text:
## Strength and weakness in computable structure theory.(English)Zbl 1485.03111

Day, Adam (ed.) et al., Computability and complexity. Essays dedicated to Rodney G. Downey on the occasion of his 60th birthday. Cham: Springer. Lect. Notes Comput. Sci. 10010, 302-323 (2017).
Summary: We survey the current results about degrees of categoricity and the degrees that are low for isomorphism as well as the proof techniques used in the constructions of elements of each of these classes. We conclude with an analysis of these classes, what we may deduce about them given the sorts of proof techniques used in each case, and a discussion of future lines of inquiry.
For the entire collection see [Zbl 1352.03004].

### MSC:

 03C57 Computable structure theory, computable model theory 03D45 Theory of numerations, effectively presented structures
Full Text:

### References:

 [1] Anderson, B.A., Csima, B.F.: Degrees that are not degrees of categoricity. Notre Dame J. Formal Logic (to appear) · Zbl 1436.03229 [2] Ash, C.J., Knight, J.: Computable structures and the hyperarithmetical hierarchy. In: Studies in Logic and the Foundations of Mathematics, vol. 144. North-Holland (2000) · Zbl 0960.03001 [3] Csima, B.: Degrees of categoricity and related notions. In: BIRS Computable Model Theory Workshop, November 2013 [4] Csima, F.B., Franklin, J.N.Y., Shore, R.A.: Degrees of categoricity and the hyperarithmetic hierarchy. Notre Dame J. Formal Logic 54(2), 215–231 (2013) · Zbl 1311.03070 [5] Csima, B.F., Harrison-Trainor, M.: Degrees of categoricity on a cone. J. Symbolic Logic (to appear) · Zbl 1386.03049 [6] Downey, R., Griffiths, E., LaForte, G.: On Schnorr and computable randomness, martingales, and machines. Math. Log. Q. 50(6), 613–627 (2004) · Zbl 1062.68064 [7] Downey, R.G., Hirschfeldt, D.R.: Algorithmic Randomness and Complexity. Springer, New York (2010) · Zbl 1221.68005 [8] Ershov, Y.: A hierarchy of sets, I. Algebra Logic 7(1), 212–232 (1968). Originally appearing in Algebra i Logika 7(1), 47–74 (1968) (Russian) [9] Ershov, Y.: On a hierarchy of sets, II. Algebra Logic 7(4), 25–43 (1968). Originally appearing in Algebra i Logika 7(4), 15–47 (1968) (Russian) [10] Ershov, Y.: On a hierarchy of sets, III. Algebra Logic 9(1), 20–31 (1970). Originally appearing in Algebra i Logika 9(1), 34–51 (1970) (Russian) · Zbl 0233.02017 [11] Fokina, E.B., Kalimullin, I., Miller, R.: Degrees of categoricity of computable structures. Arch. Math. Logic 49(1), 51–67 (2010) · Zbl 1184.03026 [12] Franklin, J.N.Y.: Schnorr trivial reals: a construction. Arch. Math. Logic 46(7–8), 665–678 (2008) · Zbl 1142.03020 [13] Franklin, J.N.Y.: Lowness and highness properties for randomness notions. In: Arai, T. et al. (ed.) Proceedings of the 10th Asian Logic Conference, pp. 124–151. World Scientific (2010) · Zbl 1193.03072 [14] Franklin, J.N.Y., Solomon, R.: Degrees that are low for isomorphism. Computability 3(2), 73–89 (2014) · Zbl 1322.03028 [15] Franklin, J.N.Y., Stephan, F.: Schnorr trivial sets and truth-table reducibility. J. Symbol. Logic 75(2), 501–521 (2010) · Zbl 1193.03073 [16] Franklin, J.N.Y., Turetsky, D.: Genericity and lowness for isomorphism (submitted) · Zbl 1435.03068 [17] Fröhlich, A., Shepherdson, J.C.: Effective procedures in field theory. Philos. Trans. R. Soc. Lond. Ser. A. 248, 407–432 (1956) · Zbl 0070.03502 [18] Harizanov, V.S.,: Pure computable model theory. In: Handbook of Recursive Mathematics, Volume 1: Recursive Model Theory, Studies in Logic and the Foundations of Mathematics, vol. 138, pp. 3–114 (1998) · Zbl 0952.03037 [19] Hirschfeldt, D., Khoussainov, B., Shore, R., Slinko, A.: Degree spectra and computable dimensions in algebraic structures. Ann. Pure Appl. Logic 115(1–3), 71–113 (2002) · Zbl 1016.03034 [20] Hirschfeldt, D.R., Shore, R.A.: Combinatorial principles weaker than Ramsey’s theorem for pairs. J. Symbol. Logic 72(1), 171–206 (2007) · Zbl 1118.03055 [21] Hirschfeldt, D.R., Shore, R.A., Slaman, T.A.: The atomic model theorem and type omitting. Trans. Am. Math. Soc. 361(11), 5805–5837 (2009) · Zbl 1184.03005 [22] Hirschfeldt, D.R., White, W.M.: Realizing levels of the hyperarithmetic hierarchy as degree spectra of relations on computable structures. Notre Dame J. Formal Logic 43(1), 51–64 (2003). 2002 · Zbl 1048.03035 [23] Knight, J.F.: Degrees coded in jumps of orderings. J. Symbol. Logic 51(4), 1034–1042 (1986) · Zbl 0633.03038 [24] Marker, D.: Non $\Sigma _n$ axiomatizable almost strongly minimal theories. J. Symbol. Logic 54(3), 921–927 (1989) · Zbl 0698.03021 [25] Montalbán, A.: Priority arguments via true stages. J. Symbol. Log. 79(4), 1315–1335 (2014) · Zbl 1353.03050 [26] Moschovakis, Y.N.: Descriptive Set Theory. Studies in Logic and the Foundations of Mathematics, vol. 100. North-Holland (1980) · Zbl 0433.03025 [27] Nies, A.: Computability and Randomness. Clarendon Press, Oxford (2009) · Zbl 1237.03027 [28] Nies, A., Stephan, F., Terwijn, S.A.: Randomness, relativization and Turing degrees. J. Symbol. Logic 70(2), 515–535 (2005) · Zbl 1090.03013 [29] Odifreddi, P.: Classical Recursion Theory. Studies in Logic and the Foundations of Mathematics, vol. 125. North-Holland (1989) · Zbl 0661.03029 [30] Odifreddi, P.: Classical Recursion Theory, Volume II. Studies in Logic and the Foundations of Mathematics, vol. 143. North-Holland (1999) · Zbl 0931.03057 [31] Sacks, G.E.: Higher Recursion Theory. Springer, New York (1990) · Zbl 0716.03043 [32] Slaman, T.A., Solovay, R.M.: When oracles do not help. In: Fourth Annual Workshop on Computational Learning Theory, pp. 379–383. Morgan Kaufman, Los Altos (1991) [33] Soare, R.: The Friedberg-Muchnik theorem re-examined. Can. J. Math. 24, 1070–1078 (1972) · Zbl 0225.02028 [34] Soare, R.I.: Recursively Enumerable Sets and Degrees. Perspectives in Mathematical Logic. Springer, New York (1987) [35] Suggs, J.: Degrees that are low for $\mathcal{C}$ isomorphism. Ph.D. thesis, University of Connecticut (2015)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.


text:
Hong Kong
Stage 1 - Stage 3

# Best Buys and Discounts II

Lesson

Everyone loves a great deal or a sale when they're shopping! However, most products today come in different varieties and are sold at more than one store. For example, say you were going to buy a new tablet device. Store A is charging $\$350$$350, with a 20%20% discount and Store B is charging \335$$335 with a $15%$15% discount. How do you work out which store will give you the best price?

## Discounts

Let's start by talking about discounts. Everyone will have seen shops advertising, "$25%$25% off," "pay less when you pay cash" and so on. These are all examples of shops offering discounts. A discount is basically a reduction in price, whether it be on goods or services. Businesses often use discount sales as a way of enticing people so it's important to be able to calculate discounts to make sure you're getting a great deal.

### Definitions

Regular price: The price before any discount is given (i.e. The non-sale price).

Sale price: The price after a discount is given.

Percentage discount: The amount of discount received expressed as a percentage (usually of the regular price but this may change depending on the question). To work out the percentage discount, we use the formula:

$\text{Percentage discount}=\frac{\text{discount }}{\text{regular price}}\times100%$Percentage discount=discount regular price×100%

### A common mistake

When working with percentages, students often subtract the percentage from the regular price. For example, if a t-shirt cost $\$30$$30 and had 20%20% off, some students may write: 30-20%3020% But this doesn't specify what they're finding 20%20% of and as such, can cause problems doing calculations. Remember we need to find 20%20% of the original price, which is maths speak for "use multiplication." ### Calculating discounts to find sales prices There are two ways of calculating discounts to find the sales price of item. We will look at both way using the tablet example above. You can use either method and they should both give you the same answer. • ##### Calculate the discount, then subtract it from the regular price Store A was charging \350$$350 with a $20%$20% discount. So to calculate the discount, we need to find $20%$20% of $\$350$$350: 350\times20%=\70350×20%=70 Then we can take the discount amount of \70$$70 away from the regular price to give us the sale price at store A:

$350-70=\$280$35070=$280

So you could buy the tablet at store A for $\$280$$280. • ##### Calculate the sales price as a percentage of the regular price This way seems a bit theoretical at first but can make for quick calculations once you get the hang of it. The regular price represents the whole or 100%100% of the price. A sale price can be expressed as the remaining percentage after a discount has been taken away. So if an item had a 10%10% discount, 100-10=9010010=90, so the sale price would be 90%90% of the original price. Let's use this method to calculate the sale price at store B, which was offering the tablet for \335$$335 with a $15%$15% discount.

So firstly, how much of the regular price would remain after a $15%$15% discount?

$100-15=85%$10015=85%

So now we can calculate $85%$85% of the regular price to give us the sale price.

$335\times85%=\$284.75$335×85%=$284.75

So even though store B had a cheaper regular price, store A offered a slightly better sales price. Switch calculation methods between the stores and you will see that you get the same answer regardless of what method you use.

### Amount per dollar

If Bill's Brand sells $200$200 grams of cement for $\$24$$24 and Bob's Brand sells 150150 grams of the same cement for \13$$13, which is better value? Let's see how much cement we would get for one dollar at each shop:

Bill's:

 $\$24$$24 == 200200 grams (divide both sides by 2424) \1$$1 $=$= $8.33$8.33... grams

So at Bill's, we'd get approximately $8.3$8.3 grams of cement for a dollar.

Bob's:

 $\$13$$13 == 150150 grams (divide both sides by 1313) \1$$1 $=$= $11.53$11.53... grams

We get approximately $11.5$11.5 grams of cement at Bob's, which is much better value than at Bill's.

#### More examples

##### Question 1

At a growers' market, durians are sold at stand A for $80$80 cents per kilogram. At stand B, each $4.5$4.5kg durian is sold for $\$4.00$$4.00. A) First, calculate the price of 4.54.5kg of durian from store A. Give your answer correct to two decimal places. Think: How do we change the price per kilo to work out the price for 4.54.5kg? Do: 4.5\times0.8=\3.604.5×0.8=3.60 B) Which store is the best buy? Think: In other words, at which store would you pay less for 4.54.5kg of durian? Do: Choose Store A, as you pay less for a 4,5kg in this store. C) What should the owner of store C charge for his 4.54.5kg durians if he wants to beat the best buy by 6%6%. Give your answer correct to two decimal places. Think: This means he wants to charge 94% of the lowest price. Do:  0.94\times3.600.94×3.60 == 3.3843.384 == \3.38$$3.38 (to $2$2 d.p.)

The owner of store C should charge $\$3.38$$3.38 for his 4.54.5kg durians. ##### Question 2 At the end of the financial year, a television, originally advertised at \3800$$3800, is discounted by $5%$5%. Valentina negotiates a further discount for paying cash, which brings the price down to $\$3285.10$$3285.10. What further percentage discount did she receive for paying cash? A) Firstly, calculate the price Valentina would have paid after receiving only the first discount. Think: How do we calculate a 5%5% discount? Do:  3800\times5%3800×5% == \190$$190 $3800-190$3800−190 $=$= $\$3610$$3610 Valentina would have paid \3610$$3610.

B) Hence calculate the percentage value of the second discount.

Think: What is the difference between the prices after the first discount and the second discount? Convert this difference into a percentage of the first reduced price.

Do:

 $3610-3285.10$3610−3285.10 $=$= $\$324.90$$324.90 \frac{324.90}{3610}\times100%324.903610×100% == 9%9% ##### Question 3 Which offers a greater discount: a single discount of 13%13% or successive discounts of 6%6% and 11%11%? Think: Successive discounts of 6%6% and 11%11% are NOT the same as a single 17%17% discount. Let's think about this theoretically by letting the original amount be xx. In other words, xx would be 100%100%. Do: A single discount of 13%13% of xx can be expressed as a decimal: \left(1-0.13\right)x=0.87x(10.13)x=0.87x In other words, it leaves 87%87% of the original amount. Now let's consider the successive discounts: \text{6% discount}=0.94x6% discount=0.94x The 11%11% discount will be calculated on this new amount: 0.89\times0.94x=0.8366x0.89×0.94x=0.8366x In other words, these successive discounts leave about 83.66%83.66% of the original amount, so this option offers a greater discount. #### More Worked Examples ##### QUESTION 4 A TV normally sells for \1792.94$$1792.94, but is currently on sale.

In each of the following scenarios, calculate the percentage discount correct to two decimal places.

1. The TV is discounted by $\$149.50$$149.50. 2. The TV is on sale for \1428.74$$1428.74.

##### QUESTION 5

The price of a heater selling for $\$234234 is initially discounted by $14%$14% and later marked up by $14%$14%.

1. Choose the expression that correctly represents the final sales price of the heater.

$234-14%+14%$23414%+14%

A

$234\times\left(\left(-14\right)%\right)\times14%$234×((14)%)×14%

B

$234\times86%\times114%$234×86%×114%

C

$234\div86%\times114%$234÷86%×114%

D
2. What is the final sales price to the nearest cent?

format: plain


text:
# Parse standard balanced group within verbatim context

by siracusa   Last Updated October 10, 2019 01:23 AM - source

Say I want to write a custom parser which reads characters in a verbatim context, where the parser argument should be split into several kinds of tokens:

• Ordinary characters, i.e. all characters beside spaces and \ (and also the } that limits the argument), which should be printed literally;
• spaces, which are just ignored;
• characters escaped by \, which are also printed literally; and
• snippets of arbitrary TeX code, interspersed in the form \{...}.

All tokens are parsed in a context where every character has catcode 12. For the TeX snippets the original catcode table should be used temporarily. Also, the argument string for parsing should be read from the current input stream, because it may include unbalanced braces.

The code below pretty much implements the above requirements, with the only limitation that the TeX snippets are only parsed correctly if preceded by an extra opening brace, i.e. \{{...}. This problem arises because the parser has to look ahead what character follows \ to decide if it is an escaped character or the beginning of a TeX snippet, thus assigning the wrong catcode to { if a balanced group follows.

Is there a way to somehow back up the missing { in the token stream such the TeX's normal argument grabber can be used for collecting the snippet code? I would really like to avoid an extra prefix to signal the begin of a group or to rebuild a balanced group parser.

Current implementation:

\documentclass{article}
\usepackage{l3cctab}
\usepackage{xcolor}

\ExplSyntaxOn

\cs_set:Npn \__foo_temp:w #1 #2 {
\exp_last_unbraced:NNf \cs_set_eq:NN #1 { \char_generate:nn { #2 } { 12 } }
}
\__foo_temp:w \c__foo_space_char     \ %
\__foo_temp:w \c__foo_lbrace_char    \{
\__foo_temp:w \c__foo_rbrace_char    \}
\__foo_temp:w \c__foo_backslash_char \\

\cctab_new:N \g__foo_orig_cctab

\cs_new_protected:Npn \foo_parse:w {
\cctab_gset:Nn \g__foo_orig_cctab { }
\cctab_begin:N \c_other_cctab
\__foo_parse_aux:w
}

\cs_new_protected:Npn \__foo_parse_aux:w #1 {
\token_if_eq_charcode:NNTF #1 \c__foo_lbrace_char
{ \__foo_parse_token:w }
{ \PackageError {foo} { Wrong~argument } }
}

\cs_new_protected:Npn \__foo_parse_token:w #1 {
\token_if_eq_charcode:NNTF #1 \c__foo_rbrace_char {
\__foo_parse_finish:
} {
\token_if_eq_charcode:NNTF #1 \c__foo_backslash_char {
\__foo_parse_esc_token:w
} {
% Print only if not a space
\token_if_eq_charcode:NNF #1 \c__foo_space_char
{ \texttt{#1} }
\__foo_parse_token:w
}
}
}

\cs_new_protected:Npn \__foo_parse_esc_token:w #1 {
\token_if_eq_charcode:NNTF #1 \c__foo_lbrace_char {
\cctab_begin:N \g__foo_orig_cctab
% >>> How to insert the missing opening brace here? <<<
\__foo_parse_balanced_group:n
} {
\texttt{#1}
\__foo_parse_token:w
}
}

\cs_new_protected:Npn \__foo_parse_balanced_group:n #1 {
\fbox { #1 }
\cctab_end:
\__foo_parse_token:w
}

\cs_new_protected:Npn \__foo_parse_finish: {
\cctab_end:
}

\ExplSyntaxOff

\begin{document}
\ExplSyntaxOn

\foo_parse:w{
$foo_\b\a\r_ \{{\textcolor{blue}{\LaTeX}} \}\}\}%{{{\\ } % ^----- extra brace here \ExplSyntaxOff \end{document}  Tags : #### Answers 1 With Dirty Tricks, of course ;-) To put it simply, you have: \foo some text}  and you want \foo to grab everything up to the }, so the code of \foo has to put a { there. \foo has to be basically \fooaux{. But TeX doesn't let you have an unbalanced { there, so you have to trick it into believing the braces are balanced. You can do it like this: \def\foo{% \expandafter\fooaux\expandafter{\iffalse}\fi} \def\fooaux#1{(#1)} \foo some text}  and the output will be (some text). TeX first replaces \foo by its definition. Then the two \expandafter will trigger the \iffalse which, being a false conditional, will eat everything to the matching \fi, which is the }. In this process the two \expandafter and the \iffalse}\fi disappear, leaving you with \fooaux{, which then matches with the some text} left in the input stream. expl3-ifying that in your code: \documentclass{article} \usepackage{l3cctab} \usepackage{xcolor} \ExplSyntaxOn \cs_set:Npn \__foo_temp:w #1 #2 { \exp_last_unbraced:NNf \cs_new_eq:NN #1 { \char_generate:nn { #2 } { 12 } } } \__foo_temp:w \c__foo_space_char \ % \__foo_temp:w \c__foo_lbrace_char \{ \__foo_temp:w \c__foo_rbrace_char \} \__foo_temp:w \c__foo_backslash_char \\ \cctab_new:N \g__foo_orig_cctab \cs_new_protected:Npn \foo_parse:w { \cctab_gset:Nn \g__foo_orig_cctab { } \cctab_begin:N \c_other_cctab \__foo_parse_aux:w } \cs_new_protected:Npn \__foo_parse_aux:w #1 { \token_if_eq_charcode:NNTF #1 \c__foo_lbrace_char { \__foo_parse_token:w } { \PackageError {foo} { Wrong~argument } } } \cs_new_protected:Npn \__foo_parse_token:w #1 { \token_if_eq_charcode:NNTF #1 \c__foo_rbrace_char { \__foo_parse_finish: } { \token_if_eq_charcode:NNTF #1 \c__foo_backslash_char { \__foo_parse_esc_token:w } { % Print only if not a space \token_if_eq_charcode:NNF #1 \c__foo_space_char { \texttt{#1} } \__foo_parse_token:w } } } \cs_new_protected:Npn \__foo_parse_esc_token:w #1 { \token_if_eq_charcode:NNTF #1 \c__foo_lbrace_char { \cctab_begin:N \g__foo_orig_cctab \exp_after:wN \__foo_parse_balanced_group:n % Brace hack: removes this V \exp_after:wN { \if_false: } \fi: } { \texttt{#1} \__foo_parse_token:w } } \cs_new_protected:Npn \__foo_parse_balanced_group:n #1 { \fbox {#1} \cctab_end: \__foo_parse_token:w } \cs_new_protected:Npn \__foo_parse_finish: { \cctab_end: } \ExplSyntaxOff \begin{document} \ExplSyntaxOn \foo_parse:w{$foo_\b\a\r_
\{\textcolor{blue}{\LaTeX}} \}\}\}%{{{\\
} %  ^----- NO extra brace here :-)

\ExplSyntaxOff
\end{document}

and the output:

Phelype Oleinik
October 10, 2019 01:14 AM


text:
# The average proportion non-conforming of 20 samples each of size 100 items is 0.12. The upper control limit for the relevant chart is ______ (round off to 2 decimal places)

This question was previously asked in
GATE PI 2019 Official Paper
View all GATE PI Papers >

## Detailed Solution

Concept:

P-chart (Proportion or Fraction Defective Chart)

• It is used to monitor and control the fraction produced in a process that is defective or non-conforming.
• It follows a binomial distribution.
• This chart is best suited in cases where inspection is carried out to classify articles as either excepted or rejected.

$$Control\;limits = \bar p \pm 3 \times \sqrt {\frac{{\bar p\;\left( {1 - \bar p} \right)}}{n}}$$

Calculation:

Given:

The average proportion, $$\bar p$$ = 0.12
Size, n = 100
$$Control\;limits = \bar p \pm 3 \times \sqrt {\frac{{\bar p\;\left( {1 - \bar p} \right)}}{n}}$$

Upper Control Limit (UCL) is given by :
$$Upper \:control\;limit = 0.12 + 3 \times \sqrt {\frac{{0.12\;\left( {1 - 0.12} \right)}}{100}}$$
$$Upper \:control\;limit = 0.12 + 3 \times \sqrt {\frac{{0.12\;\left( 0.88 \right)}}{100}}$$

∴ UCL = 0.21


text:
# Strange errot message

### 4 thoughts on “Strange errot message”

1. Tom Lindeman says:

Hi Cosmin

I just encountered an error message as follows:

……line 245, in FTPopen
SystemError: new style getargs format but argument is not a tuple
# clear __builtin__._
# clear sys.pat……

And this line 245 (and following line 246) are as follows (with some info masked):

res4 = MDM.send(‘AT#FTPOPEN="84.xxx.xx.xx","zcitynetyyyyyy","passw",0r’)
MOD.sleep(20)                   #Wait 2 sec

Never seen before, although I have used similar pieces of code before.

Regards,

Tom

1. Cosmin Buhu says:

Use " to escape double quotes, and MDM.send needs two parameters (you’re missing the timeout).

1. Tom Lindeman says:

Thanks again Cosmin,

Sorry, my mistake. I was too quick to copy from elsewhere and missed the timeout parameter here!  🙁

But actually, no need to modify the quotes. This form I have used a long time, and also worked here after inserting the missing timeout parameter.

However, to say the least, the error message is quite confusing…

Is there any place where one could see the error message explanations in complete, tuned for PythonWin?

Cheers, Tom

1. Cosmin Buhu says:

I think we must rely on Google again …


text:
# High-affinity Cu(I) chelator PSP-2 as potential anti-angiogenic agent

## Abstract

Copper is an essential trace metal that has been implicated in angiogenesis, the formation of new blood vessels. As tumor growth relies on establishing a functional capillary network for blood supply, copper chelation therapy may hold promise as an anti-cancer strategy by suppressing angiogenesis. To test the anti-angiogenic effect of PSP-2, a recently developed high affinity Cu(I) chelator with low zeptomolar dissociation constant, we utilized the endothelial cancer cell line EAhy926 and assessed changes in cell migration, proliferation, and tube formation in Matrigel. In addition, sprouting was assessed by the chicken and sheep aortic ring assay, and vascular pattern formation was studied in the chorioallantoic membrane of chicken embryos (CAM assay). While incubation with PSP-2 resulted in selective depletion of cellular copper levels, cell migration was not affected and the proliferating activity was even slightly increased. Moreover, the endothelial tube formation assay revealed significant morphological changes in the presence of PSP-2, with thicker tubular walls and an overall decreased meshes area. Similarly, the aortic ring assay and CAM assay showed that PSP-2 evokes significantly longer sprouts with smaller angles at branching points. Altogether, PSP-2 exhibits significant bioactivity at concentrations as low as 5 μM, rendering it a promising anti-angiogenic agent. As EAhy926 cells exhibit both endothelial and tumorigenic properties, the anti-angiogenic effect of PSP-2 might potentially translate also into anti-cancer activity.

## Introduction

Copper is an important micronutrient involved in many fundamental biological processes1. An adequate supply of copper is not only critical for normal cell physiology but is also important in cancer progression2. Increased copper levels have been shown to stimulate angiogenesis, the formation of new blood vessels from existing vasculature, presumably by influencing the activity of proangiogenic factors, including VEGF, bFGF, TGFβ, and angiogenin3. Taking advantage of this effect, a range of biocompatible inorganic composite materials have been developed that promote vascularization in wound-healing by slow release of copper ions4,5,6. Conversely, copper depletion has emerged as a viable strategy to suppress vascularization in tumors7,8. For example, chelation therapy with penicillamine or tetrathiomolybdate has shown promising successes in reducing tumor growth, especially when combined with other anti-cancer regimens9. In addition, copper chelation has also been reported to induce anti-fibrotic and anti-inflammatory effects8.

The pro-angiogenic activity of copper cannot be reduced to a single target; rather, copper appears to play critical roles at multiple stages of angiogenesis3. At present, it remains unclear to what extent in vivo copper chelation affects the proliferation of endothelial cells or whether it suppresses blood vessel formation through a different mechanism. The mode of action might also differ between chelators, depending on the stability and redox potentials of the corresponding copper-complexes. For example, studies with tetrathiomolybdate suggest that its affinity is sufficient to sequester copper from cuproproteins such as ceruloplasmin, serum albumin, or metallothioneins. In addition, ternary cluster formation with Atox1, a copper-transporting metallochaperone, might block delivery to the trans-Golgi network where copper is inserted into nascent cuproproteins10.

We recently reported a membrane-permeant chelator, PSP-2, which binds Cu(I) with low zeptomolar dissociation constant (logK = 20, 25 °C) and exhibits negligible affinity towards other biologically relevant divalent metal cations, including Mn(II), Fe(II), and Zn(II)11. Despite the high Cu(I) affinity, PSP-2 revealed low acute toxicity in cell culture up to a concentration of 100 µM over a period of 24 hours. In vitro studies demonstrated that the chelator can intercept intracellular copper trafficking pathways, for example, by reversing copper-induced trafficking of the Menkes protein (ATP7A) or by blocking long-term potentiation of neurons in acute mouse hippocampal tissue slices11.

The favorable properties of PSP-2 as biological copper chelator thus prompted us to assess PSP-2 as a potential anti-angiogenic agent. To this end, we chose the human endothelial cancer cell line EAhy926 as a model system12, which was previously used in the context of endothelial cell damage13,14. As an initial in vitro assessment of anti-angiogenic activity15, we explored the effect of PSP-2 on cellular proliferation and endothelial tube formation. In addition, we utilized the chorioallantoic membrane of the chicken embryo (CAM assay)16,17 to assess the anti-angiogenic activity of PSP-2 in ovo. Furthermore, its effect on vascular sprouting was studied using the aortic ring assay18. The latter is an angiogenesis model where vascular sprouts are growing outside the aortic wall. In earlier studies, the assay has been employed to demonstrate the pro-angiogenic effect of copper, supplied in the form of CuSO419, and to test the anti-angiogenic impact of drugs20,21.

Hence, we hypothesized that

1. (i)

PSP-2 reduces EAhy926 cell proliferation,

2. (ii)

PSP-2 changes the tube formation pattern in EAhy926 cells,

3. (iii)

PSP-2 diminishes the sprouting in the aortic ring assay and

4. (iv)

PSP-2 hinders normal angiogenesis in the CAM assay.

## Materials and Methods

### Chemicals

PSP-2 (1,2-bis(bis(dimethylphosphorothioylmethyl)phosphino)ethane) was synthesized as previously described11. Avastin was purchased from Kantonsapotheke Zürich (Bevacizumab 25 mg/ml, Kantonsapotheke Zürich, Switzerland).

### Cell culture

EAhy926 and murine NIH 3T3 cells were purchased from ATCC (LGC Standards GmbH, Wesel, Germany), and ATCC (Manassas, VA, USA), respectively. Passages P3-14 were used for all experiments. EAhy926 cells were grown in DMEM (Gibco, Thermo Fisher Scientific, Reinach, Switzerland) containing 10% fetal bovine serum at 37 °C, 5% CO2, 95% humidity. NIH 3T3 mouse fibroblasts were cultured in Dulbecco’s Modified Eagle’s Medium (DMEM), supplemented with 10% bovine calf serum and 1% penicillin-streptomycin, at 37 °C under an atmosphere of humidified air containing 5% CO2.

### Synchrotron X-ray fluorescence (SXRF) imaging

For SXRF imaging experiments, 3T3 mouse fibroblasts were grown on 5 × 5 mm silicon nitride windows (Silson Ltd., UK), pre-coated with 0.2 mg/mL solution of Human Fibronectin (Corning Cat. No. 354008). To increase copper levels above the limit of detection, cells were cultured in media supplemented with 50 µM CuCl2 which was removed prior to incubation with regular media containing 50 µM PSP-2 for 24 hours. Cells were then washed with PBS, fixed with a mixture of 3% paraformaldehyde and 1.5% glutaraldehyde in PBS for 20 min, washed again with PBS. The windows were air-dried overnight at room temperature by propping up on a rubber grid-mat placed in a covered sterile cell culture dish. SXRF imaging was performed at the 2-ID-D beamline of the Advanced Photon Source (Argonne National Laboratory, Argonne, USA). Cells were kept at room temperature under a He atmosphere and raster scanned through the incident beam with 10 keV excitation, which was focused to a spot size of 0.3 × 0.3 µm2 using a Fresnel zone plate22. Emission spectra were collected with a single-element fluorescence detector positioned at a 90 degree angle relative to the excitation beam, and elemental maps were generated by Gaussian fitting of the raw emission spectra using MAPS software23. Quantitative elemental densities were obtained by comparing the emission intensities from the sample to those of a thin film standard (AXO Dresden, Germany).

### Elemental quantification by total reflection x-ray fluorescence analysis (TRXF)

EAhy926 cells were cultured in DMEM supplemented with 10% fetal bovine serum and 1% penicillin-streptomycin at 37 °C under an atmosphere of humidified air containing 5% CO2. After growing cells in media supplemented with 50 µM CuCl2 for 24 h, the medium was replaced with regular medium with or without 50 µM PSP-2 and cells were incubated for another 24 h. Cells were then washed with PBS and harvested by trypsinization with trypsin-EDTA (1 mL). The cell count was determined with a hemocytometer, and cells were pelleted by centrifugation at 5000 rpm at 4 °C for 5 min. For the determination of total content of copper, the cell pellets were disrupted using 200 µL of concentrated nitric acid and digested for 48 h at room temperature. The analysis by TRXF was performed using a Bruker S2 PICOFOX spectrometer and the digested samples were standardized by addition of 1 µg/mL of Ga(III).

### Wound closure assay

EAhy926 cells were seeded into a non-coated 12 well plate (Techno Plastic Products AG, Trasadingen, Switzerland) in full media and allowed to reach confluency. Cells were scratched with a 10 µL tip (Starlab Group, Hamburg, Germany), washed three times with PBS before medium containing DMSO without and with PSP-2 at a final concentration of 5 µM was added. The wounded area was imaged at 20x magnification (Axiovert-10, Carl Zeiss AG) and analyzed at 0 h and 24 h post scratching using ImageJ (v1.47t; NIH, Bethesda, MA, USA). The size of the wound gap was measured at 3 random region of interests (ROI). Quantitative analysis of cell migration was performed based on the average wound gap from those ROIs, and the percentage of change was calculated using Eq. 1:

$$\% \,change=\frac{({\rm{average}}\,{\rm{space}}\,{\rm{at}}\,{\rm{time}}\,{\rm{0}})\,-\,({\rm{average}}\,{\rm{space}}\,{\rm{at}}\,{\rm{time}}\,16\,{\rm{h}})}{({\rm{average}}\,{\rm{space}}\,{\rm{at}}\,{\rm{time}}\,0\,{\rm{h}})}\times \,100 \%$$
(1)

### CCK-8 proliferation assay

EAhy926 cells were seeded into a non-coated 96 well plate (Techno Plastic Products AG, Trasadingen, Switzerland) in full media. After cells reached confluency, the growth medium was replaced with media containing DMSO ± PSP-2 at final concentrations of 5, 10, and 20 µM, and the cells were incubated for another 24 h. CCK-8 reagent (Sigma, Buchs SG, Switzerland) was added to the media one hour before the viability was assessed by measuring the OD at 450 nm.

### Tube formation assay

EAhy926 cells were seeded into a 6-well plate and cultured in full media until they reached confluency. The cells were serum starved for one hour, washed with PBS, and harvested. Matrigel (Corning Matrigel Growth Factor Reduced Basement Membrane Matrix; Corning, NY14831, USA) was diluted 1:1 with serum-free media and prewarmed at 37 °C in a 96 well plate. After seeding with approximately 85000 cells in serum free media on the Matrigel, the cells were incubated at 37 °C for 24 h in medium with DMSO ± PSP-2 at a final concentration of 5 µM. Two to three images per well were captured at 10x magnification, using a phase contrast inverted microscope (Axiovert-10, Carl Zeiss AG, Feldbach, Switzerland), equipped with a digital camera. Images were analyzed using the Angiogenesis Analyzer, a plugin developed for the ImageJ (Version 1.47t; NIH, Bethesda, MA, USA).[ref]

### Aortic ring assay

(A) Chicken: Fertilized Lohman white LSL chicken eggs were purchased from Animalco AG, Switzerland, and incubated at 37 °C and 65% relative humidity in an incubator for 14 days. For experiments in chicken embryos until embryonic day 14 no IACUC approval is required according to Swiss animal care guidelines (TSchV, Art. 112). On incubation day 14, the aorta was carefully extracted according to established protocols24. Briefly, chick embryo aortic arches were excised after exposure of the chicken heart. Excess tissue around the aortic arches was removed and 2–3 rings per arch were achieved by cutting the arches perpendicularly. Aortic rings were imbedded in Matrigel and exposed to 3 groups of culture media; DMEM with DMSO, DMSO + PSP-2, or avastin, a non-copper based anti-angiogenic drug (positive control). For each of the 3 groups, 10 rings were used. (B) Sheep: For the sheep aortic ring assay, sheep aorta was dissected in disks of 5 mm in diameter. For experiments with sheep aorta from dead sheep, no IACUC approval is required according to Swiss animal care guidelines (TSchV). The disks were subsequently incubated in Matrigel and exposed to 2 groups of culture media; DMEM with either DMSO or DMSO + PSP-2. For each group, n = 5 rings were used. For quantitative assessment of morphological differences upon PSP-2 treatment, the number of sprouts within a defined region of interest (ROI) were counted (n = 7) and the branch lengths between two junctions were measured based on relative length units, with n = 158 and n = 190 for DMSO and DMSO + PSP-2, respectively. Angles between branches were assessed within the same ROI, with n = 94 and n = 122 for DMSO and DMSO + PSP-2, respectively. All quantitative readouts were evaluated with Synedra View 18.0.0.7 software.

### CAM assay

Fertilized Lohman white LSL chicken eggs were purchased from Animalco AG, Switzerland, and incubated at 37 °C and 65% relative humidity in an incubator for 3.5 days. For experiments in chicken embryos until embryonic day 14 no IACUC approval is required according to Swiss animal care guidelines (TSchV, Art. 112). Then, using a drill, a window was created in the eggshell after removing 2 mL of albumen25. The window was covered with a petri dish and incubated at 37 °C for another 3.5 days as described earlier26. On incubation day (ID) 7, a 5 mM stock solution of PSP-2 in DMSO was diluted into incubation buffer to a final concentration of 5 µM, and 100 uL of this solution was dripped onto the CAM surface with a silicone ring of 1 cm in diameter, confining the area of spreading liquid (control: diluted DMSO without PSP-2). On ID 8, another 100 µL of DMSO ± 5 µM PSP-2 were applied. On ID 14, at the end of the experiment, images were acquired of the whole ring. The sample size for each group was n = 4. For quantitative assessment, the lengths of the branches were determined based on relative length units using Synedra View 18.0.0.7 software, with n = 131 and n = 86 for DMSO and DMSO + PSP-2, respectively. Angles between branches were analyzed based on n = 91 and n = 36 for DMSO and DMSO + PSP-2, respectively.

### Statistical analysis

Data were analysed with StatView 5.0.1 and SPSS Statistics Version 25. Data were checked for normal distribution with Kolmogorov-Smirnow and Shapiro-Wilk tests. If data were not normally distributed, non-parametric tests, such as Kruskal Wallis test for more than 2 groups or Mann Whitney test for 2 groups, were conducted. For normal distribution, parametric one-way analysis of variance (one-way ANOVA) was conducted for more than 2 groups. Pairwise comparison probabilities (p values) were calculated using the Fisher’s PLSD. P values less than 0.05 were considered significant. Unpaired t tests were conducted for the comparison of 2 groups. The p values were indicated for pairs that were significantly different. Values were expressed as means ± standard deviations.

All methods were carried out in accordance with the relevant Swiss animal care guidelines and regulations.

## Results

### PSP-2 reduces total cellular copper without altering zinc or iron levels

In earlier studies, we demonstrated that PSP-2 can readily traverse cell membranes and intercept copper-dependent intracellular pathways11,27. To investigate whether the effects of PSP-2 actually correspond to selective removal of cellular copper, we visualized changes in trace metal content by synchrotron x-ray fluorescence (SXRF) microscopy using 3T3 mouse fibroblasts as a model cell line (Fig. 1). Because endogenous copper levels are low, often residing at the limit of detection for single cell SXRF analysis, cells were grown in media supplemented with 50 µM CuCl2. As evident from the false-color density maps, zinc and copper showed similar distribution patterns with a pronounced nuclear localization (Fig. 1A). Upon incubation with 50 µM PSP-2 for 24 h, copper levels decreased to approach the limit of detection, while zinc levels remained unaffected (Fig. 1B). A comparison of the corresponding x-ray fluorescence emission spectra, averaged over the entire cell area, best illustrate the effect of PSP-2 (Fig. 1C,D). While the copper signal dramatically decreased, iron and zinc remained essentially unaltered. Further quantification relative to a NIST standard yielded an approximately 3-fold decrease in the cellular copper content whereas the average zinc content remained unchanged upon PSP-2 treatment (Fig. E,F). To account for differences in cell size, the integrated copper and zinc densities were normalized to the sulfur content of each cell.

To test the effect of PSP-2 with cells relevant to angiogenesis, we determined changes of the average elemental content using the human umbilical vein endothelial cancer cell line EAhy926. Elemental analysis of the average copper content by total reflection x-ray fluorescence analysis (TRXF) revealed a dramatic effect after incubating with PSP-2 for 24 h with a greater than 20-fold decrease compared to control (Fig. 2). In agreement with above SXRF data, the relative x-ray emission signals for zinc and iron remained unaffected by PSP-2 (Fig. 2A). Altogether, these data demonstrate that PSP-2 is able to selectively reduce copper levels within the complex chemical environment of a live cell.

### Wound closure and cellular proliferation assay with EAhy926 endothelial cells

The stimulating effect of copper on endothelial cell proliferation is well documented28. To explore whether PSP-2-mediated copper sequestration might affect wound healing, we performed an in vitro scratch assay with EAhy926 cells. In this assay, cell migration is monitored over time after creating a “scratch” in a confluent cell monolayer29,30. As evident from Fig. 3, after 24 hours we observed no statistically significant difference in cell migration between cells exposed to 5 µM PSP-2, supplied from a 5 mM stock solution in DMSO, and cells treated with DMSO at the same concentration after a 1:1000 dilution.

In order to assess the effect of PSP-2 on cellular proliferation, we performed a CCK-8 viability assay. While at low concentrations of PSP-2 the proliferation of EAhy926 cells was slightly enhanced, we observed no significant difference at 20 µM PSP-compared to untreated control cells (Fig. 4). Although small, the increases in cellular proliferation at 5 and 10 µM ligand concentrations were statistically significant.

### Tube formation assay with EAhy926 cells

When plated on Matrigel, endothelial cells form capillary-like mesh structures within a few hours, thus offering a convenient model for the rapid screening of angiogenesis inhibitors31. In the presence of 5 or 10 µM PSP-2, tube formation resulted in a visual change in morphology with a significant increase in wall thickness as well as a reduction in the number of tubular walls (Fig. 5, and Supporting Information, SI Fig. 1). In addition, PSP-2 treatment reduced the total meshes area at both incubation concentrations.

### Outgrowth of endothelial cells using the chicken aortic ring assay

Contrary to the effect of avastin, a potent inhibitor of angiogenesis, the outgrowth of endothelial cells in the chicken aortic ring assay was not compromised in the presence of PSP-2; however, the outgrowth pattern was distinctly different compared to control. While in the presence of DMSO alone a regular network was formed, the presence of PSP-2 led to a network with fewer branches and overall elongated vessels, which assumed a parallel alignment to each other (Fig. 6).

### CAM assay

The CAM assay has been frequently used to study pro- and anti-angiogenic effects evoked by drugs32 or biomaterials33 placed on the chorioallantoic membrane of living chicken embryo34,35. Moreover, vascularization can be assessed not only by histology ex vivo but also longitudinally at different time points by MRI in ovo26. Finally, the CAM assay is inexpensive and easy to execute. Application of PSP-2 in the CAM assay revealed a distinct change in the vascular pattern (Fig. 7). Major structural differences upon PSP-2 application were found in a reduced number of branches and in longish vessels compared to treatment with DMSO only, which was used as control.

## Discussion

In this study, we demonstrated that the high-affinity Cu(I)-chelator PSP-2 exhibits significant anti-angiogenic effects. Consistent with previous metal ion selectivity studies in aqueous buffer11, treatment of 3T3 cells with PSP-2 resulted only in a decrease of copper levels without affecting other biologically relevant transition metals such as iron and zinc (Fig. 1). Moreover, the subcellular distribution of zinc remained qualitatively unchanged as revealed by x-ray fluorescence microscopy (Fig. 1). Likewise, PSP-2 selectively removed copper from EAhy926 cells, which were chosen as a model cell line as it offers both endothelial as well as tumorigenic characteristics (Fig. 2).

Despite the ability to deplete total copper levels, PSP-2 did not reduce cellular proliferation and migration (Figs 3 and 4). At low incubation concentrations, cell proliferation even increased by approximately 10% in the presence of 5 μM PSP-2, and around 5% with 10 μM PSP-2. At a concentration of 20 μM PSP-2, however, there was no significant difference compared to untreated control cells (Fig. 4). These data contrast the effect of tetrathiomolybdate, which inhibited proliferation of human umbilical vein endothelial cells (HUVECs) already at a concentration as low as 2 µM36, or of other anti-angiogenic agents such as Sorafenib, which reduced proliferation of HUVECs by 40% at a concentration of 4 μM37. However, as cellular proliferation might be differently regulated in the EAhy926 cancer cell line, the HUVECs studies are not directly comparable.

The most striking anti-angiogenic effect of PSP-2 was observed during network formation of EAhy926 cells. In the presence of 5 μM PSP-2, we observed a pronounced wall thickening as well as a significant reduction of the total meshes area (Fig. 5). Similar effects were reported for tetrathiomolybdate36 as well as the anti-angiogenic agent arsenic oxide (As2O3)37. In the presence of PSP-2, vascular tube formation was disrupted in favor of cell accumulation at branching points to produce fewer but much thicker walls. The morphology of the networks formed in the presence of PSP-2 completely differed from the control. These results parallel earlier reports where copper chelation had no effect on endothelial process formation but resulted in almost complete inhibition of network formation38. As the tube formation assay is usually performed with HUVECs39 and includes different aspects of angiogenesis, such as endothelial cell adhesion, migration, alignment, and tube formation, it is noteworthy that PSP-2 affected tube formation of EAhy926 cells, which represent not only endothelial cell characteristics but also tumorigenic properties.

The in vitro anti-angiogenic activity of PSP-2 in the tube formation assay was further confirmed by two additional assays, the aortic ring assay as well as the CAM assay. Previous reports showed that bovine aortal cell migration was significantly increased in the presence of 1 μM CuSO419, thus underscoring the role of copper in angiogenesis3,40. In the chicken aortic ring assay, 5 μM PSP-2 did not suppress outgrowth of endothelial cells as observed with the inhibitor avastin, which was used as positive control. However, the outgrowth pattern significantly changed to yield longer sprouts with fewer branching points (Fig. 6A). Quantitative readouts in the aortic ring assay could be assessed only beyond a cell-dense area of fibroblasts. Although the number of endothelial cell sprouts was very similar, the lengths of the branches were significantly higher in the samples that were incubated with PSP-2. Moreover, PSP-2 treatment led to significantly smaller angles between the branches (Fig. 6B).

A similar vessel pattern was found in the CAM assay, where PSP-2 suppressed branching, in favor of long, parallel arranged vessels within the silicone ring (Fig. 7A). Likewise, the length of the branches was significantly increased in the samples that were treated with PSP-2, with significantly smaller angles between the branches (Fig. 7B). Taken together, both assays confirmed the results found in the tube formation assay with EAhy926 cells. Endothelial cells from the chicken aorta as well as vessel sprouting on the chorioallantoic membrane (including primarily endothelial cells), were hindered to form regular tube-like structures.

## Conclusions

In this study, we showed that PSP-2, a high-affinity Cu(I) chelator with low zeptomolar dissociation constant, selectively reduces cellular copper levels and exhibits significant anti-angiogenic activity in the tube formation assay performed with EAhy926 endothelial cancer cells. Moreover, PSP-2 treatment resulted in significantly longer sprout branches with smaller angles in the chicken aortic ring assay. Finally, PSP-2 applied in the CAM assay significantly elongated vessel branches and reduced the angle between vessels. In light of the tumorigenic nature of EAhy926, PSP-2 might harbor not only anti-angiogenic, but also anti-tumorigenic activity, worthwhile of further exploration in an animal model. In addition, further research is needed to evaluate anti-cancer effects of PSP-2 in comparison to and in combination with other anti-cancer drugs.

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Ribatti, D. The chick embryo chorioallantoic membrane as a model for tumor biology. Exp. Cell Res. 328, 314–324, https://doi.org/10.1016/j.yexcr.2014.06.010 (2014).

36. 36.

Lowndes, S., Sheldon, H., Cai, S., Taylor, J. & Harris, A. Copper chelator ATN-224 inhibits endothelial function by multiple mechanisms. Microvasc. Res. 77, 314–326, https://doi.org/10.1016/j.mvr.2009.01.003 (2009).

37. 37.

Yang, M. H. et al. Anti-angiogenic effect of arsenic trioxide in lung cancer via inhibition of endothelial cell migration, proliferation and tube formation. Oncol. Lett. 14, 3103–3109, https://doi.org/10.3892/ol.2017.6518 (2017).

38. 38.

Finney, L. et al. X-ray fluorescence microscopy reveals large-scale relocalization and extracellular translocation of cellular copper during angiogenesis. Proc. Natl. Acad. Sci. USA 104, 2247–2252, https://doi.org/10.1073/pnas.0607238104 (2007).

39. 39.

Gandin, V. et al. Homoleptic phosphino copper(I) complexes with in vitro and in vivo dual cytotoxic and anti-angiogenic activity. Metallomics 7, 1497–1507, https://doi.org/10.1039/c5mt00163c (2015).

40. 40.

Harris, E. D. A requirement for copper in angiogenesis. Nutr. Rev. 62, 60–64, https://doi.org/10.1301/nr.2004.feb.60-64 (2004).

## Acknowledgements

Financial support from the National Institutes of Health (GM67169 to CJF) is gratefully acknowledged. Use of the Advanced Photon Source was supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Contract No. DE-AC02–06CH11357.

## Author information

Authors

### Contributions

D.M.H. performed migration, proliferation, tube formation and aortic ring assays. T.M. synthesized PSP-2. S.H. and B.L. performed the S.X.R.F. and T.R.X.F. studies. D.M.H. and J.B. analyzed data. P.W. extracted chicken aortic rings. D.M.H. extracted sheep aortic rings. D.M.H. and J.B. performed CAM assay. C.J.F. made Figs 1 and 2. J.B. made Figs 3–7 and the two supporting information figures. M.C., C.J.F. and J.B. coordinated the project. D.M.H., C.J.F. and J.B. wrote, revised and edited the manuscript.

### Corresponding authors

Correspondence to Christoph J. Fahrni or Johanna Buschmann.

## Ethics declarations

### Competing Interests

The authors declare no competing interests.

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

## Rights and permissions

Reprints and Permissions

Heuberger, D.M., Harankhedkar, S., Morgan, T. et al. High-affinity Cu(I) chelator PSP-2 as potential anti-angiogenic agent. Sci Rep 9, 14055 (2019). https://doi.org/10.1038/s41598-019-50494-5

• Accepted:

• Published:


text:
zbMATH — the first resource for mathematics

Every homotopy theory of simplicial algebras admits a proper model. (English) Zbl 0994.18008
To axiomatize the notion of a “homotopy theory” D. G. Quillen introduced closed model categories [“Homotopical Algebra”, Lect. Notes Math. 43 (1967; Zbl 0168.20903)], and produced a number of examples of such, one class of which are categories of simplicial algebras. In this paper is examined the case of simplicial algebras, i.e., simplicial objects in a category of algebras associated to an algebraic theory in the sense of F. W. Lawvere [“Functorial semantics of algebraic theories”, Proc. Natl. Acad. Sci. USA 50, 869-872 (1963; Zbl 0119.25901)], and more generally the case of simplicial algebras over a multi-sorted, simplicial theory. This class of examples includes simplicial groups, rings etc. The author shows that any closed model category of simplicial algebras over an algebraic theory is Quillen equivalent to a proper closed model category.
Theorem B. Let $$T$$ be a (possibly simplicial, possibly multy-sorted) theory, and let $$T$$-alg be the corresponding category of simplicial $$T$$-algebras, equipped with a simplicial model category structure in which a map is a weak equivalence of fibrations if it is a weak equivalence or fibration of the underlying simplicial sets.
Then there exists a morphism $$S\to T$$ of simplicial theories such that: (1) the induced adjoint pair $$S$$-alg$$\leftrightarrows T$$-alg is a Quillen equivalence of model categories, and
(2) $$S$$-alg is a proper simplicial closed model category.
A corollary of this theorem is:
Theorem A. The homotopy theory of a category of simplicial algebras always admits a proper model.
The author remarks that “Whether any reasonable homotopy theory (e.g., one associated to a model category) admits a proper model is an open question”.
Theorem C. Given the hypotheses of Theorem B, suppose that in addition $$T$$-alg is a pointed category. Then $$S$$ can be chosen as in Theorem B so that $$S$$-alg is also a pointed category.
Theorem D. Given the hypotheses of Theorem B (respectively of Theorem C), the theory $$S$$ can be chosen as in Theorem B (or Theorem C) so that $$S$$-alg is a cellular model category in the sense of P. Hirschhorn [“Model categories and their localizations”, Preprint, http://www-math.mit.edu/~psh].
Corollary. For any set of maps in $$S$$-alg there is a localization model category structure with respect to this set.
Reviewer: Ioan Pop (Iaşi)

MSC:
 18G55 Nonabelian homotopical algebra (MSC2010) 55U35 Abstract and axiomatic homotopy theory in algebraic topology 18C10 Theories (e.g., algebraic theories), structure, and semantics
Full Text:
References:
 [1] B. Badzioch, Algebraic theories in homotopy theory, Preprint, 2000 · Zbl 1028.18001 [2] Bousfield, A.K.; Friedlander, E.M., Homotopy theory of γ-spaces, spectra, and bisimplicial sets, (), 80-130 · Zbl 0405.55021 [3] Boardman, J.M., Homotopy structures and the language of trees, (), 37-58 · Zbl 0242.55012 [4] Borceux, F., Categories and structures, () [5] Bousfield, A.K., The localization of spaces with respect to homology, Topology, 14, 133-150, (1975) · Zbl 0309.55013 [6] Boardman, J.M.; Vogt, R.M., Homotopy invariant algebraic structures on topological spaces, Lecture notes in math., 347, (1973), Springer Berlin · Zbl 0285.55012 [7] W.G. Dwyer, P. Hirschhorn, D.M. Kan, General abstract homotopy theory, in preparation [8] Dwyer, W.G.; Kan, D.M., Simplicial localizations of categories, J. pure appl. algebra, 17, 3, 267-284, (1980) · Zbl 0485.18012 [9] P.G. Goerss, M.J. Hopkins, Resolutions in model categories, Preprint [10] P.G. Goerss, M.J. Hopkins, Andre-Quillen (co-)homology for simplicial algebras over simplicial operads, Preprint · Zbl 0999.18009 [11] Getzler, E.; Jones, J.D.S., Operads, homotopy algebra, and iterated integrals for double loop spaces, Preprint [12] Goerss, P.G.; Jardine, J.F., Simplicial homotopy theory, (1999), Birkhäuser Basel · Zbl 0949.55001 [13] P. Hirschhorn, Localization in model categories, Preprint, http://www-math.mit.edu/ psh [14] Hovey, M., Model categories, (1999), American Mathematical Society Providence, RI · Zbl 0909.55001 [15] Kan, D.M., On c.s.s. complexes, Amer. J. math., 79, 449-476, (1957) · Zbl 0078.36901 [16] Lawvere, F.W., Functorial semantics of algebraic theories, Proc. nat. acad. sci. USA, 50, 869-872, (1963) · Zbl 0119.25901 [17] Quillen, D.G., Homotopical algebra, Lecture notes in math., 43, (1967), Springer Berlin · Zbl 0168.20903 [18] Quillen, D.G., Rational homotopy theory, Ann. of math., 90, 65-87, (1969) [19] C. Rezk, Spaces of algebra structures and cohomology of operads, Ph.D. dissertation, MIT, Cambridge, MA, 1996 [20] S. Schwede, Stable homotopy of algebraic theories, Preprint · Zbl 0897.55008 [21] J. Smith, Combinatorial model categories, in preparation
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.


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# If 42.42 = k(14 + m/50), where k and m are positive integers and m < 5

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If 42.42 = k(14 + m/50), where k and m are positive integers and m < 5 [#permalink]

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22 May 2016, 14:48
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If 42.42 = k(14 + m/50), where k and m are positive integers and m < 50, then what is the value of k + m ?

(A) 6
(B) 7
(C) 8
(D) 9
(E) 10
[Reveal] Spoiler: OA

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Re: If 42.42 = k(14 + m/50), where k and m are positive integers and m < 5 [#permalink]

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22 May 2016, 16:57
5
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Bunuel wrote:
If 42.42 = k(14 + m/50), where k and m are positive integers and m < 50, then what is the value of k + m ?

(A) 6
(B) 7
(C) 8
(D) 9
(E) 10

42.42 = 14K + Km/50...we can rewrite the number as follows:

42+ 0.42= 14K + Km/50........Since K is integer, then 42=14K..........K=3

0.42=Km/50......42/100=3m/50......m=7

k+m=3+7=10

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Re: If 42.42 = k(14 + m/50), where k and m are positive integers and m < 5 [#permalink]

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31 May 2017, 19:54
Mo2men wrote:
Bunuel wrote:
If 42.42 = k(14 + m/50), where k and m are positive integers and m < 50, then what is the value of k + m ?

(A) 6
(B) 7
(C) 8
(D) 9
(E) 10

42.42 = 14K + Km/50...we can rewrite the number as follows:

42+ 0.42= 14K + Km/50........Since K is integer, then 42=14K..........K=3

0.42=Km/50......42/100=3m/50......m=7

k+m=3+7=10

As in , how is 42.42 = 42 + 0.42?
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Re: If 42.42 = k(14 + m/50), where k and m are positive integers and m < 5 [#permalink]

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31 May 2017, 23:06
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42.42 = 42 + 0.42 = 42 + 42/100 = 42 + 21/50

Now this slightly resembles the right hand side, but still we need to change it to the form: k (14+m/50)
where k and m are positive integers. And we can see that 42 = 14*3

So, 42 + 21/50 = 3 (14 + 7/50) ... (taking out 3 common)
Hence k=3, m=7. Required sum = 3+7 = 10.

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Re: If 42.42 = k(14 + m/50), where k and m are positive integers and m < 5 [#permalink]

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01 Jun 2017, 03:10
ishitam wrote:
Mo2men wrote:
Bunuel wrote:
If 42.42 = k(14 + m/50), where k and m are positive integers and m < 50, then what is the value of k + m ?

(A) 6
(B) 7
(C) 8
(D) 9
(E) 10

42.42 = 14K + Km/50...we can rewrite the number as follows:

42+ 0.42= 14K + Km/50........Since K is integer, then 42=14K..........K=3

0.42=Km/50......42/100=3m/50......m=7

k+m=3+7=10

As in , how is 42.42 = 42 + 0.42?

It is the same when you say 3.5. It is sum of 3 plus 0.5 . Part of the number is integer number (42) and part is decimal number (0.42)

Hope it helps.
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Re: If 42.42 = k(14 + m/50), where k and m are positive integers and m < 5 [#permalink]

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04 Jun 2017, 09:06
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We have

$$\frac{4242}{100}= 14k+\frac{km}{50} => 4242 = 1400k + \frac{100km}{50} => 4200+42 = 1400*(3) + 2km => 4200 + 42 = 4200 + 2(3)m => 42=6m => m=\frac{42}{6}=7$$

Therefore,$$m+k=3+7=10$$

Option E

In the beginning, it might seems rough but practice brings all to life.

Please Kudos should you like the detailed solution.
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Re: If 42.42 = k(14 + m/50), where k and m are positive integers and m < 5 [#permalink]

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28 Jul 2017, 20:24
42.42 = 4242/100 = 2121/50 = 14K + KM/50 = (14K*50+KM)/50

=> 2121 = 14K*50 + KM = K(700+M)
3*7*101 = K(700+M)
Since M < 50

3*707=K(700+M)

K =3 M=7 => K+M = 10
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Re: If 42.42 = k(14 + m/50), where k and m are positive integers and m < 5 [#permalink]

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28 Jul 2017, 22:54
1
KUDOS
Bunuel wrote:
If 42.42 = k(14 + m/50), where k and m are positive integers and m < 50, then what is the value of k + m ?

(A) 6
(B) 7
(C) 8
(D) 9
(E) 10

42.42 = k(14+m/50 )
42 = 14k and 0.42 = km/50
-> k = 3

also
0.42 = 3*m/50
m = 7

k+m = 10

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Re: If 42.42 = k(14 + m/50), where k and m are positive integers and m < 5 [#permalink]

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18 Mar 2018, 18:44
Hi All,

This question has a number of built-in 'limitations' that you can take advantage of that will allow you to really minimize the amount of math that is required.

We're told that K and M are POSITIVE INTEGERS and M < 50.

We're also told that 42.42 = K(14+m/50), which we can rewrite as…

42.42 = 14K + KM/50

Since both variables are positive integers, K MUST be 1, 2, or 3 (otherwise the total would be larger than 42.42…

Since M < 50, then M/50 is < 1, so it's likely that K = 3. This tells us that…

.42 = 3M/50

Now the math will only take a few steps…

21 = 3M

7 = M

The value of K + M = 10

[Reveal] Spoiler:
E

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Re: If 42.42 = k(14 + m/50), where k and m are positive integers and m < 5   [#permalink] 18 Mar 2018, 18:44
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## Thinking Mathematically (6th Edition)

Published by Pearson

# Chapter 2 - Set Theory - Chapter Summary, Review, and Test - Review Exercises - Page 109: 8

#### Answer

$\in$

#### Work Step by Step

The set $\left\{1, 2, 3, 4, ..., 99, 100\right\}$ contains all the counting numbers from 1 to 100. $93$ is a counting number that is less than $100$. Thus, $93$ is an element of the given set above. Therefore, $93 \in \left\{1, 2, 3, 4, ..., 99, 100\right\}$

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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## Course Editing Guide

### Create Activities (give tasks to students)

#### How to use and modify weights in the gradebook?

• Students can view their own grades at one place.
• When grades are awarded from within a Moodle activity (assignment), scores are automatically added to the gradebook.
• Scores can also be manually added or edited (Overridden) in the gradebook.

In order for Moodle to easily aggregate scores automatically, entries in the gradebook are numeric by default. The default score for each activity is 100. You can then control the weights of individual activities to aggregate a final score for the class. If you prefer to grade only with letters no numeric scores are involved and Moodle cannot aggregate a final grade for the course.

Default calculation of the final grade in Moodle - Simple weighted mean or weighted mean

For simple weighted main, the weight of each item is simply the difference between its Maximum and Minimum grade.

Example:

A1 70/100, A2 20/80, A3 10/10, category max 100:
(0.7*100 + 0.25*80 + 1.0*10)/(100 + 80 + 10) = 0.526 --> 52.6/100
If you would like to adjust the weights of individual assignments, you need to change gradebook calculation method to "weighted mean of grades".

1.

2.
Click on the "change" link next to the aggregation method, and change the "simple weighted mean" to "weighted mean of grades" and save.

3. For "weighted mean of grades" adjust the weights next to individual assignments.

So it will work like this:

Each grade item can be given a weight to influence its importance in the overall mean. In simple terms, the "total" will be equal to the sum of the scores in each grade item, these scores being multiplied by the grade items' weights, and that sum being finally divided by the sum of the weights, as shown in this example.

Example:A1 70/100 weight 10, A2 20/80 weight 5, A3 10/10 weight 3, category max 100:
(0.7*10 + 0.25*5 + 1.0*3)/(10 + 5 + 3) = 0.625 --> 62.5/100

4. How to give extra credits for any assignment or task

How to set extra credits?
To set an item to add extra credits instead of normal grading click on EDIT > edit settings next to the item.
Tick the "extra credit" box and save.


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# w, x and y are positive integers such that w ≤ x ≤ y. If the average

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w, x and y are positive integers such that w ≤ x ≤ y. If the average [#permalink]

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w, x and y are positive integers such that w ≤ x ≤ y. If the average (arithmetic mean) of w, x and y is 20, is w > 15 ?

(1) y = 28

(2) One of the three numbers is 17
[Reveal] Spoiler: OA

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Last edited by Bunuel on 18 Oct 2017, 21:01, edited 2 times in total.
Renamed the topic.
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w, x and y are positive integers [#permalink]

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18 Oct 2017, 20:27
w, x and y are positive integers such that w ≤ x ≤ y. If the average (arithmetic mean) of w, x and y is 20, is w > 15 ?

(1) y = 28

(2) One of the three numbers is 17

it seems OA, Option B, is not correct here. Can you confirm. IMO it should be C

Given $$w+x+y=60$$, or $$w=60-x-y$$

Statement 1: implies $$w=60-x-28=32-x$$, so if $$x=16$$, then $$w=16$$ but if $$x=20$$ $$w=12$$. Hence insufficient

Statement 2: if $$w=17$$, then we have a "$$Yes$$" for our question stem

but if $$x=17$$, then $$w=60-17-y=43-y$$, now if y=30, $$w=13$$ and if $$y=27$$, then $$w=16$$. Hence insufficient

Combining 1 & 2, we get the value of two variable, hence third variable can be calculated. Sufficient

Option C

Last edited by niks18 on 18 Oct 2017, 21:02, edited 1 time in total.
w, x and y are positive integers   [#permalink] 18 Oct 2017, 20:27
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text:
# Building Regularized Logistic Regressions from Scratch with Computational Graphs in R

The R package built for this post is available on GitHub: nanxstats/logreg.

As one of the cornerstones for deep learning frameworks, automatic differentiation was briefly mentioned in our previous post. Today let’s focus on the other important piece: the computational graph. I’m only able to write something about this, thanks to Ron Triepels, the author of the recently published R package cgraph. This package offers a decent, minimalist implementation for computational graphs and automatic differentiation in native C and R.

The computational graph is a useful abstraction/framework for describing and executing computational workflows for data. In many cases, the workflows/pipelines are composed of very reusable components, such as all the types of layers in neural networks. Besides machine learning applications, another interesting example is the Common Workflow Language (CWL). CWL is widely used for describing bioinformatics workflows. The components in the workflow described in YAML or JSON formats are parsed to create a computational graph to run through later.

To me, one of the benefits of using computational graphs is that you will notice the issues immediately if there is an error compiling the graph before even running things, say, mistakes were made when describing the workflow. This may help enforce and improve code quality and reproducibility. Plus, The modularized computational components makes it particularly easy to prototype new machine learning algorithms, just like legos. On the performance side, since the data and computation flow is predictable, many aggressive optimizations are possible, for example, on parallelization.

To experience how powerful computational graph + automatic differentiation is for solving classical machine learning problems, I took a few hours to use cgraph and built an R package logreg. I implemented three methods here:

• Logistic regression
• Logistic regression with the ridge ($$\ell_2$$) penalty
• Logistic regression with the seamless-$$\ell_0$$ (SELO) penalty, a differentiable approximation of the $$\ell_0$$ penalty.

Although these methods could be implemented very differently before, the core implementation with cgraph is surprisingly similar: pretty much just replacing the loss/cost functions, as is shown below (SELO). Check out this vignette for test examples, and please feel free to leave me a message if you find any bugs in the code.

# initialize graph
graph <- cg_graph()

# initialize input (X), target (y)
input <- cg_input("input")
target <- cg_input("target")

# intialize parameters (beta, alpha)
parms <- list(
beta = cg_parameter(initialize_glorot_normal(ncol(x), 1), "beta"),
alpha = cg_parameter(initialize_constant(0), "alpha")
)

# define the model (yhat = beta X + alpha)
output <- cg_sigmoid(cg_add(cg_matmul(input, parms$beta), cg_as_numeric(parms$alpha)), "output")

# define the selo loss (y, yhat, beta)
cg_mean(crossentropy(target, output)),
cg_div(
cg_mean(cg_ln(cg_add(cg_div(cg_abs(parms$beta), cg_add(cg_abs(parms$beta), tau)), 1))),
log(2)
),
"loss"
)

# track errors
error <- rep(0, n_epochs)

# optimize parameters via sgd
for (i in 1:n_epochs) {
values <- cg_graph_run(graph, loss, list(input = x, target = y))
for (parm in parms) parm$value <- parm$value - learning_rate * grads[[parm$name]] error[i] <- values$loss
}


text:
# Reinforcement provided in a rectangular RCC beam of effective depth 500 mm is such that actual depth of neutral axis is at 300 mm from extreme compression fibre. Fe 415 steel is used and provided on tension face only. It is a:

Free Practice With Testbook Mock Tests

## Options:

1. doubly reinforced section

2. over reinforced section

3. under reinforced section

4. balanced section

### Correct Answer: Option 2 (Solution Below)

This question was previously asked in

SSC JE CE Previous Year Paper 8 (Held on 29 Oct 2020 Morning)

SSC JE CE Previous Year Paper (Held On: 29 Oct 2020 Morning)

## Solution:

Concept:

A RCC Beam section is said to be:

1. Balanced section: if Xu = Xu, lim

2. Under reinforced Section: If Xu < Xu.Lim

3. Over reinforced Section: If Xu > Xu.Lim

Where

Xu is the actual depth of Neutral Axis

Xu, Lim is the limiting depth of neutral axis, which is given as:

$${X_{u,\;Lim}} = \frac{{700}}{{1100\; + \;0.87{f_y}}} \times d$$

fy is the yield strength of material and d is the effective depth

Calculation:

Given:

x4 = 300 mm, d = 500 mm

Fe 415 steel ⇒ fy = 415 N/mm2

$$\frac{{{x_4}\;lim}}{d} = \left( {\frac{{700}}{{1100 \;+\; 87.415}}} \right) = .479$$

⇒ x4 lim = 0.479 × 500 = 239.55 mm

Since x4 > x4 lim ⇒ section is over-single C/d.