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This article may require copy editing for tone. You can assist by editing it. (May 2022) (Learn how and when to remove this template message)
2.1 Using linear regression
2.2 Using recursive formula
2.3 Using matrix inversion
3.1 Geometrical
3.2 As conditional independence test
4 Semipartial correlation (part correlation)
5 Use in time series analysis
Using linear regression[edit]
{\displaystyle \mathbf {w} _{X}^{*}}
{\displaystyle \mathbf {w} _{Y}^{*}}
{\displaystyle \mathbf {w} _{X}^{*}=\arg \min _{\mathbf {w} }\left\{\sum _{i=1}^{N}(x_{i}-\langle \mathbf {w} ,\mathbf {z} _{i}\rangle )^{2}\right\}}
{\displaystyle \mathbf {w} _{Y}^{*}=\arg \min _{\mathbf {w} }\left\{\sum _{i=1}^{N}(y_{i}-\langle \mathbf {w} ,\mathbf {z} _{i}\rangle )^{2}\right\}}
{\displaystyle \langle \mathbf {w} ,\mathbf {v} \rangle }
{\displaystyle e_{X,i}=x_{i}-\langle \mathbf {w} _{X}^{*},\mathbf {z} _{i}\rangle }
{\displaystyle e_{Y,i}=y_{i}-\langle \mathbf {w} _{Y}^{*},\mathbf {z} _{i}\rangle }
{\displaystyle {\hat {\rho }}_{XY\cdot \mathbf {Z} }={\frac {N\sum _{i=1}^{N}e_{X,i}e_{Y,i}-\sum _{i=1}^{N}e_{X,i}\sum _{i=1}^{N}e_{Y,i}}{{\sqrt {N\sum _{i=1}^{N}e_{X,i}^{2}-\left(\sum _{i=1}^{N}e_{X,i}\right)^{2}}}~{\sqrt {N\sum _{i=1}^{N}e_{Y,i}^{2}-\left(\sum _{i=1}^{N}e_{Y,i}\right)^{2}}}}}}
{\displaystyle ={\frac {N\sum _{i=1}^{N}e_{X,i}e_{Y,i}}{{\sqrt {N\sum _{i=1}^{N}e_{X,i}^{2}}}~{\sqrt {N\sum _{i=1}^{N}e_{Y,i}^{2}}}}}.}
> generalCorr::parcorMany(cbind(X,Y,Z))
nami namj partij partji rijMrji
[1,] "X" "Y" "0.8844" "1" "-0.1156"
[2,] "X" "Z" "0.1581" "1" "-0.8419"
Using recursive formula[edit]
{\displaystyle Z_{0}\in \mathbf {Z} ,}
that[citation needed]
{\displaystyle \rho _{XY\cdot \mathbf {Z} }={\frac {\rho _{XY\cdot \mathbf {Z} \setminus \{Z_{0}\}}-\rho _{XZ_{0}\cdot \mathbf {Z} \setminus \{Z_{0}\}}\rho _{Z_{0}Y\cdot \mathbf {Z} \setminus \{Z_{0}\}}}{{\sqrt {1-\rho _{XZ_{0}\cdot \mathbf {Z} \setminus \{Z_{0}\}}^{2}}}{\sqrt {1-\rho _{Z_{0}Y\cdot \mathbf {Z} \setminus \{Z_{0}\}}^{2}}}}}.}
{\displaystyle {\mathcal {O}}(n^{3})}
Note in the case where Z is a single variable, this reduces to:[citation needed]
{\displaystyle \rho _{XY\cdot Z}={\frac {\rho _{XY}-\rho _{XZ}\rho _{ZY}}{{\sqrt {1-\rho _{XZ}^{2}}}{\sqrt {1-\rho _{ZY}^{2}}}}}}
Using matrix inversion[edit]
{\displaystyle {\mathcal {O}}(n^{3})}
{\displaystyle \mathbf {V} \setminus \{X_{i},X_{j}\}}
{\displaystyle \rho _{X_{i}X_{j}\cdot \mathbf {V} \setminus \{X_{i},X_{j}\}}=-{\frac {p_{ij}}{\sqrt {p_{ii}p_{jj}}}}.}
Geometrical[edit]
The same also applies to the residuals eY,i generating a vector eY. The desired partial correlation is then the cosine of the angle φ between the projections eX and eY of x and y, respectively, onto the hyperplane perpendicular to z.[3]: ch. 7
As conditional independence test[edit]
{\displaystyle {\hat {\rho }}_{XY\cdot \mathbf {Z} }}
{\displaystyle z({\hat {\rho }}_{XY\cdot \mathbf {Z} })={\frac {1}{2}}\ln \left({\frac {1+{\hat {\rho }}_{XY\cdot \mathbf {Z} }}{1-{\hat {\rho }}_{XY\cdot \mathbf {Z} }}}\right).}
{\displaystyle H_{0}:\rho _{XY\cdot \mathbf {Z} }=0}
{\displaystyle H_{A}:\rho _{XY\cdot \mathbf {Z} }\neq 0}
{\displaystyle {\sqrt {N-|\mathbf {Z} |-3}}\cdot |z({\hat {\rho }}_{XY\cdot \mathbf {Z} })|>\Phi ^{-1}(1-\alpha /2),}
Semipartial correlation (part correlation)[edit]
Use in time series analysis[edit]
{\displaystyle \varphi (h)=\rho _{X_{0}X_{h}\,\cdot \,\{X_{1},\,\dots \,,X_{h-1}\}}.}
^ a b Baba, Kunihiro; Ritei Shibata; Masaaki Sibuya (2004). "Partial correlation and conditional correlation as measures of conditional independence". Australian and New Zealand Journal of Statistics. 46 (4): 657–664. doi:10.1111/j.1467-842X.2004.00360.x.
^ Guilford J. P., Fruchter B. (1973). Fundamental statistics in psychology and education. Tokyo: McGraw-Hill Kogakusha, LTD.
^ Rummel, R. J. (1976). "Understanding Correlation".
^ Kendall MG, Stuart A. (1973) The Advanced Theory of Statistics, Volume 2 (3rd Edition), ISBN 0-85264-215-6, Section 27.22
^ Fisher, R.A. (1924). "The distribution of the partial correlation coefficient". Metron. 3 (3–4): 329–332.
^ https://web.archive.org/web/20140206182503/http://luna.cas.usf.edu/~mbrannic/files/regression/Partial.html. Archived from the original on 2014-02-06. {{cite web}}: Missing or empty |title= (help)
^ StatSoft, Inc. (2010). "Semi-Partial (or Part) Correlation", Electronic Statistics Textbook. Tulsa, OK: StatSoft, accessed January 15, 2011.
Prokhorov, A.V. (2001) [1994], "Partial correlation coefficient", Encyclopedia of Mathematics, EMS Press
Retrieved from "https://en.wikipedia.org/w/index.php?title=Partial_correlation&oldid=1088288836"
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Growth of the Weil–Petersson diameter of moduli space
15 January 2012 Growth of the Weil–Petersson diameter of moduli space
William Cavendish, Hugo Parlier
Duke Math. J. 161(1): 139-171 (15 January 2012). DOI: 10.1215/00127094-1507312
In this paper we study the Weil–Petersson geometry of
\overline{{ℳ}_{g,n}}
, the compactified moduli space of Riemann surfaces with genus g and n marked points. The main goal of this paper is to understand the growth of the diameter of
\overline{{ℳ}_{g,n}}
g
n
. We show that this diameter grows as
\sqrt{n}
in
, and is bounded above by
C\sqrt{g}logg
g
C
. We also give a lower bound on the growth in
g
of the diameter of
\overline{{ℳ}_{g,n}}
in terms of an auxiliary function that measures the extent to which the thick part of moduli space admits radial coordinates.
William Cavendish. Hugo Parlier. "Growth of the Weil–Petersson diameter of moduli space." Duke Math. J. 161 (1) 139 - 171, 15 January 2012. https://doi.org/10.1215/00127094-1507312
Secondary: 30FXX
William Cavendish, Hugo Parlier "Growth of the Weil–Petersson diameter of moduli space," Duke Mathematical Journal, Duke Math. J. 161(1), 139-171, (15 January 2012)
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(Redirected from Deque)
"Deque" redirects here. Not to be confused with dequeueing, a queue operation.
Not to be confused with Double-ended priority queue.
In computer science, a double-ended queue (abbreviated to deque, pronounced deck, like "cheque"[1]) is an abstract data type that generalizes a queue, for which elements can be added to or removed from either the front (head) or back (tail).[2] It is also often called a head-tail linked list, though properly this refers to a specific data structure implementation of a deque (see below).
2 Distinctions and sub-types
4.1 Purely functional implementation
4.1.1 Real-time deques via lazy rebuilding and scheduling
4.1.2 Implementation without laziness
Deque is sometimes written dequeue, but this use is generally deprecated in technical literature or technical writing because dequeue is also a verb meaning "to remove from a queue". Nevertheless, several libraries and some writers, such as Aho, Hopcroft, and Ullman in their textbook Data Structures and Algorithms, spell it dequeue. John Mitchell, author of Concepts in Programming Languages, also uses this terminology.
Distinctions and sub-types[edit]
This differs from the queue abstract data type or first in first out list (FIFO), where elements can only be added to one end and removed from the other. This general data class has some possible sub-types:
An input-restricted deque is one where deletion can be made from both ends, but insertion can be made at one end only.
An output-restricted deque is one where insertion can be made at both ends, but deletion can be made from one end only.
Both the basic and most common list types in computing, queues and stacks can be considered specializations of deques, and can be implemented using deques.
The basic operations on a deque are enqueue and dequeue on either end. Also generally implemented are peek operations, which return the value at that end without dequeuing it.
Names vary between languages; major implementations include:
insert element at back inject, snoc, push Append push_back offerLast push array_push append push push_back push
insert element at front push, cons Prepend push_front offerFirst unshift array_unshift appendleft unshift push_front unshift
remove last element eject Delete_Last pop_back pollLast pop array_pop pop pop pop_back pop
remove first element pop Delete_First pop_front pollFirst shift array_shift popleft shift pop_front shift
examine last element peek Last_Element back peekLast $array[-1] end <obj>[-1] last back <obj>[<obj>.length - 1]
examine first element First_Element front peekFirst $array[0] reset <obj>[0] first front <obj>[0]
There are at least two common ways to efficiently implement a deque: with a modified dynamic array or with a doubly linked list.
The dynamic array approach uses a variant of a dynamic array that can grow from both ends, sometimes called array deques. These array deques have all the properties of a dynamic array, such as constant-time random access, good locality of reference, and inefficient insertion/removal in the middle, with the addition of amortized constant-time insertion/removal at both ends, instead of just one end. Three common implementations include:
Storing deque contents in a circular buffer, and only resizing when the buffer becomes full. This decreases the frequency of resizings.
Allocating deque contents from the center of the underlying array, and resizing the underlying array when either end is reached. This approach may require more frequent resizings and waste more space, particularly when elements are only inserted at one end.
Storing contents in multiple smaller arrays, allocating additional arrays at the beginning or end as needed. Indexing is implemented by keeping a dynamic array containing pointers to each of the smaller arrays.
Double-ended queues can also be implemented as a purely functional data structure.[3]: 115 Two versions of the implementation exist. The first one, called 'real-time deque, is presented below. It allows the queue to be persistent with operations in O(1) worst-case time, but requires lazy lists with memoization. The second one, with no lazy lists nor memoization is presented at the end of the sections. Its amortized time is O(1) if the persistency is not used; but the worst-time complexity of an operation is O(n) where n is the number of elements in the double-ended queue.
Let us recall that, for a list l, |l| denotes its length, that NIL represents an empty list and CONS(h, t) represents the list whose head is h and whose tail is t. The functions drop(i, l) and take(i, l) return the list l without its first i elements, and the first i elements of l, respectively. Or, if |l| < i, they return the empty list and l respectively.
Real-time deques via lazy rebuilding and scheduling[edit]
A double-ended queue is represented as a sextuple (len_front, front, tail_front, len_rear, rear, tail_rear) where front is a linked list which contains the front of the queue of length len_front. Similarly, rear is a linked list which represents the reverse of the rear of the queue, of length len_rear. Furthermore, it is assured that |front| ≤ 2|rear|+1 and |rear| ≤ 2|front|+1 - intuitively, it means that both the front and the rear contains between a third minus one and two thirds plus one of the elements. Finally, tail_front and tail_rear are tails of front and of rear, they allow scheduling the moment where some lazy operations are forced. Note that, when a double-ended queue contains n elements in the front list and n elements in the rear list, then the inequality invariant remains satisfied after i insertions and d deletions when (i+d) ≤ n/2. That is, at most n/2 operations can happen between each rebalancing.
Let us first give an implementation of the various operations that affect the front of the deque - cons, head and tail. Those implementation do not necessarily respect the invariant. In a second time we'll explain how to modify a deque which does not satisfy the invariant into one which satisfy it. However, they use the invariant, in that if the front is empty then the rear has at most one element. The operations affecting the rear of the list are defined similarly by symmetry.
empty = (0, NIL, NIL, 0, NIL, NIL)
fun insert'(x, (len_front, front, tail_front, len_rear, rear, tail_rear)) =
(len_front+1, CONS(x, front), drop(2, tail_front), len_rear, rear, drop(2, tail_rear))
fun head((_, CONS(h, _), _, _, _, _)) = h
fun head((_, NIL, _, _, CONS(h, NIL), _)) = h
fun tail'((len_front, CONS(head_front, front), tail_front, len_rear, rear, tail_rear)) =
(len_front - 1, front, drop(2, tail_front), len_rear, rear, drop(2, tail_rear))
fun tail'((_, NIL, _, _, CONS(h, NIL), _)) = empty
It remains to explain how to define a method balance that rebalance the deque if insert' or tail broke the invariant. The method insert and tail can be defined by first applying insert' and tail' and then applying balance.
fun balance(q as (len_front, front, tail_front, len_rear, rear, tail_rear)) =
let floor_half_len = (len_front + len_rear) / 2 in
let ceil_half_len = len_front + len_rear - floor_half_len in
if len_front > 2*len_rear+1 then
let val front' = take(ceil_half_len, front)
val rear' = rotateDrop(rear, floor_half_len, front)
in (ceil_half_len, front', front', floor_half_len, rear', rear')
else if len_front > 2*len_rear+1 then
let val rear' = take(floor_half_len, rear)
val front' = rotateDrop(front, ceil_half_len, rear)
else q
where rotateDrop(front, i, rear)) return the concatenation of front and of drop(i, rear). That isfront' = rotateDrop(front, ceil_half_len, rear) put into front' the content of front and the content of rear that is not already in rear'. Since dropping n elements takes
{\displaystyle O(n)}
time, we use laziness to ensure that elements are dropped two by two, with two drops being done during each tail' and each insert' operation.
fun rotateDrop(front, i, rear) =
if i < 2 then rotateRev(front, drop(i, rear), $NIL)
else let $CONS(x, front') = front in
$CONS (x, rotateDrop(front', j-2, drop(2, rear)))
where rotateRev(front, middle, rear) is a function that returns the front, followed by the middle reversed, followed by the rear. This function is also defined using laziness to ensure that it can be computed step by step, with one step executed during each insert' and tail' and taking a constant time. This function uses the invariant that |rear|-2|front| is 2 or 3.
fun rotateRev(NIL, rear, a)=
reverse(rear++a)
fun rotateRev(CONS(x, front), rear, a)=
CONS(x, rotateRev(front, drop(2, rear), reverse (take(2, rear))++a))
where ++ is the function concatenating two lists.
Implementation without laziness[edit]
Note that, without the lazy part of the implementation, this would be a non-persistent implementation of queue in O(1) amortized time. In this case, the lists tail_front and tail_rear could be removed from the representation of the double-ended queue.
Ada's containers provides the generic packages Ada.Containers.Vectors and Ada.Containers.Doubly_Linked_Lists, for the dynamic array and linked list implementations, respectively.
C++'s Standard Template Library provides the class templates std::deque and std::list, for the multiple array and linked list implementations, respectively.
As of Java 6, Java's Collections Framework provides a new Deque interface that provides the functionality of insertion and removal at both ends. It is implemented by classes such as ArrayDeque (also new in Java 6) and LinkedList, providing the dynamic array and linked list implementations, respectively. However, the ArrayDeque, contrary to its name, does not support random access.
Javascript's Array prototype & Perl's arrays have native support for both removing (shift and pop) and adding (unshift and push) elements on both ends.
Python 2.4 introduced the collections module with support for deque objects. It is implemented using a doubly linked list of fixed-length subarrays.
As of PHP 5.3, PHP's SPL extension contains the 'SplDoublyLinkedList' class that can be used to implement Deque datastructures. Previously to make a Deque structure the array functions array_shift/unshift/pop/push had to be used instead.
GHC's Data.Sequence module implements an efficient, functional deque structure in Haskell. The implementation uses 2–3 finger trees annotated with sizes. There are other (fast) possibilities to implement purely functional (thus also persistent) double queues (most using heavily lazy evaluation).[3][4] Kaplan and Tarjan were the first to implement optimal confluently persistent catenable deques.[5] Their implementation was strictly purely functional in the sense that it did not use lazy evaluation. Okasaki simplified the data structure by using lazy evaluation with a bootstrapped data structure and degrading the performance bounds from worst-case to amortized. Kaplan, Okasaki, and Tarjan produced a simpler, non-bootstrapped, amortized version that can be implemented either using lazy evaluation or more efficiently using mutation in a broader but still restricted fashion. Mihaesau and Tarjan created a simpler (but still highly complex) strictly purely functional implementation of catenable deques, and also a much simpler implementation of strictly purely functional non-catenable deques, both of which have optimal worst-case bounds.
Rust's std::collections includes VecDeque which implements a double-ended queue using a growable ring buffer.
In a doubly-linked list implementation and assuming no allocation/deallocation overhead, the time complexity of all deque operations is O(1). Additionally, the time complexity of insertion or deletion in the middle, given an iterator, is O(1); however, the time complexity of random access by index is O(n).
In a growing array, the amortized time complexity of all deque operations is O(1). Additionally, the time complexity of random access by index is O(1); but the time complexity of insertion or deletion in the middle is O(n).
One example where a deque can be used is the work stealing algorithm.[6] This algorithm implements task scheduling for several processors. A separate deque with threads to be executed is maintained for each processor. To execute the next thread, the processor gets the first element from the deque (using the "remove first element" deque operation). If the current thread forks, it is put back to the front of the deque ("insert element at front") and a new thread is executed. When one of the processors finishes execution of its own threads (i.e. its deque is empty), it can "steal" a thread from another processor: it gets the last element from the deque of another processor ("remove last element") and executes it. The work stealing algorithm is used by Intel's Threading Building Blocks (TBB) library for parallel programming.
^ Jesse Liberty; Siddhartha Rao; Bradley Jones. C++ in One Hour a Day, Sams Teach Yourself, Sixth Edition. Sams Publishing, 2009. ISBN 0-672-32941-7. Lesson 18: STL Dynamic Array Classes, pp. 486.
^ a b Okasaki, Chris (September 1996). Purely Functional Data Structures (PDF) (Ph.D. thesis). Carnegie Mellon University. CMU-CS-96-177.
^ Adam L. Buchsbaum and Robert E. Tarjan. Confluently persistent deques via data structural bootstrapping. Journal of Algorithms, 18(3):513–547, May 1995. (pp. 58, 101, 125)
^ Haim Kaplan and Robert E. Tarjan. Purely functional representations of catenable sorted lists. In ACM Symposium on Theory of Computing, pages 202–211, May 1996. (pp. 4, 82, 84, 124)
^ Blumofe, Robert D.; Leiserson, Charles E. (1999). "Scheduling multithreaded computations by work stealing" (PDF). J ACM. 46 (5): 720–748. doi:10.1145/324133.324234. S2CID 5428476.
Type-safe open source deque implementation at Comprehensive C Archive Network
Deque implementation in C
Multiple implementations of non-catenable deques in Haskell
Retrieved from "https://en.wikipedia.org/w/index.php?title=Double-ended_queue&oldid=1086922103"
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evalpow - Maple Help
Home : Support : Online Help : Mathematics : Evaluation : evalpow
general evaluator for expressions, which can be formal power series, polynomials, or functions
evalpow(expr)
any arithmetic expression involving formal power series, polynomials, or functions that is acceptable for power series package
The function evalpow(expr) evaluates the arithmetic expression expr and then returns an unnamed power series.
The following operators can be used:
+
-
/
^
Also, functions may be composed with each other. For example,
f\left(g\right)
The other functions that can be used in evalpow are:
powinv
powrev (reversion)
powdiff (first derivative)
powint (first integral)
powquo (quotient)
powsub (subtract)
powcoth
powsqrt(square root)
Note that the evalpow also accepts the standard forms, or the inner MAPLE forms for some of the above functions. For example, exp, or Exp for powexp, Diff for powdiff, but NOT diff.
The command with(powseries,evalpow) allows the use of the abbreviated form of this command.
\mathrm{with}\left(\mathrm{powseries}\right):
\mathrm{powcreate}\left(f\left(n\right)=\frac{f\left(n-1\right)}{n},f\left(0\right)=1\right):
\mathrm{powcreate}\left(g\left(n\right)=\frac{g\left(n-1\right)}{2},g\left(0\right)=0,g\left(1\right)=1\right):
\mathrm{powcreate}\left(h\left(n\right)=\frac{h\left(n-1\right)}{5},h\left(0\right)=1\right):
k≔\mathrm{evalpow}\left({f}^{3}+g-\mathrm{powquo}\left(h,f\right)\right):
\mathrm{tpsform}\left(k,x,5\right)
\frac{\textcolor[rgb]{0,0,1}{24}}{\textcolor[rgb]{0,0,1}{5}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{233}}{\textcolor[rgb]{0,0,1}{50}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{7273}}{\textcolor[rgb]{0,0,1}{1500}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{52171}}{\textcolor[rgb]{0,0,1}{15000}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{O}\textcolor[rgb]{0,0,1}{}\left({\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{5}}\right)
b≔\mathrm{evalpow}\left(\mathrm{Diff}\left(\mathrm{powlog}\left(1+x\right)\right)\right):
c≔\mathrm{tpsform}\left(b,x,6\right)
\textcolor[rgb]{0,0,1}{c}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{5}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{O}\textcolor[rgb]{0,0,1}{}\left({\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{6}}\right)
e≔\mathrm{evalpow}\left(\mathrm{Tan}\left(1+x\right)\right):
f≔\mathrm{tpsform}\left(e,x,3\right)
\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{≔}\frac{\textcolor[rgb]{0,0,1}{\mathrm{sin}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{1}\right)}{\textcolor[rgb]{0,0,1}{\mathrm{cos}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{1}\right)}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{\mathrm{cos}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{1}\right)\textcolor[rgb]{0,0,1}{+}\frac{{\textcolor[rgb]{0,0,1}{\mathrm{sin}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{1}\right)}^{\textcolor[rgb]{0,0,1}{2}}}{\textcolor[rgb]{0,0,1}{\mathrm{cos}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{1}\right)}}{\textcolor[rgb]{0,0,1}{\mathrm{cos}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{1}\right)}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{\mathrm{sin}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{1}\right)\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{cos}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{1}\right)\textcolor[rgb]{0,0,1}{+}\frac{{\textcolor[rgb]{0,0,1}{\mathrm{sin}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{1}\right)}^{\textcolor[rgb]{0,0,1}{2}}}{\textcolor[rgb]{0,0,1}{\mathrm{cos}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{1}\right)}\right)}{{\textcolor[rgb]{0,0,1}{\mathrm{cos}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{1}\right)}^{\textcolor[rgb]{0,0,1}{2}}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{O}\textcolor[rgb]{0,0,1}{}\left({\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{3}}\right)
g≔\mathrm{tpsform}\left(\mathrm{evalpow}\left(\mathrm{sinh}\left(x\right)\right),x,8\right)
\textcolor[rgb]{0,0,1}{g}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{6}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{120}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{5}}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{5040}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{7}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{O}\textcolor[rgb]{0,0,1}{}\left({\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{8}}\right)
h≔\mathrm{evalpow}\left(\mathrm{powadd}\left(\mathrm{powexp}\left(x\right),\mathrm{powpoly}\left(1+x,x\right),\mathrm{powlog}\left(1+x\right)\right)\right):
m≔\mathrm{tpsform}\left(h,x,8\right)
\textcolor[rgb]{0,0,1}{m}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{5}}{\textcolor[rgb]{0,0,1}{24}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{5}}{\textcolor[rgb]{0,0,1}{24}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{5}}\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{119}}{\textcolor[rgb]{0,0,1}{720}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{6}}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{103}}{\textcolor[rgb]{0,0,1}{720}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{7}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{O}\textcolor[rgb]{0,0,1}{}\left({\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{8}}\right)
powseries[inverse]
powseries[powadd]
powseries[powcreate]
powseries[powexp]
powseries[powsin]
powseries[reversion]
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Ultra-violet radiation from sunlight causes the reaction that produce
s{o}_{2}
{o}_{3}
Average MCQ Metabolism MCQ Phylum - Pisces MCQ Basics of Neurology MCQ Change of State MCQ Phylum - Aschelminthes MCQ Hydrocarbons MCQ Solution MCQ Direct TAX MCQ H.C.F and L.C.M MCQ Botany MCQ Concrete Technology and Design MCQ
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Factor the following quadratics completely.
5x^3+13x^2-6x
Start by factoring out an
x
Then factor the remaining quadratic.
6t^2-26t+8
Start by factoring out the GCF.
6x^2-24
6
. You will be left with a difference of squares.
Watch the following video if you need help with factoring quadratic expressions. Click the link at right for the full version of the video: Factoring Quadratics
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2{x}^{2}+\sqrt{3}x-1=0
Q.12. A milkman has 80 % milk in his stock of 800 litres of adulterated milk. How much 100 % pure milk is added to it so that purity is between 90 % and 95 %?
29. Solve the following system of inequalities graphically
2x+y
\le
24, x+y < 11, 2x+5y
\le
40, x,y
\ge
please solve ques 10 and 11
Q10. Reduce the equation 3x – 4y + 20 = 0 in to normal form
Q11. Solve the inequality
\frac{x +3}{x -7}\le 0
Solve graphically x+2y< equal to 8 ,2x+y> equal to 2 , x-y< equal to 1 , x,y > equal to 0
Q). Solve the following system of inequations graphically :
3x+4y\le 12, 4x+3y\le 12, x\ge 0, y\ge 0
Solve the following system of inequalities graphically , 3x+2y>equal to 24,3x+y<equal to 15x>equal to4,x,y is >equal to 0
Solve the inequality x/2<(5x-2)/3 - (7x-3)/3 and show the graph of the solution on number line
55 no. Please!
Q.55. The inequality
\left(x-1\right)\mathrm{ln}\left(2-x\right)<0
holds if x satisfies:
-log(.093*10^-2)solve with steps plzzz
Solve: x-2?x+5>2
Example 2. Solve
\frac{x-2}{x+5}>2
Find product u\mathrm{sin}g suitable identity-\phantom{\rule{0ex}{0ex}}\left(x-\frac{1}{2}\right)\left(x+\frac{1}{2}\right)\left({x}^{2}+\frac{1}{{x}^{2}}\right)\left({x}^{4}+\frac{1}{{x}^{4}}\right)
The number of irrational solutions of the equation\phantom{\rule{0ex}{0ex}}\sqrt{{x}^{2}+\sqrt{{x}^{2}+11}}+\sqrt{{x}^{2}-\sqrt{{x}^{2}+11}}=4\phantom{\rule{0ex}{0ex}}A0\phantom{\rule{0ex}{0ex}}B2\phantom{\rule{0ex}{0ex}}C4\phantom{\rule{0ex}{0ex}}D11
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1 June 2010 The tropical vertex
Mark Gross, Rahul Pandharipande, Bernd Siebert
Mark Gross,1 Rahul Pandharipande,2 Bernd Siebert3
2Department of Mathematics, Princeton Univerity
3Department Mathematik, Universität Hamburg
Elements of the tropical vertex group are formal families of symplectomorphisms of the
2
-dimensional algebraic torus. We prove that ordered product factorizations in the tropical vertex group are equivalent to calculations of certain genus zero relative Gromov-Witten invariants of toric surfaces. The relative invariants which arise have full tangency to a toric divisor at a single unspecified point. The method uses scattering diagrams, tropical curve counts, degeneration formulas, and exact multiple cover calculations in orbifold Gromov-Witten theory
Mark Gross. Rahul Pandharipande. Bernd Siebert. "The tropical vertex." Duke Math. J. 153 (2) 297 - 362, 1 June 2010. https://doi.org/10.1215/00127094-2010-025
Mark Gross, Rahul Pandharipande, Bernd Siebert "The tropical vertex," Duke Mathematical Journal, Duke Math. J. 153(2), 297-362, (1 June 2010)
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GetHostName - Maple Help
Home : Support : Online Help : Connectivity : Web Features : Network Communication : Sockets Package : GetHostName
retrieve the name of the local host
The procedure GetHostName obtains a valid node name for the local machine. It returns a string that contains some valid hostname for the machine on which the calling process is running.
A fully qualified domain name is returned if possible. If all methods to determine the name of the local host fail, then the string "localhost" is returned.
\mathrm{Sockets}[\mathrm{GetHostName}]\left(\right)
\textcolor[rgb]{0,0,1}{"be527fdac554"}
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Modular arithmetic — Wikipedia Republished // WIKI 2
This article is about the (mod n) notation. For the binary operation mod(a,n), see modulo operation.
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book Disquisitiones Arithmeticae, published in 1801.
A familiar use of modular arithmetic is in the 12-hour clock, in which the day is divided into two 12-hour periods. If the time is 7:00 now, then 8 hours later it will be 3:00. Simple addition would result in 7 + 8 = 15, but clocks "wrap around" every 12 hours. Because the hour number starts over after it reaches 12, this is arithmetic modulo 12. In terms of the definition below, 15 is congruent to 3 modulo 12, so "15:00" on a 24-hour clock is displayed "3:00" on a 12-hour clock.
[Discrete Mathematics] Modular Arithmetic
What does a ≡ b (mod n) mean? Basic Modular Arithmetic, Congruence
1 Congruence
3 Congruence classes
4 Residue systems
4.1 Reduced residue systems
5 Integers modulo n
7 Computational complexity
8 Example implementations
Congruence modulo n is a congruence relation, meaning that it is an equivalence relation that is compatible with the operations of addition, subtraction, and multiplication. Congruence modulo n is denoted:
{\displaystyle a\equiv b{\pmod {n}}.}
The parentheses mean that (mod n) applies to the entire equation, not just to the right-hand side (here b). This notation is not to be confused with the notation b mod n (without parentheses), which refers to the modulo operation. Indeed, b mod n denotes the unique integer a such that 0 ≤ a < n and
{\displaystyle a\equiv b\;({\text{mod}}\;n)}
{\displaystyle b}
{\displaystyle n}
{\displaystyle a=kn+b,}
explicitly showing its relationship with Euclidean division. However, the b here need not be the remainder of the division of a by n. Instead, what the statement a ≡ b (mod n) asserts is that a and b have the same remainder when divided by n. That is,
{\displaystyle a=pn+r,}
{\displaystyle b=qn+r,}
{\displaystyle a-b=kn,}
{\displaystyle 38\equiv 14{\pmod {12}}}
{\displaystyle {\begin{aligned}2&\equiv -3{\pmod {5}}\\-8&\equiv 7{\pmod {5}}\\-3&\equiv -8{\pmod {5}}.\end{aligned}}}
The congruence relation satisfies all the conditions of an equivalence relation:
If c ≡ d (mod φ(n)), where φ is Euler's totient function, then ac ≡ ad (mod n)—provided that a is coprime with n.
The modular multiplicative inverse is defined by the following rules:
The multiplicative inverse x ≡ a–1 (mod n) may be efficiently computed by solving Bézout's equation
{\displaystyle ax+ny=1}
{\displaystyle x,y}
—using the Extended Euclidean algorithm.
Fermat's little theorem: If p is prime and does not divide a, then a p – 1 ≡ 1 (mod p).
Euler's theorem: If a and n are coprime, then a φ(n) ≡ 1 (mod n), where φ is Euler's totient function
Wilson's theorem: p is prime if and only if (p − 1)! ≡ −1 (mod p).
Chinese remainder theorem: For any a, b and coprime m, n, there exists a unique x (mod mn) such that x ≡ a (mod m) and x ≡ b (mod n). In fact, x ≡ b mn–1 m + a nm–1 n (mod mn) where mn−1 is the inverse of m modulo n and nm−1 is the inverse of n modulo m.
Lagrange's theorem: The congruence f (x) ≡ 0 (mod p), where p is prime, and f (x) = a0 xn + ... + an is a polynomial with integer coefficients such that a0 ≠ 0 (mod p), has at most n roots.
Primitive root modulo n: A number g is a primitive root modulo n if, for every integer a coprime to n, there is an integer k such that gk ≡ a (mod n). A primitive root modulo n exists if and only if n is equal to 2, 4, pk or 2pk, where p is an odd prime number and k is a positive integer. If a primitive root modulo n exists, then there are exactly φ(φ(n)) such primitive roots, where φ is the Euler's totient function.
Quadratic residue: An integer a is a quadratic residue modulo n, if there exists an integer x such that x2 ≡ a (mod n). Euler's criterion asserts that, if p is an odd prime, and a is not a multiple of p, then a is a quadratic residue modulo p if and only if
{\displaystyle a^{(p-1)/2}\equiv 1{\pmod {p}}.}
Like any congruence relation, congruence modulo n is an equivalence relation, and the equivalence class of the integer a, denoted by an, is the set {... , a − 2n, a − n, a, a + n, a + 2n, ...}. This set, consisting of all the integers congruent to a modulo n, is called the congruence class, residue class, or simply residue of the integer a modulo n. When the modulus n is known from the context, that residue may also be denoted [a].
Reduced residue systems
Main article: Reduced residue system
Given the Euler's totient function φ(n), any set of φ(n) integers that are relatively prime to n and mutually incongruent under modulus n is called a reduced residue system modulo n.[5] The set {5,15} from above, for example, is an instance of a reduced residue system modulo 4.
The set of all congruence classes of the integers for a modulus n is called the ring of integers modulo n,[6] and is denoted
{\textstyle \mathbb {Z} /n\mathbb {Z} }
{\displaystyle \mathbb {Z} /n}
{\displaystyle \mathbb {Z} _{n}}
{\displaystyle \mathbb {Z} _{n}}
is, however, not recommended because it can be confused with the set of n-adic integers. The ring
{\displaystyle \mathbb {Z} /n\mathbb {Z} }
{\displaystyle \mathbb {Z} /n\mathbb {Z} =\left\{{\overline {a}}_{n}\mid a\in \mathbb {Z} \right\}=\left\{{\overline {0}}_{n},{\overline {1}}_{n},{\overline {2}}_{n},\ldots ,{\overline {n{-}1}}_{n}\right\}.}
{\displaystyle \mathbb {Z} /n\mathbb {Z} }
is not an empty set; rather, it is isomorphic to
{\displaystyle \mathbb {Z} }
{\displaystyle \mathbb {Z} /n\mathbb {Z} }
{\displaystyle {\overline {a}}_{n}+{\overline {b}}_{n}={\overline {(a+b)}}_{n}}
{\displaystyle {\overline {a}}_{n}-{\overline {b}}_{n}={\overline {(a-b)}}_{n}}
{\displaystyle {\overline {a}}_{n}{\overline {b}}_{n}={\overline {(ab)}}_{n}.}
{\displaystyle \mathbb {Z} /n\mathbb {Z} }
becomes a commutative ring. For example, in the ring
{\displaystyle \mathbb {Z} /24\mathbb {Z} }
{\displaystyle {\overline {12}}_{24}+{\overline {21}}_{24}={\overline {33}}_{24}={\overline {9}}_{24}}
{\displaystyle \mathbb {Z} /n\mathbb {Z} }
because this is the quotient ring of
{\displaystyle \mathbb {Z} }
{\displaystyle n\mathbb {Z} }
{\displaystyle 0\mathbb {Z} }
is the singleton set {0}. Thus
{\displaystyle \mathbb {Z} /n\mathbb {Z} }
{\displaystyle n\mathbb {Z} }
is a maximal ideal (i.e., when n is prime).
{\displaystyle \mathbb {Z} }
under the addition operation alone. The residue class an is the group coset of a in the quotient group
{\displaystyle \mathbb {Z} /n\mathbb {Z} }
, a cyclic group.[8]
{\displaystyle \mathbb {Z} /0\mathbb {Z} }
{\displaystyle \mathbb {Z} }
The ring of integers modulo n is a finite field if and only if n is prime (this ensures that every nonzero element has a multiplicative inverse). If
{\displaystyle n=p^{k}}
is a prime power with k > 1, there exists a unique (up to isomorphism) finite field
{\displaystyle \mathrm {GF} (n)=\mathbb {F} _{n}}
{\displaystyle \mathbb {Z} /n\mathbb {Z} }
The multiplicative subgroup of integers modulo n is denoted by
{\displaystyle (\mathbb {Z} /n\mathbb {Z} )^{\times }}
{\displaystyle {\overline {a}}_{n}}
(where a is coprime to n), which are precisely the classes possessing a multiplicative inverse. This forms a commutative group under multiplication, with order
{\displaystyle \varphi (n)}
In theoretical mathematics, modular arithmetic is one of the foundations of number theory, touching on almost every aspect of its study, and it is also used extensively in group theory, ring theory, knot theory, and abstract algebra. In applied mathematics, it is used in computer algebra, cryptography, computer science, chemistry and the visual and musical arts.
A very practical application is to calculate checksums within serial number identifiers. For example, International Standard Book Number (ISBN) uses modulo 11 (for 10 digit ISBN) or modulo 10 (for 13 digit ISBN) arithmetic for error detection. Likewise, International Bank Account Numbers (IBANs), for example, make use of modulo 97 arithmetic to spot user input errors in bank account numbers. In chemistry, the last digit of the CAS registry number (a unique identifying number for each chemical compound) is a check digit, which is calculated by taking the last digit of the first two parts of the CAS registry number times 1, the previous digit times 2, the previous digit times 3 etc., adding all these up and computing the sum modulo 10.
In cryptography, modular arithmetic directly underpins public key systems such as RSA and Diffie–Hellman, and provides finite fields which underlie elliptic curves, and is used in a variety of symmetric key algorithms including Advanced Encryption Standard (AES), International Data Encryption Algorithm (IDEA), and RC4. RSA and Diffie–Hellman use modular exponentiation.
In computer algebra, modular arithmetic is commonly used to limit the size of integer coefficients in intermediate calculations and data. It is used in polynomial factorization, a problem for which all known efficient algorithms use modular arithmetic. It is used by the most efficient implementations of polynomial greatest common divisor, exact linear algebra and Gröbner basis algorithms over the integers and the rational numbers. As posted on Fidonet in the 1980s and archived at Rosetta Code, modular arithmetic was used to disprove Euler's sum of powers conjecture on a Sinclair QL microcomputer using just one-fourth of the integer precision used by a CDC 6600 supercomputer to disprove it two decades earlier via a brute force search.[9]
In computer science, modular arithmetic is often applied in bitwise operations and other operations involving fixed-width, cyclic data structures. The modulo operation, as implemented in many programming languages and calculators, is an application of modular arithmetic that is often used in this context. The logical operator XOR sums 2 bits, modulo 2.
In music, arithmetic modulo 12 is used in the consideration of the system of twelve-tone equal temperament, where octave and enharmonic equivalency occurs (that is, pitches in a 1:2 or 2:1 ratio are equivalent, and C-sharp is considered the same as D-flat).
The method of casting out nines offers a quick check of decimal arithmetic computations performed by hand. It is based on modular arithmetic modulo 9, and specifically on the crucial property that 10 ≡ 1 (mod 9).
Arithmetic modulo 7 is used in algorithms that determine the day of the week for a given date. In particular, Zeller's congruence and the Doomsday algorithm make heavy use of modulo-7 arithmetic.
More generally, modular arithmetic also has application in disciplines such as law (e.g., apportionment), economics (e.g., game theory) and other areas of the social sciences, where proportional division and allocation of resources plays a central part of the analysis.
Since modular arithmetic has such a wide range of applications, it is important to know how hard it is to solve a system of congruences. A linear system of congruences can be solved in polynomial time with a form of Gaussian elimination, for details see linear congruence theorem. Algorithms, such as Montgomery reduction, also exist to allow simple arithmetic operations, such as multiplication and exponentiation modulo n, to be performed efficiently on large numbers.
Some operations, like finding a discrete logarithm or a quadratic congruence appear to be as hard as integer factorization and thus are a starting point for cryptographic algorithms and encryption. These problems might be NP-intermediate.
{\displaystyle a\cdot b{\pmod {m}}}
On computer architectures where an extended precision format with at least 64 bits of mantissa is available (such as the long double type of most x86 C compilers), the following routine is[clarification needed], by employing the trick that, by hardware, floating-point multiplication results in the most significant bits of the product kept, while integer multiplication results in the least significant bits kept:[citation needed]
{\displaystyle a^{b}{\pmod {m}}}
Boolean ring
Circular buffer
Division (mathematics)
Finite field
Legendre symbol
Modular exponentiation
Modulo (mathematics)
Multiplicative group of integers modulo n
Pisano period (Fibonacci sequences modulo n)
Primitive root modulo n
Quadratic reciprocity
Quadratic residue
Rational reconstruction (mathematics)
Reduced residue system
Serial number arithmetic (a special case of modular arithmetic)
Two-element Boolean algebra
Cyclic group
Carmichael's theorem
Chinese remainder theorem
Fermat's little theorem (a special case of Euler's theorem)
Thue's lemma
^ Weisstein, Eric W. "Modular Arithmetic". mathworld.wolfram.com. Retrieved 2020-08-12.
^ Pettofrezzo & Byrkit (1970, p. 90)
^ Long (1972, p. 78)
^ "2.3: Integers Modulo n". Mathematics LibreTexts. 2013-11-16. Retrieved 2020-08-12.
^ Sengadir T., Discrete Mathematics and Combinatorics, p. 293, at Google Books
^ "Euler's sum of powers conjecture". rosettacode.org. Retrieved 2020-11-11.
^ Garey, M. R.; Johnson, D. S. (1979). Computers and Intractability, a Guide to the Theory of NP-Completeness. W. H. Freeman. ISBN 0716710447.
John L. Berggren. "modular arithmetic". Encyclopædia Britannica.
Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, ISBN 978-0-387-90163-3, MR 0434929, Zbl 0335.10001 . See in particular chapters 5 and 6 for a review of basic modular arithmetic.
Maarten Bullynck "Modular Arithmetic before C.F. Gauss. Systematisations and discussions on remainder problems in 18th-century Germany"
Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduction to Algorithms, Second Edition. MIT Press and McGraw-Hill, 2001. ISBN 0-262-03293-7. Section 31.3: Modular arithmetic, pp. 862–868.
Anthony Gioia, Number Theory, an Introduction Reprint (2001) Dover. ISBN 0-486-41449-3.
Long, Calvin T. (1972). Elementary Introduction to Number Theory (2nd ed.). Lexington: D. C. Heath and Company. LCCN 77171950.
Pettofrezzo, Anthony J.; Byrkit, Donald R. (1970). Elements of Number Theory. Englewood Cliffs: Prentice Hall. LCCN 71081766.
"Congruence", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
In this modular art article, one can learn more about applications of modular arithmetic in art.
Modular Arithmetic and patterns in addition and multiplication tables
Geometric number theory
Transcendental number theory
Arithmetic combinatorics
Arithmetic geometry
Arithmetic topology
Arithmetic dynamics
Natural numbers
Rational numbers
Algebraic numbers
p-adic numbers
Modular arithmetic
Arithmetic functions
Quadratic forms
Modular forms
Diophantine equations
Diophantine approximation
Continued fractions
List of recreational topics
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Working capital - Wikipedia
Find sources: "Working capital" – news · newspapers · books · scholar · JSTOR (May 2014) (Learn how and when to remove this template message)
Working capital (WC) is a financial metric which represents operating liquidity available to a business, organisation, or other entity, including governmental entities. Along with fixed assets such as plant and equipment, working capital is considered a part of operating capital. Gross working capital is equal to current assets. Working capital is calculated as current assets minus current liabilities.[1] If current assets are less than current liabilities, an entity has a working capital deficiency, also called a working capital deficit and negative working capital.[2]
2 Working capital cycle
{\displaystyle {\text{Working Capital}}={\text{Current Assets}}-{\text{Current Liabilities}}}
Working capital cycle[edit]
Main article: Cash conversion cycle
Working capital management[edit]
Decision criteria[edit]
Management of working capital[edit]
^ Fernando, Jason. "Working Capital Definition". investopedia.com. Investopedia. Retrieved 20 May 2022.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Working_capital&oldid=1088879582"
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Compute active and reactive powers using voltage and current phasors - Simulink - MathWorks Nordic
Power (Phasor)
Compute active and reactive powers using voltage and current phasors
The Power (Phasor) block computes the active power P and reactive power Q of a pair of voltage-current signals. P and Q are computed as follows:
P+jQ=\frac{1}{2}\left(V×{I}^{\ast }\right)
I* = complex conjugate of I.
Sign conventions are defined as
The phasor (complex signal) voltage, in volts peak.
The phasor (complex signal) current, in amperes peak.
Returns the active power, in watts.
Returns the active power, in vars.
The power_PhasorPowerMeasurements example shows the use of the Power (Phasor) block.
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I have received in a parcel from Down your letter and most interesting enclosures and Taylor’s pamphet.2 Good heavens, what trouble you have taken for me! I am sure I am very much obliged; and the facts are most interesting to me, but I have not yet digested them.— Since writing last we have had a terrible week with my poor girl on the point of death; but she has rallied surprisingly. When we shall be able to move her home we know not as yet at all.— You ask how my experiments were tried; but they have been tried in so many manners that I hardly know how to answer. I will before very long (but all my work has been utterly stopped for a fortnight by this miserable illness) publish an account.—
When I saw you I had tried only putting a minute drop on the leaves on growing plants and observing whether or not they contracted as if over a fly.3 From many reasons I inferred that the leaves absorbed some nitrogenous element, probably some form of Ammonia; so I thought I would try under the microscope the effect of C. of Ammonia; and several other salts and substances, and it is truly wonderful how quickly a minute dose acts and produces marvellous changes in the absorbing glands and in the adjoining cells. I have tried plain water over and over again with no effect. I have over and over again kept a leaf for 1, 2, 3, 4, and 5 hours in water with no effect, and then put these same leaves in a few measured drops of very weak solutions of C. of Ammonia (all made, and re-made by myself) and the same peculiar effects were produced in one hour or 1
\frac{1}{2}
hour as is produced instantaneously by a stronger solution. Generally I have scrutinized every gland and and hair on leaf before experimenting; but it occurred to me that I might in some way affect the leaf; though this is almost impossible, as I scrutinized with equal care those that I put into distilled water (the same water being used for dissolving the C. of Ammonia). I then cut off 4 leaves (not touching them with fingers) and put in plain water and 4 other leaves into the weak solution and after leaving them for 1
\frac{1}{2}
hours I examined every hair on all 8 leaves; no changes on the 4 in water every gland and hair affected in those of Ammonia.
I had measured the quantity of weak solution and I counted the glands which had absorbed the ammonia and were plainly affected; the result convinced me that each gland could not have absorbed more than 1/64,000 or
\frac{1}{65000}
of a grain.— I have tried numbers of other experiments all pointing to the same result. Some experiments lead me to believe that very sensitive leaves are acted on by much smaller doses. Reflect how little Ammonia a plant can get growing on poor soil—yet it is nourished. The really surprising part seems to me that the effect should be visible and not under very high power; for after trying high power, I thought it would be safer not to consider any effect which was not plainly visible under
\frac{2}{3}
object glass and middle eye-piece. The effect which the C. of Ammonia produces is the segregation of the homogeneous fluid in the cells into (first) a cloud of granules and colourless fluid; and subsequently the granules coalesce into larger masses and for hours have the oddest movements coalescing, dividing, coalescing ad infinitum.4 I do not know whether you will care for these ill-written details; but as you asked I am sure I am bound to comply after all the very kind and great trouble which you have taken.
Dated by the relationship to the letter from Edward Cresy, 30 October 1860.
See letter from Edward Cresy, 30 October 1860 and enclosure. Cresy also forwarded to CD the letter from A. W. von Hofmann to Edward Cresy, 27 October 1860. The pamphlet by Alfred Swaine Taylor was probably Taylor 1860, which discussed chemical tests for arsenic and antimony. There is a note on this work in DAR 60.1: 69.
Cresy and his wife visited Down on 18 September 1860, shortly before the Darwins left for Eastbourne (Emma Darwin’s diary). CD’s notes on his observations on Drosera and Dionaea, begun in July 1860, are in DAR 54, 60.1, and 60.2.
CD’s initial observations on the cytological phenomenon of the ‘aggregation of protoplasm’ are in DAR 60.1: 77–81.
Thanks for pamphlet by A. S. Taylor.
"… we have had a terrible week with my poor girl [Henrietta] on the point of death".
Discusses experiments involving placing solutions of ammonia and other substances on leaves of plants.
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Effects of Syngas Ash Particle Size on Deposition and Erosion of a Film Cooled Leading Edge | J. Turbomach. | ASME Digital Collection
Ali Rozati,
Ali Rozati
Department of Mechanical Engineering, High Performance Computational Fluid-Thermal Sciences and Engineering Laboratory,
, Blacksburg VA 24061
Danesh K. Tafti,
Sai Shrinivas Sreedharan
Rozati, A., Tafti, D. K., and Sreedharan, S. S. (September 9, 2010). "Effects of Syngas Ash Particle Size on Deposition and Erosion of a Film Cooled Leading Edge." ASME. J. Turbomach. January 2011; 133(1): 011010. https://doi.org/10.1115/1.4000492
The paper investigates the deposition and erosion caused by Syngas ash particles in a film cooled leading edge region of a representative turbine vane. The carrier phase is predicted using large eddy simulation for three blowing ratios of 0.4, 0.8, and 1.2. Ash particle sizes of
1 μm
3 μm
5 μm
7 μm
10 μm
are investigated using Lagrangian dynamics. The
1 μm
particles with momentum Stokes number,
Stp=0.03
(based on approach velocity and leading edge diameter), follow the flow streamlines around the leading edge and few particles reach the blade surface. The
10 μm
particles, on the other hand with a high momentum Stokes number,
Stp=0.03
, directly impinge on the surface, with blowing ratio having a minimal effect. The
3 μm
5 μm
7 μm
particles with
Stp=0.03
, 0.8 and 1.4, respectively, show some receptivity to coolant flow and blowing ratio. On a number basis, 85–90% of the
10 μm
particles, 70–65% of
7 μm
5 μm
particles, 15% of
3 μm
particles, and less than 1% of
1 μm
particles deposit on the surface. Overall there is a slight decrease in percentage of particles deposited with increase in blowing ratio. On the other hand, the potential for erosive wear is highest in the coolant hole and is mostly attributed to
5 μm
7 μm
particles. It is only at
BR=1.2
10 μm
particles contribute to erosive wear in the coolant hole.
ash, blades, coolants, erosion, film flow, flow simulation, particle size, turbines
Coolants, Erosion, Particulate matter, Flow (Dynamics), Particle size, Blades, Temperature, Syngas
Deposition, Erosion and Corrosion Protection for Coal-Fired Gas Turbine
Effect of Particle Characteristics on Trajectories and Blade Impact Patterns
Selection and Operation of Gas Turbine Air Filters
The Chemistry of Saudi Arabian Sand: A Deposition Problem on Helicopter Turbine Airfoils
Proceedings of the 3rd International SAMPE Metals and Processing Conference
Experimental and Numerical Simulation of Ingested Particles in Gas Turbine Engines
AGARD (NATO) 83rd Symposium of Propulsion and Energetics Panel on Turbines
, Rotterdam, the Netherlands, Apr. 25–28.
Effects of Particle Size, Gas Temperature, and Metal Temperature on High Pressure Turbine Deposition in Land Based Gas Turbine From Various Synfuels
Large-Eddy Simulation of Leading Edge Film Cooling: Analysis of Flow Structures, Effectiveness, and Heat Transfer Coefficient
Effect of Coolant-Mainstream Blowing Ratio on Leading Edge Film Cooling Flow and Heat Transfer
Prediction of Coal Ash Thermal Properties Using Partial Least Squares Regression
Development and Application of a Dispersed Two-Phase Flow Capability in a General Multi-Block Navier Stokes Solver
,” MS thesis, Mechanical Engineering, Virginia Tech.
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Jack and Jill were each placing points on the grid shown below. Jack’s points are the full circles, and Jill’s are the open circles.
Record Jack and Jill’s points as ordered pairs.
Remember, in an ordered pair, the x-coordinate (showing the placement along the x-axis) is listed first and the
y-coordinate (showing the placement along the y-axis) is listed second, as in (x, y).
Jill's ordered pairs are
(1,1),(2,1),(2,2)
(2,3)
Can you name Jack's ordered pairs?
Give the coordinates of one more point that Jill could draw so that she has four of her points in a row.
Can you spot the possible points? Here's a clue: the point should be at the
2
value on the x-axis, so the ordered pair will look something like this
(2,y)
\left(2,4\right)
\left(2, 0\right)
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NCERT Solutions for Class 7 Math Chapter 2 - Fractions And Decimals
Mathematics NCERT Grade 7, Chapter 2: Fractions and Decimals- We have already studied about fractions and decimals. This chapter deals with the concept of multiplication and division of fractions as well as of decimals.
A proper fraction is a fraction that represents a part of a whole.
An improper fraction is a combination of a whole and a proper fraction.
The first half of the chapter deals with the Multiplication of Fractions and the later part deals with the concept of Division of Fractions. Two fractions are multiplied by multiplying their numerators and denominators separately.
Multiplication of fractions is sub-divided into the following topics:
Before moving onto decimal numbers, the division of fractions is explained. This particular topic contains the following divisions:
Division of whole number by a fraction
The same operations are given for decimal numbers.
Multiplication of decimal numbers by 10, 100, and 1000
Reciprocal of a fraction can be studied in this chapter wherein two non-zero numbers whose product with each other is 1 are called reciprocals of each other.
Division of decimal numbers by 10, 100, and 1000
For quick revision, important points of the chapter are listed in the end.
Since 42>24>14,
As 49 > 30>14,
In a “magic square”, the sum of the numbers in each row, in each column and along the diagonal is the same. Is this a magic square?
(Along the first row )
Along the first row, sum =
Along the second row, sum =
Along the third row, sum =
Along the first column, sum =
Along the second column, sum =
Along the third column, sum =
Along the first diagonal, sum =
Along the second diagonal, sum =
A rectangular sheet of paper is cm long and cm wide.
Find the perimeters of (i) ΔABE (ii) the rectangle BCDE in this figure. Whose perimeter is greater?
(i) Perimeter of ΔABE = AB + BE + EA
Perimeter of ΔABE =
\frac{177}{20}=\frac{177×3}{20×3}=\frac{531}{60}\phantom{\rule{0ex}{0ex}}\frac{47}{6}=\frac{47×10}{6×10}=\frac{470}{60}
⇒\frac{177}{20}>\frac{47}{6}
Perimeter (ΔABE) > Perimeter (BCDE)
Salil wants to put a picture in a frame. The picture is cm wide.
To fit in the frame the picture cannot be more than cm wide. How much should the picture be trimmed?
Width of picture =
Required width =
Ritu ate part of an apple and the remaining apple was eaten by her brother Somu. How much part of the apple did Somu eat? Who had the larger share? By how much?
Part of apple eaten by Ritu =
Therefore, Somu ate part of the apple.
Difference between the 2 shares =
Therefore, Ritu’s share is larger than the share of Somu by .
Video Solution for fractions and decimals (Page: 32 , Q.No.: 7)
NCERT Solution for Class 7 math - fractions and decimals 32 , Question 7
Michael finished colouring a picture in hour. Vaibhav finished colouring the same picture in hour. Who worked longer? By what fraction was it longer?
Time taken by Michael =
Time taken by Vaibhav =
Difference = = =
(i) represents addition of 2 figures, each representing 1 shaded part out of 5 equal parts. Hence, is represented by (d).
(ii) represents addition of 2 figures, each representing 1 shaded part out of 2 equal parts. Hence, is represented by (b).
(iii) represents addition of 3 figures, each representing 2 shaded parts out of 3 equal parts. Hence, is represented by (a).
(iv) represents addition of 3 figures, each representing 1 shaded part out of 4 equal parts. Hence, is represented by (c).
(i) represents the addition of 3 figures, each representing 1 shaded part out of 5 equal parts and represents 3 shaded parts out of 5 equal parts. Hence, is represented by (c).
(ii) represents the addition of 2 figures, each representing 1 shaded part out of 3 equal parts and represents 2 shaded parts out of 3 equal parts. Hence, is represented by (a).
(iii) represents the addition of 3 figures, each representing 3 shaded parts out of 4 equal parts and represents 2 fully shaded figures and one figure having 1 part as shaded out of 4 equal parts. Hence, is represented by (b)
(i) of the circles in box (a) (ii) of the triangles in box (b)
(iii) of the squares in box (c)
(i) It can be observed that there are 12 circles in the given box. We have to shade of the circles in it. As , therefore, we will shade any 6 circles of it.
(ii) It can be observed that there are 9 triangles in the given box. We have to shade of the triangles in it. As , therefore, we will shade any 6 triangles of it.
(iii) It can be observed that there are 15 squares in the given box. We have to shade of the squares in it. As , therefore, we will shade any 9 squares of it.
(a) of (i) 24 (ii) 46
(b) of (i) 18 (ii) 27
(c) of (i) 16 (ii) 36
(d) of (i) 20 (ii) 35
(c) (i)
(d) (i)
Find (a) of (i) (ii) (b) of (i) (ii)
Vidya and Pratap went for a picnic. Their mother gave them a water bottle that contained 5 litres of water. Vidya consumed of the water. Pratap consumed the remaining water.
(i) Water consumed by Vidya = of 5 litres
(ii) Water consumed by Pratap = of the total water
(i) of (a) (b) (c)
(ii) of (a) (b) (c)
(i) (a)
(ii) (a)
This is an improper fraction and it can be written as a mixed fraction as .
This is a whole number.
(i) of or of
(ii) of or of
Converting these fractions into like fractions,
Therefore, of is greater.
From the figure, it can be observed that gaps between 1st and last sapling = 3
Length of 1 gap =
Therefore, distance between I and IV sapling =
Lipika reads a book for hours everyday. She reads the entire book in 6 days. How many hours in all were required by her to read the book?
Number of hours Lipika reads the book per day =
Total number of hours required by her to read the book =
A car runs 16 km using 1 litre of petrol. How much distance will it cover using litres of petrol.
Number of kms a car can run per litre petrol = 16 km
Quantity of petrol =
Number of kms a car can run for litre petrol = = 44 km
It will cover 44 km distance by using litres of petrol.
(a) (i) Provide the number in the box , such that .
(ii) The simplest form of the number obtained in is _______.
(b) (i) Provide the number in the box , such that ?
(a) (i) As ,
Therefore, the number in the box , such that is
(ii) The simplest form of is .
(b) (i) As ,
(ii) As cannot be further simplified, therefore, its simplest form is
A proper fraction is the fraction which has its denominator greater than its numerator while improper fraction is the fraction which has its numerator greater than its denominator. Whole numbers are a collection of all positive integers including 0.
Therefore, it is an improper fraction.
Therefore, it is a proper fraction.
Therefore, it is a whole number.
(i) 0.5 or 0.05 (ii) 0.7 or 0.5 (iii) 7 or 0.7
(iv) 1.37 or 1.49 (v) 2.03 or 2.30 (vi) 0.8 or 0.88
(i) 0.5 or 0.05
Converting these decimal numbers into equivalent fractions,
It can be observed that both fractions have the same denominator.
Therefore, 0.5 > 0.05
(ii) 0.7 or 0.5
As 7 > 5,
Therefore, 0.7 >0.5
(iii) 7 or 0.7
(iv) 1.37 or 1.49
As 137 < 149,
Therefore, 1.37 < 1.49
(v) 2.03 or 2.30
As 80 < 88,
Therefore, 0.8 < 0.88
(i) 7 paise (ii) 7 rupees 7 paise (iii) 77 rupees 77 paise
(iv) 50 paise (v) 235 paise
There are 100 paise in 1 rupee. Therefore, if we want to convert paise into rupees, then we have to divide paise by 100.
(i) 7 paise =
(ii) 7 Rs 7 paise =
(iii) 77 Rs 77 paise = Rs 77.77
(iv) 50 paise
(v) 235 paise
(i) Express 5 cm in metre and kilometre
(ii) Express 35 mm in cm, m and km
(i) 200 g (ii) 3470 g (iii) 4 kg 8 g
(i) 200 g
(ii) 3470 g
(iii) 4 kg 8 g = 4.008 kg
(i) 20.03 (ii) 2.03 (iii) 200.03
(iii) 200.03
(i) 2.56 (ii) 21.37 (iii) 10.25
(iv) 9.42 (v) 63.352
(v) 63.352
Distance travelled by Dinesh = AB + BC = (7.5 + 12.7) km
Therefore, Dinesh travelled 20.2 km.
Distance travelled by Ayub = AD + DC = (9.3 + 11.8) km
Therefore, Ayub travelled 21.1 km.
Hence, Ayub travelled more distance.
Difference = (21.1 − 20.2) km
Therefore, Ayub travelled 0.9 km more than Dinesh.
Total fruits bought by Shyama = 5 kg 300 g + 3 kg 250 g
Total fruits bought by Sarala = 4 kg 800 g + 4 kg 150 g
Therefore, 28 km is 14.6 km less than 42.6 km.
(i) 0.2 × 6 (ii) 8 × 4.6 (iii) 2.71 × 5
(iv) 20.1 × 4 (v) 0.05 × 7 (vi) 211.02 × 4
(vii) 2 × 0.86
(i) 1.3 × 10 (ii) 36.8 × 10 (iii) 153.7 ×10
(iv) 168.07 × 10 (v) 31.1 × 100 (vi) 156.1 × 100
(vii) 3.62 × 100 (viii) 43.07 × 100 (ix) 0.5 × 10
(x) 0.08 × 10 (xi) 0.9 × 100 (xii) 0.03 × 1000
(i) 1.3 × 10 = 13
(ii) 36.8 × 10 = 368
(iii) 153.7 × 10 = 1537
(vi) 168.07 × 10 = 1680.7
(v) 31.1 × 100 = 3110
(vi) 156.1 × 100 = 15610
(vii) 3.62 × 100 = 362
(viii) 43.07 × 100 = 4307
(ix) 0.5 × 10 = 5
(x) 0.08 × 10 = 0.8
(xi) 0.9 × 100 = 90
(xiii) 0.03 × 1000 = 30
Distance covered in 1 litre of petrol = 55.3 km
Distance covered in 10litre of petrol = 10 × 55.3 = 553 km
Therefore, it will cover 553 km distance in 10 litre petrol.
(i) 2.5 × 0.3 (ii) 0.1 × 51.7 (iii) 0.2 × 316.8
(iv) 1.3 × 3.1 (v) 0.5 × 0.05 (vi) 11.2 × 0.15
(vii) 1.07 × 0.02 (viii) 10.05 × 1.05 (ix) 101.01 × 0.01
(x) 100.01 × 1.1
(i) 0.4 ÷ 2 (ii) 0.35 ÷ 5 (iii) 2.48 ÷ 4
(iv) 65.4 ÷ 6 (v) 651.2 ÷ 4 (vi) 14.49 ÷ 7
(vii) 3.96 ÷ 4 (viii) 0.80 ÷ 5
(i) 4.8 ÷ 10 (ii) 52.5 ÷ 10 (iii) 0.7 ÷ 10
(iv) 33.1 ÷ 10 (v) 272.23 ÷ 10 (vi) 0.56 ÷ 10
(vii) 3.97 ÷ 10
We know that when a decimal number is divided by a multiple of 10 only (i.e., 10, 100, 1000, etc.), the decimal point will be shifted to the left by as many places as there are zeroes. Since here we are dividing by 10, the decimal will shift to the left by 1 place.
(i) 4.8 ÷ 10 = 0 .48
(ii) 52.5 ÷ 10 = 5.25
(iii) 0.7 ÷ 10 = 0.07
(iv) 33.1 ÷ 10 = 3.31
(v) 272.23 ÷ 10 = 27.223
(vi) 0.56 ÷ 10 = 0.056
(vii) 3.97 ÷ 10 = 0.397
(i) 2.7 ÷ 100 (ii) 0.3 ÷ 100 (iii) 0.78 ÷ 100
(iv) 432.6 ÷ 100 (v) 23.6 ÷ 100 (vi) 98.53 ÷ 100
We know that when a decimal number is divided by a multiple of 10 only (i.e., 10, 100, 1000, etc.), the decimal point will be shifted to the left by as many places as there are zeroes. Since here we are dividing by 100, the decimal will shift to the left by 2 places.
(i) 2.7 ÷ 100 = 0.027
(ii) 0.3 ÷ 100 = 0.003
(iii) 0.78 ÷ 100 = 0.0078
(iv) 432.6 ÷ 100 = 4.326
(v) 23.6 ÷ 100 = 0.236
(vi) 98.53 ÷ 100 = 0.9853
(i) 7.9 ÷ 1000 (ii) 26.3 ÷ 1000 (iii) 38.53 ÷ 1000
(iv) 128.9 ÷ 1000 (v) 0.5 ÷ 1000
(i) 7.9 ÷ 1000 = 0.0079
(ii) 26.3 ÷ 1000 = 0.0263
(iii) 38.53 ÷ 1000 = 0.03853
(iv) 128.9 ÷ 1000 = 0.1289
(v) 0.5 ÷ 1000 = 0.0005
(i) 7 ÷ 3.5 (ii) 36 ÷ 0.2 (iii) 3.25 ÷ 0.5
(iv) 30.94 ÷ 0.7 (v) 0.5 ÷ 0.25 (vi) 7.75 ÷ 0.25
(vii) 76.5 ÷ 0.15 (viii) 37.8 ÷ 1.4 (ix) 2.73 ÷ 1.3
∴Distance covered in 1 litre of petrol =
Therefore, the vehicle will cover 18 km in 1 litre petrol.
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Developments in Alaska in 19621 | AAPG Bulletin | GeoScienceWorld
Developments in Alaska in 19621
https://doi.org/10.1306/BC743AE3-16BE-11D7-8645000102C1865D
Keith W. Calderwood; Developments in Alaska in 1962. AAPG Bulletin 1963;; 47 (6): 1200–1212. doi: https://doi.org/10.1306/BC743AE3-16BE-11D7-8645000102C1865D
During 1962, 23 exploratory wells were drilled compared with 20 in 1961. Development wells decreased from 29 to 13.
First offshore drilling commenced in 1962 from 2 barges and 1 platform in Cook Inlet basin and from 1 platform along the Alaska Peninsula. Offshore drilling in Cook Inlet was discontinued in November due to winter ice but will be resumed in early spring.
First wells were drilled on the west side of Cook Inlet.
Three new gas fields were discovered during 1962 and 1 offshore gas blow-out was burning. No oil fields were found.
A seismic crew was air-lifted to the Arctic (North) Slope and shooting commenced in the area east of Naval Petroleum Reserve No. 4. This is the first seismic work in the area since the U. S. Navy Department discontinued exploration in 1953.
Seismic mapping increased from
7314
crew-months in 1961 to
8114
crew-months in 1962 with all of the increase due to offshore sparker-gas exploder work.
Surface geologic exploration decreased 25.9%. An increase in surface geologic work and also seismic activity is expected in 1963. The Arctic (North) Slope is to receive the greatest increase. Exploratory drilling may show a slight decline and development drilling a continued decline.
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Spectral skewness for audio signals and auditory spectrograms - MATLAB spectralSkewness - MathWorks France
Spectral Skewness of Time-Domain Audio
Spectral Skewness of Frequency-Domain Audio Data
Calculate Spectral Skewness of Streaming Audio
Spectral skewness for audio signals and auditory spectrograms
skewness = spectralSkewness(x,f)
skewness = spectralSkewness(x,f,Name=Value)
[skewness,spread,centroid] = spectralSkewness(___)
spectralSkewness(___)
skewness = spectralSkewness(x,f) returns the spectral skewness of the signal, x, over time. How the function interprets x depends on the shape of f.
skewness = spectralSkewness(x,f,Name=Value) specifies options using one or more name-value arguments.
[skewness,spread,centroid] = spectralSkewness(___) returns the spectral spread and spectral centroid. You can specify an input combination from any of the previous syntaxes.
spectralSkewness(___) with no output arguments plots the spectral skewness.
If the input is in the time domain, the spectral skewness is plotted against time.
If the input is in the frequency domain, the spectral skewness is plotted against frame number.
Read in an audio file, calculate the skewness using default parameters.
skewness = spectralSkewness(audioIn,fs);
Plot the spectral skewness against time.
spectralSkewness(audioIn,fs)
Read in an audio file and then calculate the mel spectrogram using the melSpectrogram function. Calculate the skewness of the mel spectrogram over time.
skewness = spectralSkewness(s,cf);
Plot the spectral skewness against the frame number.
spectralSkewness(s,cf)
Calculate the skewness of the power spectrum over time. Calculate the skewness for 50 ms Hamming windows of data with 25 ms overlap. Use the range from 62.5 Hz to fs/2 for the skewness calculation.
skewness = spectralSkewness(audioIn,fs, ...
Plot the spectral skewness.
spectralSkewness(audioIn,fs, ...
Create a dsp.AudioFileReader object to read in audio data frame-by-frame. Create a dsp.SignalSink to log the spectral skewness calculation.
Calculate the spectral skewness for the frame of audio.
Log the spectral skewness for later plotting.
To calculate the spectral skewness for only a given input frame, specify a window with the same number of samples as the input, and set the overlap length to zero. Plot the logged data.
skewness = spectralSkewness(audioIn,fileReader.SampleRate, ...
logger(skewness)
The input to your audio stream loop has an inconsistent samples-per-frame with the analysis window of spectralSkewness.
You want to calculate the spectral skewness for overlapped data.
Specify that the spectral skewness is calculated for 50 ms frames with a 25 ms overlap.
skewness = spectralSkewness(audioBuffered,fs, ...
"power" –– The spectral skewness is calculated for the one-sided power spectrum.
"magnitude" –– The spectral skewness is calculated for the one-sided magnitude spectrum.
skewness — Spectral skewness
Spectral skewness, returned as a scalar, vector, or matrix. Each row of skewness corresponds to the spectral skewness of a window of x. Each column of skewness corresponds to an independent channel.
The spectral skewness is calculated as described in [1]:
\text{skewness}=\frac{\sum _{k={b}_{1}}^{{b}_{2}}{\left({f}_{k}-{\mu }_{1}\right)}^{3}{s}_{k}}{{\left({\mu }_{2}\right)}^{3}\sum _{k={b}_{1}}^{{b}_{2}}{s}_{k}}
spectralCentroid | spectralSpread | spectralKurtosis
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On a Boundary-Value Problem for Pseudo-Differential Operators and its Relation to Jump Processes | EMS Press
On a Boundary-Value Problem for Pseudo-Differential Operators and its Relation to Jump Processes
V. M. Glushkov Institute of Cybernetics, Kiev, Ukraine
We show the solvability of Dirichlet and Neumann boundary-value problems in
L_p(\mathbb{R}^n_+)
for an operator
A
of some special type. In case
p=2
it is possible to give another description of the trace of the domain of
A
\mathbb{R}^{n-1}
V. Knopova, On a Boundary-Value Problem for Pseudo-Differential Operators and its Relation to Jump Processes. Z. Anal. Anwend. 26 (2007), no. 1, pp. 1–24
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Hosmer–Lemeshow test - Wikipedia
1.2 Pearson chi-squared goodness of fit test
1.3 Calculation of the statistic
Logistic regression models provide an estimate of the probability of an outcome, usually designated as a "success". It is desirable that the estimated probability of success be close to the true probability. Consider the following example.
A researcher wishes to know if caffeine improves performance on a memory test. Volunteers consume different amounts of caffeine from 0 to 500 mg, and their score on the memory test is recorded. The results are shown in the table below.
n.volunteers
A.grade
proportion.A
10 450 30 1 0.03
group: identifier for the 11 treatment groups, each receiving a different dose
caffeine: mg of caffeine for volunteers in a treatment group
n.volunteers: number of volunteers in a treatment group
A.grade: the number of volunteers who achieved an A grade in the memory test (success)
proportion.A: the proportion of volunteers who achieved an A grade
The researcher performs a logistic regression, where "success" is a grade of A in the memory test, and the explanatory (x) variable is dose of caffeine. The logistic regression indicates that caffeine dose is significantly associated with the probability of an A grade (p < 0.001). However, the plot of the probability of an A grade versus mg caffeine shows that the logistic model (red line) does not accurately predict the probability seen in the data (black circles).
The logistic model suggests that the highest proportion of A scores will occur in volunteers who consume zero mg caffeine, when in fact the highest proportion of A scores occurs in volunteer consuming in the range of 100 to 150 mg.
The same information may be presented in another graph that is helpful when there are two or more explanatory (x) variables. This is a graph of observed proportion of successes in the data and the expected proportion as predicted by the logistic model. Ideally all the points fall on the diagonal red line.
The expected probability of success (a grade of A) is given by the equation for the logistic regression model:
{\displaystyle p(success)={\frac {1}{1+e^{-(b_{0}+b_{1}x_{1})}}}}
where b0 and b1 are specified by the logistic regression model:
b1 is the coefficient for x1
For the logistic model of P(success) vs dose of caffeine, both graphs show that, for many doses, the estimated probability is not close to the probability observed in the data. This occurs even though the regression gave a significant p-value for caffeine. It is possible to have a significant p-value, but still have poor predictions of the proportion of successes. The Hosmer–Lemeshow test is useful to determine if the poor predictions (lack of fit) are significant, indicating that there are problems with the model.
There are many possible reasons that a model may give poor predictions. In this example, the plot of the logistic regression suggests that the probability of an A score does not change monotonically with caffeine dose, as assumed by the model. Instead, it increases (from 0 to 100 mg) and then decreases. The current model is P(success) vs caffeine, and appears to be an inadequate model. A better model might be P(success) vs caffeine + caffeine^2. The addition of the quadratic term caffeine^2 to the regression model would allow for the increasing and then decreasing relationship of grade to caffeine dose. The logistic model including the caffeine^2 term indicates that the quadratic caffeine^2 term is significant (p=0.003) while the linear caffeine term is not significant (p=0.21).
The graph below shows the observed proportion of successes in the data versus the expected proportion as predicted by the logistic model that includes the caffeine^2 term.
The Hosmer–Lemeshow test can determine if the differences between observed and expected proportions are significant, indicating model lack of fit.
Pearson chi-squared goodness of fit test[edit]
The Pearson chi-squared goodness of fit test provides a method to test if the observed and expected proportions differ significantly. This method is useful if there are many observations for each value of the x variable(s).
For the caffeine example, the observed number of A grades and non-A grades are known. The expected number (from the logistic model) can be calculated using the equation from the logistic regression. These are shown in the table below.
The null hypothesis is that the observed and expected proportions are the same across all doses. The alternative hypothesis is that the observed and expected proportions are not the same.
The Pearson chi-squared statistic is the sum of (observed – expected)^2/expected. For the caffeine data, the Pearson chi-squared statistic is 17.46. The number of degrees of freedom is the number of doses (11) minus the number of parameters from the logistic regression (2), giving 11 - 2 = 9 degrees of freedom. The probability that a chi-square statistic with df=9 will be 17.46 or greater is p = 0.042. This result indicates that, for the caffeine example, the observed and expected proportions of A grades differ significantly. The model does not accurately predict the probability of an A grade, given the caffeine dose. This result is consistent with the graphs above.
In this caffeine example, there are 30 observations for each dose, which makes calculation of the Pearson chi-squared statistic feasible. Unfortunately, it is common that there are not enough observations for each possible combinations of values of the x variables, so the Pearson chi-squared statistic cannot be readily calculated. A solution to this problem is the Hosmer-Lemeshow statistic. The key concept of the Hosmer-Lemeshow statistic is that, instead of observations being grouped by the values of the x variable(s), the observations are grouped by expected probability. That is, observations with similar expected probability are put into the same group, usually to create approximately 10 groups.
Calculation of the statistic[edit]
The Hosmer–Lemeshow test statistic is given by:
{\displaystyle H=\sum _{g=1}^{G}\left({\frac {(O_{1g}-E_{1g})^{2}}{E_{1g}}}+{\frac {(O_{0g}-E_{0g})^{2}}{E_{0g}}}\right)=\sum _{g=1}^{G}\left({\frac {(O_{1g}-E_{1g})^{2}}{N_{g}\pi _{g}}}+{\frac {(N_{g}-O_{1g}-(N_{g}-E_{1g}))^{2}}{N_{g}(1-\pi _{g})}}\right)=\sum _{g=1}^{G}{\frac {(O_{1g}-E_{1g})^{2}}{N_{g}\pi _{g}(1-\pi _{g})}}.\,\!}
Here O1g, E1g, O0g, E0g, Ng, and πg denote the observed Y=1 events, expected Y=1 events, observed Y=0 events, expected Y=0 events, total observations, predicted risk for the gth risk decile group, and G is the number of groups. The test statistic asymptotically follows a
{\displaystyle \chi ^{2}}
distribution with G − 2 degrees of freedom. The number of risk groups may be adjusted depending on how many fitted risks are determined by the model. This helps to avoid singular decile groups.
The Pearson chi-squared goodness of fit test cannot be readily applied if there are only one or a few observations for each possible value of an x variable, or for each possible combination of values of x variables. The Hosmer-Lemeshow statistic was developed to address this problem.
Suppose that, in the caffeine study, the researcher was not able to assign 30 volunteers to each dose. Instead, 170 volunteers reported the estimated amount of caffeine they consumed in the previous 24 hours. The data are shown in the table below.
The table shows that, for many dose levels, there are only one or a few observations. The Pearson chi-squared statistic would not give reliable estimates in this situation.
The logistic regression model for the caffeine data for 170 volunteers indicates that caffeine dose is significantly associated with an A grade, p < 0.001. The graph shows that there is a downward slope. However, the probability of an A grade as predicted by the logistic model (red line) does not accurately predict the probability estimated from the data for each dose (black circles). Despite the significant p-value for caffeine dose, there is lack of fit of the logistic curve to the observed data.
This version of the graph can be somewhat misleading, because different numbers of volunteers take each dose. In an alternative graph, the bubble plot, the size of the circle is proportional to the number of volunteers.[1]
The plot of observed versus expected probability also indicates the lack of fit of the model, with much scatter around the ideal diagonal.
Calculation of the Hosmer-Lemeshow statistic proceeds in 6 steps,[2] using the caffeine data for 170 volunteers as an example.
1. Compute p(success) for all n subjects
Compute p(success) for each subject using the coefficients from the logistic regression. Subjects with the same values for the explanatory variables will have the same estimated probability of success. The table below shows the p(success), the expected proportion of volunteers with an A grade, as predicted by the logistic model.
2. Order p(success) from largest to smallest values
The table from Step 1 is sorted by p(success), the expected proportion. If every volunteer took a different dose, there would be 170 different values in the table. Because there are only 21 unique dose values, there are only 21 unique values of p(success).
3. Divide the ordered values into Q percentile groups
The ordered values of p(success) are divided into Q groups. The number of groups, Q, is typically 10. Because of tied values for p(success), the number of subjects in each group may not be identical. Different software implementations of the Hosmer–Lemeshow test use different methods for handling subjects with the same p(success), so the cut points to create the Q groups may differ. In addition, using a different value for Q will produce different cut points. The table in Step 4 shows the Q = 10 intervals for the caffeine data.
4. Create a table of observed and expected counts
The observed number of successes and failures in each interval are obtained by counting the subjects in that interval. The expected number of successes in an interval is the sum of the probability of success for the subjects in that interval.
The table below shows the cut points for the p(success) intervals selected by the R function HLTest() from Bilder and Loughin, with the number of observed and expected A and not A.
5. Calculate the Hosmer-Lemeshow statistic from the table
The Hosmer-Lemeshow statistic is calculated using the formula given in the introduction, which for the caffeine example is 17.103.
{\displaystyle H=\sum _{q=1}^{10}\left({\frac {(Observed.A-Expected.A)^{2}}{Expected.A}}+{\frac {(Observed.not.A-Expected.not.A)^{2}}{Expected.not.A}}\right)}
Compare the computed Hosmer-Lemeshow statistic to a chi-squared distribution with Q-2 degrees of freedom to calculate the p-value.
There are Q = 10 groups in the caffeine example, giving 10 – 2 = 8 degrees of freedom. The p-value for a chi-squared statistic of 17.103 with df = 8 is p = 0.029. The p-value is below alpha = 0.05, so the null hypothesis that the observed and expected proportions are the same across all doses is rejected. The way to compute this is to get a cumulative distribution function for a right-tail chi-square distribution with 8 degrees of freedom, i.e. cdf_chisq_rt(x, 8), or 1-cdf_chisq_lt(x, 8).
The Hosmer–Lemeshow test has limitations. Harrell describes several:[3]
"The Hosmer-Lemeshow test is for overall calibration error, not for any particular lack of fit such as quadratic effects. It does not properly take overfitting into account, is arbitrary to choice of bins and method of computing quantiles, and often has power that is too low."
"For these reasons the Hosmer-Lemeshow test is no longer recommended. Hosmer et al have a better one d.f. omnibus test of fit, implemented in the R rms package residuals.lrm function."
"But I recommend specifying the model to make it more likely to fit up front (especially with regard to relaxing linearity assumptions using regression splines) and using the bootstrap to estimate overfitting and to get an overfitting-corrected high-resolution smooth calibration curve to check absolute accuracy. These are done using the R rms package."
Other alternatives have been developed to address the limitations of the Hosmer-Lemeshow test. These include the Osius-Rojek test and the Stukel test.[4]
^ Bilder, Christopher R.; Loughin, Thomas M. (2014), Analysis of Categorical Data with R (First ed.), Chapman and Hall/CRC, ISBN 978-1439855676
^ Kleinbaum, David G.; Klein, Mitchel (2012), Survival analysis: A Self-learning text (Third ed.), Springer, ISBN 978-1441966452
^ "r - Evaluating logistic regression and interpretation of Hosmer-Lemeshow Goodness of Fit". Cross Validated. Retrieved 2020-02-29.
^ available in the R script AllGOFTests.R: www.chrisbilder.com/categorical/Chapter5/AllGOFTests.R.
Hosmer, David W.; Lemeshow, Stanley (2013). Applied Logistic Regression. New York: Wiley. ISBN 978-0-470-58247-3.
Alan Agresti (2012). Categorical Data Analysis. Hoboken: John Wiley and Sons. ISBN 978-0-470-46363-5.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Hosmer–Lemeshow_test&oldid=959428902"
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Implement vehicle in 3D environment - Simulink - MathWorks Deutschland
Simulation 3D Vehicle
Initial array values to scale vehicle per part, Scale
Implement vehicle in 3D environment
The Simulation 3D Vehicle block implements a vehicle with four wheels in the 3D simulation environment.
Verify that the Simulation 3D Vehicle block executes before the Simulation 3D Scene Configuration block. That way, Simulation 3D Vehicle prepares the signal data before the Unreal Engine® 3D visualization environment receives it. To check the block execution order, right-click the blocks and select Properties. On the General tab, confirm these Priority settings:
Vehicle and wheel translation, in m. Array dimensions are 5-by-3.
Translation(1,1), Translation(1,2), and Translation(1,3) — Vehicle translation along the inertial vehicle Z-down X-, Y-, and Z- axes, respectively.
Translation(...,1), Translation(...,2), and Translation(...,3) — Wheel translation relative to vehicle, along the vehicle Z-down X-, Y-, and Z- axes, respectively.
Translation=\left[\begin{array}{ccc}{X}_{v}& {Y}_{v}& {Z}_{v}\\ {X}_{FL}& {Y}_{FL}& {Z}_{FL}\\ {X}_{FR}& {Y}_{FR}& {Z}_{FR}\\ {X}_{RL}& {Y}_{RL}& {Z}_{RL}\\ {X}_{RR}& {Y}_{RR}& {Z}_{RR}\end{array}\right]
Vehicle and wheel rotation, in rad. Array dimensions are 5-by-3.
Rotation(1,1), Rotation(1,2), and Rotation(1,3) — Vehicle rotation about the inertial vehicle Z-down X-, Y-, and Z- axes, respectively.
Rotation(...,1), Rotation(...,2), and Rotation(...,3) — Wheel rotation relative to vehicle, about the vehicle Z-down X-, Y-, and Z- axes, respectively.
Rotation=\left[\begin{array}{ccc}Rol{l}_{v}& Pitc{h}_{v}& Ya{w}_{v}\\ Rol{l}_{FL}& Pitc{h}_{FL}& Ya{w}_{FL}\\ Rol{l}_{FR}& Pitc{h}_{FR}& Ya{w}_{FR}\\ Rol{l}_{RL}& Pitc{h}_{RL}& Ya{w}_{RL}\\ Rol{l}_{RR}& Pitc{h}_{RR}& Ya{w}_{RR}\end{array}\right]
Scale — Vehicle scale
Vehicle and wheel scale, dimensionless. Array dimensions are 5-by-3.
Scale(1,1), Scale(1,2), and Scale(1,3) — Vehicle scale along the inertial vehicle Z-down X-, Y-, and Z- axes, respectively.
Scale(...,1), Scale(...,2), and Scale(...,3) — Wheel scale relative to vehicle, along vehicle Z-down X-, Y-, and Z- axes, respectively.
The signal contains scale information according to the axle and wheel locations.
Scale=\left[\begin{array}{ccc}{X}_{{V}_{scale}}& {Y}_{{V}_{scale}}& {Z}_{{V}_{scale}}\\ {X}_{F{L}_{scale}}& {Y}_{F{L}_{scale}}& {Z}_{F{L}_{scale}}\\ {X}_{F{R}_{scale}}& {Y}_{F{R}_{scale}}& {Z}_{F{R}_{scale}}\\ {X}_{R{L}_{scale}}& {Y}_{R{L}_{scale}}& {Z}_{R{L}_{scale}}\\ {X}_{R{R}_{scale}}& {Y}_{R{R}_{scale}}& {Z}_{R{R}_{scale}}\end{array}\right]
Scale(1,1) Vehicle Z-down X-axis
Scale(1,2) Vehicle Z-down Y-axis
Scale(1,3) Vehicle Z-down Z-axis
Muscle car (default) | Sedan | Sport utility vehicle | Small pickup truck | Hatchback | Box truck
If you set Actor type to Passenger vehicle, use the Vehicle type parameter to specify the vehicle. This table provides links to the vehicle dimensions.
Initial vehicle and wheel translation, in m. Array dimensions are 5-by-3.
Translation(...,1), Translation(...,2), and Translation(...,3) — Initial wheel translation relative to vehicle, along the vehicle Z-down X-, Y-, and Z- axes, respectively.
The parameter contains translation information according to the axle and wheel locations.
Translation=\left[\begin{array}{ccc}{X}_{v}& {Y}_{v}& {Z}_{v}\\ {X}_{FL}& {Y}_{FL}& {Z}_{FL}\\ {X}_{FR}& {Y}_{FR}& {Z}_{FR}\\ {X}_{RL}& {Y}_{RL}& {Z}_{RL}\\ {X}_{RR}& {Y}_{RR}& {Z}_{RR}\end{array}\right]
Initial vehicle and wheel rotation, about the vehicle Z-down X-, Y-, and Z- axes.
Rotation(1,1), Rotation(1,2), and Rotation(1,3) — Initial vehicle rotation about the inertial vehicle Z-down coordinate systemX-, Y-, and Z- axes, respectively.
Rotation(...,1), Rotation(...,2), and Rotation(...,3) — Initial wheel rotation relative to vehicle, about the vehicle Z-down X-, Y-, and Z- axes, respectively.
The parameter contains rotation information according to the axle and wheel locations.
Rotation=\left[\begin{array}{ccc}Rol{l}_{v}& Pitc{h}_{v}& Ya{w}_{v}\\ Rol{l}_{FL}& Pitc{h}_{FL}& Ya{w}_{FL}\\ Rol{l}_{FR}& Pitc{h}_{FR}& Ya{w}_{FR}\\ Rol{l}_{RL}& Pitc{h}_{RL}& Ya{w}_{RL}\\ Rol{l}_{RR}& Pitc{h}_{RR}& Ya{w}_{RR}\end{array}\right]
Initial array values to scale vehicle per part, Scale — Vehicle initial scale
ones( 5, 3 ) (default) | 5-by-3 array
Initial vehicle and wheel scale, dimensionless. Array dimensions are 5-by-3.
Scale(1,1), Scale(1,2), and Scale(1,3) — Initial vehicle scale along the inertial vehicle Z-down X-, Y-, and Z- axes, respectively.
Scale(...,1), Scale(...,2), and Scale(...,3) — Initial wheel scale relative to vehicle, along vehicle Z-down X-, Y-, and Z- axes, respectively.
The parameter contains scale information according to the axle and wheel locations.
Scale=\left[\begin{array}{ccc}{X}_{{V}_{scale}}& {Y}_{{V}_{scale}}& {Z}_{{V}_{scale}}\\ {X}_{F{L}_{scale}}& {Y}_{F{L}_{scale}}& {Z}_{F{L}_{scale}}\\ {X}_{F{R}_{scale}}& {Y}_{F{R}_{scale}}& {Z}_{F{R}_{scale}}\\ {X}_{R{L}_{scale}}& {Y}_{R{L}_{scale}}& {Z}_{R{L}_{scale}}\\ {X}_{R{R}_{scale}}& {Y}_{R{R}_{scale}}& {Z}_{R{R}_{scale}}\end{array}\right]
Use the enabled parameters to specify the vehicle lights for:
Simulation 3D Vehicle with Ground Following | Simulation 3D Scene Configuration
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NPV - Anaplan Technical Documentation
The net present value of an investment enables you to know how much a future investment is worth today. You could use the NPV function to assess whether an investment is worthwhile while accounting for depreciation.
NPV(Discount rate, Cash flow, Dates, Transactions)
Discount rate (required) Number The discount rate used to calculate net present value. This argument uses percentage format, so 0.1 is equivalent to 10%.
Cash flow (required) Number A series of positive and negative values that represent cash inflow and outflow.
The date associated with each value of the Cash flow argument.
This argument is optional, but if used, you must provide the Transactions argument also.
A list of transactions.
This argument is optional, but if used, the list must be a common dimension of the Cash flow and Dates argument.
The NPV function returns a number.
How NPV is calculated
NPV is the solution to this equation:
NPV = \Sigma_{i = 1}^{N} \quad_{(1 + Rate) ^{d}i/365} ^ {\quad \quad P}
N is the number of payments in and out over the time scale
Pi is the payment in the ith period
di is the number of days from the start of the first period
Rate is the discount rate
Use NPV with the Users list
You can reference the Users list with the NPV function. However, you cannot reference specific users within the Users list as this is production data, which can change and make your formula invalid.
Example of NPV with timescale
This example uses two modules. The first module, Annual cash flow, contains Cash flow line item with a Time Scale of Year. The Cash flow line item contains both a positive and negative value.
Cash flow -100,000 0 130,000
The second module contains two line items and no other dimensions. One line item contains the Discount rate, and another line item contains a formula that uses NPV with the Discount rate and the Cash flow line item from the previous module. As the IRR function returns a single value, the result does not need a time dimension.
NPV of annual cash flow
NPV(Discount rate, 'Annual cash flow'.Cash flow)
Example of NPV with dates
In this example, two modules are used. One source module, Plant Transaction Data, is dimensioned by the Transactions and Plants lists. Transactions is on rows, and Plants is on pages, with Plant 1 selected.
The Plant Transaction Data module contains the values of transactions, dates of transactions, and a description for some transactions.
Description Cash flow Date
Transaction 01 Land purchase -100,000 1/1/2011
Transaction 02 Turbine purchase -850,000 1/1/2012
Transaction 03 Energy generation revenue 200,000 1/3/2013
Transaction 16 Final energy generation revenue 200,000 1/3/2026
The second module uses the data from the Plant Transaction Data module with the NPV function to calculate the net present value for each plant. The column for Plant 1 contains the internal rate of return for data displayed in the Plant Transaction Data module.
Net Present Value for Plant
NPV(0.1, 'Plant Transaction Data'. Cash flow, 'Plant Transaction Data'. Date, Transactions)
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Uses of Amines: Industry & Daily Life | StudySmarter
Have you ever wondered where the bright red pigment of your favourite woollen socks comes from? It's quite likely that it is derived from amines. This is just one example of the uses of amines.
This article is about the uses of amines in organic chemistry.
We'll start with a recap of what amines actually are before exploring their uses.
You'll be able to see examples of amines in industryand in real life. This will include looking at quaternary ammonium salts and azo compounds.
In Amines, we introduced you to a new type of organic molecule: amines. These are ammonia derivatives, characterised by a nitrogen atom bonded to at least one organic hydrocarbon R group. Amines can be further divided into three different types:
Primary amines contain a nitrogen atom bonded to just one R group and have the general formula NH2R.
Secondary amines contain a nitrogen atom bonded to two R groups and have the general formula NHR2.
Tertiary amines contain a nitrogen atom bonded to three R groups and have the general formula NR3.
Primary, secondary, and tertiary amines. Anna Brewer, StudySmarter Original
You can also get quaternary ammonium cations. These consist of a nitrogen atom bonded to four R groups. The nitrogen atom bonds to the fourth R group using a dative covalent bond. Quaternary ammonium ions are an important part of quaternary ammonium salts.
A quaternary ammonium cation. Anna Brewer, StudySmarter
Let's now briefly consider some of the characteristic properties of amines:
Amines are polar molecules.
Amines form hydrogen bonds, both with other amine molecules and with water. This means that they have high melting and boiling points, and that shorter-chain amines are soluble in aqueous solutions.
Amines can act as both nucleophiles and bases. Nucleophiles are electron-pair donors whilst bases are hydrogen ion acceptors. This enables many of their reactions.
Check out Amines for a more detailed look into the properties of amines. Head over to Amines Basicity if you want to learn more about their reactions as nucleophiles and bases.
Uses of amines in daily life
Now that we know what amines are, we can look at some of their day-to-day applications. After that, we'll consider their uses in industry.
Amines are found in every cell in your body in the form of proteins. Proteins are condensation polymers, made up of repeating units called amino acids. Each amino acid has both a carboxyl functional group and an amine functional group, and they join together to form a long polymer chain. This chain then folds into a specific 3D shape that is unique to each protein.
Another type of polymer involving amines is polyamides. These include nylon, Kevlar, and a variety of plastics.
Many common drugs and pharmaceuticals are amines. These include the analgesic morphine, the decongestant ephedrine, and the antidepressant amoxapine.
Amines play a role in cosmetics, such as shampoos, soaps, and shaving creams. We'll look at how they are made in just a second.
The common compound tetramethylammonium chloride, used to disinfect water, is also an amine.
Amines are the precursor to many dyes and tanning agents.
Head over to Proteins Biochemistry to learn more. You can also learn more about polyamides and other polymers in Condensation Polymers.
Uses of amines in industry
Knowing what we use amines for is well and good, but how do we make those products? It is now time to learn about two important industrial applications of amines.
Earlier, we learned that a quaternary ammonium ion consists of a nitrogen atom bonded to four organic hydrocarbon R groups. It has a permanent positive charge, which means that it can bond ionically to negatively-charged ions, forming a quaternary ammonium salt.
A quaternary ammonium salt. Anna Brewer, StudySmarter Original
Uses of quaternary ammonium salts
Quaternary ammonium salts have a few uses: conditioners, detergents, and antimicrobial agents. They're suitable because of their charge. In conditioners and fabric softeners, the positive charge of the ammonium ion is attracted to the negative charge of wet clothes or hair, and the ammonium ions form a layer on the surface. This helps keep the hair or fabric smooth and glossy. In detergents and antimicrobial agents, the positive charge of the ammonium ion is attracted to the negative charge of bacterial cell walls. This disrupts the wall and damages the cell.
In industry, we often also use amines to make diazonium salts and azo compounds. Diazonium salts contain an -N+≡N group, while azo compounds contain an N=N azo group. Producing azo compounds involves a multi-step synthesis:
Phenylamine (C6H5NH2) is reacted with nitric(III) acid (HNO2) at low temperatures to form a diazonium salt.
The diazonium salt reacts with another aromatic organic molecule to form an azo compound.
Let's investigate those steps in more detail.
Forming a diazonium salt
First of all, phenylamine reacts with nitric(III) acid at low temperatures to form a diazonium salt containing the -N≡N+ group. Nitric(III) acid is extremely reactive and so must be prepared in situ. To carry out the reaction, we mix phenylamine with a chilled solution of a strong acid, such as hydrochloric acid (HCl), and then add sodium nitrite (NaNO2). The hydrochloric acid and sodium nitrite first react to form nitric(III) acid and sodium chloride:
\mathrm{HCl}+ {\mathrm{NaNO}}_{2}\to {\mathrm{HNO}}_{2}+\mathrm{NaCl}
The nitric(III) acid formed then reacts with phenylamine and more hydrochloric acid to form a diazonium salt:
{\mathrm{C}}_{6}{\mathrm{H}}_{5}{\mathrm{NH}}_{2}+ {\mathrm{HNO}}_{2}+ \mathrm{HCl}\stackrel{<10°\mathrm{C}}{\to }{\mathrm{C}}_{6}{\mathrm{H}}_{5}{\mathrm{N}}^{+}\mathrm{N}{\mathrm{Cl}}^{-}+ 2{\mathrm{H}}_{2}\mathrm{O}
Here's a diagram to help you understand the structure of the molecules involved.
This reaction must be carried out below 10°C. If you heat the mixture, a different reaction takes place. Instead, you produce phenol (C6H5OH), nitrogen gas, and water:
{\mathrm{C}}_{6}{\mathrm{H}}_{5}{\mathrm{NH}}_{2}+ {\mathrm{HNO}}_{2}\stackrel{>10°\mathrm{C}}{\to }{\mathrm{C}}_{6}{\mathrm{H}}_{5}\mathrm{OH}+{\mathrm{N}}_{2}+ {\mathrm{H}}_{2}\mathrm{O}
Forming an azo compound
The second step of the process involves reacting the diazonium salt with another aromatic organic molecule. This is an example of a coupling reaction, forming an azo compound with the N=N azo functional group. In this reaction, the diazonium ion acts as an electrophile and substitutes into the second molecule's benzene ring.
One example is the reaction of a diazonium salt with phenol. This reaction takes place in a basic solution, typically of sodium hydroxide, and produces an azo compound with two benzene rings, one with an -OH group. These benzene rings are joined by an N=N azo bridge. It also produces an acid, which varies depending on the diazonium salt used.
Here's the equation for the reaction between a diazonium chloride salt and phenol. The structural formulae of these molecules can get a little tricky, so we've used displayed formulae to show you the reaction.
A similar reaction takes place between diazonium salts and phenylamine. This produces another type of azo compound, but this time the second benzene has an -NH2 group:
Uses of azo compounds
Finally, we'll consider the uses of azo compounds. Thanks to their two benzene rings, which are full of delocalised pi electrons, they are very stable and present with vivid colours. Azo compounds, therefore, form the basis of many dyes, including ones such as methyl orange that are used as pH indicators. They're also used in the textile industry and in tattoo inks.
Uses of Amines - Key takeaways
Amines are ammonia derivatives that contain a nitrogen atom bonded to one or more organic hydrocarbon R groups.
Amines are polar, can form hydrogen bonds, and act as both bases and nucleophiles.
Amines are found in proteins, plastics, pharmaceuticals, cosmetics, and dyes.
In industry, amines are turned into quaternary ammonium salts and azo compounds.
Quaternary ammonium salts contain a nitrogen atom bonded to four R groups and are used in detergents, conditioners, and antimicrobial agents.
Azo compounds are made from diazonium salts. These in turn are made from phenylamine and nitric(III) acid. Azo compounds are used as dyes.
Examples of amines include methylamine and phenylamine. However, we also find amines in daily life. For example, all proteins are made from amines known as amino acids, whilst many drugs such as morphine are also amines.
Amines are used in cosmetics, dyes, pharmaceuticals, and plastics.
What is the importance of amines?
Amines play important roles in many drugs, cosmetics, detergents, plastics, and antimicrobials. They also make up all proteins, which are found in every cell in our body.
Are amines acidic?
Amines aren't acidic, but basic. This means that they act as proton acceptors.
What are the physical properties of amines?
Amines can form hydrogen bonds. This means that they have high melting and boiling points.
Shorter-chain amines are soluble in water.
Final Uses of Amines Quiz
An ammonia derivative containing a nitrogen atom bonded to at least one R group.
Which of the following are true about amines?
True or false? Some proteins are not made of amines.
Compare and contrast a tertiary amine with a quaternary ammonium ion.
Both are ammonia derivatives.
Both contain a nitrogen atom bonded to R groups by covalent bonds.
The tertiary amine contains a nitrogen atom bonded to just three R groups whereas the quaternary ammonium ion contains a nitrogen atom bonded to four R groups.
The tertiary amine has a neutral charge whereas the quaternary ammonium ion has a positive charge.
The quaternary ammonium ion features a dative covalent bond.
Which of the following types of bonding are present in quaternary ammonium salts?
Covalent, dative covalent and ionic
Which of the following contain amines?
Give three uses of quaternary ammonium salts.
Explain why quaternary ammonium salts are good antimicrobial agents.
Quaternary ammonium salts contain a quaternary ammonium ion with a positive charge. This is attracted to the negative charge of bacterial cell walls. The ions disrupt the cell wall, damaging the cell.
Which of the following are made from amines?
How do you make a diazonium salt from phenylamine?
Mix phenylamine with a chilled solution of a strong acid such as hydrochloric acid (HCl)
Add sodium nitrite (NaNO2). This reacts with the hydrochloric acid to form nitric(III) acid.
The nitric acid reacts with phenylamine and more hydrochloric acid to form a diazonium salt and water.
How do you make an azo compound from a diazonium salt and phenol?
Mix the diazonium salt with phenol in basic solution, such as in sodium hydroxide. This produces an azo compound and an acid.
What is the azo group?
The reaction between a diazonium salt and phenol is a ____ reaction.
In the reaction between a diazonium salt and phenol, the diazonium salt acts as ____.
An electrophile
Name the two possible organic products formed in the reaction between phenylamine and nitric acid. Give the conditions needed for the respective reactions.
If the reaction is carried out below 10°C, then it produces a diazonium salt. If the reaction is carried out above 10°C, then it produces phenol.
of the users don't pass the Uses of Amines quiz! Will you pass the quiz?
Synthetic Routes Learn
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IMPORTANT UPDATE: This will affect the submission of uptime data and the Performance Score results for receiving a delegation.
The current sidecar tracking system will be phased out after the current cycle (Cycle 6) and be replaced by the SNARK-work-based uptime tracking system. Until further notice, while transitioning to the new system, continue running the sidecar AND SNARK-based tracking system on the SAME node.
Please follow the instructions in this post here and ask any questions in the #delegation-program channel on Discord while following along for the latest updates.
An important part of running a staking service is predicting/determining winning slots in which you can produce blocks, as well as paying out participants. The Mina protocol does not automatically payout rewards to delegates, so part of running a staking service is manually paying out participants.
This document aims to explain the different components that you should think about when managing those payouts. Specifically, this document provides an understanding of odds of winning blocks, gathering data from the ledger for later use, and computing relevant staking payout information from this data.
The coinbase reward for producing a block is 720 tokens. However, some accounts will receive 2x supercharged rewards if they contain only unlocked tokens. In the case of stake delegations, whether or not the reward is supercharged is based off of the account that won the block, not the account that is doing the staking and block production.
Dumping Staking Ledgers
In order to compute odds of winning a block for a given epoch, or to retroactively compute the coinbase reward a given account would receive, you need to have the staking ledger from that epoch. Mina daemons only keep around the staking ledger for the current epoch and the staking ledger for the next epoch, so if you want to capture a staking ledger for an epoch, you need to do it before or during that epoch.
The mina ledger export command can be used to export ledgers from a running daemon. It takes, as an argument, an identifier of the ledger you wish to export. The table below describes what each of these identifiers represent.
staking-epoch-ledger The staking ledger for the current epoch.
next-epoch-ledger The staking ledger for the next epoch (epoch after current).
staged-ledger The most recent staged ledger (from the best tip of that node).
In order to ensure you always have each staking ledger available for use after epochs have expired, we recommend exporting the staking-epoch-ledger every
\dfrac{7140\times3}{60} = 357
hours (there are
7140
slots in an epoch, and each slot is
3
minutes long).
By default, ledgers are exported as json data. See mina ledger export -help for documentation of flags which will enable other formats. When output as json, the ledger will be represented as an array of account objects. Below is an example of what an account object in json looks like.
"pk": "B62qrwZRsNkU39TrGpFwDdpRS2JaCB2yFZKMFNqLFYjcqGE5G5fWA8p",
"delegate": "B62qrwZRsNkU39TrGpFwDdpRS2JaCB2yFZKMFNqLFYjcqGE5G5fWA8p",
"token_permissions": {},
"receipt_chain_hash": "2mzbV7WevxLuchs2dAMY4vQBS6XttnCUF8Hvks4XNBQ5qiSGGBQe",
"voting_for": "3NK2tkzqqK5spR2sZ7tujjqPksL45M3UUrcA4WhCkeiPtnugyE2x",
"stake": true,
"edit_state": "signature",
"send": "signature",
"set_delegate": "signature",
"set_permissions": "signature",
"set_verification_key": "signature"
For a running staking service, you are interested in the accounts which you control and the accounts which are staking to accounts you control. As an example, if we only had one account in our staking service, we could grab all the accounts we are interested in for a ledger using a command like:
mina ledger export staking-epoch-ledger | jq "$(cat <<EOF
.[] | \
select( \
.pk == "B62qjwvvHoM9RVt9onMpSMS5rFzhy3jjXY1Q9JU7HyHW4oWFEbWpghe" or \
.delegate == "B62qjwvvHoM9RVt9onMpSMS5rFzhy3jjXY1Q9JU7HyHW4oWFEbWpghe" \
An important detail you will want to compute as a staking service is whether or not an account will receive supercharged coinbase rewards, which you will likely want to include into the staking service's weighting when paying out participants. Accounts with locked tokens (which are the accounts that do not receive supercharged staking rewards) will have a special "timing" field in their json representation. Here is an example of what this looks like:
"pk": "B62qmVHmj3mNhouDf1hyQFCSt3ATuttrxozMunxYMLctMvnk5y7nas1",
"delegate": "B62qk2ujo9BoBxCs9BFQUsv3efaJDzbJeLs4YJdZMJzJoVj69ShVdKs",
"initial_minimum_balance": "230400",
"cliff_time": "86400",
"cliff_amount": "230400",
"vesting_period": "1",
"vesting_increment": "0"
Just because an account has a "timing" field, it does not necessarily mean that account has locked tokens. Timed accounts contain locked and unlocked tokens, and the "timing" field describes the unlock schedule for the locked tokens in that account. If an account has a "timing" field, you can compute the global slot at which that account's tokens will be fully unlocked, after which point the account will receive supercharged rewards. This can be computed by determining how long a timed account will take to fully vest it's tokens, and adding that to the "cliff_time", which is the slot where the account begins vesting tokens. Here is some example python code which will compute this unlock time from an account's timing information.
vesting_amount = initial_min_balance - cliff_amount
vesting_time = vesting_amount * math.ceil(vesting_period / vesting_increment)
unlocked_time = cliff_time + vesting_time
Choosing How to Payout Rewards
It's up to each staking service to decide how they wish to compute rewards. It's important to take account supercharging and the weights of different delegators into account, depending no how you choose to payout rewards.
As an example for calculating staking service payouts, we recommend taking a look at a document put together by a member of our community over at docs.minaexplorer.com.
Odds of Winning a Block
Blocks in Mina are produced within distinct time intervals called "slots". Each account in the ledger has a chance of winning each slot on the network, which allows them to produce a block for that slot (or, in the case of delegation, allows the delegated account to produce a block on their behalf). An account computes a random number (via a Verifiable Random Function, or VRF) for each slot, and compares that random number against a required threshold to determine if they have "won" that slot and can produce a block at that time. Their chance of winning a slot is thus determined by the threshold, which is a function of their relative stake in the network.
The stake distribution that is sampled when determining the VRF threshold are contained in a special ledger called the "staking ledger". Staking ledgers are fixed ledgers from past epochs in the network (epochs are fixed spans of slots; every epoch is 7140 slots long). Using past fixed ledgers prevents malicious stakers from increasing their chances of winning during an epoch by altering the distribution of stake on the main ledger (the "staged ledger"). Every epoch, the staking ledger changes. Due to this constraint, any account on the system can only check for "winning slots" within the next 2 epochs at any given time.
Once you know the staking ledger that will be used for a given epoch, the required VRF threshold for producing blocks in that epoch can be computed for each account. We can compute a stake ratio for a given account by dividing the stake that account controls in the staking ledger by the total amount of currency in the staking ledger. Figure 1 on page 20 of the Ouroboros Genesis paper provides the raw function for computing VRF thresholds. Filling in Mina's value for the active slots coefficient
f
gives us the following function:
\phi\left(\alpha\right) \triangleq 1 - \left(\dfrac{1}{4}\right)^\alpha
. This function takes as a parameter
\alpha
, which is the stake ratio we compute for an account.
Since each VRF is compared against this threshold (which is between 0 and 1), and each VRF is a pseudo-random number between 0 and 1, the VRF threshold for a given account is essentially the probability that a slot will be won by that account. Thus, we can extrapolate this function for computing VRF thresholds to compute the mean number of blocks we can expect a given account to win within an epoch. This can be done merely by summing all the probabilities for winning each slot of an epoch, and since the probability for an entire epoch is fixed, the mathematical expression to compute this simplifies to
\phi\left(\alpha\right)\times7140
As an example, let's say there is an account on the network which has
10^6
(1 million) mina tokens in the staking ledger epoch
ep
. This same staking ledger for epoch
ep
has a total supply of
8\times10^8
(800 million) mina tokens. We can compute the odds that this account will win an individual slot in
ep
\phi\left(\dfrac{10^6}{8\times10^8}\right) = 0.0017313674
\sim0.17\%
chance that this account will win each slot during epoch
ep
. We can then compute the mean number of blocks we expect this account to produce for this epoch by multiplying the result by
7140
, giving us a probabilistic mean of
\sim12.36
blocks for the epoch.
Sending Many Transactions
See the advanced section of the sending payments page
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Hemodynamic Changes Induced by Pneumoperitoneum and Measured With ECOM | J. Med. Devices | ASME Digital Collection
Timothy Shine,
David Corda,
Stephen Aniskevich,
Stephen Aniskevich
Bruce Leone,
Bruce Leone
Neil Feinglass,
Sorin Brull,
Sorin Brull
Shine, T., Corda, D., Aniskevich, S., Leone, B., Feinglass, N., Brull, S., and Han, B. (June 3, 2011). "Hemodynamic Changes Induced by Pneumoperitoneum and Measured With ECOM." ASME. J. Med. Devices. June 2011; 5(2): 027504. https://doi.org/10.1115/1.3589225
bioelectric phenomena, biomedical measurement, cardiovascular system, haemodynamics, patient monitoring, surgery
Hemodynamics, Surgery, Bioelectric phenomena, Biomedical measurement, Cardiovascular system, Patient monitoring
Laparascopic surgery required inducing a pneumoperitoneum during surgery and anesthesia this presents unique hemodynamic challenges for the anesthetic management of patients. We monitored hemodynamic management using ECOM endotracheal tubes the parameters are derived using Bioimpedance Cardiac output, stroke volume variability, and systemic vascular resistance were measured using this technology. Pneumoperitoneum results in intra-abdominal pressure of 15–20 mm hg induced by
CO2
insufflation Hemodynamic parameters were measured using a new noninvasive device, the endotracheal cardiac output monitor (ECOM) (ConMed Corporation, Utica, NY). This monitor provides measurements—including cardiac output, systemic vascular resistance, and stroke volume variation—which were previously unavailable noninvasively. The results obtained were consistent with those found in the literature (1–4). Based on our assessment, it appears that ECOM derived hemodynamic changes are similar to those obtained invasively. Therefore, ECOM’s noninvasive method to measure cardiac output seems advantageous when considering patient safety, because it is less invasive. A better understanding of the applicability and reliability of this new technology in the clinical setting is important for patient safety.
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Create a dead-zone nonlinearity estimator object - MATLAB idDeadZone - MathWorks Switzerland
idDeadZone
Create a Default Dead-Zone Nonlinearity Estimator
Estimate a Hammerstein-Wiener Model with Dead-zone Nonlinearity
Create a dead-zone nonlinearity estimator object
NL = idDeadZone
NL = idDeadZone('ZeroInterval',[a,b])
NL = idDeadZone creates a default dead-zone nonlinearity estimator object for estimating Hammerstein-Wiener models. The interval in which the dead-zone exists (zero interval) is set to [NaN NaN]. The initial value of the zero interval is determined from the estimation data range, during estimation using nlhw. Use dot notation to customize the object properties, if needed.
NL = idDeadZone('ZeroInterval',[a,b]) creates a dead-zone nonlinearity estimator object initialized with zero interval, [a,b].
Alternatively, use NL = idDeadZone([a,b]).
idDeadZone is an object that stores the dead-zone nonlinearity estimator for estimating Hammerstein-Wiener models.
Use idDeadZone to define a nonlinear function
y=F\left(x,\theta \right)
, where y and x are scalars, and θ represents the parameters a and b, which define the zero interval.
The dead-zone nonlinearity function has the following characteristics:
\begin{array}{l}a\le x<b\text{ }F\left(x\right)=0\\ x<a\text{ }F\left(x\right)=x-a\\ x\ge b\text{ }F\left(x\right)=x-b\end{array}
For example, in the following plot, the dead-zone is in the interval [-4,4].
The value F(x) is computed by evaluate(NL,x), where NL is the idDeadZone object.
For idDeadZone object properties, see Properties.
NL = idDeadZone;
Specify the zero interval.
NL.ZeroInterval = [-4,5];
Create an idDeadZone object, and specify the initial guess for the zero-interval.
OutputNL = idDeadZone('ZeroInterval',[-0.1 0.1]);
Estimate model with no input nonlinearity.
m = nlhw(z1,[2 3 0],[],OutputNL);
[a,b] — Zero interval
Zero interval of the dead-zone, specified as a 2–element row vector of doubles.
The dead-zone nonlinearity is initialized at the interval [a,b]. The interval values are adjusted to the estimation data by nlhw. To remove the lower limit, set a to -Inf. The lower limit is not adjusted during estimation. To remove the upper limit, set b to Inf. The upper limit is not adjusted during estimation.
When the interval is [NaN NaN], the initial value of the zero interval is determined from the estimation data range during estimation using nlhw.
Option to fix or free the parameters of ZeroInterval, specified as a 2–element logical row vector. When you set an element of Free to false, the corresponding value in ZeroInterval remains fixed during estimation to the initial value that you specify.
NL — Dead-zone nonlinearity estimator object
idDeadZone object
Dead-zone nonlinearity estimator object, returned as an idDeadZone object.
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Boolean_ring Knowpia
In mathematics, a Boolean ring R is a ring for which x2 = x for all x in R, that is, a ring that consists only of idempotent elements.[1][2][3] An example is the ring of integers modulo 2.
Every Boolean ring gives rise to a Boolean algebra, with ring multiplication corresponding to conjunction or meet ∧, and ring addition to exclusive disjunction or symmetric difference (not disjunction ∨,[4] which would constitute a semiring). Boolean rings are named after the founder of Boolean algebra, George Boole.
There are at least four different and incompatible systems of notation for Boolean rings and algebras:
In commutative algebra the standard notation is to use x + y = (x ∧ ¬ y) ∨ (¬ x ∧ y) for the ring sum of x and y, and use xy = x ∧ y for their product.
In logic, a common notation is to use x ∧ y for the meet (same as the ring product) and use x ∨ y for the join, given in terms of ring notation (given just above) by x + y + xy.
In set theory and logic it is also common to use x · y for the meet, and x + y for the join x ∨ y. This use of + is different from the use in ring theory.
A rare convention is to use xy for the product and x ⊕ y for the ring sum, in an effort to avoid the ambiguity of +.
Historically, the term "Boolean ring" has been used to mean a "Boolean ring possibly without an identity", and "Boolean algebra" has been used to mean a Boolean ring with an identity. The existence of the identity is necessary to consider the ring as an algebra over the field of two elements: otherwise there cannot be a (unital) ring homomorphism of the field of two elements into the Boolean ring. (This is the same as the old use of the terms "ring" and "algebra" in measure theory.[a])
One example of a Boolean ring is the power set of any set X, where the addition in the ring is symmetric difference, and the multiplication is intersection. As another example, we can also consider the set of all finite or cofinite subsets of X, again with symmetric difference and intersection as operations. More generally with these operations any field of sets is a Boolean ring. By Stone's representation theorem every Boolean ring is isomorphic to a field of sets (treated as a ring with these operations).
Relation to Boolean algebrasEdit
Venn diagrams for the Boolean operations of conjunction, disjunction, and complement
Since the join operation ∨ in a Boolean algebra is often written additively, it makes sense in this context to denote ring addition by ⊕, a symbol that is often used to denote exclusive or.
Given a Boolean ring R, for x and y in R we can define
x ∧ y = xy,
x ∨ y = x ⊕ y ⊕ xy,
¬x = 1 ⊕ x.
These operations then satisfy all of the axioms for meets, joins, and complements in a Boolean algebra. Thus every Boolean ring becomes a Boolean algebra. Similarly, every Boolean algebra becomes a Boolean ring thus:
x ⊕ y = (x ∨ y) ∧ ¬(x ∧ y).
If a Boolean ring is translated into a Boolean algebra in this way, and then the Boolean algebra is translated into a ring, the result is the original ring. The analogous result holds beginning with a Boolean algebra.
A map between two Boolean rings is a ring homomorphism if and only if it is a homomorphism of the corresponding Boolean algebras. Furthermore, a subset of a Boolean ring is a ring ideal (prime ring ideal, maximal ring ideal) if and only if it is an order ideal (prime order ideal, maximal order ideal) of the Boolean algebra. The quotient ring of a Boolean ring modulo a ring ideal corresponds to the factor algebra of the corresponding Boolean algebra modulo the corresponding order ideal.
Properties of Boolean ringsEdit
Every Boolean ring R satisfies x ⊕ x = 0 for all x in R, because we know
x ⊕ x = (x ⊕ x)2 = x2 ⊕ x2 ⊕ x2 ⊕ x2 = x ⊕ x ⊕ x ⊕ x
and since (R,⊕) is an abelian group, we can subtract x ⊕ x from both sides of this equation, which gives x ⊕ x = 0. A similar proof shows that every Boolean ring is commutative:
x ⊕ y = (x ⊕ y)2 = x2 ⊕ xy ⊕ yx ⊕ y2 = x ⊕ xy ⊕ yx ⊕ y
and this yields xy ⊕ yx = 0, which means xy = yx (using the first property above).
The property x ⊕ x = 0 shows that any Boolean ring is an associative algebra over the field F2 with two elements, in precisely one way. In particular, any finite Boolean ring has as cardinality a power of two. Not every unital associative algebra over F2 is a Boolean ring: consider for instance the polynomial ring F2[X].
The quotient ring R/I of any Boolean ring R modulo any ideal I is again a Boolean ring. Likewise, any subring of a Boolean ring is a Boolean ring.
Any localization
{\displaystyle RS^{-1}}
of a Boolean ring R by a set
{\displaystyle S\subseteq R}
is a Boolean ring, since every element in the localization is idempotent.
The maximal ring of quotients
{\displaystyle Q(R)}
(in the sense of Utumi and Lambek) of a Boolean ring R is a Boolean ring, since every partial endomorphism is idempotent.[5]
Every prime ideal P in a Boolean ring R is maximal: the quotient ring R/P is an integral domain and also a Boolean ring, so it is isomorphic to the field F2, which shows the maximality of P. Since maximal ideals are always prime, prime ideals and maximal ideals coincide in Boolean rings.
Every finitely generated ideal of a Boolean ring is principal (indeed, (x,y) = (x + y + xy)). Furthermore, as all elements are idempotents, Boolean rings are commutative von Neumann regular rings and hence absolutely flat, which means that every module over them is flat.
Unification in Boolean rings is decidable,[6] that is, algorithms exist to solve arbitrary equations over Boolean rings. Both unification and matching in finitely generated free Boolean rings are NP-complete, and both are NP-hard in finitely presented Boolean rings.[7] (In fact, as any unification problem f(X) = g(X) in a Boolean ring can be rewritten as the matching problem f(X) + g(X) = 0, the problems are equivalent.)
Unification in Boolean rings is unitary if all the uninterpreted function symbols are nullary and finitary otherwise (i.e. if the function symbols not occurring in the signature of Boolean rings are all constants then there exists a most general unifier, and otherwise the minimal complete set of unifiers is finite).[8]
Ring sum normal form
^ When a Boolean ring has an identity, then a complement operation becomes definable on it, and a key characteristic of the modern definitions of both Boolean algebra and sigma-algebra is that they have complement operations.
^ Fraleigh (1976, p. 25,200)
^ Herstein (1975, p. 130,268)
^ "Disjunction as sum operation in Boolean Ring".
^ B. Brainerd, J. Lambek (1959). "On the ring of quotients of a Boolean ring". Canadian Mathematical Bulletin. 2: 25–29. doi:10.4153/CMB-1959-006-x. Corollary 2.
^ Martin, U.; Nipkow, T. (1986). "Unification in Boolean Rings". In Jörg H. Siekmann (ed.). Proc. 8th CADE. LNCS. Vol. 230. Springer. pp. 506–513. doi:10.1007/3-540-16780-3_115. ISBN 978-3-540-16780-8.
^ Kandri-Rody, Abdelilah; Kapur, Deepak; Narendran, Paliath (1985). "An ideal-theoretic approach to word problems and unification problems over finitely presented commutative algebras". Rewriting Techniques and Applications. Lecture Notes in Computer Science. Vol. 202. pp. 345–364. doi:10.1007/3-540-15976-2_17. ISBN 978-3-540-15976-6.
^ A. Boudet; J.-P. Jouannaud; M. Schmidt-Schauß (1989). "Unification of Boolean Rings and Abelian Groups". Journal of Symbolic Computation. 8 (5): 449–477. doi:10.1016/s0747-7171(89)80054-9.
Atiyah, Michael Francis; Macdonald, I. G. (1969), Introduction to Commutative Algebra, Westview Press, ISBN 978-0-201-40751-8
Fraleigh, John B. (1976), A First Course In Abstract Algebra (2nd ed.), Addison-Wesley, ISBN 978-0-201-01984-1
Herstein, I. N. (1975), Topics In Algebra (2nd ed.), John Wiley & Sons
McCoy, Neal H. (1968), Introduction To Modern Algebra (Revised ed.), Allyn and Bacon, LCCN 68015225
Ryabukhin, Yu. M. (2001) [1994], "Boolean ring", Encyclopedia of Mathematics, EMS Press
John Armstrong, Boolean Rings
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(Redirected from Dirac)
Paul Adrien Maurice Dirac (8 August 1902 – 20 October 1984) was a British engineer, theoretical physicist and a founder of the field of quantum physics.
See also: Dirac equation
1.1 The Principles of Quantum Mechanics (4th ed. 1958)
1.2 The Evolution of the Physicist's Picture of Nature (1963)
2 Quotes about Dirac
The very idea of God is a product of the human imagination.
Approximate practical methods of applying quantum mechanics should be developed, which can lead to an explanation of the main features of complex atomic systems without too much computation.
I think it’s a peculiarity of myself that I like to play about with equations, just looking for beautiful mathematical relations which maybe don’t have any physical meaning at all. Sometimes they do.
Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, Vol. 123, No. 792 (6 April 1929)
At the beginning of time the laws of Nature were probably very different from what they are now. Thus we should consider the laws of Nature as continually changing with the epoch, instead of as holding uniformly throughout space-time. This idea was first put forward by Milne, who worked it out on... assumptions... not very satisfying... we should expect them also to depend on position in space, in order to preserve the beautiful idea of the theory of relativity [that] there is fundamental similarity between space and time.
The Relation between Mathematics and Physics (Feb. 6, 1939) Proceedings of the Royal Society (Edinburgh) Vol. 59, 1938-39, Part II, pp. 122-129.
Bombay Lectures (1955)
As quoted in Dirac: A Scientific Biography (1990), by Helge Kragh, p. 258[1]
Interview with Dr. P. A. M. Dirac by Thomas S. Kuhn at Dirac's home, Cambridge, England, May 7, 1963
My research work was based in pictures. I needed to visualise things and projective geometry was often most useful e.g. in figuring out how a particular quantity transforms under Lorentz transf[ormation]. When I came to publish the results I suppressed the projective geometry as the results could be expressed more concisely in analytic form.
As quoted in The Cosmic Code : Quantum Physics As The Language Of Nature (1982) by Heinz R. Pagels, p. 295; also in Paul Adrien Maurice Dirac : Reminiscences about a Great Physicist (1990) edited by Behram N. Kursunoglu and Eugene Paul Wigner, p. xv
A good deal of my research work in physics has consisted in not setting out to solve some particular problems, but simply examining mathematical quantities of a kind that physicists use and trying to get them together in an interesting way regardless of any application that the work may have. It is simply a search for pretty mathematics. It may turn out later that the work does have an application. Then one has had good luck.
P.A.M. Dirac, "Pretty Mathematics," International Journal of Theoretical Physics, Vol. 21, Issue 8–9, August 1982, p. 603
The Principles of Quantum Mechanics (4th ed. 1958)[edit]
{\displaystyle {\int _{-\infty }^{\infty }{\delta \left({x}\right){d{x}}}}=1}
{\displaystyle \delta \left({x}\right)=0{\text{ for }}x\not =0}
III. Representation - 15. The δ function
The Evolution of the Physicist's Picture of Nature (1963)[edit]
"The Evolution of the Physicist's Picture of Nature" in Scientific American (May 1963)
It seems that if one is working from the point of view of getting beauty in one's equations, and if one has really a sound insight, one is on a sure line of progress. If there is not complete agreement between the results of one's work and experiment, one should not allow oneself to be too discouraged, because the discrepancy may well be due to minor features that are not properly taken into account and that will get cleared up with further development of the theory.
Just by studying mathematics we can hope to make a guess at the kind of mathematics that will come into the physics of the future. A good many people are working on the mathematical basis of quantum theory, trying to understand the theory better and to make it more powerful and more beautiful. If someone can hit on the right lines along which to make this development, it may lead to a future advance in which people will first discover the equations and then, after examining them, gradually learn how to apply them.
Quotes about Dirac[edit]
Dirac wrote the first chapter in laser optics. ~ F. J. Duarte
One of the most revered – and strangest – figures in the history of science. ~ Graham Farmelo
Well, our friend Dirac, too, has a religion, and its guiding principle is "God does not exist and Dirac is His prophet." ~ Wolfgang Pauli
Perhaps the most distinguished of 'why botherers has been Dirac (1963 Sci. American 208 May 45). He divided the difficulties of quantum mechanics into two classes, those of the first class and those of the second. The second-class difficulties were essentially the infinities of relativistic quantum field theory. Dirac was very disturbed by these, and was not impressed by the 'renormalisation' procedures by which they are circumvented. Dirac tried hard to eliminate these second-class difficulties, and urged others to do likewise. The first-class difficulties concerned the role of the 'observer', 'measurement', and so on. Dirac thought that these problems were not ripe for solution, and should be left for later. He expected developments in the theory which would make these problems look quite different. It would be a waste of effort to worry overmuch about them now, especially since we get along very well in practice without solving them.
John S. Bell, "Against 'mesurement'", Physics World (August 1990)
Dirac was the strangest man who ever visited my institute. […] During one of Dirac’s visits I asked him what he was doing. He replied that he was trying to take the square-root of a matrix, and I thought to myself what a strange thing for such a brilliant man to be doing. Not long afterwards the proof sheets of his article on the equation arrived, and I saw he had not even told me that he had been trying to take the square root of the unit matrix!
Niels Bohr, quoted in Kurt Gottfried, "P.A.M. Dirac and the Discovery of Quantum Mechanics" (2010)
F. J. Duarte, in "Introduction to Lasers" in Tunable Laser Optics (2003), p. 3
I have trouble with Dirac. This balancing on the dizzying path between genius and madness is awful.
Albert Einstein, Letter to Paul Ehrenfest, Aug. 23, 1926.
The latest and most successful creation of theoretical physics, namely Quantum Mechanics, is fundamentally different in its principles from the two programmes which we will briefly call Newton's and Maxwell's. For the quantities that appear in its laws make no claim to describe Physical Reality itself, but only the probabilities for the appearances of a particular physical reality on which our attention is fixed. Dirac, to whom, in my opinion, we owe the most logically perfect presentation of this theory, rightly points out that it appears, for example, to be by no means easy to give a theoretical description of a photon that shall contain within it the reasons that determine whether or not the photon will pass a polarizator set obliquely in its path.
Albert Einstein, in James Clerk Maxwell (1931), p. 72-73
One of the most revered – and strangest – figures in the history of science.
Graham Farmelo, "Prologue" in The Strangest Man: The Hidden Life of Paul Dirac, Mystic of the Atom (2009)
When I was a young man, Dirac was my hero. He made a breakthrough, a new method of doing physics. He had the courage to simply guess at the form of an equation, the equation we now call the Dirac equation, and to try to interpret it afterwards. Maxwell in his day got his equations, but only in an enormous mass of 'gear wheels' and so forth.
Richard Feynman, "The Reason for Antiparticles"
Before World War II there had been considerable theoretical effort directed towards the question of the self-energy of the electron. However, because of the war, interest had remained dormant. Now, the stimulus of results of Lamb and Retherford the latent interest developed into a major attack by theoretical physicists, and within a few years the problem was solved to the satisfaction of nearly everyone. (To the end of his life, however, Dirac maintained that any theory involving the subtraction of infinities was ugly, unsatisfactory and surely incomplete.)
Val L. Fitch and Jonathan L. Rosner, in: Brown, Laurie M.; Pais, Abraham; Pippard, Brian, eds. (1995). "Chapter 9. Elementary particle physics in the second half of the twentieth century". Twentieth Century Physics, Vol. II. Institute of Physics Publishing. pp. 635–794. (quote from p. 643)
Here we find a man with an almost miraculous apprehension of the structure of the physical world, coupled with gentle incomprehension of that less logical, messier world, the world of other people.
Louisa Gilder, "Quantum Leap", The New York Times, September 8, 2009
Dirac, in his first paper, in contrast to what his “hole”-theory implied, had identified the positively charged particle corresponding to the electron also with the proton. However, after Weyl had pointed out that Dirac’s hole theory led to equal masses, he changed his mind and gave the new particle the same mass as the electron.
Hubert F. M. Goenner, On the History of Unified Field Theories (2004) p. 9, footnote.
Dirac has done more than anyone this century, with the exception of Einstein, to advance physics and change our picture of the universe. He is surely worthy of the memorial in Westminster Abbey. It is just a scandal that it has taken so long.
Stephen Hawking, Dirac Memorial Address, published in Paul Dirac: The Man and His Work (1998), edited by Peter Goddard
Wolfgang Pauli, in a remark made during the Fifth Solvay International Conference (October 1927), after a discussion of the religious views of various physicists, at which all the participants laughed, including Dirac; as quoted in Teil und das Ganze (1969), by Werner Heisenberg, p. 119
There are many variant translations and paraphrases of this statement, which is an ironic play upon the Muslim statement of faith, the Shahada, often translated: "There is no god but Allah, and Muhammad is His Prophet.":
Sourced to Franz Kafka, Betrachtungen (Reflections), Number 52, ca. 1917. See, for instance, Reflections on Sin, Suffering, Hope, and the True Way.
↑ Kragh, Helge (March 30, 1990). Dirac: A Scientific Biography. p. 258. Retrieved on December 6, 2017.
P. A. M. Dirac Speaking with Friedrich Hund on Symmetry in Relativity, Quantum mechanics and Physics of Elementary Particles (Göttingen, 1982) Filmdokumente zur Zeitgeschichte, G209; Youtube link title: "Interview, Göttingen 1982".
Paul Adrien Maurice Dirac biography @Timeline of Nobel Winners
Paul Adrien Maurice Dirac MacTutor History of Mathematics archive
Paul Adrien Maurice Dirac @Genealogy - Familienforschung - Généalogie
Retrieved from "https://en.wikiquote.org/w/index.php?title=Paul_Dirac&oldid=3076551"
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Species_diversity Knowpia
Species diversity is the number of different species that are represented in a given community (a dataset). The effective number of species refers to the number of equally abundant species needed to obtain the same mean proportional species abundance as that observed in the dataset of interest (where all species may not be equally abundant). Meanings of species diversity may include species richness, taxonomic or phylogenetic diversity, and/or species evenness. Species richness is a simple count of species. Taxonomic or phylogenetic diversity is the genetic relationship between different groups of species. Species evenness quantifies how equal the abundances of the species are.[1][2][3]
Calculation of diversityEdit
Species diversity in a dataset can be calculated by first taking the weighted average of species proportional abundances in the dataset, and then taking the inverse of this. The equation is:[1][2][3]
{\displaystyle {}^{q}\!D={1 \over {\sqrt[{q-1}]{\sum _{i=1}^{S}p_{i}p_{i}^{q-1}}}}}
The denominator equals mean proportional species abundance in the dataset as calculated with the weighted generalized mean with exponent q - 1. In the equation, S is the total number of species (species richness) in the dataset, and the proportional abundance of the ith species is
{\displaystyle p_{i}}
. The proportional abundances themselves are used as weights. The equation is often written in the equivalent form:
{\displaystyle {}^{q}\!D=\left({\sum _{i=1}^{S}p_{i}^{q}}\right)^{1/(1-q)}}
The value of q determines which mean is used. q = 0 corresponds to the weighted harmonic mean, which is 1/S because the
{\displaystyle p_{i}}
values cancel out, with the result that 0D is equal to the number of species or species richness, S. q = 1 is undefined, except that the limit as q approaches 1 is well defined:[4]
{\displaystyle \lim _{q\rightarrow 1}{}^{q}\!D=\exp \left(-\sum _{i=1}^{S}p_{i}\ln p_{i}\right),}
which is the exponential of the Shannon entropy.
q = 2 corresponds to the arithmetic mean. As q approaches infinity, the generalized mean approaches the maximum
{\displaystyle p_{i}}
value. In practice, q modifies species weighting, such that increasing q increases the weight given to the most abundant species, and fewer equally abundant species are hence needed to reach mean proportional abundance. Consequently, large values of q lead to smaller species diversity than small values of q for the same dataset. If all species are equally abundant in the dataset, changing the value of q has no effect, but species diversity at any value of q equals species richness.
Negative values of q are not used, because then the effective number of species (diversity) would exceed the actual number of species (richness). As q approaches negative infinity, the generalized mean approaches the minimum
{\displaystyle p_{i}}
value. In many real datasets, the least abundant species is represented by a single individual, and then the effective number of species would equal the number of individuals in the dataset.[2][3]
The same equation can be used to calculate the diversity in relation to any classification, not only species. If the individuals are classified into genera or functional types,
{\displaystyle p_{i}}
represents the proportional abundance of the ith genus or functional type, and qD equals genus diversity or functional type diversity, respectively.
Diversity indicesEdit
Often researchers have used the values given by one or more diversity indices to quantify species diversity. Such indices include species richness, the Shannon index, the Simpson index, and the complement of the Simpson index (also known as the Gini-Simpson index).[5][6][7]
When interpreted in ecological terms, each one of these indices corresponds to a different thing, and their values are therefore not directly comparable. Species richness quantifies the actual rather than effective number of species. The Shannon index equals log(1D), that is, q approaching 1, and in practice quantifies the uncertainty in the species identity of an individual that is taken at random from the dataset. The Simpson index equals 1/2D, q = 2, and quantifies the probability that two individuals taken at random from the dataset (with replacement of the first individual before taking the second) represent the same species. The Gini-Simpson index equals 1 - 1/2D and quantifies the probability that the two randomly taken individuals represent different species.[1][2][3][7][8]
Sampling considerationsEdit
Depending on the purposes of quantifying species diversity, the data set used for the calculations can be obtained in different ways. Although species diversity can be calculated for any data-set where individuals have been identified to species, meaningful ecological interpretations require that the dataset is appropriate for the questions at hand. In practice, the interest is usually in the species diversity of areas so large that not all individuals in them can be observed and identified to species, but a sample of the relevant individuals has to be obtained. Extrapolation from the sample to the underlying population of interest is not straightforward, because the species diversity of the available sample generally gives an underestimation of the species diversity in the entire population. Applying different sampling methods will lead to different sets of individuals being observed for the same area of interest, and the species diversity of each set may be different. When a new individual is added to a dataset, it may introduce a species that was not yet represented. How much this increases species diversity depends on the value of q: when q = 0, each new actual species causes species diversity to increase by one effective species, but when q is large, adding a rare species to a dataset has little effect on its species diversity.[9]
In general, sets with many individuals can be expected to have higher species diversity than sets with fewer individuals. When species diversity values are compared among sets, sampling efforts need to be standardised in an appropriate way for the comparisons to yield ecologically meaningful results. Resampling methods can be used to bring samples of different sizes to a common footing.[10][11] Species discovery curves and the number of species only represented by one or a few individuals can be used to help in estimating how representative the available sample is of the population from which it was drawn.[12][13]
^ a b c Hill, M. O. (1973) Diversity and evenness: a unifying notation and its consequences. Ecology, 54, 427–432
^ a b c d Tuomisto, H. (2010) A diversity of beta diversities: straightening up a concept gone awry. Part 1. Defining beta diversity as a function of alpha and gamma diversity. Ecography, 33, 2-22. doi:10.1111/j.1600-0587.2009.05880.x
^ a b c d Tuomisto, H. 2010. A consistent terminology for quantifying species diversity? Yes, it does exist. Oecologia 4: 853–860. doi:10.1007/s00442-010-1812-0
^ Xu, S., Böttcher, L., and Chou, T. (2020). Diversity in biology: definitions, quantification and models. Physical Biology, 17, 031001. doi:10.1088/1478-3975/ab6754
^ Krebs, C. J. (1999) Ecological Methodology. Second edition. Addison-Wesley, California.
^ Magurran, A. E. (2004) Measuring biological diversity. Blackwell Publishing, Oxford.
^ a b Jost, L. (2006) Entropy and diversity. Oikos, 113, 363–375
^ Jost, L. (2007) Partitioning diversity into independent alpha and beta components. Ecology, 88, 2427–2439.
^ Tuomisto, H. (2010) A diversity of beta diversities: straightening up a concept gone awry. Part 2. Quantifying beta diversity and related phenomena. Ecography, 33, 23-45. doi:10.1111/j.1600-0587.2009.06148.x
^ Colwell, R. K. and Coddington, J. A. (1994) Estimating terrestrial biodiversity through extrapolation. Philosophical Transactions: Biological Sciences, 345, 101-118.
^ Webb, L. J.; Tracey, J. G.; Williams, W. T.; Lance, G. N. (1969), Studies in the Numerical Analysis of Complex Rain-Forest Communities: II. The Problem of Species-Sampling. Journal of Ecology, Vol. 55, No. 2, Jul., 1967, pp. 525-538, Journal of Ecology, British Ecological Society, JSTOR 2257891
^ Good, I. J. and Toulmin, G. H. (1956) The number of new species, and the increase in population coverage, when a sample is increased. Biometrika, 43, 45-63.
^ Chao, A. (2005) Species richness estimation. Pages 7909-7916 in N. Balakrishnan, C. B. Read, and B. Vidakovic, eds. Encyclopedia of Statistical Sciences. New York, Wiley.
Harrison, Ian; Laverty, Melina; Sterling, Eleanor. "Species Diversity". Connexions (cnx.org). William and Flora Hewlett Foundation, the Maxfield Foundation, and the Connexions Consortium. Retrieved 1 February 2011. (Licensed under Creative Commons 1.0 Attribution Generic).
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Closing the Bloom chapter - Tomorrow Blog
TL;DR: We’re shutting down our self-service carbon accounting software Bloom before we properly launched.
Since Tomorrow was founded in 2016, the chatter around the need for organisations to act and take responsibility for their impact on climate change has grown steadily. Numerous surveys point towards the fact that employees, consumers, investors and governments now expect companies to raise up to the challenge. Yet, as we investigated our own carbon footprint, it was clear that doing so was easier said than done.
We strongly believe that information precedes action, and from discussions around us, it was clear that measuring a company’s carbon footprint was a major barrier for organisations to take meaningful action. If we could drastically reduce that barrier, it could potentially pave the way for making carbon information ubiquitous for all companies, and enable a new corporate movement for climate action.
Furthermore, we believed that we had a role to play. The combination of climate knowledge and technology, and our experience with building a carbon footprint tracker app for individuals, North, meant that we knew what to build to have a climate impact, and how to build it. This became Bloom.
What Bloom became
We spent a large part of last year building Bloom, involving users in every step of the way. The vision for Bloom was to democratize corporate climate action, enabling as many companies as possible to reduce their carbon footprint. We realised the issue was that when companies wanted to take action, they relied on an archaic solution based on spreadsheets, or an imprecise solution based on surveys. There ought to be a better, more precise, and more automated solution! We therefore built Bloom to automatically collect data about an organisation’s activities (commute, business travels, catering, office supplies, electricity and heating, etc.) and based on that calculate the company’s carbon footprint. Bloom would then generate recommendations for setting reduction targets and initiatives. Lastly Bloom would then provide communication assets, which would help the company communicate in a truthful way without the risk of greenwashing . In order to increase adoption rate and maximise climate impact, Bloom should ideally come at a low price point, so price wouldn’t be a barrier for companies to get started. If we were to build a sustainable business around this with a low price point and all these capabilities, naturally the tool should be self-service as price would increase drastically if humans needed to be involved in the process. This meant that we intended Bloom to be a self-service SaaS.
As of today, Bloom integrates with hundreds of European banks, 3 bookkeeping softwares and all smart meters in France and Denmark. The continuous improvement in the product meant that we were able to calculate Tomorrow’s carbon footprint for the previous year in less than 30 minutes. We pride ourselves in having what we think are some of the best open-source emission factors for transactions and for transportation, and a relatively impressive open-source database of emission factors ranging from haddock (3.41kgCO2/kg) to a Macbook Pro (210kgCo2/unit).
You can watch a video about Bloom connecting to a bank and an example of doing carbon accounting here, and here is a video about Bloom connecting to a smart meter to get the carbon footprint of your electricity consumption here.
We opened Bloom to testers in the fall, and we gathered great feedback about what it could and could not be. At the same time, it was clear that it would not be possible to have enough revenue fast enough to cover the cost for a larger team, which we saw as necessary to build a product that lived up to our expectations. For financing Bloom, we started to look for investors.
While raising that investment round, we were challenged by prospective investors about the market viability of Bloom; investors are usually looking for revenues of about $100,000 a month 18 months after a seed fundraising. We strongly believe that revenue would be a good measure of climate impact as well, so their doubt encouraged us to dive deeper into the market opportunity.
After a couple of months of investigation, interviewing dozens of organisations, we feel confident in our decision not to continue our efforts on Bloom, as we don’t have a strong enough conviction that the product as we envisioned it can become a sustainable business and have a meaningful impact on climate change.
The lessons learnt on the way
To our surprise and disappointment, the need in the market for a product like Bloom is not as we expected. We focused on small-and-medium enterprises. Our research tells us companies that were sincerely interested in doing climate action could not be satisfied with a self-service tool like Bloom and wanted more tailored support, as they wanted to be sure to do the right thing which meant looking deeper in their carbon footprint and getting recommendations more than a self-service software would be able to do for the foreseeable future. While we genuinely enjoyed working with these companies, their size rarely afforded them the budgets required for us to make a sustainable business out of this. Similarly, our lack of expertise in general sustainability consulting meant that we would probably fall short of their expectations and be far from our comfort zone. On the other hand, we also saw that organisations who did not have a genuine interest in reducing their carbon footprint (driven mostly by external pressure) were sufficiently happy with survey-based tools (often linked with carbon offsets) or free tools provided by governments to report on their energy levels and business travels. Indeed, we’ve come to believe that there’s currently not a fast-growing market for a self-service carbon accounting software.
Interestingly enough, there was a large wave of carbon accounting software companies 10 years ago - most of them shut down, pivoted to become energy management systems or were acquired and then shut down. Back then, the idea was very similar to what is currently pitched to investors and potential customers: an archaic market based on spreadsheets that is just waiting to be disrupted. We had hoped that today was different: that the digitalisation of companies’ bookkeeping, banking, energy management software, travel management, and the large citizen protests of 2018 and 2019 meant that a ripe market was just around the corner. What we found out was that most companies have not even made it to the spreadsheets yet: a clear example of that is that most companies that have publicly announced carbon neutrality objectives in 2030 have not yet measured their carbon footprint, and most carbon neutrality pledges are achievable using carbon offsets. The majority of carbon offsets purchased today cost between
5 and
20 per ton CO2, while keeping GHG emissions consistent with a 1.5–2°C pathway would require companies to have an internal carbon tax of around $100 per ton CO2 in 2020 according to United Nations Global Compact - letting people believe that offsets can be a satisfying solution to an organisation’s footprint reduces drastically the incentive to focus on real reductions, and thus the need for precise measurement to achieve these reductions. There is a chance that a software like Bloom could thrive in the future, particularly if governments or an alliance of large corporations start requiring proper reporting of an organisation’s entire value chain (scope 3 emissions). Even then, we’re uncertain whether such a change would happen faster than carbon pricing or faster than accountants rising to the occasion, each potentially undermining the need for an independent tool. We believe that there is in the short-term a largely untapped market for helping especially large organisations in their sustainability journey as consultants. A lot more companies that are big enough to be pressured by investors and governments are much earlier than we expected in their sustainability journey.
However, we also acknowledge that our assessment may be too conservative - there are now plenty of companies providing a tool similar to Bloom targeting a similar market to what we targeted; we cheer them on and hope to witness their success.
While learning that Bloom was not viable was a tough realisation for us, we’re happy to have made the call this early in the process. Indeed, if too few companies are interested in the product that we had envisioned, it would also have meant a limited impact on climate change, which is our raison d’être.
We really want to thank our early testers, our open-source community, the team, our existing investors, prospective investors and our many supporters for their patience and their trust. We want to give especially a huge shout out to the experts who helped us understand carbon accounting in depth, the experts who helped us develop the open-source carbon models, the amazing testers who gave us great feedback and helped us fix the many bugs, and the investors who kept us on our toes and helped us leave no stone unturned. The fact that the time for an automated self-service carbon accounting software isn’t now makes us worried given the urgency of the climate situation. We are especially worried that net zero claims turn to an offsetting race and distract companies from true reduction. This is why we urge policy makers to create the framework which demands proper disclosure of reduction efforts through granular and frequent measurements.
On our end, we’ll be doubling down on making our electricity data as ubiquitous as possible, to enable companies who already wish to be future-proof to take meaningful action on their electricity footprint.
As always, we do not want our work to go to waste, so we’re eager to share our learnings with anyone who may be interested. Don’t hesitate to reach out if you’re interested in more details!
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Double Entry Book Keeping Ts Grewal Vol I 2018 for Class 12 Commerce Accountancy Chapter 4 - Admission Of A Partner
Double Entry Book Keeping Ts Grewal Vol I 2018 Solutions for Class 12 Commerce Accountancy Chapter 4 Admission Of A Partner are provided here with simple step-by-step explanations. These solutions for Admission Of A Partner are extremely popular among Class 12 Commerce students for Accountancy Admission Of A Partner Solutions come handy for quickly completing your homework and preparing for exams. All questions and answers from the Double Entry Book Keeping Ts Grewal Vol I 2018 Book of Class 12 Commerce Accountancy Chapter 4 are provided here for you for free. You will also love the ad-free experience on Meritnation’s Double Entry Book Keeping Ts Grewal Vol I 2018 Solutions. All Double Entry Book Keeping Ts Grewal Vol I 2018 Solutions for class Class 12 Commerce Accountancy are prepared by experts and are 100% accurate.
Current A/cs: Stock
Y 30,000 70,000 Less: 5% Reserve for D. Debts 5,000 70,000
1,30,000 Bill Receivalbe
Z is admitted as a new partner for 1/4th share under the following terms :
(a) Z is to introduce ₹ 1,25,000 as capital .
(c) It is found that the creditors included a sum of ₹ 7,500 which was not to be paid . But it was also found that there was a liability for compensation to Workmen amounting to ₹ 10,000.
(e) In regard to the Partners' Capital Accounts present fixed capital method is to be converted into fluctuating capital method .
You are required to prepare Revaluation Account , Partners' Capital Accounts, Bank Account and the Balance Sheet of the new firm.
Reserve for D. Debts
Liability for WCF 10,000
X’s Current A/c
Y’s Current A/c
Current A/c 37,500 27,500
as on 1st April, 2018
Creditors (1,30,000 – 7,500 – 20,000)
Bills Payable (50,000 + 20,000)
Bank (50,000 + 1,25,000 + 50,000)
X's Loan
Liability for WCF
Less: 10% Reserve for D. Debts
Rajesh and Ravi are partners sharing profits in the ratio of 3: 2 . Their Balance Sheet at 31st March , 2018 stood as:
Outstanding Rent 4,000 Stock 15,000
Capital A/cs: Prepaid Insurance 1,500
Less : Provision for D.D.
Rajesh 29,000
Raman is admitted as a new partner introducing a capital of ₹ 16,000. The new profit-sharing ratio is decided as 5 : 3 : 2 . Raman is unable to bring in any cash for goodwill . So it is decided to value the goodwill on the basis of Raman's share in the profits and the capital contributed by him. Following revaluation s are made
Provision for D. Debts
Profit on Revaluation transferred to
Rajesh Capital
Ravi Capital
(before and just went of
Goodwill)
Rajesh’s Capital
Raman’s Capital
as on March 31, 2018 after Raman’s admission
Cash (2,000 + 16,000)
Stock (15,000 – 750)
Less: Provision for D. Debts
Building (35,000 + 5,000)
Furniture (5,000 – 500)
WN1 Calculation of Sacrificing Ratio
Actual Capital of all Partners before adjustment of goodwill = Rajesh’s Capital + Ravi’s Capital + Raman’s Capital
Capitalised value on the basis of Raman’s share
Raman’s share of Goodwill
WN3 Adjustment of Raman’s share of goodwill
Rajesh and Ravi each Capital Accounts will be credited by
Raman’s Capital A/c
To Ravi’s Capital A/c
(Raman’s share of goodwill adjusted)
WN4 Distribution of Profit on Revaluation (in old ratio)
A and B are partners in a firm sharing profits in the ratio of 3 : 2 . They admit C as a partner on 1st April, 2018 on which date the Balance Sheet of the firm was:
You are required to prepare the Revaluation Account , Partners' Capital Accounts and Balance Sheet of the new firm after considering the following;
Stock (2,000 – 400)
Bank (charges)
as on April 01, 2018 after C’s admission
Building (50,000 – 3,000)
Stock (20,000 – 1,600)
Creditors (20,000 + 800)
Revaluation (Bank charges)
WN1 Sacrificing Ratio
Old Ratio (A and B) 3 : 2
WN2 Distribution of Premium for Goodwill
A and B are partners in a firm . The net profit of the firm is divided as follows : 1/2 to A , 1/3 to B and 1/6 carried to a Reserve . They admit C as a partner on 1st April, 2018 on which date , the Balance Sheet of the firm was:
Following are the required adjustments on admission of C :
(d) Creditors include a contingent liability of ₹ 4,000 , which has been decided by the court at ₹ 3,200.
₹ 2,000 due from X bad to the full extent;
₹ 4,000 due from Y insolvent , estate expected to pay only 50%.
You are required to prepare Revaluation Account , Partners' Capital Accounts and Balance Sheet of the new firm.
Creditors (4,000 – 3,200)
(4,000 × 50%)
Plan and Machinery
Stock (18,000 × 100/90)
Creditors (20,000 – 800)
Less: Prov. for D. Debts
Bank (5,000 + 30,000)
Following is the Balance Sheet of the firm, Ashirvad, owned by A , B and C who share profits and losses of the business in the ratio of 3 : 2 :1 .
B 1,20,000 Stock-in-Trade 40,000
Outstanding Salaries and wages 7,200 Cash in Hand 4,200
On 1st April, 2018, they admit D as a partner on the following conditions :
(a) D will bring in ₹ 1,20,000 as his capital and also ₹ 30,000 as goodwill premium for a quarter of the share in the future profits / losses of the firm.
(b) The values of the fixed assets of the firm will be increased by 10% before the admission of D .
(d) The future profits and losses of the firm will be shared equally by all the partners .
Pass the necessary journal entries and Prepare Revaluation Account, Partners' Capital Accounts and opening Balance Sheet of the new firm
Note: There will be no entry for the promise made by Mohan, since it is an event and not a transaction. There is another view , ₹ 3,000 is to be considered as bad debts recovered . In this situation result will be as follows :
Gain( Profit) on Revaluation
—
₹ 36,000; Capital A/cs: A
—
₹ 1,66,000; B
—
₹ 1,42,000; C
—
₹ 1,16,000; D's Capital
—
₹ 1,20,000; Balance Sheet Total
—
95,000 × 10%
2,05,000 × 10%
A’s Capital (Goodwill)
B’s Capital (Goodwill)
Revaluation (Profit)
C’s Capital (Goodwill)
as on April 1, 2018, after D’s admission
Furniture (95,000 + 9,500)
Business Premises (2,05,000+20,500)
Cash in hand (4,200 + 1,50,000)
Outstanding salaries and wages
WN2 Calculation of C’s gain in goodwill
WN3 Amount of Goodwill to be distributed between A and B (Sacrificing Partners)
WN4 Journal Entries for D’s Capital and distribution of goodwill
(D brought Capital and share of Capital)
To B’s Capital
(Gain goodwill distributed between A and B
in sacrificing ratio i.e. 3:1)
A and B are partners in a firm sharing profits and losses in the ratio of 3 : 2 . Following is their Balance Sheet as at 31st March, 2018:
(c) No entry has been passed in respect of a debt of ₹ 300 recovered by A from a customer , which was previously written off as bad in previous year . The amount is to be paid by A.
(d) Investments are taken over by B at their market value of ₹ 4,900 against cash payment .
You are required to prepare Revaluation Account, Partner's Capital Accounts and new Balance Sheet
Investment (5,000 – 4,900)
Less: 10% Provision for Doubtful Debts
X and Y are partners sharing profits and losses in the ratio of 3/4 and 1/4 . Their Balance Sheet as at 31st March, 2018 is:
Sundry Creditors 1,50,000 Bills Receivable 15,000
(c) A Provision for Doubtful Debts is to be created @ 5% on Sundry Debtors .
You are required to show Revaluation Account , Partners' Capital Accounts and Balance Sheet of the new firm.
Note: Z's Share of Goodwill ₹ 20,000 (i.e, ₹ 1,00,000
×
1/5 ) can be adjusted through Z's Current A/c. In that situation, Partners' Capital A/cs: X
—
₹ 1,87,875; Y
—
₹ 92,625; Z
—
₹ 50,000; Z's Current A/c (Dr.)
—
₹ 20,000; Balance Sheet Total
—
₹ 5,18,000.
(1,25,000 × 20%)
as on April 01, 2018 after Z’s admission
Land and Building (1,25,000 + 25,000)
Office Furniture (5,000 – 500)
Stock (1,00,000 – 10,000)
Less: 5% Provision for D. Debts
Cash in Hand (12,500 + 50,000)
WN1: Sacrificing Ratio
WN2: Calculation of Partners' Share of Goodwill
Goodwill of the firm = 1, 00,000
(Z’s share of goodwill changed from his
(Workmen’s Compensation Fund distributed)
1,73,000
On the above date , the partners decided to admit Anshu as a partner on the following terms:
(a) The new profit-sharing ratio of Deepika , Rajshree and Anshu will be 5 : 3 : 2 respectively.
(c) Anshu is unable to bring in any cash for his share of goodwill. Partners' therefore, decide to calculate the goodwill on the basis of Anshu's share in the profits and the capital contribution made by her to the firm.
(d) Plant and Machinery is to be valued at ₹ 60,000, Stock at ₹ 40,000 and the Provision for Doubtful Debts is to be maintained at ₹ 4,000. Value of Land and Building has appreciated by 20% . Furniture has been depreciated by 10%.
(e) There is and additional liability of ₹ 8,000 being outstanding salary payable to employees of the firm. This liability is not included in the outstanding liabilities , stated in the above Balance Sheet. Partners decide to show this liability in the books of account of the reconstituted firm.
Prepare Revaluation Account , Partners' Capital Accounts and Balance Sheet of Deepika , Rajshree and Anshu.
Less: Old Reserve
Furniture 10,000 × 10% 1,000 Stock (40,000 – 32,000) 8,000
Deepika Capital
Rajshree Capital
(before adjustment of Goodwill)
Anshu’s Capital (Goodwill)
as on March 31, 2018 after Anshu’s admission
Less: reserve for D. Debts
WN1: Calculation of Sacrificing Ratio
Capitalised value on the basis of Anshu’s share
Goodwill = Capitalised value − Actual Capital of all partners before adjustment of Goodwill
Anshu’s share of Goodwill
Deepika and Rajshree each will entitle for Goodwill
X and Y are partners sharing profits in the ratio of 2 : 1 . Their Balance Sheet as at 31st March, 2018 was:
(b) Z brings in ₹ 15,000 for goodwill, half of which is withdrawn by old partners .
(c) Investments are valued at ₹ 10,000 . X takes over Investments at this value.
(e) An unrecorded stock of Stationery on 31st March,2018 is ₹ 1,000.
(f) By bringing in r withdrawing cash , the Capitals of X and Y are to be made proportionate to that of Z on their profit-sharing basis.
Pass journal entries , prepare Revaluation Account , Capital Accounts and new Balance Sheet of the firm.
To Typewriter A/c
(Revaluation loss transferred to X and Y’s
Capital Account in their old ratio)
(Y withdrew excess capital after all adjustments)
X’s Capital (Goodwill)
Y’s Capital (Goodwill)
Typewriter (5,000 × 20%)
Fixed Assets (1,37,000 × 10%)
Cash (withdraw of goodwill)
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Pulse-position modulation — Wikipedia Republished // WIKI 2
Pulse-position modulation (PPM) is a form of signal modulation in which M message bits are encoded by transmitting a single pulse in one of
{\displaystyle 2^{M}}
possible required time shifts.[1][2] This is repeated every T seconds, such that the transmitted bit rate is
{\displaystyle M/T}
bits per second. It is primarily useful for optical communications systems, which tend to have little or no multipath interference.
PAM | PWM | PPM | COMMUNICATION | BSNL JE(TTA)| JTO | ENGINEERING EXAMINATIONS
Lecture - 39 Pulse Modulation Schemes - PWM and PPM
Sampling Signals Part 2 (8/10) – Pulse Modulation
2 Synchronization
3 Sensitivity to multipath interference
4 Non-coherent detection
5 PPM vs. M-FSK
6 Applications for RF communications
7 PPM encoding for radio control
An ancient use of pulse-position modulation was the Greek hydraulic semaphore system invented by Aeneas Stymphalus around 350 B.C. that used the water clock principle to time signals.[3] In this system, the draining of water acts as the timing device, and torches are used to signal the pulses. The system used identical water-filled containers whose drain could be turned on and off, and a float with a rod marked with various predetermined codes that represented military messages. The operators would place the containers on hills so they could be seen from each other at a distance. To send a message, the operators would use torches to signal the beginning and ending of the draining of the water, and the marking on the rod attached to the float would indicate the message.
In modern times, pulse-position modulation has origins in telegraph time-division multiplexing, which dates back to 1853, and evolved alongside pulse-code modulation and pulse-width modulation.[4] In the early 1960s, Don Mathers and Doug Spreng of NASA invented pulse-position modulation used in radio-control (R/C) systems. PPM is currently being used in fiber-optic communications, deep-space communications, and continues to be used in R/C systems.
One of the key difficulties of implementing this technique is that the receiver must be properly synchronized to align the local clock with the beginning of each symbol. Therefore, it is often implemented differentially as differential pulse-position modulation, whereby each pulse position is encoded relative to the previous, such that the receiver must only measure the difference in the arrival time of successive pulses. It is possible to limit the propagation of errors to adjacent symbols, so that an error in measuring the differential delay of one pulse will affect only two symbols, instead of affecting all successive measurements.
Sensitivity to multipath interference
Aside from the issues regarding receiver synchronization, the key disadvantage of PPM is that it is inherently sensitive to multipath interference that arises in channels with frequency-selective fading, whereby the receiver's signal contains one or more echoes of each transmitted pulse. Since the information is encoded in the time of arrival (either differentially, or relative to a common clock), the presence of one or more echoes can make it extremely difficult, if not impossible, to accurately determine the correct pulse position corresponding to the transmitted pulse. Multipath in Pulse Position Modulation systems can be easily mitigated by using the same techniques that are used in Radar systems that rely totally on synchronization and time of arrival of the received pulse to obtain their range position in the presence of echoes.
One of the principal advantages of PPM is that it is an M-ary modulation technique that can be implemented non-coherently, such that the receiver does not need to use a phase-locked loop (PLL) to track the phase of the carrier. This makes it a suitable candidate for optical communications systems, where coherent phase modulation and detection are difficult and extremely expensive. The only other common M-ary non-coherent modulation technique is M-ary frequency-shift keying (M-FSK), which is the frequency-domain dual to PPM.
PPM vs. M-FSK
PPM and M-FSK systems with the same bandwidth, average power, and transmission rate of M/T bits per second have identical performance in an additive white Gaussian noise (AWGN) channel. However, their performance differs greatly when comparing frequency-selective and frequency-flat fading channels. Whereas frequency-selective fading produces echoes that are highly disruptive for any of the M time-shifts used to encode PPM data, it selectively disrupts only some of the M possible frequency-shifts used to encode data for M-FSK. On the other hand, frequency-flat fading is more disruptive for M-FSK than PPM, as all M of the possible frequency-shifts are impaired by fading, while the short duration of the PPM pulse means that only a few of the M time-shifts are heavily impaired by fading.
Optical communications systems tend to have weak multipath distortions, and PPM is a viable modulation scheme in many such applications.
Applications for RF communications
Narrowband RF (radio frequency) channels with low power and long wavelengths (i.e., low frequency) are affected primarily by flat fading, and PPM is better suited than M-FSK to be used in these scenarios. One common application with these channel characteristics, first used in the early 1960s with top-end HF (as low as 27 MHz) frequencies into the low-end VHF band frequencies (30 MHz to 75 MHz for RC use depending on location), is the radio control of model aircraft, boats and cars, originally known as "digital proportional" radio control. PPM is employed in these systems, with the position of each pulse representing the angular position of an analogue control on the transmitter, or possible states of a binary switch. The number of pulses per frame gives the number of controllable channels available. The advantage of using PPM for this type of application is that the electronics required to decode the signal are extremely simple, which leads to small, light-weight receiver/decoder units (model aircraft require parts that are as lightweight as possible). Servos made for model radio control include some of the electronics required to convert the pulse to the motor position – the receiver is required to first extract the information from the received radio signal through its intermediate frequency section, then demultiplex the separate channels from the serial stream, and feed the control pulses to each servo.
PPM encoding for radio control
A complete PPM frame is about 22.5 ms (can vary between manufacturer), and signal low state is always 0.3 ms. It begins with a start frame (high state for more than 2 ms). Each channel (up to 8) is encoded by the time of the high state (PPM high state + 0.3 × (PPM low state) = servo PWM pulse width).
More sophisticated radio control systems are now often based on pulse-code modulation, which is more complex but offers greater flexibility and reliability. The advent of 2.4 GHz band FHSS radio-control systems in the early 21st century changed this further.
Pulse-position modulation is also used for communication with the ISO/IEC 15693 contactless smart card, as well as in the HF implementation of the Electronic Product Code (EPC) Class 1 protocol for RFID tags.
Pulse-amplitude modulation
Pulse-code modulation
Pulse-density modulation
Pulse-width modulation
Ultra wideband
^ K. T. Wong (March 2007). "Narrowband PPM Semi-Blind Spatial-Rake Receiver & Co-Channel Interference Suppression" (PDF). European Transactions on Telecommunications. The Hong Kong Polytechnic University. 18 (2): 193–197. doi:10.1002/ett.1147. Archived from the original (PDF) on 2015-09-23. Retrieved 2013-09-26.
^ Yuichiro Fujiwara (2013). "Self-synchronizing pulse position modulation with error tolerance". IEEE Transactions on Information Theory. 59: 5352–5362. arXiv:1301.3369. doi:10.1109/TIT.2013.2262094.
^ Michael Lahanas. "Ancient Greek Communication Methods". Archived from the original on 2014-11-02.
^ Ross Yeager & Kyle Pace. "Copy of Communications Topic Presentation: Pulse Code Modulation". Prezi.
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Write an equation for the following situation. You do not need to solve it.
Laura takes very good care of her vehicles. She owns a blue van and a red truck. Although she bought them both new, she has owned the truck for
17
years longer than she has owned the van. The sum of the ages of the vehicles is 41 years.
x
represent the age of the blue van.
What expression would represent the age of the red truck?
x + 17
is the age of the red truck.
The sum of the two ages equals
41
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(Not recommended) Numerically evaluate integral, adaptive Simpson quadrature - MATLAB quad - MathWorks América Latina
Compute Definite Integral
fcnEvals
(Not recommended) Numerically evaluate integral, adaptive Simpson quadrature
quad is not recommended. Use integral instead.
q = quad(fun,a,b)
q = quad(fun,a,b,tol)
q = quad(fun,a,b,tol,trace)
[q,fcnEvals] = quad(___)
q = quad(fun,a,b) approximates the integral of function fun from a to b using recursive adaptive Simpson quadrature:
q=\underset{a}{\overset{b}{\int }}f\left(x\right)dx
q = quad(fun,a,b,tol) specifies an absolute error tolerance tol for each subinterval, instead of the default value of 1e-6.
q = quad(fun,a,b,tol,trace) optionally turns on the display of diagnostic information. When trace is nonzero, quad shows the vector of values [fcnEvals, a, b-a, Q] during the recursion.
[q,fcnEvals] = quad(___) additionally returns the number of function evaluations fcnEvals. You can specify any of the previous input argument combinations.
{\int }_{0}^{2}\frac{1}{{x}^{3}-2x-5}dx.
First, create an anonymous function myfun that computes the integrand.
myfun = @(x) 1./(x.^3-2*x-5);
Now use quad to compute the integral. Specify the limits of integration as the second and third input arguments.
q = quad(myfun,0,2)
Alternatively, you can pass the integrand to quad by creating a function file:
y = 1./(x.^3-2*x-5);
With this method, the call to quad becomes quad(@myfun,0,2).
Integrand, specified as a function handle that defines the function to be integrated from a to b.
For scalar-valued problems, the function y = fun(x) must accept a vector argument x and return a vector result y, where y is the integrand evaluated at each element of x. This requirement generally means that fun must use array operators (.^, .*, …) instead of matrix operators (^, *, …).
Parameterizing Functions explains how to provide additional parameters to the function fun, if necessary.
Example: q = quad(@(x) exp(1-x.^2),a,b) integrates an anonymous function handle.
Example: q = quad(@myFun,a,b) integrates the function myFun, which is saved as a file.
a, b — Integration limits (as separate arguments)
Integration limits, specified as separate scalar arguments. The limits a and b must be finite.
Example: quad(fun,0,1) integrates fun from 0 to 1.
tol — Absolute error tolerance
[] or 1e-6 (default) | scalar
Absolute error tolerance, specified as a scalar. quad uses the absolute error tolerance on each subinterval in the integration. As the magnitude of tol increases, quad performs fewer function evaluations and completes the calculation faster, but produces less accurate results.
Example: quad(fun,a,b,1e-12) sets the absolute error tolerance to 1e-12.
trace — Toggle for diagnostic information
Toggle for diagnostic information, specified as a nonzero scalar. When trace is nonzero, quad displays the vector of values [fcnEvals, a, b-a, Q] for each subinterval in the recursion:
fcnEvals gives the number of function evaluations
a and b are the limits of integration
Q is the computed area of the subinterval
Example: quad(fun,a,b,1e-8,1) integrates fun from a to b with a tolerance of 1e-8 and diagnostic information turned on.
q — Value of integral
Value of integral, returned as a scalar.
fcnEvals — Number of function evaluations
Number of function evaluations, returned as a scalar.
quad implements a low order quadrature method using an adaptive recursive Simpson's rule .
[1] Gander, W., and W. Gautschi. “Adaptive Quadrature—Revisited.” BIT Numerical Mathematics 40 (2000): 84–101. https://doi.org/10.1023/A:1022318402393
quad2d | quadgk | trapz | integral | integral2 | integral3
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Molecular mass - wikidoc
The molecular mass (abbreviated Mr) of a substance, formerly also called molecular weight and abbreviated as MW, is the mass of one molecule of that substance, relative to the unified atomic mass unit u (equal to 1/12 the mass of one atom of carbon-12). Due to this relativity, the molecular mass of a substance is commonly referred to as the relative molecular mass, and abbreviated to Mr.
Although this term appears well-defined, there are varying interpretations of this definition. It is interpreted by many, including many chemists, to be a synonym of molar mass differing only in units (see average molecular mass below). This is inconsistent with a strict interpretation of the definition because it neglects that the mass of a single molecule is not the same as the average of an ensemble. A mole of molecules most often contains a variety of molecular masses due to natural isotopes and the average is usually not identical to any single molecule. The actual difference numerically is very small and only matters to physicists and a small subset of highly specialized chemists; however it is always more correct, accurate and consistent to use molar mass in any bulk stoichiometric calculations.
The average molecular mass (sometimes abbreviated as average mass) is another variation on the use of the term molecular mass. The average molecular mass is the abundance weighted mean (average) of the molecular masses in a sample. This is often closer to what is meant when "molecular mass" and "molar mass" are used synonymously and may have derived from shortening of this term. The average molecular mass and the molar mass of a particular substance in a particular sample are in fact numerically identical and may be interconverted by avogadro's number. It should be noted, however, that the molar mass is almost always a computed figure derived from the standard atomic weights, whereas the average molecular mass, in fields that need the term, is often a measured figure specific to a sample. Therefore, they often vary since one is theoretical and the other is measured. Specific samples may vary significantly from the expected isotopic composition due to real deviations from earth average isotopic abundances.
The molecular mass can be calculated as the sum of the individual isotopic masses (as found in a table of isotopes) of all the atoms in any one molecule. This is possible because molecules are created by chemical reactions which, unlike nuclear reactions, have very small binding energies compared to the rest mass of the atoms (
{\displaystyle <}
10-9) and therefore create a negligible mass defect. Note that the use of average atomic masses as found on a standard periodic table will result in an average molecular mass, whereas the use of isotopic masses will result in a molecular mass consistent with the strict interpretation of the definition, i.e. that of a single molecule. Note that any given molecule may contain any given combination of isotopes, so there may be multiple molecular masses for each chemical compound.
The molecular mass can also be measured directly using mass spectrometry. In mass spectrometry, the molecular mass of a small molecule is usually reported as the monoisotopic mass, that is, the mass of the molecule containing only the most common isotope of each element. Note that this also differs subtly from the molecular mass in that the choice of isotopes is defined. The masses used to compute the monoisotopic molecular mass are found on a table of isotopic masses and are not the same as found on a typical periodic table. The average molecular mass is often used for larger molecules since molecules with many atoms are unlikely to be composed exclusively of the most abundant isotope of each element. A theoretical average molecular mass can be calculated using the standard atomic weights found on a typical periodic table, since there is likely to be a statistical distribution of atoms representing the isotopes throughout the molecule. This however may differ from the true average molecular mass of the sample due to natural (or artificial) variations in the isotopic distributions.
The molar mass of a substance is the mass of 1 mol (the SI unit for the basis SI quantity amount of substance, having the symbol n) of the substance. This has a numerical value which is the average molecular mass of the molecules in the substance multiplied by Avogadro's constant approximately 6.022*1023. Its SI unit is kg/mol, although more usually the unit g/mol is used because in those units the numerical value equals the average molar mass in units of u.
Learning by Simulations Calculation of Molecular Formulas from Molecular Masses
Molecular Mass Calculator An online molecular mass calculator
Molecular Weight Calculator Online Molecular Weight Calculator
Molecular Weight Calculator Calculates molecular weight and elemental composition
Free online calculations for mol weight and elemental composition using ChemAxon's Marvin and Calculator Plugins - requires Java
ar:كتلة جزيئية ca:Massa molecular cs:Molární hmotnost de:Molekülmasse el:Μοριακό βάρος eo:Molekula maso ko:분자 질량 nl:Moleculaire massa sr:Молекулска тежина fi:Molekyylimassa sv:Molekylmassa
Retrieved from "https://www.wikidoc.org/index.php?title=Molecular_mass&oldid=679789"
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Approximations of stable actions on $ \Bbb {R} $-trees | EMS Press
Approximations of stable actions on
\Bbb {R}
This article shows how to approximate a stable action of a finitely presented group on an
\Bbb {R}
-tree by a simplicial one while keeping control over arc stabilizers. For instance, every small action of a hyperbolic group on an
\Bbb {R}
-tree can be approximated by a small action of the same group on a simplicial tree. The techniques we use highly rely on Rips's study of stable actions on
\Bbb {R}
-trees and on the dynamical study of exotic components by D. Gaboriau.
Vincent Guirardel, Approximations of stable actions on
\Bbb {R}
-trees. Comment. Math. Helv. 73 (1998), no. 1, pp. 89–121
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Remote Sensing | Free Full-Text | Mapping Arctic Lake Ice Backscatter Anomalies Using Sentinel-1 Time Series on Google Earth Engine
Real-Time Detection of Daytime and Night-Time Fire Hotspots from Geostationary Satellites
Monitoring the Spatiotemporal Dynamics of Aeolian Desertification Using Google Earth Engine
Robust Multipath-Assisted SLAM with Unknown Process Noise and Clutter Intensity
Pointner, G.
Austrian Polar Research Institute, c/o Universität Wien, 1010 Vienna, Austria
Department of Geoinformatics—Z_GIS, DK GIScience, Paris Lodron University of Salzburg, 5020 Salzburg, Austria
Seepage of geological methane through sediments of Arctic lakes might contribute conceivably to the atmospheric methane budget. However, the abundance and precise locations of such seeps are poorly quantified. For Lake Neyto, one of the largest lakes on the Yamal Peninsula in Northwestern Siberia, temporally expanding regions of anomalously low backscatter in C-band SAR imagery acquired in late winter and spring have been suggested to be related to seepage of methane from hydrocarbon reservoirs. However, this hypothesis has not been verified using in-situ observations so far. Similar anomalies have also been identified for other lakes on Yamal, but it is still uncertain whether or how many of them are related to methane seepage. This study aimed to document similar lake ice backscatter anomalies on a regional scale over four study regions (the Yamal Peninsula and Tazovskiy Peninsulas; the Lena Delta in Russia; the National Petroleum Reserve Alaska) during different years using a time series based approach on Google Earth Engine (GEE) that quantifies changes of
{\sigma }^{0}
from the Sentinel-1 C-band SAR sensor over time. An algorithm for assessing the coverage that takes the number of acquisitions and maximum time between acquisitions into account is presented, and differences between the main operating modes of Sentinel-1 are evaluated. Results show that better coverage can be achieved in extra wide swath (EW) mode, but interferometric wide swath (IW) mode data could be useful for smaller study areas and to substantiate EW results. A classification of anomalies on Lake Neyto from EW
\mathsf{\Delta }{\sigma }^{0}
images derived from GEE showed good agreement with the classification presented in a previous study. Automatic threshold-based per-lake counting of years where anomalies occurred was tested, but a number of issues related to this approach were identified. For example, effects of late grounding of the ice and anomalies potentially related to methane emissions could not be separated efficiently. Visualizations of
\mathsf{\Delta }{\sigma }^{0}
images likely reflect the temporal expansions of anomalies and are expected to be particularly useful for identifying target areas for future field-based research. Characteristic anomalies that clearly resemble the ones observed for Lake Neyto could be identified solely visually in the Yamal and Tazovskiy study regions. All data and algorithms produced in the framework of this study are openly provided to the scientific community for future studies and might potentially aid our understanding of geological lake seepage upon the progression of related field-based studies and corresponding evaluations of formation hypotheses. View Full-Text
Keywords: arctic; lake ice; SAR; change detection; methane; Yamal; permafrost; Google Earth Engine arctic; lake ice; SAR; change detection; methane; Yamal; permafrost; Google Earth Engine
Description: Geospatial raster data and vector data created in the frame of the study "Mapping Arctic Lake Ice Backscatter Anomalies using Sentinel-1 Time Series on Google Earth Engine" and Python code to reproduce the results.
Pointner, G.; Bartsch, A. Mapping Arctic Lake Ice Backscatter Anomalies Using Sentinel-1 Time Series on Google Earth Engine. Remote Sens. 2021, 13, 1626. https://doi.org/10.3390/rs13091626
Pointner G, Bartsch A. Mapping Arctic Lake Ice Backscatter Anomalies Using Sentinel-1 Time Series on Google Earth Engine. Remote Sensing. 2021; 13(9):1626. https://doi.org/10.3390/rs13091626
Pointner, Georg, and Annett Bartsch. 2021. "Mapping Arctic Lake Ice Backscatter Anomalies Using Sentinel-1 Time Series on Google Earth Engine" Remote Sensing 13, no. 9: 1626. https://doi.org/10.3390/rs13091626
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Full Pyramid | Toph
N, print a full pyramid of asterisks.
A full pyramid of asterisks of size
N has
N lines of asterisks. The first line has 1 asterisk, the second line has 2 asterisks, the third line has 3 asterisks and so on. Each asterisk has a space between them. The asterisks in each line are centered to make it look like a pyramid.
Here is a full pyramid of asterisks of size 4:
N (
0 < N < 100
0<N<100).
Print the full pyramid of asterisks of the given size.
Do not print any space after the last asterisk in each line.
Kickoff Programming Contest of Spring 2022
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We know that when an electron, a matter particle, collides with a positron, an antimatter particle, they annihilate each other as the energy in the two particles is carried away by two real photons to conserve energy. The same phenomenon occurs as all matter annihilates an equal quantity of antimatter. In the case of charged particles like the proton and the antiproton, their opposite charges cancel while in the case of neutral particles like the neutron and the antineutron, their opposite spins cancel. In all cases, it is the cancellation of forward moving time for matter and backward moving time for antimatter that is responsible for the annihilation process having taken place. A reactor to produce energy for commercial use has been proposed based on matter antimatter collisions.
Matter, Electron, Particle, Reactor
Irani, A. (2021) Matter-Antimatter Annihilation. Journal of High Energy Physics, Gravitation and Cosmology, 7, 474-477. doi: 10.4236/jhepgc.2021.72027.
1. The Cancellation of Time
We consider the relationship between the Electric and Magnetic fields when viewed from two different reference frames. They are given by [1]:
{{E}^{\prime }}_{x}={E}_{x}
{{E}^{\prime }}_{y}=c\gamma \left({E}_{y}/c-\beta {B}_{z}\right)
{{E}^{\prime }}_{z}=c\gamma \left({E}_{z}/c\text{+}\beta {B}_{y}\right)
{{B}^{\prime }}_{x}={B}_{x}
{{B}^{\prime }}_{y}=\gamma \left({B}_{y}+\beta {E}_{z}/c\right)
{{B}^{\prime }}_{z}=\gamma \left({B}_{z}-\beta {E}_{y}/c\right)
We pick the unprimed Electric and Magnetic fields as the rest frame for the charge q, and the primed Electric and Magnetic fields as the frame in which the charge q is moving with velocity v along the x-axis.
From Coulomb’s Law in the rest frame:
\stackrel{¯}{E}=q\times \stackrel{^}{r}/\left(4\pi {\epsilon }_{0}{r}^{2}\right)
\stackrel{¯}{B}=0
In spherical coordinates since:
\stackrel{^}{r}=\mathrm{sin}\theta \mathrm{cos}\varphi \stackrel{^}{x}+\mathrm{sin}\theta \mathrm{sin}\varphi \stackrel{^}{y}+\mathrm{cos}\theta \stackrel{^}{z}
{E}_{x}=q\mathrm{sin}\theta \mathrm{cos}\varphi /\left(4\pi {\epsilon }_{0}{r}^{2}\right)
{E}_{y}=q\mathrm{sin}\theta \mathrm{sin}\varphi /\left(4\pi {\epsilon }_{0}{r}^{2}\right)
{E}_{z}=q\mathrm{cos}\theta /\left(4\pi {\epsilon }_{0}{r}^{2}\right)
{B}_{x}={B}_{y}={B}_{z}=0
Using the above relationships between primed and unprimed Electric and Magnetic fields, we get:
{{E}^{\prime }}_{x}={E}_{x}=q\mathrm{sin}\theta \mathrm{cos}\varphi /\left(4\pi {\epsilon }_{0}{r}^{2}\right)
{{E}^{\prime }}_{y}=\gamma {E}_{y}=\gamma q\mathrm{sin}\theta \mathrm{sin}\varphi /\left(4\pi {\epsilon }_{0}{r}^{2}\right)
{{E}^{\prime }}_{z}=\gamma {E}_{z}=\gamma q\mathrm{cos}\theta /\left(4\pi {\epsilon }_{0}{r}^{2}\right)
{{B}^{\prime }}_{x}=0
{{B}^{\prime }}_{y}=\gamma \beta {E}_{z}/c=\gamma \beta q\mathrm{cos}\theta /\left(4\pi {\epsilon }_{0}{r}^{2}\right)
{{B}^{\prime }}_{z}=-\gamma \beta {E}_{y}/c=-\gamma \beta q\mathrm{sin}\theta \mathrm{sin}\varphi /\left(4\pi {\epsilon }_{0}{r}^{2}\right)
\text{TheEnergyDensity}=\text{Energy}/\frac{4}{3}\pi {r}^{3}=\frac{1}{2}{\epsilon }_{0}{\left({E}^{\prime }\right)}^{2}+{\left({B}^{\prime }\right)}^{2}/2{\mu }_{0}
\begin{array}{l}\sqrt{\text{Energy}}=\sqrt{m{c}^{2}}\\ =\pm q\sqrt{\frac{1}{24\pi {\epsilon }_{0}r}}\times \{\left({\mathrm{sin}}^{2}\theta {\mathrm{cos}}^{2}\varphi +{\gamma }^{2}{\mathrm{sin}}^{2}\theta {\mathrm{sin}}^{2}\varphi +{\gamma }^{2}{\mathrm{cos}}^{2}\theta \right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.05em}}+{\left({\gamma }^{2}-1\right)\times \left({\mathrm{cos}}^{2}\theta +{\mathrm{sin}}^{2}\theta {\mathrm{sin}}^{2}\varphi \right)\}}^{1/2}\end{array}
\gamma =1
the second term inside {( ) + ( )}1/2 becomes 0 as it is due to the Magnetic field and the first term becomes 1 as it is due to the Electric field. We get the simpler [2] equation
\pm q\sqrt{\frac{1}{24\pi {\epsilon }_{0}r}}
in this special situation as it is for the frame of reference in which the charge q is at rest.
Energy can produce both positive and negative charges +q and −q. Next let us examine what happens when a charged particle and antiparticle annihilate each other. If +q and −q both → 0 that implies m → 0 in accordance with the above equation, and that is exactly what happens when an electron and positron annihilate to produce two real photons of zero mass moving away with the same energy as the electron positron pair to conserve energy.
The question then arises why positive and negative charges in a plasma do not annihilate one another. The reason is that besides opposite charges, opposite times must also cancel out. For the electron time moves in the forward direction while for the positron time moves in the backward direction. Hence time also cancels out leading to two photons which move at the speed of light for which time stands still.
The same rules must also apply for neutral matter and antimatter annihilation because neutral matter is made up of protons, electrons, and neutrons while neutral antimatter is made up of antiprotons, positrons, and antineutrons. While neutrons and antineutrons have no charge, they have opposite spin and hence their spins must cancel each other. The cancellation of spin should also apply to neutrinos and antineutrinos when they interact since they also carry opposite spin. CRT is a symmetry where C (charge conjugation) stands for matter or antimatter charge, R stands for rotational spin clockwise or counterclockwise, and T stands for time moving forward or backward. Hence, we need cancellation of charge for charged particles and antiparticles, cancellation of spin for neutral particles and antiparticles, but cancellation of time in all cases which is the condition that must always be met for particle-antiparticle annihilation to take place.
2. Particle Antiparticle Reactors
Antimatter has been considered as a trigger mechanism for nuclear weapons, and as the possibility of creating thermonuclear energy [3]. We propose an alternative method of creating energy by shooting electron-positron or proton-antiproton beams at each other and using the energy released as a means of producing commercial energy as in a fission reactor without the radioactive byproducts. This type of energy production should be compared to the proposed fusion reactors such as magnetically confined or inertially confined fusion with the advantage of 100% conversion of the mass in the two beams into pure energy, a conversion factor that neither fission nor fusion can reproduce, and without the need for raw materials uranium-235 or plutonium-239 that would be needed for fission, or deuterium and tritium needed for fusion reactions to take place.
A reactor in which antiproton beams are made to collide with electron beams of the same velocity should be an advantageous scenario to consider since the antiproton is 1836 times heavier than the electron. Hence for the same beam volumes (equal length and radius), the ratio of the densities of the antiproton beam to the electron beam would be 5.45 × 10−4, thereby reducing the necessity of producing intense antiproton beams.
[1] Fleisch, D. (2019) A Student’s Guide to Vectors and Tensors. 182.
[2] Irani, A. (2021) Dark Energy, Dark Matter, and the Multiverse. Journal of High Energy Physics, Gravitation and Cosmology, 7, 172-190.
[3] Gsponer, A. and Jean-Pierre, H. (1987) The Physics of Antimatter Induced Fusion and Thermonuclear Explosions. In: Velarde, G. and Minguez, E. (Eds.), Proceedings of the 4th International Conference on Emerging Nuclear Energy Systems, Madrid, 30June-4 July 1986, 166-169. arXiv:physics/0507114
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Home : Support : Online Help : Connectivity : Database Package : Result : Close
close a Result module
result:-Close( )
Close frees the resources associated with result. This happens automatically when result is garbage collected; however, you can call Close to release the resources immediately.
Any descendant modules of result are closed when result is closed. (A module is a descendant of a parent module if it is returned by one of the parent module's exports or if it is a descendant of one of the parent module's descendants.)
Create a Result, res.
\mathrm{driver}≔\mathrm{Database}[\mathrm{LoadDriver}]\left(\right):
\mathrm{conn}≔\mathrm{driver}:-\mathrm{OpenConnection}\left(\mathrm{url},\mathrm{name},\mathrm{pass}\right):
\mathrm{res}≔\mathrm{conn}:-\mathrm{ExecuteQuery}\left("SELECT * FROM animals"\right):
Close res.
\mathrm{res}:-\mathrm{Close}\left(\right)
Try using res.
\mathrm{res}:-\mathrm{Next}\left(\right)
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Influence of Loading Distribution on the Off-Design Performance of High-Pressure Turbine Blades | J. Turbomach. | ASME Digital Collection
(DRDC Valcartier), Quebec City, Quebec Canada G3J 1X5t
e-mail: daniel.corriveau@drdc-rddc.gc.ca
, Ottawa, ON K1S 5B6, Canada
Corriveau, D., and Sjolander, S. A. (August 12, 2006). "Influence of Loading Distribution on the Off-Design Performance of High-Pressure Turbine Blades." ASME. J. Turbomach. July 2007; 129(3): 563–571. https://doi.org/10.1115/1.2464145
Linear cascade measurements for the aerodynamic performance of a family of three transonic, high-pressure (HP) turbine blades have been presented previously by the authors. The airfoils were designed for the same inlet and outlet velocity triangles but varied in their loading distributions. The previous papers presented results for the design incidence at various exit Mach numbers, and for off-design incidence at the design exit Mach number of 1.05. Results from the earlier studies indicated that by shifting the loading towards the rear of the airfoil an improvement in the profile loss performance of the order of 20% could be obtained near the design Mach number at design incidence. Measurements performed at off-design incidence, but still at the design Mach number, showed that the superior performance of the aft-loaded blade extended over a range of incidence from about
−5.0deg
+5.0deg
relative to the design value. For the current study, additional measurements were performed at off-design Mach numbers from about 0.5 to 1.3 and for incidence values of
−10.0deg
+5.0deg
+10.0deg
relative to design. The corresponding Reynolds numbers, based on outlet velocity and true chord, varied from roughly
4×105
10×105
. The measurements included midspan losses, blade loading distributions, and base pressures. In addition, two-dimensional Navier–Stokes computations of the flow were performed to help in the interpretation of the experimental results. The results show that the superior loss performance of the aft-loaded profile, observed at design Mach number and low values of off-design incidence, does not extend readily to off-design Mach numbers and larger values of incidence. In fact, the measured midspan loss performance for the aft-loaded blade was found to be inferior to, or at best equal to, that of the baseline, midloaded airfoil at most combinations of off-design Mach number and incidence. However, based on the observations made at design and off-design flow conditions, it appears that aft-loading can be a viable design philosophy to employ in order to reduce the losses within a blade row provided the rearward deceleration is carefully limited. The loss performance of the front-loaded blade is inferior or at best equal to that of the other two blades for all operating conditions.
turbines, blades, aerodynamics, Mach number, turbulence, Navier-Stokes equations, design engineering, turbine aerodynamics, transonic, incidence, losses
Blades, Design, Flow (Dynamics), Mach number, Turbine blades, Pressure, High pressure (Physics)
An Experimental Investigation of the Effect of Incidence on the Two-Dimensional Flow
Proceedings of the 9th International Symposium on Air Breathing Engines (ISABE)
, Athen, Greece, Sept. 4-9.
Off-Design Performance of a Linear Cascade of Turbine Blades
Shock Boundary Layer Interaction on High Turning Transonic Turbine Cascade
Flow Around the Sections of Rotor Blading of a Turbine Stage With Relatively Long Blades at Off-Design Conditions
,” Ph.D. thesis, Carleton University, Ottawa, Canada.
Experimental Investigation of Two Transonic Linear Turbine Cascades at Off-Design Conditions
The Effect of Reynolds Number and Velocity Distribution on LP Turbine Cascade Performance
Data Reduction of Wake Flow Measurements With Injection of Other Gas
,” Technical Report No. DLR-FB 95-32, DLR.
Impact of Flow Quality in Transonic Cascade Wind Tunnels: Measurements in an HP Turbine Cascade
Proceedings 23rd Int. Congress of Aeronautical Science (ICAS)
, Toronto, Canada, Sept. 8-13.
Experimental and Numerical In*vestigation on the Performance of a Family of Three HP Transonic Turbine Blades
Loss Mechanism in Turbomachines
Surface Roughness Effects on Turbine Blade Aerodynamics
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Economic value added (EVA), also known as economic profit, is a measure of a company's or project's financial success based on residual wealth, calculated as subtracting the cost of capital from operating profits. The purpose of EVA is to determine the value a company generates from the capital invested into it with the overall goal of improving the returns generated for shareholders.
There are two major ways a company can improve its economic value added (EVA): increase revenues or decrease capital costs. Revenue can be increased by raising prices or selling additional goods and services. Capital costs can be minimized in several ways, including increasing economies of scale. It is also possible for a firm to offset capital costs by choosing investments that earn more than their associated capital charges.
Economic value added (EVA) is a measure of a company's financial success determined by comparing its returns on invested capital to the cost of capital. It shows a company's economic profit.
A positive EVA indicates a company is generating wealth for shareholders whereas a negative EVA indicates that a company is not generating returns above its cost of capital.
To improve its EVA, a company can increase revenues by increasing the price for its goods or services or it can sell more goods.
A company can also increase its EVA by reducing its capital costs by improving efficiency and reaching economies of scale.
EVA was developed by Stern Value Management to measure the difference between the cost of capital and the rate of return, creating a path for companies to determine whether capital invested into the company will be a drag on assets or help it in terms of successful financial performance.
When the EVA is positive, it indicates that a company is generating economic profit. A negative EVA would show that a company is not generating wealth for shareholders from its capital commitments.
Economic value added is sometimes also referred to as shareholder value added (SVA), although some companies might make different adjustments in their NOPAT and cost of capital calculations. These are not the same as cash value added (CVA), which is a metric used by value investors to see how well a company can generate cash flow.
Formula for Economic Value Added (EVA)
The formula for EVA is as follows:
\begin{aligned}&\text{Economic Value Added}\\&\quad=\text{Net Operating Profits After Tax}\\&\qquad-(\text{Weighted Average Cost of}\\&\qquad\qquad\text{Capital}\times\text{Capital Invested})\\&\textbf{where:}\\&\text{Capital Invested}=\text{Equity}\!+\!\text{Long Term}\\&\text{Debt at the Start of the Period}\end{aligned}
Economic Value Added=Net Operating Profits After Tax−(Weighted Average Cost ofCapital×Capital Invested)where:Capital Invested=Equity+Long TermDebt at the Start of the Period
How to Increase Economic Value Added (EVA)
In the EVA formula, a firm's revenue is expressed as being equal to net operating profits after tax (NOPAT). Capital costs are traditionally estimated using a weighted average cost of capital (WACC). EVA is the result of subtracting all net capital charges from NOPAT. Below are two ways to increase EVA.
Traditional methods of increasing revenue include increasing prices and increasing the number of goods sold. Increasing prices is straightforward, a company charges more for a product or service than it did before. If costs remain the same this will then increase the profit margin. The only downside to this tactic is that certain consumers may not be willing to pay more for the same product, which could lead to a decrease in demand and, therefore, a decrease in revenue.
Selling more goods would increase revenues, as long as the cost of producing more goods to meet the increased demand doesn't outweigh the benefit. Meaning, if a company wants to improve its EVA by adding to its revenues, it must ensure the marginal revenue gain is larger than the accompanying marginal costs, including taxes. This makes sense; you would not spend $150 to earn an additional $100 in revenue.
For example, if you need to create a new factory to meet the additional demand for goods, you would need to ensure that the return on investment of the factory is greater than the WACC.
Since revenue generation is usually uncertain, it is often easier for a company to reduce its net capital costs.
Decreasing Capital Costs
Net capital costs can be lowered by reducing operating expenses, increasing marginal productivity, or liquidating capital that does not cover the cost of capital for the purpose that it is associated with.
The costs associated with research and development (R&D) should be included as part of costs invested into the company.
To reduce operating expenses, a company might renegotiate with its creditor to acquire a lower interest rate on debt, it may negotiate better terms with its suppliers, and it may be able to get better terms on rent for its office or factory space.
A company may also improve its marginal productivity by reaching economies of scale. Here, a company would be able to figure out a way to produce the same amount of goods at a lower cost, or conversely, produce more goods without a significant increase in costs. This can be achieved by arriving at improved means of efficiency, such as a better production plan or new technology.
Economic value added (EVA) is a way for companies to determine if the capital invested into the company will add value to shareholders. A positive EVA indicates that the capital invested is generating returns above the minimum required return and a negative EVA indicates the opposite.
To increase EVA, a company can increase revenues by increasing the price or the number of goods sold, as long as the marginal cost to produce more units is not above the marginal return. Companies can also decrease their capital costs by improving operational efficiency and reaching economies of scale.
How Shareholder Value Added (SVA) Works
Shareholder value added (SVA) is a measure of the operating profits that a company has produced in excess of its funding costs, or cost of capital.
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Experimental Measurements and Modeling of the Effects of Large-Scale Freestream Turbulence on Heat Transfer | J. Turbomach. | ASME Digital Collection
A. C. Nix,
A. C. Nix
, Morgantown, WV 26506-6106
e-mail: andrew.nix@mail.wvu.edu
T. E. Diller,
T. E. Diller
Nix, A. C., Diller, T. E., and Ng, W. F. (October 5, 2006). "Experimental Measurements and Modeling of the Effects of Large-Scale Freestream Turbulence on Heat Transfer." ASME. J. Turbomach. July 2007; 129(3): 542–550. https://doi.org/10.1115/1.2515555
The influence of freestream turbulence representative of the flow downstream of a modern gas turbine combustor and first stage vane on turbine blade heat transfer has been measured and analytically modeled in a linear, transonic turbine cascade. High-intensity, large length-scale freestream turbulence was generated using a passive turbulence-generating grid to simulate the turbulence generated in modern combustors after passing through the first stage vane row. The grid produced freestream turbulence with intensity of approximately 10–12% and an integral length scale of
2cm
(Λx∕c=0.15)
near the entrance of the cascade passages. Mean heat transfer results with high turbulence showed an increase in heat transfer coefficient over the baseline low turbulence case of approximately 8% on the suction surface of the blade, with increases on the pressure surface of approximately 17%. Time-resolved surface heat transfer and passage velocity measurements demonstrate strong coherence in velocity and heat flux at a frequency correlating with the most energetic eddies in the turbulence flow field (the integral length scale). An analytical model was developed to predict increases in surface heat transfer due to freestream turbulence based on local measurements of turbulent velocity fluctuations and length scale. The model was shown to predict measured increases in heat flux on both blade surfaces in the current data. The model also successfully predicted the increases in heat transfer measured in other work in the literature, encompassing different geometries (flat plate, cylinder, turbine vane, and turbine blade) and boundary layer conditions.
gas turbines, blades, heat transfer, transonic flow, boundary layer turbulence
Blades, Heat transfer, Turbulence, Heat flux, Boundary layers, Pressure, Turbine blades, Cascades (Fluid dynamics), Suction, Turbines
Augmentation of Stagnation Region Heat Transfer Due to Turbulence From an Advanced Dual-Annular Combustor
,” Paper No. ASME GT-2002-30184.
,” Paper No. ASME 98-GT-236.
Influence of Free-Stream Turbulence on Turbulence Boundary Layer Heat Transfer and Mean Profile Development, Parts I and II
Heat Transfer with Very High Free Stream Turbulence
,” Stanford University Report No. HMT-42, Stanford University, Stanford, CA.
Heat Transfer with Very High Free Stream Turbulence: Parts I & II
Correlating Friction Velocity in Turbulent Boundary Layers Subjected to Free-Stream Turbulence
,” AIAA Paper No. J-26247.
Enhanced Heat Transfer and Shear Stress Due to High Free-Stream Turbulence
Influence of Turbulence Parameters, Reynolds Number, and Body Shape on Stagnation Region Heat Transfer
Aspects of Vane Film Cooling with High Turbulence—Part I: Heat Transfer
Effect of High Freestream Turbulence with Large Scale on Blade Heat/Mass Transfer
Effect of Free-Stream Turbulence on Flat-Plate Heat Flux Signals: Spectra & Eddy Transport Velocities
A Frequency Domain Analysis: Turbine Leading Edge Region Heat Transfer
,” Paper No. ASME 97-WA/HT2.
Pestian
High Intensity, Large Length-Scale Freestream Turbulence Generation in a Transonic Cascade
Detailed Film Cooling Effectiveness and Three Component Velocity Field Measurements on a First Stage Turbine Vane Subject to High Freestream Turbulence
,” Ph.D. dissertation, The University of Texas at Austin, Austin, TX.
Investigation of Heat Transfer in a Film Cooled Transonic Turbine Cascade, Part I: Steady Heat Transfer
,” Paper No. ASME 2000-GT-202.
A Single-Plate Interferometric Study of the Unsteady Density Field in a Transonic Cascade
,” Ph.D dissertation, Virginia Tech., Blacksburg, VA.
A Frequency Domain Analysis of Surface Heat Transfer/Freestream Turbulence Interactions in a Transonic Turbine Cascade
Investigation of Heat Transfer in a Film Cooled Transonic Turbine Cascade, Part II: Unsteady Heat Transfer
Capturing Sudden Increase in Heat Transfer on the Suction Side of a Turbine Blade Using a Navier–Stokes Solver
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You have been uncommonly kind in all you have done.— You not only have saved me much trouble & some anxiety, but have done all, incomparably better than I could have done it— I am much pleased at all you say about Murray.—1 I will write either today or tomorrow to him & will send shortly a large bundle of M.S. but unfortunately I cannot for a week, as the three first chapters are in three copyists’ hands—2
I am sorry about Murray objecting to term abstract as I look at it as only possible apology for not giving References & facts in full.—but I will defer to him & you.—
I am, also, sorry about term “Natural Selection”, but I hope to retain it with Explanation, somewhat as thus,— “Through Natural Selection or the preserv-ation of favoured races”3
Why I like term is that it is constantly used in all works on Breeding, & I am surprised that it is not familiar to Murray; but I have so long studied such works, that I have ceased to be a competent judge.4
I again most truly & cordially thank you for your really valuable assistance.—
Emma comes up to London for 2 or 3 days on Friday & she proposes to come & breakfast with Lady Lyell & you on Saturday morning: I have told her 9
\frac{1}{2}
is your hour, so you need not write—5
See letter to Charles Lyell, 28 March [1859].
Entries in CD’s Account book (Down House MS) for 6 and 9 April 1859 indicate that CD paid Mr Fletcher and John Mumford for copying. The third copyist was presumably Ebenezer Norman.
The full title of the first edition of Origin reads: ‘On the origin of species by means of natural selection, or the preservation of favoured races in the struggle for life’.
See Young 1985 and Secord 1985.
Emma Darwin recorded in her diary that she went to London with Henrietta Emma Darwin on1 April 1859 and returned to Down on 4 April. On 3 April she ‘lunched with Lyells’.
Young, Robert M. 1985. Darwin’s metaphor: nature’s place in Victorian culture. Cambridge: Cambridge University Press.
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Dictionary:Absorption - SEG Wiki
1. A process whereby energy is converted into heat while passing through a medium. Absorption for seismic waves is typically about 0.25 dB/cycle and may be as large as 0.5 dB/cycle.[1]
Absorption involves change of amplitude and velocity with frequency; it is thus a mechanism (but not the only one) for attenuating high frequencies and changing waveshape (Peg-leg multiples, which do not involve absorption, produce effects that are similar.).
2. The process by which radiant energy is converted into other forms of energy.
3. The penetration of the molecules or ions of a substance into the interior of a solid or liquid.
{\displaystyle \Delta E}
{\displaystyle \lambda }
{\displaystyle {\frac {A}{A_{0}}}={\frac {\text{amplitude}}{\text{initial amplitude}}},{\frac {A_{1}}{A_{2}}}={\frac {\text{amplitude}}{\text{amplitude one cycle later}}}}
↑ Tokso¨z, M. N. and Johnston, D. H., 1982, Seismic wave attenuation: Soc. Expl. Geophys.
Retrieved from "https://wiki.seg.org/index.php?title=Dictionary:Absorption&oldid=35577"
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The debt-to-capital ratio is a measurement of a company's financial leverage. The debt-to-capital ratio is calculated by taking the company's interest-bearing debt, both short- and long-term liabilities and dividing it by the total capital. Total capital is all interest-bearing debt plus shareholders' equity, which may include items such as common stock, preferred stock, and minority interest.
The Formula for Debt-To-Capital Ratio
\text{Debt-To-Capital Ratio} = \frac{Debt}{Debt \text{ }+\text{ } Shareholders'\ Equity}
Debt-To-Capital Ratio=Debt + Shareholders′ EquityDebt
How to Calculate Debt-To-Capital Ratio
What Does Debt-To-Capital Ratio Tell You?
The debt-to-capital ratio gives analysts and investors a better idea of a company's financial structure and whether or not the company is a suitable investment. All else being equal, the higher the debt-to-capital ratio, the riskier the company. This is because a higher ratio, the more the company is funded by debt than equity, which means a higher liability to repay the debt and a greater risk of forfeiture on the loan if the debt cannot be paid timely.
However, while a specific amount of debt may be crippling for one company, the same amount could barely affect another. Thus, using total capital gives a more accurate picture of the company's health because it frames debt as a percentage of capital rather than as a dollar amount.
Measurement of a company's financial leverage, calculated by taking the company's interest-bearing debt and dividing it by total capital.
All else equal, the higher the debt-to-capital ratio, the riskier the company.
While most companies finance their operations through a mixture of debt and equity, looking at the total debt of a company may not provide the best information.
Example of How to Use Debt-To-Capital Ratio
As an example, assume a firm has $100 million in liabilities comprised of the following:
Notes payable $5 million
Bonds payable $20 million
Deferred income $3 million
Long-term liabilities $55 million
Other long-term liabilities $1 million
Of these, only notes payable, bonds payable, and long-term liabilities are interest-bearing securities, the sum of which total $5 million + $20 million + $55 million = $80 million.
As for equity, the company has $20 million worth of preferred stock and $3 million of minority interest listed on the books. The company has 10 million shares of common stock outstanding, which is currently trading at $20 per share. Total equity is $20 million + $3 million + ($20 x 10 million shares) = $223 million. Using these numbers, the calculation for the company's debt-to-capital ratio is:
Debt-to-capital = $80 million / ($80 million + $223) = $80 million / $303 million = 26.4%
Assume this company is being considered as an investment by a portfolio manager. If the portfolio manager looks at another company that had a debt-to-capital ratio of 40%, all else equal, the referenced company is a safer choice since its financial leverage is approximately half that of the compared company's.
As a real-life example, consider Caterpillar (NYSE: CAT), which has $36.6 billion in total debt as of December 2018. Its shareholders’ equity for the same quarter was $14 billion. Thus, its debt-to-capital ratio is 73%, or $36.6 billion / ($36.6 billion + $14 billion).
The Difference Between Debt-To-Capital Ratio and Debt Ratio
Unlike the debt-to-capital ratio, the debt ratio divides total debt by total assets. The debt ratio is a measure of how much of a company’s assets are financed with debt. The two numbers can be very similar, as total assets are equal to total liabilities plus total shareholder’ equity. However, for the debt-to-capital ratio, it excludes all other liabilities besides interest-bearing debt.
Limitations of Using Debt-To-Capital Ratio
The debt-to-capital ratio may be affected by the accounting conventions a company uses. Often, values on a company's financial statements are based on historical cost accounting and may not reflect the true current market values. Thus, it is very important to be certain the correct values are used in the calculation, so the ratio does not become distorted.
Caterpillar. "10-K Annual Report 2018," Page 46. Accessed Aug. 19, 2020.
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ERLANGB - Anaplan Technical Documentation
For example, you can use the ERLANGB function to ensure that a certain percentage of all requests are fulfilled.
ERLANGB(Number of servers, Arrival rate, Average duration)
The ERLANGB function returns a number, which is the probability a request is blocked.
How Erlang B is calculated
Erlang B is the solution to this equation:
ERLANGB(x,y,z) = \dfrac{\frac{a^x}{x!}}{\sum \quad_{k = x}^{k = 0} \quad \frac{a^k}{k!}}
In this example, the Call Centers list is on columns, and line items on rows. The first three line items contain the scheduled number of servers, arrival rate of requests, and average duration to fulfil requests. The fourth line item, Blocking Possibility calculates the possibility of a call being blocked using a formula.
The final two line items are a numeric line item, Required Extra Agents, to adjust the number of servers, and a formula that displays the blocking possibility after adjustment. This can be used to adjust the number of servers until the desired blocking possibility is reached (in this case, less than 5%).
Request Arrival Rate 0.76 0.93 1.4 1.2
Blocking Possibility
ERLANGB(Scheduled Number of Servers, Request Arrival Rate, Average Duration)
Extra Servers -1 12 19 46
Amended Blocking Possibility
ERLANGB(Scheduled Number of Servers + Required Extra Servers, Request Arrival Rate, Average Duration)
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Symbolic cumulative product - MATLAB cumprod - MathWorks Italia
Cumulative Product of Symbolic Vector
Cumulative Product of Each Column and Row in Symbolic Matrix
Reverse Cumulative Product of 3-D Symbolic Array
Symbolic cumulative product
B = cumprod(A) returns the cumulative product of A starting at the beginning of the first array dimension in A whose size does not equal 1. The output B has the same size as A.
If A is a matrix, then cumprod(A) returns a matrix containing the cumulative products of each column of A.
B = cumprod(___,direction) specifies the direction using any of the previous syntaxes. For instance, cumprod(A,2,'reverse') returns the cumulative product within the rows of A by working from end to beginning of the second dimension.
B = cumprod(___,nanflag) specifies whether to include or omit NaN values from the calculation for any of the previous syntaxes. cumprod(A,'includenan') includes all NaN values in the calculation while cumprod(A,'omitnan') ignores them.
Create a symbolic vector. Find the cumulative product of its elements.
\left(\begin{array}{ccccc}x& 2 x& 3 x& 4 x& 5 x\end{array}\right)
In the vector of cumulative products, element B(2) is the product of A(1) and A(2), while B(5) is the product of elements A(1) through A(5).
\left(\begin{array}{ccccc}x& 2 {x}^{2}& 6 {x}^{3}& 24 {x}^{4}& 120 {x}^{5}\end{array}\right)
Create a 3-by-3 symbolic matrix A whose all elements are x.
A = ones(3)*x
\left(\begin{array}{ccc}x& x& x\\ x& x& x\\ x& x& x\end{array}\right)
Compute the cumulative product of elements of A. By default, cumprod returns the cumulative product of each column.
\left(\begin{array}{ccc}x& x& x\\ {x}^{2}& {x}^{2}& {x}^{2}\\ {x}^{3}& {x}^{3}& {x}^{3}\end{array}\right)
To compute the cumulative product of each row, set the value of the dim option to 2.
\left(\begin{array}{ccc}x& {x}^{2}& {x}^{3}\\ x& {x}^{2}& {x}^{3}\\ x& {x}^{2}& {x}^{3}\end{array}\right)
A(:,:,1) = [x y 0; x 3 x*y; x 1/3 y];
\left(\begin{array}{ccc}x& y& 0\\ x& 3& x y\\ x& \frac{1}{3}& y\end{array}\right)
\left(\begin{array}{ccc}x& y& 3\\ 3& x& y\\ y& 3& x\end{array}\right)
Compute the cumulative product along the rows by specifying dim as 2. Specify the 'reverse' option to work from right to left in each row. The result is the same size as A.
B = cumprod(A,2,'reverse')
\left(\begin{array}{ccc}0& 0& 0\\ 3 {x}^{2} y& 3 x y& x y\\ \frac{x y}{3}& \frac{y}{3}& y\end{array}\right)
\left(\begin{array}{ccc}3 x y& 3 y& 3\\ 3 x y& x y& y\\ 3 x y& 3 x& x\end{array}\right)
To compute the cumulative product along the third (page) dimension, specify dim as 3. Specify the 'reverse' option to work from the largest page index to the smallest page index.
\left(\begin{array}{ccc}{x}^{2}& {y}^{2}& 0\\ 3 x& 3 x& x {y}^{2}\\ x y& 1& x y\end{array}\right)
\left(\begin{array}{ccc}x& y& 3\\ 3& x& y\\ y& 3& x\end{array}\right)
Create a symbolic vector containing NaN values. Compute the cumulative products.
\left(\begin{array}{ccccc}a& b& 1& \mathrm{NaN}& 2\end{array}\right)
\left(\begin{array}{ccccc}a& a b& a b& \mathrm{NaN}& \mathrm{NaN}\end{array}\right)
You can ignore NaN values in the cumulative product calculation using the 'omitnan' option.
\left(\begin{array}{ccccc}a& a b& a b& a b& 2 a b\end{array}\right)
cumprod(A,1) works on successive elements in the columns of A and returns the cumulative product of each column.
cumprod(A,2) works on successive elements in the rows of A and returns the cumulative product of each row.
'includenan' — Include NaN values from the input when computing the cumulative products, resulting in NaN values in the output.
'omitnan' — Ignore all NaN values in the input. The product of elements containing NaN values is the product of all non-NaN elements. If all elements are NaN, then cumprod returns 1.
Cumulative product array, returned as a vector, matrix, or multidimensional array of the same size as the input A.
cumsum | fold | int | symprod | symsum
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Determination of the curvature of the blow-up set and refined singular behavior for a semilinear heat equation
15 June 2006 Determination of the curvature of the blow-up set and refined singular behavior for a semilinear heat equation
Hatem Zaag1
1Département de mathématiques et applications, École Normale Supérieure
u\left(x,t\right)
, a solution of
{u}_{t}=\Delta u+|u|{}^{p-1}u
which blows up at some time
T>0
u:{\mathbb{R}}^{N}×\left[0,T\right)\to \mathbb{R}
p>1
\left(N-2\right)p<N+2
. Under a nondegeneracy condition, we show that the mere hypothesis that the blow-up set
S
\left(N-1\right)
-dimensional implies that it is
{C}^{2}
. In particular, we compute the
N-1
principal curvatures and directions of
S
. Moreover, a much more refined blow-up behavior is derived for the solution in terms of the newly exhibited geometric objects. Refined regularity for
S
and refined singular behavior of
u
S
are linked through a new mechanism of algebraic cancellations that we explain in detail
Hatem Zaag. "Determination of the curvature of the blow-up set and refined singular behavior for a semilinear heat equation." Duke Math. J. 133 (3) 499 - 525, 15 June 2006. https://doi.org/10.1215/S0012-7094-06-13333-1
Hatem Zaag "Determination of the curvature of the blow-up set and refined singular behavior for a semilinear heat equation," Duke Mathematical Journal, Duke Math. J. 133(3), 499-525, (15 June 2006)
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Class representing single-variable polynomial nonlinear estimator for Hammerstein-Wiener models - MATLAB idPolynomial1D - MathWorks Switzerland
Use idPolynomial1D to define a nonlinear function
y=F\left(x\right)
, where F is a single-variable polynomial function of x:
F\left(x\right)=c\left(1\right){x}^{n}+c\left(2\right){x}^{\left(n-1\right)}+\dots +c\left(n\right)x+c\left(n+1\right)
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Cohomology of the Hilbert scheme of points on a surface with values in representations of tautological bundles
1 November 2009 Cohomology of the Hilbert scheme of points on a surface with values in representations of tautological bundles
Luca Scala1
{X}^{\left[n\right]}
be the Hilbert scheme of
n
points on the smooth quasi-projective surface
X
{L}^{\left[n\right]}
be the tautological bundle on
{X}^{\left[n\right]}
naturally associated to the line bundle
L
X
. As a corollary of Haiman's results, we express the image
\Phi \left({L}^{\left[n\right]}\right)
{L}^{\left[n\right]}
for the Bridgeland-King-Reid equivalence
\Phi :{D}^{b}\left({X}^{\left[n\right]}\right)\to {D}_{{\mathfrak{S}}_{n}}^{b}\left({X}^{n}\right)
in terms of a complex
{C}_{L}^{•}
{\mathfrak{S}}_{n}
-equivariant sheaves in
{D}_{{\mathfrak{S}}_{n}}^{b}\left({X}^{n}\right)
and we characterize the image
\Phi \left({L}^{\left[n\right]}\otimes \cdot \cdot \cdot \otimes {L}^{\left[n\right]}\right)
in terms of the hyperderived spectral sequence
{E}_{1}^{p,q}
associated to the derived
k
-fold tensor power of the complex
{C}_{L}^{•}
. The study of the
{\mathfrak{S}}_{n}
-invariants of this spectral sequence allows us to get the derived direct images of the double tensor power and of the general
k
-fold exterior power of the tautological bundle for the Hilbert-Chow morphism, providing Danila-Brion-type formulas in these two cases. This easily yields the computation of the cohomology of
{X}^{\left[n\right]}
{L}^{\left[n\right]}\otimes {L}^{\left[n\right]}
{\Lambda }^{k}{L}^{\left[n\right]}
Luca Scala. "Cohomology of the Hilbert scheme of points on a surface with values in representations of tautological bundles." Duke Math. J. 150 (2) 211 - 267, 1 November 2009. https://doi.org/10.1215/00127094-2009-050
Luca Scala "Cohomology of the Hilbert scheme of points on a surface with values in representations of tautological bundles," Duke Mathematical Journal, Duke Math. J. 150(2), 211-267, (1 November 2009)
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Error, (in rtable/Product) use *~ for elementwise multiplication of Vectors or Matrices; use . (dot) for Vector/Matrix multiplication - Maple Help
Home : Support : Online Help : Error, (in rtable/Product) use *~ for elementwise multiplication of Vectors or Matrices; use . (dot) for Vector/Matrix multiplication
An expression involving the multiplication of Vectors and/or Matrices (possibly and/or Arrays) has been constructed using the standard multiplication operator, `⋅`, which is ambiguous.
This will happen if Vectors and/or Matrices are multiplied using a commutative multiplication operator, `⋅`:
A ≔ \left[\begin{array}{rr}\mathrm{a__11}& \mathrm{a__12}\\ \mathrm{a__21}& \mathrm{a__22}\end{array}\right]\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}
\textcolor[rgb]{0,0,1}{A}\textcolor[rgb]{0,0,1}{:=}\left[\begin{array}{cc}\mathrm{a__11}& \mathrm{a__12}\\ \mathrm{a__21}& \mathrm{a__22}\end{array}\right]
v ≔ \left[\begin{array}{c}\mathrm{v__1}\\ \mathrm{v__2}\end{array}\right]
\textcolor[rgb]{0,0,1}{v}\textcolor[rgb]{0,0,1}{:=}\left[\begin{array}{c}\mathrm{v__1}\\ \mathrm{v__2}\end{array}\right]
A\cdot v
W ≔ \mathrm{Array}\left(1..2,1..2,\left[\left[a,b\right],\left[c,d\right]\right]\right);
\textcolor[rgb]{0,0,1}{W}\textcolor[rgb]{0,0,1}{:=}\left[\begin{array}{cc}\textcolor[rgb]{0,0,1}{a}& \textcolor[rgb]{0,0,1}{b}\\ \textcolor[rgb]{0,0,1}{c}& \textcolor[rgb]{0,0,1}{d}\end{array}\right]
A W
To multiply Vectors and/or Matrices together using the standard Linear Algebra multiplication operation, use the non-commutative multiplication operator, `.` (dot):
A . A
\left[\begin{array}{cc}{\mathrm{a__11}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\mathrm{a__12}\textcolor[rgb]{0,0,1}{}\mathrm{a__21}& \mathrm{a__11}\textcolor[rgb]{0,0,1}{}\mathrm{a__12}\textcolor[rgb]{0,0,1}{+}\mathrm{a__12}\textcolor[rgb]{0,0,1}{}\mathrm{a__22}\\ \mathrm{a__11}\textcolor[rgb]{0,0,1}{}\mathrm{a__21}\textcolor[rgb]{0,0,1}{+}\mathrm{a__21}\textcolor[rgb]{0,0,1}{}\mathrm{a__22}& \mathrm{a__12}\textcolor[rgb]{0,0,1}{}\mathrm{a__21}\textcolor[rgb]{0,0,1}{+}{\mathrm{a__22}}^{\textcolor[rgb]{0,0,1}{2}}\end{array}\right]
A . v
\left[\begin{array}{c}\mathrm{a__11}\textcolor[rgb]{0,0,1}{,}\mathrm{a__11}\textcolor[rgb]{0,0,1}{}\mathrm{v__1}\textcolor[rgb]{0,0,1}{,}\mathrm{v__1}\textcolor[rgb]{0,0,1}{+}\mathrm{a__12}\textcolor[rgb]{0,0,1}{,}\mathrm{a__12}\textcolor[rgb]{0,0,1}{}\mathrm{v__2}\textcolor[rgb]{0,0,1}{,}\mathrm{v__2}\\ \mathrm{a__21}\textcolor[rgb]{0,0,1}{,}\mathrm{a__21}\textcolor[rgb]{0,0,1}{}\mathrm{v__1}\textcolor[rgb]{0,0,1}{,}\mathrm{v__1}\textcolor[rgb]{0,0,1}{+}\mathrm{a__22}\textcolor[rgb]{0,0,1}{,}\mathrm{a__22}\textcolor[rgb]{0,0,1}{}\mathrm{v__2}\textcolor[rgb]{0,0,1}{,}\mathrm{v__2}\end{array}\right]
v . v
\stackrel{\textcolor[rgb]{0,0,1}{&conjugate0;}}{\mathrm{v__1}\textcolor[rgb]{0,0,1}{,}\mathrm{v__1}}\textcolor[rgb]{0,0,1}{}\mathrm{v__1}\textcolor[rgb]{0,0,1}{,}\mathrm{v__1}\textcolor[rgb]{0,0,1}{+}\stackrel{\textcolor[rgb]{0,0,1}{&conjugate0;}}{\mathrm{v__2}\textcolor[rgb]{0,0,1}{,}\mathrm{v__2}}\textcolor[rgb]{0,0,1}{}\mathrm{v__2}\textcolor[rgb]{0,0,1}{,}\mathrm{v__2}
To multiply Vectors and/or Matrices and/or Arrays together using elementwise multiplication, use the standard multiplication operator, `⋅` followed by the "elementwise" operator, `~`:
A \cdot ~ A
\left[\begin{array}{cc}{\mathrm{a__11}}^{\textcolor[rgb]{0,0,1}{2}}& {\mathrm{a__12}}^{\textcolor[rgb]{0,0,1}{2}}\\ {\mathrm{a__21}}^{\textcolor[rgb]{0,0,1}{2}}& {\mathrm{a__22}}^{\textcolor[rgb]{0,0,1}{2}}\end{array}\right]
A \cdot ~ W
\left[\begin{array}{cc}\mathrm{a__11}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{a}& \mathrm{a__12}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{b}\\ \mathrm{a__21}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{c}& \mathrm{a__22}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{d}\end{array}\right]
Note that when multiplying Arrays together (not with Vectors or Matrices), the standard multiplication operator will result in the elementwise product, so the `~` is not necessary:
W . W
\left[\begin{array}{cc}{\textcolor[rgb]{0,0,1}{a}}^{\textcolor[rgb]{0,0,1}{2}}& {\textcolor[rgb]{0,0,1}{b}}^{\textcolor[rgb]{0,0,1}{2}}\\ {\textcolor[rgb]{0,0,1}{c}}^{\textcolor[rgb]{0,0,1}{2}}& {\textcolor[rgb]{0,0,1}{d}}^{\textcolor[rgb]{0,0,1}{2}}\end{array}\right]
Note also that implicit multiplication is interpreted based on the operands: For Vector/Matrix operands this will be interpreted as the `.` (dot - non-commutative) multiplication operator, while for Array operands this will be interpreted as the elementwise operator:
A v
\left[\begin{array}{c}\mathrm{a__11}\textcolor[rgb]{0,0,1}{}\mathrm{v__1}\textcolor[rgb]{0,0,1}{+}\mathrm{a__12}\textcolor[rgb]{0,0,1}{}\mathrm{v__2}\\ \mathrm{a__21}\textcolor[rgb]{0,0,1}{}\mathrm{v__1}\textcolor[rgb]{0,0,1}{+}\mathrm{a__22}\textcolor[rgb]{0,0,1}{}\mathrm{v__2}\end{array}\right]
W W
\left[\begin{array}{cc}{\textcolor[rgb]{0,0,1}{a}}^{\textcolor[rgb]{0,0,1}{2}}& {\textcolor[rgb]{0,0,1}{b}}^{\textcolor[rgb]{0,0,1}{2}}\\ {\textcolor[rgb]{0,0,1}{c}}^{\textcolor[rgb]{0,0,1}{2}}& {\textcolor[rgb]{0,0,1}{d}}^{\textcolor[rgb]{0,0,1}{2}}\end{array}\right]
Array, binary operators, dot, LinearAlgebra, Matrix, Vector
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Home : Support : Online Help : Science and Engineering : Units : Environments : Natural : verify
The verify Function in the Natural Units Environment
In the Natural Units environment, the global verify function is replaced by a verify function that converts any unevaluated arithmetic operators, equalities, or inequalities to their global equivalents. The first two arguments are tested to check whether they are valid unit names.
verify(3.50000003 = 'a1', 3.499999997 = 'a1', 'float(100) = boolean');
\textcolor[rgb]{0,0,1}{\mathrm{true}}
:-verify(3.50000003 = 'a1', 3.499999997 = 'a1', 'float(100) = boolean'); # unexpectedly \textcolor[rgb]{0,0,1}{\mathrm{false}}
\textcolor[rgb]{0,0,1}{\mathrm{true}}
:-verify(m, 1250/381*ft, 'units'); # this returns false as 'm' is not interpreted as a meter
\textcolor[rgb]{0,0,1}{\mathrm{false}}
verify(m, 1250/381*ft, 'units');
\textcolor[rgb]{0,0,1}{\mathrm{true}}
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REreduceorder - Maple Help
Home : Support : Online Help : Mathematics : Discrete Mathematics : Summation and Difference Equations : LREtools : REreduceorder
apply the method of reduction of order to a LRE
REreduceorder(problem, partsol)
partsol
partial solution, or list of partial solutions
This routine is used to return a new problem of reduced order from a problem and one or many partial solutions. The result is an RESol data structure of the new problem. No attempt is made to solve the reduced problem.
partsol may be a single partial solution, or a list of partial solutions. Note that it is assumed all given partial solutions are correct and valid. When a reduced problem is returned, the order of the resulting problem will be equal to the order of the original less the number of partial solutions given.
If multiple partial solutions are given, the problem is reduced recursively starting with the first solution in the list. The other solutions are then also 'reduced' to solutions of the new problem, and reducing is then called on the rest.
The command with(LREtools,REreduceorder) allows the use of the abbreviated form of this command.
\mathrm{with}\left(\mathrm{LREtools}\right):
\mathrm{REreduceorder}\left(2na\left(n+2\right)+\left({n}^{2}+1\right)a\left(n+1\right)-{\left(n+1\right)}^{2}a\left(n\right)=0,a\left(n\right),\varnothing ,\mathrm{C1}\right)
\textcolor[rgb]{0,0,1}{\mathrm{RESol}}\textcolor[rgb]{0,0,1}{}\left({\left(\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{n}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{n}\right)\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\right)\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{n}\right)}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{0}\right)\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{1}\right)\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{1}\right)}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{INFO}}\right)
\mathrm{REreduceorder}\left(a\left(n+2\right)-2a\left(n+1\right)+a\left(n\right)=0,a\left(n\right),\varnothing ,\mathrm{C1}n\right)
\textcolor[rgb]{0,0,1}{\mathrm{RESol}}\textcolor[rgb]{0,0,1}{}\left({\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{n}\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{+}\left(\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{2}\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\right)\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{n}\right)}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{0}\right)\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{0}\right)}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{INFO}}\right)
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Well-posedness - DispersiveWiki
(Redirected from Wellposed)
What is well-posedness?
By well-posedness in
{\displaystyle H^{s}}
we generally mean that there exists a unique solution u for some time T for each set of initial data in
{\displaystyle H^{s}}
, which stays in
{\displaystyle H^{s}}
and depends continuously on the initial data as a map from
{\displaystyle H^{s}}
{\displaystyle H^{s}}
. However, there are a couple subtleties involved here.
Existence. For classical (smooth) solutions it is clear what it means for a solution to exist; for rough solutions one usually asks (as a bare minimum) for a solution to exist in the sense of distributions. (One may sometimes have to write the equation in conservation form before one can make sense of a distribution). It is possible for negative regularity solutions to exist if there is a sufficient amount of local smoothing available.
Uniqueness. There are many different notions of uniqueness. One common one is uniqueness in the class of limits of smooth solutions. Another is uniqueness assuming certain spacetime regularity assumptions on the solution. A stronger form of uniqueness is in the class of all
{\displaystyle H^{s}}
functions. Stronger still is uniqueness in the class of all distributions for which the equation makes sense.
Time of existence. In subcritical situations the time of existence typically depends only on the
{\displaystyle H^{s}}
norm of the initial data, or at a bare minimum one should get a fixed non-zero time of existence for data of sufficiently small norm. When combined with a conservation law this can often be extended to global existence. In critical situations one typically obtains global existence for data of small norm, and local existence for data of large norm but with a time of existence depending on the profile of the data (in particular, the frequencies where the norm is largest) and not just on the norm itself.
Continuity. There are many different ways the solution map can be continuous from
{\displaystyle H^{s}}
{\displaystyle H^{s}}
. One of the strongest is real analyticity (which is what is commonly obtained by iteration methods). Weaker than this are various types of
{\displaystyle C^{k}}
continuity (
{\displaystyle C^{1}}
{\displaystyle C^{2}}
{\displaystyle C^{3}}
, etc.). If the solution map is C^k, then this implies that the k^th derivative at the origin is in
{\displaystyle H^{s}}
, which roughly corresponds to some iterate (often the k^th iterate) lying in
{\displaystyle H^{s}}
. Weaker than this is Lipschitz continuity, and weaker than that is uniform continuity. Finally, there is just plain old continuity. Interestingly, several examples have emerged recently in which one form of continuity holds but not another; in particular we now have several examples (critical wave maps, low-regularity periodic KdV and mKdV, Benjamin-Ono, quasilinear wave equations, ...) where the solution map is continuous but not uniformly continuous.
For a survey of LWP and GWP issues, see Ta2002.
Retrieved from "https://dispersivewiki.org/DispersiveWiki/index.php?title=Well-posedness&oldid=5554"
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Doppler spectroscopy - Wikipedia
Indirect method for finding extrasolar planets and brown dwarfs
Diagram showing how a smaller object (such as an extrasolar planet) orbiting a larger object (such as a star) could produce changes in position and velocity of the latter as they orbit their common center of mass (red cross).
Doppler spectroscopy detects periodic shifts in radial velocity by recording variations in the color of light from the host star. When a star moves towards the Earth, its spectrum is blueshifted, while it is redshifted when it moves away from us. By analyzing these spectral shifts, astronomers can deduce the gravitational influence of extrasolar planets.[1]
880 extrasolar planets (about 21.0% of the total) were discovered using Doppler spectroscopy, as of February 2020.[2]
3 Radial-velocity comparison tables
3.1 For MK-type stars with planets in the habitable zone
Exoplanets discovered by year (as of February 2014). Those discovered using radial velocity are shown in black, whilst all other methods are in light grey.
Otto Struve proposed in 1952 the use of powerful spectrographs to detect distant planets. He described how a very large planet, as large as Jupiter, for example, would cause its parent star to wobble slightly as the two objects orbit around their center of mass.[3] He predicted that the small Doppler shifts to the light emitted by the star, caused by its continuously varying radial velocity, would be detectable by the most sensitive spectrographs as tiny redshifts and blueshifts in the star's emission. However, the technology of the time produced radial-velocity measurements with errors of 1,000 m/s or more, making them useless for the detection of orbiting planets.[4] The expected changes in radial velocity are very small – Jupiter causes the Sun to change velocity by about 12.4 m/s over a period of 12 years, and the Earth's effect is only 0.1 m/s over a period of 1 year – so long-term observations by instruments with a very high resolution are required.[4][5]
Advances in spectrometer technology and observational techniques in the 1980s and 1990s produced instruments capable of detecting the first of many new extrasolar planets. The ELODIE spectrograph, installed at the Haute-Provence Observatory in Southern France in 1993, could measure radial-velocity shifts as low as 7 m/s, low enough for an extraterrestrial observer to detect Jupiter's influence on the Sun.[6] Using this instrument, astronomers Michel Mayor and Didier Queloz identified 51 Pegasi b, a "Hot Jupiter" in the constellation Pegasus.[7] Although planets had previously been detected orbiting pulsars, 51 Pegasi b was the first planet ever confirmed to be orbiting a main-sequence star, and the first detected using Doppler spectroscopy.[8]
In November 1995, the scientists published their findings in the journal Nature; the paper has since been cited over 1,000 times. Since that date, over 700 exoplanet candidates have been identified, and most have been detected by Doppler search programs based at the Keck, Lick, and Anglo-Australian Observatories (respectively, the California, Carnegie and Anglo-Australian planet searches), and teams based at the Geneva Extrasolar Planet Search.[9]
Beginning in the early 2000s, a second generation of planet-hunting spectrographs permitted far more precise measurements. The HARPS spectrograph, installed at the La Silla Observatory in Chile in 2003, can identify radial-velocity shifts as small as 0.3 m/s, enough to locate many rocky, Earth-like planets.[10] A third generation of spectrographs is expected to come online in 2017. With measurement errors estimated below 0.1 m/s, these new instruments would allow an extraterrestrial observer to detect even Earth.[11]
Properties (mass and semimajor axis) of planets discovered through 2013 using radial velocity, compared (light gray) with planets discovered using other methods.
A series of observations is made of the spectrum of light emitted by a star. Periodic variations in the star's spectrum may be detected, with the wavelength of characteristic spectral lines in the spectrum increasing and decreasing regularly over a period of time. Statistical filters are then applied to the data set to cancel out spectrum effects from other sources. Using mathematical best-fit techniques, astronomers can isolate the tell-tale periodic sine wave that indicates a planet in orbit.[7]
If an extrasolar planet is detected, a minimum mass for the planet can be determined from the changes in the star's radial velocity. To find a more precise measure of the mass requires knowledge of the inclination of the planet's orbit. A graph of measured radial velocity versus time will give a characteristic curve (sine curve in the case of a circular orbit), and the amplitude of the curve will allow the minimum mass of the planet to be calculated using the binary mass function.
The Bayesian Kepler periodogram is a mathematical algorithm, used to detect single or multiple extrasolar planets from successive radial-velocity measurements of the star they are orbiting. It involves a Bayesian statistical analysis of the radial-velocity data, using a prior probability distribution over the space determined by one or more sets of Keplerian orbital parameters. This analysis may be implemented using the Markov chain Monte Carlo (MCMC) method.
The method has been applied to the HD 208487 system, resulting in an apparent detection of a second planet with a period of approximately 1000 days. However, this may be an artifact of stellar activity.[12][13] The method is also applied to the HD 11964 system, where it found an apparent planet with a period of approximately 1 year. However, this planet was not found in re-reduced data,[14][15] suggesting that this detection was an artifact of the Earth's orbital motion around the Sun.[citation needed]
Although radial-velocity of the star only gives a planet's minimum mass, if the planet's spectral lines can be distinguished from the star's spectral lines then the radial-velocity of the planet itself can be found and this gives the inclination of the planet's orbit and therefore the planet's actual mass can be determined. The first non-transiting planet to have its mass found this way was Tau Boötis b in 2012 when carbon monoxide was detected in the infra-red part of the spectrum.[16]
The graph to the right illustrates the sine curve using Doppler spectroscopy to observe the radial velocity of an imaginary star which is being orbited by a planet in a circular orbit. Observations of a real star would produce a similar graph, although eccentricity in the orbit will distort the curve and complicate the calculations below.
This theoretical star's velocity shows a periodic variance of ±1 m/s, suggesting an orbiting mass that is creating a gravitational pull on this star. Using Kepler's third law of planetary motion, the observed period of the planet's orbit around the star (equal to the period of the observed variations in the star's spectrum) can be used to determine the planet's distance from the star (
{\displaystyle r}
) using the following equation:
{\displaystyle r^{3}={\frac {GM_{\mathrm {star} }}{4\pi ^{2}}}P_{\mathrm {star} }^{2}\,}
r is the distance of the planet from the star
Mstar is the mass of the star
Pstar is the observed period of the star
Having determined
{\displaystyle r}
, the velocity of the planet around the star can be calculated using Newton's law of gravitation, and the orbit equation:
{\displaystyle V_{\mathrm {PL} }={\sqrt {GM_{\mathrm {star} }/r}}\,}
{\displaystyle V_{\mathrm {PL} }}
is the velocity of planet.
The mass of the planet can then be found from the calculated velocity of the planet:
{\displaystyle M_{\mathrm {PL} }={\frac {M_{\mathrm {star} }V_{\mathrm {star} }}{V_{\mathrm {PL} }}}\,}
{\displaystyle V_{\mathrm {star} }}
is the velocity of parent star. The observed Doppler velocity,
{\displaystyle K=V_{\mathrm {star} }\sin(i)}
, where i is the inclination of the planet's orbit to the line perpendicular to the line-of-sight.
Thus, assuming a value for the inclination of the planet's orbit and for the mass of the star, the observed changes in the radial velocity of the star can be used to calculate the mass of the extrasolar planet.
Radial-velocity comparison tables[edit]
Star's Radial Velocity Due to the Planet
(vradial)
Jupiter 1 28.4 m/s
Neptune 0.1 4.8 m/s
Neptune 1 1.5 m/s
Super-Earth (5 M🜨) 0.1 1.4 m/s
Alpha Centauri Bb (1.13 ± 0.09 M🜨;) 0.04 0.51 m/s (1[17])note 1
Super-Earth (5 M🜨) 1 0.45 m/s
Earth 0.09 0.30 m/s
Earth 1 0.09 m/s
Ref:[18] Notice 1: Most precise vradial measurements ever recorded. ESO's HARPS spectrograph was used.[17]
note 1: unconfirmed and disputed
Planets[18]
Detectable by:
51 Pegasi b Hot Jupiter 0.05 4.23 days 55.9[19] First-generation spectrograph
55 Cancri d Gas giant 5.77 14.29 years 45.2[20] First-generation spectrograph
Jupiter Gas giant 5.20 11.86 years 12.4[21] First-generation spectrograph
Gliese 581c Super-Earth 0.07 12.92 days 3.18[22] Second-generation spectrograph
Saturn Gas giant 9.58 29.46 years 2.75 Second-generation spectrograph
Alpha Centauri Bb; unconfirmed and disputed Terrestrial planet 0.04 3.23 days 0.510[23] Second-generation spectrograph
Neptune Ice giant 30.10 164.79 years 0.281 Third-generation spectrograph
Earth Habitable planet 1.00 365.26 days 0.089 Third-generation spectrograph (likely)
Pluto Dwarf planet 39.26 246.04 years 0.00003 Not detectable
For MK-type stars with planets in the habitable zone[edit]
(MEarth)
0.10 1.0 8×10−4 M8 0.028 168 6
0.21 1.0 7.9×10−3 M5 0.089 65 21
0.47 1.0 6.3×10−2 M0 0.25 26 67
0.65 1.0 1.6×10−1 K5 0.40 18 115
The major limitation with Doppler spectroscopy is that it can only measure movement along the line-of-sight, and so depends on a measurement (or estimate) of the inclination of the planet's orbit to determine the planet's mass. If the orbital plane of the planet happens to line up with the line-of-sight of the observer, then the measured variation in the star's radial velocity is the true value. However, if the orbital plane is tilted away from the line-of-sight, then the true effect of the planet on the motion of the star will be greater than the measured variation in the star's radial velocity, which is only the component along the line-of-sight. As a result, the planet's true mass will be greater than measured.
To correct for this effect, and so determine the true mass of an extrasolar planet, radial-velocity measurements can be combined with astrometric observations, which track the movement of the star across the plane of the sky, perpendicular to the line-of-sight. Astrometric measurements allows researchers to check whether objects that appear to be high mass planets are more likely to be brown dwarfs.[4]
A further disadvantage is that the gas envelope around certain types of stars can expand and contract, and some stars are variable. This method is unsuitable for finding planets around these types of stars, as changes in the stellar emission spectrum caused by the intrinsic variability of the star can swamp the small effect caused by a planet.
The method is best at detecting very massive objects close to the parent star – so-called "hot Jupiters" – which have the greatest gravitational effect on the parent star, and so cause the largest changes in its radial velocity. Hot Jupiters have the greatest gravitational effect on their host stars because they have relatively small orbits and large masses. Observation of many separate spectral lines and many orbital periods allows the signal to noise ratio of observations to be increased, increasing the chance of observing smaller and more distant planets, but planets like the Earth remain undetectable with current instruments.
Left: A representation of a star orbited by a planet. All the movement of the star is along the viewer's line-of-sight; Doppler spectroscopy will give a true value of the planet's mass.
Right: In this case none of the star's movement is along the viewer's line-of-sight and the Doppler spectroscopy method will not detect the planet at all.
Systemic (amateur extrasolar planet search project)
^ "Catalog". exoplanet.eu/catalog/. Retrieved 2020-02-16.
^ O. Struve (1952). "Proposal for a project of high-precision stellar radial velocity work". The Observatory. 72 (870): 199–200. Bibcode:1952Obs....72..199S.
^ a b c "Radial velocity method". The Internet Encyclopedia of Science. Retrieved 2007-04-27.
^ A. Wolszczan (Spring 2006). "Doppler spectroscopy and astrometry – Theory and practice of planetary orbit measurements" (PDF). ASTRO 497: "Astronomy of Extrasolar Planets" lectures notes. Penn State University. Archived from the original (PDF) on 2008-12-17. Retrieved 2009-04-19.
^ "A user's guide to Elodie archive data products". Haute-Provence Observatory. May 2009. Retrieved 26 October 2012.
^ a b Mayor, Michel; Queloz, Didier (1995). "A Jupiter-mass companion to a solar-type star". Nature. 378 (6555): 355–359. Bibcode:1995Natur.378..355M. doi:10.1038/378355a0. ISSN 1476-4687. OCLC 01586310.
^ Brennan, Pat (July 7, 2015). "Will the real 'first exoplanet' please stand up?". Exoplanet Exploration: Planets Beyond our Solar System. Retrieved 28 February 2022.
^ R.P. Butler; et al. (2006). "Catalog of Nearby Exoplanets" (PDF). Astrophysical Journal. 646 (2–3): 25–33. arXiv:astro-ph/0607493. Bibcode:2006ApJ...646..505B. doi:10.1086/504701. Archived from the original (PDF) on 2007-07-07.
^ Mayor; et al. (2003). "Setting New Standards With HARPS" (PDF). ESO Messenger. 114: 20. Bibcode:2003Msngr.114...20M.
^ "ESPRESSO – Searching for other Worlds". Centro de Astrofísica da Universidade do Porto. 2009-12-16. Archived from the original on 2010-10-17. Retrieved 2010-10-26.
^ P.C. Gregory (2007). "A Bayesian Kepler periodogram detects a second planet in HD 208487". Monthly Notices of the Royal Astronomical Society. 374 (4): 1321–1333. arXiv:astro-ph/0609229. Bibcode:2007MNRAS.374.1321G. doi:10.1111/j.1365-2966.2006.11240.x.
^ Wright, J. T.; Marcy, G. W.; Fischer, D. A; Butler, R. P.; Vogt, S. S.; Tinney, C. G.; Jones, H. R. A.; Carter, B. D.; et al. (2007). "Four New Exoplanets and Hints of Additional Substellar Companions to Exoplanet Host Stars". The Astrophysical Journal. 657 (1): 533–545. arXiv:astro-ph/0611658. Bibcode:2007ApJ...657..533W. doi:10.1086/510553.
^ P.C. Gregory (2007). "A Bayesian periodogram finds evidence for three planets in HD 11964". Monthly Notices of the Royal Astronomical Society. 381 (4): 1607–1616. arXiv:0709.0970. Bibcode:2007MNRAS.381.1607G. doi:10.1111/j.1365-2966.2007.12361.x.
^ Wright, J.T.; Upadhyay, S.; Marcy, G. W.; Fischer, D. A.; Ford, Eric B.; Johnson, John Asher (2009). "Ten New and Updated Multi-planet Systems, and a Survey of Exoplanetary Systems". The Astrophysical Journal. 693 (2): 1084–1099. arXiv:0812.1582. Bibcode:2009ApJ...693.1084W. doi:10.1088/0004-637X/693/2/1084.
^ Weighing The Non-Transiting Hot Jupiter Tau BOO b, Florian Rodler, Mercedes Lopez-Morales, Ignasi Ribas, 27 June 2012
^ a b "Planet Found in Nearest Star System to Earth". European Southern Observatory. 16 October 2012. Retrieved 17 October 2012.
^ a b "ESPRESSO and CODEX the next generation of RV planet hunters at ESO". Chinese Academy of Sciences. 2010-10-16. Archived from the original on 2011-07-04. Retrieved 2010-10-16.
^ "51 Peg b". Exoplanets Data Explorer.
^ "55 Cnc d". Exoplanets Data Explorer.
^ Endl, Michael. "The Doppler Method, or Radial Velocity Detection of Planets". University of Texas at Austin. Retrieved 26 October 2012. [permanent dead link]
^ "GJ 581 c". Exoplanets Data Explorer.
^ "alpha Cen B b". Exoplanets Data Explorer.
^ "An NIR laser frequency comb for high precision Doppler planet surveys". Chinese Academy of Sciences. 2010-10-16. Retrieved 2010-10-16. [dead link]
Retrieved from "https://en.wikipedia.org/w/index.php?title=Doppler_spectroscopy&oldid=1083847921"
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2018 Well-Posedness and Numerical Study for Solutions of a Parabolic Equation with Variable-Exponent Nonlinearities
Jamal H. Al-Smail, Salim A. Messaoudi, Ala A. Talahmeh
We consider the following nonlinear parabolic equation:
{u}_{t}-\mathrm{d}\mathrm{i}\mathrm{v}\left(|\nabla u{|}^{p\left(x\right)-\mathrm{2}}\nabla u\right)=f\left(x,t\right)
f:\mathrm{\Omega }×\left(\mathrm{0},T\right)\to \mathbb{R}
and the exponent of nonlinearity
p\left(·\right)
are given functions. By using a nonlinear operator theory, we prove the existence and uniqueness of weak solutions under suitable assumptions. We also give a two-dimensional numerical example to illustrate the decay of solutions.
Jamal H. Al-Smail. Salim A. Messaoudi. Ala A. Talahmeh. "Well-Posedness and Numerical Study for Solutions of a Parabolic Equation with Variable-Exponent Nonlinearities." Int. J. Differ. Equ. 2018 (SI2) 1 - 9, 2018. https://doi.org/10.1155/2018/9754567
Jamal H. Al-Smail, Salim A. Messaoudi, Ala A. Talahmeh "Well-Posedness and Numerical Study for Solutions of a Parabolic Equation with Variable-Exponent Nonlinearities," International Journal of Differential Equations, Int. J. Differ. Equ. 2018(SI2), 1-9, (2018)
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Leonhard Euler - Wikiquote
Leonhard Euler (15 April 1707 – 18 September 1783) was a Swiss mathematician and physicist, considered to be one of the greatest mathematicians of all time.
1.1 Introduction to the Analysis of the Infinite (1748)
1.2 A conjecture about the nature of air (1780)
2 Quotes about Euler
Upon losing the use of his right eye; as quoted in In Mathematical Circles (1969) by H. Eves
As quoted in Calculus Gems (1992) by G. Simmons
All the greatest mathematicians have long since recognized that the method presented in this book is not only extremely useful in analysis, but that it also contributes greatly to the solution of physical problems. For since the fabric of the universe is most perfect, and is the work of a most wise Creator, nothing whatsoever takes place in the universe in which some relation of maximum and minimum does not appear. Wherefore there is absolutely no doubt that every effect in the universe can be explained as satisfactorily from final causes, by the aid of the method of maxima and minima, as it can from the effective causes themselves. Now there exist on every hand such notable instances of this fact, that, in order to prove its truth, we have no need at all of a number of examples; nay rather one's task should be this, namely, in any field of Natural Science whatsoever to study that quantity which takes on a maximum or a minimum value, an occupation that seems to belong to philosophy rather than to mathematics. Since, therefore, two methods of studying effects in Nature lie open to us, one by means of effective causes, which is commonly called the direct method, the other by means of final causes, the mathematician uses each with equal success. Of course, when the effective causes are too obscure, but the final causes are more readily ascertained, the problem is commonly solved by the indirect method; on the contrary, however, the direct method is employed whenever it is possible to determine the effect from the effective causes. But one ought to make a special effort to see that both ways of approach to the solution of the problem be laid open; for thus not only is one solution greatly strengthened by the other, but, more than that, from the agreement between the two solutions we secure the very highest satisfaction.
introduction to De Curvis Elasticis, Additamentum I to his Methodus Inveniendi Lineas Curvas Maximi Minimive Proprietate Gaudentes 1744; translated on pg10-11, "Leonhard Euler's Elastic Curves", Oldfather et al 1933
La construction d'une machine propre à exprimer tous les sons de nos paroles , avec toutes les articulations , seroit sans-doute une découverte bien importante. … La chose ne me paroît pas impossible.
It would be a considerable invention indeed, that of a machine able to mimic speech, with its sounds and articulations. … I think it is not impossible.
Letter to Friederike Charlotte of Brandenburg-Schwedt (16 June 1761)
Lettres à une Princesse d'Allemagne sur différentes questions de physique et de philosophie, Royer, 1788, p. 265
As quoted in An Introduction to Text-to-Speech Synthesis (2001) by Thierry Dutoit, p. 27; also in Fabian Brackhane and Jürgen Trouvain "Zur heutigen Bedeutung der Sprechmaschine Wolfgang von Kempelens" (in: Bernd J. Kröger (ed.): Elektronische Sprachsignalverarbeitung 2009, Band 2 der Tagungsbände der 20. Konferenz "Elektronische Sprachsignalverarbeitung" (ESSV), Dresden: TUDpress, 2009, pp. 97–107)
It will seem a little paradoxical to ascribe a great importance to observations even in that part of the mathematical sciences which is usually called Pure Mathematics, since the current opinion is that observations are restricted to physical objects that make impression on the senses. As we must refer the numbers to the pure intellect alone, we can hardly understand how observations and quasi-experiments can be of use in investigating the nature of numbers. Yet, in fact, as I shall show here with very good reasons, the properties of the numbers known today have been mostly discovered by observation, and discovered long before their truth has been confirmed by rigid demonstrations. There are many properties of the numbers with which we are well acquainted, but which we are not yet able to prove; only observations have led us to their knowledge. Hence we see that in the theory of numbers, which is still very imperfect, we can place our highest hopes in observations; they will lead us continually to new properties which we shall endeavor to prove afterwards. The kind of knowledge which is supported only by observations and is not yet proved must be carefully distinguished from the truth; it is gained by induction, as we usually say. Yet we have seen cases in which mere induction led to error. Therefore, we should take great care not to accept as true such properties of the numbers which we have discovered by observation and which are supported by induction alone. Indeed, we should use such discovery as an opportunity to investigate more exactly the properties discovered and to prove or disprove them; in both cases we may learn something useful.
Opera Omnia, ser. 1, vol. 2, p. 459 Spcimen de usu observationum in mathesi pura, as quoted by George Pólya, Induction and Analogy in Mathematics Vol. 1, Mathematics and Plausible Reasoning (1954)
Introduction to the Analysis of the Infinite (1748)[edit]
Original title: Introductio in analysin infinitorum. Translated as Introduction to Analysis of the Infinite (1988–89) by John Blanton (Book I ISBN 0387968245; Book II ISBN 0387971327 (online version).
A function of a variable quantity is an analytic expression composed in any way whatsoever of the variable quantity and numbers or constant quantities.
A conjecture about the nature of air (1780)[edit]
A conjecture about the nature of air, by which are to be explained the phenomenon which have been observed in the atmosphere (Conjectura circa naturam aeris, pro explicandis phaenomenis in atmosphaera observatis) (1870) (online version).
Quanquam nobis in intima naturae mysteria penetrare, indeque veras caussas Phaenomenorum agnoscere neutiquam est concessum: tamen evenire potest, ut hypothesis quaedam ficta pluribus phaenomenis explicandis aeque satisfaciat, ac si vera caussa nobis esset perspecta.
Quotes about Euler[edit]
He calculated without any apparent effort, just as men breathe, as eagles sustain themselves in the air. ~ François Arago
I discovered the works of Euler and my perception of the nature of mathematics underwent a dramatic transformation. ~ Alexander Stepanov
He calculated without any apparent effort, just as men breathe, as eagles sustain themselves in the air.
François Arago; Variant: Euler calculated without apparent effort, as men breathe, or as eagles sustain themselves in the wind.
The most influential mathematics textbook of ancient times is easily named, for the Elements of Euclid has set the pattern in elementary geometry ever since. The most effective textbook of the medieval age is less easily designated; but a good case can be made out for the Al-jabr of Al-Khwarizmi, from which algebra arose and took its name. Is it possible to indicate a modern textbook of comparable influence and prestige? Some would mention the Géométrie of Descartes or the Principia of Newton or the Disquisitiones of Gauss; but in pedagogical significance these classics fell short of a work by Euler titled Introductio in analysin infinitorum.
Carl B. Boyer on Euler's Introduction to the Analysis of the Infinite in "The Foremost Textbook of Modern Times" (1950)
Carl B. Boyer in "The Foremost Textbook of Modern Times" (1950)
Of no little importance are Euler's labors in analytical mechanics. ...He worked out the theory of the rotation of a body around a fixed point, established the general equations of motion of a free body, and the general equation of hydrodynamics. He solved an immense number and variety of mechanical problems, which arose in his mind on all occasions. Thus on reading Virgil's lines. "The anchor drops, the rushing keel is staid," he could not help inquiring what would be the ship's motion in such a case. About the same time as Daniel Bernoulli he published the Principle of the Conservation of Areas and defended the principle of "least action," advanced by P. Maupertius. He wrote also on tides and on sound.
Florian Cajori, A History of Mathematics (1893) "Euler, Lagrange and Laplace" p. 240.
Somebody said "Talent is doing what others find difficult. Genius is doing easily what others find impossible." ...by that definition, Euler was a genius. He could do the seemingly impossible, and he did it throughout his long and illustrious life. ...Way to Go, Uncle Leonhard!
William Dunham, "A Tribute to Euler" (Oct 14, 2008) 49:10, Clay Public Lecture, Harvard University Science Center, from the Clay Mathematics Institute.
Frederick the Great, Letters of Voltaire and Frederick the Great (1927), translated by Richard Aldington, letter 221 from Frederick to Voltaire (25 November 1777)
Euler lacked only one thing to make him a perfect genius: He failed to be incomprehensible.
Ferdinand Georg Frobenius, as quoted by William Dunham, "A Tribute to Euler" (Oct 14, 2008) 48:27, Clay Public Lecture, Harvard University Science Center, from the Clay Mathematics Institute.
Carl Friedrich Gauss, as quoted by Louise Grinstein, Sally I. Lipsey, Encyclopedia of Mathematics Education (2001) p. 235.
Following a suggestion by Daniel Bernoulli, Euler gave the first treatment of elastic lines by means of the calculus of variations in the Additamentum I to his Methodus inveniendi (1744...) which carries the title De curvis elasticus. Euler characterized the equilibrium position of an elastic line by the following variational principle: Among all curves of equal length, joining two points where they have prescribed tangents, to determine that which minimizes the value of the expression
{\displaystyle \int ds/\rho ^{2}}
{\displaystyle \rho }
is the radius of curvature]. In other words, Euler interpreted an elastic line as an inextensible curve
{\displaystyle {\boldsymbol {\zeta }}}
with a "potential energy" of
{\displaystyle \int \kappa ^{2}ds}
{\displaystyle \kappa }
{\displaystyle 1/\rho }
] being the curvature function of
{\displaystyle {\boldsymbol {\zeta }}}
, whose positions of (stable) equilibrium are characterized by the minima of the potential energy, i.e., by Johann Bernoulli's principle of virtual work. Thus the problem of the elastic line leads to the isoperimetric problem
{\displaystyle \int _{\boldsymbol {\zeta }}\kappa ^{2}ds\to min\qquad with\int _{\boldsymbol {\zeta }}ds=L}
Mariano Giaquinta, Stefan Hildebrandt, Calculus of Variations I (2004) Grundlehren der mathematischen WissenSchaften Vol. 310.
Galileo does not attempt any theory to account for the flexure of the beam. This theory, supplied by Hooke's law, was applied by Mariotte, Leibnitz, De Lahire, and Varignon, but they neglect compression of the fibres, and so place the neutral in the lower face of Galileo's beam. The true position of the neutral plane was assigned by James Bernoulli 1695, who in his investigation of the simplest case of bent beam, was led to the consideration of the curve called the "elastica." This "elastica" curve speedily attracted the attention of the great Euler (1744), and must be considered to have directed his attention to the elliptic integrals. Probably the extraordinary divination which led Euler to the formula connecting the sum of two elliptic integrals, thus giving the fundamental theorem of the addition equation of elliptic functions, was due to mechanical considerations concerning the "elastica" curve; a good illustration of the general principle that the pure mathematician will find the best materials for his work in the problems presented to him by natural and physical questions.
A. G. Greenhill, Nature (Feb. 3, 1887) Review of A History of the Theory of Elasticity, Volume 35, pp. 313-314.
Who has studied the works of such men as Euler, Lagrange, Cauchy, Riemann, Sophus Lie, and Weierstrass, can doubt that a great mathematician is a great artist? The faculties possessed by such men, varying greatly in kind and degree with the individual, are analogous with those requisite for constructive art. Not every mathematician possesses in a specially high degree that critical faculty which finds its employment in the perfection of form, hi conformity with the ideal of logical completeness; but every great mathematician possesses the rarer faculty of constructive imagination.
E. W. Hobson, "Presidential Address British Association for the Advancement of Science" (1910) in: Nature, Vol. 84, p. 290. Cited in: Moritz (1914, 182); Mathematics as a fine art
To the reader of today much in the conception and mode of expression of that time appears strange and unusual. Between us and the mathematicians of the late seventeenth century stands Leonhard Euler... He is the real founder of our modern conception. However non-rigorous he may be in details: he ends and conquers the previous epoch of direct geometric infinitesimal considerations and introduces the period of mathematical analysis according to form and content. Whatever was written after him on the logarithmic series is necessarily based no longer on the already obscured predecessors in the receding mathematical Renaissance, but on Euler's Introductio in analysin infinitorum... in which the entire seventh chapter [De Quantitabus exponentialibus ac Logarithmis] treats of logarithms.
Josef Ehrenfried Hofman, "On the Discovery of the Logarithmic Series and Its Development in England up to Cotes" (Oct., 1939) National Mathematics Magazine, Vol. 14, No. 1, pp. 37-38.
Pierre-Simon Laplace, as quoted in Calculus Gems (1992). variant: Read Euler, read Euler. He is the master of us all. As quoted by S. H. Hollingdale, "Leonhard Euler (1707-1783): A Bicentennial Tribute", Bulletin (1983) Volumes 19-20, Institute of Mathematics and Its Applications, & by Edwin Joseph Purcell, Dale E. Varberg, Calculus with Analytic Geometry (1987), Vol. 1.
He was later to write that he had made some of his best discoveries while holding a baby in his arms surrounded by playing children.
Richard Mankiewicz, in The Story of Mathematics (2000), p. 142
If we compared the Bernoullis to the Bach family, then Leonhard Euler is unquestionably the Mozart of mathematics, a man whose immense output... is estimated to fill at least seventy volumes. Euler left hardly an area of mathematics untouched, putting his mark on such diverse fields as analysis, number theory, mechanics and hydrodynamics, cartography, topology, and the theory of lunar motion. ...Moreover, we owe to Euler many of the mathematical symbols in use today, among them i, π, e, and f(x). And as if that were not enough, he was a great popularizer of science...
Euler and Ramanujan are mathematicians of the greatest importance in the history of constants (and of course in the history of Mathematics ...)
E. W. Middlemast
In 1736, during his first stay in St. Petersburg, Euler tackled the now famous problem of the seven bridges of Königsberg. His contribution to this problem is often cited as the birth of graph theory and topology.
David S. Richeson (8 March 2012). Euler's Gem: The Polyhedron Formula and the Birth of Topology. Princeton University Press. p. 100. ISBN 1-4008-3856-8.
Edward Charles Titchmarsh as quoted in Mathematical Maxims and Minims (1988) by N. Rose
Brief biography at Evansville College
Brief biography at the University of St Andrews, Scotland
The Euler Archive, Mathematics at Dartmouth
Euler and "Fermat's Last Theorem"
Euler's presentation of "The Seven Bridges of Königsberg"
Leonhard Euler at the Notable Names Database
List of topics named after Euler (at Wikipedia)
An Evening with Leonhard Euler presented by William Dunham.
Retrieved from "https://en.wikiquote.org/w/index.php?title=Leonhard_Euler&oldid=3113005"
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Square tiling - Wikipedia
Regular tiling of the Euclidean plane
3 Wythoff constructions from square tiling
4 Topologically equivalent tilings
6 Related regular complex apeirogons
Uniform colorings[edit]
This tiling is also topologically related as a part of sequence of regular polyhedra and tilings with four faces per vertex, starting with the octahedron, with Schläfli symbol {n,4}, and Coxeter diagram , with n progressing to infinity.
Wythoff constructions from square tiling[edit]
Topologically equivalent tilings[edit]
Circle packing[edit]
Related regular complex apeirogons[edit]
{\displaystyle {\tilde {A}}_{n-1}}
{\displaystyle {\tilde {C}}_{n-1}}
{\displaystyle {\tilde {B}}_{n-1}}
{\displaystyle {\tilde {D}}_{n-1}}
{\displaystyle {\tilde {G}}_{2}}
{\displaystyle {\tilde {F}}_{4}}
{\displaystyle {\tilde {E}}_{n-1}}
Retrieved from "https://en.wikipedia.org/w/index.php?title=Square_tiling&oldid=1051821090"
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Time - Simple English Wikipedia, the free encyclopedia
Time is the never-ending continued progress of existence and events. It happens in an apparently irreversible way from the past, through the present and to the future.
To measure time, we can use anything that repeats itself regularly. One example is the start of a new day (as Earth rotates on its axis). Two more are the phases of the moon (as it orbits the Earth), and the seasons of the year (as the Earth orbits the Sun).
Even in ancient times, people developed calendars to keep track of the number of days in a year. They also developed sundials that used the moving shadows cast by the sun through the day to measure times smaller than a day. Today, highly accurate clocks can measure time in less than a billionth of a second. The study of time measurement is known as horology.
The SI (International Systems of Units) unit of time is one second, written as s.[1] When used as a variable in mathematics, time is often represented by the symbol
{\displaystyle t}
In Einsteinian physics, time and space can be combined into a single concept. For more on the topic, see space-time continuum.
1.1 Things used to measure time
Units of timeEdit
1 millennium = 10 centuries = 100 decades = 200 lustrums = 250 quadrenniums = 333.33 trienniums = 500 bienniums = 1,000 years
1 century = 10 decades = 20 lustrums = 25 quadrenniums = 33.33 trienniums = 50 bienniums = 100 years
1 decade = 2 lustrums = 2.5 quadrenniums = 3.33 trienniums = 5 bienniums = 10 years
1 year = 12 months = 52 weeks = 365 days (366 days in leap years)
1 month = 4 weeks = 2 fortnights = 28 to 31 days
1 fortnight = 2 weeks = 14 days
Things used to measure timeEdit
a digital clock in Glasgow
Time of dayEdit
U.S. Naval Observatory Archived 2010-08-30 at the Wayback Machine
Time in Archived 2011-08-10 at the Wayback Machine Hexadecimal numeral system
↑ "Current definitions of the SI units". physics.nist.gov. Retrieved 2020-08-16.
Retrieved from "https://simple.wikipedia.org/w/index.php?title=Time&oldid=8163658"
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Model series RLC network - Simulink - MathWorks France
Model series RLC network
The Series RLC block models the series RLC network described in the block dialog box, in terms of its frequency-dependent S-parameters.
For the given resistance, inductance, and capacitance, the block first calculates the ABCD- parameters at each frequency contained in the vector of modeling frequencies, and then converts the ABCD- parameters to S-parameters using the RF Toolbox™ abcd2s function. See the Output Port block reference page for information about determining the modeling frequencies.
For this circuit, A = 1, B = Z, C = 0, and D = 1, where
Z=\frac{-LC{\omega }^{2}+jRC\omega +1}{jC\omega }
\omega =2\pi f
The series RLC object is a two-port network as shown in the following circuit diagram.
Resistance (Ohms) — Resistance of the series RLC network
Inductance (H) — Inductance of the series RLC network
Capacitance (F) — Capacitance of the series RLC network
General Passive Network | LC Bandpass Pi | LC Bandpass Tee | LC Bandstop Pi | LC Bandstop Tee | LC Highpass Pi | LC Highpass Tee | LC Lowpass Pi | LC Lowpass Tee | Series C | Series L | Series R | Shunt C | Shunt L | Shunt R | Shunt RLC
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Mini-Workshop: The Hauptvermutung for High-Dimensional Manifolds | EMS Press
The Mini-Workshop \emph{The Hauptvermutung for High-Dimensional Manifolds}, organised by Erik Pedersen (Binghamton) and Andrew Ranicki (Edinburgh) was held August 13th--18th, 2006. The meeting was attended by 17 participants, ranging from graduate students to seasoned veterans. The manifold Hauptvermutung is the conjecture that topological manifolds have a unique combinatorial structure. This conjecture was disproved in 1969 by Kirby and Siebenmann, who used a mixture of geometric and algebraic methods to classify the combinatorial structures on manifolds of dimension
>4
. However, there is some dissatisfaction in the community with the state of the literature on this topic. This has been voiced most forcefully by Novikov, who has written ``In particular, the final Kirby-Siebenmann classification of topological multidimensional manifolds therefore is not proved yet in the literature." (http://front.math.ucdavis.edu/math-ph/0004012) At this conference we discussed a number of questions concerning the Hauptvermutung and the structure theory of high-dimensional topological manifolds. These are our conclusions: We found nothing fundamentally wrong with the original work of Kirby and Siebenmann \cite{ks1}, which is solidly grounded in the literature. Their determination of
TOP/PL
depends on Kirby's paper on the Annulus Conjecture and his `torus trick'. It was noted that Kirby's paper is based on the well-documented work on
PL
classification of homotopy tori (Hsiang and Shaneson, Wall) and Sullivan's identification of the
PL
normal invariants with
[-,G/PL]
, but does not depend on any other work of Sullivan, documented or undocumented. This classification can be reduced to the Farrell Fibering Theorem \cite{far}, the calculation of
\pi_i(G/PL)
(Kervaire and Milnor \cite{km}), and Wall's non-simply connected surgery theory \cite{wall}. There are modern proofs determining the homotopy type of
TOP/PL
using either the bounded surgery of Ferry and Pedersen \cite{fp} or a modification of the definition of the structure set. Sullivan's determination of the homotopy type of
G/PL,
which is well-doc\-u\-ment\-ed (for instance, in Madsen and Milgram \cite{mm}) is used to determine the homotopy type of
G/TOP
and is fundamental to understanding the classification of general topological manifolds. The 4-fold periodicity of the topological surgery sequence established by Siebenmann \cite[p.283]{ks1} contains a minor error having to do with base points. This is an easily corrected error, and the 4-fold periodicity is true whenever the manifold has a boundary. The equivalence of the algebraic and topological surgery exact sequence as established by Ranicki \cite{ran} was confirmed. Sullivan's characteristic variety theorem, however it is understood, is not essential for the Kirby-Siebenmann triangulation of manifolds. The following papers have been commissioned: \begin{itemize} \item W. Browder, ``
PL
classification of homotopy tori'' \item J. Davis, ``On the product structure theorem'' \item I. Hambleton, ``
PL
classification of homotopy tori'' \item M. Kreck, ``A proof of Rohlin's theorem'' \item E.K. Pedersen, ``Determining the homotopy type of
TOP/PL
using bound\-ed surgery'' \item A. Ranicki, ``Siebenmann's periodicity theorem'' \item M. Weiss, ``Identifying the algebraic and geometric surgery sequences'' \end{itemize} The Hauptvermutung website {\it http://www.maths.ed.ac.uk/$\sim$aar/haupt} will re\-cord further developments. \begin{thebibliography}{99} \bibitem{far} F.~T.~Farrell, \emph{The obstruction to fibering a manifold over a circle}, Yale Ph.D. thesis (1967), Indiana Univ. Math. J. {\bf 21}, 315-346 (1971) \bibitem{fp} S.~Ferry and E.~K.~Pedersen, {\it Epsilon surgery}, in {\it Novikov conjectures, Index Theorems and Rigidity}, Vol. 2, LMS Lecture Notes {\bf 227}, Cambridge, 167--226 (1995) \bibitem{km} M.~Kervaire and J.~Milnor, \emph{Groups of homotopy spheres}, Ann. of Maths. {\bf 77}, 504--537 (1963) \bibitem{ks1} R.~Kirby and L.~Siebenmann, {\it Foundational essays on topological manifolds, smoothings, and triangulations}, Ann. of Maths. Studies {\bf 88}, Princeton University Press (1977) \bibitem{mm} I.~Madsen and J.~Milgram, {\it The classifying spaces for surgery and cobordism of manifolds.} Ann. of Maths. Studies {\bf 92}, Princeton University Press (1979) \bibitem{ran} A.~Ranicki, {\it Algebraic
L
-theory and topological manifolds}, Tracts in Mathematics {\bf 102}, Cambridge University Press (1992) \bibitem{wall} C.~T.~C. Wall, \emph{Surgery on {C}ompact {M}anifolds}, Academic Press (1970) \end{thebibliography}
Andrew Ranicki, Erik Kjaer Pedersen, Mini-Workshop: The Hauptvermutung for High-Dimensional Manifolds. Oberwolfach Rep. 3 (2006), no. 3, pp. 2195–2226
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Ehrhart Quasipolynomials: Algebra, Combinatorics, and Geometry | EMS Press
The mini-workshop Ehrhart Quasipolynomials: Algebra, Combinatorics, and Geometry, organised by Jes\'us De Loera (Davis) and Christian Haase (Durham), was held August 15th-21st, 2004. A small group of mathematicians and computer scientists discussed recent developments and open questions about \emph{Ehrhart quasipolynomials}. These fascinating functions are defined in terms of the lattice points inside convex polyhedra. More precisely, given a rational convex polytope
P
n
, the Ehrhart quasipolynomials are defined as
i_P ( n ) = \# \left( n P \cap {\mathbb Z}^{ d } \right)
. This equals the number of integer points inside the dilated polytope
n P = \{ nx : x \in P \}
i_P(n)
appear in a natural way in many areas of mathematics. The participants represented a broad range of topics where Ehrhart quasipolynomials are useful; e.g. combinatorics, representation theory, algebraic geometry, and software design, to name some of the areas represented. Each working day had at least two different themes, for example the first day of presentations included talks on how lattice point counting is relevant in compiler optimization and software engineering as well as talks about tensor product multiplicities in representation theory of complex semisimple Lie Algebras. Some special activities included in the miniworkshop were (1) a problem session, a demonstration of the software packages for counting lattice points {\tt Ehrhart} (by P. Clauss), {\tt LattE} (by J. De Loera et al.), and {\tt Barvinok} (by S. Verdoolaege), (2) a guest speaker from one of the research in pairs groups (by R. Vershynin),and (3) a nice expository event where each of the three mini-workshops sharing the Oberwolfach facilities had a chance to introduce the hot questions being pursued to the others. The atmosphere was always very pleasant and people worked very actively. For instance, two of the talks reported on new theorems obtained during the miniworkshop. The organizers and participants sincerely thank MFO for providing a wonderful working environment, perhaps unique around the world. We also thank G\"unter M. Ziegler for his support and encouragement. In what follows we present the abstracts of talks following the order in which talks were presented.
Jesús De Loera, Christian Haase, Ehrhart Quasipolynomials: Algebra, Combinatorics, and Geometry. Oberwolfach Rep. 1 (2004), no. 3, pp. 2071–2102
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High-resolution FFT of a portion of a spectrum - MATLAB - MathWorks España
dsp.ZoomFFT
zfftOut
Compute FFT of a Subband Using Zoom FFT
Compute Zoom FFT of Variable-Size Inputs
The dsp.ZoomFFT System object™ computes the fast Fourier Transform (FFT) of a signal over a portion of frequencies in the Nyquist interval. By setting an appropriate decimation factor D, and sampling rate Fs, you can choose the bandwidth of frequencies to analyze BW, where BW = Fs/D. You can also select a specific range of frequencies to analyze in the Nyquist interval by choosing the center frequency of the desired band.
To compute the FFT of a portion of the spectrum:
Create the dsp.ZoomFFT object and set its properties.
zfft = dsp.ZoomFFT
zfft = dsp.ZoomFFT(d)
zfft = dsp.ZoomFFT(d,Fc)
zfft = dsp.ZoomFFT(d,Fc,Fs)
zfft = dsp.ZoomFFT(Name,Value)
zfft = dsp.ZoomFFT creates a zoom FFT System object, zfft, that performs an FFT on a portion of the input signal's frequency range. The object determines the frequency range over which to perform the FFT using the specified center frequency and decimation factor values.
zfft = dsp.ZoomFFT(d) creates a zoom FFT object with the DecimationFactor property set to d.
zfft = dsp.ZoomFFT(d,Fc) creates a zoom FFT object with the DecimationFactor property set to d, and the CenterFrequency property set to Fc.
zfft = dsp.ZoomFFT(d,Fc,Fs) creates a zoom FFT object with the DecimationFactor property set to d, the CenterFrequency property set to Fc, and the SampleRate property set to Fs.
zfft = dsp.ZoomFFT(Name,Value) creates a zoom FFT object with each specified property set to the specified value. Enclose each property name in single quotes. You can use this syntax with any previous input argument combinations.
Example: zfft = dsp.ZoomFFT(2,2e3,48e3,'FFTLength',64);
Decimation factor, specified as a positive integer. This value specifies the factor by which the object reduces the bandwidth of the input signal. The number of rows in the input signal must be a multiple of the decimation factor.
Center frequency of the desired band in Hz, specified as a real scalar in the range (– SampleRate/2, SampleRate/2).
FFT length, specified as a positive integer. The FFT length must be greater than or equal to the ratio of the frame size (number of input rows) and the decimation factor, L/D. The default, [], specifies an FFT length that equals the ratio, L/D.
zfftOut = zfft(input)
zfftOut = zfft(input) computes the zoom FFT of the input. Each column of the input is treated as an independent channel. The object computes the FFT of each channel of the input signal independently over time.
Data input whose zoom FFT the object computes, specified as a vector or a matrix. The number of input rows must be a multiple of the decimation factor.
This object supports variable-size input signals, as long as the input frame size is a multiple of the decimation factor. That is, you can change the input frame size (number of rows) even after calling the algorithm. However, the number of channels (number of columns) must remain constant.
zfftOut — Zoom FFT output
Zoom FFT output, returned as a vector or matrix. If the FFT length is set to auto, the output frame size equals the input frame size divided by the decimation factor. If the object specifies the FFT length, the output frame size equals the specified FFT length. The output data type matches the input data type.
Compute FFT of the [1500 Hz 2500 Hz] subband using zoom FFT for a signal sampled at 48 kHz.
Set the center frequency to 2 kHz and the bandwidth of interest to 1 kHz. The bandwidth is centered at the center frequency. The decimation factor is the ratio of the input sample rate, 48 kHz, and the bandwidth of interest, 1 kHz. Choose an FFT length of 64. Set the input frame size to be the decimation factor times the FFT length. Create a dsp.ZoomFFT object with the specified decimation factor, center frequency, sample rate, and FFT length.
CF = 2e3;
L = D * fftlen;
zfft = dsp.ZoomFFT(D,CF,Fs,'FFTLength',fftlen);
The FFT is computed over frequencies starting at 1500 Hz and spaced by Hz apart, which is the resolution or the minimum frequency that can be discriminated. The number of frequencies at which the zoom FFT is computed equals the FFT length.
Fsd = Fs/D;
F = CF + (Fsd/fftlen)*(0:fftlen-1)-Fsd/2;
Initialize the Scope
Create an array plot to show the frequencies in F.
ap = dsp.ArrayPlot('XDataMode','Custom','CustomXData',F,...
'YLabel','z .* conj(z)','XLabel','Frequency (Hz)','YLimits',[0 1.1e3],...
'Title',sprintf('Decimation Factor = %d. Center Frequency = %d Hz. Resolution = %f Hz',D, CF,(Fs/D)/fftlen));
Create a sine wave with frequencies at 1625 Hz, 2000 Hz, and 2125 Hz.
sine = dsp.SineWave('SampleRate',Fs,'Frequency',tones,'SamplesPerFrame',L);
Pass a noisy sine wave with a sample rate of 48 kHz. Compute the zoom FFT of this sine wave in the subband [1500 Hz 2500 Hz]. Rearrange the Fourier transform by shifting the zero-frequency component to the center of the array. View the tones at 1625 Hz, 2000 Hz, and 2125 Hz in the array plot.
x = sum(sine(),2)+1e-1*randn(L,1);
ap(z.*conj(z));
The dsp.ZoomFFT object accepts variable-size inputs as long as the input is a multiple of the decimation factor. The number of input channels cannot change.
Create a dsp.ZoomFFT object with a decimation factor of 4, center frequency of 2 kHz, and an input sample rate of 48 kHz. Pass a random input with 4*64 rows and 2 columns. Vary the number of rows to 4*128 and 4*32. The resulting FFT lengths are 64, 128, and 32, respectively. The size of the outputs is [64 2], [128 2], and [32 2], respectively.
zfft = dsp.ZoomFFT(4,2e3,48e3);
y1 = zfft(randn(4*64,2));
y2 = zfft(randn(4*128,2));
Set the FFT length as 256 and pass variable-size inputs. The size of all the outputs is [256 2].
release(zfft);
zfft.FFTLength = 256;
{F}_{d}={F}_{c}-\left({F}_{s}/D\right)×floor\left(\left(D×{F}_{c}+{F}_{s}/2\right)/{F}_{s}\right)
Zoom FFT | FFT | Magnitude FFT | Short-Time FFT
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Effect of Oxidation Chemistry of Supercritical Water on Stress Corrosion Cracking of Austenitic Steels | ASME J. of Nuclear Rad Sci. | ASME Digital Collection
Bin Gong,
Bin Gong 1
Water Chemistry Laboratory,
Third Section of Huafu Road, Huayang Town, Shuangliu County
e-mail: gongbin_npic@163.com
Yanping Huang,
CNNC Key Laboratory on Nuclear Reactor Thermalhydraulics Technology,
P.O. Box 622-200, Chengdu, Sichuan 610041
e-mail: hyanping007@163.com
E. Jiang,
e-mail: jiangee@126.com
Yongfu Zhao,
Yongfu Zhao
e-mail: zhaoyongfu0127@126.com
e-mail: wwliu527@163.com
International Cooperation Department,
e-mail: zw2001amy@163.com
Manuscript received June 1, 2015; final manuscript received July 8, 2015; published online December 9, 2015. Assoc. Editor: Thomas Schulenberg.
Gong, B., Huang, Y., Jiang, E., Zhao, Y., Liu, W., and Zhou, Z. (December 9, 2015). "Effect of Oxidation Chemistry of Supercritical Water on Stress Corrosion Cracking of Austenitic Steels." ASME. ASME J of Nuclear Rad Sci. January 2016; 2(1): 011019. https://doi.org/10.1115/1.4031076
Austenitic steel is a candidate material for supercritical water-cooled reactor (SCWR). This study is to investigate the stress corrosion cracking (SCC) behavior of HR3C under the effect of supercritical water chemistry. A transition phenomenon of the water parameters was monitored during a pseudocritical region by water quality experiments at 650°C and 30 MPa. The stress–strain curves and fracture time of HR3C were obtained by slow strain rate tensile (SSRT) tests in the supercritical water at 620°C and 25 MPa. The concentration of the dissolved oxygen (DO) was
200–1000 μg/kg
, and the strain rate was
7.5×10−7/s
. The recent results showed that the failure mode was dominated by intergranular brittle fracture. The relations of the oxygen concentration and the fracture time were nonlinear.
200–500 μg/kg
of oxygen accelerated the cracking, but a longer fracture time was measured when the oxygen concentration was increased to
1000 μg/kg
. Chromium depletion occurred in the oxide layer at the tip of cracks. Grain size increased and chain-precipitated phases were observed in the fractured specimens. These characteristics were considered to contribute to the intergranular SCC.
supercritical water-cooled reactor, water chemistry, stress corrosion cracking, cladding, austenitic steel
Chemistry, Fracture (Materials), Steel, Stress corrosion cracking, Water, Water chemistry, Oxidation, Temperature, Supercritical water reactors, Oxygen
Challenges and Recent Progress in Corrosion and Stress Corrosion Cracking of Alloys for Supercritical Water Reactor Core Components
Proceedings of the 12th International Conference on Environmental Degradation of Materials in Nuclear Power Systems—Water Reactors
Ampornrat
.10.1016/j.jnucmat.2007.05.017
Proceedings of the 4th International Symposium on Supercritical Water-Cooled Reactors
Irradiation-assisted Stress Corrosion Cracking of Austenitic Alloys in Supercritical Water
Haušild
Stress Corrosion Cracking Susceptibility of Austenitic Stainless Steels in Supercritical Water Conditions
Corrosion and SCC Properties of Fine Grain Stainless Steel in Subcritical and Supercritical Pure Water
Proceedings of the NACE International Corrosion2007 Conference & EXPO, NACE International
Review of Materials Issues in Supercritical Water Oxidation Systems and the Need for Corrosion Control
Supercritical Water Radiolysis Chemistry Supercritical Water Corrosion
INL Generation IV Nuclear Energy Systems, Technical Document
, http://nuclear.inel.gov/deliverables/docs/uwnd_scw_level_ii_sep_2006_v3.pdf
The Effect of Temperature on the SSRT Behavior of Austenitic Stainless Steels in SCW
Corrosion Behavior of Hastelloy C-276 in Supercritical Water
Measurement of the Nickel/Nickel Oxide Transition in Ni-Cr-Fe Alloys and Updated Data and Correlations to Quantify the Effect of Aqueous Hydrogen on Primary Water SCC
, Schenectady, NY, .
Jay-Gerin
Power Plant Chem.
Sakaihara
Cracking Susceptibility of Ni Base Alloys and 316 Stainless Steel in Less Oxidizing or Reducing SCW
Stress Corrosion Crack Growth Rates in Type 304 Stainless Steel in Simulated BWR Environments
Quantifying the Crack Tip Oxidation Kinetics Parameters and Their Contribution to Stress Corrosion Cracking in High Temperature Water
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Padé Approximant - MATLAB & Simulink - MathWorks Italia
Padé Approximant
The Padé approximant of order [m, n] approximates the function f(x) around x = x0 as
\frac{{a}_{0}+{a}_{1}\left(xâ{x}_{0}\right)+...+{a}_{m}{\left(xâ{x}_{0}\right)}^{m}}{1+{b}_{1}\left(xâ{x}_{0}\right)+...+{b}_{n}{\left(xâ{x}_{0}\right)}^{n}}.
The Padé approximant is a rational function formed by a ratio of two power series. Because it is a rational function, it is more accurate than the Taylor series in approximating functions with poles. The Padé approximant is represented by the Symbolic Math Toolbox™ function pade.
When a pole or zero exists at the expansion point x = x0, the accuracy of the Padé approximant decreases. To increase accuracy, an alternative form of the Padé approximant can be used which is
\frac{{\left(xâ{x}_{0}\right)}^{p}\left({a}_{0}+{a}_{1}\left(xâ{x}_{0}\right)+...+{a}_{m}{\left(xâ{x}_{0}\right)}^{m}\right)}{1+{b}_{1}\left(xâ{x}_{0}\right)+...+{b}_{n}{\left(xâ{x}_{0}\right)}^{n}}.
The pade function returns the alternative form of the Padé approximant when you set the OrderMode input argument to Relative.
The Padé approximant is used in control system theory to model time delays in the response of the system. Time delays arise in systems such as chemical and transport processes where there is a delay between the input and the system response. When these inputs are modeled, they are called dead-time inputs. This example shows how to use the Symbolic Math Toolbox to model the response of a first-order system to dead-time inputs using Padé approximants.
The behavior of a first-order system is described by this differential equation
\mathrm{Ï}\frac{dy\left(t\right)}{dt}+y\left(t\right)=ax\left(t\right).
Enter the differential equation in MATLAB®.
syms tau a x(t) y(t) xS(s) yS(s) H(s) tmp
F = tau*diff(y)+y == a*x;
Find the Laplace transform of F using laplace.
F = laplace(F,t,s)
\mathrm{laplace}\left(y\left(t\right),t,s\right)-\mathrm{Ï}â\left(y\left(0\right)-sâ\mathrm{laplace}\left(y\left(t\right),t,s\right)\right)=aâ\mathrm{laplace}\left(x\left(t\right),t,s\right)
Assume the response of the system at t = 0 is 0. Use subs to substitute for y(0) = 0.
F = subs(F,y(0),0)
\mathrm{laplace}\left(y\left(t\right),t,s\right)+sâ\mathrm{Ï}â\mathrm{laplace}\left(y\left(t\right),t,s\right)=aâ\mathrm{laplace}\left(x\left(t\right),t,s\right)
To collect common terms, use simplify.
\left(sâ\mathrm{Ï}+1\right)â\mathrm{laplace}\left(y\left(t\right),t,s\right)=aâ\mathrm{laplace}\left(x\left(t\right),t,s\right)
For readability, replace the Laplace transforms of x(t) and y(t) with xS(s) and yS(s).
F = subs(F,[laplace(x(t),t,s) laplace(y(t),t,s)],[xS(s) yS(s)])
\mathrm{yS}\left(s\right)â\left(sâ\mathrm{Ï}+1\right)=aâ\mathrm{xS}\left(s\right)
The Laplace transform of the transfer function is yS(s)/xS(s). Divide both sides of the equation by xS(s) and use subs to replace yS(s)/xS(s) with H(s).
F = F/xS(s);
F = subs(F,yS(s)/xS(s),H(s))
H\left(s\right)â\left(sâ\mathrm{Ï}+1\right)=a
Solve the equation for H(s). Substitute for H(s) with a dummy variable, solve for the dummy variable using solve, and assign the solution back to H(s).
F = subs(F,H(s),tmp);
H(s) = solve(F,tmp)
\frac{a}{sâ\mathrm{Ï}+1}
The input to the first-order system is a time-delayed step input. To represent a step input, use heaviside. Delay the input by three time units. Find the Laplace transform using laplace.
step = heaviside(t - 3);
step = laplace(step)
\frac{{\mathrm{e}}^{-3âs}}{s}
Find the response of the system, which is the product of the transfer function and the input.
y = H(s)*step
\frac{aâ{\mathrm{e}}^{-3âs}}{sâ\left(sâ\mathrm{Ï}+1\right)}
To allow plotting of the response, set parameters a and tau to their values. For a and tau, choose values 1 and 3, respectively.
y = subs(y,[a tau],[1 3]);
y = ilaplace(y,s);
Find the Padé approximant of order [2 2] of the step input using the Order input argument to pade.
stepPade22 = pade(step,'Order',[2 2])
stepPade22 =
\frac{3â{s}^{2}-4âs+2}{2âsâ\left(s+1\right)}
Find the response to the input by multiplying the transfer function and the Padé approximant of the input.
yPade22 = H(s)*stepPade22
yPade22 =
\frac{aâ\left(3â{s}^{2}-4âs+2\right)}{2âsâ\left(sâ\mathrm{Ï}+1\right)â\left(s+1\right)}
Find the inverse Laplace transform of yPade22 using ilaplace.
yPade22 = ilaplace(yPade22,s)
a+\frac{9âaâ{\mathrm{e}}^{-s}}{2â\mathrm{Ï}-2}-\frac{aâ{\mathrm{e}}^{-\frac{s}{\mathrm{Ï}}}â\left(2â{\mathrm{Ï}}^{2}+4â\mathrm{Ï}+3\right)}{\mathrm{Ï}â\left(2â\mathrm{Ï}-2\right)}
To plot the response, set parameters a and tau to their values of 1 and 3, respectively.
yPade22 = subs(yPade22,[a tau],[1 3])
\frac{9â{\mathrm{e}}^{-s}}{4}-\frac{11â{\mathrm{e}}^{-\frac{s}{3}}}{4}+1
Plot the response of the system y and the response calculated from the Padé approximant yPade22.
fplot([y yPade22],[0 20])
title('Pade Approximant for dead-time step input')
legend('Response to dead-time step input',...
'Pade approximant [2 2]',...
The [2 2] Padé approximant does not represent the response well because a pole exists at the expansion point of 0. To increase the accuracy of pade when there is a pole or zero at the expansion point, set the OrderMode input argument to Relative and repeat the steps. For details, see pade.
stepPade22Rel = pade(step,'Order',[2 2],'OrderMode','Relative')
stepPade22Rel =
\frac{3â{s}^{2}-6âs+4}{sâ\left(3â{s}^{2}+6âs+4\right)}
yPade22Rel = H(s)*stepPade22Rel
yPade22Rel =
\frac{aâ\left(3â{s}^{2}-6âs+4\right)}{sâ\left(sâ\mathrm{Ï}+1\right)â\left(3â{s}^{2}+6âs+4\right)}
yPade22Rel = ilaplace(yPade22Rel)
\begin{array}{l}a-\frac{aâ{\mathrm{e}}^{-\frac{t}{\mathrm{Ï}}}â\left(4â{\mathrm{Ï}}^{2}+6â\mathrm{Ï}+3\right)}{{\mathrm{Ï}}_{1}}+\frac{12âaâ\mathrm{Ï}â{\mathrm{e}}^{-t}â\left(\mathrm{cos}\left(\frac{\sqrt{3}ât}{3}\right)-\sqrt{3}â\mathrm{sin}\left(\frac{\sqrt{3}ât}{3}\right)â\left(\frac{36âa-72âaâ\mathrm{Ï}}{36âaâ\mathrm{Ï}}+1\right)\right)}{{\mathrm{Ï}}_{1}}\\ \\ \mathrm{where}\\ \\ \mathrm{ }{\mathrm{Ï}}_{1}=4â{\mathrm{Ï}}^{2}-6â\mathrm{Ï}+3\end{array}
yPade22Rel = subs(yPade22Rel,[a tau],[1 3])
\frac{12â{\mathrm{e}}^{-t}â\left(\mathrm{cos}\left(\frac{\sqrt{3}ât}{3}\right)+\frac{2â\sqrt{3}â\mathrm{sin}\left(\frac{\sqrt{3}ât}{3}\right)}{3}\right)}{7}-\frac{19â{\mathrm{e}}^{-\frac{t}{3}}}{7}+1
fplot(yPade22Rel,[0 20],'DisplayName','Relative Pade approximant [2 2]')
The accuracy of the Padé approximant can also be increased by increasing its order. Increase the order to [4 5] and repeat the steps. The [n-1 n] Padé approximant is better at approximating the response at t = 0 than the [n n] Padé approximant.
\frac{27â{s}^{4}-180â{s}^{3}+540â{s}^{2}-840âs+560}{sâ\left(27â{s}^{4}+180â{s}^{3}+540â{s}^{2}+840âs+560\right)}
\frac{aâ\left(27â{s}^{4}-180â{s}^{3}+540â{s}^{2}-840âs+560\right)}{sâ\left(sâ\mathrm{Ï}+1\right)â\left(27â{s}^{4}+180â{s}^{3}+540â{s}^{2}+840âs+560\right)}
\frac{27â{s}^{4}-180â{s}^{3}+540â{s}^{2}-840âs+560}{sâ\left(3âs+1\right)â\left(27â{s}^{4}+180â{s}^{3}+540â{s}^{2}+840âs+560\right)}
yPade45 = ilaplace(yPade45)
\begin{array}{l}\frac{101520â\left({â}_{k=1}^{4}\frac{{\mathrm{e}}^{{\mathrm{Ï}}_{2}ât}â{\mathrm{Ï}}_{2}}{12â\left(90â{\mathrm{Ï}}_{2}+45â{{\mathrm{Ï}}_{2}}^{2}+9â{{\mathrm{Ï}}_{2}}^{3}+70\right)}\right)}{143}-\frac{2721â{\mathrm{e}}^{-\frac{t}{3}}}{1001}+\frac{172560â\left({â}_{k=1}^{4}\frac{{\mathrm{e}}^{tâ{\mathrm{Ï}}_{2}}}{{\mathrm{Ï}}_{1}}\right)}{143}+\frac{294120â\left({â}_{k=1}^{4}\frac{{\mathrm{e}}^{tâ{\mathrm{Ï}}_{2}}â{{\mathrm{Ï}}_{2}}^{2}}{{\mathrm{Ï}}_{1}}\right)}{1001}+\frac{46440â\left({â}_{k=1}^{4}\frac{{\mathrm{e}}^{tâ{\mathrm{Ï}}_{2}}â{{\mathrm{Ï}}_{2}}^{3}}{{\mathrm{Ï}}_{1}}\right)}{1001}+1\\ \\ \mathrm{where}\\ \\ \mathrm{ }{\mathrm{Ï}}_{1}=12â\left(9â{{\mathrm{Ï}}_{2}}^{3}+45â{{\mathrm{Ï}}_{2}}^{2}+90â{\mathrm{Ï}}_{2}+70\right)\\ \\ \mathrm{ }{\mathrm{Ï}}_{2}=\mathrm{root}\left({z}^{4}+\frac{20â{z}^{3}}{3}+20â{z}^{2}+\frac{280âz}{9}+\frac{560}{27},z,k\right)\end{array}
yPade45 = vpa(yPade45)
3.2418384981662546679005910164486â{\mathrm{e}}^{-1.930807068546914778929595950184ât}â\mathrm{cos}\left(0.57815608595633583454598214328008ât\right)-2.7182817182817182817182817182817â{\mathrm{e}}^{-0.33333333333333333333333333333333ât}-1.5235567798845363861823092981669â{\mathrm{e}}^{-1.4025262647864185544037373831494ât}â\mathrm{cos}\left(1.7716120279045018112388813990878ât\right)+11.595342871672681856604670597166â{\mathrm{e}}^{-1.930807068546914778929595950184ât}â\mathrm{sin}\left(0.57815608595633583454598214328008ât\right)-1.7803798379230333426855987436911â{\mathrm{e}}^{-1.4025262647864185544037373831494ât}â\mathrm{sin}\left(1.7716120279045018112388813990878ât\right)+1.0
fplot(yPade45,[0 20],'DisplayName','Pade approximant [4 5]')
The following points have been shown:
Padé approximants can model dead-time step inputs.
The accuracy of the Padé approximant increases with the increase in the order of the approximant.
When a pole or zero exists at the expansion point, the Padé approximant is inaccurate about the expansion point. To increase the accuracy of the approximant, set the OrderMode option to Relative. You can also use increase the order of the denominator relative to the numerator.
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GoG | Toph
Alice and Bob are playing GoG. GoG is a two-player board game. It consists of a grid with
n rows and
m columns. Players take alternate moves. In each move, a player can move in four directions(Up, Down, Left, and Right). If a player goes out of the grid or goes to a cell that has already been visited by any player (Alice or Bob), then the player dies. The last person alive wins.
Alice starts at the top-left cell and Bob starts at the bottom-right cell. Can you determine the winner, considering both players play optimally and Alice goes first?
T (1 \leq T \leq 100)
T(1≤T≤100), the number of test cases.
T lines each will contain the two integers
n and
m (1 \leq n, m \leq 10^5; 2 \leq n \times m \leq 10^5)
m(1≤n,m≤105;2≤n×m≤105), the dimensions of the grid for that testcase.
For each testcase, print the name of the winner in a new line.
Siddik_53rdEarliest, 6M ago
habijabiLightest, 0 B
We claim that if there are even number of cells then Bob wins, and if there are odd number of cells ...
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IsGCLTGroup - Maple Help
Home : Support : Online Help : Mathematics : Group Theory : IsGCLTGroup
attempt to determine whether a group is Lagrangian
determine whether a group is a GCLT group
IsLagrangian( G )
IsGCLTGroup( G )
G
is Lagrangian (or, a CLT-group) if it satisfies the converse of Lagrange's Theorem in the sense that it has a subgroup of order equal to every divisor of its order.
Every finite nilpotent group is Lagrangian, and a finite group is supersoluble if, and only if, each of its subgroups is Lagrangian. (Finite nilpotent groups have a much stronger property: a finite group is nilpotent if, and only if, it has a normal subgroup of order
d
, for each divisor
d
of its order.)
The class of Lagrangian groups is neither subgroup- nor quotient-closed.
The IsLagrangian( G ) command attempts to determine whether the group G is Lagrangian. It returns true if G is Lagrangian and returns false otherwise.
A GCLT-group is a finite group
G
such that, for each subgroup
H
G
p
H
G
L
G
H
, for which the index [L:H] is equal to
p
. GCLT-groups are most commonly referred to as
\mathrm{𝒥}
-groups in the literature.
Every GCLT-group is Lagrangian, but not conversely.
The IsGCLTGroup( G ) command attempts to determine whether the group G is a GCLT-group. It returns true if G is a GCLT-group, and returns the value false otherwise.
\mathrm{with}\left(\mathrm{GroupTheory}\right):
The following examples illustrate that the class of Lagrangian groups is not subgroup-closed.
\mathrm{IsLagrangian}\left(\mathrm{Symm}\left(4\right)\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\mathrm{IsLagrangian}\left(\mathrm{Alt}\left(4\right)\right)
\textcolor[rgb]{0,0,1}{\mathrm{false}}
\mathrm{IsGCLTGroup}\left(\mathrm{Symm}\left(4\right)\right)
\textcolor[rgb]{0,0,1}{\mathrm{false}}
\mathrm{IsGCLTGroup}\left(\mathrm{DihedralGroup}\left(6\right)\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
The smallest Lagrangian group that is not a GCLT-group is the direct product of a cyclic group of order
3
and the symmetric group of degree
3
G≔\mathrm{PermutationGroup}\left(\mathrm{DirectProduct}\left(\mathrm{CyclicGroup}\left(3\right),\mathrm{Symm}\left(3\right)\right)\right)
\textcolor[rgb]{0,0,1}{G}\textcolor[rgb]{0,0,1}{≔}〈\left(\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\right)\textcolor[rgb]{0,0,1}{,}\left(\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{5}\right)\textcolor[rgb]{0,0,1}{,}\left(\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{6}\right)〉
\mathrm{IsLagrangian}\left(G\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\mathrm{IsGCLTGroup}\left(G\right)
\textcolor[rgb]{0,0,1}{\mathrm{false}}
The GroupTheory[IsLagrangian] and GroupTheory[IsGCLTGroup] commands were introduced in Maple 2019.
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Ideal N-channel MOSFET for switching applications - MATLAB - MathWorks Italia
MOSFET (Ideal, Switching)
On-state voltage, Vds(Tj,Ids)
Drain-source current vector, Ids
Switch-on loss, Eon(Tj,Ids)
Switch-off loss, Eoff(Tj,Ids)
Drain-source current vector for switching losses, Ids
Ideal N-channel MOSFET for switching applications
The MOSFET (Ideal, Switching) block models the ideal switching behavior of an n-channel metal-oxide-semiconductor field-effect transistor (MOSFET).
The switching characteristic of an n-channel MOSFET is such that if the gate-source voltage exceeds the specified threshold voltage, the MOSFET is in the on state. Otherwise, the device is in the off state. This figure shows a typical i-v characteristic:
To define the I-V characteristic of the MOSFET, set the On-state behaviour and switching losses parameter to either Specify constant values or Tabulate with temperature and current. The Tabulate with temperature and current option is available only if you expose the thermal port of the block.
In the on state, the drain-source path behaves like a linear resistor with resistance, Rds_on. However, if you expose the thermal port of the block and parameterize the device using tabulated I-V data, the tabulated resistance is a function of the temperature and current.
In the off state, the drain-source path behaves like a linear resistor with low off-state conductance, Goff.
if G > Vth v == i*Rds_on; else v == i/Goff; end
G is the gate-source voltage.
Vth is the threshold voltage.
v is the drain-source voltage.
i is the drain-source current.
Rds_on is the on-state resistance.
Using the Integral Diode settings, you can include the body diode or an integral protection diode. The integral diode provides a conduction path for reverse current. For example, to provide a path for a high reverse-voltage spike that is generated when a semiconductor device suddenly switches off the voltage supply to an inductive load.
Set the Integral protection diode parameter based on your goal.
The figure shows an idealized representation of the output voltage, Vout, and the output current, Iout, of the semiconductor device. The interval shown includes the entire nth switching cycle, during which the block turns off and then on.
Switching losses are one of the main sources of thermal loss in semiconductors. During each on-off switching transition, the MOSFET parasitics store and then dissipate energy.
Switching losses depend on the off-state voltage and the on-state current. When a switching device is turned on, the power losses depend on the initial off-state voltage across the device and the final on-state current once the device is fully in its on state. Similarly, when a switching device is turned off, the power losses depend on the initial on-state current through the device and the final off-state voltage across the device when in the fully off state.
In this block, switching losses are applied by stepping up the junction temperature with a value equal to the switching loss divided by the total thermal mass at the junction. The Switch-on loss, Eon(Tj,Ids) and Switch-on loss, Eoff(Tj,Ids) parameter values set the sizes of the switching losses and they are either fixed or dependent on junction temperature and drain-source current. In both cases, losses are scaled by the off-state voltage prior to the latest device turn-on event.
As the final current after a switching event is not known during the simulation, the block records the on-state current at the point that the device is commanded off. Similarly, the block records the off-state voltage at the point that the device is commanded on. For this reason, the simlog does not report the switching losses to the thermal network until one switching cycle later.
For all ideal switching devices, the switching losses are reported in the simlog as lastTurnOffLoss and lastTurnOnLoss and recorded as a pulse with amplitude equal to the energy loss. If you use a script to sum the total losses over a defined simulation period, you must sum the pulse values at each pulse rising edge. Alternatively, you can use the ee_getPowerLossSummary and ee_getPowerLossTimeSeries functions to extract conduction and switching losses from logged data.
To enable the Variables settings for this block, set the Modeling option parameter to PS control port | Thermal port or Electrical control port | Thermal port.
Electrical conserving port associated with the source terminal.
Electrical conserving port associated with the drain terminal.
Drain-source on resistance, R_DS(on) Threshold voltage, Vth
Threshold voltage, Vth Drain-source on resistance, R_DS(on) On-state voltage, Vds(Tj,Ids)
Off-state conductance Temperature vector, Tj
Specify constant values — Use scalar values to specify the output current, switch-on loss, and switch-off loss data. This is the default parameterization method.
On-state voltage, Vds(Tj,Ids) — On-state voltage
[0, 1.1, 1.3, 1.45, 1.75, 2.25, 2.7; 0, 1, 1.15, 1.35, 1.7, 2.35, 3] V (default)
Drain-source current vector, Ids — Drain-source current vector
[ 0 10 50 100 200 400 600 ] A (default)
Drain-source currents for which the on-state voltage is defined. The first element must be zero. Specify this parameter using a vector quantity.
0.02286 J (default)
Switch-off loss — Switch-off loss
Energy dissipated during a single switch-off event. This parameter is defined as a function of temperature and final on-state output current. Specify this parameter using a scalar quantity.
Switch-on loss, Eon(Tj,Ids) — Switch-on loss
[ 0 2.9e-4 0.00143 0.00286 0.00571 0.01314 0.02286; 0 5.7e-4 0.00263 0.00514 0.01029 0.02057 0.03029 ] J (default)
Switch-off loss, Eoff(Tj,Ids) — Switch-off loss
[0, .21, 1.07, 2.14, 4.29, 9.86, 17.14; 0, .43, 1.97, 3.86, 7.71, 15.43, 22.71] * 1e-3 J (default)
Energy dissipated during a single switch-off event. This parameter is defined as a function of temperature and final on-state output current. Specify this parameter using a vector quantity.
Drain-source current vector for switching losses, Ids — Drain-source current vector for switching losses
Drain-source currents for which the switch-on loss and switch-off-loss are defined. The first element must be zero. Specify this parameter using a vector quantity.
Integral protection diode — Protection diode
Protection diode with no dynamics (default) | None | Protection diode with charge dynamics
Block integral protection diode. The default value is Protection diode with no dynamics.
-\frac{{i}^{2}{}_{RM}}{2a},
From R2021a forward, the Energy dissipation time constant parameter of the MOSFET (Ideal, Switching) block is no longer used. A step in junction temperature now reflects the switching losses. If your model contains a thermal mass directly connected to this block thermal port, remove it and model the thermal mass inside the component itself.
From R2020b forward, the MOSFET (Ideal, Switching) block has improved losses and thermal modelling options.
If you selected Voltage, current, and temperature for Thermal loss dependent on, then the thermal on-state losses are unchanged and the On-state voltage, Vds(Tj,Ids) parameter sets their values. However, the electrical on-state losses are now equal to the thermal on-state losses. Prior to R2020b, the electrical on-state losses were defined by the value of the on-state resistance.
The On-state voltage parameter is no longer used.
Diode | GTO | Ideal Semiconductor Switch | IGBT (Ideal, Switching) | N-Channel MOSFET | P-Channel MOSFET | Thyristor (Piecewise Linear)
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GetProcessID - Maple Help
Home : Support : Online Help : Connectivity : Web Features : Network Communication : Sockets Package : GetProcessID
retrieve the system process ID of the calling process
The procedure GetProcessID returns the system process identifier for the calling process. This value is a small positive integer that uniquely identifies the process on the host computer system.
\mathbf{use}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{Sockets}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{in}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{GetProcessID}\left(\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{end use}
\textcolor[rgb]{0,0,1}{43602}
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Conformal Array - MATLAB & Simulink - MathWorks Deutschland
Support for Arrays with Custom Geometry
Create Default Conformal Array
Uniform Circular Array Created from Conformal Array
Custom Antenna Array
The phased.ConformalArray object lets you model a phased array with arbitrary geometry. For example, you can use phased.ConformalArray to design:
A planar array with a nonrectangular geometry, such as a circular array
An array with nonuniform geometry, such as a linear array with variable spacing
A nonplanar array
When you use phased.ConformalArray, you must specify these aspects of the array:
Sensor element of the array
Direction normal to each array element
To create a conformal array with default properties, use this command:
array = phased.ConformalArray
phased.ConformalArray with properties:
ElementNormal: [2x1 double]
This default conformal array consists of a single phased.IsotropicAntennaElement antenna located at the origin of the local coordinate system. The direction normal to the sensor element is 0° azimuth and 0° elevation.
This example shows how to construct a 60-element uniform circular array. In constructing a uniform circular array, you can use either the phased.UCA or the phased.ConformalArray System objects. The conformal array approach is more general because it allows you to point the array elements in arbitrary directions. A UCA restricts the array element directions to lie in the plane of the array. This example illustrates how you can use the phased.ConformalArray System object™ to create any other array shape. Assume an operating frequency of 400 MHz. Tune the array by specifying the arclength between the elements to be 0.5
\lambda
\lambda
is the wavelength corresponding to the operating frequency. Array elements lie in the x-y-plane. Element normal directions are set to
\left({\varphi }_{n},0\right)
{\varphi }_{n}
is the azimuth angle of the
{n}^{th}
array element.
Set the number of elements and the operating frequency of the array.
Compute the element spacing in radians.
thetarad = deg2rad(theta);
Choose the radius so that the inter-element arclength is one-half wavelength.
arclength = 0.5*(physconst('LightSpeed')/fc);
radius = arclength/thetarad;
Compute the element azimuth angles. Azimuth angles must lie in the range
\left(-18{0}^{\circ },18{0}^{\circ }\right)
ang = (0:N-1)*theta;
ang(ang >= 180.0) = ang(ang >= 180.0) - 360.0;
array = phased.ConformalArray;
array.ElementPosition = [radius*cosd(ang);...
radius*sind(ang);...
zeros(1,N)];
array.ElementNormal = [ang;zeros(1,N)];
Show the UCA array geometry.
Plot the array response pattern at 1 GHz.
pattern(array,1e9,[-180:180],0,'PropagationSpeed',physconst('LightSpeed'),...
This example shows how to construct and visualize a custom-geometry array containing antenna elements with a custom radiation pattern. The radiation pattern of each element is constant over each azimuth angle and has a cosine pattern for the elevation angles.
Define the custom antenna element and plot its radiation pattern.
'ElevationAngles',el,...
'MagnitudePattern',repmat(elresp',1,numel(az)));
pattern(antenna,3e8,0,el,'CoordinateSystem','polar','Type','powerdb',...
Define the locations and normal directions of the elements. All elements lie in the z-plane. The elements are located at (1;0;0) , (0;1;0), and (0;-1;0) meters. The element normal azimuth angles are 0°, 120°, and -120°, respectively. All normal elevation angles are 0°.
xpos = [1 0 0];
ypos = [0 1 -1];
zpos = [0 0 0];
normal_az = [0 120 -120];
normal_el = [0 0 0];
Define a conformal array with those elements.
array = phased.ConformalArray('Element',antenna,...
'ElementPosition',[xpos; ypos; zpos],...
'ElementNormal',[normal_az; normal_el]);
Plot the positions and normal directions of the elements.
viewArray(array,'ShowNormals',true)
pattern(array,fc,az,el,'CoordinateSystem','polar','Type','powerdb',...
'Normalize',true,'PropagationSpeed',physconst('LightSpeed'))
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Monopole Floer homology for rational homology 3-spheres
1 December 2010 Monopole Floer homology for rational homology 3-spheres
Kim A. Frøyshov
Kim A. Frøyshov1
1Institut for Matematiske Fag, Aarhus Universitet
We give a new construction of monopole Floer homology for
{\text{spin}}^{c}
rational homology
3
-spheres. As applications, we define two invariants of certain
4
-manifolds with
{b}_{1}=1
{b}^{+}=0
Kim A. Frøyshov. "Monopole Floer homology for rational homology 3-spheres." Duke Math. J. 155 (3) 519 - 576, 1 December 2010. https://doi.org/10.1215/00127094-2010-060
Kim A. Frøyshov "Monopole Floer homology for rational homology 3-spheres," Duke Mathematical Journal, Duke Math. J. 155(3), 519-576, (1 December 2010)
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NCERT Solutions for Class 8 Math Chapter 8 - Comparing Quantities
-
NCERT Solution for Class 8 math - comparing quantities 119 , Question 4
\mathrm{A} = \mathrm{Rs} \left[10000{\left(1 + \frac{10}{100}\right)}^{1}\right]\phantom{\rule{0ex}{0ex}} = \mathrm{Rs} \left[10000\left(\frac{11}{10}\right)\right]\phantom{\rule{0ex}{0ex}} = \mathrm{Rs} 11000
\frac{1}{2}
\mathrm{S}.\mathrm{I}.=\mathrm{Rs}\left(\frac{11000×10×\frac{1}{2}}{100}\right)=\mathrm{Rs} 550
\therefore
\mathrm{Rs} 11000- \mathrm{Rs} 10000=\mathrm{Rs} 1000
\therefore
=\mathrm{Rs} 1000+ \mathrm{Rs} 550=\mathrm{Rs} 1550
NCERT Solution for Class 8 math - comparing quantities 134 , Question 11
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Invariant mass - Wikipedia
"Proper mass" redirects here. For the liturgical mass proper, see Proper (liturgy).
Find sources: "Invariant mass" – news · newspapers · books · scholar · JSTOR (March 2011) (Learn how and when to remove this template message)
This article may need to be rewritten to comply with Wikipedia's quality standards, as Outdated, incorrect, see talk. You can help. The talk page may contain suggestions. (February 2016)
1 Sum of rest masses
2 As defined in particle physics
3 Example: two-particle collision
3.2 Collider experiments
4 Rest energy
Sum of rest massesEdit
As defined in particle physicsEdit
{\displaystyle m_{0}^{2}c^{2}=\left({\frac {E}{c}}\right)^{2}-\left\|\mathbf {p} \right\|^{2}}
{\displaystyle m_{0}^{2}=E^{2}-\left\|\mathbf {p} \right\|^{2}.}
{\displaystyle \left(Wc^{2}\right)^{2}=\left(\sum E\right)^{2}-\left\|\sum \mathbf {p} c\right\|^{2},}
{\displaystyle W}
is the invariant mass of the system of particles, equal to the mass of the decay particle.
{\textstyle \sum E}
is the sum of the energies of the particles
{\textstyle \sum \mathbf {p} }
is the vector sum of the momentum of the particles (includes both magnitude and direction of the momenta)
{\displaystyle W^{2}=\left(\sum E_{\text{in}}-\sum E_{\text{out}}\right)^{2}-\left\|\sum \mathbf {p} _{\text{in}}-\sum \mathbf {p} _{\text{out}}\right\|^{2}.}
Example: two-particle collisionEdit
{\displaystyle {\begin{aligned}M^{2}&=(E_{1}+E_{2})^{2}-\left\|{\textbf {p}}_{1}+{\textbf {p}}_{2}\right\|^{2}\\&=m_{1}^{2}+m_{2}^{2}+2\left(E_{1}E_{2}-{\textbf {p}}_{1}\cdot {\textbf {p}}_{2}\right).\end{aligned}}}
Massless particlesEdit
The invariant mass of a system made of two massless particles whose momenta form an angle
{\displaystyle \theta }
has a convenient expression:
{\displaystyle {\begin{aligned}M^{2}&=(E_{1}+E_{2})^{2}-\left\|{\textbf {p}}_{1}+{\textbf {p}}_{2}\right\|^{2}\\&=[(p_{1},0,0,p_{1})+(p_{2},0,p_{2}\sin \theta ,p_{2}\cos \theta )]^{2}\\&=(p_{1}+p_{2})^{2}-p_{2}^{2}\sin ^{2}\theta -(p_{1}+p_{2}\cos \theta )^{2}\\&=2p_{1}p_{2}(1-\cos \theta ).\end{aligned}}}
Collider experimentsEdit
In particle collider experiments, one often defines the angular position of a particle in terms of an azimuthal angle
{\displaystyle \phi }
{\displaystyle \eta }
. Additionally the transverse momentum,
{\displaystyle p_{T}}
, is usually measured. In this case if the particles are massless, or highly relativistic (
{\displaystyle E\gg m}
) then the invariant mass becomes:
{\displaystyle M^{2}=2p_{T1}p_{T2}(\cosh(\eta _{1}-\eta _{2})-\cos(\phi _{1}-\phi _{2})).}
Rest energyEdit
The rest energy
{\displaystyle E_{0}}
of a particle is defined as:
{\displaystyle E_{0}=m_{0}c^{2},}
{\displaystyle c}
is the speed of light in vacuum.[2] In general, only differences in energy have physical significance.[3]
Landau, L.D., Lifshitz, E.M. (1975). The Classical Theory of Fields: 4-th revised English Edition: Course of Theoretical Physics Vol. 2. Butterworth Heinemann. ISBN 0-7506-2768-9. {{cite book}}: CS1 maint: multiple names: authors list (link)
Retrieved from "https://en.wikipedia.org/w/index.php?title=Invariant_mass&oldid=1089306676"
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Proper cooling of the airfoil trailing edge is imperative in gas turbine designs since this area is often one of the life limiting areas of an airfoil. A common method of providing thermal protection to an airfoil trailing edge is by injecting a film of cooling air through slots located on the airfoil pressure side near the trailing edge, thereby providing a cooling buffer between the hot mainstream gas and the airfoil surface. In the conventional designs, at the breakout plane, a series of slots open to expanding tapered grooves in between the tapered lands and run the cooling air through the grooves to protect the trailing edge surface. In this study, naphthalene sublimation technique was used to measure area averaged mass/heat transfer coefficients downstream of the breakout plane on the slot and on the land surfaces. Three slot geometries were tested: (a) a baseline case simulating a typical conventional slot and land design; (b) the same geometry with a sudden outward step at the breakout plane around the opening; and (c) the sudden step was moved one-third away from the breakout plane in the slot. Mass/heat transfer results were compared for these slots geometries for a range of blowing ratios
[M=(ρu)s∕(ρu)m]
from 0 to 2. For the numerical investigation, a pressure-correction based, multiblock, multigrid, unstructured/adaptive commercial software was used in this investigation. Several turbulence models including the standard high Reynolds number
k-ε
turbulence model in conjunction with the generalized wall function were used for turbulence closure. The applied thermal boundary conditions to the computational fluid dynamics (CFD) models matched the test boundary conditions. Effects of a sudden downward step (Coanda) in the slot on mass/heat transfer coefficients on the slot and on the land surfaces were compared both experimentally and numerically.
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Home : Support : Online Help : Mathematics : Discrete Mathematics : Ordinals : Gcd
greatest common left divisor of ordinals
Gcd(a, b, ...)
The Gcd(a, b, ...) calling sequence computes the unique greatest common left divisor of the given ordinal numbers. It returns either an ordinal data structure, a nonnegative integer, or a polynomial with positive integer coefficients.
If some of the arguments are parametric ordinals and the greatest common left divisor cannot be determined, an error is raised.
\mathrm{with}\left(\mathrm{Ordinals}\right)
[\textcolor[rgb]{0,0,1}{\mathrm{`+`}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{`.`}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{`<`}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{<=}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Add}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Base}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Dec}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Decompose}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Div}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Eval}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Factor}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Gcd}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Lcm}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{LessThan}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Log}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Max}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Min}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Mult}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Ordinal}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Power}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Split}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Sub}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{`^`}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{degree}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{lcoeff}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{log}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{lterm}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{\omega }}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{quo}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{rem}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{tcoeff}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{tdegree}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{tterm}}]
a≔\mathrm{Ordinal}\left([[\mathrm{\omega },1],[1,2],[0,1]]\right)
\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{≔}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathbf{\omega }}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}
b≔\mathrm{Ordinal}\left([[3,1],[1,1],[0,1]]\right)
\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{≔}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathbf{\omega }}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}
c≔\mathrm{Ordinal}\left([[2,1],[1,3],[0,1]]\right)
\textcolor[rgb]{0,0,1}{c}\textcolor[rgb]{0,0,1}{≔}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathbf{\omega }}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}
\mathrm{Gcd}\left(a,b,c\right)
\textcolor[rgb]{0,0,1}{\mathbf{\omega }}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}
\mathrm{Div}\left(a,\right)
{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0}
\mathrm{Div}\left(b,\right)
{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0}
\mathrm{Div}\left(c,\right)
\textcolor[rgb]{0,0,1}{\mathbf{\omega }}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0}
Any of the arguments can be a positive integer.
\mathrm{Gcd}\left(12,20,30\right)
\textcolor[rgb]{0,0,1}{2}
\mathrm{Gcd}\left(18,12·b,30·c\right)
\textcolor[rgb]{0,0,1}{6}
\mathrm{Gcd}\left(3,\mathrm{\omega }\right)
\textcolor[rgb]{0,0,1}{3}
\mathrm{Gcd}\left(3,\mathrm{\omega },\mathrm{\omega }+1\right)
\textcolor[rgb]{0,0,1}{1}
d≔\mathrm{Ordinal}\left([[2,x],[1,3],[0,1]]\right)
\textcolor[rgb]{0,0,1}{d}\textcolor[rgb]{0,0,1}{≔}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathbf{\omega }}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}
\mathrm{Gcd}\left(a,b,d\right)
\textcolor[rgb]{0,0,1}{\mathbf{\omega }}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}
e≔\mathrm{Ordinal}\left([[2,1],[1,1],[0,1]]\right)
\textcolor[rgb]{0,0,1}{e}\textcolor[rgb]{0,0,1}{≔}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathbf{\omega }}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}
\mathrm{Gcd}\left(d,e\right)
\textcolor[rgb]{0,0,1}{\mathbf{\omega }}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}
\mathrm{Div}\left(d,\right)
\textcolor[rgb]{0,0,1}{\mathbf{\omega }}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0}
\mathrm{Div}\left(e,\right)
\textcolor[rgb]{0,0,1}{\mathbf{\omega }}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0}
f≔\mathrm{Ordinal}\left([[3,1],[1,3],[0,1]]\right)
\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{≔}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathbf{\omega }}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}
\mathrm{Gcd}\left(d,f\right)
Error, (in Ordinals:-Gcd) cannot determine if x is nonzero
\mathrm{Gcd}\left(\mathrm{Eval}\left(d,x=x+1\right),f\right)
\textcolor[rgb]{0,0,1}{\mathbf{\omega }}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}
g≔\mathrm{Ordinal}\left([[4,1],[2,x+1]]\right)
\textcolor[rgb]{0,0,1}{g}\textcolor[rgb]{0,0,1}{≔}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{\cdot }\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\right)
h≔\mathrm{Ordinal}\left([[3,2],[1,y+1],[0,z]]\right)
\textcolor[rgb]{0,0,1}{h}\textcolor[rgb]{0,0,1}{≔}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathbf{\omega }}\textcolor[rgb]{0,0,1}{\cdot }\left(\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\right)\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{z}
\mathrm{Gcd}\left(g,h\right)
\textcolor[rgb]{0,0,1}{\mathbf{\omega }}\textcolor[rgb]{0,0,1}{\cdot }\left(\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\right)\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{z}
\mathrm{Div}\left(g,\right)
{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathbf{\omega }}\textcolor[rgb]{0,0,1}{\cdot }\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0}
\mathrm{Div}\left(h,\right)
{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0}
\mathrm{Gcd}\left(4,h,\mathrm{\omega }+6\right)
\textcolor[rgb]{0,0,1}{\mathrm{igcd}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{z}\right)
The Ordinals[Gcd] command was introduced in Maple 2015.
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Momentum - Citizendium
In classical mechanics, the momentum of a point particle is the mass m of the particle times its velocity v. Conventionally, momentum is indicated by the symbol p, so that
{\displaystyle \mathbf {p} \equiv m\mathbf {v} .}
Both p and v are vectors. To distinguish p from angular momentum, it is often called linear momentum.
Just as velocity, momentum is expressed with respect to a reference frame. In most applications of classical mechanics this frame is fixed to the earth (a "laboratory frame"). Einstein's theory of special relativity treats the choice of frames that are in uniform motion (inertial frames) with respect to each other.
Dimension of momentum: N⋅s (newton times second, from dp/dt = F).
2 Momentum of an N-particle system
3 Application of conservation of momentum
4 Generalized momentum
6 Electromagnetic momentum
Newton's second law states that the momentum of a particle changes in time when a force F acts on it,
{\displaystyle {\frac {d\mathbf {p} }{dt}}=m{\frac {d\mathbf {v} }{dt}}\equiv m\mathbf {a} =\mathbf {F} ,}
where the acceleration a of the particle is introduced and it is assumed—as is common in classical mechanics—that the mass is constant (independent of time). Clearly, if no force acts on the particle:
{\displaystyle {\frac {d\mathbf {p} }{dt}}=\mathbf {0} ,}
which states that the momentum of a free particle (i.e., particle on which no force is acting) is conserved.
Momentum of an N-particle system
The momentum of a system of N particles is the vector sum,
{\displaystyle \mathbf {P} =\sum _{i=1}^{N}\mathbf {p} _{i}.}
When the internal forces between the particles constituting the system satisfy Newton's third law (action = −reaction),
{\displaystyle \mathbf {F} _{ij}=-\mathbf {F} _{ji},\quad {\hbox{for}}\quad i,j=1,\ldots ,N,}
{\displaystyle {\frac {d\mathbf {P} }{dt}}=\sum _{i=1}^{N}\mathbf {F} _{i}^{\textrm {ext}},}
where we find on the right hand side the vector sum of external forces, Fexti, acting on the individual particles of the system. When the total external force is zero (either because all the individual external forces are zero, or because they sum vectorially to zero), then the total momentum of the system is conserved,
{\displaystyle {\frac {d\mathbf {P} }{dt}}=\mathbf {0} .}
Application of conservation of momentum
Think of a rocket ship floating still in outer space. Assume that no gravitational, or other, forces are acting on it. The total momentum of the ship plus filled fuel tank is zero. Then ignite the rocket engine and assume that its exhaust gases go one way (say downward). The exhaust gases have mass and obtain velocity by the combustion, so that they have momentum, Pgas, directed downward. Because the total momentum is conserved (is zero), the ship gets momentum, Pship, upward,
{\displaystyle \mathbf {P} _{\textrm {gas}}+\mathbf {P} _{\textrm {ship}}=\mathbf {0} \quad \Longrightarrow \quad -\mathbf {P} _{\textrm {gas}}=\mathbf {P} _{\textrm {ship}}=M_{\textrm {ship}}\mathbf {V} _{\textrm {ship}}.}
so that the ship will get a velocity Vship upward.
Another example: suppose you are sitting in a moving car without seat belt. Your body gets the momentum: speed of vehicle, say 50 m/h, times your body weight. Suppose the car hits something and comes to a sudden stop (a strong force is acting on the body of the car and the car obtains zero speed). On you, however, no force is acting and your momentum will be conserved. Since your body weight does not change during the collision, your body will continue going forward with the same speed, 50 m/h. As the car has now speed zero, your body will move through the interior of the car with 50 m/h.
In Lagrange mechanics, the Lagrangian L ≡ T − V plays a central role. Here T is the kinetic energy of the system and V its potential energy. The Lagrangian is defined in terms of generalized (non-Cartesian) coordinates q and generalized velocities (time derivatives of the generalized coordinates). Indicating the latter in Newton's fluxion notation, we have
{\displaystyle L(q_{1},q_{2},\dots ,q_{f};{\dot {q}}_{1},{\dot {q}}_{2},\dots ,{\dot {q}}_{f};t)\equiv T(q_{1},q_{2},\dots ,q_{f};{\dot {q}}_{1},{\dot {q}}_{2},\dots ,{\dot {q}}_{f};t)-V(q_{1},q_{2},\dots ,q_{f};{\dot {q}}_{1},{\dot {q}}_{2},\dots ,{\dot {q}}_{f};t),}
where f is the number of degrees of freedom of the system. The generalized momentum has f components defined by
{\displaystyle p_{i}\equiv {\frac {\partial L}{\partial {\dot {q}}_{i}}},\quad i=1,\dots ,f.}
The advantage of Lagrange mechanics is that it can be applied to systems with holonomic constraints (often requiring generalized coordinates), relativistic mechanics, and systems with an infinite number of degrees of freedom (fields). The Lagrange definition of momentum is a generalization in the sense that it coincides with the definition given above for Newtonian systems.
As an example consider a point particle in 3-dimensional space with
{\displaystyle T={\frac {1}{2}}mv^{2}\quad {\hbox{and}}\quad \mathbf {v} ={\dot {\mathbf {r} }}.}
Let V be a function of r = (x, y, z) only. We make the identification for this simple system (f = 3),
{\displaystyle x=q_{1},\,y=q_{2},\,z=q_{3},\,v_{x}={\dot {q}}_{1},\,v_{y}={\dot {q}}_{2},\,v_{z}={\dot {q}}_{3}\quad {\hbox{and}}\quad L=T(v_{x},v_{y},v_{x})-V(x,y,z).}
{\displaystyle p_{x}={\frac {\partial L}{\partial v_{x}}}={\frac {\partial T}{\partial v_{x}}}={\frac {1}{2}}m{\frac {\partial v^{2}}{\partial v_{x}}}=mv_{x}}
and likewise for py and pz, so that p = mv , also in the Lagrangian definition.
The relativistic Lagrangian of a material point of mass m moving with velocity v is:
{\displaystyle L=-mc^{2}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}},}
where c is the speed of light. The relativistic momentum is obtained by differentiating L with respect to the components of v,
{\displaystyle p_{\alpha }={\frac {\partial L}{\partial v_{\alpha }}}\quad \Longrightarrow \quad \mathbf {p} ={\frac {m\mathbf {v} }{\sqrt {1-{v^{2}}/{c^{2}}}}}.}
Note that if v << c then v2/c2 ≈ 0 and the relativistic momentum is (approximately) equal to the classical (non-relativistic) momentum introduced above.
Consider an electromagnetic field (E(r,t), B(r,t)). An electromagnetic momentum can be assigned to a volume V. In SI units this is defined by,
{\displaystyle \mathbf {P} _{\textrm {EM}}\equiv \epsilon _{0}\iiint _{V}\mathbf {E} (\mathbf {r} ,t)\times \mathbf {B} (\mathbf {r} ,t)\,{\textrm {d}}v,}
where ε0 is the electric constant and the cross indicates a cross product. In Gaussian units,
{\displaystyle \mathbf {P} _{\textrm {EM}}\equiv {\frac {1}{4\pi c}}\iiint _{V}\mathbf {E} (\mathbf {r} ,t)\times \mathbf {B} (\mathbf {r} ,t)\,{\textrm {d}}v,}
where c is the speed of light. Apart from a factor, the electromagnetic momentum is an integral over the Poynting vector.
It can be shown[1] that a finite volume containing electromagnetic density and electric charges satisfies a Newton type equation,
{\displaystyle {\frac {d}{dt}}(\mathbf {P} _{\textrm {EM}}+\mathbf {P} _{\textrm {mech}})=\mathbf {F} ,}
provided the above definition of electromagnetic momentum is used. The mechanical momentum of the charges, Pmech, has the definition mass times velocity given above. The force is a surface integral over Maxwell's stress tensor. The fact that the mechanical and the electromagnetic momentum appear here on equal footing justifies the name "momentum", although no mass appears in the definition PEM.
The dimension in SI units of PEM is,
[C2/(N⋅m2)] × (V/m)× T× m3 = (C2/N) × V × (V⋅s/m2) = (J/m)2 × (s/N) = N⋅s,
where C is coulomb, V is volt, N is newton, J is joule, and T is tesla. Just as the mechanical momentum, the electromagnetic momentum has the dimension N⋅s.
↑ J. D. Jackson, Classical Electrodynamics, Wiley, New York, 2nd ed. (1975). pp. 238–239
Retrieved from "https://citizendium.org/wiki/index.php?title=Momentum&oldid=538345"
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How to Convert Binary to Octal Number: 11 Steps (with Pictures)
How to Convert Binary to Octal Number
1 Converting by Hand
2 Converting Shortcuts and Variations
Binary and octal systems are different number systems commonly used in computing. They have different bases -- binary is base-two and octal base-eight -- meaning they must be grouped to convert. This, however, sounds far more complicated than this very easy conversion actually is.
Converting by Hand Download Article
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Recognize series of binary numbers. Binary numbers are simply strings of 1's and 0's, such as 101001, 001, or even just 1. If you see this kind of string it is usually binary. However, some books and teachers further denote binary numbers through a subscript "2", such as 10012, which prevents confusion with the number "one thousand and one."
This subscript denotes the "base" of the number. Binary is a base-two system, octal is base-eight.
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Group all the 1's and 0's in the binary number in sets of three, starting from the far right. There are two different binary numbers and only eight octal. Since
{\displaystyle 2^{3}=8,}
you'll need three binary numbers to designate each octal number. Start from the right to make your groups. For example, the binary number 101001 would break down to 101 001.
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Add zeros to the left of the last digit if you don't have enough digits to make a set of three. The binary number 10011011 has eight digits, which, though not a multiple of three, can still convert to octal. Just add extra zeros to your front group until it has three places. For example:
Original Binary: 10011011
Grouping: 10 011 011
Adding Zeros for Groups of Three: 010 011 011[1] X Research source
Add a 4, 2, and a 1 underneath each set of three numbers to note your placeholders. Each of the three binary numbers in a set stands for a place in the octal number system. The first number is for a 4, the second a 2, and the third a 1. To keep things straight, write these numbers underneath your sets of three binary numbers. For example:
Note, if you're looking for a shortcut, you can skip this step and just compare your sets of binary numbers to this octal conversion chart.
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If there is a one above any of your placeholders, write that number (4, 2, or 1) to start your octal numbers. If there is a one above the "4," then your octal number has a 4 in it. If there is a 0 above the one's place, the octal number does not have a one in it, so leave a blank, zero, or dash. As seen in an example:
Convert 1010100112 to octal.
Separate into threes:
Add placeholders:
Mark each places:
401 020 021[2] X Research source
Add up the new numbers in each set of three. Once you know what places are in the octal number, simply add up each set of three individually. So, for 101, which turns into 4, 0, and 1, you end up with 5 (
{\displaystyle 4+0+1=5}
). Continuing the example above:
Separate, add placeholders, and mark each place:
Add up each set of three:
{\displaystyle (4+0+1)(0+2+0)(0+2+1)=5,2,3}
Place your newly converted answers together to form your final octal number. Splitting up the binary number was just to make solving easier -- the original number was one lone string. So, now that you've converted, put everything back together to get your final answer. That's all it takes.
Separate, add placeholders, mark places, and add totals:
Put converted numbers back together:
Add a subscript 8 (like this8) to complete the conversion. There is technically no way to know if 523 refers to an octal number or a normal base-ten number without proper notation. To ensure that your teacher knows you've been doing the work well, place a subscript 8, referring to octal as a base-8 system, on your answer.
5238[3] X Research source
Converting Shortcuts and Variations Download Article
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Use a simple octal conversion chart to save time and work. This won't work on a test, but is a great choice in any other setting. Since there are only 8 possible combinations of numbers, it is actually a pretty easy chart to memorize. All you have to do is separate the numbers in groups of three, then match them with the chart in the pictures.[4] X Research source
Note how numbers 8 and 9 don't have straight conversions. In octal, these numbers do not exist, since there are only 8 digits (0-7) in a base-eight system.
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Keep the decimal where it is and work outward if you are dealing with decimals. Say you need to convert the binary number 10010.11 to an octal number. Normally, you work from right to left to group the numbers into sets of three. With the decimal, you work away from the point. So, for the numbers left of the decimal (10010), you start at the point and work left (010 010). For the numbers to the right (.11), you start from the point and work right (110). When adding zeros, always add them in the direction you're working. The final breakdown is 010 010 . 110.
101.1 → 101 . 100
1.01001 → 001 . 010 010
1001101.0101 → 001 001 101 . 010 100
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Use the octal conversion chart to convert from octal back to binary. You'll need the chart to work backward, as a simple "3" doesn't give you enough information to do the math unless you already know the octal system well and want to re-think each combination. Simply use the following chart to easily convert each octal digit into a set of three binary numbers, then ram them together:
7 → 111[5] X Research source
How can I convert 40.12 into an octal number?
See the wikiHow article on converting from decimal to octal for details.
How can I convert the 1111100001 binary number into an actual number?
Group the binary number into the group of three's. Add an extra zero to the left of the number to complete the group of three. Then follow the above procedure.
Could you write the rules of converting a binary number into octal?
First identify if it is a binary number. Always Group them into numbers of three. Add extra zero on extreme left to complete the group, don't add anywhere else.
How numbers are valued in octal number system?
As a base-eight system, each digit in an octal number has a higher value than each number in a binary system. This is because binary numbers start from base-two. Decimal and Hexadecimal systems, which are base-ten and base-sixteen respectively, have higher values per place holder.
How can I convert an octal number into binary?
There are several ways to convert octal to binary. One way is to change the octal to decimal and then change the decimal to binary. However, it doubles the work. The second way is more efficient: Start from most significant octal bit to the least significant bit or reverse and change into a three binary bit and do it until completion. For example: Octal number 125= Binary number 1010101 Explanation : 1=001 2=010 5=101 so the binary number is 001010101 = 1010101
How do I make the binary of 1001 into an octal number?
Group the binary digits as the sets of 3 digits from the left, add zeros at the left side of the binary digit remaining to have the three digit format, and arrive at or substitute the equivalent octal numbers. So, 1001 will become 001 001. Now find the octal equivalent of 11 with base 8, and that is the answer.
How do you convert binary to Hexadecimal?
Check out out wikiHow's similar article on converting to Hexadecimal, found at How to Convert Binary to Hexadecimal.
Can I put zero on the left hand side after the decimal point? For instance in the case of 11010001.10, can I put 0 after 10 on the right hand side?
You have to put zeroes only on the right-hand side after the decimal to complete grouping of numbers. In this case, 10 will become 100, which is 4 in octal base.
Why do I need to use 421 when converting binary to octal?
4, 2, and 1 are powers of 2. And binary is base 2. 2^0 = 1, 2^1 = 2, and 2^2 = 4.
How do I convert a binary number to an octal one?
I chose 10001. 1. Sort the bin_num in groups of 3.(10 001)(bin_num means binary number) 2. Add 0's if needed.(010 001)3. Add how many of each "place" has and join.(010 001, 0+2+0=2 0+0+1=1, 2 1, 21)4. Voila! You have an octal.
Take your time breaking numbers up. A big sheet of paper with lots of room is usually best.
↑ http://www.robotroom.com/NumberSystems4.html
↑ http://coolconversion.com/math/binary-octal-hexa-decimal/_binary__101010011__octal_
1. Group the digits in sets of 3.
2. Add zeros if you have less than 3 of any number.
3. Add a "421" under group of 3 as a placeholder.
4. If there's a 1 above any placeholders, write the placeholder number down.
5. Add the new numbers in each set of three.
6. Place the newly converted answers together to form the octal number.
7. Add subscript 8 to complete the conversion.
Italiano:Convertire un Numero Binario in Ottale
Español:convertir un número binario en octal
Русский:конвертировать двоичные числа в восьмеричные
Deutsch:Eine Binärzahl in eine Oktalzahl umwandeln
Français:convertir un nombre binaire en octal
Nederlands:Binair naar octaal omzetten
Bahasa Indonesia:Mengonversi Bilangan Biner ke Oktal
العربية:التحويل من النظام الثنائي إلى النظام الثماني
Tiếng Việt:Chuyển từ nhị phân sang bát phân
中文:将二进制数转换为八进制数
"I don't know how, but this article managed to take something complicated and break it down into simple solutions, easy to follow and easy to memorize. It also provided alternative methods of conversion, like the table."..." more
"I appreciate you for saying that we need to do divide the binary numbers to three. It helped me a lot since my computer science teacher didn't tell us to do that. Thank you!"..." more
"This is a very effective learning platform for various topics. These articles helped me learn Decimal, Binary, Hex and Octal where my university has a learning gap."..." more
Pushpendra Vasisth
"The steps of putting 421, then adding with the above, then adding the numbers together were helpful. It is pretty simple."..." more
"Step by Step procedure makes it easier to learn and understand, even examples were too helpful for fast learning."
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Geometry_of_numbers Knowpia
Geometry of numbers is the part of number theory which uses geometry for the study of algebraic numbers. Typically, a ring of algebraic integers is viewed as a lattice in
{\displaystyle \mathbb {R} ^{n},}
and the study of these lattices provides fundamental information on algebraic numbers.[1] The geometry of numbers was initiated by Hermann Minkowski (1910).
Best rational approximants for π (green circle), e (blue diamond), ϕ (pink oblong), (√3)/2 (grey hexagon), 1/√2 (red octagon) and 1/√3 (orange triangle) calculated from their continued fraction expansions, plotted as slopes y/x with errors from their true values (black dashes)
The geometry of numbers has a close relationship with other fields of mathematics, especially functional analysis and Diophantine approximation, the problem of finding rational numbers that approximate an irrational quantity.[2]
Minkowski's resultsEdit
{\displaystyle \Gamma }
is a lattice i{\displaystyle n}
{\displaystyle \mathbb {R} ^{n}}
{\displaystyle K}
is a convex centrally symmetric body. Minkowski's theorem, sometimes called Minkowski's first theorem, states that if
{\displaystyle \operatorname {vol} (K)>2^{n}\operatorname {vol} (\mathbb {R} ^{n}/\Gamma )}
{\displaystyle K}
contains a nonzero vector in
{\displaystyle \Gamma }
The successive minimum
{\displaystyle \lambda _{k}}
is defined to be the inf of the numbers
{\displaystyle \lambda }
{\displaystyle \lambda K}
{\displaystyle k}
linearly independent vectors of
{\displaystyle \Gamma }
. Minkowski's theorem on successive minima, sometimes called Minkowski's second theorem, is a strengthening of his first theorem and states that[3]
{\displaystyle \lambda _{1}\lambda _{2}\cdots \lambda _{n}\operatorname {vol} (K)\leq 2^{n}\operatorname {vol} (\mathbb {R} ^{n}/\Gamma ).}
Later research in the geometry of numbersEdit
In 1930-1960 research on the geometry of numbers was conducted by many number theorists (including Louis Mordell, Harold Davenport and Carl Ludwig Siegel). In recent years, Lenstra, Brion, and Barvinok have developed combinatorial theories that enumerate the lattice points in some convex bodies.[4]
Subspace theorem of W. M. SchmidtEdit
In the geometry of numbers, the subspace theorem was obtained by Wolfgang M. Schmidt in 1972.[5] It states that if n is a positive integer, and L1,...,Ln are linearly independent linear forms in n variables with algebraic coefficients and if ε>0 is any given real number, then the non-zero integer points x in n coordinates with
{\displaystyle |L_{1}(x)\cdots L_{n}(x)|<|x|^{-\varepsilon }}
lie in a finite number of proper subspaces of Qn.
Influence on functional analysisEdit
Minkowski's geometry of numbers had a profound influence on functional analysis. Minkowski proved that symmetric convex bodies induce norms in finite-dimensional vector spaces. Minkowski's theorem was generalized to topological vector spaces by Kolmogorov, whose theorem states that the symmetric convex sets that are closed and bounded generate the topology of a Banach space.[6]
Researchers continue to study generalizations to star-shaped sets and other non-convex sets.[7]
^ MSC classification, 2010, available at http://www.ams.org/msc/msc2010.html, Classification 11HXX.
^ Schmidt's books. Grötschel et alii, Lovász et alii, Lovász.
^ Cassels (1971) p. 203
^ Grötschel et alii, Lovász et alii, Lovász, and Beck and Robins.
^ Schmidt, Wolfgang M. Norm form equations. Ann. Math. (2) 96 (1972), pp. 526-551. See also Schmidt's books; compare Bombieri and Vaaler and also Bombieri and Gubler.
^ For Kolmogorov's normability theorem, see Walter Rudin's Functional Analysis. For more results, see Schneider, and Thompson and see Kalton et alii.
^ Kalton et alii. Gardner
Matthias Beck, Sinai Robins. Computing the continuous discretely: Integer-point enumeration in polyhedra, Undergraduate Texts in Mathematics, Springer, 2007.
Enrico Bombieri; Vaaler, J. (Feb 1983). "On Siegel's lemma". Inventiones Mathematicae. 73 (1): 11–32. Bibcode:1983InMat..73...11B. doi:10.1007/BF01393823. S2CID 121274024.
Enrico Bombieri & Walter Gubler (2006). Heights in Diophantine Geometry. Cambridge U. P.
J. W. S. Cassels. An Introduction to the Geometry of Numbers. Springer Classics in Mathematics, Springer-Verlag 1997 (reprint of 1959 and 1971 Springer-Verlag editions).
John Horton Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, NY, 3rd ed., 1998.
M. Grötschel, Lovász, L., A. Schrijver: Geometric Algorithms and Combinatorial Optimization, Springer, 1988
Hancock, Harris (1939). Development of the Minkowski Geometry of Numbers. Macmillan. (Republished in 1964 by Dover.)
Edmund Hlawka, Johannes Schoißengeier, Rudolf Taschner. Geometric and Analytic Number Theory. Universitext. Springer-Verlag, 1991.
Kalton, Nigel J.; Peck, N. Tenney; Roberts, James W. (1984), An F-space sampler, London Mathematical Society Lecture Note Series, 89, Cambridge: Cambridge University Press, pp. xii+240, ISBN 0-521-27585-7, MR 0808777
C. G. Lekkerkererker. Geometry of Numbers. Wolters-Noordhoff, North Holland, Wiley. 1969.
Lenstra, A. K.; Lenstra, H. W. Jr.; Lovász, L. (1982). "Factoring polynomials with rational coefficients" (PDF). Mathematische Annalen. 261 (4): 515–534. doi:10.1007/BF01457454. hdl:1887/3810. MR 0682664. S2CID 5701340.
Lovász, L.: An Algorithmic Theory of Numbers, Graphs, and Convexity, CBMS-NSF Regional Conference Series in Applied Mathematics 50, SIAM, Philadelphia, Pennsylvania, 1986
Malyshev, A.V. (2001) [1994], "Geometry of numbers", Encyclopedia of Mathematics, EMS Press
Minkowski, Hermann (1910), Geometrie der Zahlen, Leipzig and Berlin: R. G. Teubner, JFM 41.0239.03, MR 0249269, retrieved 2016-02-28
Schmidt, Wolfgang M. (1996). Diophantine approximations and Diophantine equations. Lecture Notes in Mathematics. Vol. 1467 (2nd ed.). Springer-Verlag. ISBN 3-540-54058-X. Zbl 0754.11020.
Siegel, Carl Ludwig (1989). Lectures on the Geometry of Numbers. Springer-Verlag.
Rolf Schneider, Convex bodies: the Brunn-Minkowski theory, Cambridge University Press, Cambridge, 1993.
Anthony C. Thompson, Minkowski geometry, Cambridge University Press, Cambridge, 1996.
Hermann Weyl. Theory of reduction for arithmetical equivalence . Trans. Amer. Math. Soc. 48 (1940) 126–164. doi:10.1090/S0002-9947-1940-0002345-2
Hermann Weyl. Theory of reduction for arithmetical equivalence. II . Trans. Amer. Math. Soc. 51 (1942) 203–231. doi:10.2307/1989946
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Home : Support : Online Help : System : HelpTools : Database : Remove
remove help database from the list of used databases
Remove( path )
string or a list of strings; path to an existing help database or directory with databases
The Remove command removes help database denoted by path from the list of currently used databases. If path is a directory, then all databases are removed.
\mathrm{with}\left(\mathrm{HelpTools}\right):
\mathrm{with}\left(\mathrm{Database}\right):
\mathrm{hdb}≔\mathrm{FileTools}:-\mathrm{JoinPath}\left(["maple","toolbox","UserHelp","lib","maple.help"],\mathrm{base}='\mathrm{homedir}'\right)
\textcolor[rgb]{0,0,1}{\mathrm{hdb}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{"C:\Users\jsmith\maple\toolbox\UserHelp\lib\maple.help"}
Remove user's help database from the list of active databases
\mathrm{Remove}\left(\mathrm{hdb}\right)
[\textcolor[rgb]{0,0,1}{"C:\Users\jsmith\maple\toolbox\UserHelp\lib\maple.help"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"C:\Program Files\Maple 2021\lib\maple.help"}]
The HelpTools[Database][Remove] command was introduced in Maple 18.
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Rs Aggarwal 2018 for Class 8 Math Chapter 19 - Three Dimensional Figures
Rs Aggarwal 2018 Solutions for Class 8 Math Chapter 19 Three Dimensional Figures are provided here with simple step-by-step explanations. These solutions for Three Dimensional Figures are extremely popular among Class 8 students for Math Three Dimensional Figures Solutions come handy for quickly completing your homework and preparing for exams. All questions and answers from the Rs Aggarwal 2018 Book of Class 8 Math Chapter 19 are provided here for you for free. You will also love the ad-free experience on Meritnation’s Rs Aggarwal 2018 Solutions. All Rs Aggarwal 2018 Solutions for class Class 8 Math are prepared by experts and are 100% accurate.
Write down the number of faces of each of the following figures:
(ii) Cube
(iv) Square pyramid
(v) Tetrahedron
(i) A cuboid has 6 faces, namely ABCD, EFGH, HDAE, GCBF, HDCG and EABF.
(ii) A cube has 6 faces, namely IJKL, MNOP, PLIM, OKJN, LKOP and IJNM.
(iii) A triangular prism has 5 faces (3 rectangular faces and 2 triangular faces), namely QRUT, QTVS, RUVS, QRS and TUV.
(iv) A square pyramid has 5 faces (4 triangular faces and 1 square face), namely OWZ, OWX, OXY, OYZ and WXYZ.
(v) A tetrahedron has 4 triangular faces, namely KLM, KLN, LMN and KMN.
Write down the number of edges of each of the following figures:
(ii) Rectangular pyramid
(i) A tetrahedron has 6 edges, namely KL, LM, LN, MN, KN and KM.
(ii) A rectangular pyramid has 8 edges, namely AB, AE, AD, AC, EB, ED, DC and CB.
(iii) A cube has 12 edges, namely PL, LK, KO, OP, MN, NJ, JI, IM, PM, LI, ON and KJ.
(iv) A triangular prism has 9 edges, namely QR, RS, QS, TU, TV, UV, QT, RU, and SV.
Write down the number of vertices of each of the following figures:
(iii) Tetrahedron
(i) A cuboid has 8 vertices, namely A, B, C, D, E, F, G and H.
(ii) A square pyramid has 5 vertices, namely O, W, X, Y and Z.
(iii)A tertrahedron has 4 vertices, namely K, L, M and N.
(iv) A triangular prism has 6 vertices, namely Q, R, S, T, U and V.
(i) A cube has ....... vertices, ....... edges and ....... faces.
(ii) The point at which three faces of a figure meet is known as its .......
(iii) A cuboid is also known as a rectangular .......
(iv) A triangular pyramid is called a .......
(i) A cube has 8 vertices, 12 edges and 6 faces.
Vertices: I, J, K, L, M, N, O and P
Edges : IJ, JN, NM, MI, PL, LK, KO, OP, PM, LI, KJ, and ON
Faces : MNJI, POKL, PLIM, OKJN, PONM and LKJI
(ii) The point at which the three faces of a figure meet is known as its vertex.
(iii) A cuboid is also known as a rectangular cube.
(iv) A triangular pyramid is called a tetrahedraon.
Define Euler's relation between the number of faces, number of edges and number of vertices for various 3-dimensional figures.
The Euler's relation for a three dimensional figure can be expressed in the following manner:
F-E+V=2\phantom{\rule{0ex}{0ex}}H\mathrm{ere},\phantom{\rule{0ex}{0ex}}F- \mathrm{Number} \mathrm{of} \mathrm{faces}\phantom{\rule{0ex}{0ex}}E- \mathrm{Number} \mathrm{of} \mathrm{edges}\phantom{\rule{0ex}{0ex}}V- \mathrm{Number} \mathrm{of} \mathrm{vertices}
How many edges are there in a
(iv) square pyramid?
(i) A cuboid has 12 edges, namely AD, DC, CB, BA, EA, FB, HD, DC, CG, GH, HE, and GF.
(ii) A tetrahedron has 6 edges, namely KL, LM, MN, NL , KM and KN.
(iii) A triangular prism has 9 edges, namely QR, RS, SQ, TU, UV, VT, RU, SV and QT.
(iv) A square pyramid has 8 edges, namely OW, OX, OY , OZ , WX, XY, YZ and ZW.
How many faces are there in a
(i) cube
(iv) pentagonal pyramid?
(i) A cube has 6 faces, namely IJKL, MNOP, PLIM , OKJN, POKL and MNJI.
(ii) A pentagonal prism has 7 faces, i.e. 2 pentagons and 5 rectangles, namely ABCDE, FGHIJ, ABGF, AEJF , EDIJ, DCHI and CBGH.
(iii) A tetrahedron has 4 faces, namely KLM, KLN, LMN and KMN.
(iv) A pentagonal pyramid has 6 faces, i.e. 1 pentagon and 5 triangles, namely NOPQM, SNM, SOP, SNO, SMQ and SQP.
How many vertices are there in a
(iii) pentagonal prism
(ii) A tetrahedron has 4 vertices, namely K, L, M and N.
(iii) A pentagonal prism has 10 vertices, namely A, B, C, D, E, F, G, H, I and J.
(iv) A square pyramid has 5 vertices, namely O, W, X, Y and Z.
Verify Euler's relation for each of the following:
Euler's relation is:
F-E+V=2\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}Here:\phantom{\rule{0ex}{0ex}}F- \mathrm{Number} \mathrm{of} \mathrm{faces}\phantom{\rule{0ex}{0ex}}E- \mathrm{Number} \mathrm{of} \mathrm{edges}\phantom{\rule{0ex}{0ex}}V- \mathrm{Number} \mathrm{of} \mathrm{vertices}
\mathrm{Number} \mathrm{of} \mathrm{faces} = F=2 \mathrm{squares}+4 \mathrm{rectangular}=6\phantom{\rule{0ex}{0ex}}\mathrm{Number} \mathrm{of} \mathrm{edges} = E = 12\phantom{\rule{0ex}{0ex}}\mathrm{Number} \mathrm{of} \mathrm{vertices} =V= 8\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}⇒\left(F-E+V\right)=6-12+8=2
\mathrm{Number} \mathrm{of} \mathrm{faces} = F=4\phantom{\rule{0ex}{0ex}}\mathrm{Number} \mathrm{of} \mathrm{edges} = E = 6\phantom{\rule{0ex}{0ex}}\mathrm{Number} \mathrm{of} \mathrm{vertices} =V= 4\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}⇒\left(F-E+V\right)=4-6+4=2
\mathrm{Number} \mathrm{of} \mathrm{faces} = F=2 \mathrm{triangular} + 3 \mathrm{rectangular} = 5\phantom{\rule{0ex}{0ex}}\mathrm{Number} \mathrm{of} \mathrm{edges} = E = 9 \phantom{\rule{0ex}{0ex}}\mathrm{Number} \mathrm{of} \mathrm{vertices} =V =6\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}⇒\left(F-E+V\right)=5-9+6=2
\mathrm{Number} \mathrm{of} \mathrm{faces} = F=2 \mathrm{triangular} + 3 \mathrm{rectangular} = 5\phantom{\rule{0ex}{0ex}}\mathrm{Number} \mathrm{of} \mathrm{edges} = E = 8\phantom{\rule{0ex}{0ex}}\mathrm{Number} \mathrm{of} \mathrm{vertices} =V =5\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}⇒\left(F-E+V\right)=5-8+5=2
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Solve Partial Differential Equation of Nonlinear Heat Transfer - MATLAB & Simulink Example - MathWorks Italia
Define PDE Parameters
Extract PDE Coefficients
Specify PDE Model, Geometry, and Coefficients
Find Transient Solution
This example shows how to solve a partial differential equation (PDE) of nonlinear heat transfer in a thin plate.
The plate is square, and its temperature is fixed along the bottom edge. No heat is transferred from the other three edges since the edges are insulated. Heat is transferred from both the top and bottom faces of the plate by convection and radiation. Because radiation is included, the problem is nonlinear.
The purpose of this example is to show how to represent the nonlinear PDE symbolically using Symbolic Math Toolbox™ and solve the PDE problem using finite element analysis in Partial Differential Equation Toolbox™. In this example, perform transient analysis and solve the temperature in the plate as a function of time. The transient analysis shows the time duration until the plate reaches its equilibrium temperature at steady state.
The plate has planar dimensions 1 m by 1 m and is 1 cm thick. Because the plate is relatively thin compared with the planar dimensions, the temperature can be assumed to be constant in the thickness direction, and the resulting problem is 2-D.
Assume that convection and radiation heat transfers take place between the two faces of the plate and the environment with a specified ambient temperature.
{Q}_{c}={h}_{c}\left(T-{T}_{a}\right)
{T}_{a}
is the ambient temperature,
T
is the temperature at a particular
x
y
location on the plate surface, and
{h}_{c}
is a specified convection coefficient.
{Q}_{r}=ϵ\sigma \left({T}^{4}-{T}_{a}^{4}\right)
ϵ
is the emissivity of the face and
\sigma
is the Stefan-Boltzmann constant. Because the heat transferred due to radiation is proportional to the fourth power of the surface temperature, the problem is nonlinear.
\rho {C}_{p}{t}_{z}\frac{\partial T}{\partial t}-k{t}_{z}{\nabla }^{2}T+2{Q}_{c}+2{Q}_{r}=0
\rho
is the material density of the plate,
{C}_{p}
is its specific heat,
{t}_{z}
is its plate thickness,
k
is its thermal conductivity, and the factors of two account for the heat transfer from both of its faces.
Set up the PDE problem by defining the values of the parameters. The plate is composed of copper, which has the following properties.
kThermal = 400; % thermal conductivity of copper, W/(m-K)
rhoCopper = 8960; % density of copper, kg/m^3
specificHeat = 386; % specific heat of copper, J/(kg-K)
thick = 0.01; % plate thickness in meters
stefanBoltz = 5.670373e-8; % Stefan-Boltzmann constant, W/(m^2-K^4)
hCoeff = 1; % convection coefficient, W/(m^2-K)
tAmbient = 300; % the ambient temperature
emiss = 0.5; % emissivity of the plate surface
Define the PDE in symbolic form with the plate temperature as a dependent variable
T\left(t,x,y\right)
syms T(t,x,y)
syms eps sig tz hc Ta rho Cp k
Qc = hc*(T - Ta);
Qr = eps*sig*(T^4 - Ta^4);
pdeeq = (rho*Cp*tz*diff(T,t) - k*tz*laplacian(T,[x,y]) + 2*Qc + 2*Qr)
pdeeq(t, x, y) =
2 \mathrm{eps} \mathrm{sig} \left({T\left(t,x,y\right)}^{4}-{\mathrm{Ta}}^{4}\right)-k \mathrm{tz} \left(\frac{{\partial }^{2}}{\partial {x}^{2}}\mathrm{ }T\left(t,x,y\right)+\frac{{\partial }^{2}}{\partial {y}^{2}}\mathrm{ }T\left(t,x,y\right)\right)-2 \mathrm{hc} \left(\mathrm{Ta}-T\left(t,x,y\right)\right)+\mathrm{Cp} \rho \mathrm{tz} \frac{\partial }{\partial t}\mathrm{ }T\left(t,x,y\right)
Now, create coefficients to use as inputs in the PDE model as required by Partial Differential Equation Toolbox. To do this, first extract the coefficients of the symbolic PDE as a structure of symbolic expressions using the pdeCoefficients function.
symCoeffs = pdeCoefficients(pdeeq,T,'Symbolic',true)
symCoeffs = struct with fields:
a: 2*hc + 2*eps*sig*T(t, x, y)^3
c: [2x2 sym]
f: 2*eps*sig*Ta^4 + 2*hc*Ta
d: Cp*rho*tz
Next, substitute the symbolic variables that represent the PDE parameters with their numeric values.
symVars = [eps sig tz hc Ta rho Cp k];
symVals = [emiss stefanBoltz thick hCoeff tAmbient rhoCopper specificHeat kThermal];
symCoeffs = subs(symCoeffs,symVars,symVals);
Finally, since the fields symCoeffs are symbolic objects, use the pdeCoefficientsToDouble function to convert the coefficients to the double data type, which makes them valid inputs for Partial Differential Equation Toolbox.
coeffs = pdeCoefficientsToDouble(symCoeffs)
coeffs = struct with fields:
a: @makeCoefficient/coefficientFunction
d: 3.4586e+04
f: 1.0593e+03
Now, using Partial Differential Equation Toolbox, solve the PDE problem using finite element analysis based on these coefficients.
First, create the PDE model with a single dependent variable.
Specify the geometry for the PDE model—in this case, the dimension of the square.
Define a geometry description matrix. Create the square geometry using the decsg (Partial Differential Equation Toolbox) function. For a rectangular geometry, the first row contains 3, and the second row contains 4. The next four rows contain the
x
-coordinates of the starting points of the edges, and the four rows after that contain the
y
-coordinates of the starting points of the edges.
Convert the DECSG geometry into a geometry object and include it in the PDE model.
title('Geometry with Edge Labels Displayed');
Next, create the triangular mesh in the PDE model with a mesh size of approximately 0.1 in each direction.
hmax = 0.1; % element size
title('Plate with Triangular Element Mesh');
xlabel('X-coordinate, meters');
ylabel('Y-coordinate, meters');
Specify the coefficients in the PDE model.
specifyCoefficients(model,'m',coeffs.m,'d',coeffs.d, ...
'c',coeffs.c,'a',coeffs.a,'f',coeffs.f);
Apply the boundary conditions. Three of the plate edges are insulated. Because a Neumann boundary condition equal to zero is the default in the finite element formulation, you do not need to set the boundary conditions on these edges. The bottom edge of the plate is fixed at 1000 K. Specify this using a Dirichlet condition on all nodes on the bottom edge (edge E1).
Specify the initial temperature to be 0 K everywhere, except at the bottom edge. Set the initial temperature on the bottom edge E1 to the value of the constant boundary condition, 1000 K.
Define the time domain to find the transient solution of the PDE problem.
endTime = 10000;
Set the tolerance of the solver options.
Solve the problem using solvepde. Plot the temperature along the top edge of the plate as a function of time.
plot(tlist,u(3,:));
title 'Temperature Along the Top Edge of the Plate as a Function of Time'
ylabel 'Temperature (K)'
Based on the plot, the transient solution starts to reach its steady state value after 6000 seconds. The equilibrium temperature of the top edge approaches 450 K after 6000 seconds.
Show the temperature profile of the plate after 10,000 seconds.
title(sprintf('Temperature in the Plate, Transient Solution (%d seconds)\n', ...
tlist(1,end)));
Show the temperature at the top edge at 10,000 seconds.
u(3,end)
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ComplexRootClassification - Maple Help
Home : Support : Online Help : Mathematics : Factorization and Solving Equations : RegularChains : ParametricSystemTools Subpackage : ComplexRootClassification
compute a classification of the complex roots of a polynomial system depending on parameters
ComplexRootClassification(F, d, R)
ComplexRootClassification(F, H, d, R)
ComplexRootClassification(CS, d, R)
The integer d must be positive and smaller than the number of variables.
The characteristic of R must be zero and the last d variables of R are regarded as parameters.
For a parametric algebraic system, this command computes all the possible numbers of solutions of this system together with the corresponding necessary and sufficient conditions on its parameters.
More precisely, let V be the variety defined by F. The command ComplexRootClassification(F, d, R) returns a classification of the complex roots of F depending on parameters, that is, a finite partition P of the parameter space into constructible sets such that above each part, the number of solutions of V is either infinite or constant.
If a constructible set CS is specified, the representing regular systems of CS must be square-free. The function call ComplexRootClassification(CS, d, R) returns a classification of the points of the constructible set CS, that is, a finite partition P of the parameter space into constructible sets such that above each part, the number of solutions of CS is either infinite or constant.
W
be the variety defined by the product of polynomials in H. The command ComplexRootClassification(F, H, d, R) returns a classification of the points of the constructible set V-W depending on parameters.
\mathrm{with}\left(\mathrm{RegularChains}\right):
\mathrm{with}\left(\mathrm{ConstructibleSetTools}\right):
\mathrm{with}\left(\mathrm{ParametricSystemTools}\right):
R≔\mathrm{PolynomialRing}\left([x,y,s]\right)
\textcolor[rgb]{0,0,1}{R}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{polynomial_ring}}
F≔[s-\left(y+1\right)x,s-\left(x+1\right)y]
\textcolor[rgb]{0,0,1}{F}\textcolor[rgb]{0,0,1}{≔}[\textcolor[rgb]{0,0,1}{s}\textcolor[rgb]{0,0,1}{-}\left(\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{s}\textcolor[rgb]{0,0,1}{-}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}]
The computation below shows that the input parametric system can have 1 solution or 2 distinct solutions. The corresponding conditions on the parameters are given by constructible sets.
\mathrm{CC}≔\mathrm{ComplexRootClassification}\left(F,1,R\right)
\textcolor[rgb]{0,0,1}{\mathrm{CC}}\textcolor[rgb]{0,0,1}{≔}[[\textcolor[rgb]{0,0,1}{\mathrm{constructible_set}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{\mathrm{constructible_set}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}]]
These constructible sets are printed below.
\mathrm{map}\left(x↦[\mathrm{Info}\left(x[1],R\right),x[2]],\mathrm{CC}\right)
[[[[\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{s}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{1}]]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}]\textcolor[rgb]{0,0,1}{,}[[[]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{s}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{s}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}]]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{s}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{1}]]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}]]
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Plot contours - MATLAB fcontour - MathWorks España
Plot Contours of Symbolic Expression
Plot Contours of Symbolic Function
Change Line Style, Color and Width
Plot Multiple Contour Plots on Same Figure
fcontour(f,[min max])
fcontour(f,[xmin xmax ymin ymax])
fcontour(f) plots the contour lines of symbolic expression f(x,y) over the default interval of x and y, which is [-5 5].
fcontour(f,[min max]) plots f over the interval min < x < max and min < y < max.
fcontour(f,[xmin xmax ymin ymax]) plots f over the interval xmin < x < xmax and ymin < y < ymax. The fcontour function uses symvar to order the variables and assign intervals.
fcontour(___,LineSpec) uses LineSpec to set the line style and color. fcontour doesn’t support markers.
fcontour(___,Name,Value) specifies line properties using one or more Name,Value pair arguments. Use this option with any of the input argument combinations in the previous syntaxes. Name,Value pair settings apply to all the lines plotted. To set options for individual plots, use the objects returned by fcontour.
fcontour(ax,___) plots into the axes object ax instead of the current axes object gca.
fc = fcontour(___) returns a function contour object. Use the object to query and modify properties of a specific contour plot. For details, see FunctionContour Properties.
\mathrm{sin}\left(x\right)+\mathrm{cos}\left(y\right)
over the default range of
-5<x<5
-5<y<5
. Show the colorbar. Find a contour's level by matching the contour's color with the colorbar value.
f\left(x,y\right)=\mathrm{sin}\left(x\right)+\mathrm{cos}\left(y\right)
-5<x<5
-5<y<5
f(x,y) = sin(x) + cos(y);
\mathrm{sin}\left(x\right)+\mathrm{cos}\left(y\right)
-\pi /2<x<\pi /2
0<y<5
by specifying the plotting interval as the second argument of fcontour.
fcontour(f,[-pi/2 pi/2 0 5])
{x}^{2}-{y}^{2}
as blue, dashed lines by specifying the LineSpec input. Specify a LineWidth of 2. Markers are not supported by fcontour.
fcontour(x^2 - y^2,'--b','LineWidth',2)
Plot multiple contour plots either by passing the inputs as a vector or by using hold on to successively plot on the same figure. If you specify LineStyle and Name-Value arguments, they apply to all contour plots. You cannot specify individual LineStyle and Name-Value pair arguments for each plot.
Divide a figure into two subplots by using subplot. On the first subplot, plot
\mathrm{sin}\left(x\right)+\mathrm{cos}\left(y\right)
x-y
by using vector input. On the second subplot, plot the same expressions by using hold on.
fcontour([sin(x)+cos(y) x-y])
title('Multiple Contour Plots Using Vector Inputs')
fcontour(sin(x)+cos(y))
fcontour(x-y)
title('Multiple Contour Plots Using Hold Command')
{e}^{-\left(x/3{\right)}^{2}-\left(y/3{\right)}^{2}}+{e}^{-\left(x+2{\right)}^{2}-\left(y+2{\right)}^{2}}
. Specify an output to make fcontour return the plot object.
f = exp(-(x/3)^2-(y/3)^2) + exp(-(x+2)^2-(y+2)^2);
Function: exp(- x^2/9 - y^2/9) + exp(- (x + 2)^2 - (y + 2)^2)
Change the LineWidth to 1 and the LineStyle to a dashed line by using dot notation to set properties of the object fc. Visualize contours close to 0 and 1 by setting LevelList to [1 0.9 0.8 0.2 0.1].
Fill the area between contours by setting the Fill input of fcontour to 'on'. If you want interpolated shading instead, use the fsurf function with its option 'EdgeColor' set to 'none' followed by the command view(0,90).
Create a plot that looks like a sunset by filling the contours of
erf\left(\left(y+2{\right)}^{3}\right)-{e}^{\left(-0.65\left(\left(x-2{\right)}^{2}+\left(y-2{\right)}^{2}\right)}.
f = erf((y+2)^3) - exp(-0.65*((x-2)^2+(y-2)^2));
Control the resolution of contour lines by using the 'MeshDensity' option. Increasing 'MeshDensity' can make smoother, more accurate plots while decreasing it can increase plotting speed.
Divide a figure into two using subplot. In the first subplot, plot the contours of
\mathrm{sin}\left(x\right)\mathrm{sin}\left(y\right)
. The corners of the squares do not meet. To fix this issue, increase 'MeshDensity' to 200 in the second subplot. The corners now meet, showing that by increasing 'MeshDensity' you increase the plot's resolution.
fcontour(sin(x).*sin(y))
fcontour(sin(x).*sin(y),'MeshDensity',200)
title('Increased MeshDensity = 200')
x\mathrm{sin}\left(y\right)-y\mathrm{cos}\left(x\right)
. Add a title and axis labels. Create the x-axis ticks by spanning the x-axis limits at intervals of pi/2. Display these ticks by using the XTick property. Create x-axis labels by using arrayfun to apply texlabel to S. Display these labels by using the XTickLabel property. Repeat these steps for the y-axis.
fcontour(x*sin(y)-y*cos(x), [-2*pi 2*pi])
title('xsin(y)-ycos(x) for -2\pi < x < 2\pi and -2\pi < y < 2\pi')
ax.XTickLabel = arrayfun(@texlabel, S, 'UniformOutput', false);
Create animations by changing the displayed expression using the Function property of the function handle, and then using drawnow to update the plot. To export to GIF, see imwrite.
By varying the variable i from –π/8 to π/8, animate the parametric curve isin(x) + icos(y).
fc = fcontour(-pi/8.*sin(x)-pi/8.*cos(y));
for i=-pi/8:0.01:pi/8
fc.Function = i.*sin(x)+i.*cos(y);
f — Expression or function to be plotted
[min max] — Plotting range for x and y
Plotting range for x and y, specified as a vector of two numbers. The default range is [-5 5].
[xmin xmax ymin ymax] — Plotting range for x and y
Plotting range for x and y, specified as a vector of four numbers. The default range is [-5 5 -5 5].
Axes object. If you do not specify an axes object, then the plot function uses the current axes.
The properties listed here are only a subset. For a complete list, see FunctionContour Properties.
fc — One or more function contour objects
One or more function contour objects, returned as a scalar or a vector. These objects are unique identifiers, which you can use to query and modify the properties of a specific contour plot. For details, see FunctionContour Properties.
fcontour assigns the symbolic variables in f to the x axis, then the y axis, and symvar determines the order of the variables to be assigned. Therefore, variable and axis names might not correspond. To force fcontour to assign x or y to its corresponding axis, create the symbolic function to plot, then pass the symbolic function to fcontour.
For example, the following code plots the contour of the surface f(x,y) = sin(y) in two ways. The first way forces the waves to oscillate with respect to the y axis. The second way assigns y to the x axis because it is the first (and only) variable in the symbolic function.
fcontour(f);
fcontour(f(x,y)); % Or fcontour(sin(y));
fimplicit | fimplicit3 | fmesh | fplot | fplot3 | fsurf
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Nk2001 - DispersiveWiki
Revision as of 18:29, 10 June 2009 by Jkasala (talk | contribs)
M. Nakao.
{\displaystyle L^{p}}
estimates for the wave equation and global existence for semilinear wave equations in exterior domains. Math. Annalen, 320 (2001), 11-31. MathSciNet, arXiv.
Retrieved from "https://dispersivewiki.org/DispersiveWiki/index.php?title=Nk2001&oldid=6603"
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Iteration Space - Maple Help
Home : Support : Online Help : Programming : CodeTools : Program Analysis : Iteration Space
relations on the index variables of a ForLoop
IterationSpace(loop)
The ForLoop to be analyzed
This command computes the iteration space of the ForLoop loop, which is the set of values for the indices into the arrays over all iterations of loop. This set is represented as the integer solutions to a list of non-strict inequalities. The relations are written in terms of the loop's index variables and parameters.
The returned system of inequalities has not been simplified and may represent an empty set for trivial loops.
\mathrm{with}\left(\mathrm{CodeTools}[\mathrm{ProgramAnalysis}]\right):
\left(i,j\right)
used by the given nested loop, as encoded by a list of inequalities:
p1 := proc(a, b, n)
for i from 1 to min(n + 2, 10) do
for j from i to n + 2 do
a[i, j] := b[i + 1] + a[i - 1, j - 1]
loop1 := CreateLoop(p1):
iteration_space1 := IterationSpace(loop1);
\textcolor[rgb]{0,0,1}{\mathrm{iteration_space1}}\textcolor[rgb]{0,0,1}{≔}[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{\le }\textcolor[rgb]{0,0,1}{i}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{i}\textcolor[rgb]{0,0,1}{\le }\textcolor[rgb]{0,0,1}{10}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{i}\textcolor[rgb]{0,0,1}{\le }\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{i}\textcolor[rgb]{0,0,1}{\le }\textcolor[rgb]{0,0,1}{j}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{j}\textcolor[rgb]{0,0,1}{\le }\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}]
Specifying a value for the parameter
n
, the integer solutions to the given list of inequalities can be computed:
\mathrm{iteration_space1_2}≔\mathrm{subs}\left(n=2,\mathrm{iteration_space1}\right):
\mathrm{isolve}\left(\mathrm{convert}\left(\mathrm{iteration_space1_2},\mathrm{set}\right)\right)
{\textcolor[rgb]{0,0,1}{i}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{j}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{1}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{i}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{j}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{i}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{j}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{i}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{j}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{i}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{j}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{i}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{j}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{i}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{j}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{i}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{j}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{i}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{j}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{i}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{j}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{4}}
These solutions correspond to the values of the loop variables for which the loop's statements will be executed. Note that the order of solutions returned by isolve will not necessarily match the order of the loop's execution.
Trivial Loop with an Empty Iteration Space
The body of the following loop will never be executed:
p2 := proc(a)
a[i] := a[i - 1]:
\textcolor[rgb]{0,0,1}{\mathrm{iteration_space2}}\textcolor[rgb]{0,0,1}{≔}[\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{\le }\textcolor[rgb]{0,0,1}{i}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{i}\textcolor[rgb]{0,0,1}{\le }\textcolor[rgb]{0,0,1}{1}]
Its iteration space is a list of infeasible inequalities, i.e. there are no solutions for
i
\mathrm{simplex}[\mathrm{feasible}]\left(\mathrm{iteration_space2}\right)
\textcolor[rgb]{0,0,1}{\mathrm{false}}
The CodeTools[ProgramAnalysis][IterationSpace] command was introduced in Maple 2016.
simplex[feasible]
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Create release variables
Get a value from a map variable
Share global variables with configuration objects
Revise Jira issue lists with variables
When creating release templates, you will create tasks that contain information that varies based on the release. For example, you can have one generic release template that is used for the release process of several applications. Different releases based on this template will require different application names.
You can use variables to manage information that:
Is used in several places in the release, such as the name of the application, which you can use in task descriptions and email notifications
Variables are identified by the ${ } syntax. Release supports several types of variables. Examples: text, password, number, and list.
In Release, you can create variables with different scopes:
Release variables can only be used in a specific template or release.
Global variables can be used in all templates and releases.
Folder variables can be used in all templates and releases inside a specific folder.
How to create a global variable
If you have the Edit Global Variables permission, you can create global variables in Settings > Global variables. For information about creating, editing, and deleting global variables, see Configure global variables.
How to create a release variable
If you have the Edit Template or Edit Release permission on a template or a release, respectively, you can create a release variable by:
Typing the variable name in a field in the release flow editor using the ${ } syntax
Using the Variables screen
For more information about creating, editing, and deleting release variables, see Create release variables.
How to create a folder variable
If you have the Edit folder variables permission on a folder, you can create folder variables in Design > Folders, select a folder, then select Variables (available in Release 8.6.0 and later).
For more information about creating, editing, and deleting folder variables, see Configure folder variables.
You can use variables in most fields in Release releases. Examples: in the titles of phases and tasks, in descriptions of phases, tasks, and releases, and in conditions and scripts.
While global, folder and release variables can be used in input properties of tasks, only release variables can be used in the output properties of such tasks.
You can create release variables that must be filled in before a release or task can start. For more information, see Create release variables.
You can change the values of variables in an active release, although doing so will only affect tasks that are in a planned state.
Examples of using variables inside another variable
In Release version 8.5 and later, you can use variables inside another variable. This does not apply to password types and index/key access of complex types (ex,
{list[2]} or
{map[‘key1’]})
Example 1: String variables inside String Type variable
${host} = localhost
${port} = 5516
${url} = http://${host}:${port}/
Example 2: String variables inside List Type variable
value: ['${var1}', '${var2}'],
Example 3: String variables inside Set Type variable
Example 4: String variables inside Map type variable
key: 'stringMap',
value: {'key1': '${var1}', 'key2': '${var2}'},
Using a List variable as a value in the List box variable types
You can create a List variable and use it as a possible value for a List box variable.
Create a global variable or a release variable with the type List.
When you create a new variable and select the type List box, click the button next to the Possible values to switch between a list of normal values or a variable of type List.
Select the second option and choose a List type variable.
You can use the List box variable in templates, releases, or tasks allowing users to select the values from predefined List variable.
Using lists of Applications or Environments in the List box variable types
You can choose a list of Applications or Environments to be used as possible values for a List box variable.
Please specify the run-as user and password on the template properties as those credentials will be used to retrieve the available Applications and Environments.
Note: Applications or environments will only be retrieved when the new release is being created and during the User Input task. In all other cases - such as manually changing variable values in either a Template or Release - the list of values will be empty.
Dynamically populating a List box variable type using a custom script
You can use a script to dynamically retrieve possible values inside a List box variable.
Create a new List box variable.
Select Value provider as the Value provider type.
Select the Script value provider.
Depending on the selected script, you will have different input fields. The script will be evaluated when creating a release.
Creating a custom script value provider
Creating a new script value provider is similar to the Plugin tasks. For a more detailed explanation of how to extend the Release type system, see Defining a custom task.
Example script value provider
In the synthetic.xml file, you must extend xlrelease.JythonProvider or xlrelease.GroovyProvider:
<type type="test.Test2ValueProvider" extends="xlrelease.JythonProvider" label="Sample value provider with CI ref"
description="This value provider has CI ref parameter that points to JIRA server.">
<property name="jiraServer" label="JIRA server" referenced-type="jira.Server" kind="ci" description="JIRA server to use" />
<property name="username" required="false" description="Overrides the username used to connect to the server"/>
<property name="password" password="true" required="false" description="Overrides the password used to connect to the server"/>
You can store value provider scripts in the ext or plugins directory of the Release server.
Use ext when you are developing a custom value provider. The ext directory contains custom type definitions in the synthetic.xml file. Scripts are placed in the subdirectories of ext.
The plugins directory contains bundled plugins that are packaged in a single zip file with the .jar extension.
For the value provider defined above, Release will try to find and execute the python script at this location: test/Test2ValueProvider.py. Value provider scripts must return a list of objects in the variable named result.
Note: The properties of the value provider will be injected into the script as well as a dictionary. As a result, you do not need to access the properties as valueProvider.jiraServer but use jiraServer, instead.
The following script can be used to display the title of the JIRA server passed as a parameter to the value provider defined above:
# let's connect to the provided jiraServer
from xlrelease.HttpRequest import HttpRequest
# example of request to the Jira server
req = HttpRequest(jiraServer)
result = [jiraServer["title"]]
# result = [valueProvider.jiraServer.title]
You can add the following properties to the <type> element to further customize your value provider:
This is an example of a value provider that generates a range of numbers:
<type type="test.TestValueProvider" extends="xlrelease.JythonProvider" label="Sample script value provider"
description="This value provider has two parameters for range.">
<property name="param1" label="Lower bound" default="1" description="Minimum value." required="false" />
<property name="param2" label="Upper bound" default="5" description="Maximum value." required="false" />
Place the corresponding script into test/TestValueProvider.py:
def generateRange():
t = range(long(valueProvider.param1), long(valueProvider.param2))
result = generateRange()
Special release variables
You can use the following special release variables to access the properties of a release:
${release.url}
${release.id}
External Password Variables
Starting with release 9.7, password variables values can be stored in third party secret management systems. These include:
Included in the 9.7 release are adapters that allow XL release to retrieve data from these secret managers. Secrets managed in this way can be used in password fields. Also, starting in 9.7, passwords can also be used in regular text fields if “Allow passwords in all fields” is checked on the release properties tab.
Configuring Secret Server
See the related how-to pages for your server:
Mapping an external secret.
Any global, folder or release variable can be mapped to an external secret by following these steps:
Choose a Password from the Type drop down.
Click on the keyboard icon next to the default value text field.
Choose your preconfigured Vault or Conjur server.
For Vault, you will need to add the path to the secret and the key of the secret.
For Conjur, you will need to add the path to the secret only.
Once this is saved, the variable can be used like any other password variable.
Restrictions on external password variables.
External secrets can only be mapped to password variables.
External secrets are treated as string values.
External secrets cannot be used in dictionary values.
The value from an external secret are not passed to scripts via the globalVariable, folderVariable or releaseVariable dictionary within a script.
The value from an external secret can not be retrieved from Release REST API.
The path specified in creating the external secret is not validated. It will be resolved during the release execution.
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Inverse Laplace transform - MATLAB ilaplace - MathWorks Italia
Inverse Laplace Transform of Symbolic Expression
Default Independent Variable and Transformation Variable
Inverse Laplace Transforms Involving Dirac and Heaviside Functions
Inverse Laplace Transform of Array Inputs
If Inverse Laplace Transform Cannot Be Found
Inverse Laplace Transform of Symbolic Function
ilaplace(F,transVar)
ilaplace(F,var,transVar)
ilaplace(F) returns the Inverse Laplace Transform of F. By default, the independent variable is s and the transformation variable is t. If F does not contain s, ilaplace uses the function symvar.
ilaplace(F,transVar) uses the transformation variable transVar instead of t.
ilaplace(F,var,transVar) uses the independent variable var and the transformation variable transVar instead of s and t, respectively.
Compute the inverse Laplace transform of 1/s^2. By default, the inverse transform is in terms of t.
F = 1/s^2;
Compute the inverse Laplace transform of 1/(s-a)^2. By default, the independent and transformation variables are s and t, respectively.
syms a s
F = 1/(s-a)^2;
t*exp(a*t)
Specify the transformation variable as x. If you specify only one variable, that variable is the transformation variable. The independent variable is still s.
ilaplace(F,x)
x*exp(a*x)
Specify both the independent and transformation variables as a and x in the second and third arguments, respectively.
ilaplace(F,a,x)
x*exp(s*x)
Compute the following inverse Laplace transforms that involve the Dirac and Heaviside functions:
ilaplace(1,s,t)
dirac(t)
F = exp(-2*s)/(s^2+1);
ilaplace(F,s,t)
heaviside(t - 2)*sin(t - 2)
Find the inverse Laplace transform of the matrix M. Specify the independent and transformation variables for each matrix entry by using matrices of the same size. When the arguments are nonscalars, ilaplace acts on them element-wise.
ilaplace(M,vars,transVars)
[ exp(x)*dirac(a), dirac(b)]
[ ilaplace(sin(y), y, c), dirac(1, d)*1i]
If ilaplace is called with both scalar and nonscalar arguments, then it expands the scalars to match the nonscalars by using scalar expansion. Nonscalar arguments must be the same size.
ilaplace(x,vars,transVars)
[ x*dirac(a), dirac(1, b)]
[ x*dirac(c), x*dirac(d)]
If ilaplace cannot compute the inverse transform, then it returns an unevaluated call to ilaplace.
syms F(s) t
F(s) = exp(s);
f = ilaplace(F,s,t)
ilaplace(exp(s), s, t)
Return the original expression by using laplace.
laplace(f,t,s)
Compute the Inverse Laplace transform of symbolic functions. When the first argument contains symbolic functions, then the second argument must be a scalar.
ilaplace([f1 f2],x,[a b])
[ ilaplace(exp(x), x, a), dirac(1, b)]
s (default) | symbolic variable | symbolic expression | symbolic vector | symbolic matrix
Independent variable, specified as a symbolic variable, expression, vector, or matrix. This variable is often called the "complex frequency variable." If you do not specify the variable, then ilaplace uses s. If F does not contain s, then ilaplace uses the function symvar to determine the independent variable.
t (default) | x | symbolic variable | symbolic expression | symbolic vector | symbolic matrix
Transformation variable, specified as a symbolic variable, expression, vector, or matrix. It is often called the "time variable" or "space variable." By default, ilaplace uses t. If t is the independent variable of F, then ilaplace uses x.
The inverse Laplace transform f = f(t) of F = F(s) is:
f\left(t\right)=\frac{1}{2\pi i}\underset{c-i\infty }{\overset{c+i\infty }{\int }}F\left(s\right){e}^{st}ds.
Here, c is a suitable complex number.
If any argument is an array, then ilaplace acts element-wise on all elements of the array.
To compute the direct Laplace transform, use laplace.
For a signal f(t), computing the Laplace transform (laplace) and then the inverse Laplace transform (ilaplace) of the result may not return the original signal for t < 0. This is because the definition of laplace uses the unilateral transform. This definition assumes that the signal f(t) is only defined for all real numbers t ≥ 0. Therefore, the inverse result does not make sense for t < 0 and may not match the original signal for negative t. One way to correct the problem is to multiply the result of ilaplace by a Heaviside step function. For example, both of these code blocks:
laplace(sin(t))
laplace(sin(t)*heaviside(t))
return 1/(s^2 + 1). However, the inverse Laplace transform
ilaplace(1/(s^2 + 1))
returns sin(t), not sin(t)*heaviside(t).
fourier | ifourier | iztrans | laplace | ztrans
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(Redirected from Square brackets)
{\displaystyle [4\times (3+2)]^{2}=400.}
{\displaystyle \left\langle V(t)^{2}\right\rangle =\lim _{T\to \infty }{\frac {1}{T}}\int _{-{\frac {T}{2}}}^{\frac {T}{2}}V(t)^{2}\,{\rm {d}}t.}
Retrieved from "https://en.wikipedia.org/w/index.php?title=Bracket&oldid=1088703216#Square_brackets"
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Small amplitude limit - DispersiveWiki
Small amplitude limit
The small amplitude limit for a nonlinear equation arises when considering initial position
{\displaystyle u(0)}
{\displaystyle u(0)=\epsilon f}
{\displaystyle f}
and a small parameter
{\displaystyle \epsilon >0}
, in the limit
{\displaystyle \epsilon \to 0}
. For equations which are second-order in time, such as nonlinear wave equations, one must also specify an initial velocity
{\displaystyle u_{t}(0)=\epsilon g}
For bounded times, the small amplitude limit is usually just the linear counterpart of the equation; however when analyzing long times (e.g. times comparable to
{\displaystyle 1/\epsilon }
), significant nonlinear effects may still occur in the limit.
Retrieved from "https://dispersivewiki.org/DispersiveWiki/index.php?title=Small_amplitude_limit&oldid=2904"
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Stresses and Strains in the Medial Meniscus of an ACL Deficient Knee under Anterior Loading: A Finite Element Analysis with Image-Based Experimental Validation | J. Biomech Eng. | ASME Digital Collection
, Rochester, New York 14627
Jason Snibbe,
, Beverly Hills, California
Yao, J., Snibbe, J., Maloney, M., and Lerner, A. L. (September 14, 2005). "Stresses and Strains in the Medial Meniscus of an ACL Deficient Knee under Anterior Loading: A Finite Element Analysis with Image-Based Experimental Validation." ASME. J Biomech Eng. February 2006; 128(1): 135–141. https://doi.org/10.1115/1.2132373
The menisci are believed to play a stabilizing role in the ACL-deficient knee, and are known to be at risk for degradation in the chronically unstable knee. Much of our understanding of this behavior is based on ex vivo experiments or clinical studies in which we must infer the function of the menisci from external measures of knee motion. More recently, studies using magnetic resonance (MR) imaging have provided more clear visualization of the motion and deformation of the menisci within the tibio-femoral articulation. In this study, we used such images to generate a finite element model of the medial compartment of an ACL-deficient knee to reproduce the meniscal position under anterior loads of 45, 76, and
107N
. Comparisons of the model predictions to boundaries digitized from images acquired in the loaded states demonstrated general agreement, with errors localized to the anterior and posterior regions of the meniscus, areas in which large shear stresses were present. Our model results suggest that further attention is needed to characterize material properties of the peripheral and horn attachments. Although overall translation of the meniscus was predicted well, the changes in curvature and distortion of the meniscus in the posterior region were not captured by the model, suggesting the need for refinement of meniscal tissue properties.
biological tissues, biomedical MRI, finite element analysis, physiological models, deformation, biomechanics
Anterior cruciate ligament, Deformation, Finite element analysis, Finite element model, Knee, Materials properties, Stress, Errors, Biological tissues
The Natural History of Meniscal Tears in Anterior Cruciate Ligament Insufficiency
Fate of the ACL-Injured Patient: A Prospective Outcome Study
The Incidence of Meniscal Tears Associated with Acute Anterior Cruciate Ligament Disruption Secondary to Snow Skiing Accidents Arthroscopy
Arthroscopy: J. Relat.Surg.
Medial and Lateral Meniscal Tear Patterns in Anterior Cruciate Ligament-Deficient Knees: A Prospective Analysis of 575 Tears
The Effects of Time Course After Anterior Cruciate Ligament Injury in Correlation with Meniscal and Cartilage Loss
Niciforos
Change in Meniscal Strain with Anterior Cruciate Ligament Injury and After Reconstruction
The Role of the Meniscus in the Anterior-Posterior Stability of the Loaded Anterior Cruciate-Deficient Knee: Effects of Partial versus Total Excision
Femoro-Tibial and Menisco-Tibial Translation Patterns in Patients with Unilateral Anterior Cruciate Ligament Deficiency—A Potential Cause of Secondary Meniscal Tears
A Model for the Function and Failure of the Meniscus
A Transversely Isotropic Biphasic Finite Element Model of the Meniscus
Finite Element Analysis of the Meniscus: The Influence of Geometry and Material Properties on Its Behavior
In Vivo Determination of Contact Areas and Pressure of the Femorotibial Joint Using Non-Linear Finite Element Analysis
A 3D Finite Element Model of a Knee for Joint Contact Stress Analysis During Sport Activities
Tibial Meniscal Dynamics Using Three-Dimensional Reconstruction of Magnetic Resonance Images
Tamez-Pena
The Integration of Automatic Segmentation and Motion Tracking for 4D Reconstruction and Visualization of Musculoskeletal Structure
,” IEEE Workshop on Biomedical Image Analysis, Santa Barbara, CA., pp.
The Use of Sequential MR Image Sets for Determining Tibiofemoral Motion: Reliability of Coordinate Systems and Accuracy of Motion Tracking Algorithm
Sensitivities of Medial Meniscal Motion and Deformation to Material Properties of Cartilage, Meniscus and Meniscal Attachments
, Advances in Bioengineering, ASME, b0373520.
Methodology and Apparatus to Determine Material Properties of the Knee Joint Meniscus
Menisco-Femoral Ligaments—Structural and Material Properties
Edmonds-Wilson
Spatial Distribution of Hip Capsule Structural and Material Properties
Curvature Characteristics and Congruence of the Thumb Carpometacarpal Joint: Differences Between Female and Male Joints
The Effect of Joint-Compressive Load and Quadriceps Muscle Force on Knee Motion in the Intact and Anterior Cruciate Ligament-Sectioned Knee
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Denoising Projection Data with a Robust Adaptive Bilateral Filter in Low-Count SPECT
Susumu Nakabayashi1, Takashi Chikamatsu2, Takao Okamoto3, Tatsuro Kaminaga4, Norikazu Arai2, Shinobu Kumagai2, Kenshiro Shiraishi4, Takahide Okamoto1,2, Takenori Kobayashi1, Jun’ichi Kotoku1,2*
1Graduate School of Medical Care and Technology, Teikyo University, Tokyo, Japan
2Central Radiation Division, Teikyo University Hospital, Tokyo, Japan
3Diagnostic Imaging, PET Center, Musashimurayama Hospital, Tokyo, Japan
4Department of Radiology, Teikyo University School of Medicine, Tokyo, Japan
B\left(f\right)
B\left(f\right)=\frac{1}{1+{\left(\frac{f}{{f}_{\text{c}}}\right)}^{n}}
\begin{array}{l}g\left(i,j\right)\\ =\frac{{\sum }_{m=-w}^{w}{\sum }_{n=-w}^{w}f\left(i+m,j+n\right)\mathrm{exp}\left(-\frac{{m}^{2}+{n}^{2}}{2{\sigma }_{s}^{2}}\right)\mathrm{exp}\left\{-\frac{{\left[f\left(i+m,j+n\right)-f\left(i,j\right)\right]}^{2}}{2{\sigma }_{I}^{2}}\right\}}{{\sum }_{m=-w}^{w}{\sum }_{n=-w}^{w}\mathrm{exp}\left(-\frac{{m}^{2}+{n}^{2}}{2{\sigma }_{s}^{2}}\right)\mathrm{exp}\left\{-\frac{{\left[f\left(i+m,j+n\right)-f\left(i,j\right)\right]}^{2}}{2{\sigma }_{I}^{2}}\right\}}\end{array}
g\left(i,j\right)
\left(i,j\right)
f\left(i,j\right)
{\sigma }_{s}
{\sigma }_{I}
{\sigma }_{I}
{\sigma }_{I}
f\left(i,j\right)
\mu \left(i+\stackrel{^}{k},j+\stackrel{^}{l}\right)
\left(i+\stackrel{^}{k},j+\stackrel{^}{l}\right)
\stackrel{^}{k}
\stackrel{^}{l}
\stackrel{^}{k},\stackrel{^}{l}=\mathrm{arg}{\mathrm{min}}_{k,l\in A}{\sum }_{s,t\in A}{\left[f\left(i+k+s,j+l+t\right)-f\left(i,j\right)\right]}^{2}
\left\{-1,0,1\right\}
\left(i+\stackrel{^}{k},j+\stackrel{^}{l}\right)
{\sigma }_{I}\left(i+\stackrel{^}{k},j+\stackrel{^}{l}\right)
{\sigma }_{I}
\left(i+\stackrel{^}{k},j+\stackrel{^}{l}\right)
\mu \left(i+\stackrel{^}{k},j+\stackrel{^}{l}\right)
{\sigma }_{I}\left(i+\stackrel{^}{k},j+\stackrel{^}{l}\right)
\begin{array}{l}g\left(i,j\right)\\ =\frac{{\sum }_{m=-2}^{2}{\sum }_{n=-2}^{2}f\left(i+m,j+n\right)\mathrm{exp}\left(-\frac{{m}^{2}+{n}^{2}}{2{\sigma }_{s}^{2}}\right)\mathrm{exp}\left\{-\frac{{\left[f\left(i+m,j+n\right)-\mu \left(i+\stackrel{^}{k},j+\stackrel{^}{l}\right)\right]}^{2}}{2{\sigma }_{I}^{2}\left(i+\stackrel{^}{k},j+\stackrel{^}{l}\right)}\right\}}{{\sum }_{m=-2}^{2}{\sum }_{n=-2}^{2}\mathrm{exp}\left(-\frac{{m}^{2}+{n}^{2}}{2{\sigma }_{s}^{2}}\right)\mathrm{exp}\left\{-\frac{{\left[f\left(i+m,j+n\right)-\mu \left(i+\stackrel{^}{k},j+\stackrel{^}{l}\right)\right]}^{2}}{2{\sigma }_{I}^{2}\left(i+\stackrel{^}{k},j+\stackrel{^}{l}\right)}\right\}}\end{array}
{\sigma }_{s}=1
\mu \left(i+\stackrel{^}{k},j+\stackrel{^}{l}\right)
f\left(i,j\right)
\text{NMSE}=\frac{{\sum }_{i=1}^{N}{\sum }_{j=1}^{128}{\sum }_{k=1}^{128}{\left[T\left(i,j,k\right)-R\left(i,j,k\right)\right]}^{2}}{{\sum }_{i=1}^{N}{\sum }_{j=1}^{128}{\sum }_{k=1}^{128}\text{ }R{\left(i,j,k\right)}^{2}}
R\left(i,j,k\right)
T\left(i,j,k\right)
\text{SNR}=20\mathrm{log}\frac{{\text{SD}}_{\text{image}}}{{\text{SD}}_{\text{noise}}}\left(\text{dB}\right)
{\text{SD}}_{\text{image}}
{\text{SD}}_{\text{noise}}
{Y}_{\varnothing }={q}_{\varnothing }\left(t,{f}_{e}\right)\sqrt{\frac{{V}_{e}}{tn}}
\varnothing
{q}_{\varnothing }\left(t,{f}_{e}\right)
{\sigma }_{s}
{\sigma }_{s}=1\text{\hspace{0.17em}}\text{pixel}
{\sigma }_{I}
{Y}_{0.05}=0.170
{Y}_{0.01}=0.200
{Y}_{0.05}=0.187
{Y}_{0.01}=0.219
Nakabayashi, S., Chikamatsu, T., Okamoto, T., Kaminaga, T., Arai, N., Kumagai, S., Shiraishi, K., Okamoto, T., Kobayashi, T. and Kotoku, J. (2018) Denoising Projection Data with a Robust Adaptive Bilateral Filter in Low-Count SPECT. International Journal of Medical Physics, Clinical Engineering and Radiation Oncology, 7, 363-375. https://doi.org/10.4236/ijmpcero.2018.73030
1. Groch, M.W. and Erwin, W.D. (2000) SPECT in the Year 2000: Basic Principles. Journal of Nuclear Medicine Technology, 28, 233-244.
2. Minoshima, S., Maruno, H., Yui, N., Togawa, T., Kinoshita, F., Kubota, M., et al. (1993) Optimization of Butterworth Filter for Brain SPECT Imaging. Annals of Nuclear Medicine, 7, 71-77. https://doi.org/10.1007/BF03164571
3. David, R.G., Gilland, A.B., Yaghoobi, N., Firouzabady, H. and Rustgou, F. (1988) Determination of the Optimum Filter Function for SPECT Imaging. Journal of Nuclear Medicine, 29, 643-650.
4. Lyra, M. and Ploussi, A. (2011) Filtering in SPECT Image Reconstruction. International Journal of Biomedical Imaging, 2011, Article ID: 693795. https://doi.org/10.1155/2011/693795
5. Matsutomo, N., Tanaka, T., Nagaki, A. and Sasaki, M. (2014) Validation of Noise Reduction in Iterative Reconstruction: A Simulation Phantom Study. Nihon Hoshasen Gijutsu Gakkai Zasshi, 70, 773-783. https://doi.org/10.6009/jjrt.2014_JSRT_70.8.773
6. Sayed, I.S. and Mohamed Nasrudin, N.S. (2016) Effect of Cut-Off Frequency of Butterworth Filter on Detectability and Contrast of Hot and Cold Regions in Tc-99m SPECT. International Journal of Medical Physics, Clinical Engineering and Radiation Oncology, 5, 100-109. https://doi.org/10.4236/ijmpcero.2016.51011
7. Tomasi, C. and Manduchi, R. (1998) Bilateral Filtering for Gray and Color Images. IEEE International Conference on Computer Vision, Bombay, 7 January 1998, 839-846. https://doi.org/10.1109/ICCV.1998.710815
8. Zhou, J., Zhu, H., Shu, H. and Luo, L. (2007) A Generalized Diffusion Based Inter-Iteration Nonlinear Bilateral Filtering Scheme for PET Image Reconstruction. Computerized Medical Imaging and Graphics, 31, 447-457. https://doi.org/10.1016/j.compmedimag.2007.04.003
9. Hofheinz, F., Langner, J., Beuthien-Baumann, B., Oehme, L., Steinbach, J., Kotzerke, J., et al. (2011) Suitability of Bilateral Filtering for Edge-Preserving Noise Reduction in PET. EJNMMI Research, 1, 1-9. https://doi.org/10.1186/2191-219X-1-23
10. Katayama, Y., Ueda, K., Hiura, S., Yamanaga, T., Miyoshi, H., Ohmura, M., et al. (2013) Bilateral Filter Applied to Bone Scintigraphy. Nihon Hoshasen Gijutsu Gakkai Zasshi, 69, 1363-1371. https://doi.org/10.6009/jjrt.2013_JSRT_69.12.1363
11. Buades, A., Coll, B. and Morel, J.-M. (2005) A Non-Local Algorithm for Image Denoising. IEEE International Conference on Computer Vision and Pattern Recognition, 2, 60-65.
12. Paris, S., Kornprobst, P., Tumblin, J. and Durand, F. (2008) Bilateral Filtering: Theory and Applications. Foundations and Trends in Computer Graphics and Vision, 4, 1-73. https://doi.org/10.1561/0600000020
13. Maeda, H., Yamaki, N. and Azuma, M. (2012) Development of the Software Package of the Nuclear Medicine Data Processor for Education and Research. Nihon Hoshasen Gijutsu Gakkai Zasshi, 68, 299-306. https://doi.org/10.6009/jjrt.2012_JSRT_68.3.299
14. Nagasawa, S. (2002) Improvement of the Scheffe’s Method for Paired Comparisons. Kansei Engineering International, 3, 47-56. https://doi.org/10.5057/kei.3.3_47
15. Hideyuki, T. (2014) Practical Statistical Tests Machine Learning (3) Significance Tests for Human Subjective Tests. Systems, Control and Information, 58, 514-520.
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Compute mean value of signal - Simulink - MathWorks Nordic
Mean (Variable Frequency)
Compute mean value of signal
The Mean (Variable Frequency) block computes the mean value of the signal connected to the second input of the block. The mean value is computed over a running average window of one cycle of the frequency of the signal:
\begin{array}{l}Mean\left(f\left(t\right)\right)=\frac{1}{T}\underset{\left(t-T\right)}{\overset{t}{\int }}f\left(t\right)\cdot dt\\ f\left(t\right):\text{input signal, T = 1/ frequency}\end{array}
This block uses a running average window. Therefore, one cycle of simulation must complete before the block outputs the computed mean value. For the first cycle of simulation, the output is held constant to the specified initial value.
The minimum frequency value determines the buffer size of the Variable Time Delay block used inside the block to compute the mean value. Default is 45.
Initial input (DC component)
Specify the initial value of the input during the first cycle of simulation. Default is 0.
The frequency of the signal.
Connects to the signal to be analyzed.
The mean value of the signal.
The power_MeanVariableFrequency model compares the Mean block to the Mean (Variable Frequency) block for three identical input signals. It shows that, even if the frequency of the input signals varies during the simulation, the Mean (Variable Frequency) block outputs correct values.
The model sample time is parameterized by the Ts variable with a default value of 50e-6 s. Set Ts to 0 in the command window to simulate the model in continuous mode.
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Classic definition[edit]
{\displaystyle 0\neq 1}
{\displaystyle {\frac {b}{a}}\cdot {\frac {a}{b}}={\frac {ba}{ab}}=1.}
{\displaystyle {\begin{aligned}&{\frac {a}{b}}\cdot \left({\frac {c}{d}}+{\frac {e}{f}}\right)\\[6pt]={}&{\frac {a}{b}}\cdot \left({\frac {c}{d}}\cdot {\frac {f}{f}}+{\frac {e}{f}}\cdot {\frac {d}{d}}\right)\\[6pt]={}&{\frac {a}{b}}\cdot \left({\frac {cf}{df}}+{\frac {ed}{fd}}\right)={\frac {a}{b}}\cdot {\frac {cf+ed}{df}}\\[6pt]={}&{\frac {a(cf+ed)}{bdf}}={\frac {acf}{bdf}}+{\frac {aed}{bdf}}={\frac {ac}{bd}}+{\frac {ae}{bf}}\\[6pt]={}&{\frac {a}{b}}\cdot {\frac {c}{d}}+{\frac {a}{b}}\cdot {\frac {e}{f}}.\end{aligned}}}
Constructible numbers[edit]
In antiquity, several geometric problems concerned the (in)feasibility of constructing certain numbers with compass and straightedge. For example, it was unknown to the Greeks that it is, in general, impossible to trisect a given angle in this way. These problems can be settled using the field of constructible numbers.[7] Real constructible numbers are, by definition, lengths of line segments that can be constructed from the points 0 and 1 in finitely many steps using only compass and straightedge. These numbers, endowed with the field operations of real numbers, restricted to the constructible numbers, form a field, which properly includes the field Q of rational numbers. The illustration shows the construction of square roots of constructible numbers, not necessarily contained within Q. Using the labeling in the illustration, construct the segments AB, BD, and a semicircle over AD (center at the midpoint C), which intersects the perpendicular line through B in a point F, at a distance of exactly
{\displaystyle h={\sqrt {p}}}
{\displaystyle {\sqrt[{3}]{2}}}
A field with four elements[edit]
Elementary notions[edit]
Consequences of the definition[edit]
The additive and the multiplicative group of a field[edit]
Characteristic[edit]
Subfields and prime fields[edit]
Finite fields[edit]
Constructing fields[edit]
Constructing fields from rings[edit]
Field of fractions[edit]
{\displaystyle {\frac {a}{b}}+{\frac {c}{d}}={\frac {ad+bc}{bd}}.}
{\displaystyle \sum _{i=k}^{\infty }a_{i}x^{i}\ (k\in \mathbb {Z} ,a_{i}\in F)}
Residue fields[edit]
{\displaystyle \mathbf {R} [X]/\left(X^{2}+1\right)\ {\stackrel {\cong }{\longrightarrow }}\ \mathbf {C} .}
Constructing fields within a bigger field[edit]
Field extensions[edit]
Algebraic extensions[edit]
{\displaystyle x\in F}
{\displaystyle \sum _{k=0}^{n-1}a_{k}x^{k},\ \ a_{k}\in E.}
Transcendence bases[edit]
Closure operations[edit]
Fields with additional structure[edit]
Ordered fields[edit]
{\displaystyle x_{1}^{2}+x_{2}^{2}+\dots +x_{n}^{2}=0}
Topological fields[edit]
Local fields[edit]
{\displaystyle \operatorname {Gal} \left(\mathbf {Q} _{p}\left(p^{1/p^{\infty }}\right)\right)\cong \operatorname {Gal} \left(\mathbf {F} _{p}((t))\left(t^{1/p^{\infty }}\right)\right).}
Differential fields[edit]
For a finite Galois extension, the Galois group Gal(F/E) is the group of field automorphisms of F that are trivial on E (i.e., the bijections σ : F → F that preserve addition and multiplication and that send elements of E to themselves). The importance of this group stems from the fundamental theorem of Galois theory, which constructs an explicit one-to-one correspondence between the set of subgroups of Gal(F/E) and the set of intermediate extensions of the extension F/E.[46] By means of this correspondence, group-theoretic properties translate into facts about fields. For example, if the Galois group of a Galois extension as above is not solvable (cannot be built from abelian groups), then the zeros of f cannot be expressed in terms of addition, multiplication, and radicals, i.e., expressions involving
{\displaystyle {\sqrt[{n}]{\ }}}
Invariants of fields[edit]
Model theory of fields[edit]
{\displaystyle \operatorname {ulim} _{p\to \infty }{\overline {\mathbf {F} }}_{p}\cong \mathbf {C} .}
The absolute Galois group[edit]
K-theory[edit]
{\displaystyle K_{n}^{M}(F)=F^{\times }\otimes \cdots \otimes F^{\times }/\left\langle x\otimes (1-x)\mid x\in F\setminus \{0,1\}\right\rangle .}
{\displaystyle K_{n}^{M}(F)/p=H^{n}(F,\mu _{l}^{\otimes n}).}
Linear algebra and commutative algebra[edit]
{\displaystyle x=a^{-1}b.}
This immediate consequence of the definition of a field is fundamental in linear algebra. For example, it is an essential ingredient of Gaussian elimination and of the proof that any vector space has a basis.[55]
{\displaystyle \mathbb {Z} }
Finite fields: cryptography and coding theory[edit]
Geometry: field of functions[edit]
{\displaystyle {\frac {f(x)}{g(x)}},}
Number theory: global fields[edit]
{\displaystyle F=\mathbf {Q} ({\sqrt {-d}})}
Related notions[edit]
Division rings[edit]
Retrieved from "https://en.wikipedia.org/w/index.php?title=Field_(mathematics)&oldid=1083143888"
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Approximate Analytical Solution for One-Dimensional Solidification Problem of a Finite Superheating Phase Change Material Including the Effects of Wall and Thermal Contact Resistances
2012 Approximate Analytical Solution for One-Dimensional Solidification Problem of a Finite Superheating Phase Change Material Including the Effects of Wall and Thermal Contact Resistances
Hamid El Qarnia, Fayssal El Adnani, El Khadir Lakhal
This work reports an analytical solution for the solidification of a superheating phase change material (PCM) contained in a rectangular enclosure with a finite height. The analytical solution has been obtained by solving nondimensional energy equations by using the perturbation method for a small perturbation parameter: the Stefan number,
\epsilon
. This analytical solution, which takes into account the effects of the superheating of PCM, finite height of the enclosure, thickness of the wall, and wall-solid shell interfacial thermal resistances, was expressed in terms of nondimensional temperature distributions of the bottom wall of the enclosure and both PCM phases, and the dimensionless solid-liquid interface position and its dimensionless speed. The developed solution was firstly compared with that existing in the literature for the case of nonsuperheating PCM. The predicted results agreed well with those published in the literature. Next, a parametric study was carried out in order to study the impacts of the dimensionless control parameters on the dimensionless temperature distributions of the wall, the solid shell, and liquid phase of the PCM, as well as the solid-liquid interface position and its dimensionless speed.
Hamid El Qarnia. Fayssal El Adnani. El Khadir Lakhal. "Approximate Analytical Solution for One-Dimensional Solidification Problem of a Finite Superheating Phase Change Material Including the Effects of Wall and Thermal Contact Resistances." J. Appl. Math. 2012 1 - 20, 2012. https://doi.org/10.1155/2012/174604
Hamid El Qarnia, Fayssal El Adnani, El Khadir Lakhal "Approximate Analytical Solution for One-Dimensional Solidification Problem of a Finite Superheating Phase Change Material Including the Effects of Wall and Thermal Contact Resistances," Journal of Applied Mathematics, J. Appl. Math. 2012(none), 1-20, (2012)
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Carbon Capture for Automobiles Using Internal Combustion Rankine Cycle Engines | J. Eng. Gas Turbines Power | ASME Digital Collection
Robert W. Bilger,
Robert W. Bilger
School of Aerospace, Mechanical and Mechatronic Engineering,
, New South Wales 2006, Australia
e-mail: bilger@aeromech.usyd.edu.au
, Shanghai 201804, P. R. China
e-mail: zjwu@mail.tongji.edu.cn
Bilger, R. W., and Wu, Z. (February 10, 2009). "Carbon Capture for Automobiles Using Internal Combustion Rankine Cycle Engines." ASME. J. Eng. Gas Turbines Power. May 2009; 131(3): 034502. https://doi.org/10.1115/1.3077657
Internal combustion Rankine cycle (ICRC) power plants use oxy-fuel firing with recycled water in place of nitrogen to control combustion temperatures. High efficiency and specific power output can be achieved with this cycle, but importantly, the exhaust products are only
CO2
and water vapor: The
CO2
can be captured cheaply on condensation of the water vapor. Here we investigate the feasibility of using a reciprocating engine version of the ICRC cycle for automotive applications. The vehicle will carry its own supply of oxygen and store the captured
CO2
. On refueling with conventional gasoline, the
CO2
will be off-loaded and the oxygen supply replenished. Cycle performance is investigated on the basis of fuel-oxygen-water cycle calculations. Estimates are made for the system mass, volume, and cost and compared with other power plants for vehicles. It is found that high thermal efficiencies can be obtained and that huge increases in specific power output are achievable. The overall power-plant system mass and volume will be dominated by the requirements for oxygen and
CO2
storage. Even so, the performance of vehicles with ICRC power plants will be superior to those based on fuel cells and they will have much lower production costs. Operating costs arising from supply of oxygen and disposal of the
CO2
are expected to be around 20 c/l of gasoline consumed and about $25/tonne of carbon controlled. Over all, ICRC engines are found to be a potentially competitive option for the powering of motor vehicles in the forthcoming carbon-controlled energy market.
automobiles, carbon compounds, internal combustion engines, power plants, internal combustion engines, carbon dioxide capture, vehicle power plants
Automobiles, Carbon capture and storage, Combustion, Cycles, Engines, Fuels, Oxygen, Power stations, Rankine cycle, Vehicles, Water, Carbon, Carbon dioxide, Internal combustion engines, Exhaust systems, Temperature
IPPC 2005: IPCC Special Report on Carbon Dioxide Capture and Storage
, prepared by Working Group III of the Intergovernmental Panel on Climate Change,
High Efficiency Zero Emission Power Generation Based on a High Temperature Steam Cycle
, Clearwater, FL, Mar.
Zero Release Combustion Technologies and the Oxygen Economy
Fifth International Conference on Technologies and Combustion for a Clean Environment
, Lisbon, Portugal, Jul. 12–15, pp.
Qualitative and Quantitative Comparison of Two Oxy-Fuel Power Cycles for CO2 Capture
The Internal Combustion Engine in Theory and Practice, Vol. 1: Thermodynamics, Fluid Flow, Performance
STANJAN Chemical Equilibrium Solver
”, Version 3.94, Department of Mechanical Engineering,
Fundamental Study on a Novel Gas Turbine Cycle
Design and Analysis of Zero CO2 Emission Powerplants for the Transportation Sector
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Polar anisotropy - SEG Wiki
Polar anisotropy
Polar anisotropy has a pole of rotational symmetry (hence the name). This corresponds to undeformed (and unfractured) shales, and to sequences of thin (compared to the seismic wavelength) beds of isotropic and/or polar-ansiotropic symmetry. It is also called "Transverse Isotropy" because all directions normal to the pole have the same velocities. In seismics, the corresponding elastic stiffness matrix (symmetric) has five independent components (in Voigt notation):
{\displaystyle \{c_{\alpha \beta }\}={\begin{pmatrix}c_{11}&c_{12}&c_{13}&0&0&0\\c_{12}&c_{11}&c_{13}&0&0&0\\c_{13}&c_{13}&c_{33}&0&0&0\\0&0&0&c_{44}&0&0\\0&0&0&0&c_{44}&0\\0&0&0&0&0&c_{66}\\\end{pmatrix}}}
{\displaystyle c_{12}=c_{11}-2c_{66}}
. These indices refer to coordinate directions in the natural coordinate system of the medium; the 3-direction (normally vertical) is the pole of symmetry.
The resulting expressions for the seismic velocities at any incidence angle may be found with conventional algebraic techniques; they are quite complicated. [1] [2]. The assumption of weak[3] polar anisotropy makes it feasible to analyze real data for polar anisotropy.
↑ Rudzki,M. P., 1915. Über die Theorie der Erdbebenwellen: Die Naturwissenschaften,3, 201–204.
↑ Thomsen, L., 2014. Seismic Anisotropy in Exploration and Exploitation, the SEG/EAGE Distinguished Instructor Short Course #5 Lecture Notes, 2nd Edition, Soc. Expl. Geoph., Tulsa
↑ Thomsen, L., 1986. Weak Elastic Anisotropy, Geophysics, 51(10), pp. 1954-1966.
Retrieved from "https://wiki.seg.org/index.php?title=Polar_anisotropy&oldid=158279"
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40 CFR § 403.7 - Removal credits. | CFR | US Law | LII / Legal Information Institute
40 CFR § 403.7 - Removal credits.
(1) Definitions. For the purpose of this section:
(i) Removal means a reduction in the amount of a pollutant in the POTW's effluent or alteration of the nature of a pollutant during treatment at the POTW. The reduction or alteration can be obtained by physical, chemical or biological means and may be the result of specifically designed POTW capabilities or may be incidental to the operation of the treatment system. Removal as used in this subpart shall not mean dilution of a pollutant in the POTW.
(v) NPDES permit limitations. The granting of removal credits will not cause a violation of the POTW's permit limitations or conditions. Alternatively, the POTW can demonstrate to the Approval Authority that even though it is not presently in compliance with applicable limitations and conditions in its NPDES permit, it will be in compliance when the Industrial User(s) to whom the removal credit would apply is required to meet its categorical Pretreatment Standard(s), as modified by the removal credit provision.
y=\frac{x}{1-r}
(b) Establishment of removal credits; demonstration of Consistent Removal -
(1) Definition of Consistent Removal. “Consistent Removal” shall mean the average of the lowest 50 percent of the removal measured according to paragraph (b)(2) of this section. All sample data obtained for the measured pollutant during the time period prescribed in paragraph (b)(2) of this section must be reported and used in computing Consistent Removal. If a substance is measurable in the influent but not in the effluent, the effluent level may be assumed to be the limit of measurement, and those data may be used by the POTW at its discretion and subject to approval by the Approval Authority. If the substance is not measurable in the influent, the date may not be used. Where the number of samples with concentrations equal to or above the limit of measurement is between 8 and 12, the average of the lowest 6 removals shall be used. If there are less than 8 samples with concentrations equal to or above the limit of measurement, the Approval Authority may approve alternate means for demonstrating Consistent Removal. The term “measurement” refers to the ability of the analytical method or protocol to quantify as well as identify the presence of the substance in question.
(iii) Sampling procedures: Composite.
(A) The influent and effluent operational data shall be obtained through 24-hour flow-proportional composite samples. Sampling may be done manually or automatically, and discretely or continuously. For discrete sampling, at least 12 aliquots shall be composited. Discrete sampling may be flow-proportioned either by varying the time interval between each aliquot or the volume of each aliquot. All composites must be flow-proportional to each stream flow at time of collection of influent aliquot or to the total influent flow since the previous influent aliquot. Volatile pollutant aliquots must be combined in the laboratory immediately before analysis.
(2) In addition, upon the Approval Authority's concurrence, a POTW may utilize an historical data base amassed prior to the effective data of this section provide that such data otherwise meet the requirements of this paragraph. In order for the historical data base to be approved it must present a statistically valid description of daily, weekly and seasonal sewage treatment plant loadings and performance for at least one year.
(2) The POTW must have submitted to the Approval Authority an application for pretreatment program approval meeting the requirements of §§ 403.8 and 403.9 in a timely manner, not to exceed the time limitation set forth in a compliance schedule for development of a pretreatment program included in the POTW's NPDES permit, but in no case later than July 1, 1983, where no permit deadline exists;
(e) POTW application for authorization to give removal credits and Approval Authority review -
(1) Who must apply. Any POTW that wants to give a removal credit must apply for authorization from the Approval Authority.
(v) Sludge management certification. A specific description of the POTW's current methods of using or disposing of its sludge and a certification that the granting of removal credits will not cause a violation of the sludge requirements identified in paragraph (a)(3)(iv) of this section.
(vi) NPDES permit limit certification. A certification that the granting of removal credits will not cause a violation of the POTW's NPDES permit limits and conditions as required in paragraph (a)(3)(v) of this section.
(5) Approval Authority review. The Approval Authority shall review the POTW's application for authorization to give or modify removal credits in accordance with the procedures of § 403.11 and shall, in no event, have more that 180 days from public notice of an application to complete review.
(6) EPA review of State removal credit approvals. Where the NPDES State has an approved pretreatment program, the Regional Administrator may agree in the Memorandum of Agreement under 40 CFR 123.24(d) to waive the right to review and object to submissions for authority to grant removal credits. Such an agreement shall not restrict the Regional Administrator's right to comment upon or object to permits issued to POTW's except to the extent 40 CFR 123.24(d) allows such restriction.
(f) Continuation and withdrawal of authorization -
(1) Effect of authorization.
(i) Once a POTW has received authorization to grant removal credits for a particular pollutant regulated in a categorical Pretreatment Standard it may automatically extend that removal credit to the same pollutant when it is regulated in other categorical standards, unless granting the removal credit will cause the POTW to violate the sludge requirements identified in paragraph (a)(3)(iv) of this section or its NPDES permit limits and conditions as required by paragraph (a)(3)(v) of this section. If a POTW elects at a later time to extend removal credits to a certain categorical Pretreatment Standard, industrial subcategory or one or more Industrial Users that initially were not granted removal credits, it must notify the Approval Authority.
(2) Inclusion in POTW permit. Once authority is granted, the removal credits shall be included in the POTW's NPDES Permit as soon as possible and shall become an enforceable requirement of the POTW's NPDES permit. The removal credits will remain in effect for the term of the POTW's NPDES permit, provided the POTW maintains compliance with the conditions specified in paragraph (f)(4) of this section.
(3) Compliance monitoring. Following authorization to give removal credits, a POTW shall continue to monitor and report on (at such intervals as may be specified by the Approval Authority, but in no case less than once per year) the POTW's removal capabilities. A minimum of one representative sample per month during the reporting period is required, and all sampling data must be included in the POTW's compliance report.
(4) Modification or withdrawal of removal credits -
(i) Notice of POTW. The Approval Authority shall notify the POTW if, on the basis of pollutant removal capability reports received pursuant to paragraph (f)(3) of this section or other relevant information available to it, the Approval Authority determines:
(B) That such discharge limit revisions are causing a violation of any conditions or limits contained in the POTW's NPDES Permit.
(i) The Consistent Removal claimed is reduced pursuant to the following equation:
{r}_{c}={r}_{m}\frac{8760-Z}{8760}
rm = POTW's Consistent Removal rate for that pollutant as established under paragraphs (a)(1) and (b)(2) of this section
Z = hours per year that Overflows occurred between the Industrial User(s) and the POTW Treatment Plant, the hours either to be shown in the POTW's current NPDES permit application or the hours, as demonstrated by verifiable techniques, that a particular Industrial User's Discharge Overflows between the Industrial User and the POTW Treatment Plant; and
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It is grand about Nepenthes. You are heartily welcome to notice in any way all or any of my published or unpublished results,—though I cannot remember anything published.2 I will give an abstract, as far as memory & time or rather strength serves of my chief results. But I do not know what you exactly want. I have a fair copy by copyist of my observations on Dionæa, which I lent Burdon Sanderson;3 & which you could see, but I do not suppose you could want it.— I will give my results higglety-pigglety.4
(1) Organic & inorganic objects placed on discal glands, these transmit influence to marginal tentacles which become inflected; but with this difference that if the object contains soluble nitrogenous matter they remain inflected for a longer period than if none is contained
(2) Immersion in nitrogenous organic fluid induces inflection, not so non-nitrogenous organic fluids.
(3) An organic or inorganic particle placed on a single marginal gland causes that tentacle to bend; & it is a certain & truly wonderful fact that a particle with no soluble matter weighing
\frac{1}{80,000}
of a grain suffices, although the particle is partly supported by the viscid secretion. I think it certain that the continued pressure of
\frac{1}{1,000,000}
of a grain would suffice, if wholly resting on the gland: a particle wholly resting on the viscid secretion does not act. Though so sensitive to pressure a gland may be roughly touched with a needle once, or twice, & there is no movement, but if touched thrice or four times there is inflection, for touches seem to act like pressure. All this forms a wonderful contrast to the sensitive filaments of Dionæa, which have been specialised for a touch & not for prolonged gentle pressure.
(4) All salts of Ammonia cause inflection, (& many other salts, but not all, & most acids, but not all) & I shall be bold enough to publish that
\frac{1}{20,000,000}
of a grain of crystallised phosphate of Ammonia absorbed by one gland suffices to cause it to transmit some influence to the basal & bending part of the tentacles, which sweep through a semicircle of 180o.— Ph. of Ammonia is far more powerful than the Nitrate of Amm., & the Nitrate more powerful than the Carbonate, all in causing inflection: though the latter salt causes aggregation much more quickly than the two former salts.
\frac{1}{140,000}
of a grain of the carbonate causes aggregation.—
(5) Water at 120o–125o Fah. quickly causes inflection & aggregation, (not so at higher temperatures)—
(6) You have seen what I call aggregation of the protoplasmic contents of the cells. A multitude of causes which excite inflection induce aggregation; but not all the causes. There may be aggregation without inflection. The process supervenes at the proper rate (or not at all), only when the protoplasm is uninjured (it will not occur in a ruptured cell) & in an oxygenated condition.
The protoplasm seems in so unstable a condition that almost any cause causes the granules to aggregate; for instance a minute particle of glass or hair on a gland.
(7) Objects of any kind placed on the disc not only cause the marginal tentacles to be inflected; but the glands secrete more copiously, & the secretion becomes acid: & according to Frankland the acid belongs to the acetic series & is nearest to Propionic.5
(8) I think the most interesting result is about digestion about which you know. There is now no exception to the rule that whatever substance (& I have tried many) pepsin & hydrochloric acid will digest so will Drosera; & what the former cannot digest, the latter cannot. It is marvellous to see the blade of the leaf convert itself with the overarching tentacles into a temporary stomach & pour out an acid secretion with some ferment so closely analogous to pepsin. I shd have said that B. Sanderson ascertained for me that propionic acid & its several allies can digest with pepsin.6
(9) The action of poisons is remarkable but too long,— strychnine & quinine &c are poisonous; but many deadly poisons to animals, as atropine curare, are quite innocuous
The poison of the cobra (given me by Dr Fayrer)7 is absorbed & seems to act as a pleasant stimulus to the protoplasm, for after an immersion of 50 hours in a strong solution, I never saw before the protoplasm in such vigorous spontaneous movement. What a profound difference between animal & vegetable protoplasm!
(10) Camphor is a stimulant, so that after an immersion of 3m a touch on the glands will cause the tentacles to bend, which otherwise wd not have sufficed.
(11) I dare not say anything about the lines of transmission of the motor influence from the central to the marginal tentacles: I doubt about my old experiments. I can (however) confirm fully Dr Nitschke’s statement that if a minute bit of meat be placed on one side of disc, not only do the surrounding tentacles bend, but they direct themselves to the points where the meat lies. It is very striking to put one atom of meat on one side of disc & another atom on the opposite side, & observe the positions of the inflected tentacles.8
But I shall have wearied you out.— I have no amanuensis to help me, & am very sorry for bad writing.
We go on Saturday for few days to Abinger (T. H Farrer) & thence to W. at Southampton.9
God Help you reading this letter.—
Hooker wanted to present his experimental results on the digestive ability of the tropical pitcher-plant, Nepenthes, at the meeting of the British Association for the Advancement of Science (see letter from J. D. Hooker, 18 July 1874 and n. 2).
In the summer of 1873, CD had described his work on Drosera (sundew) to John Scott Burdon Sanderson, who proposed testing for electrical changes in the leaves. CD suggested that Dionaea would be more suitable as an experimental subject (see Correspondence vol 21, letter from J. S. Burdon Sanderson, 13 August [1873], and letter to J. S. Burdon Sanderson, 15 August 1873). Burdon Sanderson had lectured on electrical phenomena associated with leaf contraction in Dionaea muscipula (the Venus fly trap) at the Royal Institution of Great Britain (see Burdon Sanderson 1874a, 1874b).
CD published the results of his investigations on Drosera in Insectivorous plants.
Edward Frankland had made the comment in his letter of 10 October 1873 (Correspondence vol 21). Acetic series: i.e. carboxylic acids. Acetic, butyric, formic, and propionic acid (now usually known as propanoic acid) are carboxylic acids.
See letter from J. S. Burdon Sanderson, 30 March [1874]).
See letter from Joseph Fayrer, 17 June 1874.
CD quoted Theodor Nitschke on this point in Insectivorous plants, p. 244 (see also Nitschke 1860, p. 240).
CD was away from home from 25 July until 24 August 1874; the first five days were spent at the home of Thomas Henry Farrer and the remainder at William Erasmus Darwin’s (see ‘Journal’ (Appendix II)).
Royal Botanic Gardens, Kew (JDH/3/6 Insectivorous plants 1873–8: 32–37)
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Barrier - Maple Help
Home : Support : Online Help : Programming : Grid Package : Barrier
block until all processes have reached this routine
The Barrier command blocks all jobs initiated by the Launch command until all processes have executed the Barrier command.
This is useful for synchronizing execution. It can be used to force node 0 to wait for all other nodes to finish computing.
This function is currently not available in "hpc" mode.
In this example nodes with a higher node number get more work to do than those with lower node numbers. In fact, node 0 has virtually nothing to do and quickly exits. Because the job is finished when node 0 is done, you will see very little or no output from the other nodes.
dosome := proc()
local i, me;
for i from 1 to 10^(me+3) do
if i mod 10^5 = 0 then
print(me,i);
print(me,"done");
\mathrm{Grid}[\mathrm{Launch}]\left(\mathrm{dosome},\mathrm{numnodes}=4\right)
0, "done"
This procedure is identical to the one above except that a call to Barrier is added at the end of the procedure. This will force all nodes to wait and synchronize around that point. Unlike the previous example, this time every node has a chance to finish.
doall:= proc()
\mathrm{Grid}[\mathrm{Launch}]\left(\mathrm{doall},\mathrm{numnodes}=4\right)
The Grid[Barrier] command was introduced in Maple 15.
Grid:-Launch
Grid:-Setup
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