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Convergence in probability: Intuition: The probability that Xn differs from the X by more than ε (a fixed distance) is 0. Contents . %%EOF The weak law of large numbers (WLLN) tells us that so long as $E(X_1^2)<\infty$, that Under the same distributional assumptions described above, CLT gives us that n (X ¯ n − μ) → D N (0, E (X 1 2)). Yes, you are right. (max 2 MiB). The concept of convergence in distribution is based on the … 6 Convergence of one sequence in distribution and another to … probability zero with respect to the measur We V.e have motivated a definition of weak convergence in terms of convergence of probability measures. We say V n converges weakly to V (writte • Convergence in probability Convergence in probability cannot be stated in terms of realisations Xt(ω) but only in terms of probabilities. It’s clear that $X_n$ must converge in probability to $0$. CONVERGENCE OF RANDOM VARIABLES . 288 0 obj <>stream Suppose B is the Borel σ-algebr n a of R and let V and V be probability measures o B).n (ß Le, t dB denote the boundary of any set BeB. Note that if X is a continuous random variable (in the usual sense), every real number is a continuity point. I will attempt to explain the distinction using the simplest example: the sample mean. This is typically possible when a large number of random eﬀects cancel each other out, so some limit is involved. It is easy to get overwhelmed. This leads to the following deﬁnition, which will be very important when we discuss convergence in distribution: Deﬁnition 6.2 If X is a random variable with cdf F(x), x 0 is a continuity point of F if P(X = x 0) = 0. Xt is said to converge to µ in probability … Suppose that fn is a probability density function for a discrete distribution Pn on a countable set S ⊆ R for each n ∈ N ∗ +. 5.2. $$Is n the sample size? Convergence in probability. R ANDOM V ECTORS The material here is mostly from • J. Deﬁnitions 2. Just hang on and remember this: the two key ideas in what follows are \convergence in probability" and \convergence in distribution." By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy, 2020 Stack Exchange, Inc. user contributions under cc by-sa. Also, Could you please give me some examples of things that are convergent in distribution but not in probability? Types of Convergence Let us start by giving some deﬂnitions of diﬁerent types of convergence. n!1 0. Convergence in distribution is the weakest form of convergence typically discussed, since it is implied by all other types of convergence mentioned in this article. Note that although we talk of a sequence of random variables converging in distribution, it is really the cdfs that converge, not the random variables. Noting that \bar{X}_n itself is a random variable, we can define a sequence of random variables, where elements of the sequence are indexed by different samples (sample size is growing), i.e. x) = 0. n!1 . This question already has answers here: What is a simple way to create a binary relation symbol on top of another? In econometrics, your Z is usually nonrandom, but it doesn’t have to be in general. 0 1. If it is another random variable, then wouldn't that mean that convergence in probability implies convergence in distribution? Definition B.1.3. X. n 2.1.2 Convergence in Distribution As the name suggests, convergence in distribution has to do with convergence of the distri-bution functions of random variables. dY. The basic idea behind this type of convergence is that the probability of an “unusual” outcome becomes smaller and smaller as the sequence progresses. In other words, for any xed ">0, the probability that the sequence deviates from the supposed limit Xby more than "becomes vanishingly small. Note that the convergence in is completely characterized in terms of the distributions and .Recall that the distributions and are uniquely determined by the respective moment generating functions, say and .Furthermore, we have an equivalent'' version of the convergence in terms of the m.g.f's I posted my answer too quickly and made an error in writing the definition of weak convergence. Then X_n does not converge in probability but X_n converges in distribution to N(0,1) because the distribution of X_n is N(0,1) for all n. 1.1 Almost sure convergence Deﬁnition 1. By using our site, you acknowledge that you have read and understand our Cookie Policy, Privacy Policy, and our Terms of Service. (4) The concept of convergence in distribtion involves the distributions of random ari-v ables only, not the random ariablev themselves. Proposition7.1Almost-sure convergence implies convergence in … Convergence in probability gives us confidence our estimators perform well with large samples. Viewed 32k times 5.$$\lim_{n \rightarrow \infty} F_n(x) = F(x),$$1.2 Convergence in distribution and weak convergence p7 De nition 1.10 Let P n;P be probability measures on (S;S).We say P n)P weakly converges as n!1if for any bounded continuous function f: S !R Z S f(x)P n(dx) ! Formally, convergence in probability is defined as And, no, n is not the sample size. h����+�Q��s�,HC�ƌ˄a�%Y�eeŊd뱰�c�BY()Yِ��\J4al�Qc��,��o����;�{9�y_���+�TVĪ:����OZC k��������� ����U\[�ux�e���a;�Z�{�\��T��3�g�������dw����K:{Iz� ��]R�؇=Q��p;���I��bJ%�k�U:"&��M�:��8.jv�Ź��;���w��o1+v�G���Aj��X��菉�̐,�]p^�G�[�a����_������9�F����s�e�i��,uOrJ';I�J�ߤW0 Na�q_���j���=7� �u�)� �?��ٌ�f5�G�N㟚V��ß x�Nk Topic 7. h�ĕKLQ�Ͻ�v�m��*P�*"耀��Q�C��.$$, $$\sqrt{n}(\bar{X}_n-\mu) \rightarrow_D N(0,E(X_1^2)).$$, $$\lim_{n \rightarrow \infty} F_n(x) = F(x),$$, https://economics.stackexchange.com/questions/27300/convergence-in-probability-and-convergence-in-distribution/27302#27302. Convergence in distribution in terms of probability density functions. Convergence in probability and convergence in distribution. And $Z$ is a random variable, whatever it may be. $$\forall \epsilon>0, \lim_{n \rightarrow \infty} P(|\bar{X}_n - \mu| <\epsilon)=1. I have corrected my post. suppose the CLT conditions hold: p n(X n )=˙! Xn p → X. d: Y n! In particular, for a sequence X1, X2, X3, ⋯ to converge to a random variable X, we must have that P( | Xn − X | ≥ ϵ) goes to 0 as n → ∞, for any ϵ > 0. Click here to upload your image Over a period of time, it is safe to say that output is more or less constant and converges in distribution. Under the same distributional assumptions described above, CLT gives us that Convergence in Probability. For example, suppose X_n = 1 with probability 1/n, with X_n = 0 otherwise. It tells us that with high probability, the sample mean falls close to the true mean as n goes to infinity.. We would like to interpret this statement by saying that the sample mean converges to the true mean.$$\forall \epsilon>0, \lim_{n \rightarrow \infty} P(|\bar{X}_n - \mu| <\epsilon)=1. However, $X_n$ does not converge to $0$ according to your definition, because we always have that $P(|X_n| < \varepsilon ) \neq 1$ for $\varepsilon < 1$ and any $n$. where $\mu=E(X_1)$. Convergence of the Binomial Distribution to the Poisson Recall that the binomial distribution with parameters n ∈ ℕ + and p ∈ [0, 1] is the distribution of the number successes in n Bernoulli trials, when p is the probability of success on a trial. %PDF-1.5 %���� $$plim\bar{X}_n = \mu,$$ The answer is that both almost-sure and mean-square convergence imply convergence in probability, which in turn implies convergence in distribution. 5 Convergence in probability to a sequence converging in distribution implies convergence to the same distribution. Knowing the limiting distribution allows us to test hypotheses about the sample mean (or whatever estimate we are generating). We write X n →p X or plimX n = X. Convergence in distribution means that the cdf of the left-hand size converges at all continuity points to the cdf of the right-hand side, i.e. most sure convergence, while the common notation for convergence in probability is X n →p X or plim n→∞X = X. Convergence in distribution and convergence in the rth mean are the easiest to distinguish from the other two. 4 Convergence in distribution to a constant implies convergence in probability. A sequence of random variables {Xn} is said to converge in probability to X if, for any ε>0 (with ε sufficiently small): Or, alternatively: To say that Xn converges in probability to X, we write: Convergence in probability gives us confidence our estimators perform well with large samples. e.g. If fn(x) → f∞(x) as n → ∞ for each x ∈ S then Pn ⇒ P∞ as n → ∞. convergence of random variables. The concept of convergence in probability is based on the following intuition: two random variables are "close to each other" if there is a high probability that their difference will be very small. To say that Xn converges in probability to X, we write. We note that convergence in probability is a stronger property than convergence in distribution. Convergence in distribution of a sequence of random variables. Although convergence in distribution is very frequently used in practice, it only plays a minor role for the purposes of this wiki. P n!1 X, if for every ">0, P(jX n Xj>") ! You can also provide a link from the web. $$\bar{X}_n \rightarrow_P \mu,$$. Your definition of convergence in probability is more demanding than the standard definition. Then define the sample mean as $\bar{X}_n$. A quick example: $X_n = (-1)^n Z$, where $Z \sim N(0,1)$. This is ﬁne, because the deﬁnition of convergence in 4 distribution requires only that the distribution functions converge at the continuity points of F, and F is discontinuous at t = 1. I understand that $X_{n} \overset{p}{\to} Z$ if $Pr(|X_{n} - Z|>\epsilon)=0$ for any $\epsilon >0$ when $n \rightarrow \infty$. Convergence in Distribution [duplicate] Ask Question Asked 7 years, 5 months ago. Convergence in probability is stronger than convergence in distribution. 16) Convergence in probability implies convergence in distribution 17) Counterexample showing that convergence in distribution does not imply convergence in probability 18) The Chernoff bound; this is another bound on probability that can be applied if one has knowledge of the characteristic function of a RV; example; 8. Convergence in probability. Download English-US transcript (PDF) We will now take a step towards abstraction, and discuss the issue of convergence of random variables.. Let us look at the weak law of large numbers. dY, we say Y n has an asymptotic/limiting distribution with cdf F Y(y). Active 7 years, 5 months ago. X a.s. n → X, if there is a (measurable) set A ⊂ such that: (a) lim. Z S f(x)P(dx); n!1: Econ 620 Various Modes of Convergence Deﬁnitions • (convergence in probability) A sequence of random variables {X n} is said to converge in probability to a random variable X as n →∞if for any ε>0wehave lim n→∞ P [ω: |X n (ω)−X (ω)|≥ε]=0. Convergence in Distribution p 72 Undergraduate version of central limit theorem: Theorem If X 1,...,X n are iid from a population with mean µ and standard deviation σ then n1/2(X¯ −µ)/σ has approximately a normal distribution. • Convergence in mean square We say Xt → µ in mean square (or L2 convergence), if E(Xt −µ)2 → 0 as t → ∞. Constant and converges in distribution is very frequently used in practice, it is safe to say X.! 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# What does orthogonal mean in the context of statistics?

In other contexts, orthogonal means "at right angles" or "perpendicular".

What does orthogonal mean in a statistical context?

Thanks for any clarifications.

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Thanks for the question. I have asked a more general one: what is so common among all cases of orthogonality. I was also interested to know how does statistical independence satisfy this property? physics.stackexchange.com/questions/67506 –  Val Jun 30 at 14:52
I am surprised that none of the answers here mention that usually it is meant in the mathematical "linear algebra" sense of the word. For example, when we speak of an "orthogonal set of variables" usually it is meant that $X^{T}X=I$ for the matrix with the set of variables $X$. "orthonormal" is used as well. –  probabilityislogic Sep 13 at 7:25
@probability "Orthogonal" has meaning for a vector space with a quadratic form $Q$: two vectors $v$ and $w$ are orthogonal if and only if $Q(v,w)=0$. "Orthonormal" means in addition that $Q(v,v)=1=Q(w,w)$. Thus "orthogonal" and "orthonormal" are not synonymous, nor are they restricted to finite matrices. (E.g., $v$ and $w$ may be elements of a Hilbert space, such as the space of $L^2$ complex-valued functions on $\mathbb{R}^3$ used in classical quantum mechanics.) –  whuber Sep 13 at 15:02

It means they are independent to each other. Independent variables are often considered to be at right angles to each other. For example on the X-Y plane the X and Y axis are said to be orthogonal because if a given point's x value changes, its y value remains the same, and vice versa (i.e. they are independent). See also http://en.wikipedia.org/wiki/Independence_(probability_theory) and http://en.wikipedia.org/wiki/Orthogonality

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Because the distinction between correlation and lack of dependence is important, equating orthogonality with independence is not a good thing to do. –  whuber Jun 20 '11 at 20:26

@Mien already provided an answer, and, as pointed out by @whuber, orthogonal means uncorrelated. However, I really wish people would provide some references. You might consider the following links helpful since they explain the concept of correlation from a geometric perspective.

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I can't make a comment because I don't have enough points, so I'm forced to speak my mind as an answer, please forgive me. From the little I know, I disagree with the selected answer by @crazyjoe because orthogonality is defined as

$$E[XY^{\star}] = 0$$

So:

If $Y=X^2$ with symmetric pdf they they are dependent yet orthogonal.

If $Y=X^2$ but pdf zero for negative values, then they dependent but not orthogonal.

Therefore, orthogonality does not imply independence.

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If X and Y are independent then they are Orthogonal. But the converse is not true as pointed out by the clever example of user497804. For the exact definitions refer to

Orthogonal : Complex-valued random variables $C_1$ and $C_2$ are called orthogonal if they satisfy ${\rm cov}(C_1,C_2)=0$

(Pg 376, Probability and Random Processes by Geoffrey Grimmett and David Stirzaker)

Independent: The random variables $X$ and $Y$ are independent if and only if $F(x,y) = F_X(x)F_Y(y)$ for all $x,y \in \mathbb{R}$

which, for continuous random variables, is equivalent to requiring that $f(x,y) = f_X(x)f_Y(y)$

(Page 99, Probability and Random Processes by Geoffrey Grimmett and David Stirzaker)

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It's most likely they mean 'unrelated' if they say 'orthogonal'; if two factors are orthogonal (e.g. in factor analysis), they are unrelated, their correlation is zero.

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The correlation coefficient is (or is naturally interpretable as) the cosine of an angle. When it is zero, what do you think the angle is? :-) Uncorrelated does not mean unrelated! –  whuber Jun 20 '11 at 14:14
I'm not saying you're wrong, but could you give me an example of something that's uncorrelated and related; or vice versa? I'm not sure I understand the difference. –  Mien Jun 20 '11 at 14:43
And yes, I know that that angle would be 90°. A right angle is orthogonal. –  Mien Jun 20 '11 at 14:57
Let $X$ be a random variable taking values in $\{-1,0,1\}$ with equal probability and let $Y=X^2$. The correlation between $X$ and $Y$ is $\rho_{X,Y}=0$, but clearly they are related: $Y$ is a function of $X$. –  Max Jun 20 '11 at 19:34
Ah yes, thank you. But the opposite isn't possible, is it (if there isn't a third variable or something similar)? –  Mien Jun 20 '11 at 20:23

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# An Inequality with Exradii and an Altitude

### Solution

We use the common notations: $a,b,c\,$ the side lengths; $\displaystyle s=\frac{a+b+c}{2}\,$ the semiperimeter; $[\Delta ABC]=S\,$ the area of $\Delta ABC.$

Let $s-a=x,\,$ $s-b=y,\,$ $s-c=z.\,$ Then

$\displaystyle \frac{1}{r_b}=\frac{y}{S},\,$ $\displaystyle \frac{1}{r_c}=\frac{z}{S},\,$ $\displaystyle \frac{1}{h_a}=\frac{y+z}{2S}.$

Denote $\displaystyle \frac{y}{S}=u\,$ and $\displaystyle \frac{z}{S}=v.\,$ The required inequality reduces to

$\sqrt{u^2+v+1}+\sqrt{v^2+u+1}\ge 2\sqrt{(u+v)^2+2(u+v)+4},$

which, after squaring becomes

$2\sqrt{(u^2+v+1)(v^2+u+1)}\ge 2uv+u+v+2.$

Let $u+v=2m\,$ and $uv=n.\,$ Squaring again leads to $(m^2-n)(8m+3)\ge 0,\,$ which is true, because, by the AM-GM inequality, $m^2\ge n.\,$

### Acknowledgment

I am grateful to Leo Giugiuc who kindly communicated to me a problem by Nguyen Viet Hung, along with a solution of his. The problem was originally posted at the mathematical inequalities facebook group.


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# Olinthus Gilbert Gregory

### Quick Info

Born
29 January 1774
Yaxley, Huntingdonshire (now Cambridgeshire), England
Died
2 February 1841
Woolwich, near London, England

Summary
Olinthus Gregory was an English mathematician, author and editor. He was one of the founders of the Royal Astronomical Society.

### Biography

Olinthus Gregory's parents are unknown but they must have been working class. His first schooling was in his home town but then he spent over ten years at Richard Weston's boarding school. Weston (1732-1806) was primarily a botanist who published a number of works on gardening and botany but he was also a competent mathematician. The training that he gave Gregory was of such a high standard that in 1793, at the age of nineteen, Gregory published Lessons, Astronomical and Philosophical. It is interesting to give the full title of the work which is so long that it is seldom quoted: Lessons, Astronomical and Philosophical: for the Amusement and Instruction of British Youth: Being an Attempt to Explain and Account for the Most Usual Appearances in Nature in a Familiar Manner, From Established Principles, the Whole Interspersed With Moral Reflections. Gregory was able to publish that text-book having acquired John Joshua Proby, earl of Carysfort, as his patron. Proby was a fellow of the Royal Society and held high political office at the time. Gregory's work, covering a wide range of topics, became a popular school-book. We quote his ideas given in the book for regulating musical time. He proposes:-
... the substitution of proper characters to denote the different kinds and velocities of musical time, instead of those vague, indefinite ones which are now in use. ... Every composer of musical airs would be of real service to the practitioner if he would point out the absolute rate at which his music is to be performed; this would be no difficult task as he would only have to mention the length of a pendulum which would make one complete vibration in the time that part of a bar called a beat was performing.
At this time Gregory was still being taught by Richard Weston who suggested to him that he contribute, as Weston did himself, to the mathematical problems in the Ladies' Diary. Here are two of the questions Gregory proposed for 1793-94:
The sum of all the chords drawn in a semicircle from one end of the diameter to those points which divide the circumference into a certain number of equal parts (together with the diameter) is 429.7877. Now if the diameter be 100, what is the number of chords and the length of each?

There is an inflexible rod, void of gravity, 26 inches long, at one end of which is suspended 1 cwt. 1 qr. $23\large\frac{1}{2}\normalsize$ lbs. of sugar, in a barrel that weighs 21 lb.; at 1 foot distance from this end hangs a weight of $8 \large\frac{1}{2}\normalsize$ lbs. and 4 inches further a weight of $6\large\frac{1}{2}\normalsize$ lbs.; also at 7 inches from the other end weight of 5 lb., and at this other end a weight of $4\large\frac{1}{2}\normalsize$ lbs. Query the point of the rod, which being made a fulcrum, these weights etc will remain in equilibrio?
In 1794 he composed a treatise on the slide rule. This was never published but it brought him to the attention of Charles Hutton. Gregory, perhaps encouraged by Hutton, began to think of an academic career. However, while studying Christianity deeply, he went though various different beliefs but felt he could not subscribe to the Thirty-Nine Articles of religion which was required before a student could matriculate. Nevertheless, he went to Cambridge around 1798 where he taught mathematics, ran a bookshop for about a year, and worked as an editor of the Cambridge Intelligencer for about six months. This radical newspaper began publishing in 1793 and ended publication in 1803. Gregory still appears as "Teacher of Mathematics, Cambridge" on the title page of his book A Treatise on Astronomy published in 1802. He dedicated the work to Charles Hutton:-
In testimony of admiration and esteem for the eminent talents and genius he has so frequently and usefully displayed in his public capacity; and for those generous and amiable qualities which shed a lustre over the path of private life ...
Gregory explains in the Preface of A Treatise on Astronomy that he has religious reason for studying the subject. While making such a study, he writes:-
... is it possible that we should not be impressed with a sense of the unlimited power, unbounded wisdom, and infinite goodness, of the adorable Creator and Governor? The great excellence of Astronomy then, even as a promoter of morality and devotion, must be admitted if it appear that it furnishes us with stronger arguments to prove the existence of a supreme, intelligent First Cause - shows more effectually his power and wisdom - or gives us more clear and just notions of his other attributes and perfections ...
His friendship with Hutton certainly stood him in good stead for Hutton recommended him for the new position of second mathematical master at the Royal Military Academy at Woolwich, a post he took up in December 1802. He married Anne Beddome (born 1789), a daughter of Boswell Brandon Beddome and Ann Wilkins, on 20 December 1809; they had three children, Annie (born 1811), Boswell Robert (born 1813) who sadly drowned in the Thames, and Charles Hutton (1817-1898) who became a well-known engineer. In May 1821 Gregory was appointed to the chair of mathematics at the Royal Military Academy, Woolwich, and held this post until he retired in June 1838. For the last ten years he had suffered from an illness which became progressively more severe. He died at his home on Woolwich Common.

In addition to his duties at Woolwich, Gregory undertook several other tasks. He edited the Gentleman's Diary from 1802 until 1819. From 1818, he edited the Ladies' Diary, was involved in the production of White's Ephemeris (second only to the Nautical Almanac as an astronomical almanac), and was in charge of acquiring the astronomical data necessary for all the almanacs published by the Worshipful Company of Stationers and Newspaper Makers of London. He was an important member of the Philosophical Society of London which grew up in the early 1800s and was officially founded in 1811 with its own rooms in Fleet Street, London. He was elected vice-president of the Society and lectured to it on ballistics. However, the Society only survived until 1815. He then became a founder member of the Royal Astronomical Society, being one of fourteen to attend the first meeting of that Society on Wednesday 12 January 1820. Of these fourteen, Gregory was elected to serve on the committee along with John Herschel, Charles Babbage, Henry Colebrooke, Thomas Colby, Daniel Moore, William Pearson, and Francis Baily. Gregory served as secretary to the Royal Astronomical Society from 1824 to 1828 and as vice-president in 1829-30. Perhaps it is worth noting at this point that Herschel considered Gregory [3]:-
... as rather too coarse in his strictures and behaviour.
Proposals to set up a secular university in London were made in 1825 and Gregory soon added his weight to the movement. By the end of the year Gregory was serving on a ten-man committee which had been set up to interview and select the university's teaching staff. His name was inscribed on the foundation stone of the new University, laid in Gower Street on 30 April 1827.

Let us now look at a few more of Gregory's publications, in addition to those already mentioned above. In 1806 he published the 2-volume work A Treatise of Mechanics, theoretical, practical and descriptive. The book was dedicated, as his earlier work, to John Joshua Proby, earl of Carysfort. Gregory wrote:-
Although the promotion of the arts and sciences has been ever deemed the proper province of the great and noble, yet it has not always been the good fortune of those who employ their time and exertions in the dissemination of knowledge to meet with a patron who is at once distinguished by his rank, his talents, and his zeal in the encouragement of useful performances. I cannot, therefore but esteem it a high honour and a source of considerable gratification that I am permitted to present a 'Treatise of Mechanics' to the public under you Lordship's protection ...
In the Preface Gregory explains why he wrote the book:-
It has of late been a too prevailing opinion in this country that a man may become celebrated as a natural philosopher and be very successful in the application of his knowledge to practical purposes, particularly in the construction of machinery, while he is completely unacquainted with the principles of mathematics. Among a variety of unpleasant consequences which have resulted fro this erroneous opinion may be reckoned the rapid decline of the mathematical sciences in Britain, the dissemination of superficial and vague notions on physical topics and the absolute necessity of having recourse to foreign publications for profound and extensive information on these subjects which we should have supposed had not experience convinced us to the contrary, Englishmen would have been proud to cultivate since they were first placed upon an unshaken basis in the 'Principia' of our own countryman, the illustrious Newton.
The first volume was divided into five books: Statics; Dynamics, Hydrostatics, Hydrodynamics, and Pneumatics. The second volume covered Practical Mechanics and Description of Machines (Alphabetically arranged).

In 1811 Gregory published Letters on the Evidences, Doctrines, and Duties of the Christian Religion. He explains in the Preface that the letters were written to a friend:-
... much younger than myself, who had a considerable acquaintance with almost all except religious subjects. He expressed much surprise that a person of my habits and pursuits in other respects should adopt the religious notions I had long entertained ...
In this work he states:-
We do not deny that the scheme of revelation has its difficulties: for if the things of nature are often difficult to comprehend, it would be strange indeed if supernatural matters were so simple, and obvious, and suited to finite capacities, as never to startle or puzzle us at all. Origen remarked, with his usual sagacity, that "he who believes the Scripture to have proceeded from him who is the Author of Nature, may well expect to find the same sort of difficulties in it as are found in the constitution of nature:" and this obviously suggests the reflection, that he, who denies the Bible to have come from God on account of these difficulties, may, for exactly the same reason, deny that the world was formed by him.
Further books by Gregory include: Elements of Plane and Spherical Trigonometry (1816), Mathematics for Practical Men (1825), Memoirs of ... John Mason Good, M.D. (1828), Memoir of the Rev. Robert Hall (1832), Aids and Incentives to the Acquisition of Knowledge (1838), and Hints to the Teachers of Mathematics (1840). The 1838 work is Gregory's Farewell Address on retiring as Professor of Mathematics, delivered on 7 June of that year. In it he addressed the first year students of the Royal Military Academy:-
The genuine object of all sound education is the development of the intellectual, the moral, and the bodily faculties of man; or, as it has been sometimes more tersely expressed, the improvement and application of head, heart, and limb. The system of education in the institution in which you have the honour to receive instruction, embraces all this. The blame will be your own, and it will through life be the subject of regret, if any of you quit this Academy without having acquired the manners of a gentleman, the principles of a man of honour and high and pure morality, the ornamental facilities of an artist, and a competent store of literary and philosophical knowledge.
We should not come away with the impression that all of Gregory's contributions came from his books. He also conducted important experiments. For example he carried out experiments with a pendulum in Shetland and also made astronomical observations there, reporting on these experiments in an article in the Philosophical Magazine in 1818. In 1823 he carried out experiments at Woolwich to determine the velocity of sound using mortars, guns and muskets which were fired at various distances from observers. He came up with the result of 1100 feet per second which is a good result (the speed of sound in dry air at 20° C is 1125 feet per second).

Gregory received many honours for his achievements. In the 1826 edition of his Mechanics text he lists himself as: Corresponding Associate of the Academy of Dijon; Honorary Member of the Literary and Philosophical Society of New York; Honorary Member of the New York Historical Society; Honorary Member of the Literary and Philosophical Society of Newcastle-upon-Tyne; Honorary Member of the Antiquarian Society of Newcastle-upon-Tyne; Honorary Member of the Cambridge Philosophical Society; Honorary Member of the Institution of Civil Engineers; etc. The 'etc.' here includes honorary memberships of societies in Bristol and Paris. He was awarded an honorary M.A. by the University of Aberdeen in 1806 and an honorary LL.D. two years later. He was elected president of the Woolwich Institution for the Advancement of Literary, Scientific and Technical Knowledge from its foundation in 1838. Gregory, together with his friend Alexander John Scott, had been the founders of the Institution.

In [6] Gregory's character is recorded as follows:-
... all he did and said was dictated by benevolence of feeling, and he was man of unbounded charity. As a Christian, he was moral and devout, and as a scholar he merited and obtained consideration of the first mathematicians of the day; his great zeal in his vocation, his parental kindness, his earnest and impressive admonitions, his entertaining, improving, and philosophical conversation, and his ever readiness to assist, will be gratefully remembered by many. He took a warm interest in the cultivation of mathematics, to which he may be said to have devoted, with indefatigable perseverance, nearly the whole of his valuable life.

### References (show)

1. Biography by Alexander Gordon, rev. Ben Marsden, in Dictionary of National Biography (Oxford University Press, Oxford, 2004). See THIS LINK.
2. A Gordon, Gregory, Olinthus Gilbert (1774-1841), Dictionary of National Biography (Oxford, 1890).
3. D P Miller, The Revival of the Physical Sciences in Britain, 1815-1840, Osiris (2) 2 (1986), 107-134.
4. Obituary: Olinthus Gregory, The Times (6 February 1841).
5. Obituary: Olinthus Gregory, Monthly Notices of the Astronomical Society of London 5 (1839-43), 81-82.
6. Gregory, Olinthus Gilbert, The English cyclopaedia: a new dictionary of universal knowledge 3 (Bradbury & Evans, 1856), 193-194.
7. A Rice, Inspiration or Desperation? Augustus De Morgan's Appointment to the Chair of Mathematics at London University in 1828, The British Journal for the History of Science 30 (3) (1997), 257-274.

### Additional Resources (show)

Other pages about Olinthus Gregory:

Other websites about Olinthus Gregory:

Written by J J O'Connor and E F Robertson
Last Update November 2010


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# Solve the linear congruence x+2y -= 1(mod 5) 2x+y -= 1(mod 5)

Question
Congruence
Solve the linear congruence
$$\displaystyle{x}+{2}{y}\equiv{1}{\left(\text{mod}{5}\right)}$$
$$\displaystyle{2}{x}+{y}\equiv{1}{\left(\text{mod}{5}\right)}$$

2021-02-06
Step 1
Given a system of linear congruences
$$\displaystyle{x}+{2}{y}\equiv{1}{\left(\text{mod}{5}\right)}$$
$$\displaystyle{2}{x}+{y}\equiv{1}{\left(\text{mod}{5}\right)}$$
Solve it.
Step 2
Multiply the first congruence by 2.
$$\displaystyle{2}{x}+{4}{y}\equiv{2}{\left(\text{mod}{5}\right)}$$
Add the preceding one with the second congruence in the system.
$$\displaystyle{4}{x}+{5}{y}\equiv{3}{\left(\text{mod}{5}\right)}$$
$$\displaystyle\Rightarrow{4}{x}\equiv{3}{\left(\text{mod}{5}\right)}$$
Step 3
Multiply the congruence by $$\displaystyle{4}^{{-{1}}}{\left(\text{mod}{5}\right)}={4}$$ to get
$$\displaystyle{16}{x}\equiv{12}{\left(\text{mod}{5}\right)}$$
$$\displaystyle\Rightarrow{x}\equiv{2}{\left(\text{mod}{5}\right)}$$
Step 4
Plugging back in x for the second equation.
$$\displaystyle{4}+{y}\equiv{1}{\left(\text{mod}{5}\right)}$$
$$\displaystyle\Rightarrow{y}\equiv-{3}{\left(\text{mod}{5}\right)}{\quad\text{or}\quad}{y}\equiv{2}{\left(\text{mod}{5}\right)}$$
Thus the solution of the system of congruences is
$$\displaystyle{x}\equiv{2}{\left(\text{mod}{5}\right)},{y}\equiv{2}{\left(\text{mod}{5}\right)}$$

### Relevant Questions

Solve the congruence equation:
2x=3 mod 5
Find all whole number solutions of the congruence equation.
$$\displaystyle{\left({2}{x}+{1}\right)}\equiv{5}\text{mod}{4}$$
Find all whole number solutions of the congruence equation.
$$(2x + 1)\equiv 5\ mod\ 4$$
Solve the set of congruences
$$\displaystyle{2}{x}\equiv{1}{\left(\text{mod}{5}\right)}$$
$$\displaystyle{x}\equiv{3}{\left(\text{mod}{4}\right)}$$
Solve the linear congruence
$$\displaystyle{7}{x}\equiv{13}{\left(\text{mod}{19}\right)}$$
$$\displaystyle{7}{x}+{3}{y}\equiv{10}{\left(\text{mod}{16}\right)}$$
Solve the following linear congruence: $$17x\ congruence\ 3(mod\ 210)$$
Solve the congruence $$\displaystyle{x}^{{20}}-{1}\equiv{0}{\left(\text{mod}{61}\right)}$$
Using Fermat's Little Theorem, solve the congruence $$\displaystyle{2}\cdot{x}^{{{425}}}+{4}\cdot{x}^{{{108}}}-{3}\cdot{x}^{{2}}+{x}-{4}\equiv{0}\text{mod}{107}$$.


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## Trigonometry (11th Edition) Clone

$$\sin\frac{x}{2}=\frac{\sqrt{13}}{4}$$
$$\cos x=-\frac{5}{8}\hspace{1.5cm}\frac{\pi}{2}\lt x\lt\pi\hspace{1.5cm}\sin\frac{x}{2}=?$$ Apply the half-angle identity for sine $$\sin\frac{x}{2}=\pm\sqrt{\frac{1-\cos x}{2}}$$ As $\frac{\pi}{2}\lt x\lt\pi$, we deduce that $\frac{\pi}{4}\lt\frac{x}{2}\lt\frac{\pi}{2}$. That area is in quadrant I, showing that the angle $\frac{x}{2}$ lies in quadrant I, where sines are positive. That means $\sin\frac{x}{2}\gt0$, so we need to choose the positive square root. $$\sin\frac{x}{2}=\sqrt{\frac{1-\cos x}{2}}$$ $$\sin\frac{x}{2}=\sqrt{\frac{1-\Big(-\frac{5}{8}\Big)}{2}}$$ $$\sin\frac{x}{2}=\sqrt{\frac{1+\frac{5}{8}}{2}}$$ $$\sin\frac{x}{2}=\sqrt{\frac{\frac{13}{8}}{2}}$$ $$\sin\frac{x}{2}=\sqrt{\frac{13}{16}}$$ $$\sin\frac{x}{2}=\frac{\sqrt{13}}{4}$$


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# Talk:Modular equivalence

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

Mod notation can be bulky and counter-intuitive. Here is an alternate mod notation that simply removes the middle line from the equivalence sign, places the modulus inside and drops the now moot 'mod'.

The following macros are defined for the Latex that follows and the second graphic shows the result:

Examples: In lieu of $x\equiv y \mod{z}$, we have $x\bm{z}y$

Rather than take both sides $\mod{m}$, we would take both sides $\bm{m}$

Taking all $m\bm{7}1\in \mathbb{Z}_{+}$ gives $\{1,8,15,\dots\}$

A non-equivalence can be written $9\bmn{4}3$

It can be chained ala $27\bm{11}5\bm{3}2$

and even be adapted to carry quotient information as in $77\bmq{13}{5}12$.

Here's how that looks when rendered:

Would $$7 \underset{3}{=} 1$$ work similarly?

--Christian Lawson-Perfect (talk) 15:06, 10 July 2021 (UTC)


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no way to change it manipulating the dtype of the data array.
 a=pyfits.open('filename.fits') a[1].header.update('TTYPE1','newname')
you need to change the header, using the update method of the right TTYPE and then write again the fits file using a.writeto.


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# 2.4 Electron Configurations (2023)

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The electron configuration of an atom is the representation of the arrangement of electrons distributed among the orbital shells and subshells. Commonly, the electron configuration is used to describe the orbitals of an atom in its ground state, but it can also be used to represent an atom that has ionized into a cation or anion by compensating with the loss of or gain of electrons in their subsequent orbitals. Many of the physical and chemical properties of elements can be correlated to their unique electron configurations. The valence electrons, electrons in the outermost shell, are the determining factor for the unique chemistry of the element.

## Introduction

Before assigning the electrons of an atom into orbitals, one must become familiar with the basic concepts of electron configurations. Every element on the Periodic Table consists of atoms, which are composed of protons, neutrons, and electrons. Electrons exhibit a negative charge and are found around the nucleus of the atom in electron orbitals, defined as the volume of space in which the electron can be found within 95% probability. The four different types of orbitals (s,p,d, and f) have different shapes, and one orbital can hold a maximum of two electrons. The p, d, and f orbitals have different sublevels, thus can hold more electrons.

As stated, the electron configuration of each element is unique to its position on the periodic table. The energy level is determined by the period and the number of electrons is given by the atomic number of the element. Orbitals on different energy levels are similar to each other, but they occupy different areas in space. The 1s orbital and 2s orbital both have the characteristics of an s orbital (radial nodes, spherical volume probabilities, can only hold two electrons, etc.) but, as they are found in different energy levels, they occupy different spaces around the nucleus. Each orbital can be represented by specific blocks on the periodic table. The s-block is the region of the alkali metals including helium (Groups 1 & 2), the d-block are the transition metals (Groups 3 to 12), the p-block are the main group elements from Groups 13 to 18, and the f-block are the lanthanides and actinides series.

Using the periodic table to determine the electron configurations of atoms is key, but also keep in mind that there are certain rules to follow when assigning electrons to different orbitals. The periodic table is an incredibly helpful tool in writing electron configurations. For more information on how electron configurations and the periodic table are linked, visit the Connecting Electrons to the Periodic Table module.

## Rules for Assigning Electron Orbitals

### Occupation of Orbitals

Electrons fill orbitals in a way to minimize the energy of the atom. Therefore, the electrons in an atom fill the principal energy levels in order of increasing energy (the electrons are getting farther from the nucleus). The order of levels filled looks like this:

1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d, and 7p

One way to remember this pattern, probably the easiest, is to refer to the periodic table and remember where each orbital block falls to logically deduce this pattern. Another way is to make a table like the one below and use vertical lines to determine which subshells correspond with each other.

### Pauli Exclusion Principle

The Pauli exclusion principle states that no two electrons can have the same four quantum numbers. The first three (n, l, and ml) may be the same, but the fourth quantum number must be different. A single orbital can hold a maximum of two electrons, which must have opposing spins; otherwise they would have the same four quantum numbers, which is forbidden. One electron is spin up (ms = +1/2) and the other would spin down (ms = -1/2). This tells us that each subshell has double the electrons per orbital. The s subshell has 1 orbital that can hold up to 2 electrons, the p subshell has 3 orbitals that can hold up to 6 electrons, the d subshell has 5 orbitals that hold up to 10 electrons, and the f subshell has 7 orbitals with 14 electrons.

Example 1: Hydrogen and Helium

The first three quantum numbers of an electron are n=1, l=0, ml=0. Only two electrons can correspond to these, which would be either ms = -1/2 or ms = +1/2. As we already know from our studies of quantum numbers and electron orbitals, we can conclude that these four quantum numbers refer to the 1s subshell. If only one of the ms values are given then we would have 1s1 (denoting hydrogen) if both are given we would have 1s2 (denoting helium). Visually, this is be represented as:

As shown, the 1s subshell can hold only two electrons and, when filled, the electrons have opposite spins.

### Hund's Rule

When assigning electrons in orbitals, each electron will first fill all the orbitals with similar energy (also referred to as degenerate) before pairing with another electron in a half-filled orbital. Atoms at ground states tend to have as many unpaired electrons as possible. When visualizing this processes, think about how electrons are exhibiting the same behavior as the same poles on a magnet would if they came into contact; as the negatively charged electrons fill orbitals they first try to get as far as possible from each other before having to pair up.

Example 2: Oxygen and Nitrogen

If we look at the correct electron configuration of the Nitrogen (Z = 7) atom, a very important element in the biology of plants: 1s2 2s2 2p3

We can clearly see that p orbitals are half-filled as there are three electrons and three p orbitals. This is because Hund's Rule states that the three electrons in the 2p subshell will fill all the empty orbitals first before filling orbitals with electrons in them. If we look at the element after Nitrogen in the same period, Oxygen (Z = 8) its electron configuration is: 1s2 2s2 2p4 (for an atom).

Oxygen has one more electron than Nitrogen and as the orbitals are all half filled the electron must pair up.

### The Aufbau Process

Aufbau comes from the German word "aufbauen" meaning "to build." When writing electron configurations, orbitals are built up from atom to atom. When writing the electron configuration for an atom, orbitals are filled in order of increasing atomic number. However, there are some exceptions to this rule.

Example 3: 3rd row elements

Following the pattern across a period from B (Z=5) to Ne (Z=10), the number of electrons increases and the subshells are filled. This example focuses on the p subshell, which fills from boron to neon.

• B (Z=5) configuration: 1s2 2s2 2p1
• C (Z=6) configuration:1s2 2s2 2p2
• N (Z=7) configuration:1s2 2s2 2p3
• O (Z=8) configuration:1s2 2s2 2p4
• F (Z=9) configuration:1s2 2s2 2p5
• Ne (Z=10) configuration:1s2 2s2 2p6

### Exceptions

Although the Aufbau rule accurately predicts the electron configuration of most elements, there are notable exceptions among the transition metals and heavier elements. The reason these exceptions occur is that some elements are more stable with fewer electrons in some subshells and more electrons in others (Table 1).

 Period 4: Period 5: Chromium: Z:24 [Ar] 3d54s1 Niobium: Z:41 [Kr] 5s1 4d4 Copper: Z:29 [Ar] 3d104s1 Molybdenum: Z:42 [Kr] 5s1 4d5 Ruthenium: Z:44 [Kr] 5s1 4d7 Rhodium: Z:45 [Kr] 5s1 4d8 Palladium: Z:46 [Kr] 4d10 Silver: Z:47 [Kr] 5s1 4d10 Period 6: Period 7: Lanthanum: Z:57 [Xe] 6s2 5d1 Actinium: Z:89 [Rn] 7s2 6d1 Cerium: Z:58 [Xe] 6s2 4f1 5d1 Thorium: Z:90 [Rn] 7s2 6d2 Gadolinium: Z:64 [Xe] 6s2 4f7 5d1 Protactium: Z:91 [Rn] 7s2 5f2 6d1 Platinum: Z:78 [Xe] 6s1 4f14 5d9 Uranium: Z:92 [Rn] 7s2 5f3 6d1 Gold: Z:79 [Xe] 6s1 4f14 5d10 Neptunium: Z:93 [Rn] 7s2 5f4 6d1 Curium: Z:96 [Rn] 7s2 5f7 6d1 Lawrencium: Z:103 [Rn] 7s2 5f14 7p1

## Writing Electron Configurations

When writing an electron configuration, first write the energy level (the period), then the subshell to be filled and the superscript, which is the number of electrons in that subshell. The total number of electrons is the atomic number, Z. The rules above allow one to write the electron configurations for all the elements in the periodic table. Three methods are used to write electron configurations:

1. orbital diagrams
2. spdf notation
3. noble gas notation

Each method has its own purpose and each has its own drawbacks.

### Orbital Diagrams

An orbital diagram, like those shown above, is a visual way to reconstruct the electron configuration by showing each of the separate orbitals and the spins on the electrons. This is done by first determining the subshell (s,p,d, or f) then drawing in each electron according to the stated rules above.

Example 4: Aluminum and Iridium

Write the electron configuration for aluminum and iridium.

SOLUTION

Aluminum is in the 3rd period and it has an atomic number of Z=13. If we look at the periodic table we can see that its in the p-block as it is in group 13. Now we shall look at the orbitals it will fill: 1s, 2s, 2p, 3s, 3p. We know that aluminum completely fills the 1s, 2s, 2p, and 3s orbitals because mathematically this would be 2+2+6+2=12. The last electron is in the 3p orbital. Also another way of thinking about it is that as you move from each orbital block, the subshells become filled as you complete each section of the orbital in the period. The block that the atom is in (in the case for aluminum: 3p) is where we will count to get the number of electrons in the last subshell (for aluminum this would be one electron because its the first element in the period 3 p-block). This gives the following:

Note that in the orbital diagram, the two opposing spins of the electron can be visualized. This is why it is sometimes useful to think about electron configuration in terms of the diagram. However, because it is the most time consuming method, it is more common to write or see electron configurations in spdf notation and noble gas notation. Another example is the electron configuration of iridium:

The electron configuration of iridium is much longer than aluminum. Although drawing out each orbital may prove to be helpful in determining unpaired electrons, it is very time consuming and often not as practical as the spdf notation, especially for atoms with much longer configurations. Hund's rule is also followed, as each electron fills up each 5d orbital before being forced to pair with another electron.

### spdf Notation

The most common way to describe electron configurations is to write distributions in the spdf notation. Although the distributions of electrons in each orbital are not as apparent as in the diagram, the total number of electrons in each energy level is described by a superscript that follows the relating energy level. To write the electron configuration of an atom, identify the energy level of interest and write the number of electrons in the energy level as its superscript as follows: 1s2. This is the electron configuration of helium; it denotes a full s orbital. The periodic table is used as a reference to accurately write the electron configurations of all atoms.

Example 5: Yttrium

Write the electronic configuration of Yttrium.

SOLUTION

Start with the straightforward problem of finding the electron configuration of the element yttrium. As always, refer to the periodic table. The element yttrium (symbolized Y) is a transition metal, found in the fifth period and in Group 3. In total it has thirty-nine electrons. Its electron configuration is as follows:

1s2 2s2 2p6 3s2 3p6 4s2 3d10 4p6 5s2 4d1

This is a much simpler and more efficient way to portray electron configuration of an atom. A logical way of thinking about it is that all that is required is to fill orbitals across a period and through orbital blocks. The number of elements in each block is the same as in the energy level it corresponds. For example, there are 2 elements in the s-block, and 10 elements in the d-block. Moving across, simply count how many elements fall in each block. Yttrium is the first element in the fourth period d-block; thus there is one electron in that energy level. To check the answer, verify that the subscripts add up to the atomic number. In this case, 2+2+6+2+6+2+10+6+2+1= 39 and Z=39, so the answer is correct.

A slightly more complicated example is the electron configuration of bismuth (symbolized Bi, with Z = 83). The periodic table gives the following electron configuration:

1s2 2s2 2p6 3s2 3p6 4s2 3d10 4p65s2 4d10 5p6 6s2 4f14 5d10 6p3

The reason why this electron configuration seems more complex is that the f-block, the Lanthanide series, is involved. Most students who first learn electron configurations often have trouble with configurations that must pass through the f-block because they often overlook this break in the table and skip that energy level. Its important to remember that when passing the 5d and 6d energy levels that one must pass through the f-block lanthanoid and actinoid series. Keeping this in mind, this "complex" problem is greatly simplified.

Another method (but less commonly used) of writing the spdf notation is the expanded notation format. This is the same concept as before, except that each individual orbital is represented with a subscript. The p, d, and f orbitals have different sublevels. The p orbitals are px, py, and pz, and if represented on the 2p energy with full orbitals would look like: 2px2 2py2 2pz2. The expanded notation for neon (Ne, Z=10) is written as follows:

1s2 2s2 2px2 2py2 2pz2

The individual orbitals are represented, but the spins on the electrons are not; opposite spins are assumed. When representing the configuration of an atom with half filled orbitals, indicate the two half filled orbitals. The expanded notation for carbon is written as follows:

1s2 2s2 2px1 2py1

Because this form of the spdf notation is not typically used, it is not as important to dwell on this detail as it is to understand how to use the general spdf notation.

### Noble Gas Notation

This brings up an interesting point about elements and electron configurations. As the p subshell is filled in the above example about the Aufbau principle (the trend from boron to neon), it reaches the group commonly known as the noble gases. The noble gases have the most stable electron configurations, and are known for being relatively inert. All noble gases have their subshells filled and can be used them as a shorthand way of writing electron configurations for subsequent atoms. This method of writing configurations is called the noble gas notation, in which the noble gas in the period above the element that is being analyzed is used to denote the subshells that element has filled and after which the valence electrons (electrons filling orbitals in the outer most shells) are written. This looks slightly different from spdf notation, as the reference noble gas must be indicated.

What is the electronic configuration of vanadium (V, Z=23)?

SOLUTION

Vanadium is the transition metal in the fourth period and the fifth group. The noble gas preceding it is argon (Ar, Z=18), and knowing that vanadium has filled those orbitals before it, argon is used as the reference noble gas. The noble gas in the configuration is denoted E, in brackets: [E]. To find the valance electrons that follow, subtract the atomic numbers: 23 - 18 = 5. Instead of 23 electrons to distribute in orbitals, there are 5. Now there is enough information to write the electron configuration:

Vanadium, V: [Ar] 4s2 3d3

This method streamlines the process of distributing electrons by showing the valence electrons, which determine the chemical properties of atoms. In addition, when determining the number of unpaired electrons in an atom, this method allows quick visualization of the configurations of the valance electrons. In the example above, there are a full s orbital and three half filled d orbitals.

## References

1. Petrucci, Ralph H et al. General Chemistry: Principles & Modern Applications Ninth Edition. , Upper Saddle River, NJ: Pearson Prentice Hall, 2007. Print
2. Sherman, Alan, Sharon J. Sherman, and Leonard Russikoff. Basic Concepts of Chemistry Fifth Edition. Boston, MA: Houghton Mifflin Company, 1992. Print.
3. IUPAC. Compendium of Chemical Terminology, 2nd ed. (the "Gold Book"). Compiled by A. D. McNaught and A. Wilkinson. Blackwell Scientific Publications, Oxford (1997). XML on-line corrected version: http://goldbook.iupac.org (2006-) created by M. Nic, J. Jirat, B. Kosata; updates compiled by A. Jenkins. ISBN 0-9678550-9-8.doi:10.1351/goldbook.
4. Scerri, Eric R. "The Electron Configuration Model, Quantum Mechanics, and Reduction." The British Journal for the Philosophy of Science 42.3 (1991): 309 -325.
5. Ostrovsky, V.N. (2004). On recent discussion concerning quantum justification of the periodic table of the elements. Foundations of Chemistry, 75, 93-116.
6. Meek, T.L., & Allen, L.C. (2002). Configuration irregularities: deviations from the madelung rule and inversion of orbital energy levels. ScienceDirect, 362(5), 362-364.

## Problems

Unless specified, use any method to solve the following problems. Answers are given in noble gas notation.

1. Find the electron configurations of the following:

1. silicon
2. tin

2. Scenario: You are currently studying the element iodine and wish to use its electron distributions to aid you in your work.

1. Find the electron configuration of iodine
2. How many unpaired electrons does iodine have?

3. Thought Questions:

1. In your own words describe how to write an electron configuration and why it is an important skill in the study of chemistry.
2. Describe the major concepts (Hunds, Pauli...etc.) and explain why each is a key part of the "tool kit" when describing electron configurations
3. Why is it possible to abbreviate electron configurations with a noble gas in the noble gas notation?

4. Identify the following elements:

1. 1s2 2s2 2p6 3s2 3p6 4s2 3d6
2. 1s2 2s2 2p6 3s2 3p6 4s2 3d10 4p6 5s2 4d7
3. 1s2 2s2 2p6 3s2 3p4
4. 1s2 2s2 2p6 3s2 3p6 4s2 3d10 4p6 5s2 4d10 5p6 6s2 4f14 5d10 6p4

5. Without using a periodic table or any other references, fill in the correct box in the periodic table with the letter of each question. (a)The element with electron configuration: 1s2 2s2 2p6 3s2 3p5; (b)A noble gases with f electrons; (c) a fifth-period element whose atoms have three unpaired p electrons; (d) First rowtransition metals having one 4s electron.

1. Find the electron configuration of the following:

a) silicon: [Ne] 3s2 3p2

b) tin: [Kr] 5s2 4d10 5p2

c) lead: [Xe] 6s2 4f14 5d10 6p2

2. Scenario: You are currently studying the element iodine and wish to use its electron distributions to aid you in your work.

a) Find the electron configuration of iodine

[Kr] 5s2 4d10 5p5

b) How many unpaired electrons does iodine have?

To find the answer we refer to part a) and look at the valence electrons. We see that iodine has 5 electrons in the p orbitals. We know that the full p orbitals will add up to 6. Using the Hund's rule and Pauli exclusion principals we can make a diagram like the following:

The answer is one.

3. Thought Questions:

a) In your own words describe how to write an electron configuration and why it is an important skill in the study of chemistry.

The first part of this question is straightforward. The second part is slightly more complicated. Because each individual's knowledge of chemistry differs, there are many answers to this question. The important aspect is that we realize that knowing electron configurations helps us determine the valence electrons on an atom. This is important because valence electrons contribute to the unique chemistry of each atom.

b) Describe the major concepts (Hunds, Pauli...etc.) and explain why each is a key part of the "tool kit" when describing electron configurations

This should also be a straightforward question, and if it seems a little difficult refer to the body of this text about these rules and how they relate to creating an electron configuration. Remember to make logical connections! We know that the main "tools" we have in writing electron configurations are orbital occupation, the Pauli exclusion principle, Hund's rule, and the Aufbau process. Orbitals are occupied in a specific order, thus we have to follow this order when assigning electrons. The Pauli exclusion principle states that no two electrons can have the same four quantum numbers . The fourth quantum number, which refers to spin, denotes one of two spin directions. This means that in one orbital there can only be two electrons and they mus have opposite spins. This is important when describing an electron configuration in terms of the orbital diagrams. Hund's rule states that electrons first occupy the similar energy orbitals that are empty before occupying those that are half full. This is especially helpful when determining unpaired electrons. The Aufbau process denotes the method of "building up" each subshell before moving on to the next; we first fill the 2s orbitals before moving to the 2p orbitals.

c) Why is it possible to abbreviate electron configurations with a noble gas in the noble gas notation?

We know that the noble gas has all of its orbitals filled; thus it can be used as a "shorthand" or abbreviated method for writing all of the electron configurations after 1s.

4. Identify the following elements:

a) 1s2 2s2 2p6 3s2 3p6 4s2 3d6

The element is iron, Fe

b) 1s2 2s2 2p6 3s2 3p6 4s2 3d10 4p6 5s2 4d7

The element is Rhodium, Rh

c) 1s2 2s2 2p6 3s2 3p4

The element is Sulfur, S

d) 1s2 2s2 2p6 3s2 3p6 4s2 3d10 4p6 5s2 4d10 5p6 6s2 4f14 5d10 6p4

The element is Polonium, Po

5. Without using a periodic table or any other references, fill in the correct box in the periodic table with the letter of each question. (a) The element with electron configuration: 1s2 2s2 2p6 3s2 3p5; (b)A noble gases with f electrons; (c) a fifth-period element whose atoms have three unpaired p electrons; (d) First row transition metals having one 4s electron.

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# 4.for function f(x)=x^2lnx (a). use lhopital's rule to find (b).find the area of the region containe

4.for function f(x)=x^2lnx

(a). use lhopital's rule to find

(b).find the area of the region contained between the graph and the x-axis-that is from x=0 to x=1 (you may wish to plot the graph of the function using maple or a calculator). Write the resulting integral in the form and use lhopital's rule to evaluate it.


text:
# zbMATH — the first resource for mathematics

## Nôno, Takayuki

Compute Distance To:
 Author ID: nono.takayuki Published as: Nôno, Takayuki; Nono, Takayuki; Nóno, Takayuki
 Documents Indexed: 86 Publications since 1949 Reviewing Activity: 88 Reviews
all top 5

#### Co-Authors

 18 single-authored 58 Mimura, Fumitake 13 Fujiwara, Fumiyo 7 Morinaga, Kakutaro 2 Sakurai, Takatoshi 2 Sato, Ryuzo 1 Yosida, Katumi
all top 5

#### Serials

 24 Tensor. New Series 15 Bulletin of the Kyushu Institute of Technology. Mathematics, Natural Science 15 Journal of Science of the Hiroshima University. Series A 4 Bulletin of Fukuoka University of Education. Part III 4 Scientiae Mathematicae Japonicae 3 Bulletin of the Kyushu Institute of Technology. Pure and Applied Mathematics 2 Journal of Mathematical Analysis and Applications 2 Journal of Science of the Hiroshima University. Series A-I 1 Bulletin of the Technical University of Istanbul 1 Hokkaido Mathematical Journal 1 Zeitschrift für Nationalökonomie
all top 5

#### Fields

 29 Mechanics of particles and systems (70-XX) 21 Dynamical systems and ergodic theory (37-XX) 20 Global analysis, analysis on manifolds (58-XX) 18 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 16 Calculus of variations and optimal control; optimization (49-XX) 15 Mechanics of deformable solids (74-XX) 9 Partial differential equations (35-XX) 6 Differential geometry (53-XX) 5 Topological groups, Lie groups (22-XX) 2 Ordinary differential equations (34-XX) 2 Manifolds and cell complexes (57-XX) 1 Linear and multilinear algebra; matrix theory (15-XX) 1 Nonassociative rings and algebras (17-XX) 1 Fluid mechanics (76-XX) 1 Operations research, mathematical programming (90-XX)

#### Citations contained in zbMATH Open

14 Publications have been cited 30 times in 22 Documents Cited by Year
On the linearization of a form of higher degree and its representation. Zbl 0049.02502
Morinaga, Kakutaro; Nono, Takayuki
1952
On the non-commutative solutions of the exponential equation $$e^Xe^Y=e^{X+Y}$$. Zbl 0056.01506
Morinaga, Kakutaro; Nôno, Takayuki
1954
On the symmetry groups of simple materials: An application of the theory of Lie groups. Zbl 0172.26603
Nono, Takayuki
1968
On the non-commutative solutions of the exponential equation $$e^xe^y=e^{x+y}$$. II. Zbl 0057.25103
Morinaga, Kakutaro; Nôno, Takayuki
1954
A new conservation law for a system of second-order differential equations. Zbl 0805.34006
Mimura, Fumitake; Nôno, Takayuki
1994
Sur l’application exponentielle dans les groupes de Lie. Zbl 0094.01502
Nôno, Takayuki
1960
Conservation laws derived from equivalent Lagrangian densities in continuum mechanics. Zbl 0611.73006
Mimura, Fumitake; Nôno, Takayuki
1986
Conservation laws in nonconservative linear elastodynamics. Zbl 0611.73007
Mimura, Fumitake; Nôno, Takayuki
1986
Conservation laws for nonconservative continuum dynamics. Zbl 0617.70015
Mimura, Fumitake; Nôno, Takayuki
1986
On stress-functions in general coordinates. Zbl 0041.10304
Morinaga, Kakutaro; Nono, Takayuki
1950
The inverse problem of Lagrangian dynamics within the context of the calculus on differential forms. Zbl 0708.58002
Mimura, Fumitake; Nôno, Takayuki
1989
On the logarithmic functions of matrices. I. Zbl 0054.00804
Morinaga, Kakutaro; Nono, Takayuki
1949
Note on the paper ”On the singularity of general linear groups”. Zbl 0083.02201
Nôno, Takayuki
1958
Sur les familles triples locales de transformations locales de Lie. Zbl 0113.15801
Nono, Takayuki
1961
A new conservation law for a system of second-order differential equations. Zbl 0805.34006
Mimura, Fumitake; Nôno, Takayuki
1994
The inverse problem of Lagrangian dynamics within the context of the calculus on differential forms. Zbl 0708.58002
Mimura, Fumitake; Nôno, Takayuki
1989
Conservation laws derived from equivalent Lagrangian densities in continuum mechanics. Zbl 0611.73006
Mimura, Fumitake; Nôno, Takayuki
1986
Conservation laws in nonconservative linear elastodynamics. Zbl 0611.73007
Mimura, Fumitake; Nôno, Takayuki
1986
Conservation laws for nonconservative continuum dynamics. Zbl 0617.70015
Mimura, Fumitake; Nôno, Takayuki
1986
On the symmetry groups of simple materials: An application of the theory of Lie groups. Zbl 0172.26603
Nono, Takayuki
1968
Sur les familles triples locales de transformations locales de Lie. Zbl 0113.15801
Nono, Takayuki
1961
Sur l’application exponentielle dans les groupes de Lie. Zbl 0094.01502
Nôno, Takayuki
1960
Note on the paper ”On the singularity of general linear groups”. Zbl 0083.02201
Nôno, Takayuki
1958
On the non-commutative solutions of the exponential equation $$e^Xe^Y=e^{X+Y}$$. Zbl 0056.01506
Morinaga, Kakutaro; Nôno, Takayuki
1954
On the non-commutative solutions of the exponential equation $$e^xe^y=e^{x+y}$$. II. Zbl 0057.25103
Morinaga, Kakutaro; Nôno, Takayuki
1954
On the linearization of a form of higher degree and its representation. Zbl 0049.02502
Morinaga, Kakutaro; Nono, Takayuki
1952
On stress-functions in general coordinates. Zbl 0041.10304
Morinaga, Kakutaro; Nono, Takayuki
1950
On the logarithmic functions of matrices. I. Zbl 0054.00804
Morinaga, Kakutaro; Nono, Takayuki
1949
all top 5

#### Cited by 29 Authors

 2 Bourgeois, Gerald 1 Alekseevskiĭ, Dmitriĭ Vladimirovich 1 Alekseevsky, Andrey V. 1 Belinfante, Johan Gijsbertus Frederik 1 de Seguins Pazzis, Clément 1 Filippov, Vladimir Mikhailovich 1 Fleury, Norbert 1 Gorbatsevich, Vladimir Vital’evich 1 Jagannathan, Ramaswamy 1 Lawton, Wayne M. 1 Markov, Yuri 1 Markova, Margarita 1 Mikheev, Pavel O. 1 Morgan, Antony J. A. 1 Perroud, Marcel 1 Popovici, Iulian 1 Rausch de Traubenberg, Michel 1 Rinehart, Robert F. 1 Sabinin, Lev Vasil’evich 1 Sahadevan, Ramajayam 1 Sato, Ryuzo 1 Savchin, Vladimir Mikhailovich 1 Shorokhov, S. G. 1 Teodorescu, Petre P. 1 Turtoi, Adriana 1 Wermuth, Edgar M. E. 1 Winternitz, Pavel 1 Yamaleev, Robert M. 1 Yaşar, Emrullah
all top 5

#### Cited in 17 Serials

 3 Linear Algebra and its Applications 2 Journal of Mathematical Physics 2 Journal of Soviet Mathematics 1 Archive for Rational Mechanics and Analysis 1 International Journal of Theoretical Physics 1 Mathematical Notes 1 Annales de l’Institut Henri Poincaré. Nouvelle Série. Section A. Physique Théorique 1 Annali di Matematica Pura ed Applicata. Serie Quarta 1 Functional Analysis and its Applications 1 Journal of Econometrics 1 Proceedings of the American Mathematical Society 1 Rendiconti del Circolo Matemàtico di Palermo. Serie II 1 Acta Applicandae Mathematicae 1 Journal of Mathematical Sciences (New York) 1 Advances in Applied Clifford Algebras 1 Mathematical Problems in Engineering 1 Nonlinear Dynamics
all top 5

#### Cited in 19 Fields

 7 Topological groups, Lie groups (22-XX) 6 Linear and multilinear algebra; matrix theory (15-XX) 4 Differential geometry (53-XX) 3 Group theory and generalizations (20-XX) 3 Ordinary differential equations (34-XX) 3 Operator theory (47-XX) 2 Nonassociative rings and algebras (17-XX) 2 Global analysis, analysis on manifolds (58-XX) 2 Mechanics of particles and systems (70-XX) 2 Mechanics of deformable solids (74-XX) 2 Quantum theory (81-XX) 1 Dynamical systems and ergodic theory (37-XX) 1 Difference and functional equations (39-XX) 1 Approximations and expansions (41-XX) 1 Harmonic analysis on Euclidean spaces (42-XX) 1 Calculus of variations and optimal control; optimization (49-XX) 1 Manifolds and cell complexes (57-XX) 1 Statistics (62-XX) 1 Game theory, economics, finance, and other social and behavioral sciences (91-XX)


text:
# ADC Operation During Sleep

The ADC module can operate during Sleep. This requires the ADC clock source to be set to the ADCRC option. When the ADCRC oscillator source is selected, the ADC waits one additional instruction before starting the conversion. This allows the SLEEP instruction to be executed, which can reduce system noise during the conversion. If the ADC interrupt is enabled, the device will wake up from Sleep when the conversion completes. If the ADC interrupt is disabled, the device remains in Sleep and the ADC module is turned off after the conversion completes, although the ON bit remains set.