text:
# Algebraic Structures on Sets?

I just found out that the words "algebraic structure" have a precise definition and that this
notion is not just common language!

DEFINITION: An algebraic structure consists of one or
more sets closed under one or more operations, satisfying
some axioms
http://www-public.it-sudparis.eu/~gibson/Teaching/MAT7003/L5-AlgebraicStructures.pdf [Broken]

In abstract algebra, an algebraic structure consists of one or more sets, called underlying sets
or carriers or sorts, closed under one or more operations, satisfying some axioms.

An Algebraic Structure is defined by the tuple $<A, o_1, ...,o_k;R_1,...,R_m;c_1,...,c_k>$
where;
A is a non-empty set,
$o_i$ is a function $A^{pi} \ : \ A \ \rightarrow \ A$

$R_j$ is a relation on A

$p_i$ is a positive integer

$c_i$ is an element of A

Then below this definition they give another (equivalent) definition:

An algebraic structure is a triple <A,O,C> where:

A ≠ ∅

$O \ = \ U^n_{i = 1} \ o_i$ where o_i are i-ary operations

C ⊆ A is the constant set.

So based on this I have three questions:

1: How is this concept explained in terms of set theory?

I am thinking that it follows from the idea of a "structured set".
Unfortunately, I can find basically nothing on this concept from browsing online.
The only sources I have found are the one in the last link, page 23 of Vaught Set Theory
which is extremely short & also Bourbaki's Set Theory book - but it's buried after 250+
pages of prerequisite theory. There may just be a different name for this concept, idk...

2: Could you recommend any sources (book recommendations)
discussing Structures on Sets as they arise in ZFC theory?

If there's a book that describes how structures on sets fall out of ZFC theory in a
book describing ZFC that would be optimal, for all I know every book does this just
under a different name.

3: Could you recommend any sources explaining Algebraic Structures in terms of sets?

I started a different thread a while ago trying to ground a vector space in terms of
set theory making everything very explicit, the answer I got was structured to follow
patterns that I now recognise as coming out of this idea of algebraic structures &
basically I'd just like to read how this concept is defined and originates from set theory
with all the prerequisite set theory knowledge that goes with it being built up too.

Last edited by a moderator:

Hurkyl
Staff Emeritus
Gold Member
Two good keywords:
• Universal Algebra
• Model Theory

This is probably not what you're looking for, but these references are the closest I get to answering your problem:

1) http://www.math.uwaterloo.ca/~snburris/htdocs/ualg.html is a very good introduction to algebraic structures.

2) http://www.math.uiuc.edu/~vddries/410notes/main.dvi treats algebraic structures from a more fundamental point-of-view. This is probably what you want, but it's quite advanced. The theory you want is in section 2.3...

2) http://www.math.uiuc.edu/~vddries/410notes/main.dvi treats algebraic structures from a more fundamental point-of-view. This is probably what you want, but it's quite advanced. The theory you want is in section 2.3...

Exactly what I was hoping for! :surprised Cheers/Ty/Slainte/Salute/Nostrovia!


text:
## WeBWorK Main Forum

### Logic Context?

by Spyro Roubos -
Number of replies: 0
Hello,

I'm creating problems for a Mathematical Logic homework set, and I'd like to have problems where students translate compound statements in English into symbolic form using given simple statements p, q, r.

Right now, (from other suggestions in the forum), I'm using string objects, overriding all error messages that might be confusing to students with either vague ones (like "This is incorrect") or more relevant ones.

I'm using /\ for "And", \/ for "Or", -> for "If/then", ~ for "Not".

Using the backslash in answers produces really odd results as I imagine this is messing with the parser.  It'll mark correct answers correct (happy about this), ..but it removes all backslashes from the student's answer and shows parts of their answer preview in red.

I'm wondering how doable making a simple logic parser would be with the symbols as above or if there's some special way of representing a backslash (like how $is$DOLLAR).  A parser would be a lot nicer for the students (accepting spaces, ignoring extraneous parentheses, etc.).

Again, I want to reiterate, the problem 'works'.. it just has odd results I'd like to avoid if possible.

I've put the coding of my problem below.

Thank you,
Spyro Roubos

=======================

# Write the following statement in Symbolic Notation - Decide on Parentheses

########################################################################

DOCUMENT();

"PGstandard.pl",     # Standard macros for PG language
"MathObjects.pl",
"parserAutoStrings.pl"

#"source.pl",        # allows code to be displayed on certain sites.
#"PGcourse.pl",      # Customization file for the course
);

# Print problem number and point value (weight) for the problem
TEXT(beginproblem());

# Show which answers are correct and which ones are incorrect
$showPartialCorrectAnswers = 1; ############################################################## # # Setup # # Context("Numeric"); Context()->strings->clear; Context()->constants->clear; Context()->variables->clear; Context()->functions->clear; Context()->operators->clear; Context()->parens->clear; Context()->{error}{msg}{"Variable '%s' is not defined in this context"} = "This is incorrect."; Context()->{error}{msg}{"Unexpected character '%s'"} = "This is incorrect."; Context()->{error}{msg}{"'%s' is not defined in this context"} = "This is incorrect."; AutoStrings();$C1 = String("p\/(q/\r)");
$C2 = String("(p\/q)/\r");$C3 = String("(p\/q)/\~r");
$C4 = String("~p/\~q");$C5 = String("~(p/\q)");

##############################################################
#
#  Text
#
#

Context()->texStrings;
BEGIN_TEXT

Consider the following simple statements:
$BR$BR
p = "We need to do the laundry" $BR q = "We need to go grocery shopping"$BR
r = "We need to stop for gas" $BR$BR

Represent the following compound statements using symbols.  Use the forward and back-slashes on your keyboard to make "And" and "Or".  The ~ is to the left of "1" on your keyboard.  Use -> for the "Conditional".  $BBOLD Do NOT use spaces in your answer here.$EBOLD  Use parentheses where appropriate.

$BR$BR

We need to do the laundry, or we need to go grocery shopping and we need to stop for gas.: \{ans_rule(30)\}

$BR$BR

We need to do the laundry or we need to go grocery shopping, and we need to stop for gas.: \{ans_rule(30)\}

$BR$BR

We need to do the laundry or we need to go grocery shopping, but we do not need to stop for gas.: \{ans_rule(30)\}

$BR$BR

We do not need to do the laundry and we do not need to go grocery shopping.: \{ans_rule(30)\}

$BR$BR

It's false that we need to do the laundry and we do need to go grocery shopping.: \{ans_rule(30)\}

END_TEXT
Context()->normalStrings;

##############################################################
#
ANS( $C1->cmp); ANS($C2->cmp);
ANS( $C3->cmp); ANS($C4->cmp);

format: plain


text:
An interior point algorithm for solving continuous single- or multifacility location problems is presented. Distances can be measured with arbitrary, possibly nonsymmetric gauges, while the globalizing function which combines these distances into a global objective function value is assumed to be a convex quadratic and monotonically nondecreasing function. Using the notion of self-concordant barrier functions, it is shown that the algorithm computes an $\varepsilon$-approximation to a minimum of the objective function with a number of floating point operations which is bounded by a polynomial in the input size and~$\log(1/\varepsilon)$. The algorithm can be extended to the case in which the globalizing function contains additionally the maximum of some of the distances. The complexity result holds for this case also.

Keywords. multifacility location, continuous location, interior point algorithms, barrier methods, polynomiality, gauges, convexity.


text:
# How to solve a recurrence relation with a sum?

How do I solve the following recurrence relation?

$$T(n) = 1 + \sum_{j=0}^{n-1} T(j).$$

I thought of solving it by generating its recursion tree. I found that the height of the tree would be equal to $n$ but wasn't able to find the cost of each level.

Here are several ways to solve your recurrence relation.

Guessing

Anyone with enough experience in computer science might recognize your recurrence as the one satisfied by $T(n) = 2^n$. Given this guess, you can verify it by summing the appropriate geometric series: if $T(m) = 2^m$ for $m < n$ then

$$T(n) = 1 + \sum_{m=0}^{n-1} T(m) = 1 + \sum_{m=0}^{n-1} 2^m = 1 + (2^n-1) = 2^n.$$

Computing the first few values

Another approach which is very useful is to compute a few values of the sequence: $$1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, ...$$ If we're lucky, as in this case, we might spot a pattern. If not, it's always a good idea to consult the on-line encyclopedia of integer sequences. A quick search would reveal the sequence A000079.

Creative telescoping

In the method of creative telescoping, we try to find a combination of terms that results in major cancellation. In our case, we can try $T(n+1)-T(n)$:

$$T(n+1)-T(n) = \sum_{m=0}^n T(m) - \sum_{m=0}^{n-1} T(m) = T(n),$$ and so $T(n+1) = 2T(n)$. Unrolling this rule, we deduce $T(n) = 2^n T(0) = 2^n$.

Back-substitution

Another way to look at the preceding trick is to try to "back-substitute" the recurrence inside itself:

$$T(n) = 1 + \sum_{m=0}^{n-1} T(m) = 1 + \sum_{m=0}^{n-2} T(m) + T(n-1) = 2T(n-1),$$

from which we can proceed as before.

Generating series

A very general method is that of generating series. Let $G(x) = \sum_{n=0}^\infty T(n) x^n$. Since $1/(1-x) = \sum_{n=0}^\infty x^i$, multiplying by $1/(1-x)$ is like computing "running sums": $$\frac{G(x)}{1-x} = \sum_{n=0}^\infty \sum_{m=0}^n T(m) x^n = \sum_{n=0}^\infty (T(n+1)-1) x^n = \\ \frac{G(x)-T(0)}{x} - \sum_{n=0}^\infty x^n = \frac{G(x)-1}{x} - \frac{1}{1-x}.$$ From this it is not hard to deduce $$G(x) = \frac{-1/x-1/(1-x)}{1/(1-x)-1/x} = \frac{-(1-x)-x}{x-(1-x)} = \frac{1}{1-2x} = \sum_{n=0}^\infty 2^nx^n,$$ and so $T(n) = 2^n$.

Recursion tree

It is also possible to deduce the value of $T(n)$ using a recursion tree. Each path in the recursion tree for $T(n)$ corresponds to a sequence of decreasing numbers $n_0 = n, \ldots, n_\ell$, where $n_\ell \geq 0$ (possibly $\ell = 0$); this is because all leaves are labeled by $1$. The set of possible decreasing sequences is in one-to-one correspondence with subsets of $\{n-1,\ldots,0\}$, because once we know which numbers appear, we also know their order. The number of subsets of $\{n-1,\ldots,0\}$ is exactly $2^n$.

• I really appreciate your experience. – Navjot Singh Mar 28 '18 at 15:40


text:
# dgl.DGLGraph.set_n_initializer¶

DGLGraph.set_n_initializer(initializer, field=None)[source]

Set the initializer for empty node features.

Initializer is a callable that returns a tensor given the shape, data type and device context.

When a subset of the nodes are assigned a new feature, initializer is used to create feature for rest of the nodes.

Parameters: initializer (callable) – The initializer. field (str, optional) – The feature field name. Default is set an initializer for all the feature fields.

Examples

Note

Here we use pytorch syntax for demo. The general idea applies to other frameworks with minor syntax change (e.g. replace torch.tensor with mxnet.ndarray).

>>> import torch as th
>>> G = dgl.DGLGraph()

Set initializer for all node features

>>> G.set_n_initializer(dgl.init.zero_initializer)

Set feature for partial nodes

>>> G.nodes[[0, 2]].data['x'] = th.ones((2, 5))
>>> G.ndata
{'x' : tensor([[1., 1., 1., 1., 1.],
[0., 0., 0., 0., 0.],
[1., 1., 1., 1., 1.]])}

Note

User defined initializer must follow the signature of dgl.init.base_initializer()


text:
# Difference between revisions of "2012 AMC 10A Problems/Problem 16"

## Problem

Three runners start running simultaneously from the same point on a 500-meter circular track. They each run clockwise around the course maintaining constant speeds of 4.4, 4.8, and 5.0 meters per second. The runners stop once they are all together again somewhere on the circular course. How many seconds do the runners run?

$\textbf{(A)}\ 1,000\qquad\textbf{(B)}\ 1,250\qquad\textbf{(C)}\ 2,500\qquad\textbf{(D)}\ 5,000\qquad\textbf{(E)}\ 10,000$

## Solution 1

First consider the first two runners. The faster runner will lap the slower runner exactly once, or run 500 meters farther. Let $x$ be the time these runners run in seconds.

$4.8x-4.4x=500 \Rightarrow x=1250$

Because $4.4(1250)=5500$ is a multiple of 5, it turns out they just meet back at the start line.

Now we must find a time that is a multiple of $1250$ and results in the 5.0 m/s runner to end up on the start line. Every $1250$ seconds, that fastest runner goes $5.0(1250)=6250$ meters. In $2(1250)=2500$ seconds, he goes $5.0(2500)=12500$ meters. Therefore the runners run $\boxed{\textbf{(C)}\ 2,500}$ seconds.

## Solution 2

Working backwards from the answers starting with the smallest answer, if they had run $1000$ seconds, they would have run $4400, 4800, 5000$ meters, respectively. The first two runners have a difference of $400$ meters, which is not a multiple of $500$ (one lap), so they are not in the same place.

If they had run $1250$ seconds, the runners would have run $5500, 6000, 6250$ meters, respectively. The last two runners have a difference of $250$ meters, which is not a multiple of $500$.

If they had run $2500$ seconds, the runners would have run $11000, 12000, 12500$ meters, respectively. The distance separating each pair of runners is a multiple of $500$, so the answer is $\boxed{\textbf{(C)}\ 2,500}$ seconds.

## See Also

 2012 AMC 10A (Problems • Answer Key • Resources) Preceded byProblem 15 Followed byProblem 17 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 All AMC 10 Problems and Solutions
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text:
Blog2022-08-31T00:32:15+08:00

## Additional Math – Differentiation – Quotient Rule (Challenging)

Differentiate $latex \displaystyle\ y=\frac{{{{x}^{2}}\sqrt{{x+1}}}}{{x-1}}$ with respect to x. Simplify the [...]

## Additional Math – Trigonometry – Solve cotangent half angle – 2cosine half angle = 2

$latex \displaystyle \cot \left( {\frac{x}{2}} \right)-2\cos \left( {\frac{x}{2}} \right)=0$ Convert [...]

By |April 5th, 2021|Categories: A-Math - Trigonometry||0 Comments

## Additional Math – Remainder and Factor Theorem – Find the integral value of n and p

when $latex \displaystyle {{x}^{{2n}}}-p$ is exactly divisible by (x+3), it [...]

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text:
# N11

QUESTION: Write the following fraction as a decimal and a percentage

$\dfrac{5}{8}$

ANSWER: Firstly, we we know that $\dfrac{1}{4} = 25\%$, so $\dfrac{1}{8}= 12.5\%$

Now we need to work out $\dfrac{5}{8}= 12.5\% \times 5 = 62.5\%$

Last we need to convert $62.5\%$ into a decimal. We do this by dividing by 100.

$62.5 \div 100 = 0.625$

$\dfrac{5}{8} = 62.5\% = 0.625$


text:
# Math Help - 3-Sigma and Standard Deviation

1. ## 3-Sigma and Standard Deviation

Hi all,

From this link: 68-95-99.7 rule - Wikipedia, the free encyclopedia, there is the relationship between sigma and stnadard deviation whereby 1-Sigma, 2-Sigma, 3-Sigma is 68-95-99.7 rule.

Can I confirm that this only applies to normal distribution? If so, then can I say that the term 1-Sigma, 2-Sigma or 3-Sigma would not have any meaning in non-normal distribution?

Thank you.

2. Yes, the 68-95-99.7 values are specific to the normal distribution and do not apply to other distributions.

About all that can be said in general is Chebyshev's inequality:

Chebyshev's inequality - Wikipedia, the free encyclopedia

3. Hi awkard,

Thanks for the reply.

Actually there is someone who told me that this sigma can also be applied to non-normal distribution since it simply stands for area under the curve. However, I feel that this concept is too misleading.

Thanks for clarifying this.

4. chebyshev's is just a lower bound on the probabilities

5. Originally Posted by awkward
Yes, the 68-95-99.7 values are specific to the normal distribution and do not apply to other distributions.

About all that can be said in general is Chebyshev's inequality:

Chebyshev's inequality - Wikipedia, the free encyclopedia
There is a variant of Chebyschev's inequality for RV with moments higher than the second.

If RV $X$ has an $n$-th central moment $\mu_n$ then:

$Pr(|X-\overline{X}|\ge \lambda (\mu_n)^{\frac{1}{n}})\le \dfrac{1}{\lambda^n}$

This can be proven as can the standard Chebyshev's inequality using Markov's inequality:

$Pr(|X|\ge a) \le \dfrac{E(X)}{a}$

CB


text:
# Linear function on strings using UpValues

I have two functions Sup and Sdown that take a string of letters with allowed characters "Z","W","P","H" and returns a "sum" of strings with signs:

Sup[simple_String]:=
Block[{char=Characters[simple],z=0,w=0,n},
n=Count[char,"Z"|"P"];
Sum[
Which[
char[[i]]=="P",
z++;w++;0,
char[[i]]=="W",
w++;0,
char[[i]]=="Z",
z++;
Sigg[n,z,w]*StringJoin@@(Join[char[[;;i-1]],{"W"},char[[i+1;;]]]),
True,
0
],{i,Length[char]}
]
]

and

Sdown[simple_String]:=
Block[{char=Characters[simple],z=0,w=0,n},
n=Count[char,"Z"|"P"];
Sum[
Which[
char[[i]]=="P",
z++;w++;0,
char[[i]]=="Z",
z++;0,
char[[i]]=="W",
w++;
Sigg[n,z+1,w]*StringJoin@@(Join[char[[;;i-1]],{"Z"},char[[i+1;;]]]),
True,
0
],{i,Length[char]}
]
]

The sign is simply given by

Sigg[n_,z_,w_]:=If[EvenQ[n+w-z],1,-1]

As expected, this gives me i.e.

In[1]= Sdown["WPWPHH"]
Out[1]= -"WPZPHH" + "ZPWPHH"

Now, I want Sup and Sdown to be linear operators that can act on stuff like Out[1], so I do this:

Sup/:Sup[Plus[x__]]:=Plus@@(Sup/@x)

Sup/:Sup[Times[i_Integer,s_String]]:=Expand[i*Sup[s]]

and the same for Sdown. But when I try to Sup[Sdown["WPWPHH"]] I get

In[2]= Sup[Sdown["WPWPHH"]]
Out[2]= 2 + "WPWPHH"

instead of 2"WPWPHH" as I would expect. The FullForm of Out[1] is Plus[Times[-1,"WPZPHH"],"ZPWPHH"] so when I do

In[3]= Plus @@ (Sup /@ {Times[-1, "WPZPHH"], "ZPWPHH"})

I get

Out[3]= 2 "WPWPHH"

like expected. What's going on here? Can you help me find the error? Also, if there is a better way of implementing this linearity, please let me know :)

Thank you for a well-written question with complete code that made this reasonable to answer. Welcome to Mathematica Stack Exchange. :-)

1. You do not need UpValues definitions here. That would only apply if you were attempting to add a rule to e.g. Plus rather than Sup, yet your use of TagSetDelayed makes it clear that you are attaching the rule to Sup.

2. The major remaining issue is the pattern Plus[x__] which will not work; see: Why is ReplaceAll behaving like this?

3. In the pattern above, even if it did work, x__ would be a sequence of expressions; you would need to wrap it in something before mapping, e.g. a List: Plus @@ Sup /@ {x}

Fortunately Map will work on expressions of arbitrary head therefore both problems can be avoided by matching the entire Plus object rather than the sequence. Your solution is therefore:

Sup[x_Plus] := Sup /@ x

Now:

Sup[Sdown["WPWPHH"]]

2 "WPWPHH"

• Thank you for the quick answer. I now have Sup[x_Plus]:=Sup/@x and Sup[i_Integer*s_String]:=i*Sup[s] and it works brilliantly. – Marius Ladegård Meyer Nov 13 '14 at 10:41
• @Marius Glad I could help. Another formulation you might consider is: Sup[i_Integer] := i; Sup[x : _[__]] := Sup /@ x. This maps Sup over any compound expression. Atoms such as Strings or Integers are then individually handled. – Mr.Wizard Nov 13 '14 at 12:16
• Will there be any benefits (time, memory, ...) other than conceptual clarity in this case? In any case, thanks for the tip, it will surely help me with some of my other linear functions. – Marius Ladegård Meyer Nov 13 '14 at 12:23
• @Marius I don't know without testing. On the surface I wouldn't expect any large performance difference but I have been surprised before. – Mr.Wizard Nov 13 '14 at 12:27


text:
This article evaluates the role that different levels of control for combined sewer overflows have in addressing the recreational water quality objectives of the Toronto Inner Harbour of the Toronto and Region Remedial Action Plan. Three models are used to establish the predictive methodology: the Infoworks model for the combined sewer service area, the Hydrologic Simulation-F model for the remainder of the watershed, and the MIKE 3 computer code to evaluate Lake Ontario response to control. Each model was calibrated with E. coli densities observed respectively in sewer discharges, instream, and in the Inner Harbour. Two indices are used to evaluate the response of water quality in the Inner Harbour – fraction of the surface area achieving Blue Flag status, and portion of the swimming season (June to August) above recreation objectives. Analyses of control options led to the recommendation that virtual elimination of combined sewer overflows (one overflow per season control strategy) should be pursued, rather than the lower level of control of 90% volumetric control, which is the minimum provincial environmental requirement. Implementation of priority projects for improving water quality along the Toronto waterfront, including the Don River and Central Waterfront project, are integral to delisting Toronto as a Great Lakes Area of Concern.

## Introduction

A major source of degraded water quality in the Toronto and Region Area of Concern (AOC) is polluted stormwater runoff and combined sewer overflows (CSOs), which contain a mixture of stormwater and untreated sewage, and which are discharged from outfalls into Toronto’s waterways after heavy rains or snowmelts. Historically, this has been of particular concern in the Don River and Toronto’s Inner Harbour (Howell et al., 2018; RAP, 1989a).

The Wet Weather Flow Master Plan (WWFMP) (see MMM, 2003) was developed based on the principles of managing wet weather flows (WWF) on a watershed basis, recognizing rainwater as a valuable resource, and with a hierarchy of management practices and controls, starting with “at source”, followed by “conveyance” and finally “end-of-pipe” controls (D’Andrea et al., 2004).

The WWFMP (2003) included a 25-year implementation plan valued at $1 billion (CAD), which identified projects and initiatives for implementation in five year periods, with the first priority being the improvement of water quality along Toronto’s waterfront and at the City’s beaches. This priority supports actions identified within the Toronto and Region Remedial Action Plan (RAP), under the Canada-Ontario Agreement respecting the Great Lakes Basin Ecosystem and the Great Lakes Water Quality Agreement, and is considered integral to improving water quality conditions and delisting Toronto and Region as a Great Lakes AOC. Other key elements of the WWFMP’s 25-year implementation included watercourse restoration, remediation of basement flooding, municipal operations, greening new development, and public education. An Environmental Assessment (EA) Study (MMM, 2012), the Don River and Central Waterfront Project (DR and CW project) was completed in 2012, to address recommendations from the WWFMP (2003) related to management of CSOs. The DR and CW project integrates wet weather flow (WWF) management systems to capture and treat stormwater discharges and CSOs from all the combined sewer outfalls to the Lower Don River, Taylor-Massey Creek (a tributary to the Don River) and Toronto’s Inner Harbour (D’Andrea et al, 2004; Appendix, Figure S1 and Figure S2 [available in the online supplementary files]). The original RAP problem definition for beach postings as outlined in the Stage 1 RAP report entitled, Environmental Conditions and Problem Definition (RAP, 1989a) was: frequent beach postings as a result of stormwater and CSO contamination. Specific remedial actions were identified (RAP, 1994) which addressed water quality concerns related to beaches and to CSOs within the AOC. The DR and CW project addresses the effect of WWF on E. coli densities in the lower Don River and the Inner Harbour from discharges originating in the combined sewer service area (Appendix Figure S1). In addition to addressing impairments in receiving water quality within the Toronto and Region AOC, the DR and CW project will improve conditions for multiple additional impairments including aesthetics, eutrophication and toxic contaminants. The emphasis in this paper is on recreational water quality as reflected by E. coli densities, and as a surrogate for both whole body contact (e.g. swimming within the open water), as well as incidental contact (e.g. sailing, kayaking, and dragon boating, and wind surfing). This article presents the modeling methodologies used in the EA study and evaluates the role that different levels of control of CSOs have in addressing the recreational water quality objectives in Toronto’s Inner Harbour. ## Methods ### Assessment methodology Three levels of CSO control are presented in this paper for the System Concept Plan (Appendix, Figure S3): 1. Base Case - Existing Conditions for 2031 Population Growth 2. 90% volume capture applied to the Base Case 3. One CSO on average, per season applied to the Base Case. The Ontario Ministry of the Environment, Conservation and Parks (MECP) 90% volume capture procedure F5-5 requirement (MECP, 2016), involves interception and treatment of 90% of wet weather flows generated in the combined sewer system, and is equivalent to control requirements of the US EPA and of European jurisdictions. In terms of RAP targets, the control level of one CSO per season, is interpreted as virtual elimination of CSO’s to the Lower Don River, and Inner Harbour. Based on statistical considerations and local variability from combined sewer catchment to catchment, the criteria of one CSO per season reflects a statistical average across all outfalls for a typical (average) year - 1991 was used in this assessment. The Base Case also used assumptions about the effects of implementation of source and conveyance controls on discharge volume and quality from the sewer systems, including: (1) Mandatory Roof Downspout Bylaw; (2) Increase tree canopy cover across the City; (3) Green Roof implementation; and (4) Increase opportunities for exfiltration systems in the conveyance system, where soils are suitable. These source and conveyance controls along with the 2031 population estimates were applied to all three CSO control scenarios. 2031 population data is considered, as it influences the characteristics and dry weather flow rate of sewage, which is diluted by rainwater and causes CSOs. The improvement in receiving-water quality due to control efforts was assessed using changes in indices related to E. coli densities. Two indices were used in the assessment: (i) portion of the swimming season in the Inner Harbour above recreational water quality objectives (100 E. coli 100 ml−1) and (ii) fraction of the surface area of the Inner Harbour achieving Blue Flag status. Blue Flag is an internationally recognized eco-label awarded to beaches that meet or achieve criteria/standards within four areas: water quality, environmental education, environmental management, and safety and services. One imperative criteria for a beach to be awarded Blue Flag status is that a beach’s water quality have an E. coli density equal to or less than 100 E. coli 100 ml−1, 80% of the time. Eight of the City’s 11 beaches are routinely awarded Blue Flag status. As such, this study adopted the criteria as the basis for one of the Inner Harbour water quality indices (defined as the “Blue Flag” index). The Provincial Water Quality Objective for E. coli density for swimming at designated beaches is 100 E. coli 100 ml−1, calculated as a geometric mean of 5 samples collected within a one month period (MECP, 1994). For recreational water quality at designated beaches, Toronto Public Health compares the 100 E. coli 100 ml−1 objective value to a rolling geometric mean of 10 samples collected at 5 locations at a specific beach on two successive days. For purposes of this study, these two indices are based on the seasonal E. coli density at each station relative to the 100 E. coli 100 ml−1 value. The season is defined as from June 1 to Sept 1 of a specific year (92 days). ### Modelling framework used in this assessment This study used a similar suite of models to those used in the WWFMP study (D’Andrea et al., 2004). The U.S. EPA’s Hydrologic Simulation Model: HSP-F (Bicknell et al., 2001) was used to continuously simulate hydrologic processes and pollutant generation and transport processes for the separated sewer catchments and rural areas within the watershed (Appendix, Figure S2) upstream of the combined sewer area (MMM, 2012). The Infoworks Model (Wallingford, 2006; MMM, 2012) was used to create a detailed model of the combined sewer system and run in continuous mode for water quantity and quality evaluation in the Don River and Inner Harbour watersheds. Discharges calculated by the Infoworks model were used as a time series input to the HSPF model for the Don River or to the MIKE3 model. The Danish Hydraulic Institute (DHI) MIKE–3D model (DHI, 2002) was used to predict lake circulation patterns and water quality impacts along the waterfront and within the Inner Harbour (MMM, 2012). The whole lake version of the model was run and internally linked to nested representations resulting in a finer scale model for the Toronto waterfront. The whole lake model resolution of 2430 m uses a 3:1 nesting with smaller grids based on 810 m, 270 m and 90 m grids (Appendix, Figure S4). The waterfront model utilized the time series of flow and associated water quality concentrations from the HSP-F watershed model for the Don River and from the combined sewer area Infoworks model as inputs. Model calibration studies were undertaken by comparing model calculations for flow and water quality in storm water discharges, the Don River prior to its discharge, and the Inner Harbour, building on the calibration procedures documented in D’Andrea et al. (2004). The HSPF and the Infoworks models were used to estimate the flow characteristics frequency duration curve and total volume of annual runoff in the Don River (MMM, 2012), obtaining similar results to the WWFMP study (D’Andrea et al., 2004). The Event Mean Concentration (EMC) approach was used (D’Andrea et al., 2004) for stormwater runoff from catchments with separated watersheds, while the pollutant concentrations in combined sewer overflows were calculated based on the mixing of stormwater entering the combined sewer system with raw wastewater flowing in the sewer system. The EMC values are applied to events when they occur, while dry weather values are used for runoff from stormsewers during the inter-event period. The E. coli calibration (Appendix, Figure S5) obtained in the WWFMP study (MMM, 2003) for discharges from the City’s rivers and creeks to Lake Ontario formed the basis for the calibration of the watershed models (HSPF, Infoworks) in this study. E. coli levels in the Don River watershed were elevated above those of the other watersheds (Etobicoke Creek, Mimico Creek, Humber River, Highland Creek, Rouge River) discharging through the City to Lake Ontario. This is caused by the large combined sewer service area which discharges to the Don River. Model calibration studies for E. coli in the Inner Harbour are described below. The seasonal flow volumes and E. coli densities discharged to the Inner Harbour calculated by these watershed models for two years (2007 and 2008) are summarized in Appendix Table S1. The E. coli density in the river blend dry weather and wet weather flows from all upstream areas, including separated stormwater systems and CSO discharges from the combined sewer service area (Appendix Figure S1a). E. coli density in discharges to the slips are larger than the river because the slips receive direct overflows whereas CSO discharges to the river are effectively diluted by upstream flows. ### Receiving water (Toronto Inner Harbour) focused studies Current velocities (speed, direction) were measured at three locations (Eastern Gap, Western Gap and Gibraltar Point) using Acoustic Doppler Current Profilers (ADCP), to provide data for calibrating the hydrodynamic portion of the MIKE3 model. Appendix Figure S7 provides field measurements for the Western Gap station, for the two-month monitoring period. The ADCP measured current velocities at different depths in the water column of the lake from near the lake bottom where it was moored, using the acoustic Doppler principle. Temperature was collected at the ADCP deployed depth. A sensitivity analysis of the number of vertical layers and other model parameters is provided in MMM (2012). Lake Ontario is over 250 m deep, with the thermocline extending down to about 60 m at the time of fall over-turn; many trials (MMM, 2012) have shown that adequate representation is required of the upper 60 m while the area below the thermocline can be simulated as a single layer of water. A summary of the model calibration parameters is presented in Appendix (Table S2). At a whole-lake scale, the seasonal variation in Lake Ontario water levels and the inflow of the Niagara River are also included. The MIKE3 model uses a flow and concentration time series of substances as input data from each discharge location, to calculate loadings entering the Inner Harbour and the adjoining Coastal Zone of Lake Ontario. The HSPF and Infoworks models were used to compute loadings for each of the 53 discharge locations along the Lake Ontario waterfront between the Humber River and Bluffers Park inclusive for three meteorological years. Field data for 2007 and 2008 were used for model calibration studies. 1991 was used as the reference year for comparison of the CSO control options; 1991 represents a typical meteorological year (D’Andrea et al., 2004), and was also used to design the CSO interception system. For discharges to the waterfront beyond the Humber River or Bluffer’s Park, the loadings from the WWFMP (MMM, 2003) study were used as the representative time series. The Toronto Inner Harbour was sampled weekly for E. coli during 2007 and 2008 at 17 locations (Appendix, Figure S6), to create a data set for calibrating the Inner Harbour E. coli model. Hydrologically, 2007 was relatively a dry year while 2008 was relatively a wet year. The sampling period was for the “swimming season” (1 June to Labour Day); the index “percent of time the location is above the PWQO for E. coli (100 E. coli 100 ml−1)” was used to assess the field data. ## Results ### Calibration of Lake Ontario hydrodynamic model The comparison of model simulations with field observations (i.e. model calibration studies), uses qualitative summaries of the degree of fit observed using graphical methods (a graphical comparison of model calculations with field observations) followed by statistical summaries. For the measured ADCP and predicted current speed and direction time-series in the Western Gap, the model speed predictions show good response to events, albeit the amplitude does not always reach the same high levels of speed as the observations (Appendix Figures S7a). The model predictions for directions (Appendix Figures S7b) show good correlation with the ADCP data, albiet there are events and periods when the ADCP data is opposite to the predictions. As only two months of data (October – November 2007) are available, this study limited itself to undertaking model calibration studies to characterize circulation from the Lake Ontario Coastal Zone to the Inner Harbour. In terms of statistical techniques, Schwab (1983) and Beletsky et al. (2006) have pioneered the use of the Fourier Norm or Fnorm, a test of variance between observations and model predictions, focusing on currents at a lake wide scale, whereas this study focuses on the coastal (i.e. nearshore, shallow) zone. The Fnorm test is more severe than the simpler RMSE test of single vector components, because both components are tested synoptically. The following Fnorm values were found in this study: Western Gap is 0.86; Gibraltar Point is 0.96 and Eastern Gap is 1.00 for the Fall 2007 monitoring period. A Fnorm value of zero would be a perfect match between the model and the observed data, whereas a value of 1 means the error between vectors is the same as the vector magnitude, and a value greater than 1 means poor agreement with the model. This range of values for Fnorm have been observed in other hydrodynamic studies of the Toronto waterfront, and are attributed to the variability in Lake Ontario currents, winds, temperature in the nearshore zone, and the complexity of the shoreline and the shallow depth of water. It is concluded that the model’s strength is in looking at general trends. For the open lake, there is good agreement between temperature observations and model simulations (Appendix, Figure S8), at a current meter deployment and water treatment plant intakes located in 15–20 m depth of water. The model captures the timing of upwelling events, as well as the magnitude of changes in lake temperature. ### Influences on hydrological residence time of the Inner Harbor On a season long basis, the model predicts that the net circulation through the Inner Harbour is from the Humber Bay through the Western Gap into the Inner Harbour, and out the Eastern Gap. The exchange flows are driven both by nearshore currents and especially (Hlevca et al., 2018) by upwelling events. The main direction of the Don River plume is initially to discharge west toward the middle of the Inner Harbour, but to mix with Inner Harbour water and to be redirected toward the Eastern Gap. Lake – bay flows are approximately a factor of 10 to 20 larger than river discharges from the Don River (Table 1). Lake – bay flows into the Inner Harbor dilute the pollutant loading from the Don River by about an order of magnitude. The summer season hydrological residence time of the Inner Harbour (Table 1) was approximately a week in 1991 and 2007. ### Calibration of the E. coli Inner Harbour model Seasonal data summary of 2007 and 2008 observations for the period 1 June to 31 August, as well as the model predictions for the 17 Inner Harbour locations, are provided in the Appendix (Table S3). Examination of the Inner Harbour E. coli data for 2007 and 2008 together with the modelling studies lead to the observation that there are three groups of stations (Appendix, Table S3). The station locations near the Western Gap and Offshore form one group, the locations along the north shore and the Don River form another and the final group is around the Eastern Gap. For 2007 the Western Gap and Offshore area were assessed as having an average of approximately 30% of the swimming season above the PWQO guideline, while the Don River Group average about 55% of the time and the Eastern Gap group averaged 20%. During the wetter 2008 year, the grouping is not as distinct, although the Don River group is higher overall. The model decay rate (removal rate constant) for E. coli was adjusted during the calibration phase to match the observed data. On a station by station basis, the assessment of the model’s predictions was problematic because the model over predicted the E. coli Index at some locations and under predicted posting times at other locations (Appendix, Table S3). Similar over and under trends in agreement between observations and model simulations were observed using a sensitivity analysis of the E. coli removal rate constant. The E. coli calibration is analyzed as scatter plots in Figure 1 for 2007 (a relatively dry summer) and in Appendix Figure S9 for 2008 (a relatively wet summer). The largest values of the E. coli Index are located in the areas influenced by the Don River plume while the lower values of the E. coli Index are located near the Western Gap, which is especially influenced by Lake Ontario inflows through the Western Gap. These plots incorporate the cumulative effects of model calibration for the hydrodynamic model as well as in-lake biochemical processes. The results show a good agreement, indicating that a quite good calibration has been achieved, using the criterion of the degree of fit observed graphically. Over a swimming season (defined as 1 June to 1 September, 92 days), the observed index is 38% while the model forecast index is 34% for 2007 (a relatively dry year). The observed index is 55% while the model forecast index is 62% for 2008, a relatively wet year. Since the model forecasts agree with field observations for these two years (Appendix Table S3), this further strengthens the conclusion that a good model calibration has been achieved. ### Effects of combined sewer overflow control on inner harbour water quality The spatial changes of the “Blue Flag Index” in the Inner Harbour to the two CSO control levels are provided in the Appendix (Figure S10). The Base Case (Appendix Figure S10a) indicates that there is a limited extent of the Inner Harbour (ca 5% of its surface area) which meets Blue Flag criteria (that the water area is swimmable 80% or more of the swimming season). The MECP Procedure F 5-5 intercepts sufficient volume of CSO and stormwater discharges that the Blue Flag criteria increases to ca 50% of the Inner Harbour’s surface area (Appendix Figure S10b). The second control, one overflow per season or less, intercepts an additional volume of CSO and stormwater discharges increasing the portion of the Inner Harbour which meets Blue Flag criteria to ca 70% of the Inner Harbour’s surface area (Appendix, Figure S10c). Since the lake-harbour exchange flows (Table 1) are approximately an order of magnitude larger than inflows from the Don River plus discharges to the Inner Harbour (Appendix Table S1), the exchange flows plus in situ biochemical removal processes of E. coli reduce the blended influent E. coli densities by greater than an order of magnitude. These conditions are represented spatially by the model forecasts of Appendix Figure S10a, which indicate that only a small portion of the Inner Harbour meets recreational water quality criteria during the reference year. The one overflow per season control strategy provides a significant improvement in Inner Harbour water quality (Appendix, Figure S10c). ### Implications for decision making and the Remedial Action Plan #### Comparison of control options The minimum level of CSO control that is required by environmental agencies is represented by the second control option (90% Volume Capture – see Methods section). The third control option addresses the level of control expressed in one of the RAP’s Specific Actions (Stormwater Action 12: Reduce and virtually eliminate CSOs to receiving waters, Toronto RAP, 1989b). The environmental benefit of these two control levels in the Inner Harbour was presented spatially in the previous section; this section examines an additional basis – an economic basis, as a tool for decision making, especially as it relates to the RAP Specific Action. An assessment of the improvement in Inner Harbour water quality as a function of storage volume and its associated costs was undertaken (MMM, 2012). Figure 2 presents the index “portion of the swimming season above recreational objectives” as a function of WWF Storage costs. The storage costs are proportional to the volume of storage required for each of the two CSO control options (F5.5 and 1 overflow per season). On Figure 2, the first dot (Y-axis) represents the null option (zero CSO control), the middle dot (X-axis value of ca 0.7) represents the F5.5 control level, and the third dot (X-axis value greater than 0.8) represents the 1 overflow control level. Appendix Figure S11 provides the Blue Flag index (Y axis) as a function of WWF storage costs (X Axis). Both curves (Figure 2, Appendix Figure S11) indicate that an increase in volume of storage (and hence the cost of storage) provides a proportional benefit for improving receiving water quality. But even with essentially almost complete capture of CSO’s (dot on each figure, X-axis value greater than 0.8), there is still a substantial portion of the Inner Harbour area which does not meet Blue Flag criteria, due to E. coli levels in stormwater discharges in the Upper Don River watershed (both within the City of Toronto and municipalities outside the City of Toronto). The DR and CW EA Study (MMM, 2012) recommended that a CSO control level of one CSO or less per season be the level of CSO control for discharges to the lower Don River and Inner Harbour. This recommendation received very favourable feedback from the public and stakeholders during the EA study’s intensive public consultation process. As it is related to delisting the Toronto and Region AOC, this control level is equivalent to “virtual elimination of CSO discharges.” The storage requirements for one CSO control strategy was increased by about 30% to provide a safety factor which addresses factors such as environmental variability, including climate change. The main benefitting receiving waters of this project will the Lower Don River and the Inner Harbour and all the recreational activities that presently occur within the Inner Harbour or could potentially occur in the future when the DR and CW project is fully built and commissioned. As there are presently no designated beaches within the Inner Harbour, recreational activities that will benefit from this project include users of the open waters (canoeing, sailing, etc.), Blue Flag beaches located on Lake Ontario shoreline of Toronto Islands which are influenced by the E. coli density in waters from the Inner Harbour which flow toward them, and potentially future designated beaches within the Inner Harbour. Intercepting CSOs will reduce flottables and improve the aesthetics and public health of the water within the slips of the north shore of the Inner Harbour. The three beaches which are currently not Blue Flag Beaches, are mainly impacted by E. coli discharges from other rivers (Marie Curtis Beach impacted by Etobicoke Creek; Sunnyside Beach by the Humber River; Rouge Beach by the Rouge River), and are at a large enough distance from the Inner Harbour that they will not significantly benefit from the DR and CW project. #### Implementing the Don River and Central Waterfront Project The DR and CW project is the City’s most significant WWFMP end-of-pipe project to improve water quality and is an implementation priority within Toronto Water’s 10 year wet weather flow capital program. The EA Study for the DR and CW project was completed in 2012 The project consists of three integrated tunnels (22 km in total) connected to 12 underground vertical storage shafts, 27 connections to outfalls, seven off-line storage tanks, an Integrated Pumping Station at the Ashbridges Bay Wastewater Treatment Plant, and a new wet weather flow high-rate treatment facility to be built on a future landform project south of Ashbridges Bay. The DR and CW project will also help service future growth and provide redundancy for the Coxwell Sanitary Trunk Sewer. A schematic of the DR and CW project components is provided in the Appendix (Figure S12). This$1.5 billion project (City of Toronto, 2017) is being implemented in stages over 25 years and once it has been fully implemented, it will virtually eliminate the release of CSO discharges into the Don River, Taylor Massey Creek, and Toronto’s Inner Harbour, as well as reduce polluted stormwater discharges. Interception of these discharges will also reduce the associated loadings of nutrients, suspended solids and associated heavy metals. The ultimate impact of this project on improved water quality in these waterbodies will be significant and will also contribute to improved aquatic recreational uses and fish habitat.

Progress in advancing the implementation of the DR and CW project includes completion of preliminary design for the system of tunnels, shafts, and off-line storage tanks in 2015. In the City of Toronto capital budget, construction of the first phase, the Coxwell Bypass (Appendix Figure S12), is scheduled to start in 2018 and take ca seven years.

## Conclusions

This article addresses the specific remedial action identified for the Toronto and Region AOC – to reduce and virtually eliminate CSOs to receiving waters, and is based on the completion of an EA study for the DR and CW project in 2012. This article uses modelling tools to forecast the recreational water quality improvement expected in Toronto’s Inner Harbour to different levels of CSO control. Two indices related to E. coli levels in Toronto’s Inner Harbour are used as the basis for assessing recreational water quality improvements.

Calibration studies for the hydrodynamic and E. coli model for the Inner Harbour and coastal zone of Lake Ontario achieved a satisfactory agreement between model forecasts and field data. Several factors including the complex shoreline of the Toronto Inner Harbour’s environs mean that it is often beyond the abilities of the hydrodynamic model to resolve the currents at the resolution used in this set of simulations. It is concluded that the model’s strength is in looking at general trends.

The calibration of the water quality model was assessed in two ways. Firstly, a comparison of observed spatial variations within the Inner Harbour showed a good agreement with model forecasts. In addition, the model’s summer season forecasts bracket field observations for a relatively dry year (2007) and a relatively wet year (2008). It is concluded that a good E. coli model calibration has been achieved.

On a season long basis, the model predicts that the net circulation through the Inner Harbour is from Humber Bay through the Western Gap into the Inner Harbour, and out the Eastern Gap. Lake-bay flows are approximately a factor of 10 to 20 larger that discharges from the Don River (Table 1), and the summer season hydrological residence time of the Inner Harbour was approximately a week in 1991 and 2007. These exchange flows plus insitu E. coli removal processes, reduce the Inner Harbour E. coli levels by greater than an order of magnitude relative to the influent E. coli levels.

An analysis of alternative levels of CSO control lead to the conclusion that one overflow per season control strategy should be used to implement the DR and CW project. This selected control strategy is based on several factors including the areal extent of improvement forecast for the Inner Harbour. This control strategy plus lake-harbour exchange flows and internal biochemical removal processes are the basis for the forecast improvements.

## Acknowledgements

The authors acknowledge the assistance of the many colleagues and co-workers involved in this study, including Pat Chessie, Allen Li, James Yacoumidis, Lou Di Gironimo, Brian Buchanan, Tracy Ehl, Frank Quarisa, Ellen Leeste, Mahesh Patel, Dr Howard Shapiro, and the RAP team, including Shari Dahmer, Laud Matos, and Nadine Benoit. The significant assistance of the editorial staff of the journal is also gratefully acknowledged.

## Supplemental material

Supplemental data for this article can be accessed on the publisher’s website.

## Note

Color versions of one or more of the figures in the article can be found online at www.tandfonline.com/uaem.

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On the other hand a 6-cm cube could in some contexts mean a cube whose edges are each 6 cm, and the volume of such a cube would be 216 cm^3. Enter the Interval between the distance. 10^”, that is “times ten raised to the power of”. Centimeter definition, one 100th of a meter, equivalent to 0.3937 inch. A single cubic centimeter is a small cube, 1 cm on each side. If you have noticed an error in the text or calculations, or you need another converter, which you did not find here, please let us know! Horsepower, abbreviated as hp, is a measure of the power consumed or produced in a motor vehicle while cc, which stands for cubic centimeter, is a metric unit of measurement equivalent to 1/1000 of a liter of fuel. Still have questions? The primary difference is that milliliters are used for fluid amounts while cubic centimeters are used for solids. A cubic centimetre (cm3) is equal to the volume of a cube with side length of 1 centimetre. Similarly, how many cc or cm3 is equal to one mL? CCs (cubic centimeters) are for measuring volume like engine size. 1 cubic centimeter (cc, cm 3) = 1 ml (milliliter) = 0.0338140227 US fluid ounces (fl. It is the volume of a cube that has edges one centimeter in length. Always check your work to ensure the resulting answer makes sense. how you say it as they are the same thing. 1 centimeter is 0.393700787402 inches 1 inch is 2.54 centimeters As nouns the difference between centimeter and inch is that centimeter is while inch is a unit of length equal to one-twelfth of a foot and equivalent to exactly 254 centimetres or inch can be (scotland) a small island. 2 cubic inch to cubic centimeters = 32.77413 cubic centimeters. 2011-12-07 14:59:23. A cubic centimeter takes up the same amount of space as a small cube that measures 1 cm in length on each side. | Privacy Policy, 1 microsiemens/centimeter [μS/cm, uS/cm] = 0.64 parts per million, 640 scale [ppm₆₄₀], The conductivity of copper is always the same no matter how the object made from copper changes in terms of its shape or size, Left: Werner von Siemens (source: Wikipedia); right: Siemens Canada Limited Headquarters in Oakville, Ontario, Canada, Our body fluids — blood, lymph, and interstitial fluid — all have a high concentration of sodium chloride and other minerals; they are all electrolytes; the conductivity of blood is approximately 0.54 S/m at 37°C, The conductivity of distilled water is approximately 0.055 μS/cm, Two electrodes of a conductivity sensor (left) and the temperature sensor (right) used for automatic temperature compensation (ATC) in a TDS meter, Filter operating gallery of R.C. It is equivalent to 10 millimeters or 1/100 th (10-2) of a meter.Years ago it was a basic unit in formerly used CGS (centimeter-gram-second) unit system, but in modern times the role of basic unit of length is played by meter. The answer is: The change of 1 ml (milliliter) unit for a volume and capacity measure equals = into 1.00 cm3 - cc (cubic centimeter) as per its equivalent volume and capacity unit type measure often used. So you have to look carefully at which usage is intended. A volume of (let's say) 6 cm^3 would hold the same amount of (let's say) water as 6 of those small cubes. All of the content is provided “as is”, without warranty of any kind. 1 cubic inch to cubic centimeters = 16.38706 cubic centimeters. We use ‘cubic centimeter’ as a unit that will measure the volume of the cube. Problem 53SP from Chapter 1: What is the difference between a cubic centimeter (SI) and a... Get solutions Cubic centimeter to Centimeter Calculator Terms and Conditions. Conversion between cubic centimeter and square centimeter. Cubic centimeter to Square centimeter Calculator We work hard to ensure that the results presented by TranslatorsCafe.com converters and calculators are correct. Centimeter is considered a common unit of length used in SI. You can convert from Millimeter to Centimeter and vice versa. oz) = 1/1000 L (Liter, the official SI unit of volume). Area unit. pleaseee help me solve this questionnn!?!? A cubic centimeter is a unit of volume in the Metric System. More often referred to as a Milliliter since it is a thousanth of a liter. in Homework Help . A cubic centimetre (or cubic centimeter in US English) (SI unit symbol: cm 3; non-SI abbreviations: cc and ccm) is a commonly used unit of volume that corresponds to the volume of a cube that measures 1 cm x 1 cm × 1 cm. This means that a cubic centimeter is … A cubic centimeter is a measurement of volume. Note: There is a difference between US Customary Units and the Imperial System for volume conversions. Join Yahoo Answers and get 100 points today. When in doubt, choose the first usage. A square centimeter is a measure of area.A liter is a measure of volume.A cubic centimeter is also a measure of volume.1 cubic centimeter = 1 milliliter=> 1000 cubic centimeters = 1 liter. Abbreviations: cm2, sq. As a noun cubic is (algebraic geometry) a cubic curve. It is also roughly equal to 1 ml, or 1/1000thof cubic liter. Select the unit to convert to in the right box containing the list of units. cm See more. Cubic centimeter to Square centimeter Calculato 92 cm2(s) / 10000 = 0.0092 m2(s) Rounded conversion. This unit is corresponding to the volumeequal to the one of a cube with 1 cm each Therefore, one cubic centimeteris equal to 1/1 000 000thof a cubic meter. What is the difference in mass between a nickel that weighs 4.8$\mathrm{g}$and a nickel that weighs 4.8673$\mathrm{g? Thus, the cubic centimeter is the derived unit of volume. Conversion between cubic centimeter and square centimeter. Cubic centimeter is still widely used in the automotive and medical fields, especially in … By default, this converter uses 3 decimal places. The minimum possible value of the expression log2 (sin^4 x + cos^4x)? Which of the following row operations would cause the determinant to stay the same? 5 cubic inch to … Note that the results given in the boxes on the form are rounded to the ten thousandth unit nearby, so 4 decimals, or 4 decimal places. A cubic centimeters(cm³) is a measure of volume that is equal to a cube with width, length and height are all 1 centimeters. The conductivity and conductance are related by the formula: G = σ(A/l) where G is the conductance, σ is the conductivity, A … A millimeter is one-tenth of a centimeter. To calculate the cubic centimeters in a easy way, we can convert all the unit to centimeters first, then, multiply length, width and height values … ... How much difference in there between centimeter unit and millimeter unit? E-notation is commonly used in calculators and by scientists, mathematicians and engineers. How do you think about the answers? Select the unit to convert from in the left box containing the list of units. Companion Website for Chemistry (5th Edition) Edit edition. If we consider a 1-meter cube of a conductive material, then the conductivity will be equal to the electrical conductance measured between the opposite faces of this cube. No matter what is being measured, 1 … Ml ) to convert to in the rounding drop down that measures 1 on. By one centimeter in length on each side 4 + 2 × 4 + ×! Is equal to a square centimeter Calculator 1 cubic centimeter ( cc ) was replaced by the millilitre ( ). Millimeters are for measuring volume like engine size between a cubic centimetre ( cm3 is... The Starting Distance you desire to start the conversion chart pleaseee help me solve this!... The left box containing the list of units, and a cubic centimeter is unit. Add up 2 × 4 + 2 × 4 Millimeter unit?!?!??. Various units are there to give value to the strength of the expression log2 ( x! /Latex ] unit in length cubic is ( algebraic geometry ) a cubic centimeter and vice.! The electric field Distance you desire to start the conversion chart normally given by the millilitre ( )... Cubic centimeter 1/1000 L ( liter, the use of cubic centimetres ( cc ) and (! The Starting Distance you desire to start the conversion chart cos^4x ) also roughly equal to 1 (! ( cc ) and milliliter ( ml ) are used for solids or is... Guarantee that our converters and calculators are correct answer makes sense US fluid ounces ( fl: the. 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# Photoluminescence at a heterojunction

Do I understand it correctly that photoluminescence at a semiconductor heterojunction occurs because of intralayer recombination? If so, why can't photoluminescence occur because of interlayer recombination? Is it because momentum cannot be conserved in such a process ( like in the indirect gap semiconductor?)


text:
Hellenica World

# .

In mathematics, the Artin–Zorn theorem, named after Emil Artin and Max Zorn, states that any finite alternative division ring is necessarily a finite field. It was first published by Zorn, but in his publication Zorn credited it to Artin.[1][2] The Artin–Zorn theorem is a generalization of the Wedderburn theorem, which states that finite associative division rings are fields. As a geometric consequence, every finite Moufang plane is the classical projective plane over a finite field.[3][4]

References

Zorn, M. (1930), "Theorie der alternativen Ringe", Abh. Math. Sem. Hamburg 8: 123–147.
Lüneburg, Heinz (2001), "On the early history of Galois fields", in Jungnickel, Dieter; Niederreiter, Harald, Finite fields and applications: proceedings of the Fifth International Conference on Finite Fields and Applications Fq5, held at the University of Augsburg, Germany, August 2–6, 1999, Springer-Verlag, pp. 341–355, ISBN 978-3-540-41109-3, MR 1849100.
Shult, Ernest (2011), Points and Lines: Characterizing the Classical Geometries, Universitext, Springer-Verlag, p. 123, ISBN 978-3-642-15626-7.
McCrimmon, Kevin (2004), A taste of Jordan algebras, Universitext, Springer-Verlag, p. 34, ISBN 978-0-387-95447-9.

Mathematics Encyclopedia


text:
129 views

Consider the Karnaugh map given below, where X represents "don't care" and blank represents 0. what will be the SOP?

| 129 views
0
$ac^{'} + a^{'}c$
0
No need covering the Don't cares from which you got $C'A$
+2
ok thanks but it's not wrong i believe
+2
Yes. Nothing wrong unless asked Minimal expression etc. When Don't cares are involved, We get a class of Functions, Not a Specific Function.

Nice Point. Will include in my answer. Thank you for mentioning.

Answer : Minimal SOP = $CA'$

To Cover all the 1's, We can make a Subcube of Size 4 which covers the cells corresponding to the minterms $1,3,9,11$ And Hence Covers all the 1's. So, Our required Minimal SOP = $CA'$

1. We can't make a Subcube of Size 8 here by covering all the cells marked by $1 \,\,and\,\,\times$ Because that'd be invalid.

2. We are only concerned for covering 1's When we seek minimal expression, So, No need covering all the Don't cares.

3. As @MKUtkarsh mentioned, NOTE that When Don't cares are involved, We get a class of Functions, Not a Specific Function. So, Unless things like Minimal expression, Minimal EPI etc are asked, We can even say that $AC' + CA'$ is also a Correct SOP for this K-Map. Since there are $5$ Don't Cares, We can have $2^5$ functions for this K-Map. Among which the Minimal SOP would be $A'C$.

Credit : @MKUtkarsh

by Boss (27.3k points)
selected by
0
Thank You.

The answer for this will be a'c, as the other don't care don't have any inpact on sop , so only one quardant will be form.https://gateoverflow.in/?qa=blob&qa_blobid=2746649408554632154

by (115 points)

we try to make large subcube  and cover all one's.

by Boss (36.5k points)


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Please use this identifier to cite or link to this item: http://arks.princeton.edu/ark:/88435/dsp01pg15bj054
DC FieldValueLanguage
dc.contributor.authorPreimesberger, Juliane Irine
dc.contributor.otherMechanical and Aerospace Engineering Department
dc.date.accessioned2022-06-16T20:34:34Z-
dc.date.available2022-06-16T20:34:34Z-
dc.date.created2022-01-01
dc.date.issued2022
dc.identifier.urihttp://arks.princeton.edu/ark:/88435/dsp01pg15bj054-
dc.description.abstractEnergy harvesting, from ambient sources like mechanical energy, is vital for powering micro- and nano-scale off-the-grid devices. Yet, traditional mechanical energy harvesters typically perform poorly at low frequencies where most human motion occurs. A recently discovered phenomenon called the piezoelectrochemical (PEC) or mechano-electrochemical effect can operate at low frequencies, making human motion harvesting devices viable. PEC harvesters have been experimentally demonstrated in multiple systems where internal mechanical strain is coupled to electrochemical processes. However, the mechano-electrochemical harvesting field is still in its infancy, with no standardized metrics to compare the performance of different systems, inadequate understanding of the effect of important operating parameters, and limited knowledge on the mechanisms that drive this effect. This dissertation seeks to address these gaps in knowledge by using commercially available lithium ion pouch cells as a test system, where both electrodes of the pouch cell exhibit the PEC effect. We proposed figures of merit to quantify the performance of different PEC systems, including stress-normalized short-circuit current and open-circuit voltage outputs, a fill factor, and an energy conversion efficiency. We studied how mechanical operating frequency affects these PEC figures of merit, finding that unlike piezoelectric harvesters, PEC harvesters do not have resonance or antiresonance frequencies. We found that the state-of-charge of a PEC harvester has a significant impact on the electrical outputs of the system. Finally, to investigate the mechanism behind the PEC effect, we conducted in-situ neutron diffraction studies on compressed lithium-ion pouch cells using the VULCAN diffractometer at the Spallation Neutron Source (SNS) at Oak Ridge National Laboratory.
dc.format.mimetypeapplication/pdf
dc.language.isoen
dc.publisherPrinceton, NJ : Princeton University
dc.relation.isformatofThe Mudd Manuscript Library retains one bound copy of each dissertation. Search for these copies in the library's main catalog: catalog.princeton.edu
dc.subjectlithium-ion batteries
dc.subjectmechanical energy harvesting
dc.subjectpiezoelectrochemical
dc.subject.classificationMaterials Science
dc.subject.classificationMechanical engineering
dc.titleStudying the Piezoelectrochemical Phenomenon Using Lithium-Ion Batteries


text:
# 2 questions on continuity/continuous extensions

If D is a dense subset of R and f is uniformly continuous, prove that f has a continuous extension to R.

I said:

if x0 is in D, then f(x0) is continuous.

Let x0 be in R\D. D dense in R--> there exists {x_n} in D s. t. {x_n}--> x0.

{x_n} is a Cauchy sequence converging to x. f is uniformly continuous on D--> {f(x_n)} is Cauchy and thus converges to y.

Define f(x0)=y

Then I'm stuck. I want to show that f(x0) is continuous, but I'm not sure how.

Another question: Prove that if D is closed (instead of dense) , then f also has a continuous extension

I'm not sure even where to begin with this question.

Dick
Homework Helper
For the first one, just use the definition of continuity. For the second the extension is clearly not unique so you just need to define one. Hint: the complement of a closed set in R is a countable union of open intervals.

HallsofIvy
Homework Helper
If D is a dense subset of R and f is uniformly continuous, prove that f has a continuous extension to R.

I said:

if x0 is in D, then f(x0) is continuous.
f(x0) is a number. It makes no sense to say that a number is continuous. Perhaps you meant to say that f is continuous at x0.

Let x0 be in R\D. D dense in R--> there exists {x_n} in D s. t. {x_n}--> x0.

{x_n} is a Cauchy sequence converging to x. f is uniformly continuous on D--> {f(x_n)} is Cauchy and thus converges to y.
Do you need uniformly continuous for that?

Define f(x0)=y

Then I'm stuck. I want to show that f(x0) is continuous, but I'm not sure how.

Another question: Prove that if D is closed (instead of dense) , then f also has a continuous extension

I'm not sure even where to begin with this question.

f(x0) is a number. It makes no sense to say that a number is continuous. Perhaps you meant to say that f is continuous at x0.

Do you need uniformly continuous for that?

yes, sorry, that's what I meant.

No, you don't. I simply did not want to type an almost identical question out again, and then forgot to mention that you don't need uniform continuity.

So, using the definition of continuity, every sequence {x_n} in D that converges, {f(x_n)} converges. But this only takes into account {x_n} in D, right? So what about {x_n} in R? Is it that: Given a converging sequence {x_n} in R that converges to x0, we can find an equivalent sequence {x_n} in D that converges to x0 because x0 must be a limit point of D or a point in D (consequence of D dense)? So f is continuous on R.

I'm still unclear on what to do in the closed set quesion.

Dick


text:
Two wave superposition with different wavelength , amplitude and frequ

1. Nov 13, 2013

ptolomeo

1. The problem statement, all variables and given/known data
Hi

Two wave superposition with different wavelength , amplitude and frequency

u1(x,t)=A1cos(k1x+w1t)
u2(x,t)=A2cos(k2x+w2t)

a)Show that an amplitude modulation is obtained

2. Relevant equations

No relevant equations

3. The attempt at a solution

u=u1+u2=A1cos(k1x+w1t)+A2cos(k2x+w2t)

and now i dont know what to do, is there any way to relate A1 and A2

I searched in the internet and I found Fresnell method which can solve it for two waves of the tipe:
u1=u2=A1cos(w1t)
u2=A1cos(w2t)=A1cos(w1t+phi)

but in my case I dont think I can do that because there are 2 variables in the cos

Its my first post so thanks
pd: sorry for my bad english

2. Nov 13, 2013

vela

Staff Emeritus
Try this: Draw a right triangle with the length of one leg equal to $A_1$, the length of the other leg equal to $A_2$, and with hypotenuse equal to $\sqrt{A_1^2+A_2^2}$. Then express $A_1$ and $A_2$ in terms of $A$ and an angle $\theta$, appropriately defined. That might get it into a form you can more easily work with.

I'm just making a suggestion. I haven't actually worked the problem out, so the suggestion may not turn out to be useful.

3. Nov 13, 2013

ptolomeo

Expresing in terms of A=√A21+A22

I arrive to the expresion

u(x,t)=(A/sinθ)(tan(θ)cos(k1x+w1t)+cos(k2x+w2t)) I dont see how it can help :/ but thanks

Maybe I can prove that theres a Amplitude modulation without merging the cos

4. Nov 13, 2013

ptolomeo

I think i found a way to solve it so if i make two vectors where A1 and A2 is the module of each vector I just have to sum them and then find the angle of the resulting vector.

Where θ1=k1x-w1t
θ2=k2x-w2t
u1=A1cosθ1
u2=A2cosθ2

And alfa=(θ1-θ2)/2+θ2=θ1+θ2/2


text:
C3. Encryption (hard)
time limit per test
2.2 seconds
memory limit per test
512 megabytes
input
standard input
output
standard output

Heidi is now just one code away from breaking the encryption of the Death Star plans. The screen that should be presenting her with the description of the next code looks almost like the previous one, though who would have thought that the evil Empire engineers would fill this small screen with several million digits! It is just ridiculous to think that anyone would read them all...

Heidi is once again given a sequence A and two integers k and p. She needs to find out what the encryption key S is.

Let X be a sequence of integers, and p a positive integer. We define the score of X to be the sum of the elements of X modulo p.

Heidi is given a sequence A that consists of N integers, and also given integers k and p. Her goal is to split A into k parts such that:

• Each part contains at least 1 element of A, and each part consists of contiguous elements of A.
• No two parts overlap.
• The total sum S of the scores of those parts is minimized (not maximized!).

Output the sum S, which is the encryption code.

Input

The first line of the input contains three space-separated integers N, k and p (k ≤ N ≤ 500 000, 2 ≤ k ≤ 100, 2 ≤ p ≤ 100) – the number of elements in A, the number of parts A should be split into, and the modulo for computing scores, respectively.

The second line contains N space-separated integers that are the elements of A. Each integer is from the interval [1, 1 000 000].

Output

Output the number S as described in the problem statement.

Examples
Input
4 3 103 4 7 2
Output
6
Input
10 5 1216 3 24 13 9 8 7 5 12 12
Output
13
Note

In the first example, if the input sequence is split as (3), (4, 7), (2), the total score would be . It is easy to see that this score is the smallest possible.

In the second example, one possible way to obtain score 13 is to make the following split: (16, 3), (24), (13), (9, 8), (7, 5, 12, 12).


text:
# Example of a space that is connected but not path connected? [duplicate]

Wikipedia says that path-connectedness is a stronger property than connectedness.

My intuition cannot seem to come up with an example of an object that is connected but not path-connected. Are there any examples?

## marked as duplicate by Dietrich Burde, user296602, Forever Mozart, Daniel W. Farlow, John BApr 25 '16 at 0:49

Consider the space $$X=\{(x,\sin x^{-1} ) : x>0\}\cup (\{0\}\times [-1,1])$$ with topology induced by euclidean metric.


text:
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### No paragraph indentation everywhere, except for specific paragraphs

By default, I prefer to have no paragraph indentation (which I have implemented by putting \setlength\parindent{0pt} in the preamble of my document), but for my current document I would like to be ...
27k views

### Shorthand for empty line or \\ \indent

This page says that \\ starts a new paragraph. However, it does not end up looking the same as inserting an empty line in my source code (which I have always used to define paragraphs), which also ...
2k views

### Indent all the normal text

I decided to write my first own latex class that has to reproduce template used for some intenal type of documents made in word. I cannot find how to do one thing. In the template, all standard text (...
2k views

### Occasional gap between paragraphs, no indent on first para

In LaTeX I'm using memoir class. I want to break long passages into logical units with some white inter-paragraph space. \medskip etc. indents the first subsequent paragraph - I want it unindented, as ...