several sixteens. Numerous numbers that are multiples of 16 include 16, 32, 48, 64, 80, 96, 112, 128, 144, 160, and 176.
What is the trick to finding multiples of a number?In mathematics, a multiple is any number that results from multiplying a given number by an integer. As an illustration, multiples of 5 include: 10, 15, 20, 25, 30, etc. 14, 21, 28, 35, 42, 49, and so on are all multiples of 7.
A number is a multiple of a number if another natural number and the given number are multiplied together.
Divide the digits into pairs starting at the end, then multiply the final group by 1, the group before it by 2, the group before that by 4, then 8, 16, and so on. Include every component. If the outcome is a multiple of 7, then so was the original number. Since 91 (137) is a multiple of 7, so is 21553.
16-digit multiples. There are 16 multiples, including 16, 32, 48, 64, 80, 96, 112, 128, 144, 160, and 176.
To learn more about multiples refer to:
https://brainly.com/question/28923509
#SPJ4
Download the dataset advertising.csv. This dataset includes sales data, as well as the budget for TV, Radio and Newspaper advertising. We are interested in predicting sales given a certain budget. a. (12 mark) Read in the dataset into a dataframe called advertising. b. (12 mark) Display the last 15 elements of the dataframe. C. (1 mark) Create three scatterplots showing the relationship of Sales with the other 3 variables. Use plot(). d. (2 marks) Create a linear model using TV to explain Sales. e. (1/2 mark) Print out the summary table of the linear model. f. (1 mark) Plot the linear model onto the graph of Sales and TV. Make the colour of the line red. TV 19.6 2.6 8.6 65.9 9.2 39.6 Radio Newspaper Sales 230.1 37.8 69.2 22.1 44.5 39.3 45.1 10.4 17.2 45.9 69.3 9.3 151.5 41.3 58.5 18.5 180.8 10.8 58.4 12.9 8.7 48.9 75 7.2 57.5 32.8 23.5 11.8 120.2 11.6 13.2 8.6 2.1 4.8 199.8 21.2 10.6 66.1 5.8 24.2 214.7 24 4 17.4 23.8 35.1 97.5 7.6 7.2 9.7 204.1 32.9 46 19 195.4 47.7 52.9 22.4 67.8 36.6 114 12.5 281.4 55.8 24.4 69.2 18.3 11.3 147.3 23.9 19.1 14.6 218.4 27.7 53.4 237.4 23.5 12.5 13.2 15.9 49.6 5.6 228.3 16.9 26.2 15.5 62.3 12.6 18.3 262.9 3.5 19.5 142.9 29.3 12.6 240.1 16.7 22.9 15.9 248.8 27.1 22.9 18.9 70.6 16 40.8 10.5 292.9 28.3 43.2 21.4 112.9 17.4 38.6 11.9 97.2 30 9.6 265.6 0.3 17.4 95.7 7.4 290.7 4.1 8.5 12.8 266.9 43.8 25.4 20.5 18 5.1 9.7 12 15 1.5 20 1.4 9.5
A summary table of the given data set is given below.
What is the summary table?You can compare typical study methodologies, conclusions, limits, etc. using a summary table. You can arrange the entries in any way you find convenient; for example, you might arrange your research according to timeliness or group entries with comparable study aims, models, or outcomes.
Only code:
d=read.csv(file.choose(),header = T) ### data file is selected after this command run
dataframe=as.data.frame(d) ### a) creating data frame
last_15=tail(data frame,15) # b) last 15 observation
last_15
str(d)
par(mfrow=c(2,2)) ## c) Plots
plot(d[,5],d[,2])
plot(d[,5],d[,3])
plot(d[,5],d[,4])
model=lm(d$sales~d$TV,data = d) # d) linear model
summary(model) ## summary of model
plot(model,color="red") ## plot model
code and output:
d=read.csv(file.choose(),header = T) ### data file is selected after this command run
dataframe=as.data.frame(d) ### a) creating data frame
last_15=tail(data frame,15) # b) last 15 observation
last_15
X TV radio newspaper sales
186 205.0 45.1 19.6 22.6
187 139.5 2.1 26.6 10.3
188 191.1 28.7 18.2 17.3
189 286.0 13.9 3.7 15.9
190 18.7 12.1 23.4 6.7
191 39.5 41.1 5.8 10.8
192 75.5 10.8 6.0 9.9
193 17.2 4.1 31.6 5.9
194 166.8 42.0 3.6 19.6
195 149.7 35.6 6.0 17.3
196 38.2 3.7 13.8 7.6
197 94.2 4.9 8.1 9.7
198 177.0 9.3 6.4 12.8
199 283.6 42.0 66.2 25.5
200 232.1 8.6 8.7 13.4
str(d)
'data. frame': 200 obs. of 5 variables:
X: int 1 2 3 4 5 6 7 8 9 10 ...
TV : num 230.1 44.5 17.2 151.5 180.8 ...
radio: num 37.8 39.3 45.9 41.3 10.8 48.9 32.8 19.6 2.1 2.6 ...
newspaper: num 69.2 45.1 69.3 58.5 58.4 75 23.5 11.6 1 21.2 ...
sales: num 22.1 10.4 9.3 18.5 12.9 7.2 11.8 13.2 4.8 10.6 ...
par(mfrow=c(2,2)) ## c) Plots
plot(d[,5],d[,2])
plot(d[,5],d[,3])
plot(d[,5],d[,4])
model=lm(d$sales~d$TV,data = d) # d) linear model
summary(model) ## summary of model
Call:
lm(formula = d$sales ~ d$TV, data = d)
Residuals:
Min 1Q Median 3Q Max
-8.3860 -1.9545 -0.1913 2.0671 7.2124
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 7.032594 0.457843 15.36 <2e-16 ***
d$TV 0.047537 0.002691 17.67 <2e-16 ***
---
Sign in. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 3.259 on 198 degrees of freedom
Multiple R-squared: 0.6119, Adjusted R-squared: 0.6099
F-statistic: 312.1 on 1 and 198 DF, p-value: < 2.2e-16
plot(model,color="red") ## plot model
Learn more about summary tables here:
https://brainly.com/question/24280486
#SPJ4
Calculate (1/x1)+(1/x2)+(1/x3), where x1,x2,x3 are the solutions of the eq. x^3 -3x +1=0
The expression can be evaluated using the property of zeros as 0.
What is an Algebraic expression?An algebraic expression can be obtained by doing mathematical operations on the variable and constant terms.
The variable part of an algebraic expression can never be added or subtracted from the constant part.
The given equation is x³ - 3x + 1 = 0.
Its zeros are given as x₁, x₂ and x₃.
As per the properties of zeros of a cubic equation, the following expressions can be written as,
x₁x₂x₃ = Constant term ÷ Coefficient of x³
⇒ x₁x₂x₃ = 1.
And, x₁x₂ + x₂x₃ + x₁x₃ = Coefficient of x² ÷ Coefficient of x³
⇒ x₁x₂ + x₂x₃ + x₁x₃ = 0.
Now, the given expression 1/x₁ + 1/ x₂ + 1/x₃ can be evaluated as below,
(x₁x₂ + x₂x₃ + x₁x₃)/x₁x₂x₃ = 0/1
⇒ 0
Hence, the given expression can be evaluated as 0.
To know more about algebraic expression click on,
brainly.com/question/953809
#SPJ4
32 x 79 =
PLEASE HELP QUICK
Answer:
2528
Step-by-step explanation:
[tex]32\times79[/tex]
Multiply:
[tex]32\times79[/tex]
[tex]=\fbox{2528}[/tex]
What is a whole number called?
Answer:
they are called whole numbers
Step-by-step explanation:
Who found zero in maths?
Answer:
Mohammed ibn-Musa al-Khowarizmi
What is the total net worth if the banker pays off the student loans? $219,869 $247,517 $233,693 $252,304.
Given the credit union banker's assets and liabilities, their total net worth is C. $233,693.
The difference between a person's long-term and short-term assets and obligations is that person's net worth (long-term and short-term).
The equity of a company, or the net assets possessed by the stockholders after all obligations have been settled, is comparable to the net value of the company.
Assets:
Car Value = $29,850
Savings Accounts = $12,409
Treasury Bonds = $10,000
Checking Account Balance = $19,419
Home Value = $194,450
Total assets = $266,128
Liabilities:
Car loan = $10,560
Student Loans = $13,824
Credit Card Balance = $8,051
Total liabilities = $32,435
Net worth = $233,693 ($266,128 - $32,435)
Thus, the banker's net worth is $233,693.
Complete question:
The assets and liabilities of a credit union banker are listed below.
What is the total net worth if the banker pays off the student loans?
A. $219,869
B. $247,517
C. $233,693
D. $252,304
(Refer to the image for the table)
To know more about net value visit: brainly.com/question/29985435
#SPJ4
Car Value = $29,850
Savings Accounts = $12,409
Treasury Bonds = $10,000
Checking Account Balance = $19,419
Home Value = $194,450
Total assets = $266,128
Liabilities:
Car loan = $10,560
Student Loans = $13,824
Credit Card Balance = $8,051
Total liabilities = $32,435
Net worth = $233,693 ($266,128 - $32,435)
Thus, the banker's net worth is $233,693.
Two piece of metal meaure1 1/6 yard and3/12 yard each. Three time the amount of metal i needed for a project. How many total yard of metal i needed?
The total number of yards needed is (4 + 1/4) yards of metal.
In this case, there are two pieces measuring (1 + 1/6) yard and (3/12) yard; adding the two, the total length is:
L = (1 + 1/6) yard + (3/12) yard
= (1 + 2/12) yard + 3/12yardd
= (1 + 5/12) yard
Since three times that amount of metal is needed, then the total amount of metal that is needed is found by multiplying the total length obtained above by three:
3×L = 3×(1 + 5/12) yard = (3 + 15/12) yard
Writing this as a mixed number:
(3 + 12/12yard + 3/12yard) = (4 + 3/12) yard = (4 + 1/4) yard
Read more on mixed numbers:
https://brainly.com/question/29112400?referrer=searchResults
#SPJ4
.A class counts the number of vehicles that pass by its school from 1:00 to 2:00 P.M. There are 3 times as many cars as trucks.
Answer:
Ok So From What You Said I Calculated This:
1:00-2:00
cars=c, trucks=t
answer=a
c x 3 = a
a÷3=t
Step-by-step explanation:
Answer:Ok So From What You Said I Calculated This:
1:00-2:00
cars=c, trucks=t
answer=a
c x 3 = a
a÷3=t
Step-by-step explanation: I rlly hope this helps please mark brainliest thanks for the points! please tell meif im wrong have a good day night!! :)
I need to know how to find the answer to this question. -15 + n = -9
Answer:
n = 6
Step-by-step explanation:
-15 + n = -9
Add 15 to both sides
n = 6
So, the answer is 6
y is the number of tickets purchased for a randomly chosen olympic event. is the random variable y discrete or continuous?
The number of tickets purchased for a randomly chosen Olympic event. is the continuous random variable.
y is the number of tickets purchased for a randomly chosen Olympics event. is the random variable y discrete or continuous?
A random variable is a numerical quantity that is generated by a random experiment.
A random variable is called discrete if it has either a finite or a countable number of possible values. A random variable is called continuous if its possible values contain a whole interval of numbers.
A random variable is a number generated by a random experiment.
A random variable is called discrete if its possible values form a finite or countable set.
A random variable is called continuous if its possible values contain a whole interval of numbers.
In Olympics , number of event are there . that are not countable.
therefor, the number of tickets purchased for a randomly chosen Olympic event. is the continuous random variable.
To know more about random variable visit
#SPJ4
A random variable is a numerical quantity that is generated by a random experiment.
A random variable is called discrete if it has either a finite or a countable number of possible values. A random variable is called continuous if its possible values contain a whole interval of numbers.
A random variable is called continuous if its possible values contain a whole interval of numbers.
The number of tickets purchased for a randomly chosen Olympic event. is the continuous random variable.
How can you describe the behavior of the graph of a polynomial function at its intercept with even number of multiplicities?
At its intercept with even number of multiplicities, the graph of a polynomial function will have a smooth curve that passes through the intercept point. The curve will loop around the intercept point, increasing and then decreasing in a cyclic manner, before repeating the pattern.
At its intercept with even number of multiplicities, the graph of a polynomial function will have a smooth curve that passes through the intercept point. This is because the multiplicities of the intercept point are even, meaning that the graph will not have a hard turn or corner at the intercept point. Instead, the graph will loop around the intercept point, gradually increasing until it reaches a peak before gradually decreasing and eventually reaching the same point it started at. This cyclic pattern will repeat itself as the graph moves away from the intercept point. The even number of multiplicities means the graph will not have any sharp turns or corners, only a smooth curve that passes through the intercept point.
Learn more about polynomial here
https://brainly.com/question/11536910
#SPJ4
What is the formula for standard deviation of a sample in R?
For the sample provided, The formula for standard deviation is slightly different than the population standard deviation. The answer is
[tex]\sqrt \frac{\sum (x-\bar x)^2}{n-1}[/tex] .
What do you mean by sample?Sampling is the process of choosing a portion of a statistical population to estimate attributes of the entire population in statistics, quality control, and survey methods. Statisticians try to get samples that are typical of the population under consideration.
What is standard deviation?The standard deviation is a statistic that expresses how much variance or dispersion there is in a group of numbers. While a high standard deviation suggests that the values are dispersed throughout a wider range, a low standard deviation suggests that the values tend to be close to the established mean.
sample size = n
sample standard deviation (s) = [tex]\sqrt \frac{\sum (x-\bar x)^2}{n-1}[/tex]
To learn more about sampling visit:
https://brainly.com/question/14893265
#SPJ4
A football team has P points.
P = 3W + D
W is the number of wins.
D is the number of draws.
a) A team has 5 wins and 3 draws.
How many points does the team have?
b) After 35 games a different team has 50 points.
14 games were draws.
How many games has this team lost?
Ridgid motions to prove
Step-by-step explanation:
Kamla tablet in other developed its to the previous yield
If x = 6, y = 4 and z = 1, evaluate x(y to the power of 2 - z)
The solution to the evaluation of the expression x(y² - z) x = 6, y = 4 and z = 1 is 90
How to solve equationEvaluate:
x(y² - z)
When
x = 6, y = 4 and z = 1
x(y² - z)
substitute the value of x, y and z
= 6(4² - 1)
Applying PEMDAS:
P = parenthesis
E = Exponential
M = Multiplication
D = Division
A = Addition
S = Subtraction
= 6(16 - 1)
= 96 - 6
= 90
In conclusion, the expression x(y² - z) when evaluated gives 90.
Read more on equation:
https://brainly.com/question/13763238
#SPJ1
i need help with this
Answer:
v = 6
w=13
Step-by-step explanation:
v = 6 because we know that both of the triangles are the same, they just rotate, but the measure is still the same.
To calculate w, we have the formula
2w + v + 11 = w - 3v +48
We already know v = 6, so put 6 in the v spot
2w + 6 + 11 = w -3(6) + 48
2w + 17 = w - 18 + 48
2w + 17 = w +30
Subtracted 17 from both sides
2w = w + 13
Subtract w from both sides
w= 13
Step-by-step explanation:
to add to the other correct answer :
the symbol means that both triangles are congruent.
and that means that when you turn then so that both can be laid on top of each other in the same direction, they would perfectly cover each other without any overlap whatsoever.
and that means that they must have the same angles and the same side lengths (and also other lengths like heights, bisectors, ...).
therefore,
v + 9 = 15
v = 6
and then
2w + v + 11 = w - 3v + 48
giving us w = 13
as the other answer already shows how.
and ED = QR, of course.
How do you describe the relationship of a square root?
Therefore , the relationship of square root can be illustrated as [tex]y^{2}[/tex] = x.
What is square root ?A number's square root is that factor of the number that, when multiplied by itself, yields the original number. Specifically, squares and square roots are exponents. Think of the number nine. This can be expressed as [tex]3^{2}[/tex] or as 3 x 3.
Here,
In math, the square root of a number x is a number y such that y2 = x, or a number y whose square
(the result of multiplication by itself, or y), is x.
As an illustration, 4 and 4 are the square roots of 16, since 42 = (4)2 = 16.
Therefore , the relationship of square root can be illustrated as [tex]y^{2}[/tex] = x
To know more about square root , visit
https://brainly.com/question/29286039
#SPJ4
X= (3, 3², -2, 6) : F(x)= 2/3 x + 3
The values of the input functions when put into the function are;
f(3) = 5
f(3²) = 9
f(-2) = 8/3
f(6) = 7
How to input values in a function?We are given the function to be represented as;
f(x) = ²/₃x + 3
Now, we are told that the input value of x are;
X = (3, 3², -2, 6)
Thus;
for f(3), we plug in the value of 3 into the function for x to get;
f(3) = ²/₃(3) + 3
f(3) = 2 + 3
f(3) = 5
for f(3²), we plug in the value of 3² into the function for x to get;
f(3²) = f(9) = ²/₃(9) + 3
f(3²) = 9
for f(-2), we plug in the value of -2 into the function for x to get;
f(-2) = ²/₃(-2) + 3
f(-2) = 8/3
for f(6), we plug in the value of -2 into the function for x to get;
f(6) = ²/₃(6) + 3
f(6) = 7
Read more about Function input values at; https://brainly.com/question/10283950
#SPJ1
Find the circumference of a semicircle with diameter d cm
Answer:
Circumference of the semi-circle is d(π/2 + 1)
Step-by-step explanation:
Circumference of a semicircle is πr + 2r
So, since r = d/2,
therefore, π*d/2 + 2*d/2 = πd/2 + d = d(π/2 + 1).
What is the median if there are 11 numbers?
The Median for n = 11 numbers will be 6th term of the Observation set.
One measure of central tendency is the median. The median is the number that lies in the middle of the specified range of integers. It separates the higher half from the lower half of a particular data sample. At least half of the observations are above or equal to the median, and at least half of the observations are below the median.
The median can be calculated as follows when there are odd numbers of observations in a data collection:
Step 1: Arrange the sample of data in ascending or descending order.
Step 2: If the number of observations (let's say n) is odd, the middle observation is the median of the given data.
Median = [tex](\frac{n+1}{2} )^{th}[/tex] term.
Here n = 11, So median will be 6th term of Observation set.
Learn more about Median:
brainly.com/question/28060453
#SPJ4
pls help me sort these from least to greatest
Answer:
53/8, 6 17/25, 6.71, 6.713
Step-by-step explanation:
6.713, 6.71, 53/8, 6 17/25
53/8 = 6.625
6 17/25 = 167/25 = 6.68
So, the order from least to greatest is: 53/8, 6 17/25, 6.71, 6.713
Given that D(x) = 2x, select all of the following that are true statements. D( x) is a direct variation. D( x) is a function. D( x) is a rule for the set of points (5, 10), (6, 12) and (-2, -4). x is the dependent variable. D(6) = 3
Answer:
Step-by-step explanation:
I reformatted the question to clarify the statements. Please check for accuracy.
Given that D(x) = 2x, select all of the following that are true statements.
1. D( x) is a direct variation.
2. D( x) is a function.
3. D( x) is a rule for the set of points (5, 10), (6, 12) and (-2, -4).
4. x is the dependent variable.
5. D(6) = 3
----------
1. D( x) is a direct variation. YES. D(x) will always be 2 times x.
2. D( x) is a function. YES. Each calculated D(x) will have a unique point.
3. D( x) is a rule for the set of points (5, 10), (6, 12) and (-2, -4). YES. Each (x,y) shows a value of y that is 2 times x, as per the equation's instructions.
4. x is the dependent variable. YES. y depends on what value is chosen for x.
5. D(6) = 3 NO. d(6) = 2*(6) or 12
How do you find potential rational zeros?
The leading coefficient and the constant term must both be divisible by the fraction's denominator in order to factor using the rational root theorem.
In plants, a stem's primary role is to aid in the transfer of water and minerals from the roots to the leaves and other sections of the plant. Additionally, it supports the buds, leaves, flowers, fruits, and branches of plants.
The rational root theorem, also known as the rational root test, is an algebraic principle that states that for a polynomial equation with one variable and integer coefficients to have a solution (root) that is a rational number, both the leading coefficient and the constant term must be divisible by the fraction's denominator.
To learn more about rational root theorem link is here
brainly.com/question/9353378
#SPJ4
pls help me again like this is hard
Answer:
10. Some of the class 4 children Amar, Bindu, Charisma, Danny are friends with some of the class 5 children Amar Diana Ekta Raghu Prem. The following table tells us which 2 children are friends with each other.
10. C11. A
12. 20 cm - 2(4cm) = 12
2(6cm length) + 2(4cm width) = 20 cm
12 + 8 = 20 cm
area = length * width
area = 6 * 4
area = 24 cm²
Step-by-step explanation:
heart and star pls <3 brainliest will be appreciated <3(っ◔◡◔)っ -{ elyna s }-How do you solve algebraic expressions Grade 7?
Algebraic expressions can be solved by using the order of operations. The order of operations is PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
To solve an algebraic expression, start by simplifying any parts inside parentheses, then move on to the exponents. Next, perform any multiplications and divisions from left to right. Finally, complete any additions and subtractions from left to right.How solve algebraic expressions Grade 7?The best way to solve algebraic expressions in Grade 7 is to use the Order of Operations (also known as PEMDAS) or BEDMAS. The Order of Operations states that you must first complete any operations inside parentheses, followed by Exponents, Multiplication and Division (from left to right), and finally Addition and Subtraction (from left to right). Once you have completed the operations in the correct order, you will have the solution to the algebraic expression.
Learn more about algebraic expressions :
https://brainly.com/question/28345080
#SPJ4
What is the solution of the system of equations system?
The solutions to systems of equations are the variable mappings such that all component equations are satisfied—in other words, the locations at which all of these equations intersect. To solve a system is to find all such common solutions or points of intersection.
A group of equations comprising one or more variables is known as a system of equations. The variable mappings that satisfy each component equation, or the points where all of these equations cross, are the solutions of systems of equations.
Calculating the unknown variables while keeping the equations balanced on both sides is the process of solving a system of equations. Finding the value of the variable that makes the condition of all the provided equations true is how we solve an equation system. There are several ways to solve a system of equations, including
graphical approachsubstitute techniqueElimination procedureCross-multiplication methodLearn more about system of equations to visit this link
https://brainly.com/question/24065247
#SPJ4
How do you know if a graph is positive or negative infinity?
You can tell if a graph is positive or negative infinity by looking at the slope of the graph. Positive infinity means the slope is increasing without bound, while negative infinity means the slope is decreasing without bound.
In order to determine if a graph is positive or negative infinity, you need to look at the slope of the graph. Slope is a measure of the steepness of a line and is calculated by looking at the change in y-values divided by the change in x-values. If the slope of the graph is increasing without bound, this means that the graph is approaching positive infinity. On the other hand, if the slope of the graph is decreasing without bound, the graph is approaching negative infinity. This means that the y-values are getting closer and closer to negative infinity as the x-values get larger. You can also look at the graph itself to get an idea of the direction of the graph. If the graph is increasing without bound, the graph is approaching positive infinity, while if the graph is decreasing without bound, it is approaching negative infinity.
Learn more about graph here
https://brainly.com/question/17267403
#SPJ4
What value of n makes the equation true?
Answer:
n=5
Step-by-step:
What divided by 10 equals 2? Well, 5 does. So, when you divide the exponents, it will equal the other side.
What is the multiple inverse of 16?
The multiple inverse of 16 is 0.0625.
What is multiple inverse?1/N or N-1 is used to denote a number's multiplicative inverse, such as the number N. It also goes by the name reciprocal.
Inverse refers to something that is the opposite. The acquired reciprocal of a number is such that its value equals identity 1 when multiplied by the original number. In other terms, it is a technique for producing identity 1 by diluting a number by itself, as in N/N = 1.
Inverse of x = 1/x or x⁻¹.
It is also called as the reciprocal of a number and 1 is called the multiplicative identity.
Let the multiple inverse of 16 is x, then
16 * x = 1
⇒ x = 1/16
⇒ x = 0.0625
To learn more about multiple inverse visit:
brainly.com/question/13715269
#SPJ4
May I ask what’s the answer to this MC question because what I've calculated is 36 to the power 222 which giant matched the above choices
Answer:
B
Step-by-step explanation:
I just answered this question. you posted it multiple times ?
again
16 = 2⁴
9 = 3²
so, we have
(2⁴)¹¹¹ × (3²)²²² = 2⁴⁴⁴ × 3⁴⁴⁴ = (2×3)⁴⁴⁴ = 6⁴⁴⁴
remember, you can only combine either the exponents of the same base, or the bases of the same exponents (like in our case here).