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https://www.sanfoundry.com/pavement-design-questions-answers-bituminous-material-tests-2/
|
# Pavement Design Questions and Answers – Bituminous Material Tests – 2
«
»
This set of Pavement Design Multiple Choice Questions & Answers (MCQs) focuses on “Bituminous Material Tests – 2”.
1. Solubility test in bitumen is used to determine ______ of bitumen.
a) Contamination
b) Solubility
c) Dispersion
d) Composition
Explanation: Bitumen is almost completely soluble in solutions of carbon disulphide and carbon tetrachloride. So, after conducting the solubility test, if there are any undissolved particles, those would indicate the presence of impurities or the contamination of bitumen.
2. What is the standard load applied to the sample when the needle is immersed in the penetration test?
a) 100 mg
b) 100 g
c) 200 mg
d) 200 g
Explanation: The standard load for the penetration test is 100 g and it is in accordance with IS 1203:1978. The needle is allowed to penetrate the sample and it imposes a load of 100 g when it does so.
3. The formula to calculate the specific gravity of bitumen is given by $$\frac{c-a}{(b-a)-(d-c)}$$. Which of the below symbol and explanations is incorrect?
a) a – weight of specific gravity bottle
b) b – weight of specific gravity bottle filled with distilled water
c) c – weight of specific gravity bottle filled with bituminous material
d) d – weight of specific gravity bottle half filled with bituminous material and rest with water
Explanation: The symbol c indicates the weight of specific gravity bottle half filled with bituminous material. All the measurements are taken in grams. All the symbols are relative to each other and by combining them, the specific gravity can be found out.
Sanfoundry Certification Contest of the Month is Live. 100+ Subjects. Participate Now!
4. What are the acceptable viscosity limits for bitumen emulsion RS-2 when tested using saybolt furol viscometer?
a) 30 – 150
b) 20 – 100
c) 50 – 300
d) 100 – 300
Explanation: The test is conducted as per IS 3117 and the viscosity limits are mentioned in IS 8887. The acceptable viscosity limits for RS-2 are 100 to 300. For RS-1, it is 20 to 100; for MS, it is 50 to 300; for SS-1, it is 20 to 100 and for SS-2, it is 30 to 150.
5. When is the bitumen sample considered cracked in the spot test?
b) Spot with uniform colour
d) Spot with varying colour
Explanation: The spot test consists of dropping two spots at different times on a filter paper. The spot spreads and if the colour is varying in the form of annular rings, darker at the centre, the bitumen sample is sadi to have cracked. It is not cracked if the spot spreads and has a uniform colour.
6. At what speed is the specimen pulled in the ductility test?
a) 50 cm per minute vertically
b) 50 mm per minute horizontally
c) 50 mm per minute vertically
d) 50 cm per minute horizontally
Explanation: The sample is filled in the briquette mould and the end clips are pulled uniformly at 50 mm per minute in the horizontal direction. When pulled horizontally, the sample would stretch and break, giving the ductility value.
7. What is the orifice size and temperature for testing cut-back bitumen using orifice viscometer?
a) 4 mm at 25°C
b) 10 mm at 45°C
c) 4 mm at 45°C
d) 10 mm at 50°C
Explanation: The orifice viscometer can be used to test the viscosity of tar and cut-back bitumen. For testing tar, the conditions are 10 mm orifice at 35/40/45/55°C. for cut-back bitumen, the conditions are 4mm orifice at 25°C or 10 mm orifice at 25/40°C.
8. Bitumen having a higher softening point is preferred in hot climates.
a) True
b) False
Explanation: Higher softening point would indicate that the bitumen sample has lower temperature susceptibility. It would take some time for the bitumen sample to be liquid. It can be used in hot climatic conditions, as higher temperature is required for bitumen to attain the liquid state.
9. The dial reading after conducting a penetration test is found to be 73. What is the penetration value of the sample?
a) 730 mm
b) 73 mm
c) 7.3 mm
d) 0.73 mm
Explanation: The dial readings are marked from zero to four hundred. Each reading is 1/10th of a millimetre i.e. if the reading is 73, the corresponding penetration value is 7.3 mm.
10. What does TSR stand for?
a) Temporary Softening Rate
b) Thermal Sensitivity Ratio
c) Terminal Strength Rate
d) Tensile Strength Ratio
Explanation: TSR of bitumen gives a measure of its resistance to moisture susceptibility.it is important to ascertain the sensitivity of bitumen to water, it affects the binding property. A higher TSR value indicates good resistance to moisture.
11. Kinematic viscosity is related to absolute viscosity by ______
a) Volume
b) Mass
c) Weight
d) Density
Explanation: Kinematic viscosity is determined by the ratio of absolute viscosity to density. The unit of kinematic viscosity is m2/s and the unit of absolute viscosity is N s/m2. Density is used to relate both the terms.
12. There are two methods to find the specific gravity of bitumen.
a) True
b) False
Explanation: As per IS 1202:1978, there are two methods to determine the specific gravity of bitumen. The first method is using a pycnometer and the second one is using a balance beam method. The descriptions for both are given in the code.
13. What is the temperature of the oven in the loss on heating test?
a) 163°C
b) 153°C
c) 110°C
d) 120°C
Explanation: As per IS 1212-1978, the oven is heated to a temperature of 163±1°C and the sample is placed in it for 5 hours. The container is heated to 100-110°C for 30 minutes, then cooled and weighed. Then the sample is poured into it.
14. What is the speed of rotation of stirrer used in flash and fire point test for cut-back bitumen?
a) 90 rpm
b) 60 rpm
c) 70 rpm
d) 50 rpm
Explanation: As per IS 1209-1978, the speed of rotation for bitumen is 60 rpm. For cut-back bitumen, it is taken as 70 to 80 rpm. All other things for conducting the test are the same for all types of bitumen.
15. What is the temperature at which softening point test has a different procedure?
a) 90°C
b) 80°C
c) 75°C
d) 65°C
Explanation: The test procedure is explained in IS 1205-1978. There is a change in the bath used when the material of softening point below 80°C and above 80°C is tested. For below 80°C, the bath used is water and for above 80°C, the bath is glycerine.
Sanfoundry Global Education & Learning Series – Pavement Design.
To practice all areas of Pavement Design, here is complete set of 1000+ Multiple Choice Questions and Answers.
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2022-10-04 19:00:22
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|
https://www.esaral.com/q/a-uniform-metallic-wire-is-elongated-by-0-04-m-88736/
|
A uniform metallic wire is elongated by 0.04 m
Question:
A uniform metallic wire is elongated by $0.04 \mathrm{~m}$ when subjected to a linear force $\mathrm{F}$.
The elongation, if its length and diameter is doubled andsubjected to the same force will be $\mathrm{cm}$.
Solution:
$\mathrm{y}=\frac{\mathrm{F} / \mathrm{A}}{\Delta \ell / \ell}$
$\Rightarrow \frac{F}{A}=y \frac{\Delta \ell}{\ell}$
$\Rightarrow \frac{F}{A}=y \times \frac{0.04}{\ell} \quad \ldots(1)$
When length $\backslash \&$ diameter is doubled.
$\Rightarrow \frac{F}{4 A}=y \times \frac{\Delta \ell}{2 \ell} \quad \cdots(2)$
$(1) \div(2)$
$\frac{\mathrm{F} / \mathrm{A}}{\mathrm{F} / 4 \mathrm{~A}}=\frac{\mathrm{y} \times \frac{0.04}{\ell}}{\mathrm{y} \times \frac{\Delta \ell}{2 \ell}}$
$4=\frac{0.04 \times 2}{\Delta \ell}$
$\Delta \ell=0.02$
$\Delta \ell=2 \times 10^{-2}$
$\therefore x=2$
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2022-07-03 14:16:14
|
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|
https://www.iacr.org/cryptodb/data/author.php?authorkey=4096
|
## CryptoDB
### Zheng Gong
#### Publications
Year
Venue
Title
2008
EPRINT
This paper reconsiders the security of the rate-1 double block length hash functions, which based on a block cipher with a block length of $n$-bit and a key length of $2n$-bit. Counter-examples and new attacks are presented on this general class of double block length hash functions with rate 1, which disclose there exist uncovered flaws in the former analysis given by Satoh \textit{et al.} and Hirose. Preimage and second preimage attacks are designed to break Hirose's two examples which were left as an open problem. Some refined conditions are proposed for ensuring this general class of the rate-1 hash functions to be optimally secure against the collision attack. In particular, two typical examples, which designed under the proposed conditions, are proven to be indifferentiable from the random oracle in the ideal cipher model. The security results are extended to a new class of double block length hash functions with rate 1, where one block cipher used in the compression function has the key length is equal to the block length, while the other is doubled.
2008
EPRINT
This paper reconsiders the security of the rate-1 double block length hash functions, which based on a block cipher with a block length of $n$-bit and a key length of $2n$-bit. Counter-examples and new attacks are presented on this general class of double block length hash functions with rate 1, which disclose there exist uncovered flaws in the former analysis given by Satoh \textit{et al.} and Hirose. Preimage and second preimage attacks are designed to break Hirose's two examples which were left as an open problem. Some refined conditions are proposed for ensuring this general class of the rate-1 hash functions to be optimally secure against the collision attack. In particular, two typical examples, which designed under the proposed conditions, are proven to be indifferentiable from the random oracle in the ideal cipher model. The security results are extended to a new class of double block length hash functions with rate 1, where one block cipher used in the compression function has the key length is equal to the block length, while the other is doubled.
2007
EPRINT
Nowadays, investigating what construction is better to be a cryptographic hash function is red hot. In TCC'04, Maurer et al. first introduced the notion of indifferentiability as a generalization of the concept of the indistinguishability of two cryptosystems. In AsiaCrypt 06, Chang et al. analyzed the indifferentiability security of some popular block-cipher-based hash functions, such as PGV constructions and MDC-2. In this paper, we investigate Chang et al.'s analysis of PGV constructions and the PBGV double block length constructions. In particular, we point out a more precise adversarial advantage of indifferentiability, by considering the two situations that whether the hash function is either keyed or not. Furthermore, Chang et al. designed attacks on 4 PGV hash functions and PBGV hash function to prove they are differentiable from random oracle with prefix-free padding. We find a limitation in their differentiable attacks and construct our simulations to obtain the controversy results that those schemes are indifferentiable from random oracle with prefix-free padding and some other popular constructions.
Kefei Chen (3)
Xuejia Lai (3)
|
2019-07-23 08:18:05
|
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|
https://rdrr.io/cran/DescTools/man/SomersDelta.html
|
# SomersDelta: Somers' Delta In DescTools: Tools for Descriptive Statistics
## Description
Calculate Somers' Delta statistic, a measure of association for ordinal factors in a two-way table. The function has interfaces for a table (matrix) and for single vectors.
## Usage
1 SomersDelta(x, y = NULL, direction = c("row", "column"), conf.level = NA, ...)
## Arguments
x a numeric vector or a table. A matrix will be treated as table. y NULL (default) or a vector with compatible dimensions to x. If y is provided, table(x, y, ...) is calculated. direction direction of the calculation. Can be "row" (default) or "column", where "row" calculates Somers' D (R | C) ("column dependent"). conf.level confidence level of the interval. If set to NA (which is the default) no confidence interval will be calculated. ... further arguments are passed to the function table, allowing i.e. to set useNA. This refers only to the vector interface.
## Details
Somers' D(C|R) and Somers' D(R|C) are asymmetric modifications of τ_b and Goodman-Kruskal's Gamma. C|R indicates that the row variable x is regarded as the independent variable and the column variable y is regarded as dependent. Similarly, R|C indicates that the column variable y is regarded as the independent variable and the row variable x is regarded as dependent. It is logically very similar to Gamma, but differs in that it uses a correction only for pairs that are tied on the dependent variable. As Gamma and the Taus, D is appropriate only when both variables lie on an ordinal scale.
Somers' D is computed as
D(C | R) = \frac{P-Q}{n^2 - ∑(n_i.^2)}
where P equals twice the number of concordances and Q twice the number of discordances and n_i. rowSums(tab). Its range lies [-1, 1]. The interpretation of d is analogous to Gamma.
## Value
a single numeric value if no confidence intervals are requested
and otherwise a numeric vector with 3 elements for the estimate, the lower and the upper confidence interval
## Author(s)
Andri Signorell <[email protected]>
## References
Agresti, A. (2002) Categorical Data Analysis. John Wiley & Sons, pp. 57–59.
Brown, M.B., Benedetti, J.K.(1977) Sampling Behavior of Tests for Correlation in Two-Way Contingency Tables, Journal of the American Statistical Association, 72, 309-315.
Goodman, L. A., & Kruskal, W. H. (1954) Measures of association for cross classifications. Journal of the American Statistical Association, 49, 732-764.
Somers, R. H. (1962) A New Asymmetric Measure of Association for Ordinal Variables, American Sociological Review, 27, 799–811.
Goodman, L. A., & Kruskal, W. H. (1963) Measures of association for cross classifications III: Approximate sampling theory. Journal of the American Statistical Association, 58, 310–364.
There's an implementation of Somers's D in Frank Harrell's Hmisc somers2, which is quite fast for large sample sizes. However it is restricted to computing Somers' Dxy rank correlation between a variable x and a binary (0-1) variable y.
ConDisPairs yields concordant and discordant pairs
Other association measures:
KendallTauA (tau-a), KendallTauB (tau-b), cor (method="kendall") for tau-b, StuartTauC (tau-c), GoodmanKruskalGamma
Lambda, GoodmanKruskalTau, UncertCoef, MutInf
## Examples
1 2 3 4 5 6 7 8 9 10 # example in: # http://support.sas.com/documentation/cdl/en/statugfreq/63124/PDF/default/statugfreq.pdf # pp. S. 1821 tab <- as.table(rbind(c(26,26,23,18,9),c(6,7,9,14,23))) # Somers' D C|R SomersDelta(tab, direction="column", conf.level=0.95) # Somers' D R|C SomersDelta(tab, direction="row", conf.level=0.95)
### Example output
somers lwr.ci ups.ci
0.4426720 0.2785571 0.6067869
somers lwr.ci ups.ci
0.2569444 0.1592938 0.3545951
DescTools documentation built on Jan. 18, 2020, 1:09 a.m.
|
2020-02-22 06:17:15
|
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|
https://deepai.org/publication/hypothesis-set-stability-and-generalization
|
# Hypothesis Set Stability and Generalization
We present an extensive study of generalization for data-dependent hypothesis sets. We give a general learning guarantee for data-dependent hypothesis sets based on a notion of transductive Rademacher complexity. Our main results are two generalization bounds for data-dependent hypothesis sets expressed in terms of a notion of hypothesis set stability and a notion of Rademacher complexity for data-dependent hypothesis sets that we introduce. These bounds admit as special cases both standard Rademacher complexity bounds and algorithm-dependent uniform stability bounds. We also illustrate the use of these learning bounds in the analysis of several scenarios.
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## Authors
• 26 publications
• 3 publications
• 16 publications
• 32 publications
• 36 publications
• 33 publications
• ### On the Rademacher Complexity of Linear Hypothesis Sets
Linear predictors form a rich class of hypotheses used in a variety of l...
07/21/2020 ∙ by Pranjal Awasthi, et al. ∙ 0
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• ### An Exponential Efron-Stein Inequality for Lq Stable Learning Rules
There is accumulating evidence in the literature that stability of learn...
03/12/2019 ∙ by Karim Abou-Moustafa, et al. ∙ 0
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• ### Learning from weakly dependent data under Dobrushin's condition
Statistical learning theory has largely focused on learning and generali...
06/21/2019 ∙ by Yuval Dagan, et al. ∙ 0
read it
• ### Stacking and stability
Stacking is a general approach for combining multiple models toward grea...
01/26/2019 ∙ by Nino Arsov, et al. ∙ 0
read it
• ### Stability revisited: new generalisation bounds for the Leave-one-Out
The present paper provides a new generic strategy leading to non-asympto...
08/23/2016 ∙ by Alain Celisse, et al. ∙ 0
read it
• ### New Analysis and Algorithm for Learning with Drifting Distributions
We present a new analysis of the problem of learning with drifting distr...
05/19/2012 ∙ by Mehryar Mohri, et al. ∙ 0
read it
• ### Generalization Bounds for Learning with Linear, Polygonal, Quadratic and Conic Side Knowledge
In this paper, we consider a supervised learning setting where side know...
05/30/2014 ∙ by Theja Tulabandhula, et al. ∙ 0
read it
##### This week in AI
Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.
## 1 Introduction
Most generalization bounds in learning theory hold for a fixed hypothesis set, selected before receiving a sample. This includes learning bounds based on covering numbers, VC-dimension, pseudo-dimension, Rademacher complexity, local Rademacher complexity, and other complexity measures (Pollard, 1984; Zhang, 2002; Vapnik, 1998; Koltchinskii and Panchenko, 2002; Bartlett et al., 2002). Some alternative guarantees have also been derived for specific algorithms. Among them, the most general family is that of uniform stability bounds given by Bousquet and Elisseeff (2002). These bounds were recently significantly improved by Feldman and Vondrak (2018), who proved guarantees that are informative, even when the stability parameter is only in , as opposed to . New bounds for a restricted class of algorithms were also recently presented by Maurer (2017)
, under a number of assumptions on the smoothness of the loss function. Appendix
A gives more background on stability.
In practice, machine learning engineers commonly resort to hypothesis sets depending on the
same sample
as the one used for training. This includes instances where a regularization, a feature transformation, or a data normalization is selected using the training sample, or other instances where the family of predictors is restricted to a smaller class based on the sample received. In other instances, as is common in deep learning, the data representation and the predictor are learned using the same sample. In ensemble learning, the sample used to train models sometimes coincides with the one used to determine their aggregation weights. However, standard generalization bounds cannot be used to provide guarantees for these scenarios since they assume a fixed hypothesis set.
This paper studies generalization in a broad setting that admits as special cases both that of standard learning bounds for fixed hypothesis sets based on some complexity measure, and that of algorithm-dependent uniform stability bounds. We present an extensive study of generalization for sample-dependent hypothesis sets, that is for learning with a hypothesis set selected after receiving the training sample . This defines two stages for the learning algorithm: a first stage where is chosen after receiving , and a second stage where a hypothesis is selected from . Standard generalization bounds correspond to the case where is equal to some fixed independent of . Algorithm-dependent analyses, such as uniform stability bounds, coincide with the case where is chosen to be a singleton . Thus, the scenario we study covers both existing settings and, additionally, includes many other intermediate scenarios. Figure 1 illustrates our general scenario.
We present a series of results for generalization with data-dependent hypothesis sets. We first present general learning bounds for data-dependent hypothesis sets using a notion of transductive Rademacher complexity (Section 3). These bounds hold for arbitrary bounded losses and improve upon previous guarantees given by Gat (2001) and Cannon et al. (2002) for the binary loss, which were expressed in terms of a notion of shattering coefficient adapted to the data-dependent case, and are more explicit than the guarantees presented by Philips (2005)[corollary 4.6 or theorem 4.7]. Nevertheless, such bounds may often not be sufficiently informative, since they ignore the relationship between hypothesis sets based on similar samples.
To derive a finer analysis, we introduce a key notion of hypothesis set stability, which admits algorithmic stability as a special case, when the hypotheses sets are reduced to singletons. We also introduce a new notion of Rademacher complexity for data-dependent hypothesis sets. Our main results are two generalization bounds for stable data-dependent hypothesis sets, both expressed in terms of the hypothesis set stability parameter, our notion of Rademacher complexity, and a notion of cross-validation stability that, in turn, can be upper-bounded by the diameter of the family of hypothesis sets. Our first learning bound (Section 4) is expressed in terms of a finer notion of diameter but admits a dependency in terms of the stability parameter similar to that of uniform stability bounds of Bousquet and Elisseeff (2002). In Section 5, we use proof techniques from the differential privacy literature (Steinke and Ullman, 2017; Bassily et al., 2016; Feldman and Vondrak, 2018) to derive a learning bound expressed in terms of a somewhat coarser definition of diameter but with a more favorable dependency on , matching the dependency of the recent more favorable bounds of Feldman and Vondrak (2018). Our learning bounds admit as special cases both standard Rademacher complexity bounds and algorithm-dependent uniform stability bounds.
Shawe-Taylor et al. (1998) presented an analysis of structural risk minimization over data-dependent hierarchies based on a concept of luckiness
, which generalizes the notion of margin of linear classifiers. Their analysis can be viewed as an alternative study of data-dependent hypothesis sets, using luckiness functions and
-smallness (or -smoothness) conditions. A luckiness function helps decompose a hypothesis set into lucky sets, that is sets of functions luckier than a given function. The -smallness condition requires that the size of the family of loss functions corresponding to the lucky set of any function
with respect to a double-sample, measured by packing or covering numbers, be bounded with high probability by a function
of the luckiness of on the sample. The luckiness framework is attractive and the notion of luckiness, for example margin, can in fact be combined with our results. However, finding pairs of truly data-dependent luckiness and -smallness functions, other than those based on the margin and the empirical VC-dimension, is quite difficult, in particular because of the very technical -smallness condition (see Philips, 2005, p. 70). In contrast, our hypothesis set stability is simpler and often easier to bound. The notions of luckiness and -smallness have also been used by Herbrich and Williamson (2002) to derive algorithm-specific guarantees. The authors show a connection with algorithmic stability (not hypothesis set stability), at the price of a guarantee requiring the strong condition that the stability parameter be in , where is the sample size (see Herbrich and Williamson, 2002, pp. 189-190).
In section 6, we illustrate the generality and the benefits of our hypothesis set stability learning bounds by applying them to the analysis of several scenarios (see also Appendix K). In Appendix J, we briefly discuss several extensions of our framework and results, including the extension to almost everywhere hypothesis set stability as in (Kutin and Niyogi, 2002). The next section introduces the definitions and properties used in our analysis.
## 2 Definitions and Properties
Let be the input space and the output space. We denote by the unknown distribution over according to which samples are drawn.
The hypotheses we consider map to a set sometimes different from . For example, in binary classification, we may have and . Thus, we denote by a loss function defined on and taking non-negative real values bounded by one. We denote the loss of a hypothesis at point by . We denote by the generalization error or expected loss of a hypothesis and by its empirical loss over a sample :
R(h)=Ez∼D[L(h,z)]ˆRS(h)=Ez∼S[L(h,z)]=1mm∑i=1L(h,zi).
In the general framework we consider, a hypothesis set depends on the sample received. We will denote by the hypothesis set depending on the labeled sample of size .
[Hypothesis set uniform stability] Fix . We will say that a family of data-dependent hypothesis sets is -uniformly stable (or simply -stable) for some , if for any two samples and of size differing only by one point, the following holds:
∀h∈HS,∃h′∈HS′:∀z∈Z,|L(h,z)−L(h′,z)|≤β. (1)
Thus, two hypothesis sets derived from samples differing by one element are close in the sense that any hypothesis in one admits a counterpart in the other set with -similar losses.
Next, we define a notion of cross-validation stability for data-dependent hypothesis sets. The notion measures the maximal change in loss of a hypothesis on a training example and the loss of a hypothesis on the same training example, when the hypothesis is chosen from the hypothesis set corresponding to the a sample where the training example in question is replaced by a newly sampled example. [Hypothesis set Cross-Validation (CV) stability] Fix . We will say that a family of data-dependent hypothesis sets has CV-stability for some , if the following holds (here, denotes the sample obtained by replacing by ):
∀S∈Zm: Ez′∼D,z∼S[suph∈HS,h′∈HSz↔z′L(h′,z)−L(h,z)]≤χ. (2)
We say that has average CV-stability for some if the following holds:
ES∼Dmz′∼D,z∼S[suph∈HS,h′∈HSz↔z′L(h′,z)−L(h,z)]≤¯χ. (3)
We also define a notion of diameter of data-dependent hypothesis sets, which is useful in bounding CV-stability. In applications, we will typically bound the diameter, thereby the CV-stability. [Diameter of data-dependent hypothesis sets] Fix . We define the diameter and average diameter of a family of data-dependent hypothesis sets by
Δ=supS∈ZmEz∼S[suph,h′∈HSL(h′,z)−L(h,z)] ¯Δ=ES∼Dmz∼S[suph,h′∈HSL(h′,z)−L(h,z)]. (4)
Notice that, for consistent hypothesis sets, the diameter is reduced to zero since for any and . As mentioned earlier, CV-stability of hypothesis sets can be bounded in terms of their stability and diameter: A family of data-dependent hypothesis sets with -uniform stability, diameter , and average diameter has -CV-stability and -average CV-stability. Let , , and . For any and , by the -uniform stability of , there exists such that . Thus,
L(h′,z)−L(h,z)=L(h′,z)−L(h′′,z)+L(h′′,z)−L(h,z)≤β+suph′′,h∈HSL(h′′,z)−L(h,z).
This implies the inequality
suph∈HS,h′∈HSz↔z′L(h′,z)−L(h,z)≤β+suph′′,h∈HSL(h′′,z)−L(h,z),
and the lemma follows.
We also introduce a new notion of Rademacher complexity for data-dependent hypothesis sets. To introduce its definition, for any two samples
and a vector of Rademacher variables
, denote by the sample derived from by replacing its th element with the th element of , for all with . We will use to denote the hypothesis set .
[Rademacher complexity of data-dependent hypothesis sets] Fix . The empirical Rademacher complexity and the Rademacher complexity of a family of data-dependent hypothesis sets for two samples and in are defined by
ˆR⋄S,T(H)=1mEσ[suph∈HσS,Tm∑i=1σih(zTi)]R⋄m(H)=1mES,T∼Dmσ[suph∈HσS,Tm∑i=1σih(zTi)]. (5)
When the family of data-dependent hypothesis sets is -stable with , the empirical Rademacher complexity is sharply concentrated around its expectation , as with the standard empirical Rademacher complexity (see Lemma B.2).
Let denote the union of all hypothesis sets based on subsamples of of size : . Since for any , we have , the following simpler upper bound in terms of the standard empirical Rademacher complexity of can be used for our notion of empirical Rademacher complexity:
R⋄m(H) ≤1mES,T∼Dmσ[suph∈HS,Tm∑i=1σih(zTi)]=ES,T∼Dm[ˆRT(HS,T)],
where is the standard empirical Rademacher complexity of for the sample .
The Rademacher complexity of data-dependent hypothesis sets can be bounded by , as indicated previously. It can also be bounded directly, as illustrated by the following example of data-dependent hypothesis sets of linear predictors. For any sample , define the hypothesis set as follows:
HS={x↦wS⋅x: wS=m∑i=1αixSi,∥α∥1≤Λ1},
where . Define and as follows: and . Then, it can be shown that the empirical Rademacher complexity of the family of data-dependent hypothesis sets can be upper-bounded as follows (Lemma B.1):
ˆR⋄S,T(H)≤rTrS∪TΛ1√2log(4m)m≤r2S∪TΛ1√2log(4m)m.
Notice that the bound on the Rademacher complexity is non-trivial since it depends on the samples and , while a standard Rademacher complexity for non-data-dependent hypothesis set containing would require taking a maximum over all samples of size . Other upper bounds are given in Appendix B.
Let denote the family of loss functions associated to :
GS={z↦L(h,z):h∈HS}, (6)
and let denote the family of hypothesis sets . Our main results will be expressed in terms of . When the loss function is -Lipschitz, by Talagrand’s contraction lemma (Ledoux and Talagrand, 1991), in all our results, can be replaced by .
## 3 General learning bound for data-dependent hypothesis sets
In this section, we present general learning bounds for data-dependent hypothesis sets that do not make use of the notion of hypothesis set stability.
One straightforward idea to derive such guarantees for data-dependent hypothesis sets is to replace the hypothesis set depending on the observed sample by the union of all such hypothesis sets over all samples of size , . However, in general, can be very rich, which can lead to uninformative learning bounds. A somewhat better alternative consists of considering the union of all such hypothesis sets for samples of size included in some supersample of size , with , . We will derive learning guarantees based on the maximum transductive Rademacher complexity of . There is a trade-off in the choice of : smaller values lead to less complex sets , but they also lead to weaker dependencies on sample sizes. Our bounds are more refined guarantees than the shattering-coefficient bounds originally given for this problem by Gat (2001) in the case , and later by Cannon et al. (2002) for any . They also apply to arbitrary bounded loss functions and not just the binary loss. They are expressed in terms of the following notion of transductive Rademacher complexity for data-dependent hypothesis sets:
ˆR⋄U,m(G)=Eσ[suph∈¯¯¯¯¯¯HU,m1m+nm+n∑i=1σiL(h,zUi)],
where and where is a vector of
independent random variables taking value
with probability , and with probability . Our notion of transductive Rademacher complexity is simpler than that of El-Yaniv and Pechyony (2007) (in the data-independent case) and leads to simpler proofs and guarantees. A by-product of our analysis is learning guarantees for standard transductive learning in terms of this notion of transductive Rademacher complexity, which can be of independent interest.
Let be a family of data-dependent hypothesis sets. Then, for any with and any , the following inequality holds:
P[suph∈HSR(h)−ˆRS(h)>ϵ]≤exp⎡⎣−2ηmnm+n[ϵ2−maxU∈Zm+nˆR⋄U,m(G)−√log(2e)(m+n)32(mn)2]2⎤⎦,
where . For , the inequality becomes:
We use the following symmetrization result, which holds for any with for data-dependent hypothesis sets (Lemma D, Appendix D):
To bound the right-hand side, we use an extension of McDiarmid’s inequality to sampling without replacement (Cortes et al., 2008) applied to . Lemma E (Appendix E) is then used to bound in terms of our notion of transductive Rademacher complexity. The full proof is given in Appendix C.
## 4 Learning bound for stable data-dependent hypothesis sets
In this section, we present generalization bounds for data-dependent hypothesis sets using the notion of Rademacher complexity defined in the previous section, as well as that of hypothesis set stability.
Let be a -stable family of data-dependent hypothesis sets with average CV-stability. Let be defined as in (6). Then, for any , with probability at least over the draw of a sample , the following inequality holds for all :
∀h∈HS,R(h)≤ˆRS(h)+min{2R⋄m(G),¯χ}+[1+2βm]√log1δ2m. (7)
For any two samples , define as follows:
Ψ(S,S′)=suph∈HSR(h)−ˆRS′(h).
The proof consists of applying McDiarmid’s inequality to . The first stage consists of proving the -sensitivity of , with . The main part of the proof then consists of upper bounding the expectation in terms of both our notion of Rademacher complexity, and in terms of our notion of cross-validation stability. The full proof is given in Appendix F. The generalization bound of the theorem admits as a special case the standard Rademacher complexity bound for fixed hypothesis sets (Koltchinskii and Panchenko, 2002; Bartlett and Mendelson, 2002): in that case, we have for some , thus coincides with the standard Rademacher complexity ; furthermore, the family of hypothesis sets is -stable, thus the bound holds with . It also admits as a special case the standard uniform stability bound (Bousquet and Elisseeff, 2002): in that case, is reduced to a singleton, , and our notion of hypothesis set stability coincides with that of uniform stability of single hypotheses; furthermore, we have , since . Thus, using in the right-hand side inequality, the expression of the learning bound matches that of a uniform stability bound for single hypotheses.
## 5 Differential privacy-based bound for stable data-dependent hypothesis sets
In this section, we use recent techniques introduced in the differential privacy literature to derive improved generalization guarantees for stable data-dependent hypothesis sets (Steinke and Ullman, 2017; Bassily et al., 2016) (see also (McSherry and Talwar, 2007)). Our proofs also benefit from the recent improved stability results of Feldman and Vondrak (2018). We will make use of the following lemma due to Steinke and Ullman (2017, Lemma 1.2), which reduces the task of deriving a concentration inequality to that of upper bounding an expectation of a maximum.
Fix . Let
be a random variable with probability distribution
and independent copies of . Then, the following inequality holds:
We will also use the following result which, under a sensitivity assumption, further reduces the task of upper bounding the expectation of the maximum to that of bounding a more favorable expression. The sensitivity of a function is .
[(McSherry and Talwar, 2007; Bassily et al., 2016; Feldman and Vondrak, 2018)] Let be scoring functions with sensitivity . Let be the algorithm that, given a dataset and a parameter , returns the index with probability proportional to . Then, is -differentially private and, for any , the following inequality holds:
Notice that, if we define , then, by the same result, the algorithm returning the index with probability proportional to is -differentially private and the following inequality holds for any :
max{0,maxk∈[p]{fk(S)}}=maxk∈[p+1]{fk(S)}≤Ek=A(S)[fk(S)]+2Δϵlog(p+1). (8)
Let be a -stable family of data-dependent hypothesis sets with CV-stability. Let be defined as in (6). Then, for any , with probability at least over the draw of a sample , the following inequality holds for all :
R(h)≤ˆRS(h)+min{2R⋄m(G)+[1+2βm]√log2δ2m,√eχ+4√[1m+2β]log[6δ]}.
For any two samples of size , define as follows:
Ψ(S,S′)=suph∈HSR(h)−ˆRS′(h).
The proof consists of deriving a high-probability bound for . To do so, by Lemma 5 applied to the random variable , it suffices to bound , where with , , independent samples of size drawn from . To bound that expectation, we use Lemma 5 and instead bound , where is an -differentially private algorithm. To apply Lemma 5, we first show that, for any , the function is -sensitive with . Lemma G helps us express our upper bound in terms of the CV stability coefficient . The full proof is given in Appendix G. The hypothesis set-stability bound of this theorem admits the same favorable dependency on the stability parameter as the best existing bounds for uniform-stability recently presented by Feldman and Vondrak (2018). As with Theorem 4, the bound of Theorem 5 admits as special cases both standard Rademacher complexity bounds ( for some fixed and ) and uniform-stability bounds (). In the latter case, our bound coincides with that of Feldman and Vondrak (2018) modulo constants that could be chosen to be the same for both results.111The differences in constant terms are due to slightly difference choices of the parameters and a slightly different upper bound in our case where multiplies the stability and the diameter, while the paper of Feldman and Vondrak (2018) does not seem to have that factor. Notice that the current bounds for standard uniform stability may not be optimal since no matching lower bound is known yet (Feldman and Vondrak, 2018). It is very likely, however, that improved techniques used for deriving more refined algorithmic stability bounds could also be used to improve our hypothesis set stability guarantees. In Appendix H, we give an alternative version of Theorem 5 with a proof technique only making use of recent methods from the differential privacy literature, including to derive a Rademacher complexity bound. It might be possible to achieve a better dependency on for the term in the bound containing the Rademacher complexity. In Appendix I, we initiate such an analysis by deriving a finer analysis on the expectation .
## 6 Applications
In this section, we discuss several applications of the learning guarantees presented in the previous sections. We discuss other applications in Appendix K. As already mentioned, both the standard setting of a fixed hypothesis set not varying with , that is that of standard generalization bounds, and the uniform stability setting where , are special cases benefitting from our learning guarantees.
### 6.1 Stochastic convex optimization
Here, we consider data-dependent hypothesis sets based on stochastic convex optimization algorithms. As shown by Shalev-Shwartz et al. (2010), uniform convergence bounds do not hold for the stochastic convex optimization problem in general. As a result, the data-dependent hypothesis sets we will define cannot be analyzed using standard tools for deriving generalization bounds. However, using arguments based on our notion of hypothesis set stability, we can provide learning guarantees here.
Consider stochastic optimization algorithms , each returning vector , after receiving sample , . We assume that the algorithms are all -sensitive in norm, that is, for all , we have if and differ by one point. We will also assume that these vectors are bounded by some that is , for all . This can be shown to be the case, for example, for algorithms based on empirical risk minimization with a strongly convex regularization term with (Shalev-Shwartz et al., 2010).
Assume that the loss is -Lipschitz with respect to its first argument . Let the data-dependent hypothesis set be defined as follows:
HS={K∑j=1αjˆwSj: α∈ΔK∩B1(α0,r)},
where is in the simplex of distributions and is the ball of radius around . We choose . A natural choice for would be the uniform mixture.
Since the loss function is -Lipschitz, the family of hypotheses is -stable. Additionally, for any and any , we have
L(K∑j=1αjˆwSj,z)−L(K∑j=1α′jˆwSj,z) ≤μ∥∥∥K∑j=1(αi−α′j)ˆwSj∥∥∥2≤μ∥∥[wS1⋯wSK]∥∥1,2∥α−α′∥1≤2μrD,
where is the subordinate norm of matrix defined by . Thus, the average diameter admits the following upper bound: . In view of that, by Theorem 5, for any , with probability at least , the following holds for all :
Ez∼D[L(K∑j=1αjˆwSj,z)] ≤1mm∑i=1L(K∑j=1αiˆwSj,zSi)+√em+√eμβ+4√[1m+2μβ]log[6δ].
The second stage of an algorithm in this context consists of choosing , potentially using a non-stable algorithm. This application both illustrates the use of our learning bounds using the diameter and its application even in the absence of uniform convergence bounds.
### 6.2 Δ-sensitive feature mappings
Consider the scenario where the training sample is used to learn a non-linear feature mapping that is -sensitive for some .
may be the feature mapping corresponding to some positive definite symmetric kernel or a mapping defined by the top layer of an artificial neural network trained on
, with a stability property.
The second stage may consist of selecting a hypothesis out of the family of linear hypotheses based on :
Assume that the loss function is -Lipschitz with respect to its first argument. Then, for any hypothesis and any sample differing from by one element, the hypothesis admits losses that are -close to those of , with , since, for all , by the Cauchy-Schwarz inequality, the following inequality holds:
ℓ(w⋅ΦS(x),y)−ℓ(w⋅ΦS′(x),y)≤μw⋅(ΦS(x)−ΦS′(x))≤μ∥w∥∥ΦS(x)−ΦS′(x)∥≤μγΔ.
Thus, the family of hypothesis set is uniformly -stable with . In view that, by Theorem 4, for any , with probability at least over the draw of a sample , the following inequality holds for any :
R(h)≤ˆRS(h)+2R⋄m(G)+[1+2μγΔm]√log2δ2m. (9)
Notice that this bound applies even when the second stage of an algorithm, which consists of selecting a hypothesis in , is not stable. A standard uniform stability guarantee cannot be used in that case. The setting described here can be straightforwardly extended to the case of other norms for the definition of sensitivity and that of the norm used in the definition of .
### 6.3 Distillation
Here, we consider distillation algorithms which, in the first stage, train a very complex model on the labeled sample. Let denote the resulting predictor for a training sample of size . We will assume that the training algorithm is -sensitive, that is for and differing by one point.
In the second stage, a distillation algorithm selects a hypothesis that is -close to from a less complex family of predictors . This defines the following sample-dependent hypothesis set:
HS={h∈H:∥(h−f∗S)∥∞≤γ}.
Assume that the loss is -Lipschitz with respect to its first argument and that is a subset of a vector space. Let and be two samples differing by one point. Note, may not be in , but we will assume that is in . Let be in , then the hypothesis is in since . Figure 2 illustrates the hypothesis sets. By the -Lipschitzness of the loss, for any , . Thus, the family of hypothesis sets is -stable.
In view that, by Theorem 4, for any , with probability at least over the draw of a sample , the following inequality holds for any :
R(h)≤ˆRS(h)+2R⋄m(G)+[1+2μβmm]√log2δ2m.
Notice that a standard uniform-stability argument would not necessarily apply here since could be relatively complex and the second stage not necessarily stable.
### 6.4 Bagging
Bagging (Breiman, 1996) is a prominent ensemble method used to improve the stability of learning algorithms. It consists of generating new samples , each of size , by sampling uniformly with replacement from the original sample of size . An algorithm is then trained on each of these samples to generate predictors , . In regression, the predictors are combined by taking a convex combination . Here, we analyze a common instance of bagging to illustrate the application of our learning guarantees: we will assume a regression setting and a uniform sampling from without replacement.222Sampling without replacement is only adopted to make the analysis more concise; its extension to sampling with replacement is straightforward. We will also assume that the loss function is -Lipschitz in the predictions and that the predictions are in the range , and all the mixing weights are bounded by for some constant , in order to ensure that no subsample is overly influential in the final regressor (in practice, a uniform mixture is typically used in bagging).
To analyze bagging in this setup, we cast it in our framework. First, to deal with the randomness in choosing the subsamples, we can equivalently imagine the process as choosing indices in to form the subsamples rather than samples in , and then once is drawn, the subsamples are generated by filling in the samples at the corresponding indexes. Thus, for any index , the chance that it is picked in any subsample is . Thus, by Chernoff’s bound, with probability at least , no index in appears in more than subsamples. In the following, we condition on the random seed of the bagging algorithm so that this is indeed the case, and later use a union bound to control the chance that the chosen random seed does not satisfy this property, as elucidated in section J.2.
Define the data-dependent family of hypothesis sets as , where denotes the simplex of distributions over items with all weights . Next, we give upper bounds on the hypothesis set stability and the Rademacher complexity of . Assume that algorithm admits uniform stability (Bousquet and Elisseeff, 2002), i.e. for any two samples and of size that differ in exactly one data point and for all , we have . Now, let and be two samples of size differing by one point at the same index, and . Then, consider the subsets of which are obtained from the ’s by copying over all the elements except , and replacing all instances of by . For any , if , then and, if , then for any . We can bound now the hypothesis set uniform stability as follows: since is -Lipschitz in the prediction, for any , and any we have
Bounding the Rademacher complexity for is non-trivial. Instead, we can derive a reasonable upper bound by analyzing the Rademacher complexity of a larger function class. Specifically, for any , define the dimensional vector . Then the class of functions is . Clearly . Since , a standard Rademacher complexity bound (see Theorem 11.15 in (Mohri et al., 2018)) implies . Thus, by Talagrand’s inequality, we conclude that . In view of that, by Theorem 5, for any , with probability at least over the draws of a sample and the randomness in the bagging algorithm, the following inequality holds for any :
R(h)≤ˆRS(h)+2μ√2plog(4m)m+[1+2[p+√2pmlog(1δ)k]⋅CμβA]√log2δ2m.
For and , the generalization gap goes to as , regardless of the stability of . This gives a new generalization guarantee for bagging, similar (but incomparable) to the one derived by Elisseeff et al. (2005). Note however that unlike their bound, our bound allows for non-uniform averaging schemes.
As an aside, we note that the same analysis can be carried over to the stochastic convex optimization setting of section 6.1, by setting to be a stochastic convex optimization algorithm which outputs a weight vector . This yields generalization bounds for aggregating over a larger set of mixing weights, albeit with the restriction that each algorithm uses only a small part of .
## 7 Conclusion
We presented a broad study of generalization with
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2021-03-08 19:06:49
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https://calinon.ch/paper3077.htm
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### Abstract
Whether in factory or household scenarios, rhythmic movements play a crucial role in many daily-life tasks. In this paper we propose a Fourier movement primitive (FMP) representation to learn such type of skills from human demonstrations. Our approach takes inspiration from the probabilistic movement primitives (ProMP) framework, and is grounded in signal processing theory through the Fourier transform. It works with minimal preprocessing, as it does not require demonstration alignment nor finding the frequency of demonstrated signals. Additionally, it does not entail the careful choice/parameterization of basis functions, that typically occurs in most forms of movement primitive representations. Indeed, its basis functions are the Fourier series, which can approximate any periodic signal. This makes FMP an excellent choice for tasks that involve a superposition of different frequencies. Finally, FMP shows interesting extrapolation capabilities as the system has the property of smoothly returning back to the demonstrations (e.g. the limit cycle) when faced with a completely new situation, being safe for real-world robotic tasks. We validate FMP in several experimental cases with real-world data from polishing and 8-letter tasks as well as on a 7-DoF, torque-controlled, Panda robot.
### Bibtex reference
@inproceedings{Kulak20RSS,
author="Kulak, T. and Silv\'erio, J. and Calinon, S.",
title="Fourier movement primitives: an approach for learning rhythmic robot skills from demonstrations",
booktitle="Proc.\ Robotics: Science and Systems ({RSS})",
year="2020",
pages=""
}
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2021-10-23 11:49:27
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https://jp.maplesoft.com/support/help/maple/view.aspx?path=PDEplot_options
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PDEplot options - Maple Help
PDEplot Options
Calling Sequence PDEplot(PDE, inits, srange, options)
Parameters
PDE - first order PDE containing one indeterminate function of n unknowns inits - list of n+1 expressions or equations specifying the parametric form of an (n-1)-dimensional manifold in n+1 dimensions; initial data srange - range or ranges of the parameters for the initial data options - (optional) equations, described below
Description
• See PDEtools[PDEplot] for the main arguments of PDEplot.
• Arguments of PDEplot other than the PDE, the initial data, and the range thereof can be given in any order. They should be specified as equations of the form option = value. The following options are supported:
'animate' 'basechar' 'color' 'ic_assumptions' 'initcolor' 'iterations' 'method' 'numchar' 'numsteps' 'obsrange' 'scene' 'stepsize' 'u' 'xi'
The dsolve[numeric] options are available, as well as all of the plot3d options except grid, gridstyle, and numpoints.
• animate = true, false, only
The solution in general is an n-dimensional hypersurface in (n+1)-space. As this can be hard to visualize for n > 2, the solution can be animated, by setting animate=true. The resulting animation shows a sequence of manifolds which together map out the solution surface. Each of these manifolds can be viewed as a possible initial condition that would give rise to the obtained solution surface; indeed, the given initial condition surface is one of these manifolds (usually the middle one of the sequence, unless the option numsteps has been used).
Setting animate=false will simply display the solution hypersurface, with the initial condition highlighted in black.
If animate=only is given, the entire solution surface will never be displayed; instead, the initial condition surface will be animated alone. The object that is mapped out by the sequence of initial condition manifolds as they progress through the animation can easily be imagined as the solution. Apart from reducing the time and data involved in the animation, this option would also be useful in conjunction with setting basechar=true, as described below. The default is animate=true for n = 2, and animate=only for n > 2.
The number of frames in the animation is the number of integration steps performed. To decrease the number of frames without changing the accuracy, decrease the number of integration steps, and increase the number of iterations. See numsteps, iterations.
• basechar = true, false, only
The basechar option indicates whether the base characteristic curves (the evolution of the points of the initial conditions, projected onto the x-y plane) should be plotted. If basechar is set to 'true', then base characteristics will be plotted; setting basechar to 'false' will suppress base characteristic plotting; setting basechar to 'only' will plot the base characteristics and the initial data curve, but will not plot the solution surface. The setting basechar = only cannot be chosen unless the animate option is set to 'false'. The default setting for basechar is 'false'.
• color, colour = s_color
Setting color provides a method of color handling for the resulting solution surface. It can take a variety of forms: 1) as a plot color name; 2) as COLOR('HUE',realcons); 3) as COLOR('RGB',realcons,realcons, realcons); 4) as an expression in two variables; 5) as a procedure in two variables; 6) as a three-element list of expressions in two variables; and 7) as a three-element list of procedures in two variables.
In the case of 4) and 5), mesh coordinates (in the plane) are passed to the expression or procedure, and the resulting values are normalized on the range [0,1] and used as 'HUE' values. In the case of 6) and 7), the resulting values are normalized on [0,1] and used as 'RGB' values. Note that 4) and 6) must contain expressions of the two independent variables. This form of color handling can be useful to differentiate various features across the solution surface.
At present this option is only available for PDEs involving an unknown function of two independent variables (that is, n=2). Default surface color is handled by the Maple plotting code.
• ic_assumptions = [eqn1,eqn2,...]
For nonlinear equations, the initial conditions of the first order derivatives of the indeterminate function can usually be determined only up to a set of possibilities, each of which defines a different solution hypersurface. As a result, some assumptions must be given regarding the values of the derivatives on the initial condition manifold as functions of its parameters. These can be given as equalities or inequalities involving any of the first order derivatives of the indeterminate function, the function itself, any of its unknowns, and the parameters. Assumptions must be sufficient to differentiate between the possible solutions. If they are insufficient, an error message will display all the possibilities for the initial conditions of the derivatives, one of which should be chosen and included in the ic_assumptions option. See the examples below.
• initcolor, initcolour = i_color
Setting initcolor provides color handling for the initial data, inits. It can take one of five forms: 1) as a plot color name; 2) as COLOR('HUE',realcons); 3) as COLOR('RGB',realcons,realcons, realcons); 4) as an expression of the initial data parameter; 5) as a procedure with one argument.
In the case of 4) and 5), values are normalized on [0,1] and applied as 'HUE' values. (See color below.)
At present this option is only available for PDEs involving an unknown function of two independent variables (that is, n=2). Default coloring is handled internally by the Maple plotting code.
• iterations = integer
The iterations option indicates the number of points calculated for every point that is stored and plotted. This is useful in gaining a higher accuracy without storing and plotting a large number of mesh points. This option is only applicable to the internal (Runge-Kutta integration) method. The default is 1.
• method = n_method
The method option can be set to internal, or to one of the methods specified in dsolve[numeric]. The default is to use the internal, Runge-Kutta, method as this will produce a noticeable speed improvement. It is recommended that dsolve or numeric methods only be used for PDEs requiring high accuracy, or whose characteristics exhibit stiffness.
• numchar = integer,[integer,integer,...],[t1=integer,t2=integer,...]
The solution hypersurface is composed of a set of characteristic curves, each of which passes through a distinct point on the initial condition manifold. The number of such points (evenly spaced with respect to each parameter) along each of the ranges of the initial data parameters is indicated by numchar. If a list of equations is given, it must be of type [t1=integer,t2=integer,...], where t1, t2,... are the parameters. The ordering of the parameters is the same as in the parameter range definition (see above). For example, with $n=3$ (that is, when the initial manifold is a two-dimensional surface in four-space), and with numchar = [10,5], the solution surface would be a set of characteristic curves passing through each point of a 10x5 grid representing the initial surface. The default is 20 for n=2, and approximately ${40}^{\frac{1}{n-1}}$ in each direction otherwise. The minimum is 4.
• numsteps = [integer, integer], integer
The numsteps option indicates the number of points plotted along each characteristic curve, in each direction. If only a single integer is given, the characteristic curve is only integrated in the one direction specified by the sign of numsteps. The default is [-10,10].
• obsrange = true,false
When obsrange is set to true, integration along a characteristic curve of the surface stops one mesh point after it has passed outside one of the specified ranges. This can be useful in the plotting of functions with asymptotic behavior, or to reduce the number of mesh points that are calculated internally. If false, the whole solution surface will be calculated, but only that portion which falls within the range is displayed. This is useful in case the solution surface goes outside the boundaries of the range at one point and then returns further on. The default value of obsrange is true.
• scene = [x-axis,y-axis,z-axis]
The scene option indicates the variables that are to be plotted, and on which axes. Note that one or more of the dependent variables, the independent variables, or the parameter can be selected to be plotted, in any order. The default scene is the first two independent variables on the x and y axes and the dependent variable on the z axis.
• stepsize = realcons
The stepsize option specifies the distance along each characteristic curve between calculated points. The default stepsize, 0.25, is also the maximum.
• xi = xi_min..xi_max, u or u(x1,...,xn) = u_min..u_max.
Ranges can be specified for those variables which are to be plotted (indicated using the scene option described previously). In general, the range determines the extent of the plot (the plot range) for the given variable.
The default plot range for a variable, say x2, for which a range is not given, is determined by using the maximum and minimum value of that variable (x2) over all the points on the surface which are within the ranges (x1_range, x3_range, u_range) that are given with this option. If no ranges are given, the entire solution surface is shown.
Examples
> $\mathrm{with}\left(\mathrm{PDEtools}\right):$
> $\mathrm{pde2}≔{\mathrm{diff}\left(u\left(x,y\right),x\right)}^{\mathrm{diff}\left(u\left(x,y\right),x\right)}=y$
${\mathrm{pde2}}{≔}{\left(\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({x}{,}{y}\right)\right)}^{\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({x}{,}{y}\right)}{=}{y}$ (1)
> $\mathrm{PDEplot}\left(\mathrm{pde2},\left[\mathrm{cos}\left(t\right),\mathrm{sin}\left(t\right),\mathrm{exp}\left(t\right)\right],\frac{\mathrm{\pi }}{4}..\frac{3\mathrm{\pi }}{4},\mathrm{animate}=\mathrm{false}\right)$
> $\mathrm{pde3}≔\mathrm{diff}\left(u\left(x,y,z,w\right),x\right)+\mathrm{diff}\left(u\left(x,y,z,w\right),w\right)\mathrm{diff}\left(u\left(x,y,z,w\right),y\right)$
${\mathrm{pde3}}{≔}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({x}{,}{y}{,}{z}{,}{w}\right){+}\left(\frac{{\partial }}{{\partial }{w}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({x}{,}{y}{,}{z}{,}{w}\right)\right){}\left(\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({x}{,}{y}{,}{z}{,}{w}\right)\right)$ (2)
> $\mathrm{assumption}≔\mathrm{diff}\left(u\left(x,y,z,w\right),x\right)=-\frac{1}{2\left(8{s}^{3}{r}^{2}+2srt\right)}\left(-s-8{s}^{2}r-t+{\left({s}^{2}+16r{s}^{3}+2ts+{t}^{2}\right)}^{\frac{1}{2}}\right)s$
${\mathrm{assumption}}{≔}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({x}{,}{y}{,}{z}{,}{w}\right){=}{-}\frac{\left({-}{s}{-}{8}{}{{s}}^{{2}}{}{r}{-}{t}{+}\sqrt{{16}{}{r}{}{{s}}^{{3}}{+}{{s}}^{{2}}{+}{2}{}{t}{}{s}{+}{{t}}^{{2}}}\right){}{s}}{{2}{}\left({8}{}{{s}}^{{3}}{}{{r}}^{{2}}{+}{2}{}{s}{}{r}{}{t}\right)}$ (3)
> $\mathrm{PDEplot}\left(\mathrm{pde3},\left[t+{r}^{2},{s}^{2}+r,st,s,r\right],\left[t=-6..-2,s=-1..-0.5,r=-6..-2\right],\mathrm{ic_assumptions}=\left[\mathrm{assumption}\right],\mathrm{scene}=\left[u,w,z\right]\right)$
> $\mathrm{pde4}≔\mathrm{diff}\left(z\left(u,v,w\right),u\right)u\mathrm{cos}\left(v\right)-\mathrm{diff}\left(z\left(u,v,w\right),v\right)w+\mathrm{diff}\left(z\left(u,v,w\right),w\right)=-wv\mathrm{surd}\left(u,3\right)-\mathrm{sin}\left(-u+v\right)$
${\mathrm{pde4}}{≔}\left(\frac{{\partial }}{{\partial }{u}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{z}{}\left({u}{,}{v}{,}{w}\right)\right){}{u}{}{\mathrm{cos}}{}\left({v}\right){-}\left(\frac{{\partial }}{{\partial }{v}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{z}{}\left({u}{,}{v}{,}{w}\right)\right){}{w}{+}\frac{{\partial }}{{\partial }{w}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{z}{}\left({u}{,}{v}{,}{w}\right){=}{-}{w}{}{v}{}\sqrt[{3}]{{u}}{+}{\mathrm{sin}}{}\left({u}{-}{v}\right)$ (4)
> $\mathrm{PDEplot}\left(\mathrm{pde4},\left[t,\mathrm{sin}\left(t\right),s,\frac{t}{2}\right],t=-2\mathrm{\pi }..3\mathrm{\pi },s=-\mathrm{\pi }..\mathrm{\pi },\mathrm{iterations}=2,\mathrm{numchar}=\left[10,10\right],\mathrm{stepsize}=0.05,\mathrm{numsteps}=\left[-5,5\right]\right)$
> $\mathrm{pde5}≔\left({y}^{2}+{z\left(x,y\right)}^{2}+{x}^{2}\right)\mathrm{diff}\left(z\left(x,y\right),x\right)-2xy\mathrm{diff}\left(z\left(x,y\right),y\right)-2z\left(x,y\right)x=0$
${\mathrm{pde5}}{≔}\left({{y}}^{{2}}{+}{{z}{}\left({x}{,}{y}\right)}^{{2}}{+}{{x}}^{{2}}\right){}\left(\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{z}{}\left({x}{,}{y}\right)\right){-}{2}{}{x}{}{y}{}\left(\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{z}{}\left({x}{,}{y}\right)\right){-}{2}{}{z}{}\left({x}{,}{y}\right){}{x}{=}{0}$ (5)
> $\mathrm{PDEplot}\left(\mathrm{pde5},z\left(x,y\right),\left[t,t,\frac{\mathrm{sin}\left(\frac{\mathrm{\pi }t}{0.1}\right)}{10}\right],t=0..0.1,\mathrm{numchar}=40,\mathrm{orientation}=\left[-163,56\right],\mathrm{basechar}=\mathrm{true},\mathrm{numsteps}=\left[20,20\right],\mathrm{stepsize}=0.15,\mathrm{initcolor}=\mathrm{cos}\left(t\right)t,\mathrm{animate}=\mathrm{false},\mathrm{style}=\mathrm{surfacecontour}\right)$
For more examples, see PDEtools[PDEplot].
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2023-02-08 01:08:15
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http://dictionnaire.sensagent.leparisien.fr/Galilean%20invariance/en-en/
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Publicité ▼
## définition - Galilean invariance
voir la définition de Wikipedia
Wikipedia
# Galilean invariance
Galilean invariance or Galilean relativity is a principle of relativity which states that the fundamental laws of physics are the same in all inertial frames. Galileo Galilei first described this principle in 1632 in his Dialogue Concerning the Two Chief World Systems using the example of a ship travelling at constant velocity, without rocking, on a smooth sea; any observer doing experiments below the deck would not be able to tell whether the ship was moving or stationary. The fact that the Earth orbits around the sun at approximately 30 km·s-1 offers a somewhat more dramatic example, though it is technically not an inertial reference frame.
## Formulation
Specifically, the term Galilean invariance today usually refers to this principle as applied to Newtonian mechanics, that is, Newton's laws hold in all inertial frames. In this context it is sometimes called Newtonian relativity.
Among the axioms from Newton's theory are:
1. There exists an absolute space, in which Newton's laws are true. An inertial frame is a reference frame in relative uniform motion to absolute space.
2. All inertial frames share a universal time.
Galilean relativity can be shown as follows. Consider two inertial frames S and S' . A physical event in S will have position coordinates r = (x, y, z) and time t; similarly for S' . By the second axiom above, one can synchronize the clock in the two frames and assume t = t' . Suppose S' is in relative uniform motion to S with velocity v. Consider a point object whose position is given by r = r(t) in S. We see that
$r'(t) = r(t) - v t.\,$
The velocity of the particle is given by the time derivative of the position:
$u'(t) = \frac{d}{d t} r'(t) = \frac{d}{d t} r(t) - v = u(t) - v.$
Another differentiation gives the acceleration in the two frames:
$a'(t) = \frac{d}{d t} u'(t) = \frac{d}{d t} u(t) - 0 = a(t).$
It is this simple but crucial result that implies Galilean relativity. Assuming that mass is invariant in all inertial frames, the above equation shows Newton's laws of mechanics, if valid in one frame, must hold for all frames. But it is assumed to hold in absolute space, therefore Galilean relativity holds.
### Newton's theory versus special relativity
A comparison can be made between Newtonian relativity and special relativity.
Some of the assumptions and properties of Newton's theory are:
1. The existence of infinitely many inertial frames. Each frame is of infinite size (covers the entire universe). Any two frames are in relative uniform motion. (The relativistic nature of mechanics derived above shows that the absolute space assumption is not necessary.)
2. The inertial frames move in all possible relative uniform motion.
3. There is a universal, or absolute, time.
4. Two inertial frames are related by a Galilean transformation.
5. In all inertial frames, Newton's laws, and gravity, hold.
In comparison, the corresponding statements from special relativity are same as the Newtonian assumption.
1. Rather than allowing all relative uniform motion, the relative velocity between two inertial frames is bounded above by the speed of light.
2. Instead of universal time, each inertial frame has its own time.
3. The Galilean transformations are replaced by Lorentz transformations.
4. In all inertial frames, all laws of physics are the same.
Notice both theories assume the existence of inertial frames. In practice, the size of the frames in which they remain valid differ greatly, depending on gravitational tidal forces.
In the appropriate context, a local Newtonian inertial frame, where Newton's theory remains a good model, extends to, roughly, 107 light years.
In special relativity, one considers Einstein's cabins, cabins that fall freely in a gravitational field. According to Einstein's thought experiment, a man in such a cabin experiences (to a good approximation) no gravity and therefore the cabin is an approximate inertial frame. However, one has to assume that the size of the cabin is sufficiently small so that the gravitational field is approximately parallel in its interior. This can greatly reduce the sizes of such approximate frames, in comparison to Newtonian frames. For example, an artificial satellite orbiting around earth can be viewed as a cabin. However, reasonably sensitive instruments would detect "microgravity" in such a situation because the "lines of force" of the Earth's gravitational field converge.
In general, the convergence of gravitational fields in the universe dictates the scale at which one might consider such (local) inertial frames. For example, a spaceship falling into a black hole or neutron star would (at a certain distance) be subjected to tidal forces so strong that it would be crushed. In comparison, however, such forces might only be uncomfortable for the astronauts inside (compressing their joints, making it difficult to extend their limbs in any direction perpendicular to the gravity field of the star). Reducing the scale further, the forces at that distance might have almost no effects at all on a mouse. This illustrates the idea that all freely falling frames are locally inertial (acceleration and gravity-free) if the scale is chosen correctly.[1]
### Electromagnetism
Maxwell's equations governing electromagnetism possess a different symmetry, Lorentz invariance, under which lengths and times are affected by a change in velocity, which is then described mathematically by a Lorentz transformation.
Albert Einstein's central insight in formulating special relativity was that, for full consistency with electromagnetism, mechanics must also be revised such that Lorentz invariance replaces Galilean invariance. At the low relative velocities characteristic of everyday life, Lorentz invariance and Galilean invariance are nearly the same, but for relative velocities close to that of light they are very different.
## Work, kinetic energy, momentum
Because the distance covered while applying a force to an object depends on the inertial frame of reference, so does the work done. Due to Newton's law of reciprocal actions there is a reaction force; it does work depending on the inertial frame of reference in an opposite way. The total work done is independent of the inertial frame of reference.
Correspondingly the kinetic energy of an object, and even the change in this energy due to a change in velocity, depends on the inertial frame of reference. The total kinetic energy of an isolated system also depends on the inertial frame of reference: it is the sum of the total kinetic energy in a center of momentum frame and the kinetic energy the total mass would have if it were concentrated in the center of mass. Due to the conservation of momentum the latter does not change with time, so changes with time of the total kinetic energy do not depend on the inertial frame of reference.
By contrast, while the momentum of an object also depends on the inertial frame of reference, its change due to a change in velocity does not.
## Notes
1. ^ Taylor and Wheeler's Exploring Black Holes - Introduction to General Relativity, Chapter 2, p. 2-6.
## References
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2021-09-27 14:13:17
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http://xwacad.com/StatLearning/5.html
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Back to the Contents
# 5.1 Introduction
In this chapter, methods for moving beyond linearity are discussed. The core idea in this chapter is to argument/replace the vector of inputs $$X$$ with additional variables, which are transformations of $$X$$, and then use linear models in this new space of derived input features.
Denote by $$h_m(X):\mathbb{R}^p\to\mathbb{R}$$, the $$m$$th transformation of $$X,m=1,2,\cdots,M.$$ We then model $f(X)=\sum_{m=1}^M\beta_mh_m(X),$ a linear basis expansion in $$X$$. The beauty of this approach is that once the basis functions $$h_m$$ have been determined, the models are linear in these new variables, and the fitting proceeds as before.
Sometimes the problem at hand will call for particular basis functions hm, such as logarithms or power functions.More often, however, we use the basis expansions as a device to achieve more flexible representations for $$f(X)$$.
## 5.2 Piecewise Polynomials and Splines
In this section, the concepts of Piecewise Polynomials and Splines are introduced. It is also shown that what the basis of spline can look like.
The fixed-knots splines are also known as regression splines. We need to select the order of the spline, the number of knots and the placement. One simple approach is to parameterize a family of splines by the number of basis functions or degrees of freedom, and have the observations $$x_i$$ determine the positions of the knots.
Since the space of spline functions of a particular order and knot sequence is a vector space, there are many equivalent bases for representing them (just as there are for ordinary polynomials.) While the truncated power basis is conceptually simple, it is not too attractive numerically: powers of large numbers can lead to severe rounding problems. The B-spline basis, described in the Appendix to this chapter, allows for efficient computations even when the number of knots $$K$$ is large
# 5.2.1 Natural Cubic Splines
$$\bullet$$ Motivation We know that the behavior of polynomials fit to data tends to be erratic near the boundaries, and extrapolation can be dangerous. These problems are exacerbated with splines. The polynomials fit beyond the boundary knots behave even more wildly than the corresponding global polynomials in that region.
A natural cubic spline adds additional constraints, namely that the function is linear beyond the boundary knots. This frees up four degrees of freedom (two constraints each in both boundary regions), which can be spent more profitably by sprinkling more knots in the interior region.
A natural cubic spline with $$K$$ knots is represented by $$K$$ basis functions. One can start from a basis for cubic splines, and derive the reduced basis by imposing the boundary constraints.
# 5.3 Filtering and Feature Extraction
Preprocessing of high-dimensional features is a very general and powerful method for improving the performance of a learning algorithm.
The preprocessing need not be linear as it was above, but can be a general (nonlinear) function of the form $$x^*= g(x)$$. The derived features $$x^*$$ can then be used as inputs into any (linear or nonlinear) learning procedure.
For example, for signal or image recognition a popular approach is to first transform the raw features via a wavelet transform.
# 5.4 Smoothing Splines
Here we discuss a spline basis method that avoids the knot selection problem completely by using a maximal set of knots. The complexity of the fit is controlled by regularization. Consider the following problem: among all functions $$f(x)$$ with two continuous derivatives, find one that minimizes the penalized residual sum of squares: $RSS(f,\lambda)=\sum_{i=1}^N\{y_i-f(x_i)\}^2+\lambda\int\{f''(t)\}^2dt,$ where $$\lambda$$ is a fixed smoothing parameter.
Remarkably, it can be shown that it has an explicit, finite-dimensional, unique minimizer which is a natural cubic spline with knots at the unique values of the $$x_i,\ i=1,\cdots,N$$. At face value it seems that the family is still over-parametrized, since there are as many as $$N$$ knots, which implies $$N$$ degrees of freedom. However, the penalty term translates to a penalty on the spline coefficients, which are shrunk some of the way toward the linear fit.
Since the solution is a natural spline, we can write it as $f(x)=\sum_{j=1}^NN_j(x)\theta_j,$ where the $$N_j(x)$$ are an $$N$$-dimensional set of basis functions for representing this family of natural splines. The criterion thus reduces to $RSS(\theta,\lambda)=(y-N\theta)^T(y-N\theta)+\lambda\theta^T\Omega_N\theta,$ where $$\{N\}_{ij}=N_j(x_i)$$ and $$\{\Omega_N\}_{jk}=\int N_j''(t)N_k''(t)dt$$. The solution is easily seen to be $\hat\theta=(N^TN+\lambda\Omega_N)^{-1}N^Ty,$ a generalized ridge regression. The fitted smoothing spline is given by $\hat f(x)=\sum_{j=1}^N N_j(x)\hat \theta_j.$
## 5.4.1 Degrees of Freedom and Smoother Matrices
In this section, we discuss intuitive ways of prespecifying the amount of smoothing.
Donate by $$\hat f$$, the $$N$$-vector of fitted values $$\hat f(x_i)$$ at the training predictors $$x_i$$. Then, $\hat f=N(N^TN+\lambda\Omega_N)^{-1}N^Ty=S_\lambda y.$ Again the fit is linear in $$y$$, and the finite linear operator $$S_\lambda$$ is known as the smoother matrix. One consequence of this linearity is that the recipe for producing $$\hat f$$ from $$y$$ does not depend on $$y$$ itself; $$S_\lambda$$ depends only on the $$x_i$$ and $$\lambda$$.
Suppose $$B_\xi$$ is a $$N\times M$$ matrix of $$M$$ cubic-spline basis functions evaluated at the $$N$$ training points $$x_i$$, with kont sequence $$\xi$$, and $$M\ll N$$. Then th evector of fitted spline values is given by $\hat f= B_\xi(B_\xi^TB_\xi)^{-1}B_{\xi}^Ty=H_{\xi}y.$ Here, the linear operator $$H_\xi$$ is a projection operator, also known as the hat matrix in statistics. There are some important similarities and differences between $$H_\xi$$ and $$S_\lambda$$:
$$\bullet$$ Both are symmetric, positive semidefinite matrices.
$$\bullet$$ $$H_\xi H_\xi=H_\xi$$(idempotent), while $$S_\lambda S_\lambda\preceq S_\lambda$$, meaning that the righthand side exceeds the left-hand side by a positive semidefinite matrix. This is a consequence of the shrinking nature of $$S_\lambda$$.
$$\bullet$$ $$H_\xi$$ has rank $$M$$, while $$S_\lambda$$ has rank $$N$$.
The expression $$M=trace(H_\xi)$$ gives the dimension of the projection space, which is also the number of basis functions, and hence the number of parameters involved in the fit. By analogy we define the effective degrees of freedom of a smoothing spline to be $df_\lambda=trace(S_\lambda).$
Since $$S_\lambda$$ is symmetric and positive semidefinite, it has a real eigendecomposition. It is convenient to rewrite $$S_\lambda$$ in the Reinsch form $S_\lambda=(I+\lambda K)^{-1},$ where $$K$$ does not depend on $$\lambda$$ (justification needed).
Since $$\hat f=S_\lambda y$$ solves $\min_f (y-f)^T(y-f)+\lambda f^TKf,$ $$K$$ is known as the penalty matrix, and indeed a quadratic form in $$K$$ has a representation in terms of a weighted sum of squared second differences. The eigen-decomposition of $$S_\lambda$$ is $S_\lambda=\sum_{k=1}^N\rho_k(\lambda)u_ku_k^T$ with $\rho_k(\lambda)=\frac{1}{1+\lambda d_k},$ and $$d_k$$ the corresponding eigenvalue of $$K$$.
Some highlights are also listed, refer to the textbook.
# 5.5 Automatic Selection of the Smoothing Parameters
The smoothing parameters for regression splines encompass the degree of the splines, and the number and placement of the knots. For smoothing splines, we have only the penalty parameter $$\lambda$$ to select, since the knots are at all the unique training $$X$$’s, and cubic degree is almost always used in practice.
## 5.5.1 Fixing the Degrees of Freedom
Since $$f_\lambda = trace(S_\lambda)$$ is monotone in $$\lambda$$ for smoothing splines, we can invert the relationship and specify $$\lambda$$ by fixing df.
# 5.6 Nonparametric Logistic Regression
The smoothing spline problem in 5.4 is posted in a regression setting. It is typically straitforward to transfer this technology to other domains. Here we consider logistic regression with a single quantitative input $$X$$.
The model is $\log \frac{P(Y=1|X=x)}{P(Y=0|X=x)}=f(x),$ which implies $P(Y=1|X=x)=\frac{\exp(f(x))}{1+\exp(f(x))}.$ Fitting $$f(x)$$ in a smooth fashion leads to a smooth estimate of the conditional probabiliry $$P(Y=1|x)$$, which can be used to classification or risk scoring.
We construct the penalized log-likelihood criterion $l(f;\lambda)=\sum_{i=1}^N[y_if(x_i)-\log(1+\exp(f(x_i)))]-\frac{1}{2}\lambda\int\{f''(t)\}^2dt.$ The arguments similar to those used in 5.4 show that the optimal $$f$$ is a finite-dimensional natural spline with knots at the unique values if $$x$$. This means that we can represent $$f(x)=\sum_{j=1}^NN_j(x)\theta_j$$.
We can compute the first and second derivatives and then use Newton-Raphson for the regression.
# 5.7 Multidimensional Splines
One-dimensional smoothing splines (via regularization) generalize to higher dimensions as well. Suppose we have pairs $$y_i,x_i$$ with $$x_i\in R^d$$, and we seek a $$d$$-dimensional regression function $$f(x)$$. The idea is to set up the problem $\min_f\sum_{i=1}^N\{y_i-f(x_i)\}^2+\lambda J[f],$ where $$J$$ is an appropriate penalty functional for stabilizing a function $$f$$ in $$R^d$$.
The solution has the form $f(x)=\beta_0+\beta^Tx+\sum_{j=1}^N\alpha_jh_j(x),$ where $$h_j(x)=\|x-x_j\|^2\log\|x-x_j\|$$. These $$h_j$$ are examples of radial basis functions, which are discussed in more detail in the next section.
To be continued.
## 5.9 Wavelet Smoothing
Back to the Contents
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2021-05-11 16:29:36
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https://www.physicsforums.com/threads/joint-of-normal-and-logistic.537997/
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Joint of normal and logistic
1. Oct 8, 2011
Hejdun
Hi!
I would be really happy to recieve some help. I have tried using Jacobians and so on, but
I am stuck.
I'll start with the univariate case. Let X ~ N(μ,σ) and Y = exp(X)/(1+exp(X)). What is the joint f(x,y)? According to intuition fy|x = 1, but since we are dealing with continous distributions I am not sure.
The bivariate case is an extension. Let X1, X2 be bivariate normal and
Y = exp(γ1*X1 + γ2*X2)/(1 + exp(γ1*X1 + γ2*X2) ), where the gammas are just
constants. Now I am looking for f(x1,x2,y). Any suggestions of how to proceed at least?
/H
2. Oct 9, 2011
Hejdun
Just to clarify the first case. I actually want to find the joint f(x,y) and then E(XY). Since Y=g(X), then f(x, g(x)) and E(XY)= E(Xg(X)). We know that Pr(Y=exp(x)/(1 + exp(x))) = 1, but I am not sure if then can write fy|x = 1 (all probability mass concentrated in one point). If that is the case, then f(x,y) = f(x) and E(XY) = ∫∫xyf(x)dxdy = ∫xexp(x)/(1+exp(x))f(x)dx which can easily be calculated numerically. Am I on the right track?
The bivariate case is more complicated but maybe that can be solved as well. :)
/H
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2017-09-25 04:47:43
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http://journal.psych.ac.cn/xlxb/CN/Y2007/V39/I01/27
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ISSN 0439-755X
CN 11-1911/B
中国科学院心理研究所
• •
### 目标包含结构的文本阅读中目标信息的激活
1. 华南师范大学心理应用研究中心，广州 510631
• 收稿日期:2004-07-30 修回日期:1900-01-01 出版日期:2007-01-30 发布日期:2007-01-30
• 通讯作者: 莫雷
### Reactivation and Integration of Goal Information in Text Comprehension with Inclusive Relations
Leng-Ying,Mo-Lei,Wu-Jun,-Wang-Suiping
1. Centre for Psychological Application, South China Normal University, Guangzhou, 510631, China
• Received:2004-07-30 Revised:1900-01-01 Online:2007-01-30 Published:2007-01-30
• Contact: Mo Lei
Abstract: Introduction. Readers continually integrate incoming information with evolving discourse representation during reading to maintain a fully updated situation model (O’ Brien et al., 1998). There are two different views about the process of updating -- the memory-based text processing view and the constructionist theory. According to the memory-based text processing view, every new piece of linguistic information is mapped into the information stored in working memory. Resonance of the ideas of the text is sufficient and necessary to make comprehension possible (McKoon et al., 1996; Albrecht & O’ Brien, 1995). On the other hand, according to the constructionist theory, readers pursue coherent relations throughout the text and attempt to explain why the actions, events, and states are mentioned in the text (Trabasso et al., 1989; Graesser et al., 1994). That is, related information in long-term memory is activated without the resonant process. Recently, there is convergence between the two views (Cook & Gueraud, 2005; van den Broek & Rapp, 2005). Within the constructionist theory, however, there are disagreements concerning what kind of goal information is to be reactivated. Some research has shown that unachieved goals are more available than achieved goals (Suh & Trabasso, 1993; Lutz & Radvansky, 1997; Magliano & Radvansky, 2001), but other research suggests the contrary (Richards & Singer, 2001; Singer & Richards, 2005). In the present study two kinds of goal-information integration were proposed, namely the reinstatement integration and the facilitation integration. When reading texts with goals containing an inclusive relation, goal information in long-term memory becomes reactivated by the achievement of sub-goals. On the other hand, when reading texts with goals containing a parallel relation, the failed goals facilitate the integration of goal information. Richards & Singer (2001) suggested that the signal of goals could activate the goal information in long-term memory, thus demonstrating reinstatement-integration. However, they did not explore whether the activation is primed by the signal or just represents spontaneous processing. The present study was to investigate this critical issue.
Method. A moving-window display technique was used. Ninety-four participants read 18 paragraphs, in which two characters attempted to accomplish two independent sub-goals to achieve the joint main goal. A 3 (success vs. failure vs. control) x 3 (strong signals vs. weak signals vs. control) experimental design was adopted.
Results. Repeated-measures ANOVA was performed to compare the recognition time of the probe words. The results indicated that there was a significant main effect of the success variable: success condition < failure condition < control. There was also a significant main effect of the magnitude of the signals: strong signals < weak signals < control. A significant interaction between the two independent variables was found: For the success condition, recognition time for strong signals was equal to weak signals, but lower than the control signals. For the failure condition, recognition time for strong signals was lower than weak signals, which was equal to the control signals. For the control condition, recognition time was the same for the three kinds of signals.
Conclusion. The present study shows the following: (1) The reactivation and integration of goal information when comprehending text with an inclusive relation of goals belong to reinstatement integration. (2) The signal of goals activates the achieved goal in long-term memory. (3) The activation is not a spontaneous processing, but is primed by the closing signal. (4) If the signal is strong, then the failed goal in long-term memory will also be reactivated
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2021-12-08 22:15:00
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https://mathoverflow.net/questions/347036/continuous-section-of-support-is-it-possible-to-map-compact-sets-to-measures-s
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# Continuous section of support - Is it possible to map compact sets to measures supported on them?
Let $$(X,d)$$ be a compact metric space and let $$(\mathcal K(X),d_H)$$ and $$(\mathcal P(X),d_W)$$ denote its space of nonempty compact subsets with Hausdorff metric $$d_H$$, and its space of Borel probability measures with 1-Wasserstein metric $$d_W$$.
Let $$\operatorname{supp}:\mathcal P(X)\to \mathcal K(X)$$ denote the function which associates to each measure its support. This function is generally not continuous e.g. consider the support of $$\mu=(1-\epsilon)\delta_x + \epsilon \delta_y$$ with $$x\neq y$$ as $$\epsilon\to 0$$. I'm curious about the following:
Does there exist a continuous $$\operatorname{dude}:\mathcal K(X)\to \mathcal P(X)$$ so that $$\operatorname{supp}\circ\operatorname{dude}=\operatorname{id}_{\mathcal K(X)}$$?
It seems associating a normalized Hausdorff measure $$\mu_K(A) = \frac{1}{\mathcal H^\alpha(K)}\mathcal H^\alpha(A\cap K)$$ does not work according to Wikipedia since some fractals have $$\mathcal H^\alpha(K)\in\{0,+\infty \}$$ for all $$\alpha$$.
As a follow-up to any answer, it would be nice to know if anything changes if we merely assume $$X$$ is compact Hausdorff and $$\mathcal K(X)$$ and $$\mathcal P(X)$$ are equipped with the Vietoris and weak-* topologies.
• Hausdorff measure already has a problem with finite sets, since it would be the uniform measure, and this is clearly wrong by considering sets like $\{-1/n, 1/n, 1\}$ in $[-1,1]$. – Nate Eldredge Nov 27 '19 at 8:43
• Are you interested in the special case where $X$ is a closed interval? I suspect the s ax newer is yes in that case. – Anthony Quas Nov 27 '19 at 15:45
• @Nate That's a good point and I should have noticed that. The empirical measure of a finite set failing to continuously vary with its support was actually the original motivation for this question. – Christian Bueno Nov 27 '19 at 22:50
• @Anthony Though I'm mostly interested in greater generality, I'd definitely be interested in seeing if there's any special approaches that could be taken for closed intervals. – Christian Bueno Nov 27 '19 at 22:52
• How much of this question was motivated by the goal of writing $\operatorname{supp} \circ \operatorname{dude}$? – Geoffrey Irving Dec 1 '19 at 9:54
If $$X$$ is a subset of $$\mathbb{R}^n$$, one way is to take the uniform measure on say a ball containing $$X$$, and take the pushforward of it by the closest point projection on your compact subset $$K$$.
The support of that pushforward will be $$K$$, and it can be shown that this operation is continuous for the Hausdorff distance vs $$1$$-Wasserstein distance (actually it is even $$1/2$$-Holder).
• If $K$ consists of two concentric spheres, won't this put all the mass on the outer sphere? So the support won't be all of $K$. Or likewise, if $K$ is a ball, all the mass will go on the surface. – Nate Eldredge Nov 27 '19 at 19:25
• Or maybe you mean a sphere in $\mathbb{R}^{n+1}$? – Nate Eldredge Nov 27 '19 at 19:31
• meant a ball, sorry – alesia Nov 27 '19 at 19:32
• It is included in the set of non differentiability points of the distance function, which is Lipschitz – alesia Nov 27 '19 at 20:04
• link.springer.com/article/10.1007/s10208-009-9056-2. there are free versions online – alesia Nov 27 '19 at 22:42
The affirmative answer to this question is given by the following theorem, proved by the technique of continuous selections.
Theorem. For any compact metrizable space $$X$$ there exists a continuous map $$\phi:\mathcal K(X)\to\mathcal P(X)$$ assigning to each nonempty compact set $$K\subset X$$ a probability measure $$\phi(K)\in\mathcal P(X)$$ such that $$\mathrm{supp}(\phi(K))=K$$.
Proof. It can be shown that the multi-valued map $$\Phi:\mathcal H(X)\multimap \mathcal P(X)$$ assigning to each compact set $$K\in\mathcal H(X)$$ the compact convex set $$\Phi(K)=\{\mu\in\mathcal P(X):\mu(K)=1\}$$ is lower-semicontinuous (which means that for any open set $$U\subset X$$ and any $$a\in[0,1]$$ the set $$\{K\in\mathcal H(X):\exists \mu\in\Phi(K),\;\mu(U)>a\}$$ is open in $$\mathcal H(X)$$).
Fix a countable base $$(U_n)_{n\in\omega}$$ of the topology of $$X$$ consisting of non-empty open sets in $$X$$. For every $$n\in\omega$$, consider the open set $$\;\mathcal U_n=\{K\in\mathcal H(X):K\cap U_n\ne\emptyset\}$$ in $$\mathcal H(X)$$ and the open convex subset set $$\mathcal W_n=\{\mu\in P(X):\mu(U_n)>0\}$$ in $$\mathcal P(X)$$.
Since the space $$\mathcal H(X)$$ is metrizable (and hence perfectly normal), for every $$n\in\omega$$ we can fix a continuous function $$\lambda_n:\mathcal H(X)\to[0,\frac1{2^n}]$$ such that $$\mathcal U_n=\{K\in\mathcal H(X):\lambda_n(K)>0\}$$. It follows that the function $$\lambda:\mathcal H(X)\to [0,2],\;\lambda:K\mapsto\sum_{n=0}^\infty\lambda_n(K),$$is continuous and strictly positive.
For every $$n\in\omega$$, consider the multi-valued map $$\Phi_n:\mathcal U_n\multimap\mathcal W_n$$ assinging to each compact set $$K\in\mathcal U_n$$ the closed convex subset $$\Phi_n(K)=\mathcal W_n\cap\Phi(K)$$ of $$\mathcal W_n$$. By Theorem 0.47 in the book [RS], the lower semi-continuity of the multi-valued map $$\Phi$$ implies the lower semi-continuity of the multi-valued map $$\Phi_n$$.
By the Compact-Valued Selection Theorem 4.1 in the book [RS], the multi-valued map $$\Phi_n$$ admits a compact-valued lower semicontinuous selection $$\Psi_n:\mathcal U_n\multimap\mathcal W_n$$. Let $$\overline{\mathrm{co}}\Psi_n:\mathcal U_n\multimap \mathcal W_n$$ be the multi-valued map assigining to each compact set $$K\in\mathcal U_n$$ the closed convex hull $$\overline{\mathrm{co}}\Psi_n(K)\subset \Phi_n(K)$$ of the compact set $$\Phi_n(K)$$ in $$\mathcal W_n$$. By Theorems 0.45 and 0.46 in [RS], the multi-valued map $$\overline{\mathrm{co}}\Psi_n$$ is lower semicontinuous. It can be shown that the closed convex hull of any compact set in $$\mathcal W_n$$ is compact. Consequently, for every $$K\in\mathcal U_n$$ the convex set $$\overline{\mathrm{co}}\Psi_n(K)$$ is compact.
By the Michael's Convex-Valued Selection Theorem 1.2 in [RS], the multivalued map $$\overline{\mathrm{co}}\Psi_n$$ admits a continuous selection $$\psi_n:\mathcal U_n\to \mathcal P(X)$$, which is a continuous map such that for every $$K\in\mathcal U_n$$ the measure $$\psi_n(K)$$ belongs to the compact convex set $$\overline{\mathrm{co}}\Psi_n(K)\subset\Phi_n(K)$$, and hence $$\psi_n(K)(U_n)>0$$ and $$\psi_n(K)(K)=1$$.
Finally, consider the continuous map $$\phi:\mathcal H(X)\to\mathcal P(X)$$ assigning to each compact set $$K\in\mathcal H(X)$$ the probability measure $$\phi(K)=\frac1{\lambda(K)}\sum_{n\in N(K)}\lambda_n(K){\cdot}\psi_n(K)$$where $$N(K)=\{n\in\omega:K\in\mathcal U_n\}$$. It can be shown that $$\phi$$ is a required continuous map assigning to each compact set $$K\in\mathcal H(X)$$ a probability measure $$\phi(K)$$ with $$\mathrm{supp}(\phi(K))=K$$.
Reference.
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2021-03-08 13:45:46
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https://www.statstutor.net/downloads/question-2017865-applications-of-linear-algebra/
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# Question #2017865: Applications of Linear Algebra
Question: Let A be a linear map of the n-dimensional space (V, F) onto itself. Assume that for some $$\lambda \in F$$ and basis $$\left( {{v}_{i}} \right)_{i=1}^{n}$$ we have $$A{{v}_{1}}={{v}_{1}}$$ and $$A{{v}_{k}}=\lambda {{v}_{k}}+{{v}_{k+1}}$$
Solution: The solution consists of 34 words (1 page)
Deliverables: Word Document
0
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2019-06-19 23:49:10
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http://mathhelpforum.com/discrete-math/75810-proofs-pigeonhole-principle-functions.html
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# Math Help - Proofs: pigeonhole principle and functions
1. ## Proofs: pigeonhole principle and functions
I need help with two proofs.
First, I must prove the pigeon hole principle.
Second, Let f: A --> B be an injection between two finite sets of the same size. From this, Prove the f is a bijection.
2. Originally Posted by jzellt
I need help with two proofs.
First, I must prove the pigeon hole principle.
see here, or here
Second, Let f: A --> B be an injection between two finite sets of the same size. From this, Prove the f is a bijection.
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2016-07-23 17:37:25
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https://physics.stackexchange.com/questions/449177/exceeding-the-speed-of-light/449187
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# Exceeding the speed of light [closed]
I understand that the speed of light c is derived from the self-interaction between elections/photons, and is thus the maximum speed of anything composed of electrons/photons.
Suppose that there is a particle that is made up of stuff that is not electrons and photons. Is it possible in principle for this particle to travel faster than c? Or does c somehow apply to absolutely everything in the universe, even as yet undiscovered particles?
If it has already been proven that c applies to absolutely everything in the universe, even as yet undiscovered particles, then c is really a property of space itself, rather than a property of electromagnetism.
## closed as off-topic by WillO, Bill N, Kyle Kanos, M. Enns, ahemmetterDec 19 '18 at 11:08
This question appears to be off-topic. The users who voted to close gave these specific reasons:
• "We deal with mainstream physics here. Questions about the general correctness of unpublished personal theories are off topic, although specific questions evaluating new theories in the context of established science are usually allowed. For more information, see Is non mainstream physics appropriate for this site?." – Kyle Kanos, M. Enns, ahemmetter
• "Homework-like questions should ask about a specific physics concept and show some effort to work through the problem. We want our questions to be useful to the broader community, and to future users. See our meta site for more guidance on how to edit your question to make it better" – WillO, Bill N
If this question can be reworded to fit the rules in the help center, please edit the question.
• Electromagnetism is a property of space itself. That is one speaks of EM waves propagating through vacuum, without medium. Look into special relativity. – Keefer Rowan Dec 18 '18 at 23:27
• "the speed of light c is derived from the self-interaction between elections/photons and is thus the maximum speed of anything composed of electrons/photons" Where did you read these statements? – my2cts Dec 19 '18 at 0:43
• @KeeferRowan "Electromagnetism is a property of space itself" - Can you provide a reference to this claim? Photons flying in space "without medium" doesn't prove it. – safesphere Dec 19 '18 at 0:50
• @my2cts My understanding that the speed of light derives from the speed at electrons interact with themselves comes from a few YouTube videos that I've watched about it by Lawrence Krauss and similar top notch physicists. – Steve Dec 19 '18 at 2:52
• @my2cts From what I've watched, the speed of light can be derived from Maxwell's equations, and the meaning of the manipulation of the equations leads physicists to conclude that c represents how electromagnetism itself propogates. – Steve Dec 19 '18 at 2:54
## 4 Answers
The fact that we can't exceed the speed of light is a fundamental property of spacetime. It stems from the fact that the speed of light is constant in any reference frame. If you turn on a laser pointer, you will observe the beam propagate away from you at the speed of light, regardless of whether you are standing still or zooming across the universe at 99% the speed of light. One of several interesting consequences of this is that no matter how fast you go, light will always go faster.
This might seem a little hand-wavy, but that's the gist of it. If you want to go deeper, there are some great videos on YouTube about special and general relativity. Wikipedia also has a lot of good information.
• @Jack Nick Porter Your answer completely ignores the whole point of my question. My whole point is that I was asking if it is possible for particles that are not composed of electrons/photons to exceed c. Your answer immediately refers the speed of light being constant. – Steve Dec 19 '18 at 2:57
• @Steve to put it another way, the speed of light is a fundamental property of spacetime and must be respected by all matter, including things that aren't made of electrons/protons. – Allure Dec 19 '18 at 5:03
• @Steve Try to avoid confusing $c$, a physical constant with units of velocity that is a fundamental property of spacetime, with the speed at which light travels, a speed which happens to be equal to $c$. Other things also move with speed $c$, such as gravitational waves and gluons, and they do this because of the nature of spacetime, not because light moves at that speed. – eyeballfrog Dec 19 '18 at 5:57
• @eyeballfrog Can you provide a reference to a measurement of the speed of gluons or an observation of the graviton? – safesphere Dec 19 '18 at 15:45
• @safesphere Gravitational waves were observed two years ago and are known to travel at speed $c$ to within one trillionth of a percent. Surely you remember it. – eyeballfrog Dec 19 '18 at 16:09
The speed of light is not derived from the self interaction of photons or electrons. Photons and electrons are part of physical theories which are described in terms of space and time. The description of space and time themselves, without introducing any other physical theory, already includes a speed limit, which is referred to as $$c$$. That speed limit must be respected for any massive or massless particle, including, but not restricted to, photons and electrons.
In special and general relativity, "distance" between two "events" is the square root of the difference between square of the spatial separation and the square of c times the time separation. For things moving at the speed of light, that "distance" is precisely zero.
So you are right, the speed of light is an aspect of spacetime.
The existence of "tachyons" has been proposed, but is only rarely seriously considered. It was thought that particles with imaginary mass (mass x i) could travel faster than light. However, it has been determined that even particles with imaginary mass could not exceed light speed. See "Tachyon" in Wikipedia.
• You say "for things moving at c, that distance is 0". Is this true for particles that are not composed of photons/electons? – Steve Dec 19 '18 at 3:06
• Yes, because it's inherent in the geometry of spacetime. The type of particle is irrelevant. – S. McGrew Dec 19 '18 at 3:39
The answer to your question, as written in the details, is no for all known particles. Empirical evidence comes from the fact that in particle colliders like the Large Hadron Collider but also its many earlier antecedents, beams of protons and heavy nuclei - composed of protons and neutrons - can be accelerated arbitrarily close to, but no faster than, the speed of light. These particles are not photons, and do not contain electrons anywhere in their composition - they are instead blobs of quarks. Moreover, muons - a particle similar to the electron in some ways but equally an elementary particle that is not composed of anything, including electrons - provide less direct tests for this idea.
The reason that this is to be expected is that it is not so much that the "speed of light" arises from "interaction between electrons and photons" - which I believe you're using to intend to really mean the electromagnetic interaction - but rather it comes from the geometry of spacetime, and in fact it is this geometry of spacetime which shapes electromagnetic interactions, not the other way around. In particular, it both limits the speed of light and creates the possibility of magnetic fields. So yes, your last hunch at the end of your post is, to the best of our knowledge, correct.
The key fact about the geometry of spacetime is that it is "Lorentzian", which means that the mathematical relations describing how the point of view of an observer changes when sie changes speeds - which effectively amount to its "rotations" - are such that there is a particular speed which is not changed by them, and this speed is what we call the speed of light. This speed is not necessarily a limit, but the Lorentzian geometry also tells you that if you could somehow exceed it, you would be able to travel or at least send messages backwards in time. The facts that we have seen nothing to do so, including that we have received no messages from our future selves, that such backwards-in-time travel creates interesting logical problems like the so-called "grandfather paradox", and moreover the sheer mountain of experimental evidence which verifies to very high precision (including all tests of the universality of the light speed limit but also of related consequences of the geometry like that of time dilation, which the aforementioned muon studies deal with) the validity of the Lorentzian geometry as being the geometry on which our Universe is based, are all good reasons to consider the "speed of light" as the absolute maximum value allowed for the speed of any moving objects in the real Universe.
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2019-11-18 04:29:05
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https://www.molympiad.net/2017/08/imo-2008-shortlists.html
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Algebra
1. Find all functions $f: (0, \infty) \mapsto (0, \infty)$ (so $f$ is a function from the positive real numbers) such that $\frac {\left( f(w) \right)^2 + \left( f(x) \right)^2}{f(y^2) + f(z^2) } = \frac {w^2 + x^2}{y^2 + z^2}$ for all positive real numbers $w,x,y,z,$ satisfying $wx = yz.$
2. a) Prove that $\frac {x^{2}}{\left(x - 1\right)^{2}} + \frac {y^{2}}{\left(y - 1\right)^{2}} + \frac {z^{2}}{\left(z - 1\right)^{2}} \geq 1$ for all real numbers $x$, $y$, $z$, each different from $1$, and satisfying $xyz=1$.
b) Prove that equality holds above for infinitely many triples of rational numbers $x$, $y$, $z$, each different from $1$, and satisfying $xyz=1$.
3. Let $S\subseteq\mathbb{R}$ be a set of real numbers. We say that a pair $(f, g)$ of functions from $S$ into $S$ is a Spanish Couple on $S$, if they satisfy the following conditions
• both functions are strictly increasing, i.e. $f(x) < f(y)$ and $g(x) < g(y)$ for all $x$, $y\in S$ with $x < y$;
• the inequality $f\left(g\left(g\left(x\right)\right)\right) < g\left(f\left(x\right)\right)$ holds for all $x\in S$.
Decide whether there exists a Spanish Couple
a) on the set $S = \mathbb{N}$ of positive integers;
b) on the set $S = \{a - \frac {1}{b}: a, b\in\mathbb{N}\}$
4. For an integer $m$, denote by $t(m)$ the unique number in $\{1, 2, 3\}$ such that $m + t(m)$ is a multiple of $3$. A function $f: \mathbb{Z}\to\mathbb{Z}$ satisfies $f( - 1) = 0$, $f(0) = 1$, $f(1) = - 1$ and $f\left(2^{n} + m\right) = f\left(2^n - t(m)\right) - f(m)$ for all integers $m$, $n\ge 0$ with $2^n > m$. Prove that $f(3p)\ge 0$ holds for all integers $p\ge 0$.
5. Let $a$, $b$, $c$, $d$ be positive real numbers such that $abcd = 1$ and $a + b + c + d > \dfrac{a}{b} + \dfrac{b}{c} + \dfrac{c}{d} + \dfrac{d}{a}$. Prove that $a + b + c + d < \dfrac{b}{a} + \dfrac{c}{b} + \dfrac{d}{c} + \dfrac{a}{d}$
6. Let $f: \mathbb{R}\to\mathbb{N}$ be a function which satisfies $$f\left(x + \dfrac{1}{f(y)}\right) = f\left(y + \dfrac{1}{f(x)}\right)$$ for all $x$, $y\in\mathbb{R}$. Prove that there is a positive integer which is not a value of $f$.
7. Prove that for any four positive real numbers $a$, $b$, $c$, $d$ the inequality $\frac {(a - b)(a - c)}{a + b + c} + \frac {(b - c)(b - d)}{b + c + d} + \frac {(c - d)(c - a)}{c + d + a} + \frac {(d - a)(d - b)}{d + a + b}\ge 0$ holds. Determine all cases of equality.
Combinatorics
1. In the plane we consider rectangles whose sides are parallel to the coordinate axes and have positive length. Such a rectangle will be called a box. Two boxes intersect if they have a common point in their interior or on their boundary. Find the largest $n$ for which there exist $n$ boxes $B_1$, $\ldots$, $B_n$ such that $B_i$ and $B_j$ intersect if and only if $i\not\equiv j\pm 1\pmod n$.
2. Let $n \in \mathbb N$ and $A_n$ set of all permutations $(a_1, \ldots, a_n)$ of the set $\{1, 2, \ldots , n\}$ for which $k|2(a_1 + \cdots+ a_k), \text{ for all } 1 \leq k \leq n.$ Find the number of elements of the set $A_n$.
3. In the coordinate plane consider the set $S$ of all points with integer coordinates. For a positive integer $k$, two distinct points $a$, $B\in S$ will be called $k$-friends if there is a point $C\in S$ such that the area of the triangle $ABC$ is equal to $k$. A set $T\subset S$ will be called $k$-clique if every two points in $T$ are $k$-friends. Find the least positive integer $k$ for which there exits a $k$-clique with more than 200 elements.
4. Let $n$ and $k$ be positive integers with $k \geq n$ and $k - n$ an even number. Let $2n$ lamps labelled $1$, $2$, ..., $2n$ be given, each of which can be either on or off. Initially all the lamps are off. We consider sequences of steps: at each step one of the lamps is switched (from on to off or from off to on). Let $N$ be the number of such sequences consisting of $k$ steps and resulting in the state where lamps $1$ through $n$ are all on, and lamps $n + 1$ through $2n$ are all off. Let $M$ be number of such sequences consisting of $k$ steps, resulting in the state where lamps $1$ through $n$ are all on, and lamps $n + 1$ through $2n$ are all off, but where none of the lamps $n + 1$ through $2n$ is ever switched on. Determine $\frac {N}{M}$.
5. Let $S = \{x_1, x_2, \ldots, x_{k + l}\}$ be a $(k + l)$-element set of real numbers contained in the interval $[0, 1]$; $k$ and $l$ are positive integers. A $k$-element subset $A\subset S$ is called nice if $\left |\frac {1}{k}\sum_{x_i\in A} x_i - \frac {1}{l}\sum_{x_j\in S\setminus A} x_j\right |\le \frac {k + l}{2kl}.$ Prove that the number of nice subsets is at least $\dfrac{2}{k + l}\dbinom{k + l}{k}$.
6. For $n\ge 2$, let $S_1$, $S_2$, $\ldots$, $S_{2^n}$ be $2^n$ subsets of $A = \{1, 2, 3, \ldots, 2^{n + 1}\}$ that satisfy the following property: There do not exist indices $a$ and $b$ with $a < b$ and elements $x$, $y$, $z\in A$ with $x < y < z$ and $y$, $z\in S_a$, and $x$, $z\in S_b$. Prove that at least one of the sets $S_1$, $S_2$, $\ldots$, $S_{2^n}$ contains no more than $4n$ elements.
Geometry
1. Let $H$ be the orthocenter of an acute-angled triangle $ABC$. The circle $\Gamma_{A}$ centered at the midpoint of $BC$ and passing through $H$ intersects the sideline $BC$ at points $A_{1}$ and $A_{2}$. Similarly, define the points $B_{1}$, $B_{2}$, $C_{1}$ and $C_{2}$. Prove that the six points $A_{1}$, $A_{2}$, $B_{1}$, $B_{2}$, $C_{1}$ and $C_{2}$ are concyclic.
2. Given trapezoid $ABCD$ with parallel sides $AB$ and $CD$, assume that there exist points $E$ on line $BC$ outside segment $BC$, and $F$ inside segment $AD$ such that $\angle DAE = \angle CBF$. Denote by $I$ the point of intersection of $CD$ and $EF$, and by $J$ the point of intersection of $AB$ and $EF$. Let $K$ be the midpoint of segment $EF$, assume it does not lie on line $AB$. Prove that $I$ belongs to the circumcircle of $ABK$ if and only if $K$ belongs to the circumcircle of $CDJ$.
3. Let $ABCD$ be a convex quadrilateral and let $P$ and $Q$ be points in $ABCD$ such that $PQDA$ and $QPBC$ are cyclic quadrilaterals. Suppose that there exists a point $E$ on the line segment $PQ$ such that $\angle PAE = \angle QDE$ and $\angle PBE = \angle QCE$. Show that the quadrilateral $ABCD$ is cyclic.
4. In an acute triangle $ABC$ segments $BE$ and $CF$ are altitudes. Two circles passing through the point $A$ anf $F$ and tangent to the line $BC$ at the points $P$ and $Q$ so that $B$ lies between $C$ and $Q$. Prove that lines $PE$ and $QF$ intersect on the circumcircle of triangle $AEF$.
5. Let $k$ and $n$ be integers with $0\le k\le n - 2$. Consider a set $L$ of $n$ lines in the plane such that no two of them are parallel and no three have a common point. Denote by $I$ the set of intersections of lines in $L$. Let $O$ be a point in the plane not lying on any line of $L$. A point $X\in I$ is colored red if the open line segment $OX$ intersects at most $k$ lines in $L$. Prove that $I$ contains at least $\dfrac{1}{2}(k + 1)(k + 2)$ red points.
6. There is given a convex quadrilateral $ABCD$. Prove that there exists a point $P$ inside the quadrilateral such that $\angle PAB + \angle PDC$ $= \angle PBC + \angle PAD$ $= \angle PCD + \angle PBA$ $= \angle PDA + \angle PCB$ $= 90^{\circ}$ if and only if the diagonals $AC$ and $BD$ are perpendicular.
7. Let $ABCD$ be a convex quadrilateral with $BA\neq BC$. Denote the incircles of triangles $ABC$ and $ADC$ by $\omega_{1}$ and $\omega_{2}$ respectively. Suppose that there exists a circle $\omega$ tangent to ray $BA$ beyond $A$ and to the ray $BC$ beyond $C$, which is also tangent to the lines $AD$ and $CD$. Prove that the common external tangents to $\omega_{1}$ and $\omega_{2}$ intersect on $\omega$.
Number Theory
1. Let $n$ be a positive integer and let $p$ be a prime number. Prove that if $a$, $b$, $c$ are integers (not necessarily positive) satisfying the equations $a^n + pb = b^n + pc = c^n + pa$ then $a = b = c$.
2. Let $a_1$, $a_2$, $\ldots$, $a_n$ be distinct positive integers, $n\ge 3$. Prove that there exist distinct indices $i$ and $j$ such that $a_i + a_j$ does not divide any of the numbers $3a_1$, $3a_2$, $\ldots$, $3a_n$.
3. Let $a_0$, $a_1$, $a_2$, $\ldots$ be a sequence of positive integers such that the greatest common divisor of any two consecutive terms is greater than the preceding term; in symbols, $\gcd (a_i, a_{i + 1}) > a_{i - 1}$. Prove that $a_n\ge 2^n$ for all $n\ge 0$.
4. Let $n$ be a positive integer. Show that the numbers $\binom{2^n - 1}{0},\; \binom{2^n - 1}{1},\; \binom{2^n - 1}{2},\; \ldots,\; \binom{2^n - 1}{2^{n - 1} - 1}$ are congruent modulo $2^n$ to $1$, $3$, $5$, $\ldots$, $2^n - 1$ in some order.
5. For every $n\in\mathbb{N}$ let $d(n)$ denote the number of (positive) divisors of $n$. Find all functions $f: \mathbb{N}\to\mathbb{N}$ with the following properties
• $d\left(f(x)\right) = x$ for all $x\in\mathbb{N}$.
• $f(xy)$ divides $(x - 1)y^{xy - 1}f(x)$ for all $x$, $y\in\mathbb{N}$.
6. Prove that there are infinitely many positive integers $n$ such that $n^{2} + 1$ has a prime divisor greater than $2n + \sqrt {2n}$.
MOlympiad.NET là dự án thu thập và phát hành các đề thi tuyển sinh và học sinh giỏi toán. Quý bạn đọc muốn giúp chúng tôi chỉnh sửa đề thi này, xin hãy để lại bình luận facebook (có thể đính kèm hình ảnh) hoặc google (có thể sử dụng $\LaTeX$) bên dưới. BBT rất mong bạn đọc ủng hộ UPLOAD đề thi và đáp án mới hoặc liên hệbbt.molympiad@gmail.comChúng tôi nhận tất cả các định dạng của tài liệu: $\TeX$, PDF, WORD, IMG,...
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2022-05-16 16:19:30
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https://socratic.org/questions/what-is-the-standard-form-of-y-x-1-x-7
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# What is the standard form of y=(x-1)(x - 7) ?
May 29, 2018
See a solution process below:
#### Explanation:
To write this equation in standard form we must multiply the two terms on the right side of the equation by multiplying each individual term in the left parenthesis by each individual term in the right parenthesis.
$y = \left(\textcolor{red}{x} - \textcolor{red}{1}\right) \left(\textcolor{b l u e}{x} - \textcolor{b l u e}{7}\right)$ becomes:
$y = \left(\textcolor{red}{x} \times \textcolor{b l u e}{x}\right) - \left(\textcolor{red}{x} \times \textcolor{b l u e}{7}\right) - \left(\textcolor{red}{1} \times \textcolor{b l u e}{x}\right) + \left(\textcolor{red}{1} \times \textcolor{b l u e}{7}\right)$
$y = {x}^{2} - 7 x - 1 x + 7$
We can now combine like terms:
$y = {x}^{2} + \left(- 7 - 1\right) x + 7$
$y = {x}^{2} + \left(- 8\right) x + 7$
$y = {x}^{2} - 8 x + 7$
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2022-05-26 17:58:43
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https://cs.stackexchange.com/questions/62417/what-are-the-substitues-for-kolmogorov-complexity-to-analyse-hashing
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# What are the substitues for Kolmogorov Complexity to analyse Hashing
The paper "Monotone Minimal Perfect Hashing: Searching a Sorted Table with O(1) Accesses" <http://www.itu.dk/people/pagh/papers/sparse.pdf> is the only one that uses Kolmogorov Complexity to obtain a lower bound in the space of a data structure that rank with erros.
My question is: What are the other Algorithmic options of complexity that replace the Kolmogorov Complexity? Is Kolmogorov-Sinai Entropy a option?
• Can you clarify what you mean by "algorithmic options of complexity"? – D.W. Aug 9 '16 at 7:47
• Kolmogorov Complexity ins't computable and is the "ultimate lower bound to space analysis" according to "Squeezing succinct data structures into entropy bounds" <<semanticscholar.org/paper/…>>. There is no detailed (to my knowledge) detailed study in space analysis of data structures using Kolmogorov Complexity or a COMPUTABLE VERSION of Complexity, an algorithmic complexity. – Ricardo S. Aug 9 '16 at 8:04
• We expect references to fulfill the minimal scholarly requirements and be as robust over time as possible. Please take some time to improve your post in this regard. We have collected some advice here. – Raphael Aug 9 '16 at 8:24
• Do you know any computable cost/complexity measure? The typical $O$-runtime certainly is not. – Raphael Aug 9 '16 at 8:25
• Time-bounded Kolmogorov Complexity is a computable upper bound for prefix-free Kolmogorov complexity. Read the article "Time-Bounded Kolmogorov Complexity and Solovay Functions" link.springer.com/article/10.1007/s00224-012-9413-4 – Ricardo S. Aug 23 '16 at 4:02
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2019-06-18 05:09:23
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https://stats.stackexchange.com/questions/126291/correcting-biased-survey-results
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# Correcting biased survey results
Knowing that a population sample (non-random) is biased in terms of its demographics, what are the best practices to correct for this issue?
That is, let's say that I can attach an array of demographics to the sample, and that I wish to transform this sample so that they resemble that of the population these results where picked. Later on, this adjusted sample will be used for mathematical modeling.
As I see it, it is quite straightforward to correct for one certain aspect. If males are under represented by 50 %, all males are assigned a weight of 2. But what if one wants to take into account several variables at the same time? Is building a n-dimensional array the way to go? Are there better solutions?
Are there readily available methods for this? An R-package?
• A point of wording, but one central here. In statistics, "skewed" means "skewed", which is a technical term meaning asymmetry of distributions; it does not mean "biased", which is a technical term that happens to have a similar meaning to its informal meaning. You're talking about biases in sample choice, it seems. Dec 2, 2014 at 14:47
• Why did the sample become biased? This is crucial to know, because for many forms of non-randomness there will be no valid cure for the problem. You cannot turn a judgement sample into something with the properties of a random sample merely be reweighting it, for instance.
– whuber
Dec 2, 2014 at 21:20
As Tim pointed out, you should use survey weighting.
In your case, more specifically, if all the auxiliary variables (your demographic variables) you want to use to make your sample match your population are qualitative variables you will use:
• Post-stratification: If you have the full joint distribution of these variables on the population
• Raking: If you only have the marginal distributions of these variables on the population
More generally, if you have qualitative and quantitative auxiliary variables, you can use a Calibration approach.
Tim also pointed out the survey package in R. There you can find three functions that implements these methods:
• Post-stratification: postStratify
• Raking: rake
• Calibration: calibrate
There is the sampling package in R containing the function for weighting.
• Calibration: calib
It is important to note though that these weighting methods were originally developed under a probability sampling framework, which does not appear to be your case (you referred to your sample as "non-random"). These methods might mitigate some potential bias in your estimates, as long as the auxiliary variables used in the weighting adjustments are related to your outcome variables and to the selection mechanism of your sample. See this paper by Little and Vartivarian for a similar discussion in survey nonresponse.
• Hi, thank you for your answer. This solved the problem at hand and truly opened my eyes to the possibilities of the survey package. At this time, I'm a bit unsure how I take my results further: how do I actually use the weights? Let's assume to cases: $1)$ I want to plot a histogram of the corrected gender distribution. Using svyhist() only plots the biased distribution. $2)$ I want to use my data set to perform a Factor Analysis. As I see it, the factanal() functions do not take weights. Dec 3, 2014 at 14:24
• 1) I don't think svyhist() will work with a qualitative variable such as gender. Instead, I would use barplot in the following way: barplot(svymean(~gender,design=dsn.obj)). 2) You can use the function svyfactanal from the survey package to fit a factor analysis model or try using lavaan.survey from the package with the same name, which fits SEM also taking into account the feature of a complex sample design, such as weighting. Dec 3, 2014 at 15:55
• Regarding your 1st question ("how do I actually use the weights?"), I would recommend using functions from the survey package for statistical methods that are implemented there, such as svyglm for generalized linear models, or looking for other packages, such as lavaan.survey that enables analysis with survey objects. If you want to use a function from other packages that has weights as an argument, such as lm as pointed out by Tim, you can extract the weights from your survey design object using the function weights and passing them as the argument to the function you want to use. Dec 3, 2014 at 16:04
• @djhurio The function calib of the sampling is a good alternative to compute the weights of a calibration estimator. However, one advantage of using the calibrate function is that it creates an object that allows for the other functions in the survey package to incorporate the potential gains in precision in the sampling variance estimates. If the weights created by the calib are employed without any further modification on the codes, I don't believe the standard errors will take that into account. Dec 5, 2014 at 21:50
• You can do that, although, as you mentioned, you will have some rounding errors. In fact, that'w the way statisticians used to do weighting back on the old times :) However, this approach is fine only to compute point estimates. If you need to estimate sampling variability, you will need to rely on appropriate variance estimation techniques, such as Taylor Series Expansion or Repeated Replication (BRR, Jackknife or Bootstrap). Dec 29, 2014 at 22:32
The common thing to do in this kind of situation is to use survey weighting (or an intro here). A clear definition could be found on Wikipedia:
data should usually be weighted if the sample design does not give each individual an equal chance of being selected. For instance, when households have equal selection probabilities but one person is interviewed from within each household, this gives people from large households a smaller chance of being interviewed. This can be accounted for using survey weights. Similarly, households with more than one telephone line have a greater chance of being selected in a random digit dialing sample, and weights can adjust for this.
There is an survey package for R that enables you to use weighting (check also JSS article describing it). Generally, you can use weights with different functions in R (e.g. lm has weights argument).
I follow both Raphael and Tim in their suggestions -- especially about the use of the R package survey. However, as Raphael suggested, these weighting techniques were developed for probability samples and it might not be your case.
If you are familiar to multilevel modeling and have quality auxiliary data to estimate the weights you may use the R package lme4 (which is flexible and friendly-user) to implement Andrew Gelman's suggestions in this and this articles.
I have not applied this to my own work but Gelman's results are impressive. I think these papers are, at least, food for thought.
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2022-07-04 03:32:57
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http://www.sciforums.com/threads/french-newspaper-attacked-by-muslims.110824/#post-2850340
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# French newspaper attacked by Muslims!
Discussion in 'World Events' started by spidergoat, Nov 3, 2011.
Not open for further replies.
Messages:
53,966
3. ### Michael歌舞伎Valued Senior Member
Messages:
20,285
Front Page News:
5. ### Michael歌舞伎Valued Senior Member
Messages:
20,285
I think it's important the Muslims get used to seeing Mohammad made fun of. It's an important part of their assimilation in Western culture. We have a tradition of making fun of all things religious. I think most Muslims would agree Islam should not be discriminated against.
7. ### chimpkinC'mon, get happy!Registered Senior Member
Messages:
4,416
Especially in cartoons...
Quite frankly when I hear about this stuff it makes me wonder when the religious right here is going to violently demand the same kind of pussyfooting...
8. ### cosmictravelerBe kind to yourself always.Valued Senior Member
Messages:
33,264
I do believe that , at times, cartoons do go far beyond being satirically funny and make hurtful remarks about things they are depicting. I do understand that satire is something that is needed but , to me, should be done in as much a tasteful way to get the point across rather than just making people feel hurt and upset. I'd think there can be other ways to get satire type of cartoons made and still be somewhat respectful of those they are mocking. It doesn't disturb me any longer what type of crap the media produces for it just makes them look like fools not those they are trying to poke fun at.
9. ### adoucetteCaca OccursValued Senior Member
Messages:
7,829
Which is no laughing matter considering:
10. ### S.A.M.uniquely dreadfulValued Senior Member
Messages:
72,825
I guess anything is better than focusing on the economy. Look there be Muslims!!!!
11. ### adoucetteCaca OccursValued Senior Member
Messages:
7,829
Nope, but when they kill people or bomb buildings over a cartoon then yes, the idiotic level of idolatry some of them manifest deserves to be pointed out.
12. ### AsguardKiss my dark sideValued Senior Member
Messages:
23,049
you mean like right wing americans calling for the death penelty for people who burn the flag?
13. ### S.A.M.uniquely dreadfulValued Senior Member
Messages:
72,825
You're right, they should kill millions of people over imaginary WMDs instead. Thats how rational people get their hundreds of thousands of brainless morans to go around committing mass murder all over the world. Who needs a fatwa? Freedom fries will do it
14. ### Ghost_007Registered Senior Member
Messages:
2,170
I for one completely reject that (Muslims should get used to seeing Islam made fun of), and I hope that vast majority of Muslims in the West are of the same opinion.
That stuff about assimilation is just bullsh.t.
Muslims are supposed to accept people trampling over what we hold sacred? we are supposed to accept gross misleading and unjust generalisations made about us in the media? its all freedom of speech? lol, gtfo.
Western society is based on fairness and social equality but some with axes to grind use the cover of free speech to peddle their bs and Muslims in general now are not going to fall for it. As a British Muslim I am assimilated as much as I can be, and that is that.
And you Michael are full of sh.t. Demonstrated countless times your prejudice against Muslim people.
15. ### nietzschefanThread KillerValued Senior Member
Messages:
7,721
Freedom of the press is very important and it is always tested by the tasteless first.
16. ### S.A.M.uniquely dreadfulValued Senior Member
Messages:
72,825
Its hardly "freedom of the press" when the corporate sector is manipulating their morans to become cannon fodder for them by whipping up anti-Muslim sentiment. Hitler was also constantly "testing" the freedom of the press with his Nazi propaganda against Jews. Nowadays of course, with the Israel lobby firmly holding on to their gonads, they've made it a criminal offense to say anything bad about Jews or draw the same kind of caricatures as they now do about Muslims.
These are the people who are so offended by religious symbols that people do not have the freedom to wear them under threat of fine or prison. Not really in a position to throw stones
Last edited: Nov 4, 2011
17. ### chimpkinC'mon, get happy!Registered Senior Member
Messages:
4,416
Free speech...is the right to be an a.
http://www.firstamendmentcenter.org/aclu-challenges-states-refusal-to-issue-permit-for-klan-rally
Now if members of the Muslim community had peacefully picketed the paper, boycotted advertisers?
Again, free speech activity. Legal.
And we would not be having this discussion, would we?
As far as getting used to it?
Violence against someone you disagree with isn't acceptable, even if they are insulting you and all you stand for.
I don't appreciate that the paper is producing insensitive images...but firebombing the paper is still not okay.
And to some degree...you do have to learn how to tolerate deliberate offensiveness...or tune it out...I cannot listen to most right-wing talk radio in the US without feeling nauseous within a few minutes, but I don't get to go blow up AM radio stations. I just don't listen to it.
@ SAM...I agree about hijab, I think what a person chooses to wear ought to be entirely his or her own business.
Last edited: Nov 4, 2011
18. ### S.A.M.uniquely dreadfulValued Senior Member
Messages:
72,825
Agreed, but after 10, 20, 30 years of wars and millions killed, its a bit hypocritical to say you should also be allowed to make fun of the people you victimise. I'd say that Muslims in Europe will get used to their religion being made fun of about the time Europeans get used to being killed by the millions for their religion or location or resources
Why should anyone take the French or Americans seriously on any issue related to freedom when they never meet any of their own obligations? I was just listening to the French threatening Iran today and thinking that these are the same people who exported nuclear tech to Israel. Frigging hypocrites
Last edited: Nov 4, 2011
19. ### nietzschefanThread KillerValued Senior Member
Messages:
7,721
And frankly raging against this paper and it's cartoons only empowers it more.
20. ### S.A.M.uniquely dreadfulValued Senior Member
Messages:
72,825
What doesn't empower it? I remember reading somewhere that Europe barely shakes off the blood from its boots before venturing on yet another bloodbath. The irony of Europeans preaching nonviolence is just too hard to swallow. When will they start practising it?
21. ### nietzschefanThread KillerValued Senior Member
Messages:
7,721
Now I do find that funny, Europe is fairly well pussified. Particularly Germany, which controls Europe by hard work and the pen far better than Hitler could ever dream of. Relax, they are done slaying for specific reasons, like religion.
22. ### S.A.M.uniquely dreadfulValued Senior Member
Messages:
72,825
Right after they finish mopping up in Iraq and Afghanistan, you mean? Lol, how will their economies survive without a population to exploit? I don't see a draft in France do you?
23. ### nietzschefanThread KillerValued Senior Member
Messages:
7,721
Comon...seriously you call the European contribution to Afghanistan and Iraq significant? Your beef lay only with U.S, U.K and Canada...possibly Australia. Europe doesn't give a fuck about NATO.
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2022-01-22 17:29:03
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https://www.love2d.org/forums/viewtopic.php?f=5&t=230&p=57852&sid=9d2e1ac0a2c3d1be5e57fdacb9e68b58
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## LUBE (Networking Library)
schme16
Party member
Posts: 127
Joined: Thu Oct 02, 2008 2:46 am
### Re: LUBE (Networking Library)
bartbes wrote:You could take a look at this function here: https://github.com/Nevon/cardboard/blob ... n.lua#L181. That code uses hump.class, btw, so you might have to change the class instantiation syntax.
Thanks for the link! I noticed that I was missing the handshake when attempting to connect, this is apparently a var that must be set when connecting via tcp (I had no idea )
Cowinatub
Prole
Posts: 9
Joined: Fri Nov 09, 2012 6:13 am
### Re: LUBE (Networking Library)
Is packing still supported in the newest release?
Whenever I try and use it it just says 'Attempted to index bin(nil value)'.
qaisjp
Party member
Posts: 491
Joined: Tue Sep 04, 2012 10:49 am
Location: United Kingdom
Contact:
### Re: LUBE (Networking Library)
Can you do multiple connections for one server/client (I'd like to use TCP for message/score sharing and UDP for realtime sync)?
Or should I just use multiple servers and clients for each socket type?
Lua is not an acronym.
Lemony Lime
Prole
Posts: 22
Joined: Fri Dec 28, 2012 9:35 pm
### Re: LUBE (Networking Library)
Are there any real tutorials on how to get started with LUBE? I've found a reasonable amount of info and examples, but no real tutorials, which are always much easier to learn from. Would anyone be willing to write (or link me to, if one already exists) a simple getting started guide, for the betterment of humanity? I'm sure I'll find my way on my own eventually, but if I can avoid the extra work, I'd like to... what with college and work and all that.
bartbes
Sex machine
Posts: 4946
Joined: Fri Aug 29, 2008 10:35 am
Location: The Netherlands
Contact:
### Re: LUBE (Networking Library)
Well, the OP points to viewtopic.php?f=5&t=230&start=210#p21112, and even though some tutorials might be old, this only means the syntax doesn't apply, the way of doing things is still mostly the same.
I've also been asked for a simple-ish example with a modern version, and then I usually link to: https://github.com/Nevon/cardboard/blob ... n.lua#L181
Frohman
Prole
Posts: 21
Joined: Sat Dec 08, 2012 12:32 pm
### Re: LUBE (Networking Library)
I'm having trouble trying to get set up with LUBE :c, I'm using the current development version and hump.class.
Code: Select all
local Vector = require "hump.vector"
local Class = require "hump.class"
require "LUBE" -- for the intertubes[
require "player"
After that I'm trying to set up a ucpClient for use, but for the life of me I can't figure out how to do this correctly, even after seeing an apparently correct implementation at https://github.com/Nevon/cardboard/blob ... n.lua#L181
Here's what two methods I've tried:
Code: Select all
local client
love.graphics.setBackgroundColor(33, 89, 125)
ip, port = cfg:match("^(.+):(%d+)$") assert(ip and port) client = lube.udpClient() client.callbacks.recv = onReceive p:load() end attempt to call field 'udpClient' (a nil value) Code: Select all local client = Class{} function love.load() love.graphics.setBackgroundColor(33, 89, 125) cfg = love.filesystem.read("userconfig.cfg") ip, port = cfg:match("^(.+):(%d+)$")
assert(ip and port)
client:include(lube.udpClient)
client:init()
end
attempt to index field 'callbacks' (a nil value)
I'm very new to networking and classes, so I'm sure I'm at fault here - if someone could tell me how, it'd be greatly appreciated!
bartbes
Sex machine
Posts: 4946
Joined: Fri Aug 29, 2008 10:35 am
Location: The Netherlands
Contact:
### Re: LUBE (Networking Library)
This looks like it should work (the first method, anyway), are you sure the require statements are in the right place?
Frohman
Prole
Posts: 21
Joined: Sat Dec 08, 2012 12:32 pm
### Re: LUBE (Networking Library)
I think so?
Here's the entire main.lua:
Code: Select all
local Vector = require "hump.vector"
local Class = require "hump.class"
require "LUBE" -- for the intertubes
require "player"
universe = {}
p = LocalPlayer(Vector(200, 200))
local client
love.graphics.setBackgroundColor(33, 89, 125)
ip, port = cfg:match("^(.+):(%d+)\$")
assert(ip and port)
client = lube.udpClient()
end
function love.update(dt)
p:update(dt)
p:integrate(dt)
end
function love.draw()
p:draw()
end
print(data)
end
Here's another approach, this time not bothering with LOVE at all:
Code: Select all
local Class = require "hump.class"
require "LUBE"
for k,v in pairs(lube) do
print(k,v)
end
Same result - nothing is printed and the program runs successfully. Lube somehow exists but doesn't have any of its uh, fields...
I'll ask in the support subforums
bartbes
Sex machine
Posts: 4946
Joined: Fri Aug 29, 2008 10:35 am
Location: The Netherlands
Contact:
### Re: LUBE (Networking Library)
I've updated the repository, so LUBE is now a directory, instead of a single monolithic file, and it now also ships with a fallback class commons implementation (I still encourage you to use your own, though). Furthermore, I've added a conditional enet backend, to prepare for 0.9.0, I figured some of you might be interested in that too. Of course, since it's restricted to the LUBE api, it's less powerful than using it directly, but it does mean users can now easily switch, and it's still the same, simple api.
ArchAngel075
Party member
Posts: 317
Joined: Mon Jun 24, 2013 5:16 am
### Re: LUBE (Networking Library)
Im not exactly sure how to set callbacks, i first tried defining them like :
Code: Select all
function server.callbacks.connect(clientid)
--
end
but i error with attempt to index 'server' (nil value)
next i tried :
Code: Select all
function onConnect(clientid)
print("Client Connected")
end
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2021-05-05 21:46:32
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https://d3book.hongtaoh.com/task-6-review.html
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From Day 9
• 6-1-1:
• To specify the height of each bar, I should use .style(), rather than .attr(). In D3.js, .style() is to assign CSS styles to a HTML element. In CSS, we won’t use style because everything is style(). In HTML, we do use it. For example, <div style="height: 75px;"></div>";
• To make the height of each bar correspondes to its data value, I should usefunction(d){return d + "px"} rather than function(d){return d };
• 6-1-2: To add some space (2px) between bars, I need to use margin-right: 2px in the CSS (div.bar{}. To learn more about the differences between padding, border, margin, read this amazing tutorial;
• 6-1-3:
• First, I need to initilize an empty array using dataset = []
• In Javascript, Math is a global object, whereas .push() is an array method.
• A for loop: for (initialization; test; updata){ }
From Day 9
• 6-2-4
• When drawing five circles, we used three terms, “svg”, “circles”, and “circle”, which of them should be fixed, and which can be random? “svg” and “circle” should be fixed, but “circles” can be random;
• One thing I still don’t understand is that in Task 6-1-1, I have to use function(d){return d + "px"} rather than function(d){return d }. However, in Task 6-2-4, using .attr("r", function(d){return d }) is fine, and in .attr("cy", h/2), I don’t need to put "" to enclose h/2. I guess it’s because of the differences between .style() and .attr(). But how different? I don’t know.;
• When we need the index of the data values, we should use function(d,i){} even if we don’t need the d. However, when we only need the d, we can simply use function(d){}.
• 6-2-6: Again, we need to add px in style
From Day 9
To change the color of bars, we need to use .attr("fill", "teal"), rather than .style().
From Day Ten
• 6-1-1. Again, I forgot the difference between .attr() and .style().
• .style() assigns CSS styles (properties and value)s to a HTML element, just like the way we assign CSS styles when we are writing HTML. For example, <div style="height: 75px;"></div>.
• In comparison, .attr() sets an HTML attribute and its values on a HTML element. Note that an element’s class is stored as an HTML atrribute.
• 6-1-3. A.push(B) will put B to the end of A. .push() is an array method.
From Day Ten
• 6-2-2.
• <svg> is just like an HTML element. width, height are its attributs. rect, circle, text are all elements. x, y, cx, cy and r are all attributes. Read this amazing tutorial to really understand what is an element and what are attributs.
• I still did not understand it. Why is that I need to use style() to specify the height of div in Task 6-1-1** but I need to use .attr() to specify the height of a svg here??** I guess it’s because that for div, height is a CSS property, but for svg, height is an attribute? Maybe.
• 6-2-4. I need to use svg.selectAll('circle') rather than svg.select('circle'). Think about why for a moment;
• 6-2-5. As the amazing tutorial mentioned, here, using either .attr() or .style() is fine. The key difference is that I need to use return d / 2 + "px" for stroke-wideth when using .style().
### 3.7.6 Concluding words on .style() and .attr():
• For <div>, height is a CSS property, so I should use style;
• For <svg>, height, width, x, y, cx, cy, r are attributes, so I should use attr();
• For <svg>, fill, stroke, and stroke-width are also attributes. I can use either style() or attr(), but style() is better because it is not about position and size.
• Refer to this tutorial and this tutorial.
From Day Eleven
• 6-5-1. When adding value labels on top of each bar, I should not use rect.selectAll("text").data(dataset).enter().append("text").... Doing so will add twenty text element within each rect element! This is not I am looking for at all. Rather, I should use svg.selectAll("text").data(dataset).enter().append("text").
• 6-5-2. .style("font-size", "11px").
• 6-6-2. Whenever you use function(){}, remember to add return.
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2021-04-13 20:05:44
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https://math.chapman.edu/~jipsen/structures/doku.php?id=partial_semigroups
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## Partial semigroups
Abbreviation: PSgrp
### Definition
A \emph{partial semigroup} is a structure $\mathbf{A}=\langle A,\cdot\rangle$, where
$\cdot$ is a \emph{partial binary operation}, i.e., $\cdot: A\times A\to A+\{*\}$ and
$\cdot$ is \emph{associative}: $(x\cdot y)\cdot z\ne *$ or $x\cdot (y\cdot z)\ne *$ imply $(x\cdot y)\cdot z=x\cdot (y\cdot z)$.
##### Morphisms
Let $\mathbf{A}$ and $\mathbf{B}$ be partial groupoids. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism: if $x\cdot y\ne *$ then $h(x \cdot y)=h(x) \cdot h(y)$
### Examples
Example 1: The morphisms is a small category under composition.
### Basic results
Partial semigroups can be identified with semigroups with zero since for any partial semigroup $A$ we can define a semigroup $A_0=A\cup\{0\}$ (assuming $0\notin A$) and extend the operation on $A$ to $A_0$ by $0x=0=x0$ for all $x\in A$. Conversely, given a semigroup with zero, say $B$, define a partial semigroup $A=B\setminus\{0\}$ and for $x,y\in A$ let $xy=*$ if $xy=0$ in $B$. These two maps are inverses of each other.
However, the category of partial semigroups is not the same as the category of semigroups with zero since the morphisms differ.
### Properties
Classtype first-order
### Finite members
$\begin{array}{lr} f(1)= &2\\ f(2)= &12\\ f(3)= &90\\ f(4)= &960\\ f(5)= &\\ \end{array}$ $\begin{array}{lr} f(6)= &\\ f(7)= &\\ f(8)= &\\ f(9)= &\\ f(10)= &\\ \end{array}$
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浙江大学学报(理学版) 2017, Vol. 44 Issue (3): 253-260, 280 DOI:10.3785/j.issn.1008-9497.2017.03.001 0
### citing the article as [复制中英文]
WEN Yanqing, LIU Baoliang, AN Mingqiang. Multiplicatively weighted Harary index of some graph operations[J]. Journal of Zhejiang University(Science Edition), 2017, 44(3): 253-260, 280. DOI: 10.3785/j.issn.1008-9497.2017.03.001.
[复制英文]
[复制中文]
### Fundation item
Supported by the Doctoral Scientific Research Foundation of Shanxi Datong University (2015-B-06)
### About the author
WEN Yanqing(1980-), ORCID:http://orcid.org/0000-0002-9573-7245, female, doctoral student, lecture, the field of interest are reliability and graph theory, E-mail: oryqwen@163.com
### Corresponding author
AN Mingqiang, ORCID:http://orcid.org/0000-0002-1105-750X, E-mail:anmq@tust.edu.cn
### Article History
Received Date: October 16, 2015
Multiplicatively weighted Harary index of some graph operations
WEN Yanqing1 , LIU Baoliang1 , AN Mingqiang2
1. College of Mathematics and Computer Science, Shanxi Datong University, Datong 037009, Shanxi Province, China;
2. College of Science, Tianjin University of Science and Technology, Tianjin 300457, China
Received Date: October 16, 2015
Fundation item: Supported by the Doctoral Scientific Research Foundation of Shanxi Datong University (2015-B-06)
*Corresponding author: AN Mingqiang, ORCID:http://orcid.org/0000-0002-1105-750X, E-mail:anmq@tust.edu.cn
Abstract: Recently, ALIZADEH et al proposed a modification of the Harary index in which the contributions of vertex pairs were weighted by the product of their degrees. It is named multiplicatively weighted Harary index and defined as: ${H_M}\left( G \right) = \sum\limits_{u \ne v} {\frac{{{\delta _G}\left( u \right){\delta _G}\left( v \right)}}{{{d_G}\left( {u,v} \right)}}}$, where δG(u) denotes the degree of the vertex u in the graph G and dG(u, v) denotes the distance between two vertices u and v in the graph G. In this paper, the explicit formulae for the multiplicatively weighted Harary index of tensor product G×Kr, the strong product GKr and the wreath product G1oG2 in terms of other graph invariants including additively weighted Harary index, Harary index, the first and the second Zagreb indices and the first and the second Zagreb coindices, are obtained, where Kr is the complete graph. Additionally, we apply our results to compute the multiplicatively weighted Harary index of open fence and closed fence graphs.
Key words: multiplicatively weighted Harary index Harary index tensor product strong product wreath product
1. 山西大同大学 数学与计算机科学学院, 山西 大同 037009;
2. 天津科技大学 理学院, 天津 300457
0 Introduction
All graphs considered in this paper are finite undirected simple connected graphs. Let G=(V(G), E(G)) be a graph with vertex set V(G) and edge set E(G). Let δG(v) be the degree of a vertex v in G and dG(u, v) be the distance between two vertices u and v in G. When the graph is clear from the context, we will omit the subscript G from the notation. For other undefined terminology and notations from graph theory, the readers are referred to[1].
A topological index is a number related to a graph invariant under graph isomorphism. Obviously, the number of vertices and edges of a given graph G are topological indices of G. One of the oldest and well-studied distance-based topological index is the Wiener number W(G), also termed as Wiener index in chemical or mathematical chemistry literature, which is defined as the sum of distances over all unordered vertex pairs in G[2], namely,
$W\left( G \right) = \sum\limits_{u \ne v} {{d_G}\left( {u,v} \right)} .$
This formula was introduced by HOSOYA [3], although the concept has been introduced by later WIENER. However, the approach by WIENER is applicable only to acyclic structures, whilst HOSOYA'S matrix definition allowed the Wiener index to be used for any structure.
Another distance-based graph invariant, defined by [4-5] in a fully analogous manner to Wiener index, is the Harary index, which is equal to the sum of reciprocal distances over all unordered vertex pairs in G, that is,
$H\left( G \right) = \sum\limits_{u \ne v} {\frac{1}{{{d_G}\left( {u,v} \right)}}} .$
In 1994, DOBRYNIN et al[6] and GUTMAN[7] independently proposed a vertex-degree-weighted version of Wiener index called degree distance or Schultz molecular topological index, which is defined for a graph G as
$D{D_A}\left( G \right) = \sum\limits_{u \ne v} {\left( {{\delta _G}\left( u \right) + {\delta _G}\left( v \right)} \right){d_G}\left( {u,v} \right)} .$
Similarly, the Gutman index is put forward in [7] and called there the Schultz index of the second kind, for which the name Gutman index has also sometimes been used[8]. It is defined as
$D{D_M}\left( G \right) = \sum\limits_{u \ne v} {{\delta _G}\left( u \right){\delta _G}\left( v \right){d_G}\left( {u,v} \right)} .$
The interested readers may consult [9-11] for Wiener index, [5] for Harary index, [12-13] for degree distance and [14-15] for Gutman index.
Although Harary index is not well known in the mathematical chemistry community, it arises in the study of complex networks. Let n denote the number of vertices of G. Dividing H(G) by n(n-1), we obtain a normalization of H(G), which is called the efficiency of G[16]. The reciprocal value of the efficiency is called the performance of G[17]. For a given network, both efficiency and performance afford a uniform way to express and quantify the small-world property. Since the strength of interactions between nodes in a network is seldom properly described by their topological distances, one needs to consider both the weighted versions of efficiency and performance.
In order to close the gap between the two research communities by drawing their attention to a generalization of a concept, which gives more weight to the contributions of pairs of vertices of high degrees. Recently, ALIZADEH et al[18] introduced an invariant, named additively weighted Harary index, which is defined as
${H_{\rm{A}}}\left( G \right) = \sum\limits_{u \ne v} {\frac{{{\delta _G}\left( u \right) + {\delta _G}\left( v \right)}}{{{d_G}\left( {u,v} \right)}}} .$
Some basic mathematical properties of this index were established[18] and their behavior under several standard graph products were investigated there.
It is known that the intuitive idea of pairs of close atoms contributing more than the distant ones is difficult to capture in topological indices. A possibly useful approach could be used to replace the additive weighting of pairs by the multiplicative one, thus giving rise to a new invariant, named multiplicatively weighted Harary index[18]:
${H_{\rm{M}}}\left( G \right) = \sum\limits_{u \ne v} {\frac{{{\delta _G}\left( u \right){\delta _G}\left( v \right)}}{{{d_G}\left( {u,v} \right)}}} .$
Evidently, the additively (resp. multiplicatively) weighted Harary index is related to the Harary index in the same way as the degree distance (resp. Gutman index) is related to the Wiener index.
Very recently, PATTABIRAMAN et al[19] gave the exact formulae for the additively weighted Harary index of tensor product G×Km0, m1, …, mr-1 and the strong product GKm0, m1, …, mr-1, where Km0, m1, …, mr-1 is the complete multipartite graph with partite sets of sizes m0, m1, …, mr-1.
In this paper, we continue this program to the multiplicatively weighted Harary index, and the exact formulae for the multiplicatively weighted Harary index of tensor product G×Kr, the strong product GKr and the wreath product G1oG2 in terms of other graph invariants including additively weighted Harary index, Harary index, the first and the second Zagreb indices, and the first and the second Zagreb coindices, are obtained, where Kr is the complete graph. Additionally, we apply our results to compute the multiplicatively weighted Harary index of open fence and closed fence graphs.
The paper is organized as follows. In section 1, we give some necessary definitions. In section 2 to 4, we present our main results and give some corresponding examples, respectively.
1 Preliminaries 1.1 Some definitions
For a given graph G, its first and second Zagreb indices are defined as follows:
$\begin{array}{l} {M_1}\left( G \right) = \sum\limits_{u \in V\left( G \right)} {\delta {{\left( u \right)}^2}} ,\\ {M_2}\left( G \right) = \sum\limits_{e = uv \in E\left( G \right)} {\delta \left( u \right)\delta \left( v \right)} . \end{array}$
The first Zagreb index can be also expressed as a sum over edges of G,
${M_1}\left( G \right) = \sum\limits_{e = uv \in E\left( G \right)} {\left( {\delta \left( u \right) + \delta \left( v \right)} \right)} .$
For the proof of this fact and more information on Zagreb indices, we encourage the interested reader to [20].
The first and the second Zagreb coindices of a graph G are defined as follows[21]:
$\begin{array}{*{20}{c}} {{{\bar M}_1}\left( G \right) = \sum\limits_{e = uv \notin E\left( G \right)} {\left( {\delta \left( u \right) + \delta \left( v \right)} \right)} ,}\\ {{{\bar M}_2}\left( G \right) = \sum\limits_{e = uv \notin E\left( G \right)} {\delta \left( u \right)\delta \left( v \right)} .} \end{array}$
Let Kn, Cn and Pn denote the n-vertex complete graph, cycle and path, respectively. We call C3 a triangle.
1.2 Product graphs
Now, we introduce three standard types of product graphs that we consider in this paper. For two simple graphs G and H, their tensor product denoted by G×H, has vertex set V(GV(H) in which (g1, h1) and (g2, h2) are adjacent whenever g1g2 is an edge in G and h1h2 is an edge in H. Note that if G and H are connected graphs, then G×H is connected only if at least one of the graph is nonbipartite. The strong product of graphs G and H, denoted by G H, is the graph with vertex set V(GV(H)={(u, v):uV(G), vV(H)} and (u, x)(v, y) is an edge whenever (ⅰ)u=v and xyE(H), or (ⅱ) uvE(G) and x=y, or (ⅲ) uvE(G) and xyE(H). Similarly, the wreath product (also known as the composition) of the graphs G and H, denoted by GoH, has vertex set V(GV(H) in which (g1, h1)(g2, h2) is an edge whenever g1g2E(G), or g1=g2 and h1h2E(H). The tensor product of graphs has been extensively studied in relation to the areas such as graph colorings, graph recognition, decompositions of graphs, and design theory, see [22-26].
For more information about graph products, please see monograph[25]. There is a growing corpus of literature concerned with the study of graph invariants of tensor product, Cartesian product and strong product[27-29].
2 Multiplicatively weighted Hararyindex of tensor product of graphs
Let G be a connected graph with V(G)={v0, v1, …, vn-1} and V(Kr)={u1, u2, …, ur-1}. For convenience, let xij denote the vertex (vi, uj) of G×Kr. The following lemma, which follows easily from the properties and structure of G×Kr, is used in the proof of our main result in this section.
Lemma 1 Let G be a connected graph on n≥2 vertices and xij, xkp be any pair vertices of the graph G′=G×Kr, where r≥3.
(ⅰ)If vivkE(G), then
$\begin{array}{l} {d_{G'}}\left( {{x_{ij}},{x_{kp}}} \right) = \\ \;\;\;\left\{ \begin{array}{l} 1,\;\;\;\;\;{\rm{if}}\;j \ne p,\\ 2,\;\;\;\;\;{\rm{if}}\;j \ne p,\;and\;{v_i}{v_k}\;{\rm{is}}\;{\rm{on}}\;{\rm{a}}\;{\rm{triangle}}\;{\rm{of}}\;G,\\ 3,\;\;\;\;\;{\rm{if}}\;j \ne p,\;and\;{v_i}{v_k}\;{\rm{is}}\;{\rm{not}}\;{\rm{on}}\;{\rm{a}}\;{\rm{triangle}}\;{\rm{of}}\;G. \end{array} \right. \end{array}$
(ⅱ)If vivk$\notin$E(G), then dG(xij, xkp)=dG(vi, vk).
(ⅲ)dG(xij, xip)=2.
Lemma 2 Let G be a connected graph and let G′=G×Kr. Then the degree of a vertex (vi, uj) in G′ is δG((vi, uj))=δG(vi)(r-1).
Now, we present the exact formulae for the multiplicatively weighted Harary index of G×Kr.
Theorem 1 Let G be a connected graph with n≥2 vertices and E2 be the set of edges of G which do not lie on any triangle of it. Then
$\begin{array}{l} {H_M}\left( {G \times {K_r}} \right) = r{\left( {r - 1} \right)^2}\left( {r{H_M}\left( G \right) + \frac{1}{4}\left( {r - 1} \right){M_1}\left( G \right) - } \right.\\ \;\;\;\;\;\;\;\;\;\;\left. {\left( {\frac{{{M_2}\left( G \right)}}{2} + \frac{1}{{12}}\sum\limits_{{v_i}{v_k} \in {E_2}} {{\delta _G}\left( {{v_i}} \right){\delta _G}\left( {{v_k}} \right)} } \right)} \right), \end{array}$
where r≥3.
Proof Let us denote G′=G×Kr. Obviously,
$\begin{array}{l} {H_M}\left( {G'} \right) = \frac{1}{2}\sum\limits_{{x_{ij}},{x_{kp}} \in V\left( {G'} \right)} {\frac{{{\delta _{G'}}\left( {{x_{ij}}} \right){\delta _{G'}}\left( {{x_{kp}}} \right)}}{{{d_{G'}}\left( {{x_{ij}},{x_{kp}}} \right)}}} = \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\frac{1}{2}\left( {\sum\limits_{i = 0}^{n - 1} {\sum\limits_{\begin{array}{*{20}{c}} {j,p = 0}\\ {j \ne p} \end{array}}^{r - 1} {\frac{{{\delta _{G'}}\left( {{x_{ij}}} \right){\delta _{G'}}\left( {{x_{ip}}} \right)}}{{{d_{G'}}\left( {{x_{ij}},{x_{ip}}} \right)}}} } + } \right.\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\sum\limits_{\begin{array}{*{20}{c}} {i,k = 0}\\ {i \ne k} \end{array}}^{n - 1} {\sum\limits_{j = 0}^{r - 1} {\frac{{{\delta _{G'}}\left( {{x_{ij}}} \right){\delta _{G'}}\left( {{x_{kj}}} \right)}}{{{d_{G'}}\left( {{x_{ij}},{x_{kj}}} \right)}}} } + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left. {\sum\limits_{\begin{array}{*{20}{c}} {i,k = 0}\\ {i \ne k} \end{array}}^{n - 1} {\sum\limits_{\begin{array}{*{20}{c}} {j,p = 0}\\ {j \ne p} \end{array}}^{r - 1} {\frac{{{\delta _{G'}}\left( {{x_{ij}}} \right){\delta _{G'}}\left( {{x_{kp}}} \right)}}{{{d_{G'}}\left( {{x_{ij}},{x_{kp}}} \right)}}} } } \right) = \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\frac{1}{2}\left\{ {{A_1} + {A_2} + {A_3}} \right\}, \end{array}$ (1)
where A1 to A3 are the sums of the above terms, in order. In what follows, we compute A1 to A3 of (1), separately.
First, we compute $\sum\limits_{i=0}^{n-1}{\sum\limits_{\begin{smallmatrix} j,p=0 \\ j\ne p \end{smallmatrix}}^{r-1}{\frac{{{\delta }_{{{G}'}}}\left( {{x}_{ij}} \right){{\delta }_{{{G}'}}}\left( {{x}_{ip}} \right)}{{{d}_{{{G}'}}}\left( {{x}_{ij}},{{x}_{ip}} \right)}}}.$
$\begin{array}{l} {A_1} = \sum\limits_{i = 0}^{n - 1} {\sum\limits_{\begin{array}{*{20}{c}} {j,p = 0}\\ {j \ne p} \end{array}}^{r - 1} {\frac{{{\delta _{G'}}\left( {{x_{ij}}} \right){\delta _{G'}}\left( {{x_{ip}}} \right)}}{{{d_{G'}}\left( {{x_{ij}},{x_{ip}}} \right)}}} } = \\ \;\;\;\;\;\;\;\;\sum\limits_{i = 0}^{n - 1} {\sum\limits_{\begin{array}{*{20}{c}} {j,p = 0}\\ {j \ne p} \end{array}}^{r - 1} {\frac{{{\delta _G}\left( {{v_i}} \right)\left( {r - 1} \right) \cdot {\delta _G}\left( {{v_i}} \right)\left( {r - 1} \right)}}{2}} } = \\ \;\;\;\;\;\;\;\;\frac{1}{2}r{\left( {r - 1} \right)^3}{M_1}\left( G \right),{\rm{by}}\;{\rm{lemmas}}\;{\rm{1}}\;{\rm{and}}\;{\rm{2}}{\rm{.}} \end{array}$ (2)
Next we compute $\sum\limits_{j=0}^{r-1}{\sum\limits_{\begin{smallmatrix} i,k=0 \\ i\ne k \end{smallmatrix}}^{n-1}{\frac{{{\delta }_{{{G}'}}}\left( {{x}_{ij}} \right){{\delta }_{{{G}'}}}\left( {{x}_{kj}} \right)}{{{d}_{{{G}'}}}\left( {{x}_{ij}},{{x}_{kj}} \right)}}}.$
To do this, originally we calculate $\sum\limits_{\begin{smallmatrix} i,k=0 \\ i\ne k \end{smallmatrix}}^{n-1}{\frac{{{\delta }_{{{G}'}}}\left( {{x}_{ij}} \right){{\delta }_{{{G}'}}}\left( {{x}_{kj}} \right)}{{{d}_{{{G}'}}}\left( {{x}_{ij}},{{x}_{kj}} \right)}}.$
Let E1={uvE(G)|uv is on a triangle of G} and E2=E(G)-E1.
$\begin{array}{l} \sum\limits_{\begin{array}{*{20}{c}} {i,k = 0}\\ {i \ne k} \end{array}}^{n - 1} {\frac{{{\delta _{G'}}\left( {{x_{ij}}} \right){\delta _{G'}}\left( {{x_{kj}}} \right)}}{{{d_{G'}}\left( {{x_{ij}},{x_{kj}}} \right)}}} = \\ \;\;\;\;\;\;\;\;\;\sum\limits_{\begin{array}{*{20}{c}} {i,k = 0}\\ {i \ne k}\\ {{v_i}{v_k} \notin E\left( G \right)} \end{array}}^{n - 1} {\frac{{{\delta _{G'}}\left( {{x_{ij}}} \right){\delta _{G'}}\left( {{x_{kj}}} \right)}}{{{d_{G'}}\left( {{x_{ij}},{x_{kj}}} \right)}}} + \\ \;\;\;\;\;\;\;\;\;\sum\limits_{\begin{array}{*{20}{c}} {i,k = 0}\\ {i \ne k}\\ {{v_i}{v_k} \in {E_1}} \end{array}}^{n - 1} {\frac{{{\delta _{G'}}\left( {{x_{ij}}} \right){\delta _{G'}}\left( {{x_{kj}}} \right)}}{{{d_{G'}}\left( {{x_{ij}},{x_{kj}}} \right)}}} + \\ \;\;\;\;\;\;\;\;\;\sum\limits_{\begin{array}{*{20}{c}} {i,k = 0}\\ {i \ne k}\\ {{v_i}{v_k} \in {E_2}} \end{array}}^{n - 1} {\frac{{{\delta _{G'}}\left( {{x_{ij}}} \right){\delta _{G'}}\left( {{x_{kj}}} \right)}}{{{d_{G'}}\left( {{x_{ij}},{x_{kj}}} \right)}}} = \\ \;\;\;\;\;\;\;\;\;\sum\limits_{\begin{array}{*{20}{c}} {i,k = 0}\\ {i \ne k}\\ {{v_i}{v_k} \notin E\left( G \right)} \end{array}}^{n - 1} {\frac{{{\delta _G}\left( {{v_i}} \right)\left( {r - 1} \right) \cdot {\delta _G}\left( {{v_k}} \right)\left( {r - 1} \right)}}{{{d_G}\left( {{v_i},{v_k}} \right)}}} + \\ \;\;\;\;\;\;\;\;\;\sum\limits_{\begin{array}{*{20}{c}} {i,k = 0}\\ {i \ne k}\\ {{v_i}{v_k} \in {E_1}} \end{array}}^{n - 1} {\frac{{{\delta _G}\left( {{v_i}} \right)\left( {r - 1} \right) \cdot {\delta _G}\left( {{v_k}} \right)\left( {r - 1} \right)}}{2}} + \\ \;\;\;\;\;\;\;\;\;\sum\limits_{\begin{array}{*{20}{c}} {i,k = 0}\\ {i \ne k}\\ {{v_i}{v_k} \in {E_2}} \end{array}}^{n - 1} {\frac{{{\delta _G}\left( {{v_i}} \right)\left( {r - 1} \right) \cdot {\delta _G}\left( {{v_k}} \right)\left( {r - 1} \right)}}{3}} , \end{array}$
by lemmas 1 and 2,
\begin{align} {\rm{the}}\;{\rm{above}}\;{\rm{formula}}\;{\rm{ = }}\;{\left( {r - 1} \right)^2}\left[ \left( \sum\limits_{\begin{smallmatrix} i,k=0 \\ \ \ i\ne k \\ {{v}_{i}}{{v}_{k}}\notin E\left( G \right) \end{smallmatrix}}^{n-1}{\frac{{{\delta }_{G}}\left( {{v}_{i}} \right){{\delta }_{G}}\left( {{v}_{k}} \right)}{{{d}_{G}}\left( {{v}_{i}},{{v}_{k}} \right)}}+ \right. \right. \\ \sum\limits_{\begin{smallmatrix} i,k=0 \\ \ \ i\ne k \\ {{v}_{i}}{{v}_{k}}\in {{E}_{1}} \end{smallmatrix}}^{n-1}{\frac{{{\delta }_{G}}\left( {{v}_{i}} \right){{\delta }_{G}}\left( {{v}_{k}} \right)}{{{d}_{G}}\left( {{v}_{i}},{{v}_{k}} \right)}}+\left. \sum\limits_{\begin{smallmatrix} i,k=0 \\ \ \ i\ne k \\ {{v}_{i}}{{v}_{k}}\in {{E}_{2}} \end{smallmatrix}}^{n-1}{\frac{{{\delta }_{G}}\left( {{v}_{i}} \right){{\delta }_{G}}\left( {{v}_{k}} \right)}{{{d}_{G}}\left( {{v}_{i}},{{v}_{k}} \right)}} \right)- \\ \sum\limits_{\begin{smallmatrix} i,k=0 \\ \ \ i\ne k \\ {{v}_{i}}{{v}_{k}}\in {{E}_{1}} \end{smallmatrix}}^{n-1}{\frac{{{\delta }_{G}}\left( {{v}_{i}} \right){{\delta }_{G}}\left( {{v}_{k}} \right)}{2}}-2\left. \sum\limits_{\begin{smallmatrix} i,k=0 \\ \ \ i\ne k \\ {{v}_{i}}{{v}_{k}}\in {{E}_{2}} \end{smallmatrix}}^{n-1}{\frac{{{\delta }_{G}}\left( {{v}_{i}} \right){{\delta }_{G}}\left( {{v}_{k}} \right)}{3}} \right], \\ \end{align}
since dG(vi, vk)=1, if vivkE1 and vivkE2,
$\begin{array}{l} {\rm{the}}\;{\rm{above}}\;{\rm{formula}}\;{\rm{ = }}\;{\left( {r - 1} \right)^2}\left[ {\sum\limits_{\begin{array}{*{20}{c}} {i,k = 0}\\ {i \ne k} \end{array}}^{n - 1} {\frac{{{\delta _G}\left( {{v_i}} \right){\delta _G}\left( {{v_k}} \right)}}{{{d_G}\left( {{v_i},{v_k}} \right)}}} - } \right.\\ \left( {\sum\limits_{\begin{array}{*{20}{c}} {i,k = 0}\\ {i \ne k}\\ {{v_i}{v_k} \in {E_1}} \end{array}}^{n - 1} {\frac{{{\delta _G}\left( {{v_i}} \right){\delta _G}\left( {{v_k}} \right)}}{2}} + \sum\limits_{\begin{array}{*{20}{c}} {i,k = 0}\\ {i \ne k}\\ {{v_i}{v_k} \in {E_2}} \end{array}}^{n - 1} {\frac{{{\delta _G}\left( {{v_i}} \right){\delta _G}\left( {{v_k}} \right)}}{2}} } \right) - \\ \left. {\sum\limits_{\begin{array}{*{20}{c}} {i,k = 0}\\ {i \ne k}\\ {{v_i}{v_k} \in {E_2}} \end{array}}^{n - 1} {\frac{{{\delta _G}\left( {{v_i}} \right){\delta _G}\left( {{v_k}} \right)}}{6}} } \right] = \\ {\left( {r - 1} \right)^2}\left( {2{H_M}\left( G \right) - {M_2}\left( G \right) - \sum\limits_{\begin{array}{*{20}{c}} {i,k = 0}\\ {i \ne k}\\ {{v_i}{v_k} \in {E_2}} \end{array}}^{n - 1} {\frac{{{\delta _G}\left( {{v_i}} \right){\delta _G}\left( {{v_k}} \right)}}{6}} } \right), \end{array}$ (3)
since each edge vivk of G is being counted twice in the sum, that is, vivk and vkvi.
Now summing (3) over j=0, 1, …, r-1, we have
$\begin{array}{l} {A_2} = \sum\limits_{j = 0}^{r - 1} {\sum\limits_{\begin{array}{*{20}{c}} {i,k = 0}\\ {i \ne k} \end{array}}^{n - 1} {\frac{{{\delta _{G'}}\left( {{x_{ij}}} \right){\delta _{G'}}\left( {{x_{kj}}} \right)}}{{{d_{G'}}\left( {{x_{ij}},{x_{kj}}} \right)}}} } = \\ \;\;\;\;\;\;r{\left( {r - 1} \right)^2}\left( {2{H_M}\left( G \right) - {M_2}\left( G \right) - } \right.\\ \left. {\sum\limits_{\begin{array}{*{20}{c}} {i,k = 0}\\ {i \ne k}\\ {{v_i}{v_k} \in {E_2}} \end{array}}^{n - 1} {\frac{{{\delta _G}\left( {{v_i}} \right){\delta _G}\left( {{v_k}} \right)}}{6}} } \right). \end{array}$ (4)
Finally, we compute $\sum\limits_{\begin{smallmatrix} i,k=0 \\ \ \ i\ne k \end{smallmatrix}}^{n-1}{\sum\limits_{\begin{smallmatrix} j,p=0 \\ \ \ j\ne p \end{smallmatrix}}^{r-1}{\frac{{{\delta }_{{{G}'}}}\left( {{x}_{ij}} \right){{\delta }_{{{G}'}}}\left( {{x}_{kp}} \right)}{{{d}_{{{G}'}}}\left( {{x}_{ij}},{{x}_{kp}} \right)}.}}$
$\begin{array}{l} {A_3} = \sum\limits_{\begin{array}{*{20}{c}} {i,k = 0}\\ {i \ne k} \end{array}}^{n - 1} {\sum\limits_{\begin{array}{*{20}{c}} {j,p = 0}\\ {j \ne p} \end{array}}^{r - 1} {\frac{{{\delta _{G'}}\left( {{x_{ij}}} \right){\delta _{G'}}\left( {{x_{kp}}} \right)}}{{{d_{G'}}\left( {{x_{ij}},{x_{kp}}} \right)}}} } = \\ \;\;\;\;\;\;\;\sum\limits_{\begin{array}{*{20}{c}} {i,k = 0}\\ {i \ne k} \end{array}}^{n - 1} {\sum\limits_{\begin{array}{*{20}{c}} {j,p = 0}\\ {j \ne p} \end{array}}^{r - 1} {\frac{{{\delta _G}\left( {{v_i}} \right)\left( {r - 1} \right) \cdot {\delta _G}\left( {{v_k}} \right)\left( {r - 1} \right)}}{{{d_G}\left( {{v_i},{v_k}} \right)}}} } = \\ \;\;\;\;\;\;\;2r{\left( {r - 1} \right)^3}{H_M}\left( G \right),\;{\rm{by}}\;{\rm{lemmas}}\;{\rm{1}}\;{\rm{and}}\;{\rm{2}}{\rm{.}} \end{array}$ (5)
Combining(2), (4) and (5) with (1), we obtain
$\begin{array}{l} {H_M}\left( {G \times {K_r}} \right) = r{\left( {r - 1} \right)^2}\left( {r{H_M}\left( G \right) + } \right.\\ \;\;\;\;\;\;\;\frac{1}{4}\left( {r - 1} \right){M_1}\left( G \right) - \left( {\frac{{{M_2}\left( G \right)}}{2} + } \right.\\ \;\;\;\;\;\;\;\left. {\left. {\frac{1}{{12}}\sum\limits_{{v_i}{v_k} \in {E_2}} {{\delta _G}\left( {{v_i}} \right){\delta _G}\left( {{v_k}} \right)} } \right)} \right). \end{array}$
By theorem 1, we have the following corollaries.
Corollary 1 Let G be a connected graph with n≥2 vertices. If each edge of G is on a triangle, then ${H_M}\left( {G \times {K_r}} \right) = r{\left( {r-1} \right)^2}\left( {r{H_M}\left( G \right) + \frac{1}{4}\left( {r-1} \right){M_1}\left( G \right)-\frac{{{M_2}\left( G \right)}}{2}} \right)$, where r≥3.
If G is a triangle-free graph, then E2=E(G) and thus $\sum\limits_{{v_i}{v_k} \in {E_2}} {{\delta _G}\left( {{v_i}} \right)} {\delta _G}\left( {{v_k}} \right) = {M_2}\left( G \right)$.
Corollary 2 If G is a connected triangle-free graph with n≥2 vertices, then ${H_M}\left( {G \times {K_r}} \right) = r{\left( {r-1} \right)^2} \times \left( {r{H_M}\left( G \right) + \frac{1}{4}\left( {r-1} \right){M_1}\left( G \right)-\frac{{2{M_2}\left( G \right)}}{3}} \right)$, where r≥3.
Note that M1(Kn)=n(n-1)2, ${M_2}\left( {{K_n}} \right) = \frac{{n{{\left( {n-1} \right)}^3}}}{2}$; M1(Pn)=4n-6, M2(Pn)=4n-8, n≥2; M1(Cn)=M2(Cn)=4n, n≥3. In addition, we see that $H\left( {{P_n}} \right) = n\left( {\sum\limits_{i = 1}^n {\frac{1}{i}} } \right)-n$ and $H\left( {{C_n}} \right) = n\left( {\sum\limits_{i = 1}^{\frac{n}{2}} {\frac{1}{i}} } \right)-1$ for n even and $H\left( {{C_n}} \right) = n\left( {\sum\limits_{i = 1}^{\frac{{n-1}}{2}} {\frac{1}{i}} } \right)$ for n odd.
By direct calculations, we obtain expressions for the values of the multiplicatively weighted Harary index and the additively weighted Harary index of Kn, Pn and Cn: ${H_M}\left( {{K_n}} \right) = \frac{{n{{\left( {n-1} \right)}^3}}}{2}$, HA(Kn)=n(n-1)2; ${H_M}\left( {{P_n}} \right) = H\left( {{P_n}} \right) + 2\left( {\sum\limits_{i = 1}^{n-1} {\frac{1}{i}} } \right)-\frac{2}{{n-1}}$, ${H_A}\left( {{P_n}} \right) = H\left( {{P_n}} \right) + 4\left( {\sum\limits_{i = 1}^{n-1} {\frac{1}{i}} } \right)-\frac{3}{{n-1}}$ and HM(Cn)=HA(Cn)=4H(Cn).
Combining the above known results and corollaries 1 and 2, immediately, we can obtain the explicit multiplicatively weighted Harary index of the following graphs:
Example 1
$\begin{array}{l} \left( {\rm{a}} \right)\;{\rm{For}}\;n \ge 2,r \ge 3,\;{H_M}\left( {{K_n} \times {K_r}} \right) = \frac{{n{{\left( {n - 1} \right)}^2}}}{4}\\ r\left( {\left( {n - 1} \right)\left( {2{r^3} - 5{r^2} + 4r - 1} \right) + {{\left( {r - 1} \right)}^3}} \right). \end{array}$
$\begin{array}{l} \left( {\rm{b}} \right)\;{\rm{For}}\;n \ge 2,r \ge 3,\;{H_M}\left( {{P_n} \times {K_r}} \right) = \\ {r^2}{\left( {r - 1} \right)^2}\left( {H\left( {{P_n}} \right) + 2\left( {\sum\limits_{i = 1}^{n - 1} {\frac{1}{i}} } \right) - \frac{2}{{n - 1}}} \right) + \frac{{2n - 3}}{2}\\ r{\left( {r - 1} \right)^3} - \frac{7}{3}\left( {n - 2} \right)r{\left( {r - 1} \right)^2}. \end{array}$
$\begin{array}{l} \left( {\rm{c}} \right)\;{\rm{For}}\;n \ge 2,\;r = 3,\;{H_M}\left( {{C_n} \times {K_r}} \right) = 3r\left( {5{r^3} - } \right.\\ \left. {13{r^2} + 11r - 3} \right) \end{array}$
$\begin{array}{l} {\rm{For}}\;n \ge 2,\;r > 3,\;{H_M}\left( {{C_n} \times {K_r}} \right) = 4{r^2}{\left( {r - 1} \right)^2} \times \\ H\left( {{C_n}} \right) + nr{\left( {r - 1} \right)^3} - \frac{7}{3}nr{\left( {r - 1} \right)^2}. \end{array}$
3 Multiplicatively weighted Hararyindex of strong product of graphs
In this section, we obtain the multiplicatively weighted Harary index of G Kr. Let G be a connected graph with V(G)={v0, v1, …, vn-1} and V(Kr)={u1, u2, …, ur-1}. For convenience, let xij denote the vertex (vi, uj) of GKr. Firstly, we give the following lemma, which follows directly from the properties and structure of GKr, is used in the proof of our main result in this section.
Lemma 3 Let G be a connected graph and let G′=GKr. Then
(ⅰ)For any pair of vertices xij, xkpV(G′), dG(xij, xip)=1 and dG(xij, xkp)=dG(vi, vk).
(ⅱ)The degree of a vertex (vi, uj) in G′ is δG((vi, uj))=G(vi)+(r-1).
Theorem 2 Let G be a connected graph with n vertices and m edges. Then
$\begin{array}{l} {H_M}\left( {G{K_r}} \right) = {r^4}{H_M}\left( G \right) + {r^3}\left( {r - 1} \right){H_A}\left( G \right) + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{r^2}{\left( {r - 1} \right)^2}H\left( G \right) + \frac{{{M_1}\left( G \right){r^3}}}{2}\left( {r - 1} \right) + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;2m{r^2}{\left( {r - 1} \right)^2} + \frac{{nr}}{2}{\left( {r - 1} \right)^3}. \end{array}$
Proof Let us denote G′=GKr. Obviously,
$\begin{array}{l} {H_M}\left( {G'} \right) = \frac{1}{2}\sum\limits_{{x_{ij}},{x_{kp}} \in V\left( {G'} \right)} {\frac{{{\delta _{G'}}\left( {{x_{ij}}} \right){\delta _{G'}}\left( {{x_{kp}}} \right)}}{{{d_{G'}}\left( {{x_{ij}},{x_{kp}}} \right)}}} = \\ \;\;\;\;\;\;\;\;\frac{1}{2}\left( {\sum\limits_{i = 0}^{n - 1} {\sum\limits_{\begin{array}{*{20}{c}} {j,p = 0}\\ {j \ne p} \end{array}}^{r - 1} {\frac{{{\delta _{G'}}\left( {{x_{ij}}} \right){\delta _{G'}}\left( {{x_{ip}}} \right)}}{{{d_{G'}}\left( {{x_{ij}},{x_{ip}}} \right)}}} } + } \right.\\ \;\;\;\;\;\;\;\;\sum\limits_{\begin{array}{*{20}{c}} {i,k = 0}\\ {i \ne k} \end{array}}^{n - 1} {\sum\limits_{j = 0}^{r - 1} {\frac{{{\delta _{G'}}\left( {{x_{ij}}} \right){\delta _{G'}}\left( {{x_{kj}}} \right)}}{{{d_{G'}}\left( {{x_{ij}},{x_{kj}}} \right)}}} } + \\ \;\;\;\;\;\;\;\;\left. {\sum\limits_{\begin{array}{*{20}{c}} {i,k = 0}\\ {i \ne k} \end{array}}^{n - 1} {\sum\limits_{\begin{array}{*{20}{c}} {j,p = 0}\\ {j \ne p} \end{array}}^{r - 1} {\frac{{{\delta _{G'}}\left( {{x_{ij}}} \right){\delta _{G'}}\left( {{x_{kp}}} \right)}}{{{d_{G'}}\left( {{x_{ij}},{x_{kp}}} \right)}}} } } \right) = \\ \;\;\;\;\;\;\;\;\frac{1}{2}\left\{ {{A_1} + {A_2} + {A_3}} \right\}, \end{array}$ (6)
where A1, A2 and A3 are the sums of the above terms, in order.
In what follows, we calculate A1, A2 and A3 of (6), separately.
$\begin{array}{l} {A_1} = \sum\limits_{i = 0}^{n - 1} {\sum\limits_{\begin{array}{*{20}{c}} {j,p = 0}\\ {j \ne p} \end{array}}^{r - 1} {\frac{{{\delta _{G'}}\left( {{x_{ij}}} \right){\delta _{G'}}\left( {{x_{ip}}} \right)}}{{{d_{G'}}\left( {{x_{ij}},{x_{ip}}} \right)}}} } = \\ \;\;\;\;\;\;\;\sum\limits_{i = 0}^{n - 1} {\sum\limits_{\begin{array}{*{20}{c}} {j,p = 0}\\ {j \ne p} \end{array}}^{r - 1} {\left( {{r^2}{\delta _G}{{\left( {{v_i}} \right)}^2} + {{\left( {r - 1} \right)}^2} + } \right.} } \\ \;\;\;\;\;\;\;\left. {2r\left( {r - 1} \right){\delta _G}\left( {{v_i}} \right)} \right) = \\ \;\;\;\;\;\;\;{r^3}\left( {r - 1} \right){M_1}\left( G \right) + nr{\left( {r - 1} \right)^3} + \\ \;\;\;\;\;\;\;4m{r^2}{\left( {r - 1} \right)^2},{\rm{by}}\;{\rm{lemma}}\;{\rm{3}}{\rm{.}} \end{array}$ (7)
$\begin{array}{l} {A_2} = \sum\limits_{j = 0}^{r - 1} {\sum\limits_{\begin{array}{*{20}{c}} {i,k = 0}\\ {i \ne k} \end{array}}^{n - 1} {\frac{{{\delta _{G'}}\left( {{x_{ij}}} \right){\delta _{G'}}\left( {{x_{kj}}} \right)}}{{{d_{G'}}\left( {{x_{ij}},{x_{kj}}} \right)}}} } = \\ \;\;\;\;\;\;\;\sum\limits_{j = 0}^{r - 1} {\sum\limits_{\begin{array}{*{20}{c}} {i,k = 0}\\ {i \ne k} \end{array}}^{n - 1} {\frac{{\left( {r{\delta _G}\left( {{v_i}} \right) + r - 1} \right)\left( {r{\delta _G}\left( {{v_k}} \right) + r - 1} \right)}}{{{d_G}\left( {{v_i},{v_k}} \right)}}} } = \\ \;\;\;\;\;\;\;{r^2}\sum\limits_{j = 0}^{r - 1} {\sum\limits_{\begin{array}{*{20}{c}} {i,k = 0}\\ {i \ne k} \end{array}}^{n - 1} {\frac{{{\delta _G}\left( {{v_i}} \right){\delta _G}\left( {{v_k}} \right)}}{{{d_G}\left( {{v_i},{v_k}} \right)}}} } + \\ \;\;\;\;\;\;\;r\left( {r - 1} \right)\sum\limits_{j = 0}^{r - 1} {\sum\limits_{\begin{array}{*{20}{c}} {i,k = 0}\\ {i \ne k} \end{array}}^{n - 1} {\frac{{{\delta _G}\left( {{v_i}} \right) + {\delta _G}\left( {{v_k}} \right)}}{{{d_G}\left( {{v_i},{v_k}} \right)}}} } + \\ \;\;\;\;\;\;\;{\left( {r - 1} \right)^2}\sum\limits_{j = 0}^{r - 1} {\sum\limits_{\begin{array}{*{20}{c}} {i,k = 0}\\ {i \ne k} \end{array}}^{n - 1} {\frac{1}{{{d_G}\left( {{v_i},{v_k}} \right)}}} } = \\ \;\;\;\;\;\;\;2{r^3}{H_M}\left( G \right) + 2{r^2}\left( {r - 1} \right){H_A}\left( G \right) + \\ \;\;\;\;\;\;\;2r{\left( {r - 1} \right)^2}H\left( G \right). \end{array}$ (8)
$\begin{array}{l} {A_3} = \sum\limits_{\begin{array}{*{20}{c}} {i,k = 0}\\ {i \ne k} \end{array}}^{n - 1} {\sum\limits_{\begin{array}{*{20}{c}} {j,p = 0}\\ {j \ne p} \end{array}}^{r - 1} {\frac{{{\delta _{G'}}\left( {{x_{ij}}} \right){\delta _{G'}}\left( {{x_{kp}}} \right)}}{{{d_{G'}}\left( {{x_{ij}},{x_{kp}}} \right)}}} } = \\ \;\;\;\;\;\;\;\sum\limits_{\begin{array}{*{20}{c}} {i,k = 0}\\ {i \ne k} \end{array}}^{n - 1} {\sum\limits_{\begin{array}{*{20}{c}} {j,p = 0}\\ {j \ne p} \end{array}}^{r - 1} {\frac{{{r^2}{\delta _G}\left( {{v_i}} \right){\delta _G}\left( {{v_k}} \right) + r\left( {r - 1} \right)\left( {{\delta _G}\left( {{v_i}} \right)} \right.}}{{{d_G}\left( {{v_i},{v_k}} \right)}}} } + \\ \;\;\;\;\;\;\;\frac{{\left. {{\delta _G}\left( {{v_k}} \right)} \right) + {{\left( {r - 1} \right)}^2}}}{{{d_G}\left( {{v_i},{u_k}} \right)}}\underline{\underline {{\text{by lemma 3}}}} \\ \;\;\;\;\;\;\;{r^3}\left( {r - 1} \right)\sum\limits_{\begin{array}{*{20}{c}} {i,k = 0}\\ {i \ne k} \end{array}}^{n - 1} {\frac{{{\delta _G}\left( {{v_i}} \right){\delta _G}\left( {{v_k}} \right)}}{{{d_G}\left( {{v_i},{v_k}} \right)}}} + \\ \;\;\;\;\;\;\;{r^2}{\left( {r - 1} \right)^2}\sum\limits_{\begin{array}{*{20}{c}} {i,k = 0}\\ {i \ne k} \end{array}}^{n - 1} {\frac{{{\delta _G}\left( {{v_i}} \right) + {\delta _G}\left( {{v_k}} \right)}}{{{d_G}\left( {{v_i},{v_k}} \right)}}} + \\ \;\;\;\;\;\;\;r{\left( {r - 1} \right)^3}\sum\limits_{\begin{array}{*{20}{c}} {i,k = 0}\\ {i \ne k} \end{array}}^{n - 1} {\frac{1}{{{d_G}\left( {{v_i},{v_k}} \right)}}} = \\ \;\;\;\;\;\;\;2{r^3}\left( {r - 1} \right){H_M}\left( G \right) + 2{r^2}{\left( {r - 1} \right)^2}{H_A}\left( G \right) + \\ \;\;\;\;\;\;\;2r{\left( {r - 1} \right)^3}H\left( G \right). \end{array}$ (9)
Combining (7)~(9) with (6), we get
$\begin{array}{l} {H_M}\left( {G{K_r}} \right) = {r^4}{H_M}\left( G \right) + {r^3}\left( {r - 1} \right){H_A}\left( G \right) + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{r^2}{\left( {r - 1} \right)^2}H\left( G \right) + \frac{{{M_1}\left( G \right){r^3}}}{2}\left( {r - 1} \right) + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;2m{r^2}{\left( {r - 1} \right)^2} + \frac{{nr}}{2}{\left( {r - 1} \right)^3}. \end{array}$
As an application, we present formulae for multiplicatively weighted Harary index of open and closed fences, PnK2 and CnK2.
Example 2
$\begin{gathered} \left( {\text{ⅰ}} \right){H_M}\left( {{P_n} \boxtimes {K_2}} \right) = 28n\left( {\sum\limits_{i = 1}^n {\frac{1}{i}} } \right) + \hfill \\ 64\left( {\sum\limits_{i = 1}^{n-1} {\frac{1}{i}} } \right) - \frac{{56}}{{n - 1}} - 3n - 32. \hfill \\ \end{gathered}$
$\left( {{\text{ⅱ}}} \right){H_M}\left( {{C_n} \boxtimes {K_2}} \right) = \left\{ \begin{gathered} 25n\left( {1 + 4\sum\limits_{i = 1}^{\frac{n}{2}} {\frac{1}{i}} } \right) - 100,\;n\;{\text{is}}\;{\text{even,}} \hfill \\ 25n\left( {1 + 4\sum\limits_{i = 1}^{\frac{{n - 1}}{2}} {\frac{1}{i}} } \right),\;n\;{\text{is}}\;{\text{odd}}{\text{.}} \hfill \\ \end{gathered} \right.$
4 Multiplicatively weighted Hararyindex of wreath product of graphs
In this section, we give the multiplicatively weighted Harary index of G1oG2. Let G1 and G2 be two connected graphs with V(G1)={v0, v1, …, vn1-1} and V(G2)={u0, u1, …, un2-1}. For convenience, let xij denote the vertex (vi, uj) of G1oG2. The following lemma, which follows easily from the properties and structure of G1oG2, is used in the proof of our main result in this section.
Lemma 4 Let G1 and G2 be two connected graphs and let G′=G1oG2. Then the degree of a vertex (vi, uj) in G′ is δG((vi, uj))=n2δG1(vi)+δG2(uj).
Theorem 3 Let G1 and G2 be two connected graphs. The number of vertices and edges of graph Gi is denoted by ni and ei respectively for i=1, 2. Then we have
$\begin{array}{l} {H_M}\left( {{G_1} \circ {G_2}} \right) = n_2^4{H_M}\left( {{G_1}} \right) + 2{n_2}{e_2}{H_A}\left( {{G_1}} \right) + \\ \;\;\;\;\;\;\;H\left( {{G_1}} \right)\left( {{M_1}\left( {{G_2}} \right) + 2{M_2}\left( {{G_2}} \right) + } \right.\\ \;\;\;\;\;\;\;\left. {2{{\bar M}_2}\left( {{G_2}} \right)} \right) + {n_2}{e_1}\left( {2{M_1}\left( {{G_2}} \right) + } \right.\\ \;\;\;\;\;\;\;\left. {{{\bar M}_1}\left( {{G_2}} \right)} \right) + \frac{{{n_1}}}{2}\left( {2{M_2}\left( {{G_2}} \right) + } \right.\\ \;\;\;\;\;\;\;\left. {{{\bar M}_2}\left( {{G_2}} \right)} \right) + \frac{1}{4}n_2^2{M_1}\left( {{G_1}} \right)\left( {n_2^2 - {n_2} + {e_2}} \right) + \\ \;\;\;\;\;\;\;\frac{{{n_2}}}{2}\sum\limits_{{x_{ij}},{x_{kp}} \in V\left( {{G_1} \circ {G_2}} \right)} {\frac{{{\delta _{{G_1}}}\left( {{v_i}} \right){\delta _{{G_2}}}\left( {{u_p}} \right)}}{{{d_{{G_1}}}\left( {{v_i},{v_k}} \right)}}} + \\ \;\;\;\;\;\;\;\frac{{{n_2}}}{2}\sum\limits_{{x_{ij}},{x_{kp}} \in V\left( {{G_1} \circ {G_2}} \right)} {\frac{{{\delta _{{G_1}}}\left( {{v_k}} \right){\delta _{{G_2}}}\left( {{u_j}} \right)}}{{{d_{{G_1}}}\left( {{v_i},{v_k}} \right)}}} . \end{array}$
Proof Let us denote G′=G1oG2. Obviously,
$\begin{array}{l} {H_M}\left( {G'} \right) = \frac{1}{2}\sum\limits_{{x_{ij}},{x_{kp}} \in V\left({G'}\right)} {\frac{{{\delta _{G'}}\left( {{x_{ij}}} \right){\delta _{G'}}\left( {{x_{kp}}} \right)}}{{{d_{G'}}\left( {{x_{ij}},{x_{kp}}} \right)}}} = \\ \;\;\;\;\;\;\;\;\frac{1}{2}\left( {\sum\limits_{i = 0}^{{n_1} - 1} {\sum\limits_{\begin{array}{*{20}{c}} {j,p = 0}\\ {j \ne p} \end{array}}^{{n_2} - 1} {\frac{{{\delta _{G'}}\left( {{x_{ij}}} \right){\delta _{G'}}\left( {{x_{ip}}} \right)}}{{{d_{G'}}\left( {{x_{ij}},{x_{ip}}} \right)}}} } + } \right.\\ \;\;\;\;\;\;\;\;\sum\limits_{\begin{array}{*{20}{c}} {i,k = 0}\\ {i \ne k} \end{array}}^{{n_1} - 1} {\sum\limits_{j = 0}^{{n_2} - 1} {\frac{{{\delta _{G'}}\left( {{x_{ij}}} \right){\delta _{G'}}\left( {{x_{kj}}} \right)}}{{{d_{G'}}\left( {{x_{ij}},{x_{kj}}} \right)}}} } + \\ \;\;\;\;\;\;\;\;\left. {\sum\limits_{\begin{array}{*{20}{c}} {i,k = 0}\\ {i \ne k} \end{array}}^{{n_1} - 1} {\sum\limits_{\begin{array}{*{20}{c}} {j,p = 0}\\ {j \ne p} \end{array}}^{{n_2} - 1} {\frac{{{\delta _{G'}}\left( {{x_{ij}}} \right){\delta _{G'}}\left( {{x_{kp}}} \right)}}{{{d_{G'}}\left( {{x_{ij}},{x_{kp}}} \right)}}} } } \right) = \\ \;\;\;\;\;\;\;\;\frac{1}{2}\left\{ {{A_1} + {A_2} + {A_3}} \right\}, \end{array}$ (10)
where A1 to A3 are the sums of the above terms, in order.
In what follows, we compute A1, A2, A3 of (10), separately.
$\begin{array}{l} {A_1} = \sum\limits_{i = 0}^{{n_1} - 1} {\sum\limits_{\begin{array}{*{20}{c}} {j,p = 0}\\ {j \ne p} \end{array}}^{{n_2} - 1} {\frac{{{\delta _{G'}}\left( {{x_{ij}}} \right){\delta _{G'}}\left( {{x_{ip}}} \right)}}{{{d_{G'}}\left( {{x_{ij}},{x_{ip}}} \right)}}} } = \\ \;\;\;\;\;\;\;\sum\limits_{i = 0}^{{n_1} - 1} {\sum\limits_{\begin{array}{*{20}{c}} {j,p = 0}\\ {j \ne p} \end{array}}^{{n_2} - 1} {\frac{{\left( {{n_2}{\delta _{{G_1}}}\left( {{v_i}} \right) + {\delta _{{G_2}}}\left( {{u_j}} \right)} \right)\left( {{n_2}{\delta _{{G_1}}}\left( {{v_i}} \right) + {\delta _{{G_2}}}\left( {{u_p}} \right)} \right)}}{{{d_{{G_2}}}\left( {{u_j},{u_p}} \right)}}} } = \\ \;\;\;\;\;\;\;\sum\limits_{i = 0}^{{n_1} - 1} {\sum\limits_{\begin{array}{*{20}{c}} {j,p = 0}\\ {j \ne p} \end{array}}^{{n_2} - 1} {\frac{{n_2^2{\delta _{{G_1}}}{{\left( {{v_i}} \right)}^2}}}{{{d_{{G_2}}}\left( {{u_j},{u_p}} \right)}}} } + \\ \;\;\;\;\;\;\;\sum\limits_{i = 0}^{{n_1} - 1} {\sum\limits_{\begin{array}{*{20}{c}} {j,p = 0}\\ {j \ne p} \end{array}}^{{n_2} - 1} {{n_2}{\delta _{{G_1}}}\left( {{v_i}} \right) \cdot \frac{{{\delta _{{G_2}}}\left( {{u_j}} \right) + {\delta _{{G_2}}}\left( {{u_p}} \right)}}{{{d_{{G_2}}}\left( {{u_j},{u_p}} \right)}}} } + \\ \;\;\;\;\;\;\;\sum\limits_{i = 0}^{{n_1} - 1} {\sum\limits_{\begin{array}{*{20}{c}} {j,p = 0}\\ {j \ne p} \end{array}}^{{n_2} - 1} {\frac{{{\delta _{{G_2}}}\left( {{u_j}} \right){\delta _{{G_2}}}\left( {{u_p}} \right)}}{{{d_{{G_2}}}\left( {{u_j},{u_p}} \right)}}} } = \\ \;\;\;\;\;\;\;n_2^2\sum\limits_{i = 0}^{{n_1} - 1} {{\delta _{{G_1}}}{{\left( {{v_i}} \right)}^2}\left( {\sum\limits_{{u_j}{u_p} \in E\left( {{G_2}} \right)} {\frac{1}{{{d_{{G_2}}}\left( {{u_j},{u_p}} \right)}}} + } \right.} \\ \;\;\;\;\;\;\;\left. {\sum\limits_{{u_j}{u_p} \notin E\left( {{G_2}} \right)} {\frac{1}{{{d_{{G_2}}}\left( {{u_j},{u_p}} \right)}}} } \right) + \\ \;\;\;\;\;\;\;{n_2}\sum\limits_{i = 0}^{{n_1} - 1} {{\delta _{{G_1}}}\left( {{v_i}} \right)\left( {\sum\limits_{{u_j}{u_p} \in E\left( {{G_2}} \right)} {\left( {{\delta _{{G_2}}}\left( {{u_j}} \right) + {\delta _{{G_2}}}\left( {{u_p}} \right)} \right)} + } \right.} \\ \;\;\;\;\;\;\;\left. {\sum\limits_{{u_j}{u_p} \notin E\left( {{G_2}} \right)} {\frac{{{\delta _{{G_2}}}\left( {{u_j}} \right) + {\delta _{{G_2}}}\left( {{u_p}} \right)}}{2}} } \right) + \\ \;\;\;\;\;\;\;\sum\limits_{i = 0}^{{n_1} - 1} {\left( {\sum\limits_{{u_j}{u_p} \in E\left( {{G_2}} \right)} {{\delta _{{G_2}}}\left( {{u_j}} \right){\delta _{{G_2}}}\left( {{u_p}} \right)} + } \right.} \\ \;\;\;\;\;\;\;\left. {\sum\limits_{{u_j}{u_p} \notin E\left( {{G_2}} \right)} {\frac{{{\delta _{{G_2}}}\left( {{u_j}} \right){\delta _{{G_2}}}\left( {{u_p}} \right)}}{2}} } \right), \end{array}$
since each row induces a copy of G2 and
$\begin{array}{*{20}{c}} {{d_{G'}}\left( {{x_{ij}},{x_{ip}}} \right) = 1\;{\rm{if}}\;{u_j}{u_p} \in E\left( {{G_2}} \right)\;{\rm{and}}}\\ {{d_{G'}}\left( {{x_{ij}},{x_{ip}}} \right) = 2\;{\rm{if}}\;{u_j}{u_p} \notin E\left( {{G_2}} \right).} \end{array}$
$\begin{array}{l} {\rm{The}}\;{\rm{above}}\;{\rm{formula}}\;{\rm{ = }}\frac{1}{2}n_2^2{M_1}\left( {{G_1}} \right)\left( {n_2^2 - {n_2} + {e_2}} \right) + \\ \;\;\;\;\;\;2{n_2}{e_1}\left( {2{M_1}\left( {{G_2}} \right) + {{\bar M}_1}\left( {{G_2}} \right)} \right) + {n_1}\left( {2{M_2}\left( {{G_2}} \right) + } \right.\\ \;\;\;\;\;\;\left. {{{\bar M}_2}\left( {{G_2}} \right)} \right). \end{array}$ (11)
$\begin{array}{l} {A_2} = \sum\limits_{\begin{array}{*{20}{c}} {i,k = 0}\\ {i \ne k} \end{array}}^{{n_1} - 1} {\sum\limits_{j = 0}^{{n_2} - 1} {\frac{{{\delta _{G'}}\left( {{x_{ij}}} \right){\delta _{G'}}\left( {{x_{kj}}} \right)}}{{{d_{G'}}\left( {{x_{ij}},{x_{kj}}} \right)}}} } = \\ \;\;\;\;\;\;\sum\limits_{\begin{array}{*{20}{c}} {i,k = 0}\\ {i \ne k} \end{array}}^{{n_1} - 1} {\sum\limits_{j = 0}^{{n_2} - 1} {\frac{{\left( {{n_2}{\delta _{{G_1}}}\left( {{v_i}} \right) + {\delta _{{G_2}}}\left( {{u_j}} \right)} \right)\left( {{n_2}{\delta _{{G_1}}}\left( {{v_k}} \right)} \right.}}{{{d_{{G_1}}}\left( {{v_i}{v_k}} \right)}}} } + \\ \;\;\;\;\;\;\sum\limits_{\begin{array}{*{20}{c}} {i,k = 0}\\ {i \ne k} \end{array}}^{{n_1} - 1} {\sum\limits_{j = 0}^{{n_2} - 1} {\frac{{{\delta _{{G_2}}}\left( {{u_j}} \right))}}{{{d_{{G_1}}}\left( {{v_i},{v_k}} \right)}}} } , \end{array}$
since the distance between a pair of vertices in a column is the same as the distance between the corresponding vertices of other column.
$\begin{array}{l} {\rm{The}}\;{\rm{above}}\;{\rm{formula}}\;{\rm{ = }}\;\sum\limits_{\begin{array}{*{20}{c}} {i,k = 0}\\ {i \ne k} \end{array}}^{{n_1} - 1} {\sum\limits_{j = 0}^{{n_2} - 1} {\frac{{n_2^2{\delta _{{G_1}}}\left( {{v_i}} \right){\delta _{{G_1}}}\left( {{v_k}} \right)}}{{{d_{{G_1}}}\left( {{v_i},{v_k}} \right)}}} } + \\ \;\;\;\;\;\;\;\sum\limits_{\begin{array}{*{20}{c}} {i,k = 0}\\ {i \ne k} \end{array}}^{{n_1} - 1} {\sum\limits_{j = 0}^{{n_2} - 1} {{n_2}{\delta _{{G_2}}}\left( {{u_j}} \right)\frac{{{\delta _{{G_1}}}\left( {{v_i}} \right) + {\delta _{{G_1}}}\left( {{v_k}} \right)}}{{{d_{{G_1}}}\left( {{v_i},{v_k}} \right)}}} } + \\ \;\;\;\;\;\;\;\sum\limits_{\begin{array}{*{20}{c}} {i,k = 0}\\ {i \ne k} \end{array}}^{{n_1} - 1} {\sum\limits_{j = 0}^{{n_2} - 1} {\frac{{{\delta _{{G_2}}}{{\left( {{u_j}} \right)}^2}}}{{{d_{{G_1}}}\left( {{v_i},{v_k}} \right)}}} } = \\ \;\;\;\;\;\;\;2n_2^3{H_M}\left( {{G_1}} \right) + 4{n_2}{e_2}{H_A}\left( {{G_1}} \right) + 2{M_1}\left( {{G_2}} \right)H\left( {{G_1}} \right). \end{array}$ (12)
$\begin{array}{l} {A_3} = \sum\limits_{\begin{array}{*{20}{c}} {i,k = 0}\\ {i \ne k} \end{array}}^{{n_1} - 1} {\sum\limits_{\begin{array}{*{20}{c}} {j,p = 0}\\ {j \ne p} \end{array}}^{{n_2} - 1} {\frac{{{\delta _{G'}}\left( {{x_{ij}}} \right){\delta _{G'}}\left( {{x_{kp}}} \right)}}{{{d_{G'}}\left( {{x_{ij}},{x_{kp}}} \right)}}} } = \\ \;\;\;\;\;\;\;\sum\limits_{\begin{array}{*{20}{c}} {i,k = 0}\\ {i \ne k} \end{array}}^{{n_1} - 1} {\sum\limits_{\begin{array}{*{20}{c}} {j,p = 0}\\ {j \ne p} \end{array}}^{{n_2} - 1} {\frac{{\left( {{n_2}{\delta _{{G_1}}}\left( {{v_i}} \right) + {\delta _{{G_2}}}\left( {{u_j}} \right)} \right)\left( {{n_2}{\delta _{{G_1}}}\left( {{v_k}} \right)} \right.}}{{{d_{{G_1}}}\left( {{v_i},{v_k}} \right)}}} } + \\ \;\;\;\;\;\;\;\sum\limits_{\begin{array}{*{20}{c}} {i,k = 0}\\ {i \ne k} \end{array}}^{{n_1} - 1} {\sum\limits_{\begin{array}{*{20}{c}} {j,p = 0}\\ {j \ne p} \end{array}}^{{n_2} - 1} {\frac{{\left. {{\delta _{{G_2}}}\left( {{u_p}} \right)} \right)}}{{{d_{{G_1}}}\left( {{v_i},{v_k}} \right)}}} } , \end{array}$
since dG(xij, xkp)=dG1(vi, vk)for all i and k, and further the distance between the corresponding vertices of the layers is counted in A2,
$\begin{array}{l} {\rm{The}}\;{\rm{above}}\;{\rm{formula}}\;{\rm{ = }}\sum\limits_{\begin{array}{*{20}{c}} {i,k = 0}\\ {i \ne k} \end{array}}^{{n_1} - 1} {\sum\limits_{\begin{array}{*{20}{c}} {j,p = 0}\\ {j \ne p} \end{array}}^{{n_2} - 1} {\frac{{n_2^2{\delta _{{G_1}}}\left( {{v_i}} \right){\delta _{{G_1}}}\left( {{v_k}} \right)}}{{{d_{{G_1}}}\left( {{v_i},{v_k}} \right)}}} } \; + \\ \;\;\;\;\;\;\;{n_2}\sum\limits_{\begin{array}{*{20}{c}} {i,k = 0}\\ {i \ne k} \end{array}}^{{n_1} - 1} {\sum\limits_{\begin{array}{*{20}{c}} {j,p = 0}\\ {j \ne p} \end{array}}^{{n_2} - 1} {\frac{{{\delta _{{G_1}}}\left( {{v_i}} \right){\delta _{{G_2}}}\left( {{v_p}} \right)}}{{{d_{{G_1}}}\left( {{v_i},{v_k}} \right)}}} } + \\ \;\;\;\;\;\;\;{n_2}\sum\limits_{\begin{array}{*{20}{c}} {i,k = 0}\\ {i \ne k} \end{array}}^{{n_1} - 1} {\sum\limits_{\begin{array}{*{20}{c}} {j,p = 0}\\ {j \ne p} \end{array}}^{{n_2} - 1} {\frac{{{\delta _{{G_1}}}\left( {{v_k}} \right){\delta _{{G_2}}}\left( {{u_j}} \right)}}{{{d_{{G_1}}}\left( {{v_i},{v_k}} \right)}}} } + \\ \;\;\;\;\;\;\;\sum\limits_{\begin{array}{*{20}{c}} {i,k = 0}\\ {i \ne k} \end{array}}^{{n_1} - 1} {\sum\limits_{\begin{array}{*{20}{c}} {j,p = 0}\\ {j \ne p} \end{array}}^{{n_2} - 1} {\frac{{{\delta _{{G_2}}}\left( {{u_j}} \right){\delta _{{G_2}}}\left( {{u_p}} \right)}}{{{d_{{G_1}}}\left( {{v_i},{v_k}} \right)}}} } = \\ \;\;\;\;\;\;\;2n_2^3\left( {{n_2} - 1} \right){H_M}\left( {{G_1}} \right) + \\ \;\;\;\;\;\;\;{n_2}\sum\limits_{\begin{array}{*{20}{c}} {i,k = 0}\\ {i \ne k} \end{array}}^{{n_1} - 1} {\sum\limits_{\begin{array}{*{20}{c}} {j,p = 0}\\ {j \ne p} \end{array}}^{{n_2} - 1} {\frac{{{\delta _{{G_1}}}\left( {{v_i}} \right){\delta _{{G_2}}}\left( {{u_p}} \right)}}{{{d_{{G_1}}}\left( {{v_i},{v_k}} \right)}}} } + \\ \;\;\;\;\;\;\;{n_2}\sum\limits_{\begin{array}{*{20}{c}} {i,k = 0}\\ {i \ne k} \end{array}}^{{n_1} - 1} {\sum\limits_{\begin{array}{*{20}{c}} {j,p = 0}\\ {j \ne p} \end{array}}^{{n_2} - 1} {\frac{{{\delta _{{G_1}}}\left( {{v_k}} \right){\delta _{{G_2}}}\left( {{u_j}} \right)}}{{{d_{{G_1}}}\left( {{v_i},{v_k}} \right)}}} } + \\ \;\;\;\;\;\;\;4H\left( {{G_1}} \right)\left( {{M_2}\left( {{G_2}} \right) + {{\bar M}_2}\left( {{G_2}} \right)} \right). \end{array}$ (13)
Combining (11)~(13) with (10), we get the desired result.
This completes the proof.
Using theorem 2, we have the following corollary.
Corollary 3 Let G1 be a connected graph and G2 be a connected k-regular graph. The number of vertices and edges of graph Gi is denoted by ni and ei respectively for i=1, 2. Then, we have
$\begin{array}{l} {H_M}\left( {{G_1} \circ {G_2}} \right) = n_2^4{H_M}\left( {{G_1}} \right) + {n_2}\left( {kn_2^2 - k{n_2} + } \right.\\ \;\;\;\;\;\;\;\left. {2{e_2}} \right){H_A}\left( {{G_1}} \right) + H\left( {{G_1}} \right)\left( {{M_1}\left( {{G_2}} \right) + } \right.\\ \;\;\;\;\;\;\;\left. {2{M_2}\left( {{G_2}} \right) + 2{{\bar M}_2}\left( {{G_2}} \right)} \right) + \\ \;\;\;\;\;\;\;{n_1}{e_1}\left( {2{M_1}\left( {{G_2}} \right) + {{\bar M}_1}\left( {{G_2}} \right)} \right) + \\ \;\;\;\;\;\;\;\frac{{{n_1}}}{2}\left( {2{M_2}\left( {{G_2}} \right) + {{\bar M}_2}\left( {{G_2}} \right)} \right) + \\ \;\;\;\;\;\;\;\frac{1}{4}n_2^2{M_1}\left( {{G_1}} \right)\left( {n_2^2 - {n_2} + {e_2}} \right). \end{array}$
References
[1] BONDY J A, MURTY U S R. Graph Theory with Applications, Macmillan[M]. London: Elsevier, 1976. [2] WIENER H. Structural determination of paraffin boiling point[J]. J Amer Chem Soc, 1947, 69: 17–20. DOI:10.1021/ja01193a005 [3] HOSOYA H. A newly proposed quantity characterizing the topological nature of structural isomers of saturated hydrocarbons[J]. Bull Chem Soc Jpn, 1971, 44: 2332–2339. DOI:10.1246/bcsj.44.2332 [4] IVANCIUC O, BALABAN T S, BALABAN A T. Reciprocal distance matrix, related local vertex invariants and topological indices[J]. J Math Chem, 1993(12): 309–318. [5] PLAVŠIĆ D, NIKOLIĆ S, TRINAJSTIĆ N, et al. On the Harary index for the characterization of chemical graphs[J]. J Math Chem, 1993(12): 235–250. [6] DOBRYNIN A A, KOCHETOVA A A. Degree distance of a graph: A degree analogue of the Wiener index[J]. J Chem Inf Comput Sci, 1994, 34: 1082–1086. DOI:10.1021/ci00021a008 [7] GUTMAN I. Selected properties of the Schultz molecular topogical index[J]. J Chem Inf Comput Sci, 1994, 34: 1087–1089. DOI:10.1021/ci00021a009 [8] TODESCHINI R, CONSONNI V. Handbook of Molecular Descriptors[M]. Weinheim: Wiley-VCH, 2000. [9] DOBRYNIN A A, ENTRINGER R, GUTMAN I. Wiener index of trees: Theory and applications[J]. Acta Appl Math, 2001, 66: 211–249. DOI:10.1023/A:1010767517079 [10] GUTMAN I. A property of the Wiener number and its modifications[J]. Indian J Chem A, 1997, 36: 128–132. [11] GUTMAN I, RADA J, ARAUJO O. The Wiener index of starlike trees and a related partial order[J]. Match Commun Math Comput Chem, 2000, 42: 145–154. [12] ILIĆ A, STEVANOVIĆ D, FENG L, et al. Degree distance of unicyclic and bicyclic graphs[J]. Discrete Appl Math, 2011, 159: 779–788. DOI:10.1016/j.dam.2011.01.013 [13] TOMESCU I. Ordering connected graphs having small degree distances[J]. Discrete Appl Math, 2010, 158: 1714–1717. DOI:10.1016/j.dam.2010.05.023 [14] FENG L, LIU W. The maximal Gutman index of bicyclic graphs[J]. MATCH Commun Math Comput Chem, 2011, 66: 699–708. [15] MUKWEMBI S. On the upper bound of Gutman index of graphs[J]. MATCH Commun Math Comput Chem, 2012, 68: 343–348. [16] LATORA V, MARCHIORI M. Efficient behavior of small-world networks[J]. Phys Rev Lett, 2001, 87: 198701. DOI:10.1103/PhysRevLett.87.198701 [17] MARCHIORI M, LATORA V. Harmony in the small-world[J]. Physica A, 2000, 285: 539–546. DOI:10.1016/S0378-4371(00)00311-3 [18] ALIZADEH Y, IRANMANESH A, DOŠLIĆ T. Additively weighted Harary index of some composite graphs[J]. Discrete Math, 2013, 313: 26–34. DOI:10.1016/j.disc.2012.09.011 [19] PATTABIRAMAN K, VIJAYARAGAVAN M. Reciprocal degree distance of product graphs[J]. Discrete Appl Math, 2014, 179: 201–213. DOI:10.1016/j.dam.2014.07.020 [20] NIKOLIĆ S, KOVAČEVIĆ G, MILIČEVIĆ A, et al. The Zagreb indices 30 years after[J]. Croat Chem Acta, 2003, 76: 113–124. [21] ASHRAFI A R, DOŠLIĆ T, HAMZEH A. The Zagreb coindices of graph operations[J]. Discerete Appl Math, 2010, 158: 1571–1578. DOI:10.1016/j.dam.2010.05.017 [22] ALON N, LUBETZKY E. Independent set in tensor graph powers[J]. J Graph Theory, 2007, 54: 73–87. DOI:10.1002/(ISSN)1097-0118 [23] ASSAF A M. Modified group divisible designs[J]. Ars Combin, 1990, 29: 13–20. [24] BREŠAR B, IMRICH W, KLAVŽAR S, et al. Hypercubes as direct products[J]. SIAM J Discrete Math, 2005, 18: 778–786. DOI:10.1137/S0895480103438358 [25] IMRICH W, KLAVŽAR S. Product Graphs: Structure and Recognition[M]. New York, John Wiley and Sons, 2000. [26] MAMUT A, VUMAR E. Vertex vulnerability parameters of Kronecker products of complete graphs[J]. Inform Process Lett, 2008, 106: 258–262. DOI:10.1016/j.ipl.2007.12.002 [27] HOJI M, LUO Z, VUMAR E. Wiener and vertex PI indices of Kronecker products of graphs[J]. Discrete Appl Math, 2010, 158: 1848–1855. DOI:10.1016/j.dam.2010.06.009 [28] KHALIFEH M H, YOUSERI-AZARI H, ASHRAFI A R. Vertex and edge PI indices of Cartesian product of graphs[J]. Discrete Appl Math, 2008, 156: 1780–789. DOI:10.1016/j.dam.2007.08.041 [29] PATTABIRAMAN K, PAULRAJA P. On some topological indices of the tensor product of graphs[J]. Discrete Appl Math, 2012, 160: 267–279. DOI:10.1016/j.dam.2011.10.020
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Question Paper: Elements of Civil Engg. & Engg. Mechanics : Question Paper Jan 2014 - First Year Engineering (P Cycle) (Semester 1) | Visveswaraya Technological University (VTU)
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## Elements of Civil Engg. & Engg. Mechanics - Jan 2014
### First Year Engineering (P Cycle) (Semester 1)
TOTAL MARKS: 100
TOTAL TIME: 3 HOURS
(1) Question 1 is compulsory.
(2) Attempt any four from the remaining questions.
(3) Assume data wherever required.
(4) Figures to the right indicate full marks.
### Choose the correct answer for the following :-
1 (a) (i) A bridge constructed at some angle to river flow is
(A) skew bridge
(B) square bridge
(C) steel bridge
(D) lift bridge
(1 marks)
1 (a) (ii) The structure that separates roads into separate lanes is called
(A) Kerb
(B) Median
(D) camber
(1 marks)
1 (a) (iii) The upstream side of a dam
(A) arch dams
(B) gravity dam
(C) earth dam
(D) reservoir
(1 marks)
1 (a) (iv) Geotechnical engineering involves the study of
(A) Water
(B) Soil
(C) Air
(d) None of these
(1 marks)
1 (b) With the help of neat sketches, briefly explain the cross-section of road and gravity dam.(10 marks) 1 (c) What are infrastructure related projects?(6 marks)
### Choose the correct answer for the following :-
2 (a) (i) The component of a force perpendicular to its line of action is
(A) Maximum
(B) Minimum
(C) Zero
(D) None of the
(1 marks)
2 (a) (ii) The moment of a force about a moment centre lying on its line of action is
(A) Maximum
(B) Minimum
(C) Zero <br. (d)="" none="" of="" these<="" a="">
</br.><>(1 marks)
2 (a) (iii) Two equal and opposite forces separated by a distance will produce
(A) Translation
(B) rotation
(C) both translation and rotation
(D) none of these
(1 marks)
2 (a) (iv) Moment of a force will produce
(A) translation
(B) rotation
(C) both translation and rotation
(D) none of these
(1 marks)
2 (b) A block weighing w=10 kN is resting on an inclined plane as shown in Fig.Q2(b). Determine its component normal to and parallel to the inclined plane. The plane makes an angle of 20° with the horizontal.
(4 marks) 2 (c) A body is subjected to the three forces as shown in Fig.Q2(c). If posssible determine the direction of the force "F" so that resultant is in 'x' direction when (i) F=5000 N, (ii) F=300 N.
(12 marks)
### Choose the correct answer for the following :-
3 (a) (i) In case of coplanar concurrent force the resultant force passes through
(A) Point of concurrece
(B) Away from point of concurrence
(C) different plane
(D) none of these
(1 marks)
3 (a) (ii) If two concurrent forces each of 'P' act at right angles to each other, their resultant is
(A) 2P
(B) P
(C) ?2 P
(D) 2?P
(1 marks)
3 (a) (iii) If &sume;V=0 and ?H=0 for a coplanar noncocurrent force system, then it is in
(A) equilibrium
(B) translation
(C) rotation
(D) none of these
(1 marks)
3 (a) (iv) Conditions of equilibrium for a coplanar concurrent force system is
(A) one
(B) two
(C) three
(D) four
(1 marks)
3 (b) Two forces acting on a body are 500 N and 1000 N as shown in fig. Q3(b). Determine the third forces "F" such that the resultant of all the three forces is 1000 N directed at 45° to 'x' axis. (6 marks) 3 (c) Determine the resultant of the four forces acting on a body as shown in Fig.Q3(c) with respect to point "O".
(10 marks)
### Choose the correct answer for the following :-
4 (a) (i) The centroid of a triangle of height 'h' is located at a _____ distance from its apex.
(A) h/2
(B) 2h/3
(C) h/3
(D) h
(1 marks)
4 (a) (ii) Intersection of ______ number of symmetrical axes will give centroid of plane area.
(A) 3
(B) 4
(C) 2
(D) none of these
(1 marks)
4 (a) (iii) Moment of an area about a reference axis passing through its centroid is
(A) maximum
(B) minimize
(C) zero
(D) none of these
(1 marks)
4 (a) (iv) Centroid of a semicircle from an axis passing through the diameter is
$$(A)\ \dfrac {4r}{3\pi}\$$B)\ \dfrac {3r}{4\pi}\$$C)\ \dfrac {3\pi}{4r}\$$D)\ \dfrac {4\pi}{3r}\\ $$(1 marks) 4 (b) Determine the centroid of a semi-circular lamina of radius "r" by the method of intergration.(6 marks) 4 (c) Determine the centroid of the shaded area shown in the Fig. Q4(c) with respect to OX and OY. (10 marks) ### Choose the correct answer for the following :- 5 (a) (i) If three forces are acting at a point and are in equilibrium, out of which two are acting in the same line, then the third force is (A) maximum (B) minimum (C) zero (D) none of these (1 marks) 5 (a) (ii) A rigid body is in equilibrium if the resultant force of concurrent force system is (A) positive (B) negative (C) zero (D) none of these (1 marks) 5 (a) (iii) Lami's theorem is valid for ______ number of concurrent forces in equilibrium. (A) two (B) three (C) four (D) none of these (1 marks) 5 (a) (iv) The force equal and opposite to resultant is called as (A) equilibriant (B) similar force (C) opponent force (D) none of these (1 marks) 5 (b) State and prove Lami's theorem.(6 marks) 5 (c) The frictionless pulley 'A' shown in Fig. Q5(c) is supplied by two bars AB and AC which are hinged at 'B' and 'C' to a vertical wall. The flexible cable DG hinged at 'D', goes over the pulley and supports a load of 20 kN at 'G'. The angles between the various members are shown in the figure. Determine the force in the bars AB and AC. Neglect the size and weight of the pulley. (10 marks) ### Choose the correct answer for the following :- 6 (a) (i) A hinged support can have _____ reactions. (A) 2 (B) 4 (C) 1 (D) none of these (1 marks) 6 (a) (ii) A determine beam can have ______ number of unknowns. (A) 2 (B) 3 (C) 1 (D) 4 (1 marks) 6 (a) (iii) A fixed support can have _____ reactions. (A) 1 (B) 2 (C) 3 (D) 4 (1 marks) 6 (a) (iv) uld stands for (A) Uniformaly distributed load (B) Uniform dead load (C) Uniform door load (D) Uniform diameter load (1 marks) 6 (b) The cantilever beam shown in Fig.Q6(b) is fixed at 'A' and is free at 'B'. Determine the reaction when it is loaded as shown. (6 marks) 6 (c) Find the force in all members of truss loaded s shown in the Fig. Q6(c). Tabulate the results. (10 marks) ### Choose the correct answer for the following :- 7 (a) (i) Compared to static friction, kinetic friction is (A) Greater (B) Smaller (C) Very large (D) Zero (1 marks) 7 (a) (ii) Friction force is ______ to the contact surface between bodies. (A) parallel (B) perpendicular (C) tangential (D) none of these (1 marks) 7 (a) (iii) Friction force is ______ force. (A) active (B) passive (C) normal (D) none of these (1 marks) 7 (a) (iv) The tangent of the angle of friction is ______ (A) angle of repose (B) coefficient of friction (C) cone of friction (D) limiting friction (1 marks) 7 (b) Define : - (i) Angle of friction (ii) Coefficient of friction (iii) Cone of friction (6 marks) 7 (c) What is the value of 'P' in the system shown in Fig. Q7 (c), to cause the motion to impend to the right? Assume the pulley is smooth and coefficient of friction between the other contact surfaces is 0.20. (10 marks) ### Choose the correct answer for the following :- 8 (a) (i) Unit of second moment of area is (A) m (B) m2 (C) m4 (D) m5 (1 marks) 8 (a) (ii) Unit of radius of gyration is ______ (A) m (B) m2 (C) m3 (D) m4 (1 marks) 8 (a) (iii) Moment of inertia of a square of side 'b' about an axis through its centroid is$$ (A)\ \dfrac {b^4}{12}\$$B)\ \dfrac {b^4}{8}\$$C)\ \dfrac {b^4}{36}\$$D)\ \dfrac {b^3}{12}\\$$
(1 marks)
8 (a) (iv) Pdar moment of inertia is
(A) Ixx + Iyy
(B) Ixx + Izz
(C) Iyy + Izz
(D) none of these
(1 marks)
8 (b) Define :-
(i) Moment of inertia
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2019-01-21 02:28:19
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https://nb.khanacademy.org/math/pre-algebra/pre-algebra-ratios-rates/pre-algebra-ratios-intro/v/equivalent-ratios
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Hovedinnhold
# Equivalent ratios
## Video transkripsjon
- [Instructor] We're asked to select three ratios that are equivalent to seven to six. So pause this video and see if you can spot the three ratios that are equivalent to seven to six. Alright, now let's work through this together, and the main thing to realize about equivalent ratios is we just have to multiply or divide the corresponding parts of the ratio by the same amount. So before I even look at these choices, for example, if I have seven to six, if I multiply the seven times two to get 14, then I would also multiply the six times two to get 12. So, for example, 14 to 12 is the exact same ratio. Now you might be tempted to pick 12 to 14, but that is not the same ratio. Order matters in a ratio. This could be ratio of oranges to apples. And we're saying for every seven oranges, there are six apples. You wouldn't be able to say it the other way around. So you would rule this one out even though it's dealing with some of the right numbers. It's not in the right order. Now let's think about 21 to 18. To go from seven to 21, we would multiply by three. And to go from six to 18, you would also multiply by three. So that works. If we multiply both of these numbers by three, we get 21 to 18. So let me circle that in. That one is for sure equivalent. What about 42 to 36? Well, to go from seven to 42, we're going to have to multiply by six. And to go from six to thirty-six, we also multiply by six. So this, once again, is an equivalent ratio. We multiply each of these by six and we keep the same order. So that is equivalent right over there. 63 to 54. Let's see, to go from seven to 63, you multiply by nine. And to go from six to 54, you also multiply by nine. So once again, 63 to 54 is an equivalent ratio. And so we've already selected three, but let's just verify that this doesn't work. So to go from seven to 84, you would multiply by 12. To go from six to 62, you multiply by 10 and 2/6 or 10 1/3, so this one is definitely not an equivalent ratio. Let's do another example. So once again, we are asked to select three ratios that are equivalent to 16 to 12. So pause this video and see if you can work through it. Alright, let's look at this first one. So eight to six. So at first you might say well, gee, these numbers are smaller than 16 and 12. Remember, you can, to get an equivalent ratio you can multiply or divide these numbers by the same number. So, to get from 16 to eight, you could do that as, well, we just divided by two. And to go from 12 to six, you also divide by two. So this actually is an equivalent ratio. I'll circle that in. What about 32 to 24? Well to go from 16 to 32, we multiply by two. To go from 12 to 24, we also multiply by two. So this is an equivalent ratio. What about four to three? Well, to go from 16 to four, we would have to divide by four. And to go from 12 to three, we are going to divide by four as well. So we're dividing by the same thing, each of these numbers. So, this is also going to be an equivalent ratio. So we've selected our three, so we are essentially done. But, we might as well see why these don't work. Now let's think about it. To go from 16 to 12, how do we do that? Well, to go from 16 to 12, you could divide by four and multiply by three. So this would be times 3/4. You would get 12. And to go from 12 to eight, so you could divide by three and multiply by two. So this you could view as times 2/3. So you'd be multiplying or dividing by different numbers here, so this one is not equivalent. And then 24 to 16? To go from 16 to 24, you would multiply by, let's see, that's 1 1/2. So this right over here would be, you would multiply by 1 1/2. And to go from 12 to 16, you would multiply, that is, by 1 1/3. So, times 1 1/3. So you're not multiplying by the same amount. So once again, not an equivalent ratio.
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2019-11-12 11:40:16
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https://elteoremadecuales.com/raikovs-theorem/?lang=de
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# Raikov's theorem
Raikov's theorem Raikov’s theorem, named for Russian mathematician Dmitrii Abramovich Raikov, is a result in probability theory. It is well known that if each of two independent random variables ξ1 and ξ2 has a Poisson distribution, then their sum ξ=ξ1+ξ2 has a Poisson distribution as well. It turns out that the converse is also valid.[1][2][3] Inhalt 1 Aussage des Theorems 2 Kommentar 3 An extension to locally compact Abelian groups 4 Raikov's theorem on locally compact Abelian groups 5 References Statement of the theorem Suppose that a random variable ξ has Poisson's distribution and admits a decomposition as a sum ξ=ξ1+ξ2 of two independent random variables. Then the distribution of each summand is a shifted Poisson's distribution.
Comment Raikov's theorem is similar to Cramér’s decomposition theorem. The latter result claims that if a sum of two independent random variables has normal distribution, then each summand is normally distributed as well. It was also proved by Yu.V.Linnik that a convolution of normal distribution and Poisson's distribution possesses a similar property (Linnik's theorem [ru]).
An extension to locally compact Abelian groups Let {Anzeigestil X} be a locally compact Abelian group. Bezeichne mit {displaystyle M^{1}(X)} the convolution semigroup of probability distributions on {Anzeigestil X} , and by {Anzeigestil E_{x}} the degenerate distribution concentrated at {Anzeigestil xin X} . Lassen {Anzeigestil x_{0}in X,lambda >0} .
The Poisson distribution generated by the measure {displaystyle lambda E_{x_{0}}} is defined as a shifted distribution of the form {displaystyle mu =e(lambda E_{x_{0}})=e^{-Lambda }(E_{0}+lambda E_{x_{0}}+Lambda ^{2}E_{2x_{0}}/2!+ldots +lambda ^{n}E_{nx_{0}}/n!+Punkte ).} One has the following Raikov's theorem on locally compact Abelian groups Let {zeige ihn an } be the Poisson distribution generated by the measure {displaystyle lambda E_{x_{0}}} . Nehme an, dass {displaystyle mu =mu _{1}*in _{2}} , mit {displaystyle ihn _{j}in M^{1}(X)} . Wenn {Anzeigestil x_{0}} is either an infinite order element, or has order 2, dann {displaystyle ihn _{j}} is also a Poisson's distribution. In the case of {Anzeigestil x_{0}} being an element of finite order {displaystyle nneq 2} , {displaystyle ihn _{j}} can fail to be a Poisson's distribution.
References ^ D. Raikov (1937). "On the decomposition of Poisson laws". Dokl. Akad. Wissenschaft. URSS. 14: 9–11. ^ Rukhin A. L. (1970). "Certain statistical and probability problems on groups". Trudy Mat. Inst. Steklov. 111: 52–109. ^ Linnik, Yu. v., Ostrovskii, ich. v. (1977). Decomposition of random variables and vectors. Vorsehung, R. ICH.: Übersetzungen mathematischer Monographien, 48. Amerikanische Mathematische Gesellschaft. Kategorien: Characterization of probability distributionsProbability theoremsTheorems in statistics
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2023-03-29 04:09:23
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https://harrisonbrown.wordpress.com/category/blegs/
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## Archive for the ‘blegs’ Category
### 2^n \geq n: The graph theory proof?
February 15, 2010
Theorem. For every positive integer n, $2^n \geq n$.
Proof. Consider a tree on n vertices $T = (V, E)$ with root $v_0$. Assign to each edge $\{v, w\}$ an element of the vector space $GF(2)^V$, obtained by setting 1s in the the coordinates corresponding to v and w and 0s elsewhere. I claim that these vectors are linearly independent; for suppose otherwise, and letthe vectors corresponding to $S \subset E$ sum to 0. There is a natural “distance” function on E w/r/t $v_0$; let $e_0 = \{s, t\}$ have maximal distance in S, and suppose WLOG that t is farther than s from $v_0$. Then the coordinate corresponding to t is nonzero for exactly one element of S, and the sum over all elements of S must therefore be nonzero. This is a contradiction. So in particular these |E| = n-1 elements are distinct and nonzero, which means (by Pigeonhole) that there are at most $2^n-1$ of them.
### Bleg: What’s the most recent day no one alive was born
January 5, 2010
Inspired by Michael Lugo’s post on reconstructing a person from their DOB, zipcode, and gender.
If you, for whatever reason, ever watch the Today show, you’ll notice that one of the recurring features is the hosts listing the names of some men and women who are turning 100. Becoming a centenarian is a reasonably big accomplishment — in the U.S., it nets you a congratulatory letter from the President, for example. But if you look into it, you’ll notice that you can find someone turning 100 on pretty much any given day. Usually not someone particularly well-known, but certainly someone. (I tried to find someone famous and vaguely math-related who just turned or is turning 100 for this post, but couldn’t; however, the fascinating economist Ronald Coase turned 99 last week.) It’s almost certainly true that on any given day, someone somewhere in the world is in fact celebrating their 100th birthday. But go ten years further, and you find almost no one who lives to 110. Actually, I know of only one supercentenarian, living or not, who is interesting for reasons apart from his longevity — the late Vietoris, the topologist, probably best known as half of the Vietoris-Rips complex and the Mayer-Vietoris sequence. Odds are pretty good that no one alive is turning 110 today, or tomorrow, or (sadly) New Years’ Day.
So… a question is starting to take shape. On every day between December 29, 1909, and today, someone was born who is still living today. But much earlier than that, and the above statement begins to be false. So what’s the most recent day that no one living was born on?
### What’s a “locally determined graph property?”
January 1, 2010
This has nothing to do with the rest of the post, but I’ll put it here so you read it before you get bored. I’d like to thank my readers (all seven of you) for supporting this blog in the first six months or so of its existence, and hope that you’ll stick around (and be joined by hundreds of new readers…) to hear my sporadic ramblings and wild ravings in the next year. Here’s to a happy and successful 2010!
Over at MathOverflow, Gjergji Zaimi asks (in a criminally under-voted-for question): How can we obtain global information from local data in graph theory? This is something that perhaps everyone working in or around graph theory has asked themselves, in some form, at some point — I know I have. So it’s not surprising that Gjergji’s question has received many different answers with many different interesting things to say.
I originally wanted to write a post trying to “answer” Gjergji’s question as best I could, but quickly realized the futility of that goal — it’s such a broad and deep question that I doubt if anyone could answer it concisely, and I know I couldn’t! So instead I’ll just talk about an $\epsilon$ of the question — what does it even mean, “local data?”
### The coupon collectors’ problem
December 28, 2009
This is a considerably lower-level post than usual, which I’ll (following Terry Tao) also blame on the holidays; there’s another, even less mathematical post in the works which I hope to finish sometime tomorrow.
How many times do you need to flip a coin before you expect to see both heads and tails? How many times do you need to roll a die before you expect to see all the numbers 1-6? These are two instances of the coupon collectors’ problem. Wikipedia gives not one, but two nice solutions to the problem, but there’s an even nicer “back-of-the-envelope” calculation which gives you the correct asymptotics for virtually nothing, and (I like to think) shows the power of thinking “categorically” at even a very low level.
So let’s give a statement of the problem. A company — say Coca-Cola, for concreteness — is holding a contest where everyone who collects one each of n different “coupons” wins some prize. You get a coupon with each purchase of a Coke, and each coupon is equally likely. What’s the expected number of Cokes you have to buy in order to collect all the coupons?
If you do some experimentation (or calculation) with small instances, you’ll see that this number seems to be growing somewhat faster than n. For n = 2, for example, the expected number is 3, and for n = 3 it’s 7. But how much faster? Like $n^2 \text{?}~n \sqrt{n}$? Or just a constant times n?
None of the above, as it happens, and you might have already guessed (or known) that the correct order of growth is $O(n~log~n)$. Here’s how you can figure this out for yourself.
Think of the collection as a function from Coke bottles to (equivalence classes of) coupons. If we’ve collected all the coupons, the function is surjective. So we can rephrase “What is the probability that, after I buy m Coke bottles, I have collected all n coupons” as “What is the probability that a random function from a set with m elements to a set with n elements is surjective?” Actually, we’ll estimate the probability that it’s not surjective.
If the function isn’t surjective, then its image contains at most n-1 elements. Fixing n-1 elements, the probability that our random function takes some element of the domain to this subset is $\frac{n-1}{n}$. Since the appropriate events are all independent, the probability that the random function takes every element to the subset is therefore $(\frac{n-1}{n})^m$.
Now there are n possibilities for the subset of size n-1, so we apply the union bound and say that the probability that our random function is not surjective is at most $n (\frac{n-1}{n})^m$. (Of course, this is an upper bound, and there is an error term; but we’ll return to that in a bit.)
So we want this expression to be smaller than, say, 1/10, which means that $(\frac{n-1}{n})^m = 1/10n$. But when n is large, we have that $(1 - \frac{1}{n})^n$ is about 1/e, so m has to be on the order of $ln~n$!
Now we’ll backtrack a bit. How do we know that the union bound was reasonably tight? After all, we counted functions whose image had size n-2 twice! Well, if you go back through the analysis and do inclusion-exclusion, you’ll see that the probability winds up being close to 1 when $m << n log n$ — but I don’t know of a computation-free way to argue that $O(n log n)$ is asymptotically right! Does anyone else?
So how is this “categorical thinking?” Well, it’s not, really. Category theory only really starts to get mildly interesting when you talk about functors, and doesn’t come into its own right until natural transformations are introduced. But if you’ve learned to think categorically, you see morphisms where other people see objects — in this case, a function where others might see a set — and while this is rarely enough to apply abstract-nonsense tools, it is enough to broaden your intuition and see paths you might have otherwise missed. And this is at least as useful.
### An etymological question
December 18, 2009
Galois was of course the first to highly successfully use the notion of a field. However, if ones reads his papers they’ll see that he never explicity gave the concept of an algebraic structure closed under addition, subtraction, commutative multiplication, and division a name. Dedekind would be the first to do that; he gave the name Körper, or “body,” to what we’d today call a number field. A couple of decades later, E.H. Moore of Chicago would introduce the term “field” in English.
“Körper” caught on fairly quickly among Continental mathematicians, giving us the French corps, and from there it spread to Spanish and Portuguese; in the other direction, the German mutated into Hungarian “test” and Polish “ciało”, both essentially with the same meaning of “body.”
However, in Italian and most of the Slavic languages, the word for “field” is also the agricultural term. This means that the algebraic terminology didn’t solidify until considerably later, probably between the World Wars at earliest. This is understandable; while both Italy and Russia had strong mathematical communities around the turn of the last century, they were somewhat isolated and if nothing else had relatively fewer top-tier algebraists than the French or, especially, the German schools.
What’s really curious is the following: In both Italian and Russian, as I mentioned, the word for English “field” is a literal translation of “field.” In pretty much every language, the word for “ring” can also refer to a thing that you wear on your finger. But in Italian and (several of) the Slavic languages — and in these languages alone, as far as I know — the word for “skew field”, or “division ring”, translates to English as “body”! This seems to me to be a rather exceptional situation — surely either a modification of “ring” or of “field” will do, as in every other language, but it seems not to be the case. So there are two open problems here:
1. Explain the situation that caused “field” to replace “body” to refer to a commutative division ring, but not to refer to a division ring in general, in Italian and Russian.
2. Are there any other examples of crufty terminology that’s unique to one or two languages (or closely-related language families?)
### More on graphs and digraphs
November 16, 2009
So this question’s been bugging me ever since I first thought it up, and I figured (in the spirit of MaBloWriMo, which by now is pretty much dead on this blog) that I’d ask about it here — I need to give Math Overflow a break.
The question concerns adjoint functors, which I don’t understand half as well as I’d like, but enjoy thinking about anyway. One of the (many!) motivating examples that adjoint functors generalize is the common “free/forgetful” dichotomy. For instance, there’s a functor from the category of groups (say) to the category of sets, which is defined by simply “forgetting” the group structure and giving back the underlying set. This functor doesn’t have an inverse, of course; that would make the two categories isomorphic, which is way too much to expect. Nor does it have an “inverse up to natural transformation.” That would make the categories equivalent, which is almost as good as isomorphism. But it does have the next-best thing after that: a functor in the opposite direction which comes with a natural isomorphism on some hom-sets. This is the free functor, that assigns to each set the free group on that set. These functors are called adjoint functors.
### Generalized LYM inequalities
November 6, 2009
I’ve been thinking about this problem ever since an old post of Qiaochu’s first raised the question, and I’ve been frustrated by my inability to solve it. I could post it on MO, but I sort of already have, and anyway it raises questions which are too ill-formed right now to be right for MO. So anyway, here we go:
A lot of problems in extremal combinatorics correspond to finding large antichains in partially ordered sets. (By the way, all posets in this post will be assumed to have a least element.) Classically speaking, Dilworth’s theorem completely characterizes the size of antichains in posets; however, this is often tricky to apply, since it’s not always clear whether a partition into chains is minimal. In addition, it’s sometimes the case (particularly with infinite posets) that there are infinitely long antichains, but a nontrivial bound should still be attainable. The way to get by both of these obstacles is to assign weights to the elements of the poset, and rather than looking for large antichains, we look for antichains with high total weight.
The classical example of this solves the problem of finding the largest antichain in the lattice of subsets of a given finite set — the content of Sperner’s theorem. (more…)
### Quasi-bleg: Why are there bump functions?
September 4, 2009
When I was learning analysis (beyond, say, first-year calculus), one of the facts that most surprised me was the fact that there are functions that were smooth (i.e., infinitely differentiable) and yet compactly supported. Of course, I didn’t think about it with that phrasing; there’s a pretty simple geometric interpretation of smoothness for most functions one encounters in calculus (actually, one rarely sees differentiable functions that aren’t smooth!) Specifically, if a function isn’t smooth at $x$, then there’s some sort of a “kink” at that point, or at least “around” that point.
Is this justified? Well, not totally, but let’s give a couple of examples to at least show why it’s a good first approximation. (more…)
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2017-08-17 08:03:18
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http://nanoscale.blogspot.com/2008/06/simple-numbers.html
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## Friday, June 06, 2008
### Simple numbers
So, if crude oil futures cost at least $126/42 gallon barrel these days, doesn't that imply that the raw starting material for gasoline already costs (once you factor in the time delay between futures contracts and refining)$3/gallon? This suggests to me that the "correct" price for gasoline in the US should be closer to $5-6/gallon, when the refining catches up with futures. (That doesn't even touch on issues about how much of the crude oil pricing is due to speculation vs. actual supply & demand, or how much of this is due to the effective weak dollar policies of the US central bank.) #### 13 comments: Anonymous said... The US has lived a dream of cheap gas for a very long time. No wonder why SUVs and bigger vehicles are (maybe, were) popular. In most countries gas is no way near the US price. We'll have to see how they deal with it, but I think we'll have the$5 gallon soon.
Anonymous said...
this assumes the yield from crude is <100%. Seems reasonable, but for example most pumps mention 10% ethanol. But I'll ask my friend.
Andrew said...
This comment has been removed by the author.
Andrew said...
Yield of gasoline is only around 50%, though the total yield of marketable products from a barrel of oil is more like 120%-- about 50 gallons of product from 42 gallons of oil. Gasoline is not the most expensive of these, but still, as Doug points out, close to $3 per gallon goes to crude oil costs, and gas should continue to rise, especially with the weak dollar. Here's what a barrel of oil gets you. sylow said... Doug, you are right. When crude hits 150$, gas should be 5$in USA nationwide(coastal areas will be higher than that). Jim Cramer said that on CNBC yesterday too. Gas price is lagging crude futures at the moment. descendent said... Here's what I don't get: The US was in (arguably) a similar Energy problem in the late 70's. But once the crisis subsided, the interest in developing renewable energy kinda fizzled. Fast forward 30 years, and where is all the alternative energy technology the world has been developing since the 1st energy crisis? 30 years to work on this problem and what do we have to show for it besides nuclear power plants (which the US still hasn't fully embraced) and better fuel economy (which in some ways exacerbated the problem)? --end futile rant-- CarlBrannen said... There's been steady improvements in biofuels of the evolutionary sort. They're getting bad press from big oil and OPEC press flack right now, but that should die down as the bad harvests reverse and food comes down in price. Anonymous said... There are predictions of$12 per gallon gasoline in the next twenty years, as demand continues to outstrip supply. This figure apparently includes biofuels as a factor.
Doug Natelson said...
carlbrannen - not to sound too pessimistic here, but what makes you think food prices will come down, particularly when fuel prices are so strongly coupled to production and distribution?
CarlBrannen said...
Doug, here's the latest estimates from the USDA: www.fas.usda.gov/psdonline/circulars/production.pdf
Some places are down (especially with rice), others are up, but overall, this will be a year of new world record grain harvests. For example, Ukraine wheat production is up 50% from last year. China soybean up 19% and their corn and wheat to make new records. High grain prices make high grain production.
Total world grain production (from the above link):
2006/2007) 2004.6
2007/2008) 2114.1 preliminary
2008/2009) 2161.9 projected
In case you don't believe US Dept. of Agriculture figures, the UN is saying the same thing; record harvests for 2008. Do a google search. Learn more. Ignore the news articles; the news media is for entertainment, not information. Scary stories make good entertainment. Any day now we're going to starve because of the lack of bees, LOL.
Along this line, I'm arguing with a particle physicist about the economics and physics of ethanol. I'm stunned at the difficulty in explaining even the simplest facts about the industry. See for yourself:
www.backreaction.blogspot.com/2008/06/biofuel.html
I'm the VP of engineering at a company that owns an ethanol plant. It's my business to know as much as possible about these things. I'm paid to learn about it just like a physicist is paid to learn about physics. If you have any questions about the subject I'll be glad to answer them. To make sure I hear your questions, post them on my blog devoted to the subject:
www.ethanolfuel.wordpress.com
I am reminded about something Feynman said just before receiving the Nobel prize, on the subject of outsiders discussing a topic they know nothing about:
"It's because somebody knows something about it that we can't talk about physics. It's the things that nobody knows anything about that we can discuss."
Doug Natelson said...
Carl - Thanks for the links. Interesting reading material. I have two points.
1) Just because we have record harvests world-wide doesn't mean that my prices at the store will fall. US milk production is up this year over last year by around 2% (see here), but the price in my supermarket is still up quite a bit over a year ago. Yes, higher prices imply producers will try to boost supply to get more profits; still, given the costs of production, increased demand, and the weakening dollar, I don't think it's obvious what the net result will be.
Even if feed prices come down and milk production goes up, shipping milk to me in Texas from Wisconsin will be quite a bit more expensive if diesel fuel (whatever the source) costs $7/gal. 2) No need to get defensive, dude. I'm not knocking biofuels, I'm not blaming you or the ethanol industry for higher food prices, and there's no need to go out of your way to call me arrogant and ignorant. Doug Natelson said... FWIW, I should have said (in my earlier comment) "...when fuel prices are so strongly coupled to food production and distribution?". CarlBrannen said... Doug, if there's enough bad weather around enough of the world we could get a bad harvest and people will starve. On the other hand, high food prices for US consumers are mostly about high transportation costs due to oil price rises. The dollar amount of corn or wheat in your diet is negligible. In fact, total world agriculture per person is only around$220 per person per year, or less than a dollar per day. These are the basic prices of food (including meat); they do not include the surcharges for things like paying a waiter to serve it to you or a store to have it convenient, or a factory to package it with some nasty chemicals to make it taste better, etc.
Along that line of thought, despite the world-wide shortage of rice, the local Safeway just put one of my staples, Rice-A-Roni, on sale at a buck a box. A box has about 7 ounces of rice, if I recall.
I don't follow rice prices, what are they, as much as $15 per bushel? A bushel of rice should be something like 60 pounds so that's$0.25 per pound; my box of RaR has about 12 cents worth of rice (and that's with record high rice prices).
The weakening dollar will raise all prices, food and everything else. So as that happens it won't be so noticeable as food price rising per se. In fact, the price of scrap metal has gotten so high that companies like mine are very concerned; druggies are stealing and destroying expensive equipment for its scrap value.
And I should add that the bad weather in the US is worrying me.
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2017-12-11 04:15:05
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http://mathoverflow.net/feeds/question/107894
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central limit theorem for binomial random variable - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T09:36:47Z http://mathoverflow.net/feeds/question/107894 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/107894/central-limit-theorem-for-binomial-random-variable central limit theorem for binomial random variable metrics 2012-09-23T09:41:21Z 2012-09-23T10:26:33Z <p>I'm confused about applying central limit theorem to Bernoulli random variables. Let $X_i=\frac{n}{\sqrt{n-1}}(Z_i - \frac{1}{n})$ where $Z_i$ is iid Bernoulli($\frac{1}{n}$). Then $E[X_i]=0$ and $Var(X_i) = 1$. Thus, it seems that standard Lindberg-Levy CLT can be applied to $S_n = \frac{1}{\sqrt{n}} \sum_{i-1}^n X_i$ which is a linear function of a binomial random variable. But the moment generating function of $S_n$ doesn't converge to that of standard normal, and the convergency works only when the probability parameter of the Bernoulli function is $1/T^{1-\alpha}$ $0 < \alpha < 1$. I read a couple of textbook and couldn't find if any further condition is required to apply CLT to $iid$ variables with finite mean and variance. What's wrong with applying CLT to $S_n$?</p> http://mathoverflow.net/questions/107894/central-limit-theorem-for-binomial-random-variable/107896#107896 Answer by ansobol for central limit theorem for binomial random variable ansobol 2012-09-23T10:26:33Z 2012-09-23T10:26:33Z <p>In your example, unlike the CLT, you deal not with <em>one</em> sequence of independent random variables $X_i$, but with a two-parameter family $X^{(n)}_i$, where for example $X_1^{(n)}$ and $X_1^{(m)}$ have different statistics for $m \neq n$ and cannot be both denoted simply by $X_1$. Therefore as $n$ grows, the sum $S_n$ does not merely grow term by term as in the standard CLT, but is replaced with an entirely new sum for each $n$. No surprise that this particular scaling gives a Poisson distribution; other scalings of $X^{(n)}_i$ may give yet other non-normal distributions.</p>
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2013-05-25 09:36:39
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https://economics.stackexchange.com/questions/5097/interpreting-how-graphs-of-cobb-douglas-utility-functions-how-does-mrs-vary-as
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# Interpreting how graphs of Cobb-Douglas utility functions. How does MRS vary as we vary $\alpha$?
Suppose I have the following Cobb-Douglas function $$U(x,y) = x^\alpha y^{1-\alpha} = 1$$ where $\alpha \in [0,1]$.
$$MRS = -\frac{U_x}{U_y} = - \frac{\alpha}{1-\alpha} \frac{y}{x}$$ $$\frac{\partial MRS}{\partial \alpha} = -\frac{1}{(1-\alpha)^2}\frac{y}{x}$$
So suppose I have the following set of graphs:
Here I just picked some values. Red is $\alpha = \frac{1}{4}$, blue is $\alpha = \frac{1}{2}$, black is $\alpha = \frac{3}{4}$
I understand how the steepness and flatness of the different curves change as I vary $\alpha$. But I am confused as to what $\frac{\partial MRS}{\partial \alpha}$ tells me about the graph. In particular, $\frac{\partial MRS}{\partial \alpha}$ is dependent on $\alpha$. So even though I know $\frac{\partial MRS}{\partial \alpha}$ tells me how much $MRS$ changes as I change alpha....changing $\alpha$ changes $\frac{\partial MRS}{\partial \alpha}$...so I am very confused!
My Question
What is $\frac{\partial MRS}{\partial \alpha}$ telling me about the graph? Since $\frac{\partial MRS}{\partial \alpha}$ is dependent on $\alpha$, how does this affect things?
I think it is important to note that $MRS(x,y)$ is a function. There is exactly one indifference curve passing through $(x,y)$. $MRS(x,y)$ shows the steepness of this curve at point $(x,y)$. Then $\frac{\partial MRS(x,y)}{\partial \alpha}$ would show how much steeper the indifference curve passing through $(x,y)$ gets at this point if you change the parameter $\alpha$.
As YM'fr already stated in his answer, the interpretation of this is how the marginal substition rate (the rate at which she would be willing to trade) of a consumer in possesion of the goods package $(x,y)$ would change.
If $\alpha$ rises, the utility puts more weights to the $x$. Then you must give up more $x$ for one $y$ (for the same utility). Your MRS, for a given $y$, increases in absolute value. Graphically:
If you set $y$ and $x$, the slope is lower for higher $\alpha$ (be careful to only change one thing at a time).
Concerning your "loop" problem, look at the function $1/x$, the first derivative is also a function of $x$, like the 2nd and so on. What matter is the value at one particular x.
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2019-12-06 15:31:39
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https://www.physicsforums.com/threads/fouriers-law-of-heat-conduction.631679/
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# Fouriers law of heat conduction.
1. Aug 27, 2012
### guitar24
If I have the equation -k*dT/dr=0
do I get rid of the constant k and integrate, or do i integrate first and keep the constant?
2. Aug 27, 2012
### rock.freak667
You can divide it, otherwise you'd just replace C/k with some other constant.
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2017-10-19 23:06:04
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http://mathhelpforum.com/algebra/195032-logs-past-paper-question.html
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# Math Help - logs past paper question
1. ## logs past paper question
log2 x = log 4x +6
this is a question my teacher gave us, the she gave this the first equation on starting the question which is below
log2x = log2x +6 / log24
how did my teacher got this, i need to know because i am presently studing for an exam
PS the 2 and 4 is subscrip SORRY AND final log24 only the 2 is subscript
2. ## Re: logs past paper question
Originally Posted by mathkid12
log2 x = log 4x +6
this is a question my teacher gave us, the she gave this the first equation on starting the question which is below
log2x = log2x +6 / log24
how did my teacher got this, i need to know because i am presently studing for an exam
PS the 2 and 4 is subscrip SORRY AND final log24 only the 2 is subscript
1. You can convert logs of a certain base into logs of another base by using:
$\log_a(b)=\frac{\log_c(b)}{\log_c(a)}$
2. Re-write the given equation:
$\log_2(x)=\log_4(x)+6~\implies~ \\ \log_2(x)=\frac{\log_2(x)}{\log_2(4)}+6~\implies~ \\ \log_2(x)=\frac{\log_2(x)}{2}+6$
3. Collect like terms:
$\frac12 \log_2(x)=6~\implies~\log_2(x)=12~\implies~x=2^{12 }=4096$
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2015-08-01 10:59:30
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http://www.finderchem.com/math-question-help-formula-formation-basic.html
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2016-10-25 12:21:51
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# 6th rate ship of the line
### 6th rate ship of the line
The AubreyâMaturin series of novels by Patrick O'Brian features the sixth-rate ship HMS Surprise as the frigate captained by Jack Aubrey. Torpedo-boat destroyers, torpedo boats, and similar vessels were not rated. Structurally, these were two-deckers with a … These entry-level war ships are very mobile, and relatively faster to their predecessors. Could be propelled by oars or sail. an officer holding the substantive rank of captain) as their commander. Historical category for Royal Navy vessels, based on number of guns, actual historical frigate of the same name, Sixth-rate ships at the Royal Navy website, https://en.wikipedia.org/w/index.php?title=Sixth-rate&oldid=970250984, Short description is different from Wikidata, Creative Commons Attribution-ShareAlike License, This page was last edited on 30 July 2020, at 05:41. The high end of the fifth rate would include two-deckers of 40- or 44-guns (from 1690) or even the demi-batterie 32-gun and 36-gun ships of the 1690–1730 period. A first-, second- or third-rate ship was regarded as a "ship-of-the-line". Vision of the Seas - 4.3811. on the upper deck of a sloop or post ship, thus providing its main battery), such carronades were counted. A sixth rate's range went from 4–18 to 20–28 (after 1714 any ship with fewer than 20 guns was unrated).[1]. In a weaker crew there would be a large proportion of 'landsmen', adults who were unused to the sea. Rating was not the only system of classification used. In February 1817 the rating system changed. See Celebrity Equinox's 2021 to 2022 schedule and popular upcoming cruise itineraries on Cruise Critic. The number and weight of guns determined the size of crew needed, and hence the amount of pay and rations needed. A sixth rate carried about 23 marines, while in a strong crew the bulk of the rest were experienced seamen rated 'able' or 'ordinary'. 1. Looking for Celebrity Equinox itineraries? Accessible Staterooms Our accessible staterooms are designed with wider doors, roll-in showers, grab bars, and other special features for guests with mobility issues and other … "The earliest naval list in which any classification of ships appears, is dated 1546, and it divides the fifty-eight ships of Henry VIII's Navy, according to their "quality, into... 'ships, galliasses, pinnaces, and cow barges. Vessels might also carry other guns that did not contribute to the rating. Their rating is indicative of their power, with First Rates being the heaviest, most powerful -- but also most cumbersome -- ships; 6th rates are the opposite, being relatively weak, but fast and manoeuvreable. ^* The smaller fourth-rates, primarily the 50-gun ships, were, from 1756 on, no longer classified as ships of the line. Another list, dated 1612, divides them into... 'ships royal, measuring from twelve hundred to eight hundred tons; middling ships, from eight hundred to six hundred tons; small ships, three hundred and fifty tons; and pinnaces, from two hundred and fifty to eight tons. When the carronades replaced or were in lieu of carriage-mounted cannon they generally counted in arriving at the rating, but not all were, and so may or may not have been included in the count of guns, though rated vessels might carry up to twelve 18-, 24- or 32-pounder carronades. [1], The earliest categorisation of Royal Navy ships dates to the reign of King Henry VIII. The larger of the unrated vessels were generally all called sloops, but that nomenclature is quite confusing for unrated vessels, especially when dealing with the finer points of "ship-sloop", "brig-sloop", "sloop-of-war" (which really just meant the same in naval parlance as "sloop") or even "corvette" (the last a French term that the British Navy did not use until the 1840s). They carried a crew of about 650 men. 4: 160. The Polish Legion is a line infantry unit, able to give fire or charge home with bayonets. That is, a change of one unit in x corresponds to a change of 0.4 units in y. They combined … As with the rest of the navy units - its battle value is worth its price. The formal system of dividing up the Navy's combatant warships into a number or groups or "rates", however, only originated in the very early part of the Stuart era, with the first lists of such categorisation appearing around 1604. From about 1660 the classification moved from one based on the number of men to one based on the number of carriage guns a ship carried. Also some of the guns were removed from a ship during peacetime service, to reduce the stress on the ship's structure, which is why there was actually a distinction between the wartime complement of guns (and men) and the lower peacetime complement—the figure normally quoted for any vessel is the highest (wartime) establishment. First rates were the largest and most powerful ships but were expensive to build and run and there … Vessels were sometimes classified according to the substantive rank of her commanding officer. Notable exceptions to this rule were ships such as the Santisima Trinidad of Spain, which had 140 guns and four gun decks (the Spanish and French had different rating systems from those of Britain). the maximum breadth of the vessel. [5] The recommendation from the Board of Admiralty to the Prince Regent was dated 25 November 1816, but the Order in Council establishing the new ratings was issued in February 1817. Lieutenant-commanders, lieutenants, ensigns, or warrant officers might command unrated vessels, depending on the size of the vessel.[6]. {\displaystyle {\frac {k\times b\times {\frac {1}{2}}b}{94}}} Vessels with fewer than three masts were unrated sloops, generally two-masted vessels rigged as snows or ketches (in the first half of the 18th century), or brigs in succeeding eras. Description: it is a decent ship. The rated number of guns often differed from the number a vessel actually carried. Ship Turning Rate. My library Surprise began her life in 1794 as a French ship, under the name of Unité. CCSS.Math.Content.6.RP.A.2 Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship. From that date, the first rate comprised all ships carrying 110 guns and upwards, or the complement of which consisted of 1,000 men or more; the second rate included one of HM's royal yachts, and otherwise comprised all ships carrying under 110 guns but more than 80 guns, or the complements of which were under 1,000 but not less than 800 men; the third rate included all the rest of HM's royal yachts and "all such vessels as may bear the flag of pendant of any Admiral Superintendent or Captain Superintendent of one of HM's Dockyards", and otherwise comprised all ships carrying at most 80 guns but not less than 60 guns, or the complements of which were under 800 but not less than 600 men; the fourth rate comprised all frigate-built ships of which the complement was not more than 600 and not less than 410 men; the fifth rate comprised all ships of which the complement was not more than 400 and not less than 300 men; the sixth rate consisted of all other ships bearing a captain. And they also ask us to figure out what the equation of this line actually is. Ships of the Line. Look up rates for new shipments and inland tariffs. The smaller sixth rates with between 20 and 24 guns, still all ship-rigged and sometimes flush-decked vessels, were generally designated as post ships. These were too small to be formally counted as frigates (although colloquially often grouped with them), but still required a post-captain (i.e. The Surprise was portrayed in the 2003 film Master and Commander which was adapted from the novels. Allure of the Seas - 4.515. There was a further major change in the rating system in 1856. caledonia sold for £48,000 It is based on the actual historical frigate of the same name, formerly the French Unité, which was captured and renamed by the Royal Navy in 1796. The number of Midshipmen in a ship was fixed by the rating of the ship and it was at the discretion of the Captain as to who was carried so positions usually went to the children of relations, acquaintances or former shipmates. Therefore, one should not change a measurement in "tons burthen" into a displacement in "tons" or "tonnes". Since not big enough to stand in the line of battle, were often called frigates, though not classed as frigates by the Royal Navy. The ship speed is the base speed without any modifiers through woods or modules. , where Dates of service, name changes, previous and next incarnations, dimensions, armament, commanders, officers and crewmen, actions, battles, sources However, these were gradually phased out, as the low freeboard (i.e., the height of the lower deck gunport sills above the waterline) meant that in rough weather it was often impossible to open the lower deck gunports.[4]. Pepys's original classification was updated by further definitions in 1714, 1721, 1760, 1782, 1801 and 1817 (the last being the most severe, as it provided for including in the count of guns the carronades that had previously been excluded). Since I’ve already got a first rate, I’m going to do a 6 th rate frigate. It thus encompassed ships with up to 30 guns in all. × The sixth rate ship of the line is a small frigate, carrying 32 guns with a crew of 95. This is the smallest vessel in the system set out by Samuel Pepys (the very same fellow who is now famous for his sometimes rude diaries), and … Ships of the Line are heavy 4th-1st rates; well armored, well armed with a great quantity of guns of high caliber designed to fight and survive the line of battle- hence their reference as ships of the line. The largest third rates, those of 80 guns, were likewise three-deckers from the 1690s until the early 1750s, but both before this period and subsequent to it, 80-gun ships were built as two-deckers. Theoretically they initially joined as First Class Volunteers for a period before being appointed as … k This was only on the basis of their roughly-estimated size and not on their weight, crew or number of guns. [citation needed] Soon afterwards, the structure was again modified, with the term rank now being replaced by rate, and the former small ships now being sub-divided into fourth, fifth and sixth rates. While a fourth rate was defined as a ship of the line, fifth and the smaller sixth rates were never included among ships-of-the-line. One therefore needs to distinguish between the established armament of a vessel (which rarely altered) and the actual guns carried, which might happen quite frequently for a variety of reasons; guns might be lost overboard during a storm, or "burst" in service and thus useless, or jettisoned to speed the ship during a chase, or indeed removed down into the hold in order to use the ship (temporarily) as a troop transport, or for a small vessel, such as the schooner HMS Ballahoo, to lower the centre of gravity and thus improve stability in bad weather. They are among the best at sea – or anywhere. They were generally classified, like all smaller warships used primarily in the role of escort and patrol, as "cruisers", a term that covered everything from the smaller two-deckers down to the small gun-brigs and cutters. Sixth Rate This wooden model kit is a reproduction of the 22-gun British 6th Rate Frigate HMS Cyane. Although the rating system described was only used by the Royal Navy, other major navies used similar means of grading their warships. b a large, finely carved and well presented early 19th century napoleonic french prisoner of war bone ship model for a first rate ship of the line traditionally identified as h.m.s. 1 [4], The smaller fourth rates, of about 50 or 60 guns on two decks, were ships-of-the-line until 1756, when it was felt that such 50-gun ships were now too small for pitched battles. At this time the combatant ships of the "Navy Royal"[Note 2] were divided up according to the number of men required to man them at sea (i.e. For instance, when the commanding officer of a gun-brig or even a cutter was a lieutenant with the status of master-and-commander, the custom was to recategorise the vessel as a sloop. In the British Royal Navy, a second-rate was a ship of the line … [dubious – discuss] The royal ships were now graded as first rank, the great ships as second rank, the middling ships as third rank, and the small ships as fourth rank. Slightly worse than Fourth Rate Ship of the Line but it's faster and more agile. was the length, in feet, from the stem to the sternpost, and Through the early modern period, the term "ship" referred to a vessel that carried square sails on three masts. 1st Rate: 100 to 120: 3: 2nd Rate: 90 to 98: 3: 3rd Rate: 64 to 80: 2: 4th Rate: 48 to 60: 2: Frigate. The first movement towards a rating system may be seen in the 15th century and the first half of the 16th century, when the largest carracks in the Navy (such as the Mary Rose, the Peter Pomegranate and the Henri Grâce à Dieu) were denoted "great ships". {\displaystyle b} Young Guard: Infantry 步兵: 800: 200: 98 ratings 個評分 Great unit 非常推薦 Disciplined and inspirational, the men of the Young Guard are elite soldiers of the highest order. We serve dishes made with the freshest ingredients, and our menus reflect regional flavors from around the world. It also indicated whether a ship was powerful enough to stand in the line of battle. ^* The smaller sixth-rates were often popularly called frigates, though not classed as "frigates" by the Admiralty officially. The other quarterdeck officers were the chaplain and a Royal Marines lieutenant. For example, the French Navy used a system of five rates ("rangs") which had a similar purpose. The division of the navy into 'rates' appears for the first time in a table drawn up by Charles I., in 1626, and entitled,—'The New Rates for Seaman's monthly wages, confirmed by the Commissioners of His Majesty's Navy, according to His Majesty's several rates of ships, and degrees of officers.' Graph the line that represents a proportional relationship between y and x with a unit rate 0.4. On Koningsdam, there are 18 of these cabins; the line's newest ship, Nieuw Statendam, has even more Large Interior staterooms, each of which offers up to a whopping 266 square feet. The Aubrey–Maturin series of novels by Patrick O'Brian features the sixth-rate ship HMS Surprise as the frigate captained by Jack Aubrey. Aubrey served on her as a Midshipman, before being presumably promoted to Lieutenant. So let me get my scratch pad out and we could think about it. For example, "This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar." When these carracks were superseded by the n… In the rating system of the Royal Navy used to categorise sailing warships, a sixth-rate was the designation for small warships mounting between 20 and 28 carriage-mounted guns on a single deck, sometimes with smaller guns on the upper works and sometimes without. Of unrated vessels, the category of sloops comprised all vessels commanded by commanders; next followed all other ships commanded by lieutenants, and having complements of not less than 60 men; finally were "smaller vessels, not classed as above, with such smaller complements as the Lords Commissioners of the Admiralty may from time to time direct". By the end of the 18th century, the rating system had mostly fallen out of common use (although technically it remained in existence for nearly another century), ships of the line usually being characterized directly by their nominal number of guns, the numbers even being used as the name of the type, as in "a squadron of three seventy-fours". Sailing vessels with only two masts or a single mast were technically not "ships", and were not described as such at the time. Explorer of the Seas - 4.3812. [4], The rating system did not handle vessels smaller than the sixth rate. Sixth-rate ships typically had a crew of about 150â240 men, and measured between 450 and 550 tons. For those with mobility issues and other disabilities, our attentive crew is always on hand to take the hassle out of getting on and off the ship. The rating system of the Royal Navy formally came to an end in the late 19th century by declaration of the Admiralty. Anthem of the Seas cabins - 4.514. A series of major changes to the rating system took effect from the start of January 1817, when the carronades carried by each ship were included in the count of guns (previously these had usually been omitted); the first rate from that date included all of the three-deckers (the adding in of their carronades had meant that all three-deckers now had over 100 guns), the new second rate included all two-deckers of 80 guns or more, with the third rate reduced to two-deckers of fewer than 80 guns. Symphony of the Seas cabins - 4.533. Great Ships (the rest of the ships in the previous "great ships" grouping) mounting 38–40 guns; This page was last edited on 20 January 2021, at 02:28. Most were phased out without replacement, although a few lasted in auxiliary roles until after 1815. The middle of the 18th century saw the introduction of a new fifth-rate type—the classic frigate, with no ports on the lower deck, and the main battery disposed solely on the upper deck, where it could be fought in all weathers. Fourth Rate The first movement towards a rating system may be seen in the 15th century and the first half of the 16th century, when the largest carracks in the Navy (such as the Mary Rose, the Peter Pomegranate and the Henri Grâce à Dieu) were denoted "great ships". These vessels could perhaps be considered comparable to the light cruisers and destroyers of more recent times, respectively. Examples of a ship of the line: Sixth Rate: Sailing warship with 20-30 guns (1779). Converted merchant vessels that were armed and equipped as cruisers were of the second rate if over 6000 tons, and of the third rate if over 1000 and less than 6000 tons. Historical category for Royal Navy vessels, based on number of guns, First, second and third rates (ships of the line), Royal Navy rating system in force during the Napoleonic Wars, Galliasses, not to be confused with the Mediterranean vessel, The term Royal Navy was only introduced after the Restoration of King. HMS Calcutta, a converted East Indiamen, served most of her career as a 56 gun 4th rate. Admiral's Flagship, 5th Rate. From 1778, however, the most important exception was the carronade. b The Navy did retain some fourth rates for convoy escort, or as flagships on far-flung stations; it also converted some East Indiamen to that role. The rating of a ship was of administrative and military use. Liberty of the Seas - 4.457. Sixth-rate ships were generally useful as convoy escorts, for blockade duties and the carrying of dispatches; their small size made them less suited for the general cruising tasks the fifth-rate frigates did so well. Captains commanded ships of the first rate; captains or commanders commanded ships of the second rate; commanders or lieutenant-commanders commanded ships of the third rate; lieutenant-commanders or lieutenants commanded ships of the fourth rate. In the first half of the 18th century the main battery guns were 6-pounders, but by mid-century these were supplanted by 9-pounders. The ship armor is for a ship of oak wood type and no armor related modules, to see the armor of the ship you have to go into the ship page itself. A 28-gun ship would have about 19 officers; commissioned officers would include the captain, and two lieutenants; warrant officers would include the master, ship's surgeon, and purser. The table specified the amount of monthly wages a seaman or officer would earn, in an ordered scheme of six rates, from "first-rate" to "sixth-rate", with each rate divided into two classes, with differing numbers of men assigned to each class. Only the larger sixth-rates (those mounting 28 carriage guns or more) were technically frigates. [1], Samuel Pepys, then Secretary to the Admiralty, revised the structure in 1677 and laid it down as a "solemn, universal and unalterable" classification. During the Napoleonic Wars, the Royal Navy increased the number of sloops in service by some 400% as it found that it needed vast numbers of these small vessels for escorting convoys (as in any war, the introduction of convoys created a huge need for escort vessels), combating privateers, and themselves taking prizes.[4]. The fifth rates at the start of the 18th century were generally "demi-batterie" ships, carrying a few heavy guns on their lower deck (which often used the rest of the lower deck for row ports) and a full battery of lesser guns on the upper deck. The first and second rates were three-deckers; that is, they had three continuous decks of guns (on the lower deck, middle deck and upper deck), usually as well as smaller weapons on the quarterdeck, forecastle and poop. Until that date, carronades only "counted" if they were in place of long guns; when the carronades replaced "long" guns (e.g. 6 to 14: 1 . [1], The earliest rating was based not on the number of guns, but on the established complement (number of men). Shop for the latest online womens dresses, sweaters, outerwear, tops, bottoms, bags, shoes, jewelry, watches & accessories from … All the other third rates, with 74 guns or less, were likewise two-deckers, with just two continuous decks of guns (on the lower deck and upper deck), as well as smaller weapons on the quarterdeck, forecastle and (if they had one) poop. On the whole the trend was for each rate to have a greater number of guns. Regardless of armament, sixth-rates were known as "post ships" because, being rated, they were still large enough to have a post-captain in command instead of a lieutenant or commander.[2]. A well trained crew can handle the vessel with great efficiency, and with a good commander can make quick work of larger war ships. "[2]:128[q 1]. The Surprise was portrayed in the 2003 film Master and Commander which was adapted from the novels. Second-rate and third-rate are also used as adjectives to mean that something is of inferior quality. HMS Cyane was a Royal Navy Banterer-class sixth-rate post ship of nominally 22 guns, built in 1806 at Topsham, near Exeter, England.She was ordered in January 1805 as HMS Columbine but renamed Cyane on 6 … - So we have different definitions for d of t on the left and the right and let's say that d is distance and t is time, so this is giving us our distance as a function of time, on the left, it's equal to 3t plus one and you can see the graph of how distance is changing as a function of time here is a line and just as a review from algebra, the rate of change of a line… × b Oasis of the Seas - 4.438. Captured by HMS Inconstant she was renamed (since the Royal Navy already possessed a ship called Unité, taken just a week previously). Examples of such weapons would include mortars, howitzers or boat guns, the boat guns being small guns intended for mounting on the bow of a vessel's boats to provide fire support during landings, cutting out expeditions, and the like. Ship armour. Examples of a sixth rate: Skiff: A small flat-bottomed ship's boat, having a sharp pointed bow and a square stern. Ship of the line. Third Rate. As part of our efforts to reach our environmental goals, iPhone 12 Pro and iPhone 12 Pro Max do not include a power adapter or EarPods.Included in the box is a USB‑C to Lightning cable that supports fast charging and is compatible with USB‑C power adapters and computer ports. Later in her career, she was u… Finally came 4th rates, two-deckers with 50 guns, essentially heavy frigates. 6th Floor, GPHA Towers, Harbour Area, P.O. Harmony of the Seas cabins - 4.542. For instance, Pepys allowed a first rate 90–100 guns, but on the 1801 scheme a first rate had 100–120. The main cause behind this declaration focused on new types of gun, the introduction of steam propulsion and the use of iron and steel armour which made rating ships by the number of guns obsolete. Recent reports of COVID-19 on cruises highligh… The term first-rate has passed into general usage, as an adjective used to mean something of the best or highest quality available. Walkthrough of a 6th rate frigate I figure the best way to get my point across is to do a step by step of how I build a ship. [1] The rest of the men were the crew, or the 'lower deck'. The new carronades were generally housed on a vessel's upperworks—quarterdeck and forecastle—some as additions to its existing ordnance and some as replacements. Freedom of the Seas - 4.496. At the low end of the fourth rate one might find the two-decker 50-gun ships from about 1756. Fleets: Austria, France, Spain, Ottoman Empire, Marat Confederation, Prussia, Republic of Poland, Sweden, … When these carracks were superseded by the new-style galleons later in the 16th century, the term "great ship" was used to formally delineate the Navy's largest ships from all the rest. No specific connection with the size of the ship or number of armaments aboard was given in this 1626 table, and as far as is known, this was related exclusively to seaman pay grades. From c.1650 the burthen of a vessel was calculated using the formula Introduced in the late 1770s, the carronade was a short-barreled and relatively short-range gun, half the weight of equivalent long guns, and was generally mounted on a slide rather than on trucks. 28-gun sixth rates were classed as frigates, those smaller as 'post ships', indicating that they were still commanded by a full ('post') captain, as opposed to sloops of 18 guns and less under commanders. End in the line: sixth rate: Sailing warship with 20-30 guns ( )... Ship '' referred to a vessel 's upperworks—quarterdeck and forecastle—some as additions its! French ship, under the name of Unité figure out what the equation of this line is... A line infantry unit, able to give fire or charge home with bayonets on cruise.. 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Wooden model kit is a joyful celebration and our menus reflect regional flavors from around the world rate! Fourth rate the ship speed is the burthen tonnage ( bm ) celebration and our specialty always..., served most of her commanding officer corresponds to a change of 0.4 units in y the. The second comprised the post ships '' of between 20 and 24 guns ''. ( 1779 ) specialty restaurants always deliver an extraordinary experience on wooden benches number! Modern period, the rating of a ship 's boat, having a sharp pointed and! Of Sloop-of-war '' included any unrated combatant vessel—in theory, the term even extended bomb! Scheme a first rate, I ’ ve already got a first rate had 100–120 Area. Was no exact correlation between formal gun rating and the 28 gun designs were of... 'S Flagship, 5th rate ) and were classed as frigates Wars there was no exact between! Board a Princess® cruise ship is a joyful celebration and our specialty restaurants always an! They combined … HMS Calcutta, a converted East Indiamen, served most of her career as ... Highligh… Looking for Celebrity Equinox itineraries such carronades were counted guns that determined a ship the. Q 1 ], the most numerous line-of-battle ships were the crew, or the 'lower '. Popularly called frigates, though not classed as ship '' referred a. Regional flavors from around the world perhaps be considered comparable to the light cruisers and destroyers of more recent,! Royal Navy ships dates to the sea among the best or highest quality available 6-pounders, by. The larger sixth rates were never included among ships-of-the-line cruisers and destroyers of recent. ) which had a crew of about 150â240 men, and our menus reflect regional flavors from around world. Or highest quality available: Sloop-of-war find the two-decker 50-gun ships from 1756! Figure out what the equation of this line actually is Harbour Area, P.O similar means of grading their.... All of fourth rate the ship speed is the burthen tonnage ( bm ) ships with up to 30 in! In the 2003 film Master and Commander which was adapted from the.. At sea – or anywhere therefore, one should not change a measurement in tons '' .
Categories : Uncategorized
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2021-04-15 16:27:21
|
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https://www.rdocumentation.org/packages/cvAUC/versions/1.1.0
|
# cvAUC v1.1.0
0
0th
Percentile
## Cross-Validated Area Under the ROC Curve Confidence Intervals
This package contains various tools for working with and evaluating cross-validated area under the ROC curve (AUC) estimators. The primary functions of the package are ci.cvAUC and ci.pooled.cvAUC, which report cross-validated AUC and compute confidence intervals for cross-validated AUC estimates based on influence curves for i.i.d. and pooled repeated measures data, respectively. One benefit to using influence curve based confidence intervals is that they require much less computation time than bootstrapping methods. The utility functions, AUC and cvAUC, are simple wrappers for functions from the ROCR package.
# cvAUC
The cvAUC R package provides a computationally efficient means of estimating confidence intervals (or variance) of cross-validated Area Under the ROC Curve (AUC) estimates.
In binary classification problems, the AUC is commonly used to evaluate the performance of a prediction model. Often, it is combined with cross-validation in order to assess how the results will generalize to an independent data set. In order to evaluate the quality of an estimate for cross-validated AUC, we obtain an estimate of its variance.
For massive data sets, the process of generating a single performance estimate can be computationally expensive. Additionally, when using a complex prediction method, the process of cross-validating a predictive model on even a relatively small data set can still require a large amount of computation time. Thus, in many practical settings, the bootstrap is a computationally intractable approach to variance estimation. As an alternative to the bootstrap, a computationally efficient influence curve based approach to obtaining a variance estimate for cross-validated AUC can be used.
The primary functions of the package are ci.cvAUC and ci.pooled.cvAUC, which report cross-validated AUC and compute confidence intervals for cross-validated AUC estimates based on influence curves for i.i.d. and pooled repeated measures data, respectively. One benefit to using influence curve based confidence intervals is that they require much less computation time than bootstrapping methods. The utility functions, AUC and cvAUC, are simple wrappers for functions from the ROCR package.
Erin LeDell, Maya L. Petersen & Mark J. van der Laan, "Computationally Efficient Confidence Intervals for Cross-validated Area Under the ROC Curve Estimates." (In Review)
## Install cvAUC
You can install:
• the latest released version from CRAN with
install.packages("cvAUC")
• the latest development version from GitHub with
if (packageVersion("devtools") < 1.6) {
install.packages("devtools")
}
devtools::install_github("ledell/cvAUC")
## Using cvAUC
Here is a quick demo of how you can use the package. In this example we do the following:
• Load an i.i.d. data set with a binary outcome.
• We will use 10-fold cross-validation, so we need to divide the indices randomly into 10 folds. In this step, we stratify the folds by the outcome variable. Stratification is not necessary, but is commonly performed in order to create validation folds with similar distributions. This information is stored in a 10-element list called folds. Below, the function that creates the folds is called .cvFolds.
• For the vth iteration of the cross-validation (CV) process, fit a model on the training data (i.e. observations in folds {1,...,10}\v) and then using this saved fit, generate predicted values for the observations in the vth validation fold. The .doFit function below does this procedure. In this example, we the Random Forest algorithm.
• Next, this function is applied across all folds to generate predicted values for each validation fold.
• These predicted values is stored in vector called predictions.
• Lastly, we use the ci.cvAUC function to calculate CV AUC and to generate a 95% confidence interval for this CV AUC estimate.
First, we define a few utility functions:
.cvFolds <- function(Y, V){
# Create CV folds (stratify by outcome)
Y0 <- split(sample(which(Y==0)), rep(1:V, length=length(which(Y==0))))
Y1 <- split(sample(which(Y==1)), rep(1:V, length=length(which(Y==1))))
folds <- vector("list", length=V)
for (v in seq(V)) {folds[[v]] <- c(Y0[[v]], Y1[[v]])}
return(folds)
}
.doFit <- function(v, folds, train){
# Train & test a model; return predicted values on test samples
set.seed(v)
ycol <- which(names(train)==y)
params <- list(x = train[-folds[[v]],-ycol],
y = as.factor(train[-folds[[v]],ycol]),
xtest = train[folds[[v]],-ycol])
fit <- do.call(randomForest, params)
pred <- fit$test$votes[,2]
return(pred)
}
This function will execute the example:
iid_example <- function(train, y = "V1", V = 10, seed = 1) {
# Create folds
set.seed(seed)
folds <- .cvFolds(Y = train[,c(y)], V = V)
# Generate CV predicted values
cl <- makeCluster(detectCores())
registerDoParallel(cl)
predictions <- foreach(v = 1:V, .combine="c",
.packages=c("randomForest")) %dopar% .doFit(v, folds, train)
stopCluster(cl)
predictions[unlist(folds)] <- predictions
# Get CV AUC and 95% confidence interval
runtime <- system.time(res <- ci.cvAUC(predictions = predictions,
labels = train[,c(y)],
folds = folds,
confidence = 0.95))
print(runtime)
return(res)
}
Load a sample binary outcome training set into R:
train_csv <- "http://www.stat.berkeley.edu/~ledell/data/higgs_10k.csv"
train <- read.table(train_csv, sep=",")
Run the example:
library(randomForest)
library(doParallel)
library(cvAUC)
res <- iid_example(train = train, y = "V1", V = 10, seed = 1)
print(res)
# $cvAUC # [1] 0.7813759 # #$se
# [1] 0.004534395
#
# $ci # [1] 0.7724886 0.7902631 # #$confidence
# [1] 0.95
## cvAUC Performance
For the example above (10,000 observations), it took ~0.2 seconds to calculate the cross-validated AUC and the influence curve based confidence intervals. This was benchmarked on a 2.3 GHz Intel Core i7 processor using cvAUC package version 1.1.0.
For bigger (i.i.d.) training sets, here are a few rough benchmarks:
• 100,000 observations: ~0.5 seconds
• 1 million observations: ~13.0 seconds
## Functions in cvAUC
Name Description AUC Area Under the ROC Curve ci.cvAUC Confidence Intervals for Cross-validated Area Under the ROC Curve (AUC) Estimates cvAUC Cross-validated Area Under the ROC Curve (AUC) cvAUC-package Cross-Validated Area Under the ROC Curve Confidence Intervals ci.pooled.cvAUC Confidence Intervals for Cross-validated Area Under the ROC Curve (AUC) Estimates for Pooled Repeated Measures Data admissions Data set: Simulated Admissions Data with Binary Outcome adherence Data set: Simulated Pooled Repeated Measures Data No Results!
|
2020-09-19 08:26:09
|
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|
https://www.construct.net/tutorials/building-android-apps-apks-in-construct-3-19
|
12
# Building Android apps (APKs) in Construct 3
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Published 12 Oct, 2017
1,938 words
~8-13 minutes
Construct 3 has its own mobile app build service, which can build your mobile apps for you. This feature is only available to subscribers. Here's how to get Construct 3 to build an APK (an Android app) for you.
You might want to catch up on the tutorials supporting multiple screen sizes and touch controls to make sure your project is ready for use on mobile. It's also easy to test on mobile using Construct 3's Remote Preview feature.
Also, make sure you fill out everything in the About section of Project Properties, such as the author, app ID, version and so on. All this information will be used by the exported app.
## Export
Once you're satisfied your project works well on mobile and is ready to be exported, select MenuProjectExport and pick the Android (Cordova) option.
When the Cordova options dialog appears, there's a dropdown list labelled Android build. If you just want to test your project as a mobile app, choose Debug APK. If you're ready to publish your app to the Google Play store, choose Unsigned release APK. (Note Construct currently cannot sign release APKs for you.)
## Building the APK
When you continue, your project will be uploaded to the app build service for building. This can take a while. You can carry on working in Construct while the app is uploaded, built and downloaded again. The status of the build will be shown in the lower-left corner of the window.
When the build finishes, a dialog will pop up with a "Build finished!" message, and a link to download the resulting APK.
## Testing a debug APK
Follow these steps if you chose the Debug APK option. Otherwise skip to the next section for information about signing a release APK.
### Enable developer mode
You can't run debug APKs on an Android device until you enable Developer Mode. Follow these steps to enable it:
1. Open Settings
2. Find the About section and tap it
3. Find the Build number. This might be in a sub-section, like Software information, or More.
4. Tap the Build number repeatedly until you see a notification that developer mode is enabled.
Now you can install debug APKs! It will also show a new Developer options section in Settings, but you don't need that for this guide.
### Transfer the APK to your device
Once you've downloaded the .apk file, it's on your computer, but it needs to be on your Android device. There are a number of ways you can transfer the file to your device. Here's a couple of ideas:
• Upload it to a cloud storage service like Google Drive, Dropbox or OneDrive. Then use the mobile app to download it again on the device.
• Connect your device to your computer with a cable and transfer it across to the device's storage.
• Copy the file to a USB drive, and then connect the USB drive to the Android device. (You might need a USB On-The-Go cable.)
• Construct 3 works on Android, so you can actually do the build in Construct 3 on an Android device and have the APK download to the device directly.
• Look for an Android app that supports transferring files from a computer.
• If all else fails, try emailing it to yourself in an attachment and open the email on the device.
### Run the APK
Once you tap the APK file on your Android device, you should see a screen asking if you want to install it. Tap Install and after a moment you should see a success message and an option to open the app. (It'll also be in the apps section if you want to run it later.) Open the app and you should see your game running!
## Signing a release APK
Follow these steps if you chose the Unsigned release APK option. Otherwise see the previous section.
Release APKs are intended for publishing to the Google Play store. Before you can publish it, you must align it (to optimise it) then sign it (to securely verify you as the author). You can't run a release APK on your device until it is signed. Depending on the device you may also need to enable developer mode (see the previous section), or enable permission to install apps from unknown sources, before you can install the release APK. This is because it doesn't come directly from the Google Play Store. (Installing an APK manually is also known as side-loading.)
### Tips on using command lines
You'll need to run some programs from the command line to follow these steps. On macOS and Linux it's referred to as the Terminal. Older versions of Windows use the Command prompt, but in the latest versions of Windows 10 it's been replaced with Powershell. Normally you can just search for these names as an app and run it. Many systems have shortcuts to open them as well, such as holding Shift and right-clicking in a folder on Windows. Here are some more tips to help you follow these steps:
• The command line has a current directory, which is always displayed. This is where files are written to and read from by default, if you use a relative name like file.txt in a command.
• A command to run a program is the filename of the program followed by any options (called arguments in a command line).
• Usually if a path has a space in it, e.g. C:\Program Files, it must be surrounded in double quotes to work correctly, e.g. "C:\Program Files".
• In Windows PowerShell, if the path to the program itself has a space in it, prefix it with &. For example if you get an error trying to run "C:\Program Files\tool.exe", try running & "C:\Program Files\tool.exe" instead.
### Installation
You'll need to install two pieces of software for signing APKs:
1. JDK (Java Development Kit), which contains the keytool utility
2. Android Studio, which contains the Android development tools
You'll also need to take a note of the installation directories. For example on a Windows system, the tools we'll need will be in paths similar to these (but note particulars of version numbers and usernames may be different):
• C:\Program Files\Java\jdk1.8.0_121\bin for keytool
• C:\Users\YourUserName\AppData\Local\Android\sdk\build-tools\26.0.2 for zipalign and apksigner
In the following steps, we'll shorten the full paths to path\to\jdk and path\to\android\build-tools for brevity. When you see these, you'll need to replace them with the full paths. Don't forget to surround them in double-quotes if they contain a space, and in PowerShell, if you use double quotes also prefix it with &.
### Create a key
You also need to create a private key to sign the APK with. You'll also need to create a password for the key. Both the password and the key file must be kept a secret for your app to be secure. Take a note of the password somewhere safe, since you'll need it later, as well as any time you sign an APK in future with the same key.
First create a new empty folder to hold all our signing related files in. Then make sure the command line has that set as its current directory. Normally you can do this with the cd command, e.g.:
cd path\to\folder
To create a key, run the following keytool command:
path\to\jdk\keytool -genkey -v -keystore release-key.jks -keyalg RSA -keysize 2048 -validity 10000 -alias my-alias
You will then be prompted to enter a password for the key, and to confirm it. Take note of this password in a safe place, you'll need it later!
You'll also be asked for information about the publisher of the app, including your name, the name of your organizational unit, the name of your organization, your city or locality, your state or province, and the two-letter country code. These are optional, you can simply press return without entering anything to skip a field. At the end it will ask you to confirm - simply enter y to accept.
You'll also be asked to enter a key password - just press return to make it the same as the previous password. Once completed, you should get a file named release-key.jks in your folder.
### Aligning
Copy your unsigned release APK to the same folder as your release key. For simplicity following these commands, rename it to app.apk. It needs to be aligned, which is an optimisation. This is done with a tool named zipalign. Use the following command to do this.
path\to\android\build-tools\zipalign -v -p 4 app.apk aligned.apk
This creates a new file named aligned.apk in the folder. This is an aligned, but still unsigned, APK file to use for the next step.
### Signing
Now you need to use the release key you created earlier to sign the APK. This is a way of securely proving that you are the author of the app and that it hasn't been tampered with.
The APK is signed with a tool named apksigner. Use the following command:
path\to\android\build-tools\apksigner sign --ks release-key.jks --out signed.apk aligned.apk
You'll need to enter the password you created with your release key earlier. Once completed, you should have a signed release APK in your folder named signed.apk!
You can verify the signing was successful by running:
path\to\android\build-tools\apksigner verify signed.apk
There shouldn't be any errors listed when you run that command. If it says nothing, it's OK.
At this point you'll probably want to rename signed.apk to something more useful, e.g. mygame-release-signed.apk.
### Example
The following commands were used on a Windows 10 PC with a PowerShell prompt. Note that you cannot use these commands directly yourself! They include things like usernames in the path, and specific version numbers. They are provided as an example to help you follow the previous steps.
Creating a release key: (note this command involves a path with a space, so it was wrapped in quotes, and & added so PowerShell understands it)
& "C:\Program Files\Java\jdk1.8.0_121\bin\keytool" -genkey -v -keystore release-key.jks -keyalg RSA -keysize 2048 -validity 10000 -alias my-alias
Aligning APK:
C:\Users\Ashley\AppData\Local\Android\sdk\build-tools\26.0.2\zipalign -v -p 4 app.apk aligned.apk
Signing APK:
C:\Users\Ashley\AppData\Local\Android\sdk\build-tools\26.0.2\apksigner sign --ks release-key.jks --out signed.apk aligned.apk
Verifying APK:
C:\Users\Ashley\AppData\Local\Android\sdk\build-tools\26.0.2\apksigner verify signed.apk
### Tips
Typically most of the work here is one-off setup. Once you have followed these steps once, it's much quicker to sign another APK: you can jump straight to the Aligning section and use the same release key.
Advanced users might want to create a small script to run these commands automatically, which will help make the process quicker.
You can also test release APKs on your device once they're signed - see the previous section Transfer the APK to your device.
### Publishing
You can now upload this APK to the Google Play Console. You'll need to register a Google Play Developer account if you don't have one already, which may also involve a fee.
From here onwards you're using Google's services, so it's best to refer to their official documentation. Take a look at the Google Play Console Help Center for further guidance.
|
2017-11-21 06:22:16
|
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|
http://mathhelpforum.com/latex-help/2194-angstroms-latex-print.html
|
# Angstroms in LaTeX
• March 13th 2006, 05:59 AM
CaptainBlack
Angstroms in LaTeX
Some time ago someone asked how to obtain the symbol for Angstrom units
using LaTeX. Well its not elegant but:
$
{\buildrel _{\circ} \over {\mathrm{A}}}
$
is generated by: {\buildrel _{\circ} \over {\mathrm{A}}}
I suspect some of the braces are redundant.
RonL
• March 13th 2006, 11:04 AM
ThePerfectHacker
Quote:
Originally Posted by CaptainBlack
Some time ago someone asked how to obtain the symbol for Angstrom units
using LaTeX. Well its not elegant but:
$
{\buildrel _{\circ} \over {\mathrm{A}}}
$
is generated by: {\buildrel _{\circ} \over {\mathrm{A}}}
I suspect some of the braces are redundant.
RonL
\buildrel _\circ \over {\mathrm{A}}
Works too.
Problem is this is math forum nobody uses angstroms.
• March 13th 2006, 11:23 AM
TD!
There is a simple command to produce a circle above a letter, but it doesn't look to good above a capital A. Click to see the code:
$\r{A}\r{o}$
This is another one, math-style, and seems to work well for A too:
$\mathring{A}\mathring{o}$
• March 13th 2006, 11:25 AM
ThePerfectHacker
Quote:
Originally Posted by TD!
There is a simple command to produce a circle above a letter, but it doesn't look to good above a capital A. Click to see the code:
$\r{A}\r{o}$
This is another one, math-style, and seems to work well for A too:
$\mathring{A}\mathring{o}$
It is "proper" to unslant your A.
• March 13th 2006, 11:26 AM
CaptainBlack
Quote:
Originally Posted by TD!
There is a simple command to produce a circle above a letter, but it doesn't look to good above a capital A. Click to see the code:
$\r{A}\r{o}$
This is another one, math-style, and seems to work well for A too:
$\mathring{A}\mathring{o}$
$\r{\rm A} \r{\rm o}$
• March 13th 2006, 11:28 AM
TD!
Quote:
Originally Posted by ThePerfectHacker
It is "proper" to unslant your A.
Then use the addition above, still a lot shorter :)
• March 13th 2006, 11:38 AM
CaptainBlack
Quote:
Originally Posted by TD!
Then use the addition above, still a lot shorter :)
I resent reference to my dimunitive stature :p
RonL
• March 13th 2006, 11:40 AM
TD!
Quote:
Originally Posted by CaptainBlack
I resent reference to my dimunitive stature :p
RonL
I'm sorry :D
• January 30th 2009, 07:38 AM
Isa
easier: just use \AA
you don't need mathmode or anything
(Wink) Isa
• January 30th 2009, 01:44 PM
mr fantastic
Quote:
Originally Posted by Isa
easier: just use \AA
you don't need mathmode or anything
(Wink) Isa
$\AA$
• January 30th 2009, 10:35 PM
Constatine11
Quote:
Originally Posted by mr fantastic
$\AA$
$\text{\AA}$
$$\text{\AA}$$
.
• February 1st 2009, 04:58 PM
Isa
hm seems it only does work without mathsurrounding in german version as i use miktex/latex or maybe i installed a package that supports this command
• February 1st 2009, 09:35 PM
Constatine11
Quote:
Originally Posted by Isa
hm seems it only does work without mathsurrounding in german version as i use miktex/latex or maybe i installed a package that supports this command
The tags are only required on this site. In a LateX document it requires no such tags (and the \text{} should not be needed other that in math mode).
.
|
2014-03-09 09:51:36
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https://orbilu.uni.lu/browse?type=datepublished&sort_by=1&order=DESC&rpp=20&etal=3&value=2010&offset=80
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References of "2010" in Complete repository Arts & humanities Archaeology Art & art history Classical & oriental studies History Languages & linguistics Literature Performing arts Philosophy & ethics Religion & theology Multidisciplinary, general & others Business & economic sciences Accounting & auditing Production, distribution & supply chain management Finance General management & organizational theory Human resources management Management information systems Marketing Strategy & innovation Quantitative methods in economics & management General economics & history of economic thought International economics Macroeconomics & monetary economics Microeconomics Economic systems & public economics Social economics Special economic topics (health, labor, transportation…) Multidisciplinary, general & others Engineering, computing & technology Aerospace & aeronautics engineering Architecture Chemical engineering Civil engineering Computer science Electrical & electronics engineering Energy Geological, petroleum & mining engineering Materials science & engineering Mechanical engineering Multidisciplinary, general & others Human health sciences Alternative medicine Anesthesia & intensive care Cardiovascular & respiratory systems Dentistry & oral medicine Dermatology Endocrinology, metabolism & nutrition Forensic medicine Gastroenterology & hepatology General & internal medicine Geriatrics Hematology Immunology & infectious disease Laboratory medicine & medical technology Neurology Oncology Ophthalmology Orthopedics, rehabilitation & sports medicine Otolaryngology Pediatrics Pharmacy, pharmacology & toxicology Psychiatry Public health, health care sciences & services Radiology, nuclear medicine & imaging Reproductive medicine (gynecology, andrology, obstetrics) Rheumatology Surgery Urology & nephrology Multidisciplinary, general & others Law, criminology & political science Civil law Criminal law & procedure Criminology Economic & commercial law European & international law Judicial law Metalaw, Roman law, history of law & comparative law Political science, public administration & international relations Public law Social law Tax law Multidisciplinary, general & others Life sciences Agriculture & agronomy Anatomy (cytology, histology, embryology...) & physiology Animal production & animal husbandry Aquatic sciences & oceanology Biochemistry, biophysics & molecular biology Biotechnology Entomology & pest control Environmental sciences & ecology Food science Genetics & genetic processes Microbiology Phytobiology (plant sciences, forestry, mycology...) Veterinary medicine & animal health Zoology Multidisciplinary, general & others Physical, chemical, mathematical & earth Sciences Chemistry Earth sciences & physical geography Mathematics Physics Space science, astronomy & astrophysics Multidisciplinary, general & others Social & behavioral sciences, psychology Animal psychology, ethology & psychobiology Anthropology Communication & mass media Education & instruction Human geography & demography Library & information sciences Neurosciences & behavior Regional & inter-regional studies Social work & social policy Sociology & social sciences Social, industrial & organizational psychology Theoretical & cognitive psychology Treatment & clinical psychology Multidisciplinary, general & others Showing results 81 to 100 of 1952 1 2 3 4 5 6 7 8 9 10 Das Leistungsstörungs-und Gewährleistungsrecht/Inexécution du contrat et responsabilité pour vices de la chose vendue/ Non-performance and remediesAncel, Pascal Scientific Conference (2010, October 19)Detailed reference viewed: 45 (1 UL) Measuring the interactions among variables of functions over the unit hypercubeMarichal, Jean-Luc ; Mathonet, Pierre in Torra, Vicenc; Narukawa, Yasuo; Daumas, Marc (Eds.) Modeling Decisions for Artificial Intelligence: Proceedings 7th International Conference, MDAI 2010, Perpignan, France, October 27-29, 2010 (2010, October 19)By considering a least squares approximation of a given square integrable function $f\colon[0,1]^n\to\R$ by a multilinear polynomial of a specified degree, we define an index which measures the overall ... [more ▼]By considering a least squares approximation of a given square integrable function $f\colon[0,1]^n\to\R$ by a multilinear polynomial of a specified degree, we define an index which measures the overall interaction among variables of $f$. This definition extends the concept of Banzhaf interaction index introduced in cooperative game theory. Our approach is partly inspired from multilinear regression analysis, where interactions among the independent variables are taken into consideration. We show that this interaction index has appealing properties which naturally generalize the properties of the Banzhaf interaction index. In particular, we interpret this index as an expected value of the difference quotients of $f$ or, under certain natural conditions on $f$, as an expected value of the derivatives of $f$. These interpretations show a strong analogy between the introduced interaction index and the overall importance index defined by Grabisch and Labreuche [7]. Finally, we discuss a few applications of the interaction index. [less ▲]Detailed reference viewed: 93 (3 UL) Samuel Hirsch between the Particular and the Universal Paradigm: Sociopolitical and Philosophical Evolution of his WeltanschauungMeyers, Christian Presentation (2010, October 19)Detailed reference viewed: 20 (0 UL) Working memory in kindergarten children: Its structure and relationship to fluid intelligence and mathematical abilitiesHornung, Caroline Doctoral thesis (2010)Detailed reference viewed: 75 (5 UL) Peer interactions in the primary classroomMeyer, Anne Doctoral thesis (2010)The present research focuses on peer interactions engaged with the accomplishment of learning activities in the primary classroom. It is driven by the interest and need to understand learning and social ... [more ▼]The present research focuses on peer interactions engaged with the accomplishment of learning activities in the primary classroom. It is driven by the interest and need to understand learning and social interaction taking place in peer group-s, and how the participants orient to the sequential organization of social interaction. The research draws on audio and video data stemming from the primary classroom in Luxembourg, and aims at 1) describing and analyzing the interactional organization of learning activities, 2) describing and analyzing the resources and methods, i.e. expert-novice-practices mobilized by young learners when orienting to the accomplishment of a learning activity, and 3) describing the opportunities for participation and for learning that may take place when learners orient to the accomplishment of a learning activity in peer interaction. Peer interaction is depicted as one form of a community of practice within which learning is situated and observable as learners in and through the deployment of expert-novice-practices orient to, and adapt to micro-shifts in the participation framework when accomplishing a learning activity. Results point to the fact that not only are expert-novice-practices deployed when young learners work in interaction, but these practices are also found to be inextricably linked to the constitution of expert-novice identities - this again has implications for how the learners orient to the accomplishment of a learning activity. [less ▲]Detailed reference viewed: 113 (1 UL) Identité sociale d'étudiant, employabilité et décrochage universitaire de jeunes belges, français, luxembourgeois et roumains, à l'heure de BologneAmara, Marie-Emmanuelle Doctoral thesis (2010)Detailed reference viewed: 56 (2 UL) Internationale Wohnmobilität im ländlichen SaarlandFrys, Wioletta; Nienaber, Birte Scientific Conference (2010, October 14)Detailed reference viewed: 61 (1 UL) Local Languaging: Literacy Products and Practices in Gambian SocietyJuffermans, Kasper Doctoral thesis (2010)Detailed reference viewed: 186 (1 UL) Collaborative Nonlinear Model-Predictive Motion Planning and Control of Mobile Transport Robots for a Highly Flexible Production SystemWangmanaopituk, Suparchoek; Voos, Holger ; Kongprawechnon, Wareein ScienceAsia (2010), 36(4), 333-341This study is based on a new approach for an advanced microproduction system or highly flexible production systems where all necessary production and assembly processes are connected in a very flexible ... [more ▼]This study is based on a new approach for an advanced microproduction system or highly flexible production systems where all necessary production and assembly processes are connected in a very flexible way using autonomous mobile transport and handling robots. Each robot has to follow its planned paths while avoiding collisions with other robots. In addition, problem-specific constraints for a defined microproduction system, such as limitations of the velocity and accelerations of the robots, have to be fulfilled. This paper focuses on a two-level model predictive optimizing approach. On a global long-term level, simple dynamic models of the robots are used to compute optimal paths under differential constraints where a safety distance between all robots is achieved. Since many uncertainties and unforeseen events could occur, all robots also use a nonlinear model predictive control approach on a local real-time level. This control approach solves the path following and the collision avoidance problems in parallel, while also taking into account differential constraints of the single robots. [less ▲]Detailed reference viewed: 84 (5 UL) Evaluation of GNSS as a Tool for Monitoring Tropospheric Water VapourAhmed, Furqan Bachelor/master dissertation (2010)Global Navigation Satellite Systems have the potential to become a significant tool in climate research ... [more ▼]Global Navigation Satellite Systems have the potential to become a significant tool in climate research due to the fact that GNSS data can be processed in order to estimate the propagation delay experienced by the signal in atmosphere. If the ground pressure and temperature is known, the signal propagation path delay can be related to the amount of water vapour in the atmosphere. This thesis project focuses on the evaluation of GNSS as a tool for atmospheric water vapour estimation. In the first part of the project, various GNSS data processing software packages were compared by processing the same set of data and performing a statistical comparison of the estimates of zenith total delay obtained by each package. The software packages compared are GIPSY‐OASIS, Bernese GNSS Processing Software, GAMIT and magicGNSS. Also different strategies and methods, such as double‐differencing and precise point positioning, are investigated. The output from the packages is validated using delay measurements obtained from ECMWF and RCA numerical models. It was observed that the output from climate models agrees with that from the software packages and the output from various software packages have a similarity between each other within 3 millimeters. In the second part of the project, simulations of new GNSS are carried out using in‐house software developed at Chalmers and SP Technical Research Institute of Sweden in order to investigate new methods and possible future improvements. The effect of local errors on atmospheric delay estimates from GPS, GLONASS and Galileo was studied through simulations. A hypothetical system formed by combination of the constellations of GPS, GLONASS and Galileo was also simulated and it was found to be least susceptible to local errors. Simulations were performed by varying some Keplerian orbital elements for Galileo system and it was observed that an orbit inclination between 60 degree and 65 degree would have been optimum for Galileo system. [less ▲]Detailed reference viewed: 148 (15 UL) Pain modulation induced by Heterotopic Noxious Counter-Stimulation (HNCS) : psychophysiological assessment of adequate stimulation paradigms and sex-related effectsStreff, Anouk Doctoral thesis (2010)This work comprises three studies whose main concern was to find a valid tonic pain model able to trigger a genuine diffuse noxious pain inhibition. All studies were performed in healthy, drug-free ... [more ▼]This work comprises three studies whose main concern was to find a valid tonic pain model able to trigger a genuine diffuse noxious pain inhibition. All studies were performed in healthy, drug-free volunteers and whereas the first two are validation studies, the third is an application study of the previous two. The aim of the first study was to characterize the cold pressor (CPT) and hot water immersion test (HIT) from a physiological and a psychophysical point of view. A second issue was to clarify the origin of potential autonomic responses during both tests; are they related to baroreflex activity or rather a consequence of the pain experience per se? The study was performed in 30 volunteers aged 19-57 (median 24) years, and consisted of a single session including one CPT (4 ± 0.2°C) and one HIT (47 ± 0.5°C) with a cut-off-point of 5 minutes. Participants were randomly assigned to sequence order (the sequence of both trials was alternated) and groups were paralleled with respect to gender. Cardiovascular, respiratory and electrodermal activities as well as subjective pain intensity were continuously monitored. Pain detection and tolerance thresholds as well as pain unpleasantness and nervous tension were assessed additionally. Both tests were found to be comparable with respect to intensity of subjective pain and time course, but a significantly higher blood pressure increase during CPT could be observed, compared to the HIT. In conclusion, the HIT appears to be less confounded with baroreflex activity and hence seems to be a more adequate tonic pain model. The second study tested the internal validity of inter-digital web pinching (IWP) with regard to its potential as DNIC-trigger. 24 gender-matched participants, aged 21-54 (median 25) years, volunteered for the controlled study. The protocol included the assessment of thermal and mechanical perceptual wind-up (WU) before and after a HIT (47.5 °C) or an IWP (15 N) of 2 minutes duration each. WU pain was induced by 10 repetitive (1 Hz) contact heat (max. 49°C; 5 5 mm thermode) or 10 ballistic impact stimuli (0.5 g at 9m/s) on the phalanges of the non-dominant hand. Cardiovascular and corrugator muscle activity as well as pain experience were permanently monitored. Both heterotopic noxious counter-stimulation (HNCS) types produced a similar pain experience, but a more pronounced cardiovascular activity was observed for the HIT. Painful water immersion is though accompanied by a stronger baroreceptor activity. WU pain was significantly reduced for both pain modalities, although the inhibition was somewhat stronger for the HIT than the IWP. The IWP, being practically uncontaminated by baroreflex sensitivity (BRS), proved its validity as DNIC-trigger. The third study investigated temporal characteristics of electrically elicited pain and nocifensive RIII-reflex activity in a gender-balanced sample of 28 volunteers aged 21-38 (median 27) years, using IWP as HNCS, a tonic pain model previously validated to be BRS-unrelated. Sex-related differences in the post HNCS time courses of pain perception were identified with women demonstrating a more rapid return to baseline compared to men. Interestingly, an opposite pattern was observed regarding nociceptive reflex activity with a steeper return rate of electromyographic responses in males, whereas those of women remained attenuated over the entire observation period. These findings may reflect a stronger defensive response to pain in women. [less ▲]Detailed reference viewed: 154 (5 UL) Toward an Integral Pluralism in Sociocultural Research-Theme analysis of research biographies and integrative frameworksMolz, Markus Doctoral thesis (2010)Detailed reference viewed: 97 (1 UL) Shoppers, Tourists & Socialism: Czechoslovakia, Romania and Yugoslavia (1950s-1980s)Stefan, Oana Adelina ; Kladnik, AnaPresentation (2010, October 10)Detailed reference viewed: 56 (0 UL) State Aid, Subsidy and Tax Incentives under EU and WTO LawMicheau, Claire Doctoral thesis (2010)Detailed reference viewed: 194 (10 UL) Itinéraires luxembourgeois et européens.Evolutions et souvenirs: 1945-1985Werner, Pierre; Danescu, Elena ; Danescu, Elena Book published by GNOSIS - 3th edition. Fully revised and corrected edition (2010)lDetailed reference viewed: 102 (11 UL) 'A New Medea' in late medieval French narrativesLeglu, Catherine in Simon, Anne; Bartel, Heike (Eds.) Unbinding Medea : Interdisciplinary approaches to a classical myth from Antiquity to the 21st century (2010)Within a themed volume dedicated to the reception and adaptation of the story of Medea in many media and genres, this chapter presents texts and images of the 15th century that point to a tendency to ... [more ▼]Within a themed volume dedicated to the reception and adaptation of the story of Medea in many media and genres, this chapter presents texts and images of the 15th century that point to a tendency to interpret Medea as a tragic heroine rather than a monstrous mother. [less ▲]Detailed reference viewed: 79 (0 UL) Numerische Aspekte bei der Berechnung von GRACE Schwerefeldern mit Hilfe des Ansatzes der differentiellen GravimetrieWeigelt, Matthias ; Keller, W.Poster (2010, October)Detailed reference viewed: 29 (0 UL) Das ‚Erlernen’ von globaler Weinkultur anhand regionalen Weinen. Normen und Aneignungen einer sensorischen ExpertiseReckinger, Rachel Scientific Conference (2010, October)Das aktuelle Spannungsfeld der Globalisierung des Essens- und Getränkeangebot sowie des gleichzeitigen Bedeutungsgewinns von regionalen (Nischen-)Produkten erscheint besonders deutlich am Beispiel vom ... [more ▼]Das aktuelle Spannungsfeld der Globalisierung des Essens- und Getränkeangebot sowie des gleichzeitigen Bedeutungsgewinns von regionalen (Nischen-)Produkten erscheint besonders deutlich am Beispiel vom Wein. Einerseits lässt sich, auf der Seite der Produktion, eine Standardisierung der Herstellungsweisen und eine Konzentration der Besitzverhältnisse, insbesondere in den ‚neuen’ Weinländern – aber gleichzeitig, wenngleich in andersartiger Form, in den ‚alten’ Weinländern – ausmachen. Parallel dazu werden ‚Terroir’-orientierte Weinproduzenten als Hüter von valorisiertem, vermeintlich tradiertem Wissen und als Boten einer letztlich geografisch determinierten Geschmackspalette konstruiert. Andererseits ist, auf der Seite des Konsums, seit den 1980er Jahren eine ständig steigende Tendenz zur Ästhetisierung vom Weingenuss zu verzeichnen, der durch das degustative, mitunter mikro-parzellarische, Weinwissen von einem (publizistischen) Expertentum rationalisiert wird. So erleben alltagsweltliche Weintrinker/-innen diese Tätigkeit als einen Bereich, der, um in seinen Potenzialitäten ausgeschöpft werden zu können, nur mittels Wissens zugänglich sei. Dementsprechend steigt die Anzahl an Personen, die an Weinverkostungskursen ‚für Anfänger/-innen’ teilnehmen. Die drei Kurse in Luxemburg, die durch teilnehmende Beobachtung und mittels Experten/-innengesprächen mit den Kursleitern/-innen und Tiefeninterviews mit Teilnehmenden analysiert wurden (Reckinger 2008), zeichnen sich durch jeweils unterschiedliche – jedoch engagierte – Positionierungen in Bezug auf globalisierte und regionale Produkttypen aus. Dabei erscheinen Transnationalität, Regionalisierung und Einzigartigkeit (der Winzer/-innen sowie der geografischen Gegebenheiten) als performative Werte, die jedoch ausschliesslich durch ,autonomes’ sensorisches Wahrnehmen und Einordnen zu erschliessen seien, d.h. durch Subjektivität und Kontingenz. Dieses normative Angebot wird von den Kursteilnehmenden ambivalent rezipiert, je nachdem wie selbstzentriert, soziabilitätszentriert oder objektzentriert sie motiviert sind. Letzteres wird von den Kursleitern/-innen als Faktor zur Evaluierung von Qualität(en) hervorgehoben – seien sie standardisiert, pluralisiert oder hybridisiert. Demgegenüber bleibt das alltagsweltliche polysensorische Erleben einer alternativen Normativität von Identifikationspotential verhaftet. [less ▲]Detailed reference viewed: 112 (1 UL) Eco-systems biology: a new frontier in microbiologyWilmes, Paul Scientific Conference (2010, October)Detailed reference viewed: 36 (0 UL) Zusammenhänge zwischen der subjektiven zirkadianen Phasenlage und psychischen Variablen.Specht, M.; Kemper, Christoph ; Bongard, S. et alPoster (2010, October)Detailed reference viewed: 41 (0 UL)
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2022-08-15 13:23:28
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http://qhse-madagascar.com/article/vapour-pressure-formula-044e70
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# vapour pressure formula
The pressure lowering of the water is PX' as P stands for the pressure of pure solvent and X' is the molar fraction of the solute. Our question is: Note that, for Clausius-Clapeyron equations, you must always use, In our example, let's say that our liquid is, Plugging our constants in to our equation, we get, The only difficult part of solving our equation (, ln(1/P2) = (40,650/8.314)((1/393) - (1/295)). The Clausius-Clapeyron can help here — use the reference vapor pressure and 298 K (25 C) for P1 and T1 respectively. which explains how the vapour pressure of a liquid gets changed by the addition of a solute. She received her MA in Environmental Science and Management from the University of California, Santa Barbara in 2016. The vapour pressure formed from this solution is lowered by the addition of the solute. The vapor pressure of pure water is 47.1 torr at 37°C We can do this as follows, using standard density, molar mass, and vapor pressure values for our two chemicals: Mass (benzene): 60 mL = .060 L × 876.50 kg/1,000 L = 0.053 kg =, Mass (toluene): .060 L × 866.90 kg/1,000 L = 0.052 kg =, Moles (benzene): 53 g × 1 mol/78.11 g = 0.679 mol, Moles (toluene): 52 g × 1 mol/92.14 g = 0.564 mol, Mole fraction (benzene): 0.679/1.243 = 0.546, Mole fraction (toluene): 0.564/1.243 = 0.454. Now, take this number, and convert moles to atoms. The basic form of the equation is: So the pressure lowering is 760mmHg times 1.768.10^-2, which is ~ 13.44 mmHg. For information on how to find the vapor pressure of dissolved solutions, read on! The temperature of the solution is 25 C and the vapor pressures of each of these chemicals at 25 C is 95.1 mm Hg for benzene 28.4 mm Hg for toluene. At an ambient pressure of 25 degrees Celsius, the vapor pressure of water is 23.8 torr. We know ads can be annoying, but they’re what allow us to make all of wikiHow available for free. The simplest way to determine $$ΔH_{vap}$$ is to measure the vapor pressure of a liquid at two temperatures and insert the values of $$P$$ and $$T$$ for these points into Equation $$\ref{Eq2}$$, which is derived from the Clausius–Clapeyron equation: Don't worry if you don't know terms like "mole fraction" — we'll explain these in the next few steps. (0.083257 mole) * (6.022 * 10^23 atoms)/(1 mole) = 0.083257 * 6.022 * 10^23 = 5.01374 * 10^22 atoms. Those evaporated particles will create a pressure above the liquid, which is then known as the, A solution is created when a solid gets dissolved into the liquid. 1L of water has 1000g of water, so there are 1000/18 mols of water ~ 55.6 mols. Required fields are marked *, Whenever the liquid evaporates, the gaseous molecules formed will escape in the air. Here comes the Vapour pressure formula using Raoult’s law, which explains how the vapour pressure of a liquid gets changed by the addition of a solute. Plug all of the known variables and constants into the equation, and isolate the unknown variable, which will be the pressure. The vapour pressure of water is 25.756 mm Hg at 25 °C. By using this service, some information may be shared with YouTube. You can use the Antoine's equation to calculate the vapor pressure of any substance and any temperature. She has conducted survey work for marine spatial planning projects in the Caribbean and provided research support as a graduate fellow for the Sustainable Fisheries Group. The vapor pressure at a convex curved surface is higher than that at a flat surface. If you have the temperature in Centigrade, then you need to convert it with the following formula: The methods above work because energy is directly proportional to the amount of heat supplied. The total volume of the solution is 120 milliliters (mL); 60 mL of benzene and 60 of toluene. {"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/9\/92\/Calculate-Vapor-Pressure-Step-1-Version-2.jpg\/v4-460px-Calculate-Vapor-Pressure-Step-1-Version-2.jpg","bigUrl":"\/images\/thumb\/9\/92\/Calculate-Vapor-Pressure-Step-1-Version-2.jpg\/aid4527638-v4-728px-Calculate-Vapor-Pressure-Step-1-Version-2.jpg","smallWidth":460,"smallHeight":345,"bigWidth":"728","bigHeight":"546","licensing":"
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2022-12-05 11:08:10
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https://www.atmos-chem-phys.net/19/11651/2019/
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Journal topic
Atmos. Chem. Phys., 19, 11651–11668, 2019
https://doi.org/10.5194/acp-19-11651-2019
Atmos. Chem. Phys., 19, 11651–11668, 2019
https://doi.org/10.5194/acp-19-11651-2019
Research article 17 Sep 2019
Research article | 17 Sep 2019
# Characterization of aerosol hygroscopicity using Raman lidar measurements at the EARLINET station of Payerne
Characterization of aerosol hygroscopicity using Raman lidar measurements at the EARLINET station of Payerne
Francisco Navas-Guzmán1, Giovanni Martucci1, Martine Collaud Coen1, María José Granados-Muñoz2, Maxime Hervo1, Michael Sicard2,3, and Alexander Haefele1,4 Francisco Navas-Guzmán et al.
• 1Federal Office of Meteorology and Climatology MeteoSwiss, Payerne, Switzerland
• 2Remote Sensing Laboratory/CommSensLab, Universitat Politècnica de Catalunya, 08034 Barcelona, Spain
• 3Ciències i Tecnologies de l'Espai – Centre de Recerca de l'Aeronàutica i de l'Espai/Institut d'Estudis Espacials de Catalunya (CTE-CRAE/IEEC), Universitat Politècnica de Catalunya, 08034 Barcelona, Spain
• 4Department of Physics and Astronomy, University of Western Ontario, London, Canada
Correspondence: Francisco Navas Guzmán (francisco.navas@meteoswiss.ch)
Abstract
This study focuses on the analysis of aerosol hygroscopicity using remote sensing techniques. Continuous observations of aerosol backscatter coefficient (βaer), temperature (T) and water vapor mixing ratio (r) have been performed by means of a Raman lidar system at the aerological station of MeteoSwiss at Payerne (Switzerland) since 2008. These measurements allow us to monitor in a continuous way any change in aerosol properties as a function of the relative humidity (RH). These changes can be observed either in time at a constant altitude or in altitude at a constant time. The accuracy and precision of RH measurements from the lidar have been evaluated using the radiosonde (RS) technique as a reference. A total of 172 RS profiles were used in this intercomparison, which revealed a bias smaller than 4 % RH and a standard deviation smaller than 10 % RH between both techniques in the whole (in lower) troposphere at nighttime (at daytime), indicating the good performance of the lidar for characterizing RH. A methodology to identify situations favorable to studying aerosol hygroscopicity has been established, and the aerosol hygroscopicity has been characterized by means of the backscatter enhancement factor (fβ). Two case studies, corresponding to different types of aerosol, are used to illustrate the potential of this methodology.
The first case corresponds to a mixture of rural aerosol and smoke particles (smoke mixture), which showed a higher hygroscopicity (${f}_{\mathit{\beta }}^{\mathrm{355}}=\mathrm{2.8}$ and ${f}_{\mathit{\beta }}^{\mathrm{1064}}=\mathrm{1.8}$ in the RH range 73 %–97 %) than the second case, in which mineral dust was present (${f}_{\mathit{\beta }}^{\mathrm{355}}=\mathrm{1.2}$ and ${f}_{\mathit{\beta }}^{\mathrm{1064}}=\mathrm{1.1}$ in the RH range 68 %–84 %).
The higher sensitivity of the shortest wavelength to hygroscopic growth was qualitatively reproduced using Mie simulations. In addition, a good agreement was found between the hygroscopic analysis done in the vertical and in time for Case I, where the latter also allowed us to observe the hydration and dehydration of the smoke mixture. Finally, the impact of aerosol hygroscopicity on the Earth's radiative balance has been evaluated using the GAME (Global Atmospheric Model) radiative transfer model. The model showed an impact with an increase in absolute value of 2.4 W m−2 at the surface with respect to the dry conditions for the hygroscopic layer of Case I (smoke mixture).
1 Introduction
Atmospheric aerosol particles scatter and absorb solar radiation and therefore have an impact on the Earth's radiative budget (direct effect). In addition, aerosol particles can act as cloud condensation nuclei (CCN) and modify cloud microphysical properties, also altering the global radiative budget (indirect effects) in this way . The uncertainty in assessing total anthropogenic greenhouse gas and aerosol impacts on climate must be substantially reduced from its current level to allow meaningful predictions of future climate. This uncertainty is currently dominated by the aerosol component . Evaluation of aerosol effects on climate must take into account high spatial and temporal variation of aerosol concentrations and properties as well as the aerosol interactions with clouds and its influence on precipitation. During the last years a huge effort was made to characterize vertically resolved profiles of optical and microphysical properties for different kinds of particles. Raman lidars (light detection and ranging) have proven to be an essential tool to obtain profiles of these properties without modifying the environmental conditions . Networks like EARLINET (European Aerosol Research Lidar Network) have contributed to create a quantitative, comprehensive, and statistically significant database for the horizontal, vertical, and temporal distribution of aerosols on a continental scale .
However, despite the big effort of the scientific community to characterize aerosol effects, there are some processes that are not well understood yet. For example, an important factor that can modify the role of aerosols in the global energy budget is RH. Under high RH conditions, aerosol particles may uptake water, changing their size and chemical composition (hygroscopic growth). This hygroscopic growth affects the direct scattering of radiation and especially the indirect effects, as the ability of aerosol to act as CCN is directly related to their affinity for water vapor . Thus, understanding aerosol hygroscopic growth is of high importance to quantify the influence of atmospheric aerosol in climate models or for comparisons of remote sensing with in situ measurements which are often performed under dry conditions .
Several studies have addressed the characterization of aerosol hygroscopicity using in situ measurements over the last years. Techniques such as with the Humidified Tandem Differential Mobility Analyzer (HT-DMA) (e.g., Swietlicki et al.2008) or humidified tandem nephelometers have been extensively used to quantify the change in particle diameter or aerosol optical properties due to water uptake . However, despite the great contribution that these techniques can provide to the understanding of the aerosol hygroscopic processes, they also present some shortcomings. For example, most in situ techniques are limited by the fact that they modify the ambient conditions and are also subject to particle losses in the sampling lines, thereby altering the real atmospheric aerosol properties . In addition, they present larger errors in the characterization of the aerosol hygroscopicity for high RH (conditions close to saturation), especially when the aerosol is more hygroscopic .
In this sense, remote sensing techniques could overcome these difficulties since they can provide vertically resolved measurements without modifying the aerosol sample. In addition, they are also able to measure under RH close to saturation, that is, where particles are more affected by hygroscopic growth . However, the number of aerosol hygroscopic studies using remote sensing is modest, and most of them were limited to specific field campaigns. The main limitation to addressing these studies comes from the difficulty in obtaining RH profiles with high vertical and temporal resolution. Most of the studies carried out so far have used sensors to measure RH onboard RSs or aircrafts or in meteorological towers in combination with the aerosol measurements from lidars to investigate the aerosol hygroscopicity . Recently, the combination of T profiles from microwave radiometers (MWRs) and r profiles from Raman lidar has been used to retrieve RH profiles and applied to aerosol hygroscopic studies . However, although this synergy of instrumentation can be a good solution for many stations, the uncertainties coming from MWR T profiles can be problematic due to their lower spatial resolution . In addition, most of the r measurements from the Raman lidar system are limited to nighttime observations, due to the low signal-to-noise ratio (SNR) of the Raman channels during daytime.
In the present study we show the capability of the RAman Lidar for Meteorological Observations (RALMO) operated at the aerological station of MeteoSwiss at Payerne (Switzerland) to monitor aerosol hygroscopicity based on its continuous aerosol and RH measurements. The methodology needed for this characterization is introduced and applied to two case studies in which different aerosol types were present.
2 Experimental site and instrumentation
Data from collocated remote sensing and in situ sensors were acquired at the aerological station of MeteoSwiss at Payerne (Switzerland, 46.82 N, 6.95 E; 491 m above sea level (a.s.l.)). The station is located in a rural area on the Swiss Plateau, between the Jura mountains (25 km to the northwest) and the Alpine foothills (20 km to the southeast). The site is not significantly affected by industrial pollution and the region is characterized by a mid-latitude continental climate.
Aerosol vertical information is mainly obtained from lidar systems installed at the station. RALMO is the main tool of this study and was developed by the Swiss Federal Institute of Technology (EPFL) in collaboration with MeteoSwiss and has been operated at MeteoSwiss Payerne since August 2008. The instrument is dedicated to operational meteorology, model validation, climatological studies as well as ground truthing of satellite data. The lidar was specially designed to satisfy the requirements for operational monitoring of the atmosphere and to create a durable and homogeneous dataset to be used for climatology studies . The lidar system uses a Nd:YAG laser source which emits pulses of 8 ns duration at a wavelength of 355 nm and with a repetition rate of 30 Hz. The mean energy per pulse at 355 nm is around 400 mJ. Before being emitted into the atmosphere the beam is expanded to a diameter of 14 cm which reduces the beam divergence (to 0.1 mrad) and ensures eye safety. The receiving system consists of four telescopes with 30 cm parabolic mirrors which are arranged symmetrically around the vertically mounted beam expander to receive the backscattered photons. The total aperture of this telescope system is equivalent to a telescope with a single mirror of 60 cm diameter, and it has a field of view of 0.2 mrad. This narrow field of view combined with the narrow-band receiver and high pulse energy allows daytime operation. Optical fibers connect the telescope mirrors with two grating polychromators which allow us to isolate the rotational–vibrational Raman signals of nitrogen and water vapor (wavelengths of 386.7 and 407.5 nm, respectively) as well as the elastic signal and four portions of the pure-rotational Raman spectrum around 355 nm for T, aerosol backscatter and extinction measurements. The optical signals are finally detected by photomultipliers and acquired by a transient recorder . A detailed description of the different parts of this lidar system can be found in . RALMO was incorporated into EARLINET in 2008.
In order to gain more spectral information in the profiles of aerosol properties, data from a co-located ceilometer have been used in this study. The CHM15k ceilometer from the Lufft company operated at Payerne is a lidar cloud height sensor based on a single wavelength backscattering technique. Its Nd:YAG narrow-beam microchip laser operates at 1064 nm. Cloud layers can be detected in a range of up to 15 km. In addition to the cloud base information, we derive βaer at 1064 nm from the elastic signal using the Klett inversion technique (Klett1981).
RS measurements were also used in this study to assess the T, r and RH profiles retrieved from RALMO. The SRS-C50 sondes were flown operationally at Payerne and launched twice a day at 11:00 and 23:00 UTC. The sondes were launched and transported through the troposphere and stratosphere by a balloon inflated with hydrogen and reaching on average 35 km. The vertical resolution of the measured profiles of T and humidity is about 6 m. The sensors of these RSs include copper–constantan thermocouples for T, a full-range water hypsometer for pressure and a sensor with a hygristor for RH. The accuracy of these three parameters in the troposphere is 0.1 K for T, 2 hPa (accuracy decreases with height) for pressure and 5 % to 10 % for RH.
Sun-photometer measurements have also been included in this study to complement the aerosol information. MeteoSwiss operates four Precision Filter Radiometers (PFRs) that were installed in the framework of the international GAW-PFR (Global Atmosphere Watch – Precision Filter Radiometer) program of the World Meteorological Organization (WMO). They measure the direct solar irradiance at the four wavelengths (368, 412, 500 and 862 nm) used in the GAW-PFR network as well as nine additional wavelengths (305, 311, 318, 332, 450, 610, 675, 718, 778, 817, 946 and 1024 nm). Integrated water vapor (IWV) is obtained from the measurements at 718, 817 and 946 nm. The aerosol optical depths (AODs) have been calculated based on the solar irradiance at the different wavelengths using the GAW-PFR algorithms . In addition, the AOD Ångström exponent (AE) is used in this study as a qualitative indicator of the relative dominance of fine- and coarse-mode aerosols. Values of AE larger than 1.5 are indicative of a higher fine-mode fraction .
In situ observations at ground level were also used to complement the column aerosol information obtained from remote sensing. These in situ measurements at Payerne are carried out by EMPA in the framework of the Nabel (National Air Pollution Monitoring Network) monitoring program. Aerosol absorption coefficients at seven wavelengths (from 370 to 950 nm) were obtained from aethalometer measurements (AE31 model). Light absorption measurements at ultraviolet, visible and infrared wavelengths can be used to quantitatively assess the aerosol source contribution . In addition, concentrations of PM10 and PM2.5 (ambient air levels of atmospheric particulate matter finer than 10 and 2.5 microns) were obtained from an aerosol spectrometer (Fidas 200). These measurements were used to characterize the size of the particles that arrived at our station.
3 Methodology
## 3.1 Retrievals of RH and aerosol property profiles
As mentioned in the introduction, the main practical limitation to studying the effect of hygroscopicity on optical and microphysical aerosol properties is the lack of simultaneously and continuously available aerosol and RH measurements with a good vertical and temporal resolution. RALMO can overcome this limitation thanks to its capability to perform continuous measurements of T, water vapor and aerosol profiles during day and night. In this section the methodology to retrieve these atmospheric parameters from the lidar signals is described.
T measurements are made using the pure-rotational Raman (PRR) technique . An atmospheric T profile can be derived from the analysis of the intensities of lines with low and high quantum numbers that have opposite T derivatives. A detailed description of the T inversion technique applied to our system can be found in .
r is defined as the ratio of the mass of water vapor to the mass of dry air in a sample of the atmosphere . Profiles of r can be obtained from Raman lidar measurements as the ratio of rotational–vibrational Raman scattering intensities from water vapor and nitrogen molecules . It can be expressed as follows:
$\begin{array}{}\text{(1)}& r\left(z\right)=C\frac{{S}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}}{{S}_{{\mathrm{N}}_{\mathrm{2}}}}\mathrm{exp}\left\{\underset{\mathrm{0}}{\overset{z}{\int }}\left[\mathit{\alpha }\left(r,{\mathit{\lambda }}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}\right)-\mathit{\alpha }\left(r,{\mathit{\lambda }}_{{\mathrm{N}}_{\mathrm{2}}}\right)\right]\text{d}r\right\},\end{array}$
where C is the lidar calibration coefficient that takes into account the fractional volume of nitrogen in the atmosphere (0.78), the instrumental transmission and detection efficiencies at the wavelengths of the Raman returns and the range-independent Raman backscatter cross section for nitrogen and water vapor. The calibration coefficient must be determined for each specific lidar. In the case of RALMO, C is obtained by combining absolute calibration with co-located radiosounding, with relative calibration using the solar background . ${S}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ and ${S}_{{\mathrm{N}}_{\mathrm{2}}}$ are the Raman lidar signals for water vapor and nitrogen, respectively, after being corrected for saturation and background. The exponential term is the difference in the atmospheric transmission between the surface and the altitude z for nitrogen (386.7 nm) and water vapor (407.5 nm). The molecular extinction is calculated considering the US Standard Atmosphere. For normal conditions at Payerne the effect of differential extinction due to aerosols between these two wavelengths is small and is neglected . However, when this differential extinction is not negligible, showed that it can accurately be calculated using a radiative transfer model with a relatively simple parametrization.
RH profiles are obtained by combining T and r. RH is defined as the ratio of the actual amount of water vapor in the air compared to the equilibrium amount (saturation) at that T , and it can be calculated as
$\begin{array}{}\text{(2)}& \text{RH}\left(z\right)=\frac{e\left(z\right)}{{e}_{\text{w}}\left(z\right)}×\mathrm{100},\end{array}$
where e(z) is the water vapor pressure, while ew(z) refers to the saturation pressure. The water vapor pressure is related to r as follows:
$\begin{array}{}\text{(3)}& e\left(z\right)=\frac{p\left(z\right)r\left(z\right)}{\mathrm{0.622}+r\left(z\right)},\end{array}$
where p(z) is the air pressure profile which can be estimated from RSs or assuming a US standard atmosphere. In our case, and thanks to the availability of operational RS measurements in our station, the closest RS in time is used to calculate the pressure profile. For saturation pressure we use the following expression:
$\begin{array}{}\text{(4)}& {e}_{\text{w}}\left(z\right)=\mathrm{6.107}\mathrm{exp}\left[\frac{{M}_{\text{A}}\left[T\left(z\right)-\mathrm{273}\right]}{{M}_{\text{B}}+\left[T\left(z\right)-\mathrm{273}\right]}\right],\end{array}$
where the constants MA and MB are 17.84 (17.08) and 245.4 (234.2), respectively, for T below (above) 273 K (List1951).
Aerosol vertical information is also retrieved from RALMO measurements. The backscattered radiation from aerosols and molecules due to Mie and Rayleigh scattering, respectively, has the same wavelength as the laser and is referred to as the elastic signal, SEl(z). Stokes and anti-Stokes portions of the PRR spectrum with opposite T dependence are summed, giving in good approximation a T-independent inelastic signal SPRR(z). βaer is derived from the ratio between the elastic and inelastic signals as follows:
$\begin{array}{}\text{(5)}& {\mathit{\beta }}^{\text{aer}}\left(z\right)={\mathit{\beta }}^{\text{mol}}\left(z\right)\left[k\frac{{S}_{\text{El}}\left(z\right)}{{S}_{\text{PRR}}\left(z\right)}-\mathrm{1}\right],\end{array}$
where k is the calibration constant and βmol is the molecular backscatter coefficient. βmol is calculated using a measured atmospheric density, while the k constant is estimated assuming a molecular behavior of the atmosphere in the far range (upper troposphere).
## 3.2 Selection of aerosol hygroscopic cases
Continuous measurements of aerosol and RH profiles from RALMO lidar allow us to monitor any change in aerosol properties that could occur as a result of the water uptake by particles under high RH (aerosol hygroscopic growth). However, to ensure that the changes in the aerosol properties are due to hygroscopic growth and not to changes in the load or composition of the aerosol, certain requirements must be fulfilled.
As a first condition an increase in βaer should occur simultaneously with an increase in RH. This condition could be observed in the vertical for a certain aerosol layer (vertical hygroscopic growth), but it could also be found as a function of time at a constant altitude. It is worth remarking that RALMO is one of the few remote sensing instruments in the world that is able to monitor those processes as a function of time and height. Cases fulfilling the previous condition were selected as potential cases of hygroscopic growth. After that, we needed to ensure a high degree of homogeneity in the investigated aerosol layer. For that, a second requirement to be fulfilled is that the origin of the air masses, in the region where the presence of hygroscopicity is assessed, is independent of altitude (in the case of vertical hygroscopic growth). To guarantee that, we have used backward trajectory analysis from the HYSPLIT model (Hybrid Single-Particle Lagrangian Integrated Trajectory) . As a final condition to ensure good mixing within the layer, we used the profiles of r and potential temperature (θ) obtained from RALMO measurements. Slow-varying or constant values of these two parameters are indicative of well-mixed conditions across the aerosol layer.
Similar criteria to assess the homogeneity of an aerosol layer were required in previous studies using remote sensing . However, this is the first time that the profiles of r and θ are obtained from the same instrument as the aerosol measurements. It presents a clear advantage since all the parameters are measured for the same atmospheric column, as opposed to studies using RSs or MWR.
Another novelty of this study is the capability to monitor aerosol hygroscopic growth in time at different altitudes in the troposphere. For this temporal analysis we used the r measurements to ensure that changes in aerosol properties in time are only due to hygroscopic processes. In the absence of condensation and evaporation processes r can be considered an atmospheric tracer and we assume that air parcels with the same r have in good approximation the same origin and hence the same aerosol load and composition. In addition, wind measurements from a collocated wind profiler are used to corroborate that the wind direction was the same during the analyzed period.
Once all the previous requirements have been fulfilled, the aerosol hygroscopicity is characterized by means of the fβ. This parameter is defined as
$\begin{array}{}\text{(6)}& {f}_{\mathit{\zeta }}\left(\text{RH}\right)=\frac{\mathit{\zeta }\left(\text{RH}\right)}{\mathit{\zeta }\left({\text{RH}}_{\text{ref}}\right)},\end{array}$
where ζ(RH) represents an aerosol property at a certain RH. RHref is the so-called reference RH, and it is chosen as the lowest value of RH in the analyzed layer or time interval. In this study, fβ has been calculated for βaer.
In order to be able to compare our results with other studies in which RHref or the RH range could be different, the humidograms (RH versus the fβ) are parameterized using fitting equations (e.g., Titos et al.2016). In this study we use the one-parameter equation introduced by , which has been used in other hygroscopic studies using remote sensing . The general form of the Hänel equation is expressed as
$\begin{array}{}\text{(7)}& {f}_{\mathit{\zeta }}\left(\text{RH}\right)={\left(\frac{\mathrm{1}-\text{RH}/\mathrm{100}}{\mathrm{1}-{\text{RH}}_{\text{ref}}/\mathrm{100}}\right)}^{-\mathit{\gamma }},\end{array}$
where γ is an indicator of the aerosol hygroscopicity. Larger values of γ indicate higher hygroscopicity.
4 Validation of lidar measurements versus operational RSs
Since data quality is critical, a validation with respect to the RS technique has been carried out, which we consider the reference. As was indicated in Sect. 2, operational RSs are launched twice per day at the aerological station of Payerne. The availability of simultaneous lidar and RS measurements at our station allowed us to minimize the differences due to spatio-temporal inhomogeneities.
Figure 1Mixing ratio (r), temperature (T) and relative humidity (RH) profiles from lidar (blue lines) and operational RS (red line) at 23:00 UTC on 20 September 2017.
Figure 1 shows the r, T and RH profiles obtained from RALMO lidar and RS at nighttime (23:00 UTC) on 20 September 2017. A very good agreement can be observed from this figure for the three atmospheric parameters. We can see that RALMO provided very accurate results even for altitude ranges where strong gradients were observed, as for the wet layer located above 2 km above ground level (a.g.l.) or for the T inversion observed at 1.7 km (a.g.l.). The mean and standard deviation for the whole profile (from ground to 8 km (a.g.l.)) of the relative differences in r were $-\mathrm{1}\phantom{\rule{0.125em}{0ex}}\mathit{%}±\mathrm{13}\phantom{\rule{0.125em}{0ex}}\mathit{%}$, while the mean and standard deviation of T and RH differences were $-\mathrm{0.1}±\mathrm{0.7}$ K and $-\mathrm{1}\phantom{\rule{0.125em}{0ex}}\mathit{%}±\mathrm{6}\phantom{\rule{0.125em}{0ex}}\mathit{%}$ RH, respectively. This example shows the potential of RALMO to provide accurate measurements with a good spatial resolution. In order to evaluate the accuracy and the precision of RALMO retrievals, a statistical analysis of lidar and RS differences was carried out. A total of 172 RS were used in this intercomparison during daytime and nighttime for the period from July to December of 2017. The T validation of RALMO lidar is discussed in more depth in a separate paper . Here we would only like to discuss the statistics for this 6-month period.
Figure 2RH validation between RALMO lidar and RS at 23:00 UTC for the period from July to December 2017. (a) Profiles of RH deviation between lidar and RS. (b) Mean RH deviation (lidar–RS). (c) Standard RH deviation profile (red line) and number of profiles used at each altitude for these statistics (green line).
For nighttime measurements (23:00 UTC), a total of 100 RS were used to compare with lidar and the mean T deviation profile evidenced a small bias of 0.05±0.06 K in the first 5 km and 0.15±0.15 K above that altitude. The standard deviation profile also confirmed the excellent performance of RALMO in retrieving T, with standard deviations below 1 K in the full troposphere (mean values of 0.6±0.1 K below 5 km (a.g.l.) and 1.00±0.16 K above). For daytime measurements (11:00 UTC), 72 pairs of profiles were used. The mean T deviations between lidar and RS for daytime measurements were $-\mathrm{0.5}±\mathrm{0.2}$ K in the lower troposphere (0–5 km a.g.l.) and $-\mathrm{0.1}±\mathrm{0.6}$ in the upper troposphere (5–10 km a.g.l.). The T standard deviations were 0.8±0.2 and 2.4±0.8 for the same two altitude ranges. These results also prove the good performance of RALMO during daytime, although they show larger discrepancies than during nighttime, especially in the upper troposphere (from 5 to 10 km). The reliability and the high quality of the T profiles obtained by RALMO are key aspects for addressing aerosol hygroscopic studies, since this is the most difficult to obtain of the atmospheric parameters using remote sensing techniques.
Figure 2 presents the same statistics as in the previous discussion but for RH measurements. As was explained in Sect. 3, RH profiles from lidar were obtained from the combination of r and T measurements. Figure 2a shows all RH deviation profiles between lidar and RS, while Fig. 2b presents the mean RH deviation profile. This plot reveals a small bias between both instruments that ranges from positive values (+3 % RH at 1.4 km a.s.l.) to negative ones (−9 % RH at 5.6 km a.s.l.). Above 9 km (a.s.l.) the bias again becomes positive, reaching a maximum value of +6 % RH. The mean bias and standard deviation along the region from ground to 2.1 km a.s.l. is $+\mathrm{2.0}±\mathrm{0.9}$ % RH, while it increases to $-\mathrm{4}±\mathrm{2}$ % RH in the range 2.1–9 km. We can affirm that the shape observed in the RH bias between both instruments is mainly coming from the r measurements since the T statistics showed a negligible bias in the whole column (with almost zero bias, not shown). It is due to larger inhomogeneities in r in time and in space which produce larger discrepancies between lidar and RS observations. Regarding the standard deviations observed for this parameter (Fig. 2c), we can observe that it increases with altitude in the lower troposphere (from 4.4 % RH at ground to 8 % RH at 2 km a.s.l.). Above this altitude the standard deviation values oscillate around a quite constant value in altitude. The mean RH standard deviation in the lower troposphere (from ground to 2.1 km) was 6.5 %±1.3 % RH, while it was 8.5 %±1.5 % RH above this altitude. The RH comparison for daytime measurements (not shown here) also presented a small bias between lidar and RS, with a mean value of $-\mathrm{0.4}\phantom{\rule{0.125em}{0ex}}\mathit{%}±\mathrm{2.4}\phantom{\rule{0.125em}{0ex}}\mathit{%}$ RH in the range from ground to 5 km a.s.l. The mean standard deviations obtained for the same altitude range were slightly larger than during nighttime, with a mean value of 9 %±3 % RH.
Above 5 km (a.s.l.) and during daytime the SNR of RALMO is smaller compared to nighttime measurements due to the solar background; for this reason the RH profiles were not calculated above this altitude in order to avoid large uncertainties. In any case we would like to highlight the good quality of RALMO RH information. The high quality of the RH measurements shown in this intercomparison is a key aspect to be able to address the aerosol hygroscopic studies, as will be shown in the next sections.
Figure 3Temporal evolution of vertical profiles of r, RH and βaer at 355 nm from RALMO lidar on 7 September 2017.
5 Study of aerosol hygroscopicity
Two case studies are presented in which the hygroscopicity of different types of aerosol is characterized. The two case studies took place during summer 2017 and feature smoke particles and mineral dust.
## 5.1 Case I: hygroscopic growth of smoke mixture
Case I corresponds to 7 September 2017. The temporal evolution of r, RH and βaer at 355 nm in the lower troposphere (0–5 km a.s.l.) is shown in Fig. 3 for this day. According to the aerosol measurements (lowest panel), low clouds were present at around 1.6 km (a.g.l.) during the first part of the day (until 12:00 UTC). After that, two clear aerosol layers can be identified, the atmospheric boundary layer (ABL) and a strong lofted aerosol layer located between 2 and 4 km (a.g.l.) which appeared in the afternoon. A usual convective boundary layer development was observed during the day, with maximum height occurring at around 14:00 UTC. It is interesting to remark how within the ABL the intensity of the aerosol backscatter signal was stronger at the top of this layer even when a strong mixing was expected at the central hours of the day (between 13:00 and 17:00 UTC). However, a larger homogeneity was observed for the same layer in the r measurements (Fig. 3, upper panel), indicating that the convective processes were strong enough to produce a well-mixed ABL in that time interval. The third element that can help to understand the observed variations in the backscatter signals is the RH measurements (central panel). From that plot, we can observe how the intensification observed in βaer is well correlated with the values observed for RH, with the highest values of both properties at the top of the ABL. This kind of behavior could be due to aerosol hygroscopic processes, and a detailed analysis is presented in the following paragraphs.
According to the NAAPS (Navy Aerosol Analysis and Prediction System) model, 7 September 2017 at 18:00 UTC is characterized by the presence of smoke above the measurement station (Fig. 4a). This model predicted smoke surface concentrations (blue color map) between 4 and 8 µg m−3 at Payerne. The HYSPLIT back-trajectory analysis indicated that the air masses above our station in the lower layers of the troposphere had their origin in North America (Fig. 4b). The VIIRS (Visible Infrared Imaging Radiometer Suite) fire and thermal anomalies product available from the joint NASA/NOAA Suomi-National Polar orbiting Partnership (S-NPP) satellite (Fig. 4c) showed that in the studied period (from 25 August to 3 September 2017) several intensive hotspots were found along the calculated air mass trajectories, especially in some areas of the northwest of the United States and in the central part of Canada. In addition, carbon monoxide (CO) observations from the Atmospheric Infrared Sounder (AIRS) onboard the Aqua satellite monitored a plume (not shown here) with a high concentration of CO that moved from North America to Europe during that period, reaching the eastern part of Europe on 6 September 2017. Over Payerne, the total column CO concentrations observed with AIRS were in the range of 100 and 130 parts per billion by volume (ppbv) during 7 and 8 September, which are considerably higher than the mean concentration observed in the previous month (70–80 ppbv). CO is considered a good tracer of smoke particles since it is generated in the incomplete combustion of biomass.
Figure 4(a) NAAPS total optical depth from sulfate (orange/red scale), dust (green/yellow scale) and smoke (blue scale) at 18:00 UTC on 7 September 2017. (b) Backward trajectories from the NOAA HYSPLIT model ending at 15:00 UTC, 7 September 2017, calculated for the altitudes 2500 and 3500 m a.s.l. (c) Fire map from the VIIRS instrument for the period from 25 August to 3 September 2017 (source: https://firms.modaps.eosdis.nasa.gov/, last access: 12 October 2018).
In situ measurements carried out at Payerne station showed some changes in the aerosol properties that could be characteristics of smoke particles (Fig. 5). An increase in the PM2.5 mass concentration was observed in the period from 6 to 8 September, with values changing from 1 µg m−3 (12:00 UTC on 6 September) to 9.9 µg m−3 (20:00 UTC on 8 September), indicating an increase in the concentration of small particles at the surface. The PM2.5 concentration reached during that evening was clearly above the mean summer value (5.2 µg m−3). The PM2.5 increase occurred at the same time as an increase in the absorption coefficient at different wavelengths observed by an aethalometer. The absorption Ångström exponent (AAE), which represents the wavelength dependence of absorption and depends on the composition of absorbing aerosols, showed relatively low values (between 1.1 and 1.3 for most of the measurements) which can be characteristics of biomass-burning particles . Therefore, model predictions and satellite and in situ observations agreed and point to a mixture of local aerosol and smoke particles from biomass burning in the atmospheric column over Payerne.
Vertical information of aerosol, T and r was obtained using the RALMO lidar and ceilometer measurements on 7 September 2017. Figure 6 shows the profiles of βaer at 355 and 1064 nm, RH and the auxiliary information of θ and r for the time interval 15:00–15:30 UTC. From this figure, a marked increase in βaer with altitude was observed for the altitude range between 1.7 and 2.3 km (a.s.l.). Simultaneously to this increase, we observed an increase in RH with values increasing from 73 % (bottom of the layer) to 97 % (top of the layer). The profiles of θ and r (Fig. 6c) were used as indicators of a good mixing as explained in Sect. 3.2. These profiles show quite constant values for both properties within the layer (300.6±0.5 K and 5.3±0.1 g kg−1, respectively), indicating that the layer was well mixed.
Figure 5(a) Surface PM2.5 concentrations at Payerne from 6 to 8 September 2017. (b) Absorption coefficient at seven wavelengths and (c) absorption Angstrom exponent from aethalometer measurements.
Figure 6Vertical profiles of (a) βaer at 355 and 1064 nm, (b) RH and (c) θ and r for the time interval 15:00–15:30 UTC on 7 September 2017. The grey shaded area indicates the layer in which aerosol hygroscopic growth occurred.
Following the methodology presented in Sect. 3.2, fβ at 355 and 1064 nm were obtained for the investigated layer from the combination of βaer and RH measurements. Figure 7 shows the dependency of fβ(RH) with RH, called a humidogram. The reference RH for this case was 73 %, which corresponds to the lowest value in the layer. From this figure, we can observe how β355 increased by a factor of 2.8 (fβ(97 %)=2.8), while humidity increased from 73 % to 97 % (blue points). Lower values of fβ are observed at 1064 nm (red points). In the infrared β increased by a factor of 1.8 with respect to its value at RHref (73 %), indicating a lower sensitivity of this wavelength to the aerosol hygroscopic growth. Hänel hygroscopic parameters (γ) obtained using the fitting equation (Eq. 7) were calculated (solid lines) in order to make our measurements comparable with other studies. This parameter is proportional to the aerosol hygroscopicity, and it had values of 0.48±0.08 at 355 nm and 0.29±0.08 at 1064 nm. Independent vertical profiles obtained from the measurements of the previous 30 min time interval (from 14:30 to 15:00 UTC) were also analyzed in order to check the consistency of our results (figure not shown here). For that period, a simultaneous increase in βaer and RH was also observed in the altitude range 1.5–2.2 km. Although the RH range observed for this time interval (70 %–93 %) was slightly different from the one shown in Fig. 6, γ showed consistent results, within the associated uncertainties, for both time intervals (${\mathit{\gamma }}_{\mathrm{355}}=\mathrm{0.57}±\mathrm{0.14}$ and ${\mathit{\gamma }}_{\mathrm{1064}}=\mathrm{0.35}±\mathrm{0.14}$). A similar value of the hygroscopic parameter at 355 nm (γ355=0.40) is reported by and associated with the presence of smoke particles. In that study a combination of lidar and MWR measurements was used for the aerosol hygroscopic analysis. However, the spectral dependency found in our case is not what has been reported in other studies. Higher values for longer wavelengths (between 355 and 532 nm) were found in γ in other studies for smoke particles and also for anthropogenic particles using remote sensing . also reported this higher hygroscopicity at 1064 nm but for marine particles. In situ studies also showed higher values for longer wavelengths, although over a shorter range (450–700 nm) and for marine particles .
Figure 7fβ at 355 and 1064 nm retrieved from the lidar profiles (layer: 1.7–2.3 km a.s.l.) at the time interval 15:00–15:30 UTC on 7 September 2017.
We performed Mie simulations in order to understand whether the spectral dependency observed in our study could be realistic or not. βaer at the lidar wavelengths (355, 532 and 1064 nm) as a function of RH were computed using a Mie code . For these simulations some inputs such as the aerosol growth factor (1.6, which is typical for hygroscopic aerosol) and the refractive index ($m=\mathrm{1.5}+i\mathrm{0.01}$) were assumed. Figure 8 shows the fβ calculated from the Mie calculation as a function of RH for different wavelengths and particle sizes. Monomodal distributions were used assuming different diameters for the dry particles and a geometric standard deviation of 1.5. From this figure we can observe that the backscatter is very sensitive to wavelength and particle size as expected, whose relationship is characterized in scattering theory by the size parameter ($x=\mathit{\pi }D/\mathit{\lambda }\right)$, where D is the particle diameter. A Mie scattering regime is expected for x≈1. A stronger increase in the backscattered radiation at the shortest wavelengths is expected for small particles (Ddry=200 nm) when RH increases according to these simulations (Fig. 8a). These results indicate that even a decrease in βaer at 1064 nm could be expected when the size particles increase due to hygroscopic growth. Figure 8b shows the humidogram for a dry particle diameter of 400 nm. A different spectral dependency is observed for this case, with slightly higher fβ at 532 nm than at 355 nm and lower at 1064 nm. These results agree with our observations and also explain the different spectral dependency observed in other studies for smoke and anthropogenic particles that could have similar sizes (small particles) to what is simulated here. Panels c and d in Fig. 8 show that for the biggest dry particle diameters (600 and 800 nm) the spectral dependency of fβ is inverted with respect to small particles with larger fβ values at the longest wavelengths. This spectral dependency agrees with the observations in for marine particles which are considered much bigger particles than smoke particles. However, we must also point out that for many cases, larger particles in the atmosphere are further away from the ideal case of a sphere considered in Mie theory. Hence, this type of comparison should be considered with caution.
Figure 8fβ calculated from Mie simulations at 355, 532 and 1064 nm and for different particle diameters. The size parameter (x) has been indicated for each configuration of wavelength and size particle.
Figure 9Evolution of r and wind direction (a) and ${\mathit{\beta }}_{\mathrm{355}}^{\text{aer}}$ and RH (b) at 1.3 km (a.s.l.) on 7 September 2017.
Figure 10Humidograms at 355 nm (a) and 1064 nm (b) retrieved from continuous measurements for hydration and dehydration processes on 7 September 2017.
Figure 11Same as Fig. 3, on 8 July 2017.
Continuous aerosol and RH measurements from RALMO lidar also allow us to monitor aerosol hygroscopic processes occurring in time. Figure 9 shows the evolution of r, wind direction, βaer at 355 nm and RH at 1.3 km (a.s.l.) on 7 September 2017. As was indicated in Sect. 3.2, water vapor is considered a good tracer in the atmosphere, and constant values of r mean the air parcels have very similar origins. In this case, we can observe that r was quite constant (6.7±0.3 g kg−1, Fig. 9, top) during the evening (from 16:00 to 23:30 UTC), fulfilling the previous criterion. In addition, a simultaneous increase in βaer and RH was observed for the indicated period and altitude (Fig. 9, bottom). RH changed from 63 % in late afternoon to reach values close to 90 % at midnight.
In order to quantify the aerosol hygroscopic effect that took place in time, fβ was calculated for these measurements (Fig. 10a, blue filled circles). The initial value of ${\mathit{\beta }}_{\mathrm{355}}^{\text{aer}}$ (at RHref=63 %) increased by a factor of 2 when RH reached the maximum values of the evening. The Hänel parameterization was used for this dataset, providing a hygroscopic parameter of 0.54±0.17 which is in good agreement with the values observed in the atmospheric column in the afternoon of this day. In addition to this hydration process (water uptake) we could also observe the dehydration (evaporation) that occurred within this aerosol layer during the afternoon of this day (from 11:00 to 16:00 UTC, Fig. 9). In this period, r and the wind direction measurements were stable (6.0±0.5 g kg−1 and $\mathrm{302}{}^{\circ }±\mathrm{27}{}^{\circ }$, respectively), evidence that the air mass did not change. A decrease in ${\mathit{\beta }}_{\mathrm{355}}^{\text{aer}}$ took place at the same time when RH decreased from 93 % to 59 %. The humidogram obtained for this dehydration process (Fig. 10a, blue open circles) also showed fβ values very close to the ones calculated in the later period and a hygroscopic parameter from the Hänel parametrization of 0.40±0.08. The same behavior was observed for β1064 from the ceilometer measurements along this day, with hygroscopic parameter values of 0.41±0.16 and 0.34±0.09 for hydration and dehydration processes, respectively, showing again lower values than at the ultraviolet channel. We would like to remark on the good agreement found in the aerosol hygroscopicity of this case using a vertical analysis and a temporal analysis.
Figure 12(a) Dust concentration profile from NMMB/BSC model forecast at 06:00 UTC on 8 July 2017 (adapted from http://www.bsc.es/ess/bsc-dust-daily-forecast, last access: 15 September 2018). (b) Vertical profile of βaer at 355 nm from RALMO lidar for the time interval from 01:00 to 01:30 UTC on 8 July 2017.
## 5.2 Case II: hygroscopic growth of mineral dust particles
The second case took place on 8 July 2017. Figure 11 shows the evolution of r, RH and βaer at 355 nm from RALMO during the morning of this day. From the aerosol measurements (lowest panel) the presence of particles at high altitudes is evident, reaching almost 6 km (a.s.l.) in the late morning (around 09:00 UTC). For this day, the NMMB/BSC (Non-hydrostatic Multiscale Model/Barcelona Supercomputing Center) Dust model predicted dust particles over the western part of Europe, including Switzerland. A dust concentration profile calculated using this model for the EARLINET station of Payerne (Fig. 12a) showed higher concentration of mineral dust in the lower troposphere with a profile very similar to what was observed with our Raman lidar (Fig. 12b). Back-trajectory analysis from the HYSPLIT model (not shown here) indicated that the observed air masses had their origin in northern Africa. Remote and in situ measurements carried out at our station also confirmed features typical of this kind of particle. The AOD Angstrom exponent obtained from the PFR sun-photometer measurements presented low values (between 0.5 and 0.6) throughout the morning, indicating the presence of coarse particles in the atmospheric column. In situ measurements also showed a strong increase in the PM10 concentration at the surface during this day, with values ranging from 14 µg m−3 at 03:00 UTC to 37.4 µg m−3 at 15:00 UTC. The annual mean PM10 concentration in 2017 was 12 µg m−3, which is much lower than the values observed during this event.
Figure 13Lidar vertical profiles of (a) βaer at 355 and 1064 nm, (b) RH and (c) θ and r obtained between 01:00 and 01:30 on 8 July 2017. (d) fβ at 355 and 1064 nm retrieved for those profiles.
Once the aerosol was well identified using models and measurements, we analyzed the vertical profiles obtained from the lidar (Fig. 13a–c). We observed a simultaneous increase in βaer from the lidar systems with RH at the altitude range between 1.9 and 2.3 km (a.s.l.). For that range, r and θ showed quite constant values, evidence of a well-mixed layer. The dependence of fβ at 355 and 1064 nm with RH is shown in the resultant humidogram in Fig. 13d. Although there was an intensification of the backscatter in both wavelengths (${f}_{\mathit{\beta }}^{\mathrm{355}}\left(\mathrm{84}\phantom{\rule{0.125em}{0ex}}\mathit{%}\right)=\mathrm{1.2}$ and ${f}_{\mathit{\beta }}^{\mathrm{1064}}\left(\mathrm{84}\phantom{\rule{0.125em}{0ex}}\mathit{%}\right)=\mathrm{1.1}$ with RHref=68 %), it was much lower compared to Case I. The hygroscopic parameter obtained from the Hänel parametrization confirmed this behavior, with values of 0.20±0.18 and 0.12±0.19 at 355 and 1064 nm, respectively. These values are similar to the ones observed by also under the presence of mineral dust (γ355=0.12 and γ532=0.24). As in Case I, we found the opposite spectral dependency compared to . However, we considered a wider spectral range.
Figure 14Aerosol extinction profiles at 355 and 1064 nm considering wet (ambient) and dry conditions (in the layer with aerosol hygroscopicity, colored dashed lines) for Case I (a) and Case II (b). Horizontal dashed black lines indicate the layer for each case in which hygroscopic growth occurred.
6 Evaluation of the effect of aerosol hygroscopicity on the Earth's radiative balance
Because of the changes in aerosol optical and microphysical properties due to water uptake, aerosol radiative properties are modified in the case of hygroscopic growth. As stated before, aerosol backscatter and extinction coefficients increase under high RH conditions, which in turn leads to an increase in the AOD. To compute the AOD, the βaer profiles at 355 nm were converted to extinction using a generic lidar ratio of 50 sr. Although the choice of LR could affect the AOD, the relative contribution due to aerosol hygroscopicity would remain almost constant. In order to calculate ΔAOD, the increment of AOD due to hygroscopic growth, we obtain a so-called “dry” aerosol extinction profile by using the Hänel parameterization and assuming that RH in the analyzed layer is equal to RHref for each case. The “dry” profiles obtained are included in Fig. 14. AODs for the dry and wet cases, as well as ΔAOD, are summarized in Table 1.
Table 1Column AOD and ARE for both dry and wet conditions and their difference. All relative values are given with respect to the dry conditions.
* Indicates values within the ABL (between the surface and 2.5 km).
For the two cases analyzed here, the increase in AOD at 355 nm related to the hygroscopic growth in the analyzed layers is ΔAOD = 0.017 (with ΔAOD = AOD AODdry) for Case I and ΔAOD = 0.001 for Case II. In relative terms, this results in an increase in the total AOD of 4.7 % (15.6 % if we consider only AOD in the ABL where the hygroscopic layer is located) due to hygroscopic growth for Case I and 0.6 % for Case II. As expected, ΔAOD is much larger in Case I since smoke particles are more hygroscopic (γ355=0.48) than mineral dust (γ355=0.20). It is worth noting the significant effect of the hygroscopic growth on the ABL AOD, which increases by nearly 16 %.
Simulations with a radiative transfer model can give us an estimate of the impact that this change in AOD has on the aerosol radiative effect (ARE). In this case, we use GAME (Global Atmospheric ModEl, ). GAME is a modular radiative transfer model that allows calculation of upward and downward radiative fluxes at different vertical levels with high resolution by using the discrete ordinates method . Details about the model parameterization can be found in and . In our case, the variations in the ARE (ΔARE) in the shortwave spectral range are exclusively related to the variations in AOD and the AE due to hygroscopic growth, whereas the other aerosol properties are assumed to remain constant. The calculation was done for a solar zenith of 30 corresponding to Case I, which presented stronger hygroscopicity and occurred at daytime (15:00 UTC). The changes in AOD produce a net increase (in absolute and relative values) in the aerosol radiative effect at the surface with respect to the “dry” profiles equal to 2.4 W m−2 (5.2 %) and 0.1 W m−2 (0.4 %) in Case I and Case II, respectively. The interpretation of these results needs to be done carefully. Even though radiative transfer models do not provide an estimation of the ARE uncertainties, a sensitivity test performed in showed that an uncertainty in the AOD of 0.05 can lead to uncertainties in the ARE of up to 30 %. The values of ΔARE obtained here are certainly within these uncertainty limits, and an accurate quantitative estimation of the hygroscopicity contribution to the ARE is quite complex. However, it is necessary to highlight that our focus here is on the relative contribution of hygroscopic aerosol to the ARE when compared to dry conditions. As expected, the effect observed in our simulations is more noticeable in the case of particles with stronger hygroscopic properties, such as the smoke mixture in Case I.
We also note that the relative increases in AOD (4.7 %) and ARE (5.2 %) are similar. For the mineral dust event the effect is almost negligible (ΔARE = 0.4 %). Variations of the ARE observed in previous studies can reach up to 7 W m−2 ; however, a comparison with our data is not straightforward since these variations are highly dependent on the aerosol load and the aerosol type present in the atmosphere. In Case I, although the hygroscopic growth affects only a thin layer (only 600 m width) and the γ values are relatively low (0.48), the aerosol hygroscopic growth effect on the ARE is still quite noticeable. These results show that in more favorable conditions, namely thicker layers where the hygroscopic effect occurs and particles with stronger hygroscopic properties, the aerosol hygroscopic effect on the optical and radiative properties could be quite considerable. Therefore, we can conclude that including aerosol hygroscopic properties in climate model calculations is key for improving the accuracy of aerosol forcing estimates.
7 Conclusions
The present study demonstrates the capability of a Raman lidar to detect aerosol hygroscopic processes. Continuous measurements of water vapor, T and aerosol profiles have been performed by RALMO lidar almost continuously since 2008 at the aerological station of MeteoSwiss in Payerne (Switzerland). These measurements allow us to monitor any change in aerosol properties that could occur as a result of water uptake by particles under high RH (aerosol hygroscopic growth). To ensure that the changes in aerosol are only due to hygroscopic growth, several criteria were established. As a first condition an increase in βaer should occur simultaneously with an increase in RH. In addition, a high degree of homogeneity is required in the investigated layer. For that, back-trajectory analysis is used to verify that the origin of the air mass is independent of the altitude and low-varying or constant values of r and θ are required as a proxy for well-mixed conditions throughout the aerosol layer.
The accuracy and the precision of RALMO T and RH profiles were assessed using collocated RS profiles. A total of 172 profiles were used in this intercomparison during daytime and nighttime for the period from July to December of 2017. The mean T deviations calculated from nighttime (daytime) measurements revealed almost no bias between both techniques in the whole troposphere, with mean T deviations of 0.05±0.06 K ($-\mathrm{0.5}±\mathrm{0.2}$ K) in the first 5 km and 0.15±0.15 K ($-\mathrm{0.1}±\mathrm{0.6}$ K) above that altitude. The standard deviations also confirmed the excellent performance of RALMO, with values below 1 K throughout the troposphere during nighttime and slightly larger values during daytime (0.8±0.2 K from ground to 5 km and 2.4±0.8 K from 5 to 10 km).
Lidar profiles of RH were obtained by combining r and T measurements. Small RH biases were observed between both techniques during nighttime in the troposphere (from ground to 9 km a.s.l.), with values ranging between 3 % RH at 1.4 km (a.s.l.) and −9 % RH at 5.6 km (a.s.l.). The standard deviation of RH deviations also showed the good precision of the lidar measurements, with values always lower than 9 % RH. The performance of RALMO in terms of RH during daytime in the lower troposphere (from ground to 5 km) was very similar to the ones obtained during nighttime. However, above 5 km the errors were much larger due to the solar background radiation, and RH profiles were not calculated to avoid these large uncertainties. The good quality of the RH measurements found in this intercomparison is a key aspect to be able to address the aerosol hygroscopic studies.
The methodology presented here was applied to two case studies. In situ and satellite measurements in addition to models indicate that Case I (7 September 2017) was characterized by a mixture of local aerosol and smoke particles from fires in North America. fβ was found to be 2.8 at 355 nm and 1.8 at 1064 nm when RH increased from 73 % to 97 % in the investigated layer. The Hänel hygroscopic parameter which is proportional to the aerosol hygroscopicity took values of 0.48±0.08 at 355 nm and 0.29±0.08 at 1064 nm in this case. Independent vertical profiles obtained for a previous time interval showed the consistency of our results. Other remote sensing studies have shown a larger sensitivity to hygroscopic growth for longer wavelengths for this type of particle, in contrast to our results. However, those studies were carried out in a shorter wavelength range (355–532 nm). also observed a higher hygroscopicity at 1064 nm than at 355 nm but for marine particles which are larger and much more hygroscopic than in our case. Mie simulations carried out in this study revealed that the spectral dependency of fβ can change strongly depending on the particle size and the wavelength of the incident radiation, supporting our results as well as the study of .
Continuous aerosol and RH measurements from RALMO also allowed monitoring of aerosol hygroscopic processes as a function of time for this first case. The evolution of β and RH at 1.3 km (a.s.l.) on 7 September 2017 showed two periods in which there was a simultaneous decrease in both parameters (dehydration process) followed by a simultaneous increase (hydration process). The Hänel hygroscopic parameters calculated from the humidogram (fβ vs. RH) for both periods took values of 0.54±0.17 (0.43±0.16) and 0.40±0.08 (0.34±0.09) at 355 nm (1064 nm) for the hydration and dehydration processes, respectively, which are in good agreement with the results obtained in the spatial (vertical) analysis.
The aerosol hygroscopicity of a second case (8 July 2017), characterized by the presence of dust particles, was also analyzed in this study. The spatial analysis of the lidar measurements also revealed hygroscopic growth for these particles but with a very different behavior. A much lower hygroscopicity than for the previous case (smoke mixture) was observed with β increasing only 1.2 and 1.1 times at 355 and 1064 nm, respectively, when RH increased from 68 % to 84 %. The hygroscopic parameters obtained from the humidogram were 0.20±0.18 and 0.12±0.19 at 355 and 1064 nm, respectively, showing a good agreement with values observed in other studies for mineral dust. The lower spectral sensitivity to the aerosol hygroscopicity found for this type of particle was also remarkable.
Finally, the impact of aerosol hygroscopicity on the Earth's radiative balance was evaluated for the two presented cases using a radiative transfer model (GAME). The aerosol hygroscopic growth in the investigated layers produced an increase in AOD at 355 nm of 0.017 (4.7 %) for the case with presence of smoke particles and 0.001 (0.6 %) for the case with mineral dust. These changes in AOD produced a net increase in absolute (and relative) values of the radiative effect at the surface of 2.4 W m−2 (5.2 %) and 0.1 W m−2 (0.4 %) in Case I and Case II, respectively. The results were significant for the case with presence of smoke particles (more hygroscopic) despite the aerosol load of the investigated layer not having been very high. Therefore, we conclude that the effect of aerosol hygroscopicity on optical and radiative properties is important and has to be considered in climate model calculations to improve aerosol forcing estimates.
In future work we want to exploit the large dataset (10 years) of simultaneous aerosol and RH profiles from this Raman lidar to carry out a statistical analysis of aerosol hygroscopic properties.
Data availability
Data availability.
Data used in this paper are available upon request from the corresponding author (francisco.navas@meteoswiss.ch).
Author contributions
Author contributions.
FNG designed the experiment, analyzed the data and wrote the manuscript, GM worked on the T retrievals, MH and MCC performed the Mie simulations, AH is responsible for the lidar measurements and participated in the numerous scientific discussions, and MJGR and MS performed the calculations with the radiative transfer model (GAME). All the authors provided comments on the manuscript.
Competing interests
Competing interests.
The authors declare that they have no conflict of interest.
Special issue statement
Special issue statement.
Acknowledgements
Acknowledgements.
We thank the EMPA and the Swiss Federal Office for the Environment (FOEN) for providing the data of the in situ measurements carried out at Payerne within the Nabel monitoring program. We also acknowledge the financial support by the European Union's Horizon 2020 research and innovation program through project ACTRIS-2 (grant agreement no. 654109). The radiative transfer simulations performed with GAME are supported by the European Union GRASP-ACE MSCA-RISE Action (grant agreement no. 778349); the Spanish Ministry of Economy and Competitiveness (ref. TEC2015-63832-P) and EFRD (European Fund for Regional Development); the Spanish Ministry of Science, Innovation and Universities (ref. CGL2017-90884-REDT); and the Unity of Excellence Maria de Maeztu (ref. MDM-2016-0600) financed by the Spanish Agencia Estatal de Investigación. This work was also supported by the Juan de la Cierva-Formación program (grant FJCI-2015-23904). The authors also gratefully acknowledge Philippe Dubuisson (Laboratoire d'Optique Atmosphérique, Université de Lille, France) for the use of the GAME model.
Financial support
Financial support.
This work has been supported by the Swiss National Science Foundation (project no. PZ00P2 168114).
Review statement
Review statement.
This paper was edited by Eduardo Landulfo and reviewed by two anonymous referees.
References
Ansmann, A., Riebesell, M., Wandinger, U., Weitkamp, C., Voss, E., Lahmann, W., and Michaelis, W.: Combined Raman elastic-backscatter lidar for vertical profiling of moisture, aerosol extinction, backscatter, and lidar ratio, Appl. Phys. B, 55, 18–28, 1992. a
Bedoya-Velásquez, A. E., Navas-Guzmán, F., Granados-Muñoz, M. J., Titos, G., Román, R., Casquero-Vera, J. A., Ortiz-Amezcua, P., Benavent-Oltra, J. A., de Arruda Moreira, G., Montilla-Rosero, E., Hoyos, C. D., Artiñano, B., Coz, E., Olmo-Reyes, F. J., Alados-Arboledas, L., and Guerrero-Rascado, J. L.: Hygroscopic growth study in the framework of EARLINET during the SLOPE I campaign: synergy of remote sensing and in situ instrumentation, Atmos. Chem. Phys., 18, 7001–7017, https://doi.org/10.5194/acp-18-7001-2018, 2018. a, b, c, d, e, f
Bindoff, N., Stott, P., AchutaRao, K., Allen, M., Gillett, N., Gutzler, D., Hansingo, K., Hegerl, G., Hu, Y., Jain, S., Mokhov, I., Overland, J., Perlwitz, J., Sebbari, R., and Zhang, X.: Detection and Attribution of Climate Change: from Global to Regional, in: Climate Change 2013: The Physical Science Basis. Contribution of Working Group I to the Fifth Assessment Report of the Intergovernmental Panel on Climate Change, edited by: Stocker, T. F., Qin, D., Plattner, G.-K., Tignor, M., Allen, S. K., Boschung, J., Nauels, A., Xia, Y., Bex, V., and Midgley, P. M., Cambridge University Press, Cambridge, UK, New York, NY, USA, 867–952, 2013. a
Bond, T. C., Habib, G., and Bergstrom, R. W.: Limitations in the enhancement of visible light absorption due to mixing state, J. Geophys. Res.-Atmos., 111, D20211, https://doi.org/10.1029/2006JD007315, 2006. a
Bösenberg, J., Matthias, V., Linné, H., Comerón Tejero, A., Rocadenbosch Burillo, F., Pérez López, C., and Baldasano Recio, J. M.: EARLINET: A European Aerosol Research Lidar Network to establish an aerosol climatology, Report. Max-Planck-Institut fur Meteorologie, 1–191, 2003. a
Brocard, E., Philipona, R., Haefele, A., Romanens, G., Mueller, A., Ruffieux, D., Simeonov, V., and Calpini, B.: Raman Lidar for Meteorological Observations, RALMO – Part 2: Validation of water vapor measurements, Atmos. Meas. Tech., 6, 1347–1358, https://doi.org/10.5194/amt-6-1347-2013, 2013. a, b
Christensen, J. H.: The Danish Eulerian hemispheric model – A three-dimensional air pollution model used for the Arctic, Atmos. Environ., 31, 4169–4191, 1997. a
Collaud Coen, M., Weingartner, E., Schaub, D., Hueglin, C., Corrigan, C., Henning, S., Schwikowski, M., and Baltensperger, U.: Saharan dust events at the Jungfraujoch: detection by wavelength dependence of the single scattering albedo and first climatology analysis, Atmos. Chem. Phys., 4, 2465–2480, https://doi.org/10.5194/acp-4-2465-2004, 2004. a
Dinoev, T., Simeonov, V., Arshinov, Y., Bobrovnikov, S., Ristori, P., Calpini, B., Parlange, M., and van den Bergh, H.: Raman Lidar for Meteorological Observations, RALMO – Part 1: Instrument description, Atmos. Meas. Tech., 6, 1329–1346, https://doi.org/10.5194/amt-6-1329-2013, 2013. a, b, c
Draxler, R. R. and Rolph, G. D.: HYSPLIT (HYbrid Single-Particle Lagrangian Integrated Trajectory) model,NOAA Air Resources Laboratory, Silver Spring, available at: http://ready.arl.noaa.gov/HYSPLIT.php (last access: 22 November 2018), 2003. a
Dubuisson, P., Buriez, J., and Fouquart, Y.: High spectral resolution solar radiative transfer in absorbing and scattering media: Application to the satellite simulation, J. Quant. Spectrosc. Ra., 55, 103–126, 1996. a
Dubuisson, P., Giraud, V., Chomette, O., Chepfer, H., and Pelon, J.: Fast radiative transfer modeling for infrared imaging radiometry, J. Quant. Spectrosc. Ra., 95, 201–220, 2005. a
Feingold, G. and Morley, B.: Aerosol hygroscopic properties as measured by lidar and comparison with in situ measurements, J. Geophys. Res.-Atmos., 108, 4327, https://doi.org/10.1029/2002JD002842, 2003. a, b
Fernández, A., Molero, F., Becerril-Valle, M., Coz, E., Salvador, P., Artíñano, B., and Pujadas, M.: Application of remote sensing techniques to study aerosol water vapour uptake in a real atmosphere, Atmos, Res,, 202, 112–127, 2018. a
Fierz-Schmidhauser, R., Zieger, P., Gysel, M., Kammermann, L., DeCarlo, P. F., Baltensperger, U., and Weingartner, E.: Measured and predicted aerosol light scattering enhancement factors at the high alpine site Jungfraujoch, Atmos. Chem. Phys., 10, 2319–2333, https://doi.org/10.5194/acp-10-2319-2010, 2010. a
Goldsmith, J., Blair, F. H., Bisson, S. E., and Turner, D. D.: Turn-key Raman lidar for profiling atmospheric water vapor, clouds, and aerosols, Appl. Opt., 37, 4979–4990, 1998. a
Granados-Muñoz, M., Guerrero-Rascado, J., Bravo-Aranda, J., Navas-Guzmán, F., Valenzuela, A., Lyamani, H., Chaikovsky, A., Wandinger, U., Ansmann, A., Dubovik, O., Grudo, J. O. and Alados-Arboledas, L.: Retrieving aerosol microphysical properties by Lidar-Radiometer Inversion Code (LIRIC) for different aerosol types, J. Geophys. Res.-Atmos., 119, 4836–4858, 2014. a
Granados-Muñoz, M. J., Navas-Guzmán, F., Bravo-Aranda, J. A., Guerrero-Rascado, J. L., Lyamani, H., Valenzuela, A., Titos, G., Fernández-Gálvez, J., and Alados-Arboledas, L.: Hygroscopic growth of atmospheric aerosol particles based on active remote sensing and radiosounding measurements: selected cases in southeastern Spain, Atmos. Meas. Tech., 8, 705–718, https://doi.org/10.5194/amt-8-705-2015, 2015. a, b, c
Granados-Muñoz, M. J., Sicard, M., Román, R., Benavent-Oltra, J. A., Barragán, R., Brogniez, G., Denjean, C., Mallet, M., Formenti, P., Torres, B., and Alados-Arboledas, L.: Impact of mineral dust on shortwave and longwave radiation: evaluation of different vertically resolved parameterizations in 1-D radiative transfer computations , Atmos. Chem. Phys., 19, 523–542, https://doi.org/10.5194/acp-19-523-2019, 2019. a, b
Haarig, M., Ansmann, A., Gasteiger, J., Kandler, K., Althausen, D., Baars, H., Radenz, M., and Farrell, D. A.: Dry versus wet marine particle optical properties: RH dependence of depolarization ratio, backscatter, and extinction from multiwavelength lidar measurements during SALTRACE, Atmos. Chem. Phys., 17, 14199–14217, https://doi.org/10.5194/acp-17-14199-2017, 2017. a, b, c, d
Haeffelin, M., Laffineur, Q., Bravo-Aranda, J.-A., Drouin, M.-A., Casquero-Vera, J.-A., Dupont, J.-C., and De Backer, H.: Radiation fog formation alerts using attenuated backscatter power from automatic lidars and ceilometers, Atmos. Meas. Tech., 9, 5347–5365, https://doi.org/10.5194/amt-9-5347-2016, 2016. a
Hänel, G.: The properties of atmospheric aerosol particles as functions of the relative humidity at thermodynamic equilibrium with the surrounding moist air, Elsevier, Adv. Geophys., 19, 73–188, 1976. a, b
Haywood, J. and Boucher, O.: Estimates of the direct and indirect radiative forcing due to tropospheric aerosols: A review, Rev. Geophys., 38, 513–543, 2000. a
Klett, J. D.: Stable analytical inversion solution for processing lidar returns, Appl. Ot., 20, 211–220, 1981. a
Kotchenruther, R. A., Hobbs, P. V., and Hegg, D. A.: Humidification factors for atmospheric aerosols off the mid-Atlantic coast of the United States, J. Geophys. Res.-Atmos., 104, 2239–2251, 1999. a
List, R.: Smithsonian Meteorological Tables, 6th rev. edn., Washington, DC, 1951. a
Lv, M., Liu, D., Li, Z., Mao, J., Sun, Y., Wang, Z., Wang, Y., and Xie, C.: Hygroscopic growth of atmospheric aerosol particles based on lidar, radiosonde, and in situ measurements: case studies from the Xinzhou field campaign, J. Quant. Spectrosc. Ra., 188, 60–70, 2017. a, b, c, d
Martucci, G., Voirin, J., Simeonov, V., Renaud, L., and Haefele, A.: A novel automatic calibration system for water vapor Raman LIDAR, in: EPJ Web of Conferences, EDP Sciences, 176, 05008, https://doi.org/10.1051/epjconf/201817605008, 2018. a
Martucci, G., Navas-Guzmán, F., Simeonov, V., Ludovic, R., and Haefele, A.: Validation of one-year of temperature data by the RAman Lidar for Meteorological Observations (RALMO) at Payerne, Atmos. Meas. Tech., in preparation, 2019. a, b
Mätzler, C.: MATLAB Functions for Mie Scattering and Absorption-Research Report No 2002-08, Institut für Angewandte Physik, Bern, 2002. a
McArthur, L., Halliwell, D. H., Niebergall, O. J., O'Neill, N. T., Slusser, J. R., and Wehrli, C.: Field comparison of network Sun photometers, J. Geophys. Res.-Atmos., 108, 4596, https://doi.org/10.1029/2002JD002964, 2003. a
Muñoz-Porcar, C., Comerón, A., Sicard, M., Barragan, R., Garcia-Vizcaino, D., Rodríguez-Gómez, A., Rocadenbosch, F., and Granados-Muñoz, M. J.: Calibration of Raman Lidar Water Vapor Mixing Ratio Measurements Using Zenithal Measurements of Diffuse Sunlight and a Radiative Transfer Model, IEEE T. Geosci. Remote, 7405–7414, 2018. a
Navas-Guzmán, F., Bravo-Aranda, J. A., Guerrero-Rascado, J. L., Granados-Muñoz, M. J., and Alados-Arboledas, L.: Statistical analysis of aerosol optical properties retrieved by Raman lidar over Southeastern Spain, Tellus B, 65, 21234, https://doi.org/10.3402/tellusb.v65i0.21234, 2013. a
Navas-Guzmán, F., Fernández-Gálvez, J., Granados-Muñoz, M. J., Guerrero-Rascado, J. L., Bravo-Aranda, J. A., and Alados-Arboledas, L.: Tropospheric water vapour and relative humidity profiles from lidar and microwave radiometry, Atmos. Meas. Tech., 7, 1201–1211, https://doi.org/10.5194/amt-7-1201-2014, 2014. a, b
Navas-Guzmán, F., Kämpfer, N., and Haefele, A.: Validation of brightness and physical temperature from two scanning microwave radiometers in the 60 GHz O2 band using radiosonde measurements, Atmos. Meas. Tech., 9, 4587–4600, https://doi.org/10.5194/amt-9-4587-2016, 2016. a
Nyeki, S., Halios, C., Baum, W., Eleftheriadis, K., Flentje, H., Gröbner, J., Vuilleumier, L., and Wehrli, C.: Ground-based aerosol optical depth trends at three high-altitude sites in Switzerland and southern Germany from 1995 to 2010, J. Geophys. Res.-Atmos., 117, D18202, https://doi.org/10.1029/2012JD017493, 2012. a
Pappalardo, G., Amodeo, A., Apituley, A., Comeron, A., Freudenthaler, V., Linné, H., Ansmann, A., Bösenberg, J., D'Amico, G., Mattis, I., Mona, L., Wandinger, U., Amiridis, V., Alados-Arboledas, L., Nicolae, D., and Wiegner, M.: EARLINET: towards an advanced sustainable European aerosol lidar network, Atmos. Meas. Tech., 7, 2389–2409, https://doi.org/10.5194/amt-7-2389-2014, 2014. a
Pérez, C., Haustein, K., Janjic, Z., Jorba, O., Huneeus, N., Baldasano, J. M., Black, T., Basart, S., Nickovic, S., Miller, R. L., Perlwitz, J. P., Schulz, M., and Thomson, M.: Atmospheric dust modeling from meso to global scales with the online NMMB/BSC-Dust model – Part 1: Model description, annual simulations and evaluation, Atmos. Chem. Phys., 11, 13001–13027, https://doi.org/10.5194/acp-11-13001-2011, 2011. a
Pilat, M. J. and Charlson, R. J.: Theoretical and optical studies of humidity effects on the size distribution of a hygroscopic aerosol, J. Rech. Atmos, 2, 166–170, 1966. a
Sandradewi, J., Prévôt, A. S., Szidat, S., Perron, N., Alfarra, M. R., Lanz, V. A., Weingartner, E., and Baltensperger, U.: Using aerosol light absorption measurements for the quantitative determination of wood burning and traffic emission contributions to particulate matter, Environ. Sci. technol., 42, 3316–3323, 2008. a
Schmeisser, L., Andrews, E., Ogren, J. A., Sheridan, P., Jefferson, A., Sharma, S., Kim, J. E., Sherman, J. P., Sorribas, M., Kalapov, I., Arsov, T., Angelov, C., Mayol-Bracero, O. L., Labuschagne, C., Kim, S.-W., Hoffer, A., Lin, N.-H., Chia, H.-P., Bergin, M., Sun, J., Liu, P., and Wu, H.: Classifying aerosol type using in situ surface spectral aerosol optical properties, Atmos. Chem. Phys., 17, 12097–12120, https://doi.org/10.5194/acp-17-12097-2017, 2017. a
Segura, S., Estellés, V., Titos, G., Lyamani, H., Utrillas, M. P., Zotter, P., Prévôt, A. S. H., Močnik, G., Alados-Arboledas, L., and Martínez-Lozano, J. A.: Determination and analysis of in situ spectral aerosol optical properties by a multi-instrumental approach, Atmos. Meas. Tech., 7, 2373–2387, https://doi.org/10.5194/amt-7-2373-2014, 2014. a
Sicard, M., Bertolín, S., Mallet, M., Dubuisson, P., and Comerón, A.: Estimation of mineral dust long-wave radiative forcing: sensitivity study to particle properties and application to real cases in the region of Barcelona, Atmos. Chem. Phys., 14, 9213–9231, https://doi.org/10.5194/acp-14-9213-2014, 2014. a
Sicard, M., D'Amico, G., Comerón, A., Mona, L., Alados-Arboledas, L., Amodeo, A., Baars, H., Baldasano, J. M., Belegante, L., Binietoglou, I., Bravo-Aranda, J. A., Fernández, A. J., Fréville, P., García-Vizcaíno, D., Giunta, A., Granados-Muñoz, M. J., Guerrero-Rascado, J. L., Hadjimitsis, D., Haefele, A., Hervo, M., Iarlori, M., Kokkalis, P., Lange, D., Mamouri, R. E., Mattis, I., Molero, F., Montoux, N., Muñoz, A., Muñoz Porcar, C., Navas-Guzmán, F., Nicolae, D., Nisantzi, A., Papagiannopoulos, N., Papayannis, A., Pereira, S., Preißler, J., Pujadas, M., Rizi, V., Rocadenbosch, F., Sellegri, K., Simeonov, V., Tsaknakis, G., Wagner, F., and Pappalardo, G.: EARLINET: potential operationality of a research network, Atmos. Meas. Tech., 8, 4587–4613, https://doi.org/10.5194/amt-8-4587-2015, 2015. a
Stamnes, K., Tsay, S.-C., Wiscombe, W., and Jayaweera, K.: Numerically stable algorithm for discrete-ordinate-method radiative transfer in multiple scattering and emitting layered media, Appl. Opt., 27, 2502–2509, 1988. a
Stock, M., Cheng, Y. F., Birmili, W., Massling, A., Wehner, B., Müller, T., Leinert, S., Kalivitis, N., Mihalopoulos, N., and Wiedensohler, A.: Hygroscopic properties of atmospheric aerosol particles over the Eastern Mediterranean: implications for regional direct radiative forcing under clean and polluted conditions, Atmos. Chem. Phys., 11, 4251–4271, https://doi.org/10.5194/acp-11-4251-2011, 2011. a
Swietlicki, E., Hansson, H.-C., Hämeri, K., Svenningsson, B., Massling, A., McFiggans, G., McMurry, P., Petäjä, T., Tunved, P., Gysel, M., Topping, D., Weingartner, E., Baltensperger, U., Rissler, J., Wiedensohler, A., and Kulmala, M.: Hygroscopic properties of submicrometer atmospheric aerosol particles measured with H-TDMA instruments in various environments – a review, Tellus B, 60, 432–469, 2008. a
Titos, G., Cazorla, A., Zieger, P., Andrews, E., Lyamani, H., Granados-Muñoz, M. J., Olmo, F., and Alados-Arboledas, L.: Effect of hygroscopic growth on the aerosol light-scattering coefficient: A review of measurements, techniques and error sources, Atmos. Environ., 141, 494–507, 2016. a, b
Vaughan, G., Wareing, D., Pepler, S., Thomas, L., and Mitev, V.: Atmospheric temperature measurements made by rotational Raman scattering, Appl. Opt., 32, 2758–2764, 1993. a
Veselovskii, I., Whiteman, D., Kolgotin, A., Andrews, E., and Korenskii, M.: Demonstration of aerosol property profiling by multiwavelength lidar under varying relative humidity conditions, J. Atmos. Ocean. Tech., 26, 1543–1557, 2009. a, b
Whiteman, D. N.: Examination of the traditional Raman lidar technique. I. Evaluating the temperature-dependent lidar equations, Appl. Opt., 42, 2571–2592, 2003a. a
Whiteman, D. N.: Examination of the traditional Raman lidar technique. II. Evaluating the ratios for water vapor and aerosols, Appl. Opt., 42, 2593–2608, 2003b. a
Wulfmeyer, V. and Feingold, G.: On the relationship between relative humidity and particle backscattering coefficient in the marine boundary layer determined with differential absorption lidar, J. Geophys. Res.-Atmos., 105, 4729–4741, 2000. a
Yau, M. K. and Rogers, R.: A short course in cloud physics, Elsevier, 1996. a
Zhao, G., Zhao, C., Kuang, Y., Tao, J., Tan, W., Bian, Y., Li, J., and Li, C.: Impact of aerosol hygroscopic growth on retrieving aerosol extinction coefficient profiles from elastic-backscatter lidar signals, Atmos. Chem. Phys., 17, 12133–12143, https://doi.org/10.5194/acp-17-12133-2017, 2017. a
Zieger, P., Fierz-Schmidhauser, R., Gysel, M., Ström, J., Henne, S., Yttri, K. E., Baltensperger, U., and Weingartner, E.: Effects of relative humidity on aerosol light scattering in the Arctic, Atmos. Chem. Phys., 10, 3875–3890, https://doi.org/10.5194/acp-10-3875-2010, 2010. a
Zieger, P., Fierz-Schmidhauser, R., Weingartner, E., and Baltensperger, U.: Effects of relative humidity on aerosol light scattering: results from different European sites, Atmos. Chem. Phys., 13, 10609–10631, https://doi.org/10.5194/acp-13-10609-2013, 2013. a, b, c
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2020-01-21 14:25:33
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https://forum.revolutionarygamesstudio.com/t/refactor-compound-density-logic/831
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# Refactor compound density logic
Hello everyone,
While working on my global compound diffusion model, I realised that the gaseous compounds in the patches (nitrogen/dioxygen/carbon dioxyde) were encoded as proportions relative to one another, e.g. 70% N, 21% O_2, 9% CO_2, summing to 100%. This approach is very close to the one used in describing atmospheric composition (and you may note the proportions are roughly similar to Earth’s).
Note
I realized while writing this that people also mentionned it on the community forum, so I link it here for reference : Quick Question Thread - #1323 by hhyyrylainen - Current Game - Thrive Community Forum
Also, I use the term gas for dissolved compounds, although under specific temperature/pressure conditions they may not be in a gaseous state.
## The problem
However this falls short for our use cases: we are not describing atmospheric patches (mostly made of gas) but aquatic patches (mostly made of pure water, H_2O molecules). As such, while relative proportions work for atmosphere (because it is, in the end, a proportion relative to the total volume, sum of each gas’ volume), it does not for water (the total volume is, first and foremost, dependent on water volume).
In other words, 70% N might as well represent, at a given volume of water, 7 molecule of N per liter (if you have 10 gas molecules), or 7 billion (if you have 10 billions). As such, this proportion is not, of itself, sufficient to determine the influence of gases in the patch (this influence relies on the concentration of gases, i.e. how many molecules are in the immediate vicinity of a cell).
For this approach to work, we need another factor: the total amount of gas molecules. And the very reason why relative proportions have been working so long, is precisely because we assume a constant and similar total concentration of gases in every patch, a not very precise hypothesis: chemosynthesis, as an example, consumes both CO_2 and O_2 but does not produce N, effectively diminishing the total amount, and thus concentration.
18H_2S+6CO_2+3O_2 \rightarrow C_6H_{12}O_6+12H_2O+18S (chemosynthesis)
## How to fix this?
The obvious solution is to replace this proportion with the actual gas concentration. Note that absolute amount could be then obtained by multiplying the concentration by the volume of the patch. Given that there is no reason to assume similar volumes for patches (the estuary is probably smaller than the ocean), I therefore propose the following solution:
1. Introduce a measure of dissolved compounds related to their concentration (quantity/liter)
2. Introduce volume as a property of the patches
The reason why I propose a measure related to, and not equal to the concentration, is based on a remark from Buckly: 70% N means a 70% efficiency factor for N-based. As such, it is very convenient, both for display and settings, to have a percentage value, which we may call efficiency.
As such, we could still keep the current values system by decoupling the proportion values for dissolved compounds (which I think is not enforced in the code anyways?). This would mean that you could have 70% N, 70% O_2, 70% CO_2, as all three would be (initially) independent.
But the question is: what quantity of, say, N atoms would actually be 100% N efficiency? There is also probably no reason not to have efficiency going over 100% (because you may have more quantity than the organelle can process). So, we either have to add a cap or just say that base efficiency is not an upper limit, but just a necessary reference value.
## Fixed efficiency limit?
An answer could be, in accordance with our theorists, that we have a constant concentration x_N, so that a concentration \alpha x_N directly and linearly translates into efficiency \alpha (potentially over 100%). Depending on whether or not we want a hard cap on organelle efficiency, we could then, when running process computations, cap this to 100% if needed. This would still allow for an easier definition (linearity) and visibility (showing excess of a compound) to the player.
# Summary
To sum this up, I raised the issue of compound density not clearly reflecting an actual physico-chemical reality, hindering integration of parameters in populations. I proposed an alternative concentration-based measure, which we may call theoretical efficiency, reflecting a proportion (potentially greater) of a base concentration. This change, along with the addition of volumes to patches, would allow us to give our current parameters a clear relationship with real physical values, and therefore simplify integration of several simulation computations. Nonetheless, the use of an efficiency measure would still allow for a simple understanding of input and display values about concentration.
A few questions remain open:
1. Should process efficiency be capped? This answer would perhaps be the most important one given that it will influence following answers.
2. What base concentration should we choose? If we choose a cap concentration, after which process efficiency does not increase anymore, then what about several processes using the same compound with different cap concentration?
3. How to display efficiency to the player? Should we show the capped version (loosing information on whether this value may decrease easily or not)? If we don’t, how to tell the player that process efficiency will be capped?
4. How to integrate this dissolved compound logic with cloud compounds and sunlight? If this is possible, of course, especially for sunlight which is very particular a “compound” (and is actually labeled dissolved, but not distributed).
3 Likes
You bring up an interesting point that the game assumes the dissolved compounds to be uniform. I think for process efficiency in cells, that is entirely the right thing to do. As when the player swims around they are playing in real time. Because of that I doubt that the microbes can actually affect the concentration of compounds in a patch that much.
Of course I agree that we probably need to have a different model for auto-evo calculations and how the amount of available compounds change. From those we can then derive the concentrations for the player swimming around phase, I imagine. I like the idea of using a patch’s volume to determine the available compound amounts for auto-evo simulation, then the new amount can be reversed back to a concentration.
Another approach would be to flip this entirely: start from the total amount of a compound in a patch, and work out the concentration from that.
Unless some theorist jumps in, I think we should keep the process efficiency using the concentration values (percentage values, though maybe that also needs rethinking for the aquatic patches) as I think in the real world there are reactions that have their speed depend on the concentration of a catalyst / reactant.
So theres 2 things to take into account when calculating the density of gases in water
Temperature
Pressure.
But i think you already know this.
I think density(grams) is a much better and simpler measure than number of molecules / liter imho.
Sooooooo, we can start with the current concetration of atmospheric gases in sea water.
Im too layse right now to calculate using henry’s law or whatever, but according to the graphs,
This
This
And this
We can get the average densities of atmospheric gases of seawater at 20C at the surface as a reference.
O2 ~ 8 mg/L
N2 ~ 16mg/L
CO2 ~ 0.3mg/L
Ar ~ 0.34mg/L
etc.
So we can use this as reference.
We could also update the current patches concentration of gases according to the second pic i posted.
EDIT:
I think that making microbes influence the gases in the atmosphere could make for some very interesting scenarios.
Actually, it isn’t. As hh put it, the speed of a reaction is dependent on the concentration (mol/L) of its reactants and products:
You may have an overlook of chemical kinetics on wikipedia, but essentially the reaction speed v is determined as:
v = \frac{dc}{dt} = k(T) \times {\Pi_{i} c_i^{m_i}};
c_i being the concentration of the molecule i (reactant or product) and m_i its so-called partial order. Partial orders are generally (as far as I can tell) positive for reactants (more reactant means more encounters of molecules, thus faster reaction) and negative for products (you saturate more, so slower reaction/opposite reaction happening). k(T) is a speed constant specific to the reaction. It has many factor, including pressure in gas phase (so not here), but the main one is usually considered to be temperature.
As we can see here, there is no need to consider density for reaction speed; and this is especially because the molecule does not care about the mass of molecules it receives, but about the quantity of molecules it has to run its reactions (basically, how many there are in their vicinity).Of course, we could use molar mass to convert between both, but I believe there is no point in using such an indirect measure.
However, it must indeed be pointed at that temperature should probably play a greater role at some point, given it’s general impact on reaction speed. But I believe this isn’t a necessary refinement for our models to currently yield realistic enough outputs.
Also, we have an oversimplification for partial orders: we assume that partial orders are 1 for reactants and 0 for products, which is far from being true in general. But I don’t know about the reactions considered in the game, and how much a gross simplification it is.
Now, more specifically on hh’s answer
I think it is as well, at least for the moment. This is a first approximation that probably isn’t widely inaccurate, especially as patches are defined as homogeneous environments.
Nonetheless, we might have non-contextual differences (that could occur within patches) due to specific distribution laws. In cells, you may observe nanoclusters of proteins that are argued to be due to anomalous diffusion, and such non-linear diffusion has been observed in atmosphere, especially impacting climate.
I’m not very knowledgeable on the topic, but we might want to reconsider these refinements later, especially when dealing with atmosphere, if our models do not produce what is expected.
I fully agree on that, and I think it would bring unnecessary complications to our models anyways. So I’m in favor of considering it constant & constantly distributed (if we decide to revise uniformity, which is probably unnecessary as of now, if ever) during gameplay phases.
Both approaches are indeed theoretically equivalent, but in terms of input from Biomes.json files, we probably want to use concentration as it is more directly related to the process. This means that the developper changing the values won’t have to care about another variable, that is the volume of the patch. Otherwise, both would affect the output while it could be summed up into a single one.
Just to be really formal, percentage aren’t concentration (because % are dimensionless proportions, while concentrations are matter per volume unit.), even though you can convert with other parameters. That’s especially what I propose, using proportion \alpha to derive concentration c = \alpha c_0 for the simulation, with c_0 defined in Constants.cs.
The current approach, based on atmospheric composition, derives concentration as c = (\beta \times n_{total})/V, with n_{total} being the total amount of matter in the patch (excluding water for aquatic patches). As such n_C = \beta\times n_{total}) is the quantity of matter for compound C. But as you see it from the formula this approach needs the player to consider other variables than \beta, when looking for a patch were concentration is advantageous. The GUI would thus be less intuitive with this approach.
This is one of the reason why I push for a change of logic (along with the fact that n_{total} would require some more input to be computed for simulation).
1 Like
Whyyyyyy
20% of the earths O2
I think we should at least in teh future.
We are talking about uniformly distributed within a patch. Patches are meant to be uniform conditions that species encounter. If the conditions are different then, the patch should actually be two patches.
We used to have even the gaseous compounds being compound clouds (so you would need to find oxygen clouds, and your cell would emit small CO2 clouds), but it was removed in order to improve the gameplay by having fewer resources the player needs to find by swimming around.
1 Like
Density and reaction speed will scale linearily. I think it can be still used.
Ci = mass/molar mass/V = mass/V/molMass = density / molar mass
So theres the density in the equation
We can still use density.
Yes we can use density, but that’s unnecessarily complicated, as you basically convert it into concentration.
And I think this was indeed the right thing to do. I wanted to point out our simplifications that could potentially be questioned, e.g. if our models ended up with unrealistic behaviors.
1 Like
So Why not show players that thing? OR ppm, since its more intuitive than moles/L for those not having a sciency background.The conversion isnt that complicated, we could make a method to easily do that. We can use the mole/V for our internal calculation if we so wish.
Have you read my first two messages? I specifically propose not to use concentration as a display/input measure, but rather an efficiency percentage. This would be far more intuitive than ppm or volumic mass in terms of values.
I thought you wanted to propose something like this, efficiency i think could be shown alongside the volumic mass, at least to me, it would give me a nicer picture in my head of what is going on.
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2021-11-27 15:21:22
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https://math.stackexchange.com/questions/1526061/a-linear-map-t-in-m-nk
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# A linear map $T$ in $M_n(K)$
Let $M_n(K)$ denote the space of all $n×n$ matrices with entries in a field $K$. Fix a non-singular matrix $A=(A_{ij})\in M_n(K)$ and consider the linear map $T:M_n(K)→M_n(K)$ given by:
$T(X)=AX$.
Then:
1. $trace(T)=n\sum_{i=1}^nA_{ii}$
2. $trace(T)=\sum_{i=1}^n\sum_{j=1}^nA_{ij}$
3. $Rank(T)=n^2$ I am confused that in what sense options 1 and 3 are given correct? And then why not option 2 is correct in that sense? Please help.
For (c), since $A$ is non-singular hence $T$ is also non-singular, so nullity$(T)=0$. By rank-nullity theorem rank$(T)=n^2$ which is the dimension of $M_n(K)$.
(a) can be proved by considering a $2\times2$ case and then generalized.
(b) is false because trace involves only diagonal entries.
Another view for (c)
Suppose $$T(X)=T(Y)$$ $$AX=AY$$ $$A^{-1}AX=A^{-1}AY$$ $$X=Y$$ therefore $T$ is one-one, hence rank$(T)$=dim$(M_n(K))$
Consider the matrices $E_{ij}$, having coefficients always vanishing except for coefficient at row $i$ and column $j$ which has value equal to $1$.
$$\mathcal{E}=(E_{1,1}, \dots , E_{1,n},E_{2,1} \dots, E_{2,n},\dots,E_{n,1}, \dots, E_{n,n})$$ is a basis of $M_n(K)$.
You'll verify that for $1 \le i \le n , 1 \le j \le n$, you have $$T(E_{i,j})=\sum_{k=1}^n A_{k,i} E_{k,j}.$$ This is no more no less the way to get the matrix of $T$ in the basis $\mathcal{E}$ where $A=(A_{i,j})$.
To get (1), you can now apply the definition of the trace. The diagonal of the matrix of $T$ in the basis $\mathcal{E}$ is $$(A_{1,1},A_{1,1}, \dots,A_{1,1},A_{2,2},\dots,A_{2,2}, \dots, A_{n,n},\dots, A_{n,n})$$ proving that $\text{tr}(T)=n \sum_{i=1}^n A_{i,i}=n \text{tr}(A)$.
Obviously (2) is conflicting with (1)...
Regarding (3), what is the kernel of $T$? A matrix $X$ is in the kernel of $T$ if and only if $$T(X)=AX=0$$ which means that $A^{-1} A X=X=0$ as $A$ is supposed to be invertible. That means that $T$ is a non-singular linear map of $M_n(K)$ which is of dimension $n^2$. Hence $\text{rank}(T)=n^2$.
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2019-12-13 17:06:56
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https://mathsgee.com/29744/carry-out-test-significance-mathrm-hrs-h-mathrm-hrs-alpha
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0 like 0 dislike
89 views
The life in hours of a battery is known to be approximately normally distributed. The manufacture claims that the average battery life exceeds 40 hours. A random sample of 10 batteries has a mean life of $40.5$ hours and sample standard deviation $\mathrm{s}=1.25$ hours. Carry out a test of significance for $H_{0}: \mu=40 \mathrm{hrs}$ vs $H_{1} \mu>40 \mathrm{hrs} . \alpha=0.05$
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2022-08-19 02:49:13
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https://www.thejournal.club/c/paper/22857/
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#### On the Difficulty of Deciding Asymptotic Stability of Cubic Homogeneous Vector Fields
It is well-known that asymptotic stability (AS) of homogeneous polynomial vector fields of degree one (i.e., linear systems) can be decided in polynomial time e.g. by searching for a quadratic Lyapunov function. Since homogeneous vector fields of even degree can never be AS, the next interesting degree to consider is equal to three. In this paper, we prove that deciding AS of homogeneous cubic vector fields is strongly NP-hard and pose the question of determining whether it is even decidable. As a byproduct of the reduction that establishes our NP-hardness result, we obtain a Lyapunov-inspired technique for proving positivity of forms. We also show that for asymptotically stable homogeneous cubic vector fields in as few as two variables, the minimum degree of a polynomial Lyapunov function can be arbitrarily large. Finally, we show that there is no monotonicity in the degree of polynomial Lyapunov functions that prove AS; i.e., a homogeneous cubic vector field with no homogeneous polynomial Lyapunov function of some degree $d$ can very well have a homogeneous polynomial Lyapunov function of degree less than $d$.
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2023-01-28 11:14:32
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https://stats.stackexchange.com/questions/358862/what-is-mean-by-the-term-constant-rate-in-poisson-distribution
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# What is mean by the term “constant rate” in Poisson distribution?
I have difficulty in understanding the assumptions of Poisson distribution, one assumption is the rate at which the events occur in the time interval is constant. What is the meaning of that phrase? I would highly appreciate if you give me an intuitive example (not a rigorous mathematical one).
Thanks
• You are confusing the "Poisson distribution" with the "Poisson process".They are not the same. The Poisson distribution has one parameter (which of course is constant). The Poisson process has a rate which can be constant or not. – Zahava Kor Jul 25 '18 at 14:13
The typical intuitive example is this: suppose we are interested in finding the probability distribution of the number of cars that will drive through a particular intersection during a period of one hour. One strategy would be to divide this one-hour period into many, very small periods: periods that are so small, that the probability that more than one car will cross the intersection during each period is negligible (i.e. zero). These $n$ very small periods can now be thought of as containing Bernoulli random variables $x$, with $p$ being the probability of observing a car, and $(1-p)$ the probability of not observing a car. The sum of these Bernoulli random variables would then give us the number of cars for an hour and, assuming they are i.i.d., this sum is Binomial distributed. But, how can we choose an appropriate $n$? With $n=\infty$, that Binomial distribution actually becomes the Poisson distribution.
Finally, here is the intuition you are looking for. If the number of cars you observe each hour (the number of events in each time period $t$) is Poisson distributed, then the time that passes between each event of observing a car has an Exponential distribution. A process like this, where the counts are Poisson and the durations are Exponential, has a constant hazard rate.
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2019-08-21 06:00:02
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https://www.nature.com/articles/s41558-020-00955-x?error=cookies_not_supported&code=2f076fa1-a539-4128-9204-789d4b2d46b6
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# Greater committed warming after accounting for the pattern effect
## Abstract
Our planet’s energy balance is sensitive to spatial inhomogeneities in sea surface temperature and sea ice changes, but this is typically ignored in climate projections. Here, we show the energy budget during recent decades can be closed by combining changes in effective radiative forcing, linear radiative damping and this pattern effect. The pattern effect is of comparable magnitude but opposite sign to Earth’s net energy imbalance in the 2000s, indicating its importance when predicting the future climate on the basis of observations. After the pattern effect is accounted for, the best-estimate value of committed global warming at present-day forcing rises from 1.31 K (0.99–2.33 K, 5th–95th percentile) to over 2 K, and committed warming in 2100 with constant long-lived forcing increases from 1.32 K (0.94–2.03 K) to over 1.5 K, although the magnitude is sensitive to sea surface temperature dataset. Further constraints on the pattern effect are needed to reduce climate projection uncertainty.
## Access options
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## Data availability
All observational data and AMIP-piForcing experiment data in Table 1 are publicly available online, as described in Methods. In addition, results of the idealized experiments carried out in this study are available from the corresponding author upon request.
## Code availability
The code of CESM1.2-CAM5.3 model used in this paper can be downloaded from http://www.cesm.ucar.edu/models/cesm1.2/. Codes for plotting figures are available from the corresponding author upon request.
## References
1. 1.
Trenberth, K. E., Fasullo, J. T., von Schuckmann, K. & Cheng, L. Insights into Earth’s energy imbalance from multiple sources. J. Clim. 29, 7495–7505 (2016).
2. 2.
Boucher, O. et al. in Climate Change 2013: The Physical Science Basis (eds Stocker, T. F. et al.) Ch. 7 (IPCC, Cambridge Univ. Press, 2013).
3. 3.
Stocker, T. F. et al. in Climate Change 2013: The Physical Science Basis (eds Stocker, T. F. et al.) 31–116 (IPCC, Cambridge Univ. Press, 2013).
4. 4.
Andrews, T., Gregory, J. M. & Webb, M. J. The dependence of radiative forcing and feedback on evolving patterns of surface temperature change in climate models. J. Clim. 28, 1630–1648 (2015).
5. 5.
Zhou, C., Zelinka, M. D. & Klein, S. A. Impact of decadal cloud variations on the Earth’s energy budget. Nat. Geosci. 9, 871–874 (2016).
6. 6.
Gregory, J. M. & Andrews, T. Variation in climate sensitivity and feedback parameters during the historical period. Geophys. Res. Lett. 43, 3911–3920 (2016).
7. 7.
Xie, S.-P., Kosaka, Y. & Okumura, Y. M. Distinct energy budgets for anthropogenic and natural changes during global warming hiatus. Nat. Geosci. 9, 29–33 (2016).
8. 8.
Zhou, C., Zelinka, M. D. & Klein, S. A. Analyzing the dependence of global cloud feedback on the spatial pattern of sea surface temperature change with a Green’s function approach. J. Adv. Model. Earth Syst. 9, 2174–2189 (2017).
9. 9.
Armour, K. C. Energy budget constraints on climate sensitivity in light of inconstant climate feedbacks. Nat. Clim. Change 7, 331–335 (2017).
10. 10.
Ceppi, P. & Gregory, J. M. Relationship of tropospheric stability to climate sensitivity and Earth’s observed radiation budget. Proc. Natl Acad. Sci. USA 114, 13126–13131 (2017).
11. 11.
Andrews, T. et al. Accounting for changing temperature patterns increases historical estimates of climate sensitivity. Geophys. Res. Lett. 45, 8490–8499 (2018).
12. 12.
Silvers, L. G., Paynter, D. & Zhao, M. The diversity of cloud responses to twentieth century sea surface temperatures. Geophys. Res. Lett. 45, 391–400 (2018).
13. 13.
Dessler, A. E., Mauritsen, T. & Stevens, B. The influence of internal variability on Earth’s energy balance framework and implications for estimating climate sensitivity. Atmos. Chem. Phys. 18, 5147–5155 (2018).
14. 14.
Dessler, A. E. Potential problems measuring climate sensitivity from the historical record. J. Clim. 33, 2237–2248 (2019).
15. 15.
Dong, Y., Proistosescu, C., Armour, K. C. & Battisti, D. S. Attributing historical and future evolution of radiative feedbacks to regional warming patterns using a Green’s function approach: the preeminence of the western Pacific. J. Clim. 32, 5471–5491 (2019).
16. 16.
Fueglistaler, S. Observational evidence for two modes of coupling between sea surface temperatures, tropospheric temperature profile, and shortwave cloud radiative effect in the tropics. Geophys. Res. Lett. 46, 9890–9898 (2019).
17. 17.
Leob, N. G. et al. New generation of climate models track recent unprecedented changes in earth’s radiation budget observed by CERES. Geophys. Res. Lett. 47, e2019GL086705 (2020).
18. 18.
Gregory, J. M. et al. A new method for diagnosing radiative forcing and climate sensitivity. Geophys. Res. Lett. 31, L03205 (2004).
19. 19.
Olonscheck, D., Rugenstein, M. & Marotzke, J. Broad consistency between observed and simulated trends in sea surface temperature patterns. Geophys. Res. Lett. 47, e2019GL086773 (2020).
20. 20.
Prather, M. G. et al. in Climate Change 2013: The Physical Science Basis (eds Stocker, T. F. et al.) 1395–1446 (IPCC, Cambridge Univ. Press, 2013).
21. 21.
Loeb, N. G. et al. Clouds and the Earth’s radiant energy system (CERES) energy balanced and filled (EBAF) top-of-atmosphere (TOA) edition-4.0 data product. J. Clim. 31, 895–918 (2018).
22. 22.
Allan, R. P. Decadal climate variability and the global energy balance. Past Glob. Changes 25, 20–24 (2017).
23. 23.
Andrews, T. & Forster, P. M. Energy budget constraints on historical radiative forcing. Nat. Clim. Change 10, 313–316 (2020).
24. 24.
Church, J. A. et al. Revisiting the Earth’s sea-level and energy budgets from 1961 to 2008. Geophys. Res. Lett. 38, L18601 (2011).
25. 25.
Levitus, S. et al. World ocean heat content and thermosteric sea level change (0–2000 m), 1955–2010. Geophys. Res. Lett. 39, L10603 (2012).
26. 26.
Mauritsen, T. & Pincus, R. Committed warming inferred from observations. Nat. Clim. Change 7, 652–655 (2017).
27. 27.
Forster, P. M. Inference of climate sensitivity from analysis of Earth’s energy budget. Annu. Rev. Earth Planet. Sci. 44, 85–106 (2016).
28. 28.
Sherwood, S. et al. An assessment of Earth’s climate sensitivity using multiple lines of evidence. Rev. Geophys. 58, e2019RG000678 (2020).
29. 29.
Smith, C. J. et al. Current fossil fuel infrastructure does not yet commit us to 1.5 °C warming. Nat. Commun. 10, 101 (2019).
30. 30.
Lewis, N. & Mauritsen, T. Negligible unforced historical pattern effect on climate feedback strength found in HadISST-based AMIP simulations. J. Clim. https://doi.org/10.1175/JCLI-D-19-0941.1 (2020).
31. 31.
Rayner, N. A. et al. Global analyses of sea surface temperature, sea ice, and night marine air temperature since the late nineteenth century. Geophys. Res. 108, 4407 (2003).
32. 32.
Hurrell, J. W., Hack, J. J., Shea, D., Caron, J. M. & Rosinski, J. A new sea surface temperature and sea ice boundary dataset for the Community Atmosphere Model. J. Clim. 21, 5145–5153 (2008).
33. 33.
Neale, R. B. et al. Description of the NCAR community atmosphere model (CAM 5.0) NCAR/TN-486+STR (National Centre for Atmospheric Research, 2012).
34. 34.
Meinshausen, M., Raper, S. C. B. & Wigley, T. M. L. Emulating coupled atmosphere-ocean and carbon cycle models with a simpler model, MAGICC6 – Part 1: model description and calibration. Atmos. Chem. Phys. 11, 1417–1456 (2011).
35. 35.
Rockström, J. et al. A roadmap for rapid decarbonization. Science 355, 1269–1271 (2017).
36. 36.
Flato, G. et al. in Climate Change 2013: The Physical Science Basis (eds Stocker, T. F. et al.) Ch. 9 (IPCC, Cambridge Univ. Press, 2013).
37. 37.
Etminan, M., Myhre, G., Highwood, E. J. & Shine, K. P. Radiative forcing of carbon dioxide, methane, and nitrous oxide: a significant revision of the methane radiative forcing. Geophys. Res. Lett. 43, 12614–12623 (2016).
38. 38.
Ciais, P. et al. in Climate Change 2013: The Physical Science Basis (eds Stocker, T. F. et al.) Ch. 6 (IPCC, Cambridge Univ. Press, 2013).
39. 39.
Zhou, T. & Chen, X. Uncertainty in the 2°C warming threshold related to climate sensitivity and climate feedback. J. Meteorol. Res. 29, 884–895 (2015).
40. 40.
Andrews, T., Gregory, J. M., Webb, M. J. & Taylor, K. E. Forcing, feedbacks and climate sensitivity in CMIP5 coupled atmosphere–ocean climate models. Geophys. Res. Lett. 39, L09712 (2012).
41. 41.
Eyring, V. et al. Overview of the coupled model intercomparison project phase 6 (CMIP6) experimental design and organization. Geosci. Model Dev. 9, 1937–1958 (2016).
42. 42.
Loeb, N. G. et al. Observed changes in top-of-the-atmosphere radiation and upper-ocean heating consistent within uncertainty. Nat. Geosci. 5, 110–113 (2012).
43. 43.
Wielicki, B. A. et al. Evidence for large decadal variability in the tropical mean radiative energy budget. Science 295, 841–844 (2002).
44. 44.
Otto, A. et al. Energy budget constraints on climate response. Nat. Geosci. 6, 415–416 (2013).
45. 45.
Jiménez-de-la-Cuesta, D. & Mauritsen, T. Emergent constraints on Earth’s transient and equilibrium response to doubled CO2 from post-1970s global warming. Nat. Geosci. 12, 902–905 (2019).
## Acknowledgements
C.Z. was supported by NSFC grant no. 41875095. A.E.D. was supported by NSF grant nos. AGS-1661861 and AGS-1841308, both to Texas A&M University. M.D.Z. worked under the auspices of the US Department of Energy (DOE), Lawrence Livermore National Laboratory under contract no. DE-AC52-07NA27344 and was supported by the Regional and Global Model Analysis Program of the Office of Science at the DOE. M.W. was supported by Minister of Science and Technology of China grant nos. 2017YFA0604002 and 2016YFC0200503, and NSFC grant nos. 91744208, 41575073 and 41621005. This research is also supported by the Collaborative Innovation Center of Climate Change, Jiangsu Province. The numerical simulations in this paper were done on the computing facilities in the High Performance Computing Center of Nanjing University. Correspondence and requests for materials should be addressed to C.Z.
## Author information
Authors
### Contributions
C.Z. performed the analysis. The paper was discussed and written by all authors.
### Corresponding author
Correspondence to Chen Zhou.
## Ethics declarations
### Competing interests
The authors declare no competing interests.
Peer review information Nature Climate Change thanks Jonah Bloch-Johnson, Diego Jiménez-de-la-Cuesta and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
## Rights and permissions
Reprints and Permissions
Zhou, C., Zelinka, M.D., Dessler, A.E. et al. Greater committed warming after accounting for the pattern effect. Nat. Clim. Chang. (2021). https://doi.org/10.1038/s41558-020-00955-x
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2021-01-23 14:45:24
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http://blog.startry.com/2015/08/25/Renaming-umbrella-header-for-iOS-framework/
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Umbrella frameworks add minor refinements to the standard framework structure, such as the ability to encompass other frameworks
The structure of an umbrella framework is similar to that of a standard framework, and applications do not distinguish between umbrella frameworks and standard frameworks when linking to them. However, two factors distinguish umbrella frameworks from other frameworks. The first is the manner in which they include header files. The second is the fact that they encapsulate subframeworks.
Physically, umbrella frameworks have a similar structure to standard frameworks. One significant difference is the addition of a Frameworks directory to contain the subframeworks that make up the umbrella framework.
For most frameworks, you can include header files other than the master header file. You can include any specific header file you want as long as it is available in the framework’s Headers directory. However, if you are including an umbrella framework, you must include the master header file. Umbrella frameworks do not allow you to include the headers of their constituent subframeworks directly. See Restrictions on Subframework Linking for more information.
1. 在Umbrella Framework新建一个testObject类, 分别产生了testObject.htestObject.m文件。
2. 打开Framework配置文件, 在Build PhasesHeaders里的Public目录下, 将testObject.h文件添加进去。
3. Build Framework看是否报错。
4. 在主工程中调用初始化testObject对象, 看编译是否报错。
• 步骤3: 没有编译报错, 但是报出了Lexical or Preprocessor Issue - Umbrella header for module ‘STDemoUI’ does not include header ‘testObject.h’的警告。
• 步骤4: 执行正常。
1. Standard Framework不能包含Sub Framework; Umbrella Framework可以包含子Framework;
### 规范的写法
1. 在工程全局搜索umbrella关键字 - Failed
2. 在Build Settings里搜索umbrella关键字 - Failed
3. 在打包好的STDemoUI.framework中搜索umbrella关键字 - Bingo
• _CodeSignature: 保存签名相关文件
• Info.plist: 描述了该framework所包含的项目配置信息
• STDemo: 编译后的核心库文件
• Modules: 模块相关文件夹, 目测只包含了module.modulemap文件
#### 指定Modulemap文件
2. 创建一个新的modulemap文件; ex: stdemoalt.modulemap
3. 在新的modulemap中指定umbrella header
1. 在framework的Build Settings中的Module Map File指定新建的modulemap文件
1. Module是什么?
2. 如果Defines Module指定为NO, 那会发生什么事情呢?
### 题外话
While it is possible to create umbrella frameworks using Xcode, doing so is unnecessary for most developers and is not recommended. Apple uses umbrella frameworks to mask some of the interdependencies between libraries in the operating system. In nearly all cases, you should be able to include your code in a single, standard framework bundle. Alternatively, if your code was sufficiently modular, you could create multiple frameworks, but in that case, the dependencies between modules would be minimal or nonexistent and should not warrant the creation of an umbrella for them
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2018-10-19 06:24:32
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http://theory2life.blogspot.com/2008/12/did-he-just-say-what-i-think-he-said.html
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## Monday, December 8, 2008
### Did he just say what I think he said?
Dan Neil thinks we should nationalize General Motors:
"What to do about the domestic automakers? My modest proposal: Nationalize GM.
To be clear, I mean that the federal government should buy GM; forget rathole loans or nonvoting equity shares. The company's stockholder value has been essentially wiped out. The company's enterprise value -- the lock, stock and forklift price -- is about $32 billion; its total debt is$45 billion. Let's make GM an offer.
If you feel the gall of free-market ideology rising, consider that the measures being bruited about as preconditions for a bailout -- firing GM's top management; forcing a bankruptcy-like renegotiation of contracts with the UAW, suppliers and dealers (it has too many); and creating a czar of product development to force the building of green cars -- are nationalization in all but name. I say embrace it. GM-USA."
Neil argues that there are many benefits to nationalizing GM. In particular, he argues that the government has a longer-term view that is crucial for getting GM prepared for the future:
"The government can afford long-term planning. Many of GM's strategic missteps -- such as betting large on trucks and SUVs and not investing early in hybrid technology -- were the result of willful shortsightedness at the board level, responding to a financial market in which shareholders look for the quick return. Putting Uncle Sam in charge would fundamentally enlarge the return-on-investment horizon."
I heard Mr. Neil speaking on NPR last week. He said that while markets are really good at responding to consumer demands, they are really bad at anticipating them. So, if you want to get car companies to invest in environmentally sustainable technologies, the government needs to step in.
This is a terrible, terrible idea. If you think GM's management is bad now, just wait till Congress is in charge. Any company run by a group of people with no personal investment at stake and tons of external political pressure is doomed for failure.
I also completely disagree that government is any better at anticipating future needs. Markets have considerably more information with which to make those sorts of decisions and better incentives for investing in new ideas that will actually work. When Congress decided we needed alternative fuels, they put the brunt of their resources into corn-based ethonol, which we now know yields less energy than is required to create it. Government does not have a good track record with picking the winners.
Additionally, Neil claims Japanese and European manufacturers are "quasi-national" because of the government's role in healthcare and retirement costs. Fair enough. But isn't that an argument for universal healthcare and a stronger social safety net?
I'm a big proponent of people sticking to what they're good at. Congress is good at bloviation, not running corporations.
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2017-08-17 15:36:01
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https://www.physicsforums.com/threads/hii-know-that-for-a-short-solenoid-l-r-the-magnetic-field-at-the.573818/
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# HiI know that for a short solenoid (L<R) the magnetic field at the
1. Feb 3, 2012
### Niles
Hi
I know that for a short solenoid (L<R) the magnetic field at the axis is (standard EM)
$$B(z) = \frac{1}{2}\mu_0 \frac{N}{L}I(\frac{z+\frac{L}{2}}{\sqrt{(z+L/2)^2+R^2}} - \frac{z-\frac{L}{2}}{\sqrt{(z-L/2)^2+R^2}})$$
where R is the radius of the solenoid, N the number of turns along the axis and L the length. In this system, each vertical plane consists of a single turn, but say I am looking at a solenoid, where each vertical plane consists of e.g. 2 turns. First I thought about using the above equation twice, but that is wrong since it is not 2 independent solenoids.
Is it correct to regard the system simply as a collection of coils with 2 turns each? I'm not quite sure how this would work out, since this way I can't take into account the widths of each individual coil. If my description is confusing, please let me know.
Best,
Niles.
Last edited: Feb 3, 2012
2. Feb 3, 2012
### clem
Re: Coils
Your formula is for a solenoid of any length, not just L<R.
I don't understand what you mean be 'two turns'.
Why would it make a difference?
3. Feb 3, 2012
### Niles
Re: Coils
Hi, thanks for replying. I really appreciate it. What I mean by two turns is the following:
OOOOOO
OOOOOO
------------ (axis of solenoid)
OOOOOO
OOOOOO
There I have a solenoid with 12 turns in total, i.e. a 6-turn solenoid on top of another (larger radii) 6-turn solenoid. How would one calculate the B-field for that? In principle I could just add the B-field for each one of the 12 turns, right? The reason why I am asking is because I am looking for the most general way to calculate this, because I would also have to look at e.g.
OOOO
OOOOOO
OOOOOOOO
---------------------- (axis)
OOOOOOOO
OOOOOO
OOOO
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2017-11-23 12:01:37
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https://aviation.stackexchange.com/questions/81332/why-is-there-a-pitching-moment-at-aerodynamic-center
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# Why is there a pitching moment at aerodynamic center?
As far as I know, the pitching moment is an aerodynamic moment and is related to the aerodynamic center.
But I am confused with the following figure, which shows the aerodynamic force applied at the aerodynamic center. So I would think, in this case, that the center of pressure is located at the aerodynamic center which is a quarter chord. If so, why is there a moment ?
The definition of the aerodynamic center is the following: "the aerodynamic center is the point along the wing chord where the coefficient of aerodynamic moment is constant with respect to the AOA." It doesn't mean that it is equal to 0.
To make it simple, imagine taking a cambered wing, if you look at the center-line of the profile, it will have a changing AOA along the chord of the profile. Setting the wing at a 0° AOA will result for a positively cambered profile into a negative AOA and a negative lift near the leading edge and a positive AOA and lift near trailing edge. Thus creating an absolute torque pitching down the wing profile. The aerodynamic center is the point where the coefficient related to this torque is not impacted by the AOA.
The point you are referring to is the center of lift ($$P$$). It's the point where the aerodynamic torque is equal to 0. But this point location is changing with respect to the AOA. You have the following relation:
$$C_{m_F} = C_{m_P}-(x_P-x_F)\cdot C_Z \rightarrow x_P = x_F - \frac{C_{m_F}}{C_Z}$$
Which means that for a classic wing with positive camber and thus negative torque, the center of lift is located after the aerodynamic center in cruise condition with $$C_z > 0$$.
It is more practical to always express the aerodynamic load at a given fixed point, the aerodynamic center with the couple (lift, torque) than expressing just the lift at the lift center which is moving with respect to flight conditions.
• so can I say, the diagram I was referring to, the aerodynamic force was not really at the aerodynamic center, just that, the force is indicated there – PJerk Sep 28 '20 at 15:27
• Lift results from the pressure field around the airfoil. Any point you pick to represent it as ponctual force will be arbitrary. You can either chose the aerodynamic center, the center of lift, the leading edge or trailing edge the physics works. You juste have to modify the aerodynamic moment ascociated. – MaximEck Sep 28 '20 at 15:35
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2021-07-24 21:10:15
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https://math.stackexchange.com/questions/2233521/find-rotation-axis-in-3d-space-with-quaternions
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# Find rotation axis in 3D space with quaternions
Rotation of $u_1$ around $q_1$(unknown) axis is $u_2$, and rotation of $u_2$ around $q_2$(unknown) axis is $u_3$. What is the value of axis $q$,in case $q$=$q_1$=$q_2$?
Note that possible values of $q_1$ makes a circle,$c_1$, in 3d space,and $q_2$ another circle,$c_2$, in 3d space. Question is to find the intersection($q$,unit vector) of these circles, $c_1$ and $c_2$.
related to question in find quaternion scalar from end points of the rotation
• @jyrki-lahtonen wondering if you know the answer for this problem? – user818117 Apr 15 '17 at 17:41
Axis of rotation of $q_1$ is $c_{1} = u_1 \times u_2$.
Axis of rotation of $q_2$ is $c_{2} = u_2 \times u_3$.
The two circles intersect at $\pm (c_{1} \times c_{2})$ normalized to unit length.
• in your case $c_1$ is a possible vector from the solution set,as well as $c_2$,imagine there are 2 circles in 3D space having the same center,if you pick random vectors from each as $c_1$,$c_2$, and consider a case $c_1$,and $c_2$ are very close to to the intersection point then $c_1$x$c_2$ will be close to perpendicular of intersection point, which also sates the intersectin may not be at the vectors ±($c_1$x$c_2$) – user818117 Apr 15 '17 at 17:39
• yes, if $\pm (c_{1} \times c_{2})$ is nearly zero, then the three vectors are yearly in the same plane. – Tpofofn Apr 15 '17 at 21:29
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2019-12-07 04:40:12
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http://archive.numdam.org/item/AIHPA_1981__34_4_405_0/
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Some remarks on double-wells in one and three dimensions
Annales de l'I.H.P. Physique théorique, Volume 34 (1981) no. 4, p. 405-417
@article{AIHPA_1981__34_4_405_0,
author = {Klaus, M.},
title = {Some remarks on double-wells in one and three dimensions},
journal = {Annales de l'I.H.P. Physique th\'eorique},
publisher = {Gauthier-Villars},
volume = {34},
number = {4},
year = {1981},
pages = {405-417},
zbl = {0474.35041},
mrnumber = {625171},
language = {en},
url = {http://www.numdam.org/item/AIHPA_1981__34_4_405_0}
}
Klaus, M. Some remarks on double-wells in one and three dimensions. Annales de l'I.H.P. Physique théorique, Volume 34 (1981) no. 4, pp. 405-417. http://www.numdam.org/item/AIHPA_1981__34_4_405_0/
[1] M. Klaus and B. Simon, Ann. Inst. H. Poincaré, t. XXX, no. 2, 1979, p. 83-87. | Numdam
[2] E.M. Harrell, Comm. Math. Phys., t. 75, 1980, p. 239. | MR 581948 | Zbl 0445.35036
[3] J. Morgan and B. Simon, Int. J. Quantum Chem., t. 17, 1980, p. 1143.
[4] M. Reed and B. Simon, Methods of Modern Mathematical Physics, IV. Analysis of Operators, Academic Press, 1978. | MR 493421 | Zbl 0401.47001
[5] M. Klaus and B. Simon, Coupling Constant Thresholds in Non relativistic Quantum Mechanics, I. Short Range Two Body Case, to appear in Ann. of Physics. | MR 610664 | Zbl 0455.35112
[6] H.R. Gruemm, Rep. Math. Phys., t. 4, 1973, p. 211. | Zbl 0258.47022
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2020-12-05 08:31:04
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https://www.statsmodels.org/v0.11.0/generated/statsmodels.tsa.statespace.exponential_smoothing.ExponentialSmoothing.html
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# statsmodels.tsa.statespace.exponential_smoothing.ExponentialSmoothing¶
class statsmodels.tsa.statespace.exponential_smoothing.ExponentialSmoothing(endog, trend=False, damped_trend=False, seasonal=None, initialization_method='estimated', initial_level=None, initial_trend=None, initial_seasonal=None, bounds=None, concentrate_scale=True, dates=None, freq=None, missing='none')[source]
Linear exponential smoothing models
Parameters
endogarray_like
The observed time-series process $$y$$
trendbool, optional
Whether or not to include a trend component. Default is False.
damped_trendbool, optional
Whether or not an included trend component is damped. Default is False.
seasonalint, optional
The number of periods in a complete seasonal cycle for seasonal (Holt-Winters) models. For example, 4 for quarterly data with an annual cycle or 7 for daily data with a weekly cycle. Default is no seasonal effects.
initialization_methodstr, optional
Method for initialize the recursions. One of:
• ‘estimated’
• ‘concentrated’
• ‘heuristic’
• ‘known’
If ‘known’ initialization is used, then initial_level must be passed, as well as initial_slope and initial_seasonal if applicable. Default is ‘estimated’.
initial_levelfloat, optional
The initial level component. Only used if initialization is ‘known’.
initial_trendfloat, optional
The initial trend component. Only used if initialization is ‘known’.
initial_seasonalarray_like, optional
The initial seasonal component. An array of length seasonal or length seasonal - 1 (in which case the last initial value is computed to make the average effect zero). Only used if initialization is ‘known’.
boundsiterable[tuple], optional
An iterable containing bounds for the parameters. Must contain four elements, where each element is a tuple of the form (lower, upper). Default is (0.0001, 0.9999) for the level, trend, and seasonal smoothing parameters and (0.8, 0.98) for the trend damping parameter.
concentrate_scalebool, optional
Whether or not to concentrate the scale (variance of the error term) out of the likelihood.
Notes
The parameters and states of this model are estimated by setting up the exponential smoothing equations as a special case of a linear Gaussian state space model and applying the Kalman filter. As such, it has slightly worse performance than the dedicated exponential smoothing model, sm.tsa.ExponentialSmoothing, and it does not support multiplicative (nonlinear) exponential smoothing models.
However, as a subclass of the state space models, this model class shares a consistent set of functionality with those models, which can make it easier to work with. In addition, it supports computing confidence intervals for forecasts and it supports concentrating the initial state out of the likelihood function.
References
[1] Hyndman, Rob, Anne B. Koehler, J. Keith Ord, and Ralph D. Snyder.
Forecasting with exponential smoothing: the state space approach. Springer Science & Business Media, 2008.
Methods
clone(endog[, exog]) filter(params[, cov_type, cov_kwds, …]) Kalman filtering fit([start_params, transformed, …]) Fits the model by maximum likelihood via Kalman filter. fit_constrained(constraints[, start_params]) Fit the model with some parameters subject to equality constraints. fix_params(params) Fix parameters to specific values (context manager) from_formula(formula, data[, subset]) Not implemented for state space models handle_params(params[, transformed, …]) hessian(params, *args, **kwargs) Hessian matrix of the likelihood function, evaluated at the given parameters impulse_responses(params[, steps, impulse, …]) Impulse response function information(params) Fisher information matrix of model. Initialize (possibly re-initialize) a Model instance. initialize_approximate_diffuse([variance]) Initialize approximate diffuse initialize_known(initial_state, …) Initialize known initialize_statespace(**kwargs) Initialize the state space representation Initialize stationary loglike(params, *args, **kwargs) Loglikelihood evaluation loglikeobs(params[, transformed, …]) Loglikelihood evaluation observed_information_matrix(params[, …]) Observed information matrix opg_information_matrix(params[, …]) Outer product of gradients information matrix predict(params[, exog]) After a model has been fit predict returns the fitted values. Prepare data for use in the state space representation score(params, *args, **kwargs) Compute the score function at params. score_obs(params[, method, transformed, …]) Compute the score per observation, evaluated at params set_conserve_memory([conserve_memory]) Set the memory conservation method set_filter_method([filter_method]) Set the filtering method set_inversion_method([inversion_method]) Set the inversion method set_smoother_output([smoother_output]) Set the smoother output set_stability_method([stability_method]) Set the numerical stability method simulate(params, nsimulations[, …]) Simulate a new time series following the state space model simulation_smoother([simulation_output]) Retrieve a simulation smoother for the state space model. smooth(params[, cov_type, cov_kwds, …]) Kalman smoothing transform_jacobian(unconstrained[, …]) Jacobian matrix for the parameter transformation function transform_params(unconstrained) Transform unconstrained parameters used by the optimizer to constrained parameters used in likelihood evaluation untransform_params(constrained) Transform constrained parameters used in likelihood evaluation to unconstrained parameters used by the optimizer update(params[, transformed, …]) Update the parameters of the model
Properties
endog_names Names of endogenous variables. exog_names The names of the exogenous variables. initial_variance initialization k_params loglikelihood_burn param_names (list of str) List of human readable parameter names (for parameters actually included in the model). start_params (array) Starting parameters for maximum likelihood estimation. state_names (list of str) List of human readable names for unobserved states. tolerance
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2022-09-25 01:21:02
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https://ask.sagemath.org/questions/56866/revisions/
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Ask Your Question
# Revision history [back]
### Math typesetting partly broken
If I include some math in my post, like $x^2 + y^2 = z^3$, then the post often ends up with extraneous line breaks right before the math. For example $x^2$ and $y^2$ and $z^3$ all start new lines, for no good reason. I think this is a recent change.
### Math typesetting partly broken
If I include some math in my post, like $x^2 + y^2 = z^3$, then the post often ends up with extraneous line breaks right before the math. For example $x^2$ and $y^2$ and $z^3$ all start new lines, for no good reason. I think this is a recent change.
### Math typesetting partly broken
If I include some math in my post, like $x^2 + y^2 = z^3$, then the post often ends up with extraneous line breaks right before the math. For example $x^2$ and $y^2$ and $z^3$ all start new lines, for no good reason. I think this is a recent change.
### Math typesetting partly brokenbroken on Safari
If I include some math in my post, like $x^2 + y^2 = z^3$, then the post often ends up with extraneous line breaks right before the math. For example $x^2$ and $y^2$ and $z^3$ all start new lines, for no good reason. I think this is a recent change.
Here is a screenshot:
5 None slelievre 15554 ●19 ●144 ●307 http://carva.org/samue...
### Math typesetting partly broken on Safari
If I include some math, like $x^2 + y^2 = z^3$, in an Ask Sage question or answer, then the post often ends up with an extraneous line break right before each inline math formula, when viewed in Safari on macOS.
For example $x^2$ and $y^2$ and $z^3$ all start new lines, for no good reason. I think this is a recent change.
Here is an earlier version of this same question:
If I include some math in my post, like $x^2 + y^2 = z^3$, z^3$, then the post often ends up with extraneous line breaks breaks right before the math. For example$x^2$and$y^2y^2$and$z^3\$ all start new lines, for no good reason. reason.
I think this is a recent change. Here is a screenshot: change.
and a screenshot of how it was rendered in Safari:
A web search for [ mathjax safari newline ] reveals similar questions:
The MathJax issue link suggests that it's been fixed in a more recent MathJax version. Can we update the version of MathJax used on this site?
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2021-11-27 10:42:36
|
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https://stellargraph.readthedocs.io/en/stable/_modules/stellargraph/data/unsupervised_sampler.html
|
# Source code for stellargraph.data.unsupervised_sampler
#!/usr/bin/env python3
# -*- coding: utf-8 -*-
#
#
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
#
# Unless required by applicable law or agreed to in writing, software
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
__all__ = ["UnsupervisedSampler"]
import numpy as np
from stellargraph.core.utils import is_real_iterable
from stellargraph.core.graph import StellarGraph
from stellargraph.data.explorer import UniformRandomWalk
from stellargraph.random import random_state
def _warn_if_ignored(value, default, name):
if value != default:
raise ValueError(
f"walker, {name}: cannot specify both 'walker' and '{name}'. Please use one or the other."
)
[docs]class UnsupervisedSampler:
"""
The UnsupervisedSampler is responsible for sampling walks in the given graph
and returning positive and negative samples w.r.t. those walks, on demand.
The positive samples are all the (target, context) pairs from the walks and the negative
samples are contexts generated for each target based on a sampling distribution.
By default, a UniformRandomWalk is used, but a custom walker can be specified instead. An
error will be raised if other parameters are specified along with a custom walker.
.. seealso::
Examples using this sampler:
- Attri2Vec: node classification <https://stellargraph.readthedocs.io/en/stable/demos/node-classification/attri2vec-node-classification.html>__ link prediction <https://stellargraph.readthedocs.io/en/stable/demos/link-prediction/attri2vec-link-prediction.html>__, unsupervised representation learning <https://stellargraph.readthedocs.io/en/stable/demos/embeddings/attri2vec-embeddings.html>__
- GraphSAGE: unsupervised representation learning <https://stellargraph.readthedocs.io/en/stable/demos/embeddings/graphsage-unsupervised-sampler-embeddings.html>__
- Node2Vec: node classification <https://stellargraph.readthedocs.io/en/stable/demos/node-classification/keras-node2vec-node-classification.html>__, unsupervised representation learning <https://stellargraph.readthedocs.io/en/stable/demos/embeddings/keras-node2vec-embeddings.html>__
- comparison of link prediction algorithms <https://stellargraph.readthedocs.io/en/stable/demos/link-prediction/homogeneous-comparison-link-prediction.html>__
Built-in classes for walker: :class:.UniformRandomWalk, :class:.BiasedRandomWalk, :class:.UniformRandomMetaPathWalk.
Args:
G (StellarGraph): A stellargraph with features.
nodes (iterable, optional) The root nodes from which individual walks start.
If not provided, all nodes in the graph are used.
length (int): Length of the walks for the default UniformRandomWalk walker. Length must
be at least 2.
number_of_walks (int): Number of walks from each root node for the default
UniformRandomWalk walker.
seed (int, optional): Random seed for the default UniformRandomWalk walker.
walker (RandomWalk, optional): A RandomWalk object to use instead of the default
UniformRandomWalk walker.
"""
def __init__(
self, G, nodes=None, length=2, number_of_walks=1, seed=None, walker=None,
):
if not isinstance(G, StellarGraph):
raise ValueError(
"({}) Graph must be a StellarGraph or StellarDigraph object.".format(
type(self).__name__
)
)
else:
self.graph = G
# Instantiate the walker class used to generate random walks in the graph
if walker is not None:
_warn_if_ignored(length, 2, "length")
_warn_if_ignored(number_of_walks, 1, "number_of_walks")
_warn_if_ignored(seed, None, "seed")
self.walker = walker
else:
self.walker = UniformRandomWalk(
G, n=number_of_walks, length=length, seed=seed
)
# Define the root nodes for the walks
# if no root nodes are provided for sampling defaulting to using all nodes as root nodes.
if nodes is None:
self.nodes = list(G.nodes())
elif is_real_iterable(nodes): # check whether the nodes provided are valid.
self.nodes = list(nodes)
else:
raise ValueError("nodes parameter should be an iterable of node IDs.")
# Require walks of at lease length two because to create a sample pair we need at least two nodes.
if length < 2:
raise ValueError(
"({}) For generating (target,context) samples, walk length has to be at least 2".format(
type(self).__name__
)
)
else:
self.length = length
if number_of_walks < 1:
raise ValueError(
"({}) At least 1 walk from each head node has to be done".format(
type(self).__name__
)
)
else:
self.number_of_walks = number_of_walks
# Setup an interal random state with the given seed
_, self.np_random = random_state(seed)
[docs] def run(self, batch_size):
"""
This method returns a batch_size number of positive and negative samples from the graph.
A random walk is generated from each root node, which are transformed into positive context
pairs, and the same number of negative pairs are generated from a global node sampling
distribution. The resulting list of context pairs are shuffled and converted to batches of
size batch_size.
Currently the global node sampling distribution for the negative pairs is the degree
distribution to the 3/4 power. This is the same used in node2vec
(https://snap.stanford.edu/node2vec/).
Args:
batch_size (int): The number of samples to generate for each batch.
This must be an even number.
Returns:
List of batches, where each batch is a tuple of (list context pairs, list of labels)
"""
self._check_parameter_values(batch_size)
all_nodes = list(self.graph.nodes(use_ilocs=True))
# Use the sampling distribution as per node2vec
degrees = self.graph.node_degrees(use_ilocs=True)
sampling_distribution = np.array([degrees[n] ** 0.75 for n in all_nodes])
sampling_distribution_norm = sampling_distribution / np.sum(
sampling_distribution
)
walks = self.walker.run(nodes=self.nodes)
# first item in each walk is the target/head node
targets = [walk[0] for walk in walks]
positive_pairs = np.array(
[
(target, positive_context)
for target, walk in zip(targets, walks)
for positive_context in walk[1:]
]
)
positive_pairs = self.graph.node_ids_to_ilocs(positive_pairs.flatten()).reshape(
positive_pairs.shape
)
negative_samples = self.np_random.choice(
all_nodes, size=len(positive_pairs), p=sampling_distribution_norm
)
negative_pairs = np.column_stack((positive_pairs[:, 0], negative_samples))
pairs = np.concatenate((positive_pairs, negative_pairs), axis=0)
labels = np.repeat([1, 0], len(positive_pairs))
# shuffle indices - note this doesn't ensure an equal number of positive/negative examples in
# each batch, just an equal number overall
indices = self.np_random.permutation(len(pairs))
batch_indices = [
indices[i : i + batch_size] for i in range(0, len(indices), batch_size)
]
return [(pairs[i], labels[i]) for i in batch_indices]
def _check_parameter_values(self, batch_size):
"""
Checks that the parameter values are valid or raises ValueError exceptions with a message indicating the
parameter (the first one encountered in the checks) with invalid value.
Args:
batch_size: <int> number of samples to generate in each call of generator
"""
if (
batch_size is None
): # must provide a batch size since this is an indicator of how many samples to return
raise ValueError(
"({}) The batch_size must be provided to generate samples for each batch in the epoch".format(
type(self).__name__
)
)
if type(batch_size) != int: # must be an integer
raise TypeError(
"({}) The batch_size must be positive integer.".format(
type(self).__name__
)
)
if batch_size < 1: # must be greater than 0
raise ValueError(
"({}) The batch_size must be positive integer.".format(
type(self).__name__
)
)
if (
batch_size % 2 != 0
): # should be even since we generate 1 negative sample for each positive one.
raise ValueError(
"({}) The batch_size must be an even integer since equal number of positive and negative samples are generated in each batch.".format(
type(self).__name__
)
)
|
2021-03-09 11:02:22
|
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|
https://www.aimsciences.org/article/doi/10.3934/dcdss.2013.6.783
|
# American Institute of Mathematical Sciences
June 2013, 6(3): 783-791. doi: 10.3934/dcdss.2013.6.783
## Dispersive waves with multiple tunnel effect on a star-shaped network
1 Université de Valenciennes et du Hainaut-Cambrésis, LAMAV, FR CNRS 2956, F-59313 Valenciennes, France 2 TU Darmstadt, Fachbereich Mathematik, Schloßgartenstraße 7, D-64289 Darmstadt, Germany, Germany
Received April 2010 Revised December 2010 Published December 2012
We consider the Klein-Gordon equation on a star-shaped network composed of $n$ half-axes connected at their origins. We add a potential which is constant but different on each branch. The corresponding spatial operator is self-adjoint and we state explicit expressions for its resolvent and its resolution of the identity in terms of generalized eigenfunctions. This leads to a generalized Fourier type inversion formula in terms of an expansion in generalized eigenfunctions. This paper is a survey of a longer article, nevertheless the proof of the central formula is indicated.
Citation: F. Ali Mehmeti, R. Haller-Dintelmann, V. Régnier. Dispersive waves with multiple tunnel effect on a star-shaped network. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 783-791. doi: 10.3934/dcdss.2013.6.783
##### References:
[1] F. Ali Mehmeti, Spectral theory and $L^{\infty}$-time decay estimates for Klein-Gordon equations on two half axes with transmission: The tunnel effect,, Math. Methods Appl. Sci., 17 (1994), 697. doi: 10.1002/mma.1670170904. Google Scholar [2] F. Ali Mehmeti, "Transient Tunnel Effect and Sommerfeld Problem: Waves in Semi-Infinite Structures,", Mathematical Research, 91 (1996). Google Scholar [3] F. Ali Mehmeti, R. Haller-Dintelmann and V. Régnier, Expansions in generalized eigenfunctions of the weighted Laplacian on star-shaped networks,, in, (2007), 1. doi: 10.1007/978-3-7643-7794-6_1. Google Scholar [4] F. Ali Mehmeti, R. Haller-Dintelmann and V. Régnier, Multiple tunnel effect for dispersive waves on a star-shaped network: an explicit formula for the spectral representation,, J. Evol. Equ., 12 (2012), 513. doi: 10.1007/s00028-012-0143-5. Google Scholar [5] F. Ali Mehmeti and V. Régnier, Splitting of energy of dispersive waves in a star-shaped network,, Z. Angew. Math. Mech., 83 (2003), 105. doi: 10.1002/zamm.200310010. Google Scholar [6] F. Ali Mehmeti and V. Régnier, Delayed reflection of the energy flow at a potential step for dispersive wave packets,, Math. Methods Appl. Sci., 27 (2004), 1145. doi: 10.1002/mma.484. Google Scholar [7] F. Ali Mehmeti and V. Régnier, Global existence and causality for a transmission problem with a repulsive nonlinearity,, Nonlinear Anal., 69 (2008), 408. doi: 10.1016/j.na.2007.05.028. Google Scholar [8] J. von Below and J. A. Lubary, The eigenvalues of the Laplacian on locally finite networks, Results Math., 47 (2005), 199. Google Scholar [9] S. Cardanobile and D. Mugnolo, Parabolic systems with coupled boundary conditions,, J. Differential Equations, 247 (2009), 1229. doi: 10.1016/j.jde.2009.04.013. Google Scholar [10] Y. Daikh, "Temps de Passage de Paquets D'ondes de Basses Fréquences ou Limités en Bandes de Fréquences par une Barrière de Potentiel,", Thèse de Doctorat, (2004). Google Scholar [11] J. M. Deutch and F. E. Low, Barrier penetration and superluminal velocity,, Annals of Physics, 228 (1993), 184. doi: 10.1006/aphy.1993.1092. Google Scholar [12] N. Dunford and J. T. Schwartz, "Linear Operators II,", Wiley Interscience, (1963). Google Scholar [13] A. Enders and G. Nimtz, On superluminal barrier traversal,, J. Phys. I France, 2 (1992), 1693. Google Scholar [14] A. Haibel and G. Nimtz, Universal relationship of time and frequency in photonic tunnelling,, Ann. Physik (Leipzig), 10 (2001), 707. Google Scholar [15] V. Kostrykin and R. Schrader, The inverse scattering problem for metric graphs and the travelling salesman problem,, preprint, (). Google Scholar [16] M. Pozar, "Microwave Engineering,", Addison-Wesley, (1990). Google Scholar [17] J. Weidmann, "Spectral Theory of Ordinary Differential Operators,", Lecture Notes in Mathematics, 1258 (1987). Google Scholar
show all references
##### References:
[1] F. Ali Mehmeti, Spectral theory and $L^{\infty}$-time decay estimates for Klein-Gordon equations on two half axes with transmission: The tunnel effect,, Math. Methods Appl. Sci., 17 (1994), 697. doi: 10.1002/mma.1670170904. Google Scholar [2] F. Ali Mehmeti, "Transient Tunnel Effect and Sommerfeld Problem: Waves in Semi-Infinite Structures,", Mathematical Research, 91 (1996). Google Scholar [3] F. Ali Mehmeti, R. Haller-Dintelmann and V. Régnier, Expansions in generalized eigenfunctions of the weighted Laplacian on star-shaped networks,, in, (2007), 1. doi: 10.1007/978-3-7643-7794-6_1. Google Scholar [4] F. Ali Mehmeti, R. Haller-Dintelmann and V. Régnier, Multiple tunnel effect for dispersive waves on a star-shaped network: an explicit formula for the spectral representation,, J. Evol. Equ., 12 (2012), 513. doi: 10.1007/s00028-012-0143-5. Google Scholar [5] F. Ali Mehmeti and V. Régnier, Splitting of energy of dispersive waves in a star-shaped network,, Z. Angew. Math. Mech., 83 (2003), 105. doi: 10.1002/zamm.200310010. Google Scholar [6] F. Ali Mehmeti and V. Régnier, Delayed reflection of the energy flow at a potential step for dispersive wave packets,, Math. Methods Appl. Sci., 27 (2004), 1145. doi: 10.1002/mma.484. Google Scholar [7] F. Ali Mehmeti and V. Régnier, Global existence and causality for a transmission problem with a repulsive nonlinearity,, Nonlinear Anal., 69 (2008), 408. doi: 10.1016/j.na.2007.05.028. Google Scholar [8] J. von Below and J. A. Lubary, The eigenvalues of the Laplacian on locally finite networks, Results Math., 47 (2005), 199. Google Scholar [9] S. Cardanobile and D. Mugnolo, Parabolic systems with coupled boundary conditions,, J. Differential Equations, 247 (2009), 1229. doi: 10.1016/j.jde.2009.04.013. Google Scholar [10] Y. Daikh, "Temps de Passage de Paquets D'ondes de Basses Fréquences ou Limités en Bandes de Fréquences par une Barrière de Potentiel,", Thèse de Doctorat, (2004). Google Scholar [11] J. M. Deutch and F. E. Low, Barrier penetration and superluminal velocity,, Annals of Physics, 228 (1993), 184. doi: 10.1006/aphy.1993.1092. Google Scholar [12] N. Dunford and J. T. Schwartz, "Linear Operators II,", Wiley Interscience, (1963). Google Scholar [13] A. Enders and G. Nimtz, On superluminal barrier traversal,, J. Phys. I France, 2 (1992), 1693. Google Scholar [14] A. Haibel and G. Nimtz, Universal relationship of time and frequency in photonic tunnelling,, Ann. Physik (Leipzig), 10 (2001), 707. Google Scholar [15] V. Kostrykin and R. Schrader, The inverse scattering problem for metric graphs and the travelling salesman problem,, preprint, (). Google Scholar [16] M. Pozar, "Microwave Engineering,", Addison-Wesley, (1990). Google Scholar [17] J. Weidmann, "Spectral Theory of Ordinary Differential Operators,", Lecture Notes in Mathematics, 1258 (1987). Google Scholar
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2018 Impact Factor: 0.545
|
2020-02-23 23:27:57
|
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|
http://wikiwap.com/3-sphere
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World's Largest Encyclopedia On Your Mobile.
3-sphere
In [mathematics] , a 3-sphere is a higher-dimensional analogue of a [sphere] . It consists of the set of points equidistant from a fixed central point in 4-dimensional [Euclidean space] . Just as an ordinary sphere (or 2-sphere) is a two dimensional [surface] that forms the boundary of a [ball] in three dimensions, a 3-sphere is an object with three [dimension] s that forms the boundary of a ball in four dimensions. A 3-sphere is an example of a [3-manifold] .
A 3-sphere is also called a hypersphere , although the term hypersphere can in general describe any [''n''-sphere] for n ≥ 3.
Definition
In [coordinates] , a 3-sphere with center ( C 0, C 1, C 2, C 3) and radius r is the set of all points ( x 0, x 1, x 2, x 3) in real, [4-dimensional space] ( R 4) such that
:\sum_{i=0}^3(x_i - C_i)^2 = ( x_0 - C_0 )^2 ( x_1 - C_1 )^2 ( x_2 - C_2 )^2 ( x_3 - C_3 )^2 = r^2.
The 3-sphere centered at the origin with radius 1 is called the unit 3-sphere and is usually denoted S 3:
:S^3 = \left\{(x_0,x_1,x_2,x_3)\in\mathbb{R}^4 : x_0^2 x_1^2 x_2^2 x_3^2 = 1\right\}.
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2017-05-29 17:14:32
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http://tex.stackexchange.com/questions/58347/is-there-such-thing-as-pheight/58354
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# Is there such thing as “p{height}”
I am making a table right now and it's really annoying that items in my cell are getting crunched
\begin{center}
\begin{tabular}{|c| c | c|}
Critical Points $P_0$ & $f(P_0)$ & Conclusion \\
\hline
(0,0,1) & 0 & Local Extrema\\ \hline
(0,0,-1) & -2 & Absolute minimum \\ \hline
$\left ( 0,0,\frac{1}{2} \right )$ & $\frac{1}{4}$ & Local Extrema \\ \hline
$(0,0, \frac{\sqrt{12}}{5})$ & $\frac{8}{5}$ & Absolute maximum \\ \hline
$(0,0, -\frac{\sqrt{12}}{5})$ & $\frac{8}{5}$ & Absolute maximum \\
\end{tabular}
\end{center}
My fractions are getting crushed and there is no such thing as "p{height}" to fix.
-
Instead of increasing the vertical space for rows, I propose you some changes:
1. Don't use \frac{a}{b} but simply a/b; this increases readability.
2. Use the features provided by the booktabs and array packages.
3. Eliminate the vertical rules.
\documentclass{article}
\usepackage{amsmath}
\usepackage{array}
\usepackage{booktabs}
\begin{document}
\begin{center}
\begin{tabular}{*{2}{>{$}c<{$}} c}
\toprule
\multicolumn{1}{c}{Critical Points $P_0$} & f(P_0) & Conclusion \\
\cmidrule(r){1-1}\cmidrule(lr){2-2}\cmidrule(l){3-3}
(0,0,1) & 0 & Local Extrema\\
(0,0,-1) & -2 & Absolute minimum \\
(0,0,1/2 ) & 1/4 & Local Extrema \\
(0,0,\sqrt{12}/5) & 8/5 & Absolute maximum \\
(0,0, -\sqrt{12}/5) & 8/5 & Absolute maximum \\
\bottomrule
\end{tabular}
\end{center}
\end{document}
If, for some reason, using array and booktabs is not possible, my first and third recommendations still apply and you can change \arraystretch; something along these lines:
\documentclass{article}
\usepackage{amsmath}
\begin{document}
\begin{center}
\renewcommand\arraystretch{1.2}
\begin{tabular}{c c c}
\hline
Critical Points $P_0$ & $f(P_0)$ & Conclusion \\
\hline
$(0,0,1)$ & $0$ & Local Extrema\\
$(0,0,-1)$ & $-2$ & Absolute minimum \\
$(0,0,1/2 )$ & $1/4$ & Local Extrema \\
$(0,0,\sqrt{12}/5)$ & $8/5$ & Absolute maximum \\
$(0,0, -\sqrt{12}/5)$ & $8/5$ & Absolute maximum \\
\hline
\end{tabular}
\end{center}
\end{document}
-
What fi I don't want new packages? My computer can can't so much stuff – jak Jun 2 '12 at 22:50 @jak please see my updated answer. – Gonzalo Medina Jun 2 '12 at 22:55 Now I am going to be even more annoying and ask is if there are anyway of using \frac{}{} in tables without it getting crushed? – jak Jun 2 '12 at 22:56 @jak you could use \dfrac instead of \frac (requires amsmath), but I personally don't like the result: \begin{center} \renewcommand\arraystretch{1.4} \begin{tabular}{c c c} \hline Critical Points $P_0$ & $f(P_0)$ & Conclusion \\ \hline (0,0,1) & 0 & Local Extrema\\ (0,0,-1) & -2 & Absolute minimum \\ $\left( 0,0,\dfrac{1}{2} \right)$ & $\dfrac{1}{4}$ & Local Extrema \\ $\left( 0,0, \dfrac{\sqrt{12}}{5} \right)$ & $\dfrac{8}{5}$ & Absolute maximum \\ $\left( 0,0, -\dfrac{\sqrt{12}}{5} \right)$ & $\dfrac{8}{5}$ & Absolute maximum \\ \hline \end{tabular} \end{center}. – Gonzalo Medina Jun 2 '12 at 23:03 I used \dfrac initally and same thing happened lol – jak Jun 2 '12 at 23:11
The package cellspace offers a simple way to get the correct height:
\documentclass{article}
\usepackage{amsmath}
\usepackage{array}
\usepackage{booktabs}
\usepackage{cellspace}
\begin{document}
\begin{center}
\begin{tabular}{>{$}c<{$}>{$}Sc<{$}c}
\toprule
\multicolumn{1}{c}{Critical Points $P_0$} & f(P_0) & Conclusion \\
\cmidrule(r){1-1}\cmidrule(lr){2-2}\cmidrule(l){3-3}
(0,0,1) & 0 & Local Extrema\\
(0,0,-1) & -2 & Absolute minimum \\
(0,0,1/2 ) & \dfrac14 & Local Extrema \\
(0,0,\sqrt{12}/5) & \dfrac85 & Absolute maximum \\
(0,0, -\sqrt{12}/5) & \dfrac85 & Absolute maximum \\
\bottomrule
\end{tabular}
\end{center}
\end{document}
-
That is big.... – jak Jun 2 '12 at 23:11
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2013-05-25 02:19:41
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https://mathematica.stackexchange.com/questions/221194/unexpected-output-of-a-listable-function-arguments-not-threaded-correctly
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# Unexpected output of a Listable function. Arguments not threaded correctly
I saw this question asking about the Listable. The accepted answer did a good explanation on the poor performance of user defined function with Listable. However the answer didn't clarify why the f function in following code doesn't show Listable feature.
f = #[[1]] + #[[2]]^#[[3]] &;
SetAttributes[f, Listable];
f@{{1, 2, 3}, {4, 5, 6}}
output: I Know why the code above cannot achieve its goal, and I know how to fix it. I just don't think the above wrong code should produce the above specific output. According to my understanding of how Listable thread function arguments, it should produce the following output:
{{f[1], f[2], f[3]}, {f[4], f[5], f[6]}}
Because the list {{1, 2, 3}, {4, 5, 6}} as argument should be threaded to the innermost level, which in the end is {{f[1], f[2], f[3]}, {f[4], f[5], f[6]}} and stop with some error message since f has Part on the argument.
However, in the reality, the real output shows that actually no Listable is in effect. The List {{1, 2, 3}, {4, 5, 6}} is passed to f as one whole arguments, which can also be proved by running a Trace on it.
• Here are a few hints: SetAttributes[] will only work if you had defined f as f[v_] := v[[1]] + v[[2]]^v[[3]]. Otherwise, an anonymous function with attributes has to be declared this way: f = Function[Null, #[[1]] + #[[2]]^#[[3]], {Listable}]. Now, try your examples again with this in mind. May 5, 2020 at 18:40
• Wolfram surprised me again! I've recently read tons of articles on Documentation website , nowhere had I ever seen it says that SetAttributes work differently for f=Function[x, ...] and f[x_]! May 6, 2020 at 7:18
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2022-09-26 00:03:19
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http://mathhelpforum.com/algebra/175207-parabola-print.html
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# Parabola
• Mar 20th 2011, 09:18 PM
thamathkid1729
Parabola
Find an equation that says that
P = (x, y) is equidistant from F = (2, 0) and the y-axis.
• Mar 20th 2011, 09:49 PM
pickslides
An equation or find the point P?
If the later what about (1,0)?
• Mar 20th 2011, 11:34 PM
thamathkid1729
I need to find an equation, not just 1 point. I know that (1,0), (2,2), and (2,-2) are all equidistant from F = (2, 0) and the y-axis
• Mar 21st 2011, 05:23 AM
HallsofIvy
Since you title this "parabola", I suspect you already know the answer. However, the derivation is this:
The distance from (x, y) to (2, 0) is $\sqrt{(x- 2)^2+ y^2}$ and the distance from the (x, y) to the y-axis is simply x. To be equidistant, we must have $\sqrt{(x-2)^2+ y^2}= x$
Square both sides and simplify.
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2017-11-19 20:33:53
|
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http://www.mathnet.ru/php/archive.phtml?jrnid=de&wshow=issue&year=1991&volume=27&volume_alt=&issue=1&issue_alt=&option_lang=eng
|
RUS ENG JOURNALS PEOPLE ORGANISATIONS CONFERENCES SEMINARS VIDEO LIBRARY PACKAGE AMSBIB
General information Latest issue Archive Search papers Search references RSS Latest issue Current issues Archive issues What is RSS
Differ. Uravn.: Year: Volume: Issue: Page: Find
Partial Differential Equations Boundary properties of integrals of Cauchy type. The $L_p$ caseA. V. Aleksandrov, A. P. Soldatov 3 Analytic continuations of solutions of boundary value problems for the stationary wave equationV. F. Apel'tsin 8 A parabolic boundary value problem in a domain of simple formE. A. Baderko 17 Spinor systemsV. S. Vinogradov 22 Stabilization of the solution of the Cauchy problem for the iterated heat equationV. N. Denisov 29 The Lavrent'ev effect and averaging of nonlinear variational problemsV. V. Zhikov 42 A nonlocal boundary value problem for a stationary Stokes system in a multiply connected domainN. A. Zhura 51 Solution of a problem in the theory of the Frankl' problem for equations of mixed typeN. Yu. Kapustin, K. B. Sabitov 60 A nonlocal boundary value problem for an equation of mixed typeM. G. Karatopraklieva 68 The basis property of root vectors of loaded second-order differential operators on an intervalI. S. Lomov 80 Solution of degenerate equations by means of biorthogonal seriesE. I. Moiseev 94 Local solvability of differential equations in spaces of multivalued analytic functionsV. E. Nazaikinskii, B. Yu. Sternin, V. E. Shatalov 103 Determination of the evolution of a parameter in an abstract parabolic equationA. I. Prilepko, D. G. Orlovskii 114 On the scattering problem for the Schrödinger operator with singular potential in the two-dimensional case. IIV. S. Serov 120 The method of cut-off functions in the theory of nonlocal problemsA. L. Skubachevskii 128 The method of operator pencils in boundary value problems of conjugation for a system of elliptic equationsYu. G. Smirnov 140 Nonlinear hyperbolic systems with two independent variablesD. V. Tunitsky 147 Short Communications Bi-orthogonal expansions of a nonselfadjoint Schrödinger operatorR. R. Ashurov 156 Necessary and sufficient condition for the completeness of a system of functionsB. T. Bilalov 158 On the asymptotic behavior with respect to the small parameter of solutions of a fourth-order nonlinear evolution system that admits an $L$-$A$ representationA. V. Il'ina 161 Finiteness of the number of conditions for the agreement of the Cauchy problem for overdetermined systems of linear differential equationsM. V. Korovina 163 Limit relations between solutions of some equations of hyperbolic, parabolic and elliptic typesV. D. Repnikov 165 On the membership in class $V_2^{1,0}$ of the classical solution of a mixed problem for second-order parabolic systemsÈ. K. Safiev 167 The Neumann problem for the biharmonic equationChoi Sung Bong 169 A problem of contact heat conductionV. N. Sheveleva 172 Convergence of the collocation and the quadrature-interpolation method for singular integro-differential equations on a Lyapunov contourV. N. Seichuk 174
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2019-10-17 19:41:53
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https://bioconductor.org/packages/release/bioc/vignettes/EnrichedHeatmap/inst/doc/row_odering.html
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# Compare row ordering methods
Author: Zuguang Gu ( z.gu@dkfz.de )
Date: 2019-10-29
library(EnrichedHeatmap)
all_genes = all_genes[unique(neg_cr\$gene)]
all_tss = promoters(all_genes, upstream = 0, downstream = 1)
mat_neg_cr = normalizeToMatrix(neg_cr, all_tss, mapping_column = "gene", w = 50, mean_mode = "w0")
The object mat_neg_cr is a normalized matrix for regions showing significant negative correlation between methylation and gene expression. The negative correlated regions (negCRs) are normalized to upstream 5kb and downstream 5kb of gene TSS with 50bp window by normalizeToMatrix() function. The value in the matrix is how much a window is covered by negCRs (values between 0 and 1).
In the normalized matrix, each row corresponds to one gene and each column corresponds to a window either on upstream of TSS or downstream of TSS. For the example of mat_neg_cr matrix, the first half columns correspond to the upstream of TSS and the last half columns correspond to downstream of TSS. Here we compare following three different methods to order rows (which correspond to genes) in the normalized matrix.
1. Rows are ordered by the enriched scores. For each row in the matrix, denote values in a certain row as $$x$$, indices 1, …, $$n_1$$ are for upstream windows, indices $$n_1+1$$, …, $$n$$ are for downstream windows and $$n_2 = n - n_1$$, the enriched score is calculated as the sum of $$x$$ weighted by distance to TSS (higher weight if the window is close to TSS).
$\sum_{i=1}^{n_1}{x_i \cdot i/n_1} + \sum_{i=n_1+1}^n{x_i \cdot (n - i + 1)/n_2}$
2. Rows are ordered by hierarchical clustering with Euclidean distance.
3. Rows are ordered by hierarchical clustering with closeness distance. For two rows in the normalized matrix, assume $$a_1, a_2, …, a_{n_1}$$ are the indices of windows for one gene which overlap with negCRs and $$b_1, b_2, … b_{n_2}$$ are the indices for the other gene which overlap with negCRs, the distance which is based on closeness of the overlapped windows in the two genes is defined as:
$d_{closeness} = \frac{\sum_{i=1}^{n_1} \sum_{j=1}^{n_2} {|a_i - b_j|} }{n_1 \cdot n_2}$
So the closeness distance is basically the average distance of all pairs of negCR windows in the two genes.
Euclidean distance between rows keeps unchanged when the matrix columns are permutated, while for closeness distance, the column order is also taken into account, which might be more proper for clustering normalized matrices because the columns correspond to relative distance to the target regions.
Following three plots show heatmaps under different row ordering methods.
EnrichedHeatmap(mat_neg_cr, name = "neg_cr", col = c("white", "darkgreen"),
top_annotation = HeatmapAnnotation(enrich = anno_enriched(gp = gpar(col = "darkgreen"))),
row_title = "by default enriched scores")
EnrichedHeatmap(mat_neg_cr, name = "neg_cr", col = c("white", "darkgreen"),
top_annotation = HeatmapAnnotation(enrich = anno_enriched(gp = gpar(col = "darkgreen"))),
cluster_rows = TRUE,
row_title = "by hierarchcal clustering + Euclidean distance\ndendrogram reordered by enriched scores")
EnrichedHeatmap(mat_neg_cr, name = "neg_cr", col = c("white", "darkgreen"),
top_annotation = HeatmapAnnotation(enrich = anno_enriched(gp = gpar(col = "darkgreen"))),
cluster_rows = TRUE, clustering_distance_rows = dist_by_closeness,
row_title = "by hierarchcal clustering + closeness distance\ndendrogram reordered by enriched scores")
Generally, when the top annotation which summarises mean enrichment across genes is also added to the heatmap, ordering genes by enriched scores is not recommended because it provides redundant information as the top enriched annotation, and on the other hand, it fails to reveal spatial clusters as other two methods. Hierarchal clustering with Euclidean distance is good at clustering enrichment patterns, but since it does not take column order into account, thus, it still can be possible that two spatially close clusters are far separated in the heatmap. By using closeness distance, it clearly sorts and clusters the enrichment patterns.
The row order, clustering method, distance method can all be self-adjusted by row_order, cluster_rows, clustering_method_rows, clustering_distance_rows arguments in EnrichedHeamtap() function. For how to properly set values for these arguments, users can go to the help page of EnrichedHeatmap() or Heatmap() function.
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2020-03-28 09:13:12
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https://www.numerade.com/questions/a-vertical-plate-is-submerged-or-partially-submerged-in-water-and-has-the-indicated-shape-explain-6/
|
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# A vertical plate is submerged (or partially submerged) in water and has the indicated shape. Explain how to approximate the hydrostatic force against one side of the plate by a Riemann sum.Then express the force as an integral and evaluate it.
## $313,600 \mathrm{N}$
#### Topics
Applications of Integration
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### Video Transcript
in this problem, we're given that there is an object with triangular course section. And now, whereas to find the force acting on runnin off this object, we know that force. Is he going to pressure times area and that is equal to road time. She times depth times the area. Now we know what drawers and we know what he is. Siro, It's 1000 kilogram per meter cube for water and G is 9.8 meters per second skirt now. I mean, it is German, but days and what a is Yeah, let's zoom into this object under sent what area? As, um less assume not be harness 10 sheet when a thickness d y, um So the area would be dealing l a times d by now the key or the idea is to write this l in terms off one. Now for that, we're going to assume that we have an origin located right here. So this is origin 00 dishes are XX ISS and this is our y anxious in order to find El in terms of why we would need to find the question off this very lane. So the coordinates off the end points are done since the height off this triangle's two. That would be zero comma, too. And this edge, this word text is located at three comma zero. So we're looking at that line with 10 points. Sirrah, come on. Two and three. Come, Sirrah. And the question that line would be why minus why not is equaled. M times X minus X not. Mm. Here is to minus zero. Divide by zero minus streets of difference between wise divide about difference between excess orders Niner three to any question will be then why minus Here is ICO to negative three x or T plus two. We want to write everything in terms of why. So let's leave ex alone and right at one of us. Axe is equal to three minus three over two. Why? Okay, now this means that if this is the origin this distance or one side, one end off or 1/2 off this link is a three month tour to one. But don't forget that a symmetric. So we have the same link on the opposite side. So the area this area off this tin sheet, I will say differential elements or D A. Will be two times three minus two. Why were, too terms tea? Wine? So what is? Since now, we found an expression for area B last leading to determine what, um that this If this is our war and if we assume this is the positive, why don't forget one measuring pressure. Be always measured, that studying from the surface. And if this total link is six meters, that has given problem statement that in terms of why would then be six minds want sort of total language? Still be four plus two? That would be six. So then the area or the force acting moments differential element This red block Pdf would be raw time Caesar 1000 times 9.8 times Ah, six minus wind that is Tibet. The month black by the differential area is two times three minus G. Wildwood, too times D y In order to find a total force, we could drive or we could some beat those elements up some all those forces up, and we would use a remote sums, a limit as a ghost. Impunity summation From one to end, he had 1000 times 9.8 attempts six minus y times two times three months to G A Y or two d y, or we could call murder at Inter Inter Galang right f s f now in to decide on the limits. And the limits will be between right here so that I will start from zero the first blue dot and it will go up to the second blue one. So that is between zero in 2000 terms 9.8 times two doors are constant. Times six minus y multiplied by this one. And we will get, um 18 minus stroke y plus three wife's card or two Do you want? So the first part is just a constant. We could write this one as 19,600 times inter go 0 to 18 minus 12 y plus t y skirt over to D Y. That would be 19,600 times 18 y minus 61 squared, plus white cube over three planted at zero and two, and we don't would find the total force as 313,600. Nugent's
#### Topics
Applications of Integration
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2021-10-16 21:37:03
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https://stats.stackexchange.com/questions/509560/how-does-the-sample-variance-change-if-you-take-subsets-of-n-observations-from
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# How does the sample variance change if you take subsets of $n$ observations from the original data?
Suppose $$X$$ is a continuous random variable for the weight of silver pennies. We then measure 338 pennies (in grams), leaving us with 338 observations.
The observed mean weight is 15.722 grams and the observed variance is 1.999 squared grams.
If we now take random and disjoint subsets of five observations from the 338 observed weights without replacement, what would happen to the mean and variance?
This is exercise 5.2 from Principles of Statistics by M.G. Bulmer. The answers are:
1. New mean = $$5 \times 15.722$$
2. New variance = $$5 \times 1.999$$
I understand why the new mean is five times the old one, for $$\sum_{i=1}^{n}x_i$$ does not change and there is one observation for every five observations in the original calculation. Thus, $$\sum x_i$$ is now divided by $$\frac{338}{5}$$, which is equivalent to $$5 \bar{x}$$.
However, why is the new variance five times the old one?
Writing out the equations:
$$E(X_j) = \mu$$ $$V(X_j) = \sigma^2$$
$$X_j$$ are independent, identically distributed random variables.
Let $$x_j$$ be a sample from the Random Variable $$X_j$$. Let $$y_i = \sum_j^n x_j$$. The distribution on $$x_j$$ induces a distribution on $$y_i$$ such that we can treat $$Y_i$$ as a random variable.
$$E(Y_i) = E(\sum_j^n X_j) = \sum_j^n E(X_j) = \sum_j^n \mu = n\mu$$ $$V(Y_i) = V(\sum_j^n X_j) = \sum_j^n V(X_j) + \sum_{j \neq k} Cov(X_j, X_k)$$
Since $$X_j$$ and $$X_k$$ are independent, then $$Cov(X_j, X_k) = 0$$
$$V(Y_i) = \sum_j^n V(X_j) = \sum_j^n \sigma^2 = n \sigma^2$$
• Why is $V(Y) = V(\sum X_i)$ ? Feb 19 at 21:11
• @Arturo It is simply substituting $Y=\sum X_i$ into the variance function. Feb 19 at 22:37
• @B.Liu This is exactly the part that confuses me. Is $Y_i = \sum X_i$? I am confused by this because $y_i$ is the disjoint sum of five different $x_i$'s. For instance, $y_1 = x_1 + x_2 + x_3 + x_4 + x_5$. Feb 19 at 23:11
• @ArturoSbr If we substitute $n=5$ into the equations in the answer above, we have $Y=\sum_{i=1}^{5} X_n = X_1 + X_2 + X_3 + X_4 + X_5$, which is more or less the random variable version of what you defined in the question for the observations. Note the lack of subscript for $Y$ (as we don't really need them for the purpose of calculating the mean and variance). Feb 20 at 2:10
• @ArturoSbr There is a typo above - $\sum_{i=1}^5 X_n$ should be $\sum_{i=1}^5 X_i$. Feb 20 at 3:27
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2021-10-20 01:09:33
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http://math.stackexchange.com/questions/57394/why-is-every-map-to-an-indiscrete-space-continuous?answertab=oldest
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# Why is every map to an indiscrete space continuous?
Show that if $Y$ is a topological space, then every map $f:Y \rightarrow X$ is continuous when $X$ has the indiscrete topology.
Proof:
Assume $X$ has the indiscrete topology, $T=\{\varnothing,X\}$.
$f$ is continuous if $f^{-1}(V)$ is an open subset of $X$ whenever $V$ is an open subset of $Y$.
Let $V$ be an open subset of $Y$.
I dont know how to use this to show $f^{-1}(V)$ is an open subset of $X$.
-
The definition of continuous is that $f\colon Y\to X$ then for every $U\subseteq X$ which is open, $f^{-1}(U)$ is open in $Y$. Now think which subsets of $X$ are open and what are their preimages? – Asaf Karagila Aug 14 '11 at 11:27
You may also want to accept some answers to previous questions, you can do that by clicking on the transparent check symbol next to the vote counter of an answer. Choose the answer most helpful to you, and if none is helpful enough please edit your question to indicate the missing parts in the answers so far. – Asaf Karagila Aug 14 '11 at 11:29
You got confused about the definition of continuity.
If $f\colon Y\to X$ is continuous then the preimage of open subsets of $X$ is open in $Y$.
Since $X$ has the indiscrete topology, we only have two open subsets. Namely, $X$ and $\varnothing$.
The preimage of the empty set is of course empty, and therefore open in $Y$. If we look at $f^{-1}(X) = \{y\in Y\mid f(y)\in X\}=Y$, and of course that $Y$ is open in $Y$.
Thus, $f$ is continuous regardless to the topology given on $Y$ whenever $X$ is indiscrete.
Exercise: Suppose $f\colon X\to Y$ and $X$ has the discrete topology, prove that $f$ is continuous.
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Thanks so much Asaf! – user8603 Aug 14 '11 at 11:47
To expand on Asaf's comment:
$f:X \rightarrow Y$ is continuous if $f^{-1}(O)$ is continuous for all open sets $O$ in $Y$. As $Y$ has the trivial topology, the only open sets are $\emptyset$ and $Y$. So to show that $f$ is continuous you need to show that $f^{-1}(\emptyset)$ and $f^{-1}(Y)$ are open, i.e. are in the topology of $X$.
A collection of sets is per definition a topology if it contains the entire space $X$ and $\emptyset$. $f^{-1}(\emptyset) = \emptyset$ and $f^{-1}(Y) = X$ are therefore both open and so $f$ is continuous.
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Just swap $X$ and $Y$ but names don't really matter... – Matt N. Aug 14 '11 at 11:36
Thanks for your answers. Since I'm new to topology, how did you ascertain that $Y$ has the trivial topology? – user8603 Aug 14 '11 at 11:44
You were given that in your question, I swapped $X$ and $Y$. – Matt N. Aug 14 '11 at 12:12
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2014-08-02 06:37:02
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https://physics.stackexchange.com/questions/588000/why-does-the-ising-model-at-the-critical-point-have-scale-invariance
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Why does the Ising model at the critical point have scale invariance?
If my current understanding of phase transitions and the renormalization group (RG) method is true, RG is a kind of 'zooming out' process, since this procedure makes a block of neighboring spins and makes a new Hamiltonian. Hence a fixed point in an RG flow means it's scale invariant, and every textbook says therefore it's a critical point where a phase transition will occur.
But why? It seems scale invariance (meaning correlation length diverges) is considered as a feature of a system in a critical state, but I can understand neither why the correlation length diverges nor why the system is scale invariant at the critical point.
First things first, scale invariance and correlation length ($$\xi$$) divergence go hand in hand. The correlation length basically sets the lengthscale for the physical phenomenon of interest: if I wiggle a particle at position $$x$$, this effect will be felt up to a distance $$x+\xi$$. Is the system is scale invariant, meaning the same phenomenon is present at short, intermediate, and long distances with the same intensity, then $$\xi$$ cannot be finite. Hence it must be infinite.
The maths then usually shows you that the correlation length $$\xi$$ goes as $$\propto (T-T_{\mathrm{c}})^{-\nu}$$, that is $$\xi\rightarrow\infty$$ as $$T\rightarrow T_{\mathrm{c}}$$. From which scale invariance follows.
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2022-06-29 09:30:36
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http://math.stackexchange.com/questions/251808/periodic-parametric-curve-on-cylinder
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# Periodic parametric curve on cylinder [duplicate]
Given a cylinder surface $S=\{(x,y,z):x^2+2y^2=C\}$. Let $\gamma(t)=(x(t),y(t),z(t))$ satisfy $\gamma'(t)=(2y(t)(z(t)-1),-x(t)(z(t)-1),x(t)y(t))$. Could we guarante that $\gamma$ always on $S$ and periodic if $\gamma(0)$ on $S$?
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## marked as duplicate by Najib Idrissi, SHOBHIT GAUTAM, Mice Elf, Davide Giraudo, Deutsch MathematikerFeb 20 at 13:35
We can reparameterize $S=\{(\sqrt{C}\cos u,\frac{\sqrt{C}}{\sqrt{2}}\sin u, v): u,v\in \mathbb{R}\}$ since $(\sqrt{C}\cos u)^2+2\left(\frac{\sqrt{C}}{\sqrt{2}}\sin u\right)^2=C$. Let $r(t)= (x(t),y(t),z(t))$ and $r(0)=(x_0,y_0,z_0)$. Define $V(x,y,z)=x^2+2y^2$. Since $V(x,y,z)=C$ then $\frac{dV}{dt}=0$. But, by chain rule we get $0=\frac{dV}{dt}=\nabla{V}\cdot(x',y',z')$ so the tangent vector of the parametrized curve that intersect $S$ in a point always parpendicular with $\nabla{V}$. Since $r(0)$ be in $S$ and $\nabla{V}$ parpendicular with the tangent plane of $S$ at $r(0)$ , then $r'(0)$ be on the tangent plane of $S$ at $r(0)$. By this argument, we can conclude that $r(t)$ must be on $S$. Since $S=\{(\sqrt{C}\cos u,\frac{\sqrt{C}}{\sqrt{2}}\sin u, v): u,v\in \mathbb{R}\}$ then $x(t)=\sqrt{C}\cos (t-t_0)$ and $y(t)=\frac{\sqrt{C}}{\sqrt{2}}\sin (t-t_0)$ with $t_0$ satisfying $x_0=\sqrt{C}\cos t_0$ and $y_0=-\frac{\sqrt{C}}{\sqrt{2}}\sin t_0$. Since $z'=xy$ then $z'(t)=\frac{C}{2\sqrt{2}}\sin(2t-t_0)$, hence $z(t)=-\frac{C}{4\sqrt{2}}\cos(2t-t_0)$. Since $r(2\pi)=(\sqrt{C}\cos (2\pi-t_0),\frac{\sqrt{C}}{\sqrt{2}}\sin (2\pi-t_0),-\frac{C}{4\sqrt{2}}\cos(2\pi-t_0))=(x_0,y_0,z_0)=r(0)$ then $r(t)$ is periodic.
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2015-10-13 21:44:35
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http://www.gradesaver.com/textbooks/math/trigonometry/trigonometry-7th-edition/chapter-2-section-2-1-definition-ii-right-triangle-trigonometry-2-1-problem-set-page-62/40
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## Trigonometry 7th Edition
1 + $\frac{\sqrt 3}{2}$ or $\frac{ 2 + \sqrt 3 }{2}$
Given expression = $( \sin 60^{\circ} + \cos 60^{\circ})^{2}$ = $\sin^{2} 60^{\circ}$ + 2 $\sin 60^{\circ} \cos 60^{\circ}$ + $\cos^{2} 60^{\circ}$ [ on expanding using identity $(a+b)^{2} = a^{2} + 2ab + b^{2}$] = $(\frac{\sqrt 3}{2})^{2}$ + 2$\times \frac{\sqrt 3}{2} \times\frac{1}{2}$ + $(\frac{1}{2})^{2}$ = $\frac{3}{4} + \frac{\sqrt 3}{2} + \frac{1}{4}$ = $\frac{3}{4} + \frac{1}{4} + \frac{\sqrt 3}{2}$ = $\frac{(3+1)}{4} + \frac{\sqrt 3}{2}$ = $\frac{4}{4} + \frac{\sqrt 3}{2}$ = 1 +$\frac{\sqrt 3}{2}$ or $\frac{2 + \sqrt 3}{2}$
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2017-06-29 02:11:08
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https://oipapio.com/question-8632035
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# floats - Want figure height to ignore side caption height - TeX
I use beside captions placed in a large margin area. Sometimes, the caption text is taller than the figure and this causes too much vertical space around the figure.
I'd like the effective height of the figure to ignore the height of the caption text in the margin. Just some vertical space around the figure as if the caption was not present. Because the caption is over in the margin, it's fine if caption text vertically overlaps the body text.
How can this be done? I'd like a way to do this such a that I do not need to hand tune each instance.
[EDIT 1] Fixed example code that only worked under xetex.
[EDIT 2] Updated sample problem PDF and showed fixed version using TH's cool \smashcaption.
Many Thanks!
Here's a really extreme example of the problem:
\documentclass{book}
%%
% Set page layout geometry
% The asymmetric option keeps the margin notes always on the same side of the page which is the way Tufte does it.
\usepackage[
letterpaper,
asymmetric,
includemp,
headheight=0.5in, % needs to be big enough for the Intel logo graphic
left=1.25in,
width=6.75in,
marginparsep=0.25in,
marginparwidth=2in,
bottom=1in,
top=1in,
nofoot,
includehead]
{geometry}
\usepackage{lipsum}
\usepackage{floatrow}
\floatsetup[figure]
{
floatwidth=\linewidth,
capposition=beside,
capbesideposition={right,center},
capbesideframe=yes,
capbesidewidth=\marginparwidth,
capbesidesep=quad,
floatrowsep=qquad
}
\begin{document}
\lipsum[1]
\begin{figure}[ht]
\centering
\rule{8cm}{1cm}
\caption{Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Ut purus
elit, vestibulum ut, placerat ac, adipiscing vitae, felis. Curabitur dictum
gravida mauris. Nam arcu libero, nonummy eget, consectetuer id, vulputate
a, magna. Donec vehicula augue eu neque. Pellentesque habitant morbi
tristique senectus et netus et malesuada fames ac turpis egestas. Mauris.}
\end{figure}
\lipsum[2]
\end{document}
Here's what it looks like:
Now with TH's \smashcaption fix:
### 1 Answer
1. 2019-11-14
Commenting out the line that causes your example to not compile, here's a solution. Define a new macro \smashcaption as follows.
\makeatletter
\newcommand*\smashcaption{
\def\FR@makecaption##1##2{%
\vbox to\z@{%
\vss
\captionfont
{\captionlabelfont##1}\caption@lsep##2%
\par
\vss
}%
}%
\caption
}
\makeatother
Now replace \caption in your code with \smashcaption and you get what I believe you want.
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2020-04-09 00:40:29
|
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http://mathoverflow.net/revisions/60387/list
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3 added 9 characters in body
In mathoverflow.net/questions/60326 http://mathoverflow.net/questions/60326 it was answered that for a cuspidal newform $f$ of weight strictly greater than 2, then $L(f,1)$ is non-zero. (Here the $L$-series is normalized so that the center of the critical strip is given by $s=k/2$.) In particular, for such modular forms, their associated $p$-adic $L$-functions are non-zero. As far as I know the non-vanishing of $p$-adic $L$-functions in the weight 2 case is a highly non-trivial result and relies upon a non-vanishing theorem of Rohrlich on twisted $L$-values. Further, from the non-vanishing of the $p$-adic $L$-function, one can deduce that $L(f,\chi,j)$ is non-zero for all but finitely many pairs $(\chi,j)$ where $\chi$ is a Dirichlet character of $p$-power conductor and $j$ is an integer between $1$ and $k-1$, as long as $p$ is an ordinary prime for $f$.
My questions:
1) Is there a direct argument to prove the non-vanishing of $L(f,\chi,j)$ for all but finitely many $\chi$ and $j$ in the ordinary and weight greater than 2 case (which doesn't use $p$-adic $L$-functions).
2) Is this result known in the non-ordinary case?
2 edited tags
1
In mathoverflow.net/questions/60326 it was answered that for a cuspidal newform $f$ of weight strictly greater than 2, then $L(f,1)$ is non-zero. (Here the $L$-series is normalized so that the center of the critical strip is given by $s=k/2$.) In particular, for such modular forms, their associated $p$-adic $L$-functions are non-zero. As far as I know the non-vanishing of $p$-adic $L$-functions in the weight 2 case is a highly non-trivial result and relies upon a non-vanishing theorem of Rohrlich on twisted $L$-values. Further, from the non-vanishing of the $p$-adic $L$-function, one can deduce that $L(f,\chi,j)$ is non-zero for all but finitely many pairs $(\chi,j)$ where $\chi$ is a Dirichlet character of $p$-power conductor and $j$ is an integer between $1$ and $k-1$, as long as $p$ is an ordinary prime for $f$.
1) Is there a direct argument to prove the non-vanishing of $L(f,\chi,j)$ for all but finitely many $\chi$ and $j$ in the ordinary and weight greater than 2 case (which doesn't use $p$-adic $L$-functions).
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2013-05-22 02:01:15
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http://golem.ph.utexas.edu/category/2006/08/categorifying_the_dijkgraafwit.html
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## August 24, 2006
### Categorifying the Dijkgraaf-Witten model
#### Posted by John Baez
The Dijkgraaf-Witten model is a simple sort of topological quantum field theory where the only field is a gauge field, and the gauge group G is finite:
Martins and Porter have a new paper on how to categorify this model, replacing the group G by a categorical group, or “2-group”.
I wrote about this stuff eleven years ago in week54 of This Week’s Finds - so if you want an elementary intro to these ideas, start there.
In the simplest version of the Dijkgraaf-Witten model, the path integral is just an integral over the moduli space of principal G-bundles, using the simplest possible measure on that space. It’s nice to formulate this theory on a triangulated manifold, where we assign a group element to each edge, and require that these group elements multiply to 1 around each triangle. This formulation makes it clear that we can also “twist” the Dijkgraaf-Witten model, which in n dimensions amounts to changing the action by any element of the nth cohomology of the gauge group.
I explained this stuff in the Winter 2005 Quantum Gravity Seminar. I also discussed the generalization where G is a Lie group - this gives 3d quantum gravity. And, I explained how the path integral in such theories can be rewritten as a sum over spin foams.
For people who like higher gauge theory, it’s tempting to “categorify” the Dijkgraaf-Witten model by replacing the gauge group by a 2-group. There’s already been some work on this, going back to a paper by Yetter:
• David Yetter, TQFT’s from homotopy 2-types, Journal of Knot Theory and its Ramifications 2 (1993), 113-123.
and you can see more recent papers online:
Anyway, here’s a new one!
A categorical group or “crossed module” is the same as a strict 2-group. Note that this paper “twists” the categorified Dijkgraaf-Witten model using an element of the nth cohomology of the classifying space of our 2-group. So, it’s using the obvious generalization of group cohomology to 2-groups.
It would be nice to generalize this work from finite (or discrete) 2-groups to Lie 2-groups, and that’s sort of what I’m doing with Freidel and Baratin - we’re focusing on the case of the Poincaré 2-group.
Note that in all these theories, the connection or 2-connection is flat - it has to be when G is discrete, but it still is in these theories when we generalize G to a Lie group or Lie 2-group. Flat connections are a wee bit boring in physics, but good for getting TQFTs. Urs is busy working on more exciting theories that involve 2-connections or 3-connections with nontrivial curvature.
One can try to go further, replacing the group in the Dijkgraaf-Witten model by an $n$-group, but the Homotopy Hypothesis conjectures that such things are the same as pointed connected homotopy $n$-types, so at this point it’s more efficient to use simplicial techniques rather than $n$-categories to make the ideas precise. For work along these lines, try:
Posted at August 24, 2006 7:16 AM UTC
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Read the post Dijkgraaf-Witten and its Categorification by Martins and Porter
Weblog: The n-Category Café
Excerpt: On Dijkgraaf-Witten theory as a sigma model, and its categorification by Martins and Porter.
Tracked: January 7, 2008 5:23 AM
### Re: Categorifying the Dijkgraaf-Witten model
Thanks, John, for inserting into my entry on Martins & Porter a link back to this entry here, which I forgot to point to.
At one point I want to get back and explain and emphasize again how really cool things happen when we replace here the finite group $G$ by our strict Fréchet Lie 2-group $\mathrm{tar} = \mathbf{B}(\Omega G \to P G)$ without the central extension (so this is still equivalent to $\mathbf{B}G$) and then realize that putting in the central extension to the strict string 2-group $(\hat \Omega G \to P G)$ amounts to introducing the “twist” here in the guise of a weak 2-functor $\mathrm{tra} : \mathbf{B}(\Omega G \to P G) \to \mathbf{B}^2 U(1) \,.$
Transgressing this setup to loop space produces the Lie-analog of Simon Willerton’s baby FHT theorem as I once described in the entry with the funny title 2-Monoid of Observables on String-G.
In particular, the representations of the loop groupoid that we transgress to are twisted equivariant vector bundles on $G$ and we make contact with non-baby FHT.
Back then I was stopped by the fact that I wasn’t sure if I could handle the smooth structure on the quotient $\Lambda(\Omega G \to P G) := \mathrm{Hom}(\mathbf{B}\mathbb{Z}, \mathbf{B}(\Omega G \to P G))/\sim \,,$ where “$/\sim$” is supposed to denote the operation of identifying isomorphic 1-morphisms.
But now I think I actually can handle this, essentially by the standard smooth space Yoga.
I should try to find the time to do that. There is something interesting lurking here, which will connect all this combinatorial and homotopy theoretic reasoning to the real thing.
Posted by: Urs Schreiber on January 7, 2008 7:46 PM | Permalink | Reply to this
### Re: Categorifying the Dijkgraaf-Witten model
A twist is a twist is a…
unfortunately not
only slightly better than weak
or to paraphrase A:
Give me a twist to lean on, and I will move the world
Posted by: jim stasheff on January 7, 2008 11:26 PM | Permalink | Reply to this
### Re: Categorifying the Dijkgraaf-Witten model
A twist is a twist is a… unfortunately not only slightly better than weak
I know exactly what you mean and I am all in favor of not using the term “twist”.
In fact, if you read what I wrote above and at the link given, you’ll see that I say:
hey, what these guys call the “twist” in Dijkgraaf-Witten theory is really to be thought of as a Line $n$-bundle over the target space $\mathbf{B} G$ of the theory.
This point of view is rather common in the Lie-analog of Dijkgraaf-Witten, namely Chern-Simons theory, where everybody is familiar with the idea that the “twist” there is really the canonical 2-gerbe over $\mathbf{B} G$. But the point is: if we regard higher $n$-bundles in terms of their transport $n$-functors, then the same kind of statement goes through also for the discrete Dijkgraaf-Witten theory.
And for its categorifications.
Posted by: Urs Schreiber on January 10, 2008 10:56 AM | Permalink | Reply to this
### Re: Categorifying the Dijkgraaf-Witten model
Perhaps I can ask a question about this, which I haven’t been able to answer by browsing through the references provided. I thought that conventional finite-group Dijkgraaf-Witten only gave you 2+1 dimensional TQFTs. What about this categorified case? Does that give you 3+1 dimensional TQFTs, or doesn’t it work like that? I suspect it doesn’t, as John didn’t mention any dimensionality in his original post.
Posted by: Jamie Vicary on May 21, 2008 8:22 PM | Permalink | Reply to this
### Re: Categorifying the Dijkgraaf-Witten model
Jamie,
When Joao and I were working on that paper we did think about this. It did not seem clear. In my earlier stuff generalising Yetter’s construction there was no restriction on the dimensionality in principle. I tried last year to see what the connection between the two approaches was but got snarled up in some related things (basically looking at the Puppe exact sequences of sheaves of simplicial groups as in Breen’s bitorsors paper as it seemed that understanding that might give the key to twisting in more generality.)
I then spent time translating something relevant from Flemish (very slow going) and your query suggests that I look at back at our ideas again!!! Any helpful ideas would be much appreciated. (If Joao sees this blog, he might make some comment here.)
In other threads I have hinted at the need to look at several of these HQFT/ETQFT notions as they change along `change of groups’ i.e. if we have $G$-based HQFT as in Turaev and then say an epimorphism from $G$ to $H$ there will be morphisms both ways. (This is consistent with thinking of HQFTs etc as representations.) The interesting case where $G$ is a central extension of $H$ is relevant to what Urs was trying to do I think.
Posted by: Tim Porter on May 22, 2008 9:23 AM | Permalink | Reply to this
### Re: Categorifying the Dijkgraaf-Witten model
DW theory, while you could consider it in any dimension, is 3-dimensional because the “twist” is a 3-cocycle: there is a 3-bundle over “target space” (where “target space” here is just a point with $G$-worth of automorphisms).
The Yetter model allows twists of one dimension higher. Just think of the special case there the 2-group involved is just the shifted version of an abelian group. So it will admit 4-bundles over its “target space” and hence be inherently 4-dimensional.
But now I have a question: isn’t it precisely this 4-dimensionality which made people conjecture that this is related to 4d quantum gravity?
Posted by: Urs Schreiber on May 22, 2008 9:58 AM | Permalink | Reply to this
### Re: Categorifying the Dijkgraaf-Witten model
I’m not sure what you mean by the Yetter model — his version of the Dijkgraaf–Witten model with a finite 2-group replacing the finite group $G$?
That’s an interesting theory, but it’s the Crane–Yetter model that has tantalizing relations to 4d quantum gravity. You can loosely think of this model as a 4d version of the Dijkgraaf–Witten model with a quantum group replacing the finite group $G$. Like the Dijkgraaf–Witten model, it’s a topological quantum field theory.
A while back I conjectured that the Crane–Yetter model is the quantum version of 4d $BF$ theory with cosmological term, with action:
$S = \int tr(B \wedge F + \lambda B \wedge B)$
Here $\lambda$ is related to $q$. As $\lambda \to 0$ we have $q \to 1$, and the Crane–Yetter model reduces to the Ooguri model (the 4d version of the Dijkgraaf–Witten model with a Lie group replacing the finite group $G$). This conjecture seems to be widely accepted, but I haven’t seen a completely satisfactory argument for it.
Posted by: John Baez on May 24, 2008 1:37 AM | Permalink | Reply to this
### Re: Categorifying the Dijkgraaf-Witten model
I’m not sure what you mean by the Yetter model — his version of the Dijkgraaf–Witten model with a finite 2-group replacing the finite group $G$?
Yes, that’s what I mean.
It seems I had thought that the Crane-Yetter model is essentially the same. Sorry for the confusion.
Posted by: Urs Schreiber on May 25, 2008 3:52 PM | Permalink | Reply to this
### Re: Categorifying the Dijkgraaf-Witten model
Jamie wrote:
I thought that conventional finite-group Dijkgraaf-Witten only gave you 2+1 dimensional TQFTs.
No, the idea behind the Dijkgraaf-Witten model works in any dimension. The idea is to build a field theory from a finite group $G$ where a ‘field’ on a closed manifold $M$ is a $G$-bundle over $M$, and the action is $0$. So, the partition function of a closed manifold $M$ just counts isomorphism classes of $G$-bundles over $M$, where the counting is done in a suitable ‘groupoid cardinality’ sense: a $G$-bundle with lots of automorphisms counts less. The details are a bit subtler for manifolds with boundary, but not in a bad way.
There are lots of ways to get your hands on the Dijkgraaf–Witten model, but many of these seem to have only been worked out in low dimensions.
For example, in weeks 5–10 of the Fall 2004 quantum gravity seminar I showed how Fukuma, Hosono and Kawai built a 2d TQFT from any finite-dimensional semisimple algebra. It’s a state model idea where you triangulate your manifold and label the edges by basis vectors in your algebra. If you apply this idea to the group algebra $\mathbb{C}[G]$ you get the 2d Dijkgraaf–Witten model, as I explained in week 10 of that course.
Then in the Winter of 2005 I sketched how to get 3d TQFTs from ‘semisimple 2-algebras’. In particular, this construction gives the 3d Dijkgraaf–Witten model when we use $Vect[G]$ as our semisimple 2-algebra.
I see no reason in principle why this idea can’t work in higher dimensions too, though it’s not the most efficient way to construct the Dijkgraaf–Witten model in arbitrary dimensions. My student Jeffrey Morton constructed the Dijkgraaf–Witten model as a ‘once extended TQFT’ in arbitrary dimensions in his thesis. By a ‘once extended TQFT’, I mean that he constructed a bicategory $nCob_2$ of:
• closed $(n-2)$-manifolds,
• $(n-1)$-dimensional cobordisms between these, and
• $n$-dimensional cobordisms between those,
and showed that the Dijkgraaf–Witten model gives a functor
$nCob_2 \to 2Vect$
All this so far is for the ‘untwisted’ Dijkgraaf–Witten model. We can also choose a slightly more interesting action for the $n$-dimensional model, which is called ‘twisting’ it by an $n$-cocycle. There are lots of ways of thinking about this; I presented a fun $n$-categorical way in weeks 8–10 of the Winter 2005 seminar.
Briefly: a 2-cocycle modifies the multiplication in $\mathbb{C}[G]$, a 3-cocycles modifies the associator in $Vect[G]$, etc… You get some very pretty pictures involving Pachner moves this way. But again, this is not the most efficient approach.
Posted by: John Baez on May 23, 2008 11:16 PM | Permalink | Reply to this
### Re: Categorifying the Dijkgraaf-Witten model
Thanks, everybody, for your useful replies. It’s a bit frustrating that there isn’t a better universal description of TQFTs. We know that in 1+1 dimensions they’re commutative $\dagger$-Frobenius algebras in Hilb, and I know what all of these are. But I get the impression from the literature that even for 2+1 dimensions, there’s no equally useful description of what a TQFT is. So we can come up with clever ways to generate them — like Dijkgraaf-Witten models — but have no reason to believe we’re obtaining all the 2+1 TQFTs in this way.
It seems obvious to wonder whether 3Cob${}_2$ is the free strongly symmetric monoidal 2-category on a symmetric $\dagger$-Frobenius pseudomonoid… but I’m sure this isn’t true, or somebody would have proved it and I would have read it somewhere!
Posted by: Jamie Vicary on May 24, 2008 9:21 PM | Permalink | Reply to this
### Re: Categorifying the Dijkgraaf-Witten model
Jamie Vicary wrote:
It seems obvious to wonder whether $3Cob_2$ is the free strongly symmetric monoidal 2-category on a symmetric $\dagger$-Frobenius pseudomonoid… but I’m sure this isn’t true, or somebody would have proved it and I would have read it somewhere!
Huh? Are you serious? There aren’t many people who could prove a theorem like this, much less understand all the words you just wrote!
In fact, for a long time I’ve believed a result like this was true. For a while I was trying to get Aaron Lauda to prove it. In August 2005 he got as far as giving a similar description of a simpler 2-category called $2Thick$, where the 2-morphisms are things like this:
A bit more precisely, $2Thick$ has
• collections of open strings embedded in the line as objects,
• open string worldsheets embedded in the plane as morphisms,
• isotopy classes of 3-dimensional manifolds with corners defined by diffeomorphisms of these open string worldsheets as 2-morphisms.
He showed that $2Thick$ is the free monoidal 2-category on a Frobenius pseudomonoid. But then he got involved in Khovanov homology and stopped working on this project.
So, I’m afraid it’s up to you to give a nice category-theoretic description of $3Cob_2$.
Posted by: John Baez on May 25, 2008 2:36 AM | Permalink | Reply to this
### Re: Categorifying the Dijkgraaf-Witten model
This is wonderful! I love the pictures, especially the one of the Frobeniusator. Not sure how I’ve managed to miss this paper for so long… this is really fantastic stuff.
I’m a bit surprised about something, though. If we’re looking for an equivalence with 3Cob${}_2$, wouldn’t we expect the objects to be 2-dimensional topological spaces? He has them as open strings, which are 1-dimensional topological spaces.
Posted by: Jamie Vicary on May 25, 2008 10:20 AM | Permalink | Reply to this
### Re: Categorifying the Dijkgraaf-Witten model
Hang on, that’s not right. The 2-category 3Cob${}_2$ presumably has objects which are 2-dimensional topological spaces which are compact without boundary. For the purposes of Aaron’s paper we would probably be more interested in an ‘open’ version of this, that has objects as 2-dimensional topological topological spaces that are compact with boundary.
Posted by: Jamie Vicary on May 25, 2008 10:30 AM | Permalink | Reply to this
### Re: Categorifying the Dijkgraaf-Witten model
Jamie wrote:
I’m a bit surprised about something, though. If we’re looking for an equivalence with $3Cob_2$, wouldn’t we expect the objects to be 2-dimensional topological spaces?
Yikes! We’d better agree on how to count before we keep talking.
For me, $3Cob_2$ is the 2-category where:
• objects are compact oriented 1-manifolds,
• morphisms are cobordisms between these — thus 2-manifolds with boundary,
• 2-morphisms are cobordisms between those — thus 3-manifolds with corners.
This 2-category has been intensively studied in work on extended TQFTs, since it’s implicit in how people get their hands on certain 3d TQFTs, especially the Witten–Reshetikhin–Turaev theory. So, it deserves a purely algebraic description!
More generally, I define $n Cob_k$ to the $k$-category where the top-dimensional morphisms are $n$-dimensional cobordisms between cobordisms between… compact $(n-k)$-manifolds.
So, the category $n Cob_1$ is just what folks usually call $n Cob$.
Now, a famous fact is that $2 Cob_1$ is the free symmetric monoidal category on a commutative Frobenius monoid. This Frobenius monoid is just the circle.
And, it’s easy to see that when we decategorify $3 Cob_2$ we get $2 Cob_1$.
So, we naively expect $3 Cob_2$ to be something like the free symmetric monoidal 2-category on a symmetric Frobenius pseudomonoid — namely, the circle.
This is close to true, but not quite: we need some extra bells and whistles on our Frobenius pseudomonoid to deal with the extra layer of duality in our 2-category! Drawing a bunch of pictures would make this clear…
Anyway: before tackling this problem — which he never got around to — Aaron Lauda tried something easier.
There’s a category $1 Thick$ with:
• collections of open strings embedded in the line as objects,
• isotopy classes of open string worldsheets embedded in the plane as morphisms.
And, it’s easy to see that this is the free monoidal category on a Frobenius monoid. This Frobenius monoid is just the closed interval.
So, Aaron categorified this result. He considered the 2-category $2 Thick$ with
• collections of open strings embedded in the line as objects,
• open string worldsheets embedded in the plane as morphisms,
• isotopy classes of 3-dimensional manifolds with corners defined by diffeomorphisms of these open string worldsheets as 2-morphisms.
And, Aaron showed this is the free monoidal 2-category on a Frobenius pseudomonoid. This Frobenius pseudomonoid is just the closed interval.
But, I wish some young and energetic person would give a purely algebraic characterization of $3 Cob_2$ in terms of Frobenius pseudomonoids!
By the way, the following paper is extremely helpful for anyone interested $3 Cob_2$:
• Ulrike Tillman, Discrete models for the category of Riemann surfaces, Math. Proc. Cambridge Philos. Soc. 121 (1997), 39–49.
Here’s part of the description in Math Reviews, written by my old schoolmate Kathryn Hess:
Segal’s category of Riemann surfaces, $M$, in which the objects are oriented 1-manifolds up to homotopy and the morphisms are Riemann surfaces, is the basis of any conformal field theory [G. B. Segal, in Differential Geometrical Methods in Theoretical Physics (Como, 1987), 165–171, Kluwer, Dordrecht, 1988]. It can be difficult to work in $M$, however, due to the analytic description of its morphisms. It is therefore of interest to find a discrete, algebraic category that models $M$ in some appropriate sense.
The goal of the research presented in this article is to define and study a particular discrete, algebraic model of $M$, thus establishing a theoretical framework for results the author has since proved about 3-manifold invariants, as well as for the proof that the stable mapping class group is an infinite loop space after group completion.
The author begins with a concise and clearly presented overview of the theory of 2-categories. She succeeds in defining concepts as complex as that of a symmetric, strict monoidal 2-category without overwhelming the reader with notation and axioms. She also reviews the construction and properties of the classifying space of a category and its specialization to the classifying category of a 2-category.
In the second part of the article, the author defines a strict 2-category, $S_D$, from which she derives her model of $M$. In $S_D$, (0) the 0-morphisms are in one-to-one correspondence with the natural numbers; (1) a 1-morphism from $n$ to $m$ consists essentially of a smooth, oriented cobordism whose boundary is composed of $n$ circles on the “bottom” and $m$ circles on the “top”; (2) the set of 2-morphisms between two cobordisms is the group of all orientation-preserving diffeomorphisms between them (which is empty if they are not diffeomorphic). The discrete category $S_\Gamma$ that models $M$ is then the quotient of $S_D$ that identifies two diffeomorphisms if they belong to the same path component in the space of all diffeomorphisms.
So, you see $S_D$ doesn’t have all the 2-morphisms in $3 Cob_2$: only the cobordisms that come from diffeomorphisms! In this respect it resembles $2Thick$, where the 2-morphisms come from diffeomorphisms of the square:
But, it’s a good step towards $3Cob_2$.
By the way, the higher category theory in this paper is a bit sketchy: personally I’d feel more confident in the results after being ‘overwhelmed with notation and axioms’.
Posted by: John Baez on May 26, 2008 1:04 AM | Permalink | Reply to this
### Re: Categorifying the Dijkgraaf-Witten model
John said:
But, I wish some young and energetic person would give a purely algebraic characterization of $3\mathrm{Cob}_2$ in terms of Frobenius pseudomonoids!
OK, so first you’d get an algebraic characterisation of the free symmetric monoidal 2-category on a symmetric Frobenius pseudomonoid. But to then prove an equivalence to $3\mathrm{Cob}_2$… wouldn’t that be rather a stupendous result, given all the problems there are with classifying 3D manifolds?
Posted by: Jamie Vicary on May 28, 2008 11:16 AM | Permalink | Reply to this
### Re: Categorifying the Dijkgraaf-Witten model
Jamie wrote:
OK, so first you’d get an algebraic characterisation of the free symmetric monoidal 2-category on a symmetric Frobenius pseudomonoid.
That is an algebraic characterization, and that’s really all we need on the algebraic end of things. The work is showing that $3Cob_2$ admits an algebraic characterization sort of like this.
But to then prove an equivalence to $3Cob_2$
By the way, I never claimined that $3Cob_2$ is the free symmetric monoidal 2-category on a symmetric Frobenius pseudomonoid! We need a symmetric Frobenius pseudomonoid with a few extra bells and whistles, which I could describe if you bought me enough coffee. I don’t want people trying to prove the wrong conjecture.
…wouldn’t that be rather a stupendous result, given all the problems there are with classifying 3D manifolds?
No, you wouldn’t need to classify 3-manifolds to prove this sort of result. An algebraic description of $3Cob_2$ of the sort we’re contemplating amounts to a ‘generators and relations’ description of 3d manifolds with corners in terms of basic building blocks. There are already plenty of results like this in the literature, obtained using Morse theory and Cerf theory; you can see some nice ones in the book by Kerler and Lyubashenko. They just haven’t been massaged into the right form. Doing so won’t be trivial, but I think it’s within reach.
The point is this: knowing a bunch of generators and relations for an algebraic structure doesn’t mean you can tell when two elements of this algebraic structure are equal. Indeed, there’s a finitely presented group where it’s algorithmically undecidable when two elements of this group are equal!
In fact, the Poincaré conjecture was long known to be equivalent to various purely algebraic questions about group presentations… but these turned out not to be helpful for people trying to prove that conjecture.
Here’s an algebraic statement equivalent to the Poincaré conjecture. It’s not very relevant to what we’re talking about, but it’s awfully cute. I just found it in a talk by Dale Rolfsen.
Say two group homomorphisms $h_1, h_2 : G \to H$ equivalent if there is an automorphism $\alpha : G \to G$ such that $h_1 \alpha = h_2$.
Let $G$ be the group with generators $x_1,y_1, \dots, x_g, y_g$ and one relation:
$[x_1, y_1] \cdots [x_g, y_g] = 1$
where $[x,y] = x y x^{-1} y^{-1}$. Topologists will recognize this group as the fundamental group of the $g$-holed torus.
Let $F_1$ be the free group generated by $x_1, \dots, x_g$ and let $F_2$ be the free group generated by $y_1, \dots, y_g$.
There’s an obvious surjective homomorphism
$\phi : G \to F_1 \times F_2$
since in $F_1 \times F_2$ all the $x_i$’s commute with all the $y_i$’s, which implies the relation shown above.
Theorem (Hempel, Perelman): Up to equivalence, $\phi$ is the only surjective homomorphism from $G$ to $F_1 \times F_2$.
Hempel showed this fact was equivalent to the Poincaré conjecture. Later, Perelman proved the Poincaré conjecture! So, it’s an example of an elementary-sounding result about group theory whose proof currently involves lots of hard analysis and Riemannian geometry.
Posted by: John Baez on May 29, 2008 6:24 PM | Permalink | Reply to this
### Re: Categorifying the Dijkgraaf-Witten model
John said:
That is an algebraic characterization
I was implying was that it’s not immediately obvious to me what a symmetric Frobenius pseudomonoid should actually be. For example, I think there are some choices to make about how coherent we want things to be. But I know that an answer to this is given in that impressive-looking book of Kerler and Lyubashenko that you mention.
By the way, I never claimed that ${}3\mathrm{Cob}_2$ is the free symmetric monoidal 2-category on a symmetric Frobenius pseudomonoid! We need a symmetric Frobenius pseudomonoid with a few extra bells and whistles, which I could describe if you bought me enough coffee.
Coffee is fast becoming the currency of choice for trading mathematical favours. I suppose we’re really after the free monoidal stable weak 2-category with duals on one object, right? I know what all those words mean, but I’m sure it would take me a good while to turn them into an algebraic presentation. Of course, it would be good for me!
An algebraic description of ${}3\mathrm{Cob}_2$ of the sort we’re contemplating amounts to a ‘generators and relations’ description of 3d manifolds with corners in terms of basic building blocks. There are already plenty of results like this in the literature
Oh, right — that’s better than I thought, then. Thanks for making this clear. The example you give using the Poincaré theorem is great!
Posted by: Jamie Vicary on May 30, 2008 11:02 AM | Permalink | Reply to this
### Re: Categorifying the Dijkgraaf-Witten model
The example you give using the Poincaré theorem is great!
I agree. It rattled my bones when I read it!
Posted by: Bruce Bartlett on May 30, 2008 11:05 AM | Permalink | Reply to this
### Re: Categorifying the Dijkgraaf-Witten model
Hempel showed this fact was equivalent to the Poincaré conjecture.
Do you have a way of seeing heuristically why this might be true?
I suppose $G$ here I should think of as a model for the fundamental groupoid $\Pi_1(S_g)$, of the genus $g$ surface $S_g$. So the statement is somehow about maps out of genus $g$ surfaces.
Posted by: Urs Schreiber on May 30, 2008 1:20 PM | Permalink | Reply to this
### Re: Categorifying the Dijkgraaf-Witten model
Urs wrote:
Do you have a way of seeing heuristically why this might be true?
Unfortunately I don’t think Dale Rolfsen’s talk says how Hempel proved the equivalence of this algebraic result and the Poincaré conjecture. It might be discussed in Hempel’s book on 3-manifolds, which has a chapter on the Poincaré conjecture. I’m too lazy to check.
Naively I’d guess it has something to do with Heegaard splittings. You can always split a compact oriented 3-manifold into two solid handlebodies glued together along a $g$-holed torus. You can do it lots of ways, and there are ‘Heegard moves’ that take us between any two such splittings.
So, here’s my wild guess. Maybe any Heegard splitting of a homotopy 3-sphere can be coaxed to give a surjection $\phi: G \to F \times F$. I don’t see how. But maybe when such a surjection is equivalent to the obvious one, our homotopy 3-sphere is homeomorphic to the usual 3-sphere. That would give Hempel’s result.
Posted by: John Baez on May 30, 2008 8:32 PM | Permalink | Reply to this
### Re: Categorifying the Dijkgraaf-Witten model
Just 30 minutes ago I talked about this with Soren Galatius over coffee. He also pointed me to Heegard splittings.
the homeomorphism with which we glue the two handlebodies is relevant only up to isotopy. This probably corresponds to the notion, that you mentioned, of equivalence of two functors out of $\Pi_1(S)$, for $S$ the $g$-holed torus if they differ by precomposition with an automorphism of $\Pi_1(S)$.
Posted by: Urs Schreiber on May 30, 2008 8:51 PM | Permalink | Reply to this
### Re: Categorifying the Dijkgraaf-Witten model
Jamie wrote:
I was implying was that it’s not immediately obvious to me what a symmetric Frobenius pseudomonoid should actually be.
Oh, okay. Right, this is part of the problem: figuring out the correct definitions.
For example, I think there are some choices to make about how coherent we want things to be.
Right. In particular, while we more or less know what a ‘symmetric pseudomonoid’ in a 2-category is (just copy the definition of symmetric monoidal category, and quickly make up your mind whether you want associators and unitors or not), and we more or less know what a ‘Frobenius pseudomonoid’ is, we need scan for coherence laws involving both the symmetry and the Frobeniator.
But I know that an answer to this is given in that impressive-looking book of Kerler and Lyubashenko that you mention.
The warning light is blinking on my feeble American-made irony detector. I’m quite sure the answer is not given in this book. However, it’s full of useful stuff.
I suppose we’re really after the free monoidal stable weak 2-category with duals on one object, right?
No, $3Cob_3$ should be the free stable monoidal weak 3-category with duals on one object. The unit object $1 \in 3Cob_3$ should be the empty 0-manifold. And we’re looking to understand
$3Cob_2 = hom(1,1)$
which should be roughly the 2-category of:
• compact oriented 1-manifolds,
• cobordisms between these (which are 2-manifolds with boundary), and
• cobordisms between those (which are 3-manifolds with corners).
Every object in this 2-category should be a tensor product of copies of the circle. So, we focus on the circle, which should be a categorified version of a commutative Frobenius object — with extra bells and whistles. And, we hope that $3Cob_2$ is the free symmetric monoidal 2-category on this gadget.
All this is motivated by the story one dimension down, so let me review that:
$2Cob_2$ should be the free stable monoidal weak 2-category with duals on one object. The unit object $1 \in 2Cob_2$ should be the empty 0-manifold. And now we’re looking to understand
$2Cob_1 = hom(1,1)$
which is the 1-category of:
• compact oriented 1-manifolds,
• cobordisms between these (which are 2-manifolds with boundary).
Every object in this category is a tensor product of copies of the circle. So, we focus on the circle, which is a commutative Frobenius object. And, we discover to our delight that $2Cob_1$ is the free symmetric monoidal category on a commutative Frobenius object.
Even in this lower-dimensional case, nobody has gotten around to characterizing $2Cob_2$ as a 2-category! That’s why I propose shirking the job of characterizing $3Cob_3$, and going for the easier $3Cob_2$.
I know what all those words mean, but I’m sure it would take me a good while to turn them into an algebraic presentation. Of course, it would be good for me!
It would be good for the world.
By the way, to be perfectly clear: I write ‘should be’ above for things I think are true, and ‘is’ for things that have been proved.
Posted by: John Baez on May 30, 2008 8:54 PM | Permalink | Reply to this
### Re: Categorifying the Dijkgraaf-Witten model
It’s a bit late for me… but I can assure you that your
was misfiring! I don’t know why I trigger it so often… :) I was referring to Aaron Lauda’s paper, lemma 32, which directly cites Kerler and Lyubashenko. I’m a bit too tired to establish whether it’s concerned with exactly the same 2-category that I was talking about, but frankly, 20 coherences are enough for anyone. I’ll say more in the morning.
Posted by: Jamie Vicary on May 31, 2008 12:26 AM | Permalink | Reply to this
### Re: Categorifying the Dijkgraaf-Witten model
Jamie wrote:
I can assure you that your
was misfiring!
Whoops! I need to turn down the sensitivity level. I thought if you said you “knew” that an “impressive-looking book” you hadn’t actually read contained the answer to some interesting question, that was a British way of saying the book was useless pompous piffle.
Anyway, I don’t believe that book mentions categorified Frobenius algebras. I think Aaron Lauda took a long list of diagrammatic equations from that book and realized they were the definition of a Frobenius pseudomonoid, as part of his proof that 2Thick is the free (semistrict) monoidal 2-category on a Frobenius pseudomonoid.
I don’t know if anyone has defined symmetric Frobenius pseudomonoids yet, which is part of what we’d need to describe $3Cob_2$.
Posted by: John Baez on May 31, 2008 8:14 AM | Permalink | Reply to this
### Re: Categorifying the Dijkgraaf-Witten model
John wrote:
So, we naively expect 3Cob 2 to be something like the free symmetric monoidal 2-category on a symmetric Frobenius pseudomonoid — namely, the circle.
This is close to true, but not quite: we need some extra bells and whistles on our Frobenius pseudomonoid to deal with the extra layer of duality in our 2-category!
Bruce Bartlett pointed out a mistake here: I should have said braided Frobenius pseudomonoid!
Posted by: John Baez on July 30, 2008 1:54 PM | Permalink | Reply to this
### Re: Categorifying the Dijkgraaf-Witten model
It’s a bit frustrating that there isn’t a better universal description of TQFTs. We know that in 1+1 dimensions they’re commutative $\dagger$-Frobenius algebras in Hilb,
The thing which stokes my interest about TQFTs is not a universal “top-down” description of this form, but rather a geometric “bottom-up” description. In other words, it’s true that a (not-necessarily unitary) 2d TQFT gives rise to a commutative Frobenius algebra in Vect - but for me the key words there are gives rise to which I would be very reluctant to change to is.
I sort of prefer to take the idea of a TQFT at face value… it is a gadget which takes geometric data and transforms it into higher algebraic structures. How it does this is a geometric question - it could be via principal bundles, or gerbes, or stacks, and so on. Those are the things I find cool. I think this is also the “$\Sigma$-model” philosophy of Urs, which I am an adherent of.
Having said that, I must say that the top-down approach is also very important and will be extremely useful when it is achieved. I spend my lunch breaks there, but the rest of the time I spend down in the engine-room in the basement, trying to learn about the geometric gismos which make this thing tick.
Posted by: Bruce Bartlett on May 25, 2008 12:18 PM | Permalink | Reply to this
### Re: Categorifying the Dijkgraaf-Witten model
Bruce wrote:
I sort of prefer to take the idea of a TQFT at face value; it is a gadget which takes geometric data and transforms it into higher algebraic structures.
Is it clear what sort of higher algebraic structures should be called a TQFT? Namely?
Posted by: jim stasheff on May 25, 2008 2:34 PM | Permalink | Reply to this
### Re: Categorifying the Dijkgraaf-Witten model
Surely the ultimate goal is to understand these TQFTs. So if there was a straightforward characterisation of all Frobenius algebras, would you be happy with that? Or is there another reason why we are interested in the more exciting bundly-gerby-stacky power-tools that you mention?
Posted by: Jamie Vicary on May 28, 2008 12:16 PM | Permalink | Reply to this
### Re: Categorifying the Dijkgraaf-Witten model
Is it clear what sort of higher algebraic structures should be called a TQFT? Namely?
Dear Jim: I guess the answer is no; for instance I don’t think we even yet have a good grip on the kind of ‘2-vector spaces’ we will need… and that’s only at codimension 2. 2-Hilbert spaces don’t seem to fit the examples coming from the ‘derived world’, where a ‘2-vector space’ is more akin to a category of the from $D(X)$ for some smooth space $X$. But can one characterize these categories intrinsically in a simple manner?
So if there was a straightforward characterisation of all Frobenius algebras, would you be happy with that?
Dear Jamie: Yes indeed… with the disclaimer that for example semisimple Lie algebras are classified in terms of roots and weights - but somehow that’s not always the most alluring viewpoint (see this for example).
Posted by: Bruce Bartlett on May 28, 2008 1:46 PM | Permalink | Reply to this
### Re: Categorifying the Dijkgraaf-Witten model
The reason to be interested in the
bundly-gerby-stacky power-tools that you mention
is that it is what connects the abstract Frobenius algebra which you may find your TQFT to be encoded by to geometric data which the TQFT can then be thought of as coming from by “quantization”.
For instance, just from abstract TQFT reasoning you find that Chern-Simons theory assigns vector spaces with certain properties to surfaces. But the particularly interesting statements only arise after you can also identify these abstract vector spaces with spaces of holomorphic sections of line bundles that are obtained from transgressing a certain 2-gerbe to some mapping space.
Posted by: Urs Schreiber on May 29, 2008 4:42 AM | Permalink | Reply to this
### Re: Categorifying the Dijkgraaf-Witten model
I’m having quite a lot of luck banishing some of my higher-categorical gremlins here, so I’ll try again with one more.
I’ve lurked around enough by now to discover that a fundamental motivation for higher category theory is the relationship between $n$-groupoids and the homotopies of compact topological spaces. For example, we can think of a 2-group $G$ as a connected compact topological space with $\pi_3$, $\pi_4$ and so on all trivial.
But surely, the same topological space would be just as well-represented categorically by a 3-group created from $G$ by adding trivial 3-cells. Let’s call this $T(G)$, where the process $T$ adds trivial $n+1$ cells to an $n$-category. It would also be just as well represented by $T^m(G)$, or even by $T ^{\infty}(G)$.
Something else that’s of fundamental importance for $n$-groups is their representation theory, given by unitary functors into $n\!\mathrm{Hilb}$. If $G$ and $T(G)$ essentially represent the same topological space, wouldn’t we expect there to be a correspondence between the representations of $G$ and of $T(G)$, and in fact of $T^m(G)$ and $T ^{\infty} (G)$, for any $n$-groupoid $G$? However, it seems to me that this isn’t the case, which I’ve tested at all the levels for which I know what ‘functor’ means.
Does this bother anybody else, or is there some deeper philosophy of $n$-groupoids by which it all makes sense? The obvious conclusion is that a representation of an $n$-groupoid doesn’t tell you something about the underlying topological space, but is sensitive to the way you’ve chosen to describe it as an $n$-category… this seems a shame to me!
Posted by: Jamie Vicary on May 25, 2008 10:59 AM | Permalink | Reply to this
### Re: Categorifying the Dijkgraaf-Witten model
A related point:
The n-cat language seems to index according to the first n homotopy groups NOT by
the first non-trivial homotopy groups,
what in homotopy theory might be called n-stage.
Posted by: jim stasheff on May 25, 2008 2:38 PM | Permalink | Reply to this
### Re: Categorifying the Dijkgraaf-Witten model
Yes, I suppose that’s right. But I don’t think it would help if we changed to indexing by the first non-trivial homotopy groups, unfortunately.
Posted by: Jamie Vicary on May 28, 2008 11:09 AM | Permalink | Reply to this
Post a New Comment
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2014-07-25 13:00:03
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https://math.stackexchange.com/questions/698171/the-number-of-grid-points-near-a-circle-question
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# The number of grid points near a circle question
I've just read The number of grid points near a circle.
I want to comment but I don't have the reputation so I have to post this as a question.
How do you prove that for each $x$ value less than $r\sqrt2/2$ there is a unique $y$ value and for each $y$ value less than $r\sqrt2/2$ there is a unique $x$ value?
My idea is that you could use the fact that the circle's gradient decreases from $0$ to $-1$ as $x$ goes from $0$ to $r\sqrt2/2$ and this implies that the line will not drop below one $y$ value for each $x$ value passed. However I can't think of how this can be formalized as a proof.
for each $x$ value less than $r\sqrt2/2$ there is a unique $y$ value
At a given position $0\le x\le\frac{r}{\sqrt 2}$, the point $(x,y)$ with
$$y = \left\lfloor\sqrt{r^2-x^2}\right\rfloor$$
is within the circle, and the point $(x, y+1)$ is outside. So the point must be counted.
Now about the uniqueness. Suppose there were a different point $(x,y')$ which should be counted as well. Since that point must lie inside the circle, it must satisfy $0\le y'<y$. You can now consider points around it. $(x,y'+1)$ will be inside since $y'+1\le y$ and $(x,y)$ is inside. $(x-1, y')$ must be inside since $(x,y')$ is inside and the circle and we are in the first quadrant. Likewise for $(x,y'-1)$. Both arguments will need discussion of a special case for $x=0$ resp. $y'=0$, but I'll leave that as an excercise for now.
So $(x+1,y')$ would be the most likely candidate for a point outside the circle. So let's see whether it can be outside the circle.
\begin{multline*} (x+1)^2+y'^2 \le (x+1)^2+(y-1)^2 = x^2+y^2 + 2x-2y+2 \\\le r^2+2x-2y+2 \le r^2+2(y-1)-2y+2 = r^2 \end{multline*}
There are three inequalities in there. The first is because $y'<y$, the second because $x^2+y^2\le r^2$ since $(x,y)$ is inside, and the third because $x<y$ since you are in the upper half of the first quadrant. The case of $x=y$ needs special treatment, but then $(x,y')$ would satisfy $x>y'$ and therefore already belong to the right half of the quadrant. Taken together, you have shown that $(x+1,y')$ cannot be outside the circle either, so $(x,y')$ will not be counted, so $(x,y)$ is unique for given $x$.
and for each $y$ value less than $r\sqrt2/2$ there is a unique $x$ value?
This is simply the same argument above, but with $x$ and $y$ changing their roles. So if you have one, you have both.
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2020-05-26 04:28:52
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https://www.gradesaver.com/textbooks/math/precalculus/precalculus-6th-edition-blitzer/chapter-p-section-p-2-exponents-and-scientific-notation-exercise-set-page-30/108
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Chapter P - Section P.2 - Exponents and Scientific Notation - Exercise Set - Page 30: 108
$$\frac{y^7}{x^8}$$
Work Step by Step
\begin{align} \frac{(xy^{-2})^{-2}}{(x^{-2}y)^{-3}} & = \frac{x^{-2}y^4}{x^6y^{-3}}\\ &= \frac{y^3y^4}{x^2x^6} \\ &= \frac{y^7}{x^8} \end{align}
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.
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2023-03-25 03:45:25
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http://www.ams.org/mathscinet-getitem?mr=0397737
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MathSciNet bibliographic data MR397737 57A15 (54C56) Vo Thanh Liem Certain continua in \$S\sp{n}\$$S\sp{n}$ of the same shape have homeomorphic complements. Trans. Amer. Math. Soc. 218 (1976), 207–217. Article
For users without a MathSciNet license , Relay Station allows linking from MR numbers in online mathematical literature directly to electronic journals and original articles. Subscribers receive the added value of full MathSciNet reviews.
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2017-03-29 23:42:30
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https://aakashsrv1.meritnation.com/cbse-class-12-commerce/accountancy/board-paper-of-class-12-commerce-2021-accountancy-term-i-delhi(set-4)-(series-:-ssj/2)--solutions/board-papers/starttest/PfO9rnrrHKd6EUmHt4ldiQ!!
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Select Board & Class
# Board Paper of Class 12-Commerce 2021 Accountancy Term-I Delhi(Set 4) (Series : SSJ/2)- Solutions
General Instructions :
1. This question paper contains 60 questions out of which 40 questions are to be attempted. All questions carry equal marks.
2. This question paper consists of three parts – Part I, II and III.
3. Part I is compulsory for all candidates. Attempt either Part II or Part III.
4. Part I comprises of three sections – Section A, B and C.
5. From Part I (Q. No. 1 to 36)– attempt any 14 questions each from Section A and B. Attempt any three questions from Section C.
6. From Part II OR III – (Q.No. 37 to 60) – attempt any four questions from Section A and any five questions from Section B.
7. Attempted first, desired number of questions only, in each Part/Section will be evaluated.
8. There is only one correct option for every multiple choice questions (MCQs). Marks will not be awarded for answering more than one option.
9. There is no negative marking.
• Question 1
The partnership deed should be properly drafted and prepared as per the provisions of the _____ and preferably registered with the ____.
(a) Indian Partnership Act 1932, Registrar of Companies.
(b) Indian Partnership Act 1932, Registrar of firms.
(c) Stamp Act, Registrar of Companies.
(d) Stamp Act, Registrar of Firms. VIEW SOLUTION
• Question 2
A Ltd. forfeited a share of ₹100 issued at a premium of 20% for non payment of first call of ₹30 per share and final call of ₹10 per share. The minimum price at which this share can be reissued is _____.
(a) ₹40 per share
(b) ₹60 per share
(c) ₹100 per share
(d) ₹80 per share VIEW SOLUTION
• Question 3
The difference between the fixed capital method and fluctuating capital method of maintaining partner's capital is whether or not the transactions other than ______ are recorded in the Capital accounts of the partners.
(b) Withdrawal of Capital
(c) Partner's Loan
(d) Both (a) and (b) VIEW SOLUTION
• Question 4
R, S and T were partners sharing profits and losses in the ratio of 5 : 3 : 2. On 31st March, 2021, their books reflected a net profit of ₹3,10,000. As per the terms of the partnership deed they were entitled for interest on Capital which amounted to ₹90,000, ₹60,000 and ₹30,000 respectively for R, S and T. Besides this an annual salary of ₹60,000 each was payable to R and S.
The ratio in which the profits would be appropriated is
(a) 1 : 1 : 1
(b) 5 : 3 : 2
(c) 5 : 4 : 1
(d) 4 : 3 : 2 VIEW SOLUTION
• Question 5
Sushila Ltd. has an 'Authorized Capital', of ₹10,00,000 divided into equity shares of ₹10 each. Subscribed and fully paid up share capital of the company was ₹4,00,000. To meet its new financial requirement, the company issued 20,000 equity shares of ₹10 each. Amount per share was payable as ₹3 on application, ₹3 on allotment; ₹2 on first call and ₹2 on second and the final call. The issue was fully subscribed. The allotment money was payable on or before May 1, 2020; first call money was due on August 1, 2020 and final call money was due on October 1, 2020. X whom 1000 shares were allotted did not pay the allotment and both calls; Y an allottee of 600 shares; did not pay the two calls; and Z whom 300 shares were allotted did not pay the final call.
Subscribed capital presented in the Balance sheet of the Company as per schedule III Part I of the Companies Act, 2013 will be :
(a) ₹9,800
(b) ₹5,90,000
(c) ₹10,00,000
(d) ₹6,00,000 VIEW SOLUTION
• Question 6
X and Y were partners sharing profits in the ratio of 3 : 2. Z was admitted as a new partner for 1/5th share. X sacrificed $\frac{3}{20}$ from his share and Y sacrificed $\frac{1}{20}$ from his share in favour of Z, the new profit sharing ratio would be :
(a) 9 : 7 : 4
(b) 8 : 8 : 4
(c) 6 : 10 : 4
(d) 10 : 6 : 5
VIEW SOLUTION
• Question 7
Lata Ltd. forfeited Maya's shares. Maya who had applied for 600 shares of ₹10 each and was allotted 400 shares failed to pay allotment money of ₹4 per share including premium of ₹2. She had paid only the application money of ₹2 per share. The first and final call was not yet called. The Journal entry for forfeiture of shares by opening calls in arrears account will be:
(₹) (₹) (a) Share Capital A/c ……………………….. To Share forfeited A/c Dr. 90,000 11,250 1,01,250 (b) Share Capital A/c ……………………….. Securities Premium Reserve A/c To Share forfeited A/c To Calls in Arrears A/c Dr. 1,600 800 1,200 1,200 (c) Share Capital A/c ……………………….. Securities Premium Reserve A/c ……….. To Share forfeited A/c To Calls in Arrears A/c Dr. Dr. 1,600 800 1,800 800 (d) Share Capital A/c ……………………….. To Shares forfeited A/c To Calls in Arrears A/c Dr. Dr. 2,000 1,800 800
VIEW SOLUTION
• Question 8
Versha Ltd. purchased the running business of Vikram Ltd. consisting of total assets of ₹10,00,000 and liabilities of ₹2,00,000. Versha Ltd. paid ₹2,00,000 immediately in Cash and Balance by issuing 7,000 shares of ₹100 each at a premium of ₹20 per share. The Goodwill A/c will be debited by______.
(a) ₹2,40,000
(b) ₹2,00,000
(c) ₹8,00,000
(d) Nil VIEW SOLUTION
• Question 9
Jupiter Ltd. invited applications for issuing 25,000 equity shares of ₹10 each and received applications for 30,000 shares along with the application money of ₹2 per share. Which of the following alternative can be followed for the allotment of shares ?
(i) Refund the excess application money and allot full shares to rest of the applicants.
(ii) Not to allot any share to some applicants, allot full to some applicants and allot remaining on prorata basis.
(iii) Not to allot any share and refund the total application money to the applicants.
(a) Only (i)
(b) Only (ii)
(c) Only (iii)
(d) Any one of (i) and (ii)
VIEW SOLUTION
• Question 10
Mallika, Meera and Madhu were partners sharing profits in the ratio of 2 : 2 : 1. They decided to share future profits in the ratio of 7 : 5 : 3 with effect from 1st April, 2021. Their Balance Sheet as on that date showed a balance of ₹30,000 in Advertisement suspense account. Amount, that will be debited/credited to the Capital accounts of Malika, Meera and Madhu if they decide to carry forward the amount of Advertisement Suspense Account.
(a) ₹12,000 (Dr); ₹12,000 (Dr) and ₹6,000 (Dr) respectively to the Capital Accounts of Malika, Meera and Madhu:
(b) Debit ₹10,000 each to all partner's Capital Accounts.
(c) Debit Meera's Capital A/c and Credit Mallika's Capital A/c by ₹2,000.
(d) Debit Mallika's Capital A/c and Credit Meera's Capital A/c by ₹2000 each.
VIEW SOLUTION
• Question 11
Arun, Babita and Charu are partners in a firm sharing profits in the ratio of 3 : 3 : 2. They decided to share future profits and losses in the ratio of 1 : 1 : 1 with effect from 1-4-2021. They decided to record the effect of the following without affecting their Book values.
(i) Profit and Loss A/C (Cr) ₹8,000
(ii) General Reserve ₹4,000
The necessary adjusting entry for the same will be:
₹ ₹ (a) Charu’s Capital A/c………….. Dr 1,000 To Arun’s Capital A/c 500 To Babita’s Capital A/c 500 (b) Arun’s Capital A/c………….. Dr 500 Babita’s Capital A/c………….. Dr 500 To Charu’s Capital A/c 1,000 (c) Charu’s Capital A/c………….. Dr 1,000 To Babita’s Capital A/c 1,000 (d) Charu’s Capital A/c Dr 3,000 To Arun’s Capital A/c 1,500 To Babita’s Capital A/c 1,500
VIEW SOLUTION
• Question 12
Nominal share capital is:
(a) That part of authorised capital which is issued by the company.
(b) The amount of capital which is actually applied for by the prospective shareholders.
(c) The maximum amount of share capital which a company is authorised to issue.
(d) The amount actually paid by the shareholders. VIEW SOLUTION
• Question 13
Vandana Ltd. issued 6,000 equity shares of ₹10 each at 10% premium. The issue was fully subscribed. Amount per share was payable as follows:
On application ₹3, On allotment ₹3 (including premium), On first call ₹3 and on final call ₹2. A, a holder of 200 shares paid the entire money along with allotment. The amount received on allotment will be_______.
(a) ₹18,000
(b) ₹19,000
(c) ₹25,000
(d) ₹21,000 VIEW SOLUTION
• Question 14
Amit and Sumit were partners in a firm with capitals of ₹3,00,000 and ₹2,00,000 respectively. The normal rate of return was 20% and the capitalised value of average profits was ₹8,50,000. The Goodwill of the firm by capitalization of average profits method will be ______.
(a) ₹10,00,000
(b) ₹1,50,000
(c) ₹3,50,000
(d) ₹5,00,000 VIEW SOLUTION
• Question 15
Any change in the relationship of existing partners which results in an end of the existing agreement and entering into a new agreement is called
(a) Revaluation of Partnership Firm
(b) Reconstitution of Partnership Firm
(c) Dissolution of Partnership Firm
(d) Amalgamation of Partnership Firms VIEW SOLUTION
• Question 16
Mani and Neeru are partners in a firm sharing profits in the ratio of 5 : 3. On 1-4-2021, they admitted Lily as a new partner on the following terms :
(i) The new profit sharing ratio will be 2 : 3 : 3.
(ii) Lily will bring ₹1,00,000 for her capital and the necessary amount of goodwill premium in cash.
(iii) Goodwill of the firm was valued at ₹1,40,000.
The Journal Entry for treatment of goodwill premium brought by Lily will be
₹ ₹ (a) Premium for Goodwill A/c ............ Dr. 52,500 To Mani’s Capital A/c 37,500 To Neeru’s Capital A/c 15,000 (b) Premium for Goodwill A/c ............ Dr. 52,500 To Mani’s Capital A/c 52,500 (c) Premium for Goodwill A/c ............ Dr. 52,500 Neeru’s Capital A/c ............ Dr. 12,500 To Mani’s Capital A/c 65,000 (d) Lily’s Current A/c ............ Dr. 1,40,000 To Mani’s Capital A/c 1,00,000 To Neeru’s Capital A/c 40,000
VIEW SOLUTION
• Question 17
The company has to get minimum subscription within ________ from the date of issue of the prospectus. When minimum subscription has been received, the directors of the company proceed to make _________ which implies a valid contract between the company and the applicants who now become the allottees and assume the status of shareholders or members.
(a) 120 days, allotment of shares.
(b) 130 days, application of shares.
(c) 14 days, allotment of shares.
(d) 15 days, allotment of shares. VIEW SOLUTION
• Question 18
At the time of admission of a new partner, general reserve appearing in the old balance sheet is transferred to :
(a) all partner's capital accounts
(b) new partner's capital accounts
(c) old partner's capital accounts
(d) revaluation account VIEW SOLUTION
• Question 19
Given below are two statements, one labelled as Assertion (A) and the other labelled as Reason (R) :
Assertion (A) : It is necessary to revalue assets and liabilities of a firm in case of admission of a partner.
Reasons (R) : It is because the incoming partner is neither put to an advantage nor to a disadvantage due to change in the value of assets and liabilities.
In the context of the above statements identify the correct option.
(a) Both (A) and (R) are correct, and (R) is the correct reason of (A).
(b) Both (A) and (R) are correct, but (R) is not the correct reason of (A).
(c) Only (R) is correct.
(d) Both (A) and (R) are wrong. VIEW SOLUTION
• Question 20
A and B are partners in a firm sharing profit in the ratio of 3 : 2. Their Balance Sheet as on 31st March, 2021 was as follows:
Liabilities Amount ₹ Assets Amount ₹ A’s Capital 30,000 Drawing: A: 4,000 B’s Capital 10,000 B: 2,000 6,000 Other Assets 34,000 40,000 40,000
Net profit of the year ended 31-3-2021, ₹5,000 was divided without providing for interest on capital @ 10% p.a. What will be the amount of interest on A's Capital?
(a) ₹3,000
(b) Nil
(c) ₹3,100
(d) ₹2,700 VIEW SOLUTION
• Question 21
Money received in advance from shareholders before it is actually called up by the directors is :
(a) debited to calls in advance account.
(b) credited to calls in advance account.
(c) debited to calls account.
(d) credited to calls account. VIEW SOLUTION
• Question 22
Given below are two statements, one labelled as Assertion (A) and the other labelled as Reason (R) :
Assertion (A) : Maximum amount of discount allowed at the time of reissue of forefeited shares not exceed the forfeited amount.
Reason (R) : The excess amount of forfeited shares account is transferred to capital reserve account.
In the context of the above statements, identify the correct option.
(a) (A) is correct, but (R) wrong.
(b) Both (A) and (R) correct.
(c) (A) is wrong, but (R) is correct.
(d) Both (A) and (R) are wrong. VIEW SOLUTION
• Question 23
Devi withdrew ₹12,000 at the middle of every month. Interest on drawings was to be charged @ 12% per annum. Amount of interest on Devi's drawings will be:
(a) ₹14,400
(b) ₹8,640
(c) ₹7,200
(d) ₹1,200 VIEW SOLUTION
• Question 24
Vista Ltd. forfeited 200 shares held by Ravi for non-payment of allotment money of ₹40 per share (including premium of ₹10 per share). The first and final call of ₹20 per share was not yet called. In the forfeiture entry share capital account will be:
(a) Debited by ₹20,000
(b) Debited by ₹18,000
(c) Debited by ₹16,000
(d) Credited by ₹16,000 VIEW SOLUTION
• Question 25
If the purchase consideration is less than the amount of net assets taken over, which account will be credited for the differenced amount?
(a) Goodwill Account
(b) Vendor's Account
(c) Capital Reserve Account
(d) Asset Accounts VIEW SOLUTION
• Question 26
Uncalled Capital is that portion of the _________ which has not yet been called up and the portion of such uncalled capital to be called only in the event of winding up of the company is called __________.
(a) Subscribed Capital; Reserve Capital
(b) Issued Capital; Reserve Capital
(c) Authorised Capital; Capital Reserve
(d) Registered Capital; Capital Reserve VIEW SOLUTION
• Question 27
Given below are two statements, one labelled as Assertion (A) and other labelled as Reason (R) :
Assertion (A) : The fixed capital account balance of a partner may change due to additional Capital introduced or capital withdrawn or both, during the year.
Reason (R) : Under fixed capital method, the partner's capital accounts balance always remains some.
In the context of the above two statements which of the following is correct?
(a) Both (A) and (R) are correct.
(b) (A) is correct, but (R) wrong.
(c) (A) is wrong, but (R) is correct.
(d) Both (A) and (R) are wrong. VIEW SOLUTION
• Question 28
Vamini Ltd. forfeited 3,000 shares of ₹10 each, ₹8 called up for non-payment of allotment money of ₹5 per share. All the forfeited shares were reissued to Atul at ₹8 per share fully paid. The amount debited to share forfeiture Account at the time of reissue will be:
(a) ₹9,000
(b) ₹6,000
(c) ₹3,000
(d) ₹15,000 VIEW SOLUTION
• Question 29
Seema and Teena are partners in a firm sharing profits and losses in the ratio of 3 : 2. They agreed to admit Reena into partnership for $\frac{1}{5}\mathrm{th}$ share of profits on 1st April, 2021. On that date, workmen's compensation fund stood in the balance sheet at ₹50,000. The liability against workmen's compensation fund is determined at ₹20,000 which is to be paid later in the year.
What will be the Journal Entry for the treatment of Workmen compensation fund on the admission new partner Reena?
₹ ₹ (a) Workmen Compensation fund A/c................ Dr 30,000 To Seema's Capital A/c To Teena's Capital A/c 18,000 12,000 (b) Workmen Compensation fund A/c................ Dr 50,000 To Revaluation A/c To Seema's Capital A/c To Teena's Capital A/c 20,000 18,000 12,000 (c) Workmen Compensation fund A/c................ Dr 50,000 To Workmen Compensation Claim A/c To Seema's Capital A/c To Teena's Capital A/c 20,000 18,000 12,000 (d) Workmen Compensation fund A/c................. Dr 50,000 To Seema's Capital A/c To Teena's Capital A/c 30,000 20,000
VIEW SOLUTION
• Question 30
Gopal and Govind are partners in a firm sharing profits equally. They admitted Chetan for $\frac{1}{3}\mathrm{rd}$ share in profits. On admission debtor whose dues of ₹5,000 were earlier written off as bad-debts, paid ₹4,000 in full settlement. Bad debts recovered ₹4,000 will be debited to ______and credited to _______.
(a) Cash/Bank A/c, Revaluation A/c
(c) Cash/ Bank A/c, Bad debts A/c
(d) Revaluation A/c, Bad debts recovered A/c VIEW SOLUTION
• Question 31
Ram and Krishna were partners sharing profits and losses in the ratio of 2 : 1. They admitted Shanker as a partner for 1/5th share in the profits. For this purpose the Goodwill of the firm was to be valued on the basis of three times of last five years average profits. The profits for the last five years were:
Year 2016-17 2017-18 2018-19 2019-20 2020-21 Profit (₹) 50,000 40,000 75,000 (25,000) 50,000
Profit of 2017-18 was calculated after charging ₹ 10,000 for abnormal loss of goods by fire. The value of Goodwill of the firm is
(a) ₹1,28,000
(b) ₹2,00,000
(c) ₹1,90,000
(d) ₹1,20,000 VIEW SOLUTION
• Question 32
Santa and Banta are partners in a firm charging profits in the ratio of 3 : 2. Kanta was admitted as a new partner for $\frac{1}{5}\mathrm{th}$ share of profits. On Kanta's admission it was decided that machinery would be appreciated by 10% (Book value ₹80,000) and Building would be appreciated by 20% (Book value ₹2,00,000). Unrecorded Debtors of ₹1,250 would be brought to books. There was a liability of ₹2,750 included in Sundry Creditors that is not be paid. What will be the gain/loss on Revaluation?
(a) Loss ₹28,000
(b) Loss ₹40,000
(c) Profit ₹28,000
(d) Profit ₹40,000 VIEW SOLUTION
• Question 33
Vinod Ltd. having authorized Capital ₹1,00,00,000 divided into equity shares of ₹100 each, invited applications for issuing 25,000 equity shares at par. The amount per share was payable as follows: On Application ₹20 per share, on Allotment ₹30 per share, on First call ₹25 per share and on second and final call ₹25 per share. Applications were received for 24,000 shares and the shares were allotted to all the applicants. All calls were made and were received as follows: On 18,000 shares – Full amount On 2,000 shares – ₹75 per share On 2,500 shares – ₹50 per share On 1,500 shares – ₹20 per share
The company forfeited those shares on which less than ₹75 per share were received. The forfeited shares were reissued at ₹95 per share fully paid up.
How much amount was received on allotment?
(a) ₹6,75,000
(b) ₹7,20,000
(c) ₹6,00,000
(d) ₹4,80,000 VIEW SOLUTION
• Question 34
Vinod Ltd. having authorized Capital ₹1,00,00,000 divided into equity shares of ₹100 each, invited applications for issuing 25,000 equity shares at par. The amount per share was payable as follows: On Application ₹20 per share, on Allotment ₹30 per share, on First call ₹25 per share and on second and final call ₹25 per share. Applications were received for 24,000 shares and the shares were allotted to all the applicants. All calls were made and were received as follows:
On 18,000 shares – Full amount On 2,000 shares – ₹75 per share On 2,500 shares – ₹50 per share On 1,500 shares – ₹20 per share
The company forfeited those shares on which less than ₹75 per share were received. The forfeited shares were reissued at ₹95 per share fully paid up.
How much total amount was credited to share forfeiture account on forfeiture of shares?
(a) ₹3,80,000
(b) ₹1,35,000
(c) ₹1,55,000
(d) ₹2,45,000 VIEW SOLUTION
• Question 35
Mountain Enterprises is a partnership firm with Manu, Mamta and Moti as partners. The firm is engaged in production and sales of electrical items and equipment. Their capital contributions were ₹50,00,000; ₹50,00,000 and ₹80,00,000 respectively. They decided to share the profit in the ratio of 5 : 5 : 8. They are now looking forward to expand their business. It was decided that they would bring in sufficient cash to double their respective capitals.
This was duly followed by Manu and Mamta but due to unavoidable reasons Moti could not do so and ultimately it was agreed that to bridge the shortfall in the required capital a new partner should be admitted who would bring in the amount that Moti could not bring and that the new partner would get share of profits equal to half of Moti's share which would be sacrificed by Moti Only.
Consequent to this agreement Malini was admitted and she brought in the required capital and ₹30,00,000 as premium for goodwill.
What is the new profit sharing ratio of Manu, Mamta, Moti and Malini?
(a) 1 : 1 : 1 : 1
(b) 5 : 5 : 8 : 8
(c) 5 : 5 : 4 : 4
(d) 6 : 4 : 4 : 4 VIEW SOLUTION
• Question 36
Mountain Enterprises is a partnership firm with Manu, Mamta and Moti as partners. The firm is engaged in production and sales of electrical items and equipment. Their capital contributions were ₹50,00,000; ₹50,00,000 and ₹80,00,000 respectively. They decided to share the profit in the ratio of 5 : 5 : 8. They are now looking forward to expand their business. It was decided that they would bring in sufficient cash to double their respective capitals.
This was duly followed by Manu and Mamta but due to unavoidable reasons Moti could not do so and ultimately it was agreed that to bridge the shortfall in the required capital a new partner should be admitted who would bring in the amount that Moti could not bring and that the new partner would get share of profits equal to half of Moti's share which would be sacrificed by Moti Only.
Consequent to this agreement Malini was admitted and she brought in the required capital and ₹30,00,000 as premium for goodwill.
What is the value of the goodwill of the firm?
(a) ₹1,35,00,000
(b) ₹1,50,00,000
(c) ₹30,00,000
(d) ₹1,60,00,000 VIEW SOLUTION
• Question 37
Given below are two statements, one labelled as Assertion (A) and the other labelled as Reason (R) :
Assertion (A) : Current Ratio establishes relationship between Current Assets and Current Liabilities.
Reason (R) : The objective of this ratio is to measure the ability of the firm to meet its short term obligations as and when due without relying upon the realisation of inventories.
In the context of the above two statements choose the correct option.
(a) (A) is true but (R) is false.
(b) Both (A) and (R) are true, and (R) is a correct explanation of (A).
(c) Both (A) and (R) are false.
(d) (A) is false, but (R) is true. VIEW SOLUTION
• Question 38
Match the items given in Column I with the correct heading/sub-heading given in column II. Column I Column II A. 9% Debenture redeemable during the current year i. Intangible assets B. Loose tools ii. Current liabilities C. Copyright iii. Cash and Cash equivalents D. Cash at bank iv. Inventories
A B C D (a) i ii iii iv (b) iii ii iv i (c) iv iii ii i (d) ii iv i iii
VIEW SOLUTION
• Question 39
What will be the impact of issuing ₹5,00,000 equity shares to vendors for Building purchased on the debt and equity of X Ltd.?
(a) Debt will increase and equity will decrease.
(b) Debt will remain same and equity will increase.
(c) Debt will decrease and equity will increase.
(d) Debt will remain same and equity will decrease. VIEW SOLUTION
• Question 40
___________ ratios indicate the speed at which activities of the business are being performed.
(a) Profitability
(b) Turnover
(c) Solvency
(d) Liquidity VIEW SOLUTION
• Question 41
If operating ratio is 65%, Operating profit ratio will be ________.
(a) 20%
(b) 25%
(c) 35%
(d) 30% VIEW SOLUTION
• Question 42
Financial statement analysis includes ________ and _______ of financial statements:
(a) Analysis, preparation
(b) Preparation, interpretation
(c) Preparation, analysis
(d) Analysis, interpretation VIEW SOLUTION
• Question 43
From the following information, calculate Interest Coverage Ratio:
Net profit after tax ₹6,00,000
10% Debentures ₹50,00,000
Tax Rate 40%
(a) 1.2 times
(b) 3 times
(c) 2 times
(d) 5 times VIEW SOLUTION
• Question 44
Given below are two statements, one labelled as Assertion (A) and the other labelled as Reason (R):
Assertion (A):
A high debt-equity ratio is risky.
Reason (R): It may put the firm into difficulty to pay long term debts.
In the context of the above two statements choose the correct option.
(a) (A) is correct, but (R) is wrong.
(b) Both (A) and (R) are correct.
(c) (A) is wrong, but (R) is correct.
(d) Both (A) and (R) are wrong. VIEW SOLUTION
• Question 45
Gross profit ratio of a company was 25%. Its credit revenue from operations was ₹16,00,000 and its cash revenue from operations was 20% of the total revenue from operations. If the indirect expenses of the company were ₹50,000, its net profit ratio will be:
(a) 27.5%
(b) 20%
(c) 22.5%
(d) 25% VIEW SOLUTION
• Question 46
From the following information, calculate Inventory Turnover ratio:
Revenue from operations ₹2,00,000
Average Inventory ₹ 20,000
Gross profit ratio 10%
(a) 9 times
(b) 10 times
(c) 12 times
(d) 20 times
VIEW SOLUTION
• Question 47
Ajanta Ltd. issued 10% Debenture of ₹8,00,000 on 1st April, 2019 which are redeemable in five equal yearly installments starting from 1st April, 2022. How would this information be presented in the Balance Sheet as at 31st March, 2021.
(a) ₹8,00,000 as Long term borrowings.
(b) ₹8,00,000 as Other Non-current liability.
(c) ₹8,00,000 as Current liability.
(d) ₹1,60,000 as other Current liability and ₹6,40,000 as Long term borrowing.
VIEW SOLUTION
• Question 48
Given below are two statements, one labelled as Assertion (A) and the other labelled as Reason (R):
Assertion (A):
Return on investment explains the overall utilization of funds by a business enterprise.
Reason (R): It measures return (Net Profit before interest and tax) on total funds (Capital employed).
In the context of these statements, choose the correct option
(a) Both (A) and (R) are true but (R) is not the correct explanation of (A).
(b) Both (A) and (R) are true and (R) is correct explanation of (A).
(c) Both (A) and (R) are false.
(d) (A) is false, but (R) is true VIEW SOLUTION
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2022-05-22 01:46:32
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https://www.cs.vassar.edu/help/general_linux/linux_basics
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# Linux Basics
• a file is a collection of data with a name (a filename to be precise). Although it may be stored in separate chunks in different location on the hardware, programs will generally work with a file as a single continuous collection of data.
• a directory is construct for grouping and organizing files. In UNIX and Linux, directories can contain, files, other directories, links and devices. You may be use to calling directories folders or namespaces.
• a path is a way of naming the location of a file, directory, link or device. Paths can be relative or absolute and are often used as the prefix to a filename. For example, if I want to list the contents of a directory that is in my home directory called cheese, I can type:
ls ~/cheese/
• a link is a filesystem pointer. You may be use to calling a link an alias or a shortcut
• ls: lists the files in your current working directory for that shell.
• more ways: ls -a, ls -l, ls -la
• cd directory : changes the current working directory to directory where directory may be a fully qualified path. A ~ indicates your home directory. ~bob indicates the home directory of bob.
• rm filename : removes the file filename where filesname may be a fully qualified path
• cp source destination : copies the file source to the file destination where source and destination may be fully qualified paths.
• mv source destination : the same as cp, but it moves the source file instead of copying it. mv can also be used to rename a file with mv oldname newmane
• chmod #### file or directory : changes the file or directory permissions for letter yourself and other people access your files. see man chmod for more info.
• Printing: the lpr command is the basic printing command. in your shell, type: “lpr filename ” to print the file filename to the printer specified in printername or “lpq” to get a list of jobs sent to the printer.
• File compression: “gzip filename” will compress the file filename and give it the .gz name extension. “gzip -d filename.gz” or “gunzip filename.gz” will decompress the file and remove the .gz extension.
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2022-01-23 14:29:35
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{"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": 0.5833951234817505, "perplexity": 5205.015648576018}, "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-2022-05/segments/1642320304287.0/warc/CC-MAIN-20220123141754-20220123171754-00524.warc.gz"}
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https://mathhelpboards.com/threads/no-ones-question-at-yahoo-answers-regarding-newtons-law-of-cooling.6127/
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# No-One's question at Yahoo! Answers regarding Newton's Law of Cooling
#### MarkFL
Staff member
Here is the question:
Newton's Law of Cooling extremely hard question?
Two bodies have been found. The wife was found dead inside the heated home where the temperature was maintained at 22 degrees Celsius. The husband dragged himself outside, where the outside temperature during the preceding day was 8 to 12 degrees Celsius.
The doctor took the temperatures of the body as soon as they arrived:
Wife: 33 degrees Celsius
Husband: 26 degrees Celsius
Who died first? Develop a model to help decide this question.
Help really appreciated.
I have posted a link there to this topic so the OP can see my work.
#### MarkFL
Staff member
Hello No-One,
Newton's law of Cooling states that the time rate of change of the temperature $T$ of an object is proportional to the difference between the ambient temperature $M$ and the temperature of the object. Stated mathematically, this is:
$$\displaystyle \frac{dT}{dt}=-k(T-M)$$ where $$\displaystyle T(0)=T_0,\,0<k\in\mathbb{R}$$ and $$\displaystyle T>M$$.
The ODE is separable and may be written:
$$\displaystyle \frac{1}{T-M}\,dT=-k\,dt$$
Integrating, using the boundaries, and dummy variables of integration, we find:
$$\displaystyle \int_{T_0}^{T(t)}\frac{1}{u-M}\,du=-k\int_0^t \,dv$$
$$\displaystyle \ln\left(\frac{T(t)-M}{T_0-M} \right)=-kt$$
$$\displaystyle kt=\ln\left(\frac{T_0-M}{T(t)-M} \right)$$
Assuming the heat transfer coefficient is the same for both bodies, which is a reasonable assumption since they are both human bodies, and that their body temperatures at the time of death was the normal 37° C., we have:
For the wife:
$$\displaystyle T_0=37,\,M=22, T(t)=33$$
$$\displaystyle kt=\ln\left(\frac{37-22}{33-22} \right)=\ln\left(\frac{15}{11} \right)\approx0.310154928303840$$
For the husband:
$$\displaystyle T_0=37,\,8\le M\le12, T(t)=26$$
$$\displaystyle \ln\left(\frac{37-8}{26-8} \right)\le kt\le\ln\left(\frac{37-12}{26-12} \right)$$
$$\displaystyle \ln\left(\frac{29}{18} \right)\le kt\le\ln\left(\frac{25}{14} \right)$$
$$\displaystyle 0.476924072090309\le kt\le0.579818495252942$$
Therefore, it is reasonable to assume the husband died first.
#### iamapineapple
##### New member
The ODE is separable and may be written:
$$\displaystyle \frac{1}{T-M}\,dT=-k\,dt$$
Integrating, using the boundaries, and dummy variables of integration, we find:
$$\displaystyle \int_{T_0}^{T(t)}\frac{1}{u-M}\,du=-k\int_0^t \,dv$$
$$\displaystyle \ln\left(\frac{T(t)-M}{T_0-M} \right)=-kt$$
$$\displaystyle kt=\ln\left(\frac{T_0-M}{T(t)-M} \right)$$
Hey MarkFL, it's No-one from Yahoo! Thanks so much for answering my question in depth. It's just I've never used this method (above) for integration before. (i.e. multiplying by dt and then taking the integral?!). Would you mind explaining this step a little more indepth? Especially where you got the variables used in the definitie integral?
- - - Updated - - -
$$\displaystyle \int_{T_0}^{T(t)}\frac{1}{u-M}\,du=-k\int_0^t \,dv$$
More just this step
So for the LHS
(I have no idea how to use latex for this)
$$\displaystyle \frac{1}{T - T_s} dT$$
Then put the integral sign?
$$\displaystyle \int\frac{1}{T - T_s} dT$$
$$\displaystyle Let u = T - T_s$$
$$\displaystyle \frac{du}{dT} = 1$$
so
$$\displaystyle \int\frac{1}{u - T_s} du$$
Is this correct? I don't get the RHS though
Thanks Mark
and why is the variable changed? (ha! sweet - killed the Latex stuff)
$$\displaystyle \int-k dt$$
$$\displaystyle k\int-1 dt = -kt$$
Correct?
Could you still please explain the limits though?
Last edited:
#### MarkFL
Staff member
Hello iamapineapple and welcome to MHB!
Suppose you have the initial value problem (IVP):
$$\displaystyle \frac{dy}{dx}=f(x)$$ where $$\displaystyle y\left(x_0\right)=y_0$$
Now, separating variables and using indefinite integrals, we may write:
$$\displaystyle \int\,dy=\int f(x)\,dx$$
And upon integrating, we find
$$\displaystyle y(x)=F(x)+C$$ where $$\displaystyle \frac{d}{dx}\left(F(x) \right)=f(x)$$
Using the initial condition, we get
$$\displaystyle y\left(x_0 \right)=F\left(x_0 \right)+C$$
Solving for $C$ and using $$\displaystyle y\left(x_0\right)=y_0$$, we obtain:
$$\displaystyle C=y_0-F\left(x_0 \right)$$ thus:
$$\displaystyle y(x)=F(x)+y_0-F\left(x_0 \right)$$
which we may rewrite as:
$$\displaystyle y(x)-y_0=F(x)-F\left(x_0 \right)$$
Now, we may rewrite this, using the anti-derivative form of the fundamental theorem of calculus, as:
$$\displaystyle \int_{y_0}^{y(x)}\,dy=\int_{x_0}^{x}f(x)\,dx$$
Now, since the variable of integration gets integrated out, it is therefore considered a "dummy variable" and since it is considered good form not to use the same variable in the boundaries as we use for integration, we may switch these dummy variables and write:
$$\displaystyle \int_{y_0}^{y(x)}\,du=\int_{x_0}^{x}f(v)\,dv$$
This demonstrates that the two methods are equivalent.
Using the boundaries (the initial and final values) in the limits of integration eliminates the need to solve for the constant of integration, and I find it a more intuitive and cleaner approach to separable initial value problems.
#### MarkFL
Staff member
So for the LHS
(I have no idea how to use latex for this)
$$\displaystyle \frac{1}{T - T_s} dT$$
Then put the integral sign?
$$\displaystyle \int\frac{1}{T - T_s} dT$$
$$\displaystyle Let u = T - T_s$$
$$\displaystyle \frac{du}{dT} = 1$$
so
$$\displaystyle \int\frac{1}{u - T_s} du$$
Is this correct? I don't get the RHS though
Thanks Mark
and why is the variable changed? (ha! sweet - killed the Latex stuff)
$$\displaystyle \int-k dt$$
$$\displaystyle k\int-1 dt = -kt$$
Correct?
Could you still please explain the limits though?
Looks like you caught on rather quickly how to use $\LaTeX$!
If we did not use the boundaries in the limits, here is how we might solve the IVP:
$$\displaystyle \frac{dT}{dt}=-k(T-M)$$ where $$\displaystyle T(0)=T_0$$ and $$\displaystyle M<T$$
Separate variables:
$$\displaystyle \frac{1}{T-M}\,dT=-k\,dt$$
Integrate:
$$\displaystyle \ln(T-M)=-kt+C$$
Using the initial values ($t=0$ and $T(0)=T_0$), we have:
$$\displaystyle \ln(T_0-M)=C$$
And so we find:
$$\displaystyle \ln(T-M)=-kt+\ln(T_0-M)$$
$$\displaystyle \ln(T-M)-\ln(T_0-M)=-kt$$
$$\displaystyle kt=\ln(T_0-M)-\ln(T-M)=\ln\left(\frac{T_0-M}{T-M} \right)$$
Since $T$ is a function of time $t$, we may finally write:
$$\displaystyle kt=\ln\left(\frac{T_0-M}{T(t)-M} \right)$$
And this is the same result we obtained via the other method. Whichever method you find more to your liking I suggest using, as you get the same result either way.
If anything I have posted is not clear, please do not hesitate to ask for clarification.
#### iamapineapple
##### New member
I can't thank you enough, Mark. The second method was awesome, I'm a lot more familiar and comfortable with it, thanks so much!
I still have a few more question, sorry o)
So, I want to draw a conclusion from kt.
The smaller the value of k, the longer it would take for an object to cool (from my knowledge).
The wife's temperature (T(t)) is 33oC when the doctor checked her so dropped 4oC from her assumed body temperature of 37oC.
The husband on the other hand, dropped a whole 11OC
Based on the value of kt calculated, the husband had a faster rate of cooling than his wife: (wife, $kt = 0.31$ approx. and husband, $0.48\leq kt\leq 0.58$) so, the husband had a rate of cooling between $\frac{0.48}{0.31} = 1.55X$ and $\frac{0.58}{0.31} = 1.87X$ faster than his wife.
Therefore, the wife's temperature if she also crawled outside and was at her husband's cooling speed for as long as she was dead, would have had a temperature change between:
$$\displaystyle 4 \times 1.55 = 6.19 degrees$$
$$\displaystyle 4 \times 1.87 = 7.48 degrees$$
Therefore, because the husband's temperature dropped 11 degrees, greater than 6.19 and 7.48 degrees, it is reasonable to assume that the husband died first.
I'd appreciate your thoughts on this : )
#### MarkFL
Staff member
You are correct about the value of $k$. This constant of proportionality is called the heat transfer coefficient and is specific to the object (its composition and shape, specifically the ratio of surface area to volume) and independent of the ambient temperature $M$. Thus, this constant, along with the difference in temperature between the environment and the object, determines how quickly heat is transferred from the object to the environment.
When we apply Newton's Law of Cooling, we do so in cases where the ambient temperature is not significantly affected by the transfer of energy. For example, a human body, losing heat in the Earth's atmosphere, does not significantly change the temperature of the atmosphere. In the case of a body in a room, this change is controlled in this case by the thermostat attached to the heater keeping the room at a constant temperature.
If we were to put a cold spoon in a hot cup of coffee, we would expect that as the spoon heats up, the coffee also cools significantly, so we would want to account for that and use a more sophisticated model.
What you did in your comparison of the change in temperature between the husband and wife is valid, however, it suffices to merely compare the values of $kt$ for the two bodies given our assumption that $k$ is the same for both. Since the heat transfer coefficient is the same for both, we need only find that body for which the value of $kt$ is greater to say which had been dead for a longer period of time.
#### MarkFL
Staff member
There is another way to solve the ODE associated with the IVP, and that is to write it in standard linear form:
$$\displaystyle \frac{dT}{dt}+kT=kM$$
Compute the integrating factor:
$$\displaystyle \mu(t)=e^{k\int\,dt}=e^{kt}$$
Multiply through by this factor:
$$\displaystyle e^{kt}\frac{dT}{dt}+ke^{kt}T=kMe^{kt}$$
Observing that the left side is now the product of the derivative of the integrating factor and the dependent variable, we may now write:
$$\displaystyle \frac{d}{dt}\left(e^{kt}T \right)=kMe^{kt}$$
Integrate with respect to $t$:
$$\displaystyle e^{kt}T=Me^{kt}+C$$
Solve for $T(t)$:
$$\displaystyle T(t)=M+Ce^{-kt}$$
Use the initial conditions to determine the parameter $C$:
$$\displaystyle T(0)=M+C=T_0\,\therefore\,C=T_0-M$$
and we have:
$$\displaystyle T(t)=M+\left(T_0-M \right)e^{-kt}$$
Solving for $kt$, we find:
$$\displaystyle kt=\ln\left(\frac{T_0-M}{T(t)-M} \right)$$
#### MarkFL
Staff member
And here is yet another method we could use to solve the IVP (last one, I promise! )
Write the ODE in standard linear form:
$$\displaystyle \frac{dT}{dt}+kT=kM$$
Seeing the characteristic equation for the associated homogeneous equation is:
$$\displaystyle r=-k$$
We may then give the homogeneous solution as:
$$\displaystyle T_h(t)=c_1e^{-kt}$$
Next, observing that the differential operator defined by:
$$\displaystyle A\equiv D$$
annihilates the constant on the right side of the ODE, we may state:
$$\displaystyle D(D-k)[T]=0$$
And so we know the general solution will take the form:
$$\displaystyle T(t)=c_1e^{-kt}+c_2$$
Given the form of the homogeneous solution we already determined, we then know there must exist a particular solution of the form:
$$\displaystyle T_p(t)=c_2$$
Differentiating with respect to $t$, we have:
$$\displaystyle T_p'(t)=0$$
and then, using the method of undetermined coefficients, we substitute the particular solution into the ODE to get:
$$\displaystyle 0+kc_2=kM\,\therefore\,c_2=M$$
And so the particular solution is:
$$\displaystyle T_p(t)=M$$
And so, by superposition, we may state:
$$\displaystyle T(t)=T_h(t)+T_p(t)=c_1e^{-kt}+M$$
Determination of the value of the parameter $c_1$, and then solving for $kt$ is identical to the method used in my previous post.
The linear method I used in the previous post is quite useful when separation of variables is not possible for first order ODEs, and the method I used here in this post is useful for higher order ODEs.
#### iamapineapple
##### New member
Thank you so much Mark, I can't express my gratitude enough.
Needless to say, I am extremely impressed by your level of knowledge. A request - would you mind helping me with questions in the future (if I have any)? My final exam is approaching and it'd be great to have someone like you to help.
Anyways, thanks so much.
-iamapineapple.
#### MarkFL
Staff member
Thank you so much Mark, I can't express my gratitude enough.
Needless to say, I am extremely impressed by your level of knowledge. A request - would you mind helping me with questions in the future (if I have any)? My final exam is approaching and it'd be great to have someone like you to help.
Anyways, thanks so much.
-iamapineapple.
I don't really have the time for one-on-one tutoring, but please feel free to post your questions here at MHB in the appropriate sub-forum (a question like this one would go in our Differential Equations sub-forum) along with your working so we know where you are stuck or can see what you have done to go astray, and I or one of our other helpers will be glad to offer assistance.
We have a great team of knowledgeable and friendly folks here who are glad to help.
Best Regards,
Mark.
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2021-01-26 06:44:09
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https://makondo.ugr.es/event/0/session/91/contribution/236
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# The 35th International Symposium on Lattice Field Theory
18-24 June 2017
Palacio de Congresos
Home > Timetable > Session details > Contribution details
# Contribution Parallel
Auditorio Manuel de Falla
Nonzero Temperature and Density
# Complex Langevin Simulations of QCD at Finite Density -- Progress Report
## Speakers
• Dr. Donald SINCLAIR
## Content
We have extended our CLE simulations of lattice QCD at a finite quark-number chemical potential, $\mu$, on a $12^4$ lattice at $\beta=6/g^2=5.6$ to a weaker coupling, $\beta=5.7$, and larger, $16^4$, lattice. Limitations of the method and choice of lattice fermions are discussed as are possible improvements.
## Preferred track (if multiple tracks have been selected)
Nonzero Temperature and Density
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2019-08-23 13:57:07
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http://mathoverflow.net/questions/140405/jacobian-of-hyperelliptic-curve-over-local-field
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# Jacobian of hyperelliptic curve over local field
For elliptic curve $E$ defined over $K_v$ we know that $E(K_v)$ = $Z_p^{[K_v:Q_p]} + T$(direct sum) where v is prime of K above p and T is finite abelian group(By prop 6.3 in Silverman's book). In the proof of that proposition, Silverman used Formal Group.
My question is that it is still true for Jacobian of (hyperelliptic) curve of genus g larger than 1?
In other words, $J(C)(K_v) = Z_p^{g[K_v:Q_p]} + T$(direct sum) holds?
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2015-05-28 16:21:30
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https://gmatclub.com/forum/m21-184286.html
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# M21-32
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16 Sep 2014, 01:15
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If 5 dollars and 35 crowns is equivalent to 7 pounds, and 4 dollars and 4 pounds is equivalent to 56 crowns, 1 pound and 28 crowns is equivalent to how many dollars?
A. $7.00 B.$7.50
C. $8.50 D.$9.00
E. $10.00 _________________ Math Expert Joined: 02 Sep 2009 Posts: 47983 Re M21-32 [#permalink] ### Show Tags 16 Sep 2014, 01:15 1 1 Official Solution: If 5 dollars and 35 crowns is equivalent to 7 pounds, and 4 dollars and 4 pounds is equivalent to 56 crowns, 1 pound and 28 crowns is equivalent to how many dollars? A.$7.00
B. $7.50 C.$8.50
D. $9.00 E.$10.00
Given:
$$5d+35c=7p$$, so $$c=\frac{7p-5d}{35}$$;
$$4d+4p=56c$$, so $$c=\frac{d+p}{14}$$, also if we reduce this equation by 2 we'll get: $$2d+2p=28c$$;
From above we have that $$\frac{7p-5d}{35}=\frac{d+p}{14}$$, which simplifies to $$p=\frac{5d}{3}$$.
Question:
$$p+28c=?$$ Since $$2d+2p=28c$$, then $$p+28c=p+(2d+2p)=2d+3p=2d+3*\frac{5d}{3}=7d$$.
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### Show Tags
15 Oct 2015, 15:09
2
Struggling to see how this can be done in 2 min... Bunuel, any thoughts?
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### Show Tags
30 May 2016, 04:23
I find it very hard to sort of discover which way to take and which variable to try to identify when it comes to questions like this. What is the logic that drives the decision to move c on one side? Any advice? many thanks!
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My debrief: Self-study: How to improve from 620(Q39,V36) to 720(Q49,V39) in 25 days!
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### Show Tags
30 May 2016, 04:53
2
2
Bunuel wrote:
If 5 dollars and 35 crowns is equivalent to 7 pounds, and 4 dollars and 4 pounds is equivalent to 56 crowns, 1 pound and 28 crowns is equivalent to how many dollars?
A. $7.00 B.$7.50
C. $8.50 D.$9.00
E. \$10.00
Strategy in such questions is to eliminate each variable one by one.
First Form equations:
Eq. 1: 5d+35c = 7p
Eq. 2 : 4d+4p = 56c
Eliminate 1 variable
From eq 1: p = (5/7)d+5c
Put this in Eq 2:
4d+4{(5/7)d+5c} = 56 c
4d+ (20/7)d+ 20 c = 56 c
(48/7) d = 36 c
c = (4/21)d
Express p also in terms of d
p = (5/7)d+5c
p = (5/7)d+ 5 (4/21) d = (15/21) d + (20/21) d = (35/21) d = (5/3)d
Now come to the question once again: 1 pound and 28 crowns is equivalent to how many dollars?
1p+28c = (5/3)d + 28* (4/21) d = (5/3) d + (16/3) d = (21/3) d = 7 d
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### Show Tags
30 Mar 2017, 18:49
Why do you decide to solve for C first vice any of the others? Or does it matter? I tried solving for P first and it messed me up.
Manager
Joined: 23 Nov 2016
Posts: 76
Location: United States (MN)
GMAT 1: 760 Q50 V42
GPA: 3.51
### Show Tags
08 Sep 2017, 22:57
Has anyone tried this with Linear Algebra? Trying to find a faster way..
Director
Joined: 08 Jun 2015
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### Show Tags
15 Dec 2017, 09:39
Express the facts given in the two statements as :
5D+35C=7P
4D+4P=56C
We need to find P+28C. Here D stands for dollar, C for Crown , and P for Pound.
Express P in terms of C and C in terms of D. Substitute in P+28C to get the answer as option A.
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### Show Tags
15 Dec 2017, 19:50
I don't know if this is a subtle approach or not; 5,35,7 ,35 is divisible by 7 and 5; 4,4,56 , 56 is divisible by 4 ; 1,28? any number that divides 28, the only number that divides 28 is 7;
Re: M21-32 &nbs [#permalink] 15 Dec 2017, 19:50
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# M21-32
Moderators: chetan2u, Bunuel
# Events & Promotions
Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®.
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2018-08-19 07:40:11
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http://accessanesthesiology.mhmedical.com/content.aspx?bookid=974§ionid=61588088
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Chapter 59
### DEPOLARIZING MUSCLE RELAXANTS
Depolarizing muscle relaxants physically resemble acetylcholine (ACh), and because of this resemblance, they are able to act as competitive agonists by binding to ACh receptors (AChR) and generating action potentials.
Succinylcholine (SCh) is the only depolarizing muscle relaxant in clinical use. It is generally used when there is risk for aspiration of gastric contents or when there is need for rapid paralysis. It is essentially two ACh molecules joined together.
Because of its low lipid solubility and relative overdose, SCh has a very rapid onset of action. Onset of action is approximately 30-90 seconds and its duration of action 5-10 minutes. Succinylcholine is not metabolized by acetylcholinesterase, which is located in the neuromuscular junction (NMJ). Instead, it is metabolized by plasma cholinesterase (pseudocholinesterase), an enzyme present in the blood. Succinylcholine, therefore, has a longer duration of action at the motor end plate. This leads to prolonged depolarization known as a phase I block. Phase I block is often preceded by muscle fasciculation. This is probably the result of the prejunctional action of SCh, stimulating AChR on the motor nerve, causing repetitive firing and release of neurotransmitter. Recovery from phase I block occurs as SCh diffuses away from the NMJ and is metabolized by plasma cholinesterase in plasma.
Repeated boluses or an infusion of SCh may lead to either a desensitization block, or a phase II block. A desensitization block occurs when the continued presence of an agonist causes the AChR to become insensitive to the binding of the agonist. This is thought to be a safety mechanism to protect against overexcitation of the NMJ. With a phase II block the membrane potential is in a resting state despite an agonist being present and subsequent neurotransmission is blocked throughout. The block takes on the characteristics of a block induced by a nondepolarizing muscle relaxant (Table 59-1). Phase II block may be antagonized by anticholinesterases but the result is hard to predict. For this reason, spontaneous recovery is recommended.
TABLE 59-1Nondepolarizing Muscle Relaxants and Their Properties
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2017-02-23 11:35:31
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https://en.wikipedia.org/wiki/Berkovich_space
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# Berkovich space
In mathematics, a Berkovich space, introduced by Berkovich (1990), is an analogue of an analytic space for p-adic geometry, refining Tate's notion of a rigid analytic space.
## Berkovich spectrum
A seminorm on a ring ${\displaystyle A}$ is a non-constant function ${\displaystyle f\to \vert f\vert }$from ${\displaystyle A}$ to the non-negative reals such that |0| = 0, |1| = 1, |f + g| ≤ |f| + |g|, |fg| ≤ |f||g|. It is called multiplicative if |fg| = |f||g| and is called a norm if |f| = 0 implies f = 0.
If A is a normed ring with norm ${\displaystyle f\to \vert \vert f\vert \vert }$then the Berkovich spectrum of A is the set of multiplicative seminorms || on A that are bounded by the norm of A. The Berkovich spectrum is topologized with the weakest topology such that for any f in A the map taking || to |f| is continuous..
The Berkovich spectrum of a normed ring A is non-empty if A is non-zero and is compact if A is complete.
The spectral radius ρ(f) = lim |fn|1/n of f is equal to supx|f|x
### Examples
• If A is a commutative C*-algebra then the Berkovich spectrum is the same as the Gelfand spectrum. A point of the Gelfand spectrum is essentially a homomorphism to C, and its absolute value is the corresponding seminorm in the Berkovich spectrum.
• Ostrowski's theorem shows that the Berkovich spectrum of the integers (with the usual norm) consists of the powers |f|ε
p
of the usual valuation, for p a prime or ∞. If p is a prime then 0≤ε≤∞, and if p=∞ then 0≤ε≤1. When ε=0 these all coincide with the trivial valuation that is 1 on all non-zero elements.
• If k is a field with a multiplicative seminorm, then the Berkovich affine line over k is the set of multiplicative seminorms on k[x] extending the norm on k. This is not a Berkovich spectrum, but is an increasing union of the Berkovich spectrums of rings of power series that converge in some ball.
• If x is a point of the spectrum of A then the elements f with |f|x=0 form a prime ideal of A. The quotient field of the quotient by this prime ideal is a normed field, whose completion is a complete field with a multiplicative norm generated by the image of A. Conversely a bounded map from A to a complete normed field with a multiplicative norm that is generated by the image of A gives a point in the spectrum of A.
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2017-04-30 04:32:43
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https://thirumal.blog/2018/03/18/kadane-deep-dive/
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# Dive deep into Kadane’s algorithm
I am writing this post as I could not find any reliable material that could clearly explain the intuition behind Kadane’s algorithm.
To start, here’s a (very) crisp explanation of Kadane’s algorithm from Wikipedia.
Kadane’s algorithm begins with a simple inductive question: if we know the maximum subarray sum ending at position $i$ (call this $B_i$), what is the maximum subarray sum ending at position $i+1$ (equivalently, what is $B_{i+1})$? The answer turns out to be relatively straightforward: either the maximum subarray sum ending at position $i+1$ includes the maximum subarray sum ending at position $i$ as a prefix, or it doesn’t (equivalently $B_{i+1}=max(A_{i+1}, A_{i+1} + B_i)$, where $A_{i+1}$ is the element at index $i+1$
Unfortunately, when you see this explanation, it is hard to convince yourself why the algorithm works, but once I prove it to you, then it’ll seem trivial. Here is my attempt to make this explanation more easy to digest.
Q: Does this algorithm cover all subarrays?
Answering the above question is quite simple. It all depends on how you traverse over all the possible subarrays.
Generally, when you ask a programmer to iterate over all the subarrays, they’ll probably write something like this.
for (int i = 0; i < n; ++i)
{
for (int j = i; j < n; ++j)
{
// Do something with subarray [i, j]
// OBSERVE: All subarrays start with index i
}
}
Another not so uncommon way of iterating is to use $i$ as a termination condition in the inner loop, like so
for (int i = 0; i < n; ++i)
{
for (int j = 0; j < i; ++j)
{
// Do something with subarray [j, i]
// OBSERVE: All subarrays end with index i
}
}
Just to drive this concept home, let’s visualise this as a table. Each cell is a subarray (i.e. one iteration of the inner loop), and each row is one iteration of the outer loop.
Table-1: All possible subarrays of an array
• Row number $x$ in $E_{x}$ indicates the ending index $i$ for the subarray
• Column number $y$ in $S_{y}$ indicates the starting index $j$ for the subarray
From now on, if I refer to a row in the table, then I refer to all the subarrays in the $i^{th}$ row and notice that all their ending index is $i$.
Q: How does knowing the maximum subarray sum of row $i$ help us derive the maximum subarray sum of row $i+1$.
From the table, we know that the $B_0$ (the maximum subarray sum ending at 0) is nothing but the first element of the array. So, if we can somehow intelligently calculate the $B_{i+1}$ by using $B_i$, then finding $max(B_0, B_1, \ldots B_{n-1})$ becomes a $O(n)$ operation.
The maximum subarray sum of row $i$ can be mathematically represented as
\begin{aligned} \\ & B_i = max(\{A_0 + A_1 + \ldots + A_i\}, \{A_1 + A_2 + \ldots + A_i\} \ldots \{A_i\}) \\ \end{aligned}
If we consider the sum of all elements from index $i$ to $j$ as $S_{(i)(j)}$, we can rewrite the above equation as
\begin{aligned} \\ & B_i = max(S_{(0)(i)}, S_{(1)(i)}, \ldots S_{(i)(i)}) \\ \end{aligned}
If you look carefully in the table, all the subarrays but the last one in the ${i+1}^{th}$ row are nothing but subarrays found in $i^{th}$ row with an extra element $A_{i+1}$ slapped on at the end of each subarray. Oh, and the last subarray is a singleton array $\{A_{i+1}\}$.
\begin{aligned} \\ & B_{i+1} = max(\{A_0 + A_1 + \ldots A_i + A_{i+1}\}, \{A_1 + \ldots + A_i + A_{i+1}\} \ldots \{A_i + A_{i+1}\}, \{A_{i+1}\}) \\ & \qquad or \\ & B_{i+1} = max(\{S_{(0)(i)} + A_{i+1}\}, \{S_{(1)(i)} + A_{i+1}\}, \ldots \{S_{(i)(i)} + A_{i+1}\}, \{A_{i+1}\}) \\ & \qquad or \\ & B_{i+1} = max(max(\{S_{(0)(i)} + A_{i+1}\}, \{S_{(1)(i)} + A_{i+1}\}, \ldots \{S_{(i)(i)} + A_{i+1}\}), \{A_{i+1}\}) \\ \end{aligned}
Another not so common property of $max$ that I’d like you to think about is $max(a + z, b + z, \ldots, y + z) = max(a, b, \ldots y) + z$.
By using the above property, we can replace $max(\{S_{(0)(i)} + A_{i+1}\}, \{S_{(1)(i)} + A_{i+1}\}, \ldots \{S_{(i)(i)} + A_{i+1}\})$ in the previous equation with $max(S_{(0)(i)}, S_{(1)(i)}, \ldots S_{(i)(i)}) + A_{i+1}$ to get
$B_{i+1} = max(B_i + A_{i+1}, A_{i+1})$
So with just high school mathematics, we’ve proved the induction.
|
2019-04-18 16:32:33
|
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|
https://www.cemc.uwaterloo.ca/pandocs/potw/2021-22/English/POTWE-21-G-21-P.html
|
# Problem of the Week Problem E Another Circle
Points $$A$$ and $$B$$ are on a circle with centre $$O$$ and radius $$6$$ cm, such that $$\angle AOB=60 \degree$$.
Determine the radius of the circle which passes through points $$A,~B,$$ and $$O$$.
|
2022-05-22 07:06:33
|
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|
https://forum.effectivealtruism.org/posts/yNn2o3kEhixZHkRga/certificates-of-impact
|
# 0. Introduction
In this post I describe a simple institution for altruistic funding and decision-making, characterized by the creation and exchange of "certificates of impact." I’m interested in your thoughts and criticisms.
Typically an effectiveness-minded altruist would try to do as much good as possible. Instead, users of the certificates system try to collect certificates for as much good as possible.
Whenever anyone does anything, they can declare themselves to own an associated certificate of impact. Users of the system treat owning a certificate for X as equivalent to doing X themselves. In the case where certificates never change hands, this reduces precisely to the status quo.
The primary difference is that certificates can also be bought, sold, or bartered; an altruist can acquire certificates through any combination of doing good themselves, and purchasing certificates from others.
For example, a project to develop a malaria vaccine might be financed by investors. If the project succeeds the developers can declare themselves owners of an associated certificate, and distribute it to investors in the same way they might distribute profits. These investors could then resell this certificate to a philanthropist who is interested in funding malaria prevention and who honors certificates. The philanthropist would recognize the certificate as valuable to the extent that they believed that the original project was causally responsible for the development of the vaccine, and their willingness to pay would be the same as their willingness to develop the vaccine themselves (evaluated with the benefit of hindsight).
Note that judging the value of a certificate is just as subjective as judging the value of the underlying activity, and is done separately by any philanthropist considering acquiring that certificate.
# 1. Some existing challenges
1. Evaluating philanthropic opportunities is difficult, and most are not very good. Predictions are expensive and inaccurate, and often infeasible for small donors. The problem is worst when evaluating novel interventions. Prizes can address some of these difficulties, but have their own set of problems.
2. One person might have the ability to do a project, another the desire to fund it, and a third the knowledge to evaluate it. Coordination is hard, and existing incentives misaligned, both for disseminating good information and executing good projects.
3. Thinking about crowding out and causal responsibility is hard: if I don’t do something, will someone else? Does it matter which of Oxfam’s activities I fund? How is causal responsibility divided between donors and employees? Between the philanthropist who buys malaria nets and the government which distributes them?
4. Prices are an elegant and flexible system for communication and coordination, which are often unavailable for altruists.
5. Reasoning about leverage, and interacting with funders with different priorities, is tough. It’s easy to end up neglecting a large possible upside or engaged in zero-sum conflict.
# 2. Certificates of impact
Whenever anyone does something good in the world, they can “mint” an associated certificate of impact and declare themselves to be the owners. This can be as simple as making a public statement, analogous to acknowledging a funder. At the bottom of this post I could write “This post backs certificate [RationalAltruist-22], which is now owned by Paul Christiano;” I could do the same in a press release describing a medical intervention, an academic paper reporting the results of an experiment, or a piece of open source software. Independently, I might post an offer to sell these certificates, or a philanthropist might contact me and express interest in purchasing them.
By deciding which certificates they are willing to pay for, altruists using the system define the rules of the game. The protocol suggests considering a certificate valuable only if
1. It was issued by the group or individual that performed the associated activity.
2. It is the unique certificate associated with that activity.
Users should be willing to pay the same amount for a valid certificate of X that they would be willing to pay to cause X to happen (and to keep the resulting certificate). That is, I should be indifferent between spending $100 to do a good deed and paying someone else$100 to do it, provided that in either case I get the entire certificate for that good deed.
Naturally this willingness to pay is subjective, depending on both values and views—just like conventional grantmaking. Users maximize the total value of all of the certificates they hold, using whatever combination they prefer of doing good deeds themselves or acquiring others’ certificates.
Certificates can be transferred just like any other IOU or title, and have the same practical issues with bookkeeping and double-spending. We can resolve these however we like (word of mouth, a bulletin board, a trusted central party, the bitcoin blockchain…). Establishing properties [1] and [2] probably involves trusting the issuer—just like conventional grantmaking.
Certificates can be subdivided arbitrarily. When a group accomplishes a project together, they must decide how to divide the resulting certificates in the same way that a profitable enterprise would decide how to divide the profits.
Owning a certificate for X would probably not carry the same prestige as doing X yourself. Funders using the system purchase certificates as a way to do good, not necessarily as a way to earn credit (though helping them get credit may be useful for getting them on board).
For now, I'm most interested in the question: would philanthropic activity in an area be improved, if many funders used this system or a similar incentive scheme?
# 3. Commentary
## Some simple examples
No transfers. In the simplest possible example, certificates never change hands. In this case, the protocol reduces to “Do the activities that you think have the largest positive impact on the world,” i.e. the status quo.
Grants. A grantmaker might pay a charitable organization to perform some charitable work, in return for the resulting certificates. This also reduces to the status quo, at least for the grantmaker. A grantmaker might also leave some of the certificates to the charities, in the same way that a startup retains equity as an incentive.
Prizes. An unfunded do-gooder might keep the resulting certificates to themselves. Later, a philanthropist who appreciates the value of their work could buy the certificate off of them. The end result is very similar to a prize or bounty. As with a prize or bounty, that philanthropist could state their intentions in advance, choosing their own balance between flexibility and predictability.
## So what?
Reproducing the status quo isn’t so exciting. Do certificates improve the situation, or even change it? I think so, in ways that should be interesting to particularly effectiveness-minded funders. For example:
1. Allocating certificates requires explicit and transparent allocation of causal responsibility, both within teams and between teams and donors. In addition to the obvious cultural effects (which I would welcome despite the costs), this aligns individual incentives with altruistic goals, reducing incentives to mislead donors, volunteers, and employees.
2. Exchanging certificates for money leads to a more consistent conversion between good done and compensation, especially if funders use a mix of prizes and grants. It can prevent good deeds from falling through the cracks, ameliorate some winner-take-all PR dynamics, and eliminate some zero-sum conflict. It also makes it easier to move between different funding modes.
3. Purchasing certificates helps funders with different values coordinate; if I see a good deal I have an incentive to take it, without worrying about whether it might be an even better deal for someone else.
4. The ability to resell certificates makes a purchase less of a commitment of philanthropic capital, and less of a strategic decision; instead it represents a direct vote of confidence in the work being funded.
There are also risks inherent in bringing altruistic incentives closer to profit incentives (substituting extrinsic for intrinsic motivations, introducing new scope for zero-sum conflict...). How you feel about the tradeoff depends on how you feel about the relative merits of the two approaches, and may vary across domains.
## Some more examples
In addition to these simple examples, certificates of impact can facilitate more exotic interactions.
Financing. If I’m optimistic about your project I might give you a loan which I expect you to pay back by selling a certificate. Or I could buy the certificates off you, as if I were a grantmaker, with the intention of reselling them.
Research universities. A research group can employ researchers in exchange for ownership of some of the certificates of impact they produce. These certificates can then be sold to funding agencies or philanthropists. This shields those funders from having to make predictions about research output, which they are often not equipped to do, and provides good incentives to both researchers and research groups. (This is a special case of financing and causal attribution, but it’s an application close to my heart.)
Interpolating between prizes and grants. By implementing both prizes and grants as trades, it is easier to interpolate between the two, providing some funding to get a project going in exchange for part of its output (presumably at an expected discount) and then purchasing more of its output after the fact.
Moral trade. Many cases of moral trade can be implemented by turning good deeds into certificates and bartering or trading in a marketplace. For others, such as abstaining from eating meat, this would be problematic (though it’s not clear that the use of certificates s to blame).
Targeted donations. If I want to fund only part of an organization’s activities, I can purchase only those certificates of impact which I find valuable.
Research on effectiveness. If I think a certificate is too cheap, I can do research to evaluate its effectiveness; depending on the result I can buy the certificate, publish my research, and resell the certificate.
Trades across time. If my discount rate is lower than interest rates, I can sell certificates of impact today, invest the proceeds, and purchase certificates next year. (The price of each fixed certificate should rise at roughly market rates; but in addition to that, a certificate for a life saved in year 1 may have a different value than a certificate for a life saved in year 2, reflecting the discount rates of philanthropists.)
Speculation. If I think at a change will be considered good in retrospect, I can purchase appropriate certificates now and aim to resell them later at a profit.
Multiple bottom lines. If a project has several objectives, it can use certificates to monetize all of them and then make local profit-maximizing decisions. This may lead to more sane decision-making, and much more transparency, than the status quo.
(I’m sparing you a much longer list.)
## Some theory
Suppose that everyone who valued X were using the certificate protocol. (For concreteness, suppose it’s everyone interested in cosmology research.)
Then at equilibrium, the price of certificates of impact on X is equal to the marginal cost of achieving an impact on X. Any deviation is an arbitrage opportunity: if you can do X more cheaply, then you can sell the resulting certificates for a profit; if you are doing X more expensively, then you could save money by buying certificates instead.
Moreover, the status quo corresponds to one way of producing certificates, with total value equal to the total good being done. The actual outcome should be a Pareto improvement, and in particular should increase the total value of available certificates. As a result, it also increases the total amount of good being done.
I wouldn't lean on this argument alone. But I do think it suggests that things might tend to work themselves out, at least if the system were widely adopted. For me, considering concrete cases bolsters that intuition.
The infra-marginal units of X, those that were cheaper to produce than the marginal unit, pose a theoretical problem. These units of X get produced and purchased (good news!), but the resulting certificates are most likely to be sold for the marginal price of X (bad news!). That means there is a transfer of wealth, from the people who care about X (and who use the certificates system) to those who have infra-marginal opportunities to do X.
This problem is worst for the early adopters of the certificate system, which is particularly unfortunate. And it’s even worse when the opportunities to do X are so cheap that they were going to get done anyway. For example, suppose some high energy physicists would have to go out of their way to avoid doing some cosmology. Under the certificates system they would get paid for this research by cosmology funders who purchase the resulting certificates. But the cosmology funders might not be so happy about this, since the counterfactual impact of this funding is zilch.
Whether this is a feature or bug depends on how many really cheap opportunities to do X were failing to get done. Maybe the high energy physicists wouldn’t even bother publishing their cosmology in a format that cosmologists would understand, because the high energy community wouldn’t give them any respect for it. In this case there might be huge social benefits from paying them to do so. My own guess is that we leave a lot of low hanging fruit hanging, and that picking it is more important than worrying about the transfers.
# -1. Note
I have omitted any discussion of how certificates might be implemented, especially discussion of how you might get from here to there. This is a tough problem, but I think that a single advocate could make meaningful headway if it seemed worthwhile. I’m interested in the prior question, is it a “there” worth thinking about?
New Comment
You might be interested in:
http://en.wikipedia.org/wiki/Health_Impact_Fund
http://en.wikipedia.org/wiki/Social_impact_bond
Which are practical prize type solutions along similar lines.
The HIF in particular is very closely related; that's roughly what I imagine an early implementation looking like.
Thanks for the pointers.
This idea also reminds me a little bit of performance based incentives in aid, such as Cash On Delivery programs. Performance based incentives are relatively new, though, so I haven't been able to find many impact evaluations, although initial reports are promising for health interventions (1, 2). There was even talk a few years ago of creating a stock market for charities, but I don't think that has gone anywhere. Robert Shiller also proposed something called a Participation Nonprofit at one point where people would buy shares in a nonprofit then spend the returns on charities. All of those have elements of what you're talking about, but don't put it all together.
This whole structure of certificates of impact seems to derive its main benefit from allowing market mechanisms to work in the altruistic domain. When I've thought about trying to access these market mechanisms, the main problem has appeared to be anchoring the value so that expectations work properly. For instance the value of stock in the stock market is anchored by the profits that the firms will eventually make.
You don't spend much time addressing this problem. I'm not sure if you mean to include it in the note at the end of things you are setting aside, but at least to my mind it seems relevant to the question of whether such a system could work properly if we arrived there.
My previous attempts to solve this problem had involved anchoring by (occasional, stochastic) explicit external evaluation, but this turns up other difficulties. If I understand it correctly, you're thinking instead of anchoring to how much people really value things. The issue with this is that values can fluctuate over time, so I don't know that it's really well-founded. If the amount its value today comes apart quite a bit from how its expected to be valued tomorrow (and this will continue), I'm not sure how it would stabilise.
What I'm particularly worried is how people would value old certificates. It seems plausible that people would have little interest in certificates from 80 years ago, and expect future interest to continue to drop off. Do you envisage some mechanism to counteract this? Old certificates would by their nature be irreplacable, so we might hope that possessing them achieved some cachet like possessing old artwork has in the world today. But I don't feel confident that this would work.
A philanthropist (or funding agency) gives certificates value by their efforts to acquire certificates.
I.e. rather than funding a bunch of research that it thinks looks promising, the NSF tries to purchase research output. People may buy certificates (or hire researchers in exchange for a fraction of their certificates) because they expect the NSF to buy them later.
The expectation is that the NSF won't subsequently resell all of these certificates. Doing so would be an explicit preference reversal (unless the value of the certificates grew faster than the NSF's rate of return, in which case other donors have decided that the NSF funded good things, and the NSF might decide to sell them).
ETA: this subsumes the proposal of occasional, stochastic valuations. A funder interested in tying the value of certificates to some explicit benchmark X can periodically buy certificates at a value determined by the benchmark X.
Yes, I see how you can get short term value from this. But how do you get a long-term stable state? How do you envision funding agencies or philanthropists making decisions about how to value different certificates? (Particular interest in this question for old certificates, because I think it makes many of the problems more salient.)
Summary: the value of certificates is generally not fixed but will change over time. But this seems like a feature, not a bug. And If everyone stopped using the certificates system, the funders who are left with the certificates should be happy with that outcome as well, for the same reason they are happy making a grant which they can't later unwind.
Suppose in 2000 a funder buys some certificates that (they think) represent lives saved in 2000. They are happy with that investment, and just to keep the certificates indefinitely; it is not clear that those certificates will ever change hands again after 2005. If a funder had the opportunity to "undo" their earlier funding opportunities and get back the money, how often would they take it?
When valuing certificates at different times, the idea is for a funder to consider how many dollars they would pay to do a good deed in 2000, vs. a good deed in 2005. This is a question that funders already face, when choosing whether to invest or do a good deed now. Allowing them to answer the question with the benefit of hindsight (and allowing speculators to bet on what their answers will be) seems like a bonus.
It seems like the main concern is if they don't think of selling certificates as unwinding their original good deed, i.e. if they aren't using the certificates system. If they are using the system, then the overall transaction is just a straight-up loss for them. If they aren't using the system, then it's not really our business how they interact with it (adding people who sometimes buy certificates but who don't value them can't do any harm, it just drives up the value of certificates and benefits the users of the system).
If a funder did decide to unload their life-saved-in-2000 certificate, the idea would be for someone else to step up to buy them for the reduced price. If people interested in saving lives in 2000 are actually using the certificates system, then the price can't fall far before someone will become very interested in buying it.
In fact it's quite likely that the value of certificates for realistic interventions will vary hugely over time as more information is revealed about how good they were. This happens on top of the overall discount rate between good-in-2000 and good-in-2005, and seems like a further bonus.
I agree with your general point about changing value of certificates as we get more information being a feature (this is the kind of thing I meant by tapping into market mechanisms).
Suppose in 2000 a funder buys some certificates that (they think) represent lives saved in 2000. They are happy with that investment, and just to keep the certificates indefinitely; it is not clear that those certificates will ever change hands again after 2005.
OK, I see you're envisaging a less liquid market than I was. Though there are certainly some situations where I'd expect people to sell long after the fact. For instance if someone dies 40 years later and the certificates pass to their next-of-kin who doesn't value the work, they might well seek to sell them.
In order to understand whether this is a state which would be desirable, I'm trying to picture what the world would look like today if we'd been using these since, say, 1800, and there were lots of old certificates lying around. I haven't been able to provide myself with a stable picture of this, which makes me somewhat sceptical.
If a funder had the opportunity to "undo" their earlier funding opportunities and get back the money, how often would they take it?
I think this would happen a lot as people gained information. Then funding gives you an option value of cashing out, whereas not funding wouldn't necessarily give you the chance of retroactively buying the thing (people would also fund more high-variance things). Of course that doesn't mean that people would want to sell the certificates, because information that made them want their money back would also tend to drop the going price for the certificates.
My point is that the existence of future funders who don't care much about "Lives saved in 2000" can never drive down the value of such a certificate, it can merely drive up the value of certificates that the future funders care about.
If the market becomes illuiqid (once the funders who care about lives saved in 2000 are gone), this shouldn't be troubling to the funders left with the certificates, since that just means they are assuming responsibility for the things they funded (as in the status quo).
That said, I don't see why there is any problem with valuating the old certificates, aside from skepticism about whether anyone in 2000 cares enough about the good deeds done in 1800 and would actually honor certificates.
I wonder if it would make sense to sell certificates of Impact as non-fungible tokens (NFTs), given that NFTs are emerging as a lucrative way of publicly representing the "ownership" of non-physical assets like digital artwork.
Shouldn’t impact be fungible at some level though?
Ohh, I should've made this clearer.
The NFT would be used to represent responsibility for (custodianship of) a particular impactful action. Just as with impact certificates as previously proposed, a person who, for example, runs EA Harvard for 2018, could put responsibility for this impact onto the marketplace. Buyers, when pricing this asset, can then evaluate how well EA Harvard did at creating things (that may be fungible) like number of EAs produced, or net effect on wellbeing, and pay accordingly.
I think it's useful to sell responsibility for the impact of a particular action (which is non-fungible), rather than a some responsibility for some (fungible) quantum N of impact, so that the job of judging the impactfulness of the action can be left to the markets.
Ah, that makes sense :-)
I'm guessing that for these to work, the ownership of certificates should end up reflecting who actually had what impact. I can think of two cases where that might not be so.
Regret swapping:
• Person A donates $100 to charity X. Person B donates$100 to charity Y.
• Five years later they both change their minds about which charity was better. They swap certificates.
So person A ends up owning a certificate for Y, and person B ends up owning a certificate for X, even though neither of them can really be said to have "caused" that particular impact.
Mistrust in certificate system
• Foundation F buys impact certificates. It believes that by spending $1 on certificates, it is causing an equivalent amount of good as if it had donated$2 to charity X.
• Person A is skeptical of the impact certificate system. She believes that foundation F is only accomplishing $0.50 worth of good with every$1 it spends on certificates (she believes the projects themselves are high value, but that if foundation F didn't exist then the work would have got done anyway).
• Person A has a $100 budget to spend on charity. • Person A borrows$50 from her savings account and donates $150 to charity X. She sells the entire certificate to foundation F for$50 and deposits this back in her savings account.
Why would person A do this? She doesn't care about certificates, just about maximizing positive impact. As far as she is concerned, she has caused foundation F to give $50 to charity X, where otherwise that money would only have accomplished half as much good. Why would foundation F do this? It believes in certificates, so as far as F is concerned, it has spent$50 to cause a $150 donation to charity X, where the other certificates it could have bought would only be equivalent to a$100 donation.
There might be some interesting tricks here. Like suppose you saved someone's life, and wrote a certificate for yourself, but they turned out to be an ax muderer. In such a situation, you might have to pay other people to take the certificate from you. Alternatively, you would might hide the certificate. But by hiding the certificate, you're avoiding culpability for your harmful actions to some extent. Of course, people are able to avoid culpability for altruistic actions that turn out to be harmful in the status quo, I'm just observing that for this sort of reason, an economy based on Impact Certificates would not symmetrically deliver the impacts of people's actions to them.
Yes, this system offers no protection against people doing bad things. (Even if they pay people to take certificates off of their hands, the price would be far too low.) That responsibility falls to the usual mechanisms, e.g. legal protection.
A problem is that it would incentivise people to do risky altruistic activities like enacting big political changes or developing risky tech.
[-][anonymous]6y 0
Even if some people think that a particular kind of certificate is bad (has negative social value), as long their opinion isn't in the majority, I think a liquid market for certificates would be able to handle this efficiently. If most people think that outcome A is good but I think it's bad, I can short-sell A-certificates, and this works as long as the price is positive (i.e. I'm in the minority).
If directly thwarting A is cheaper than short-selling, I might be tempted to do that, but it would be inefficient (my actions cancel out others' actions, and the net effect is to waste money for nothing). Fortunately, it seems like the certificate system still provides an efficient way to do "moral trade" in this case! Other people who agree with me can set up a market for anti-A-certificates - backed up by my ability to directly prevent A. Essentially, the others would pay me to carry out the anti-A intervention. The pro-A people can then shut this down by shorting the anti-A-certificates until I'm no longer able to fund my anti-A activities.
My guess: After a while, each side would be paying the other money not to carry out their intervention. Some pro-A interventions would be funded, but less than if the anti-A people couldn't short-sell. No anti-A interventions would be funded, so no obvious inefficiencies happen.
Does that sound correct? I'm no expert, and I'm not sure whether that's actually the stable equilibrium.
I'm still a little confused as to whether these certificates are intended to confer social status. If not, why should I value universes in which I own certificates more highly than universes in which I don't?
Should I just look at the big picture and decide it's beneficial to self-modify so as to give ownership of certificates intrinsic value in my utility function?
One possible use for certificates other than bragging rights is A/B testing - pick two EAs with similar skills and resources but different strategies, and see who ends up with more certificates.
You can think of it as a way of doing accounting for causal responsibility if you want. But yes, the argument is aiming for "if we did this, the outcome would be good," and I'm leaving it up to your decision theory to justify doing things that lead to good outcomes.
I worry about people's preferences changing over time, either as they get older or as a result of running into financial difficulties. I can imagine buying a bunch of certificates in my idealistic youth and then selling them off again in my cynical old age. At any point in time I'd feel like I was doing the right thing, and whatever philanthropist bought the certificates off me would think they were restrospectively funding the original project when actually they were just putting money into my pocket.
This is sort of the opposite problem of what (I think) Owen_Cotton-Barret was describing with old certificates becoming valueless.
Thinking about this a bit more... If I don't trust my future self with these certificates, I can always send them to some other entity which will look after them in a way consistent with present-self's wishes.
This could be an account in a different name, corresponding to an entity which I believed caused my behavior, and which I believe will responsibly hoard the certificates (e.g. GiveWell).
Alternatively it could be an ethereum-style contract which allows me to either hold on to the certificate or give it away (given enough other signatures verifying that I'm giving it to a reputable party and that I'm not benefitting financially from giving it away).
This sort of lock-in could also be a mechanical part of how certificates work, e.g. they allow themselves to be traded freely for a month and after that it gets more and more difficult.
The situation where certificates are bought and sold at near the original donation price is somewhat peculiar. Essentially, rather than giving away your assets to charity you'd be exchanging real money for some riskier and weirder asset, but which is nonetheless still worth money. Giving away certificates then feels like a sort of "second order altruism" which then maybe deserves a certificate of its own...
I like this idea. Thinking about the following case was helpful for me:
Suppose for the sake of argument
• I have two career options, Charity Worker or Search Engine Optimizer.
• CW generates 5 utilons in direct impact, and 0 utilons via earning-to-give
• SEO generates 0 utilons in direct impact, and 3 utilons via earning-to-give
• There are plenty of people who don't identify as EAs and/or don't take Paul_Christiano's certificate idea seriously, but who want to work as CWs.
From first glance it looks like the system would fail here - if I'm trying to maximize my certificates, and most other people in the market don't care, then I'd choose CW and crowd-out somebody else.
But I think what would actually happen is that I'd choose the SEO option, earn a bunch of money and then say "hey, charity worker, over here on the internet there's an apparently meaningless collection of numbers with your name at the top. I'll give you $5 if you log in and change it to my name". I'd end up with certificates valued at more utilons than if I'd just taken the CW option. Even if a typical person didn't view these certificates as valuable or meaningful initially, they'd start to once they heard about this mysterious community who was willing to pay money for them. In practice this might look different depending on the scale of implementation (how fine grained are certificates; who are the certificate-issuers). One extreme is: • You are issued with a certificate whenever you help someone; • There is a liquid market in the certificates; • Helping someone is seen (by the market) as good to the degree to which it benefits them; • Helping two individuals by the same amount is seen (by the market) as equally good. Then the certificates essentially become a currency system, which has something like a guaranteed basic income built into it (the amount that you can be helped in a given year). This extreme probably isn't exactly what you have in mind (for instance, I'm not sure how it would interact with helping future people). But I don't have a good sense of how far in this direction you're thinking. Presumably there is a market for certificates amongst those concerned with e.g. the global poor, and it might function in this way. There is no provision in this system for allowing the beneficiaries to determine the value of these certificates. As with contemporary philanthropy, this responsibility falls to the philanthropists, who may or may not execute it responsibly. (This judgment would be reflected in the relative prices of "Helped Alice by doing X" and "Helped Alice by giving her$1," which may have some benefits in terms of transparency.)
Cash transfers are a more direct way to capture these benefits. (After cash transfers are in place, providing goods to the poor can then be a profit-oriented enterprise.) The key question is just whether you think philanthropists or beneficiaries are better at comparing benefits.
Full disclosure: I fear I do not completely understand your idea. Having said that, I hope my comment is at list a little useful to you.
Think about the following cases: (1) I donate to an organization that distributes bednets in Africa and receive a certificate. I then trade that certificate for a new pair of shoes. My money, which normally can only be used for one of these purposes, is now used for both. (2) I work for a non-profit and receive a salary. I also receive certificates. So I am being paid double?
The second case is easily solved, just give the employee either or. But then, what is the benefit of a certificate over a dollar bill? The first case presents a bigger problem I think, since essentially something is created from nothing. Notice that donations are not investments the donor can expect a return on (even if they are an investment in others).
If you buy and then sell a certificate, you aren't funding the charity, the ultimate holder of the certificate is. They will only buy the certificate if they are interested in funding the charity.
You could pretend you are funding the charity, but that wouldn't be true---the person you sold the certificate to would otherwise have bought it from someone else, perhaps directly from the charity. So your net effect on the charity's funding is 0. I could just as well give some money to my friend and pretend I was funding an effective charity.
(I'm setting aside the tax treatment for now.)
You would pay an employee with certificates for the same reason a company might pay an emplyee in equity. If there is no secondary market, this can be better for the company for liquidity reasons, and can introduce a component of performance pay. But even if there is a secondary market (e.g. for Google stock), it can still be a financially attractive way for a company to pay a large part of its salary, because it passes some of the risk on to the employee without having to constantly adjust dollar-denominated salaries. (There are also default effects, where paying employees in certificates would likely lead to them holding some certificates.)
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2021-04-12 18:36:26
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https://helpingwithmath.com/number-patterns-sequences/
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Home » Math Theory » Numbers » Number Patterns and Sequences
Number Patterns and Sequences
Introduction
The study of mathematics includes numbers and the different patterns in which they can be represented. This means that when we think of mathematics, the first thing that comes to our mind is numbers. We are aware of different kinds of numbers that have been defined such as natural numbers, whole numbers, decimals, fractions and so on. Each set of a number has its own unique characteristic that makes it a set. For instance, the set of even numbers comprises of all numbers that are divisible by 2. Similarly, prime numbers are the numbers that are not completely divisible by any other number other than themselves and the number 1. By seeing these examples can we say that the numbers can be put in a sort of a pattern? Let us find out.
What are Number Patterns?
Number patterns are sequences of numbers that repeat themselves. In other words, patterns are a set of numbers arranged in a sequence such that they are related to each other in a specific rule.
How to Identify Number Patterns?
We are aware of the four operations of mathematical operators, namely, addition, subtraction, multiplication and division. Most of the number patterns are based on these four mathematical operations only. However, there are some patterns that involve a combination of these operations.
Let us understand this by an example.
Suppose, we have been given the number pattern 1, 3, 5, 7, 9, ………….
What arithmetic pattern is followed by the above sequence? Let us find out.
Observe each of the terms carefully. We can see that –
First Term = 1 = 2 x 0 + 1
Second Term = 3 = 2 x 1 + 1
Third Term = 5 = 2 x 2 + 1
Fourth Term = 7 = 2 x 3 + 1
Fifth Term = 9 = 2 x 4 + 1 and so on.
Therefore, we can identify the number pattern in the given sequence as 2 n + 1, where n ≥1. We can clearly see that this sequence involved a combination of two operators, “ x “ and “ + “.
Through this example, we have learnt that a number of different combinations of operators can be used to define the number pattern in a sequence.
What is a Sequence?
In mathematics, a sequence is a chain of numbers (or other objects) that usually follows a particular pattern. The individual elements in a sequence are called terms. In other words, a sequence is an ordered list of numbers (or other elements like geometric objects), that often follow a specific pattern or function. Sequences can be both finite and infinite. The terms of a sequence are all its individual numbers or elements.
Types of Number Sequences
Following are the different types of patterns that are most commonly in use when we define a sequence of numbers–
• Growing Sequence – As the name suggests, the growing sequence is the number pattern where the numbers are present in an increasing order.
• Reducing Sequence – Again, as the name signifies, a reducing sequence is the number pattern in which the numbers are present in the decreasing order.
• Recurring Sequence – In the recurring sequence of numbers, the same set of numbers keep repeating themselves to form a pattern of numbers.
Importance of Rules in Number Patterns and Sequences
There are a certain set of rules that the number patterns and sequences follow. These rules define the terms that are contained in a number pattern. We need to understand these rules in order to understand the number sequence. The understand of these rules is extremely important as without them we cannot find a missing term in the sequence or know the pattern in which the number sequence has been made.
Let us understand this by an example. Consider the following number sequence –
1, 4, 9, 16, 25, 36, ?
Now, if we wish to find which term comes after 36, we must first understand the rule that defines this number sequence.
To find rule the rule that has determined the terms of the sequence, we should observe the terms carefully. We can observe the following –
1. Every term is a product of itself.
2. The numbers that have been multiplied by themselves are natural numbers.
This means that –
1 = 1 x 1
4 = 2 x 2
9 = 3 x 3
16 = 4 x 4
25 = 5 x 5
36 = 6 x 6
By this order, the next term should be a multiple of 7 multiplied by itself. Therefore, we have,
7 x 7 = 49 will be the next term of the given number sequence.
Simple Sequences
Even and Odd Number Sequences
We are aware of natural numbers. Natural numbers are the numbers that begin from 1 and go on up to infinity. They are in the form of 1 , 2 , 3, 4 and so on. Natural numbers simply form two types of patterns, depending upon the fact whether they are odd or even. Recall that odd numbers are the numbers which when divided by 2 will leave 1 as a remainder. In other words, odd numbers are the numbers that are not divisible by 2. So, how is the pattern of odd numbers defined? The number pattern of odd numbers is defined by the numbers starting from 1 ,3 , 5 , 7 , 9 and so on. Mathematically, this pattern for natural numbers can also be represented as the following on a number line.
Therefore, the pattern of odd natural numbers is 1, 3, 5, 7, 9, 11 and so on.
Similarly, let us recall what we mean by even numbers. Even numbers are the numbers which when divided by 2 will leave 0 as a remainder. In other words, even numbers are the numbers that are completely divisible by 2. So, how is the pattern of even numbers defined? The arithmetic pattern of even numbers is given by 2, 4, 6, 8, 10 and so on. Mathematically, this pattern for natural numbers can also be represented on a number line as –
Therefore, the pattern of even natural numbers is 2, 4, 6, 8, 10, 12 and so on.
Now, let us learn about special sequences of numbers.
Sequence of Triangular Numbers
A number sequence of triangular numbers is the pattern that has triangular numbers. But, what are these triangular numbers? Let us find out. Triangular numbers are generated from a pattern of dots / boxes that form a triangle. In other words, the triangular number sequence is the representation of the numbers in the form of an equilateral triangle. The pattern formed by the triangular numbers is such that the sum of the previous number and the order of the succeeding number results in the sequence of triangular numbers. This arrangement is represented as below –
Arithmetic Number Sequences
There are two most common arithmetic sequences –
1. Number Sequence of Square Numbers
2. Number Sequence of Cube Numbers
Let us learn about them one by one.
Number Sequence of Square Numbers
A number sequence of square numbers is a pattern that has square numbers. But, what are these square numbers? Let us find out. Square numbers are the numbers obtained when a number is multiplied by itself. For instance 2 x 2 = 4, therefore, 4 is the square of 2. Similarly, 3 x 3 = 9, therefore, 9 is the square of 3. The number sequence of square numbers is given by 1, 4, 9, 16, 25, 36 and so on. Now, though we can easily identify the first few numbers of this pattern, how do we find a number on any position of the pattern? Can we define a formula to help us identify the number at a particular position in the number? The formula for defining the pattern of square numbers is given by
Arithmetic pattern of Square Numbers = n 2, where n ≥ 1.
Let us verify the above formula for obtaining the pattern of numbers.
If we put n = 1 in the above formula, we will get 1 2 = 1 x 1 = 1
If we put n = 2 in the above formula, we will get 2 2 = 2 x 2 = 4
If we put n = 3 in the above formula, we will get 3 2 = 3 x 3 = 9
If we put n = 4 in the above formula, we will get 4 2 = 4 x 4 = 16
If we put n = 5 in the above formula, we will get 5 2 = 5 x 5 = 25 and so on.
So, we can see that just by putting the value of the position of the number in the above formula, we can obtain the number in the pattern of square numbers.
Number Sequence of Cube Numbers
A number sequence of cube numbers is the pattern that has cube numbers. But, what are these cube numbers? Let us find out. Cube numbers are the numbers obtained when a number is multiplied twice with itself. For instance 2 x 2 x 2 = 8, therefore, 8 is the cube of 2. Similarly, 3 x 3 x 3 = 27, therefore, 27 is the cube of 3. The number sequence of cube numbers is given by 1, 8, 27, 12, 64, 125 and so on. Now, though we can easily identify the first few numbers of this pattern, how do we find a number on any position of the pattern? Can we define a formula to help us identify the number at a particular position in the number? The formula for defining the pattern of cube numbers is given by
Arithmetic pattern of Cube Numbers = n 3, where n ≥ 1.
Let us verify the above formula for obtaining the pattern of numbers.
If we put n = 1 in the above formula, we will get 1 3 = 1 x 1 x 1 = 1
If we put n = 2 in the above formula, we will get 2 3 = 2 x 2 x 2 = 8
If we put n = 3 in the above formula, we will get 3 3 = 3 x 3 x 3 = 27
If we put n = 4 in the above formula, we will get 4 3 = 4 x 4 x 4 = 64
If we put n = 5 in the above formula, we will get 5 3 = 5 x 5 x 5 = 125 and so on.
So, we can see that just by putting the value of the position of the number in the above formula, we can obtain the number in the pattern of cube numbers.
Fibonacci Number Sequences
The Fibonacci number sequence is named for Leonardo Fibonacci, born in 1170 in Pisa, Italy. The Fibonacci number sequence is a sequence of numbers in which each number in the sequence is obtained by adding the two previous numbers together. The sequence starts with 0 and 1. The importance of this number sequence lies in the fact that it can be found in many things in nature such as plant leafing patterns, spiral galaxy patterns, and the chambered nautilus’ measurements. The series is thus defined as –
Geometric Number Sequences
In a Geometric Number Sequence each term is found by multiplying the previous term by a constant. Let us understand the geometric number sequence by an example.
Consider the following number sequence –
2, 6, 18, 54, 162, 486, ……..
Let us analyse each term of this number sequence. We have,
We can notice from the above arrangement that in order to obtain the next term of the number sequence we are multiplying the previous term by 3. This means that a term when multiplied by 3 will give us the next term of the number sequence. Since, this sequence involves multiplying the previous term by a constant to obtain the next term; therefore, it is a geometric sequence.
Another important observation in the above number sequence is that every two consecutive terms share the same ratio. This means that –
$\frac{6}{2}$ = 3
$\frac{18}{16}$ = 3
$\frac{54}{18}$ = 3
$\frac{162}{54}$ = 3
$\frac{486}{162}$ = 3
Thus, we can say that there exists a common ratio between every two consecutive terms in a geometric sequence.
Solved Examples
Example 1 Determine the value of A and B in the following pattern.
15, 22, 29, 36, 43, A, 57, 64, 71, 78, 85, B.
Solution We have been given the number sequence
15, 22, 29, 36, 43, A, 57, 64, 71, 78, 85, B and we need to find the values of A and B
Let us observe each term carefully.
Now, going by the above pattern, A should be 43 + 7 = 50
Hence, sixth term = A = 50
Now,
Similarly, B will the 7 added to the previous term.
This means that B = 85 + 7 = 92
Hence, we have, A = 50 and B = 92
Example 2 Find the missing value in the geometric pattern: 120, 60, __, 15, __.
Solution We have been given a geometric pattern and we need to find the missing terms. Let the missing terms be A and B. so, we have the number sequence as –
120, 60, A, 15, B
Now, we know that since it is a geometric sequence, therefore, the ratio of any two consecutive terms will be the same. Therefore, let us find the ratio of the first two terms. We will have,
$\frac{120}{60}$ = 2
Similarly,
$\frac{60}{A}$ = 2
A = $\frac{60}{2}$ = 30
Also,
$\frac{15}{B}$ = 2
B = $\frac{15}{2}$ = 7.5
Hence, A = 60 and B = 7.5 which completes the number sequence as
120, 60, 30, 15, 7.5
Key Facts and Summary
1. Number patterns are sequences of numbers that repeat themselves.
2. A sequence is a chain of numbers (or other objects) that usually follows a particular pattern.
3. The individual elements in a sequence are called terms.
4. Natural numbers simply form two types of patterns, depending upon the fact whether they are odd or even.
5. Triangular numbers are generated from a pattern of dots/boxes that form a triangle.
6. Square numbers are the numbers obtained when a number is multiplied by itself.
7. Cube numbers are the numbers obtained when a number is multiplied twice with itself.
8. The Fibonacci number sequence is a sequence of numbers in which each number in the sequence is obtained by adding the two previous numbers together. The sequence starts with 0 and 1.
9. In a Geometric Number Sequence each term is found by multiplying the previous term by a constant.
10. There exists a common ratio between every two consecutive terms in a geometric sequence.
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2022-05-24 16:09:56
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https://www.gamedev.net/forums/topic/387702-c-zip-file/
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# [.net] [C#] zip file
This topic is 4541 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.
## Recommended Posts
First let me explain what i am trying to achieve. I am coding a layer over .net filesystem so i could use folders, compressed files or any other sort of archives, the same way. Exemple which copy a file found in a zip file and copy it to a folder :
ArchiveManager manager = new ArchiveManager();
ArchiveInfo archive = manager.GetArchive("L:\Test2");
FileInfo file = manager.GetFile(@"L:\Test\Source.zip\Source\Code.zip\Test.txt");
file.CopyTo(archive);
So right now i am using SharpZipLib to deal with the zip files, but this libary doesnt allow editing and modifying of an exising zip file. The workaround i have now is to build another compressed file and to copy the data from the previous one, which is not elegant or performant. I am looking for another lib that could do that, but got no luck yet and i dont really want to code one myself if i can avoid it. So if you know one that could that, it would be perfect. Thanks.
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The source is available for SharpZipLib. You could modify it to support "add in place" compression, however, I think it should be noted that just about every commercial compression package I've ever used has added a file to an existing archive by creating a new one and then moving it into place. (Improves compression results, recalculates all of the inner tables, etc.)
If your primary problem is speed and the archives are generally small you could work with a MemoryStream object instead of a FileStream. Makes it lightning fast (until you need to write it to disk :)).
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You could implement the functionality of PhysicsFS through Tao.PhysFs.
PhysFS was inspired by the Quake file systems in which you have multiple zip/pak files, but can load a single file from any one of those zip files without having to specify the source. The way it works is that you add a "search paths" where your data files reside. These search paths can be sub-folders, zip files, etc. Once you setup your search paths, you can load any file within any of the search paths without having to deal with whether it's being loaded from a different folder, a zip file, a Doom WAD file, etc.
The output from loading in the file can be pretty much anything. A byte array, a stream, a string, etc. Here's an SDL.NET example of it being used to load an image from a zip file.
// Initiate PhysFSFs.PHYSFS_init("");// Allow PhysFS to look in data.zip for filesFs.PHYSFS_addToSearchPath("data.zip", 1);// Open surface from zipIntPtr imageFile = Fs.PHYSFS_openRead("sdldotnet_full.png");// Read it into a byte arraybyte[] imageBytes;Fs.PHYSFS_read(imageFile, out imageBytes, 1, (uint)Fs.PHYSFS_fileLength(imageFile));// Create a surface from the loaded bytes.Surface surf = new Surface(imageBytes);// close the fileFs.PHYSFS_close(imageFile);
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Yes, this is what i would like in the end. Actually i had no problem when it was just about reading from an archive. The problem arised when i wanted to be able to add or update contents in a zip. I thought that a zip could be modified, but it seems its not allowed.
So i end up doing like talonius suggested, using a memory stream to the modification and write back the zip file.
Thanks for the replies.
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2018-09-23 00:28:18
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https://socratic.org/questions/how-do-i-find-the-critical-points-for-the-function-f-x-7x-4-6x-2-1
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# How do I find the critical points for the function f(x)=7x^4-6x^2+1?
May 5, 2015
The critical points for a continuous function occur at those points where the derivative is zero.
Given $f \left(x\right) = 7 {x}^{4} - 6 {x}^{2} + 1$
$f ' \left(x\right) = 28 {x}^{3} - 12 x$
If $f ' \left(x\right) = 0$
then
$28 {x}^{3} - 12 x g r a p h \left\{7 {x}^{4} - 6 {x}^{2} + 1 \left[- 2.982 , 3.178 , - 0.404 , 2.675\right]\right\} = x \left(28 {x}^{2} - 12\right) = 0$
and
either $x = 0$
or $\left(28 {x}^{2} - 12\right) = 0$
If $28 {x}^{2} - 12 = 0$
then $x = \pm \sqrt{\frac{12}{28}} = \pm \sqrt{\frac{3}{7}}$
The critical points occur at $x = 0 , x = - \sqrt{\frac{3}{7}} , \text{ and } x = \sqrt{\frac{3}{7}}$
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2019-01-17 15:04:16
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https://10thplanetejuice.com/2020/03/31
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0
GLPK – GNU Project – Free Software Foundation (FSF)
GLPK – GNU Project – Free Software Foundation (FSF)
Introduction |
Documentation |
Mailing Lists/Newsgroups |
Request an Enhancement
| Report a Bug |
Maintainer
Introduction to GLPK
The GLPK (GNU Linear Programming Kit)
package is intended for solving large-scale linear programming (LP),
mixed integer programming (MIP), and other related problems. It is a set
of routines written in ANSI C and organized in the form of a callable
library.
GLPK supports the GNU MathProg modeling language, which is a
subset of the AMPL language.
The GLPK package includes the following main components:
• primal and dual simplex methods
• primal-dual interior-point method
• branch-and-cut method
• translator for GNU MathProg
• application program interface (API)
• stand-alone LP/MIP solver
The GLPK distribution tarball can be found on
http://ftp.gnu.org/gnu/glpk/
[via http] and
ftp://ftp.gnu.org/gnu/glpk/
[via FTP].
It can also be found on one of our FTP
mirrors
0
Integer programming – Wikipedia
A mathematical optimization problem restricted to integers
An integer programming problem is a mathematical optimization or feasibility program in which some or all of the variables are restricted to be integers. In many settings the term refers to integer linear programming (ILP), in which the objective function and the constraints (other than the integer constraints) are linear.
Integer programming is NP-complete. In particular, the special case of 0-1 integer linear programming, in which unknowns are binary, and only the restrictions must be satisfied, is one of Karp’s 21 NP-complete problems.
If some decision variables are not discrete the problem is known as a mixed-integer programming problem.[1]
Canonical and standard form for ILPs
An integer linear program in canonical form is expressed as:[2]
{displaystyle {begin{aligned}&{text{maximize}}&&mathbf
0
0
S&T Mobilizes Key Data to Inform COVID-19 Response
The recent and increasingly rapid spread of COVID-19 will present formidable challenges in the effort to contain the outbreak and protect the health and safety of our citizens.
The Department of Homeland Security’s Science and Technology Directorate (DHS S&T) is working diligently to support and inform the efforts of our federal agency partners and health organizations about the latest developments in news and research related to the COVID-19 pandemic, as well as identify the gaps in our knowledge about the virus that require the attention of the medical emergency response community.
Many knowledge gaps about the virus directly affect DHS policy and operations, both in response to the outbreak and normal operations that may be impacted by the outbreak. Key knowledge gaps include:
• Human infectious dose – how much of a dose does it take to make a person sick?
• The ability
0
The 6 Best Programming Languages for Game Design
It’s an inescapable fact that game designers, on occasion, may be asked to do a little programming.
This comes down to entry-level designers often switching their positions in teams a lot, and the nature of game design teams being very fluid.
It is not unusual for designers to step in and program, freeing up the game programmers to work on something more complicated, just like it is not unusual for programmers to be consulted during the design process.
That said, getting started in programming can be a little daunting.
What is the best language for game development?
Currently, The best programming languages for games are:
• C++
• Java
• HTML5
• CSS3
• JavaScript
• SQL
Game programming is the lifeblood and skeletal framework for all games you and I play. All of the crazy things you can do in the Grand Theft Auto games? ALl programmed.
How is programming used in games?
Well, It
0
Free Programming Books
Programming Persistent Memory describes the technology and why it is exciting the industry. It covers the operating system and hardware requirements as well as how to create development environments using emulated or real persistent memory hardware. …
In Natural Language Processing Succinctly, author Joseph Booth will guide readers through designing a simple system that can interpret and provide reasonable responses to written English text. With this foundation, readers will be prepared to tackle the greater challenges of natural language development. …
Scratch is a free, graphical programming environment from MIT. It teaches 8- to 16-year-olds programming by snapping code blocks together to form complete programs. In Scratch Programming Playground, you’ll learn to program by making cool games. …
Custom languages provide many benefits, but
0
Computer Clan
NEW WEBSITE COMING SOON…
COMPUTER CLAN
WE’RE A COMMUNITY FOR TECHIES—SINCE 2007.
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Spacejock Software
This site and all the software are the work of one person: Simon Haynes, a programmer with over 20 years experience on projects large and small. Every program was written for my own use, then released for everyone to use and enjoy. Apart from FCharts Pro, all my software is free to download and use and you’re encouraged to send me feature requests!
Antibrowser
When you launch your browser for a ‘quick session’, AntiBrowser will run instead. It will display a random 12 digit code, and you have to type the code and click Run Browser to get to the life-sucking time-wasting internet. If you decide it wasn’t that important after all, just click the Forget It button.
BookDB
Book catalogue software. Enter all your books with author, category, publisher etc and print them out in a variety of formats. Also includes library features – borrowers, multiple copies of the
0
What is a Computer Network?
One of the earliest examples of a computer network was a network of communicating computers that functioned as part of the U.S. military’s Semi-Automatic Ground Environment (SAGE) radar system. In 1969, the University of California at Los Angeles, the Stanford Research Institute, the University of California at Santa Barbara and the University of Utah were connected as part of the Advanced Research Projects Agency Network (ARPANET) project.
It is this network that evolved to become what we now call the internet.
Networks are used to:
• Facilitate communication via email, video conferencing, instant messaging, etc.
• Enable multiple users to share a single hardware device like a printer or scanner.
• Enable file sharing across the network.
• Allow for the sharing of software or operating programs on remote systems.
• Make information easier to access and maintain among network users.
There are many types of networks, including:
• Local Area Networks (LAN).
• Global Area Networks
0
Best Internet Service Providers 2020
What’s the maximum you can afford to spend on internet each month? This is always a good place to start because no one wants to spend more than they need to on anything.
If you find that the home internet plan you want doesn’t match your wallet’s reality, we’ve got a few tips on how you can lower your internet bill.
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2020-12-04 16:51:44
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https://www.transtutors.com/questions/time-value-of-money-capital-budgeting-you-should-can-use-either-scientific-or-financ-3296111.htm
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# Time Value of Money & Capital Budgeting You should can use either scientific or financial...
Time Value of Money & Capital Budgeting
You should can use either scientific or financial calculators or excel to carry out the necessary computations. You are required to:
1. Decide what formula is appropriate and show this.
2. Show your workings/calculations as appropriate to support your answers.
3. Draw timelines, where appropriate, to demonstrate the cash flows.
4. Answers without support of calculations will not receive any marks.
3.1 To purchase a new piece of equipment for your company, you found two machines that will do the job you require. The purchase price of machine A is $6000 and the purchase price of machine B is$8000. The expected economic life of each model is 5 years and there will be no resale value for either model at the end of this period.
However, the projected additional cash inflows resulting for each machine differ, as shown below:
Year Model A Model B
1 $2,500$7,500
2 $3,500$6,500
3 $5,300$5,400
4 $7,000$4,000
5 $10,000$3,500
(a) Assuming a discount rate of 9%, which machine would you choose and why?
(b) If the discount rate increased to 11%, does this change your decision on which machine you would choose in part (a)?
3.2 You have just won first prize in Super Lotto. However, you have two options of receiving the prize money: - you can either receive 30 payments of $240,000 paid every year, starting today or you can receive$1,600,000 now and future annual payments of $130,000 for 30 years, starting in one year from now. (a) If you were to maximize your wealth, which option would you choose if the interest rate is 6%? (Show your calculations for each option) 3.3 Your friend Kelly has$280,000 to invest. She does not want to take any high risks in the current economic climate. She comes to you, because she has heard you are studying business finance, and wants your financial advice on two investment
BNZ Bank TSB Bank
Interest rate
4.3 % 4.00%
Compound interval Annual Quarterly
(every 12 months) (every 3 months)
To show Kelly which is the best investment, calculate the final amount under each scenario over 3 years if she invested with
(a) the BNZ Bank (show formulae/calculations)
(b) the TSB Bank (show formulae/calculations)
(c) Which investment is better at the end of the 3 years, and by what amount?
3.4 You are planning to retire in 30 years’ time. You estimate that to live at your desired standard of living at that time, you need to accumulate $1,500,000 in your retirement fund. To do so, you plan to make equal end-of-month deposits into an account paying 3.5 % annual interest. (a) What will your end of monthly deposits have to be to accumulate the$1,500,000 by the end of 30 years?
(b) At retirement time you have decided that you will be happy that your fund ($1,500,000) provides an income in perpetuity. This assures you of an acceptable income on top of your national pension from the NZ government. i) What is this annual amount in perpetuity if the interest rate is 4%? Assume that this is paid in annual amounts, starting at the end of the first year of your retirement. Ignore income taxes. ii) If your national pension is$450 per week (before tax), what will be your total annual income of pension and perpetuity payment (before taxes)?
3.5 Assuming an annual interest rate of five percent (5%), calculate the present value of the following streams of yearly (annual) payments:
(a) $22500 per year forever, with the first payment one year from today. (b)$25000 per year forever, with the first payment three years from today.
(c) $50000 per year, for ten years, the first payment starting at the end of two years. 3.6 Your company has entered a contract to purchase a warehouse. The purchase price is$7,750,000. This purchase is to be financed by a short term (4-year) loan from your bank. What monthly repayment is required, with an annual interest rate, fixed at 10% over the period of the loan?
3.7 Peach Paving invests $1.0m today on a new construction project. The project will generate annual cash-flows of$150,000 in perpetuity. The appropriate discount rate is 10%.
a) Calculate the payback period.
b) If Peach Paving’s cutoff is 10 years, should the project be accepted?
c) Calculate the discounted payback period.
d) Calculate the NPV of this construction project.
3.8 Victory Inc. has the following additional cash-flows (in thousands) for a small project. Assume the investment would be made today and the additional cash is all received at the end of each year:
Year Cash-flows
0 -5000
1 1800
2 2000
3 1800
a) Calculate the internal rate of return, IRR, on the project.
b) If the appropriate discount rate for this type of project is 10%, should the project be accepted? Prove this by calculating the NPV.
3.9. You must decide whether to purchase new capital equipment. The cost of the equipment is $50,000. This equipment will produce incremental cash flows as below over the expected life of this equipment. At the end of the 8 years you can expect to sell the equipment for$3000. The appropriate discount rate is 10%. Should you proceed with this project
Year Cash-flow
1 7000
2 9000
3 10000
4 10000
5 10000
6 10000
7 12500
8 13750
3.10 The Big Burrito is planning to purchase a touch screen order system for its drive-through window that would allow customers to select their order as soon as they arrive. This would reduce customer wait times and increase order accuracy. The touch screen and software would cost Big Burrito $180,000 and last five years. System maintenance and licensing costs would be$5000 in year one, increasing by 6% year-on-year. The touchscreen and software has no resale value at the end of the five years. In addition to the improved customer service, the Big Burrito would gain two benefits. Firstly, they could reduce their workforce by one person to save $32000 per year. Secondly, they expect drive through sales to increase by$20000 for year one, $25000 for year two,$35000 for year 3, $45000 for year 4 and$60000 for year 5 due to improved customer perception and increased through-put. The cost of goods sold is 35 % of the incremental sales. Ignoring working capital, depreciation and taxes, should Big Burrito make this touchscreen investment? The appropriate discount rate is 16%.
Marks awarded: set up spreadsheet to show incremental sales income (2), annual maintenance costs (2), labour savings (2), cost of goods sold (3), for each year to get an annual net cash-flow and calculate the PV of the net cash-flow calculate the NPV and advise decision on the investment project
3.11 Evaluate the following three projects, using the profitability index (PI) rule. Assume the cost of capital is 15%.
Cash Flows Liquidate Recondition Replace
Initial cash outflow -$100,000 -$500,000 -$1,000,000 Year 1 cash inflow 50,000 100,000 500,000 Year 2 cash inflow 60,000 200,000 500,000 Year 3 cash inflow 75,000 250,000 500,000 a) Rank these projects by their PIs. b) If these projects are independent, which one would you accept according to the PI criterion? c) If these projects are mutually exclusive, which would you accept according to PI criterion? d) Apply the NPV criterion to the projects, rank them according to their NPVs, and indicate which you would accept if they are independent and mutually exclusive. e) Compare and contrast your answer from part (c) with your answer to part (d) for the mutually exclusive case. Explain this result 3.12 Reynard Enterprises is attempting to evaluate the feasibility of investing$85,000 in a machine having a five-year life. The cost of capital is 12%.The firm has estimated the incremental cash inflows associated with the new machine (shown in the table below):
Year Cash Inflow
1 \$18,000
2 22,500
3 27,000
4 31,500
5 36,000
a) Calculate the payback period for the proposed investment
b) Calculate the NPV for the proposed investment
c) Calculate the IRR for the proposed investment
d) Evaluate the acceptability of the proposed investment using NPV and IRR. What recommendation would you make relative to implementation of the project? Why?
## Recent Questions in Financial Accounting
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2020-02-20 08:40:53
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http://mathhelpforum.com/trigonometry/30888-trig-bicycle-help.html
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Math Help - Trig bicycle help
1. Trig bicycle help
ddddddddd
2. Originally Posted by flip510
ddddddddd
And the reason for this post is?
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2016-02-08 02:50:29
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http://mathhelpforum.com/advanced-math-topics/62727-orthogonal-trajectories.html
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# Math Help - Orthogonal Trajectories
1. ## Orthogonal Trajectories
"Find the orthogonal trajectories of the family of curves $x^2+y^2=2cx$.
I know the steps: 1) solve for c...okay... $c=(x^2+y^2)/2x$.
2)Find y', and plug the c back into it.
3) set dy/dx equal to the negative reciprocal of y' from pt. 2, and solve for y. This is the family of orth. traj.
My problem is finding y' for part 2. Well, I can find y', but getting it into a form so that I can solve for y in part 3.
$y=±(2cx-x^2)^(1/2)$
y'= $1/(2sqrt((x^2+y^2)/x-x^2)*((x^2+y^2)/x-2x)$
Where should I go from here? Thanks!
2. Originally Posted by lakesfan210
"Find the orthogonal trajectories of the family of curves $x^2+y^2=2cx$.
I know the steps: 1) solve for c...okay... $c=(x^2+y^2)/2x$.
2)Find y', and plug the c back into it.
3) set dy/dx equal to the negative reciprocal of y' from pt. 2, and solve for y. This is the family of orth. traj.
My problem is finding y' for part 2. Well, I can find y', but getting it into a form so that I can solve for y in part 3.
$y=±(2cx-x^2)^(1/2)$
y'= $1/(2sqrt((x^2+y^2)/x-x^2)*((x^2+y^2)/x-2x)$
Where should I go from here? Thanks!
Have you been taught implicit differentiation:
$2x + 2y \frac{dy}{dx} = 2c \Rightarrow \frac{dy}{dx} = \frac{c - x}{y}$.
|
2014-03-13 15:34:08
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https://bz.costsproject.org/2581-calibration-function.html
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# Calibration function
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## Area of Expertise - Analytical chemistry
The calibration function y = f (x) is obtained from the signals of the measurements (e.g. absorbance) of one or more standards of known concentration. The standards should include the expected concentration range of the samples.
See also: calibration
## Learning units in which the term is dealt with
### Measures of a distribution30 min.
#### ChemistryAnalytical chemistryChemometrics
Relationship between confidence interval, analysis and calibration function with detection, determination and detection limits
## Chronoamperometry
This is a relaxation method with a potential jump and registration of the changing electrolysis current. First, a potential is applied to the working electrode at which the analyte is not yet converted. With a sudden change in the potential to a new value that is constant over time, the oxidation or reduction of the analyte begins and an electrochemical current begins to flow. This current has its maximum value immediately after the potential jump and then drops. The course over time is described by the Cotrell equation.
• I - electrolytic current
• z - number of electrons transferred
• F - Faraday's constant (96,486 As / mol)
• D - diffusion constant (among other things dependent on the viscosity of the solution and the size of the diffusing particles)
• A - electrode surface
• t - time
• c - initial concentration of the reacted substance
The product $I cdot sqrt$ is constant for the investigated substance in a certain period of time during the measurement and is dependent on the initial concentration c, the diffusion constant D. and the number of electrons transferred z (Change in the oxidation state of the substance). Accordingly, the Cotrell equation can be used to calculate the initial concentration or the change in the oxidation state or the diffusion constant. & # 911 & # 93
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2022-08-17 02:21:04
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https://www.gradesaver.com/textbooks/math/algebra/algebra-2-1st-edition/chapter-1-equations-and-inequalities-1-4-rewrite-formulas-and-equations-1-4-exercises-skill-practice-page-30/6
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## Algebra 2 (1st Edition)
We solve the equation for $b_1$: $$2A = (b_1+b_2)h \\ \frac{2A}{h} =b_1+b_2 \\b_1= \frac{2A}{h} -b_2$$ Option A is correct.
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2019-10-15 03:29:06
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https://electronics.stackexchange.com/questions/3703/avr-debugwire-in-linux
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# AVR debugwire in Linux
So, I have hooked up a ATtiny88, and am programming it with the Dragon AVR using ISP. I have also set up the build toolchain, using avrdude, and the gnu avr tools. Everything is working great.
Now I would like to do in-circuit debugging using the advertised debugwire which is also connected to the ISP and which the dragon supports. But which tools do I use? I see there is a avr-gdb, but it seems that I needs some kind of simulator, however I would like to debug in-circuit on the real MCU. Is this possible?
• I'm also interested in how one actually uses debugwire. I keep seeing it in the manuals for my chips, but haven't had occasion to use it yet. – vicatcu Jul 26 '10 at 20:22
Have a look at avarice. It's man page also has something to say about debugwire. I don't know if that'll be good news or bad, though.
• You are right, it really seems that avarice does support both the AVR Dragon and debugWire debugging. There is however the drawback, that to enable debugging with the debugWire, the reset pin fuse has to be change do debugWire mode, which means that ISP is no longer possible, leaving only the option of reflashing the device using high voltage programming. :/ – bjarkef Jul 26 '10 at 22:40
• There is a workaround for this problem: You can reprogram the fuses using debugWire/avarice. So, after your debug session, just reset the DWEN fuse with avarice and you've got ISP back. – markus_b Apr 15 '11 at 17:49
## enable debugwire
enable with avrdude (fuse for attiny88):
avrdude -c dragon_isp -P usb -p attiny88 -v -U hfuse:w:0xd9:m
## compilation
• must be compiled with -ggdb or great (--gdb3) but doesn't seem to help with macros
• no optimisations
COMPILE = $(GCC_PATH) -ggdb3 -Wall -Wextra$(OPTIMIZATION) -std=gnu11 -flto -mmcu=$(DEVICE) -DF_CPU=$(CLOCK)
need main.hex and main.elf
## debugging
start avarice:
avarice -g -w -P attiny88 :4242
then start gdb:
avr-gdb main.elf
and connect:
target remote localhost:4242
## breakpoints
only sw breakpoints with debugwire, so if need breakpoints use:
asm('break');
## switch back to spi/icsp mode
NB. VTG/VCC pin (2 on header) must be connected to chip supply for this to work!
avrdude -c dragon_isp -P usb -p attiny88 -v -U hfuse:w:0xdd:m
• This looks spot-on, but please include relevant commands in your answer before the link goes down. – Dmitry Grigoryev Jun 7 '17 at 11:20
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2019-05-24 07:20:17
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https://design.tutsplus.com/courses/designing-a-sci-fi-galaxy-for-film/lessons/integrating-the-planet-into-the-environment
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FREELessons: 7Length: 40 minutes
• Overview
• Transcript
# 2.3 Integrating the Planet Into the Environment
In this lesson I will show you how to integrate the planet into the environment using layer effects.
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2023-02-06 22:28:23
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https://ampl.com/api/latest/java/class-structure.html
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# Class structure¶
AMPL API library consists of a collection of classes to interact with the underlying AMPL interpreter and to access its inputs and outputs. It uses generic collections to represent the various entities which comprise a mathematical model. The structure of these entities is explained in this section.
The main class used to interact with AMPL, instantiate and interrogate the models is AMPL. One object of this class represents an execution of an AMPL translator, and is the first class that has to be instantiated when developing a solution based on AMPL API. It allows the interaction with the underlying AMPL translator, issuing commands, getting diagnostics and controlling the process.
The model entities are represented by a set of classes, schematized in figure Model entities classes overview. These classes represent the optimisation model being created and allow some manipulation and data assignments operations on such entities and will be presented more in detail in the section Modelling entities classes.
Model entities classes overview
## AMPL class¶
For all calculations, AMPL API uses an underlying AMPL execution engine, which is wrapped by the class AMPL. Thus, one instance of this class is the first object to be created when writing a program which uses the AMPL API library. The object is quite resource-heavy, therefore it should be explicitly closed as soon as it is not needed anymore, with a call to AMPL.close.
All the model creation and structural alteration operations are to be expressed in AMPL language through the AMPL main object; moreover, the class provides access to the current state represented via the classes derived from Entity, as shown in section Modelling entities classes and provides several other functionalities ( Java reference at AMPL classes).
The functions can be split in three groups: direct AMPL interaction, model interrogation and commands.
### Direct interaction with AMPL¶
The methods available to input AMPL commands are AMPL.eval, AMPL.read and AMPL.readData; they send the strings specified (or the specified files) to the AMPL engine for interpretation. Their async versions: AMPL.evalAsync, AMPL.readAsync and AMPL.readDataAsync, permit the calling program to continue the execution while the underlying AMPL process is busy in some time consuming operation, and to define a callback to be executed when the operation is over.
### Model interrogation¶
Evaluating AMPL files or statements creates various kind of entities in the underlying AMPL process. To get the Java (or, in general, programmatic) representation of such entities, the programmer can follow two main courses.
Once the desired entities have been created, it is possible to use their properties and methods to manipulate the model and to extract or assign data. Updating the state of the programmatic entities is implemented lazily and uses proper dependency handling. Communication with the underlying engine is therefore executed only when an entity’s properties are being accessed and only when necessary. An entity is invalidated (needs refreshing) if one of the entities it depends from has been manipulated or if a generic AMPL statement evaluation is performed (through AMPL.eval or similar routines). This is one of the reasons why it is generally better to use the embedded functionalities (e.g. fixing a variable through the corresponding API function call) than using AMPL statements: in the latter case, the API invalidates all entities, as the effects of such generic statements cannot be predicted. Refreshing is transparent to the user, but must be taken into account when implementing functions which access data or modify entities frequently.
### Commands and options¶
Some AMPL commands are encapsulated by functions in the AMPL class for ease of access. These comprise AMPL.solve and others. To access and set options in AMPL, the functions AMPL.setOption and AMPL.getOption are provided. Together with their type-safe alternatives (e.g. AMPL.getBoolOption and AMPL.setBoolOption), these functions provide an easier programmatic access to the AMPL options. In general, when an encapsulation is available for an AMPL command, the programmatic access to it is to be preferred to calling the same command using AMPL.eval.
### Output and errors handling¶
The output from the AMPL translator is handled implementing the interface OutputHandler. The method OutputHandler.output is called at each block of output from the translator. The current output handler can be accessed and set via AMPL.getOutputHandler and AMPL.setOutputHandler; the default output handler prints each block to stdout.
Error handling is two-faced:
The default implementation of the error handler throws exceptions on errors and prints the warnings to stdout.
## Modelling entities classes¶
This group of classes represents the basic entities of an AMPL optimisation model: variables, constraints, objectives, parameters and sets. They are used to access the current state of the AMPL translator (e.g. to find the values of a variable), and to some extent they can be used for data input (e.g. assign values to a parameter, fix a variable).
Objects of these classes cannot be created programmatically by the user: the model creation and structural modification is handled in AMPL (see section AMPL class), through the methods AMPL.eval and AMPL.read. The two base classes are Entity and Instance.
The classes derived from Entity represent algebraic entites (e.g. a variable indexed over a set in AMPL), and are implemented as a map from an object (number, string or tuple) to an Instance which allow access to its instances (method get). The case of scalar entities (like the AMPL entity defined by var x;) is handled at Entity level, and will be illustrated in the paragraph regarding instances below. The derived classes are: Variable, Constraint, Parameter, Objective and Set.
Any object of a class derived from Instance represents a single instance of an algebraic entity (e.g. the value of a variable for a specific value of its indexing set). The derived classes are: VariableInstance, ConstraintInstance, ObjectiveInstance and SetInstance. The composition of these classes can be described as shown below:
Relationship between Entity and Instance
The UML diagram in figure Relationship between Entity and Instance illustrates that each Entity (algebraic entity in AMPL) can contain various Instance (instances in AMPL), while each Instance has to be part of exactly one Entity. The exact methods and properties of the entity depend on the particular kind of entity under consideration (i.e. variable, constraint, parameter).
As an example, the class Variable has functionalities like Variable.fix and Variable.unfix, which would fix or unfix all instances which are part of the algebraic entity, and its corresponding instance class VariableInstance has properties like VariableInstance.value and VariableInstance.dual (together with instance level fix and unfix methods).
The class Constraint has functionalities like Constraint.drop and Constraint.restore, and its instance level class ConstraintInstance properties like ConstraintInstance.body and ConstraintInstance.dual (and methods like drop and restore for the single instance).
Note that the class Parameter, which represent an algebraic parameter, does not have an instance level class; its instances are represented by objects instead (typically double numbers or strings).
The instances can be accessed from the parent Entity through the function Entity.get; all data corresponding to the entity can be accessed through the instances, but the computational overhead of such kind of access is quite considerable. To avoid this, the user can gain bulk data access through a DataFrame object; reference to these object can be obtained using Entity.getValues methods. In case of scalar entities (e.g. the entity declared in AMPL with the statement var x;), all the instance specific methods are replicated at Entity level, to allow the code fragment value = x.value(); instead of the more explicit value = x.get().value(). See example below:
double value;
AMPL ampl = new AMPL();
try{
ampl.eval("var x;");
Variable x = ampl.getVariable("x");
value = x.get().value(); // Access through explicit reference to the instance
} finally {
ampl.close();
}
Indexed entities are central in modelling via AMPL. This is why the Entity.get method has various overloads and can be used in multiple ways, to adapt to specific use cases. These will be presented below, by mean of some Java examples.
Scalar Entities -> Entity.get In general, as seen above, access to an instance of a scalar entity is not needed, as all functionalities of the instance are replicated at entity level in this case. Anyway, to gain explicit access to an instance, the function Entity.get can be used, as shown below.
ampl.eval("var x;");
VariableInstance x = ampl.getVariable("x").get();
Indexed Entities -> Entity.get(Object... key) and Entity.get(Tuple key). To gain access to instances in indexed entities, these functions can be used, depending on the context. For specialised conversion of indices, see the function Tuple.join.
See the examples below (note that the results of the four acesses are identical).
• Each item is a index value: Each item passed to the function is interpreted as the value of one of its indices
• The (only) item is an array containing all the indices
• The (only) item is a Tuple representing all the indices
• Indices values are available in mixed formats (tuples, arrays and single elements)
ampl.eval("var x{1..2, 4..5, 7..8};");
// Each item an index
VariableInstance x = ampl.getVariable("x").get(1,4,7);
// The item is an array
x = ampl.getVariable("x").get(new Object[]{1,4,7});
// The item is a tuple
Tuple t = new Tuple(1,4,7);
x = ampl.getVariable("x").get(t);
// Mixed indices types, use Tuple.join to create a Tuple
t = Tuple.join(new Tuple(1), new Object[]{4}, 7);
x = ampl.getVariable("x").get(t);
The currently defined entities are obtained from the various get methods of the AMPL object (see section AMPL class). Once a reference to an entity is created, the entity is automatically kept up-to-date with the corresponding entity in the AMPL interpeter. That is, if a reference to a newly created AMPL variable is obtained by means of AMPL.getVariable, and the model the variable is part of is then solved by means of AMPL.solve, the values of the instances of the variable will automatically be updated. The following Java code snippet should demonstrate the concept.
AMPL ampl = new AMPL();
try{
ampl.eval("var x;");
ampl.eval("maximize z: x;");
ampl.eval("subject to c: x<=10;");
Variable x = ampl.getVariable("x");
// At this point x.value() evaluates to 0
System.out.println(x.value()); // prints 0
ampl.solve();
// At this point x.value() evaluates to 10
System.out.println(x.value()); // prints 10
} finally {
ampl.close();
}
## Relation between entities and data¶
The entities and instances in AMPL store data (numbers or strings) and can be indexed, hence the instances available depend on the values in the indexing set(s). The order in which these indexing sets is handled in the AMPL entities is not always consistent with the ordering in which the data for such sets is defined, so it is often desirable, even when interested in only data (decoupled from the AMPL entities) to keep track of the indexing values which corresponds to each value.
Moreover, when dealing with AMPL entities (like Variable), consistency is guaranteed for every instance. This means that, if a reference to an instance is kept and in the underlying AMPL interpreter the value of the instance is changed, the value read from the instance object will be always consistent with the AMPL value and, if an instance is deleted in AMPL, an exception will be thrown when accessing it. This has the obvious benefit of allowing the user to rely on the values of the instances, but has a price in terms of computational overhead. For example, accessing in this way the value of 1000 instances:
AMPL ampl = new AMPL();
try {
ampl.eval(
"set A := 1..1000;" +
"param c{i in A} default 0; " +
"var x{i in 1..1000} := c[i];");
// Enumerate through all the instances of c and set their values
Parameter c = ampl.getParameter("c");
for (int i = 1; i <= c.numInstances(); i++)
c.set(i, i * 1.1);
// Enumerate through all the instances and print their values
Variable x = ampl.getVariable("x");
for (int i = 1; i <= x.numInstances(); i++)
System.out.println(x.get(i).value());
} finally {
ampl.close();
}
will check at each access if the referenced instance is valid or not, resulting in a computational overhead. Moreover, in a multi-threaded environment (like when using AMPL.evalAsync), the value of the underlying collection of instances could be be changed by the interpreter while the main program is iterating through them, leading to undetermined results.
To ease data communication and handling, the class DataFrame is provided. Its usage is two-fold:
• It allows definition of data for multiple parameters in one single call to the underlying interpterer
• It decouples data and entities, reducing the computational overhead and risks related to concurrency
DataFrame objects should therefore be used in these circumnstances, together with the methods AMPL.setData and Entity.getValues, as shown in the code below:
// Create a new dataframe with one indexing column (A) and another column (c)
DataFrame df = new DataFrame(1, "A", "c");
for (int i = 1; i <= 1000; i++)
AMPL ampl = new AMPL();
try {
ampl.eval(
"set A;" +
"param c{i in A} default 0;"+
"var x{i in A} := c[i];");
// Assign data to the set A and the parameter c in one line
ampl.setData(df, "A");
// Get the variable x
Variable x = ampl.getVariable("x");
// From the following line onwards, df is uncoupled from the
// modelling system,
df = x.getValues();
} finally {
ampl.close();
}
// Enumerate through all the instances and print their values
for (Object[] row: df)
System.out.format("%f %f%n", row[0], row[1]);
The underlying AMPL interpreter does not need to be open when using the dataframe object, but it maintains all the correspondance between indexing set and actual value of the instances.
Simplified access to scalar values, like the value of a scalar variable or parameter or, in general, any AMPL expression that can be evaluated to a single string or number, is possible using the convenience method AMPL.getValue. This method will fail if called on an AMPL expression which does not evaluate to a single value. See below for an example:
AMPL ampl = new AMPL();
try {
ampl.eval("var x{i in 1..3} := i;");
ampl.eval("param p symbolic := 'test';");
ampl.eval("param pp := 4;");
// x2 will have the value 2
Object x2 = ampl.getValue("x[2]");
// p will have the value "test"
Object p = ampl.getValue("p");
// pp will have the value 4
Object pp = ampl.getValue("pp");
} finally {
ampl.close();
}
## Note on variables suffixes¶
For AMPL versions prior to 20150516, there was a glitch with v.lb, v.ub, v.lslack, v.uslack, and v.slack where v is a variable instantiated without need of presolve and after one or more other variables have been instantiated. Example:
var x <= 0;
var y <= 0;
display y.lb;
display x.ub;
# x.ub was wrong (with separate display commands)
# but all went well with "display y.lb, x.ub;"
|
2018-02-24 18:01:23
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https://vroomlab.wordpress.com/publications-presentations/
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Publications & Presentations
Experimental and Computational Mathematics
Geometric Proofs
“A Constructive Proof of Feuerbach’s Theorem Using a Computer Algebra System” Proceedings of 21th Conference on Applications of Computer Algebra, Kalamata, Greece, July 2015, p.47
Feuerbach’s Theorem states that the midpoints of the three sides, the base points of the three heights, and the midpoints of the line segments between the corners of a triangle and the intersection of the heights are on a circle. This talk offers a constructive proof. It is known that algebraic expression
$x^2+y^2+ex+dy+f=0\quad\quad\quad(1)$
represents a circle centered at $(-d/2, -e/2)$ with radius
$r = \frac{d^2+e^2-4f}{4}\quad\quad\quad(2)$
provide (2) is positive. Three points among nine stated in the theorem are chosen to form a system of linear equations from (1). The values of $d, e$ and $f$ are determined by solving the equations. With the solution, (2) is shown to be positive which implies (1) indeed represents a circle. We then proceed to verify that the coordinates of the remain six points satisfy (1). Hence all nine points are on the same circle.
A Constructive Proof of Euler’s Line Theorem Using a Computer Algebra System”
Mathematical Association of America Indiana Section Conference, Franklin College, Spring 2016
Euler’s Line Theorem states that in every triangle, the intersection of the medians, the intersection of the heights, and the center of the circumscribed circle are on a straight line. This talk offers an algebraic and algorithmic proof with the aid of a Computer Algebra System’s (CAS) symbolic computation capabilities. We will use Omega, a free online CAS Explorer in this presentation.
“An Algebraic Approach to Geometric Proof Using a Computer Algebra System” Proceedings of 19th Conference on Applications of Computer Algebra. Edited by Jose Luis Galan Garcia. Malaga, Spain, July 2013, pp. 197-200
Geometric proof is often considered to be a challenging subject in mathematics. The traditional approach seeks a tightly knitted sequence of statements linked together by strict logic to prove that a theorem is true. Moving from one statement to the next in traditional proofs often demands clever, if not ingenious reasoning. An algebraic approach to geometric proof, however, aims to computationally produce values that imply the thesis statement of the theorem. This presentation will demonstrate the algebraic approach to geometric proof through examples.
Maxima and Minima without Calculus
Solving Kepler’s Wine Barrel Problem Without Calculus” Mathematical Association of America Indiana Section Conference, via Zoom, Fall 2020
Kepler conducted numerical studies on his “wine barrel problem” in order to find the maximum volume held by the cylinder shaped barrels with fixed diagonal. This problem was subsequent solved analytically after the invention calculus. For volume $V=\pi d^2h/4 -\pi h^3/16$ where $d$ stands for the fixed diagonal of the cylinder, what positive value of height $h$ gives the largest volume of $V$? This talk presents an alternative solution to this famous problem by applying a special theorem without using calculus.
Boosting Rocket Performance Without Calculus” 25th Conference on Applications of Computer Algebra, Montréal, Canada, July 16-20, 2019
École de technologie supérieure
Boosting a two-stage rocket’s flight performance is a non-trivial optimization problem typically solved by calculus. This presentation will show an alternative way that only requires high school mathematics, with the help of a computer algebra system (CAS). This non-calculus approach places more emphasis on problem solving through mathematical thinking, as all symboliccalculations are carried out by the CAS. It also makes a range of interesting problems readily tackled with minimum mathematical prerequisites.
Solving Maximization/Minimization Problems by Elementary Means” Mathematical Association of America Indiana Section Conference, Indiana University East, Spring 2013
This presentation illustrates a systematic approach in which maximization/minimization problems can be solved without calculus. First, we will demonstrate how problems can be cast in certain forms so that the extreme values can be obtained immediately. Next, we will apply AM-GM inequality and its corollaries to solve problems that appear to be solvable only by calculus. The non-calculus approach makes a range of very interesting problems available to a wider audience, and at a much earlier stage of their studies in mathematics and other sciences.
Exploratory Analysis and Computation
Solving Parameter Estimation Problem Using Least Square-based Algorithms” Joint work with David Deng, Joint Mathematics Meetings (JMM), Denver, Jan 15-18, 2020
In today’s application of Data Science, a common problem concerns the estimation of unknown parameters in the governing differential equations. This presentation illustrates a machine learning approach to estimate and validate the parameters of two population models, namely, the Malthus model and Verhulst’s Logistic model.
The parameter of growth rate in the Malthus model is estimated using a least square-based algorithm applied to a training dataset with historical population data. When validating the resulting model using more contemporary dataset, we discovered inconsistency in the curve fitting of the original model.
To resolve this inconsistency, we turn to Verhulst’s Logistic model, whose governing differential equation is nonlinear, with two parameters – growth rate and saturation level. After transforming the model equation with derivatives computed through forward, centered and backward finite difference schemes, we obtain the initial values of the estimation from the least square-based algorithm. These initial values are used to extrapolate the final estimation of parameters through a second iteration of least square-based estimation. The resulting model with the estimated parameters is then ready to make predictions of population in the future.
Maximizing the Final Speed of a Two-Stage Rocket Using a Computer Algebra System” Joint work with David Deng, Mathematical Association of America Indiana Section Conference, Wabash College, Fall 2019
A rocket consists of a payload of mass $P$ propelled by two stages of masses $m_1$ (first stage) and $m_2$ (second stage), both with structural factor $1-e$. The exhaust speed of the first stage is $c_1$, and of the second stage $c_2$. The initial total mass, $m_1+m_2$ is fixed. The ratio $b = P/(m_1+ m_2)$ is assumed very small. According to the multi-stage rocket flight equation ([1]), the final speed of a two-stage rocket is
$v_f = -c_1\log(1-em_1/(m_1+m_2+P)) - c_2\log(1-em_2/(m_2+P))\quad\quad\quad(1)$
Let $a = m_2 /( m_1+m_2)$, so that (1) becomes
$v_f = -c_1 \log(1-(e-ea)/(2+b)) - c_2\log(1-ea/(a+b))\quad\quad\quad(2)$
where $0 < a < 1, b > 0, 0 < e < 1, c_1 > 0, c_2 > 0$.
We seek an appropriate value of $a$ to maximize $v_f$.
The above rocket performance optimization problem is solved with the help of a Computer Algebra System (CAS) ([2]). We found that the value of $a$ for a maximum final speed is
$\sqrt{c_2b/c_1} + O(b)$
and the maximum speed is
$-(c_1+c_2)\log(1-e)-2e\sqrt{c_1c2b}/(1-e)+O(b)$.
We compute the first order Taylor series of $a$ and $v_f$ in b to replace their complex expressions respectively. Results produced by the CAS are validated extensively through rigorous mathematical analysis.
A Non-Iterative Method for Solving Nonlinear Equations” Proceedings of 24th Conference on Applications of Computer Algebra, Santiago de Compostela, June 18–22, 2018, p.99
Newton-Raphson method is the most commonly used iterative method for finding the root(s) of a real-valued function or nonlinear systems of equations. However, its convergence is often sensitive to the error in its initial estimation of the root(s). This talk will present a non-iterative method that mitigates non-convergence. An auxiliary initial-value problem of ordinary differential equation(s) is generated by a Computer Algebra System first, then integrated numerically over a closed interval. The solution(s) to the original systems of nonlinear equations is obtained non-iteratively at the end of the interval. A proof of the theorem serving as the base for this new method is presented at the talk. Several examples will illustrate its guaranteed convergence, a clear advantage over the Newton-Raphson method.
Refuting a Conjecture On $x^{n}-1$ Using a Computer Algebra System” Mathematical Association of America Tri-Section Meeting of Illinois, Indiana, and Michigan Sections Conference, Valparaiso University, Spring 2018
There was a conjecture stating that the absolute value of a non-zero coefficient in the factors of $x^n-1$ is always 1. This presentation refutes this conjecture by a counter example using a computer algebra system (CAS). Furthermore, this talk will pose a similar problem to either prove or refute another conjecture regarding the uniqueness of a solution found by CAS.
Mira’s Wing – Mathematical Art Generated by a Computer Algebra System” Art Exhibit,Mathematical Association of America Tri-Section Meeting of Illinois, Indiana, and Michigan Sections Conference, Valparaiso University, Spring 2018
Generating Power Summation Formulas Using a Computer Algebra System” Book of Abstracts, 23rd Conference on Application of Computer Algebra, Jerusalem, Israel, July 2017, p. 35
Mathematical induction is often used in classroom to prove various Power Summation Formulas such as
$\sum\limits_{i=1}^{n}i = \frac{n(n+1)}{2}$
$\sum\limits_{i=1}^{n}i^2 = \frac{n(n+1)(2n+1)}{6}$
$\sum\limits_{i=1}^{n}i^3 = \frac{n^2(n+1)^2}{4}$
However, how the formulas are obtained in the first place is rarely discussed.
In this presentation, we will construct the Power Summation Formulas. Specifically, a recursive algorithm is derived and its implementation in Computer Algebra generates the formulas. A closer look at this algorithm also reveals the generated formulas can also be obtained by solving an initial-value problem of difference equation symbolically.
Computer-Algebra-Aided Chebyshev Methods for Ordinary Differential Equations”
Book of Abstracts, 23rd Conference on Application of Computer Algebra, Jerusalem, Israel, July 2017, p. 45
The solution of ordinary differential equation can be approximated by a linear combination of well-known basis functions. Using the Chebyshev Polynomials as the basis functions, the approximation can be expressed as
$y(x) = \sum\limits_{r=0}^{\infty}a_rT_r(x)\quad\quad\quad(1)$
where the $T_r(x)$‘s are Chebyshev Polynomials of degree $r$, and $a_r$‘s are the coefficients to be determined. In practice, we seek the approximation using a truncated expression of (1), namely,
$y(x) = \sum\limits_{r=0}^{n}a_rT_r(x)\quad\quad\quad(2)$
An online Computer Algebra System (CAS) is used to generate and subsequently solve a system of equations concerning a finite number of $a_r$’s. The use of CAS allows the retention of more $a_r$’s in (2). It also obviates the need for the traditional pad and pencil computations. Examples will be given to illustrate this approach in solving initial value problems, boundary value problems as well as eigenvalue problems for ordinary differential equations whose coefficients and other terms are themselves polynomials.
Prove Inequalities by Solving Maximum/Minimum Problems Using a Computer Algebra System“ Book of Abstracts, 20th Conference on Applications of Computer Algebra, Edited by Robert H. Lewis. New York City, US, July 2014, p. 47
This presentation offers an alternative to traditional approaches to proving non-trivial inequalities, such as applying AM-GM, Cauchy, Ho ̈lder and Minkowski inequalities. This alternative approach, as demonstrated by various examples, establishes the validity of an inequality through solving a maximization/minimization problem by commonly practiced procedures in Calculus. Since the procedures are algorithmic, a Computer Algebra System (CAS) can carry out the computation efficiently.
Omega: A Free Computer Algebra System Explorer for Online Education” Proceedings of 19th Conference on Applications of Computer Algebra, Edited by Jose Luis Galan Garcia. Malaga, Spain, July 2013, pp. 50-54
Online courses have become the medium of choice for students who cannot otherwise attend in a traditional classroom setting. Online distance education allows students to take courses in the convenience of their home or office, at their leisure, free of the distractions of campus life, without commute, while at the same time being provided with almost instantaneous access to the instructor and course materials. However, the lack of access to Computer Algebra System, traditionally exists only in computer labs on campus can be a significant detriment to online classes. In this presentation, we will introduce Omega, a free online Computer Algebra System (CAS) Explorer that provides user with immense power in both symbolic and numeric computing. To use Omega, only a web browser is required. User composes and submits mathematical query using a calculator-like graphical user interface. Upon submission, the query is processed by Omega’s CAS engine, and the result is displayed in text or graphic formats. Omega can be accessed from desktop/laptop computers, ipad/tablets, and smartphones. It is compatible with all major web browsers. Different CAS can be plugged in as Omega’s underlying engine.
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2020-10-31 19:09:59
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http://mobile-news.com.ua/maggie-beer-wbv/9efd66-computer-logic-pdf
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# computer logic pdf
11.12.2020
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endobj /LastChar 255 <]>> File Name: Computer Structure And Logic Pdf.pdf Size: 5569 KB Type: PDF, ePub, eBook Category: Book Uploaded: 2020 Nov 18, 19:39 Rating: 4.6/5 from 843 votes. 864.58 849.54 1162.04 849.54 849.54 687.5 312.5 581.02 312.5 562.5 312.5 312.5 546.87 Our primary focus is on uni-versal computers, which are computers that can perform all possible mechan- << �+vwg���aa�\P��SZ�s�%���^}���~����w��?������{�*G]�pJV+Fσ��alD�z�zq����+�P�D^�J{n��T�W|̰��e:@R�\�����, > z. /F3 25 0 R Ivan Flores Computer Logic (Prentice-Hall Electrical Engineering Series) Prentice-Hall Inc. 1960 Acrobat 7 Pdf 25.9 Mb. /Font 17 0 R Logic plays a fundamental role in computer science. 747.79 666.2 639 768.28 734.02 353.24 503.01 761.22 611.8 897.21 734.02 761.57 666.2 13 0 obj 1.1 Motivation for the Study of Logic In the early years of this century symbolic or formal logic became quite popular with philoso- >> 1.2 Judgments and Proofs Since logic programming computation is proof search, to study logic pro-gramming means to study proofs. 761.57 679.62 652.77 734.02 707.17 761.57 707.17 761.57 707.17 571.17 543.98 543.98 777.78 777.78 611.11 798.47 656.81 526.53 771.39 527.78 718.75 594.87 844.52 544.52 endobj 26 0 obj 1.1 Motivation for the Study of Logic In the early years of this century symbolic or formal logic became quite popular with philoso- endobj << 500 500 500 500 500 500 500 500 500 500 300 300 300 750.01 500 500 750.01 726.86 The syntax of propositional logic is composed of propositional symbols, logical connectives, and parenthesis. intuitionistic logic in an introductory text, the inevitably cost being a rather more summary treatment of some aspects of classical predicate logic. >> 0000133018 00000 n 815.96 748.3 679.62 728.67 811.28 765.79 571.17 652.77 598.03 757.63 622.79 552.77 510.86 249.64 275.77 484.74 249.64 772.08 510.86 458.62 510.86 484.74 354.13 359.36 endstream 812.98 724.82 633.85 772.35 811.28 431.86 541.2 833.04 666.2 947.27 784.08 748.3 endobj /LastChar 255 /Widths[1000 500 500 1000 1000 1000 777.78 1000 1000 611.11 611.11 1000 1000 1000 Algorithm has ceased to be used as a variant form of the older word. endobj /Encoding 7 0 R 380.78 380.78 979.16 979.16 410.88 514 416.31 421.41 508.79 453.82 482.64 468.86 /Name/F5 /Type/Font The output is a boolean function of inputs. Overview of logic design, algorithms, computer organization and assembly language programming and computer engineering technology. 95. An appendix on second-order logic will give the reader an idea of the advantages and limitations of the systems of first-order logic used in Chapters 2-4, and will provide an introduction to an area of much current interest. Theoretical foundations and analysis. 63. Sign in The Apollo Guidance Computer image in Section 1.2.3 was released by NASA and is in the public domain. The traffic light in Section 2.1 is from iStock-Photo, and the rotary traffic signal is from the Wikimedia Commons. endobj Computer Architecture Is Different… • Age of discipline • 60 years (vs. five thousand years) • Rate of change • All three factors (technology, applications, goals) are changing • Quickly • Automated ... • SRAM/logic: optimized for speed (used for processors) In more recent times, this algebra, like many algebras, has proved useful as a design tool. 815.96 815.96 271.99 299.19 489.58 489.58 489.58 489.58 489.58 792.66 435.18 489.58 Here we will look at the basic building blocks used to manipulate this 0-1 information. The relationship between the input and the output is based on a certain logic. 761.57 720.6 543.98 707.17 734.02 734.02 1006.01 734.02 734.02 598.37 271.99 489.58 /FontDescriptor 48 0 R /FirstChar 33 << 543.98 516.78 707.17 516.78 516.78 435.18 489.58 979.16 489.58 489.58 0 611.8 815.96 A logic is a language. x�bb�bbŃ3� ���ţ�1�x4>�� +� 46 0 obj 761.57 720.6 543.98 707.17 734.02 734.02 1006.01 734.02 734.02 598.37 271.99 489.58 The benefits of this technology are many for both the user and the enterprise. EEL 3701C Digital Logic and Computer Systems , 4 Credits . These process signals which represent true or false. >> 555.44 505.03 556.53 425.23 527.77 579.51 613.42 636.57 0 0 0 0 0 0 0 0 0 0 0 0 0 We adopt here the approach by Martin-Lo¨f [3]. ]o�̑\ b*H/1 0k͉�?��\��.��$��S��CV�,�T��M���. 499.29 748.93 748.93 249.64 275.77 458.62 458.62 458.62 458.62 458.62 693.31 406.37 �Y�1���XL� �����=J+���'z�M����E�,â$ ?��m���cc,n�Iq��r2��P��n��a?6�c�t��dX���ֳ��B��@��+0���Ǎ�$SP�N��e�P��8�/��J�+���"1�%|��ՂI��1f� ���8�)�خ�0�|�1V2�ݨVI��N�=$�H�~r����\��5�~OكD˰@��a�(y����0ϱ�������&&�|u� �"�*��a~��S��cm�U�;����?6'\˅���t�?8��#�����. Logic and Computational Thinking is a free online course from Microsoft that will give you and introduction to logic, critical thinking and analytical reasoning. Compound Propositions “PANDQ” has four lines, since the two variables can be set in four different ways: P Q PAND Q T T T T F F F T F F F F According to this table, the proposition “PAND Q” is true only when Pand Qare both true. endobj 0000002187 00000 n << Solution Manual of Digital Logic And Computer Design 2nd Edition Morris Mano /ProcSet[/PDF/Text/ImageC] Logic and Computer Design Fundamentals, Global Edition 19. price $44. It is also called discrete data because the items counted can be identified. Inductive logic is a very difficult and intricate subject, partly because the /FirstChar 33 CS429 Slideset 5: 7 Logic Design xڅXK����W{ �V$J�#l�]É�6�I|����i �GG����������eH���z|��8AFQph�6���>%�Lü��}��Y�C�*x���{�/�w�$OE����� stream << ��9!�S�(;�QRtN=��H�XW������~?�zW��6�C0�_.=28��sL���,%�/���gu� �� "H�{�H�[�?���CwV1f�^�,�ô+�ll�/�r8(���'�����0ަ�t�� 7/��',߸\���f3b�,2�B��.F��}o|N�ܱ��s��#�o�W?�(+� 0000002265 00000 n endstream Reasoning about situations means constructing arguments about them; we … This technology is now available on rugged enterprise class hand held computers and bar code scanners. /FontDescriptor 45 0 R Computer Vision API Account. 0000004266 00000 n 0000000016 00000 n z. A logic gate is a building block of a digital circuit.Most logic gates have two inputs and one output and are based on Boolean algebra. 22 0 obj Fanned systemen worden actief gekoeld door cross-case ventilatie met een ontwerp om de luchtstroming te maximaliseren, terwijl onze fanless computers schade of storing voorkomen die ontstaan is door stof, vuil en trillingen. stream Propositional Logic . << Propositional logic is a good vehicle to introduce basic properties of logic. 500 500 0 613.43 800.01 750.01 676.86 650.01 726.86 700.01 750.01 700.01 750.01 700.01 AND Gate /ProcSet[/PDF/Text/ImageC] Logic is used consistently in the development of computer software and understanding the basics of logic and the construction of arguments is key to writing successful code. 0000003519 00000 n Logic (from Greek: λογική, logikḗ, 'possessed of reason, intellectual, dialectical, argumentative') is the systematic study of valid rules of inference, i.e. A digital device counts discrete data. Although he studied logic as a basis for functional programming 30 0 obj logic, and execute the algorithms by proof search. The great thing about Boolean logic is that, once you get the hang of things, Boolean logic (or at least the parts you need in order to understand the operations of computers) is outrageously simple. 15 0 obj x�S0�30PHW S�\ � 550.01 500 500 450 412.51 400.01 325 525.01 450 650.01 450 475.01 400 500 1000.01 pdf. 5. /Subtype/Type1 /BaseFont/PDTZBD+CMSL12 << 36 0 obj As you know, all information inside a computer is processed and stored as 0-1 bits. ECE REVIEW Electronics | Computer/ Logic Historical computers COMPUTER FUNDAMENTALS Evolution In general, digital computers 271.99 326.39 271.99 489.58 489.58 489.58 489.58 489.58 489.58 489.58 489.58 489.58 Computer Logic Organization Tutorial in PDF - You can download the PDF of this wonderful tutorial by paying a nominal price of$9.99. /FontDescriptor 24 0 R Project. << /Filter[/FlateDecode] /Differences[0/Gamma/Delta/Theta/Lambda/Xi/Pi/Sigma/Upsilon/Phi/Psi/Omega/ff/fi/fl/ffi/ffl/dotlessi/dotlessj/grave/acute/caron/breve/macron/ring/cedilla/germandbls/ae/oe/oslash/AE/OE/Oslash/suppress/exclam/quotedblright/numbersign/sterling/percent/ampersand/quoteright/parenleft/parenright/asterisk/plus/comma/hyphen/period/slash/zero/one/two/three/four/five/six/seven/eight/nine/colon/semicolon/exclamdown/equal/questiondown/question/at/A/B/C/D/E/F/G/H/I/J/K/L/M/N/O/P/Q/R/S/T/U/V/W/X/Y/Z/bracketleft/quotedblleft/bracketright/circumflex/dotaccent/quoteleft/a/b/c/d/e/f/g/h/i/j/k/l/m/n/o/p/q/r/s/t/u/v/w/x/y/z/endash/emdash/hungarumlaut/tilde/dieresis/Gamma/Delta/Theta/Lambda/Xi/Pi/Sigma/Upsilon/Phi/Psi/Omega/ff/fi/fl/ffi/ffl/dotlessi/dotlessj/grave/acute/caron/breve/macron/ring/cedilla/germandbls/ae/oe/oslash/AE/OE/Oslash/suppress/Gamma/Delta/Theta/Lambda/Xi/Pi/Sigma/Upsilon/Phi/Psi/.notdef/.notdef/Omega/ff/fi/fl/ffi/ffl/dotlessi/dotlessj/grave/acute/caron/breve/macron/ring/cedilla/germandbls/ae/oe/oslash/AE/OE/Oslash/suppress/dieresis/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/dieresis] Canonical A truth tableis a table of all possible sets of inputs alongside its output. However, the precise definition is quite broad, and literally hundreds of logics have been studied by philosophers, computer scientists and mathematicians. stream For example, Chapter 13 shows how propositional logic can be used in computer circuit design. To simplify the logic circuit using Boolean algebra techniques, construct the simplified circuit and verify the truth table for the simplified expression. endobj REGISTER TRANSFER LANGUAGE AND MICROOPERATIONS: Computer Organization pdf Notes. Understanding Logic and Computer Design for All Audiences. Computer Logic services local people and businesses who have computing problems. 777.78 275 1000 666.67 666.67 888.89 888.89 0 0 555.56 555.56 666.67 500 722.22 722.22 Boolean Expression 3. Some of the key areas of logic that are particularly significant are computability theory (formerly called recursion theory), modal logic and category theory.The theory of computation is based on concepts defined by logicians and mathematicians such as Alonzo Church and Alan Turing. Computer Science Dept Va Tech October 2003 ©2003 McQuain WD & Keller BJ Logic Gates 4 OO Software Design and Construction 2-input Logic Gate Hierarchy It is sensible to view each of the 2-input logic gates as a specialized sub-type of a generic logic gate (a base type) which has 2 input wires and transmits its output to a single output wire. Computer Logic and Symbolic Reasoning ~ Wainaina MACHINE LEARNING Journal of Symbolic Computation, 8(5), 101--140 (1989). 271.99 489.58 271.99 271.99 489.58 543.98 435.18 543.98 435.18 299.19 489.58 543.98 mathematical procedure, the computer’s stock in trade. 0 Any ‘formal system’ can be considered a logic … Course Objectives /FontDescriptor 39 0 R Digital (discrete) data Is data obtained by counting. Digital logic courses or programs allow students to gain hands-on experience by building computer hardware through the use of algorithms and simple inputs. Once this is done, the connectors will be available to integrate the Computer Vision API in Logic Apps. For example, consider the following: 43 0 obj /Length 758 777.78 777.78 777.78 777.78 777.78 1000 1000 777.78 777.78 1000 0 0 0 0 0 0 0 0 0 249.64 301.89 249.64 458.62 458.62 458.62 458.62 458.62 458.62 458.62 458.62 458.62 Inductive logic investigates the process of drawing probable (likely, plausi-ble) though fallible conclusions from premises. >> How many of these do you really need? >> 458.62 249.64 458.62 249.64 249.64 458.62 510.86 406.37 510.86 406.37 275.77 458.62 First, we treat propositional symbols merely as a set of some symbols, for our purposes we'll use letters of the Roman and Greek alphabets, and refer to the set of all symbols as Prop {\displaystyle {\text{Prop}}} : 1. endobj 271.99 299.19 516.78 271.99 815.96 543.98 489.58 543.98 516.78 380.78 386.22 380.78 0000002881 00000 n 489.58 489.58 271.99 271.99 271.99 761.57 462.38 462.38 761.57 734.02 693.4 707.17 /Type/Font Scanned by artmisa using Canon DR2580C + flatbed option Addeddate 2013-05-30 01:57:54 Identifier ComputerLogic Identifier … >> H���ko�0���+��m�a.��J ��L��n���� e�!ζ��J2�MBF��}���T�,7pvF��d����[��Ѕ6F�@�6[�}U�����a�kY��l��)��п�6� �0� #�;?_,S���:e�F;�K2��v�}z ��x& �|�5\����k��q� N���Z[��γ��?��l�ܺ�ɠw�k��jK��kܡ�ݸ�PC,U�!��z>#���ʡ� /Type/Font /BaseFont/ISNYNK+CMR17 Sign in. /Type/Font 761.57 679.62 652.77 734.02 707.17 761.57 707.17 761.57 707.17 571.17 543.98 543.98 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 937.5 312.5 343.75 562.5 562.5 562.5 562.5 562.5 849.54 500 574.07 812.5 875 562.5 I was amazed when I looked through it for the first time. >> (ps) (pdf) Snyder, W. and Gallier, J. Computers exist in a wide range of forms, and thousands of computers are hid-den in devices we use everyday but don’t think of as computers such as cars, phones, TVs, microwave ovens, and access cards. Rules govern how these elements can be written together. LOGIC OPERATIONS AND TRUTH TABLES Digital logic circuits handle data encoded in binary form, i.e. >> 489.58 489.58 271.99 271.99 761.57 489.58 761.57 489.58 516.89 734.02 743.86 700.54 902.77 552.77 902.77 844.44 319.44 436.11 436.11 552.77 844.44 319.44 377.77 319.44 /BaseFont/FKVIQP+CMCSC10 /Type/Font /Filter[/FlateDecode] Computing with Logic Gates How are these logic functions actually computed in hardware? 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 562.5] 631.13 775.5 745.29 602.19 573.89 665.01 570.83 924.41 812.64 568.11 670.19 380.78 319.44 319.44 613.33 580 591.11 624.44 557.78 535.55 641.11 613.33 302.22 424.44 {�BB���E�8��������s�mRfjWx��P3Nϭr9V94p�+�;Z�~Ö����;�A8���fOv��A�,(~��ÝS-�1�����4gQ/B�����Kը��q��\�����zM����yH��ԭ�"��m, x��f���}��C�U�Eq�+��j�c�9 /FontDescriptor 9 0 R /Type/Encoding �8�6��H�J]��$P���؆�f�gQй�i�� 诟�Hy>��1o4��oG�uuc�'�j���b���R��e�K���_o�N�ܚ���Ak5� /Subtype/Type1 endstream endobj 694 0 obj <>/Size 681/Type/XRef>>stream An introduction to applying predicate logic to testing and verification of software and digital circuits that focuses on applications rather than theory. x��V]o�0}߯��H��o�1K�}hڪ��i�8�%�# S��gc��(�:i��nO��sϽ�DY�e�m�_�Eo�9� L%��e�E ��z��8��:N0�r�ͪ�t�o���4[�^��/.��+L�;m���?�[��@a�-�x1G��B�R%����Shó,�1�5&N�u�)�� /Name/F1 Recommended: Prior programming experience . Course Pre-Requisites / Co-Requisites . 772.08 719.84 641.07 615.35 693.31 667.59 719.84 667.59 719.84 667.59 525.41 499.29 552.77 552.77 319.44 319.44 523.61 302.22 424.44 552.77 552.77 552.77 552.77 552.77 /F2 13 0 R >> Certainly classical predicate logic is the basic tool of 628.21 719.84 680.45 510.86 667.59 693.31 693.31 954.53 693.31 693.31 563.11 249.64 << /LastChar 255 endobj /Widths[271.99 489.58 815.96 489.58 815.96 761.57 271.99 380.78 380.78 489.58 761.57 /Subtype/Type1 Logic design, Basic organization of the circuitry of a digital computer.All digital computers are based on a two-valued logic system—1/0, on/off, yes/no (see binary code).Computers perform calculations using components called logic gates, which are made up of integrated circuits that receive an input signal, process it, and change it into an output signal. 6 0 obj x��Oo�0����43�7ޱ[;uZ�ha�l;0⦖T�L�����S�m��l�ɋ����M���e� ������K����b��� 2Q� �*�����W��=�q��{EP��v�q�گm��q�)ZcR�A^-��&o��*oM���ʬѥY�*o�^��TP�}�;34�a1B�ԭ�s 9.1 Logic gates A large number of electronic circuits (in computers, control units, and so on) are made up of logic gates. M. Huth and M. Ryan, “Logic in Computer Science – Modeling and Reasoning about systems”, Second Edition, Cambridge University Press, 2004-Ref8.pdf - Google Drive The gate responds continuously to changes in input with a small delay.$89.99 A Logical Approach to Discrete Math (Texts and Monographs in Computer Science) 23. /Type/Font 695 0 obj <>stream /Widths[609.72 458.21 577.08 808.91 505.03 354.16 641.43 979.16 979.16 979.16 979.16 /Encoding 7 0 R /BaseFont/TUDDSB+CMR12 PDF | On Jan 1, 1990, Steve Reeves and others published Logic for computer science | Find, read and cite all the research you need on ResearchGate /LastChar 255 418.98 581.02 880.79 675.93 1067.13 879.63 844.91 768.52 844.91 839.12 625 782.41 851.38 813.88 405.55 566.66 843.05 683.33 988.88 813.88 844.44 741.66 844.44 799.99 458.62 667.59 719.84 458.62 837.18 941.67 719.84 249.64 0 0 0 0 0 0 0 0 0 0 0 0 0 ECE REVIEW Electronics | Computer/ Logic Historical computers COMPUTER FUNDAMENTALS Evolution In … 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 271.99] View Computer and Logic Gates.pdf from EEE 21 at Far Eastern University. Gates are digital (t wo state) circuits … /LastChar 127 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 489.58] /FontDescriptor 12 0 R 681 0 obj <> endobj Instruction codes. 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 500] 37 0 obj /Length 648 /FirstChar 33 489.58 489.58 271.99 271.99 271.99 761.57 462.38 462.38 761.57 734.02 693.4 707.17 Computer Logic is an IT Services and Computer Repair store in Casper, Wyoming. 49 0 obj >> Logic and Computer Design Fundamentals, Global 5th Edition, (PDF) is a comprehensive up-to-date textbook that makes logic design, computer design, and digital system design available to students of all levels. z. Syntax: the rules about how to form formulas; this is usually the easy part of a logic. /F3 25 0 R /BaseFont/LPAMUM+CMSY10 /Widths[300 500 800.01 755.21 800.01 750.01 300 400 400 500 750.01 300 350 300 500 /Filter[/FlateDecode] >> The readings taken by digital devices are very accurate. 0000002150 00000 n • “Computer Architecture is the science and art of selecting and interconnecting hardware components to create computers that meet functional, performance and cost goals.” - WWW Computer Architecture Page • An analogy … /ProcSet[/PDF/Text/ImageC] Logic In this chapter, we introduce propositional logic, an algebra whose original purpose, dating back to Aristotle, was to model reasoning. 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 489.58] OTHER LOGIC PAPERS OF INTEREST Snyder, W. and Gallier, J. Higher-Order Unification Revisited: Complete Sets of Transformations. /FirstChar 0 0000000609 00000 n Propositional symbols: A set Prop {\displaystyle {\text{Prop}}} of some symbols. M. Huth and M. Ryan, “Logic in Computer Science – Modeling and Reasoning about systems”, Second Edition, Cambridge University Press, 2004-Ref8.pdf - Google Drive False represents 0, and true represents 1. 271.99 326.39 271.99 489.58 489.58 489.58 489.58 489.58 489.58 489.58 489.58 489.58 /Widths[271.99 489.58 815.96 489.58 815.96 761.57 271.99 380.78 380.78 489.58 761.57 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 635.55 513.33 746.66 613.33 635.55 557.78 635.55 602.22 457.78 591.11 613.33 613.33 It is an electronic circuit having one or more than one input and only one output. 0 675.93 937.5 875 787.04 750 879.63 812.5 875 812.5 875 812.5 656.25 625 625 937.5 $233.32 Digital Fundamentals (10th Edition) 79. price$ 74. /Type/Font ��%#�Xb^4;H&\ ����������{W�gys��fqGpZap �,�O�@���A� ?�� /FirstChar 33 trailer 20 0 obj Computer Vision API Account. /Name/F4 In order to use the Computer Vision API connectors in the Logic Apps, first an API account for the Computer Vision API needs to be created. For example p , q , r , … {\displaystyl… /ProcSet[/PDF/Text/ImageC] (ps) (pdf) Snyder, W. and Gallier, J. /FirstChar 33 The most common symbols used to represent logic gates are shown below. 706.58 628.21 602.09 726.27 693.31 327.61 471.48 719.44 575.97 850.05 693.31 719.84 688.43 700.01 738.43 663.43 638.43 756.72 726.86 376.86 513.43 751.86 613.43 876.86 In addition to propositional and predicate logic, it has a particularly thorough treatment of temporal logic and model checking. /Font 27 0 R << /BaseFont/FTLVRA+CMMI12 Sign in. It does not provide means to determine the validity (truth or false) of atomic statements. /FontDescriptor 42 0 R The Register Transfer language. 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 458.62] 611.1 786.1 813.88 813.88 1105.55 813.88 813.88 669.44 319.44 552.77 319.44 552.77 At this level, the major components are functional units or subsystems that correspond to specific pieces of hardware built from the lower level building blocks. What is Computer Architecture? 0000001684 00000 n Once this is done, the connectors will be available to integrate the Computer Vision API in Logic Apps. /Encoding 7 0 R 342.59 875 531.25 531.25 875 849.54 799.77 812.5 862.27 738.43 707.18 884.26 879.63 600.01 550.01 575.01 862.51 875.01 300 325 500 500 500 500 500 814.82 450 525.01 700.01 700.01 500 863.43 963.44 750.01 250 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 stream Computer Logic WHAT IS DIGITAL? xref << << endobj ��ppW7n"� �&�����O�;�K�4E� �!�?uDfByN�[����e��w���#�ДMh��Ҏ(_u������j;��S6U�*%���97��B��>���n��$�u)��+�P�=�� Webster’s II New Riverside University Dictionary 1984. %PDF-1.4 %���� ����^Ձt %@�k�.��Jxґ|:K��:�uk�Zh�����T�~P�0�l髖'E����;:�Ng�3�$)A���{X@"d�@E��R�Wtt00���:OP���%:������� � ĒUA���؝ $�z����9Cl���> /Subtype/Type1 552.77 552.77 552.77 552.77 552.77 552.77 552.77 552.77 552.77 552.77 552.77 319.44 z. 10 0 obj /Encoding 37 0 R << /Type/Encoding 563.65 334.03 405.09 509.25 291.66 856.47 584.48 470.71 491.43 434.14 441.26 461.22 Logic and Computer Design Fundamentals, Global 5 th Edition, (PDF) is a comprehensive up-to-date textbook that makes logic design, computer design, and digital system design available to students of all levels. 835.55 613.33 613.33 502.22 552.77 1105.55 552.77 552.77 552.77] << … the relations that lead to the acceptance of one proposition (the conclusion) on the basis of a set of other propositions ().More broadly, logic is the analysis and appraisal of arguments. 813.88 494.44 915.55 735.55 824.44 635.55 974.99 1091.66 844.44 319.44 319.44 552.77 Another way of stating this: induc-tive logic investigates arguments in which the truth of the premises makes likely the truth of the conclusion. endobj Computer Logical Organization refers to the level of abstraction above the digital logic level, but below the operating system level. << 600.01 300 500 300 500 300 300 500 450 450 500 450 300 450 500 300 300 450 250 800.01 >> >> Binary logicdealing with “true” and “false” comes in handy to describe the behaviour of these circuits: 0is usually associated with “ false ” and 1with “ true.” endobj >> x�-�;�0�w~��D"n�)y��F6� %T�TU�� ��Y>�RB�ac���h��� %PDF-1.2 The Logic of Computer Programming Our exposition is divided between a basic text, given in an ordinary type font and secondary notes interspersed throughout the text in a smaller font. 25 0 obj The pic-ture of Grace Hopper in Chapter 3 is from the Computer … 19 0 obj stream 625 500 625 513.31 343.75 562.5 625 312.5 343.75 593.75 312.5 937.5 625 562.5 625 To obtain Boolean expression for the output based on the given logic diagram 2. Onze Industriële computers zijn gebouwd voor gebruik in de meest uitdagende omgevingen en helpen uitval te verminderen en defecten te voorkomen. OTHER LOGIC PAPERS OF INTEREST Snyder, W. and Gallier, J. Higher-Order Unification Revisited: Complete Sets of Transformations. /LastChar 255 x�bb�b`�JB �� 543.98 516.78 707.17 516.78 516.78 435.18 489.58 979.16 489.58 489.58 0 611.8 815.96 signals that have only two values, 0and 1. << 50 0 obj Today, few computer pro-grams can be mechanically certified to be free of “bugs.” The 833.34 750 833.34 416.67 666.67 666.67 777.78 777.78 444.45 444.45 444.45 611.11 /Filter[/FlateDecode] See product details. 0000001341 00000 n 31 0 obj >> endstream endobj 682 0 obj <>/Outlines 69 0 R/Metadata 117 0 R/PieceInfo<>>>/Pages 114 0 R/PageLayout/SinglePage/OCProperties<>/OCGs[683 0 R]>>/StructTreeRoot 119 0 R/Type/Catalog/LastModified(D:20080430180613)/PageLabels 112 0 R>> endobj 683 0 obj <>/PageElement<>>>/Name(Background)/Type/OCG>> endobj 684 0 obj <>/ColorSpace<>/Font<>/ProcSet[/PDF/Text/ImageC]/Properties<>/ExtGState<>>>/Type/Page>> endobj 685 0 obj [/ICCBased 693 0 R] endobj 686 0 obj <> endobj 687 0 obj <>stream endstream A third 0,32 Mb Memor 1 Data Sheet ~ Polish. Logic design, Basic organization of the circuitry of a digital computer.All digital computers are based on a two-valued logic system—1/0, on/off, yes/no (see binary code).Computers perform calculations using components called logic gates, which are made up of integrated circuits that receive an input signal, process it, and change it into an output signal. /F2 13 0 R Digital Logic And Computer Design By M. Morris Mano (2nd Edition).pdf - Google Drive. Computer Registers Computer instructions – Instruction cycle. In fact, the book is quite remarkable endobj /Subtype/Type1 444.45 444.45 444.45 444.45 500 500 388.89 388.89 277.78 500 500 611.11 500 277.78 Laboratory. 677.78 761.95 689.72 1200.9 820.49 796.11 695.56 816.67 847.5 605.56 544.64 625.83 Amazed when i looked through it for the first time certainly classical predicate logic here we will at... Apollo Guidance Computer image in Section 2.1 is from the Wikimedia Commons of logic ; this specification can be in! 101 -- 140 ( 1989 ), like many algebras, has proved useful as a basis functional. Brings this broadly recognized source to modern standards by making sure that all information contemporary! And model checking recent times, this algebra, like many algebras, proved. Amazed when i looked through it for the output based on this, logic micro,. Looked through it for the first time Arithmetic logic shift unit digital logic level, below. 5Th Edition brings this broadly recognized source to modern standards by making sure that all information is and... Particularly thorough treatment of some symbols in different ways: 1 it for the based! And sound are all stored as 1s and 0s in your Computer in an text... Its atomic components from premises cost being a rather more summary treatment of temporal logic model... Propositional logic is a logic courses or programs allow students to gain hands-on experience by building Computer hardware through use! Is contemporary and relevant tableis a table of all possible Sets of Transformations of logic... Pdf Notes the older word looked through it for the output based on this, logic gates are shown.... Logic as a basis for functional programming logic gates are the basic building used! This, logic gates are named as and gate, NOT gate etc Edition brings this broadly source. Inductive logic is a very difficult and intricate subject, partly because items... Does NOT provide means to study logic pro-gramming means to study Proofs source modern! The inevitably cost being a rather more summary treatment of some aspects of classical logic! More summary treatment of some aspects of classical predicate logic is a very and. I was amazed when i looked through it for the output is based on this computer logic pdf logic gates are as... And Monographs in Computer Science ) 23 is an exceptional book introductory text, graphics, and sound are stored. Logic design, algorithms, Computer Organization and assembly language programming and Computer Systems, 4 Credits Organization pdf.... 2Nd Edition ).pdf - Google Drive responds continuously to changes in input with a delay! Approach to discrete Math ( Texts and Monographs in Computer circuit design Systems, 4 Credits programming Computation is search. Be used as a variant form of the older word this: logic... Is an electronic circuit having one or more than one input and only output! Objectives Computer logical Organization refers to the level of abstraction above the logic. Arithmetic Mircro operatiaons, logic gates are the basic building blocks of digital. Language and MICROOPERATIONS: Computer Organization and assembly language programming and Computer design M.... The easy part of a logic when i looked through it for the based. Contemporary and relevant Computer Architecture Organization refers to circuitry that transforms bits, opposed. Bar code scanners: the rules about how to form formulas ; this is done, the ’... To determine the validity ( truth or false ) of atomic statements 5th Edition computer logic pdf this broadly source. Readings taken by digital devices are very accurate was released by NASA and is in the domain! At Far Eastern University does NOT provide means to study Proofs in the public domain Symbolic Computation 8... -- 140 ( 1989 ) about how to form formulas ; this is done the. Every terminal is in the public domain digital system treatment of some symbols logic and checking... \Displaystyle { \text { Prop } } of some symbols circuitry that transforms bits, as opposed to bits. Allows you to evaluate the validity of compound statements given the validity compound. Stock in trade Computer circuit design of a logic Computer Engineering technology as and gate, NOT gate.... Who have computing problems use of algorithms and simple inputs process of drawing probable ( likely, plausi-ble though... Validity ( truth or false ) of atomic statements are many for the. The precise definition is quite broad, and the output based on the given logic diagram 2 tableis table!, like many algebras, has proved useful as a design tool but below the operating system.. A variant form of the two binary conditions false ( high ) true. The precise definition is quite broad, and literally hundreds of logics have been studied by,!, graphics, and the rotary traffic signal is from iStock-Photo, and the rotary traffic signal is iStock-Photo! Using Boolean algebra techniques, construct the simplified expression, like many algebras, has proved useful as basis. Construct the simplified circuit and verify the truth table for the first time the process of drawing (. On this, logic gates are shown below different ways: 1 Arithmetic shift! Broadly recognized source to modern standards by making sure that all information is contemporary relevant! Vision API in logic Apps common symbols used to manipulate this 0-1 information shows how propositional is. He studied logic as a variant form of the older word an introductory text, the connectors will available. The public domain Prentice-Hall Inc. 1960 Acrobat 7 pdf 25.9 Mb this: induc-tive logic the! H/1 0k͉�? ��\��.��$ ��S��CV�, �T��M��� of atomic statements canonical a truth tableis table! Held computers and bar code scanners ps ) ( pdf ) Snyder, W. Gallier! Likely the truth table for the output is based on this, logic gates are named and! 79. price \$ 74 signal is from the Wikimedia Commons can be represented in different ways:.. Once this is done, the connectors will be available to integrate the ’! ‘ logic ’, they mean either propositional logic is a very difficult and intricate subject partly! Pdf ) Snyder, W. and Gallier, J. Higher-Order Unification Revisited: Complete of... Study Proofs some symbols logic services local people and businesses who have computing problems the formulas, mainly terms! Logic courses or programs allow students to gain hands-on experience by building hardware. Certainly classical predicate logic is composed of propositional logic can be identified bits as... By Huth and Ryan is an electronic circuit having one or more than one input and only output. Memory TRANSFER, Bus and memory TRANSFER, Bus and memory TRANSFER, Bus and memory TRANSFER, Bus memory... { \displaystyle { \text { Prop } } } of some aspects of classical predicate logic, it you! Electronic circuit having one or more than a language, it has inference.! Elements can be represented in different ways: 1 circuit using Boolean techniques... } computer logic pdf } of some symbols of mathematical procedure, the connectors will be to! The level of abstraction above the digital logic level, but below the operating system.! This 0-1 information we adopt here the Approach by Martin-Lo¨f [ 3.. Possible Sets of Transformations Higher-Order Unification Revisited: Complete Sets of inputs alongside its output, Bus and memory,. The gate responds continuously to changes in input with a small delay Computer Organization and assembly language and.
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