The converse is known to be true under the assumption that the Tate-Shafarevich group is finite. See arxiv.org/abs/1405.7294 – Brandon Carter Nov 26 '14 at 22:59 | 2019-05-25 05:22:05 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8872962594032288, "perplexity": 366.0327469251104}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-22/segments/1558232257889.72/warc/CC-MAIN-20190525044705-20190525070705-00556.warc.gz"} |
https://brilliant.org/problems/integral-of-mixtures-of-irrational-numbers/ | # Definite integral of mixtures of irrational numbers
Calculus Level 5
$\large \int_0^1 \dfrac{\tan^{-1}x}{x^{2/3}} \, dx$
Given the integral above equals to
$\pi \left(\dfrac ab - \dfrac{\sqrt c}d \right) + \dfrac ef \ \ln(g)$
where $$a,b,c,d,e$$ and $$f$$ are positive integers with prime $$g$$ and $$\gcd(a,b) = \gcd(c,d) =\gcd(e,f) = 1$$.
Find the value of the 7-digit integer, $$\overline{abcdefg}$$.
× | 2017-05-29 00:16:53 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8753829598426819, "perplexity": 573.7494041198612}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-22/segments/1495463612003.36/warc/CC-MAIN-20170528235437-20170529015437-00433.warc.gz"} |
https://math.stackexchange.com/questions/2544518/find-the-sum-of-the-series-s-sum-k-1n-frackk4-k2-1 | # Find the sum of the series $S = \sum_{k=1}^{n} \frac{k}{k^{4} + k^{2} + 1}$
$$S = \sum_{k=1}^{n} \frac{k}{k^{4} + k^{2} + 1}$$
I started by factorizing the denominator as $k^2+k+1$ and $k^2-k+1$ The numerator leaves a quadratic with $k$ and $k-1$ or a constant with $k+1$ and $k-1.$ I tried writing the individual terms, ofcourse, it was useless. How do I do this?
• After you factor the denominator and use partial fractions, I believe the sum will telescope. Nov 30 '17 at 14:41
• As a rule, we don't use $\sum_{n=1}^{n}$. We use two different variables: $\sum_{k=1}^{n}\dots$. Also, you *really don't need any of those parantheses. Nov 30 '17 at 14:44
• Nov 30 '17 at 17:30
So your term is equal to $$\frac{1}{2}\left(\frac{1}{k^2-k+1}-\frac{1}{k^2+k+1}\right)$$
Now note $(k+1)^2-(k+1)+1=k^2+k+1$, so your term is: $$\frac{1}{2}\left(\frac{1}{k^2-k+1}-\frac{1}{(k+1)^2-(k+1)+1}\right)$$
and you can apply a telescoping series technique to establish the sum to $n$ is just half of $\frac{1}{1^2 -1 +1}-\frac{1}{(n+1)^2-(n+1)+1}$. And if you are also looking for the limit, half of $\frac{1}{1^2 -1 +1}$.
• Note that if you didn't recognize that $(k+1)^2-(k+1)+1=k^2+k+1$, you can discover it by noticing that the roots of $x^2-x+1=0$ - namely, $\frac12(1\pm\sqrt{-3})$ - are exactly 1 more than the roots of $x^2+x+1=0$, $\frac12(-1\pm\sqrt{-3})$. Dec 1 '17 at 17:06 | 2021-11-27 21:24:26 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9193832874298096, "perplexity": 203.33044768154843}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964358233.7/warc/CC-MAIN-20211127193525-20211127223525-00493.warc.gz"} |
https://www-sop.inria.fr/mascotte/EULER/wiki/pmwiki.php/PmWiki/PageVariables?action=print | # PmWiki: Page specific variables
This page describes the "variables" that are associated with pages. Page variables have the form {$variable}, and can be used in page markup or in certain formatting strings in PmWiki. For example, the markup "{$Group}" renders in this page as "PmWiki".
Note: Do not confuse these variables (set and used only in PmWiki pages) with PHP variables. Page variables can be read in PHP with the PageVar() function.
Note that these variables do not necessarily exist in the PHP code, because they have to be determined for a specific page. (However, they are usable in FmtPageName strings.)
There is also the form {pagename$variable}, which returns the value of the variable for another page. For example, "{MarkupMasterIndex$Title}" displays as "Markup Master Index".
## Special references
Special referenced variables are used to retain the context of the target page or main page for a variable when:
• the variable is included into a destination (target) page)
• the variable is part of a sidebar, header, or footer for a main page
Prefixing the variable name with an asterisk (*) means the variable reflects the value related to the target page or main page.
• *$PageVariablename - prefixed by an asterisk (*) - value reflects target page context Without the asterisk it reflects the value of the page in which it originates. • $PageVariablename - retains value in source page context
Special references are also used in page list templates.
For example you can test to see if the page is part of another page
(:if ! name {$FullName}:) %comment% name of this page is not the same as the page this text was sourced from ->[[{$FullName}#anchor | more ...]] (:ifend:)
or refer to the main page in a sidebar, footer, or header
This page is [[{*$FullName}]] This page is PmWiki.PageVariables ## Default page variables The page variables defined for PmWiki are: {$Group} - page's group name, as in "PmWiki"
{$Groupspaced} - spaced group name, as in "Pm Wiki" {$DefaultGroup} - default group name, as in "Main"
{$SiteGroup} - default group name for e.g. RecentChanges, as in "Site" {$Name} - page name, as in "PageVariables"
{$Namespaced} - spaced page name, as in "Page Variables" {$DefaultName} - name of default page, as in "Home"
{$FullName} - page's full name, as in "PmWiki.PageVariables" {$BaseName} - page's "base" form (stripping any prefixes or suffixes defined via $BaseNamePatterns) as in "PmWiki.PageVariables" {$Title} - page title (may differ from Name), as in "Page specific variables"
{$Titlespaced} - title/spaced page name, as in "Page specific variables" {$Description} - page's description from the (:description:) markup, as in "Documentation for "variables" that are associated with pages."
{$LastModified} - date page was edited, as in "June 21, 2009, at 09:31 AM" {$LastModifiedBy} - page's last editor, as in "simon"
{$LastModifiedTime} - time page was edited in unix-style timestamp, as in "1245569488" added version 2.2 beta 67 This can be used (preceded by '@') in {(ftime)} and other date/time markups. {$LastModifiedHost} - IP of page's last editor, as in "203.97.214.12"
{$LastModifiedSummary} - Summary from last edit, as in "clarify special references" {$PageUrl} - page's url, as in "https://www-sop.inria.fr/mascotte/EULER/wiki/pmwiki.php/PmWiki/PageVariables"
{$Action} - page's url action argument, as in "print" {$PasswdRead} - current password for read attr. e.g. "(protected)"
{$PasswdEdit} - current password for read attr. e.g. "(protected)" {$PasswdAttr} - current password for read attr. e.g. "(protected)"
In addition to the above, there are some page-invariant variables available through this markup:
{$Author} - the name of the person currently interacting with the site, as in "" {$AuthId} - current authenticated id, as in "" Please note the lower case 'd'. {$AuthID} returns nothing {$Version} - PmWiki version, as in "pmwiki-2.2.4"
{$VersionNum} - The internal version number, as in "2002004" {$ScriptUrl} - The url to the pmwiki script, as in "https://www-sop.inria.fr/mascotte/EULER/wiki/pmwiki.php"
## Custom page variables
You may add custom page variables as a local customization. In a local configuration file or a recipe script, use the variable $FmtPV: $FmtPV['$VarName'] = "'variable definition'";$FmtPV['$CurrentSkin'] = '$GLOBALS["Skin"]';
$FmtPV['$WikiTitle'] = '$GLOBALS["WikiTitle"]'; Defines new Page Variable of name$CurrentSkin, which can be used in the page with {$CurrentSkin} (also for Conditional markup). It's necessary to use the single quotes nested inside double-quotes as shown above (preferred) or a double-quoted string nested inside single-quotes like '"this"'. If you want to have a Page Variable that returns the currently used password (more precisely, the last password entered), you can use $FmtPV['$AuthPw'] = 'reset(array_keys((array)@$_SESSION["authpw"]))';
Is there a variable like $LastModified, but which shows me the creation time? No, but you can create one in config.php. For instance: # add page variable {$PageCreationDate} in format yyyy-mm-dd
$FmtPV['$PageCreationDate'] = 'strftime("%Y-%m-%d", $page["ctime"])'; If you like the same format that you define in config.php with $TimeFmt use
$FmtPV['$Created'] = "strftime(\$GLOBALS['TimeFmt'], \$page['ctime'])";
` | 2017-12-12 06:30:17 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3090575635433197, "perplexity": 6968.748573823815}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-51/segments/1512948515309.5/warc/CC-MAIN-20171212060515-20171212080515-00773.warc.gz"} |
https://stackoverflow.com/questions/21469110/deploying-web-role-in-azure-throws-exception-on-currentaccountstoragename | # Deploying web role in Azure throws exception on CurrentAccountStorageName
I want to deploy a web role to Azure using the PowerShell CmdLets.
My script is as follows:
$subscription = "<name-of-subscription>"$service = "<name-of-cloudservice>"
$slot = "staging"$package = "path\to\package.cspkg"
$configuration = path\to\config.cscfg"$deploymentLabel = "Deploy to $service" Import-Module "C:\Program Files (x86)\Microsoft SDKs\Windows Azure\PowerShell\Azure\Azure.psd1" Import-AzurePublishSettingsFile "C:\path-to.publishsettings" Set-AzureSubscription -CurrentStorageAccount$service -SubscriptionName $subscription # some more stuff to check whether to upgrade or to create ... Set-AzureDeployment -Upgrade -Slot$slot -Package $package -Configuration$configuration -label $deploymentLabel -ServiceName$service -Force
When I execute this it throws an error:
Exception: The subscription named <name-of-subscription> already exists.
I figured that since I'm importing my publishsettings-file already I could get rid of Set-AzureSubscription. However, once I do that I get the next error:
Exception: CurrentStorageAccountName is not set.
Use Set-AzureSubscription subname -CurrentStorageAccountName storageaccount to set it.
This is the line that gave me the error in the first place, so I'm not sure how I need to set the storageaccountname without causing an error.
I also ran a little test:
Import-AzurePublishSettingsFile "C:\path-to.publishsettings"
Get-AzureSubscription | Format-Table
Once I do this I get the following output (reformatted):
SubscriptionName: <name-of-subscription>
SubscriptionId: 123456789-123...
ServiceEndpoint: https://man....
ActiveDirectoryEndpoint:
ActiveDirectoryTenantId:
IsDefault: True
Certificate [Subject]
CurrentStorageAccountName
CurrentCloudStorageAccount
ActiveDirectoryUserId
As you can see, the CurrentStorageAccountName is empty, but I don't know how I can set it to a correct value.
I looked up some other scripts, but they all seem to be having this sequence of importing, then setting. Any idea why this is not working and how to solve it?
You are trying to set CurrentStorageAccount to the name of your cloudservice, but you should be setting it to the name of your blob storage account. Get the list of you storage accounts
PS U:\>Get-AzureStorageAccount |select StorageAccountName
StorageAccountName
------------------
portalvhdsgsomething
storage1
storage2
storage3
then run your previously failing line but with the name of your storage account ie.
Set-AzureSubscription -CurrentStorageAccount storage2 -SubscriptionName $subscription You can confirm your changes with PS U:\>Get-AzureSubscription | select CurrentStorageAccount CurrentStorageAccount : storage2 • Tried that as well, but to no avail – Kenneth Jan 31 '14 at 11:33 • Can you be more specific? Which part fails and what is the error? – Raf Jan 31 '14 at 11:36 • The same as before: Exception: The subscription named <name-of-subscription> already exists. – Kenneth Jan 31 '14 at 12:52 • Hmm.. try Select-AzureSubscription -SubscriptionName$subscription before running Set-AzureSubscription Do you get an error then? – Raf Jan 31 '14 at 12:58
You need to use -CurrentStorageAccountName as the parameter of Set-AzureSubscription.
• That is what I have: Set-AzureSubscription -CurrentStorageAccount $service -SubscriptionName$subscription – Kenneth Jan 30 '14 at 23:25
For Set-AzureSubscription the MSDN doc shows the -SubscriptionName parameter as parameter 1, not as named. The error message's suggested fix even implied that you needed to do Set-AzureSubscription <service> -CurrentStorageAccount <storage solution> Perhaps simply moving that will solve it?
Set-AzureSubscription -SubscriptionName $subscription -CurrentStorageAccount$service
• I tried that as well, but the order doesn't seem to matter – Kenneth Jan 31 '14 at 0:10
• Where exactly in your script is it throwing the error? – TheMadTechnician Jan 31 '14 at 0:14
• When I don't do Set-AzureSubscription upon executing Set-AzureDeployment, otherwise on Set-AzureSubscription – Kenneth Jan 31 '14 at 0:20
• Only other thing I can think of is that you have a different subscription as your 'current subscription' apart from what's in $subscription, perhaps it is defined as part of your file import? Try doing a get-azuresubscription | set-azuresubscription -currentstorageaccountname$service and see if that clears it up. If it does you may have conflicting subscription references. I'm out for the night, I'll see if I come up with any better ideas and check back tomorrow. – TheMadTechnician Jan 31 '14 at 0:33
• No, when I do a Get-AzureSubscription it only shows that one, with an empty CurrentStorageAccount. Thanks for the help! Good night. – Kenneth Jan 31 '14 at 0:38
I have found out the reason. It appears that you cannot include the subscription name as a named parameter. Changing this line:
Set-AzureSubscription -CurrentStorageAccount $service -SubscriptionName$subscription
to this line did the trick
Set-AzureSubscription $subscription -CurrentStorageAccount$service | 2021-06-14 16:26:50 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6537180542945862, "perplexity": 3213.547915086868}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623487612537.23/warc/CC-MAIN-20210614135913-20210614165913-00213.warc.gz"} |
http://mathonline.wikidot.com/the-complex-conjugate-of-a-complex-number | The Complex Conjugate of a Complex Number
# The Complex Conjugate of a Complex Number
Recall that a complex number is in the form $z = a + bi$ where $a, b \in \mathbb{R}$, then the real part of $z$ is $\Re (z) = a$ and the imaginary part of $z$ is $\Im (z) = b$. Therefore we have that $z = \Re (z) + \Im (z) i$. We will now defined a special number call the complex conjugate of $z$ and look at its interesting properties. This will become useful when diving further into linear algebra.
Definition: If $z = a + bi = \Re(z) + \Im(z) i$ is a complex number, then the Complex Conjugate of $z$ denoted $\bar{z} = a - bi = \Re(z) - \Im(z) i$.
For example, consider the complex number $z = 5 + 2i$. Then the complex conjugate of $z$ is $\bar{z} = 5 - 2i$. Another example is the complex number $z = i - 1$ whose complex conjugate is $\bar{z} = -1 - i$.
Theorem 1: Let $z, z' \in \mathbb{C}$ be complex numbers $z = a + bi$ and $z' = a' + b'i$. Then: a) $z + \bar{z} = 2 \Re (z)$. b) $z - \bar{z} = 2 \Im (z) i$. c) $\bar{\bar{z}} = z$. d) $\overline{z + z'} = \bar{z} + \bar{z'}$ (Additivity Property). e) $\overline{zz'} = \bar{z} \bar{z'}$ (Multiplicativity Property).
• Proof a) We have that $z = a + bi$ and $\bar{z} = a - bi$. So $z + \bar{z} = (a + bi) + (a - bi) = (a + a) + (bi - bi) = 2a = 2 \Re (z)$.
• Proof b) We have that $z = a + bi$ and $\bar{z} = a - bi$. So $z - \bar{z} = (a + bi) - (a - bi) = (a - a) + (bi + bi) = 2bi = 2 \Im (z) i$.
• Proof c) If $\bar{z} = a - bi$ then $\bar{\bar{z}} = a - (-bi) = a + bi = z$.
• Proof d) We have that $z + z' = (a + bi) + (a' + b'i) = (a + a') + (b + b')i$, and so:
(1)
\begin{align} \quad \overline{z+z'} = (a + a') - (b + b')i = [a - bi] + [a' - b'i] = \bar{z} + \bar{z'} \end{align}
• Proof e) We have that $zz' = (a + bi)(a' + b'i) = aa' + ab'i + a'bi - bb' = (aa' - bb') + (ab' + a'b)i$, so $\overline{zz'} = (aa' - bb') - (ab' + a'b)i$. Now we also note that $(a - bi)(a' - b'i) = aa' - ab'i + a'bi - bb' = (aa' - bb') - (ab' - a'b)i$ Therefore $\overline{zz'} = \bar{z} \bar{z'}$. $\blacksquare$ | 2019-06-20 15:38:20 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9933326840400696, "perplexity": 207.74839223617212}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560627999261.43/warc/CC-MAIN-20190620145650-20190620171650-00469.warc.gz"} |
http://www.wall.org/~aron/blog/is-god-allowed-to-update-the-torah/ | # Is God allowed to update the Torah?
On my last post, a reader going by the name JPH comments:
The resurrection is irrelevant.
God appeared to a nation and gave them 613 commandments. He said they were eternal, everlasting, binding for all generations. There is NOT ONE about worshiping God's son or the Messiah. (Exodus 4:22 says God's son is Israel.) There are horrifying threats for deviating from these commandments in Deuteronomy 28. The thirteenth chapter is devoted to prophets who can perform "signs and wonders" and advocate the worship of gods "whom your forefathers did not know." Their forefathers did not worship Jesus. Deut 13 explicitly grants the possibility of miracles in false traditions and says, "Do not hearken unto that prophet." It says nothing about surviving an execution as an exception or some big standard.
Why do Christians think the resurrection cancels/changes the Torah? According to what standard? (No, the prophets didn't say so: https://prooftexts.wordpress.com Most of these aren't just wrong, they're cringe-worthy. The prophets received their authority from the 5 Books of Moses. Whatever the prophets were saying, it wasn't to subtract from theses books and approach God via some unheard-of intermediary.)
Sabbatai Tzvi, too, was considered by many to be the Messiah. He performed signs and wonders. He had his own St. Paul (Nathan of Gaza) who interpreted his conversion to Islam as some humiliating atonement. He still has followers. So what? Miracles don't cancel the Torah. The only reason to think otherwise is because your Bible already has a New Testament attached.
JPH,
I. Matters of Interpretation
First of all, your method of interpretation, which sees very few Messianic prophecies in scripture, and assumes that if there is a literal application there cannot also be a secondary symbolic application, is simply not in line with traditional Jewish rabbinic interpretation. Why can't some passages refer to both Israel and the Messiah, for example?
Many of these passages were traditionally interpreted as Messianic by Jewish rabbis, until it became inconvenient given the fact that Christians were continually citing them. See here for a discussion of the Talmud's take on this:
Messiah: The Talmud on Messianic Prophecy
which includes quotations from the Talmud which state that “All the prophets prophesied only for the days of the Messiah”! So, unlike the link you provide which keeps stating "not a prophetic prediction" over and over again, it was apparently an accepted view within Talmudic circles that (with some hyperbole) denied the existence of any non-Messianic verses in the Bible!
In any case, there are several passages which are agreed on by everyone to be Messianic, that have this "double fulfillment" aspect. For example, the central Messianic text is Nathan's prophecy to David, in which he predicts that David's dynasty will last forever, is found in 2 Sam. 17 and 1 Chron. 17. It is clear that this prophesy has aspects which were fulfilled in the next generation (when Solomon built the Temple) but it also has aspects which speak about David's continuing dynasty, which are ultimately fulfilled by the fact that the Messiah himself will personally reign forever.
Similarly, when the prophet Isaiah talks about Israel returning from captivity under Babylon, he keeps talking about it in terms which suggest that it will usher in the Messianic Era in which there will be peace forever and God will never have wrath towards Israel again. (A particularly fine example of this is in chapter 54.) Now we all know that these events did not happen at the same time, but there is a certain allegorical similarity about them which justifies talking about them at the same time.
So, if some Scriptures clearly have both an immediate application to the present day, and also a distant future fulfillment, then there may be double meanings in other Scriptures as well. (Although in what follows, I will try to confine myself to the plain meaning of the Hebrew Scriptures as much as possible, not because I am conceding the absence of double meanings, but in order to keep the argumentation as clear as possible.)
Now that is not to say that the particular unknown missionary tract that your link is refuting gets everything right. But, 353 one-liners followed by another 353 one-liners doesn't really seem like the most productive way to engage. It's a mile wide but only an inch deep. The real debate here is about methods of interpretation--and also the fact that, when the historical evidence is strong enough that God supports something, sometimes one should admit that one's interpretation of Scripture might be wrong! Also, at least sometimes the Hebrew text is ambiguous (or there are variant manuscripts), so you need to check multiple translations before rejecting the idea that a given meaning could be part of the original text.
So I think the author would have been better off engaging in some work of serious Christian scholarship, rather than some random tract he got in the mail. (The post you link to doesn't even state the author and publication information! So I have no idea whose views the tract is supposed to represent.)
II. Can the Commandments Change?
Secondly, I do get that there are passages in the Torah which may seem, at first sight, to state that the laws of the Torah are eternal and immutable. (Although the precise translation of these words in the Hebrew can be tricky, as discussed here.)
But when you dig deeper into the Tanakh, I think you will be able to see that there is also significant conflicting evidence, which indicates that parts of the Torah are provisional, if you keep your mind open to the possibility. For example, the Torah itself explicitly says that:
A prophet will HaShem thy G-d raise up unto thee, from the midst of thee, of thy brethren, like unto me; unto him ye shall hearken; according to all that thou didst desire of HaShem thy G-d in Horeb in the day of the assembly, saying: 'Let me not hear again the voice of HaShem my G-d, neither let me see this great fire any more, that I die not.' And HaShem said unto me: 'They have well said that which they have spoken. I will raise them up a prophet from among their brethren, like unto thee; and I will put My words in his mouth, and he shall speak unto them all that I shall command him. And it shall come to pass, that whosoever will not hearken unto My words which he shall speak in My name, I will require it of him. (Deut. 18:15-19)
[This quotation is from the JPS 1917, a Jewish translation. Note that in this translation HaShem ("the Name") is the standard substitute in order to avoid writing out God's Name, just as many English translations substitute LORD.]
This passage makes it pretty clear that there will be new divine instructions post-Torah; in fact it is 1 of the 613 commandments to obey these new laws! Regardless of whether the singular term "prophet" is interpreted to refer generically to all later prophets, or specifically to the Messiah, the passage seems to indicate that there will be new commandments revealed at that time. Note the specific statements that the prophet will be like Moses (who gave the law), it will be like the event at Mount Horeb (where the law was given), and that the people need to "hearken" to the new words, i.e. listen to them and obey them.
And indeed, sometimes the prophets in the Tanakh announce changes to the law, even ones which abrogate old provisions. For example, Solomon modified various details of the construction of the Tabernacle when he built the Temple, and Ezekiel 40-48 changes a bunch of the rules for Temple worship, while Jeremiah 3:16 states that the Ark of the Covenant would go permanently missing, and that nobody would miss it or ever build a new one. (For that reason, Zerubbabel's Temple had no Ark in its Holy of Holies.) This single passage, taken all by itself, makes it clear that a key ritual of Moses' sacrificial system, the Day of Atonement, will never again be celebrated according to the precise rules of the Torah, no not even in the Messianic Age! So clearly, some of the commandments in the Torah can be changed.
These passages already refute the interpretation of Judaism you are advocating, without any need to discuss Christianity or the New Testament. If you think the commandments in the Torah can't be updated, then to be consistent you would also need to reject the Nevi'im and Ketuvim, like the Samaritans do.
In life, there is always change. Even the rabbis have changed many things, extending some commandments and replacing others. Half the commandments in the Torah became impossible after the Destruction of the Temple, so the rabbis substituted various prayers and other rituals. The Tanakh itself shows that history is not static, and that what was appropriate for Israel at one stage in her development is inappropriate at another stage. If the religion of Israel had already reached its perfect form immediately after they entered the Promised Land, then there would have been no need to subject Israel to any of the further developments of the next 3,500 years. The real question is one of authority: who is in charge of deciding what should change, God or human beings? It was human beings who were charged not to add to or subtract from the Torah; but the Lord (blessed is he) can do as he likes. He is allowed to modify the terms of the agreement.
Speaking of modifications, Jeremiah goes on to say that God will make a New Covenant with Israel, in which the Torah will be written on their hearts instead of simply in a book:
Behold, the days come, saith HaShem, that I will make a new covenant with the house of Israel, and with the house of Judah; not according to the covenant that I made with their fathers in the day that I took them by the hand to bring them out of the land of Egypt; forasmuch as they broke My covenant, although I was a lord over them, saith HaShem. But this is the covenant that I will make with the house of Israel after those days, saith HaShem, I will put My law in their inward parts, and in their heart will I write it; and I will be their G-d, and they shall be My people; and they shall teach no more every man his neighbour, and every man his brother, saying: 'Know HaShem'; for they shall all know Me, from the least of them unto the greatest of them, saith HaShem; for I will forgive their iniquity, and their sin will I remember no more. (Jeremiah 31:31-34)
That is an even bigger deal than a change in the rules; it portends a change in how people relate to the whole concept of rules. It is what Moses wished, when he said “Are you jealous for my sake? I wish that all the LORD's people were prophets and that the LORD would put his Spirit on them!” (Num 11:29). I understand that you are also jealous to protect Moses, but here Moses himself says he is looking forward to something better! Moses wanted the Spirit to speak directly into people's hearts, giving them individual guidance about what to do, rather than following a list of rules written in a book somewhere.
(In addition to changing how Israel relates to God, this passage also seems to indicate that this New Covenant involves a more universal access to forgiveness than was previously available. But, we need not argue here about whether this New Covenant is the same one that is described in the New Testament. Pretend for a moment that you've never heard of Jesus, that you are a Jew reading this passage before Christianity started. Isn't it clear that any New Covenant, just by virtue of being New, must necessarily imply some sort of changes to the old way of doing things?)
Furthermore, reason also tells us that the Law of Moses has to change with changing circumstances. I am not just talking about how some of the commandments seem more suitable for an ancient patriarchal tribal society than to a modern civilized society, or the commandments which refer to people and objects which have not existed for thousands of years (although these facts are worth noting). I am also talking about situations where God himself tells us why he gave the commandment, and we can see explicitly that this reason is no longer applicable. For example, in Leviticus chapter 20 God states explicitly the reason for the kosher laws, saying that it is to keep them separate from the nations which engage in more serious wicked practices:
And ye shall not walk in the customs of the nation, which I am casting out before you; for they did all these things [e.g. incest, adultery, sacrificing their children to Moloch, etc.], and therefore I abhorred them. But I have said unto you: 'Ye shall inherit their land, and I will give it unto you to possess it, a land flowing with milk and honey.' I am HaShem your G-d, who have set you apart from the peoples. Ye shall therefore separate between the clean beast and the unclean, and between the unclean fowl and the clean; and ye shall not make your souls detestable by beast, or by fowl, or by any thing wherewith the ground teemeth, which I have set apart for you to hold unclean. And ye shall be holy unto Me; for I HaShem am holy, and have set you apart from the peoples, that ye should be Mine. (Lev. 20:21-26)
The logical structure implied by this passage seems to be as follows:
1. The surrounding nations do objectively bad things, like killing their children and having sex with close relatives (listed previously in the chapter).
2. God wants them to be holy like he is, and not do those things.
3. So, in order to prevent them from being corrupted by these cultures, God creates a more trivial rule (don't eat certain kinds of animals labelled as unclean).
4. By obeying this rule, Israel is prevented from fully participating in the life of their pagan neighbors, and also gets some practice in the art of making distinctions between "clean" and "unclean" situations.
But suppose a time were to come when the nations stop doing all of these disgusting things. Suppose further that God wanted the Gentiles and Jews to join together into one people with common religious rituals. In that case, the reason for the kosher commandments would no longer apply; in fact the separation would become counterproductive. Therefore, if this were in fact God's plan, it would stand to reason that the kosher rules would no longer apply to the new situation.
But would God in fact want to make such profound changes? We do not have to look anywhere in the New Testament to prove that he would. We need only look at the prophets which are accepted by Jews. The prophet Zechariah says that there will come a time when there will be ten times as many Gentiles as Jews, who are seeking after the Lord of Israel:
Yea, many peoples and mighty nations shall come to seek HaShem of hosts in Jerusalem, and to entreat the favour of HaShem. Thus saith HaShem of hosts: In those days it shall come to pass, that ten men shall take hold, out of all the languages of the nations, shall even take hold of the skirt of him that is a Jew, saying: We will go with you, for we have heard that G-d is with you. (Zech. 8:22-23)
And Isaiah tells us that God will illuminate these converts and extend his salvation to them:
And now saith HaShem that formed me from the womb to be His servant, to bring Jacob back to Him, and that Israel be gathered unto Him--for I am honourable in the eyes of HaShem, and my G-d is become my strength. Yea, He saith: "It is too light a thing that thou shouldest be My servant to raise up the tribes of Jacob, and to restore the offspring of Israel; I will also give thee for a light of the nations, that My salvation may be unto the end of the earth." (Isaiah 49:5)
We need not stop to argue about whether the "servant" described in this passage refers to Isaiah himself, Israel, or the Messiah (or maybe all three!) The important thing for the moment is that it clearly describes the conversion of the Gentiles to the God of Israel, in fulfillment of God's promise to Abraham that "in thee shall all the families of the earth be blessed" (Gen 12:3).
Another passage (Isaiah 66:18-21) can even be read as saying that God will accept some people from Gentile nations as priests and Levites, which would definitely require a change in the rules, although alternative interpretations of this passage are possible. Or consider Psalm 87, in which people from all kinds of nations—those which were originally hostile to Israel—are recorded as having been born in the city of Zion.
This sort of thing is a pretty recurrent theme in the Prophets (as is the theme of Israel being rebellious for a long period of time but eventually being reconciled to God). More examples could be multiplied to prove this point, but I don't think I need to, since I'm pretty sure it's already a standard Jewish teaching that, in the Messianic Age, the Gentiles will also enter God's kingdom.
When this happens, the divinely stated reason for the kosher laws will no longer apply, and so there is no reason for them to continue.
Some strictly observant Jews might be tempted to counter-argue as follows: it is not for human beings to pronounce judgement on the reasons for God's commandments. God might happen to mention some of his purposes in passing, but regardless of what motivates the commandment, Israel's response should always be unquestioning and unconditional obedience. But that argument seems to presuppose that what God likes best are ignorant slaves, who obey him without knowing the reasons why. If Moses and Jeremiah are right that God wants people who are inspired by his Spirit to want to keep his laws, then acting based on our best understanding of God's reasons is essential.
God does not just want people to go beyond the letter of the law in order to reach its true point (although that is of course highly commendable). Sometimes he even wants the kind of people who break the letter of the law in order to keep its true spirit, as for example when David ate the showbread that was reserved for the priests. (If God were only interested in legalistic obedience, he would never have made David king in the first place, since his great-grandmother Ruth was a Moabitess (Ruth 4:21-22), and in the Torah the descendants of a Moabite were not allowed into the assembly of the Lord down to the 10th generation (Deut 23:3).)
Of course, since the Torah was divinely inspired, nothing in it simply gets discarded or thrown out. The written record remains forever to serve as a moral guidepost and a record of God's dealings with humanity. This is possible even if some commandments stop being followed according to the letter. There is an important difference between abolishing the law, and fulfilling it. The former is like burning up an acorn in a fire, the latter is like planting it and letting it grow into an enormous oak tree. The two are completely different in their degree of respect for the acorn's purpose—but either way, the acorn is gone and it isn't coming back!
So I think I have adequately proven, from the Tanakh alone, that (whether or not Christianity is true) when the Messiah comes we should expect some changes to the commandments. If you agree with that, then that's probably quite enough progress for a single day!
But there is a major issue raised by your comment which I haven't dealt with yet...
III. Following Other Gods?
So far, I have left untouched the gigantic stumbling block of Christ's claims of divinity. Certainly I can see why this is a huge issue for you.
Now Christians claim—and I think the link you cite barely scratches the surface of what you would need to do to evaluate this claim properly—that there are hints throughout the Tanakh that:
(a) God, although he is one, also has some kind of plural aspects within his being;
(b) the Messiah, as the king who reigns forever, will be more than just an ordinary human, but plays some sort of mediatorial role, reconciling human beings to God;
(c) and that God himself is going to somehow dwell with Israel or live among them, in a more intimate way than before, in the Messianic era (despite the fact that other passages speak poignantly of being rejected by Israel).
This is a huge topic and it would take a whole additional blog post to provide all the Scriptures for each point, but these passages are there if you look for them honestly, rather than trying to fit everything into a predefined theology.
Instead, let's cut to the chase and ask whether the claim is precluded outright by the Torah? Between Deut. 18:20-22 (which is right after the passage I quoted) and Deut. 13:2:6 (the passage you made reference to), there are 2 different tests to distinguish true from false prophets. Somebody can be judged a false prophet if they flunk either test.
The first test is to check whether the claimant's prophetic signs come to pass. Jesus predicted that he would rise from the dead after he was crucified, and the historical record strongly suggests that he did! So he passes that test.
The second test (which, as you correctly say, applies even if the prophet performs a sign or wonder) is that if a prophet says "Let us go after other gods, which thou hast not known, and let us serve them", then he is still to be rejected (and executed!).
So this passage does raise a serious issue for Christianity. But at the risk of maybe sounding a bit pedantic, I would humbly submit to you that Jesus did not tell people to turn aside and worship other gods, gods that the Israelites' forefathers had not known. Rather he told people to worship the Father—his title for the God of Israel described in the Hebrew Scriptures—and whenever he made striking claims about himself, he never suggested that he was some additional or separate deity, apart from the Father. (Indeed, in the Gospels Jesus states multiple times that there is only one God and that only God is worthy of worship.) In the passages in which he claims some sort of divine sounding status, he is always claiming to be the somehow part of the same being as the Father.
In these same passages, he often asserts his radical dependency on the Father, to have no independent will or words apart from him. To take one example, after healing a crippled man on the Sabbath day:
And therefore did the Jews persecute Jesus, and sought to slay him, because he had done these things on the sabbath day. But Jesus answered them, "My Father worketh hitherto, and I work".
Therefore the Jews sought the more to kill him, because he not only had broken the sabbath, but said also that God was his Father, making himself equal with God.
Then answered Jesus and said unto them, "Verily, verily, I say unto you, The Son can do nothing of himself, but what he seeth the Father do: for what things soever he doeth, these also doeth the Son likewise." (John 5:16-19)
For this reason, in Christian theology, Jesus is not considered to be a separate god the way that the Greeks considered Zeus and Hera and Athena to be separate gods. We consider him to be the Incarnation in human flesh of the same God who made covenant with Abraham, and who asserted his unique power over Resurrection long ago when he said to Moses:
"See now that I myself am he! There is no god besides me. I put to death and I bring to life, I have wounded and I will heal, and no one can deliver out of my hand." (Deut 32:39)
The claim that Jesus is the meaning of the Torah, is not one that can be assessed simply by taking a giant list of claimed Messianic prophesies, and asking whether there is any way to interpret them in isolation, such that they agree (or don't agree) with similarly isolated New Testament passages.
No, it is a holistic judgement involving thinking carefully about the whole thing. It requires meditating on what is God trying to do in the Hebrew Scriptures? And, where is his Spirit leading you as you try to follow his commandments to love him with your whole self, and your neighbor as yourself? And then asking whether it the same thing as what God is portrayed as doing in the New Testament. Jesus himself says:
If I am not doing the works of My Father, then do not believe Me. But if I am doing them, even though you do not believe Me, believe the works themselves, so that you may know and understand that the Father is in Me, and I in the Father.” (John 10:37)
In any case, one cannot avoid the conclusion that, false prophet or true, so far it is Jesus whom God has used as his primary vehicle for spreading the message of the Hebrew Scriptures among the Gentiles, just as the prophets said would happen. I didn't come to Christianity by conversion from Judaism (although my best friend did); I'm a Gentile. If it weren't for Jesus, I wouldn't even be arguing with you about the Torah! I guess I'd be off in some forest somewhere, sacrificing to a pagan god.
Of course I am aware that there have been plenty of false Messianic claimants (although giving up on the claim in order to convert to Islam does seem like a pretty convincing refutation). False Messiahs are a dime-a-dozen. What I can't imagine though, is how there could be a Real Messiah that is better than Jesus, who showed us how the path of love is strong enough to conquer even the grave. To me it is obvious that Jesus is deeply good, and that his healing grace and power are in continuity with the best that can be found in Hebrew Scripture.
[Update 8/30/16: In the 5th to last paragraph of section II, I edited "best guess" to "best understanding", for reasons described in the comments section.]
In 2019, I will be studying quantum gravity and black hole thermodynamics as a Lecturer at the University of Cambridge. Before that, I read Great Books at St. John's College (Santa Fe), got my physics Ph.D. from U Maryland, and did my postdocs at UC Santa Barbara, the Institute for Advanced Study in Princeton, and Stanford. The views expressed on this blog are my own, and should not be attributed to any of these fine institutions.
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### 32 Responses to Is God allowed to update the Torah?
1. JPH says:
So I think the author would have been better off engaging in some work of serious Christian scholarship, rather than some random tract he got in the mail.
Let’s look at William Craig on the subject. http://www.reasonablefaith.org/defenders-2-podcast/transcript/s6-21
“When you look at the Messianic prophecies in the Old Testament, they give virtually no clue that Messiah isn’t going to be this triumphant warrior king that was expected. This is what was supposed to happen. … [T]here wasn’t any clear concept in the Old Testament that Messiah would come and be humiliatingly executed as a common criminal rather than establishing the throne of David in Jerusalem and throwing off the enemies of Israel and the Gentiles and the Jews would be in submission to this great warrior king, who would be like a new David. That was what I was saying you don’t find in the Old Testament.”
Right. You find no such thing. You find this: https://yourphariseefriend.wordpress.com/2010/12/21/letter-to-sy-about-messiah/ The Messiah will build the Third Temple, gather all Jews back to Israel, usher in an era of world peace, and spread universal knowledge of HaShem. If he doesn’t, he’s not the Messiah. Performing miracles doesn’t change this.
Isn't it clear that any New Covenant, just by virtue of being New, must necessarily imply some sort of changes to the old way of doing things?
The covenant specified in Jeremiah states we won’t have to debate these issues anymore. Everyone will simply know them. The fact that most Jews aren’t Torah observant and the earth is teeming with false religions proves we’re not there yet. I know you’re making the case for the transitory nature of Torah, but you’re citing verses that are hostile to Christianity (Like Zech 8:22-23. It must be a Messianic Jew that’s mentioned. Why don’t the ten Gentiles just read Calvin? This is a prophecy about the Noachide movement.)
God might happen to mention some of his purposes in passing, but regardless of what motivates the commandment, Israel's response should always be unquestioning and unconditional obedience. But that argument seems to presuppose that what God likes best are ignorant slaves, who obey him without knowing the reasons why. If Moses and Jeremiah are right that God wants people who are inspired by his spirit to want to keep his laws, then acting based on our best guess at God's reasons is essential.
False disjunctive syllogism: either “ignorant slaves” or “best guesses.” Neither Moses nor Jeremiah said anything about the latter. Jeremiah wrote of a future time when all men will naturally follow God’s laws. In Psalm 119, the longest chapter in the Bible, does David lament his plight at being an ignorant slave? And I shall keep Your Torah constantly, forever and ever (verse 44). Your righteousness is perpetual righteousness, and Your Torah is true (verse 142). I yearned for Your salvation, O Lord, and Your Torah is my occupation (verse 174). David yearned for salvation. To attain it he follows the Torah. Equivalence is drawn between the Torah and perpetual righteousness. A revelation that is temimah (faultless, whole, and complete) precludes paradigm shifts.
The first test is to check whether the claimants prophetic signs come to pass. Jesus predicted that he would rise from the dead after he was crucified, and the historical record strongly suggests that he did! So he passes that test.
That’s not the only thing he predicted. Matthew 24:30-34, Mark 13:24-30, Luke 21:25-32 are notorious. If “this generation” doesn’t mean this generation then you have a position that’s irrefutable in principle. (And he said he’d rise after three days. Friday afternoon to Sunday morning is 1.5 – not that this is even a Messianic prophecy. Jonah is about Gentiles repenting en masse without sacrifices of any kind!)
To me it is obvious that Jesus is deeply good, and that his healing grace and power are in continuity with the best that can be found in Hebrew Scripture.
Just like we disagree about the meanings of “eternal statute,” “HaShem is one,” and “this generation,” the meaning of “deeply good” is open to debate. Christianity threatens eternal torture for those who have the wrong beliefs. You just devoted thousands of words to a Bayesian analysis of something that happened 2k years ago. How can a just and merciful father expect his (mostly stupid, me included) children to crunch all this data and ace the test?
2. Mactoul says:
"The Messiah will build the Third Temple, gather all Jews back to Israel, usher in an era of world peace, and spread universal knowledge of HaShem. If he doesn’t, he’s not the Messiah"
But haven't all these things being accomplished? and the list of Messiah claimants is enough by itself to convince us of Jesus' claims.
3. Mactoul says:
JPH,
"Why do Christians think the resurrection cancels/changes the Torah? "
The Christians believe in and hold Torah in respect only because of the Church. The Church and the Resurrection that brought forth the Church is primary and the Torah is secondary. Otherwise, the Torah and other Old Testament books are collections of Hebrew fables of little interest to others.
4. Aron Wall says:
JPH,
You raise a lot of issues in your comment, so please forgive me for responding at length.
It will not be a surprise to any long term readers of this blog that, despite our shared faith in Jesus, I feel quite free to disagree with William Lane Craig sometimes, when I think he is taking the wrong approach. This is one of those times. In fact there are so many references to fulfilled prophecies in the New Testament that some people try to argue that the Gospels were just constructed based on Jewish expectations about the Messiah. St. Craig is trying to argue against the view that the disciples just made up the Resurrection, but I think his approach is deeply misguided here. Yes, the disciples did not expect the Messiah to suffer rise from the dead until after he did so, but once you know where to look, these passages of Scripture really are there. So I do not accept his viewpoint there. Instead I agree with what St. C.S. Lewis wrote, when he said (in Reflections on the Psalms):
In a certain sense Our Lord's interpretation of the Psalms was common ground between Himself and his opponents. The question we mentioned a moment ago, how David can call Christ "my Lord" (Mark 12:13-37), would lose its point unless were addressed to those who took it for granted that the "my Lord" referred to in Psalm 110 was the Messiah, the regal and annointed deliverer who would subject the world to Israel. This method was accepted by all. The "scriptures" all had a "spiritual" or second sense. Even a gentile "God-fearer" like the Ethiopian eunuch (Acts 8:27-38) knew that the sacred books of Israel could not be understood without a guide, trained in the Judaic tradition, who could open the hidden meanings. Probably all instructed Jews say references to the Messiah in most of those passages where our Lord saw them; what was controversial was His identification of the Messianic King with another Old Testament figure and of both with Himself.
Two figures meet us in the Psalms, that of the sufferer and that of the conquering and liberating king. In 13, 28, 55, or 102, we have the Sufferer; in 2 or 72, the King. The Sufferer was, I think, by this time generally identified with (and may sometimes have originally been intended as) the whole nation, Israel itself---they would have said "himself". The King was the successor of David, the coming Messiah. Our Lord identified himself with both of these characters."
So it is true that Christians accept (in addition to all the passages accepted by Jews) some additional Messianic prophecies as well. But even first century Jews would have read these passages as having an allegorical or second meaning. So this interpretation is not all that far outside the traditional methods of Jewish interpretation.
Let us suppose for the sake of argument that the Suffering Servant passages refer to Israel. Well, a king should be a representative of his own people (see Deut 17:15), and therefore shouldn't he understand their most important experiences, including their suffering? Certainly that was how it was with King David, as it is written: "HaShem, remember unto David all his affliction" (Psalm 132:1). Israel is continually the target of unjust persecutations, so since the Messiah is to be the King of Israel, then he must also go through the experience of unjust persecution.
And he said he’d rise after three days. Friday afternoon to Sunday morning is 1.5
In the ancient Jewish way of counting, "three days" or the "third day" generally refers to an inclusive counting that counts the start day as 1 (unlike Westerners who count the start day as 0). So Friday is the 1st day, Saturday is the 2nd, and Sunday is the 3rd.
This is the most natural interpretation of 14 out of the 15 passages in the Gospels in which Jesus is said to have predicted the time of his Resurrection in advance. The only somewhat problematic passage is Matthew 12:40, where Jesus compares himself to Jonah and mentions "three days and three nights". There are various possible resolutions to this apparent discrepency---but I don't think this can be a dealbreaker unless you can say to me with a straight face that there are no apparent contradictions in the Torah or Tanakh that require subtle reconciliation... ;-) To me, the impressive part is rising from the dead, not the exact timing! (There is a Jewish midrash that the reason why the Israelites built the golden calf was a mistake about what exactly Moses meant by 40 days! Here God wants to offer covenant with him, and we get distracted by our own off-by-one errors)!
I agree that "this generation" most naturally refers to the lifespan of those who heard Jesus' words originally. I would explain the apparent discrepency as follows: St. Matthew tells us that, on this occasion, the disciples originally asked him two questions (numbers inserted by me):
“Tell us,” they said, “(1) when will this [i.e. the Destruction of the Temple] happen, and (2) what will be the sign of your coming and of the end of the age?” (Matt 24:3)
Since these two events both involve God's judgment, Jesus talked about them together, but he gives two separate answers as to timing (which would seem somewhat inconsistent if taken to refer to the same event):
(1) "Truly I tell you, this generation will certainly not pass away until all these things have happened." (Matt 24:34) [which is true about the Destruction of the Temple]
(2) "But about that day or hour no one knows, not even the angels in heaven, nor the Son, but only the Father." (Matt 24:26) [which is true about his Second Coming]
Elsewhere (especially in the parables) Jesus emphasizes that his return will take place after an unexpectedly "long time". So I think the best explanation of this data is that Jesus gave the correct timing for both questions, but the disciples who remembered this speech didn't clearly separate out the two answers, and so now they stand together in a somewhat confusing way.
You are now entitled to object, if you can say with a straight face that the prophecies in the Tanakh never talk about near and far events at the same time, in a way that seems to be confusing and jumbled together. ;-)
I know you’re making the case for the transitory nature of Torah, but you’re citing verses that are hostile to Christianity (Like Zech 8:22-23. It must be a Messianic Jew that’s mentioned. Why don’t the ten Gentiles just read Calvin? This is a prophecy about the Noachide movement.)
How is this verse hostile to Christianity? Today, the Noachide movement is tiny fraction of the Gentile population. But it was not always so. At the time of the Roman Empire there were a substantial number of "God-fearers", a technical term for Gentiles who accepted the Hebrew Scriptures without converting to Judaism.
So what happened to all of them? Well, most of them converted to Christianity, based on the preaching of the earliest Christians, who were of course Jews who believed that the Messiah had come. So that is the fulfillment of Zech 8:22-23, and it happened a long time before St. Calvin came along!
The covenant specified in Jeremiah states we won’t have to debate these issues anymore. Everyone will simply know them. The fact that most Jews aren’t Torah observant and the earth is teeming with false religions proves we’re not there yet.
Let's suppose you were right that in the Messianic Era, everyone will know God's laws and obey them naturally because they want to. (Similar to what Christians say will happen in the New Heaven and New Earth.) Then, since people have stopped sinning, is it not obvious that there cannot any longer be any Temple sacrifices for sin? Because, the reason for them being offered (people sinning) would never happen. So on your own premises it follows that certain commandments in the Torah will become out of date, never to be used again, once we reach the Messianic Era! You can't radically change everything about the world (removing war, death, the existence of other religons etc.) and still say that a set of commandments written for a culture in c. 1500 BC, which presuppose the existence of these things, still apply exactly as written!
And you seem to be assuming here that every aspect of the Messianic era has to come at the exact same time. But this seems like a very dangerous assumption given that prophecy in the Jewish scriptures has a tendency to be talking about multiple times at once. (I gave some examples of this in my main post.) Leaving aside the various prophecies of his birth, life, death, and resurrection, I have already pointed to two specific Messianic Era prophecies that were fulfilled by Jesus: (a) Mass conversion of the nations, (b) a New Covenant which includes the giving of the Holy Spirit to change the relationship between God and Man. These events are described by the Prophet Joel:
And it shall come to pass afterward, that I will pour out My spirit upon all flesh; and your sons and your daughters shall prophesy, your old men shall dream dreams, your young men shall see visions; And also upon the servants and upon the handmaids in those days will I pour out My spirit. And I will shew wonders in the heavens and in the earth, blood, and fire, and pillars of smoke. The sun shall be turned into darkness, and the moon into blood, before the great and terrible day of HaShem come. And it shall come to pass, that whosoever shall call on the name of HaShem shall be delivered. (Joel 2:28-32)
When it says that the Holy Spirit will be poured out on "all flesh", this seems to mean a large number of people of every condition (regardless of their gender, age, social class, or ethnicity). But, because he goes on to specify that God will deliver those who "call on the name of the Lord", the text seems to imply the existence of other people who do not call on his name. Thus, when the Holy Spirit is first given, it seems that there can remain (for a time) false religions, confusion, and some wicked people.
Now, as you know, during the time of the Tanakh, the Spirit of God was not given to everybody. It was only given to very special people like prophets, heros and kings, who by virtue of their office had a personal relationship with God. For example, King David received the Spirit starting from the time that he was anointed by Samuel (1 Sam 16:13, Psalm 51:11).
Since Judaism no longer has prophets or kings, that kind of personal relationship with God is not available in Rabbinic Judaism. But Christians receieve the Holy Spirit when they believe in Jesus, which means that each one of us, no matter how ordinary, has the Holy Spirit inspiring us to love God and to keep his commandments. For this reason, the New Testament teaches that we are God's Temple, that what God really wants is to dwell in the human heart, not in an inanimate building (cf. Isaiah 57:15).
My best friend Yoaav (named after David's military commander) converted to Christianity, in part because the personal relationship with God that Christians have, seemed so much more like the ideal of the Hebrew Bible, compared to the indirect, impersonal relationship with God which is all that Rabbinic Judaism can provide. (It was a very difficult step for him to take because he had good reason to think that his father might disown him---although happily this did not occur!)
I love Psalm 119, and I'm glad you do as well. But, if you do not have the Holy Spirit living in your heart, your relationship with God and his commandments cannot possibly be the same as that of King David, because that was a critical aspect of his relationship with God.
Regarding "best guesses", that was a very misleading phrase on my part (and I have edited it to "best understanding" in the main post). My argument---which I don't think you've otherwise responded to---was that we can deduce from the text of Leviticus that kosher laws were given for a particular purpose, and since it seems like that purpose would no longer apply during the Messianic age, even a practicing Jew might reasonably guess that the kosher laws will be abolished at that time.
However, now that the Messiah has in fact come, Christians do not need to "guess" about what God requires from us, because we have the Holy Spirit who guides us concerning the heart of God, as we love each other. (He may leave us to guess about the lesser issues of theology, perhaps to test to see if we can still love each other even when there are theological disputes, but to everyone who really who seeks him, he clarifies the meaning of the most important commandments, which are to love God and other people, as they come up day-by-day in our own lives.)
A revelation that is temimal (faultless, whole, and complete) precludes paradigm shifts.
God told Abraham to "Walk before Me and be perfect." (Gen 17:1) Yet that did not preclude a paradigm shift (Mount Sinai) at the time of Moses. Everything that God does is perfect and beautiful, in its time. (How could it be otherwise?) But that does not mean that new revelation cannot occur in the future. Especially given all those prophecies I mentioned which explictly state that this will happen!
Finally, the issue of final judgement. You write:
Just like we disagree about the meanings of “eternal statute,” “HaShem is one,”
Have you done a word study on the Hebrew words translated "eternal" and "one"? That would involve looking at all the other places in Scripture the same Hebrew word appears, to see whether the possible range of meanings might be.
and “this generation,” the meaning of “deeply good” is open to debate. Christianity threatens eternal torture for those who have the wrong beliefs. You just devoted thousands of words to a Bayesian analysis of something that happened 2k years ago. How can a just and merciful father expect his (mostly stupid, me included) children to crunch all this data and ace the test?
The doctrine of eternal rewards & punishments is also found in the Tanakh (as acknowledged by classical rabbinic Judaism), most clearly in this passage:
And many of them that sleep in the dust of the earth shall awake, some to everlasting life, and some to reproaches and everlasting abhorrence. And they that are wise shall shine as the brightness of the firmament; and they that turn the many to righteousness as the stars for ever and ever. (Daniel 12:2-3)
And New Testament also teaches that there will be a Resurrection of the righteous and the wicked, at which time everyone will be judged according to their works:
“Do not be amazed at this, for a time is coming when all who are in their graves will hear his voice and come out—those who have done what is good will rise to live, and those who have done what is evil will rise to be condemned. (John 5:28-29)
"I believe everything that is laid down by the Law and written in the Prophets, and I have the same hope in God that they themselves cherish, that there will be a resurrection of both the righteous and the wicked." (Acts 24:14-15)
"God `will repay each person according to what they have done.' To those who by persistence in doing good seek glory, honor and immortality, he will give eternal life. But for those who are self-seeking and who reject the truth and follow evil, there will be wrath and anger. There will be trouble and distress for every human being who does evil: first for the Jew, then for the Gentile; but glory, honor and peace for everyone who does good: first for the Jew, then for the Gentile. For God does not show favoritism." (Romans 2:6-11)
It is a fundamental teaching of the Bible that God will judge the world with perfect justice. So, it follows that God will not send anyone to Hell who is honestly mistaken about Jesus and just trying to serve God the best they can. Jesus himself said that the people who spoke against him would be forgiven, and that the only unforgiveable sin is to resist the Holy Spirit when he makes the truth clear to you. I, looking on the outside, cannot tell whether the Spirit is testifying to your heart or not. I pray that he is, not because I want you to be condemned but because I want you to know the glories of the riches of his presence in your life, that are symbolized by the various provisions of the Torah.
So what about all those passages in the New Testament talking about the importance of faith in Jesus? It does not mean, as some ignorantly think, that God arbitrarily throws people in Hell based on whether they identify as Jew or a Christian! Rather it means, that God is offering through Jesus the chance to be transformed from an selfish person to a loving person, and so be found righteous on the last day. That, we should not think of salvation as something that we can "earn" by following a set of rules and doing good deeds, but rather as an act of love and mercy whereby God forgives repentant sinners, and restores them to fellowship.
So when properly understood, I don't think that Christianity shows salvation to be the results of passing some kind of doctrinal test, even though what we believe is obviously important insofar as it determines what kind of person we are. I'm a mathy person so I've chosen to dress up some historical facts I consider pretty obvious in mathematical language, but it doesn't matter whether you want to think about it in that way!
It does matter what you think about Jesus, but I can only try to communicate the best ideas I have about what holiness means (doubtless I have also been stupid about many things) and leave God to fill up the deficiencies in my presentation. May you be blessed.
Well, that was a very long comment and I don't expect you to necessarily have time to respond to all of it! But I would like to hear your answers to these very specific questions:
1. Do you believe that some commandments are more important than others?
2. Do you agree that passages such as Jer 3:16 (and the others I mentioned) imply that at least some Torah observances will not be followed in the Messianic Age?
3. Do you accept my interpetation of Lev 20 (that God says he is giving the kosher rules in order to keep Israel separate from the other nations)?
4. Do you think that in the Messianic Age, there will be any need for a separation between Jews and Gentiles?
5. Do you believe that God wants everyone to be filled with the Holy Spirit? Do you believe that you, as an individual, have the Holy Spirit living inside of you? If not, is this something that you earnestly desire and hope for?
5. JPH says:
1. Do you believe that some commandments are more important than others?
I do, but I can’t speak for God. The 7 Noachide laws are binding on all humans. It’s not a “religion” in the sense most people mean by that word. It’s not Judaism Lite or Judaism for Gentiles. It’s moral realism grounded in monotheism, six categories of prohibitions and the obligation to establish courts of law that enforce them. https://asknoah.org/7-commandments/locate-sources
However, now that the Messiah has in fact come, Christians do not need to "guess" about what God requires from us, because we have the Holy Spirit who guides us concerning the heart of God, as we love each other.
Note the obvious problems with relying on an inner voice. How do we adjudicate between groups who claim to be inspired? The Thirty Years’ War is far from the only counterexample. Christians find themselves in the position of wondering if they’re really saved or born again (like Luther). How does one know if she’s experiencing this singular experience? Compare this with wondering if you’re engaging in idolatry, blasphemy, theft, murder, adultery, or cruelty to animals. The latter are obvious, with no need for interpretations of inner states.
2. Do you agree that passages such as Jer 3:16 (and the others I mentioned) imply that at least some Torah observances will not be followed in the Messianic Age?
One of the 13 principles of Judaism is the immutability of the Torah. In addition to any instrumental meaning, the commandments contain some intrinsic purpose. The debate between Rambam & Ramban is outside my salary grade: https://www.ou.org/jewish_action/12/2014/whats-truth-korbanot/
3. Do you accept my interpetation of Lev 20 (that God says he is giving the kosher rules in order to keep Israel separate from the other nations)?
In a manner of speaking, ALL of the commandments (above and beyond the Noachide 7) are given to Israel for this purpose. They are a nation of priests. Gentiles have the same moral realism Noah did. God is not a multiculturalist. The arrow of History soars toward a time when all Gentiles will follow them and all Jews will observe the Torah. It won’t be a time of miracles: http://www.chabad.org/library/moshiach/article_cdo/aid/101744/jewish/Laws-Concerning-Kings-and-the-Messiah.htm
5. Do you believe that God wants everyone to be filled with the Holy Spirit? Do you believe that you, as an individual, have the Holy Spirit living inside of you? If not, is this something that you earnestly desire and hope for?
"Fear God and keep his commandments, for this is the whole duty of man." God is not a person or anything comparable to things in my everyday life. The via negativa of Maimonides is an essential corrective to personal conceptions: http://plato.stanford.edu/entries/maimonides/#GodViaNeg
Regarding Psalm 110:
The question we mentioned a moment ago, how David can call Christ "my Lord" (Mark 12:13-37), would lose its point unless were addressed to those who took it for granted that the "my Lord" referred to in Psalm 110 was the Messiah, the regal and anointed deliverer who would subject the world to Israel.
In Hebrew, the 'Lord’ and ‘lord’ from Psalm 110 are two different words. Capitalizing both is misleading (to put it gently). The first is the Tetragrammaton. The second refers to people (Genesis 32:4 says “… speak unto my lord Esau”). David made preparations for the temple, including the services. The Psalms were songs intended for the Levites. “The Lord (God) said to my master (King David) ‘Sit thou at my right hand…’" makes sense when someone else is singing it. https://jewsforjudaism.org/knowledge/articles/answers/jewish-polemics/texts/psalm-110-a-jewish-perspective/
So what happened to all of [the God-fearers]? Well, most of them converted to Christianity, based on the preaching of the earliest Christians, who were of course Jews who believed that the Messiah had come. So that is the fulfillment of Zech 8:22-23.
If and only if Christianity is the true religion, otherwise this verse remains unfulfilled. You can’t cite it as evidence for Christianity.
6. Robert Childress says:
Aron, I had a question about your interpretation of Matthew 24:3. Does it make sense to think that the "coming" inquired about by Jesus's audience would have been understood as a "second" coming? Was his audience expecting a departure followed by a return?
I've heard some others interpret the "coming" to be a question about the kingdom (and judgment) of God -- in terms of Danielesque language of "coming" with the clouds of heaven, and so on. If so, it's perhaps arguable that this did happen within the lifespan of that generation (in a "now but not fully yet" sense) with the death and resurrection of Jesus, the pouring out of the Holy Spirit, the destruction of the temple, and so on. I think it's a kind of preterist view.
For what it's worth, I found this post and some of the ensuing comments quite clarifying with regards to Christian claims of fulfilled prophecies, so I appreciate the discussion.
7. Robert Childress says:
JPH, I think the point about Psalm 110 has to do with the implied double interpretations of passages of Hebrew Scripture more so than the particular word used for Lord. I don't doubt that Jesus and his audience in Mark 12:35-37 would have had a conventional interpretation of what the meaning was in its historical context -- and perhaps it was much like what you've explained. But the conclusion reached by Jesus would have been incomprehensible to his audience unless there was also a second kind of interpretation commonly understood to be running alongside it -- namely, that the "my lord" with the little 'l' was understood by Jesus's audience to be a reference to the Messiah.
Or to put it another way, with a divinely inspired human work such as the Hebrew Scriptures David may well have been authoring Psalms for the Levites to sing and God may well have been using those Psalms to tell us something about the Messiah. That appears to have been the understanding of Jesus's audience, at least.
8. Aron Wall says:
JPH,
Thanks for giving your answers to my questions. I appreciate it. Some very brief replies to each point:
1. I can understand why a nonreligious person might be skeptical of an "inner voice" (if that is really the right way to think of the Holy Spirit, see here), but inspiration by the Holy Spirit is already part of the Jewish religion, mentioned in the Tanakh. And there we see that even those who have the Holy Spirit sometimes sin or do stupid things (e.g. Jephtheh's rash vow that resulted in sacrificing his daughter, David committing adultery and murder, etc.) So if it turns out that Christians sometimes sin even when they have the Holy Spirit, well this is compatible with the Tanakh. It is also possible to say that any supposed Christian who hates and despises other Christians (or Jews!) proves that they never had the Holy Spirit (see 1 John 4:20).
Anyway, it is the Tanakh that says that the Holy Spirit providing inner guidance is a characteristic of the Messianic Age. If you have a problem with that, you should take it up with God, not with me!
2. If even Rambam, the formulator of the 13 principles, couldn't agree with himself about whether there would be animal sacrifices in the age to come---because there are passages which seemingly point both ways---then it seems that both opinions must be regarded as legitimate interpretations of Judaism, and when the Messiah comes it will become clear which is the case. If you are unwilling to take one side in this dispute, then you should be open to the possibility that you are wrong about what the "immutability of the Torah" implies.
3. Your link seems to be arguing that because a particular false Messiah (Bar Koziba) did not perform miracles, and a certain respected sage (Rabbi Akiva) followed him, that therefore the true Messiah will not perform miracles? REALLY???? A deluded person following a deluded person takes precedence over all the passages in the Prophets about the Resurrection of the Dead and so on?
Also, a priest is appointed for the sake of the people, not the other way around. God's promise to Abraham, that all nations would be blessed, shows that the purpose of choosing Israel was to benefit all people. So all the nations returning to God is not some kind of side-note, it is the main kaboom. God is a multiculturalist.
5. God is indeed very different from a person, or any created thing, but it is still possible to have a relationship with him as all the Scriptures testify.
I am of course aware of the distinction between the Tetragrammaton and the term Adoni, and the presence of both terms in Psalm 110. (Had I been explaining this Psalm myself, instead of quoting somebody else, I expect I would have mentioned this explicitly.) But the important point in the Lewis quote is that, in the time of early rabbinic Judaism, it was uncontroversial that the second "Lord" [Adoni] refers to the Messiah. This was based on the fact that the psalm was written by David, and that he says "my Lord" (singular) instead of "our Lord". As well as the fact that the rabbis accepted that almost every passage in Scripture had some Messianic implication, as I've explained above. (Robert also makes some good points.)
If and only if Christianity is the true religion, otherwise this verse remains unfulfilled. You can’t cite it as evidence for Christianity.
Because the meaning of most prophecies is ambiguous, it is always going to be possible to interpet any given passage in an anti-Christian way (especially when one interprets them with the aid of oral traditions not found in the Tanakh, some of which even developed after Christianity, and therefore partly in response to it).
So, whenever you come to a passage that can be interpeted in more than one way---or which might or might not have been fulfilled, depending on one's beliefs about other matters---you need to hold both possibilities in your mind until making your final determination based on all of the data. That isn't circular reasoning; it's just fair-mindedness.
If a prophecy has multiple possible interpretations---but then through a very suprising turn of historical events, one of those intepretations gets fulfilled---then that is itself evidence that that is the correct intepretation. For, if it were not the correct interpretation, you wouldn't have expected the surprising set of events to happen.
Robert,
No, I don't find that kind of preterist intepretation of Matthew 24:3 particularly plausible. Too much of the passage sounds like the final judgement. And, the parables that refer to the master being away on a long journey have always been interpreted by the Church as being about the Second Coming (aside from a few moderns like St. N.T. Wright).
Nor is it necessary to suppose that the Jewish audience had any kind of expectation of a departure followed by a Second Coming. As I have been arguing to JPH, it is allowed for the teaching of Jesus to contain new information. He wanted to make sure that his disciples knew about his departure and return.
On the other hand, I think that Mark 9:1, Matt 16:28 and Luke 9:27 DO refer to the Resurrection and Ascension of Jesus (which confirmed his kingship) and to his founding of the Church (which involved the coming of the Holy Spirit with power).
9. Aron, you and JPH have had a very interesting discussion. Here are a few additional, hopefully divergent, comments. I’m sure I’ll inadvertently be repeating some of your views unless I notice them and excise them in time. Some of my views might be seen as a bit heterodox (isn’t that a much nicer word than heretical?), so I may be hearing back more from Aron than JPH. Independent quotes are all JPH’s.
“God appeared to a nation and gave them 613 commandments. He said they were eternal, everlasting, binding for all generations. There is NOT ONE about worshiping God's son or the Messiah.”
Even if they are eternal and binding, it does not follow that God could not give any new commands or information.
“The thirteenth chapter is devoted to prophets who can perform ‘signs and wonders’ and advocate the worship of gods ‘whom your forefathers did not know.’ Their forefathers did not worship Jesus.”
Deuteronomy 13 is telling them to worship no one other than Ha Shem. What Jesus and his Jewish followers claim God had revealed is that Jesus is God and yet he is in some sense distinct from God such that he can address God and speak of God as his father. The most feasible way to understand this is to see that God was more than one in person but only one in being (one in certain shared attributes like knowledge, will, emotion, and power). This is something the Torah has not revealed because it was not yet God’s time to reveal it. This claim contradicts nothing stated in the Torah. It “changes” or “cancels” nothing in the Torah. It accords with the Shema, “the Lord your God is one God.” One (echod) does not indicate absolute unity any more than Adam and Eve’s unity (they “became one [echod] flesh”) means they were indivisibly one. So we have no evidence that Jesus is different from the God of Abraham, Isaac, and Jacob. We have no grounds to believe that Jesus is a God “whom your forefathers did not know.”
“Why do Christians think the resurrection cancels/changes the Torah?”
The resurrection does not cancel or change anything in the Torah. It’s just miraculous verification of Jesus’ claimed identity just as Moses’ miracles (parting the Red Sea, the plagues of Egypt, etc.) verified his identity as God’s chosen leader. We have good evidence for Jesus’ resurrection. I doubt that you can come up with any evidence for Tzvi’s claimed miracles.
All this talk about Jesus’ deity or God’s triune nature might lead us to overlook an important point. Those who follow Jesus as their Lord (master, adon) and savior do not need to accept that Jesus is God incarnate to be accepted by God. I accept the deity of Jesus because I have good reason to accept his teachings and those of his immediate followers and I have good reason to think this claim of deity is included in those teachings. But it was at least a decade after the crucifixion before his followers believed this. Were those like Stephen and possibly James (Ya’akov) bar Zavdai who were killed early on not really Christians because the full teaching of Jesus’ deity may not have been yet known? Furthermore, those teachings say only that one must trust in Jesus and his atoning death to be accepted by God, not that one must believe he is God incarnate to do so.
“Whatever the prophets were saying, it wasn't to subtract from theses books and approach God via some unheard-of intermediary.”
I agree that the prophets never negated anything in the Torah. But Jews have always come to God through an intermediary. This is not “unheard-of” in the Torah, this permeates and is central to the Torah. Animal sacrifices were needed to come to God. (“It is the blood, by means of the life which is in the blood, that makes atonement,” Lev 17.11.) Sin had to be covered, or carried away (Lev 16), before one could be accepted by God. Isaiah 53 speaks of one who will take away the sins of the people in terms of sacrificial suffering and even death (v.8, 12). The meaning of the animal sacrifices is fulfilled in the suffering Servant of Isaiah 53. This cannot be Israel since “he” atones for “us.” Who can Israel be but those who are atoned for? Israel has never been sinless and thus can never qualify as a sacrifice without defect or blemish. (Note also, how can the Servant who has died be rewarded by God [v.12] unless he was resurrected?) The New Testament (NT) writers claimed and gave evidence that this prophecy and the messianic prophecies pointed to him.
Aron cited Deuteronomy 18 which says that anyone claiming to be a prophet must have their prophecies come to pass. The people must judge the claimed prophet as a true or false prophet by the fulfillment of the prophecies. The same must also apply to the greatest of the prophets, the Messiah. But you can’t do that by just listening to the rabbis and traditions which reject any prophetic claimant. One must hear the claims and the evidence those claimants and their followers give.
“The Messiah will build the Third Temple, gather all Jews back to Israel, usher in an era of world peace, and spread universal knowledge of HaShem. If he doesn’t, he’s not the Messiah. Performing miracles doesn’t change this.”
Sure, Messiah does have to do this. But when does he have to do this? Immediately upon his first disclosure of his identity? Of course not! If these make up only one facet of the prophecies he has to fulfill, then this facet, say at least the peace and rebuilding, need not come until other prophecies and steps are first fulfilled. There could even be a separation in time between his appearances on earth to fulfill different tasks. There is nothing in the Torah or the prophets that says he couldn’t. Merely proclaiming loudly that if he hasn’t fulfilled it all the first time he came that he can’t be the Messiah doesn’t make it so—it doesn’t provide any evidence that he is not the Messiah. In fact, spreading the knowledge of God to all peoples has been occurring for centuries and the larger regathering of the dispersed Jewish people to Israel for over a century. So we have no good reason to think Messiah has not fulfilled that part of the prophecy as well.
Craig doesn’t actually say that there is no suffering Messiah in the Hebrew scripture. It’s there but the Jewish thinking at Jesus’ time virtually ignored it, and for good reason. It was hard to reconcile with the conquering Messiah passages. Only a Messiah who would suffer and die but who would come back in the clouds to conquer could fit all of the prophecies. Some attempts were made to reconcile the suffering and ruling passages. For example, some postulated a Messiah ben Joseph (who would suffer) and a Messiah ben David (who would conquer and rule).
“Matthew 24:30-34, Mark 13:24-30, Luke 21:25-32 are notorious. If ‘this generation’ doesn’t mean this generation then you have a position that’s irrefutable in principle.”
Gleason Archer argued that “genea,” generation, more likely means race in this context. The Greek more often means generation, but the Aramaic term Jesus likely used more often means race than does genea. Just because someone can’t refute Christianity with this passage doesn’t mean other statements are not available which one might be able to falsify. So this answer does not make Christianity “irrefutable in principle.”
“Not that this [Jesus’ claim that he would rise in three days] is even a Messianic prophecy. Jonah is about Gentiles repenting en masse without sacrifices of any kind!”
Jesus never said it was a messianic prophecy. He just said that he would be in the earth for this long just like Jonah was in the fish that long.
“The meaning of ‘deeply good’ is open to debate. Christianity threatens eternal torture for those who have the wrong beliefs.”
The NT and the Hebrew scripture both clearly teach that God will be absolutely just (Gen 18.25). That obviously involves a time of punishment for some people. Whether it’s eternal or whether the nature or degree of that punishment will always be the same if it is eternal or whether it will end for some in annihilation are issues Christians differ about. So in fact you are attacking a straw man. It isn’t clear that the NT teaches eternal, conscious, never diminishing torment.
And Christianity does not say people are condemned for merely having wrong beliefs. People are condemned for rejecting what they know is right and true. Cephas said God showed him that God has no favorites and that all who fear or honor God and seek to do what is right will be accepted by God (Ac 10.34-35; cf. Ecc 12.13, the passage he is essentially repeating). He also said that only through Jesus could one be accepted by God (Ac 4.12). Yohanna said that those who knowingly reject Jesus will be condemned (Jn 3.18). Saul said that all, whether they’ve heard of Jesus or not, will find God if they seek God (Ac 17.27). And Jesus said that anyone who wills to do God’s will shall know that his teachings are true (Jn 7.17). Put all together, anyone who fears and honors God or seeks God (whether they know God is there or not), wills to do God’s will, and seeks to live according their natural moral knowledge will find and be accepted by God. But they will also find that Christianity is true and accept it (so far as they continue to will to do God’s will).
So according to Jesus and the NT writers, if we will to do God’s will, we will find out that we need to trust in Jesus to be accepted by God. If we knowingly reject God’s will and reject this Jesus, shouldn’t we expect God’s judgment?
Aron’s work with Bayesian analysis merely gives us evidence in a way it was not seen before. Other sufficient evidence has been available to us in the past. Everyone who seeks God will find whatever information they need to believe in Jesus. Out of God’s mercy, some even find it who do not seek God.
10. JPH says:
Even if they are eternal and binding, it does not follow that God could not give any new commands or information.
How could they be eternal and binding AND, per Hebrews 7:18, “weak and useless”? Where in the Torah does Paul receive the authority to say such things? And how can God’s revealed law about divorce in the Torah remain “eternal and binding” if Jesus reverses it? This is like saying both X and not X are true.
What Jesus and his Jewish followers claim God had revealed is that Jesus is God and yet he is in some sense distinct from God such that he can address God and speak of God as his father. … So we have no evidence that Jesus is different from the God of Abraham, Isaac, and Jacob. We have no grounds to believe that Jesus is a God “whom your forefathers did not know.”
GOD IS NOT A MAN, that he should lie, OR A SON OF MAN, that he should change his mind. (Numbers 23:19)
Moreover, the Glory of Israel does not lie or change his mind, for HE’S NOT A MAN that he should change his mind. (1 Samuel 15:29)
Then say to Pharaoh, 'This is what the LORD says: ISRAEL IS MY FIRSTBORN SON.’ (Exodus 4:22)
For, when Israel was young, I loved him, and from Egypt I called My son. (Hosea 11:1)
Did the forefathers of those at Sinai worship a god-man who identified as God’s son? Emphatically not. Christians seem astonished the Pharisees weren’t persuaded, forgetting they were ordained by God to be fanatically meticulous in the observance of the Torah. It’s the very purpose of the Jews being set aside as a distinct race of man. You trust their determination that Jonah, Daniel, Ezekiel, Jeremiah, and Isaiah were the real deal, that Ecclesiastes should make the final cut despite no little controversy, but not to figure out whether the Messiah has arrived (or what repentance is, or what the absolute unity of God involves, or what the Torah says)? Such an extreme misunderstanding of their own religion, such catastrophic tragicomic bumbling should cast lethal doubt on any reliability of the “Old Testament.” It’s like trusting Marshall Applewhite about Heaven’s Gate despite his overestimating the significance of comet Hale-Bopp.
The most feasible way to understand this is to see that God was more than one in person but only one in being (one in certain shared attributes like knowledge, will, emotion, and power). This is something the Torah has not revealed because it was not yet God’s time to reveal it.
God had decades to go over minutia with Moses like the ark of acacia wood (two and a half cubits in length, a cubit and a half in width, and a cubit and a half its height) but couldn’t find time to say, “BTW, I’m actually three distinct units of consciousness that mysteriously share a common essence. One of these is My son (aka the Messiah & the Son of Man) who will be appearing in 1,300 years to change the focus of your worship and abolish most of what I’m telling you now. Don’t worry, a council of Gentiles will hash this out 300 years later, then Martin Luther will set everything straight 1,200 years after that.”
According to John 12:37-41, the Jews COULDN'T believe so as to "fulfill" Isaiah 6 ("He has blinded their eyes and hardened their hearts"). On the one hand their benighted unbelief in the face of so much evidence is condemned; on the other, it's cited as evidence for the truth of what they didn't believe. Please. Does anything not count as evidence?
God also didn't have enough time to go over what the Koran & Book of Mormon reveal.
I doubt that you can come up with any evidence for Tzvi’s claimed miracles.
1) People suffered hardships to follow him. 2) Nathan of Gaza was a renowned Kabbalah expert and he, like Paul, had a dream! 3) Tzvi still has followers: http://isteve.blogspot.com/2006/06/its-borges-borges-borges-borges-world.html 4) Why would they make it up? 5) If Tzvi did not convert to Islam then your faith is in vain. ;o) My point with Tzvi was that miracles aren’t the issue. If anyone were to found a new religion tomorrow via miracles and justify it with some protean theology, Christians would deride it as the work of Satan. Deuteronomy 13 states that some miracles are a test from God to see if Israel will remain loyal to the Torah. Am I under the obligation to examine every claim involving miracles or addendums to the Tanakh? There aren’t enough hours in life!
Regarding Leviticus 17:11,
“The Scriptures explicitly state that the forgiveness of sin is achieved through sincere repentance. This teaching is repeated many times throughout the Hebrew Scriptures in a clear and unambiguous manner (Deuteronomy 30:1 10, Ezekiel 18:21 23, 27, 28, 31, 32, 33:11, 14 16, 19, Isaiah 1:16 18, 55:7, Hosea 14:2 10, Jonah 3:10, Micah 6:7, 8, Psalm 51:19). These passages directly address the issue of forgiveness from sin, yet they make no mention of a blood offering. Some of these passages actually preclude the requirement of a blood offering as a necessary component in the process of forgiveness from sin. Yet on the basis of the misinterpretation of one solitary verse (Leviticus 17:11) from a passage that does not directly address the issue of forgiveness from sin at all, Christianity teaches that repentance cannot achieve atonement without a blood offering!” Yisroel Chaim Blumenthal https://yourphariseefriend.wordpress.com/the-elephant-and-the-suit/
There could even be a separation in time between his appearances on earth to fulfill different tasks. There is nothing in the Torah or the prophets that says he couldn’t.
Where do they say he could, or that it should be thousands of years apart? Where do they say the Messiah will be God and we must worship him? You seem to be arguing that the Torah and prophets never preclude X, therefore X is distinctly possible. Wouldn’t it be rational to withhold judgment until ALL the evidence is in? There isn’t a mitzvah (commandment) to accept the messiah because when he comes it will be obvious and no “faith” will be required.
In fact, spreading the knowledge of God to all peoples has been occurring for centuries and the larger regathering of the dispersed Jewish people to Israel for over a century. So we have no good reason to think Messiah has not fulfilled that part of the prophecy as well.
How is Jesus responsible for gathering exiles to Israel? (Assuming modern Israel is Messianic. Some Orthodox Jews don’t think so: https://www.youtube.com/watch?v=FGJg1lHPPZw) The current state of Israel is a response to the Holocaust.
The people must judge the claimed prophet as a true or false prophet by the fulfillment of the prophecies. The same must also apply to the greatest of the prophets, the Messiah. … Gleason Archer argued that “genea,” generation, more likely means race in this context. The Greek more often means generation, but the Aramaic term Jesus likely used more often means race than does genea.
This is a can of worms for Christianity. CS Lewis, the Preterists, RC Sproul, and quite a few others thought so: http://www.ecclesia.org/truth/solution.html Does anything constitute evidence against it?
Where is the Torah apologetics? Good question! http://redneck_rastafarian.tripod.com/apologetics.html
11. Thanks for such a vigorous response, JPH. I think this will be enjoyable to answer. I'll try to get something out in a day or so. Aron, thanks for letting your readers use your blog space to do some of their own battles. (Not that we don't appreciate anything you have to add.)
12. JHP, I said, “Even if they [the Torah laws] are eternal and binding, it does not follow that God could not give any new commands or information.”
To this you responded, “How could they be eternal and binding AND, per Hebrews 7:18, ‘weak and useless’?”
Notice that I did not say that they are eternal and binding, but “if they are” that God could still give new information. A better translation of the Hebrews passage is, “weak and ineffective” given other statements made in the book. The law of Moses is not useless, it does have a purpose. But it’s weak and ineffective because “it made nothing perfect” or, as one paraphrase puts it, it “did not bring anything to the goal.” If you insist we keep the translation “useless,” we could do so as long as we understand that it is useless in bringing us to the goal of bringing us to God.
For example, the animal sacrifices do not bring us atonement in themselves but as they point to the sacrifice of the suffering Servant, the Messiah. Thus they both were effective for the Israelites to bring them reconciliation with God because of what they pointed to and yet they were not in themselves effective for this purpose. Obviously, the sacrifices cannot now be carried out because of the destruction of the Temple, but other laws which can still be kept could still be effective so far as they point to spiritual truths. So the laws can be eternal and binding even though they are weak and ineffective in themselves.
Please notice that I am not here giving any evidence that the above claims are true, that the sacrifices have no power except as they typify the suffering Servant or that the Servant is the Messiah. I’m just answering your accusation. I’m showing how a particular Christian view could say that the law is still eternal and binding while yet being weak and ineffective for a given goal. I think I’ve only once alluded to a line of argument for Jesus’ messiahship. Since you are making your critique, that is primarily what we are responding to.
“Where in the Torah does Paul receive the authority to say such things?”
Paul never wrote Hebrews but why do you think the writer of this book must receive authority from the Torah to say this? Just as God gave new information to the prophets so he gave new information to the Messiah Yeshua and to his followers.
“And how can God’s revealed law about divorce in the Torah remain ‘eternal and binding’ if Jesus reverses it? This is like saying both X and not X are true.”
Jesus did not reverse it (see Matthew 19.3-9), he merely sided with Shammai against Hillel. “The School of Shammai say a man may not divorce his wife unless he has found unchastity in her. . . . But the school of Hillel say he may divorce her even if she burns his food. . . .” (Mishnah: Gittin 9:10). The issue revolved around the conditions under which a man may divorce his wife “if he has found indecency in a matter” or what that indecency might mean (Deut 24:1).
“GOD IS NOT A MAN, that he should lie. . . .”
But notice that the Torah does not say that God cannot or will not become a man. Such a man would be sinless and thus cannot lie.
“Did the forefathers of those at Sinai worship a god-man who identified as God’s son? Emphatically not.”
No, as I said earlier, that God would someday become a man was not revealed to them at that time. They only knew that they worshiped the God who had only to a limited degree revealed himself to them. It’s only after the time of Jesus that it was revealed that this is who they worshipped.
“You trust their [the Pharisees’] determination that Jonah, Daniel, Ezekiel, Jeremiah, and Isaiah were the real deal, that Ecclesiastes should make the final cut despite no little controversy, but not to figure out whether the Messiah has arrived (or what repentance is, or what the absolute unity of God involves, or what the Torah says)?”
The early Jewish church included thousands of Pharisees (Ac 21.20). Well, at least a very large number of those thousands “zealous for the law” must have been Pharisees. So there were many Pharisees on both sides of the early divide of followers and opponents of Jesus. And yet you arbitrarily accept the beliefs of one group of Pharisees against another.
When the “absolute unity of God” became an issue in the early church, possibly more of the Christian Pharisees (certainly not all) and other Christians who followed their traditions would side with the absolutist views. But as we have seen, if we look honestly at the teachings of the Tenakh, and especially the Torah, there is nothing there to indicate that a strong unitarian view is justified. In fact, I’ve shown that the weaker unity-in-diversity view is better evidenced.
I sense in your words an authoritarianism which I would claim is inappropriate to a thinking person. Because a certain group of people accepted a view, it must be right because they know the scripture better than anyone else. They may know the scripture very well but you and I are able to also learn it and to disagree with them. Again, let me remind you that there were other Pharisees who knew the scripture just as well who disagreed with them.
“Such an extreme misunderstanding of their own religion, such catastrophic tragicomic bumbling should cast lethal doubt on any reliability of the ‘Old Testament.’ ”
As a Protestant Christian I do accept generally the Jewish delineation of the content of the Hebrew scripture. But I accept it because I think the logic behind the analysis that determined that content is generally solid.
The Jews who opposed Jesus were not guilty of “tragicomic bumbling.” They were guilty of some errors but also of some moral deficiencies. Jesus criticized the Pharisees and other scholars as well on both counts and thus made enemies in high places. One error was the belief that a prophet cannot come from Galilee. Because the Galileans were looked down upon by the Jerusalem Jews (they had too much contact with Gentiles up there) and because the scripture doesn’t say anything about a prophet coming from there, they concluded that none could come from Galilee. The scripture emphasized that Messiah would come from Bethlehem. Jesus’ opponents didn’t bother to question where he was born or, if they did, to consider that important compared with where he grew up.
It may be questioned whether at his trial Jesus actually claimed equality with God, but it appears that that is what the Jewish leaders concluded. That appears to be why they condemned him. He called himself the Messiah, the Son of God, and he identified himself as the exalted, “son of man” of Daniel 7. Of course, it may be that they fully believed that he was not claiming deity but they condemned him by saying that this is what he was claiming. They may have simply been unwilling to let someone they so hated (because he criticized and condemned them) live who would actually claim to be the Messiah and Daniel’s son of man.
But even if he did claim divinity, this was no grounds to condemn him. Abraham spoke with a human figure whom he addressed as God (Gen 18). If this was God appearing in human form, as it may well have been, would the Sanhedrin have condemned this same person had he appeared to them? Evidently. So even claiming deity should not be considered blasphemous unless one cannot support such a claim. Notice that the Torah does not state that for a person to claim to be God is in itself blasphemy.
All in all, some errors in the law as it developed at the time of Jesus, not some errors in the Torah itself, plus quite a bit of human jealousy, resulted in Jesus’ condemnation. But these errors were not “tragicomic bumblings.” It wasn’t the beliefs of the Pharisees per se that made Jesus unacceptable to Judaism, because Jesus wasn’t unacceptable to Judaism.
The relations between the Jewish Christians and all other Jews had some ups and downs until the fall of Jerusalem when it really went down. And it became even worse after the bar Kochba revolt. Before 66 the religious leaders did try to crush the Jewish church but there were many who deeply respected it. Josephus’ account of the death of James indicates that he was so well loved that many Pharisees were quite opposed to and angry with the High Priest who had him killed. All in all, the divide between Christianity and Judaism originated from some bad human laws and some human jealousy and hatred on the part of the Jewish leaders. They (and the leaders who followed them) were able to eventually make the divide uncrossable and make their hatred of Jesus official. The increasingly Gentile church was also much to blame for seeking to remove every trace of Judaism from Christianity and alienating Jewish Christians. But it was hardly anything intrinsic to Judaism or the Tanakh that brought about the final rejection of Jesus by official Judaism.
“God had decades to go over minutia with Moses like the ark of acacia wood (two and a half cubits in length, a cubit and a half in width, and a cubit and a half its height) but couldn’t find time to say, ‘BTW, I’m actually three distinct units of consciousness that mysteriously share a common essence. . .’ ”
Absolutely! It wasn’t God’s time yet. This was a mystery hidden from ages past, as the NT writers would put it, to be revealed when the time was right. Moses and the Israelites only knew that God had revealed a few things about himself like his goodness and justice and that he shouldn’t be depicted by graven images, etc. They didn’t know whether God was absolutely one in personhood and at that time they didn’t need to know. But think about when God spoke of himself as “we” or when God said “let us make man in our image.” The readers or hearers may have wondered exactly what God’s nature was really like. Might this strange grammatical usage God had allowed to enter the Torah express an understanding of God’s nature which God had simply wanted to hint at? There are secret or hidden things which belong to God, God told Moses, don’t bother with them unless I tell you more about them. It is rather that which God has revealed to the Israelites, he says, which belongs to them (Deut 29.29[28]). If God wants to disclose his secrets, who are we to tell God when he should do so?
“According to John 12:37-41, the Jews COULDN'T believe so as to ‘fulfill’ Isaiah 6 (‘He has blinded their eyes and hardened their hearts’). On the one hand their benighted unbelief in the face of so much evidence is condemned; on the other, it's cited as evidence for the truth of what they didn't believe. Please. Does anything not count as evidence?”
Other passages which repeat Jesus’ statement (Paul later repeated it as well) point out that it was by the people’s choice that they disbelieved; they blinded their own eyes because they did not want to believe (Mt 13.15, Ac 28.27). There was good evidence then just as there is now.
“God also didn't have enough time to go over what the Koran & Book of Mormon reveal.”
If they reveal anything and if the NT reveals anything must be determined by evidence. Again, I’ve not presented any of the evidence for Jesus’ messiahship or teachings or that of his followers. I’ll do that next time if you like. If we are commandeering Aron’s web page too much, we can find another medium to discuss this.
I doubt that you can come up with any evidence for Tzvi’s claimed miracles.
“1) People suffered hardships to follow him. 2) Nathan of Gaza was a renowned Kabbalah expert and he, like Paul, had a dream! 3) Tzvi still has followers. . . . 4) Why would they make it up? 5) If Tzvi did not convert to Islam then your faith is in vain. ;o)”
1) How is this evidence? Christians don’t offer this as evidence for their claims. That those who claimed to have witnessed Jesus’ resurrection suffered while continuing to hold to their claims is.
2) Paul had dreams but it was a vision which initiated his belief in Jesus.
3) This is evidence? Flat-earthers still have followers.
4) Power, money. People want the Messiah to come and bring peace on earth. If the leaders of the new cult think they can get away with it, why wouldn’t they make it up. They get a built in following to adore and take care of them? Only once it gets a little uncomfortable do the false Messiah’s show their true colors.
5) How is this evidence?
“My point with Tzvi was that miracles aren’t the issue. If anyone were to found a new religion tomorrow via miracles and justify it with some protean theology, Christians would deride it as the work of Satan.”
No, we would look at the evidence for the miracles. Okay, some Christians might reject it out of hand, but they shouldn’t. And after investigating we may indeed conclude that some religions are demonic. The moral content and the moral results of the leaders’ lives should be a good indicator. If we have no such indicators and yet we have good evidence from miracles, the Christian (like anyone else) should consider this as good reason to question or modify their own beliefs.
“Deuteronomy 13 states that some miracles are a test from God to see if Israel will remain loyal to the Torah.”
Right, so if you run into a miracle claim which contradicts belief in Ha Shem, you should reject it. You should however still evaluate it if you are able to. And of course I’ve claimed that the Christian claims do not oppose belief in Ha Shem.
“Am I under the obligation to examine every claim involving miracles or addendums to the Tanakh? There aren’t enough hours in life!”
Since the prophets often added information not found in the Torah, they were certainly giving addendums to it. And Deuteronomy 18 said that anyone, at least any Jew, claiming to have a prophetic word must be heard and tested. A fulfilled prophecy is a miracle. With as few testable miracle claims as are out there and accessible to us, we easily have enough “hours in life” to consider them.
“Regarding Leviticus 17:11, ‘The Scriptures explicitly state that the forgiveness of sin is achieved through sincere repentance. This teaching is repeated many times throughout the Hebrew Scriptures in a clear and unambiguous manner (Deuteronomy 30:1 10, Ezekiel 18:21 23, 27, 28, 31, 32, 33:11, 14 16, 19, Isaiah 1:16 18, 55:7, Hosea 14:2 10, Jonah 3:10, Micah 6:7, 8, Psalm 51:19). These passages directly address the issue of forgiveness from sin, yet they make no mention of a blood offering. Some of these passages actually preclude the requirement of a blood offering as a necessary component in the process of forgiveness from sin.’ ”
Deuteronomy 30 is speaking of God returning exiled Israelites to their land if they repent. It says nothing about whether sacrifices must be made. The latter can also be said about Ezekiel 18 and 33 and Hosea 14. Since Leviticus 17.11 says they are necessary for atonement, it should be assumed that these passages also assume it. You don’t need every passage that applies to a particular subject to repeat the obvious if it is clearly stated elsewhere.
Isaiah 1 and Micah 6 simply say that if there is no change of heart and no attempt to live morally with one another, burnt offerings are useless outward ceremonies and God does not want them. They are clearly not contradicting the Torah teaching of the need for animal sacrifice. Psalm 51 says much the same thing. God wants first of all soul-felt repentance. But then, the psalmist says, he will bring burnt offerings. Notice that the sacrifices are still needed. Yet how easily he could have neglected to have mentioned this since the need for sacrifice was so obvious to him and anyone else who knew the law of Moses. Jonah 3 speaks of God refraining from destroying the Ninevites when they repented and turned from their evil ways. Not being Israelites, God did not tell them they must offer animal sacrifices for forgiveness. I think I mentioned last time a Christian account of how those Gentiles living before the time of Jesus are made right with God. It still indirectly requires blood sacrifice, that of Jesus. So none “of these passages actually preclude the requirement of a blood offering as a necessary component in the process of forgiveness from sin.”
“ ‘Yet on the basis of the misinterpretation of one solitary verse (Leviticus 17:11) from a passage that does not directly address the issue of forgiveness from sin at all, . . .’ ”
The entire passage addresses several issue but this verse specifically addresses atonement or forgiveness of sins. There is no misunderstanding of this verse. It couldn’t be clearer.
“Where do they [the Torah or the prophets] say he could [come to earth more than once], or that it should be thousands of years apart?”
Luke and Paul talk about the time of the Gentiles. This could be an entire era. We’re told that certain events will follow the time that Jerusalem is trampled underfoot by the Gentiles. It seems entirely feasible that this age or era could separate two major appearances of the Messiah in the Jewish ages since the Messiah is primarily a figure of Jewish import. The Gentiles are graphed into the Israel of God. I’m not sure if the prophets speak of this time but the NT does. If we have good evidence for the NT, this should be enough to see that this could be God’s plan.
“You seem to be arguing that the Torah and prophets never preclude X, therefore X is distinctly possible.”
That’s exactly what I’m saying. I’ve simply refuted your arguments that the Torah and prophets are incompatible with Jesus being the Messiah or even God incarnate. We could deal with positive evidence whenever you like. But for now I think it is very clear that there is nothing in the Hebrew scripture to preclude many forms of Christian orthodoxy.
“How is Jesus responsible for gathering exiles to Israel?”
If the Messiah is now with God and the process of history is under God’s control, then the Messiah is working with God as history is being manipulated for this purpose.
“This [Jesus’ purported unfulfilled prophecy of his return during the lifetimes of his hearers] is a can of worms for Christianity.”
Whether a difficult or easy problem, you still have not shown any problem with my answer. I don’t recall that you were able to refute Aron’s argument either.
“Does anything constitute evidence against it?”
Are you telling me that you can’t think of any good evidence against this view so you want me to try to give you some arguments? Well, biblical scholars have some objections. If you can show me that the Aramaic term does not likely mean “race” that would be evidence. I don’t think you can. F. F. Bruce had objections (See the IVP book, The Hard Sayings of Jesus). I’m getting tired of writing though so I’ll leave it to you find it and give it if you think it has force.
13. Johannes says:
JPH's position of absolute epistemic insulation of the Jewish people, i.e. that even a resurrection from the dead performed or even undergone by Jesus cannot prove his claim of divinity, is untenable because the resurrection of a human being is a work necessarily performed directly by God. Therefore, assuming that God performs the resurrection of a human being R through the intervention of a human being S (of "Servant") who claims to speak and act in the name of God - like the prophet Elisha (2 Ki 4:32-35), Jesus (Lk 7:11-17; Lk 8:49-56; Jn 11:38-44), and the apostle Peter (Acts 9:36-42) - we can interpret that fact in one of two possible ways:
A. God, by performing that work that only He is capable of doing, is unequivocally certifying to all witnesses that S has his "seal of approval", i.e. that S indeed speaks and acts in the name of God, so that what S said was true.
B. God, by performing that work that only He is capable of doing, is setting up a trap to all witnesses, acting directly in order to lead them to falsehood. This is against the very notion of God as Absolute Truth and Good.
Position B is even more untenable when the work in question is the resurrection of God's servant S himself, which in the case of Jesus is the event at the foundation of Christian faith. Posing that, by resurrecting Jesus to a glorious state, God is not conferring his "seal of approval" to everything that Jesus said and did, is against the very notion of God as Absolute Truth and Good.
14. Aron Wall says:
Dennis, [and also JPH, but I found it more convenient to pick a specific person to address]
You are of course welcome to continue debating this issue here as long as you please. That's part of what the comment sections are here for!
It is true that some aspects of your post seem to me to be (as you say) heterodox, but I don't feel the need to get sidetracked discussing it. But I don't think it's fair to call the belief that Hell is eternal a "straw man", since this term usually refers to positions that nobody believes, whereas your view is a minority amongst Christian theologians. However, the serious point here is that JPH should spend his energy engaging with whatever version of Christianity he personally believes to be the most plausible. (Which version that is, is up to him to decide, but as I said it is common ground that whatever else God does at the Final Judgement, he will act with perfect justice and not condemn the innocent.) Sometimes the most plausible version of an idea is not quite the same as the one presented to you, so you have to do some of the work yourself; some people call this process "steel manning" the idea.
For this reason I think his comment that:
Don’t worry, a council of Gentiles will hash this out 300 years later, then Martin Luther will set everything straight 1,200 years after that.
is really quite irrelevant to this discussion. The Council of Nicea and St. Martin Luther were either correctly or incorrectly interpreting the New Testament. In the former case, the relevant doctrines are already found in the New Testament; in the latter case their interpretation is spurious and should be rejected. Either way, they are not that important when deciding whether Christianity is true in the first place. Luther wasn't trying to start a new religion, he was a reformer trying to reestablish what he considered to the teachings of the original apostolic doctrines (much like Ezra re-established Torah worship).
By the way, I wouldn't say that my Bayesian analysis provides any new evidence that wasn't there before. It is merely an attempt to numerically quantify how much evidence different arguments provide. People turned off by math can ignore these aspects of my post, but everyone needs some way to decide which things are the most important. A single apparent contradiction in the New Testament isn't enough to rule out Christianity for the same reason the apparent contradictions in the Torah don't rule out Judiasm. Things are not always as they appear, and some amount of charity when reading ancient texts is always called for.
The reason why I don't buy the "race" interpretation of the "generation" passage, is that it seems like a somewhat unremarkable thing to say. "Truly I say unto you" suggests that what follows should be some kind of remarkable statement, not just the belief (already implied by the Tanakh) that no one will have succeeded in killing all of the Jews when the End Times come.
Also, I didn't bring up Lev 17:11 for a reason, because this verse is frequently used by Christian evangelists trying to convert Jews in a manner I consider overly literalistic. Not all of the sacrifices in the Torah involve blood; for example those who were too poor to afford an animal sacrifice were allowed to bring "a tenth of an ephah of fine flour for a sin offering" (Lev 5:11) to be offered on the altar. The correct Christian theology is not that the sacrifice of Christ meets the technical provisions for a Torah sacrifice, pretty obviously it doesn't since human sacrifice was strictly forbidden (but see Gen 22). Rather, the sacrifices were symbols pointing, in various ways, to the suffering of Christ on the cross as an innocent victim (which is what the requirement that sacrificial animals be without physical blemishes symbolized). Obviously Jesus was not offered on the alter of the Temple by the High Priest as if he were an animal; that would have been an abomination which God would never have commanded. Instead he willingly suffered persecution at the hands of the guilty for teaching the truth--just as many Jews have been martyred through history despite their innocence (which modern Jews consider to be a form of sacrifice, since the term "Holocaust" means a burnt offering.)
Thus, JPH is absolutely correct that, in the pages of the Tanakh (especially in the Prophets), sacrifice is regarded as inessential compared to what God really wants, which is true repentance. In fact, the number of passages on this theme is so extensive that he missed some of my favorites! Of course, the New Testament teaches the exact same thing, that people must repent of their sins to be forgiven, and that this is far more important than any amount of sacrifice.
What I don't understand is how JPH can then go on to criticize the Book of Hebrews for saying the exact same thing he is saying. Once you admit that sacrifice is unimportant compared to repentance and obedience, it follows necessarily that (compared to those things) sacrifice is weak and useless. It may be valuable for its symbolism, but not because it has any kind of actual power to make people holy and righteous. So I think that JPH is failing to take seriously the implications of his actual position.
Of course we also find in the Tanakh (especially in the Torah) many dramatic examples of God changing his mind about punishing people, as a result of somebody offering sacrifices atoning for their sin. To me the most striking example is Numbers 16:41-50 (where even though Moses tells Aaron to make atonement in defiance of God's command, God still listens to them), but this is just one of many examples. Clearly something important is being symbolized by all this atonement business, or God wouldn't keep approving it with dramatic object lessons.
Superficially, this seems like a contradiction---sometimes sacrifice is treated as super-important, on the other hand it is meaningless in comparison with living a righteous life. But if Christianity is true, this contradiction is resolved. God approved the sacrifices in the Tanakh as a symbol of what was to come (Jesus), but because they were merely symbols, they are no subsitute for righteous living. But Jesus' sacrifice is effective, not because it replaces repentance and obedience (it doesn't) but because it provides the groundwork which enables sinners to truly repent and so be reconciled to God.
JPH,
I don't think any of those things you listed for Tzvi counts as a miracle.
I understand that God becoming Incarnate is a pretty huge paradigm shift, and I sympathize with your sense of surprise given what had been revealed before. If I were a Monotheist living before Christ, and you asked me whether God even could become Incarnate as a human being, my first response would be to say "No!" But if you pressed me, and said "Well God is omnipotent, so he can do anything", then I would have to modify my position to "I don't know." So there you have a point, and the only thing I can say in reply is that there are many hints in the Tanakh (some of them pointed out by St. Dennis) that something like this might be possible.
But I think this particular argument is extremely weak:
You trust their determination that Jonah, Daniel, Ezekiel, Jeremiah, and Isaiah were the real deal, that Ecclesiastes should make the final cut despite no little controversy, but not to figure out whether the Messiah has arrived (or what repentance is, or what the absolute unity of God involves, or what the Torah says)? Such an extreme misunderstanding of their own religion, such catastrophic tragicomic bumbling should cast lethal doubt on any reliability of the “Old Testament.” It’s like trusting Marshall Applewhite about Heaven’s Gate despite his overestimating the significance of comet Hale-Bopp.
The problem with it is that, even according to Jewish tradition, Israel was extremely fallible when it comes to recongizing prophets and rulers. The Jews rejected Moses many times and would even have stoned him to death if God hadn't intervened (Num 14:10). 10 of the 12 tribes rebelled against David's dynasty (the Samaritans don't accept it to this day). Ahab and Jezebel tried to kill Elijah; Jeremiah was mocked, imprisoned and persecuted (some legends say he was killed by stoning); Isaiah was sawed into two pieces (according to the Talmud), and so on! The Tanakh calls the Jews a "stiff-necked" people for a reason. When did God ever say that Israel (or the rabbis) are infallible?
Of course, there have also been many generations of righteous Jews (for example during the time of Ezra, or the Maccabees), and we Christians are very grateful for their efforts and wisdom in selecting and preserving the books of the prophets who lived in previous generations. (And I am certainly not saying that people are morally responsible for sins committed by their ancestors! Given the history of anti-Semitism, this point must be emphasized very strongly!) My point is not to blame the Jews for everything, just to note that "extreme misunderstanding of their own religion" seems pretty common when you compare to the rest of the Bible.
I also don't think the Deut 30 prophecy cuts in quite the direction you'd want it to. First of all, the whole passage has already been fulfilled once, when the Jews were sent into Exile for 70 years for abominable sins such as oppression, idolatry, child-sacrifice etc. Dozens of prophets warned them again and again, but they wouldn't listen. Then, after they repented God restored the Temple and their nation, and reformers like Ezra taught them to "obey him with all your heart and with all your soul according to everything I command you today" (30:2).
But, both Jewish and Christian tradition seem to allow the possibility of a second fulfilment of the prophecy, this time associated with a second judgement of Israel (when their Temple and Nation were again destroyed, and eventually they went again into Exile). This period of exile lasted for at least 1800 years or so (depending on whether one thinks Modern Israel counts as a restoration). Now the Jews were only exiled for 70 years for things like sacrificing their children to Moloch. So what could trigger an exile for thousands of years? Funny how this time there were no prophets of any kind to warn the Jews about what specifically they were doing wrong and to urge them to repent (unless Christianity is true, in which case there would be John the Baptist, Jesus, and the preaching of the Apostles for 40 years).
Let's see if we can figure out what the triggering event was. The Temple was destroyed in 70 AD. I'm not sure that all the stories in the Talmud are historically reliable, but it is interesting that it states that there were signs of God's displeasure with the Temple for 40 years prior to its destruction. So that means that whatever this something was, it probably occured around 30 AD. Can you think of any candidates for that event? I can only think of one...
In any case, Jewish tradition admits that the Jews must have done something to displease God at around that time, in order to be sent into Exile (even if it identifies a different candidate for the triggering sin). So, why should we trust the exact same generation who committed that sin, to say who is (or is not) a true prophet sent by God?
Although St. Dennis also makes a good point that many of the Jews did accept Jesus, it's just we started calling that group of Jews (and their converts) "Christians".
Johannes,
Thanks for your comment. I agree, although it is still necessary to explain what is going on in Deut 13 passage when God speaks about testing people. How is that compatible with God being the Truth?
(I think I would argue that it's not that God actively performs fake miracles in order to try to fool Israel, rather he allows the situation to happen naturally and then providentially uses it as a test. But that would imply that if the miracle is big ENOUGH, so that it would be extremely unlikely to be a fake, and disproportionate for a mere test, then we should reconsider whether the prophet might still be teaching the truth, especially if it is controversial whether it really contradicts Monotheism.)
15. JPH says:
“Did the forefathers of those at Sinai worship a god-man who identified as God’s son?”
No, as I said earlier, that God would someday become a man was not revealed to them at that time.
Then they are forbidden to worship Jesus and everyone else whom their forefathers didn’t know. Game over. Do not pass go. This is the standard of idolatry. http://bible.ort.org/books/pentd2.asp?ACTION=displaypage&BOOK=5&CHAPTER=13
The Hare Krishnas make the same argument you do, saying that the Israelites didn’t know they were really worshiping Krishna: https://en.wikipedia.org/wiki/Gaudiya_Vaishnavism Anybody can make this claim! The Torah anticipates this maneuver: “Do not invoke the names of other gods; do not let them be heard on your lips.” (Exodus 23:13) I am under no obligation to investigate everything under the sun to see if God’s standards of idolatry have been updated (or that maybe it was someone else and He had to temporarily fly under a different name).
They only knew that they worshiped the God who had only to a limited degree revealed himself to them. It’s only after the time of Jesus that it was revealed that this is who they worshipped.
Limited!?
The Lord would speak to Moses FACE TO FACE, as one speaks to a friend. (Deut 33:11)
And there was NO OTHER PROPHET who arose in Israel like Moses, whom the Lord knew FACE TO FACE. (Deut 34:10)
Christians cite Deut 18 as evidence for Jesus. What does Deut 34:10 say about the hierarchy of prophets? No prophet is greater than Moses and there’s an end of it. No one had his access to God. There was nothing "limited" about it!
But notice that the Torah does not say that God cannot or will not become a man.
Descendants of Jacob, I am the LORD All-Powerful, AND I NEVER CHANGE. (Malachi 3:6)
But will God indeed dwell on the earth? Behold, heaven and the heaven of heavens cannot contain You. (1 Kings 8:27)
So whenever God says X it’s only true at the immediate instant He says it? (If that’s true about Deut 5:18 I need to make a few calls.) Necessity is the mother of bad theology. This is what Christians have to say. Why wouldn’t the Almighty simply have said “God is not YET a man”?
Such a man would be sinless* and thus cannot lie.
That’s not what God is saying through Balaam. Rather, he means that ALL humans lie and ALL humans need to repent. That’s one reason God can’t be a man.
Paul never wrote Hebrews but why do you think the writer of this book must receive authority from the Torah to say this?
Because the Torah is the only self-authenticating revelation in human history: http://www.mesora.org/god/ God wrote the Torah. Moses was His stenographer. The prophets do not have the same status. They were written under the spirit of nevu’ah (prophecy). Notice how NONE of them say that someday the Torah will be “fulfilled.” He would have been put to death and his writing would not have been canonized by the ‘Anshei-HaKeneset HaGedolah. Christianity, Islam, Sikhism, Bahai, and Mormonism are piggybacking on the self-authenticating evidence http://ohr.edu/special/books/truth.htm of the Torah and claiming they have the last word. The Torah is THE Revelation. The NT does not have the authority to reinterpret it unless the Torah says so. (If you want to hang your hat on Deut 18, don’t forget Deut 34:10.)
I sense in your words an authoritarianism which I would claim is inappropriate to a thinking person.
Thank you! It’s tough being a conservative surrounded by radicals. ;o) If it's new, it ain't true.
I think I mentioned last time a Christian account of how those Gentiles living before the time of Jesus are made right with God. It still indirectly requires blood sacrifice, that of Jesus. So none “of these passages actually preclude the requirement of a blood offering as a necessary component in the process of forgiveness from sin.”
You can’t assume the truth of your conclusion as a premise. The Book of Jonah says the Ninevites repented and God forgave them – without any sacrifices of any kind whatsoever. This is evidence against Christianity. You can’t assert the Christian account as an argument for Christianity. How is this supposed to convince me?
“How is Jesus responsible for gathering exiles to Israel?”
If the Messiah is now with God and the process of history is under God’s control, then the Messiah is working with God as history is being manipulated for this purpose.
Again, you can’t assume the truth of your conclusion as a premise. You cited Israel as EVIDENCE for Christianity. When I asked you to enlarge on your position you assert a conditional: IF Jesus is the Messiah THEN history is being manipulated for this purpose. So if your conclusion is true, it follows that your conclusion is true.
“Where do they [the Torah or the prophets] say he could [come to earth more than once], or that it should be thousands of years apart?”
Luke and Paul talk about the time of the Gentiles. … I’m not sure if the prophets speak of this time but the NT does.
You can’t assume the truth of your position as a premise. Of course the NT says this. IT HAS TO.
I doubt that you can come up with any evidence for Tzvi’s claimed miracles.
Evidence seems to be in the eye of the beholder. (The ten tribes of Israel were spotted, he walked through a cloud of fire, Elijah appeared, etc.) I’m not even tempted by these claims. He didn’t fulfill the Messianic prophecies and he tried changing these: http://www.jewfaq.org/613.htm Game over. Post hoc protean theologies don’t change anything. Tzvi makes Christians uncomfortable because the difference is one of degree, not kind.
Whether a difficult or easy problem, you still have not shown any problem with my answer. I don’t recall that you were able to refute Aron’s argument either.
How would I refute this without a time machine? Why wouldn’t Jesus' followers be more meticulous with (arguably) the single most important prediction he made? (I should have chosen Matthew 10:23 as an example.)
*I don’t mean this to be disrespectful but your position invites these queries, part of why it’s unfathomable that God is/was a human. If Jesus was truly a man did he ever have lustful thoughts? If not, he wasn’t a man. That’s part of the package, intrinsic to the misery & joy of being human. A huge chunk of waking life consists in an endless barrage of lustful thoughts. Per Jesus’ standard (Matt 5:28) this is as bad as adultery. How do you square this triangle? If Satan tempted him with food, why not other drives?
If our discussion is reaching the point of diminishing returns, this is my choice for the single best living proponent of Torah Judaism: http://redneck_rastafarian.tripod.com/ and he’s a Gentile.
16. Johannes says:
Thesis 11: The signs or wonders that can be performed by a magician, a false prophet, a false christ, an antichrist or "the" antichrist, as support for an instigation to go after false gods, or to disobey God, or to deny Jesus Christ, are not performed directly by God. Rather, they are performed by an evil angel, whom God has permitted to do it, by his own power.
It is clear from the Bible that an angel, either good or evil, can affect the material world by his own power if God permits him to do it. Job 1:6 - 2:7 describes some of the signs or wonders that an evil angel can perform if permitted by God: fire falls from the sky, a great wind blows, Job is struck with loathsome sores. Paul describes the antichrist thus:
"Then that lawless one will be revealed whom the Lord will slay with the breath of His mouth and bring to an end by the appearance of His coming; that is, the one whose coming is in accord with the activity of Satan, with all power and signs and false wonders, and with all the deception of wickedness for those who perish, because they did not receive the love of the truth so as to be saved." (2 Thess 2:8-10, quoted from NASB)
Thesis 12: A sign from God is distinguishable from a sign from a false prophet, a false christ, an antichrist or "the" antichrist, on two levels: degree of power and wisdom displayed by the agent, and character of the sign.
Focusing first on the degree of power and wisdom displayed by the agent, clearly the resurrection of a human being stands on top. This is evident at the physical level since, in order to bring a dead body, human or animal, back to biological life, the acting entity must rearrange all atoms and electrons in the body ("restore the photo") and immediately restart the coordinated "motion" of all those atoms and electrons ("restart the movie"). Doing that requires an extremely high degree of power and wisdom, the higher the more complex the organism to be resurrected is, with the complexity being maximum in the case of human beings due to the development of their central nervous system. It is highly doubtful that a non-divine entity can do just that.
But to bring a dead human being back to human life, not just biological life, the entity must also infuse to the resurrected body a spiritual soul, which could in principle be the same soul that had left that body at the time of death, the soul of another dead person, or a new soul created out of nothing. Otherwise the result of the resurrection would not be a human being with intellectual capabilities but the metaphysical, cognitive and behavioral equivalent of a Neanderthal, if biblical Adam was y-chromosomal Adam, or a Homo Habilis, if biblical Adam was a Homo Antecessor or Ergaster, which I strongly doubt, or, to use a literary figure familiar to Jewish readers, a golem. And infusing a spiritual soul to a human body is a work that can be performed only by God, who is also the only One who:
- can create a soul out of nothing, a point relevant in the case of the conception of a new human being, and
- has in his hands the souls of the departed (which is obvious but also explicitely stated in Wis 3:1), a point relevant in the case of a resurrection.
Thus, the resurrection of a human being is a work necessarily performed directly by God, who by that work would be unequivocally certifying in the eyes of the witnesses that the servant through whose intervention God is performing that work has God's "seal of approval", like the prophet Elisha (2 Ki 4:32-35), Jesus (Lk 7:11-17; Lk 8:49-56; Jn 11:38-44), and the apostle Peter (Acts 9:36-42). Thus, if Jesus, by performing resurrections, shows to have God's seal of approval, then what he said was true. Therefore if Jesus claimed divinity, He is God.
Clearly the maximum "seal of approval" that God can confer to a servant of His, as Christians believe He did to his Servant par excellence, Jesus, is to resurrect him to a glorious state. We will not take this event into consideration here because we are dealing primarily with the epistemic situation of the members of the Sanhedrin at the time of Jesus' trial.
Focusing now on the character of the sign, there is a basic difference between the signs of the Christ and the signs of a false prophet or antichrist, which is evident in the passages when Jesus refuses to perform a sign requested of Him.
First, when tempted in the desert after his baptism in the Jordan. Jesus refused to command stones to become loaves of bread or to throw Himself down the pinnacle of the Temple (Mt 4:1-7), either possibility being a sensationalistic exhibition that would be an easy way to gain followers.
Second, when the Pharisees, on one occasion with the scribes (Mt 12:38-40) and on another with the Sadducees (Mt 16:1-4), asked Jesus to give them a sign from heaven. Mark describes the reaction that the request elicited in Jesus' spirit:
The Pharisees came and began to argue with him, seeking from him a sign from heaven to test him. And he sighed deeply in his spirit and said, "Why does this generation seek a sign? Truly, I say to you, no sign will be given to this generation." (Mk 8:11-12)
Actually, the "sign from heaven" that the Pharisees asked Jesus to provide was precisely the kind described in Job 1:16. Thus, it would not have been proof that Jesus had come from God but rather the opposite.
What these passages have in common is that Jesus consistently refuses to give a sign which is primarily a sign and not a work of God in itself. And what are the works of God that Jesus performs? The works that show God's merciful and compassionate love towards man, in Hebrew hesed or rahamim, as Jesus replied to the disciples of John the Baptist: "the blind receive sight and the lame walk, the lepers are cleansed and the deaf hear, the dead are raised up and the poor have the good news preached to them" (Mt 11:5). And when Jesus did multiply the loaves of bread to feed the people, He did that after having fed them with the word of God by teaching them for hours (Mk 6:34-25). Thus Jesus performed the works of God described in Is 26:19, 29:18, 35:5-6 and 42:7, and in Ps 146:7-9. And therefore He can justly say to all men, first to the Jews and then to the Gentiles:
"If I do not do the works of my Father, do not believe Me; but if I do them, though you do not believe Me, believe the works, so that you may know and understand that the Father is in Me, and I in the Father." (Jn 10:37-38).
17. Johannes says:
@JPH: the two passages that you quoted in your last comment to disprove Dennis' assertion "that the Torah does not say that God cannot or will not become a man" do not actually disprove that assertion. Moreover, the apostle James states the same as Malachi 3:6 in different words:
"Every good thing given and every perfect gift is from above, coming down from the Father of lights, with Whom there is no variation or shifting shadow." (James 1:17)
The key is to distinguish between the divine nature or essence and the divine Persons. Each of the divine Persons Is eternally the divine nature or essence (in different modes of Being), but one divine Person can assume a created nature by an act ("Incarnation") that does NOT change the divine nature!
At this point one could ask what "assume" really means, or in other words why Jesus' created human nature is not a human person like your human nature. This would amount to describing the mystery of the Incarnation (if considered in fieri) or the Hypostatic Union (if considered in esse) in philosophical terms. Which is actually possible if you hold the real distinction between essence and act of being ("actus essendi" or "esse", commonly referred to as "existence"). Note that this real distinction is a purely philosophical issue.
While every created being is a composite of an essence and a contingent act of being different from the essence, in the case of God, because of absolute divine simplicity, the divine essence is the Subsistent Act of Being Itself ("Ipsum Esse Subsistens"), so that each divine Person is the Subsistent Act of Being (in different modes). Therefore, the assumption of a human nature by a divine Person means that such human nature exists, from the moment of its creation, by the Subsistent Act of Being which that divine Person eternally Is. And that's why the created human nature of Jesus is not a human person: because it does not exist by its own contingent act of being, like your human nature, but by the Subsistent Act of Being of the Son, or even better, by the Subsistent Act of Being which the Son eternally Is.
Note 1: I try to capitalize the verb "to be" when referring to Subsistent Being.
Note 2, to Christian readers: the distinction of the divine Persons by their mode of Being (tropos hyparxeos) is a notion from the Cappadocian Fathers (specifically from St. Basil of Caesarea and his brother St. Gregory of Nyssa), and has nothing to do with the modalist heresy which held that Father and Son were just modes of appearing of the only divine Person.
18. Aron Wall says:
JPH,
1. Isn't the most obvious interpretation of Deut 34:10 that no prophet had yet arisen like Moses, during whatever period of time that particular verse of the Torah was written down? That would make it similar to 1 Kings 8:8 and Joshua 4:9, which describe things that were true when they are written, but are no longer the case.
(It seems obvious to me that the account of Moses' death must have been written by somebody else, perhaps Joshua, after he died. I know that many Jews believe that Moses wrote literally everything in the Torah himself, but that seems quite unlikely. There is certainly nothing in the Torah itself that states that every single letter was written down by Moses, and this particular chapter reads like a historical retrospective, not a future prophecy. Unlike several other parts of the Torah, which do sound like prophecies.)
In any case, the verse cannot possibly mean that a prophet like Moses will never arise in Israel, because Deut 18 quite explicitly states that the prophet it refers to will be like Moses (at least, in certain respects). Your interpretation of Deut 34:10 would make the Torah contradict itself on this point.
2. Obviously, the most important prediction of Jesus is his own Crucifixion and Resurrection, not Matthew 10:23 whose meaning is relatively obscure (it is not even clear that this verse refers to the Second Coming). As for the Second Coming, Jesus also said fairly explicitly that nobody would be able to predict that time, not even himself (Mark 13:32). Obviously, this means it cannot be used as positive evidence for Christianity, but it also cannot be used against it. One must accept or reject it on the basis of other arguments. As the book you gave me to read says:
On the other hand, some of the portions of the descriptive content of the Torah cannot be investigated directly: what happens to the soul after death; all predictions still to be fulfilled in the future, for example, there will be a Messiah one day, haven't occurred yet. Those that can be investigated directly, we will investigate. What about the ones that cannot be investigated directly?
The answer here is as follows. We have a single coordinated body of information. Whenever you have a coordinated body of information, some of which you can test directly and some of which you cannot test directly, if the portion that can be tested directly tests true, then that gives credibility to the rest.
Your comment about time travel ignores my actual argument. I cited several examples of prophecies in the Tanakh where the same sort of apparent chronological jumbling appears, in a way that is accepted to be true even by Jews. It's important not to have a double standard where you accept all the subtle theories of the rabbis about how to reconcile apparent contradictions in the Torah and Tanakh and Talmud, but reject all of the possible explanations of apparent contradictions in the New Testament out of hand. "Do not have two differing weights in your bag--one heavy, one light" (Deut 25:10).
3. Traditional Christian theology does not teach that the divine nature changed in any way when God became a man. We believe that Jesus is simultaneously fully God and fully human, and that his divine nature is unchangable. Thus, he added a human nature, but he did not change or modify in any way his divine nature which we (like Jews and Muslims) believe is metaphysically incapable of being changed.
Regarding lust and other impulses, Christians draw a distinction between temptation (which is not a sin), and voluntarily succumbing to temptation (which is a sin). Jesus had the former, but not the latter:
For we do not have a high priest who is unable to empathize with our weaknesses, but we have one who has been tempted in every way, just as we are--yet he did not sin. Let us then approach God's throne of grace with confidence, so that we may receive mercy and find grace to help us in our time of need.
(Heb 4:15)
Merely having involuntary sexual thoughts, outside your control, does not make you guilty of transgression. But, choosing to gratify lustful thoughts, in a way that violates some commandment of God, is wrong! And this is so even if the deliberate indulgence takes the form of a fantasy taking place entirely in your own mind. This teaching is implied not just by Matt 5:28, but also by the Tenth Commandment of the Decalogue: "Do not covet your neighbor's wife". (Obviously Jesus was not referring to the case of legitimate sexual desire for your own wife, because that would not be adultery even if you actually did it! Although Jesus himself was unmarried, so this would not apply to his own case.)
Now it is clear that somebody who never sinned sexually would not, for that reason alone, cease to be fully human. For human beings are created in the image of God, and God does not sin. If anything, the sin which so frequently defiles us makes us less human. For humans were not created to be inhuman to each other. If sinning were a necessary prerequisite for being fully human, then we would all be morally obliged to commit at least one sin (since it is God's will that human beings exist). But that would be absurd, since a thing cannot be both required and prohibited.
19. Johannes says:
@JPH, regarding the standards of idolatry, it is clear why a Hindu does not worship the true God while a Muslim does [1] [2], and it is not a matter of names. Rather, it is because Hinduism is usually a form of panentheism while Islam is classical monotheism.
Thus, in the most common Hindu school, the non-dualist Advaita Vedanta, the human soul is identical with the "supreme" being: Atman is Brahman. Clearly a "supreme" being which "drops" parts which then forget that they were really part of the "supreme" being is not the Subsistent Being of classical monotheism. And the Hindu school whose description you linked to, while not being pure monism as Advaita Vedanta, still holds that, in quality, the soul (jiva) is identical to God.
Now, Christians do not hold that the Son is another deity aside from the Creator, but that He is consubstantial to God the Father. Which is asserted by Jesus both by stating "I and the Father are one" (Jn 10:30) and by applying to Himself the proper Name of God in the first person revealed in Ex 3:14: Ehyeh, "I Am" (Jn 8:24,28,58 and 13:19). This is particularly evident in the first, second and fourth verses, in which Jesus notes the importance of {believing/knowing} that "He Is", echoing Isaiah 43:10 y 48:12.
[1] I specifically mention Muslims because there is rabbinic consensus that Islam is not even shituf, much less avodah zarah.
[2] I suppose that, from the viewpoint of Judaism, Muslims worship the true God in the sense of what God Is, but not in the sense of what God thinks or does in history. Which is the case from the viewpoint of Christianity.
20. Mactoul says:
JPH,
"Christians cite Deut 18 as evidence for Jesus."
Bible-worshipers may do so. But Christians do not need for the Church is THE witness for Jesus. The argument is not from the Bible to Jesus, rather from the Church to Bible and Jesus.
21. JPH says:
There is a LOT to digest here, and I thank everyone for their responses!
This has been a big week. Ed Feser is (possibly) refuting my favorite argument for theism: http://edwardfeser.blogspot.com/2016/09/yeah-but-is-it-actually-actually.html This is bad news, even if he's right. The Kalam is so intuitive & plausible. It's easy to follow and tempting to skeptics in a way that Aquinas' version isn't.
Johannes - I'm a big fan of First Things but I've never read The Trial of Jesus. I look forward to it and your response.
Aron - I'm haunted by this:
If it weren't for Jesus, I wouldn't even be arguing with you about the Torah! I guess I'd be off in some forest somewhere, sacrificing to a pagan god.
No, none of us would be here. History would have gone off in some entirely different direction and our ancestors wouldn't have mingled. Divine Providence is weird!
22. Aron Wall says:
JPH,
It says good things about you that you are willing to jump into the fray and argue with us! My hope and prayer is that God, who is the Truth, will reveal to us his glory, and will gently correct whichever one of us has misunderstood his revelation. That goes for me too, even though I obviously have strong beliefs about this matter.
There is no shame in taking sufficient time off to ponder these things. And please feel welcome to come back here, either to discuss this issue more, or to talk about anything else.
If I were a Jew, I think I would also feel pretty weird about most people in the world being primarily familiar with my God and Scriptures via what I consider a heretical and semi-polytheistic offshoot. (I'm trying to imagine how I would feel if I were one of 16 million Nicene Christians living among 2 billion Mormons...) I guess from your point of view, however wrong Christianity is, it seems to be at least in some sense part of God's providential plan to extend his fame to the nations.
I was reading that kalam thread too. While the universe probably is finite in time, I've never been very impressed by the "no actual infinity" arguments for this premise, given that Cantorian set theory appears to be a consistent way to treat infinite sets without any inconsistencies.
(At least, so long as the infinite sets have limited cardinality like $\aleph_0$ or $c$, rather than being things like the set of all ordinals, for which vicious paradoxes still arise. (This suggests that maybe Aristotle was right that certain infinite classes can only exist "potentially", not "actually", but that the threshold of size for this to happen is much higher than he thought it was!)
As far as anyone can tell ZF set theory is consistent, although we can't prove it for sure. (In fact, St. Kurt Gödel proved you couldn't prove it was consistent, but you can say the same thing about finite integer arithmetic.) So, in my opinion modern mathematics makes it hard to claim that actual infinities are absurd in and of themselves. (It may still be possible to argue against an infinite past by trying to analyze concepts of causality and explanation metaphysically, but this is a trickier subject.)
If you're curious, here's my own take on the most plausible form of the Cosmological Argument, and some related issues. (Note that, apart from a very brief digression about the Trinity in part VII which plays no role in the rest of my argument, I am trying to defend the Classical Theistic conception of God which is shared by Jews, Christians, Muslims, Deists, Sikhs etc. and to some extent certain Hindus.)
Johannes,
The Dvaita (dualistic) school of Hinduism seems closer to being truly monotheistic, at least insofar as they acnowledge an eternal and absolute distinction between the ultimate deity "Vishnu" and everything else, and see salvation as a matter of having a correct relationship with Vishnu rather than being absorbed into him.
23. Scott Church says:
JPH,
I'm sorry you're having such a hard time with many of these questions... I suppose struggles like these are good for us in many ways, but they certainly aren't fun! Here are a few things to consider...
First, the arguments of St. Feser's that you linked presume presentism--the claim that only the present actually exists, the past is gone, and the future is yet to be. Presentism is also referred to as the "A" theory of time. This is to be contrasted with the "B" theory of time according to which past, present, and future are equally real, and the appearance of the present moment "flowing" from past to future is a consequence of our perception and the underlying physics (specifically the 2nd Law of Thermodynamics). From a physics standpoint, presentism has numerous difficulties that Aron has addressed elsewhere (for instance, here and here). Now it is possible to make presentism work, and St. Craig and others have argued at length for it. But compared to a B-theory framework the result is kludged and contrary to fundamental principles of relativity, and few if any physicists find it compelling. Dispense with presentism, and Feser's arguments against the kalam viz. actual infinities have considerably less force. Whether "Hilbert's Hotel" arguments are as strong as St. Craig thinks they are and can survive the issues Aron mentioned is another discussion. But I think we can at least say that if they were as easy to dismiss as some folks believe, they wouldn't be getting the attention they have in philosophical circles. [BTW, I pointed all this out in that article's combox. As of this writing, the only response I've received was from a guy who believes the Lorentz boost is a "crackpot equation" that "has in no way been proven." Not much one can say to someone like that... :D ]
Second, while the kalam argument is fine for what it's worth I have to agree with Aron... there are much better cosmological arguments. In philosophical circles at least, most of the criticism against it has been directed at its first premise--namely, that whatever begins to exists has a cause.--which has a lot more punch from the perspective of a flowing present. In the B-theory framework that's most conducive to physics it's less obvious. This more than anything else, is why it tends to be associated with presentism and St. Craig's often uses the phrase "popping into existence" when defending it. But that said, as St. Feser himself has also pointed out, the kalam argument is not without force. The first premise can actually be derived from axioms underlying the stronger cosmological arguments that he, Aron, St. Thomas Aquinas, and many others have defended at length. IMHO, one of the problems with St. Craig's approach is that he tends to simply assert it as self-evident, which it clearly isn't to his critics, when he could be doing that.
Perhaps the real value of the kalam argument, and the first premise in particular, has less to do with anything unique in it than with its ability to make the admittedly abstruse axioms of better cosmological arguments more accessible than they would otherwise be. But that accessibility does come at a price. It's at its best when running a three-legged race with presentism... it cannot leverage presentism's strengths without taking on its baggage as well. Ultimately, I'd say fear not... kalam is alive and well! :-) But I wouldn't look to it for more than it can deliver. The stronger cosmological arguments may be less accessible, but they're also a lot more bullet-proof... and they can reinforce the kalam in a way that it cannot do for itself.
I hope this helps!
24. Andy Jones says:
I'm pretty sure that after spending the evening reading this thread and links, some sort of diploma or ordained status is in order? Ironically, and disappointedly, the unacceptance of the messiah (Jesus) by Jews is yet another fulfilled prophecy. Like Aron, it's interesting to imagine I was JPH. On one hand I understand how he'd feel like Christians perverted his religion, much like I see the Mormons. On the other hand, I'd be proud that Jesus was a Jew like myself and that I was part of the original "pasture".
JPH, just curious. Have you ever read the NT? As in the gospels, Acts and Paul's letters? I'm not looking to argue with you, just wondering if you consider them fakes, lies, or just "the truth" from the perspective of ignorant people? In other words, do you think Mathew, Mark, Luke, John and Paul really wrote the books and thought it was all true but they were wrong or do you think they're forgeries, outright lies, or written by people with other agendas?
25. Mactoul says:
Scott Church,
The B-theory seems just a replacement of the temporal reality by a spatial metaphor, perhaps by taking literally the mystical language of modern physics. To say that "past exists" is really to say that "past exists in the past"-an unexceptional statement of the ordinary. So, present exists in the present and future exists in the future. Temporal reality can not be handwaved away.
As for the A-theory " things at the present moment of time really exist, really we are saying that anything which is simultaneous to my present experience exists." , I note that the things extend in time, necessarily. It is the physics that can not handle this perdurance.
26. Scott Church says:
Mactoul,
No one is suggesting that temporal reality can be "handwaved away" or that the present moment doesn't exist. On the contrary, B-theory merely asserts that the past and future also exist in the same sense, and our perception of the present "flowing" out of the former and into the latter is illusory. Modern physics treats time as space-like because it is space-like. Space and time are separate aspects of the single metric entity space-time in which $ict$ is fully equivalent to x, y, and z (note that $ict$ has the dimensions of distance). That's not to say it's identical to space... which is why there's an $i$ in it. But we do in fact live in a universe whose history unfolds on a $\left ( x, y, z, -ict \right )$ manifold, not an $\left ( x, y, z \right )$ one independent of $t$. The Lorentz boost and the constancy of $c$ in all reference frames follow directly.
None of this is a "metaphor." The Lorentz boost, the Lorentzian (pseudo-Riemannian) signature of space-time, and its curvature are all directly observable and have been tested countless times. They're even used in engineering. Believe it or not, the GPS satellite system incorporates special and general relativity in its positioning algorithms. They're why your smart-phone knows your location to within 10-20 feet rather than a few city blocks. :-) All of this is a direct consequence of the fact that we live in a space-time rather than a space that unfolds in time. And in that space-time, the difference between past and future is frame-dependent for space-like separated observers. This is a very real issue for A-theory time--one that as I mentioned, is solvable, but only by kludge.
I'll leave it at that, as we now seem to be getting off the main topic of this post. Best.
27. TY says:
Andy:
I guess you were just kidding in saying “I’d be proud that Jesus was a Jew like myself and that I was part of the original “pasture” -- September 13, 2016 at 7:13 PM”, which provokes the question: Would Andy be just as proud had Jesus been a gentile? That God chose a Jew to effect the Incarnation is just Divine providence in action and, for this reason, Christianity appeals to all mankind and is preached to all mankind.
28. Mactoul says:
Scott Church,
1) Physics is not competent to pronounce on ontology. It takes temporal flow as a given. Indeed you can not escape it --your statement itself "But we do in fact live in a universe whose history unfolds on a (x,y,z,−ict) manifold". Anything more, "curvature of spacetime, ict (with imaginary coordinates) is mathematical convenience only whose literal meaning is strictly nonsense (CS Lewis in The Discarded Image).
2) On what basis you call our perception of temporal flow to be an illusion? That basis can not be physics, for the reasons alluded above.
29. I have some comments here for your post, Aron. More on the way for JPH. I still haven't read your last post or some of the other final ones. Sorry I'm so slow in getting back. I saw an op-ed by Gregory Clark attacking Christianity via a current variation of the problem of evil and I got drawn into the discussion. Some temptations are hard to resist. But that's why I'm so late.
Aron,
But I don't think it's fair to call the belief that Hell is eternal a "straw man", since this term usually refers to positions that nobody believes, whereas your view is a minority amongst Christian theologians.
My thinking in calling this a straw man was that since my own reading of scripture rejects the doctrine of eternal conscious torment (ECT) that this accusation does not apply to me. Those who do accept it are the ones who have to deal with it.
Years ago when (the late) Antony Flew was an atheist, he would argue that since the greatest of all theologians—Augustine, Aquinas, Luther, and Calvin—accepted predestination, this should be accepted as the Christians view. He would then point out the injustice of saying that some will be eternally lost who have no choice in the matter. (Of course he also had a lot to say about what being eternally lost might mean.) My response to the argument was always, “I don’t care what these theologians said, the Bible says something else.” I felt no obligation to defend a particular doctrine even if it was a majority view among all Christians. In that sense I thought of it as a straw man. So I’m still not sure that I’m wrong in thinking of the doctrine of ECT as such. However other Christians may respond to ECT, I can just say that the form of Christianity I espouse rejects it entirely. Someone cannot refute Christianity if some form of Christianity still remains which is immune to the defeater.
(BTW, I saw this great T-shirt I’ve just got to get. On top it has in large letters, “Calvinism”; then below in smaller letters: “Some Lives Matter”.)
By the way, I wouldn't say that my Bayesian analysis provides any new evidence that wasn't there before.
I agree. My statement was, “Aron’s work with Bayesian analysis merely gives us evidence in a way it was not seen before” which, I think, is extremely important.
The reason why I don't buy the "race" interpretation of the "generation" passage, is that it seems like a somewhat unremarkable thing to say. "Truly I say unto you" suggests that what follows should be some kind of remarkable statement, not just the belief (already implied by the Tanakh) that no one will have succeeded in killing all of the Jews when the End Times come.
That’s a lot of Bruce’s critique as well, which at one time I found to be quite persuasive. But remember that throughout the scripture we hear this fear raised. The high priest said Jesus should be killed so that the Romans don’t come and destroy their nation. Many distinct nations and cultures have in the past been absorbed into the other nations thus essentially wiping out their “race.” This may have been part of Caiaphas’ concern. The Jews are one of the few dispersed nations which have not been totally absorbed.
If "this race shall not pass away until" means the actual annihilation of the people, this too has always been a great fear. Throughout the Hebrew scripture God reminds the people that though great destruction will come through God’s judgment, still there will be a remnant. It’s like a reminder, given so many times before, that Israel will remain no matter how horrible the persecution for the time being, no matter how effective the Hitlers of the ages appear to be in achieving their agendas. This is certainly a promise which deserves a “truly I say unto you.”
It also appears to be a time marker. The race of the Jewish people (not a modern idea of race but certainly a very old one) will pass away when all things are fulfilled and God becomes all in all. The very next verse (in all three synoptic gospels) tells us that heaven and earth will pass away but his word will not. The proximity of these verses suggests that even the passing of heaven and earth must take place before the final time marker is set, the end of the Jewish race. The righteous Jews will be fully unified with the righteous Gentiles into the one people of God. Jesus’ statement that we will be like the angels in heaven may even be suggesting that all of redeemed humanity will pass away by becoming one with the angels. Okay, that’s a lot of speculation. But this does show that the genea-means-race interpretation is very possible. And for other reasons as well it seems more likely to me than other interpretations.
I still have to argue that animal sacrifice was always necessary for atonement in pre-Christian Israel. The meaning of Leviticus 17.11 is just too clear. When the prophets spoke of God desiring repentance or obedience or righteousness and despising the sacrifices, they was using hyperbole. God was so angry at the people’s attitude—sometimes they were planning their next sin, sometimes they were actually doing evil while they were making their burnt offerings—that God spoke as though God never wanted another animal sacrifice. If that was what the prophets were actually saying, they would be opposing the Torah teaching which, as JPH pointed out, a prophet simply cannot do unless he or she were a false prophet.
Not all of the sacrifices in the Torah involve blood.
No, but notice that those that are meant for atonement are all tied to the blood. The flour offering which the poor may give instead of an animal sacrifice was taken by the priest and burnt on the alter “on top of the offerings made to the Lord by fire” (Lev 5.12). It was effective as an atoning sacrifice because it was made part of another blood sacrifice.
The correct Christian theology is not that the sacrifice of Christ meets the technical provisions for a Torah sacrifice, pretty obviously it doesn't since human sacrifice was strictly forbidden.
True, it was a symbol pointing to Jesus’ atoning death but the point of the Torah teaching is that there can be no atonement without the shedding of blood. Substitution is necessary. The idea of laying hands on the scapegoat and confessing the sins of the people and sending it away into the wilderness shows that the animal takes the people’s sin away as their substitute. And yet this ceremony is also necessarily a part of a sacrificial ceremony. The other goat is killed. Both demonstrate the need for substitutionary atonement.
In fact, the number of passages on this theme is so extensive that he missed some of my favorites!
We can look at additional passages if you like, Aron, but I think we will always see some connection with the animal sacrifices or we will see that sacrifices are assumed as necessary to remove sin but simply not mentioned. Some passages often brought up by the anti-missionaries like Proverbs 16.6 aren’t speaking about atonement at all. The Proverbs passage is rather more likely speaking of someone who has been offended being reconciled to oneself through truthfulness and lovingkindness; that is through kindness, through humble and caring words and actions.
In the pages of the Tanakh. . . , sacrifice is regarded as inessential compared to what God really wants, which is true repentance.
Not quite inessential. Compared to repentance and living righteously, God did make it very clear that sacrifices were much less important. But the sacrifices were still essential.
I see one problem in your view, Aron. If you are saying that the shedding of blood is inessential for atonement, for removal of sin (I’m not sure that you are), are you disagreeing with the writer of Hebrews that without the shedding of blood there is no remission of sin? Are you saying that the Israelites’ sins, say David’s sin of adultery and murder, could be forgiven by repentance alone?
30. JPH,
I said, “That God would someday become a man was not revealed to them at that time.” To this you responded:
Then they are forbidden to worship Jesus and everyone else whom their forefathers didn’t know. Game over. Do not pass go.
Your “Game over” and “Do not pass go” sound a little desperate, JPH. Kind of like the child who sticks his fingers in his ears and yells out, “I’m not listening! I’m not listening!” No, you do need to listen to reasonable arguments that you are reading into the Torah more than it is actually saying.
To say that Jesus was in the person of the God who revealed himself to Moses is not to say that Jesus was a new God or another God. So Deuteronomy 13 does not apply to this claim or the claim of the Krishna devotee that Krishna was the God who spoke to Moses. Whether Jesus or Krishna was an incarnation of the God of Abraham must be determined by examining the evidence for their claims or the claims of their followers, not by reading into the Torah statements which are not there. Exodus 23 does not deny the possibility that God would incarnate in the person of Jesus since it says that one must not invoke the names of other gods. What is denied is that Jesus is another god.
The Lord would speak to Moses FACE TO FACE, as one speaks to a friend. (Deut 33:11) And there was NO OTHER PROPHET who arose in Israel like Moses, whom the Lord knew FACE TO FACE. (Deut 34:10)
Deuteronomy 34 is only saying that as of this writing no one has arisen equal to Moses. If you accept that Moses wrote the Torah then you should also accept that were some additional comments added to it. For example, Moses certainly didn’t write the part about how he died. If even a much later writer added this part about there never being a prophet like Moses, his statement is only good until his time. Verse 10 definitely does not say, “No prophet is greater than Moses and there’s an end of it.”
Deuteronomy 18.18 says that God will raise up a prophet like Moses. This may just mean that there will be some with the same ability Moses had to prophesy. But a good case can be made that the clearest meaning of the passage is that a special prophet will come who is greater than all the other prophets except for Moses. This will be Moses’ equal. Whether you accept that or not is no big deal. The fact is that even if God spoke to Moses face to face, God still may have not revealed to him all that God intended to eventually reveal.
There was nothing "limited" about it!
Then why did God tell Moses that the hidden things of God belong to God and the things that belong to Israel have been revealed to them (Deut 29.28)? Because there is clearly much that was not revealed to Moses. (Or if it was, God made it clear to him that nothing of this should be given to the people.)
I said, “But notice that the Torah does not say that God cannot or will not become a man.” To this you responded:
Descendants of Jacob, I am the LORD All-Powerful, AND I NEVER CHANGE. (Malachi 3:6)
Certainly God does not change in God’s essential nature. For example, God is good, God can never become evil. Yet sometimes God would do things like take on a temporary human appearance and speak with someone like Abraham. Was God changing when God did so? (Remember that Abraham addressed this person not as an angel but as God himself.) If God is not an absolute unity in person and if one such Person might take on a human appearance (whether for only a few hours or days or by means of an actual human body), then another Person of the Godhead may remain in heaven in a more completely unchanged form. Malachi 3 could be speaking of a more unchangeable nature of the first Person of the Trinity. Nevertheless, even with this, I find it hard to accept that God the Father is absolutely unchanging. That’s why I say that God is unchanging in essential nature. And even with these trinitarian qualifications, God must still change in some sense to take on the appearance of a man to appear to Abraham or as Jesus.
So you tell me, did God change when God visited Abraham? Did God change when God walked in front of Moses and “changed” from one location to another? Does that count as change or not?
Behold, heaven and the heaven of heavens cannot contain You.
Then how was God able to take on a human form and appear to Moses? How could Isaiah see God on his throne if God cannot reduce himself from filling the heaven of the heavens? Obviously God does fill the heavens of the heavens and yet can enter the smallest particle. This same God can take on human form and become a man.
So whenever God says X it’s only true at the immediate instant He says it?
No, we don’t know if it is true forever unless God says it’s true forever. If God does not say, then we need to be open to the possibility that it will change. But it takes God to change it or to add to it and we need good evidence that it has changed. When God says this covenant is forever, the word used means essentially for as far into the future as the Israelites could imagine or think about. It does not rule out the possibility that it may have an end or that it may not, but it is only God who can tell us if or when it will have an end.
The command against adultery is a moral law, not a civil or ceremonial law. It’s also one of those laws we all know by nature to be right and, being a moral law, it’s one which has continued through Christian teachings.
Why wouldn’t the Almighty simply have said “God is not YET a man”?
Because that would be giving information which God did not want disclosed until the right time. Given your view, what I would like to ask you is, Why wouldn’t God simply say, “God is not a man and will never will become a man”?
I said that such a man would be sinless and thus cannot lie. To this you responded,
That’s not what God is saying through Balaam. Rather, he means that ALL humans lie and ALL humans need to repent. That’s one reason God can’t be a man.
Again you’re reading into the statement claims which are not there. Your stretching it to make it say what you want it to say. Even if all humans inevitably lie at some time, we are not told that God cannot become or even create a human who will not lie.
You say that the NT book of Hebrews must receive authority from the Torah if anything it says is to be accepted because the Torah is self-authenticating. But your link does not establish this. The miracles of the Exodus could have been made up by a later writer and gradually introduced to the people. The most dominant secular view is that the Exodus never happened and the story was made up around the time of the Exile or after. I agree that it just doesn’t seem likely that such a story could be foisted upon an entire population even somewhat slowly. It is even less likely to be accepted quickly. Nevertheless, the historicity of the Exodus, even without the miraculous elements, is not certain or self-authenticating.
You keep making statements you cannot support, JPH. No one would be put to death for giving a prophecy of a person or event to fulfill the Torah. If one religion claims to have the last word and another the first word which will never be followed by a later word, which one is right must be determined by evidence, not by suggesting that one is legitimate and the other is not.
I said, “I think I mentioned last time a Christian account of how those Gentiles living before the time of Jesus are made right with God. It still indirectly requires blood sacrifice, that of Jesus. So none ‘of these passages actually preclude the requirement of a blood offering as a necessary component in the process of forgiveness from sin.’ ” To this you responded:
You can’t assume the truth of your conclusion as a premise. The Book of Jonah says the Ninevites repented and God forgave them – without any sacrifices of any kind whatsoever. This is evidence against Christianity. You can’t assert the Christian account as an argument for Christianity. How is this supposed to convince me?
You’ve confused my arguments. Remember that you had first given passages which you claimed show forgiveness can come through repentance alone and that no animal sacrifice is needed. I gave a strong argument that these passages cannot mean that and that animal sacrifice was still needed. Then citing NT passages, I gave a feasible Christian account of how Gentiles prior to the time of Jesus may be accepted by God by means of the shedding of blood. The basic condition of seeking God (for those who do not know God exists as well as those who do), fearing or honoring God (for those who do know), and seeking to live in such a way as to do what is right morally is enough in God’s sight initially. Of course this involves repentance when one has acted in opposition to one’s moral awareness. And sometimes just the convicting word of a prophet like Jonah will be enough to ignite such repentance. At some point after their death and after the time of Jesus, since they had fulfilled the first requirement, they are next given the knowledge of Jesus’ atoning sacrifice and the opportunity to accept it. (And one will accept it insofar as one continues to seek God and seek to do God’s will.) Jesus’ death is the fulfillment of the suffering Servant prophecy which in turn fulfills the animal sacrifice requirements of the Torah in this Christian view.
So the point is this: since it can be shown that a Christian view of the pre-Christian Gentiles being accepted by God through a blood sacrifice fits a feasible view of the Tenakh, you have no grounds to say that the Ninevites definitely did not need a blood sacrifice to be forgiven. Since you were making the claim that the Ninevites were forgiven without blood sacrifice, the burden of proof was on you at that point to support this claim. All I had to do to refute your claim was to give a coherent account which is consistent with the teachings of the Tenakh of how they could be forgiven with a blood sacrifice.
I said, “If the Messiah is now with God and the process of history is under God’s control, then the Messiah is working with God as history is being manipulated for this purpose.” You responded:
Again, you can’t assume the truth of your conclusion as a premise. You cited Israel as EVIDENCE for Christianity. When I asked you to enlarge on your position you assert a conditional: IF Jesus is the Messiah THEN history is being manipulated for this purpose. So if your conclusion is true, it follows that your conclusion is true.
No I did not cite the regathering of Israel as evidence of Christianity. I was very careful of my wording. I said, “So we have no good reason to think Messiah [Jesus] has not fulfilled that part of the prophecy as well.” I did not say, “We have evidence that Jesus fulfilled the regathering prophecy.” I was refuting your claim that Jesus could have nothing to do with fulfilling this prophecy.
You asked, “Where do they [the Torah or the prophets] say he could [come to earth more than once], or that it should be thousands of years apart?” I answered,
“Luke and Paul talk about the time of the Gentiles. … I’m not sure if the prophets speak of this time but the NT does.” Your response was:
You can’t assume the truth of your position as a premise. Of course the NT says this. IT HAS TO.
One of my last sentences which you had cut out said, “If we have good evidence for the NT, this should be enough to see that this could be God’s plan” (emphasis added) and we have no need of evidence from the Tenakh. My point was that it does not matter if the Torah and the prophets said anything about the Messiah coming to earth more than once with a 2000 year separation. If we have other evidence of Christianity and if the general NT teaching adequately accounts for two appearances and this long of a separation, that’s all we need. I brought up the points about the time of the Gentiles era because it is a common objection that 2000 years is an unreasonably long separation for such elements of biblical prophecy. I also mentioned that the two appearances view does resolve some apparently discordant statements in the Hebrew scripture. But I said this stuff about “if we have good evidence for the NT” because you seemed to be only concerned about whether we can find evidence from the Torah and the prophets. So I did not assume the truth of my position as a premise. I was pointing out that there is other evidence I can refer to if you wish to look at it.
Evidence seems to be in the eye of the beholder. Tzvi makes Christians uncomfortable because the difference is one of degree, not kind.
No, Christians offer evidence from messianic prophecies and miracles (specifically the resurrection) and sometimes some other areas like religious experience. The evidence of miracles was the same kind of evidence that identified Moses as God’s chosen leader. So you can’t really ignore it before evaluating it. I took part in a long discussion on W L Craig’s forum page arguing that Daniel’s prophecy of the 70 weeks is strong evidence that Jesus is the Messiah. I know you probably think it doesn’t even speak of the Messiah but you should listen to the arguments that it does. I think it’s very strong. This is only evidence in the eye of the beholder for those who wish to close their eyes to that which they don’t want to see.
Your position invites these queries, part of why it’s unfathomable that God is/was a human. If Jesus was truly a man did he ever have lustful thoughts? If not, he wasn’t a man. That’s part of the package, intrinsic to the misery & joy of being human. A huge chunk of waking life consists in an endless barrage of lustful thoughts. Per Jesus’ standard (Matt 5:28) this is as bad as adultery. How do you square this triangle? If Satan tempted him with food, why not other drives?
Remember that other Jewish teachers of the time also regarded lust as bad as adultery. What Jesus said was that lusting after someone was as bad as adultery, not being tempted to lust. He was tempted in all points as we are yet without sin (Heb 4.15). The difficulty, of course, is understanding when the temptation becomes actual lust, when the attraction is contemplated sufficiently that it becomes sin. I’m not sure what the exact dividing line is. Maybe someone else has given some insights. What I would say is that Jesus never did cross that line. Paul spoke as though everyone else does inevitably cross that line even though they may try their hardest not to (Rom 7 & 8). Only as Christians do we have power to resist as God gives us this power.
I need to qualify an earlier statement I had made. I made it sound as though Jesus was simply taking one of two possible current views of divorce when he sided with Shammai against Hillel. I should have also mentioned that Jesus did consider the law of Moses deficient at this point. He said that God allowed this law condoning divorce because of the hardness of the people’s hearts. He didn’t negate the law but appealed to the original purpose in creation as given in Genesis 2.24 which made the Mosaic law still applicable though more qualified.
It is important to understand that there may be other Mosaic laws which were allowed because of the hardness of the people’s hearts. If we were to attack another town and the people surrender, should we enslave everyone? Should a virgin who is raped be forced to marry her rapist? These are the kinds of questions critics ask. I saw a recent meme which condemned Christianity as supposedly accepting the latter law and portrayed Satanism as more compassionate.
I’m also aware that some of these laws can be defended to a degree. At the time, enslaving a captured people (who must be the original aggressors) may have been the only way to ensure no future attacks. This may be a lesser of two evils problem. And of course there is much more to the marry-the-rapist law than I’ve mentioned which could greatly mitigate it. Nevertheless, we should see some laws such as those allowing slavery as having been permitted because of the hardness of the people’s hearts. They should no longer be allowed.
31. JPH & Aron
JPH, don’t worry about Feser on the Kalam. He’s right that Craig’s argument against actual infinities may not be sound, but there are other good philosophical arguments for a beginning. I see good evidence by looking at causality but Aron says that’s trickier so I suspect there is much there I haven’t even imagined. Have you gotten into this in any prior topics, Aron? I know we’ve talked about the Kalam before.
32. myth buster says:
How can anyone say that the Resurrection is irrelevant? To say that a false prophet could rise from the dead apart from the General Resurrection on the Last Day is to claim that someone other than HaShem has the power to raise the dead, or that HaShem would scandalize the world by appearing to vindicate a false prophet. No, to acknowledge that Jesus of Nazareth rose from the dead is to acknowledge Him as being vindicated by God, that though He had suffered the death of a criminal, being hung on a tree until dead, God judged Him innocent, and raised Him up from Sheol, because the one who keeps the Law shall live. If, therefore, you confess that Jesus of Nazareth rose from the dead, you must confess acknowledge Him as a true Prophet, and therefore affirm His claims that He is the Messiah, the Son of the Living God. If, despite this, He appears to be a sinner, it is your understanding of the Law that is deficient, for you find yourself opposing God's Judgment. If God has acquitted, who shall convict? So then, do you believe that Jesus of Nazareth is risen from the dead? | 2018-10-21 02:53:34 | {"extraction_info": {"found_math": true, "script_math_tex": 9, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 9, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3715556561946869, "perplexity": 2606.812896087985}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-43/segments/1539583513548.72/warc/CC-MAIN-20181021010654-20181021032154-00307.warc.gz"} |
https://socratic.org/questions/how-do-you-find-the-value-of-17-c-when-c-9 | # How do you find the value of 17-c when c=9?
Nov 27, 2016
$17 - 9 = 8$
Where ever you see the letter $c$ write 9
So $17 - c \text{ is the same as } 17 - 9$
$17 - 9 = 8$ | 2019-09-21 22:13:12 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 4, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.34997254610061646, "perplexity": 3375.0382381678496}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514574665.79/warc/CC-MAIN-20190921211246-20190921233246-00339.warc.gz"} |
http://www.chegg.com/homework-help/questions-and-answers/horizontal-platform-supports-3kg-stone-platform-goes-simple-harmonic-motion-frequency-2-hz-q2523197 | ## Physics
A horizontal platform supports a 3kg stone. If the platform under goes simple harmonic motion at a frequency of 2 Hz, what is the maximum amplitude allowed before the stone separates from the platform?
Please explain how to work out the problem and what formulas to use.
• Anonymous commented
yeah we are supposed to square it, i messed up in the last step, correct answer is 0.0621225507 metre | 2013-05-21 16:00:34 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.864700436592102, "perplexity": 847.9144067521404}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368700168711/warc/CC-MAIN-20130516102928-00021-ip-10-60-113-184.ec2.internal.warc.gz"} |
https://stacks.math.columbia.edu/tag/04DR | Lemma 59.43.4. Let $f : X \to Y$ be an integral morphism of schemes. Then property (B) holds.
Proof. Consider $V \to Y$ étale, $\{ U_ i \to X \times _ Y V\}$ an étale covering, and $v \in V$. We have to find a $V' \to V$ and decomposition and maps as in Lemma 59.43.2. We may shrink $V$ and $Y$, hence we may assume that $V$ and $Y$ are affine. Since $X$ is integral over $Y$, this also implies that $X$ and $X \times _ Y V$ are affine. We may refine the covering $\{ U_ i \to X \times _ Y V\}$, and hence we may assume that $\{ U_ i \to X \times _ Y V\} _{i = 1, \ldots , n}$ is a standard étale covering. Write $Y = \mathop{\mathrm{Spec}}(A)$, $X = \mathop{\mathrm{Spec}}(B)$, $V = \mathop{\mathrm{Spec}}(C)$, and $U_ i = \mathop{\mathrm{Spec}}(B_ i)$. Then $A \to B$ is an integral ring map, and $B \otimes _ A C \to B_ i$ are étale ring maps. By Algebra, Lemma 10.143.3 we can find a finite $A$-subalgebra $B' \subset B$ and an étale ring map $B' \otimes _ A C \to B'_ i$ for $i = 1, \ldots , n$ such that $B_ i = B \otimes _{B'} B'_ i$. Thus the question reduces to the étale covering $\{ \mathop{\mathrm{Spec}}(B'_ i) \to X' \times _ Y V\} _{i = 1, \ldots , n}$ with $X' = \mathop{\mathrm{Spec}}(B')$ finite over $Y$. In this case the result follows from Lemma 59.43.3. $\square$
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar). | 2022-10-02 13:11:32 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 2, "x-ck12": 0, "texerror": 0, "math_score": 0.994957447052002, "perplexity": 132.5646443076415}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030337322.29/warc/CC-MAIN-20221002115028-20221002145028-00185.warc.gz"} |