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190400 | https://math.stackexchange.com/questions/461111/are-parallel-vectors-always-scalar-multiple-of-each-others | Stack Exchange Network
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Are parallel vectors always scalar multiple of each others?
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I read this in a tutorial of a university course :
We note that the vectors V, cV are parallel, and conversely, if two vectors are parallel (that is, they have the same direction), then one is a scalar multiple of the other.
Q1. There is an implication in the statement that two vectors are parallel if they are in same direction. Isn't it half right ? I mean in 3D space, two lines could not be in same direction and still be parallel right ?
Q2. If the above statement doesn't hold, then saying
one is a scalar multiple of the other.
is also wrong ?
linear-algebra
vector-spaces
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asked Aug 6, 2013 at 13:43
Amit TomarAmit Tomar
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$\begingroup$ "Parallel" is defined as "have the same direction" (up to negatives). $\endgroup$
Daniel Fischer
– Daniel Fischer
2013-08-06 13:45:22 +00:00
Commented Aug 6, 2013 at 13:45
1
$\begingroup$ @DanielFischer You mean to say 'Parallel lines' and 'Parallel vectors' are two different things ? $\endgroup$
Amit Tomar
– Amit Tomar
2013-08-06 13:46:28 +00:00
Commented Aug 6, 2013 at 13:46
$\begingroup$ The statement is correct, and is usually the definition of 'parallel'. Edit: @AmitTomar YES. This is correct. $\endgroup$
preferred_anon
– preferred_anon
2013-08-06 13:46:29 +00:00
Commented Aug 6, 2013 at 13:46
$\begingroup$ Agree with @Daniel Fischer, except where it's simply defined as being linearly dependent. $\endgroup$
Jonathan Y.
– Jonathan Y.
2013-08-06 13:47:08 +00:00
Commented Aug 6, 2013 at 13:47
3
$\begingroup$ You're confusing concept of parallel lines and parallel vectors. Vectors are defined as so called class or relation of equivalence, therefore WLOG all vectors start from one point. From this standpoint, parallel vectors are always have either same or opposite directions. $\endgroup$
Kaster
– Kaster
2013-08-06 13:48:28 +00:00
Commented Aug 6, 2013 at 13:48
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Parallel vectors on a $K$-Vector space $V$, by definiton, means: $$u \parallel v :\Leftrightarrow \exists \lambda \in K: \lambda \cdot u = v$$ Also, Parallel lines are defined by parallelicity of their respective direction vectors, wich, when fixing $0_V$ as an element of the line, implies equality of the two lines.
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answered Aug 6, 2013 at 13:59
AlexRAlexR
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Okay, third time's the charm.
First, two vectors are parallel when one is a scalar multiple of each other: given $\mathbf{u}$, $\mathbf{v}$ vectors, they are parallel if there exists $\lambda \neq 0$ such that $\mathbf{u} = \lambda \mathbf{v}$. Geometrically, this can be interpreted as follows: if $\lambda$ is positive, then the two vectors (which, remember, must both be drawn starting at the origin) point in the same direction and thus overlap. If $\lambda$ is negative, then they point in opposite directions.
Now, vectors are not the same as lines. The obvious distinction is that vectors are arrows that start at the origin (the point $(0,0,0)$ in $\mathbb{R}^3$, for example) and end somewhere, while lines are, well, lines, and they can be anywhere in space and have any direction. Any line can be described by a vector $\mathbf{v}$, called its direction vector, and a point $P$ through which the line passes. We define two lines to be parallel when their direction vectors are parallel.
You mention that two lines could not be in the same direction and still be parallel. According to our definition, that's wrong. What you're thinking of is called skew lines; they are lines that don't intersect but are also not parallel.
To answer your question, then, the statement you quote is correct. Two vectors are parallel if and only if one is a nonzero multiple of the other.
I hope this helps. If you have any doubts still, then maybe posting a drawing of what you're thinking of could help.
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edited Aug 6, 2013 at 15:57
answered Aug 6, 2013 at 13:47
JavierJavier
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$\begingroup$ How are lines parallel if their direction vectors aren't linearly dependent? $\endgroup$
Jonathan Y.
– Jonathan Y.
2013-08-06 13:48:23 +00:00
Commented Aug 6, 2013 at 13:48
$\begingroup$ @user142526: I din't word that very well. Let me fix it. $\endgroup$
Javier
– Javier
2013-08-06 13:48:57 +00:00
Commented Aug 6, 2013 at 13:48
$\begingroup$ @JavierBadia those are called skew lines, not parallel. $\endgroup$
Kaster
– Kaster
2013-08-06 13:54:08 +00:00
Commented Aug 6, 2013 at 13:54
$\begingroup$ @Kaster: I think I may be misunderstanding the OP's question. Let me ask. $\endgroup$
Javier
– Javier
2013-08-06 14:05:20 +00:00
Commented Aug 6, 2013 at 14:05
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There is an implicit assumption in this discussion that the vector space is over the real numbers and that λ is real. If u = λv and λ is complex, according to the definition you are giving u and v would be "parallel". But geometrically that is no longer true. They could be at any angle to one another.
So at the least it might be wise to define "parallel" as meaning u = λv where λ is a real number?
However, the underlying field does not need to be either the real or complex numbers -- it can be any field at all; and the vectors do not have to be arrows; so what would "parallel" mean then?
I think this aproach would work better: Let u be a vector in the space X and W the collection of all vectors in X which are orthogonal to u. I would say u and v are "parallel" if (v,w) [inner product] = 0 for every w in W; i.e. that v is also orthogonal to every vector in W.
There are some details to attend to here. For example, we need to show that W is itself a subspace of X, with dimension one less than that of X (not hard to do). And u itself is the basis for a one-dimensional subspace of X. So now what we're saying that v is "parallel" to u if v is in the subspace generated by u.
It seems to me this approach removes the geometric considerations from the problem, and clarifies the notion of "parallel" (if you want this notion) for any vector space.
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answered Aug 7, 2013 at 2:24
Betty MockBetty Mock
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190401 | https://extapps.ksc.nasa.gov/Reliability/Documents/170505_Risk_Failure_Probability_and_Failure_Rate.pdf | Risk, Failure Probability, and Failure Rate 1 170505 Tim.Adams@NASA.gov Terminology 1. Risk is the: a. Potential of losing or gaining something of value (e.g., life, property, performance, schedule, or cost). b. Effect of uncertainty on objectives. (Ref. ISO 31000, 2009) 2. In terms of loss, a risk statement contains three elements (e.g., as in three columns in a table), namely: a. Scenario, what can go wrong? b. Likelihood, what is the probability it will happen? c. Consequence, what is the impact if it did happen? 3. Reliability is the: a. Probability b. An item (e.g., system, subsystem) will perform its intended function with no failure c. For a stated mission time (or number of demands or load) d. Under stated environmental conditions. Risk vs. Reliability 1. For an item of interest, the probability used in: a. Risk is the probability of failure, denoted 𝑷𝒇. 𝑃𝑓 is not a failure rate (see page 3). b. Reliability is the probability of success, denoted 𝑷𝒔. 𝑃𝑠 is not one minus the failure rate. 2. Fundamental math rule: 𝑷𝒇+ 𝑷𝒔= 𝟏. 𝑃𝑓= 1 −𝑃𝑠 and 𝑃𝑠= 1 −𝑃𝑓 are the complements. 3. When one type of probability is known, use the complement to find the other probability. 4. A risk matrix that is quantitative (as opposed to qualitative, labels instead of measures) uses the complement of reliability as the likelihood axis and the complement of safety as the consequence axis. Types of Data and Methods Commonly Used to Make a Probability of Failure or Failure Rate 1. Demand-based (pass-fail events) – an item (e.g., starter solenoid) successfully completed its mission upon demand. The life data for this item answers “how many” and is discrete data. The binomial probability distribution models this item when the events are independent and the fixed probability of failure (𝑝𝑓) is: a. 𝑝𝑓= 𝑓𝑎𝑖𝑙𝑢𝑟𝑒 𝑐𝑜𝑢𝑛𝑡 𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑎𝑡𝑡𝑒𝑚𝑝𝑡𝑠 based on classical statistics. b. 𝑝𝑓= 𝑓𝑎𝑖𝑙𝑢𝑟𝑒 𝑐𝑜𝑢𝑛𝑡+0.5 𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑎𝑡𝑡𝑒𝑚𝑝𝑡𝑠+1 based on one common version of Bayesian statistics (see page 2). 2. Time-based (hours, cycles, miles) – an item (e.g., tire) successfully operated for y hours until it failed. The life data for this item answers “how much” and is continuous data. The exponential (a special case of the Weibull) probability distribution models this item when the failure rate (λ) is constant over time and is: a. 𝜆= 𝑓𝑎𝑖𝑙𝑢𝑟𝑒 𝑐𝑜𝑢𝑛𝑡 𝑡𝑜𝑡𝑎𝑙 𝑟𝑢𝑛 𝑡𝑖𝑚𝑒 based on classical statistics. b. 𝜆= 𝑓𝑎𝑖𝑙𝑢𝑟𝑒 𝑐𝑜𝑢𝑛𝑡+0.5 𝑡𝑜𝑡𝑎𝑙 𝑟𝑢𝑛 𝑡𝑖𝑚𝑒 based on one version common of Bayesian statistics (see next page 2). 3. Failure due to variation – an item failed not as a function of time but due to static stress. That is, the item failed because its variable stress (load) exceeded its variable strength (capacity). The Stress-Strength Interference method calculates the probability of failure (𝑝𝑓) which can be associated with the overlap (interference, intersection) in the stress and strength distributions. Note: A safety factor or the safety margin are not sufficient to address failures due to the variation in the item’s stress and the strength. Risk, Failure Probability, and Failure Rate 2 170505 Tim.Adams@NASA.gov Failure Rate Formulas Based on Bayesian Statistics1 Data Type Demand Based (failure on demand) Time Based (failure while operating) Failure Probability and Failure Rate Formulas2,3 𝑝𝑓= 𝑓𝑎𝑖𝑙𝑢𝑟𝑒 𝑐𝑜𝑢𝑛𝑡+ 0.5 𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑎𝑡𝑡𝑒𝑚𝑝𝑡𝑠+ 1 𝜆= 𝑓𝑎𝑖𝑙𝑢𝑟𝑒 𝑐𝑜𝑢𝑛𝑡+ 0.5 𝑡𝑜𝑡𝑎𝑙 𝑟𝑢𝑛 𝑡𝑖𝑚𝑒 Prior Distribution4 Beta distribution with αprior = 0.5 and βprior = 0.5 being a Jeffreys Prior Gamma distribution with αprior = 0.5 and βprior = 0 being a Jeffreys Prior Likelihood Function Binomial distribution Poisson distribution Posterior Distribution5 Beta distribution with parameters αpost = x + αprior and βpost = n - x + βprior where x is failure count and n is number of demands. The mean of the beta distribution is 𝛼 𝛼+𝛽. Gamma distribution with parameters αpost = x + αprior and βpost = t + βprior where x is failure count and t is total run time. The mean of the gamma distribution is 𝛼 𝛽. NASA PRA Procedures Guide6 Page C-6 (pdf page 364) Page C-11 (pdf page 369) NASA Handbook on Bayesian Inference7 Page 34 (pdf page 54) Page 40 (pdf page 60) Footnotes: 1 Bayesian statistics quantitatively combines human belief (a subjectively-based probability distribution) with operational or test data (an objectively-based probability distribution). 2 When the failure count is zero, these two Bayesian-based formulas are commonly used. 3 When the failure count is zero and the data type is time-based, one method in classical statistics calculates the failure rate using: 𝜆= 1/3 𝑡𝑜𝑡𝑎𝑙 𝑟𝑢𝑛 𝑡𝑖𝑚𝑒. 4 A Jeffreys Prior is used when there is insufficient information to form an informed prior distribution. Thus, the Jefferys Prior is referred to as a noninformative prior and is intended to convey little prior belief or information. A noninformative prior allows the data (described by the likelihood function) to speak for themselves. 5 A Bayesian-based failure-rate formula is the mean (average) of its posterior distribution. This mean is commonly called the point Bayes’ estimate. A posterior distribution is derived from Bayes’ Theorem (Bayes-Laplace Theorem). This Theorem uses a prior distribution (to represent the value of the failure rate as a belief or best estimate prior to collecting field data) and a likelihood function (the failure distribution for field data that was collected after the stated belief). The posterior distribution is shifted in the direction of the likelihood function that was used. 6 Source: 7 Source: Risk, Failure Probability, and Failure Rate 3 170505 Tim.Adams@NASA.gov Illustration: Failure Rate vs. Failure Probability Question: What is the probability of a flood(s) occurring in a ten-year period for an area of land that is classified by the National Flood Insurance Program (NFIP) as being in a 100-year floodplain? Method 1 – Duration is continuous data (i.e., clock time, t, is a non-negative real number): Assume the “100-year floodplain” means: The hazard rate or failure rate (λ) is one flood every 100 years, this rate remains constant over time (t), and t is any non-negative real number {𝑡∈ℝ | 𝑡≥0}. Since 𝜆= 1/100 is a constant or fixed rate over time, the exponential distribution, a continuous probability distribution, can be used as the math model. This model has no memory of previous failures (floods). The probability of success or reliability form of the exponential distribution is 𝑅(𝑡) = 𝑒−(𝑡 𝜃), where 𝜃 is the average or mean time between failure (MTBF) and the reciprocal of λ. Since 𝜆= 1 100 , then 𝜃= 100. The probability of success (no flood event) during a 10-year period is 𝑅(10) = 𝑒−( 10 100) = 0.904837. The probability of failure (at least one flood event) during a 10-year period is 1 - 0.904837 = 0.095163 ≈ 9.5%. In Excel, the two previous steps can be worked as one using the complement of success space or the cumulative distribution function (failure space): =1-EXP(-1/10010) or =EXPON.DIST(10,1/100,TRUE). A related math model is the complement of the cumulative Poisson. Let the count of failure events (x) be zero and the mean be the product of time and the failure rate. Use =1-POISSON.DIST(0,101/100,TRUE). Method 2 – Duration is discrete data (i.e., number of successes, x, and trials, n, are non-negative integers): Assume the “100-year floodplain” means: There is a probability (p) of one flood every 100 years, this probability is the same from year to year, and year counts (no floods in x years for the duration of n years) are non-negative integers where 𝑥≤𝑛. In addition, call this the probability of failure (𝑝𝑓), the probability of a one flood in one year. Since 𝑝𝑓= 1/100 is the same each year, each year is independent of one another, the year count is fixed and not infinite, and there are exactly two mutually exclusive outcomes (success and failure) for each year, the binomial distribution can be used to obtain the probability of observing x successes in n independent trials. The probability of success or reliability form of the binomial distribution for obtaining exactly 𝑥 number of successes (no-flood years) in 𝑛 trials (years) with a given probability of success where 𝑝𝑠= 1 −𝑝𝑓 is: 𝑏(𝑥, 𝑛, 𝑝𝑠) = ( 𝑛!
𝑥! (𝑛−𝑥)!) (𝑝𝑠)𝑥(1 −𝑝𝑠)𝑛−𝑥 The overall probability of failure being the probability of one or more flood events (years) in 10 trials (years) uses the complement of the above the formula where 𝑥= 10, 𝑛= 10, 𝑎𝑛𝑑 𝑝𝑠= 0.99. In Excel, the previous step can be worked as =1-BINOM.DIST(10,10,0.99,FALSE) resulting in 0.095618 ≈ 9.6%. An alternative method with the binomial distribution in success space is the cumulative form. In this form, let 𝑥= 9 (for at most 9 flood-free years out of 10 years) being =BINOM.DIST(9,10,0.99,TRUE). Comments on the two methods: These methods do not exactly agree since the Poisson and binomial distributions have an asymptotic relationship. The Poisson distribution approximates the binomial distribution when n is large and p is small. The exponential distribution is a special case of the Poisson when the number of events in the interval associated with a process equals zero. The next page graphically compares the two above methods. Note: The exact case of the binomial in failure space simplifies to (1 −𝑝𝑓)𝑛 when 𝑥= 0 (i.e., no failure or flood in every year or trial). In this case, 𝒆−𝝀𝒕≈(𝟏−𝒑𝒇)𝒏, when 𝜆= 𝑝𝑓 𝑎𝑛𝑑 𝑡= 𝑛. Risk, Failure Probability, and Failure Rate 4 170505 Tim.Adams@NASA.gov Purpose: Plot the cumulative distribution functions (CDFs) for the exponential distribution (a continuous distribution, with a mean = 100 years, and time truncated at 10 years) and the binomial distribution (a discrete probability distribution using ps=0.99 with n=10 independent trials or years). Interpretation: When the exponential’s t = 10 and the binomial’s n= 10, these two math models intersect at essentially the same value on the vertical axis (0.095) which means: “In a 100-year floodplain, there is a 9.5% probability of failure (at least one flood) in any 10-year period.” |
190402 | https://wwwusers.ts.infn.it/~milotti/Didattica/Fisica3Matematica/Atomo%20di%20Bohr.pdf | 1 Edoardo Milotti 16/11/2009 Lo spettro dell'idrogeno atomico e il modello di Bohr degli atomi idrogenoidi Questa nota riassume brevemente la storia della scoperta della serie di Balmer e del modello di Bohr dell'atomo di idrogeno. 1. Gli spettri atomici Nello spettro del sole si osservano un gran numero di righe scure, note con il nome di righe di assorbimento: sono come le impronte digitali delle specie atomiche che si trovano nel sole. In effetti, prendendo un gas ed eccitandolo per mezzo di una scarica elettrica, viene emessa luce, e si trova che diversi elementi chimici emettono luce che ha righe spettrali differenti, ad esempio l'azoto ha il seguente spettro di emissione mentre l'ossigeno ha uno spettro differente Si trova quando lo stesso gas viene interposto tra la sorgente di luce bianca ed uno spettroscopio, si vedono righe di assorbimento che corrispondono alle righe di emissione: si tratta quindi dello stesso fenomeno. Gli spettri mostrati sono notevolmente diversi e hanno righe di emissione sparse senza un apparente ordine: è possibile capire le regole di emissione e assorbimento della luce da parte degli atomi? 2. La serie di Balmer delle righe spettrali dell'idrogeno atomico Johann Jakob Balmer era un matematico svizzero (1825-‐1898), che diede un singolare contributo alla fisica 2 All'età di sessant'anni, Balmer trovò una singolare formula per la lunghezza d'onda della luce emessa dall'idrogeno atomico λ = h m2 m2 −n2 per n = 2, h = 3.6456×10-7 m, m = 3, 4, 5, 6, ... e chiamò h il "numero fondamentale dell'idrogeno". La formula mostrò di avere potere predittivo, quando venne identificata una riga di assorbimento dell'idrogeno non nota in precedenza. Johann Jakob Balmer Si scoprì poi che la formula di Balmer è un caso particolare di una formula più generale scoperta da Johannes Rydberg 1 λ = RH 1 n1 2 −1 n2 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ si ottiene la serie di Balmer prendendo n1 = 2 . Con altri valori di n1 si trovano le lunghezze d'onda di altre serie di righe spettrali dell'atomo di idrogeno. Lo spettro qui sopra mostra le quattro righe di emissione visibili nella serie di Balmer dell'idrogeno atomico 3 3. Il modello di Bohr dell'atomo di idrogeno Bohr ha costruito il suo modello dell’atomo di idrogeno nel 1913, in competizione con altri fisici che stavano cercando di svelare il mistero della stabilità delle orbite elettroniche. Secondo l’elettromagnetismo classico, gli elettroni in orbita intorno al nucleo atomico dovrebbero irraggiare – e quindi perdere energia –perché si muovono con moto accelerato. Alla fine gli elettroni dovrebbero precipitare sul nucleo, e per questo Thomson aveva elaborato negli anni precedenti un modello atomico in cui gli elettroni erano distribuiti uniformemente sulla superficie di un nucleo atomico duro, sferico. Gli esperimenti di Rutherford avevano però dimostrato che ciò non era possibile, che l’atomo doveva essere costituito da un piccolo nocciolo positivo – il nucleo – circondato da elettroni su orbite relativamente lontane. Niels Bohr è stato il primo a capire come un’idea della nuova meccanica quantistica poteva essere utilizzata per risolvere il problema. Nel 1913 la meccanica quantistica vera e propria non era ancora nata, c’erano solo delle idee ancora vaghe che dovevano essere sistemate in un contesto coerente. Nel 1900 Max Planck aveva pubblicato il suo importante articolo sulla radiazione di corpo nero, in cui aveva introdotto una forma di quantizzazione della frequenza della luce emessa da un corpo nero (questo significa che non sono ammesse tutte le frequenze per la luce ma solo multipli interi di una frequenza fondamentale). Nel 1905 Albert Einstein aveva dimostrato che le misure fatte sull’effetto fotoelettrico (l’emissione di elettroni da parte di una superficie metallica quando viene colpita dalla luce) potevano essere spiegate assumendo che la luce fosse costituita da quanti (da fotoni) e che ciascuno di essi avesse un energia proporzionale alla frequenza ν della luce Eγ = hν , dove h è proprio la costante introdotta da Planck nel 1900 . Negli stessi anni de Broglie aveva anche fatto vedere che gli elettroni possono essere descritti in un certo senso come “onde” che possiedono una lunghezza d’onda inversamente proporzionale alla loro quantità di moto. L'ipotesi di quantizzazione che sta alla base del modello è che un'orbita elettronica attorno al nucleo atomico debba essere necessariamente lunga un numero intero di lunghezze d'onda elettroniche, e cioè 2πr = nλ Un esempio è mostrato in figura 4 Se ora ci ricordiamo della relazione di de Broglie λ = h mv e introduciamo la definizione = h 2π , otteniamo una semplice espressione per il modulo del momento angolare mvr = n Possiamo interpretare quest'ultima relazione come una quantizzazione del momento angolare, che assume solo valori definiti. Dalla quantizzazione del momento angolare si trova la velocità in funzione di r ed n: v = n mr Uguagliando la forza di attrazione elettrostatica tra nucleo (nucleo con Z protoni, carica nucleare uguale a Zqe ) ed elettrone (carica −qe ), e la forza centripeta, si ottiene un’altra equazione, che permette di eliminare v e risolvere per r: Zqe 2 4πε0r2 = mv2 r si trova quindi 1 r n = Zqe 2m 4πε02 · 1 n2 ; r n = 4πε02 Zqe 2m n2 5 dove r n indica il raggio dell’n-‐esima orbita. Si noti che r n è proporzionale al quadrato di n. Da questa soluzione si trova anche la velocità associata all’n-‐esima orbita vn = n m ·1 r n = Zqe 2 4πε0·1 n Prendiamo ora l'espressione dell'energia totale associata all’n-‐esima orbita, che è la somma dell'energia cinetica e dell'energia potenziale (elettrostatica): En = Un + Tn = −Zqe 2 4πε0r n + 1 2 mvn 2 ; allora, sostituendo i valori di raggio e velocità trovati sopra, otteniamo En = −Zqe 2 4πε0r n + 1 2 mvn 2 = −m Zqe 2 4πε0 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 2 1 n2 + 1 2 m Zqe 2 4πε0 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 1 n2 = + m 2 Zqe 2 4πε0 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 1 n2 = −1 2 Z 2qe 4m 4πε0 ( ) 2 n22 = − qe 4m 2 4πε0 ( ) 2 2 ⋅Z 2 n2 = −R∞⋅Z 2 n2 dove R∞= qe 4m 2 4πε0 ( ) 2 2 = 1 Rydberg ≈13.605 eV . Come si vede sono ammissibili solo certi valori dell'energia, e l’intero n è detto numero quantico principale dell'atomo di idrogeno; inoltre il raggio rB = 4πε02 qe 2m ≈5 ⋅10−11m è il raggio di Bohr dell'atomo di idrogeno. Si noti che il raggio che corrisponde ad un certo n è dato da r n = 4πε0n22 Zqe 2m = n2rB Z e quindi il raggio dell'orbita dell'elettrone cresce velocemente al crescere di n. In particolare per valori di n vicini a 100, il raggio dell'orbita elettronica è prossimo 6 a 1 µm, ed ha quindi delle dimensioni decisamente "grandi" su scala atomica: atomi di idrogeno con n così elevati sono stati prodotti in laboratorio, e sono detti atomi di Rydberg. Quando un elettrone si sposta da un’orbita più alta ad un’orbita più bassa perde energia: questa energia può venire emessa sotto forma di un quanto di luce, con un’energia data dalla relazione di Einstein. Analogamente un elettrone può assorbire un fotone e passare ad un’orbita più alta. Ora si noti che ΔEn1,n2 = R∞⋅ 1 n1 2 −1 n2 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ per una transizione tra il livello n1 e il livello n2. Le transizioni che ci interessano avvengono con assorbimento o emissione di luce, cioè di fotoni di energia ΔEn1,n2 = hν . La figura seguente è una fotografia tratta dal libro di G. Herzberg: "Atomic spectra and atomic structure" (Dover, New York 1945) e mostra una fotografia dello spettro di emissione dell'idrogeno atomico nel visibile e nel vicino ultravioletto Anche la figura seguente è tratta dal libro di G. Herzberg e mostra i membri della serie di Balmer (cioè le transizioni da n1 > 2 a n2 = 2) dalla settima linea al continuo. H∞ è la posizione teorica del limite della serie (cioè la transizione da n1 = ∞ a n2 = 2) 7 Si noti ora che se si eccita fortemente un atomo, si può estrarre un elettrone dell’orbita più interna, e uno degli elettroni dell’orbita immediatamente più alta può riempire il vuoto emettendo un fotone la cui energia è hν = E1 −E2 = R∞⋅1 1 −1 22 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟Z 2 vale a dire ν = 3R∞ 4h ·Z (la radice quadrata della frequenza della luce emessa è proporzionale a Z). Questa relazione è stata trovata per la prima volta sperimentalmente da H. G. J. Moseley nel 1913, e per questo si chiama legge di Moseley. La figura seguente mostra la radice quadrata della frequenza della radiazione emessa in funzione di Z, e si vede che la legge di Moseley è verificata molto bene (sono mostrate due diverse classi di transizioni atomiche, per entrambe la legge funziona molto bene, proprio come previsto dal modello di Bohr). Il modello di Bohr è solo un modello classico con una quantizzazione del momento angolare orbitale introdotta in modo artificioso, ma funziona straordinariamente bene. La soluzione quantistica del problema dell’atomo di idrogeno che si ottiene dalla equazione di Schrödinger mostra che i valori che 8 abbiamo trovato con tanta facilità sono effettivamente – stupefacentemente – corretti, e che rappresentano dei valori medi. Il modello di Bohr ha avuto un’enorme influenza nei primi anni dello sviluppo della meccanica quantistica, e Bohr ha ricevuto il premio Nobel per questo lavoro nel 1922 (qui sotto è riportata l’introduzione della Nobel Lecture di Bohr, la figura che illustra la validità della legge di Moseley è tratta anch’essa dal testo di Bohr) Nota bibliografica: la storia delle origini della meccanica quantistica e del suo velocissimo sviluppo negli anni tra il 1920 e il 1940 è narrata con grande efficacia nel libro di Gino Segrè “Faust a Copenhagen” (Il Saggiatore, 2009). La lettura di questo libro è consigliata a tutti gli studenti che vogliano farsi un’idea di questo periodo entusiasmante della fisica e che vogliano conoscere le personalità giovani, creative ed indipendenti dei grandi fisici che hanno scoperto i principi fondamentali della meccanica quantistica. |
190403 | https://www.facebook.com/groups/562876904080091/posts/2224515861249512/ | The Collatz conjecture | # Introduction to p-adic math | Facebook
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Introduction to 2-adic numbers and the Collatz conjecture
Summarized by AI from the post below
The Collatz conjecture ·
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G Tony Jacobs
· July 22, 2024 ·
Introduction to p-adic math
The concepts of 2-adic valuations and 2-adic numbers sometimes come up when people study Collatz. I don't think these are particularly familiar topics for a lot of people, so here's a post that can possibly serve as an introduction to those ideas.
There are three related concepts to talk about here: p-adic valuations, the p-adic absolute value, and p-adic numbers. The symbol 'p' is meant to represent any prime number of our choice, and when we're talking about Collatz, we usually take p=2. Therefore, I'm going to talk about the 2-adic valuation, the 2-adic absolute value, and 2-adic numbers. Just be aware that all of these concepts can be adapted for any other prime number.
2-adic valuations
The 2-adic valuation of a rational number q, denoted v2(q), is the exponent of 2 in q's prime factorization. Here are the first few natural numbers, and their 2-adic valuations:
v2(1) = 0
v2(2) = 1
v2(3) = 0
v2(4) = 2
v2(5) = 0
v2(6) = 1
v2(7) = 0
v2(8 ) = 3
v2(9) = 0
v2(10) = 1
This "valuation" tells us how many times we can divide our number by 2 before reaching an odd number. The number 160 has a large 2-adic valuation, because it has a lot of 2's "in it". We can divide by 2 five times, until we finally reach an odd number:
160/2 = 80
80/2 = 40
40/2 = 20
20/2 = 10
10/2 = 5
Therefore, v2(160)=5.
We can also write down prime factorizations for non-integer rational numbers, with the difference being that some exponents can be negative. The number 21/80 has prime factorization:
2^(-4) 3^(1) 5^(-1) 7^(1)
so its 2-adic valuation is -4, while its 5-adic valuation is -1, and its 3-adic and 7-adic valuations are each equal to 1.
Finally what's the 2-adic valuation of 0? We don't really think of 0 as having a prime factorization, but we can still ask: How many times can we divide by 2 until we reach an odd value? Well, let's check:
0/2 = 0
0/2 = 0
0/2 = 0
.....
Hmm. This isn't going anywhere. We can divide by 2 as many times as we like, and the answer will never be odd, so we declare:
v2(0)=infinity.
The 2-adic absolute value
Here's where it gets a bit weirder. We're familiar with the absolute value of a number being defined as its non-negative difference from 0:
abs(x) = x – 0, or 0 – x, whichever one isn't negative
and this absolute value measures a number's "size" in a way that is pretty familiar. That's great, and useful in many contexts, but it is not the only way to define the "size" of a number.
In principle, we could define "size" however we want, and just decide arbitrarily that the number 5 seems huge, and the number 13 seems tiny. However, if we're going to define an alternative absolute value, it makes sense to require that it should have certain nice properties.
In particular:
We want the absolute value of every number to always be positive if the number isn't 0, and we want the absolute value of 0 to equal 0.
We want the absolute value of a×b to be the product of the absolute values of a and b individually.
We want the absolute value of a+b to be no greater than the sum of the absolute values of a and b individually.
The usual absolute value satisfies these properties, and they don't seem like too much to ask for.
With these requirements, there are only certain things we can do. We can declare the absolute value of every single number to be 1, and that's.... boring. But it works. That's called the "trivial absolute value", and there's not much math we can do with it.
If we want to use something that's not the usual absolute value, and we don't want to use the trivial one, then there's only one other direction we can go: We can pick a prime number p, and define a p-adic absolute value for any rational number. I'll use abs2(n) to denote the 2-adic absolute value of n. It goes like this:
abs2(n) = 2^(–v2(n)) when n is not 0
and
abs2(0) = 0
Using this definition, the 2-adic absolute value of any odd integer is 1, because that's 2^0. The 2-adic absolute value of the numbers 2, 6, 10, 14, etc. is equal to 1/2, because all of these numbers have 2-adic valuations equal to 1. The 2-adic absolute value of 4, or 12, or 20, or 28, equals 1/4. For 8 and 24, it equals 1/8. We talked above about how v2(160)=5, so abs2(160)=1/32.
Going in the other direction, abs2(1/2) = 2, abs2(7/16)=4, and abs2(105/1024)=10.
This probably seems bizarre, if not perverse. Why on Earth would we choose to define "size" in such a counter-intuitve way? Well..... It leads to some good (and useful!) mathematics. In particular, it lets us build the 2-adic number system, which is an alternative to the so-called "real number system".
2-adic numbers
To begin seeing how 2-adic numbers work, first think of infinite decimals. It's ok to write infinitely many digits to the right of a decimal point, because each digit represents something smaller than the digit before, so we end up with a sum that doesn't grow out of hand.
I mean, if we try to add 1+1+1+1+..... forever, then we have a problem. However, if we want to add 1+1/10+1/100+1/1000+..., that's much more reasonable. Doing that, we see that 1.11111.... equals the sum of a geometric series, so we apply the formula:
1.1111.... = 1+1/10+1/100+1/100+... = 1/(1 - 1/10) = 10/9
That's using the usual absolute value, which tells us that 1/100 is smaller than 1/10. However, if were were using the 2-adic absolute value, then we would have 1/100 > 1/10, and this sum wouldn't work!
What would work, on the other hand, would be a sum involving infinitely many numbers that get smaller and smaller 2-adically. Thus, 1+2+4+8+16+32+... is a nice convergent sum in the 2-adic numbers, and its value is calculated using the same formula:
...+32+16+8+4+2+1 = 1/(1-2) = -1.
Ok, that was weird. How could this be negative, and why did I write the "dot dot dot" at the left? Well, we're going to write 2-adic numbers in binary notation, and I'll always put a single dot after the units digit. Thus, we write:
1 = 1.
2+1 = 11.
4+2+1 = 111.
and
...+32+16+8+4+2+1 = ...111111.
It's like a repeating decimal, but it repeats off to the left, instead of off to the right, because in this strange new world, binary digits further to the left represent "smaller" numbers.
In general, ordinary positive integers, written as 2-adic numbers, are indistinguishable from familiar binary notation. However, negative numbers and fractions look weird. In the 2-adics, we have:
-1 = .....11111.
-2 = .....11110.
-13 = .....1110011.
-1/3 = .....010101.
1/3 = .....0101011.
If you want to see these actually "working", take the expression for -1, and add 1 to it: you should get 0. Try taking the expression for 1/3, and multiplying it by 3 (11 in binary). You should get 1.
Anyway, since 2-adic numbers are basically binary numbers with extra features, they are a natural home for the Collatz function. Several papers have been published analyzing how the Collatz function acts when we apply it to 2-adic numbers, and some of the results are significant.
Invitation to explore - questions welcome
I hope that this post serves as a reasonable introduction to the topic, and I will be happy to answer questions in the comments. For the most part, the practical upshot is that v2(n), the 2-adic valuation of n, is a very convenient way to write down a property of a number that we might find useful. Going beyond that, and thinking about the 2-adic absolute value, and how it leads to the 2-adic number system, is a task best approached cautiously, but know that some explorers of this strange world have reported some very interesting findings!
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Mark Vance
160/2 = 80
80/2 = 40
40/2 = 20…
See more
1y
Anabel Costa Donaghue
That was a nce post
1y
M Reza Dwi Prasetiawan
I dont get it about p-adic absolute value, can you give an example?. To me, this looks abstract.
1y
G Tony Jacobs replied
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5 Replies
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190404 | https://www.algebrahouse.com/examples/when-will-two-trains-meet | Ask a Question
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| When will two trains meet 6/17/2014 3 Comments Two trains leave a station, 304 miles apart, at the same time and travel toward each other. One train travels at 105 miles per hour, while the other travels at 85 miles per hour. How long will it take for the two trains to meet? Distance = rate x time d = rt One train distance = d rate = 105 time = t {the two trains left at same times} d = 105t {distance = rate x time} Other train distance = d rate = 85 time = t {the two trains left at same times} d = 85t {distance = rate x time} 105t + 85t = 304 {the two trains' combined distance equals 304} 190t = 304 {combined like terms} t = 1.6 {divided each side by 190} the two trains will meet in 1.6 hours which is 1 hour and 36 minutes - Algebra House 3 Comments Makenah Giarrusso 9/1/2023 11:04:39 am This makes more sense Reply Algebra House 9/1/2023 11:16:23 am Thank you! Reply train speed test online link 9/3/2023 04:05:32 pm This blog on "When Will Two Trains Meet" is a fantastic resource for anyone looking to understand the intriguing dynamics of train travel and the mathematics behind predicting their meeting points. The author's clear and concise explanations, accompanied by real-world examples, make complex concepts easily digestible for readers of all levels. I particularly appreciated the inclusion of practical tips and formulas that can be applied in various scenarios, making it not only an enlightening read but also a valuable reference. Whether you're a student studying physics or simply a curious train enthusiast, this blog is a must-read that will leave you with a newfound appreciation for the fascinating world of locomotion. Reply Your comment will be posted after it is approved. Leave a Reply. | Latest Videos Categories All All Word Problems Basic Math Combining Like Terms Distributive Property Equations Of Lines Exponents Factoring Foil Method Functions Geometry Graphing Imaginary Numbers Inequalities Linear Equations Literal Equations Miscellaneous Percents Pythagorean Theorem Quadratic Equations Radicals Simplifying Slopes And Intercepts Solving Equations System Of Equations Word Problems Age Word Problems Distance Word Problems Geometry Word Problems Integers Word Problems Misc. Word Problems Mixture Word Problems Money Word Problems Number Archives December 2023 January 2022 November 2021 March 2021 February 2021 December 2020 October 2019 March 2019 February 2019 January 2019 February 2018 January 2018 December 2017 November 2017 September 2017 March 2017 February 2017 December 2016 November 2016 October 2016 July 2016 January 2016 December 2015 November 2015 October 2015 September 2015 August 2015 June 2015 April 2015 February 2015 January 2015 December 2014 November 2014 August 2014 June 2014 March 2014 February 2014 January 2014 December 2013 November 2013 October 2013 September 2013 August 2013 July 2013 May 2013 April 2013 March 2013 February 2013 January 2013 December 2012 July 2012 April 2012 March 2012 February 2012 January 2012 December 2011 November 2011 October 2011 September 2011 July 2011 June 2011 May 2011 April 2011 March 2011 February 2011 January 2011 October 2010 September 2010 June 2010 May 2010 April 2010 March 2010 |
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AP Physics 2 Featured Question: Charged Particle in a Magnetic Field
Question
Consider a charged particle moving through a magnetic field that is not necessarily uniform. The particle follows a path that is not always parallel to the magnetic field’s direction. The magnetic force is the only force that acts on the particle.
(a) For this part, assume that the particle does not lose energy to electromagnetic effects due to the effects of acceleration and that the charge moves with a non-relativistic initial speed.
The particle’s direction of motion will change but not its speed. This occurs regardless of its own charge or motion or the magnetic field’s strength or orientation. Briefly explain why this is.
If the magnetic field is uniform in both strength and direction and the particle’s initial motion is perpendicular to the magnetic field direction, the particle will travel in a circular path. Explain why this occurs.
(b) The particle is initially located at point P in the plane of the page and moving to the right with speed v. The magnetic field that the particle moves through is generated by a long, straight wire carrying a current I to the right, located in the plane of the page below point P.
Is the sign of the charge positive or negative? Explain how you arrived at your answer.
The particle moves along the dotted path, which is entirely within the plane of the page (i.e. the dotted path is not a helix). This is due to the fact that the magnetic field generated by a long, straight, current-carrying wire is not uniform in strength. In a clear, coherent, paragraph-length response which may include equations and/or additional figures, explain why the particle’s path appears as it does above rather than being circular. Be sure to address specific features of the magnetic field’s strength and how that field strength affects the motion of the charge.
Solution
(a-i) 2 points
The student shows understanding that the magnetic force on the particle is at all times perpendicular to the direction of the particle’s velocity.
The student shows understanding that a net force that acts perpendicular to an object’s velocity cannot change the object’s speed. The student may use work and energy principles to make this argument but this is not necessary.
Example:
A magnetic field can only exert force on the charge perpendicular to the charge’s velocity vector. But force must be acute with velocity to speed an object up or obtuse to slow an object down. Since the force is perpendicular to velocity, the charge can’t change speed.
Example:
Magnetic force is always perpendicular to velocity. If a force is perpendicular to an object’s motion, no work is done by that force. Therefore, the magnetic force cannot change give or take kinetic energy from the object so the object cannot speed up or slow down.
Alternate Solution
The student makes a general declaration that magnetic fields cannot perform work on an object.
The student recognizes that net work on an object changes the object’s kinetic energy, so no net work means no change in the particle’s kinetic energy or speed.
Example:
Magnetic fields never do work. Since work causes an object’s kinetic energy to change, no net work means no change in speed.
(a-ii) 1 point
The student shows understanding that the radius-of-curvature of an object’s path depends on the strength of the centripetal force, and states that a uniform magnetic field produces a constant force on the particle.
(b-i) 2 points
The particle has a positive charge. This conclusion must be reached to earn both points.
The student chooses a location on the particle’s dotted path, and correctly indicates the direction of velocity and magnetic force at that point. (If a “high point” on the dotted line is chosen, such as P, the velocity is to the right and the force is down, “toward the center” of the curvature. If a “low point” closest to the wire is chosen on the dotted line, the velocity is to the left and the force is up.)
The student recognizes that the wire’s magnetic field is directed out of the page in the region where the particle is moving.
If either of these ideas is stated, one point is earned. Both points are only earned if the student connects their velocity, magnetic field, and net force directions to the correct charge sign.
(b-ii) 5 points
The student recognizes that the radius of curvature of the particle’s path decreases as the particle moves closer to the wire.
The student explains that the magnetic field strength is stronger closer to the wire. The student may use the equation as part of their explanation of this idea.
The student explains that the stronger the magnetic field, the stronger the magnetic force acting on the particle. The student may use the equation as part of their explanation of this idea.
The student states or implies that the magnetic force provides the centripetal force for the particle. The student furthermore explains that a stronger centripetal force results in a shorter radius of curvature for an object. The student may use the equation as part of their explanation of this idea.
For a logical, relevant, and internally consistent response that addresses the require argument or question asked, and follows the guidelines described in the published requirements for the paragraph-length response.
Example:
The path at point P is very straight, but close to the wire the path is sharply curved. This is because the magnetic field close to the wire is stronger, exerting more force on the charge. Because magnetic field is always perpendicular to speed, the magnetic force can only “steer” the charge’s direction. A stronger force is like turning the steering wheel harder on a car—it makes the path sharper and have lower radius. This is what occurs close to the wire where the magnetic field is strong, but far from the wire where the field is weak, the charge doesn’t change direction as quickly and the path has a larger radius.
Commentary
This problem was inspired by Free Response #5 on the released free response questions for the 2016 AP Physics 1 exam (P1-2016-FR05 for short). On that problem, a vertical string holds a standing wave that has different wavelengths in different places due to the differing tensions in the string. We often teach standing waves in media where the wave speed is the same throughout; this results in standing wave patterns having the same-sized node-antinode distance. In the same way, we often teach the motion of a charge through a magnetic field only in the special case where the magnetic field is uniform. This problem tests whether students understand qualitatively how magnetic field's affect the radius-of-curvature of a charged particle's path even as the magnetic field varies from place to place.
The problem begins with a statement that magnetic force cannot change the speed of a charge. The problem does this so that students can use this fact later as part of their paragraph-length response, even if they do not know why this is the case (much as part (a) of P1-2016-FR05 directly states that the tension is greater near the top of the rope and asks why, so that students who do not know why can still use this fact later in part (b)). The question about why a uniform field results in a circular path is set in place in order to prepare students to compare and contrast the uniform circular motion of a charge in a uniform B-field with the non-uniform, non-circular motion of a charge in a non-uniform B-field that occurs on part (b).
Then the question moves on to a charge in the vicinity of a long, straight wire. The first part of this challenge asks students to state the sign of the charge and explain how they arrived at their answer. Many teachers simply drill students on problems such as "here is the charge's sign, velocity direction, and the direction of the magnetic field that the charge is in; now tell me the direction of the magnetic force on the charge." While this is a valuable an unavoidable part of teaching magnetism, it is highly unlikely that students will see such a direct question about applying the right-hand-rule on a Physics 2 exam. Instead, students will need to consider a more complex situation and be able to take from the situation three of these things (charge sign, charge velocity direction, magnetic field direction, and direction of magnetic force) and determine the fourth.
The second part of the charge-and-wire is the reason why we have this problem: to determine whether the student can understand principles and relationships in the context of a topic that they have learned but a situation that they are unlikely to have encountered before. In this case, the student must combine the following ideas together to explain why the charge follows the dotted-line path:
The path has small radius of curvature close to the wire and long radius of curvature far away. Students can be taught during AP Physics 1 to recognize long and short radius-of-curvature paths. This can be done in the context of roads and driving; draw a curve on the board and say that the curve represents a road as seen from above. Draw the curve so that there are clear places where the radius of curvature is large and where it is small and ask students to explain where cars can go faster or where they must go slower.
The magnetic field is stronger at locations closer to the wire. This is usually taught as part of a typical magnetism unit using the equation for magnetic field due to a straight wire.
A stronger magnetic field exerts a stronger force on a charge. This is taught as part of teaching the equation . Note that students need to know that only B can affect the value of F, since a magnetic field cannot change the speed v of a particle and the charge depends on the particle's properties.
The magnetic force is perpendicular to the motion, so the magnetic force provides the centripetal force on the particle.
The stronger the centripetal force on an object, the "tighter the turn" (smaller radius-of-curvature path) the particle can make. This again should be introduced in Physics 1, again in the context of roads with different radii-of-curvature at different places. Students should be able to understand that more force is required to make a tighter turn because the motion changes more rapidly in these cases.
Quiz
Four particles move in the vicinity of a long wire carrying current I. The particles have masses and charges as shown in the table. The wire and all particles are in the plane of the page as shown in the diagram. The arrow on each particle represents that particle’s initial motion. The grid can be used to quantify the locations and relative speeds of the particles. The only force that acts on each particle due to the wire’s magnetic field.
(a) Which two particles experience magnetic forces in the same direction at the instant shown in the diagram? Explain how you arrived at your answer.
(b) Rank the particles according to the magnitudes of the forces acting on that particle from strongest to weakest. Put the symbol > or = between each letter of your ranking.
Strongest Force ____ ____ _____ _____ Weakest Force
Justify your answer quantitatively, using equations.
(c) Rank the particles according to the magnitudes of each particle’s acceleration from greatest to least.
Justify your answer quantitatively, using equations.
Greatest Acceleration ____ ____ _____ _____ Least Acceleration
Each particle makes a circular motion in the wire’s magnetic field. All of the circular motions have radii that are very small compared to the particle’s distance from the wire, so that the magnetic field around each circle can be assumed to be uniform.
(d) Which particle or particles make a counterclockwise circular motion? Explain how you arrived at your answer.
(e) Rank the particles according to the radius of each particle’s circular motion from greatest to least.
Justify your answer quantitatively, using equations.
Greatest Radius ____ ____ _____ _____ Least Radius
(f) Rank the particles according to the period of each particle’s circular motion from longest to shortest.
Justify your answer quantitatively, using equations.
Longest Period ____ ____ _____ _____ Shortest Period
Solutions
17 total points.
(a)
Above the wire, the magnetic field is out of the page. The magnetic field is into the page below the wire.
Particle P feels leftward force (positive, moving down, out of the page magnetic field).
Particle N feels upward force (negative, moving left, into the page magnetic field).
Particle D feels downward force (positive, moving right, out of the page magnetic field).
Particle A feels leftward force (positive, moving up, into the page magnetic field).
(1 point) P and A feel the same magnetic force.
(1 point) Citing a hand rule.
(b)
(1 point) The correct ranking is N = D > A > P
(1 point) Indicating that the magnetic field is proportional to 1/y, where y is the distance of the charge from the wire. Using the equation would satisfy this requirement.
(1 point) Indicating that .
(1 point) Putting these equations together to find that force is proportional to . Stating the full equation satisfies the last three points.
(c)
(1 point) The correct ranking is N > D > P > A
(1 point) Stating and using Newton’s Second Law, . Using does NOT earn this point.
(1 point) Using the previous expression for force to find acceleration proportional to . Stating the full equation satisfies the last three points.
(d)
(1 point) A is the only particle that moves counterclockwise.
(1 point) The student indicates that the force direction must be 90o counterclockwise from the velocity direction for counterclockwise circular motion, or some other convincing argument.
(e)
(1 point) The correct ranking is D = A > P = N
(1 point) Setting the magnetic force equal to the centripetal force, , or setting the acceleration found in (c) equal to .
(1 point) Solving for radius to find and finding that radius is proportional to or coming up with a full equation for r that shows mvy in the numerator and q in the denominator.
(f)
(1 point) The correct ranking is A > P = D > N
(1 point) Using the equation .
(1 point) Solving for period to that period is proportional to or some other equation that shows this relationship.
Author
John Frensley
Prosper High School
Prosper, Texas
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190406 | https://pmc.ncbi.nlm.nih.gov/articles/PMC2525492/ | Non-Traditional Vectors for Paralytic Shellfish Poisoning - PMC
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. 2008 Jun 10;6(2):308–348. doi: 10.3390/md20080015
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Non-Traditional Vectors for Paralytic Shellfish Poisoning
Jonathan R Deeds
Jonathan R Deeds
1 US Food and Drug Administration Center for Food Safety and Applied Nutrition, 5100 Paint Branch Parkway, College Park, Maryland, 20723, USA
Find articles by Jonathan R Deeds
1,, Jan H Landsberg
Jan H Landsberg
2 Fish and Wildlife Research Institute, Florida Fish and Wildlife Conservation Commission, 100 Eighth Avenue Southeast, St. Petersburg, Florida, 33712, USA
Find articles by Jan H Landsberg
2,, Stacey M Etheridge
Stacey M Etheridge
1 US Food and Drug Administration Center for Food Safety and Applied Nutrition, 5100 Paint Branch Parkway, College Park, Maryland, 20723, USA
Find articles by Stacey M Etheridge
1, Grant C Pitcher
Grant C Pitcher
3 Marine and Coastal Management, Cape Town, South Africa
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3, Sara Watt Longan
Sara Watt Longan
4 State of Alaska Department of Environmental Conservation, Anchorage, AK, USA
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4
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1 US Food and Drug Administration Center for Food Safety and Applied Nutrition, 5100 Paint Branch Parkway, College Park, Maryland, 20723, USA
2 Fish and Wildlife Research Institute, Florida Fish and Wildlife Conservation Commission, 100 Eighth Avenue Southeast, St. Petersburg, Florida, 33712, USA
3 Marine and Coastal Management, Cape Town, South Africa
4 State of Alaska Department of Environmental Conservation, Anchorage, AK, USA
✉
Authors to whom correspondence should be addressed; Tel.: +1-301-436-1474; Fax: +1-301-436-2624; E-mail: jonathan.deeds@fda.hhs.gov (J.R. Deeds); or Tel.: +1-727-896-8626; Fax: +1-727-823-0166; E-mail: jan.landsberg@myfwc.com (J.H. Landsberg).
Received 2008 Mar 14; Revised 2008 Jun 3; Accepted 2008 Jun 3; Collection date 2008 Mar.
© 2008 by the authors; licensee Molecular Diversity Preservation International, Basel, Switzerland
This article is an open-access article distributed under the terms and conditions of the Creative Commons Attribution license (
PMC Copyright notice
PMCID: PMC2525492 PMID: 18728730
Abstract
Paralytic shellfish poisoning (PSP), due to saxitoxin and related compounds, typically results from the consumption of filter-feeding molluscan shellfish that concentrate toxins from marine dinoflagellates. In addition to these microalgal sources, saxitoxin and related compounds, referred to in this review as STXs, are also produced in freshwater cyanobacteria and have been associated with calcareous red macroalgae. STXs are transferred and bioaccumulate throughout aquatic food webs, and can be vectored to terrestrial biota, including humans. Fisheries closures and human intoxications due to STXs have been documented in several non-traditional (i.e. non-filter-feeding) vectors. These include, but are not limited to, marine gastropods, both carnivorous and grazing, crustacea, and fish that acquire STXs through toxin transfer. Often due to spatial, temporal, or a species disconnection from the primary source of STXs (bloom forming dinoflagellates), monitoring and management of such non-traditional PSP vectors has been challenging. A brief literature review is provided for filter feeding (traditional) and non-filter feeding (non-traditional) vectors of STXs with specific reference to human effects. We include several case studies pertaining to management actions to prevent PSP, as well as food poisoning incidents from STX(s) accumulation in non-traditional PSP vectors.
Keywords: saxitoxins, STXs, paralytic shellfish poisoning, PSP, saxitoxin puffer fish poisoning, SPFP, non traditional vectors, gastropods, crustaceans, puffer fish, public health
1. Paralytic Shellfish Toxins and Sources
Neurotoxic paralytic shellfish toxins, which comprise saxitoxin and saxitoxin related compounds (STXs), are responsible for the sometimes fatal toxic seafood-related syndromes, paralytic shellfish poisoning (PSP) and saxitoxin puffer fish poisoning (SPFP). These compounds are produced by bloom-forming microalgae – mainly marine dinoflagellates -- approximately ten Alexandrium species, Gymnodinium catenatum, and Pyrodinium bahamense -- and freshwater or brackish cyanobacteria, Anabaena circinalis, A. lemmermannii, Aphanizomenon gracile, A. issatschenkoi (as A. flos-aquae), Cylindrospermopsis raciborskii, Lyngbya wollei, Planktothrix sp., and Rivularia sp. STXs comprise saxitoxin and at least 21 derivatives that in various combinations and concentrations have been associated with PSP. No natural toxigenic dinoflagellate or cyanobacteria population has been found to contain all naturally occurring STX derivatives (Table 1). The toxin profile (i.e., the toxin components produced) is considered by some to be characteristic of the microalgal strain or species [2–3], but this finding has not been consistent among all species in all areas. Some of the STX derivatives are highly toxic (as sodium channel-blocking agents in mammals) and include the carbamate toxins, saxitoxin (STX), neosaxitoxin (NEO), and gonyautoxins (GTX1-4). The decarbamoyl analogues (dcSTX, dcNEO, dcGTX1-4) and the deoxydecarbamoyl analogues (doSTX, doGTX2, doGTX3) are of intermediate toxicity. The least toxic derivatives are the N-sulfocarbamoyl toxins, B1 (GTX5), B2 (GTX6), and C1–C4 [1, 4]. Although not usually associated with PSP, Cochlodinium polykrikoides (as Cochlodinium type ’78) has been shown to produce two unique, zinc-bound, NEO-like compounds . In 1977, Cochlodinium sp. was implicated in PSP outbreaks in Venezuela , but corroborative evidence is lacking.
Table 1.
Microalgal sources of saxitoxins and saxitoxin derivatives (no reference is made to other toxins produced by these species).
| Species | Saxitoxin and derivatives | References |
:---:
| Dinoflagellates |
| Alexandrium acatenella | STX | 7–9 |
| Alexandrium andersoni | STX, NEO | 10–11 |
| Alexandrium angustitabulatum | unknown toxin composition | 12 |
| Alexandrium catenella | STX, GTX1–4, NEO,B1–2, C1–4 | 13–21 |
| Alexandrium cohorticula | STX, GTX1–4 | 22–23 |
| Alexandrium fundyense | STX, NEO, GTX1–4, C1–2, B1 | 9, 24–25 |
| Alexandrium minutum (= A. lusitanicum) | GTX1–4 | 20, 26–30 |
| Alexandrium ostenfeldii | GTX2–3, B2, C1–2 | 31–34 |
| Alexandrium tamarense | STX, NEO, GTX1–4, B1, C1, C2, C4 | 9, 21, 35–40 |
| Alexandrium tamiyavanichi | STX, GTX1–4, B1, C1–4 | 41–42 |
| Cochlodinium polykrikoides (= C. heterolobatum, Cochlodinium type’78) | zinc-bound carbamoyl hydroxy NEO | 5 |
| Gymnodinium catenatum | STX, NEO, trace GTX2–3, B1–2, C1–4 | 30, 41, 43–46 |
| Pyrodinium bahamense | STX, NEO, B1–B2 | 41, 47–58 |
| |
| Cyanobacteria |
| Anabaena circinalis | STX, GTX1–4, C1–C2, dcGTX2–3 | 3, 59–66 |
| Anabaena lemmermannii | STX | 67 |
| Aphanizomenon gracile | STX, NEO | 68 |
| Aphanizomenon issatschenkoi (as A. flosaquae) | NEO, STX | 69–77 |
| Cylindrospermopsis raciborskii | STX, NEO, GTX2–3 | 78–79 |
| Lyngbya wollei | dcSTX, dcGTX2–3, acetylated STX analogues | 80–81 |
| Planktothrix sp. | STX | 82 |
| Rivularia sp. | GTX2, GTX4 | 83 |
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Numerous microalgal species have been documented to produce STXs and all are potentially human health risks via the food chain. However, the sources of the majority of PSP reports are the marine dinoflagellates Alexandrium tamarense, A. fundyense, A. catenella, Gymnodinium catenatum, and Pyrodinium bahamense1 [84–85]. Because STXs are also produced by freshwater cyanobacteria, there is a potential for STXs to be transferred through the freshwater food web and pose a risk to human consumers of freshwater products (e.g. mollusks) contaminated by these toxins . STX(s) composition and concentration can vary amongst microalgal species and strains; with geographical location, with environmental factors, and under different experimental conditions [25, 39, 87–88]. Because the toxin profiles of STX-producing dinoflagellate species differ, the exposure dose and the proportion of highly toxic STX derivatives to which animals are exposed will also vary [89–90].
STXs are present in a wide range of aquatic organisms and they have been documented to occur when dinoflagellates were apparently absent . Knowledge of the widespread distribution of STXs and results of a series of experimental studies has led to the conclusion that in some cases dinoflagellates are not the only source of STXs . Although still not definitively proven, a bacterial origin for STXs has been proposed, and bacteria may play a role in the production of STXs in certain dinoflagellate species [22, 92–97].
STXs are highly lethal, having an LD 50 in mice (intraperitoneally [i.p]) of 10_μg/kg (as compared to an LD 50 for sodium cyanide at 10 mg/kg . STXs are potent neurotoxins that bind to site 1 on the voltage-dependent sodium channel, block the influx of sodium into excitable cells, and restrict signal transmission between neurons. Symptoms of PSP are paresthesia and numbness, first around the lips and mouth and then involving the face and neck; muscular weakness; sensation of lightness and floating; ataxia; motor incoordination; drowsiness; incoherence; progressively decreasing ventilatory efficiency; and in high doses, respiratory paralysis and death [98–99].
2. Traditional Vectors of Saxitoxins to Human Consumers
Most humans who experience PSP have consumed toxic bivalves , but occasionally, non traditional vectors such as toxic gastropods and crustaceans , and rarely toxic fish [52, 100] are implicated (see section 3). Numerous fatal cases of PSP have been reported globally but the successful implementation of programs monitoring for the presence of both STX-producing microalgae and the presence of STXs in shellfish in many countries has helped to minimize public health risks. To our knowledge, all documented human PSP cases have been caused by toxic marine dinoflagellates; for the most part, the geographical distribution of such PSP outbreaks has been related to the global distribution of the various STX-producing species and their toxigenic strains .
Because PSP outbreaks typically result from the consumption of toxic marine shellfish, most studies on STXs concern those vector species that are edible, economic resources. Globally, STXs have been documented in numerous species of mollusks, primarily bivalves, and extensive reviews are available on their toxic occurrence, distribution, exposure, biotransformation, and effects [84, 101–104]. Only a brief literature survey of STXs in traditional bivalve vectors will be provided here.
STX was first isolated from toxic Washington butterclams, Saxidomus gigantea [105–106]. In the USA, the first red tide bloom that led to a major PSP outbreak occurred in September 1972 from southern Maine to Cape Ann, Massachusetts. Blue mussels (Mytilus edulis) and softshell clams (Mya arenaria) were most susceptible to STX(s) accumulation and were the most toxic bivalves. Northern quahogs (Mercenaria mercenaria) did not accumulate toxins, even in areas where blue mussels and softshell clams had high STX(s) levels. Eastern oysters (Crassostrea virginica) had very low STX levels . In a few isolated areas, softshell clams and blue mussels remained toxic until April 1973 , but it is not known whether this was due to slow depuration or a re-occurrence of toxic cells.
The fate and distribution of STXs in bivalves varies according to harmful algal bloom (HAB) characteristics; environmental conditions; prior history of exposure; species, intrapopulation, and individual variability; uptake dynamics and detoxification mechanisms; anatomical localization and retention; physiological breakdown or biotransformation mechanisms; and differences in initial toxicity of dinoflagellates [84, 101, 103–104, 109–115]. Differences between bivalve species in the ability to accumulate STXs have been correlated with each species’ in vitro nerve sensitivity to STX and ability to continue actively feeding during toxic blooms [116–117]. Some bivalves demonstrate resistance to STXs contributing to an increased risk of PSP in humans .
Bivalves retain STXs for different lengths of time, and the toxic components retained vary; knowledge about these differences aids in the management and prevention of PSP. Some species depurate toxins rapidly whereas others are slow to depurate. A range of STX toxicity levels is found in different bivalve species. Extremely high STX concentrations have been found in the mussels Mytilus trossulus and M. edulis, in softshell clams and Washington butterclams, and in the scallops Patinopecten yessoensis and Placopecten magellanicus. In other bivalves, such as northern quahogs and oysters, Crassostrea spp., STXs are at low levels or are absent [84, 89, 104]. Depuration times also vary between species. Most species can eliminate STXs within weeks [84, 101], whereas Washington butterclams, sea scallops (P. magellanicus), and Atlantic surfclams (Spisula solidissima), are known to retain high levels of toxins for long periods of time (from months to more than five years) [102, 109, 120].
The toxin profiles of toxic bivalves and associated PSP risks to human consumers vary depending upon the toxigenicity of the dinoflagellate species to which the mollusks are exposed. For example, in general, bivalves exposed to Alexandrium tamarense, A. catenella, and A. minutum accumulate high GTX levels, whereas bivalves exposed to Pyrodinium bahamense and G. catenatum accumulate very low levels of GTX . Bivalve toxin profiles also vary by geographic region, by season, and by the distribution of toxic components in different tissues [2, 102, 109–110, 120–122]. The location and deposition weight of toxin components in the various bivalve organs vary between species. For example, in the scallops P. magellanicus and P. yessoensis, the majority of the toxins are concentrated in the digestive gland, and while toxicity levels in the gills, gonads, and adductor muscles are typically less than the regulatory action level of 80 μg STXeq/100g, concentrations in gills and gonads have on occasion been above regulatory limits . Since toxins are not readily accumulated in the adductor muscle of scallops, when this is the only part of the shellfish consumed, they are usually considered safe for public consumption, even in the presence of toxic algae .
Because they naturally ingest a variety of dinoflagellate species and strains, bivalves are exposed to a variety of toxic components. Knowledge of which toxins are deposited in which tissues and how they are biotransformed at each trophic level may be critical for determining the public health risk associated with the consumption of different shellfish species and their consumable tissues. For example, Atlantic surfclams and sea scallops are naturally exposed in New England to STXs associated with Alexandrium spp. STXs are typically stored in the tissues of these species, whereas other potentially poisonous substances such as the carbamate-derivative gonyautoxins are converted to less toxic compounds. The ability to convert carbamate toxins to their corresponding nontoxic decarbamoyl derivatives has been demonstrated in a few bivalves, such as Atlantic surfclams; Pacific littleneck clams, Protothaca staminea; and the Japanese clams Peronidia venulosa and Mactra chinensis [1, 110, 117, 124]. Because of public health concerns and the development of safety protocols, it is critical that we understand the dynamics of toxin distribution in different species, particularly in edible tissues.
3. Non-Traditional Vectors of Saxitoxins to Human Consumers
Non-bivalve invertebrates, the primary focus of this review, have increasingly been documented to accumulate STXs and have been implicated in PSP incidents. Amongst the mollusks, apart from traditional bivalve vectors, gastropods (Table 2) and rarely cephalopods (the octopus Abdopus sp. ), accumulate STXs apparently without any obvious ill effects [126, 127].
Table 2.
Maximum STX concentrations, microalgal sources, and global PSP reports in gastropods.
| Gastropod species and presumptive microalgal source | Common name | Maximum STX(s) concentration | Incident | Location | Reference |
:---: :---: :---: |
| Alexandrium acatenella |
| Polinices lewisii | Lewis moon snail | 176–600 μg STX eq./100g tissue | | British Columbia, Canada | 129 |
| |
| Alexandrium catenella |
| Adelomelon ancilla | Volute | toxic | | Chile | 85 |
| Argobuccinum sp. | Whelk | Stomach 5629 μg STX eq./100g tissue; Muscle 92 μg STX eq./100g tissue | | | |
| Concholepas concholepas | Barnacle rock shell | toxic | | | |
| Trophon sp. | Trophon | toxic | | | |
| Nassarius sp. | Nassa mud snail (dog whelk) | 9 μg STX eq./100g tissue | | Washington, USA | 85 |
| Neptunea spp. | | 200–250 MU 100 g−1 whole individuals | | Alaska, USA | 85 |
| Thais sp. | Oyster drill | 23 μg STX eq./100g tissue (GTX 2 and GTX 3 only) | | Washington, USA | 85 |
| Thais lamellosa | Oyster drill | Whole animal positive | | | |
| Thais lima | Oyster drill | Whole animal 180 μg STX eq./100g tissue | | | |
| |
| Alexandrium tamarense |
| Littorina sitkana | Sitka periwinkle | Trace whole animal | | Washington, USA | 85 |
| Lunatia heros (as Polinicies heros) | Northern moon snail | 1450 μg STX eq./100g tissue | 2 cases PSP | Massachusetts, USA | 130 |
| Buccinum undatum | Waved whelk | whole body 608 μg STX eq./100g tissue; digestive gland 1600 μg STX eq./100g tissue 3337 μg STX eq./100g tissue | 12 cases PSP, 4 fatalities Illnesses and deaths | Quebec, Canada Gulf of Maine, USA | 85, 131 85, 132 |
| Crepidula fornicata | Slipper limpet | 46–58 μg STX eq./100g tissue | | | |
| Colus stimpsoni | Stimpson’s colus | toxic | | | |
| Lunatia heros (=Euspira heros, Polinices heros) | Northern moon snail | 2922 μg STX eq./100g tissue | | | |
| Neptunea decemcostata | Ten-ridged whelk | Raw~3000–4000, steamed 1060 μg STX eq./100g tissue | | | |
| Thais lapillus | Purpura | 34 μg STX eq./100g tissue | | | |
| Lunatia heros (=Euspira heros, Polinices heros) | Northern moon snail | 247 μg STX eq./100g tissue | | Gulf of St. Lawrence, Canada | 133 |
| Adelomedon brasiliana | Volute | 28 MU g−1 whole | | Argentina | 134 |
| Zidona angulata | Volute | 210 MU g−1 viscera; 25 MU g−1 foot; 17 MU g−1 mucus | Mild case of PSP | | |
| Busycon spp. | Whelk | 50–500 MU 100 g−1 | | Quebec, Canada | 85 |
| |
| Rapana venosa | Veined rapa whelk | 11.4 MU g−1 viscera | | Hiroshima Bay, Japan | 135 |
| |
| Gymnodinium catenatum |
| Haliotis tuberculata | Abalone | 467 μg STX eq./100g muscle | | Spain | 83 |
| |
| Pyrodinium bahamense |
| Lambis lambis | Spider conch | ND – 175 MU 100 g−1 whole | Several PSP cases | Sabah, Malaysia | 136–137 |
| Oliva vidua fulminans | Olive | 2525 MU 100 g−1 whole | 5 human fatalities; 8 cases of PSP | Malaysia | 136–138 |
| Natica sp. | “Tekuyong” | 71–876 MU 100 g−1 | | Borneo | 139–140 |
| Unknown origin |
| Nassarius siguijorensis | Nassa | 370 MU 100 g−1 | | Daya Bay, Guangdong Province | 141 |
| Nassarius succinctus | Nassa | | 68 cases of PSP, March–Aug 1979; 1 fatality and 7 hospitalized | Zhejiang Povince, China | 128, 142 |
| Nassarius spp. | Nassa | | 50 PSP cases, 3 fatalities, April– May 2002 55 PSP cases, 1 fatality; summer 2004 | Fujian Province, China Yin Chuan city, China | 128 128 |
| Nassarius spp. | Nassa | 107,413 MU 100 g−1 | | Zhoushan Islands, China | 128 |
| Charonia lampas | Trumpet shell | 17.5 MU g−1 digestive gland | | Galicia, Spain | 143 |
| Natica lineata | Lined moon shell | PSP toxins | | Taiwan | 144 |
| Natica vitellus | Calf moon shell | | | | |
| Niotha clathrata | Basket shell | PSP, GTX-3 | | | 144–145 |
| Neptunea arthritica | Arthritic neptune | GTX 1–4, neoSTX, STX | | Sanriku coast, Japan | 146 |
| Tectus fenestratus | Fenestrate top shell | 18.7 μg STX eq./100g tissue | | Northwest Australia | 147 |
| Tectus nilotica maxima | Top shell | 5.0 MU g−1 whole | | Ishigaki Island, Japan | 52 |
| Tectus pyramis | Top shell | 19 MU g−1 whole | | Ishigaki Island, Japan | 52 |
| Turbo argyrostoma | Turban shell | 20 MU g−1 whole | | Ishigaki Island, Japan | 52 |
| Turbo marmorata | Turban shell | 4.2 MU g−1 whole | | Ishigaki Island, Japan | 52 |
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MU = mouse units (1MU = 0.18 μgSTX)
Presumed, genus and species name not given by author.
3.1 Gastropods
Molluscan gastropods including oysterdrills, volutes, whelks, periwinkles, moon snails, conch, slipper limpets, and turban shells (Table 2) accumulate STXs primarily acquired through predation (in many cases of toxic bivalves) [85, 128].
Because gastropods are able to bioaccumulate high concentrations of STXs, they are a significant risk to human consumers, and have been the cause of multiple fatalities, particularly in the Far East (Table 2). In gastropods, STXs are typically concentrated in the digestive gland but some species such as the moon snail, Lunatia heros, concentrate toxin in the muscle tissue . Variability in toxicity is also a function of species differences in predatory habits, differential acquisition of toxins by individuals, sporadic feeding, their ability to move away from toxin sources, and because gastropods are slow to depurate toxins .
3.1.1 Case Study 1: STXs in Abalone
Unlike filter-feeding bivalve mollusks, gastropods such as abalone (other common names: ormer and perlemon) feed by scavenging, predation, and grazing. Their diet primarily consists of kelp and other seaweeds, making them unlikely candidates for PSP. However, there have been reports of PSP toxins in abalone off the northwest coast of Spain [83, 148–149] and the west and south coasts of South Africa .
Spain
STXs were first detected in the Galician abalone Haliotis tuberculata in 1991. Subsequently, abalone in this region was affected by toxin concentrations sufficiently high enough to enforce indefinite closure of the industry in 1993 . dcSTX was the most abundant toxin reported in abalone, followed by low concentrations of STX [83, 148–149]. The source of these toxins remains unknown. The dinoflagellates Gymnodinium catenatum and Alexandrium minutum are the common STX(s) producers in this region; however, they do not display temporal or geographical distributions corresponding to that of abalone toxicity [83, 151]. Also, the toxin profile of these potential sources differs from that of the abalone; although biotransformation may be responsible for this discrepancy. The authors postulated that cyanobacteria may be the source of the toxin and they report measurable STX(s) concentrations for the cyanobacterium Rivularia sp. It is noteworthy that no other PSP problems were reported for other mollusks or crustaceans in this region . Anatomical distribution showed high toxicity in the epipodial fringe [149, 151] with as much as 2.6 times more toxin in the epithelium compared to the foot. Toxicity generally increased with increasing abalone size. Depuration of toxin in abalone did not occur during three months of monitoring cultured abalone fed a variety of macroalgae . No other abalone PSP reports have been reported for this area in the recent literature. However, there was a report by Huchette and Clavier (2004) that indicated the abalone fishery reopened in Spain in 2002, but was closed again shortly thereafter due to an oil spill.
South Africa
Abalone harvesting represents an old fishery in South Africa and currently this fishery includes recreational, subsistence and commercial harvesting. In addition to wild harvest, the 1990’s represented a period of movement towards land-based abalone farms. In 1999, STXs were detected in abalone from two farms located along the west coast of South Africa . Subsequent testing also found the presence of toxins in wild abalone. Throughout this evaluation of abalone PSP, toxicity was tested using the AOAC mouse bioassay with levels reported from below the limit of detection to greater than 1600 μg STX eq/100 g. For most of these cases, analysis was conducted on the entire animal; however, some samples were separated into specific body parts to examine anatomical distribution. As with other organisms, there appears to be large variability in toxicity between individuals (5–10 fold variability reported). Observations of detached and paralyzed abalone in the wild were made and analysis confirmed the presence of STXs. Pitcher et al. found a geographical gradient in toxicity with the highest toxicity observed in abalone from the north and a general decrease southwards. The notable distinction in toxin composition for South African abalone compared to those from Spain is that only STX was detected in the former . This profile is different from the known STX(s) source (Alexandrium catenella) and other vectors (e.g. mussels Mytilus galloprovincialis) in the area [150, 153]. Given the toxin profile differences and the feeding behavior of abalone, it is uncertain what the source of STXs is to the abalone. Further investigations by Etheridge et al. indicated the putative source of toxins to be from their natural diet, the macroalgal kelp Eklonia mamixa. Depuration studies suggest that either abalone can retain the toxins for long periods of time or the toxin was still present during the studies. Pitcher et al. (2001) found that abalone retained toxins for at least seven months with no apparent decline in toxicity when kept under controlled laboratory experiments with kelp as the diet. Controlled feeding experiments were conducted using juvenile abalone (2 cm in length, average wet weight 0.6 ± 0.3 g) and demonstrated that depuration did not occur when abalone were either fed kelp or were starved. However, depuration rates of 6.3 μg per 100g per day were observed when abalone was fed artificial feed. Initial toxicity in the abalone was 160 ± 38 μg STX eq per 100g and after being fed artificial feed for two weeks toxin levels decreased to 72.3 ± 12.5 μg STX eq per 100. Thus, it is possible that feeding farmed abalone artificial feed prior to market could reduce the risk of PSP. Toxin distribution among abalone tissues demonstrates differential uptake and compartmentalization. Thus, the contribution of each tissue to total toxin burden is a function of both its absolute toxicity and relative weight contribution. Pitcher et al. found moderate amounts in the foot and viscera and high amounts in the epipodial fringe. Given the high surface area of the epipodial fringe, it contributes significantly to the total toxin burden. Abalone is often marketed with the foot for human consumption; therefore, it has been suggested that scrubbing and/or removing epithelial tissue could decrease toxicity to safe levels for consumption. This could potentially be used as a strategy to reduce toxin levels prior to market.
Periodic PSP events still occur along the west coast of South Africa. In many cases this has resulted in prevention of exporting live abalone. However, shucking and scrubbing (i.e. removing the epithelial layer of the abalone) decreased toxicity to safe levels (aggregate toxicity < 80 μg/100 g whole animal). For example, Pitcher et al. found that toxicity levels in the foot and epipodial fringe (one of the largest reservoirs of STXs containing > 800 μg/100 g in some cases) both decreased significantly (approximately 6 to 9-fold) when scrubbed. Currently, testing for toxins is done regularly under the South African Shellfish Sanitation Program run by Marine and Coastal Management under an MOU. When traces of toxin are detected, sampling frequency increases and farms in the affected area can be prevented from exporting. Again, shucking, scrubbing and cleaning remain processing options (e.g. canning) that can be used to safely market abalone from this region.
3.1.2 Case Study 2: STXs in Whelks and Moon Snails
Japan
During surveillance on the toxicity of invertebrates such as bivalves inhabiting the coasts of Hiroshima Bay in 2001 and 2002, the carnivorous gastropod rapa whelk Rapana venosa, collected in the estuary of Nikoh River, was found to contain toxins which showed paralytic actions in mice; the maximum toxicities (as STXs) were 4.2 MU/g (May 2001) and 11.4 MU/g (April 2002). This equated to total toxicities of 224 and 206 MU/viscera for these specimens (1MU = 0.18 μgSTX). In an attempt to identify the toxic principle(s) in this gastropod, the viscera were extracted with 80% ethanol acidified with acetic acid, followed by defatting with dichloromethane. The aqueous layer obtained was treated with activated charcoal and then applied to a Sep-Pak C18 cartridge. The unbound toxic fraction was analyzed by high-performance liquid chromatography. The gastropod toxin was rather unexpectedly identified as STXs. It was comprised of GTX3, GTX2, and STX as the major components, which accounted for approximately 91 mol% of all components along with STXs Cl and C2, which are N-sulfocarbamoyl derivatives. Judging from their toxin patterns, it was suggested that the STX(s) toxification mechanism of the gastropod was phytoplankton, such as Alexandrium tamarense, transferred to and accumulated in filter-feeders such as the short-necked clam, and then transferred to this carnivorous whelk through predation .
New England, USA
Several species of moon snail and whelk are also known to accumulate STXs and such gastropods are often prohibited for harvesting in waters of the states of Maine and Massachusetts, USA. Closures in waters off the coast of Maine are made by the Department of Marine Resources and are posted on their website ( [accessed 3 March, 2008]). The moon snail of interest in this area is Lunatia heros, and the whelks impacted by closures are of the family Muricidae and Buccinidae. In Maine state waters, harvesting of moon snails and the whelk Buccinum undatum is closed as a precaution whenever the blue mussel Mytilus edulis exceeds the regulatory limit for STXs, due to the observation that if there are any bivalves carrying STXs then any co-occurring carnivorous gastropods will be toxic as well (D. Couture, pers. comm.). The Division of Marine Fisheries is responsible for the safety of seafood harvested in Massachusetts state waters and their closures can be found on their website ( [accessed 3 March, 2008]). Off the coast of Massachusetts, closures are often in effect for the lobed moon snail Polinices duplicatus and the northern moon snail L. heros, as well as the channeled whelk Busycon canaliculatum and the knobbed whelk B. carica (M. Hickey, pers. comm.). Notably, harvesting closures are often extended for moon snails longer than for other species because they accumulate higher levels of toxin by feeding on toxic bivalves. Certain carnivorous mollusks also appear to retain toxins for longer periods of time than the source bivalves. For example, an extensive Alexandrium fundyense bloom occurred off the coast of New England in 2005 resulting in PSP closures of vast regions in state and federal waters . The U.S. Food and Drug Administration is responsible for the safety of seafood harvested in federal waters and they began sampling shellfish in the impacted areas during the 2005 bloom. Sampling continued in 2006 and toxicity levels above the action level were still being detected for moon snails and whelk from federal waters off the coast of Massachusetts (Table 3). In that region, the only other species that remained toxic was the sea scallop (P. magellanicus), in the viscera (Table 3). Sea scallops are known to retain toxins in viscera for long periods of time compared to other co-occurring species . These data demonstrate the need to monitor toxicity for these non-traditional seafood products, even after bloom conditions have dissipated.
Table 3.
Shellfish collected from New England, USA, federal waters in 2006. All testing was done by H 3 STX receptor binding assay. Highlighted results indicate individuals above the action level (80 μg STX eq./100g tissue). M = male, F = female; LOD = below detection limit.
| Sampling Date | Common Name | Scientific Name | Number of Animals | Sampling Coordinates | STX eq. (μg/100g) |
:---: :---: :---: |
| 7-8-06 | Ocean Quahog | Arctica islandica | 8 | 41 00.183N | 7.2 |
| 70 44.543W |
| 7-8-06 | Ocean Quahog | Arctica islandica | 3 | 41 06.476N | 11.6 |
| 70 27.150W |
| 7-9-06 | Whelk | Busycon sp. | 3 | 41 25.057N | 234.3 |
| 70 02.751W |
| 7-9-06 | Atlantic Surfclam | Spisula solidissima | 3 | 41 25.057N | 15.6 |
| 70 02.751W |
| 7-9-06 | Blue Mussels | Mytilus edulus | 12 | 41 23.836N | 19.5 |
| 69 53.954W |
| 7-9-06 | Blue Mussels | Mytilus edulus | 12 | 41 23.836N | 26.3 |
| 69 53.954W |
| 7-9-06 | Northern Moon Snail | Lunatia heros | 3 | 41 26.084N | 265.5 |
| 70 03.000W |
| 7-9-06 | Northern Moon Snail | Lunatia heros | 7 | 41 23.836N | 321.0 |
| 69 53.954W |
| 7-10-06 | Sea Scallops | Placopecten magellanicus | 9 | 42 09.865N | 228.8 |
| 70 18.279W |
| 7-10-06 | Sea Scallop viscera (F) | Placopecten magellanicus | 1 | 42 09.865N | 93.6 |
| 70 18.279W |
| 7-10-06 | Sea Scallop viscera (M) | Placopecten magellanicus | 1 | 42 09.865N | 131.9 |
| 70 18.279W |
| 7-11-06 | Ocean Quahog | Arctica islandica | 11 | 42 12.025N | <LOD |
| 70 22.017W |
| 7-11-06 | Sea Scallop | Placopecten magellanicus | 6 | 42 11.391N | 50.6 |
| 70 19.700W |
| 7-11-06 | Northern Moon Snails | Lunatia heros | 6 | 42 11.391N | 318.9 |
| 70 19.700W |
| 7-11-06 | Ocean Quahogs | Arctica islandica | 12 | 42 12.025N | <LOD |
| 70 22.017W |
| 7-11-06 | Blue Mussels | Mytilus edulus | 9 | 42 12.025N | 5.0 |
| 70 22.017W |
| 7-11-06 | Atlantic Surfclam | Spisula solidissima | 2 | 42 11.391N | 16.1 |
| 70 19.700W |
| 7-11-06 | Ocean Quahog | Arctica islandica | 5 | 42 12.025N | 12.0 |
| 70 22.017W |
| 7-11-06 | Ocean Quahog | Arctica islandica | 4 | 42 11.391N | 0.2 |
| 70 19.700W |
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Number of whole animals homogenized to form representative sample.
For sea scallops only combined viscera and gonad tested, unless otherwise indicated.
3.2 Crustaceans
Among non-filter feeding, non-molluscan species, STXs have been found most commonly in xanthid crabs (Table 4) [156–159]. In some cases, toxins were believed to be derived from the calcareous alga Jania sp., consumed by the crabs . STXs have also been found in other crab species, lobsters, crayfish, penaeid shrimp, barnacles (Table 4) and a few other crustacea [85, 147].
Table 4.
Maximum STX concentrations, microalgal sources, and geographical reports of STXs in crustaceans.
| Crustacean species and presumptive microalgal source | Common name | Maximum STX(s) concentration | Location | Reference |
:---: :---:
| Alexandrium catenella |
| Cancer magister | Dungeness crab | 72 μg STX eq./100g viscera | Washington, USA | 85 |
| Cancer productus | Red rock crab | 285 μg STX eq./100g viscera 27 μg STX eq./100g muscle | Washington, USA | 161 |
| Fabia subquadrata | Pea crab | 32 μg STX eq./100g whole crabs | Washington, USA | 85 |
| Hemigrapsus nudus | Purple shore crab | 44 μg STX eq./100g whole body minus legs and carapace | Washington, USA | 161 |
| Hemigrapsus oregonensis | Green shore crab | 31 μg STX eq./100g whole | Washington, USA | 161 |
| Pagurus sp. | Hermit crab | 35 μg STX eq./100g whole crabs | Washington, USA | 85 |
| Pugettia producta | Northern kelp crab | 146 μg STX eq./100g eggs; 1710 μg STX eq./100g viscera; 48 μg STX eq./100g muscle | Washington, USA | 161 |
| Balanus spp. | Barnacles | 84 μg STX eq./100g whole | Washington, USA | 161 |
| |
| Alexandrium tamarense |
| Anonyx sarsi | Gammarid amphipod | 180 μg STX eq./100g (tissue not specified) | St.Lawrence estuary, Canada | 162 |
| Cancer borealis | Jonah crab | 56 μg STX eq./100g (tissue not specified) | Maine, USA | 85 |
| Homarus americanus | American lobster | 1512 μg STX eq./100g hepatopancreas (bioassay); 961 μg STX eq./100g hepatopancreas (HPLC); 69 μg STX eq./100g meat (HPLC) | Bay of Gaspe, Canada | 162 |
| |
| Pyrodinium bahamense |
| ND | Crab | 339 MU 100 g−1 | Brunei Darussalam | 141 |
| ND | Mangrove crabs | 239 MU 100 g−1 guts; 175 MU 100 g−1 gills | Sabah, Malaysia | 138 |
| Portunus pelagicus | Blue manna crab | 175 MU 100 g−1 whole crab; 288 MU 100 g−1 gills; 328 MU 100 g−1 guts 1.8 μg STX eq./100g whole | Sabah, Malaysia Northwest Australia | 138 147 |
| Panulirus versicolor | Painted spiny lobster | 175 MU 100 g−1 whole lobster; 175 MU 100 g−1 body only | Sabah, Malaysia | 138 |
| Panulirus longipes | Longlegged spiny lobster | 211 MU 100 g−1 whole lobster; 177 MU 100 g−1 head and legs | Sabah, Malaysia | 138 |
| ND | Penaeid shrimp | 175 MU 100 g−1 frozen tails; 268 MU 100 g−1 body only | Sabah, Malaysia | 138 |
| ND | Penaeid shrimp “Udang” | 190 MU 100 g−1 | Brunei Darussalam | 141 |
| |
| Unknown origin |
| Hemigrapsus sanguineus | Asian shore crab | 0.16 MU g−1 hepatopancreas | Sanriku coast, Japan | 146 |
| Metopograpsus frontalis | Mangrove shore crab | 10.0 μg STX eq./100g whole | Northwest Australia | 147 |
| Pachygrapsus crassipes | Striped shore crab | 0.10 MU g−1 hepatopancreas | Sanriku coast, Japan | 146 |
| Percnon planissimum | Sally lightfoot crab | 7.4 MU g−1 whole | Ishigaki Island, Japan | 52 |
| Pilumnus pulcher | Hairy crab | 80 μg STX eq./100g whole | Northwest Australia | 147 |
| Pilumnus vespertilio | Hairy crab | 120 μg STX eq./100g whole 6.1 MU g−1 whole | Northwest Australia Ishigaki Island, Japan | 147 52 |
| Schizophrys aspera | Eyelash spider crab | 2.3 MU g−1 whole | Ishigaki Island, Japan | 52 |
| Telmessus acutidens | Edible shore crab | 2723 μg STX eq./100g viscera | Fukushima Prefecture, Japan | 163,164 |
| Actaeodes tomentosus | Xanthid crab | 130 MU g−1 whole | Ishigaki Island, Japan | 52 |
| Atergatis floridus | Xanthid crab | Positive STX, NEO, GTX2 16,611 μg STX eq./100g whole 490 MU g−1 whole Positive GTX 1–4 | Fiji Islands Northwest Australia Ishigaki Island, Japan | 165 147 52 |
| Atergatopsis germaini | Xanthid crab | Positive GTX 3, NEO, STX | Taiwan | 167 |
| Demania reynaudi | Xanthid crab | Positive GTX 3–4, NEO | Taiwan | 166 |
| Eriphia scabricula | Xanthid crab | 180 MU g−1 whole | Ishigaki Island, Japan | 52 |
| Eriphia sebana | Xanthid crab | Positive STX, NEO, GTX1, GTX2 | Great Barrier Reef, Australia | 168 |
| Euzanthus exsculptus | Xanthid crab | 29 μg STX eq./100g whole | Northwest Australia | 147 |
| Lophozozymus octodentatus | Xanthid crab | 23 μg STX eq./100g whole | Northwest Australia | 147 |
| Lophozozymus pictor | Xanthid crab | 18.9 MU g−1 whole crab Positive GTX | Australia Taiwan | 169 170 |
| Neoxanthias impressus | Xanthid crab | 147 μg STX eq./100g whole 10 MU g−1 whole | Northwest Australia Ishigaki Island, Japan | 147 52 |
| Platypodia granulosa | Xanthid crab | 110 MU g−1 whole | Ishigaki Island, Japan | 52 |
| Platypodia pseudogranulosa | Xanthid crab | 10 μg STX eq./100g whole | Northwest Australia | 147 |
| Xanthias lividus | Xanthid crab | Positive GTX | Taiwan | 171 |
| Zosimus aeneus | Xanthid crab | Positive STX, NEOSTX, GTXI-3 Positive GTX 660 MU g−1 whole 108,000 μg STX eq./100g chelae muscle; 720 μg STX eq./100g cephalothorax muscle 78 μg STX eq./100g whole 259 MU g−1 whole crab | Fiji Islands Taiwan Ishigaki Island, Japan Japan Northwest Australia Philippines | 165 171 52 172 147 173 |
| Procambarus clarkii | Red swamp crayfish | 0.23 MU g−1 hepatopancreas | Sanriku, Japan | 146 |
| Carcinoscorpius rotundicauda | Mangrove horseshoe crab | STX | Thailand | 174 |
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MU = mouse units (1MU = 0.18 μgSTX); ND = no data
Many macrocrustaceans, including lobsters, are able to tolerate and hence concentrate extremely high levels of STXs. Lobsters can accumulate STXs by preying on, among other species, blue mussels which can have maximum toxicities of up to 23,000 μg STX eq/100g . Jiang et al. (2006) demonstrated the transfer and metabolism of STXs from the scallop Chlamys nobilis to spiny lobsters Panulirus stimpsoni. When experimentally fed with toxic viscera of C. nobilis, the hepatopancreas of P. stimpsoni showed the same toxin profile as that of the scallop, including GTX1–3, C1+C2 and B1, and dcGTX2+3. In spiny lobsters depurated with non-toxic squid, the mildly toxic N-sulfocarbamoyl toxins tended to transform into more highly toxic carbamates. After being fed toxic C. nobilis for six days, spiny lobsters selectively accumulated N-sulfocarbamoyl toxins with low toxicity. The concentration of dcGTX (2+3) in P. stimpsoni decreased significantly and was not detectable after six days depuration, which was likely due to their initial low level of toxicity.
Xanthid crabs can harbor toxins in their tissues at concentrations (Table 4) that would be fatal to other animals . Maximum toxin levels of more than 16,000 μg STX eq/100g were found in the xanthid crab Atergatis floridus in Australia, even though the majority of samples contained less than 80 μg STX/100g . In Japan, an individual Zosimus aeneus contained nearly 16,500 Mouse Units (MU) per g , which is equivalent to 300,000 μg STX eq/100g [105, 161]. Several species of xanthid crabs produce a hemolymph protein, saxiphilin, that binds with STX and which may confer some resistance to possible toxic effects . This mechanism may explain why some xanthid crab species appear to tolerate exceptionally high levels of toxins . When a mixture of GTX2 and GTX 3 in 3% NaCl was injected into the right chela of A. floridus, the rate of dissipation within the crab was fairly high and suggested that high concentrations of toxin are not accumulated in all species .
3.2.1 Case Study 3: STXs in Crabs
East Timor
In October 2000, an adult male died within hours of ingesting a xanthid crab Zosimus aeneus (Xanthidae) . A second, yet uneaten specimen of Z. aeneus from the same meal contained a total toxicity of 162.8 μg STX eq/100g tissue (comprising GTX2, GTX3, NEO, dcSTX, and STX); these same toxins were identified in the gut contents, blood, liver and urine of the victim. Metabolism of STXs occurred with the ingested crab harboring GTX2, GTX3 and STX, whereas NEO, dcSTX and STX dominated the STXs in the victim's urine. The STX(s) composition in the gut contents, in both their identity and proportion, was intermediate between the eaten crab and the urine suggesting that toxin conversion commenced in the victim's gut. The victim's meal did not consist solely of the toxic crab eaten and the possibility of other food items acting in a synergistic manner with the consumed STXs cannot be discounted. As well as STXs, xanthid crabs are known to harbor tetrodotoxin (TTX) and palytoxin [180–182].
Japan
Oikawa et al. [163–164, 183–184] showed that the edible crab Telmessus acutidens both accumulated and retained STXs after consuming contaminated mussels (Mytilus galloprovincialis) in Japan. STXs in two shore crab species, T. acutidens and Charybdis japonica, were compared with the toxin in the prey mussel M. galloprovincialis and causative dinoflagellates Alexandrium tamarense, all having been collected at Onahama, Fukushima Prefecture, in the northern part of Japan. When the toxicities were detected in mussels by mouse bioassays, 73.7% of the sampled T. acutidens were toxic in the hepatopancreas. Charybdis japonica was also expected to be a possible vector species, but only small quantities of STXs were detected in eight specimens of the crab by HPLC analysis. The difference in STX(s) accumulation in both T. acutidens and C. japonica was then investigated at Onahama, Fukushima Prefecture, from 2002 to 2005. The level of toxin accumulation in the hepatopancreas of T. acutidens corresponded to that of mussels when examined on a yearly basis. In 2003, some crabs had a high toxicity of approximately 1000 MU, which compares to one-third of the human minimum lethal dose. Therefore, it was concluded by the authors, that T. acutidens should be monitored as a vector species of PSP toxins. The toxin profile of T. acutidens was also investigated. Because an increase in highly toxic species of STXs with a decrease in low toxic species, such as N-sulfocarbamoyl-11-hydroxysulfate toxins, was not clearly observed between consecutive samples, toxin transformation in T. acutidens was considered to have a minimal impact on toxicity. STXs were also detected in several specimens of C. japonica, but the highest toxicity was only 7.4 MU/g in the hepatopancreas. Lastly, accumulation and depuration rates of STXs in the crab T. acutidens were investigated by feeding toxic and non-toxic mussels under laboratory controlled conditions. The crab accumulated toxins in the hepatopancreas in proportion to the amount of toxic mussels they ingested, and the toxicity in the crab hepatopancreas became 3.2 fold of that in the prey mussels after 20 days of feeding. During depuration, a fast reduction of the total toxicity was observed in the crab, and the retention rate of the toxicity after five days depuration with feeding of non-toxic mussels was 45.8 +/− 18.7%. The reduction of the toxicity was moderated in the later period of depuration, and the retention rates of the total toxicity after 10 and 20 days were 54.1 +/− 29.8% and 14.5 +/− 9.0%, respectively. The toxin profiles in the crab and mussel were investigated by high performance liquid chromatography, and reductive conversions of the toxins were observed when the toxins were transferred from the mussel to the crab. Consequently, high concentrations of GTX2, GTX3, and STX that were not detected in the prey mussels were found in the crab.
Alaska, USA
Although not thoroughly recorded in the scientific literature, the State of Alaska, Division of Environmental Health, Food Safety and Sanitation Program has been observing elevated levels of STXs in viscera from several species of commercially harvested crabs for years (Figure 1). PSP is endemic to the coastal communities of the State of Alaska . The high frequency of STX producing dinoflagellates coupled with an extensive seafood harvesting industry prompted the state to establish a STX monitoring program. Most commercially harvested crab in Alaska is landed in the open waters of the Bering Sea, but limited harvesting does occur in areas where PSP toxicity is commonly seen in filter-feeding bivalves. In these areas, high regional and species variability in crab STX(s) content exists, with Dungeness crab (Cancer magister) from Kodiak Island appearing to be a consistent food safety concern (Figure 1, 2). To protect public safety, the State of Alaska Food Safety and Sanitation Program, Department of Environmental Conservation, and the Department of Fish and Game perform both pre-season environmental sampling and in season monitoring of both harvesting areas and harvested product. A conservative action level of 70 μg STX eq. /100g viscera (FDA regulatory action level = 80 μg STX eq. /100g tissue) has been established above which product cannot be marketed either live or whole cooked but must be eviscerated at the processing facility where it is landed ( [accessed 3 March, 2008]). Due to the success of this monitoring program, no reports of PSP due to the consumption of commercially harvested crab have been reported even though visceral concentrations exceeding 500 μg STX eq./100 g have been observed almost yearly in some areas (Figure 2b).
Figure 1.
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(A) Map of the state of Alaska, U.S.A. indicating collection sites for crab STX testing. (B) Number of samples above and below the regulatory action limit of 80 μg STX eq./100 g tissue for all species of commercially harvested crab in Alaska between 1992–2004, broken down by major testing area. All values are for crab viscera only. Sample = 1 crab. All testing done by AOAC mouse bioassay.
Figure 2.
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(A) Number of samples above and below the regulatory action limit of 80 μg STX eq./100 g tissue for commercially harvested crab in Alaska between 1992–2004 for all testing areas, broken down by crab type: Dungeness (Cancer magister), Tanner: (Chionoecetes opilio and Chionoecetes bairdi), King Crab: (Red, Paralithodes camtschaticus; Blue, Paralithodes platypus; Brown, Lithodes aequispinus), and Miscellaneous (including minced, viscera, and Hair Crab, Erimacrus isenbeckii). (B) Total STXs (in μg STX equivalents/100 g viscera) for all commercially harvested crab species in Alaska for all areas from 1992–2004.
3.3 Other invertebrates
Other, non-molluscan invertebrates that accumulate STXs include annelid tubeworms Eudistylia sp. , and echinoderm starfish Asterias amurensis, Astropecten scoparius, A. polyacanthus, and Pisaster ochraceus [161, 186–187]. Thus far, these species have not been implicated in PSP cases.
3.4 Fish
Although not usually targeted, STXs have been incidentally found in numerous species of fish (Table 5). As with shellfish, because STXs are water soluble compounds, researchers believed that cold-blooded vertebrates such as finfish did not typically accumulate STX(s) nor were fish negatively affected by STXs . However, the transport of STXs through the food chain and the vectoring and accumulation of toxins through zooplankton have been identified as important mechanisms by which toxins become available to higher trophic levels such as fish [188–193].
Table 5.
Maximum STX concentrations, microalgal sources, and geographical reports of STXs in various fish tissues and species.
| Fish species and presumptive microalgal source | Common name | Maximum STX(s) concentration | Location | Reference |
:---: :---:
| Alexandrium fundyense |
| Scomber scombrus | Atlantic mackerel | 209 μg STX eq./100g liver; 367 μg STX eq./100g liver | Bay of Fundy; Gulf of St. Lawrence | 194–195 |
| |
| Alexandrium tamarense |
| Scomber japonicus | Chub mackerel | 2800 μg STX eq./100g muscle; 500 μg STX eq./100g liver; 72 μg STX eq./100g gills | Argentina | 196 |
| |
| Pyrodinium bahamense |
| Rastrelliger sp. | Short mackerel | 99 MU 100 g−1 tissue | Brunei Darussalam | 141 |
| Sardinella sp. | Sardinella | 99 MU 100 g−1 tissue 572 μg STX eq./100g guts | Brunei Darussalam Sabah, Malaysia | 141 139 |
| Sphoeroides nephelus | Southern puffer fish | 1,443 μg STX eq./100g liver; 14,571 μg STX eq./100g muscle | USA | 58 |
| Sphoeroides testudineus | Checkered puffer fish | 51.1 μg STX eq./100g liver; 104.3 μg STX eq./100g muscle | USA | 58 |
| Sphoeroides spengleri | Bandtail puffer fish | 364.5 μg STX eq./100g muscle | USA | 58 |
| |
| Unknown origin |
| Cololabis saira | Pacific saury | 0.14 MU g−1 viscera | Iwate, Japan | 146 |
| Gadus macrocephalus | Pacific cod | 0.10 MU g−1 viscera; 0.10 MU g−1 intestine | Iwate, Japan | 146 |
| Lamna ditropis | Salmon shark | 0.17 MU g−1 liver | Iwate, Japan | 146 |
| Oncorhynchus keta | Chum salmon | 1.53 MU g−1 liver; 0.69 MU g−1 viscera | Iwate, Japan | 146 |
| Scarus (= Ypsiscarus) ovifrons | Knobsnout parrotfish | 0.26 MU g−1 liver; 1.58 MU g−1 intestine | Iwate, Japan | 146 |
| Arothron firmamentum | Starry toadfish | 740 MU g−1 ovary | Japan | 197 |
| A. hispidus | White-spotted puffer | Positive STX in liver, muscle, skin, and intestine | Philippines | 198 |
| A. mappa | Map puffer | Positive STX in liver, muscle, skin, and intestine | Philippines | 198 |
| A. manillensis | Narrow-lined puffer | Positive STX in liver, muscle, skin, and intestine | Philippines | 198 |
| A. nigropunctatus | Black spotted puffer | Positive STX in liver, muscle, skin, and intestine | Philippines | 198 |
| A. reticularis | Reticulated puffer | Positive STX in liver, muscle, skin, and intestine | Philippines | 198 |
| A. stellatus | Starry toadfish | Positive STX in liver, muscle, skin, and intestine | Philippines | 198 |
| Chelonodon patoca | Milk-spotted puffer | 22.0 MU g−1 muscle; 40 MU g−1 skin; 12.0 MU g−1 liver; 2.8 MU g−1 ovary (data shown as mean) Positive STX in liver, muscle, skin, and intestine | Bangladesh Philippines | 199 198 |
| Colomesus asellus | Amazon puffer | 53.2 MU whole body | Brazil | 200 |
| Takifugu pardalis | Panther puffer | Positive for STX in liver | Japan | 201 |
| T. poecilonotus | Fine patterned puffer | Positive for STX in liver, ovary and digestive tract | Japan | 202 |
| T. radiates | Puffer | Positive for STX in liver | Japan | 202 |
| T. vermicularis | Purple puffer | Positive for STX in liver, ovary and digestive tract | Japan | 202 |
| Tetraodon cutcutia | Ocellated puffer | 7.6 MU g−1 muscle; 20 MU g−1 skin; 6.0 MU g−1 liver; 5.6 MU g−1 ovary (data shown as mean) 182 MU 100 g−1 skin; 238 MU 100 g−1 muscle; 106 MU 100 g−1 liver | Thailand Bangladesh | 199 203 |
| T. cochinchinensis (as T. fangi) | Puffer | Positive for STX whole body | Thailand | 204 |
| T. suvatii | Arrowhead puffer | 191 MU g−1 muscle; 230 MU g−1 skin; 174 MU g−1 liver; 117 MU g−1 egg | Thailand | 205 |
| T. turgidus | Brown puffer | <2 MU g−1 muscle; 37 MU g−1 skin; <2 MU g−1 liver; 27 MU g−1 ovary | Cambodia | 206 |
Open in a new tab
MU = mouse units (1MU = 0.18 μgSTX)
As they are in bivalves, the toxic profiles of STXs that accumulate in fish are likely to be partially determined by species-specific differences in the bioconversion process or are dependent upon the variety and toxin profiles of their toxic prey species. During July 1988, a small bloom of Alexandrium fundyense occurred in southwestern Bay of Fundy, New Brunswick, Canada. The highest concentration in a surface-water sample was 7.5 x 10 3 cells/L. Concentrations of STXs in Atlantic mackerel, Scomber scombrus, liver extracts were measured by mouse bioassay and ranged from 40–209 μg STXeq/100g wet weight. By far the dominant component in mackerel liver was STX except in a few fish where NEO was also dominant. GTX2 and GTX3, and rarely B2, were also detectable. The difference between the toxic profiles of the fish and A. fundyense was attributed to the variety of toxic prey consumed by the fish. The fact that mackerel accumulate STXs demonstrates the transfer of these toxins up the food chain [194–195, 207]. Atlantic mackerel in the Gulf of Lawrence retained STX, GTX2, and GTX3 all year round and progressively accumulated STXs throughout their life, likely vectored via zooplankton feeding on toxic Alexandrium .
Fish, with the exception of puffer fish (see case study 5 below) are not usually vectors for STX(s) transfer if humans only eat the muscle. Accumulation of STXs is usually confined to the fish’s gut, and either certain species perish before detectable amounts of toxin appear in the muscle [207–208] or negligible concentrations of toxins accumulate in the muscle. In experimental studies, several fish species challenged with oral (LD 50 = 400–750 μg/kg body weight) or i.p. (intraperitoneal) (4–12 μg/kg body weight) doses of STX showed similar symptoms: loss of equilibrium; gasping; reduced locomotor activity; short, irregular, hyperactive periods; and death within one hour. Heavy accumulation of STX was confined to the gut (340–840 μg per 100g tissue), while STX occurred in the muscle tissues at a level an order of magnitude lower than in the gut . Kwong et al. exposed green-lipped mussels Perna viridis and black sea bream Acanthopagrus schlegeli to toxic Alexandrium fundyense to evaluate the accumulation, distribution, transformation, and elimination of STXs in controlled experimental conditions. Mussels were fed A. fundyense for seven days followed by three weeks of depuration, and the fish were fed toxic clams for five days followed by two weeks of depuration. The fish viscera accumulated most of the STXs. In the fish, the ratio of C1/C2 was 3.0 times (p< 0.01) higher when compared to the mussel tissues, indicating that conversion from C2 to C1 might have occurred when the toxin was transferred from the clams to the fish. Jiang et al. investigated the transmission and transformation of STXs from A. tamarense to the cladoceran Moina mongolica and subsequently to the larval fish Sciaenops ocellatus. STXs were transferred to S. ocellatus when they preyed upon STX(s)-containing M. mongolica. During the experimental period, A. tamarense, M. mongolica and the larval fish’s digestive glands contained C1 and C2 toxins, while the viscera of S. ocellatus contained NEO. The proportion of C2 to C1 toxins increased when STXs were transferred from A. tamarense to M. mongolica, but in the subsequent transfer from M. mongolica to S. ocellatus the proportion of C1 to C2 toxins increased. During depuration, the contents of C1 and C2 toxins in fish larvae decreased with the duration of depuration, but NEO remained relatively constant. The present results indicated that, using a cladoceran as the vector, STXs can be transferred from toxic algae to a high trophic level fish and metabolized in the fish. Future work should address the metabolic characteristics of STXs in cladocerans and the end result when they are transferred to fishes.
3.4.1 Case study 4: STXs in planktivorous fish
Far East
With the puffer fish exception, because STXs do not typically accumulate in fish muscle, humans who consume only the muscle are unlikely to become intoxicated. However, those those who consume whole fish and eat the viscera are likely to become sick. In 1976 in Brunei, 14 nonfatal PSP cases were associated with the consumption of the planktivorous fish Rastrelliger sp. during a bloom of Pyrodinium bahamense var. compressum . One PSP incident in 1983 in Indonesia involved 191 cases and four human fatalities due to the consumption of the planktivorous clupeoid fish Sardinella spp. and Selaroides leptolepis. In a second incident in November 1983, 45 people became ill after consuming fish and suffered numbness, dizziness, and tingling sensations of the lips, tongue, and throat. Although no known toxic dinoflagellate was associated with the event , PSP was highly suspected . STXs with toxin profiles similar to Pyrodinium bahamense have been confirmed in gut contents of Sardinella sp. from Brunei and in PSP incidents involving Pyrodinium, toxic shellfish, and fish that were reported from the Philippines . These incidents likely occurred because it is customary in south-east Asia to eat small fish whole, including any potentially toxic viscera [211–212].
3.4.2 Case study 5: STXs in puffer fish
One exception to the general rule that STXs tend not to accumulate to levels associated with human intoxication in fish muscle is in members of the family Tetraodontidae (puffer fish) (Table 5) inhabiting marine and freshwater habitats. STX was first described as a minor component of highly toxic (with TTX) Takifugu pardalis livers in Japan . Soon after, STX was confirmed as a minor component in the additional Japanese species T. poecilonotus and T. vermicularis , and as a major toxin in Arothron firmamentum . STXs were found to be the sole toxic component in a range of freshwater puffer fish, some responsible for human poisoning events, in Thailand, Bangladesh, Brazil, and in Cambodia (Table 5). Seven species of marine puffer fish in the Philippines (Table 5) were found to contain both STXs and TTX, with STXs being the dominant toxin in several species .
Florida, USA
Puffer fish became an important source of protein on the east coast of the United States during the Second World War, and supported a commercial fishery in the decades that followed. The primary species landed was the northern puffer fish (Sphoeroides maculatus) but limited numbers of the southern puffer (S. nephelus), primarily from Florida, was also harvested . The industry was centered in the mid-Atlantic states of Virginia, Maryland, New York, and New Jersey with > 6,000 metric tons landed in 1965 (National Marine Fisheries Service Statistics and Economics Division, personal communication). Fish were marketed dressed and skinned under the name “sea squab”. Although the commercial puffer fish industry has steadily declined since the 1970’s, today being only harvested as by-catch, domestic puffer fish can still be found in some U.S. fresh fish markets. In addition, an average of > 500,000 fish was caught annually between 1981 and 2003 by recreational anglers in the U.S. where they are easily obtained by a range of gear including hook and line (National Marine Fisheries Service Statistics and Economics Division, personal communication ).
In January 2002, the poison control center in Tampa, Florida, USA received a report of a man hospitalized with symptoms of numbness and tingling of the hands, vomiting, and diarrhea after consuming puffer fish caught during a recreational fishing trip near Titusville, located on the northern Indian River Lagoon (IRL) on Florida’s central east coast . After additional reports of patients with symptoms of neurological illness from Virginia and New Jersey, all associated with what was believed to be S. nephelus originating from the northern IRL, uneaten fish muscle samples from the New Jersey incident sent to the Canadian Institute for Marine Biosciences by the New Jersey Department of Health were, surprisingly, found to contain no detectable TTX but to contain significant amounts of STX, with lesser amounts of the STX congeners B1, and dcSTX . This same combination of toxins was confirmed in meal remnants from two separate poisoning events in 2004 . In total, 28 cases of SPFP were reported from 2002–2004 -- all due to fish originating from the northern IRL . These were the first reports of STXs both in Florida marine waters and in indigenous puffer fish in the U.S. In April 2002, the Florida Fish and Wildlife Conservation Commission (FWC) placed a ban on the commercial and recreational harvesting of all puffer fish species for the entire IRL. At the same time, the FWC initiated intensive sampling for STXs in multiple species of aquatic biota in Florida’s coastal waters with emphasis on the IRL. Partial results of this sampling were reported [58, 217]. Analysis of IRL puffer fish found concentrations of STXs in muscle often well in excess of the 80 μg STX eq./100 g tissue regulatory action limit set for shellfish. After extended monitoring, STX concentrations in puffer muscle in certain regions of the IRL remained well above the action limit. As a result, the puffer fishing ban in the IRL was made indefinite in June 2004. Based on toxin profiles and abundance in the IRL during the first SPFP reports in 2002, Landsberg et al. suggested the dinoflagellate Pyrodinium bahamense, not reported to produce STXs in Florida waters prior to 2002, as the putative toxin source.
Deeds et al. confirmed that S. nephelus from the northern IRL contained elevated concentrations of STX in muscle (1770 ± 159 μg STX/100g tissue) compared to liver (609 ± 432 μg STX/100g tissue), with only low to non-detectable amounts of TTX in all tissues tested. The additional IRL puffer species S. testudineus (checkered puffer) and S. spengleri (bandtail puffer), known to only occur further south in the lagoon system, were found to contain significantly greater concentrations of TTX compared to STX in all tissues (maximum concentration for TTX found in S. testudineus livers 6076 ± 3283 μg TTX/100g tissue – maximum concentration of STX found in S. spengleri livers 74 ± 42 μg STX/100g tissue). This work confirmed S. nephelus, a species not associated with toxicity in the IRL prior to these events , as the likely cause of all 28 cases of SPFP originating from the IRL during 2002–2004. These events on the east coast of the U.S. represented the first confirmed cases of puffer fish poisoning due solely to STX in North America.
4. Conclusion
In comparison to non-traditional (i.e. non-filter feeding) vectors for PSP, more is known about STX sources, routes of exposure, species specific and population specific sensitivities, depuration rates, compartmentalization, and biotransformations in filter-feeding bivalves. As a result, monitoring and management of traditional bivalve vectors for PSP are, in many cases, highly successful and result in the protection of public health. Due to a lack of basic knowledge on the source(s) and fate of STXs in non-traditional vectors, human intoxications due to the consumption of these species are often more unpredictable, and resource closures are often longer and sometimes indefinite. With the apparent expansion in STX producing microrganisms world-wide, an ever-increasing demand for seafood, and the emergence of seafood as an economic commodity for export, particularly in developing countries, more study is required on STX sources, distribution, and fate in these non-traditional PSP vectors to assure both public safety and consumer confidence on local, national, and international scales.
Footnotes
1
Steidinger et al. (1980) distinguished P. bahamense var. compressa from P. bahamense var. bahamense based on morphological, dimensional, and toxicological characteristics. P. bahamense var. compressa was reported to be the toxic variety responsible for Indo-Pacific PSP events while P. bahamense var. bahamense was reported to be non-toxic. Landsberg et al. (2006) reported for the first time STX(s) production from P. bahamense in Florida, USA putatively responsible for several cases of SPFP. Badylak et al. (2004) confirmed that the P. bahamense variety occurring in Florida waters still conformed to Steidinger et al. (1980) description of P. bahamense var. bahamense except that it now appeared to be toxic. The description of P. bahamense var. compressa vs. P. bahamense var. bahamense is currently being re-evaluated (K. Steidinger, personal communication), therefore throughout the text P. bahamense without varietal designation is used.
Samples Availability: Available from the authors.
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1. Paralytic Shellfish Toxins and Sources
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190407 | http://www.cs.cmu.edu/~hzarnani/fm-rg/cmuonly/podelski04complete.pdf | A Complete Method for the Synthesis of Linear Ranking Functions Andreas Podelski and Andrey Rybalchenko Max-Planck-Institut f¨ ur Informatik Saarbr¨ ucken, Germany Abstract. We present an automated method for proving the termina-tion of an unnested program loop by synthesizing linear ranking func-tions. The method is complete. Namely, if a linear ranking function exists then it will be discovered by our method. The method relies on the fact that we can obtain the linear ranking functions of the program loop as the solutions of a system of linear inequalities that we derive from the program loop. The method is used as a subroutine in a method for prov-ing termination and other liveness properties of more general programs via transition invariants; see [PR03].
1 Introduction The verification of termination and other liveness properties of programs is a difficult problem. It requires the discovery of invariants and ranking functions to prove the termination of program loops.
We present a complete and efficient method for the synthesis of linear ranking functions for unnested program loops whose guards and update statements use linear arithmetic expressions. We have implemented the method. Preliminary experiments show that the method is efficient not only in theory but also in practice.
Roughly, the method works as follows. Given a program loop for which we want to find a linear ranking function, we construct a corresponding system of linear inequalities over rationals. As we show, the solutions of this system encode the linear ranking functions of the program loop. That is, we can check the existence of a linear ranking function by constraint solving. If it exists, a linear ranking function can be constructed from a solution of the system of linear inequalities, a solution that we obtain by constraint solving. If the system has no solutions then (and only then) a linear ranking function does not exist.
As a consequence of our approach, one can use existing highly-optimized tools for linear programming as the engine in a complete method (to our knowledge the first) for the synthesis of linear ranking functions.
We admit unnested program loops with nondeterministic update statements.
This is potentially useful to model read statements. It is strictly required in the context where we employ our method, described next.
In a work described elsewhere [PR03], we show that one can reduce the test of termination and other liveness properties (in the presence of fairness B. Steffen and G. Levi (Eds.): VMCAI 2004, LNCS 2937, pp. 239–251, 2004.
c ⃝Springer-Verlag Berlin Heidelberg 2004 240 A. Podelski and A. Rybalchenko assumptions) to the test of termination of unnested program loops. That is, we use the algorithm described in this paper as a subroutine in the software model checking method for liveness properties via transition invariants proposed in [PR03]. The experiments that we present in this paper stem from this context.
2 Unnested Program Loops We formalize the notion of unnested program loops by a class of programs that are built using a single “while” statement and that satisfy the following condi-tions: – the loop condition is a conjunction of atomic propositions, – the loop body may only contain update statements, – all update statements are executed simultaneously.
We call this class simple while programs. Pseudo-code notation for the programs of this class is given below.
while (Cond1 and . . . and Condm) do Simultaneous Updates od We consider the subclass of simple while programs built using linear arith-metic expressions over program variables.
Definition 1. A linear arithmetic simple while (LASW ) program over the tuple of program variables x = (x1, . . . , xn) is a simple while program such that: – program variables have integer domain, – every atomic proposition in the loop condition is a linear inequality over (unprimed) program variables: c1x1 + · · · + cnxn ≤c0, – every update statement is a linear inequality over unprimed and primed pro-gram variables a′ 1x′ 1 + · · · + a′ nx′ n ≤a1x1 + · · · + anxn + a0.
Note that we allow the left-hand side of an update statement to be a linear expression over program variables, and that an update can be nondeterministic, e.g., x′+y′ ≤x+2y−1. This is necessary, because we use simple while programs, and LASW programs in particular, to approximate the transitive closure of a transition relation (see Section 4).
We define a program state to be a valuation of program variables. The set of all program states is called the program domain. The transition relation de-noted by the loop body of an LASW program is the set of all pairs of program states (s, s′) such that the state s satisfies the loop condition, and (s, s′) satisfies A Complete Method for the Synthesis of Linear Ranking Functions 241 each update statement. A trace is a sequence of states such that each pair of consecutive states belongs to the transition relation of the loop body.
We observe that the transition relation of a LASW program can be expressed by a system of inequalities over unprimed and primed program variables. The translation procedure is straightforward. For the rest of this paper, we assume that an LASW program over the tuple of program variables x = (x1, . . . , xn) (treated as a column vector) can be represented by the system (AA′) x x′ ≤b of inequalities. We identify an LASW program with the corresponding system of inequalities.
Example 1. The following program loop with nondeterministic updates while (i −j ≥1) do (i, j) := (i −Nat, j + Pos) od is represented by the following system of inequalities.
−i + j ≤−1 −i + i′ ≤0 j −j′ ≤−1 Note that the relations between program variables denoted by the nondetermin-istic update statements i := i−Nat and j := j +Pos, where Nat and Pos stand for any nonnegative and positive integer number respectively, can be expressed by the inequalities i′ ≤i and j′ ≥j + 1.
3 The Algorithm We say that a simple while program is terminating if the program domain is well-founded by the transition relation of the loop body of the program, i.e., if there is no infinite sequence {si}∞ i=1 of program states such that each pair (si, si+1), where i ≥1, is an element of the transition relation.
The following theorem allows us to use linear programming over rationals to test the existence of a linear ranking function, and thus to test a sufficient condition for termination of LASW programs. The corresponding algorithm is shown in Figure 1.
Theorem 1. A linear arithmetic simple while program given by the system (AA′) x x′ ≤b is terminating if there exist nonnegative vectors over rationals 242 A. Podelski and A. Rybalchenko input program (AA′) x x′ ≤b begin if exists rational-valued λ1 and λ2 such that λ1, λ2 ≥0 λ1A′ = 0 (λ1 −λ2)A = 0 λ2(A + A′) = 0 λ2b < 0 then return(“Program Terminates”) else return(“Linear ranking function does not exist”) end.
Given λ1 and λ2, solutions of the systems above, define r def = λ2A′, δ0 def = −λ1b, and δ def = −λ2b. A linear ranking function ρ is defined by ρ(x) def = rx if exists x′ such that (AA′) x x′ ≤b, δ0 −δ otherwise.
Fig. 1. Termination Test and Synthesis of Linear Ranking Functions.
λ1 and λ2 such that the following system is satisfiable.
λ1A′ = 0 (1a) (λ1 −λ2)A = 0 (1b) λ2(A + A′) = 0 (1c) λ2b < 0 (1d) Proof. Let the pair of nonnegative (row) vectors λ1 and λ2 be a solution of the system (1a)–(1d). For every x and x′ such that (AA′) x x′ ≤b, by assumption that λ1 ≥0, we have λ1(AA′) x x′ ≤λ1b. We carry out the following sequence of transformations.
λ1(Ax + A′x′) ≤λ1b λ1Ax + λ1A′x′ ≤λ1b λ1Ax ≤λ1b by (1a) λ2Ax ≤λ1b by (1b) −λ2A′x ≤λ1b by (1c) A Complete Method for the Synthesis of Linear Ranking Functions 243 From the assumption λ2 ≥0 follows λ2(AA′) x x′ ≤λ2b. Then, we continue with λ2(Ax + A′x′) ≤λ2b λ2Ax + λ2A′x′ ≤λ2b −λ2A′x + λ2A′x′ ≤λ2b by (1c) We define r def = λ2A′, δ0 def = −λ1b, and δ def = −λ2b. Then, we have rx ≥δ0 and rx′ ≤rx−δ for all x and x′ such that (AA′) x x′ ≤b. Due to (1d) we have δ > 0.
We define a function ρ as shown in Figure 1. Any program trace induces a strictly descending sequence of values under ρ that is bounded from below, and the difference between two consecutive values is at least δ. Since no such infinite sequence exists, the program is terminating.
⊓ ⊔ The theorem above states a sufficient condition for termination. We observe that if the condition applies then a linear ranking function, i.e., a linear arith-metic expression over program variables which maps program states into a well-founded domain, exists. The following theorem states that our termination test is complete for the programs with linear ranking functions.
Theorem 2. If there exists a linear ranking function for the linear arithmetic simple while program with nonempty transition relation then the termination condition of Theorem 1 applies.
Proof. Let the vector r together with the constants δ0 and δ > 0 define a linear ranking function. Then, for all pairs x and x′ such that (AA′) x x′ ≤b we have rx ≥δ0 and rx′ ≤rx −δ.
By the non-emptiness of the transition relation, the system (AA′) x x′ ≤b has at least one solution. Hence, we can apply the ‘affine’ form of Farkas’ lemma (in [Sch86]), from which follows that there exists δ′ 0 and δ′ such that δ′ 0 ≥δ0, δ′ ≥δ, and each of the inequalities −rx ≤−δ′ 0 and −rx + rx′ ≤−δ′ is a nonnegative linear combination of the inequalities of the system (AA′) x x′ ≤b.
This means that there exist nonnegative rational-valued vectors λ1 and λ2 such that λ1(AA′) x x′ = −rx λ1b = −δ′ 0 and λ2(AA′) x x′ = −rx + rx′ λ2b = −δ′.
After multiplication and simplification we obtain λ1A = −r λ1A′ = 0 λ2A = −r λ2A′ = r, from which equations (1a)–(1c) follow directly. Since δ′ ≥δ > 0, we have λ2b < 0, i.e., the equation (1d) holds.
⊓ ⊔ 244 A. Podelski and A. Rybalchenko The following corollary is an immediate consequence of Theorems 1 and 2.
Corollary 1. Existence of linear ranking functions for linear arithmetic simple while programs with nonempty transition relation is decidable in polynomial time.
Not every LASW program has a linear ranking function (see the following example).
Example 2. Consider the following program.
while (x ≥0) do x := −2x + 10 od The program is terminating, but it does not have a linear ranking function.
For termination proof consider the following ranking function into the domain {0, . . . , 3} well-founded by the less-than relation <.
ρ(x) def = 1 if x ∈{0, 1, 2}, 2 if x ∈{4, 5}, 3 if x = 3, 0 otherwise.
It can be easily tested that the system (1a)–(1d) is not satisfiable for the LASW program −1 0 2 1 −2 −1 x x′ ≤ 0 10 −10 .
By Theorem 2, this implies that no linear ranking function exists for the program above.
⊓ ⊔ The following example illustrates an application of the algorithm based on The-orem 1.
Example 3. We prove termination of the LASW program from Example 1. The program translates to the system (AA′) x x′ ≤b, where: A def = −1 1 −1 0 0 1 , A′ def = 0 0 1 0 0 −1 , x def = i j , b def = −1 0 −1 .
A Complete Method for the Synthesis of Linear Ranking Functions 245 Let λ1 = (λ′ 1, λ′ 2, λ′ 3) and λ2 = (λ′′ 1, λ′′ 2, λ′′ 3). The system (1a)–(1d) is feasible, it has the following solutions: λ′ 2 = λ′ 3 = λ′′ 1 = 0, λ′ 1 = λ′′ 2 = λ′′ 3, λ′ 1, λ′′ 2, λ′′ 3 > 0.
Since the system is feasible the program is terminating. We construct a linear ranking function following the algorithm in Figure 1. We define r def = λ2A′, δ0 def = −λ1b, and δ def = −λ2b, and obtain r = (λ′ 1 −λ′ 1), δ0 = δ = λ′ 1. Taking λ′ 1 = 1 we obtain the following ranking function.
ρ(i, j) = i −j if i −j ≥1, 0 otherwise.
4 Application to General Programs In this section we illustrate how our method for proving termination of program loops can be used in the software model checking method for liveness properties via transition invariants proposed in [PR03]. That method applies to general-purpose programs (imperative, concurrent, . . . ); it is different from other ap-proaches to special classes of infinite-state systems, e.g. [BS99]. We then provide experimental results obtained by applying the transition invariants approach for proving termination of singular value decomposition program.
Software model checking for liveness properties is a new approach for the au-tomated verification of liveness properties of infinite-state systems by the com-putation of transition invariants. A transition invariant is an over-approximation of the transitive closure of the transition relation of the system. The presenta-tion of a transition invariant as nothing but a finite set of unnested program loops. One can characterize the validity of a liveness property via the existence of transition invariants [PR03]. Namely, the liveness property is valid if each of the unnested program loops is terminating.
That is, the general method for the verification of liveness properties de-scribed in [PR03] is parameterized by an algorithm that tests whether each unnested program loop in the transition invariant is terminating. i.e., a proce-dure implementing a termination test for simple while programs.
Proving termination of simple while programs built using linear arithmetic expressions is required for the verification of a large class of software systems, e.g., liveness properties for mutual exclusion protocols (bakery, ticket), termi-nation proofs of imperative programs (sorting algorithms, numerical algorithms dealing with matrices).
4.1 Sorting Program This example illustrates the approach from [PR03] and the role of simple while programs.
246 A. Podelski and A. Rybalchenko We consider the program shown in Figure 2 implementing a sorting algo-rithm. For legibility, we concentrate on the skeleton shown on the right, which int n,i,j,A[n]; i=n; l1: while (i>=0) { j=0; l2: while (j<=i-1) { if (A[j]>=A[j+1]) swap(A[j],A[j+1]); j=j+1; } i=i-1; } l1: if (i>=0) j=0; l2: if (i-j>=1) { j=j+1; goto l2; } else { i=i-1; goto l1; } Fig. 2. Sorting program and its skeleton.
consists of the statements st1, st2, st3.
l1: if (i>=0) { (i,j):=(i,0); goto l2; } / st1 / l2: if (i-j>=1) { (i,j):=(i,j+1); goto l2; } / st2 / l2: if (i-j<1) { (i,j):=(i-1,j); goto l1; } / st3 / We read, for example, the first program statement as: if the current program location is labeled by l1 and the “if” condition is satisfied then update the variables according to the update expressions and change the current label la-bel to l2. Note that the updates are performed simultaneously (“concurrent” assignments in [Dij76]).
Each of the ‘simple’ programs below must be read as a one-line program.
l1: if (true) { (i,j):=(Any,Any); goto l2; } / a1 / l2: if (true) { (i,j):=(Any,Any); goto l1; } / a2 / l1: if (i>=0) { (i,j):=(i-Pos,Any); goto l1; } / a3 / l2: if (i>=0) { (i,j):=(i-Pos,Any); goto l2; } / a4 / l2: if (i-j>=1) { (i,j):=(i-Nat,j+Pos); goto l2; } / a5 / Note the nondeterministic update expressions, e.g., after execution of i:=Any the value of variable i could by any integer, the update i:=i-Pos decrements the value of i by at least one.
We notice that st1 is approximated by a1, st2 by a5 and st3 by a2. This means that every transition induced by execution of the statement st1 can also be achieved executing a single step of a1. In fact, every sequence of pro-gram statements is approximated by one of a1, . . . , a5. We say that the set {a1, . . . , a5} is a transition invariant in our terminology.
A Complete Method for the Synthesis of Linear Ranking Functions 247 For example, every sequence of program statements that leads from l2 to l2 is approximated by a4 if it passes through l1, and by a5 otherwise. The following table assigns to each ‘simple’ program the set of sequences of program statements that it approximates. All non-assigned sequences are not feasible.
a1 st1(st2|st3st1)∗ a2 (st2|st3st1)∗st3 a3 st1(st2|st3st1)∗st3 a4 (st2|st3st1)∗st3st1(st2|st3st1)∗ a5 st2+ According to the formal development in [PR03], the transition invariant above is ‘strong enough’ to prove termination, which means: each of its ‘sim-ple’ programs, viewed in isolation, is terminating.
Termination is obvious for ‘simple’ programs that do not refer to a loop in the control flow graph, like the ‘simple’ programs a1 and a2. The ‘simple’ programs of the form ln: if (cond) { updates; goto ln; } translate to a while loop ln: while (cond) { updates; }.
The ‘simple’ programs that translate to a while loop are in fact simple while programs whose termination proofs we study in this paper.
Next, we describe an application of the transition invariants method with termination test in Figure 1 as a subroutine for checking whether a transition invariant is strong enough. We prove termination of a program implementing singular value decomposition algorithm.
4.2 Program with Unbounded Nondeterminism We consider the program shown in Figure 3. It has a nondeterministic choice at the location labeled by l. The value of the variable y is chosen nondeter-ministically in the first branch. Termination proof for this program requires a lexicographic ranking function. The program translates to the statements st1 and st2: l: if (x>=0) { (x,y):=(x-1,Any); goto l; } / st1 / l: if (y>=0) { (x,y):=(x,y-1); goto l; } / st2 / The transition invariant computed by our tool consists of the following ‘sim-ple’ programs.
l: if (x>=0) { (x,y):=(x-Pos,Any); goto l; } / a1 / l: if (y>=0) { (x,y):=(Any,y-Pos); goto l; } / a2 / 248 A. Podelski and A. Rybalchenko int x, y; l: if () { if (x>=0) { x--; read(y); } } else { if (y>=0) y--; } goto l: l if (x>=0) { x--; read(y); } if (y>=0) y--; Fig. 3. Program with unbounded nondeterminism.
Both ‘simple’ programs, viewed in isolation, are terminating. Hence, the program with unbounded nondeterminism, shown in Figure 3 is terminating.
4.3 Singular Value Decomposition Program We considered an algorithm for constructing the singular value decomposition (SVD) of a matrix. SVD is a set of techniques for dealing with sets of equations or matrices that are either singular or numerically very close to singular [PTVF92].
A matrix A is singular if it does not have a matrix inverse A−1 such that AA−1 = I, where I is the identity matrix.
Singular value decomposition of the matrix A whose number of rows m is greater or equal to its number of columns n is of the form A = UWV T , where U is an m × n column-orthogonal matrix, W is an n × n diagonal matrix with positive or zero elements (called singular values), and the transpose matrix of an n × n orthogonal matrix V . Orthogonality of the matrices U and V means that their columns are orthogonal, i.e., U T U = V V T = I.
The SVD decomposition always exists, and is unique up to permutation of the columns of U, elements of W and columns of V , or taking linear combinations of any columns of U and V whose corresponding elements of W are exactly equal.
SVD can be used in numerically difficult cases for solving sets of equations, constructing an orthogonal basis of a vector space, or for matrix approxima-tion [PTVF92].
We proved termination of a program implementing the SVD algorithm based on a routine described in [GVL96]. The program was taken from [PTVF92]. It is written in C and contains 163 lines of code with 42 loops in the control-flow graph, nested up to 4 levels.
A Complete Method for the Synthesis of Linear Ranking Functions 249 We used our transition invariant generator to compute a transition invariant for the SVD program. Proving the transition invariant to be strong enough required testing termination of 219 LASW programs.
We applied our implementation of the algorithm on Figure 1, which was done in SICStus Prolog [Lab01] using the built-in constraint solver for linear arithmetic [Hol95]. Proving termination required 800 ms on a 2.6 GHz Xeon computer running Linux, which is in average 3.6 ms per each LASW program.
5 Related Work The verification of termination and other liveness properties of programs requires the discovery of invariants as well as of ranking functions to prove the termina-tion of program loops. Here, we relate our work not to methods for the automated discovery of invariants (see e.g. [Kar76,CH78,BBM97]), but to the more closely related topic of methods for the automated synthesis of ranking functions, a topic that has received increasing attention in the last years [GCGL02,CS01, DGG00,Mes96,MN01,CS02,SG91].
As a first general remark, a major difference between our work and all the others lies in the fact that we obtain a completeness result.
A heuristic-based approach for discovery of ranking functions is described in [DGG00]. It inspects the program source code for ranking function candidates.
This method restricted to programs where the ranking function is exhibited already in the source code.
The algorithm in [CS01] extracts a linear ranking function of an unnested program loop by manipulating polyhedral cones; these represent the transition relation of the loop and the loop invariant. Their approach depends on the strength of the invariant generator, which they call in a subroutine to propose bounded linear arithmetic expression. The algorithm requires exponential space in the worst case. A generalization of that algorithm described in [CS02] for programs with complex control structures detects linear ranking functions for strongly connected components in the control-flow graph of more general pro-grams. In both cases the algorithm is restricted to bounded nondeterminism.
Moreover, it cannot handle loops with non-monotonic decrease, such as in while (x>=0) {x=x+1; x=x-1;}.
The method for discovery of nonnegative linear combinations of bound ar-gument sizes for proving termination of logic programs in [SG91] relies on auto-matically inferred inter-argument constraints. The duality theory of linear pro-gramming is applied to discover combinations that decrease during top-down execution of recursive rules; the determined combinations are bounded from below since argument sizes are always positive. This method was applied for inferring termination of constraint logic programs [Mes96], and in systems for inferring termination of logic programs [MN01,GCGL02]. Carried over into the context of imperative program loops, the inference of inter-argument constraints corresponds to calls to the invariant generator, as in [CS01]; the same restrictions as mentioned above apply.
250 A. Podelski and A. Rybalchenko 6 Conclusion We have presented the to our knowledge first complete algorithm for the synthe-sis of linear ranking functions for a small but natural and well-motivated class of programs, namely, unnested program loops built using linear arithmetic ex-pressions (LASW programs). The method is guaranteed to find a linear ranking function, and therefore to prove termination, if a linear ranking function exists.
The existence of a linear ranking function for an LASW program is equivalent to the satisfiability of the system of linear inequalities derived from the program.
The termination check for LASW programs is a subroutine in the auto-mated method for the verification of termination and other liveness properties of general-purpose programs via the computation of transition invariants [PR03].
We have implemented the proposed algorithm using an efficient implemen-tation of a solver for linear programming over rationals [Hol95]. We applied our implementation to prove termination of a singular value decomposition program, which required termination proofs for 219 LASW programs. This and other ex-periments indicate the practical potention of the algorithm.
Considering future work, we would like to find a characterization of LASW programs that do always have linear ranking functions, i.e., for which our al-gorithm decides termination. Another direction of work is to handle unnested program loops built using expressions other than linear arithmetic.
Acknowledgments. We thank Bernd Finkbeiner, Konstantin Korovin, and Uwe Waldmann for comments on this paper. We thank Ramesh Kumar for his help with the SVD example.
References [BBM97] Nikolaj Bjørner, Anca Browne, and Zohar Manna. Automatic generation of invariants and intermediate assertions. Theoretical Computer Science, 173(1):49–87, 1997.
[BS99] Olaf Burkart and Bernhard Steffen. Model checking the full modal mu-calculus for infinite sequential processes. Theoretical Computer Science, 221:251–270, 1999.
[CH78] Patrick Cousot and Nicolas Halbwachs.
Automatic discovery of linear restraints among variables of a program. In Proc. of POPL 1978: Symp.
on Principles of Programming Languages, pages 84–97. ACM Press, 1978.
[CS01] Michael Colon and Henny Sipma. Synthesis of linear ranking functions. In Tiziana Margaria and Wang Yi, editors, Proc. of TACAS 2001: Tools and Algorithms for the Construction and Analysis of Systems, volume 2031 of LNCS, pages 67–81. Springer-Verlag, 2001.
[CS02] Michael Colon and Henny Sipma. Practical methods for proving program termination. In Ed Brinksma and Kim Guldstrand Larsen, editors, Proc.
of CAV 2002: Computer Aided Verification, volume 2404 of LNCS, pages 442–454. Springer-Verlag, 2002.
A Complete Method for the Synthesis of Linear Ranking Functions 251 [DGG00] Dennis Dams, Rob Gerth, and Orna Grumberg. A heuristic for the au-tomatic generation of ranking functions.
In Workshop on Advances in Verification (WAVe’00), pages 1–8, 2000.
[Dij76] Edsger W. Dijkstra. A Discipline of Programming. Prentice Hall Series in Automatic Computation. Prentice Hall, Englewood Cliffs, NJ, 1976.
[GCGL02] Samir Genaim, Michael Codish, John P. Gallagher, and Vitaly Lagoon.
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of VMCAI 2002: Verification, Model Checking, and Abstract Interpreta-tion, volume 2294 of LNCS, pages 126–138. Springer-Verlag, 2002.
[GVL96] Gene H. Golub and Charles F. Van Loan. Matrix Computations; 3rd edi-tion. Johns Hopkins Univ Press, 3rd edition, 1996.
[Hol95] Christian Holzbaur. OFAI clp(q,r) Manual, Edition 1.3.3. Austrian Re-search Institute for Artificial Intelligence, Vienna, 1995. TR-95-09.
[Kar76] Michael Karr. Affine relationships among variables of a program. Acta Informatica, 6:133–151, 1976.
[Lab01] The Intelligent Systems Laboratory.
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Swedish Institute of Computer Science, PO Box 1263 SE-164 29 Kista, Sweden, October 2001. Release 3.8.7.
[Mes96] Fred Mesnard. Inferring left-terminating classes of queries for constraint logic programs. In Michael J. Maher, editor, Proc. of JICSLP 1996: Joint Int. Conf. and Symp. on Logic Programming, pages 7–21. MIT Press, 1996.
[MN01] Fred Mesnard and Ulrich Neumerkel. Applying static analysis techniques for inferring termination conditions of logic programs. In Patrick Cousot, editor, Proc. of SAS 2001: Symp. on Static Analysis, volume 2126 of LNCS, pages 93–110. Springer-Verlag, 2001.
[PR03] Andreas Podelski and Andrey Rybalchenko. Software model checking of liveness properties via transition invariants. Technical report, Max-Plank-Institut f¨ ur Informatik, 2003.
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190408 | https://www.youtube.com/watch?v=hZgNnWZs-EY | Chemical Equilibrium | Chapter 15 - Chemistry: The Central Science
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Chapter 15 of Chemistry: The Central Science (15th Global Edition) introduces chemical equilibrium, the state in which the forward and reverse reactions occur at equal rates, resulting in constant concentrations of reactants and products. The chapter begins by describing the dynamic nature of equilibrium and how it can be achieved from either the reactant or product side. Students learn to write equilibrium constant expressions (K_c for concentrations, K_p for gases) and understand the relationship between them through the ideal-gas law. The significance of the magnitude of K is discussed, indicating whether reactants or products are favored at equilibrium. The chapter covers how to manipulate equilibrium constants for reversed or combined reactions and how to use them for quantitative problem solving. Reaction quotient (Q) is introduced to predict the direction of shift toward equilibrium. Le Châtelier’s principle is explained in detail, showing how changes in concentration, pressure, volume, and temperature disturb equilibrium and how the system responds to reestablish balance. The effect of catalysts on equilibrium position versus rate is clarified. Students also practice solving equilibrium problems using ICE (Initial–Change–Equilibrium) tables for a variety of chemical systems, including weak acid ionization, gas-phase reactions, and industrial processes like the Haber synthesis of ammonia. By the end, students understand how to analyze, predict, and manipulate equilibrium conditions for both theoretical and practical applications in chemistry and engineering.
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Chemistry The Central Science Chapter 15 summary, chemical equilibrium chemistry, dynamic equilibrium definition, equilibrium constant Kc Kp, reaction quotient Q prediction, Le Châtelier’s principle examples, ICE tables equilibrium problems, concentration pressure temperature effects, equilibrium shifts chemistry, catalysts and equilibrium position, equilibrium in gas phase reactions, weak acid ionization equilibrium, Haber process ammonia synthesis, manipulating equilibrium constants, equilibrium calculations chemistry, equilibrium problems with solutions, chemistry textbook chapter summaries, Brown LeMay Bursten Murphy Woodward Stoltzfus
Transcript:
Welcome curious minds to another deep dive. Have you ever really thought about balance? Not, you know, just something perfectly still like a statue. I mean, something more dynamic. Think about a football game. You've got what, 11 players per side on the field. That number stays the same, but the players themselves constantly swapping out, moving around. It's balanced, but it's definitely not static. Or even simpler, think about a closed water bottle. Water evaporates. Water condenses. The amount of vapor in there hits a steady point, but molecules are always jumping between liquid and gas. Dynamic balance. So, what if chemical reactions work the same way? Deep down at the molecular level, they find this incredible balance, too. Not by stopping, but by uh the forward and reverse happening at the exact same pace. Today, our mission is to really unpack this fascinating world of chemical equilibrium. We'll explore how reactions reach the state where everything looks constant, but trust me, there's a whole lot of action still going on. We're diving into the core ideas from chemistry, the central science, looking at real world stuff, and making sure this fundamental concept really clicks for you. Exactly. Our goal here is to demystify this. It's absolutely critical in chemistry. We'll cover what it is, how we actually put a number on it, and uh maybe most importantly, how we can predict what will happen and even how we can sort of nudge reactions in the direction we want. get ready for some hopefully some real aha moments connecting chemistry to the world around us. Okay, so this molecular dance it's happening but just knowing that isn't quite enough for a chemist is it? We need to measure this stuff. How do you quantify that balance? Right? So fundamentally chemical equilibrium is reached when the forward reaction that's reactants turning into products and the reverse reaction products turning back into reactants happen at precisely the same speed. The rates are equal. This means the overall or net change in how much stuff you have the concentrations becomes zero. So yeah, it looks like the reaction has just stopped. Okay. But it hasn't. It's just balanced a constant molecular turn. Let's make that really concrete. Your source has this classic example. Colorless N204 gas turning into brown NO2 gas. Yes. The nitrogen tyroxide and nitrogen dioxide. Right. So you seal some colorless N204 in a tube, maybe warm it up a bit. It starts turning brown. Then the brown color just stops getting darker. What's actually going on there? Did it just run out of steam or is something else happening? That's a perfect example because it looks static, but it's absolutely not. The reaction is reversible. N204 gas can break apart into two molecules of NO2 gas and two NO2s can bump into each other and reform N204. We write it with that double arrow. N204G equals 2 NO2G. Okay. At equilibrium, the speed at which N204 breaks down is exactly matched by the speed at which NO2 molecules recombine. So the brown color which comes from the NO2 stays constant. But individual molecules are constantly swapping identities. It's truly dynamic. Forward rate equals reverse rate. Got it? So the big takeaway, concentrations become constant. Yes, but it's in a closed system and the reaction is still humming along in both directions. Dynamic, not static. Okay, now we understand the what. How do chemists put a number on it? Quantify that balance point. Well, this takes us to the law of mass action. This was figured out way back in 1864 by Goldberg and wage. It gives us a mathematical way to describe the ratio of products to reactants at equilibrium. For a general reaction, say uh little a moles of reactant A plus B moles of B go to D moles of product D plus E moles of E. So, A plus BB B plus DD plus E. Right? The equilibrium constant which we call KC the C stands for concentration is calculated like this. You take the concentrations of the products each raised to the power of its coefficient in the balanced equation. Okay. So D to the power D to the power E. Exactly. And you divide that by the concentrations of the reactants also raised to their coefficients. So A to the A B to the B. That expression KC XD DD A gives you a specific number that represents the equilibrium position at a certain temperature. A snapshot of the ratio. Okay. Now let's connect this to the real world. Your sources mention the Habber process for making ammonia. N2 gas plus 3 H2 gas makes two NH3 gas. This one reaction, it's huge, isn't it? It's got this incredible story behind it, feeding billions, but also well playing a role in war. How does one reaction carry so much weight? It's absolutely foundational. Ammonia is the key ingredient for nitrogen fertilizers. Without the habber process, feeding the current global population would be well, basically impossible. Crop yields depend heavily on it. Wow. But yeah, the history's complex. Fritz Hay were developed in around 1912 called Bosch figured out how to do it on an industrial scale and during World War Iy's access to natural nitrates needed for explosives was cut off. The habos process allowed them to make ammonia from air than nitrates keeping their munitions production going. Some argue it prolonged the war and then Habber himself brilliant chemist got the Nobel Prize for this work. But he was also deeply involved in developing chemical weapons later which obviously casts a shadow. It really highlights how science can be used for vastly different ends. A really powerful example. Okay, back to the chemistry. This KC value, this equilibrium constant. It it really is constant for a specific temperature, right? It doesn't matter how you start the reaction precisely. You could start with only N2 and H2 or only NH3 or some random mix. If you let it reach equilibrium at a certain temperature, the ratio described by Casey will always be the same. The N204 NO2 example you mentioned, the textbook data shows that whether they started with pure N204 or pure NO2, once it settled, the KC value consistently came out around 212 at that temperature. The system finds that balance. That's quite something. And you mentioned KC uses concentrations. What about gases? Isn't pressure more common? Good point. For gasphase reactions, we often use KC, where P stands for partial pressures. You calculate it the same way as KC but using the partial pressures of the gases instead of their molar concentrations. And there's a simple mathematical relationship between KP and KC using the ideal gas law involves the change in the number of moles of gas in the reaction uh represented by own. The formula is KP equals KCRT. You might also notice that we usually don't write units for K. Technically, it's because the expression uses something called activities, which are like effective concentrations or pressures relative to a standard state. This makes K dimensionless, a pure number. Helps compare different reactions, right? Activities. Okay, so we have this number K. What does the size of K actually tell us? Is a big K good? A small K bad? The size is incredibly informative. If K is large, let's say much greater than one, K1, it means the equilibrium lies to the right, meaning meaning products dominate at equilibrium. You'll have much more product than reactant once the reaction settles. Your source gives the example COG plus CL2G. It's Casey is huge like 4.56 by 109. That reaction strongly favors making the product CO2. So if you want lots of product, big K is good. Definitely. Conversely, if K is small much less than one K1, the equilibrium lies to the left. That means reactants dominate. You won't get much product formed before equilibrium is reached. Okay, that makes sense. Now, does it matter how we write the equation? If I flip it around, is K the same? Ah, absolutely crucial point. It matters a lot how you write it. The value of K is tied directly to the specific balanced equation you're using. If you reverse the equation, the new equilibrium constant KA is the reciprocal of the original. So, KO21 K remember our N204 example, N204 O2, NO2 had KC equals.20. If we were the reverse, 2 NO2 equals N204. The new KC is 1.2. 212 which is about 4.72. Oh, so you always have to specify the balanced equation when you give a K value. Always. Got it. What if you say double all the coefficients? If you multiply all the coefficients in a balanced equation by some factor, let's say N, then the new equilibrium constant is the original K raised to the power of N. So new original N. Okay. Power rule. And one more. If you add two reactions together to get an overall reaction, you multiply their individual K values to get the K for the overall reaction. It's kind of like Hess's law for eny but with multiplication instead of addition. Interesting connection. Okay. What about reactions that aren't just gases? What if you have solids or liquids involved? Do they just get ignored in the K expression? That's called heterogeneous equilibria. When you have substances in different phases and yes, pure solids and pure liquids are emitted from the equilibrium constant expression. Why is that? Because their concentrations or more accurately their activities are essentially constant. Think about the density of a solid block or a pure liquid. It doesn't change significantly even if the amount changes. Since K tracks the ratio of things whose concentrations can change as equilibrium is approached, the constant concentrations of solids and liquids get sort of absorbed into the value of K itself. Their activity is taken as one. So they're there but they don't affect the ratio calculation. Exactly. For example, if you have solid calcium carbonate decomposing, can CO3 equals CaO plus CO2. The equilibrium expression is just KC CO2 or KP equals PCO2. The solids don't appear. It simplifies things, but it's a key rule to remember. Definitely a point to watch out for. Okay, so knowing K seems really powerful. It tells us where the balance lies. Can we use it to predict things? Like, if I mix stuff together, can K tell me which way the reaction will go? Absolutely. That's one of its main uses. First, if you know all the concentrations at equilibrium, calculating K is easy. Just plug the numbers into the expression we talked about, right? But often you might know the initial amounts you put in and maybe measure just one concentration once it reaches equilibrium. From that, you can figure out K. This is where the IC table comes in really handy. IC stands for initial change equilibrium. Okay, IC table. How does that work? You make a little table. First row is your initial concentrations. Second row is the change in concentrations needed to reach equilibrium. Often you use X based on this stochometry. Third row is the equilibrium concentration which is just initial plus change. I see. So you set up the table, use the known equilibrium value to solve for X, figure out all the equilibrium concentrations and then plug those into the KC expression to calculate K. It's a standard problem solving technique. Okay, that sounds systematic. Now what about predicting direction? Say I mix some reactants and products, but I don't know if it's reached equilibrium yet. How can I tell? For that we use something called the reaction quotient. Q. Q like K. Exactly like K in form. You calculate Q. Use the exact same expression as K, but you plug in the concentrations you have right now at any point in time, not necessarily at equilibrium. Ah, okay. So Q is like a snapshot. K is the destination. Perfect analogy. Then you compare Q to K. If Q, your current ratio of products to reactants is too small. The reaction needs to shift, right? make more products, use up reactants to reach the equilibrium ratio K. Okay, Q less than K shifts, right? If Q K, congratulations, your system is already at equilibrium. No net change will occur. And if Q K, your product to reactant ratio is too high. The reaction needs to shift left, make more reactants, use up products to get back down to the equilibrium ratio K. Q greater than K shifts left. Simple comparison. Got it. It's a very useful tool for predicting reaction direction. And the final piece, can we use K to actually calculate all the equilibrium concentrations if we only know where we start? Yes, that's often the ultimate goal. You know your initial concentrations and the value of K at that temperature. You set up your IC table again expressing the equilibrium concentrations in terms of X. Then you plug those equilibrium expressions with X in them into the K expression. This usually gives you an equation you need to solve for X. Sometimes it's straightforward. Sometimes, well, sometimes it involves solving a quadratic equation. All the algebra returns. It does. Yeah. But solving for X tells you exactly how much the concentrations changed. So, you can calculate the exact amounts of everything present once equilibrium is established. It shows the real predictive power of K. This is fascinating. So, equilibrium isn't just a static endpoint. It's predictable, quantifiable. But can we influence it? M if a reaction reaches equilibrium can we like push it one way or the other especially if we want more product like in the habber process absolutely and this is where Leette's principle comes in it's a cornerstone of understanding equilibrium basically it says if you take a system that's already at equilibrium and you disturb it by changing concentration pressure or temperature the system will shift its equilibrium position to try and counteract that disturbance it pushes back it pushes back chemistry's way of trying to restore balance you could say. Okay, let's break that down. First, disturbance, change in concentrations. What if we just dump in more reactant or maybe pull out some product as it forms? Great questions. If you add more reactant, the system says, "Whoa, too much reactant." And it shifts to the right to consume some of that added reactant and make more product, counteracting the addition. Exactly. And if you remove a product, this is key. Industrially, the system says, "Hey, where did the product go?" and it shifts to the right again trying to replace the product that was removed. Think about the habber process. N2 + 3 H2= 2 NH3. If you continuously remove the ammonia NH3 as it's made, ah you force it to keep making more. Precisely. Removing the product pulls the equilibrium constantly to the right dramatically increasing the overall yield of ammonia. It's a clever application of the principle. Very clever. But does this messing with concentrations change the actual value of K? Excellent point. No, it does not. Changing concentrations shifts the position of the equilibri, meaning the actual amounts of each substance change, but the ratio defined by K remains constant at that temperature. The system just finds a new set of concentrations that still satisfy the same K value. Okay, K stays put. What about the next disturbance? Changing pressure or volume, especially for gases, if we squeeze the container. For gas reactions, pressure and volume changes matter if the number of gas molecules changes during the reaction. If you decrease the volume, which increases the pressure, the system will shift in the direction that produces fewer moles of gas. Why? Right? To reduce the number of gas particles bouncing around, which helps to lower the pressure, counteracting your squeeze. Okay? Less volume, fewer gas moles, right? And if you increase the volume, decreasing pressure, the system shifts toward the side with more moles of gas, trying to fill that extra space and bump the pressure back up a bit. So back to Habber, N2 + 3 H2 = 2 NH3. That's four moles of gas on the left, only two on the right. Exactly. So if you increase the pressure, decrease volume, the equilibrium shifts to the right, favoring ammonia production because that reduces the total number of gas molecules. High pressure is crucial for the Habber process yield. Makes sense. What if the moles of gas are equal on both sides? Then pressure changes have no effect on the equilibrium position. Like H2 plus I2= 2 HI. Two moles of gas on each side. Squeezing it doesn't offer a relief direction. Oh, and one more thing. Adding an inert gas like argon at constant volume doesn't shift the equilibrium. It increases the total pressure, sure, but it doesn't change the partial pressures of the reacting gases. So K remains satisfied. Important distinction. And again, does pressure change K itself? Nope. Just like concentration changes, pressure volume changes shift the position, but do not alter the value of K at a given temperature. Okay, K is proving quite resilient. But now the big one, changing temperature. Does K finally budge here? Yes. Temperature is the one factor that changes the actual value of the equilibrium constant K. Uh-huh. How does that work? The easiest way to think about it is to treat heat as a reactant or product. For an endothermic reaction, one that absorbs heat, heat is like a reactant. reactants plus heat products. If you increase the temperature, you're adding heat reactant. Lehat says the system will shift right to consume that added heat, making more products. So for endothermic reactions, increasing T increases the value of K. More heat drives it forward, bigger K. Okay. Your source mentions a cobalt complex that's pink in cold water and turns blue when heated. CoH262 plus pink plus CL co4 to blue plus H2O. Heating shifts it right. blue meaning the forward reactions endothermic and K increases with temperature. Cool visual. What about exothermic reactions? For exothermic reactions, ones that release heat, heat is like a product. Reactants products plus heat. If you increase the temperature, you're adding heat product. The system shifts left to consume that heat, making more reactants. So for exothermic reactions, increasing T decreases the value of K. Adding heat drives it backward. Smaller K. Precisely. Lowering the temperature would have the opposite effect in both cases. The deep reason is that temperature changes affect the rates of the forward and reverse reactions differently because they usually have different activation energies. This unequal effect on rates changes their ratio which is K K= K forward. Okay, so temperature is the K changer. Last piece, catalysts, we use them all the time. Do they shift equilibrium change K? Great question and the answer is a clear no. Catalysts do not change the equilibrium position or the value of K. So what do they do? They speed things up. A catalyst lowers the activation energy barrier for both the forward and the reverse reactions and it does so by the same amount. Ah equally for both directions. Exactly. So both forward and reverse reactions get faster but their ratio which determines K doesn't change. The result, the system reaches equilibrium much faster, but the final destination, the equilibrium composition is exactly the same as it would be without the catalyst. It's like paving the road, not changing the destination city. Perfect analogy. You get there quicker, but you end up in the same place. And this is vital for industry, right? Like back to habber. Absolutely critical. The habber process equilibrium is actually more favorable at lower temperatures since it's exothermic. Lower T means higher K. But at low temperatures, the reaction is incredibly slow. Finding a good catalyst was the breakthrough. It allowed the reaction to proceed at a reasonable rate at intermediate temperatures. A compromise between getting a decent equilibrium yield K value and getting it fast enough to be practical. Saved huge amounts of energy, too. Another great example is in catalytic converters in cars. Making nitric oxide N is endothermic. So, high engine temperatures favor its formation. That's bad. It's a pollutant, right? When the exhaust cools, the N decomposition back to N2 and O2 is slow. The catalytic converter contains catalysts like platinum, palladium that speed up that decomposition reaction at exhaust temperatures, converting harmful NO back into harmless nitrogen and oxygen before it leaves the tailpipe. Brilliant chemistry at work. Wow. Okay, we've really covered a lot from just the idea of this dynamic balance through quantifying it with K, predicting shifts with Q and then manipulating it with Lettier's principal concentration, pressure, temperature catalysts. It really gives you a toolkit for understanding reactions. It really does. And hopefully you see this isn't just abstract theory. It's fundamental to so many processes that shape our world. From making fertilizers and materials to controlling pollution. Understanding equilibrium is understanding a huge part of practical chemistry. This dive into chemistry, the central science, really lays that groundwork. So maybe the next time you see something that looks perfectly still, whether it's chemical or not, perhaps pause and wonder, is it truly static? Or could there be a hidden dynamic equilibrium humming away beneath the surface, a constant dance maintaining that appearance of calm? Something to think about. |
190409 | https://www.convertunits.com/molarmass/KH2PO4 | Molecular weight of KH2PO4
KH2PO4 molecular weight
Molar mass of KH2PO4 = 136.085541 g/mol
This compound is also known as Potassium Diphosphate.
Convert grams KH2PO4 to moles
or
moles KH2PO4 to grams
Molecular weight calculation:
39.0983 + 1.007942 + 30.973761 + 15.99944
Percent composition by element
Element: Hydrogen
Symbol: H
Atomic Mass: 1.00794
of Atoms: 2
Mass Percent: 1.481%
Element: Oxygen
Symbol: O
Atomic Mass: 15.9994
of Atoms: 4
Mass Percent: 47.027%
Element: Phosphorus
Symbol: P
Atomic Mass: 30.973761
of Atoms: 1
Mass Percent: 22.761%
Element: Potassium
Symbol: K
Atomic Mass: 39.0983
of Atoms: 1
Mass Percent: 28.731% |
190410 | https://nationalzoo.si.edu/animals/mangrove-snake | Mangrove snake | Smithsonian's National Zoo and Conservation Biology Institute
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Exhibit: Reptile Discovery Center
Mangrove snake
Boiga dendrophila
The mangrove snake is a slender black snake with narrow, yellow bands along its body and around its face. This nocturnal species spends its days coiled among mangrove branches in riverine forests or overhanging coastal waters.
Fact Sheet
Conservation
Physical Description
This slender snake is primarily black with yellow scales around its face and bands around its body. The yellow bands do not generally join over the back or under the belly. The mangrove snake's eyes are large with narrow, vertical slits, earning it another common name—the yellow-ringed cat snake. The eye structure allows it to see better at night when it is active and on the hunt.
The mangrove snake is mildly venomous with rear fangs. Like other members of the Colubrid family, it has a Duvernoy's gland, which is distinct from the venom gland and composed of cells that produce saliva. The Duvernoy's gland is located on the posterior end of the eye with a duct that connects to the rear fangs, which are enlarged and have grooves into which the venom flows. This is a common adaptation of rear-fanged snakes. The fangs are angled backward to assist in biting and holding onto prey, though sometimes the prey needs to be chewed for the venom to be properly injected.
The mangrove snake's toxin, called denmotoxin, is especially useful when hunting their primary prey, birds. In a 2006 study published in the Journal of Biological Chemistry, scientists found that the muscles of mice were not as susceptible to the mangrove snake's immobilizing venom as the muscles of birds. The snake's venom is not lethal to humans but can cause painful swelling and discoloration of the skin.
Native Habitat
Mangrove snakes are widely distributed across southeast Asia, including Indonesia, Malaysia, Thailand, Singapore, Vietnam, Cambodia and the Philippines. It is native, but not common, to Singapore. A population of mangrove snakes was also accidentally introduced to Texas.
Nine subspecies of mangrove snake are currently recognized by taxonomists. However, debate continues as to whether some of these are distinct species, or whether more subspecies have yet to be determined. The nine recognized subspecies occupy different parts of the snake's range and exhibit slight color variations.
Mangrove snakes are aptly named for the areas they inhabit: mangrove forests, riverine areas and lowland forests. They spend most of their time basking on tree branches 30 meters (100 feet) or higher but descend to the forest floor at night to hunt.
Food/Eating Habits
These snakes hunt a wide variety of prey, including other snakes, lizards and frogs. Most frequently, they forage for small mammals, such as bats and birds (and their eggs).
Reproduction and Development
Mangrove snakes lay their eggs in tree hollows. Their clutch size averages 10 eggs but ranges from four to 15. After a gestation period of about 45 days, the young hatch. The hatchlings are about 8 inches (20 centimeters) long and are similar in color to adults.
Help this Species
If you see a snake in the wild, leave it alone and encourage others to do the same. Don’t assume it is a venomous species, and don’t attack it if it doesn’t pose a threat to your safety. Tell your friends and family about the eco-services that snakes provide, such as keeping rodent populations in check.
Share the story of this animal with others. Simply raising awareness about this species can contribute to its overall protection.
Protect local waterways by using fewer pesticides when caring for your garden or lawn. Using fertilizers sparingly, keeping storm drains free of litter and picking up after your pet can also improve watershed health.
Smithsonian's National Zoo and Conservation Biology Institute. (n.d.). Mangrove snake. Retrieved September 28, 2025, from
Fun Facts
The mangrove snake has rear fangs and is mildly venomous.
The mangrove snake's eyes are large with narrow, vertical slits, earning it another common name—the yellow-ringed cat snake.
Conservation Status
lc Least Concern
nt Near Threatened
vu Vulnerable
en Endangered
cr Critically Endangered
ew Extinct in the Wild
ex Extinct
dd Data Deficient
ne Not Evaluated
Taxonomic Information
Class: Reptilia
Order: Squamata
Family: Colubridae
Genus and Species: Boiga dendrophila
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190411 | https://en.wikipedia.org/wiki/Ceva%27s_theorem | Jump to content
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Contents
(Top)
1 Proofs
1.1 Using triangle areas
1.2 Using barycentric coordinates
2 Generalizations
3 See also
4 References
5 Further reading
6 External links
Ceva's theorem
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From Wikipedia, the free encyclopedia
Geometric relation between line segments from a triangle's vertices and their intersection
For other uses, see Ceva (disambiguation).
In Euclidean geometry, Ceva's theorem is a theorem about triangles. Given a triangle △ABC, let the lines AO, BO, CO be drawn from the vertices to a common point O (not on one of the sides of △ABC), to meet opposite sides at D, E, F respectively. (The segments AD, BE, CF are known as cevians.) Then, using signed lengths of segments,
In other words, the length XY is taken to be positive or negative according to whether X is to the left or right of Y in some fixed orientation of the line. For example, AF / FB is defined as having positive value when F is between A and B and negative otherwise.
Ceva's theorem is a theorem of affine geometry, in the sense that it may be stated and proved without using the concepts of angles, areas, and lengths (except for the ratio of the lengths of two line segments that are collinear). It is therefore true for triangles in any affine plane over any field.
A slightly adapted converse is also true: If points D, E, F are chosen on BC, AC, AB respectively so that
then AD, BE, CF are concurrent, or all three parallel. The converse is often included as part of the theorem.
The theorem is often attributed to Giovanni Ceva, who published it in his 1678 work De lineis rectis. But it was proven much earlier by Yusuf Al-Mu'taman ibn Hűd, an eleventh-century king of Zaragoza. Ibn Hűd's work, however, had fallen into oblivion, and was rediscovered only in 1985.
Associated with the figures are several terms derived from Ceva's name: cevian (the lines AD, BE, CF are the cevians of O), cevian triangle (the triangle △DEF is the cevian triangle of O); cevian nest, anticevian triangle, Ceva conjugate. (Ceva is pronounced Chay'va; cevian is pronounced chev'ian.)
The theorem is very similar to Menelaus' theorem in that their equations differ only in sign. By re-writing each in terms of cross-ratios, the two theorems may be seen as projective duals.
Proofs
[edit]
Several proofs of the theorem have been created. Two proofs are given in the following.
The first one is very elementary, using only basic properties of triangle areas. However, several cases have to be considered, depending on the position of the point O.
The second proof uses barycentric coordinates and vectors, but is somehow[vague] more natural and not case dependent. Moreover, it works in any affine plane over any field.
Using triangle areas
[edit]
First, the sign of the left-hand side is positive since either all three of the ratios are positive, the case where O is inside the triangle (upper diagram), or one is positive and the other two are negative, the case O is outside the triangle (lower diagram shows one case).
To check the magnitude, note that the area of a triangle of a given height is proportional to its base. So
Therefore,
(Replace the minus with a plus if A and O are on opposite sides of BC.) Similarly,
and
Multiplying these three equations gives
as required.
The theorem can also be proven easily using Menelaus's theorem. From the transversal BOE of triangle △ACF,
and from the transversal AOD of triangle △BCF,
The theorem follows by dividing these two equations.
The converse follows as a corollary. Let D, E, F be given on the lines BC, AC, AB so that the equation holds. Let AD, BE meet at O and let F' be the point where CO crosses AB. Then by the theorem, the equation also holds for D, E, F'. Comparing the two,
But at most one point can cut a segment in a given ratio so F = F’.
Using barycentric coordinates
[edit]
Given three points A, B, C that are not collinear, and a point O, that belongs to the same plane, the barycentric coordinates of O with respect of A, B, C are the unique three numbers such that
and
for every point X (for the definition of this arrow notation and further details, see Affine space).
For Ceva's theorem, the point O is supposed to not belong to any line passing through two vertices of the triangle. This implies that
If one takes for X the intersection F of the lines AB and OC (see figures), the last equation may be rearranged into
The left-hand side of this equation is a vector that has the same direction as the line CF, and the right-hand side has the same direction as the line AB. These lines have different directions since A, B, C are not collinear. It follows that the two members of the equation equal the zero vector, and
It follows that
where the left-hand-side fraction is the signed ratio of the lengths of the collinear line segments AF and FB.
The same reasoning shows
Ceva's theorem results immediately by taking the product of the three last equations.
Generalizations
[edit]
The theorem can be generalized to higher-dimensional simplexes using barycentric coordinates. Define a cevian of an n-simplex as a ray from each vertex to a point on the opposite (n − 1)-face (facet). Then the cevians are concurrent if and only if a mass distribution can be assigned to the vertices such that each cevian intersects the opposite facet at its center of mass. Moreover, the intersection point of the cevians is the center of mass of the simplex.
Another generalization to higher-dimensional simplexes extends the conclusion of Ceva's theorem that the product of certain ratios is 1. Starting from a point in a simplex, a point is defined inductively on each k-face. This point is the foot of a cevian that goes from the vertex opposite the k-face, in a (k + 1)-face that contains it, through the point already defined on this (k + 1)-face. Each of these points divides the face on which it lies into lobes. Given a cycle of pairs of lobes, the product of the ratios of the volumes of the lobes in each pair is 1.
Routh's theorem gives the area of the triangle formed by three cevians in the case that they are not concurrent. Ceva's theorem can be obtained from it by setting the area equal to zero and solving.
The analogue of the theorem for general polygons in the plane has been known since the early nineteenth century. The theorem has also been generalized to triangles on other surfaces of constant curvature.
The theorem also has a well-known generalization to spherical and hyperbolic geometry, replacing the lengths in the ratios with their sines and hyperbolic sines, respectively.
See also
[edit]
Projective geometry
Median (geometry) – an application
Circumcevian triangle
Menelaus's theorem
Triangle
Stewart's theorem
Cevian
References
[edit]
^ Holme, Audun (2010). Geometry: Our Cultural Heritage. Springer. p. 210. ISBN 978-3-642-14440-0.
^ Benitez, Julio (2007). "A Unified Proof of Ceva and Menelaus' Theorems Using Projective Geometry" (PDF). Journal for Geometry and Graphics. 11 (1): 39–44.
^ a b c Russell, John Wellesley (1905). "Ch. 1 §7 Ceva's Theorem". Pure Geometry. Clarendon Press.
^ Alfred S. Posamentier and Charles T. Salkind (1996), Challenging Problems in Geometry, pages 177–180, Dover Publishing Co., second revised edition.
^ Follows Hopkins, George Irving (1902). "Art. 986". Inductive Plane Geometry. D.C. Heath & Co.
^ Landy, Steven (December 1988). "A Generalization of Ceva's Theorem to Higher Dimensions". The American Mathematical Monthly. 95 (10): 936–939. doi:10.2307/2322390. JSTOR 2322390.
^ Wernicke, Paul (November 1927). "The Theorems of Ceva and Menelaus and Their Extension". The American Mathematical Monthly. 34 (9): 468–472. doi:10.2307/2300222. JSTOR 2300222.
^ Samet, Dov (May 2021). "An Extension of Ceva's Theorem to n-Simplices". The American Mathematical Monthly. 128 (5): 435–445. doi:10.1080/00029890.2021.1896292. S2CID 233413469.
^ Grünbaum, Branko; Shephard, G. C. (1995). "Ceva, Menelaus and the Area Principle". Mathematics Magazine. 68 (4): 254–268. doi:10.2307/2690569. JSTOR 2690569.
^ Masal'tsev, L. A. (1994). "Incidence theorems in spaces of constant curvature". Journal of Mathematical Sciences. 72 (4): 3201–3206. doi:10.1007/BF01249519. S2CID 123870381.
Further reading
[edit]
Hogendijk, J. B. (1995). "Al-Mutaman ibn Hűd, 11the century king of Saragossa and brilliant mathematician". Historia Mathematica. 22: 1–18. doi:10.1006/hmat.1995.1001.
External links
[edit]
Menelaus and Ceva at MathPages
Derivations and applications of Ceva's Theorem at cut-the-knot
Trigonometric Form of Ceva's Theorem at cut-the-knot
Glossary of Encyclopedia of Triangle Centers includes definitions of cevian triangle, cevian nest, anticevian triangle, Ceva conjugate, and cevapoint
Conics Associated with a Cevian Nest, by Clark Kimberling
Ceva's Theorem by Jay Warendorff, Wolfram Demonstrations Project.
Weisstein, Eric W. "Ceva's Theorem". MathWorld.
Experimentally finding the centroid of a triangle with different weights at the vertices: a practical application of Ceva's theorem at Dynamic Geometry Sketches, an interactive dynamic geometry sketch using the gravity simulator of Cinderella.
"Ceva theorem", Encyclopedia of Mathematics, EMS Press, 2001
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190412 | https://www.ixl.com/math/lessons/multiplication-as-repeated-addition | Skill previewSearch shortcut:
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Multiplication as repeated addition
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When you have equal groups and want to find the total, you can use repeated addition or multiplication.
Imagine you are buying bananas. The bananas come in bunches of 3. You buy 5 bunches:
To find the total number of bananas, you could add five 3s using repeated addition.
Or, you could show this same problem using multiplication!
3 times 5 means the same thing as adding 3 five times.
You can solve any multiplication problem using repeated addition. Just rewrite the problem and add.
So,
4
×
6
=
24
!
Try some practice problems!
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DC Generator Losses: Copper, Iron, and Mechanical Loss Types Explained
Losses in Dc generator
Original title:
Losses IN DC Generator
Course
Bachelor of Science in Electrical Engineering (BSEE)
510 documents
University
Technological University of the Philippines
Academic year:2021/2022
Uploaded by:
Anonymous Student
Technological University of the Philippines
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LOSSES IN DC GENERATOR
Copper Loss These losses occur in armature and field copper windings. Copper losses consist of Armature copper loss, Field copper loss and loss due to brush contact resistance.
o Armature copper loss = I𝑎2R𝑎 ▪ This loss contributes about 30 to 40% to full load losses. The armature copper loss is variable and depends upon the amount of loading of the machine.
o Field copper loss = I𝑓 2R𝑓 ▪ In the case of a shunt wounded field, field copper loss is practically constant. It contributes about 20 to 30% to full load losses.
o Brush contact resistance also contributes to the copper losses. Generally, this loss is included into armature copper loss.
Iron Loss As the armature core is made of iron and it rotates in a magnetic field, a small current gets induced in the core itself too. Due to this current, eddy current
loss and hysteresis loss occur in the armature iron core. Iron losses are also called as Core losses or magnetic losses.
o Hysteresis loss ▪ This is due to the reversal of magnetization of the armature core. When the core passes under one pair of poles, it undergoes one complete cycle of magnetic reversal. The frequency of magnetic reversal is given by f = P∙N/120 (where, P = no. of poles and N = Speed in rpm).
▪ The loss depends upon the volume and grade of the iron, frequency of magnetic reversals and value of flux density.
where, η = Steinmetz hysteresis constant V = volume of the core in m 3
Let: A = ηβ𝑚1.
P
120
V
𝐖𝒉 = 𝐀𝐍
The core is not made solid, it is made of thin sheets of metal that when stacked into pile, you can create a solid cylinder out of it. It is also separated with each other via lamination or insulation. The purpose of doing this is to confine the eddy current into a respective sheets of material of a core.
The thickness of lamination is used for the minimization of eddy current loss.
where, K = eddy current constants f = frequency t = thickness of lamination β𝑚𝑎𝑥 = maximum flux density V = volume of the core in m 3
𝜙 = 𝑐 ; β𝑚 = 𝑐
Let: B = Kβ𝑚 2
P 2
1202
t 2 V 2
𝐖𝒆 = 𝐁𝐍𝟐
Note:
With regards to the formula of the hysteresis and the eddy current loss, it can be concluded that they are rotational loss because it is depending on the speed and the square of the speed of the rotation of the generator.
Core loss will not exist if the generator does not rotate.
𝐖𝒊 (𝐢𝐫𝐨𝐧 𝐥𝐨𝐬𝐬) = 𝐖𝒉 (𝐡𝐲𝐬𝐭𝐞𝐫𝐞𝐬𝐢𝐬 𝐥𝐨𝐬𝐬) + 𝐖𝒆 (𝐞𝐝𝐝𝐲 𝐜𝐮𝐫𝐫𝐞𝐧𝐭 𝐥𝐨𝐬𝐬)
Mechanical Losses Mechanical losses consist of the losses due to friction in bearings and commutator. Air friction loss of rotating armature also contributes to these. These losses are about 10 to 20% of full load losses.
Note:
The generator has two bearings (inboard and outboard). These two have frictions also the commutator with respect to the carbon brush.
Mechanical Loss also sometimes called friction and windage loss.
o Stray Losses ▪ In addition to the losses stated above, there may be small losses present which are called as stray losses or miscellaneous losses. These losses are difficult to account. They are usually due to inaccuracies in the designing and modeling of the machine. Most of the times, stray losses are assumed to be 1% of the full load. o Stray Power Losses (SPL) ▪ Accumulated losses of Iron losses and mechanical losses. They are considered as constant loss of dc generator.
Note:
Stray power loss is a rotational loss but a combination of iron loss and mechanical loss.
𝐒𝐏𝐋 = 𝐖𝒊 + 𝐌𝐋
SPL is an iron loss and friction loss due to mechanical.
Note:
Efficiency can be determined because of losses.
The higher the losses, the lower the performance.
Efficiency in DC Generator
Maximum Efficiency
i. e. Shunt Generator
I𝑓 < I𝐿 ≪ I𝑎
I𝑓 = 0 ; I𝑎 = I𝐿
η =
P𝑜𝑢𝑡
P𝑜𝑢𝑡 + I𝑎2R𝑎 + I𝑓 2R𝑓 + SPL
× 100%
Let: P𝑘 = constant loss
In shunt generator: P𝑘 = SPL + I𝑓 2R𝑓
η =
I𝐿V𝑇
I𝐿V𝑇 + I𝑎2R𝑎 + P𝑘
× 100%
d dI𝐿
η =
d dI𝐿
[
I𝐿V𝑇
I𝐿V𝑇 + I𝐿 2R𝑎 + P𝑘
× 100%]
Recall: d u v
= vdu − udv v 2
dη dI𝐿
= (I
𝐿V𝑇 + I𝑎2R𝑎 + P𝑘)(V𝑇dI𝐿) − (I𝐿V𝑇)(V𝑇dI𝐿 + 2I𝐿R𝑎dI𝐿) (I𝐿V𝑇 + I𝐿 2R𝑎 + P𝑘) 2
= 0
I𝐿V𝑇 + I𝐿 2R𝑎 + P𝑘 − I𝐿V𝑇 − 2I𝐿 2R𝑎 = 0
P𝑘 − I𝐿 2R𝑎 = 0
P𝑘 = I𝐿 2R𝑎
Since: I𝑎 = I𝐿
∴ P𝑘 = I𝑎2R𝑎
Condition under the maximum efficiency: constant loss = variable loss
η𝑚𝑎𝑥 =
P𝑜𝑢𝑡 𝑚𝑎𝑥
P𝑜𝑢𝑡 𝑚𝑎𝑥 + I𝑎2R𝑎 + P𝑘
× 100%
η𝑚𝑎𝑥 =
P𝑜𝑢𝑡 𝑚𝑎𝑥
P𝑜𝑢𝑡 𝑚𝑎𝑥 + 2P𝑘
× 100%
P𝑜𝑢𝑡 𝑚𝑎𝑥 = I𝐿 𝑚𝑎𝑥V𝑇
Controlling the voltage in DC Generator
% VR (voltage regulation) = V𝑁𝐿
(no load voltage) − V𝐹𝐿 (full load voltage) V𝐹𝐿 (full load voltage)
× 100%
It is not good for a voltage regulation to be high and is only ranging from 0-10%.
V𝑇 = E𝑔 − I𝑎R𝑎
From the equation, VT is dependent on the load.
If the load increases, the load and armature current also increases so as its voltage drop resulting from the terminal voltage to decrease and this what is being called the loading effect.
To maintain the voltage to be constant on the load side, Eg should be manipulated. As:
E𝑔 = K𝜙𝑝N
With the constant speed and constant K, flux is the only solution left. Since:
𝜙𝑝 ∝ I𝑓
If the field will be controlled, so as the flux and the Eg.
Field rheostat manipulates I𝑓. It limits the flow of current.
The smaller the voltage regulation, the better since the wire used is good and there are only least amount of resistance.
Example:
tan 𝛼 = tan 𝛼 240 − V𝐻𝐿 25
= 240 − 220
50
240 − V𝐻𝐿 = 20(25)
50
= 10
V𝐻𝐿 = 230 V
%VR = 240 − 220
220
= 20
220
= 0 or 9%
η𝑚𝑎𝑥 =
(687)(600)
(687)(600) + 2(14700)
× 100% = 93%
A 10 kW, 250 volt, 6 Pole, d-c shunt generator runs at 1000rpm, when delivering full load. The armature has 534 lap connected conductors. Full load copper loss is 0. The total brush drop is 1 volt. Determine the flux per pole. Neglect shunt current.
Solution:
𝜙𝑝 = E
𝑔(60)(𝑎)
PNZ
E𝑔 = V𝑇 + I𝑎R𝑎 + V𝑏𝑐 = 251 + I𝑎R𝑎 → (1)
I𝑓 ≈ 0
I𝑎 = I𝐿 + I𝑓
I𝑎 = I𝐿 = P
𝑜𝑢𝑡 V𝑇
= 10000
250
= 40 A
P𝐶𝑢 = P𝑎 (armature copper loss) + P𝑓 (field copper loss) P𝑓 ≈ 0 P𝐶𝑢 = P𝑎 = I𝑎2R𝑎 = 640
R𝑎 =
640
(40) 2
= 0 Ω
E𝑔 = 251 + (40)(0) = 267 V
𝑎 = pm = 6(1) = 6
𝜙𝑝 =
267(60)(6)
6(1000)(534)
= 𝟎. 𝟎𝟑 𝐖𝐛
A 500V dc shunt motor draws a line current of 5 Amps, on light load. If the armature resistance is 0 ohm and field resistance is 200 ohms, determine the efficiency of the machine running as generator, delivering a load of 40 Amp.
Note:
Light load is pertaining to no load.
The motor is running idle. There is no mechanical load.
Solution:
As a motor:
Note:
The current is flowing in to the circuit. The input is electrical.
Back e.m., Eb is the same as Eg.
In the motor, the terminal voltage is now greater than the Eb.
E𝑏 𝑁𝐿 = V𝑇 − I𝑎 𝑁𝐿R𝑎
I𝑎 𝑁𝐿 = I𝐿 𝑁𝐿 − I𝑓 = 5 − 500
200
= 2 A
E𝑏 𝑁𝐿 = 500 − 2(0) = 499 V
P𝐶𝑢 = (42) 2 (0) + (
200
)
2 (200) = 1520 W
η =
P𝑜𝑢𝑡
P𝑜𝑢𝑡 + P𝐶𝑢 + SPL
× 100% =
20000
20000 + 1520 + 1249.
× 100% = 𝟖𝟕. 𝟖𝟑%
In a dc machine, the total iron loss is 8KW at its rated speed and excitation. If excitation remains the same but speed is reduced by 25%, the iron loss is found to be 5 KW. Calculate the hysteresis and eddy current losses at (a) full speed ( b) half speed.
Note:
If the excitation remains the same, the flux is constant.
Solution:
𝜙𝑝 = 𝑐
N 2 = 0 1
@ 𝐍𝟏 − 𝐫𝐚𝐭𝐞𝐝 𝐬𝐩𝐞𝐞𝐝:
Let: N 1 = 1 rpm
W𝑖 1 = 8000 = Wℎ 1 + W𝑒 1
Wℎ = AN ; W𝑒 = BN 2
8000 = AN 1 + BN1 2 = A + B → (1)
@ 𝐍𝟐 = 𝟎. 𝟕𝟓𝐍𝟏 = 𝟎. 𝟕𝟓 𝐫𝐩𝐦
W𝑖 2 = 5000 = Wℎ 2 + W𝑒 2 = AN 2 + BN2 2
5000 = 0 + 0 → (2)
Solving eq. (1) and (2) simultaneously:
A = 2666. B = 5333.
(a) Wℎ 1 & W𝑒 1
Wℎ 1 = AN 1 = 2666(1) = 𝟐𝟔𝟔𝟔. 𝟔𝟕 𝐖
W𝑒 1 = BN 12 = 5333(1) 2 = 𝟓𝟑𝟑𝟑. 𝟑𝟑 𝐖
W𝑖 1 = 2666 + 5333 = 8000 W
(b) @ 𝐡𝐚𝐥𝐟 𝐬𝐩𝐞𝐞𝐝:
Let: N 3 = 1 2
N 1 = 0 rpm
Wℎ 3 = AN 3 = 2666(0) = 𝟏𝟑𝟑𝟑. 𝟑𝟑 𝐖
W𝑒 3 = BN3 2 = 5333(0) 2 = 𝟏𝟑𝟑𝟑. 𝟑𝟑 𝐖
The hysteresis and eddy current losses of a DC machine running at 1000 rpm are 250 watts and 100 watts, respectively. If the flux remains constant, at what speed will the total iron loss be halved?
Solution: @ N 1 = 1000 rpm Wℎ1 = 250 W W𝑒1 = 100 W
W𝑖1 = 250 + 100 = 350 W
if 𝜙𝑝 = 𝑐: Wℎ = AN, W𝑒 = BN 2
@N 2 =?
W𝑖2 = 1 2
W𝑖1 = 350
2
= 175 W
@ N 1 :
Wℎ1 = AN 1 ; A = 250
1000
= 0.
W𝑒1 = BN 2 ; B =
100
(1000) 2
= 1 × 10−
@ N 2 :
W𝑖2 = Wℎ2 + W𝑒
R𝑎 =
18000
0 − 18000 − (2)
2 (60) − 500
(152) 2
= 0 Ω
@ 75% load: P𝑜𝑢𝑡 = 0(18000) = 13500 W ; I𝐿 ′ = 0(150) = 𝟏𝟏𝟐. 𝟓 𝐀
I𝑎′ = I𝐿 ′ + I𝑓 = 112 + 2 = 114 A
η0 𝐹𝐿 =
P𝑜𝑢𝑡
P𝑜𝑢𝑡 + I𝑎2R𝑎 + I𝑓 2R𝑓 + SPL
× 100%
η0 𝐹𝐿 =
13500
13500 + (114) 2 (0) + (2) 2 (60) + 500
× 100% = 𝟖𝟔. 𝟒𝟓%
@ 50 load:
I𝐿𝐻𝐿 = 150 2
= 𝟕𝟓 𝐀 ; P𝑜𝑢𝑡𝐻𝐿 = 75(120) = 9000 W
I𝑎 𝐻𝐿 = I𝐿 𝐻𝐿 + I𝑓 = 75 + 2 = 77 A
η𝐻𝐿 =
9000
9000 + (77) 2 (0) + (2) 2 (60) + 500
× 100% = 𝟖𝟔. 𝟖𝟓%
20 kW, 200 V shunt generator is running at full load. The prime mover output given to the generator is 30 hp. If the armature and shunt field resistance are 0 ohm and 50 ohms respectively, determine the iron and friction loss.
Solution: P𝑖𝑛 = P𝑜𝑢𝑡 + I𝑎2R𝑎 + I𝑓 2R𝑓 + SPL
I𝐿 = P
𝑜𝑢𝑡 V𝑇
= 20000
200
= 100 A
I𝑎 = 100 + 200
50
= 104 A
I𝑓 = 4 A
SPL = P𝑖𝑛 − P𝑜𝑢𝑡 − P𝑐𝑢 = 30(746) − 20000 − (104) 2 (0) − (4) 2 (50) = 𝟏𝟎𝟑𝟗. 𝟐 𝐖
A 420 kW 600-volt 580 rpm long shunt generator has an armature resistance of 0 ohm, series field resistance of 0 ohm, shunt field resistance of 60 ohms and SPL of 12 kW. Calculate the necessary driving torque at rated load in N-m.
Solution:
T𝑖𝑛 = 9 P 𝑖𝑛 N
P = 2πNT 60
; T = 60
2π
× P
N
P𝑖𝑛 = P𝑜𝑢𝑡 + P𝑐𝑢 + SPL
I𝑎 = I𝐿 + I𝑓 = 420000
600
+ 600
60
= 710 A
P𝑖𝑛 = 420000 + (710) 2 (0 + 0) + (10) 2 (60) + 12000 = 458,164 W
T𝑖𝑛 = 9 (458, 164
580
) = 𝟕𝟓𝟒𝟑. 𝟗𝟏 𝐍 ⋅ 𝐦
A 40 kW, 440 V, long shunt compound dc generator has full load efficiency of 88%. If the resistance of the armature and interpoles is 0 ohm and that of the series and shunt fields 0 ohm and 240 ohms respectively, calculate the combined bearing friction, windage and core loss of the machine.
Solution:
P𝑖𝑛 = P
𝑜𝑢𝑡 η
= 40000
0.
= 45,454 W
DC Generator Losses: Copper, Iron, and Mechanical Loss Types Explained
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DC Generator Losses: Copper, Iron, and Mechanical Loss Types Explained
Course: Bachelor of Science in Electrical Engineering (BSEE)
510 documents
University: Technological University of the Philippines
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LOSSES IN D C GENERATOR
Copper Loss
These losses occur in armatur e and field copper windings. Copper
losses consist of Armature c opper loss, Field copper loss and loss due to brush
contact resistan ce.
o Armature c opper loss=I 𝑎
2 R 𝑎
▪This los s contributes about 30 t o 40% to full load losses. The ar m ature
copper loss i s v ariable and dep ends upon the amount of loading of t he
machine.
o Field cop per loss =I 𝑓
2 R 𝑓
▪In the case of a shunt woun ded field, field copper loss is practicall y
constant. It contr ibutes about 20 to 30% to full load losse s.
o Brush contact resistance al so contributes to the copper losses. G enerally,
this loss is inclu ded into ar mature copp er loss.
Iron Loss
As the arma ture core is m ade of iron and it rotates in a m agnetic field, a small
current gets induced in the core its elf too. Due to this current, eddy current
loss and hyste resis loss occur in the armature iron core. Iron losses are also
called as Core l osses or magnetic losses.
o Hysteresis l oss
▪This is due to the reversal of magnetization of the armature core. When
the core pa sses under one pair of pol es, it undergoe s one complete
cycle of magne tic rever sal. The frequen cy of magnetic reversal is given
by f = P∙N/120 (where, P = no. of poles and N = Speed in rpm).
▪The loss depends upon the volume and grade o f t he iron, frequency of
magnetic rever sals and value of flux density.
where, η= Steinmetz hystere sis constant
V= volume of the co re in m 3
Let: A=ηβ 𝑚
1.6 P
120 V
𝐖 𝒉=𝐀𝐍
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The core is not made solid, i t is made of thin sheets of metal that when stacked in to
pile, you can cre ate a solid cylinder out of i t. It is also sepa rated with ea ch other via
lamination or insulation. The purpo se of doing this is to confine the eddy curr ent into a
respective sheet s of material of a core.
The thickne ss of lamination is used for the minimization o f eddy current loss.
where, K = eddy current constants
f = frequency
t = thickness of lamination
β 𝑚𝑎𝑥= maximum flux densi ty
V = volume of the core in m 3
𝜙=𝑐 ;β 𝑚=𝑐
Let: B=Kβ 𝑚
2 P 2
120 2 t 2 V 2
𝐖 𝒆=𝐁𝐍 𝟐
Note:
With regards to the formula of the hysteresi s and the edd y current lo ss, it can be
concluded that the y are rotationa l loss because it is depen ding on the speed and the
square of the speed of the rota tion of the gener ator.
Core loss will n ot exist if the generator does not rotate.
𝐖 𝒊(𝐢𝐫𝐨𝐧 𝐥𝐨𝐬𝐬)=𝐖 𝒉(𝐡𝐲𝐬𝐭𝐞𝐫𝐞𝐬𝐢𝐬 𝐥𝐨𝐬𝐬)+𝐖 𝒆(𝐞 𝐝𝐝𝐲 𝐜𝐮𝐫𝐫𝐞𝐧𝐭 𝐥𝐨𝐬𝐬)
Mechanical Losses
Mechanical los ses consist of the losses due to f riction in bearings and commu tator.
Air fri ction los s of ro tating a rm ature also contributes to these.
These losse s are about 10 to 20% of full load lo sses.
Note:
The generator has two bearing s (inboard and outboard). T hese two have frictions also
the commutator with respect to the carbon brush.
Mechanical Los s also some times calle d friction a nd winda ge loss.
o Stray Losses
▪In addi tion t o the losses stated abov e, there may be small losses
present which are called as stray losses or miscellaneous losses.
These losses are difficult to accou nt. They ar e usually due to
inaccuracies in the designing and mo deling of the machine. Most of
the times, stray losses are assume d to be 1% o f the full lo ad.
o Stray Powe r Losses (SPL)
▪Accumulated losses of Iron losses and mec hanical losses. T hey are
considered a s constant loss o f dc generator.
Note:
Stray power lo ss is a rotational lo ss but a combination of iron loss and mecha nic al
loss.
𝐒𝐏𝐋=𝐖 𝒊+𝐌𝐋
SPL is an iron lo ss and friction lo ss due to me chanical.
Note:
Efficiency can be determined because of losses.
The higher the lo sses, the lower the performan ce.
Efficiency i n DC Gene rator
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Maximum Efficie ncy
i.e. Shunt Generator
I 𝑓<I 𝐿≪I 𝑎
I 𝑓=0 ;I 𝑎=I 𝐿
η=P 𝑜𝑢𝑡
P 𝑜𝑢𝑡+I 𝑎
2 R 𝑎+I 𝑓
2 R 𝑓+SPL×100%
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190414 | https://en.wikipedia.org/wiki/Stars_and_bars_(combinatorics) | Jump to content
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Contents
(Top)
1 Statements of theorems
1.1 Theorem one
1.2 Theorem two
2 Proofs via the method of stars and bars
2.1 Theorem one proof
2.2 Example
2.3 Theorem two proof
2.4 Example
3 Relation between Theorems one and two
4 Further examples
4.1 Example 1
4.2 Example 2
4.3 Example 3
4.4 Example 4
5 Relation to generating functions
6 See also
7 References
8 Further reading
Stars and bars (combinatorics)
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From Wikipedia, the free encyclopedia
Graphical aid for deriving some concepts in combinatorics
In combinatorics, stars and bars (also called "sticks and stones", "balls and bars", and "dots and dividers") is a graphical aid for deriving certain combinatorial theorems. It can be used to solve a variety of counting problems, such as how many ways there are to put n indistinguishable balls into k distinguishable bins. The solution to this particular problem is given by the binomial coefficient , which is the number of subsets of size k − 1 that can be formed from a set of size n + k − 1.
If, for example, there are two balls and three bins, then the number of ways of placing the balls is . The table shows the six possible ways of distributing the two balls, the strings of stars and bars that represent them (with stars indicating balls and bars separating bins from one another), and the subsets that correspond to the strings. As two bars are needed to separate three bins and there are two balls, each string contains two bars and two stars. Each subset indicates which of the four symbols in the corresponding string is a bar.
Six configurations of two balls in three bins and their star and bar representations
| Bin 1 | Bin 2 | Bin 3 | String | Subset of {1,2,3,4} |
| 2 | 0 | 0 | ★ ★ | | | {3,4} |
| 1 | 1 | 0 | ★ | ★ | | {2,4} |
| 1 | 0 | 1 | ★ | | ★ | {2,3} |
| 0 | 2 | 0 | | ★ ★ | | {1,4} |
| 0 | 1 | 1 | | ★ | ★ | {1,3} |
| 0 | 0 | 2 | | | ★ ★ | {1,2} |
Statements of theorems
[edit]
The stars and bars method is often introduced specifically to prove the following two theorems of elementary combinatorics concerning the number of solutions to an equation.
Theorem one
[edit]
For any pair of positive integers n and k, the number of k-tuples of positive integers whose sum is n is equal to the number of (k − 1)-element subsets of a set with n − 1 elements.
For example, if n = 10 and k = 4, the theorem gives the number of solutions to x1 + x2 + x3 + x4 = 10 (with x1, x2, x3, x4 > 0) as the binomial coefficient
where is the number of combinations of n − 1 elements taken k − 1 at a time.
This corresponds to compositions of an integer.
Theorem two
[edit]
For any pair of positive integers n and k, the number of k-tuples of non-negative integers whose sum is n is equal to the number of multisets of size k − 1 taken from a set of size n + 1, or equivalently, the number of multisets of size n taken from a set of size k, and is given by
For example, if n = 10 and k = 4, the theorem gives the number of solutions to x1 + x2 + x3 + x4 = 10 (with x1, x2, x3, x4 ) as
where the multiset coefficient is the number of multisets of size n, with elements taken from a set of size k.
This corresponds to weak compositions of an integer. With k fixed, the numbers for n = 0, 1, 2, 3, ... are those in the (k − 1)st diagonal of Pascal's triangle. For example, when k = 3 the nth number is the (n + 1)st triangular number, which falls on the second diagonal, 1, 3, 6, 10, ....
Proofs via the method of stars and bars
[edit]
Theorem one proof
[edit]
The problem of enumerating k-tuples whose sum is n is equivalent to the problem of counting configurations of the following kind: let there be n objects to be placed into k bins, so that all bins contain at least one object. The bins are distinguished (say they are numbered 1 to k) but the n objects are not (so configurations are only distinguished by the number of objects present in each bin). A configuration is thus represented by a k-tuple of positive integers.
The n objects are now represented as a row of n stars; adjacent bins are separated by bars. The configuration will be specified by indicating the boundary between the first and second bin, the boundary between the second and third bin, and so on. Hence k − 1 bars need to be placed between stars. Because no bin is allowed to be empty, there is at most one bar between any pair of stars. There are n − 1 gaps between stars and hence n − 1 positions in which a bar may be placed. A configuration is obtained by choosing k − 1 of these gaps to contain a bar; therefore there are configurations.
Example
[edit]
With n = 7 and k = 3, start by placing seven stars in a line:
★ ★ ★ ★ ★ ★ ★
Fig. 1: Seven objects, represented by stars
Now indicate the boundaries between the bins:
★ ★ ★ ★ | ★ | ★ ★
Fig. 2: These two bars give rise to three bins containing 4, 1, and 2 objects
In general two of the six possible bar positions must be chosen. Therefore there are such configurations.
Theorem two proof
[edit]
In this case, the weakened restriction of non-negativity instead of positivity means that we can place multiple bars between stars and that one or more bars also be placed before the first star and after the last star. In terms of configurations involving objects and bins, bins are now allowed to be empty.
Rather than a (k − 1)-set of bar positions taken from a set of size n − 1 as in the proof of Theorem one, we now have a (k − 1)-multiset of bar positions taken from a set of size n + 1 (since bar positions may repeat and since the ends are now allowed bar positions). An alternative interpretation in terms of multisets is the following: there is a set of k bin labels from which a multiset of size n is to be chosen, the multiplicity of a bin label in this multiset indicating the number of objects placed in that bin. The equality can also be understood as an equivalence of different counting problems: the number of k-tuples of non-negative integers whose sum is n equals the number of (n + 1)-tuples of non-negative integers whose sum is k − 1, which follows by interchanging the roles of bars and stars in the diagrams representing configurations.
To see the expression directly, observe that any arrangement of stars and bars consists of a total of n + k − 1 symbols, n of which are stars and k − 1 of which are bars. Thus, we may lay out n + k − 1 slots and choose k − 1 of these to contain bars (or, equivalently, choose n of the slots to contain stars).
Example
[edit]
When n = 7 and k = 5, the tuple (4, 0, 1, 2, 0) may be represented by the following diagram:
★ ★ ★ ★ | | ★ | ★ ★ |
Fig. 3: These four bars give rise to five bins containing 4, 0, 1, 2, and 0 objects
If possible bar positions are labeled 1, 2, 3, 4, 5, 6, 7, 8 with label i ≤ 7 corresponding to a bar preceding the ith star and following any previous star and 8 to a bar following the last star, then this configuration corresponds to the (k − 1)-multiset {5,5,6,8}, as described in the proof of Theorem two. If bins are labeled 1, 2, 3, 4, 5, then it also corresponds to the n-multiset {1,1,1,1,3,4,4}, also as described in the proof of Theorem two.
Relation between Theorems one and two
[edit]
Theorem one can be restated in terms of Theorem two, because the requirement that each variable be positive can be imposed by shifting each variable by −1, and then requiring only that each variable be non-negative.
For example:
with
is equivalent to:
with
where for each .
Further examples
[edit]
Example 1
[edit]
If one wishes to count the number of ways to distribute seven indistinguishable one dollar coins among Amber, Ben, and Curtis so that each of them receives at least one dollar, one may observe that distributions are essentially equivalent to tuples of three positive integers whose sum is 7. (Here the first entry in the tuple is the number of coins given to Amber, and so on.) Thus Theorem 1 applies, with n = 7 and k = 3, and there are ways to distribute the coins.
Example 2
[edit]
If n = 5, k = 4, and the k bin labels are a, b, c, d, then ★|★★★||★ could represent either the 4-tuple (1, 3, 0, 1), or the multiset of bar positions {2, 5, 5}, or the multiset of bin labels {a, b, b, b, d}. The solution of this problem should use Theorem 2 with n = 5 stars and k – 1 = 3 bars to give configurations.
Example 3
[edit]
In the proof of Theorem two there can be more bars than stars, which cannot happen in the proof of Theorem one.
So, for example, 10 balls into 7 bins gives configurations, while 7 balls into 10 bins gives configurations, and 6 balls into 11 bins gives configurations.
Example 4
[edit]
The graphical method was used by Paul Ehrenfest and Heike Kamerlingh Onnes—with symbol ε (quantum energy element) in place of a star and the symbol 0 in place of a bar—as a simple derivation of Max Planck's expression for the number of "complexions" for a system of "resonators" of a single frequency.
By complexions (microstates) Planck meant distributions of P energy elements ε over N resonators. The number R of complexions is
The graphical representation of each possible distribution would contain P copies of the symbol ε and N – 1 copies of the symbol 0. In their demonstration, Ehrenfest and Kamerlingh Onnes took N = 4 and P = 7 (i.e., R = 120 combinations). They chose the 4-tuple (4, 2, 0, 1) as the illustrative example for this symbolic representation: εεεε0εε00ε.
Relation to generating functions
[edit]
The enumerations of Theorems one and two can also be found using generating functions involving simple rational expressions. The two cases are very similar; we will look at the case when , that is, Theorem two first. There is only one configuration for a single bin and any given number of objects (because the objects are not distinguished). This is represented by the generating function
The series is a geometric series, and the last equality holds analytically for |x| < 1, but is better understood in this context as a manipulation of formal power series. The exponent of x indicates how many objects are placed in the bin.
Each additional bin is represented by another factor of ; the generating function for k bins is
: ,
where the multiplication is the Cauchy product of formal power series.
To find the number of configurations with n objects, we want the coefficient of (denoted by prefixing the expression for the generating function with ), that is,
: .
This coefficient can be found using binomial series and agrees with the result of Theorem two, namely .
This Cauchy product expression is justified via stars and bars: the coefficient of in the expansion of the product
is the number of ways of obtaining the nth power of x by multiplying one power of x from each of the k factors. So the stars represent xs and a bar separates the xs coming from one factor from those coming from the next factor.
For the case when , that is, Theorem one, no configuration has an empty bin, and so the generating function for a single bin is
: .
The Cauchy product is therefore , and the coefficient of is found using binomial series to be .
See also
[edit]
Gaussian binomial coefficient
Partition (number theory)
Twelvefold way
Dirichlet-multinomial distribution
References
[edit]
^ Batterson, J. Competition Math for Middle School. Art of Problem Solving.
^ Flajolet, Philippe; Sedgewick, Robert (June 26, 2009). Analytic Combinatorics. Cambridge University Press. ISBN 978-0-521-89806-5.
^ "Art of Problem Solving". artofproblemsolving.com. Retrieved 2021-10-26.
^ Feller, William (1968). An Introduction to Probability Theory and Its Applications. Vol. 1 (3rd ed.). Wiley. p. 38.
^ Ehrenfest, Paul; Kamerlingh Onnes, Heike (1914). "Simplified deduction of the formula from the theory of combinations which Planck uses as the basis of his radiation theory". Proceedings of the KNAW. 17: 870–873. Retrieved 16 May 2024.
^ Ehrenfest, Paul; Kamerlingh Onnes, Heike (1915). "Simplified deduction of the formula from the theory of combinations which Planck uses as the basis of his radiation theory". The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. Series 6. 29 (170): 297–301. doi:10.1080/14786440208635308. Retrieved 5 December 2020.
^ Planck, Max (1901). "Ueber das Gesetz der Energieverteilung im Normalspectrum". Annalen der Physik. 309 (3): 553–563. Bibcode:1901AnP...309..553P. doi:10.1002/andp.19013090310.
^ Gearhart, C. (2002). "Planck, the Quantum, and the Historians" (PDF). Phys. Perspect. 4 (2): 170–215. Bibcode:2002PhP.....4..170G. doi:10.1007/s00016-002-8363-7. Retrieved 16 May 2024.
Further reading
[edit]
Pitman, Jim (1993). Probability. Berlin: Springer-Verlag. ISBN 0-387-97974-3.
Weisstein, Eric W. "Multichoose". Mathworld -- A Wolfram Web Resource. Retrieved 18 November 2012.
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6: Moles
General, Organic, and Biochemistry with Problems, Case Studies, and Activities
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Fri, 28 Jul 2023 15:31:43 GMT
6.3: Converting between Grams and Moles
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6.2: Formula Mass
6.4: Case Study- The Mole Concept in Clinical Laboratory Analysis
Page ID 449294
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Table of contents
1. Learning Objectives
2. Formula Mass
1. Formula Mass for Covalent Substances
1. Example 6.3.1: Computing Molecular Mass for a Covalent Compound
1. Solution
2. Exercise 6.3.1)
2. Formula Mass for Ionic Compounds
1. Example 6.3.2: Computing Formula Mass for an Ionic Compound:_Computing_Formula_Mass_for_an_Ionic_Compound)
1. Solution
2. Exercise 6.3.2)
The Mole
Link to Learning
Example 6.3.3: Deriving Moles from Grams for an Element
Solution
Exercise 6.3.3
Example 6.3.4: Deriving Grams from Moles for an Element
Solution
Exercise 6.3.1
Example 6.3.5: Deriving Number of Atoms from Mass for an Element
Solution
Exercise 6.3.5
Example 6.3.6: Deriving Moles from Grams for a Compound
Solution
Exercise 6.3.6
Example 6.3.7: Deriving Grams from Moles for a Compound
Solution
Exercise 6.3.7
Example 6.3.8: Deriving the Number of Atoms and Molecules from the Mass of a Compound
Solution
Exercise 6.3.8
How Sciences Interconnect: Counting Neurotransmitter Molecules in the Brain
Footnotes
Learning Objectives
By the end of this section, you will be able to:
Calculate formula masses for covalent and ionic compounds
Define the amount unit mole and the related quantity Avogadro’s number Explain the relation between mass, moles, and numbers of atoms or molecules, and perform calculations deriving these quantities from one another
Many argue that modern chemical science began when scientists started exploring the quantitative as well as the qualitative aspects of chemistry. For example, Dalton’s atomic theory was an attempt to explain the results of measurements that allowed him to calculate the relative masses of elements combined in various compounds. Understanding the relationship between the masses of atoms and the chemical formulas of compounds allows us to quantitatively describe the composition of substances.
Formula Mass
An earlier chapter of this text described the development of the atomic mass unit, the concept of average atomic masses, and the use of chemical formulas to represent the elemental makeup of substances. These ideas can be extended to calculate the formula mass of a substance by summing the average atomic masses of all the atoms represented in the substance’s formula.
Formula Mass for Covalent Substances
For covalent substances, the formula represents the numbers and types of atoms composing a single molecule of the substance; therefore, the formula mass may be correctly referred to as a molecular mass. Consider chloroform (CHClA 3), a covalent compound once used as a surgical anesthetic and now primarily used in the production of tetrafluoroethylene, the building block for the "anti-stick" polymer, Teflon. The molecular formula of chloroform indicates that a single molecule contains one carbon atom, one hydrogen atom, and three chlorine atoms. The average molecular mass of a chloroform molecule is therefore equal to the sum of the average atomic masses of these atoms. Figure 6.3.1 outlines the calculations used to derive the molecular mass of chloroform, which is 119.37 amu.
Figure 6.3.1: The average mass of a chloroform molecule, CHCl 3, is 119.37 amu, which is the sum of the average atomic masses of each of its constituent atoms. The model shows the molecular structure of chloroform.
Likewise, the molecular mass of an aspirin molecule, CA 9HA 8OA 4, is the sum of the atomic masses of nine carbon atoms, eight hydrogen atoms, and four oxygen atoms, which amounts to 180.15 amu (Figure 6.3.2).
Figure 6.3.2: The average mass of an aspirin molecule is 180.15 amu. The model shows the molecular structure of aspirin, C 9 H 8 O 4.
Example 6.3.1: Computing Molecular Mass for a Covalent Compound
Ibuprofen, CA 13HA 18OA 2, is a covalent compound and the active ingredient in several popular nonprescription pain medications, such as Advil and Motrin. What is the molecular mass (amu) for this compound?
Solution
Molecules of this compound are composed of 13 carbon atoms, 18 hydrogen atoms, and 2 oxygen atoms. Following the approach described above, the average molecular mass for this compound is therefore:
Exercise 6.3.1
Acetaminophen, C 8 H 9 NO 2, is a covalent compound and the active ingredient in several popular nonprescription pain medications, such as Tylenol. What is the molecular mass (amu) for this compound?
Answer
151.16 amu
Formula Mass for Ionic Compounds
Ionic compounds are composed of discrete cations and anions combined in ratios to yield electrically neutral bulk matter. The formula mass for an ionic compound is calculated in the same way as the formula mass for covalent compounds: by summing the average atomic masses of all the atoms in the compound’s formula. Keep in mind, however, that the formula for an ionic compound does not represent the composition of a discrete molecule, so it may not correctly be referred to as the “molecular mass.”
As an example, consider sodium chloride, NaCl, the chemical name for common table salt. Sodium chloride is an ionic compound composed of sodium cations, Na+, and chloride anions, Cl−, combined in a 1:1 ratio. The formula mass for this compound is computed as 58.44 amu (see Figure 6.3.3).
Figure 6.3.3: Table salt, NaCl, contains an array of sodium and chloride ions combined in a 1:1 ratio. Its formula mass is 58.44 amu.
Note that the average masses of neutral sodium and chlorine atoms were used in this computation, rather than the masses for sodium cations and chlorine anions. This approach is perfectly acceptable when computing the formula mass of an ionic compound. Even though a sodium cation has a slightly smaller mass than a sodium atom (since it is missing an electron), this difference will be offset by the fact that a chloride anion is slightly more massive than a chloride atom (due to the extra electron). Moreover, the mass of an electron is negligibly small with respect to the mass of a typical atom. Even when calculating the mass of an isolated ion, the missing or additional electrons can generally be ignored, since their contribution to the overall mass is negligible, reflected only in the nonsignificant digits that will be lost when the computed mass is properly rounded. The few exceptions to this guideline are very light ions derived from elements with precisely known atomic masses.
Example 6.3.2: Computing Formula Mass for an Ionic Compound
Aluminum sulfate, Al 2(SO 4)3, is an ionic compound that is used in the manufacture of paper and in various water purification processes. What is the formula mass (amu) of this compound?
Solution
The formula for this compound indicates it contains Al 3+ and SO 4 2− ions combined in a 2:3 ratio. For purposes of computing a formula mass, it is helpful to rewrite the formula in the simpler format, Al 2 S 3 O 12. Following the approach outlined above, the formula mass for this compound is calculated as follows:
Exercise 6.3.2
Calcium phosphate, CaA 3(POA 4)A 2, is an ionic compound and a common anti-caking agent added to food products. What is the formula mass (amu) of calcium phosphate?
Answer
310.18 amu
The Mole
The identity of a substance is defined not only by the types of atoms or ions it contains, but by the quantity of each type of atom or ion. For example, water, H 2 O, and hydrogen peroxide, H 2 O 2, are alike in that their respective molecules are composed of hydrogen and oxygen atoms. However, because a hydrogen peroxide molecule contains two oxygen atoms, as opposed to the water molecule, which has only one, the two substances exhibit very different properties. Today, sophisticated instruments allow the direct measurement of these defining microscopic traits; however, the same traits were originally derived from the measurement of macroscopic properties (the masses and volumes of bulk quantities of matter) using relatively simple tools (balances and volumetric glassware). This experimental approach required the introduction of a new unit for amount of substances, the mole, which remains indispensable in modern chemical science.
The mole is an amount unit similar to familiar units like pair, dozen, gross, etc. It provides a specific measure of the number of atoms or molecules in a sample of matter. One Latin connotation for the word “mole” is “large mass” or “bulk,” which is consistent with its use as the name for this unit. The mole provides a link between an easily measured macroscopic property, bulk mass, and an extremely important fundamental property, number of atoms, molecules, and so forth. A mole of substance is that amount in which there are 6.02214076×10 23.
Figure 6.3.4: Each sample contains 6.022 10 23 atoms —1.00 mol of atoms. From left to right (top row): 65.4 g zinc, 12.0 g carbon, 24.3 g magnesium, and 63.5 g copper. From left to right (bottom row): 32.1 g sulfur, 28.1 g silicon, 207 g lead, and 118.7 g tin. (credit: modification of work by Mark Ott)
The molar mass of any substance is numerically equivalent to its atomic or formula weight in amu. Per the amu definition, a single 12 C atom weighs 12 amu (its atomic mass is 12 amu). A mole of 12 C weighs 12 g (its molar mass is 12 g/mol). This relationship holds for all elements, since their atomic masses are measured relative to that of the amu-reference substance, 12 C. Extending this principle, the molar mass of a compound in grams is likewise numerically equivalent to its formula mass in amu (Figure 6.3.5).
Figure 6.3.5: Each sample contains 6.02 10 23 molecules or formula units—1.00 mol of the compound or element. Clock-wise from the upper left: 130.2 g of C 8 H 17 OH (1-octanol, formula mass 130.2 amu), 454.4 g of HgI 2 (mercury(II) iodide, formula mass 454.4 amu), 32.0 g of CH 3 OH (methanol, formula mass 32.0 amu) and 256.5 g of S 8 (sulfur, formula mass 256.5 amu). (credit: Sahar Atwa)
| Element | Average Atomic Mass (amu) | Molar Mass (g/mol) | Atoms/Mole |
--- --- |
| C | 12.01 | 12.01 | 6.022 10 23 |
| H | 1.008 | 1.008 | 6.022 10 23 |
| O | 16.00 | 16.00 | 6.022 10 23 |
| Na | 22.99 | 22.99 | 6.022 10 23 |
| Cl | 35.45 | 35.45 | 6.022 10 23 |
While atomic mass and molar mass are numerically equivalent, keep in mind that they are vastly different in terms of scale, as represented by the vast difference in the magnitudes of their respective units (amu versus g). To appreciate the enormity of the mole, consider a small drop of water weighing about 0.03 g (see Figure 6.3.6). Although this represents just a tiny fraction of 1 mole of water (~18 g), it contains more water molecules than can be clearly imagined. If the molecules were distributed equally among the roughly seven billion people on earth, each person would receive more than 100 billion molecules.
Figure 6.3.6: The number of molecules in a single droplet of water is roughly 100 billion times greater than the number of people on earth. (credit: “tanakawho”/Wikimedia commons)
Link to Learning
The mole is used in chemistry to represent 6.022 10 23 of something, but it can be difficult to conceptualize such a large number. Watch this video and then complete the “Think” questions that follow. Explore more about the mole by reviewing the information under “Dig Deeper.”
The relationships between formula mass, the mole, and Avogadro’s number can be applied to compute various quantities that describe the composition of substances and compounds, as demonstrated in the next several example problems.
Example 6.3.3: Deriving Moles from Grams for an Element
According to nutritional guidelines from the US Department of Agriculture, the estimated average requirement for dietary potassium is 4.7 g. What is the estimated average requirement of potassium in moles?
Solution
The mass of K is provided, and the corresponding amount of K in moles is requested. Referring to the periodic table, the atomic mass of K is 39.10 amu, and so its molar mass is 39.10 g/mol. The given mass of K (4.7 g) is a bit more than one-tenth the molar mass (39.10 g), so a reasonable “ballpark” estimate of the number of moles would be slightly greater than 0.1 mol.
The molar amount of a substance may be calculated by dividing its mass (g) by its molar mass (g/mol):
The factor-label method supports this mathematical approach since the unit “g” cancels and the answer has units of “mol:”
4.7 g K(mol K 39.10 g K)=0.12 mol K
The calculated magnitude (0.12 mol K) is consistent with our ballpark expectation, since it is a bit greater than 0.1 mol,
Exercise 6.3.3
Beryllium is a light metal used to fabricate transparent X-ray windows for medical imaging instruments. How many moles of Be are in a thin-foil window weighing 3.24 g?
Answer
0.360 mol
Example 6.3.4: Deriving Grams from Moles for an Element
A liter of air contains 9.2 10−4 mol argon. What is the mass of Ar in a liter of air?
Solution
The molar amount of Ar is provided and must be used to derive the corresponding mass in grams. Since the amount of Ar is less than 1 mole, the mass will be less than the mass of 1 mole of Ar, approximately 40 g. The molar amount in question is approximately one-one thousandth (~10−3) of a mole, and so the corresponding mass should be roughly one-one thousandth of the molar mass (~0.04 g):
In this case, logic dictates (and the factor-label method supports) multiplying the provided amount (mol) by the molar mass (g/mol):
9.2×10−4mol Ar(39.95 g Ar mol Ar)=0.037 g Ar
The result is in agreement with our expectations, around 0.04 g Ar.
Exercise 6.3.1
What is the mass of 2.561 mol of gold?
Answer
504.4 g
Example 6.3.5: Deriving Number of Atoms from Mass for an Element
Copper is commonly used to fabricate electrical wire (Figure 6.3.7). How many copper atoms are in 5.00 g of copper wire?
Figure 6.3.7: Copper wire is composed of many, many atoms of Cu. (credit: Emilian Robert Vicol)
Solution
The number of Cu atoms in the wire may be conveniently derived from its mass by a two-step computation: first calculating the molar amount of Cu, and then using Avogadro’s number (N A) to convert this molar amount to number of Cu atoms:
Considering that the provided sample mass (5.00 g) is a little less than one-tenth the mass of 1 mole of Cu (~64 g), a reasonable estimate for the number of atoms in the sample would be on the order of one-tenth N A, or approximately 10 22 Cu atoms. Carrying out the two-step computation yields:
5.00 g Cu(mol Cu 63.55 g Cu)(6.022×10 23 Cu atoms mol Cu)=4.74×10 22 Cu atoms
The factor-label method yields the desired cancellation of units, and the computed result is on the order of 10 22 as expected.
Exercise 6.3.5
A prospector panning for gold in a river collects 15.00 g of pure gold. How many Au atoms are in this quantity of gold?
Answer
4.586×10 22 Au atoms
Example 6.3.6: Deriving Moles from Grams for a Compound
Our bodies synthesize protein from amino acids. One of these amino acids is glycine, which has the molecular formula C 2 H 5 O 2 N. How many moles of glycine molecules are contained in 28.35 g of glycine?
Solution
Derive the number of moles of a compound from its mass following the same procedure used for an element in Example 6.3.3:
The molar mass of glycine is required for this calculation, and it is computed in the same fashion as its molecular mass. One mole of glycine, C 2 H 5 O 2 N, contains 2 moles of carbon, 5 moles of hydrogen, 2 moles of oxygen, and 1 mole of nitrogen:
The provided mass of glycine (~28 g) is a bit more than one-third the molar mass (~75 g/mol), so the computed result is expected to be a bit greater than one-third of a mole (~0.33 mol). Dividing the compound’s mass by its molar mass yields:
28.35g glyeine(mol glycine 75.07g glyeine)=0.378 mol glycine
This result is consistent with the rough estimate,
Exercise 6.3.6
How many moles of sucrose, C 12 H 22 O 11, are in a 25-g sample of sucrose?
Answer
0.073 mol
Example 6.3.7: Deriving Grams from Moles for a Compound
Vitamin C is a covalent compound with the molecular formula C 6 H 8 O 6. The recommended daily dietary allowance of vitamin C for children aged 4–8 years is 1.42 10−4 mol. What is the mass of this allowance in grams?
Solution
As for elements, the mass of a compound can be derived from its molar amount as shown:
The molar mass for this compound is computed to be 176.124 g/mol. The given number of moles is a very small fraction of a mole (~10−4 or one-ten thousandth); therefore, the corresponding mass is expected to be about one-ten thousandth of the molar mass (~0.02 g). Performing the calculation yields:
1.42×10−4mol vitamin C(176.124 g vitamin C mol vitamin C)=0.0250 g vitamin C
This is consistent with the anticipated result.
Exercise 6.3.7
What is the mass of 0.443 mol of hydrazine, N 2 H 4?
Answer
14.2 g
Example 6.3.8: Deriving the Number of Atoms and Molecules from the Mass of a Compound
A packet of an artificial sweetener contains 40.0 mg of saccharin (C 7 H 5 NO 3 S), which has the structural formula:
Given that saccharin has a molar mass of 183.18 g/mol, how many saccharin molecules are in a 40.0-mg (0.0400-g) sample of saccharin? How many carbon atoms are in the same sample?
Solution
The number of molecules in a given mass of compound is computed by first deriving the number of moles, as demonstrated in Example 6.3.6, and then multiplying by Avogadro’s number:
Using the provided mass and molar mass for saccharin yields:
0.0400 gCA 7HA 5NOA 3S(mol CA 7HA 5NOA 3S 183.18 gCHA 5NOA 3S)(6.022×10 23CA 7HA 5NOA 3S molecules 1 mol CA 7HA 5NOA 3S)=1.31×10 20CA 7HA 5NOA 3S molecules
The compound’s formula shows that each molecule contains seven carbon atoms, and so the number of C atoms in the provided sample is:
1.31×10 20CA 7HA 5NOA 3S molecules(7 C atoms 1CA 7HA 5NOA 3S molecule)=9.17×10 20 C atoms
Exercise 6.3.8
How many C 4 H 10 molecules are contained in 9.213 g of this compound? How many hydrogen atoms?
Answer
9.545 × 10 22 molecules C 4 H 10
9.545 × 10 23 atoms H
How Sciences Interconnect: Counting Neurotransmitter Molecules in the Brain
The brain is the control center of the central nervous system (Figure 6.3.8). It sends and receives signals to and from muscles and other internal organs to monitor and control their functions; it processes stimuli detected by sensory organs to guide interactions with the external world; and it houses the complex physiological processes that give rise to our intellect and emotions. The broad field of neuroscience spans all aspects of the structure and function of the central nervous system, including research on the anatomy and physiology of the brain. Great progress has been made in brain research over the past few decades, and the BRAIN Initiative, a federal initiative announced in 2013, aims to accelerate and capitalize on these advances through the concerted efforts of various industrial, academic, and government agencies (more details available at www.whitehouse.gov/share/brain-initiative).
Figure 6.3.8: (a) A typical human brain weighs about 1.5 kg and occupies a volume of roughly 1.1 L. (b) Information is transmitted in brain tissue and throughout the central nervous system by specialized cells called neurons (micrograph shows cells at 1600× magnification).
Specialized cells called neurons transmit information between different parts of the central nervous system by way of electrical and chemical signals. Chemical signaling occurs at the interface between different neurons when one of the cells releases molecules (called neurotransmitters) that diffuse across the small gap between the cells (called the synapse) and bind to the surface of the other cell. These neurotransmitter molecules are stored in small intracellular structures called vesicles that fuse to the cell membrane and then break open to release their contents when the neuron is appropriately stimulated. This process is called exocytosis (see Figure 6.3.9). One neurotransmitter that has been very extensively studied is dopamine, C 8 H 11 NO 2. Dopamine is involved in various neurological processes that impact a wide variety of human behaviors. Dysfunctions in the dopamine systems of the brain underlie serious neurological diseases such as Parkinson’s and schizophrenia.
Figure 6.3.9: (a) Chemical signals are transmitted from neurons to other cells by the release of neurotransmitter molecules into the small gaps (synapses) between the cells. (b) Dopamine, C 8 H 11 NO 2, is a neurotransmitter involved in a number of neurological processes.
One important aspect of the complex processes related to dopamine signaling is the number of neurotransmitter molecules released during exocytosis. Since this number is a central factor in determining neurological response (and subsequent human thought and action), it is important to know how this number changes with certain controlled stimulations, such as the administration of drugs. It is also important to understand the mechanism responsible for any changes in the number of neurotransmitter molecules released—for example, some dysfunction in exocytosis, a change in the number of vesicles in the neuron, or a change in the number of neurotransmitter molecules in each vesicle.
Significant progress has been made recently in directly measuring the number of dopamine molecules stored in individual vesicles and the amount actually released when the vesicle undergoes exocytosis. Using miniaturized probes that can selectively detect dopamine molecules in very small amounts, scientists have determined that the vesicles of a certain type of mouse brain neuron contain an average of 30,000 dopamine molecules per vesicle (about 5×10−20 mol or 50 zmol). Analysis of these neurons from mice subjected to various drug therapies shows significant changes in the average number of dopamine molecules contained in individual vesicles, increasing or decreasing by up to three-fold, depending on the specific drug used. These studies also indicate that not all of the dopamine in a given vesicle is released during exocytosis, suggesting that it may be possible to regulate the fraction released using pharmaceutical therapies.
Footnotes
1Omiatek, Donna M., Amanda J. Bressler, Ann-Sofie Cans, Anne M. Andrews, Michael L. Heien, and Andrew G. Ewing. “The Real Catecholamine Content of Secretory Vesicles in the CNS Revealed by Electrochemical Cytometry.” Scientific Report 3 (2013): 1447, accessed January 14, 2015, doi:10.1038/srep01447.
This page titled 6.3: Converting between Grams and Moles is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform.
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6.2: Formula Mass
6.4: Case Study- The Mole Concept in Clinical Laboratory Analysis
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190416 | https://mathbitsnotebook.com/Algebra2/Sequences/SSGauss.html | | | | | | | | | | | | | | | | | |
--- --- --- --- --- --- --- --- |
| | | | | | | | | | | | | | | | | --- --- --- --- --- --- --- | | | | | --- | | | Gauss on Arithmetic SequencesMathBitsNotebook.com Topical Outline | Algebra 2 Outline | MathBits' Teacher Resources Terms of Use Contact Person: Donna Roberts | Carl Friedrich Gauss (1777-1855) was a German mathematician who contributed in many fields of mathematics and science and is touted as one of history's most influential mathematicians. | | | --- | | As the story goes, when Gauss was a young boy, he was given the problem to add the integers from 1 to 100. Remember that there were no calculators in those days! As the other students struggled with this lengthy addition problem, Gauss saw a different way to attack this problem. He listed the first 50 terms, and then listed the second fifty terms in reverse order beneath the first set. You can think of it as he "wrapped" the series back onto itself. | | Gauss them added the paired values, noticing that the sums were all the same value (101). Since he had 50 such pairs, he multiplied 101 times 50 and obtained the sum of the integers from 1 to 100 to be 5050. Now, Gauss's discovery works nicely as long as you have an even number of terms in your series. But what happens to the "wrapped" pairings if the series has 25 terms? Well, Gauss' discovery would need a bit of tweaking. If the number of terms is odd, do not split the series in half. Simply list the ENTIRE series forward, then list the entire series in reverse and add the pairs. In this situation, you will need to multiply the sum by the number of pairs and then divide by two, since you are actually working with 2 complete series. By observing the series from BOTH directions simultaneously, Gauss was able to quickly solve the problem and establish a relationship that we still use today when working with arithmetic series. Let's generalize what Gauss actually did. Consider the following: | | | --- | | a1 + a2 + a3 + . . . + an-1 + an | Let's represent 1+2+3+...+100 in subscripted a notation | | a1 + an | All of Gauss' wrapped pairs have a sum of 101. While any pair could be used, it will be easier if we choose the first pair on the left, the addition of the first and last term, as these terms are usually more readily available. | | | Gauss multiplied this sum times 50, which is HALF the number of terms in his sequence. So, we need to multiply times half of the number of terms in the sequence, which is represented by n. Is this starting to look familiar? | | | Thus, was born a formula for the sum of n terms of an arithmetic sequence. | This relationship of examining a series forward and backward to determine the value of a series works for any arithmetic series. You may see the formula written as: Sum, Sn, of n terms of an arithmetic series. The first formula is Gauss' formula referencing n to be even. The second formula is a more general formula implying n to be even or odd. Algebraically, both formulas are equivalent. Example 1: (Even Number of Terms) Find the sum of 2 + 4 + 6 + 8 + 10 + 12 + 14 + 16. Example 2: (Odd Number of Terms) Find the sum of 2 + 4 + 6 + 8 + 10 + 12 + 14 + 16 + 18. Why is Gauss' pairing up the terms in an arithmetic series always giving the same sums? If you examine the graphic above, you can see that if you add the first and last terms you get 18. If you add the second term to the seventh term you will also get 18 because you are +2 and -2 away from the first and last terms. Thus, you have moved +2 - 2 = 0 away from 18. Remember that in an arithmetic series, the common difference is constant and this pattern of adding and subtracting the same value as the terms are paired will continue. All sums will be 18. | | | NOTE: The re-posting of materials (in part or whole) from this site to the Internet is copyright violation and is not considered "fair use" for educators. Please read the "Terms of Use". | Topical Outline | Algebra 2 Outline | MathBitsNotebook.com | MathBits' Teacher Resources Terms of UseContact Person: Donna Roberts Copyright © 2012-2025 MathBitsNotebook.com. All Rights Reserved. | | |
190417 | https://epicentre.org.za/chancroids/?srsltid=AfmBOopbasVCCSraSDAidwHx4hOEgtM5Fr0QK48nyTZOvPoyE2gEgCKH | The Ultimate Guide To Chancroid - Epicentre
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The Ultimate Guide To ChancroidAimee Zuccarini2025-06-20T11:28:43+02:00
The Ultimate Guide To Chancroid
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What Is Chancroid?
Chancroid is a sexually transmitted infection (STI) caused by the bacterium Haemophilus ducreyi. This infection usually results in the development of painful, open ulcers in the genital area, which can be quite uncomfortable.
In many cases, it is also associated with swollen lymph nodes in the groin, a condition referred to as “buboes.” These swollen lymph nodes can be particularly painful and may take weeks or months to resolve if not treated properly. The ulcers and buboes can lead to complications such as scarring, tissue damage, or the formation of abnormal passages (fistulas) in the genital area (Ogale et al., 2023; Ahmed et al., 2022)
Common Sexually Transmitted Infections
HPV
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Genital Herpes
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Chlamydia
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Gonorrhoea
View
HIV/AIDS
View
Syphilis
View
Trich
View
Viral Hepatitis
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Male Urethritis Syndrome
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Mycoplasma Genitalium
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Mycoplasma Hominis
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Ureaplasma Urealyticum
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Chancroid
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Quick Facts About Chancroid
Symptoms: The disease usually shows itself as painful, open genital ulcers, often with swollen lymph nodes in the groin (buboes).
Easy to Spread: Chancroid is passed along through sexual contact, meaning it’s a STI. Even a single sexual encounter can lead to the infection if one partner has it.
Incubation Period: Symptoms typically appear 4 to 7 days after exposure to the bacteria.
Complications: If left untreated, chancroid can lead to scarring, tissue damage, and the formation of abnormal passages (fistulas) in the genital area. It can also increase a person chances of getting HIV .
Diagnosis: You can test for the cause of Chancriod, the bacterium Haemophilus ducreyi, through the large STI panel
How Do You Get Chancroid?
Sexual Contact is the Main Way: The most common way to catch chancroid is through sexual contact – it’s a STI. This means you can get it from vaginal, anal, or oral sex with someone who is infected (Ahmed et al., 2022).
It Only Takes One Encounter: You don’t need to have multiple sexual encounters to get chancroid. Even a single unprotected sexual encounter with an infected person can pass on the bacteria (Ogale et al., 2023).
The Bacteria Sneaks In Through Small Cuts: Haemophilus ducreyi, the bacteria that causes chancroid, can enter the body through tiny breaks or abrasions in the skin. These small cuts can happen during sexual activity, especially if there’s any friction (Ahmed et al., 2022).
It Spreads Easily in Dense Communities: In places with high rates of unprotected sex (like among sex workers or in certain high-risk groups), chancroid can spread more quickly. The more sexual contacts you have, the higher your chance of exposure (Ogale et al., 2023).
Asymptomatic Spread: Some people with chancroid may not even know they have it because they don’t show symptoms, or the symptoms are mild. But they can still pass the infection on to others, even if they aren’t experiencing sores or pain (Ahmed et al., 2022).
More Likely If You Have Other STIs: If you already have another sexually transmitted infection, like syphilis or herpes, your skin may be more vulnerable. This makes it easier for Haemophilus ducreyi to get in and cause chancroid (Ogale et al., 2023).
HIV and Chancroid Go Hand in Hand: If you’re HIV-positive, you’re more likely to get chancroid if exposed to it. The sores from chancroid can make it easier for HIV to enter the body, which is why people with HIV should be extra cautious (Ahmed et al., 2022).
How To Protect Yourself Against Chancroid
To protect yourself against Chancroid, the most effective measure is to use condoms consistently during vaginal, anal, or oral sex. Condoms act as a barrier, significantly reducing the risk of transmission.
Limiting the number of sexual partners and engaging in a monogamous relationship with an uninfected partner can also lower your chances of contracting the infection. Regular STI testing is important, especially if you have multiple sexual partners or engage in unprotected sex, as it helps detect infections early and prevent their spread.
If you or your partner notice any symptoms of chancroid, such as painful genital ulcers or swollen lymph nodes, avoid sexual contact until treated. Encourage partner testing and treatment to ensure both of you remain infection-free.
Additionally, avoiding risky sexual behaviors, like unprotected sex with commercial sex workers or multiple partners, can further reduce your exposure. Lastly, staying informed about STIs and knowing how to identify symptoms and seek timely treatment are key to protecting your sexual health and preventing the spread of chancroid (Ahmed et al., 2022; Ogale et al., 2023).
Symptoms of Chancroid
| Symptom | Men | Women |
:---
| Genital Ulcer | Painful, shallow ulcer with well-defined edges, often located on the penis (prepuce or frenulum) | Painful, shallow ulcer with well-defined edges, typically on the vulva, cervix, or perianal region |
| Lymphadenopathy (Buboes) | Swollen, painful lymph nodes in the groin (can rupture if untreated) | Swollen, painful lymph nodes in the groin (can rupture if untreated) |
| Ulcer Location | Usually on the penis or surrounding area | Often on the vulva, cervix, or perianal region |
| Complications | Can lead to scarring, phimosis (tight foreskin), or fistula formation | Can cause vaginal scarring, rectal fistulas, or complications in childbirth |
| Discharge | Pus or bleeding may come from the ulcer when scraped | Pus or bleeding may come from the ulcer when scraped |
| Atypical Presentations | Rare extragenital lesions (sores, ulcers, or other abnormal skin changes), such as on the thighs or fingers | Rare extragenital lesions (sores, ulcers, or other abnormal skin changes), including on the inner thighs or breasts |
After Exposure, When Will Chancroid Symptoms Start?
After exposure to Haemophilus ducreyi, the bacterium that causes chancroid, symptoms typically begin to appear 4 to 7 days after sexual contact with an infected person. This period is known as the incubation period.
During this time, the bacteria enter the body through small cuts or abrasions in the skin, and symptoms like painful genital ulcers and swollen lymph nodes (buboes) start to develop. However, some people may not notice symptoms right away, and in rare cases, the symptoms may take slightly longer to appear.
What Happens If You Ignore Chancroid?
| Complications | Men | Women |
:---
| Persistent Ulcers | The ulcers can remain open for weeks to months, causing ongoing pain and discomfort. | The ulcers can remain open, leading to prolonged pain and a risk of secondary infections |
| Buboes (Swollen Lymph Nodes) | Swollen lymph nodes (buboes) can become larger and more painful, sometimes rupturing and discharging pus. | Swollen lymph nodes can become painful and may rupture, leading to infections in the surrounding tissues. |
| Scarring | Untreated chancroid can lead to scarring, especially on the penis, which may result in phimosis (a condition where the foreskin cannot be retracted). | Untreated chancroid can cause vaginal scarring, leading to complications during childbirth or increased pain during sex. |
| Fistulas | Can develop between the genitals and other areas like the urethra, leading to abnormal passageways. | Can develop between the vagina and rectum (rectovaginal fistulas), causing discomfort and possible incontinence. |
| Increased Risk of HIV | Chancroid ulcers create an open entry point for HIV, increasing the likelihood of acquiring or transmitting the virus. | Chancroid increases the risk of HIV transmission, particularly if the ulcers are not treated, leading to higher vulnerability. |
| Secondary Infections | The open sores can become infected with other bacteria, leading to more severe infections. | Open sores may become infected with other bacteria, increasing the risk of pelvic infections and reproductive health issues. |
| Infertility Risks | Rare, but untreated infection could lead to complications that affect fertility. | Untreated chancroid may increase the risk of pelvic inflammatory disease (PID), which can lead to infertility. |
Ignoring Chancroid can lead to both immediate and long-term health complications in both men and women. It’s essential to seek treatment to avoid the pain, discomfort, and more serious issues that can arise if the infection is left untreated.
Is Chancroid Treatable?
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Acids and Bases (Spectator Ions Identification)
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In an acid reaction with water, or other solvents, how do we know which atom(s) from a compound reacted with that solvent are the spectator ions? Are they the ones soluble in the solvent? Is the way in determining this the same as the way we usually do it in a precipitation reaction?
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Spectator ions in any kind of reaction are those that do not take part in the interaction.
In precipitation reactions, they are those not forming the least soluble salt.
In acid-base reactions, they are those not manifesting in the given environment either acidic, either basic behaviours. Typically cations of alkali metal, or alkali earth metals, or anions of strong acids.
In redox reaction, they are those not being oxidized/reduced nor being produced by it.
For particular cases, you must be familiar with chemistry of involved compounds.
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edited Sep 9, 2020 at 2:28
answered Sep 8, 2020 at 3:39
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190419 | https://magoosh.com/english-speaking/reported-speech-rules-in-english/ | Skip to content
Talking about what someone else has already said, also known as reported speech, involves a few special grammar rules in English.
How you form reported speech will largely depend on what was said and when it was said. Unfortunately, you can’t always repeat back what you hear verbatim (using exactly the same words)!
So, how should you report speech in English? What are the grammar rules that dictate these indirect speech patterns? Finally, what are some examples of reported speech? We will answer all of these questions and more, but first, let’s take a look at exactly what is meant by “reported speech.”
Prefer to watch this lesson on video? Here’s our full length tutorial on Reported Speech Rules in English:
Reported Speech Rules in English:
What is reported speech?
Reported speech simply refers to statements that recount what someone else has already said or asked. For example, let’s say that you and your two friends went to the movies. As you’re leaving the movie theater, the following conversation takes place:
Friend #1: That movie was really scary!
You: I know, right?
Friend #2: What did he say?
You: He said that the movie was really scary.
The last sentence is what is known as “reported speech,” because you reported something that someone else said. In most cases, a statement of reported speech uses verbs like “say” or “tell,” though you can also use verbs like “state,” “proclaim,” or “announce,” depending on the context of the original statement.
In any case, this is just one example of reported speech in the simple past tense. Different rules apply based on the verb tense and the content of the statement. First, let’s look at how reported speech statements work in the simple present tense:
Reporting Statements in the Simple Present Tense
If you report a statement using the simple present tense (say, tell, etc), then you can also leave the original statement in the present tense. Here are a few examples:
I like basketball -> They say that they like basketball.
He wants to visit Paris -> He tells me that he wants to visit Paris.
I watch TV every day -> She says she watches TV every day.
As you can see, both the reporting verb and the reported verb remain in the simple present tense. It is also important to note that, regardless of the tense, the word “that” is completely optional in reported speech. The meaning stays the same with or without it.
Reporting Statements in Other Tenses
Generally, when the reporting verb is in the simple past tense, we change the reported verb as well. For example:
Statement: I feel sad.
Reported Speech: He said he felt sad.
Since reported speech is reported after the fact, the reporting verb is usually in the simple past tense. This means that you will usually need to change the tense of the second clause. For example:
| | | |
---
| Tense | Statement | Reported Speech |
| Simple Present | I like oranges. | He said that he liked oranges. |
| Present Continuous | I am swimming. | She said that she was swimming. |
| Present Perfect | I have seen the movie. | He said that he had seen the movie. |
| Simple Past | I forgot to bring my lunch. | She said that she forgot her lunch OR she said that she had forgotten her lunch. |
| Past Continuous | I was looking for the train station. | He said that she had been looking for the train station. |
| Past Perfect | I had finished the letter before they arrived. | She said that she had finished the letter before they arrived. |
| Simple Future | I will move to New York. | He said that he would move to New York. |
| Future Continuous | I will be hanging out with someone. | She said that she would be hanging out with someone. |
| Future Perfect | I will have forgotten about it by tomorrow. | He said that he would have forgotten about it by tomorrow. |
| Present Perfect Continuous | I have been waiting in line. | She said that she had been waiting in line. |
| Past Perfect Continuous | I had been exercising more often. | He said that he had been exercising more often. |
| Future Perfect Continuous | By next month, I will have been a nurse for 10 years. | She said that by next month, she will have been a nurse for 10 years. |
How to Change Tenses in Reported Speech
As you can see, the rules governing how to report speech can vary based on the tense of the original statement. Generally, you can’t go wrong if you follow these guidelines (from the original statement to reported speech):
Simple Present -> Simple Past
Present Continuous -> Past Continuous
Present Perfect -> Past Perfect
Simple Past -> Simple Past OR Past Perfect
Past Continuous -> Past Perfect Continuous
Simple Future -> “will” becomes “would”
Future Continuous -> “will” becomes “would”
Future Perfect -> “will” becomes “would”
Present Perfect Continuous -> Past Perfect Continuous
Past Perfect Continuous -> Past Perfect Continuous
Future Perfect Continuous -> Future Perfect Continuous
That said, there are some exceptions in the present tense. For example, if the original statement is comprised of general information that is unchanging, you don’t need to report it in the past tense. Here are a few examples:
Simple Present: Water freezes at 0 degrees Celsius. -> He said that water freezes at zero degrees Celcius.
Present Continuous: The planet is rotating around the sun. -> She said that the planet is rotating around the sun.
Present Perfect: Human beings have always liked dogs. -> He said that human beings have always liked dogs.
Reporting Questions
Reporting statements is relatively straightforward, as it usually just requires the second clause to change tense (sometimes not even that). However, reporting questions is more complex. First of all, when you report a question, you cannot just repeat the original question. Instead, you must turn it into a statement. Here’s an example question:
Do you have a lighter?
If you want to report this question later, you’ll need to change it, like so:
They asked me if I had a lighter.
Thankfully, once you learn the guidelines for reporting statements, you can apply many of the same rules to reporting questions. All of the tense changes are the same:
Simple Present: Do you like to read? -> He asked if I liked to read.
Note: For “Yes/No” questions, we change “do” or “does” to “if.”
Present Continuous: Are you running errands today? -> She asked if I was running errands today.
Present Perfect: Have you spoken to her? -> He asked if I had spoken to her.
Simple Past: Did you believe the story? -> She asked if I believed the story.
Past Continuous: How were you behaving? -> He asked me how I was behaving.
Simple Future: Will you go shopping later? -> She asked me if I would go shopping later.
Future Continuous: Will you be cooking tonight? -> He asked me if I would be cooking tonight.
Future Perfect: Will you have received your diploma by then? -> She asked if I would have received my diploma by then.
Present Perfect Continuous – Have you been doing your homework? -> He asked me if I had been doing my homework.
Past Perfect Continuous – How long had you been sleeping? -> She asked me how long I had been sleeping.
Future Perfect Continuous – Will you have been travelling? -> He asked if I would have been travelling.
Requests and Demands
To keep things simple, requests are treated the same as questions when reported to someone else. For example:
Please sit down. -> He asked me to sit down.
Could you open the door for me? -> She asked if I could open the door for her?
Would you mind holding my bag? -> He asked if I would mind holding his bag.
However, if someone demands something, we generally report the speech using “told” instead of “asked” or “said.” Here are some commands in reported speech:
Be quiet! -> She told me to be quiet.
Don’t touch that! -> He told me not to touch that.
Brush your teeth. -> She told me to brush my teeth.
Finally, when reporting speech, you must always consider the time in which the original statement was made. If a time is mentioned within the statement, you will also have to consider how that time relates to the current moment.
You have a doctor’s appointment on Tuesday.
For example, let’s say that the statement above was reported to you a few days prior, but you reported it to someone else on Monday (the day before the appointment). You could say either of the following:
She told me that I have a doctor’s appointment on Tuesday, or
She told me that I have a doctor’s appointment tomorrow.
Here are a few more time conversions to help you with reported speech:
Call your father right now. -> She told me to call my father right then.
I saw you at the movies last night. -> He said he saw me at the movies the night before.
Were you at school last week? -> She asked if I had been at school the week prior.
Can I talk to you tomorrow? -> He asked if he could talk to me the next day.
Reported Speech Exercises
Now that you have a better understanding of reported speech in English, it’s time to practice! Fortunately, there are a number of ways to practice reported speech in daily conversation. So, here are a few free online resources to help you get the hang of it:
Reported Speech Statements
Reported Speech Questions
B1 Grammar Reported Speech Quiz
B2 Grammar Reported Speech Quiz
Lastly, if you’d like to learn more about reported speech or find a highly qualified English tutor online to help guide you, visit Magoosh Speaking today!
Matthew Jones
Matthew Jones is a freelance writer with a B.A. in Film and Philosophy from the University of Georgia. It was during his time in school that he published his first written work. After serving as a casting director in the Atlanta film industry for two years, Matthew acquired TEFL certification and began teaching English abroad. In 2017, Matthew started writing for dozens of different brands across various industries. During this time, Matthew also built an online following through his film blog. If you’d like to learn more about Matthew, you can connect with him on Twitter and LinkedIn!
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190420 | https://www.cliffsnotes.com/study-notes/22369555 | Calculating Increasing Intervals in Calculus Functions - CliffsNotes
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Calculating Increasing Intervals in Calculus Functions
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Question 1 Given: F(x) =Z x 0 sin(t 3 )dt for 0≤x≤2 Find:The intervals on which F(x) is increasing. Solution:A function F(x) is increasing where its derivative F 0(x) is posi- tive. By the Fundamental Theorem of Calculus: F 0(x) = sin(x 3 ) Now we need to find where sin(x 3) is positive for 0≤x≤2. The sine function is positive in the intervals: (2 kπ,(2 k+ 1)π) We need to determine the values of x such that x 3 falls within these intervals. For k= 0: 0< x 3 < π 0< x <3 √ π For k= 1: 2 π < x 3 <3 π 3√2 π < x <3√ 3 π Only the intervals that fall within 0≤x≤2 are valid. Let's calculate: 3 √ π≈1.464 3 √ 2 π≈1.837 3√ 3 π≈2.144 So,F(x) is increasing in the intervals: (0,3 √ π) (3 √ 2 π,2) Question 2 Given: R(t) = 0.002(t-4.72)3 + 20.21 Find:Approximate the value of R 50 0 R(t)dt using a left Riemann sum with five equal subintervals. Determine if this approximation is less than or greater than the true value and explain why. 1
Solution:First, we divide the interval [0,50] into five equal subintervals: Δ t=50-0 5 = 10 The left Riemann sum formula: Z 50 0 R(t)dt≈4 X i=0 R(t i )Δ t where t i = 0,10,20,30,40. Z 50 0 R(t)dt≈10[R(0) +R(10) +R(20) +R(30) +R(40)] Calculate R(t) at these points: R(0) = 0.002(0-4.72)3 + 20.21 = 0.002(-104.448) + 20.21 = 19.600104 R(10) = 0.002(10-4.72)3+ 20.21 = 0.002(5.28)3 + 20.21 = 20.532023 R(20) = 0.002(20-4.72)3+ 20.21 = 0.002(15.28)3 + 20.21 = 24.486451 R(30) = 0.002(30-4.72)3+ 20.21 = 0.002(25.28)3 + 20.21 = 41.019186 R(40) = 0.002(40-4.72)3+ 20.21 = 0.002(35.28)3 + 20.21 = 87.738589 Sum these values: 19.600104 + 20.532023 + 24.486451 + 41.019186 + 87.738589 = 193.376353 So, the approximate value is: 10×193.376353 = 1933.76353 Because R(t) is strictly increasing, the left Riemann sum will be less than the true value of the integral. Question 3 Given: F(x) =Z x 0 sin(t 3 )dt for 0≤x≤2 Find:Approximate F(1) using the trapezoidal rule with four equal subdi- visions of [0,1]. Solution:Divide [0,1] into four equal subintervals: Δ x=1-0 4 = 0.25 2
The trapezoidal rule formula: Z 1 0 sin(t 3 )dt≈Δ x 2 [f(0) + 2 f(0.25) + 2 f(0.5) + 2 f(0.75) +f(1)] Calculate f(t) = sin(t 3) at these points: f(0) = sin(0 3 ) = sin(0) = 0 f(0.25) = sin(0.25 3 ) = sin(0.015625)≈0.015625 f(0.5) = sin(0.5 3 ) = sin(0.125)≈0.124674 f(0.75) = sin(0.75 3 ) = sin(0.421875)≈0.409156 f(1) = sin(1 3 ) = sin(1)≈0.841471 Plug these values into the formula: 0.25 2 [0 + 2(0.015625) + 2(0.124674) + 2(0.409156) + 0.841471] = 0.125[0 + 0.03125 + 0.249348 + 0.818312 + 0.841471] = 0.125[1.940381] = 0.242548 Rounded to three decimal places: 0.243 3
Page 1 of 3
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Blank Class Note 7-7 - Fall 24.pdf Manijeh Bahreini TECHNIQUES OF INTEGRATION 7.7 Approximate Integration Sometimes it is impossible to find the exact value of definite integrals. For example we need to find approximate 2 1 values of definite integrals, it is impossible to evaluate the fol University of Illinois, Urbana Champaign MATH 231
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190421 | https://www.mathsisfun.com/geometry/pythagoras-general.html | Generalizations of Pythagoras' Theorem
Pythagoras' Theorem
Let's start with a quick refresher of the traditional well-known Pythagoras' Theorem.
Pythagoras' Theorem says that, in a right angled triangle:the square of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b).
a2 + b2 = c2
You can learn more about Pythagoras' Theorem and review its algebraic proof.
Pythagoras' Theorem in 3D
The world we live in has three dimensions, so what would happen if we consider the Pythagorean Theorem in 3D?
Well, the Theorem still holds, and we would have something like this:
The square of the distance c from the bottom-most left front corner to the top-most right back corner of this cuboid whose sides are x, y and z, is:
c2 = x2 + y2 + z2
And this is part of a pattern that extends onwards into any number of dimensions. For the n-th dimension, we have:
c2 = a12 + a22 + ... + an2
So we can generalize Pythagoras' Theorem, going from 2D to 3D and up until any number of dimensions.
Law of Cosines
What if the triangle does not have a right angle?
For any triangle:
a, b and c are sides. C is the angle opposite to side c The Law of Cosines (also called the Cosine Rule) says:
c2 = a2 + b2 − 2ab cos(C)
It has a2, b2 and c2, and an extra term: 2ab cos(C)
Learn how to use it and find out more at Law of Cosines!
Pythagoras' Theorem and Areas
Do they need to be 2D squares on the triangle's sides?
What about semicircles?
Read more at Pythagoras' Theorem and Areas.
Higher Exponents?
Finally, another type of generalization is to try higher exponents:
an + bn = cn n>2
This is a fascinating area of research
Example: n=3 and a, b and c whole numbers
Can we find values of a, b and c that make this true?
a3 + b3 = c3
In geometry this is the same as asking:
Using only integer sides, can a cube have the same volume as two smaller cubes?
Can we? Your Turn! To answer this, start with Diophantine Equations and Fermat's Last Theorem.
Pythagorean Theorem Pythagorean Theorem Algebraic Proof Pythagorean Theorem in 3D Geometry Index
Copyright © 2024 Rod Pierce |
190422 | https://math.stackexchange.com/questions/86963/finding-interval-where-inequality-holds?rq=1 | calculus - finding interval where inequality holds - Mathematics Stack Exchange
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finding interval where inequality holds
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For which x x the following inequality is true:
(1−x)n≤1−n x 2,(1−x)n≤1−n x 2,
where n n is natural number?
This is inequality opposite to Bernoulli inequality. For sure x x should be between zero and 1 1. x=0 x=0 is fine. But what is the upper bound for x x?
calculus
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edited Nov 30, 2011 at 5:15
user17762
asked Nov 30, 2011 at 4:58
DavidDavid
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This is what I can get so fare: (1-x)^n<=e^{-nx}<=1-\frac{nx}{2}. Let y=e^{-nx}+\frac{nx}{2}-1<=0. y'=-ne^{-nx}+\frac n2=0 if x=\frac{ln 2}{n}. So, x \in [0, \frac{ln 2}{n}].David –David 2011-11-30 05:11:09 +00:00 Commented Nov 30, 2011 at 5:11
Why should x x be less than 1? The lhs will go to −∞−∞ as x x goes to ∞∞ much faster than the rhs. Thus, for a high enough value of x x the inequality will hold.tards –tards 2011-11-30 05:13:09 +00:00 Commented Nov 30, 2011 at 5:13
Yes. But I wanted to see what happends with x arround zero.David –David 2011-11-30 05:16:26 +00:00 Commented Nov 30, 2011 at 5:16
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For n=1 n=1 it is true for all x≥0 x≥0. If you are interested in x x close to 0 0, you might think to ignore the terms in x 3 x 3 and higher, getting 1−n x+n(n−1)2 x 2≤1−n x 2 1−n x+n(n−1)2 x 2≤1−n x 2 or x<2 n−1 x<2 n−1, but that is not so close. For n=4 n=4 the solution is about x<0.456311 x<0.456311, for n=6 n=6, x<0.291 x<0.291. Alpha makes quick work of these.
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answered Nov 30, 2011 at 5:51
Ross MillikanRoss Millikan
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190423 | https://www.tutorchase.com/answers/ib/biology/what-is-the-importance-of-the-hydrogen-bonds-in-dna-structure | Revision Platform
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What is the importance of the hydrogen bonds in DNA structure?
Hydrogen bonds in DNA structure are crucial for maintaining the double helix structure and enabling DNA replication.
Hydrogen bonds are the forces of attraction that hold the two strands of DNA together, forming the iconic double helix structure. Each DNA molecule consists of two strands, each made up of a sequence of nucleotides. These nucleotides are composed of a sugar, a phosphate group, and a nitrogenous base. The nitrogenous bases are adenine (A), thymine (T), cytosine (C), and guanine (G). Adenine always pairs with thymine, and cytosine always pairs with guanine. This is known as complementary base pairing.
The complementary base pairs are held together by hydrogen bonds. Adenine and thymine are connected by two hydrogen bonds, while cytosine and guanine are connected by three. These bonds are weak enough to allow the DNA strands to separate during DNA replication, but strong enough to hold the strands together under normal conditions. This balance is crucial for the stability of the DNA molecule.
Moreover, the hydrogen bonds contribute to the specificity of base pairing. The unique shape and chemical properties of each base allow it to form hydrogen bonds only with its complementary base. This ensures that the genetic information is accurately copied during DNA replication. Any errors in base pairing can lead to mutations, which can have serious consequences for the organism.
In addition, the hydrogen bonds help to protect the genetic information. The double helix structure, with the base pairs held together by hydrogen bonds in the centre, shields the genetic information from chemical reactions that could alter it. This is important for maintaining the integrity of the genetic code.
IB Biology Tutor Summary: Hydrogen bonds play a key role in DNA's structure, holding the two strands together in a double helix. They enable the specific pairing of bases (A with T, C with G), crucial for DNA replication and protecting genetic information. These bonds strike a balance, being strong enough to maintain DNA's structure yet weak enough to allow replication, ensuring the accuracy and stability of our genetic code.
Answered by Raymond - University of Hong Kong: PhD in Molecular Biology
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190424 | https://radiologykey.com/chorionicity-of-multiple-gestations/ | Chorionicity of Multiple Gestations | Radiology Key
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Chorionicity of Multiple Gestations
Abstract
Chorionicity refers to the number of placentas in a multiple gestation; amnionicity refers to the number of amniotic cavities. Prenatal determination of chorionicity and amnionicity is essential in the clinical management of multiple gestations. In the first trimester, chorionicity can be determined by counting the number of gestational sacs, and the number of yolk sacs can be used to predict amnionicity. In the late first trimester and early second trimester, systematic evaluation of placental number, fetal gender, and insertion of the intertwin membrane into the placenta allows accurate prenatal diagnosis of chorionicity. In dichorionic pregnancies, the intertwin membrane remains thick with a triangular projection of placenta known as the twin peak or lambda sign, which is visible between the layers of the dividing membrane. In monochorionic pregnancies, the intertwin membrane inserts directly into the placenta forming a characteristic “T” sign.
Keywords
chorionicity, twin, multiple gestation
Introduction
In 2014, the twin birth rate in the United States rose to an all-time high of 33.9 per 1000 live births. Multiple gestations result from either the fertilization of multiple ova or the division of a single fertilized ovum into more than one fetus. The terms monozygotic and dizygotic refer to the number of ova leading to a multifetal gestation. A monozygotic pregnancy results from a single fertilized ovum that divides into more than one fetus. In contrast, dizygotic pregnancies originate from the fertilization of two separate ova and are genetically dissimilar.
In contrast to zygosity, which refers to the genetic constitution of a twin pregnancy, the terms chorionicity and amnionicity describe the placentation and membrane composition of a pregnancy. More so than zygosity, the determination of chorionicity and amnionicity is essential in the clinical management of multiple gestations because monochorionic multiples, whether diamniotic or monoamniotic, are at increased risk of adverse outcomes.
Normal Anatomy
General Anatomic Descriptions
Chorionicity refers to the number of placentas in a multiple gestation; amnionicity refers to the number of amniotic cavities. The term dichorionic refers to a multiple gestation with two distinct placental disks (or two chorions), whereas the term monochorionic refers to a pregnancy with a single placental disk (or one chorion). Similarly, a pregnancy with two distinct amniotic cavities is described as diamniotic, and a pregnancy with a single amniotic cavity is monoamniotic. Depending on the number of chorions and amnions, twin pregnancies can be described as dichorionic diamniotic, monochorionic diamniotic, or monochorionic monoamniotic.
Detailed Description of Specific Areas
Normal Variants
Monochorionic twins occur spontaneously in 0.4% of the general population. Studies have reported that the frequency of monozygotic twinning may be more than 10 times higher in pregnancies after fertility treatment.
In contrast, the frequency of dizygotic twins varies by maternal age, parity, family history, maternal weight, nutritional state, and race. Some of these risk factors lead to an increased frequency of dizygotic twinning through increases in serum concentrations of follicle-stimulating hormone and luteinizing hormone, which increase the likelihood of multiple ovulations in a single menstrual cycle. Because gonadotropin levels are increased with advancing maternal age and use of infertility medications, both delayed childbearing and assisted reproductive technologies are associated with an increased frequency of dizygotic twinning. In addition, the incidence of spontaneous dizygotic twins varies by race, with highest rates among certain populations in Africa and relatively lower rates in whites and Asians. Other factors associated with an increased likelihood of dizygotic twinning include maternal family history of twin gestations and increasing maternal height and weight.
Dizygotic pregnancies are almost always dichorionic diamniotic, with each fetus having its own placenta and amniotic cavity. In contrast, chorionicity of monozygotic gestations is determined by the time at which division of the fertilized ovum occurs, and monozygotic gestations can be dichorionic diamniotic, monochorionic diamniotic, or monochorionic monoamniotic. If twinning occurs during the first 2 to 3 days, it precedes the separation of cells that eventually become the chorion and results in a monozygotic dichorionic diamniotic pregnancy. After approximately 3 days, twinning cannot split the chorionic cavity, and from that time forward, a monochorionic placenta results. If the split occurs between the third and eighth days, a monochorionic diamniotic pregnancy develops. Between the eighth and twelfth days, the amnion has already formed, and the pregnancy is monochorionic monoamniotic if twinning occurs. Embryonic cleavage between the 13th and 15th days results in conjoined twins within a single amnion and chorion; beyond that point, twinning does not occur. Among monozygotic twin gestations, approximately one-third are dichorionic diamniotic, whereas almost two-thirds are monochorionic diamniotic, and less than 1% are monochorionic monoamniotic.
By definition, essentially all monochorionic twins are monozygotic. In contrast, among spontaneously conceived same-sex dichorionic twins, approximately 18% are monozygotic. There have been several case reports of twin pregnancies conceived using assisted reproductive technology that resulted in dizygotic monochorionic gestations ; the mechanism of this phenomenon is not yet understood.
Differential Considerations
Compared with singletons, multiples face an increased risk of preterm delivery, growth disorders, and maternal complications. In addition to these general risks, multiples also face increased risk depending on the zygosity, chorionicity, and amnionicity of the pregnancy.
Monochorionic twins have a worse prognosis than dichorionic twins because of numerous complications unique to the twinning process and to monochorionic placentation. Monozygotic twins, whether monochorionic or dichorionic, have a significantly higher incidence of congenital anomalies than singletons or dizygotic twins. A primary concern in twin gestations with monochorionic placentation is twin transfusion syndrome, characterized by an unequal distribution of the blood flow across the shared placenta of two fetuses. Although all monochorionic twins share a portion of their vasculature, only approximately 15% to 20% develop this condition. Untreated, twin transfusion syndrome is associated with a 60% to 100% mortality rate for both twins.
There is an increased risk of fetal loss with monochorionic twins across all gestational ages and an increased risk of complications secondary to fetal demise. Specifically, owing to vascular anastomoses within the monochorionic placenta, hemodynamic changes associated with the death of one fetus result in an approximately 20% risk of multicystic encephalomalacia in the surviving twin, and an increased risk of preterm delivery. The literature has suggested that the risk of intrauterine death of one or both twins is higher in monochorionic than dichorionic pregnancies. In a study of 1000 consecutive twin pairs, monochorionic diamniotic twins had a higher risk of stillbirth compared with dichorionic diamniotic twins overall and at each gestational age after 24 weeks ; this increased risk of fetal loss persisted in “apparently normal” monochorionic diamniotic twins unaffected by growth abnormalities, congenital anomalies, or twin transfusion syndrome.
Other unique but rare problems that occur in monochorionic pregnancies include cord entanglement in monoamniotic twins, conjoined twins, and twin reversed arterial perfusion sequence, also known as acardiac twinning( Chapter 163). Monoamniotic gestations are associated with increased perinatal mortality secondary to cord entanglement. Previous studies reported a fetal mortality rate of greater than 50%, but more recent studies indicate a perinatal mortality rate ranging from 10% to 21%.
Pertinent Imaging Considerations
Given the impact that chorionicity and amnionicity have on pregnancy outcome, determination of placentation and membrane composition is vital to guide prenatal care of a twin pregnancy. Although chorionicity and amnionicity can be definitively determined postnatally by gross and histologic evaluation of the placenta and fetal membranes, prenatal diagnosis is preferable to allow appropriate prenatal evaluation and intervention. Prenatal determination of chorionicity and amnionicity is possible using ultrasound (US) ( Table 158.1).
TABLE 158.1
ULTRASOUND CHARACTERISTICS OF CHORIONICITY AND AMNIONICITY
| | EARLY FIRST TRIMESTER | LATER GESTATION |
:---:
| Gestational Sacs | Intertwin Membrane | Yolk Sacs | Gender | Placentas | Intertwin Membrane | Membrane Insertion | Membrane Layers |
| Dichorionic-diamniotic | 2 | Thick | 2 | Discordant or concordant | 2 | Thick | Twin peak or lambda sign | 4 |
| Monochorionic-diamniotic | 1 | Thin | 2 | Concordant | 1 | Thin | T sign | 2 |
| Monochorionic-monoamniotic | 1 | None | 1 | Concordant | 1 | None | NA | NA |
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Oxidation Number v.s. Formal Charge – MCAT Content
Table of Contents Learning
8 minutes remaining - you got this!
I. Introduction to Oxidation Number and Formal Charge
Oxidation Number
Formal Charge
II. Calculating Oxidation Numbers and Formal Charges
Oxidation Number Calculation Rules
Formal Charge Calculation
III. Importance of Oxidation Number and Formal Charge in Reactions
IV. Comparing Oxidation Number and Formal Charge
V. Practical Applications
VI. Bridge/Overlap
A. Electron Configurations
B. Chemical Bonding
C. Reaction Mechanisms
VII. Wrap-Up and Key Terms
Key Terms
VIII. Practice Questions
Sample Practice Question 1
Sample Practice Question 2
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Understanding the concepts of oxidation number and formal charge is essential in general chemistry. These concepts help predict how molecules interact and react. They also help us understand the behavior of atoms in molecules during chemical reactions.
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I. Introduction to Oxidation Number and Formal Charge
Oxidation Number
An oxidation number (or oxidation state) represents the total number of electrons an atom gains or loses to form a chemical bond. It helps identify the electron density around an atom in a molecule. Knowing the oxidation number is important because it tells us about the atom's electron control in a molecule.
Formal Charge
The formal charge is the charge an atom would have if all the electrons in a covalent bond were shared equally between the bonded atoms. It helps to determine the most stable Lewis structure for a molecule. By knowing the formal charge, we can predict a molecule's most likely arrangement of atoms.
II. Calculating Oxidation Numbers and Formal Charges
To fully understand oxidation numbers and formal charges, it's important to know how to calculate them accurately. This section explains the rules and provides examples.
Oxidation Number Calculation Rules
Elemental Form: The oxidation number of an atom in its elemental form is always 0. For example, O₂ or N₂.
Monatomic Ions: The oxidation number is equal to the charge of the ion. For instance, Na⁺ has an oxidation number of +1.
Oxygen: Usually has an oxidation number of -2, except in peroxides (like H₂O₂) where it is -1.
Hydrogen: Generally has an oxidation number of +1 when bonded to non-metals and -1 when bonded to metals.
Fluorine: Always has an oxidation number of -1 in its compounds.
Sum Rule: The sum of oxidation numbers in a neutral molecule is 0; in an ion, it equals the ion's charge.
Example:Calculate the oxidation number of sulfur in H₂SO₄.
Hydrogen (H) =+1, so 2 H = +2
Oxygen (O) =-2, so 4 O = -8
Let sulfur be x:
The equation:
2(+1) + x + 4(-2) = 0
Solving:
2 + x - 8 = 0
x = +6
Formal Charge Calculation
Formal Charge = Valence electrons - ½ (Bonding electrons) — Non-bonding electrons
Example: Determine the formal charge of oxygen in H₂O.
Valence electrons of O = 6
Bonding electrons (2 bonds with H) = 4
Non-bonding electrons (2 lone pairs) = 4
Formal charge:
6-4/2-4 = 0
III. Importance of Oxidation Number and Formal Charge in Reactions
Understanding how oxidation numbers and formal charges work is crucial in predicting the outcome of chemical reactions. This section explores their roles in different types of reactions.
Oxidation Numbers in Redox Reactions: Oxidation numbers help identify which species are oxidized and reduced in redox reactions. Oxidation involves an increase in oxidation number, while reduction involves a decrease.
Example:Consider the reaction:
2Mg (s) + O 2 (g) → 2MgO (s)
Mg goes from 0 (elemental) to +2 (MgO).
O₂ goes from 0 (elemental) to -2 (MgO).
Mg is oxidized, O₂ is reduced.
Formal Charges in Lewis Structures: Formal charges help determine molecules' most stable Lewis structure. This is crucial for predicting molecular geometry and reactivity.
Example:Compare possible Lewis structures for CO₂.
Structure 1: O=C=O, where each O is 0 and C is 0.
Structure 2: O≡C-O, where the left O is -1, C is +1, and the right O is 0.
The first structure is more stable as the formal charges are minimized.
IV. Comparing Oxidation Number and Formal Charge
While oxidation numbers and formal charges serve different purposes, they often lead to similar conclusions about the electron distribution in molecules. However, oxidation numbers are used mainly in redox chemistry, while formal charges are used for drawing and evaluating Lewis structures.
Example:In sulfate ion (SO₄²⁻):
The oxidation number of sulfur is +6.
Formal charges: Each oxygen typically has a formal charge of -1, and sulfur has +2.
V. Practical Applications
Oxidation numbers and formal charges are not just theoretical concepts. They have practical applications in various fields of chemistry, and this section highlights some of these applications.
Oxidation Numbers in Industrial Chemistry: Used in processes like refining metals and manufacturing chemicals. For example, understanding oxidation states is essential for efficient conversion in the production of sulfuric acid.
Formal Charges in Organic Synthesis: Crucial for predicting organic compounds' most likely structures and reactivity. For example, minimizing formal charges in designing pharmaceuticals helps ensure stability and effectiveness.
VI. Bridge/Overlap
Understanding oxidation numbers and formal charges is fundamental for grasping more complex topics in chemistry. These include:
A. Electron Configurations
Knowledge of how electrons are distributed in atoms and molecules helps predict oxidation states and formal charges.
B. Chemical Bonding
Understanding these concepts aids in learning about covalent, ionic, and metallic bonds, as well as molecular geometry.
C. Reaction Mechanisms
Helps in understanding how and why chemical reactions occur, which is essential for fields like biochemistry and pharmacology. For example, in nucleophilic substitution and electrophilic addition reactions.
VII. Wrap-Up and Key Terms
Oxidation numbers help us track electron transfer in reactions, which is key for redox processes. Formal charges help us determine the most stable structures of molecules. Both concepts are vital for understanding chemical reactivity and stability. They are essential tools for anyone studying or working in chemistry.
Key Terms
Oxidation Number: Indicates electron loss or gain.
Formal Charge: Indicates electron sharing in bonds.
Redox Reactions: Involve changes in oxidation states.
Lewis Structures: Help visualize molecular structure and stability.
VIII. Practice Questions
Sample Practice Question 1
What is the oxidation number of nitrogen in NO₃⁻?
A. +3
B. +4
C. +5
D. +6
Click to reveal answer
Ans. C
The total oxidation state of the nitrate ion (NO₃⁻) must equal -1. Oxygen has an oxidation number of -2. With three oxygens, the total is -6. Nitrogen must have an oxidation number of +5 (-6 + 5 = -1) to balance this.
Sample Practice Question 2
Calculate the formal charge of chlorine in ClO₄⁻.
A. -1
B. 0
C. +1
D. +3
Click to reveal answer
Ans. D
Chlorine has 7 valence electrons. In ClO₄⁻, it shares electrons with four oxygens (8 bonding electrons), and there are no lone pairs in chlorine. Formal charge: 7−8/2−0=+3.
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190426 | https://dspace.mit.edu/bitstream/handle/1721.1/72670/Shavit_Lock%20cohorting.pdf;sequence=1 | MIT Open Access Articles
Lock cohorting: A general technique for designing NUMA locks
The MIT Faculty has made this article openly available. Please share
how this access benefits you. Your story matters.
Citation: David Dice, Virendra J. Marathe, and Nir Shavit. 2012. Lock cohorting: a general
technique for designing NUMA locks. In Proceedings of the 17th ACM SIGPLAN symposium on
Principles and Practice of Parallel Programming (PPoPP '12). ACM, New York, NY, USA, 247-256.
As Published:
Publisher: Association for Computing Machinery (ACM)
Persistent URL:
Version: Author's final manuscript: final author's manuscript post peer review, without
publisher's formatting or copy editing
Terms of use: Creative Commons Attribution-Noncommercial-Share Alike 3.0 Lock Cohorting: A General Technique for Designing NUMA Locks
David Dice, Virendra J. Marathe
Oracle Labs
dave.dice@oracle.com, virendra.marathe@oracle.com
Nir Shavit
MIT and Tel-Aviv University
shanir@csail.mit.edu
Abstract
Multicore machines are quickly shifting to NUMA and CC-NUMA architectures, making scalable NUMA-aware locking algorithms, ones that take into account the machines’ non-uniform memory and caching hierarchy, ever more important. This paper presents lock cohorting , a general new technique for designing NUMA-aware locks that is as simple as it is powerful. Lock cohorting allows one to transform any spin-lock algo-rithm, with minimal non-intrusive changes, into scalable NUMA-aware spin-locks. Our new cohorting technique allows us to easily create NUMA-aware versions of the TATAS-Backoff, CLH, MCS, and ticket locks, to name a few. Moreover, it allows us to derive a CLH-based cohort abortable lock, the first NUMA-aware queue lock to support abortability. We empirically compared the performance of cohort locks with prior NUMA-aware and classic NUMA-oblivious locks on a syn-thetic micro-benchmark, a real world key-value store application
memcached , as well as the libc memory allocator. Our results demonstrate that cohort locks perform as well or better than known locks when the load is low and significantly out-perform them as the load increases.
Categories and Subject Descriptors D.1.3 [ Programming Tech-niques ]: Concurrent Programming
General Terms Algorithms, Design, Performance
Keywords NUMA, hierarchical locks, spin locks
1. Introduction
In coming years, as multicore machines grow in size, one can ex-pect an accelerated shift towards distributed non-uniform memory-access (NUMA) and cache-coherent NUMA (CC-NUMA) archi-tectures. 1 Such architectures, examples of which include Intel’s multi-socket Nehalem-based systems and Oracle’s 4-socket 256-way Niagara-based systems, consist of collections of computing cores with fast local memory (e.g. caches shared by cores on a sin-gle multicore chip), communicating with each other via a slower inter-chip communication medium. Access by a core to the local memory, and in particular to a shared local cache, can be several times faster than access to the remote memory or cache lines resi-dent on another chip .
1
We use the term NUMA broadly, noting that it includes Non-Uniform Communication Architecture (NUCA) machines as well.
Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. PPoPP’12, February 25–29, 2012, New Orleans, Lousiana, USA. Copyright c©2012 ACM [to be supplied]. . . $10.00
Dice , and Radovi´ c and Hagersten were the first to iden-tify the benefits of designing locks that improve locality of ref-erence on CC-NUMA architectures by developing NUMA-aware locks: general-purpose mutual-exclusion locks that encourage thr-eads with high mutual cache locality to acquire the lock consecu-tively, thus reducing the overall level of cache coherence misses when executing instructions in the critical section. Specifically, these designs attempt to minimize lock migration . We say a lock
L migrates if two threads running on a different NUMA clusters (nodes) acquire L one after the other. Radovi´ c and Hagersten introduced the hierarchical backoff lock
(HBO): a test-and-test-and-set lock augmented with a new backoff scheme to reduce cross-interconnect contention on the lock vari-able. Their hierarchical backoff mechanism allows the backoff de-lay to be tuned dynamically, so that when a thread notices that an-other thread from its own local cluster owns the lock, it can re-duce its delay and increase its chances of acquiring the lock next. This algorithm’s simplicity makes it quite practical. However, be-cause the locks are test-and-test-and-set locks, they incur invalida-tion traffic on every modification of the shared global lock variable, which is especially costly on NUMA machines. The issue of fair-ness that arises because threads backoff with different delays can be addressed, but requires more tuning parameters, which invari-ably makes the lock’s performance highly unreliable. Luchangco et al. overcame these drawbacks by introduc-ing HCLH, a hierarchical version of the CLH queue-lock . The HCLH algorithm collects requests on each cluster into a local CLH-style queue, and then has the thread at the head of the queue inte-grate each cluster’s queue into a single global queue. This avoids the overhead of spinning on a shared location and eliminates fair-ness and starvation issues. The algorithm’s drawback is that it forms the local queues of waiting threads by having each thread per-form an atomic register-to-memory-swap (SWAP) operation 2 on the shared head of the local queue, which becomes a contention bottleneck, implying that the thread merging the local queue into the global one must either wait for a long period (10s of microsec-onds) or globally merge an unacceptably short local queue. More recently, Dice et al. showed that one could overcome the synchronization overhead of HCLH locks by collecting local queues using the flat-combining technique of Hendler et al. , and then splicing them into the global queue. The resulting NUMA-aware FC-MCS lock outperforms previous locks by at least a factor of 2, but uses significantly more memory and is relatively compli-cated. In summary, the HBO lock has the benefit of being simple, but is unfair, and requires significant application and platform dependent tuning. Both HCLH and FC-MCS are fair and deliver much better performance, but are rather complex, and it is therefore question-able if they will be of general practical use.
2
On some architectures the SWAP operation is emulated using a compare-and-swap instruction loop. This paper presents lock cohorting , a new general technique for turning practically any kind of spin-lock or spin-then-block lock into a NUMA-aware lock that allows sequences of threads – local to a given node/cluster – to execute consecutively with little overhead and requiring very little tuning beyond the locks used to create the cohort lock. Apart from providing a new set of high performance NUMA-aware locks, the important benefit of lock cohorting is that it is a general transformation, not simply another NUMA-aware locking algorithm. This provides an important software engineering advan-tage: programmers do not have to adopt new and unfamiliar locks into their system. Instead, they can apply the lock cohorting trans-formation to their existing locks. This will hopefully allow them to enhance the performance of their locks (by improving locality of reference, enabled by the NUMA-awareness property of cohort locks), while preserving many of the original properties of what-ever locks their application uses.
1.1 Lock Cohorting in a Nutshell
Say we have a spin-lock of type G that is thread-oblivious , that is, allows the acquiring thread to differ from the releasing thread, and another spin-lock of type S that has the cohort detection property :a thread releasing the lock can detect if it has a non-empty cohort of threads concurrently attempting to acquire the lock. We convert a collection of locks S and G into a single NUMA-aware lock by having a single thread-oblivious global lock G and by associating each NUMA cluster i with a distinct local lock Si
that has the cohort detection property. We say a cohort lock is locked if and only if its global lock G is locked. Locks S and G
can be of different types. For example, S could be an MCS queue-lock and G a simple test-and-test-and-set backoff lock (BO) as depicted in Figure 1. To access the critical section a thread must hold both the local lock Si of its cluster, and the global lock G.However, the trick is that given the special properties of S and G,once some thread in a cluster acquires G, ownership of the cohort lock can be passed in a deadlock-free manner from one thread in cluster to the next using the local lock Si, without releasing the global lock. To maintain fairness, the global lock G is at some point released by some thread in the cohort (not necessarily the one that acquired it), allowing a cohort of threads from another cluster Sj to take ownership of the lock. In more detail, each thread attempting to enter the lock’s critical section first acquires its local lock Si, and based on the state of the local lock, decides if it can immediately enter the critical section or must compete for G. A thread T leaving the critical section first checks if it has a non-empty cohort (some local thread is waiting on Si). If so, it will release Si without releasing G, having set the state of Si to indicate that this is a local release. On the other hand, if its local cohort is empty, T will release G and then release Si, setting Si’s state to indicate that the global lock has been released. This indicates to the next local thread that acquires Si that it must re-acquire G before it can enter the critical section. The cohort detection property is therefore necessary in order to prevent a deadlock situation in which a thread leaves the local lock, without releasing the global lock, when there is no subsequent thread in the cohort, so the global lock may never be released. The cohort lock’s overall fairness is easily controlled by decid-ing when a cluster gives up the global lock. A simple cluster-local policy is to give up the local lock after an allowed number consec-utive local accesses. We note that a cohort lock constructed from unfair underlying locks will itself be unfair, but if the underlying locks are fair then the fairness of a cohort lock is determined by the policy that decides when a cohort releases the global lock. If a cohort retains ownership of the global lock for extended periods then throughput may be improved but at a cost in fairness. The benefit of the lock cohorting approach is that sequences of local threads accessing the lock are formed at a very low cost. Once a thread in a cluster has acquired the global lock, control of the global lock is subsequently passed among contending threads within the cluster – the cohort – with the efficiency of a local lock. In other words, the common path to entering the critical section is the same as a local version of the lock of type S with fairness, as we said, easily controlled by limiting the number of consecu-tive local lock transfers allowed. This contrasts sharply with the complex coordination mechanisms that create such sequences in the previous top performing HCLH and FC-MCS locks, and the platform-dependent, load-dependent, and application-dependent performance tuning required for the HBO lock.
1.2 Cohort Lock Designs and Performance
It is easy to find efficient locks that are thread-oblivious: the BO or ticket locks have this property, and since the global lock is not expected to be highly contended, they can easily serve as the global locks. With respect to the cohort detection property, there are locks such as the MCS queue lock of Mellor-Crummey and Scott that provide cohort detection by design: each spinning thread’s record in the queue has a pointer installed by its successor. There are however locks, for example, BO locks, that require us to in-troduce an explicit cohort detection mechanism to allow releasing threads to determine if other cohort threads are attempting to ac-quire the lock. More work is needed when the lock algorithms are required to be abortable. In an abortable lock, simply detecting that there is a successor is not enough to allow a thread to release the local lock but not the global lock. One must make sure there is a viable successor, that is, one that will not abort after the thread releases the local lock, as this might leave the global lock deadlocked. As we show, one can convert the BO lock (which is abortable by design) and the abortable CLH lock into abortable (NUMA-aware) cohort locks, which to our knowledge, are the first set of NUMA-aware abortable queue-locks. We tested our new lock cohorting transformation on an Ora-cle SPARC Enterprise T5440 T M Server, a 256-way 4-socket mul-ticore machine. Our tests show several variations of cohort NUMA-aware locks that outperform all prior algorithms, and in some situ-ations are over 60% more scalable than FC-MCS, the most scalable NUMA-aware lock in the literature. Furthermore, unlike FC-MCS, we found that cohort lock designs are simple to implement and re-quire significantly lower space than FC-MCS. Our novel abortable NUMA-aware lock, the first of its kind, outperforms the HBO lock (abortable by definition) and the abortable CLH lock by about a factor of 6. Our experiments with memcached demonstrate that in some configuration settings cohort locks can improve the application’s performance by over 25%, without degrading perfor-mance on all other configurations. Finally, our libc allocator ex-periments demonstrate how cohort locks can directly benefit multi-threaded programs and significantly boost their cluster-level ref-erence locality both for accesses by the allocator to allocation metadata and for accesses by the application to allocated memory blocks. In experiments conducted on a memory allocator stress test benchmark , cohort locks allow the benchmark to scale up to nearly a factor of 6X, while all other reported locks provided a scalability gain restricted to about 50%. We describe our construction in detail in Section 2, both our general approach and seven specific example lock transformations. We provide an experimental evaluation in Section 4.
2. The Lock Cohorting Transformation
In this section, we describe our new lock cohorting transformation in detail. Assume that the system is organized into clusters (nodes) Cluster 1 MCS tail 3
Global BO Lock
Threads 1B and 1C add themselves to the local MCS queue by swapping the tail pointer. 1C hasn’t directed the pointer of 1B to its node yet.
11A acquires local MCS lock and proceeds to acquire the global lock
41A wishes to leave, sees that it points to successor 1B, so it releases lock by setting 1B’s node state to enter and returns
Cluster 2 MCS tail
ls global local busy busy
Thread 1C
myNode
Thread 1B
myNode
Thread 1A
myNode
Thread 2C
myNode
Thread 2B
myNode
Thread 2A
myNode
22A acquires local lock, sees tail is null, so it spins on global lock held by 1A
5
busy global
2A will acquire global lock upon release by 1A. Then passes control down to 2B, 2C, etc
Figure 1. A NUMA-aware C-BO-MCS lock for two clusters. A thread spins if its node state is busy , and can enter the critical section if the state is local release . A thread attempts to take the global lock if it sees the state set to global release or if it is added as the first in the queue (setting a null tail pointer to its own record). of computing cores, each of which has a large cache that is shared among the cores local to that cluster, so that inter-cluster commu-nication is significantly more expensive than intra-cluster commu-nication. We use the term cluster to capture the collection of cores, and to make clear that they could be cores on a single multicore chip, or cores on a collection of multicore chips (nodes) that have proximity to the same memory or caching structure; it all depends on the size of the NUMA machine at hand. We will also assume that each cluster has a unique cluster id known to all threads running on the cluster.
2.1 Designing a Cohort Lock
We describe lock cohorting in the context of spin-locks, although it could be as easily applied to blocking-locks. We assume the standard model of shared memory based on execution histories . A lock is an object providing mutual exclusion with lock and
unlock methods, implemented in shared memory, and having the usual safety and liveness properties (see ). At a minimum we will require that the locks considered here will provide mutual ex-clusion and be deadlock-free. In addition, we define the following properties:
Definition A lock x is thread-oblivious , if in a given execution history, for a lock method call of x by a given thread, it allows the matching unlock method call (the next unlock of x that follows in the execution history) to be executed by a different thread.
Definition A lock x provides cohort detection if one can add a new predicate method alone? to x so that in any execution history, if there is no other thread concurrently executing a lock method on
x, alone? will return true .Note that we follow the custom of not using linearizability as a correctness condition when defining our lock implementations. In particular, our definition of alone? refers to the behavior of concurrent lock method calls and says if rather than iff so as to allow false-positives: there might be a thread executing the lock operation and alone? (not noticing it) could still return true. False-positives are a performance concern but do not affect correctness. False-negatives, however, could result in loss of progress. This weaker definition is intended to allow for very relaxed and efficient implementations of alone? .We construct a NUMA-aware cohort lock by having each clus-ter i on the NUMA machine have a local instance Si of a lock that has the cohort detection property, and have an additional shared thread-oblivious global lock G. Locks Si, i ∈ { 1 . . . n } (where n
is the number of clusters in the NUMA system), and G can be of different types, for example, the Si could be slight modifications of MCS queue-locks and G a simple test-and-test-and-set back-off lock (BO) as depicted in Figure 1. The lock method of a thread in cluster i in a cohort lock operates as follows. The state of the lock Si is modified so that it has a different detectable state indicating if it has a local release or a
global release .1. Call lock on Si. If upon acquiring the lock the lock method detects that the state is:
• A local release : proceed to enter the critical section.
• A global release : proceed to call the lock method of the global lock G. Once G is acquired, enter the critical section. We define a special may-pass-local predicate on the local lock
Si and the global lock G. The may-pass-local predicate indicates if the lock state is such that the global lock should be released. This predicate could, for example, be based on how long the global lock has been continuously held on one cluster or on a count of the number of times the local lock was acquired in succession in a local release state. It defines a tradeoff between fairness and performance, as typically the shorter successive access time may-pass-local grants to a given cohort, the more it loses the benefit of locality of reference in accessing the critical section. Given this added may-pass-local predicate, the unlock method of a thread in cluster i in a cohort lock operates as follows. 1. Call the alone? method and may-pass-local on Si.
• If both return false : call the unlock method of Si, setting the release state to local release . The next owner of Si can directly enter the critical section.
• Otherwise: call the unlock method of the global lock G.Once G is released, call the unlock method of Si, setting the release state to global release .As can be seen, the state of the lock upon release indicates to the next local thread that acquires Si if it must acquire G or not, and allows a chain of local lock acquisitions without the need to access the global lock. The immediate benefit is that sequences of local threads accessing the lock are formed at a very low cost: once a thread in a cluster has acquired the global lock, ownership is passed among the cluster’s threads with the efficiency of a local lock. This reduces overall cross-cluster communication and increases intra-cluster locality of reference when accessing data within the critical section.
3. Cohort Lock Designs
Though most locks can be used in the cohort locking transfor-mation, we briefly explain six specific constructions here: The first four are non-abortable (do not support timeouts ) locks and the last two are abortable (timeout capable) locks. Of the non-abortable locks, we first present a simple test-and-test-and-set backoff lock (which we will refer to as the BO lock) based co-hort lock that employs a BO lock globally and local BO locks per NUMA cluster. We refer to this lock as the C-BO-BO lock. The second lock is a similar combination of ticket locks , which we call the C-TKT-TKT lock. The third is a combination of a global BO lock, and local MCS locks per NUMA cluster. The last non-abortable lock contains MCS locks both globally and locally. For the abortable locks, we first present an abortable variant of the C-BO-BO lock, which we call the A-C-BO-BO lock, and then we present an abortable cohort lock comprising of an abortable global BO lock and abortable local CLH locks , which we call the A-C-BO-CLH lock.
3.1 The C-BO-BO Lock
In the C-BO-BO lock, the local and global locks are both simple BO locks. The BO lock is trivially thread-oblivious. However, we need to augment the local BO lock to enable cohort detection by exposing the alone? method. Specifically, to implement the alone?
method we need to add an indicator to the local BO lock that a successor exists. To that end we add to the lock a new successor-exists boolean field. This field is initially false , and is set to true by a thread immediately before it attempts to CAS the test-and-test-and-set lock state. Once a thread succeeds in the CAS and acquires the local lock, it writes false to the successor-exists field, effectively resetting it. The alone? method will check the successor-exists
field, and if it is true , a successor must exist since it was set after the reset by the local lock winner. Alone? returns the logical complement of successor-exists .The lock releaser uses the alone? method to determine if it can correctly release the local lock in local release state. If it does so, the following lock owner of the local lock implicitly inherits ownership of the global BO lock. Otherwise, the local lock is in the global release state, in which case, the new local lock owner must acquire the global lock as well. Notice that it is possible that another successor thread executing lock exists even if the field is
false , simply because the post-acquisition reset of successor-exists
by the local lock winner could have overwritten the successor’s setting of the successor-exists field. This type of incorrect-false result observed in successor-exists is allowed – it will at worst cause an unnecessary release of the global lock, but not affect correctness of the algorithm. However, incorrect-false conditions can result in greater con-tention at the global lock, which we would like to avoid. To that end, a thread that spins on the local lock also checks the successor-exists flag, and sets it back to true if it observes that the flag has been reset (by the current lock owner). This is likely to lead to ex-tra contention on the cache line containing the flag, but most of this contention does not lie in the critical path of the lock acquisition operation. Furthermore, intra-cluster write-sharing typically enjoys low latency, mitigating any ill-effects of contention on cache lines that might be modified by threads on the same cluster. These obser-vations are confirmed in our empirical evaluation.
3.2 The C-TKT-TKT Lock
The C-TKT-TKT lock has the ticket lock as both the local lock, as well as the global lock. A traditional ticket lock consists of two counters: request and grant . A thread intending to acquire the lock first atomically increments the request counter and then spins, waiting for the grant counter to contain the incremented
request value. The lock releaser subsequently releases the lock by incrementing the grant counter. The ticket lock is trivially thread-oblivious; a thread can incre-ment the request and another thread can correspondingly increment the grant counter. Cohort detection is also easy in the ticket lock; all the thread needs to do is determine if the request and grant counters match, and if not, it means that there are more requesters waiting to acquire the lock. In C-TKT-TKT, a thread first acquires the local ticket lock, and then the global ticket lock. To release the C-TKT-TKT lock, the owner first determines if it has any cohorts that may be waiting to acquire the lock. The alone? method is a simple check to see if the
request and grant counters are the same. If not, it means that there are additional requests posted by waiting cohort threads. In that case, the owner informs the next cohort in line that it has inherited the global lock by setting a special top-granted field that residees in the local ticket lock. 3 It then releases the local ticket lock by incrementing the grant counter. If the request and grant counters are the same, the owner releases the global ticket lock and then the local ticket lock (without setting the top-granted field).
3.3 The C-BO-MCS Lock
The design of the C-BO-MCS lock, depicted in Figure 1, is also straightforward. The BO lock is a simple test-and-test-and-set lock with backoff, and is therefore thread-oblivious by definition: any thread can release a lock taken by another. We remind the reader that an MCS lock consists of a list of records, one per thread, ordered by their arrival at the lock’s tail
variable. Each thread adds its record to the lock by performing a swap on a shared tail . It then installs a successor pointer from the record of its predecessor to its record in the lock. The predecessor, upon releasing the lock, will follow the successor pointer and notify the thread of the lock release by writing to a special state field in the successor’s record. The MCS lock can be easily adapted to be the local cohort de-tecting lock as follows. We implement the alone? method by simply checking if a thread’s record has a non-null successor pointer. The release state is augmented so that instead of simple busy and re-leased states, the state field encodes busy , release local or release global . Each thread will initialize its record state to busy unless it encounters a null tail pointer, indicating it has no predecessor, in which case it is in the release global state and will access the global lock. With these modifications, the global BO lock and local modified MCS locks can be plugged into the cohort lock protocol to deliver a NUMA-aware lock.
3.4 The C-MCS-MCS Lock
The C-MCS-MCS lock comprises a global MCS lock and lo-cal MCS locks. The cohort detection mechanism of the local MCS locks is the same as in C-BO-MCS. So the implementation of the local MCS lock remains the same. However, the thread-obliviousness aspect is somewhat more interesting. A key property of MCS is what is called local spinning , where a thread spin-waits on its MCS queue node, and is informed by its predecessor thread that is has become the lock owner. There-after, the thread can enter the critical section, and release the lock by transferring lock ownership to its queue node’s successor. The thread can subsequently do whatever it wants with its MCS queue node; it usually deallocates it. In order to make the global MCS lock thread-oblivious, the thread that enqueues its MCS queue node in the global MCS lock’s queue cannot always get its node back im-mediately after it releases the C-MCS-MCS lock – the node has to be preserved in the MCS queue so as to let another cohort thread re-lease the lock. We enable this feature by using thread-local pools of MCS queue nodes. A thread that posts a request node in the global MCS lock must get a free node from its local pool. On releasing
3The top-granted flag is reset by the thread that observed it set and took possession of the the local ticket lock.
the global lock, the lock releaser can return the node to the original thread’s pool. This circulation of MCS queue nodes can be done very efficiently and does not impact performance of the lock . With this extra modification we achieve a thread-oblivious MCS lock, which can be combined with the local MCS locks that are enabled with cohort detection to deliver the NUMA-aware C-MCS-MCS lock.
3.5 The C-TKT-MCS Lock
The C-TKT-MCS lock combines local MCS queue locks with a global ticket lock. We believe this lock combines the best of C-TKT-TKT and C-MCS-MCS: First, because the global lock is a ticket lock, it does not contain the complexity of circulating queue nodes between threads as in C-MCS-MCS. Second, since the local locks are MCS locks instead of ticket locks, the C-TKT-MCS lock retains their local-spinning property. As we shall see in Section 4, having local MCS locks indeed helps C-TKT-MCS to scale better than C-TKT-TKT.
3.6 Abortable Cohort Locks
The property of abortability in a mutual exclusion lock enables threads to abandon their attempt of acquiring the lock while they are waiting to acquire the lock. Abortability poses an interesting difficulty in cohort lock construction. Even if the alone? method, which indicates that a cohort thread is waiting to acquire the lock, returns false (which means that there exists a cohort thread wait-ing to acquire the lock), all the waiting cohort threads may subse-quently abort their attempts to acquire the lock. This case, if not handled correctly, can easily lead to a deadlock, where the global lock is in the acquired state, and the local lock has been handed off to a cohort thread that no longer exists, and may not appear in the future either. Thus, we must strengthen the requirements of the lock cohorting transformation with respect to the cohort detection property: if
alone? returns false , then some thread concurrently executing the local lock method will not abort before completing the local lock
method call. Notice that a thread that completed acquiring the local lock with the release local lock state cannot be aborted since by definition it is in the critical section.
3.6.1 The A-C-BO-BO Lock
The A-C-BO-BO lock is very similar to the C-BO-BO lock that we described earlier, with the difference that aborting threads also reset the successor-exists field in the local lock to inform the local lock releaser that a waiting thread has aborted. Each spinning thread reads this field while spinning, and sets it in case it was recently reset by an aborting thread. Like the C-BO-BO lock, in A-C-BO-BO, the local lock releaser checks to see if the successor-exists
flag was set (which indicates that there exist threads in the local cluster that are spinning to acquire the lock). If the successor-exists
flag was set, the releaser can release the local BO lock by writing
release local into the BO lock. 4
However, at this point the releaser must double-check the
successor-exists field to determine if it was cleared during the time the releaser released the local BO lock. If so, the releaser conser-vatively assumes that there may be no other waiting cohort, and atomically changes the local BO lock’s state to global release , and then releases the global BO lock.
4Note that the BO lock can also be in 3 states: release global (which is the default state, indicating that the lock is free to be acquired, but the acquirer must thereafter acquire the global BO lock to execute the critical section),
busy (indicating that the lock is acquired by some thread), and release local
(indicating that the next acquirer of the lock implicitly inherits ownership of the global BO lock).
3.6.2 The A-C-BO-CLH Lock
The A-C-BO-CLH lock has a BO lock as its global lock (which is trivially abortable), and an abortable variant of the CLH lock (A-CLH) as its local lock. Like the MCS lock, the A-CLH lock also consists of a list of records, one per thread, ordered by the arrival of the threads at the lock’s tail. To acquire the A-C-BO-CLH lock, a thread first must acquire its local A-CLH lock, and then explicitly or implicitly acquire the global BO lock. Because we build on the A-CLH lock, we will first briefly review it as presented by Scott . The A-CLH lock leverages the property of “implicit” CLH queue predecessors, where a thread that enqueues its node in the CLH queue spins on its predecessor node to determine if it has become the lock owner. An aborting thread marks its CLH queue node as aborted by simply making its predecessor explicit in the node (i.e. by writing the address of the predecessor node to the prev field of the thread’s CLH queue node). The successor thread that is spinning on the aborted thread’s node immediately notices the change and starts spinning on the new predecessor found in the aborted node’s prev field. The successor also returns the aborted CLH node to the corresponding thread’s local pool. The local lock in our A-C-BO-CLH builds on the A-CLH lock. For local lock handoffs, much like the A-CLH lock, the A-C-BO-CLH leverages the A-CLH queue structure in its cohort detection scheme. A thread can identify the existence of cohorts by checking the A-CLH lock’s tail pointer. If the pointer does not point to the thread’s node, it means that a subsequent request to acquire the lock was posted by another thread. However, now that threads can abort their lock acquisition attempts, this simple check is not sufficient to identify any “active” cohorts, because the ones that enqueued their nodes may have aborted, or will abort. In order to address this problem, we introduce a new successor-aborted flag in the A-CLH queue node. We colocate the successor-aborted flag with the prev field of each node so as to ensure that both are read and modified atomically. Each thread sets this flag to
false , and its node’s prev field to busy , before enqueuing the node in the CLH queue. An aborting thread atomically (with a CAS) sets its node’s predecessor’s successor-aborted flag to true to inform its predecessor that it has aborted (the thread subsequently updates its node’s prev field to make the predecessor explicitly visible to the successor). While releasing the lock, a thread first checks its node’s successor-aborted flag to determine if the successor may have aborted. If not, the thread can release the local lock by atomically (using a CAS
instruction) setting its node’s prev field to the release local state (just like the release in C-BO-MCS). This use of a CAS coupled with the colocation of prev and successor-aborted fields ensures that the successor thread cannot abort at the same time. The suc-cessor can then determine that it has become the lock owner. If the successor did abort (indicated by the successor-aborted flag), the thread releases the global BO lock, and then sets its node’s state to
release global .Our use of a CAS instruction to do local lock handoffs seems quite heavy-handed. And we conjecture that indeed it would be counter-productive if the CAS induced cache coherence traffic be-tween NUMA clusters. However, since the CAS targets memory that is likely to already be resident in cache of the local cluster in writable state, the cost of local transactions is quite low – equiv-alent to a store instruction hitting the L2 cache on the system we used for our empirical evaluation.
3.7 Bounding Local Lock Handoff Rates
All the locks described above are deeply unfair, and with even modest amounts of contention can easily lead to thread starvation. To address this problem, we add a may-pass-local method that increments a simple counter of the number of times threads in a cohort have consecutively acquired the lock in a release local state. If the counter crosses a threshold (64) in our experiments, the lock releaser releases the global lock, and then releases the local lock, transitioning it to the release global state. This simple solution appears to work very effectively for all our algorithms.
4. Empirical Evaluation
We evaluated cohort locks, comparing them with the traditional, as well as the more recent NUMA-aware locks, on multiple levels: First we conducted several experiments on microbenchmarks that stress test these locks in several ways. This gives us a good insight into the performance characteristics of the locks. Second, we inte-grated these locks in memcached , a popular key-value data store application, to study their impact on real world workload settings. Third, we modified the libc memory allocator to study the effects of cohort locks on allocation intensive multi-threaded applications; we present results of experiments on a microbenchmark . Our microbenchmark evaluation clearly demonstrates that co-hort locks outperform all prior locks by at least 60%. Additionally, the abortable cohort locks scale vastly better (by a factor of 6) than the state-of-the-art abortable locks. Furthermore, cohort locks im-proved the performance of memcached by about 20% for write-heavy workloads. Finally, our libc allocator experiments demon-strate that simply replacing the lock used by the default Solaris al-locator with a cohort lock can significantly boost cluster-level ref-erence locality for accesses by the allocator to allocation metatdata and for accesses by the application to allocated blocks, resulting in improved performance for multi-threaded application that make heavy use of memory allocation services. In our evaluation we compare the performance of our non-abortable and abortable cohort locks with existing state-of-the-art locks in the respective categories. Specifically, for our microbench-mark study, we present throughput results for our C-BO-BO, C-TKT-TKT, C-BO-MCS, C-TKT-MCS and C-MCS-MCS cohort locks. We compare these with MCS (as a base line NUMA-oblivious lock), and other NUMA-aware locks, namely, HBO , HCLH , and FC-MCS . We also evaluated our abortable co-hort locks (namely, A-C-BO-BO and A-C-BO-CLH) by compar-ing them with an abortable version of HBO, and the abortable CLH lock .
Memcached uses pthread locks for synchronization. To test our locks with memcached , we decided to adhere to the policy of not changing the memcached sources or its binary. This choice is facilitated by the fact that the pthread library is dynamically linked to the application. So we can easily use a Solaris LD PRELOAD
interpose library that installs any kind of lock we want under the
pthread library API. The scalability results for memcached are reported in Section 4.2. For the libc allocator experiments, we used the same interpose library to inject our locks into the allocator. We implemented all of the above algorithms in C and compiled them with the GCC 4.4.1 at optimization level -O3 in 32-bit mode. The experiments were conducted on an Oracle T5440 series ma-chine which consists of 4 Niagara T2+ SPARC chips, each chip containing 8 cores, and each core containing 2 pipelines with 4 hardware thread contexts per pipeline, for a total of 256 hardware thread contexts, running at a 1.4 GHz clock frequency. Each chip has a 4MB L2 cache, and each core has a shared 8KB L1 data cache. For all the NUMA-aware locks, a Niagara T2+ chip is the NUMA clustering unit, so in all we had 4 NUMA clusters.
Memcached was evaluated using a standard client application called memaslap , which is a part of a larger suite of memcached
applications called libmemcached . Results reported were averaged over 3 test runs.
0
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HBO
HCLH
FC-MCS
C-BO-BO
C-TKT-TKT
C-BO-MCS
C-TKT-MCS
C-MCS-MCS
Figure 2. The graph shows the average throughput in terms of number of critical and non-critical section pairs executed per sec-ond. The critical section accesses two distinct cache blocks (incre-ments 4 integer counters on each block), and the non-critical sec-tion is an idle spin loop of up to 4 microseconds.
4.1 Microbenchmark Evaluation 4.1.1 Scalability
We constructed what we consider to be a reasonable representa-tive microbenchmark, LBench , to measure the scalability of vari-ous lock algorithms. LBench launches a specified number of iden-tical threads. Each thread loops as follows: acquire a central shared lock, access shared variables in the critical section, release the lock, and then execute a non-critical work phase of about 4 microsec-onds. The critical section reads and writes shared variables resid-ing on two distinct cache lines. At the end of a 60 second measure-ment period the program reports the aggregate number of iterations completed by all threads as well as statistics related to the distri-bution of iterations completed by individual threads, which reflects gross long-term fairness. Finally, the benchmark can be configured to tally and report lock migrations. Figure 2 depicts the performance of the non-abortable locks on LBench. (We conducted other experiments varying the critical sec-tion length, non-critical section length, and number of cache lines accessed within the critical section, but observed similar results to those reported). As a baseline to compare against, we measured the throughput of the MCS lock, which is a classic scalable queue lock. This lock performed the worst because it does not leverage refer-ence locality, which is critical for good performance on NUMA architectures. The HCLH, HBO and FC-MCS locks perform as ex-pected – with FC-MCS generally performing the best among the three. HBO’s performance, which is better than HCLH in this work-load, is highly sensitive to the underlying workload, and is gener-ally very unstable (as we will see in Section 4.2). Our C-BO-BO lock scales very well, approaching the perfor-mance of FC-MCS. Because it is based on the BO lock, C-BO-BO is sensitive to backoff parameters – different workloads might require different backoff parameters for the best possible perfor-mance. However, this sensitivity is related only to the parameters associated with local backoff locks, unlike HBO, where the backoff parameters need to be tuned for both the local and remote backoffs. Under C-BO-BO we expect that the global lock will remain lightly contended; and in fact, in our implementation, threads contending at the global BO lock continuously spin on it and never backoff, much like the “bare bones” test-and-test-and-set lock. Our C-TKT-TKT lock scales even better (generally 30-40% better than the prior state-of-the-art NUMA-aware lock, FC-MCS). C-BO-MCS scales 1
3
7
116 32 64 96 128 160 192 224 256
L2 coherence misses per CS
of Threads
MCS
HBO
HCLH
FC-MCS
C-BO-BO
C-TKT-TKT
C-BO-MCS
C-TKT-MCS
C-MCS-MCS
Figure 3. The graph shows the average number of L2 cache co-herence misses per critical section for the experiment in Figure 2 (lower is better). The Y-axis is in log scale. the best with scalability 60% better than FC-MCS, whereas C-TKT-MCS and C-MCS-MCS trail slightly behind C-BO-MCS. In all the tests reported in this paper, the allowable maximum number of consecutive local lock handoffs for cohort locks was limited to a constant (64). As described in Section 3.7, this bound is necessary to avoid the deep unfairness that the basic cohort locks can possibly generate in an application. We conducted microbench-mark tests (not reported in this paper) on cohort lock versions with-out the local handoff limits and found that generally the deeply un-fair versions out-scale the fair versions by about 10% during high contention loads. However, in our tests we found that, for LBench, the unfair versions typically led to local lock handoffs in the order of hundreds of thousands before the lock was acquired by a re-mote thread/cohort. Thus, we believe that the cost of 10% is small to avoid the potential problem of gross long-term unfairness and starvation.
4.1.2 Locality of Reference
Figure 3 provides the key explanation for the observed scalability results. Recall that each chip (a NUMA cluster) on our experimen-tal machine has an L2 cache shared by all cores on that chip. Fur-thermore, the latency to access a cache block in the local L2 cache is much lower than the latency to access a cache block on a remote L2 cache (on our test machine, remote L2 access is approximately 4 times slower than local L2 access during light loads). The latter also involves bus transactions that can adversely affect the latency during high loads, further compounding the cost of remote L2 ac-cesses. That is, remote L2 accesses always incur latency costs even if the interconnect is otherwise idle, but they can also induce in-terconnect channel contention if the system is under heavy load. Figure 3 reports the L2 coherence miss rates collected during the scalability experiment. These are the local L2 misses that were ful-filled by a remote L2, which represents the local to remote lock handoff events and related data movement. MCS has a high L2 coherence miss rate because it is the fairest among all the locks, and does not prioritize local lock acquisition requests over remote requests. Interestingly, HCLH also has a high miss rate, which clearly explains its performance in Figure 2. We can attribute the high miss rate in HCLH to its complexity and high rates of accesses to shared lock metadata, which translates to lower rate of batching of requests coming from the same NUMA cluster. HBO shows a very good miss rate until the number of threads is substantially high (64), after which, the miss rate deterio-rates considerably. In our experiments with HBO, we observed that its backoff parameters are highly sensitive to the underlying work-
0
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of Threads
MCS
HBO
HCLH
FC-MCS
C-BO-BO
C-TKT-TKT
C-BO-MCS
C-TKT-MCS
C-MCS-MCS
Figure 4. A closer look at the throughput results of Figure 2 for low contention levels (1 to 16 threads). load; the HBO lock results discussed here are for a version whose backoff parameters were tuned for this microbenchmark (we will show later that these same backoff parameters hurt the performance of memcached ). The L2 miss rate in FC-MCS degrades gradually, but consistently, with increasing thread count. All our cohort locks have significantly lower (by a factor of two or greater – note that the Y-axis is in log scale) L2 miss rates than all other reported locks. This is because the cohort locks consis-tently provide long sequences of successive accesses to the lock from the same NUMA cluster (cohort), which accelerates the criti-cal section performance by reducing inter-core coherence transfers for data accessed within and by the critical section. There are two reasons for longer cohort sequences (or, more generally, batches )in cohort locks, compared to prior NUMA-aware locks, such as FC-MCS. First, the simplicity of cohort locks streamlines the in-struction sequence of batching requests in the cohort, which makes the batching more efficient. Second, and more importantly, a cohort batch can dynamically “grow” during its execution without the in-terference by threads from other cohorts. As an example, consider a cohort of 3 threads T 1, T 2, and T 3 (ordered in that order in a lo-cal MCS queue M in a C-BO-MCS lock). Let T 1 be the owner of the global BO lock. So T 1 can enter its critical section, and hand-off the BO lock (and the local MCS lock) to T 2 on exit. The BO lock will eventually be forwarded to T 3. However, note that in the meantime, T 1 can return and post another request after T 3’s re-quest in M . If T 3 holds the lock during this time, it ends up hand-ing it off to T 1 after it is done executing its critical section. This dynamic growth aspect of our cohort locks can significantly boost local handoff rates when there is sufficient cluster-local contention. This dynamic batch growth aspect in cohort locks contrasts with the more “static” approach of other NUMA-aware locks, which in turn gives more power to cohort locks to enhance the locality of ref-erence of the critical section. In our experiments, we have observed that the batching rate in all the locks is inversely proportional to the lock migration rate and observed coherence traffic reported in Figure 3, and that the batching rate in cohort locks increases more quickly with contention compared to other locks.
4.1.3 Low Contention Performance
We believe that for a highly scalable lock to be practical, it must also perform efficiently at low or zero contention levels. At face value, the hierarchical nature of cohort locks appears to suggest that they will be expensive at low contention levels. That is because each thread must acquire both the local and the global locks, which become a part of the critical path in low contention or contention free scenarios. To understand this cost we took a closer look at 0
50
100
150
200
116 32 64 96 128 160 192 224 256
% Standard Deviation
of Threads
MCS
HBO
HCLH
FC-MCS
C-BO-BO
C-TKT-TKT
C-BO-MCS
C-TKT-MCS
C-MCS-MCS
Figure 5. The graph shows the standard deviation in percentage points of per-thread throughput from the average throughput re-ported in Figure 2 (the lower the standard deviation, the more fair the lock is in practice). the scalability results reported in Figure 2 with an eye toward performance at low contention levels. Figure 4 zooms into that part of Figure 2. Interestingly, we observed that the performance of all the cohort locks was competitive with all other locks that do not need to acquire locks at multiple levels in the hierarchy (viz. MCS, HBO, and FC-MCS). On further reflection, we note that the extra cost of multi-level lock acquisitions in cohort locks withers away as background noise in the presence of non-trivial work in the critical and non-critical sections. In principle, one can devise contrived scenarios (for example, where the critical and non-critical sections are empty) to show that cohort locks might perform worse than other locks at low contention levels. However, we believe that such scenarios are most likely unrealistic or far too rare to be of any consequence. Even if one comes up with such a scenario, we can add the same “bypass the local lock” optimization that was employed in FC-MCS to minimize the flat combining overhead at low thread counts.
4.1.4 Fairness
Given that cohort locks are inherently unfair (which is the key “fea-ture” that all NUMA-aware locks harness and leverage to enhance locality of reference for better performance), we were interested in quantifying that unfairness. To that end, we report more data from the experiment reported in Figure 2 on the standard devia-tion of per-thread throughput from the average throughput of all the threads. The results are shown in Figure 5. These results give us a sense of how far each thread progressed during its one minute of execution in a test run. We found HBO to be the least fair lock, where some threads executed only a handful of critical sections, while others completed millions of critical sections. The next most unfair lock, to our surprise, was C-BO-MCS. (Recall that the cohort locks contain a constant limit of 64 local handoffs after which a lock releaser must release the global and local locks.) On further reflection, the reason for this unfairness is clear – the global BO lock in C-BO-MCS is unfair. After a thread releases the global BO lock, causing the cache block of the BO lock to be invalidated from other caches, and go to modified state in the releaser’s cache, it immediately releases the local lock, which is also quickly detected by the next thread waiting to acquire the local lock. This next local thread, identifying that it must acquire the global BO lock, almost instantly attempts to do so, and usually succeeds because the BO lock’s cache block is in its (and the last releaser’s) L1 or L2 cache. Hence the obvious unfairness arises from unfairness in cache coherence arbitration.
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Throughput/sec
Number of Threads
A-CLH
A-HBO
A-C-BO-BO
A-C-BO-CLH
Figure 6. Abortable lock average throughput in terms of number of critical and non-critical sections executed per second. The crit-ical section accesses two distinct cache blocks (increments 4 inte-gers counters on each block), and the non-critical section is an idle spin loop of up to 4 micro seconds. We also observe that the standard deviation for C-TKT-MCS and C-MCS-MCS are low, because their global locks (ticket and MCS) are comparatively fair. C-BO-BO is fairer than C-BO-MCS because the interval be-tween a lock releaser releasing the global lock and the next local lock acquirer attempting to acquire the global BO lock is inflated because the acquirer must first acquire the local BO lock. The ex-tended interval increases the window in which a remote thread/co-hort can acquire the global BO lock. All other locks, MCS, HCLH, FC-MCS, and C-TKT-TKT are also fair, as expected, with the standard deviation well under 5% (the deviation of FC-MCS spikes to about 20% at 16 threads, but is reasonably low at other concurrency levels).
4.1.5 Abortable Lock Performance
Our abortable lock experiments in Figure 6 make an equally com-pelling case for cohort locks. Our cohort locks (A-C-BO-BO and A-C-BO-CLH) outperform the best prior abortable lock (A-CLH) and an abortable variant of HBO (called A-HBO in Figure 6, where a thread aborts its lock acquisition by simply returning a failure flag from the lock acquire operation) by up to a factor of 6. Since lock handoff is a “local” operation in A-C-BO-CLH involving just the lock releaser (that uses a CAS to release the lock) and the lock acquirer (just like a CLH lock), A-C-BO-CLH scales significantly better than A-C-BO-BO, where threads incur significant contention with other threads on the same NUMA cluster to acquire the local BO lock. (For these and other unreported experiments, the abort rate was lower than 1%, which we believe is a reasonable rate.)
4.2 Memcached
Memcached is a popular open-source, high-performance, dis-tributed in-memory key-value data store that is typically used as a caching layer for databases in high-performance applications.
Memcached has several high profile users including Facebook, LiveJournal, Wikipedia, Flickr, Youtube, Twitter, etc. In memcached , the key-value pairs are stored in a huge hash ta-ble, and all server threads access this table concurrently. Access to the entire table is mediated through a single lock (called the
cache lock ). The cache lock is known to be a contention bottle-neck , which we believe makes it a good candidate for the eval-uation of cohort locks. Among other things, the memcached API contains two fundamental operations on key-value pairs: get (that returns the value for a give key) and set (that updates the value of # pthread locks Fib-BO MCS HBO HBO (tuned) FC-MCS C-BO-BO C-TKT-TKT C-BO-MCS C-TKT-MCS C-MCS-MCS
11.00 0.89 0.99 0.83 1.01 0.83 0.99 0.82 0.81 0.77 0.95
43.06 3.17 3.15 1.58 3.37 2.70 3.11 3.09 3.09 3.05 2.99
84.37 4.48 4.47 1.96 4.43 4.25 3.48 4.46 4.45 4.49 4.45
16 4.55 4.59 4.60 2.55 4.58 4.47 2.56 4.56 4.60 4.58 4.53
32 4.47 4.57 4.53 3.05 4.53 4.18 3.03 4.54 4.57 4.56 4.55
64 4.40 4.54 4.45 3.37 4.51 4.26 2.98 4.52 4.49 4.51 4.44
96 4.39 4.50 4.46 3.37 4.52 4.32 2.97 4.48 4.50 4.52 4.46
128 4.39 4.49 4.47 3.39 4.52 4.28 2.98 4.46 4.49 4.53 4.46
(a) 90% get s and 10% set s
11.00 1.04 1.15 0.97 0.93 0.95 1.14 1.00 0.92 1.12 1.11
42.84 3.21 3.30 1.45 3.23 2.73 2.98 3.10 3.19 2.98 3.16
83.55 4.63 4.51 1.73 4.75 4.00 2.46 4.53 4.51 4.47 4.32
16 3.56 4.95 4.93 2.17 5.18 4.51 2.59 5.08 5.05 5.03 4.97
32 3.42 4.93 4.77 2.57 5.10 3.92 2.79 4.99 5.04 4.95 4.94
64 3.29 4.81 4.45 2.86 5.08 3.93 2.67 4.88 4.88 4.79 4.74
96 3.32 4.80 4.47 2.84 5.07 3.94 2.69 4.84 4.86 4.78 4.71
128 3.32 4.81 4.24 2.84 5.09 3.90 2.68 4.87 4.85 4.71 4.68
(b) 50% get s and 50% set s
11.00 1.05 1.03 1.02 1.22 1.00 1.03 0.97 1.05 1.06 1.13
42.62 3.03 2.74 1.43 2.95 2.44 2.82 2.65 2.66 2.57 2.60
82.74 3.80 3.62 1.76 4.23 3.21 2.27 3.80 3.80 3.74 3.61
16 2.77 4.08 3.92 1.99 4.76 3.62 2.50 4.54 4.52 4.48 4.30
32 2.67 4.08 3.94 2.27 4.86 3.31 2.53 4.81 4.70 4.70 4.50
64 2.59 3.89 3.62 2.49 4.63 3.34 2.44 4.47 4.41 4.40 4.23
96 2.62 3.92 3.65 2.49 4.64 3.35 2.45 4.44 4.40 4.38 4.23
128 2.59 3.94 3.51 2.49 4.67 3.33 2.44 4.46 4.47 4.30 4.20
(c) 10% get s and 90% set s
Table 1. Scalability results (in terms of speedup over single thread runs that use pthread locks) for memcached for (a) read-heavy (90% get
operations, and 10% set operations), (b) mixed (50% get operations, and 50% set operations), and (c) write-heavy (10% get operations, and 90% set operations) configurations. the given key). These are the most frequently used API calls by
memcached client applications. To generate load for the memcached server, we use memaslap, a load generation and benchmarking tool for memcached . memaslap is a part of the standard client library (called libmemcached) for
memcached . memaslap generates a memcached workload con-sisting of a configurable mixture of get and set requests. We ex-perimented with a wide range of get -set mixture ratio, ranging from configurations with 90% get s and 10% set s (representing read-heavy workloads) to configurations with 10% get s and 90%
set s (representing write-heavy workloads). The read-heavy work-loads are the norm for memcached applications. The write-heavy workloads, however uncommon, do exist. Examples of write-heavy workloads include servers that continuously collect huge amounts of sensor network data, or servers that constantly update statis-tical information on a large collection of items. These applica-tions at times can exhibit bi-modal behavior, alternating between write-heavy and read-heavy phases, collecting and processing large amounts of data respectively. As discussed earlier, we used an interpose library to inject our locks under the pthread s API used by memcached . For our experiments, we ran an instance of memcached on the T5440 server, and an instance of memaslap on another 128-way Niagara II machine. We varied the thread count for memcached from 1 to 128 (the maximum number of threads permitted by memcached ). We ran the memaslap client with 32 threads for all tests so as to keep the load generation high and constant. For each test, the memaslap clients were executed for one minute, after which the throughput, in terms of operations per second, was reported. Table 1 shows the relative performance of memcached while it was configured to use the different locks. The figure contains 3 tables for three different get -set proportions, each representing read-heavy, mixed, and write-heavy loads respectively. Each entry in each table is normalized to the performance of pthread locks at 1 thread. The first column in all the tables represents the number of
memcached threads used in the test. The second column reports the performance of pthread locks. The remaining columns report the performance of memcached when used with MCS, a test-and-test-and-set lock with Fibonnaci backoff (Fib-BO), the HBO lock (representing prior NUMA-aware locks), a tuned version of HBO (we had to add this version because the default version did not scale well), and all the non-abortable cohort locks discussed in Section 3. Note that the fine tuning for the HBO lock was done on the local and remote backoff parameters. The version that we first used in the experiments (column titled HBO) had the backoff parameters tuned for our microbenchmark experiments. As is clear from all three tables, these did not work well on memcached . This clearly demonstrates the instability of HBO’s performance. For read-heavy loads (Table 1 (a)), the performance of all the locks except HBO and C-BO-BO is identical, with all locks en-abling over 5X scaling. For loads with moderately high set ratios (Table 1 (b)), we observe that all the spin locks except HBO and C-BO-BO significantly outperform pthread locks, and are generally competitive with each other. For write-heavy loads (Table 1 (c)), the NUMA-aware locks clearly out-scale the NUMA-oblivious locks by at least 20%. The untuned HBO and C-BO-BO locks scale poorly in all configurations. It appears that C-BO-BO suffers be-cause of contention on the local BO locks, whereas HBO suffers with contention on the central lock. FC-MCS performs better than HBO and C-BO-BO, but worse than all other spinlocks.
4.3 malloc
Memory allocation and deallocation is a common operation appear-ing frequently in all kinds of applications. A vast number of C/C++ programs use libc ’s malloc and free functions for managing their dynamic memory requirements. These functions are thread-safe, but on Solaris the default allocator relies on synchronization via a single lock to guarantee thread safety. For memory intensive multi-threaded applications that use libc ’s malloc and free functions, this lock can quickly become a contention bottleneck. Consequently, we found it to be an attractive evaluation tool for cohort locks. We modified libc ’s malloc.c file to use pthread locks, and injected our locks in the code via the interpose library discussed previously. We used the mmicro benchmark to test various lock algo-rithms via the interpose library. In the benchmark, each thread re-peatedly allocates a block of memory (size 64 bytes), initializes it thrds pthread locks fib-BO MCS HBO HBO (tuned) FC-MCS C-BO-BO C-TKT-TKT C-BO-MCS C-TKT-MCS C-MCS-MCS
1198 211 197 206 206 190 197 195 191 191 183
2197 237 224 204 231 231 223 220 214 218 208
4125 258 271 206 300 288 253 326 252 307 249
8145 294 307 230 382 322 320 486 326 456 432
16 151 318 307 244 420 327 483 592 513 576 564
32 149 323 307 248 291 329 783 839 941 827 814
64 149 302 303 259 151 328 883 1011 1183 1001 952
128 146 225 290 263 73 321 932 884 1120 863 822
255 142 139 277 257 38 264 926 695 961 682 651
Table 2. Scalability results of the malloc experiment (in terms of malloc -free pairs executed per millisecond. (by writing to the first 4 words of it), and subsequently frees it. Each test runs for 10 seconds and reports the aggregate number of mal-loc -free pairs completed in that interval. We add an artificial delay after each of the calls to malloc and free functions. This delay is a configurable parameter; we injected a delay of about 4 microsec-onds, which enables some concurrency between the thread execut-ing the critical sections ( malloc or free ), and the threads waiting in the delay loop. The results of the tests appear in Table 2, showing that cohort locks outperform all the other locks. While the other locks scale the benchmark’s throughput by up to a factor of 2X, the scalability with cohort locks ranges between a factor of 5X and 6X. There are two reasons for this impressive scalability of cohort locks: First, they tend to effectively batch requests coming from the same NUMA cluster, thus improving the lock handoff latency. The second reason has to do with the recycling of memory blocks deal-located by threads: The libc allocator maintains a single splay tree
of free nodes of various sizes (it also maintains lists of small – 40 bytes or less – memory blocks used for small size requests). Since
mmicro requests 64 byte blocks, all the requests go to the splay tree. A newly inserted node always goes to the root of the tree, and as a result, the most recently deallocated memory blocks tend to be reallocated more often (allocation is done by returning the first matching block in the splay tree). Thus a small number of tree nodes (and their respective memory blocks) are continuously cir-culated between threads. The tree node cache lines are updated on every delete ( malloc ) and insert ( free ). Additionally, the allocated memory blocks are also updated by the benchmark. All these writes play a crucial role in the performance of the underlying locks used by the allocator. Because all the cohort locks create large batches of consecutive requests coming from the same NUMA cluster, they manage to recycle blocks in the same cluster for extended periods. In contrast, for all other locks, a block of memory migrates more frequently between NUMA clusters, thus leading to greater coher-ence traffic, and the resulting performance degradation. While highly scalable allocators exist and have been described at length in the literature, selection of such allocators often entails making tradeoffs such as footprint against scalability. In part be-cause of such concerns the default on Solaris remains the simple single-lock allocator. By employing cohort locks under the default
libc allocator we can improve the scalability of applications but without forcing the user or developer to confront the issues and decisions related to alternative allocators. FC-MCS does not show any significant improvements over prior locks. The performance of HBO continues to be unstable: The first HBO column in Table 2 shows the libc allocator’s performance with the backoff parameters picked from our earlier microbench-mark experiments, while the second HBO column, titled: HBO (tuned), uses the parameters tuned for good performance on mem-cached . In this case, the tuned version of HBO scales better than the untuned version up to modest levels of contention. However, the performance dramatically deteriorates with higher contention. In contrast, cohort locks are vastly more stable across a broad swath of workloads. This property of “parameter parsimony” makes co-hort locks a significantly more attractive choice for deployment in real world applications.
5. Conclusion
The growing size of multicore machines is likely to shift the de-sign space in the NUMA and CC-NUMA direction, requiring a sig-nificant rehash of existing concurrent algorithms and synchroniza-tion mechanisms. This paper tackles the most basic of the multi-core synchronization algorithms, the lock, presenting a simple new lock design approach – lock cohorting – fit for NUMA machines. The wide range of cohort locks we presented in the paper, along with their empirical evaluation, demonstrates that lock cohorting is not only a simple approach to NUMA-aware lock construction, but also a powerful one that delivers locks that out-scale prior locks by significant margins, while remaining competitive at low contention levels.
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www.memcached.org . A. Agarwal and M. Cherian. Adaptive backoff synchronization tech-niques. SIGARCH Comput. Archit. News , 17:396–406, April 1989. T. Craig. Building FIFO and priority-queueing spin locks from atomic swap. Technical Report TR 93-02-02, University of Washington, Dept of Computer Science, February 1993. D. Dice. US Patent # 07318128: Wakeup affinity and locality . D. Dice and A. Garthwaite. Mostly Lock Free Malloc. In Proceedings of the 3rd International Symposium on Memory Management , pages 163–174, 2002. D. Dice, V. Marathe, and N. Shavit. Flat Combining NUMA Locks. In Proceedings of the 23rd ACM Symposium on Parallelism in Algo-rithms and Architectures , 2011. D. Hendler, I. Incze, N. Shavit, and M. Tzafrir. Flat Combining and the Synchronization-Parallelism Tradeoff. In Proceedings of the 22nd ACM Symposium on Parallelism in Algorithms and Architectures ,pages 355–364, 2010. M. Herlihy and N. Shavit. The Art of Multiprocessor Programming .Morgan Kaufmann, 2007. J. Mellor-Crummey and M. Scott. Algorithms for scalable synchro-nization on shared-memory multiprocessors. ACM Trans. Computer Systems , 9(1):21–65, 1991. M. Pohlack and S. Diestelhorst. From Lightweight Hardware Trans-actional Memory to LightWeight Lock Elision. In Proceedings of the 6th ACM SIGPLAN Workshop on Transactional Computing , 2011. Z. Radovi´ c and E. Hagersten. Hierarchical Backoff Locks for Nonuni-form Communication Architectures. In HPCA-9 , pages 241–252, Anaheim, California, USA, Feb. 2003. M. Scott and W. Scherer. Scalable queue-based spin locks with timeout. In Proc. 8th ACM SIGPLAN Symposium on Principles and Practices of Parallel Programming , pages 44–52, 2001. M. L. Scott. Non-blocking timeout in scalable queue-based spin locks. In Proceedings of the twenty-first annual symposium on Principles of distributed computing , PODC ’02, pages 31–40, New York, NY, USA, 2002. ACM. Victor Luchangco and Dan Nussbaum and Nir Shavit. A Hierarchical CLH Queue Lock. In Proceedings of the 12th International Euro-Par Conference , pages 801–810, 2006. |
190427 | https://ntrs.nasa.gov/api/citations/20180007747/downloads/20180007747.pdf | Michael J. Radke Case Western Reserve University, Cleveland, Ohio Nathan S. Jacobson Glenn Research Center, Cleveland, Ohio Monte Carlo Simulation of a Knudsen Effusion Mass Spectrometer Sampling System NASA/TM—2016-219118 October 2016 NASA STI Program . . . in Profi le Since its founding, NASA has been dedicated to the advancement of aeronautics and space science. The NASA Scientifi c and Technical Information (STI) Program plays a key part in helping NASA maintain this important role.
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NASA STI Program Mail Stop 148 NASA Langley Research Center Hampton, VA 23681-2199 National Technical Information Service 5285 Port Royal Road Springfi eld, VA 22161 703-605-6000 This report is available in electronic form at and NASA/TM—2016-219118 1 Monte Carlo Simulation of a Knudsen Effusion Mass Spectrometer Sampling System Michael J. Radke Case Western Reserve University Cleveland, Ohio 44106 Nathan S. Jacobson National Aeronautics and Space Administration Glenn Research Center Cleveland, Ohio 44135 Summary Knudsen flow is easily simulated with a Monte Carlo method. In this study a Visual Basic for Excel (VBA) code is developed to simulate the molecular beam from a vaporizing solid. The system at NASA Glenn Research Center uses the “restricted collimation” method of Chatillon and colleagues, which consists of two apertures between the effusion cell and the ionizer. The diameter of the first aperture is smaller than the diameter of the effusion cell orifice, so the ionizer effectively “sees” only inside the effusion cell. The code is able to calculate the transmission coefficient through the cell orifice, through the cell orifice and the first aperture, and through the cell orifice and first and second apertures. Calculated transmission coefficients through the cell orifice are compared with tabulated values to validate the code. Then transmission coefficients are calculated through the cell orifice and both apertures to the ionizer. This allows the geometry (aperture spacing and diameters) of the sampling system to be optimized. Calculated transmission factors are also compared to literature values calculated via an analytic method. 1.0 Introduction Knudsen effusion mass spectrometry (KEMS) is a well-established technique for sampling high-temperature vapors from a Knudsen cell (Refs. 1 to 3). A typical Knudsen cell is illustrated in Figure 1. The sample cell is heated uniformly, and an equilibrium vapor develops above the condensed phase. The effusion orifice has a well-defined geometry and directs a molecular beam composed of this vapor into the ionizer of a mass spectrometer. The requirements for Knudsen sampling include a cell orifice with a Knudsen number (ratio of the mean free path to orifice diameter) of 10 or greater. Thus molecule wall collisions dominate over molecule-molecule collisions, and the vapor in the molecular beam is representative of the vapor above the condensed phase. Recently, Chatillon and coworkers have discussed a novel method of sampling from a Knudsen cell (Refs. 4 to 6). Their method is termed “restricted collimation” and involves two collimating apertures above the cell effusion orifice. The first is in the plate that separates the Knudsen cell chamber from the ionizer. Chatillon terms this the “field aperture.” The field aperture is necessarily smaller than the Knudsen cell orifice, so that the ionizer effectively “sees” only the inside of the Knudsen cell. The second is right below the ionizing filament and is termed the “source aperture.” The advantages of this restricted collimation configuration is that background is minimized. This is illustrated in Figure 2. Such sampling is particularly useful in multi-cell instruments, to limit cross-over of molecular beams from adjacent cells. Chatillon and coworkers have modeled restricted collimation (Refs. 4 to 6) with equations analogous to the decay of light intensity. Molecular beams are analogous to transmission of light, a stream of photons. Many of the equations developed for light are readily applicable to these molecular beams. First, note that the flux decays as 1/a2, where a is the distance from the source to the receiver. The basic equations for flux received from a radiating element are derived in the paper and textbook by Walsh (Refs. 7 and 8). The problem germane to the collimation of a molecular beam in mass spectrometer is in NASA/TM—2016-219118 2 Figure 1.—Knudsen cell. Figure 2.—Comparison of Knudsen cell restricted collimation to traditional molecular beam definition (adapted from Ref. 2). Note that the traditional method may also sample species evaporating from the cell lid and heat shields. the transmission of the beam from one disc source to another co-axial disc source. The fraction of molecules F that leave the radiating disc and arrive at the receiving disk is given by 2 1 2 2 2 1 2 2 2 2 1 2 2 2 2 1 2 1 2 4 ) ( ) ( ) , , ( r r r c r r c r r c r r F (1) Here c is the distance between the two disks, r1 is the radius of the radiating disk, and r2 is radius of the receiving disk. The quantity F is the unit of solid angle that reaches the second aperture (Ref. 7). The two KEMS instruments at the NASA Glenn Research Center have been adapted for restricted collimation. The configuration from the cell to the ionizer is illustrated in Figures 3(a) and (b), for the magnetic sector instrument (Ref. 2) and quadrupole instrument, respectively. For both instruments, the Knudsen cell furnace chamber and ionizer chamber is separated by a copper plate. The field aperture in this plate is adjustable with a secondary piece, as shown in Figure 4. This is aligned with the source aperture using a laser. Knudsen flow through pipes has been described analytically by many investigators (Refs. 9 and 10). Such descriptions of Knudsen flow are based on the analogous behavior of molecular flow to light and the cosine distribution law for vaporizing species. In 1960, Davis published a Monte Carlo simulation of Knudsen flow in pipes (Ref. 11). Since then, many investigators have further extended this Monte Carlo approach (Refs. 11 to 17). Today, because of the wide availability of high-speed desktop computers with multicore processors, Monte Carlo simulation is one of the easiest and most flexible ways to describe Knudsen flow. NASA/TM—2016-219118 3 Figure 3.—Restricted collimation sampling for Glenn Knudsen effusion mass spectrometers. (a) Magnetic sector instrument (adapted from Ref. 2). (b) Quadrupole instrument. NASA/TM—2016-219118 4 Figure 4.—Copper plate that separates Knudsen cell furnace chamber and ionization chamber in both Knudsen effusion mass spectrometry instruments at NASA Glenn. Flange in center contains field aperture and is adjustable in plane of flange. In this report a Monte Carlo Knudsen flow simulation for pipes is adapted to model the vaporization of the condensed phase in the cell through the cell orifice and the two apertures. The code was adapted from an earlier FORTRAN code (Ref. 17) to Microsoft Visual Basic for Excel (VBA). Thus the code is readily run on any personal computer. It is very flexible, so that the various aperture diameters and distances can be adjusted to maximize molecular beam transmission. Apertures can be added or removed to calculate basic transmission through a channel or transmission through several apertures. Additionally, the pipe cross section can be changed from cylindrical to rectangular. Initially, major equations are derived. A description of the code follows, with a model of the systems at Glenn Research Center (Figs. 3(a) and (b)). Finally, the calculated transmission factors for various configurations are compared with tabulated values, determined from analytical expressions and experiments. Nomenclature used in this report is listed in Appendix A. 2.0 Derivation of Equations for Starting Point and Trajectories In this section the major equations are derived from solid geometry relations. The approach of Davis and others are followed (Refs. 11 and 17). 2.1 Solid Angles The first step to describe a trajectory of a vaporizing molecule is to define a solid angle. Consider the hemisphere shown in Figure 5 where the angle with the x-axis varies uniformly from 0 to /2 radians and the angle in the y,z-plane varies from 0 to 2 radians. An infinitesimal element of the solid angle, d, is given by d d sin d (2) NASA/TM—2016-219118 5 Figure 5.—Element of solid angle, d. Starting point for trajectory of vaporizing molecule is “p.” We can define the solid angle as the surface area of a portion of a unit sphere. In our case only is specified and varies randomly from 0 to 2 radians, so this area becomes an infinitesimal band around the sphere, as shown in Figure 5. Thus the above expression simplifies to d sin 2 d (3) Suppose we want the amount of solid angle from 0 to . This is found by the integration: cos 1 2 d sin 2 d 0 (4) 2.2 Cosine Distributions Although is a uniformly distributed random variable between 0 and 2 radians, must follow a known cosine distribution. In other words, the elemental flux of molecules, dNθ leaving a plane into a solid angle element d with angle to the surface normal is proportional to the cosine of that angle: d cos dN (5) Putting Equation (3) into Equation (5) above gives d sin cos 2 dN (6) So the question is how to generate this distribution in a computer code. Following Davis (Ref. 11), we generate two uniformly distributed random numbers R between 0 and 1 and then select the larger of the two. This means that the larger the value of R is, the higher the probability of selecting it. If P(R) is the probability of selecting R, then R R R R P d 2 d (7) Let R = cos, so now we have d sin cos 2 d d cos d cos 2 cos d cos P (8) This is the distribution required given in Equation (5). This distribution is used to determine the initial trajectory of the simulated molecules as well as their new trajectories after wall collisions. NASA/TM—2016-219118 6 2.3 Direction Cosines In order to define the trajectory of a molecule, we need to assign it direction cosines. These are simply the cosines of the angles , , and that a vector makes with the x-, y-, and z-axes, respectively. The direction cosines can be expressed in terms of the spherical coordinates and , discussed previously: cos cos sin sin cos sin cos cos 3 2 1 (9) The cosine of is derived via the procedure discussed in Section 3.2. As noted, varies randomly from 0 to 2. Von Neumann (Ref. 18) describes a method to derive trigonometric functions for angles varying uniformly between 0 and 2 without using the more computer-time-consuming functions (e.g., trigonometric functions and square roots). Consider a rectangle formed by random numbers f and g, which both vary uniformly between 0 and 1: 2 2 2 2 cos sin g f g g f f (10) The square roots in the denominators can be eliminated by using half-angle formulae: cos sin 2 sin sin 2 1 2 cos cos 2 (11) In order to have vary from 0 to 2, must vary from 0 to . Thus define in a rectangle with sides 1 0 1 1 1 0 1 2 2 1 1 g R g f R R f (12) Finally, Equations (10) to (12) are combined to yield: 2 2 2 1 2 1 2 2 2 1 2 2 2 1 1 2 1 2 2 sin 1 2 1 2 cos R R R R R R R R (13) 3.0 Method to Determine Trajectory Endpoints The program traces the trajectory of each molecule. A distance criterion is used to determine if the molecule (1) strikes the wall, (2) escapes from the top, or (3) returns to the bottom. In the event of a wall collision, another trajectory is calculated. First the equations are derived for a cylinder and then extended to cover the additional collimating apertures in the system. An option for rectangular apertures is also discussed. 3.1 Simulation Geometry The geometry of the simulation and its dimensions are selected by the user with the user interface shown in Figure 6. Each of the dimensions listed are variables to be entered in the code. NASA/TM—2016-219118 7 Figure 6.—Two-dimensional representation of simulated restricted collimation sampling system in Knudsen effusion mass spectrometer. Figure 7.—Trajectory of molecule vaporizing from cylinder base. (a) Dimensions describing first flight of molecule. (b) Base of cylinder. 3.2 Initial Trajectory Figure 7 illustrates a simple cylinder that has Cartesian coordinates defined as indicated: x is the center axis of the cylinder, and y and z are in a plane perpendicular to that center axis. This represents the orifice of the cell. The cosine distribution defining molecules initial trajectories must be oriented about the x-axis, and thus the first set of direction cosines is given as sin cos cos sin sin cos cos cos 3 2 1 (14) Here 1 is the direction cosine for the x-axis, 2 is the direction cosine for the y-axis, and 3 is the direction cosine for the z-axis. NASA/TM—2016-219118 8 Having defined the directions a molecule leaves a surface or a plane, now define the first trajectory. Consider the base of the cylinder. Define a completely random starting point. Place the origin at the center of the cylinder base, and set the radius of the cylinder equal to one. Then select a point 1 1 1 0 1 2 1 1 u R R u ys (15) 1 1 1 0 1 2 2 2 v R R v zs subject to the condition that 1 2 2 s s z y (16) where Equation (16) is the equation for a circle in the y,z-plane, representing an emissive surface at the bottom of the channel. The first “flight” of a molecule is illustrated in Figure 7(a). The first question to be answered is whether the molecule strikes the wall or leaves the orifice. Following Davis (Ref. 11), this is done with a distance criterion. Two distances are calculated in the direction of the previously defined direction cosines—the distance from the base of the channel to the wall and the distance from the base to the top of the channel. Whichever distance is smaller determines the event that occurs: a wall collision or an escape from the channel. This method of distance comparison was determined to be much more efficient than computing and comparing angles of molecule trajectories, without sacrificing simulation accuracy. First consider the distance from the base of the cylinder to the wall, DW, as shown in Figure 7(a). Figure 7(b) illustrates projection to the base of the cylinder. From this figure, 2 2 2 2 2 2 cos DW h u h DW (17) where h is the height of the collision point on the cylinder. The value of u (the length of the base in the right triangle formed with DW and h) can be expressed in terms of the angle between A and B (Fig. 7(b)) by the law of cosines: 2 2 2 2 2 2 2 2 2 2 2 cos ) ( cos ) ( cos cos 2 1 cos 2 z y z DW z y DW y B A z y z y B A B A u B A (18) Now determine u in terms of the direction cosines and x and y: ) cos cos )( ( 2 ) ( 1 2 2 2 z y DW z y u (19) Simplify this with the notation of Davis (Ref. 11): 2 2 2 2 2 cos 1 cos cos cos cos 1 P P z y Q z y D (20) NASA/TM—2016-219118 9 This gives a quadratic: P D P Q P Q DW D DW Q DW P 2 2 2 ) ( 2 ) ( 0 (21) The distance to the wall of the cylinder, scaled to the y-direction cosine, is given by P D P Q P Q SW 2 2 2 1 (22) The distance to the top of the cylinder, scaled to the x-direction cosine, is simple and given by 1 cos L L ST (23) where L is the length of the channel. As noted, the distance criterion is used to determine the event. 3.3 Trajectory After Wall Collision The coordinates of the wall collision point are determined from the direction cosines: cos ) ( cos ) ( cos ) ( old old DW z z DW y y DW x (24) Once the molecule strikes the wall, it reflects diffusely off the surface with a cosine distribution about the normal to the tangent at that point (ynew, znew). The molecule does not undergo specular reflection—and simply obey the law of reflection—because the surface of the channel is microscopically rough. A cosine distribution about the normal to the point of reflection is obtained by rotating the original coordinate system (which has the origin at the center of the base of the cylinder and the x-axis along the center of the cylinder). The first step in doing so is to rotate the x-axis into the negative y-axis and the y-axis into the negative x-axis so 3 3 1 2 2 1 (25) The next step is to rotate the y- and z-axes by an angle of transformation, ξ. The equations for rotation of the point (y1,z2) to the new coordinate system ) , ( 2 1 z y are given by 1 1 1 1 1 1 1 1 sin cos sin cos x x z y z z y y (26) Next cos ξ and sin ξ can be calculated as 2 2 2 2 cos sin z y z z y y (27) NASA/TM—2016-219118 10 Note that the actual coordinates of the impact point are not being rotated. The only transformation that occurs is of the direction cosines, which define the new trajectory of the molecule after collision. The values of y1 and z1 are only used to calculate sin ξ and cos ξ. 2 new 3 new 1 3 1 new 3 new 1 3 1 3 1 1 1 cos ) (sin ) (cos cos ) (cos ) (sin cos y z z y z y (28) Also note that cos has been defined by Equation (24) above. Thus, Equation (28) defines a set of direction cosines such that the molecule desorbs after collision with a cosine distribution about the axis perpendicular to the tangent of the desorption point. Now the issue is to determine the distance from the desorption point to (1) another wall collision, (2) the top of the channel, or (3) the bottom of the channel. Again the shortest distance is taken as the event. The distance to the wall is easily calculated from Equations (17) and (18). Now since the point is on the edge of the cylinder, y2 + z2 = 1, then D = 0 and the scale distance to the wall (Eq. (21)) simplifies to P Q SW 2 ) ( (29) The equation for the scaled distance to the top is 1 ) ( x L ST (30) The equation for the scaled distance to the bottom is 1 ) ( x SB (31) If the calculated distance is negative, the molecule will never intersect with the object of interest, and the distance is assigned a very large number. As before, the shortest distance determines the event. If the molecule strikes a wall, a new collision is simulated like before. If the molecule passes out the bottom of the channel, the molecule is counted as returning to the start, and a new molecule’s trajectory is simulated. Finally, if the molecule passes out the top of the channel, it is counted as an escape, and its trajectory can be traced through additional channels or apertures. 3.4 Trajectory Through Additional Aperture(s) A molecule will pass through an aperture (with negligible thickness) if its radial distance from the central axis at the x-position of the aperture is less than the radius of the aperture. In other words, 2 ap 2 2 r z y (32) Based on Figure 7(a), the equation for the scale distance to the first aperture from the last collision is 1 1 ) ( dist.
G L x SA (33) where (x – L) represents the remaining distance the molecule has to travel to escape the channel from its last collision and G, the distance between the top of the channel and the aperture. We can then calculate NASA/TM—2016-219118 11 the y and z positions of the molecule at the x-position of the aperture using the molecule’s direction cosines and the previously calculated scale factor from Equation (32): SA z SA y 3 2 (34) Substituting Equations (34) into Equation (32) yields 2 ap 2 3 2 2 r SA SA (35) If the inequality is satisfied, the molecule passes through the aperture. This process can be repeated for as many apertures as necessary, with the only change being the distance used to calculate the scale factor SA. For example, for the second aperture shown in Figure 6, the distance becomes (x – L) + G + H, where H is the distance between the two apertures. 3.5 Rectangular Channels and Apertures The simulation procedure for rectangular channels and apertures is largely the same as for round ones, the major exception being the distance to wall collisions. First, calculate the distance from the starting point to y and z coordinates of the channel walls, illustrated in Figure 7: 3 2 ) 2 / ( ) 2 / ( z b SZ y a SY (36) As for cylindrical channels, the shorter of the two distances determines the wall collision. The coordinates of the collision are obtained by simply multiplying each of the direction cosines by the scale factor (SY or SZ) and adding these values to the previous coordinates. Additionally, after a wall collision, the new cosine distribution simply needs to be rotated π/2 radians, such that the central axis of the distribution is normal to the surface on which the collision occurred. This is equivalent to setting y and z in Equation (27) to the appropriate values—either the half-width of the channel or zero. For a molecule striking the wall at y = a/2 this would mean z = 0, and 3 1 2 cos cos cos (37) For apertures, the only difference is that a molecule successfully passes through when the position of a molecule in the y,z-plane is less than the half-width of the channel in either coordinate: 2 2 b z a y (38) rather than the check in Equation (31). 4.0 Results First a series of calculations on simple cylindrical and rectangular channels is presented to compare to tabulated values and thereby verify proper operation of the code. Appendix B shows the input and output screens. Then the transmission factors for a typical restricted collimation mass spectrometer system are calculated and compared to the analytical approach used by Chatillon and coworkers (Refs. 4 and 6). NASA/TM—2016-219118 12 4.1 Calculated Transmission Factors Transmission factors through cylindrical and rectangular channels have been calculated for a variety of geometries and show good agreement with tabulated transmission factors, as shown in Tables I and II (Refs. 19 and 20). Figure 8 shows the angular distribution of molecules effusing from cylindrical channels with various length-to-radius ratios (l/r). To create these plots, molecules are sorted into 1 bins according to the angle they make with the x-axis as they exit the channel. These counts are then normalized to the sine of the small increment of solid angle in Equation (2) and plotted on polar axes. The l/r of the channel has a strong effect on the distribution of exiting molecules, with large l/r creating a narrow beam of molecules, as opposed to the near ideal Cosine Law distribution of very low l/r. TABLE I.—SELECTED TRANSMISSION FACTORS FOR CYLINDRICAL EFFUSION ORIFICES Length-to-radius ratio, L/r Simulation results, W Tabulated values,a W Difference, percent 0.2 0.90873 0.9092 −0.052 2 0.51322 0.5136 −0.074 4 0.35682 0.3589 −0.580 8 0.22542 0.2316 −2.668 20 0.10943 0.1135 −3.586 100 0.02549 0.0258 −1.202 aSource: Clausing (Ref. 19). TABLE II.—TRANSMISSION FACTORS FOR KNUDSEN CELL RECTANGULAR EFFUSION ORIFICES Column length per width, l/b Transmission factor Difference, percent Simulation result, W Tabulated value,a W 0.1 0.91314 0.9131 0.004 1 0.53643 0.5363 0.024 2 0.37831 0.3780 0.820 4 0.24236 0.2424 −0.017 10 0.12001 0.1195 0.427 40 0.03465 0.0346 0.145 aSource: Santeler and Boeckmann (Ref. 20). Figure 8.—Cosine distribution emerging from channel of different length-to-radius ratio l/r. NASA/TM—2016-219118 13 4.2 Calculated Properties for a Restricted Collimation Knudsen Effusion Mass Spectrometer Table III contains the calculated properties of the KEMS geometry used at NASA Glenn shown in Figure 2 using 108 simulated molecules. A1 and A2 refer the lower and upper apertures, respectively. Only 11 percent of molecules effusing directly from the pipe do not undergo any wall collisions. However, the additional collimating apertures, A1 and A2, increase the proportion of molecules passing directly from the source to the mass spectrometer ionizer to 90 percent. Figures 9(a) to (c) shows the views of the molecular path in a restricted collimation. In this simulation, 106 molecules were run. The vaporizing surface is at the bottom of the pipe. Figure 9(a) is a view of the first 50 molecular trajectories in the pipe. Figure 9(b) is a view of the whole beam and Figure 9(c) is a view from the top. The open circles are the molecules that made it through to the ionizer. In Figure 10, the results of this method are compared to those from the analytical method used by Chatillon and coworkers (Ref. 6) to describe a restricted collimation system. The transmission factor, W, through the last aperture is plotted versus the source (upper) aperture diameter. As described by Chatillon and coworkers (Ref. 6), the diameter of the lower orifice necessarily changes for geometry considerations. The geometry they used was reproduced as close as possible for this calculation. Vaporizing solid: 2.0-mm diameter at base of orifice Cell orifice: 2.0-mm diameter; 2.0-mm thickness Lower aperture: Diameter from Table 1 of Reference 6; 0.01-mm thickness; 10 mm from cell orifice Upper aperture: Diameter from Table 1 of Reference 6; 0.01-mm thickness; 50 mm from cell orifice TABLE III.—CALCULATED PROPERTIES FOR TYPICAL KNUDSEN EFFUSION MASS SPECTROMETRY CELL GEOMETRY AT NASA GLENN [Length-to-radius ratio l/r = 5.33.] Input values Pipe length, lpipe = 4.0 mm Diameter at lower aperture, dA1 = 1 mm Distance between top of channel and lower aperture, G = 25.0 mm Pipe diameter, dpipe = 1.5 mm Diameter at upper aperture, dA2 = 2 mm Distance between lower and upper apertures, H = 37.7 mm Simulation results Location Transmission factor Average number of collisions, c ̄ Percent without collisions, nc Pipe Wpipe = 0.2977 c ̄ pipe = 7.36 ncpipe = 11.2 Lower aperture WA1 = 3.5×10–4 c ̄ A1 = 7.9×10–4 ncA1 = 86.4 Upper aperture WA2 = 1.6×10–4 c ̄ A2 = 3.4×10–4 ncA2 = 90.1 Figure 9.—Views of molecular trajectory from vaporizing surface to ionizer in Knudsen effusion mass spectrometer with restricted collimation. (a) Collisions within pipe. (b) Paths through apertures. (c) View from top. Circles are molecules that make it through upper orifice. NASA/TM—2016-219118 14 Figure 10.—Transmission factor of Knudsen effusion mass spectrometer with restricted collimation as function of upper aperture diameter. The calculations herein show order-of-magnitude agreement with those of Chatillon and coworkers (Refs. 4 to 6). They also show maxima, as do those of Chatillon et al. The slightly different maxima with the two approaches needs to be explored further. The calculations of Chatillon and coworkers are based on the attenuation in beam strength described by Equation (1) and an integration over the solid angle shown in Figure 2(a). This approach treats the process more like “plug flow,” and there are likely fundamental differences between this and the Monte Carlo approach where each trajectory is treated individually. Further refinements to the Monte Carlo code are the ability to handle large sample sizes and the incorporation of molecule-molecule collisions. The code in VBA seems to be limited to ~1108 trajectories. A more appropriate computer language would be necessary to increase the sample size. Molecule-molecule collisions can be incorporated by having the molecule randomly changing direction after traversing a mean free path. These changes can be incorporated into subsequent versions. 5.0 Conclusion The existing Monte Carlo simulation allows for the fast (106 molecules in ~30 s) and accurate simulation of basic Knudsen effusion mass spectrometry (KEMS) sampling geometries, allowing for the optimization of orifice sizes and spacings in order to increase transmission and decrease the average number of collisions. This approach was verified with calculation of known transmission factors for simple cylindrical and rectangular orifices. The code has calculated angular distributions of molecules emerging from an orifice and has demonstrated the advantages of a cylindrical orifice to form a directed flow. Finally, the calculations were compared to the analytical analyses of Chatillon et al. for a restricted collimation KEMS system. Future directions were also discussed. NASA/TM—2016-219118 15 Appendix A—Nomenclature A, B vectors describing position of segment u a channel or cylinder width (y-direction) b channel or cylinder width (z-direction) c distance between two apertures c average number of collisions D, Q, P variables defined by coordinates and direction cosines for convenience d channel diameter DW distance molecule travels to wall collision F fraction of molecules leaving the radiating disk f random numbers forming right triangle to generate trigonometric functions of angle δ G distance between top of channel and lower aperture H distance between lower and upper apertures h height of collision point on cylinder L channel length l channel length N flux of molecules vaporizing in angle θ nc percent of molecules that escape without collisions P probability p starting point for vaporizing molecule r1,2 radius of radiating and receiving disk, respectively rap aperture radius Ri random number SW, SA, SB, ST variables defining trajectory distances SY, SZ distances from starting point to y and z coordinates of rectangular channel walls, respectively u length of base in right triangle defined by DW and h W transmission factor (i.e., Clausing factor) ys,zs coordinates of molecule first vaporizing from sample surface ynew,znew point where molecule strikes cylinder wall yold,zold initial coordinates of molecule’s trajectory α, β, γ angles that describe molecule’s trajectory after wall collision θ altitude, angle with x-axis μ1,2,3 direction cosines for the x-, y-, and z-axes, respectively ξ angle of transformation of y- and z-axes after molecule collides with cylinder wall φ azimuth angle in y,z-plane Ψ angle between vectors A and B ω solid angle NASA/TM—2016-219118 17 Appendix B—Input and Output Screens The input and output screens for the Visual Basic for Excel (VBA) code are shown here. Input Screen NASA/TM—2016-219118 18 Output Screen NASA/TM—2016-219118 19 References 1. Drowart, J.; and Goldfinger, P.: Investigation of Inorganic Systems at High Temperature by Mass Spectrometry. Angew. Chem. Int. Ed., vol. 6, no. 7, 1967, pp. 581–596. 2. Copland, E.H.; and Jacobson, N.S.: Measuring Thermodynamic Properties of Metals and Alloys With Knudsen Effusion Mass Spectrometry. NASA/TP—2010-216795, 2010. 3. Chupka, William A.; and Ingrham, Mark G.: Direct Determination of the Heat of Sublimation of Carbon With the Mass Spectrometer. J. Phys. Chem., vol. 59, no. 2, 1955, pp. 100–104. 4. Morland, P.; Chatillon, C.; and Rocabois, P.: High-Temperature Mass Spectrometry Using the Knudsen Effusion Cell. I.—Optimization of Sampling Constraints on the Molecular Beam. High Temp. Mat. Sci., vol. 37, no. 3, 1997, pp. 167–187. 5. Chatillon, Christian, et al.: High-Temperature Mass Spectrometry With the Knudsen Cell: II. Technical Constraints in the Multiple-Cell Method for Activity Determinations. High Temp-High Press, vol. 34, 2002, pp. 213–233. 6. Nuta, Ioana; and Chatillon, Christian: Knudsen Cell Mass Spectrometry Using Restricted Molecular Beam Collimation. I. Optimization of the Beam From the Vaporizing Surface. Rapid Commun. Mass Spectrom., vol. 29, no. 1, 2015, pp. 10–18. 7. Walsh, John W.T.: Radiation From a Perfectly Diffusing Circular Disc (Part I). Proc. Phys. Soc. London, vol. 32, 1920, pp. 59–71. 8. Walsh, John W.T.: Photometry. Constable & Co., London, 1958, pp. 136–144. 9. DeMarcus, W.C.: The Problem of Knudsen Flow. Part 2. Solution of Integral Equation With Probability Kernels. Union Carbide and Carbon Corp., Oak Ridge, TN, 1956. 10. Grimley, Robert T.; Wagner, L.C.; and Castle, Peter M.: Angular Distributions of Molecular Species Effusing From Near-Ideal Orifices. J. Phys. Chem., vol. 79, no. 4, 1975, pp. 302–308. 11. Davis, D.H.: Monte Carlo Calculation of Molecular Flow Rates Through a Cylindrical Elbow and Pipes of Other Shapes. J. Appl. Phys., vol. 31, 1960, pp. 1169–1176. 12. Ward, J.W.; Mulford, R.N.R.; and Bivins, R.L.: Study of Some of the Parameters Affecting Knudsen Effusion. II. A Monte Carlo Computer Analysis of Parameters Deduced From Experiment. J. Chem. Phys., vol. 47, 1967, pp. 1718–1723. 13. Ward, John W.; and Fraser, Malcolm V.: Some of the Parameters Affecting Knudsen Effusion. IV. Monte Carlo Calculations of Effusion Probabilities and Flux Gradients for Knudsen Cells. J. Chem. Phys., vol. 49, 1968, pp. 3743–3750. 14. Ward, John W.; and Fraser, Malcolm V.: Study of Some of the Parameters Affecting Knudsen Effusion. VI. Monte Carlo Analyses of Channel Orifices. J. Chem. Phys., vol. 50, 1969, pp. 1877–1882. 15. Ward, John W.; Bivins, Robert L.; and Fraser, Malcolm V.: Monte Carlo Simulation of Specular and Surface Diffusional Perturbations to Flow From Knudsen Cells. J. Vac. Sci. Technol., vol. 7, 1970, pp. 206–210. 16. Ward, John W.; Fraser, Malcolm V.; and Bivins, Robert L.: Monte Carlo Analysis of the Behavior of Divergent Conical Effusion Orifices. J. Vac. Sci. Technol., vol. 9, 1972, pp. 1056–1061. 17. Jacobson, N.S.: Diffusion of Gases in Capillaries. Ph.D. Thesis, Lawrence Berkeley Lab, 1981. 18. von Neumann, John: Various Techniques Used in Connection With Random Digits. National Bureau of Standards Mathematics Series, vol. 12, 1951, pp. 36–38. 19. Clausing, P.: The Flow of Highly Rarefied Gases Through Tubes of Arbitrary Length. J. Vac. Sci. Technol., vol. 8, 1971, pp. 636–646. 20. Santeler, D.J.; and Boeckmann, M.D.: Combining Transmission Probabilities of Different Diameter Tubes. J. Vac. Sci. Technol. A, vol. 5, 1987, pp. 2493–2496. |
190428 | https://webbook.nist.gov/chemistry/uv-vis/ | UV/Vis Database User's Guide
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UV/Vis Database User's Guide
V. Talrose, A.N. Yermakov, A.N. Leskin, A.A. Usov, A.A. Goncharova, N.A. Messineva, N.V. Usova, M.V. Efimkina, E.V. Aristova
Institute of Energy Problems of Chemical Physics, Russian Academy of Sciences
Sources of data
The overwhelming majority of spectra are taken from original scientific papers with the precise references. Some of the data are taken from several published collections. These collections include:
"Atlas of spectra of aromatic and hetrocyclic compounds", Koptyug, V.A. (editor), Science, Siberian Department of AS USSR, Novosibirsk, volumes 28-35 (1984-1986).
"Absorption spectra in the visible region", Lang, L. (editor) volumes 1-20, Budapest, 1959-1975.
"UV atlas of organic compounds", published in collaboration with the Photoelectric Spectrometry Group, London, and the Institute for Spectrochemie un Angewandte Sspectroskopie, Dortmund, volumes 1-5, London, Butterworths and Weinheim, Chemie, 1966-1971.
"Ultraviolet absorption spectra of aromatic hydrocarbons", Kusov, M.M., Shimanko, N.A., and Shishkina, M.V., AS USSR, Moscow.
"Ultraviolet absorption spectra of heteroorganic compounds", Bol'shakov, G.V., Vatago, V.S., and Agrest, F.B., Chemistry, Leningrad Department, Leningrad, 1969.
No data was used from "Ultraviolet spectra of aromatic compounds", Friedel, R.A., and Milton, M.O., U.S. Bureau of Mines, Bruceton PA, John Wiley and Sons, New York, 1951. 60 of the 600 substances in the publication were examined and found to often be different from spectra in later literature and have narrower wavelength disaption.
Data treatment
The UV/Vis spectra collected are taken mainly in the liquid phase (this reflects the nature of the literature the spectra are abstracted from). Consequently the data on the solvent used are included. In some rare cases when data published were obtained in the gas or vapor phases just such spectra were included in the collection.
It is typical for literature on UV/Vis spectra to contain a shortened form mentioning only absorption for some spectra plot peculiarities (couples of logarithm ε and wavelength). Data are only used if the source contains a spectrum in graph form or (a very rare case) a detailed digital form.
For the X and Y axes, nm and logarithm ε (the logarithm is base 10) are accepted as being used predominately in the recent literature for UV spectra presentation. Any other units, were recalcualted to match this convention. If the absorbency data in a literature source can not be presented in quantitative form, the source was omitted.
The spectrometer used is described strictly as stated in the original paper. The effective spectral resolution claimed for the measurements is treated likewise.
Auxiliary data
UV ID Registry number in the UV/Vis data collection.MELTING POINT Celsius scale data, updated with information from CRC Hanbdbook of Chemistry and Physics, David R. Lide (Editor-in-Chief), 78 th edition 1997-1998, CRC Press, Boca Raton, FL.BOILING POINT Same as melting point data. Abbreviations "sub." and "dec." mean sublimation and decomposition.SOVLENT/PHASE As stated in the original source. When only pH is indicated it means the solvent is water. The abbreviation "n.s.g." means no data about the solvent are given in the original source.INSTRUMENT As stated in the original source. Abbreviation "n.i.g." means no data about the instrument are given in the original source.
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190429 | https://rwu.pressbooks.pub/bio103/chapter/introduction-to-cells/ | Skip to content
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Chapter 7. Introduction to Cells
Chapter Outline
Type your examples here.
7.1 Studying Cells
7.2 Prokaryotic Cells
7.3 Eukaryotic Cells
7.4 The Endomembrane System
7.5 The Cytoskeleton
7.6 Connections Between Cells
Introduction
Close your eyes and picture a brick wall. What is the basic building block of that wall? A single brick, of course. Like a brick wall, your body is composed of basic building blocks, and the building blocks of your body are cells.
Your body has many kinds of cells, each specialized for a specific purpose. Just as a home is made from a variety of building materials, the human body is constructed from many cell types. For example, epithelial cells protect the surface of the body and cover the organs and body cavities within. Bone cells help to support and protect the body. Cells of the immune system fight invading bacteria. Additionally, blood and blood cells carry nutrients and oxygen throughout the body while removing carbon dioxide. Each of these cell types plays a vital role during the growth, development, and day-to-day maintenance of the body. In spite of their enormous variety, however, cells from all organisms—even ones as diverse as bacteria, onion, and human—share certain fundamental characteristics.
7.1 | Studying Cells
Learning Objectives
By the end of this section, you will be able to:
Describe the role of cells in organisms.
Compare and contrast light microscopy and electron microscopy.
Summarize cell theory.
A cell is the smallest unit of a living thing. A living thing, whether made of one cell (like bacteria) or many cells (like a human), is called an organism. Thus, cells are the basic building blocks of all organisms.
Several cells of one kind that interconnect with each other and perform a shared function form tissues, several tissues combine to form an organ (your stomach, heart, or brain), and several organs make up an organ system (such as the digestive system, circulatory system, or nervous system). Several systems that function together form an organism (like a human being). Here, we will examine the structure and function of cells.
There are many types of cells, all grouped into one of two broad categories: prokaryotic and eukaryotic. For example, both animal and plant cells are classified as eukaryotic cells, whereas bacterial cells are classified as prokaryotic. Before discussing the criteria for determining whether a cell is prokaryotic or eukaryotic, let’s first examine how biologists study cells.
7.1.1 Microscopy
Most individual cells cannot be seen with the naked eye, so scientists use microscopes (micro- = “small”; -scope = “to look at”) to study them. A microscope is an instrument that magnifies an object. Most photographs of cells are taken with a microscope, and these images can also be called micrographs.
Light Microscopes
To give you a sense of cell size, a typical human red blood cell is about eight millionths of a meter, or eight micrometers (abbreviated as 8 μm) in diameter; the head of a pin of is about two thousandths of a meter (2 mm) in diameter. That means about 250 red blood cells could fit across the head of a pin.
Most commonly used microscopes are classified as light microscopes (Figure 7.2a). Visible light passes and is bent through the lens system to enable the user to see the specimen. Light microscopes are advantageous for viewing living organisms, but since individual cells are generally transparent, their components are not distinguishable unless they are colored with special stains..
The optics of a light microscope’s lenses change the orientation of the image that the user sees. A specimen that is right-side up and facing right on the microscope slide will appear upside-down and facing left when viewed through a microscope, and vice versa. Similarly, if the slide is moved left while looking through the microscope, it will appear to move right, and if moved down, it will seem to move up. This occurs because microscopes use two sets of lenses to magnify the image. Because of the manner by which light travels through the lenses, this system of two lenses produces an inverted image (dissecting microscopes, work in a similar manner, but include an additional magnification system that makes the final image appear to be upright).
Light microscopes commonly used in the undergraduate college laboratory magnify up to approximately 400 times. Two parameters that are important in microscopy are magnification and resolving power. Magnification is the process of enlarging an object in appearance. Resolving power is the ability of a microscope to distinguish two adjacent structures as separate: the higher the resolution, the better the clarity and detail of the image. When oil immersion lenses are used for the study of small objects, magnification is usually increased to 1,000 times.
Electron Microscopes
In contrast to light microscopes, electron microscopes (Figure 7.2b) use a beam of electrons instead of a beam of light. Not only does this allow for higher magnification and, thus, more detail (Figure 7.3), it also provides higher resolving power. The method used to prepare the specimen for viewing with an electron microscope kills the specimen. Electrons have short wavelengths (shorter than photons) that move best in a vacuum, so living cells cannot be viewed with an electron microscope.
In a scanning electron microscope, a beam of electrons moves back and forth across a cell’s surface, creating details of cell surface characteristics. In a transmission electron microscope, the electron beam penetrates the cell and provides details of a cell’s internal structures. As you might imagine, electron microscopes are significantly more bulky and expensive than light microscopes.
Cytotechnologist
A Pap smear is a common medical test in which a small sample of cells are taken from the uterine cervix of a patient and sent to a medical lab. A professional called a cytotechnologist then stains the cells and examines them for any changes that could indicate cervical cancer or a microbial infection (Figure 7.4).
Cytotechnologists (cyto- = “cell”) study cells via microscopic examinations and other laboratory tests. They are trained to determine which cellular changes are within normal limits and which are abnormal. Their focus is not limited to cervical cells; they study cellular specimens that come from all organs. When they notice abnormalities, they consult a pathologist, who is a medical doctor who can make a clinical diagnosis. Cytotechnologists play a vital role in saving people’s lives. When abnormalities are discovered early, a patient’s treatment can begin sooner, which usually increases the chances of a successful outcome.
7.1.2 Cell Theory
The microscopes we use today are far more complex than those used in the 1600s by Antony van Leeuwenhoek, a Dutch shopkeeper who had great skill in crafting lenses. Despite the limitations of his now-ancient lenses, van Leeuwenhoek observed the movements of protista (a type of single-celled organism) and sperm, which he collectively termed “animalcules.”
In a 1665 publication called Micrographia, Robert Hooke coined the term “cell” for the box-like structures he observed when viewing cork tissue through a lens. In the 1670s, van Leeuwenhoek discovered bacteria and protozoa. Later advances in lenses, microscope construction, and staining techniques enabled other scientists to see some components inside cells. By the late 1830s, botanist Matthias Schleiden and zoologist Theodor Schwann were studying tissues and proposed the unified cell theory, which states that all living things are composed of one or more cells, the cell is the basic unit of life, and new cells arise from existing cells.
7.2 | Prokaryotic Cells
Learning Objectives
By the end of this section, you will be able to:
Name examples of prokaryotic and eukaryotic organisms.
Compare and contrast prokaryotic cells and eukaryotic cells.
Describe the relative sizes of different kinds of cells.
Explain why cells must be small.
Cells fall into one of two broad categories: prokaryotic and eukaryotic. Only the single-celled organisms of the domains Bacteria and Archaea are classified as prokaryotes. Cells of animals, plants, fungi, and protists are all eukaryotes, and are made up of eukaryotic cells.
7.2.1. Components of Prokaryotic Cells
All cells share four common components: 1) a plasma membrane, an outer covering that separates the cell’s interior from its surrounding environment; 2) cytoplasm, consisting of a jelly-like cytosol within the cell in which other cellular components are found; 3) DNA, the genetic material of the cell; and 4) ribosomes, which synthesize proteins. However, prokaryotes differ from eukaryotic cells in several ways.
A prokaryote is a simple, single-celled (unicellular) organism that lacks a nucleus, or other highly organized membrane-bound organelle (pro- = “before”; -kary- = “kernel,” which refers to the nucleus). Prokaryotic DNA is found in a central part of the cell called the nucleoid (Figure 7.5).
The prokaryotes, which include bacteria and archaea, are mostly single-celled organisms that, by definition, lack membrane-bound nuclei and other organelles. A bacterial chromosome is a covalently closed circle that, unlike eukaryotic chromosomes, is not organized around histone proteins. The central region of the cell in which prokaryotic DNA resides is called the nucleoid. In addition, prokaryotes often have abundant plasmids, which are shorter circular DNA molecules that may only contain one or a few genes. Plasmids can be transferred independently of the bacterial chromosome during cell division and often carry traits such as antibiotic resistance.
Most prokaryotes have a peptidoglycan cell wall and many have a polysaccharide capsule (Figure 7.5). The cell wall acts as an extra layer of protection, helps the cell maintain its shape, and prevents dehydration. The capsule enables the cell to attach to surfaces in its environment. Some prokaryotes have flagella, pili, or fimbriae. Flagella are used for locomotion. Pili are used to exchange genetic material during a type of reproduction called conjugation. Fimbriae are used by bacteria to attach to a host cell or other surface.
Microbiologist
The most effective action anyone can take to prevent the spread of contagious illnesses is to wash his or her hands. Why? Because microbes (organisms so tiny that they can only be seen with microscopes) are ubiquitous. They live on doorknobs, money, your hands, and many other surfaces. If someone sneezes into his hand and touches a doorknob, and afterwards you touch that same doorknob, the microbes from the sneezer’s mucus are now on your hands. If you touch your hands to your mouth, nose, or eyes, those microbes can enter your body and could make you sick.
However, not all microbes (also called microorganisms) cause disease; most are actually beneficial. You have microbes in your gut that make vitamin K. Other microorganisms are used to ferment beer and wine.
Microbiologists are scientists who study microbes. Microbiologists can pursue a number of careers. Not only do they work in the food industry, they are also employed in the veterinary and medical fields. They can work in the pharmaceutical sector, serving key roles in research and development by identifying new sources of antibiotics that could be used to treat bacterial infections.
Environmental microbiologists may look for new ways to use specially selected or genetically engineered microbes for the removal of pollutants from soil or groundwater, as well as hazardous elements from contaminated sites. These uses of microbes are called bioremediation technologies. Microbiologists can also work in the field of bioinformatics, providing specialized knowledge and insight for the design, development, and specificity of computer models of, for example, bacterial epidemics.
7.2.2 Cell Size
At 0.1 to 5.0 μm in diameter, prokaryotic cells are significantly smaller than most eukaryotic cells, which have diameters ranging from 10 to 100 μm (Figure 7.6). The small size of prokaryotes allows ions and organic molecules that enter them to quickly diffuse to other parts of the cell. Similarly, any wastes produced within a prokaryotic cell can quickly diffuse out. This is not the case in eukaryotic cells, which have developed different structural adaptations to enhance intracellular transport.
Small size is necessary for all cells, whether prokaryotic or eukaryotic. Not all cells are spherical in shape, but most tend to approximate a sphere. The formula for the surface area of a sphere is 4πr2, while the formula for its volume is 4πr3/3. Thus, as the radius of a cell increases, its surface area increases as the square of its radius, but its volume increases as the cube of its radius. Therefore, as a cell increases in size, its surface area-to-volume ratio decreases. This same principle would apply if the cell had the shape of a cube (Figure 7.7). If the cell grows too large, the plasma membrane will not have sufficient surface area to support the rate of diffusion required for the increased volume. In other words, as a cell grows, it becomes less efficient. One way to become more efficient is to divide; another way is to develop organelles that perform specific tasks. These adaptations lead to the development of more sophisticated cells called eukaryotic cells.
Concept Check
Prokaryotic cells are much smaller than eukaryotic cells.
What advantages might small cell size confer on a cell?
What advantages might large cell size have?
7.3 | Eukaryotic Cells
Learning Objectives
y the end of this section, you will be able to:
Describe the structure of eukaryotic cells.
Compare animal cells with plant cells.
State the role of the plasma membrane.
Summarize the functions of the major cell organelles.
Unlike prokaryotic cells, eukaryotic cells have a membrane-bound nucleus, numerous membrane-bound organelles that have specialized functions, and several, rod-shaped chromosomes. Because a eukaryotic cell’s nucleus is surrounded by a membrane, the word eukaryote means “true nucleus.”
Eukaryotic cells have a more complex structure than prokaryotic cells. Organelles (“little organs”) allow different functions to be compartmentalized in different areas of the cell. Before turning to organelles, let’s first examine two important components of the cell: the plasma membrane and the cytoplasm.
7.3.1 The Plasma Membrane
Like prokaryotes, eukaryotic cells have a plasma membrane (Figure 7.8), consisting of a phospholipid bilayer with embedded proteins, that separates the internal contents of the cell from its surrounding environment. The plasma membrane controls the passage of organic molecules, ions, water, and oxygen into and out of the cell. Wastes (such as carbon dioxide and ammonia) also leave the cell by passing through the plasma membrane.
The plasma membranes of cells that specialize in absorption are folded into fingerlike projections called microvilli (Figure 7.9). Such cells are typically found lining the small intestine, the organ that absorbs nutrients from digested food. People with celiac disease have an immune response to gluten, which is a protein found in wheat, barley, and rye. The immune response damages microvilli, preventing absorption of nutrients and leading to malnutrition, cramping, and diarrhea. Patients with celiac disease must follow a gluten-free diet.
7.3.2 The Cytoplasm
The cytoplasm is the entire region of a cell between the plasma membrane and the nuclear envelope. It is made up of organelles suspended in the gel-like cytosol, the cytoskeleton, and various chemicals (Figure 7.10). Even though the cytoplasm consists of 70 to 80 percent water, it has a semi-solid consistency, which comes from the proteins within it. Glucose and other simple sugars, polysaccharides, amino acids, nucleic acids, fatty acids, and ions are also found in the cytoplasm. Many metabolic reactions take place in the cytoplasm.
7.3.3 Organelles
The cytoplasm of eukaryotic cells is highly compartmentalized into structures called organelles. Like organs in your body, each type of organelle has a specialized function. The organelles and other cellular structures found in typical animal (Figure 7.10a) and plant (Figure 7.10b) cells are diagrammed below.
The Nucleus
Typically, the nucleus is the most prominent organelle in a cell. The nucleus (plural = nuclei) houses the cell’s DNA and directs the synthesis of ribosomes and proteins. Let’s look at it in more detail (Figure 7.11).
The Nuclear Envelope
The nuclear envelope is a double-membrane structure that constitutes the outermost portion of the nucleus (Figure 7.11). Both the inner and outer membranes of the nuclear envelope are phospholipid bilayers. The nuclear envelope is punctuated with pores that control the passage of ions, molecules, and RNA between the nucleoplasm and cytoplasm. The nucleoplasm is the semi-solid fluid inside the nucleus.
Chromatin and Chromosomes
Chromosomes are structures within the nucleus that are made up of DNA, the hereditary material. You may remember that in prokaryotes, DNA is organized into a single circular chromosome. In eukaryotes, chromosomes are linear structures. Every eukaryotic species has a specific number of chromosomes in the nuclei of its body’s cells. For example human cells have 46 chromosomes. Chromosomes are only visible and distinguishable from one another when the cell is getting ready to divide. When the cell is in the growth and maintenance phases of its life cycle, proteins are attached to chromosomes, and they resemble an unwound, jumbled bunch of threads. These unwound protein-chromosome complexes are called chromatin ( Figure 7.12).
The Nucleolus
We already know that the nucleus directs the synthesis of ribosomes, but how does it do this? Some chromosomes have sections of DNA that encode ribosomal RNA (rRNA). The darkly staining area within the nucleus called the nucleolus (plural = nucleoli) aggregates the ribosomal RNA with associated proteins to assemble ribosomal subunits that are then transported out through the pores in the nuclear envelope to the cytoplasm.
Ribosomes
Ribosomes are the cellular structures responsible for protein synthesis. They are large complexes of protein and rRNA, consisting of a large and a small subunit (Figure 7.13). When viewed through an electron microscope, ribosomes appear either as clusters or as single, tiny dots that float freely in the cytoplasm. They may be attached to the inside of the plasma membrane or the outside of the endoplasmic reticulum or nuclear envelope (Figure 7.11). Ribosomes bind to mRNA and translate the code provided by the sequence of the nitrogenous bases in the mRNA into a specific order of amino acids in a protein.
Because proteins synthesis is an essential function of all cells, ribosomes are found in practically every cell. Ribosomes are particularly abundant in cells that synthesize large amounts of protein. For example, the pancreas is responsible for creating several digestive enzymes and the cells that produce these enzymes contain many ribosomes.
Mitochondria
Mitochondria (singular = mitochondrion) are often called the “powerhouses” or “energy factories” of a cell because they are responsible for making adenosine triphosphate (ATP), the cell’s main energy-carrying molecule. ATP is made using the chemical energy found in glucose and other nutrients by the process of cellular respiration. In mitochondria, this process uses oxygen and produces carbon dioxide as a waste product. In fact, the carbon dioxide that you exhale with every breath comes from the breakdown of nutrients you eat inside mitochondria.
Mitochondria are oval-shaped, double membrane organelles (Figure 7.14) that have their own ribosomes and DNA. Each membrane is a phospholipid bilayer embedded with proteins. The inner membrane has folds called cristae that increase surface area. The space between the two membranes is called the intermembrane space and the space inside the inner membrane is called the matrix. ATP synthesis takes place on the inner membrane.
Peroxisomes
Peroxisomes are small, spherical organelles enclosed by single membranes. They carry out oxidation reactions that break down fatty acids and amino acids. They also detoxify many poisons that may enter the body. (Many of these oxidation reactions release hydrogen peroxide, H2O2, which would be damaging to cells; however, when these reactions are confined to peroxisomes, enzymes safely break down the H2O2 into oxygen and water.) For example, alcohol is detoxified by peroxisomes in liver cells. Glyoxysomes, which are specialized peroxisomes in plants, are responsible for converting stored fats into sugars.
Vesicles and Vacuoles
Vesicles and vacuoles are membrane-bound sacs that function in storage and transport. Other than the fact that vacuoles are somewhat larger than vesicles, there is a very subtle distinction between them: The membranes of vesicles can fuse with either the plasma membrane or other membrane systems within the cell. Additionally, some agents such as enzymes within plant vacuoles break down macromolecules. The membrane of a vacuole does not fuse with the membranes of other cellular components. Some types of cells have contractile vacuoles, which contract to expel water from the cell.
7.,3.4 Animal Cells versus Plant Cells
At this point, you know that each eukaryotic cell has a plasma membrane, cytoplasm, a nucleus, ribosomes, mitochondria, peroxisomes, and in some, vacuoles, but there are some striking differences between animal and plant cells. While both animal and plant cells have microtubule organizing centers (MTOCs), animal cells also have centrioles associated with the MTOC: a complex called the centrosome. Animal cells each have a centrosome and lysosomes, whereas plant cells do not. Plant cells have a cell wall, chloroplasts and other specialized plastids, and a large central vacuole, whereas animal cells do not.
The Centrosome
The centrosome is a microtubule-organizing center from which all microtubules in the cell originate. It is found near the nucleus of animal cells. Each centrosome contains a pair of centrioles, which lie perpendicular to each other (Figure 7.15). Each centriole is a cylinder of nine triplets of microtubules.
The centrosome replicates itself before a cell divides, and the centrioles appear to have some role in pulling the duplicated chromosomes to opposite ends of the dividing cell. However, the exact function of the centrioles in cell division isn’t clear, because cells that have had the centrosome removed can still divide, and plant cells, which lack centrosomes, are capable of cell division.
Lysosomes
Animal cells have another set of organelles not found in plant cells: lysosomes. Lysosomes are the cell’s “garbage disposal.” Enzymes inside lysosomes aid the breakdown of proteins, polysaccharides, lipids, nucleic acids, and even worn- out organelles. These enzymes are active at a much lower pH than that of the cytoplasm. Therefore, the pH within lysosomes is more acidic than the pH of the cytoplasm. Proton pumps in the lysosome membrane pump protons in to maintain the low pH inside the lysosome. In plant cells, digestive processes take place in vacuoles.
The Cell Wall
If you examine Figure 7.10b, the diagram of a plant cell, you will see a structure external to the plasma membrane called the cell wall. The cell wall is a rigid covering that protects the cell, provides structural support, and gives shape to the cell. While the chief component of prokaryotic cell walls is peptidoglycan, the major organic molecule in the plant cell wall is cellulose (Figure 7.16), a polysaccharide made up of glucose units. When you bite into a raw vegetable, like celery, it crunches because you are tearing the rigid cell walls of the celery cells with your teeth. Fungus and some protist cells also have cell walls, which are made of other structural molecules.
Chloroplasts
Like mitochondria, chloroplasts have their own DNA and ribosomes, but chloroplasts have an entirely different function. Chloroplasts are plant cell organelles that carry out photosynthesis. Photosynthesis is the series of reactions that use carbon dioxide, water, and light energy to make glucose and oxygen. This is a major difference between plants and animals; plants (autotrophs) are able to make their own food, like sugars, while animals (heterotrophs) must ingest their food.
Like mitochondria, chloroplasts have outer and inner membranes, but within the space enclosed by a chloroplast’s inner membrane is a set of interconnected and stacked fluid-filled membrane sacs called thylakoids (Figure 7.17). Each stack of thylakoids is called a granum (plural = grana). The fluid enclosed by the inner membrane that surrounds the grana is called the stroma. The fluid inside the thylakoids is called the thylakoid space. Light energy is harvested in the thylakoid membranes and sugar is made in the stroma.
Chloroplasts contain a green pigment called chlorophyll, which captures the light energy that drives the reactions of photosynthesis. Like plant cells, photosynthetic protists also have chloroplasts. Some bacteria perform photosynthesis, but their photosynthetic pigments are not relegated to an organelle.
The Central Vacuole
Plant cells each have a large central vacuole that occupies most of the area of the cell (Figure 7.8b). The central vacuole plays a key role in regulating the cell’s concentration of water in changing environmental conditions. If you forget to water a plant for a few days, it wilts because when environmental water levels are low, water moves out of the central vacuoles and cytoplasm. As the central vacuole shrinks, it leaves the cell wall unsupported, resulting in wilting. The central vacuole also supports the expansion of the cell. When the central vacuole holds more water, the cell gets larger without having to invest a lot of energy in synthesizing new cytoplasm.
Endosymbiosis
We have mentioned that both mitochondria and chloroplasts contain DNA and ribosomes. Have you wondered why? Strong evidence points to endosymbiosis as the explanation.
Symbiosis is a relationship in which organisms from two separate species depend on each other for their survival. Endosymbiosis (endo- = “within”) is a mutually beneficial relationship in which one organism lives inside the other. Endosymbiotic relationships abound in nature. We have already mentioned that microbes that produce vitamin K live inside the human gut. This relationship is beneficial for us because we are unable to synthesize vitamin K. It is also beneficial for the microbes because they are protected from other organisms and from drying out, and they receive abundant food from the environment of the large intestine.
Scientists have long noticed that bacteria, mitochondria, and chloroplasts are similar in size. We also know that bacteria have DNA and ribosomes, just as mitochondria and chloroplasts do. Scientists believe that host cells and bacteria formed an endosymbiotic relationship when the host cells ingested both aerobic and autotrophic bacteria (cyanobacteria) but did not destroy them. Through many millions of years of evolution, these ingested bacteria became more specialized in their functions, with the aerobic bacteria becoming mitochondria and the autotrophic bacteria becoming chloroplasts.
7.4 | The Endomembrane System
Learning Objectives
By the end of this section, you will be able to:
List the components of the endomembrane system.
Recognize the relationship between the endomembrane system and its functions.
The endomembrane system (endo = “within”) is a group of membranes and organelles (Figure 7.18) in eukaryotic cells that works together to modify, package, and transport lipids and proteins. It includes the nuclear envelope, lysosomes, and vesicles, which we’ve already mentioned, as well as the endoplasmic reticulum and Golgi apparatus. Although not technically within the cell, the plasma membrane is sometimes included in the endomembrane system because it interacts with the other endomembranous organelles.
7.4.1 The Endoplasmic Reticulum
The endoplasmic reticulum (ER) is a series of interconnected membranous sacs and tubules that collectively modifies proteins and synthesizes lipids. However, these two functions are performed in separate areas of the ER: the rough ER and the smooth ER, respectively.
The hollow portion of the ER tubules is called the lumen or cisternal space. The membrane of the ER, which is a phospholipid bilayer embedded with proteins, is continuous with the nuclear envelope.
Rough ER
The rough endoplasmic reticulum (RER) is so named because ribosomes attached to its cytoplasmic surface give it a studded appearance when viewed through an electron microscope (Figure 7.19). These bound ribosomes are in the process of translating proteins directly into the lumen of the RER.
Three types of proteins are made on bound ribosomes: proteins that will end up secreted from the cell: proteins that will end up inserted into the plasma membrane of the cell, and proteins that will end up inside organelles in the cell. What these three classes of proteins have in common is that they will not end up in the cytosol of the cell. From the moment that they start to be synthesized, these proteins are sequestered inside membrane-bound compartments and will never touch the cytosol.
Once the newly synthesized proteins are deposited into the lumen of the RER, they undergo structural modifications, such as folding or addition of side chains. If the modified proteins are not destined to stay in the RER, they will travel by small, membrane-bound transport vesicles that bud from the RER’s membrane to the Golgi apparatus (Figure 7.18).
Since the RER is engaged in modifying proteins that will be secreted from the cell, the RER is abundant in cells that secrete proteins, such as liver cells, for example.
Smooth ER
The smooth endoplasmic reticulum (SER) is continuous with the RER but has few or no ribosomes on its surface (Figure 7.19). Functions of the SER include synthesis of carbohydrates, lipids, phospholipids, and steroid hormones and detoxification of medications and poisons. The SER takes on different functions depending on the needs of the cell. For example, in muscle cells, a specialized SER called the sarcoplasmic reticulum stores calcium ions that are needed to trigger the coordinated contractions of the muscle cells. Cells that make a lot of lipids have a large amount of SER. For example, Leydig cells in mammalian testes produce steroid hormones such as testosterone and therefore have abundant SER.
7.4.2 The Golgi Apparatus
Vesicles containing proteins and lipids bud from the ER and transport their contents to the Golgi apparatus (GA). In the GA, proteins and lipids are sorted, packaged, and tagged so that they wind up in the right place. For this reason, the GA is sometimes called the post office of the cell. The GA is a series of flattened membranes, each forming a separate compartment The receiving side of the Golgi apparatus is called the cis face. The opposite side is called the trans face (Figure 7.20).
Transport vesicles from the ER fuse with the cis face and empty their contents into the lumen of the GA. The proteins and lipids travel from compartment to compartment by vesicle. As they travel through the GA, they undergo further modifications. The most frequent modification is the addition of short chains of sugar molecules. They may also be tagged with phosphate groups or other small molecules so that they can be routed to their proper destinations.
Finally, the modified and tagged proteins are packaged into vesicles that bud from the trans face of the Golgi. Some of these vesicles deposit their contents into other organelles of the cell. Other vesicles fuse with the plasma membrane and their contents end up outside the cell or inserted into the plasma membrane.
Cells that engage in a great deal of secretory activity, such as cells of the salivary glands that secrete digestive enzymes, have an abundance of GA. In plant cells, the Golgi apparatus has the additional role of synthesizing polysaccharides, some of which are incorporated into the cell wall and some of which are used in other parts of the cell.
Cardiologist
Heart disease is the leading cause of death in the United States. This is primarily due to sedentary lifestyles and poor diets. Heart failure is just one of many disabling heart conditions. Heart failure occurs when the heart cannot pump with sufficient force to transport oxygenated blood to all the vital organs.
Heart failure occurs when the endoplasmic reticula of cardiac muscle cells do not function properly. As a result, an insufficient number of calcium ions are available to trigger a sufficient contractile force.
Cardiologists (cardi- = “heart”) are doctors who specialize in treating heart diseases. Cardiologists can diagnose heart failure via physical examination, results from an electrocardiogram (a test that measures the electrical activity of the heart), a chest X-ray to see whether the heart is enlarged, and other tests. If heart failure is diagnosed, the cardiologist will typically prescribe appropriate medications and recommend a reduction in table salt intake and a supervised exercise program.
7.4.3 Lysosomes
Lysosomes are also part of the endomembrane system. Lysosomes are small, spherical compartments that function as the digestive and organelle-recycling facility of animal cells, They contain hydrolytic enzymes that digest non-functioning organelles, macromolecules, and pathogens (disease-causing organisms). The enzymes in lysosomes are called acid hydrolases because they catalyze hydrolysis reaction and their optimal pH is around 4.5. Lysosomes contain proton pumps in their membranes that pump hydrogen ions into the lumen, thereby lowering the pH. This elegant mechanism allows the enzymes to be safely made and transported through the endomembrane system in an inactive state. They become active only when they encounter the acidic environment inside the lysosome.
Lysosomes also use their hydrolytic enzymes to destroy pathogens that enter the cell. A good example of this occurs in a type of white blood cells called macrophages, which are part of your body’s immune system. In a process known as phagocytosis, macrophages engulf pathogens into a vesicle. The vesicle containing the pathogen fuses with a lysosome, and the lysosome’s enzymes destroy the pathogen (Figure 7.21).
7.5 | The Cytoskeleton
Learning Objectives
By the end of this section, you will be able to:
Describe the cytoskeleton.
Compare the roles of microfilaments, intermediate filaments, and microtubules.
Compare and contrast cilia and flagella.
Summarize the differences among the components of prokaryotic cells, animal cells, and plant cells.
If you were to remove all the organelles from a cell, would the plasma membrane and the cytoplasm be the only components left? No. Within the cytoplasm, there would still be ions and organic molecules, plus a network of protein fibers that help maintain the shape of the cell, secure some organelles in specific positions, allow cytoplasm and vesicles to move within the cell, and enable cells within multicellular organisms to move. Collectively, this network of protein fibers is known as the cytoskeleton. There are three types of fibers within the cytoskeleton: microfilaments, intermediate filaments, and microtubules (Figure 7.22). Here, we will examine each.
7.5.1 Microfilaments
Of the three types of protein fibers in the cytoskeleton, microfilaments are the thinnest. They function in cellular movement and are made of two intertwined strands of a globular protein called actin (Figure 7.23). For this reason, microfilaments are also known as actin filaments.
Microfilaments provide some rigidity and shape to cells. They can depolymerize (disassemble) and reform quickly, thus enabling a cell to change its shape and move. Microfilaments are also involved in cellular movement, such as cell division in animal cells and cytoplasmic streaming, which is the circular movement of the cell cytoplasm, in plant cells. Actin also helps muscle cells contract.
7.5.2 Intermediate Filaments
Intermediate filaments are made of several strands of fibrous proteins that are wound together (Figure 7.24). These elements of the cytoskeleton get their name from the fact that their diameter is between those of microfilaments and microtubules. The function of intermediate filaments is purely structural. They bear tension, thus maintaining the shape of the cell, and create a supportive scaffolding to anchor the nucleus and other organelles in place.
Intermediate filaments are the most diverse group of cytoskeletal elements. Several types of fibrous proteins are found in the intermediate filaments, including keratin, the fibrous protein that strengthens your hair, nails, and skin.
7.5.3 Microtubules
As their name implies, microtubules are small hollow tubes. The walls of the microtubule are made of dimers of α-tubulin and β-tubulin, two globular proteins (Figure 7.25). Microtubules are the widest components of the cytoskeleton. They help the cell resist compression, provide a track along which vesicles move through the cell, and pull replicated chromosomes to opposite ends of a dividing cell. Like microfilaments, microtubules can dissolve and reform quickly.
Microtubules are also the structural elements of flagella, cilia, and centrioles. In fact, in animal cells, the centrioles are the microtubule-organizing center. In eukaryotic cells, flagella and cilia are quite different structurally from their counterparts in prokaryotes, as discussed below.
Flagella and Cilia
Flagella (singular = flagellum) are long, hair-like structures that extend from the plasma membrane and are used to move an entire cell. When present, a cell has just one flagellum or a few flagella. Cilia (singular = cilium) are short, hair-like structures that are used to move entire cells or to move substances along the outer surface of the cell. For example, the cells lining the ovarian tubes have cilia that move the ovum toward the uterus, and the cells lining the respiratory tract have cilia that move mucus toward your nostrils. When cilia are present, they extend along the entire surface of the plasma membrane.
Despite their differences in length and number, flagella and cilia share a common structural arrangement of microtubules called a “9 + 2 array.” A single flagellum or cilium is made of a ring of nine microtubule pairs surrounding a single microtubule pair in the center (Figure 7.26).
Table 7.1 Components of prokaryotic and eukaryotic cells.
| | | | | |
--- ---
| Component | Function | In Pro-karyotes? | In Animal Cells? | In Plant Cells? |
| Plasma membrane | Separates cell from external environment; controls passage of organic molecules, ions, water, oxygen, and wastes in and out of cell | Yes | Yes | Yes |
| Cytoplasm | Provides turgor pressure to plant cells as fluid inside the central vacuole; site of many metabolic reactions; medium in which organelles are found | Yes | Yes | Yes |
| Nucleus | Cell organelle that houses DNA and directs synthesis of ribosomes and proteins | No | Yes | Yes |
| Nucleolus | Darkened area within the nucleus where ribosomal subunits are synthesized. | No | Yes | Yes |
| Ribosomes | Protein Synthesis | Yes | Yes | Yes |
| Mitochondria | ATP production/cellular respiration | No | Yes | Yes |
| Peroxisomes | Oxidizes and thus breaks down fatty acids and amino acids, and detoxifies poisons | No | Yes | Yes |
| Vesicles and Vacuoles | Storage and transport; digestive function in plant cells | No | Yes | Yes |
| Centrosome | Unspecified role in cell division in animal cells; source of microtubules in animal cells | No | Yes | No |
| Lysosome | Digestion of macromolecules; recycling of worn-out organelles | No | Yes | No |
| Cell Wall | Protection, structural support and maintenance of cell shape | Yes | No | Yes |
| Chloroplasts | Photosynthesis | No | No | Yes |
| Endoplasmic Reticulum | Modifies proteins and synthesizes lipids | No | Yes | Yes |
| Golgi Apparatus | Modifies, sorts, tags, packages, and distributes lipids and proteins | No | Yes | Yes |
| | Maintains cell’s shape, secures organelles in specific positions, allows cytoplasm and vesicles to move within cell, and enables unicellular organisms to move independently | Yes | Yes | Yes |
| | Cellular locomotion | | Some | Rarely – in sperm |
| | Cellular locomotion, movement of particles along extracellular surface of plasma membrane, and filtration | Some | Some | No |
7.6 | Connections Between Cells
Learning Objectives
By the end of this section, you will be able to:
Describe the extracellular matrix.
List examples of the ways that plant cells and animal cells communicate with adjacent cells.
Summarize the roles of tight junctions, desmosomes, gap junctions, and plasmodesmata.
You already know that a group of similar cells working together is called a tissue. As you might expect, if cells are to work together, they must communicate with each other, just as you need to communicate with others if you work on a group project. Let’s take a look at how cells communicate with each other.
7.6.1 Extracellular Matrix of Animal Cells
Most animal cells release materials into the extracellular space. The primary components of these materials are proteins, and the most abundant protein is collagen. Collagen fibers are interwoven with carbohydrate-containing protein molecules called proteoglycans. Collectively, these materials are called the extracellular matrix (Figure 7.27). Not only does the extracellular matrix hold the cells together to form a tissue, but it also allows the cells within the tissue to communicate with each other. How can this happen?
Cells have protein receptors on the extracellular surfaces of their plasma membranes. When a molecule within the matrix binds to the receptor, it changes the molecular structure of the receptor. The receptor, in turn, changes the conformation of the microfilaments positioned just inside the plasma membrane. These conformational changes induce chemical signals inside the cell that reach the nucleus and turn “on” or “off” the transcription of specific sections of DNA, which affects the production of associated proteins, thus changing the activities within the cell.
Blood clotting provides an example of the role of the extracellular matrix in cell communication. When the cells lining a blood vessel are damaged, they display a protein receptor called tissue factor. When tissue factor binds with another factor in the extracellular matrix, it causes platelets to adhere to the wall of the damaged blood vessel, stimulates the adjacent smooth muscle cells in the blood vessel to contract (thus constricting the blood vessel), and initiates a series of steps that stimulate the platelets to produce clotting factors.
7.6.2 Intercellular Junctions
Adjacent cells can also communicate with each directly through intercellular junctions. Plasmodesmata are junctions between plant cells; tight junctions, gap junctions, and desmosomes occur between animal cells.
Plasmodesmata
In general, plasma membranes of neighboring plant cells cannot touch one another because they are separated by the cell wall that surrounds each cell. Plasmodesmata (singular = plasmodesma) are numerous channels that pass between cell walls of adjacent plant cells, connect their cytoplasm, and enable materials to be transported from cell to cell (Figure 7.28). This allows plants to transfer water and other soil nutrients from its roots, through its stems, and to its leaves.
Tight Junctions
A tight junction is a watertight seal between two adjacent animal cells (Figure 7.29A). The cells are held tightly against each other by proteins, preventing materials from leaking between the cells. Tight junctions are typically found in tissues that line internal organs and cavities. For example, the tight junctions of the epithelial cells lining your urinary bladder prevent urine from leaking out into the extracellular space. Tight junctions in the gut normally prevent food from leaking between cells into your body. However, these junctions can be regulated to allow some substances to pass through.
Desmosomes
Also found only in animal cells are desmosomes, which act like spot welds between adjacent epithelial cells (Figure 7.29B). Proteins in the plasma membrane connect to intermediate filaments to create desmosomes. Effectively, the cytoskeletons of the two cells are linked together so they cannot easily be pulled apart. Skin is an example of a tissue with numerous desmosomes. This allows you to pull on your skin without it ripping apart.
Gap Junctions
Gap junctions in animal cells are similar to plant cell plasmodesmata. They are channels between adjacent cells that allow for the transport of ions, nutrients, and other substances that enable cells to communicate (Figure 7.29C). Structurally, however, gap junctions and plasmodesmata differ.
Gap junctions develop when a set of six proteins (called connexins) in the plasma membrane arrange themselves in a ring, called a connexon. When the pores of connexons in adjacent cells align, they form a channel between the two cells. Gap junctions are particularly important in cardiac muscle. The electrical signal for the muscle to contract is passed efficiently through gap junctions, allowing the heart muscle cells to contract in unison. |
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Geometric Sequence with formula for kth term.
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The problem looks like this
Sequence
−712,21316,−17128,4051024,−12158192
Thanks to the final two numbers, I was able to recognize the ratio38. I know this is a geometric sequence, so the ratio would be 38k. Given that it oscillates back and forth and it starts on a negative number, I would think that my formula would include a −1k+1 for n≥0.
Since I have my original value a=−712 and my ration 38k as well as my oscillator −1k+1, I would think that my function would be
ak=712⋅(38)k(−1)k+1.
Does this look correct?
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edited Nov 13, 2017 at 8:14
Michael Rozenberg
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asked Nov 13, 2017 at 5:05
PodoPodo
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It is incorrect to write −1k+1 if you mean (−1)k+1, and I corrected that in this question.
Michael Hardy
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Michael Hardy
11/13/2017 05:39:51
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Nov 13, 2017 at 5:39
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If you mean ak=712(38)k(−1)k+1 then your formula is wrong for k=1.
It should be
ak=−712(−38)k−1.
If you want that a0=−712 then it should be
ak=−712(−38)k.
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edited Nov 13, 2017 at 5:29
answered Nov 13, 2017 at 5:20
Michael RozenbergMichael Rozenberg
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Is that fine if we consider the 0th case? That would give us a -1 for the exponent, wouldn't it? So a0 = -7.5 (-(3/8))^-1, which wouldn't give us -7.5?
Podo
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Podo
11/13/2017 05:24:06
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Nov 13, 2017 at 5:24
And yes, that is what I mean, i'll change it in my question
Podo
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Podo
11/13/2017 05:25:37
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Nov 13, 2017 at 5:25
@Jeffrey Dilley I think that a1=−712. I added something. See now.
Michael Rozenberg
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Michael Rozenberg
11/13/2017 05:30:05
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Nov 13, 2017 at 5:30
looks good! Thank you :)
Podo
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Podo
11/13/2017 05:30:49
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Nov 13, 2017 at 5:30
@Jeffrey Dilley You are welcome!
Michael Rozenberg
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Michael Rozenberg
11/13/2017 05:31:44
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The ratios of successive terms (second over first) are
−38,−38,−38,−38
and you indeed have a geometric progression of common ratio −38.
Hence
ak=a0rk=−712(−38)k.
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answered Nov 13, 2017 at 8:25
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190431 | https://study.com/academy/lesson/algebraic-identities-definition-examples.html | Algebraic Identities | Definition & Examples | Study.com
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Algebraic Identities | Definition & Examples
Instructor Michelle KimbroughShow bio
Michelle has taught high school math for five years. She has a Bachelors Degree and Masters Degree in Mathematics from Southern Illinois University and Texas A&M respectively. She is a highly certified 7th-12th grade math teacher in the state of Tennessee. She has tutored and taught many math subjects including Algebra 1 and 2, Geometry, Statistics, Pre-calculus, Trigonometry, and Calculus.
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Learn what an identity is in algebra and identify common algebraic identities. See examples of algebraic identities and use math identities to solve problems.
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Table of Contents
What is an Identity in Algebra?
Algebraic Identity: Examples
Additional Algebraic Identities
Proofs of Algebraic Identities
Using Algebraic Identities to Solve Example Problems
Lesson Summary
Show
What is an Identity in Algebra? -------------------------------
In mathematics, equations are often used. It is important to understand what an identity in math is and how it differs from just any mathematical equation. An identity in math is a statement or equation that holds true no matter which values are chosen for the variables. Therefore, an algebraic identity will be true for any value chosen for the variables, whereas many algebraic equations are only true for certain values. With this in mind, all algebraic identities are equations because they are equal on both sides, independent of the chosen variables; however, not all equations are identities. Only algebraic equations that are true for ALL variables may be considered identity equations. There are many algebraic identities that can be used to help re-write or manipulate algebraic expressions and polynomials. Polynomials are expressions made of sums or differences of two or more terms. Due to the nature of an algebraic identity being true for all values of variables, these manipulations can help in factoring squares/cubes of binomials or multiplying binomials together. Factoring can be helpful to simplify expressions and/or solve for variables. The algebraic identities are crucial to help solve, factor, and re-write expressions.
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Algebraic Identity: Examples ----------------------------
There are a number of commonly used algebraic identities. Algebraic identities are the mathematical equations that hold true for any set of variables. There are commonly used identities that are useful to simplify expressions raised to a power.
Second Power Algebraic Identities
There are many second power algebraic identities that are useful in raising binomials to a power. Binomials are the sum or difference of two terms. These can be seen in the first two identities. Identity 3 and 4 are useful to understand multiplying binomials and factoring to get binomials.
Identity 1: The Square of a Sum (Binomial): (a+b)2=a 2+2 a b+b 2
Identity 2: The Square of a Difference (Binomial): (a−b)2=a 2−2 a b+b 2
Identity 3: The Difference of Two Squares: a 2−b 2=(a+b)(a−b)
Identity 4: (x+a)(x+b)=x 2+(a+b)x+a b
Identity 5: The Square of a Sum (Trinomial): (a+b+c)2=a 2+b 2+c 2+2 a b+2 b c+2 a c
Third Power Algebraic Identities
Identity 6: The Cube of a Sum (Binomial): (a+b)3=a 3+b 3+3 a b(a+b)
Identity 7: The Cube of a Difference (Binomial): (a−b)3=a 3−b 3−3 a b(a−b)
Identity 8: The Sum of Two Cubes: a 3+b 3=(a+b)(a 2−a b+b 2)
Identity 9: The Difference of Two Cubes: a 3−b 3=(a−b)(a 2+a b+b 2)
Additional Algebraic Identities -------------------------------
Along with the previous nine identities noted, there are additional identities that are useful in algebra. In algebra, when multiplying two binomials together, the "FOIL" method is often used. This stands for "First, Outer, Inner, Last," as seen in the image below. This identity follows the pattern of multiplying the first terms, outer terms, inner terms, and last terms. It results in Identity 10.
Image 1: FOIL Method
Identity 10: The "FOIL" Method: (a+b)(c+d)=a c+a d+b c+b d
These identities are the most commonly used in algebra level mathematics. However, the identities continue to extend to higher levels of math and are used for various purposes. An extension of the square of a sum and cube of a sum is the binomial identity. This identity allows a generalization for raising a binomial to any power.
Idenity 11: Binomial Theoerem: (a+b)n=Σ k=0 n(n k)a n−k b k
Proofs of Algebraic Identities ------------------------------
The algebraic identities are true for all values of variables which allows each identity to be mathematically proven. There are multiple ways to prove these identities; however, here are a few examples.
Example 1: Prove Identity 4: (x+a)(x+b)=x 2+(a+b)x+a b
This proof will be an algebraic proof.
Given:
(x+a)(x+b)
This can be re-written as
x(x+b)+a(x+b) by the distributive property.
Next by multiplying, we have
x 2+x b+a x+a b.
Then using factoring, we have
x 2+x(b+a)+a b.
Using the symmetric property to re-write, results in
x 2+(a+b)x+a b.
This proves Identity 4.
Example 2: Prove Identity 1: The Square of a Sum (Binomial): (a+b)2=a 2+2 a b+b 2.
This can be proven algebraically in a method very similar to Example 1. However, the following proof is a geometry proof of this identity.
Starting with (a+b)2, a square can be constructed with each side having a length of a+b.
Image 2: Square of side length a+b
Using this square of side lengths a+b, it is now possible to split the square into four rectangles with side lengths a and b. The rectangles are as follows: a by a, a by b, b by b, and b by a (dimensions shown in green, blue, red, and yellow, respectively).
Image 3: Square split into 4 rectangles
Using this image, the areas may be added together to represent the whole square which is (a+b)2. Finding each individual area, it is seen that we have the following:
Figure 4: Each rectangle area shown
Now by adding the areas together, it is seen that (a+b)2=a 2+a b+b a+b 2.
It is known by the symmetric property that a b=b a. Therefore we have (a+b)2=a 2+2 a b+b 2, which proves the square of sum identity.
Using Algebraic Identities to Solve Example Problems ----------------------------------------------------
As stated before, algebraic identities allow the opportunity to solve many problems. A few examples are provided to show how algebraic identities may be used.
Example 1: Solve for the value of x: (x+5)(x+3)=15.
To start, we will use Identity 4 to re-write the left side:
x 2+8 x+15=15.
Now the subtraction property of equality states that we can subtract 15 from both sides, obtaining:
x 2+8 x=0.
Using this result, we can factor to solve:
x(x+8)=0 x=0,x+8=0.
This gives the result of x=0,x=−8.
Example 2: Solve for a: a 2−8 a+16=25.
Notice that the left side of this equation matches Identity 2: The Square of Difference Identity. Therefore, it is possible to re-write the left side of the equation using this identity. This results in the following:
(a−4)2=25.
Now, using algebraic properties, we can solve for a. First, by taking the square root of both sides to obtain:
a−4=±5.
Using the addition property of equality, we can add 4 to each side to solve the following:
a=9,a=−1.
Lesson Summary --------------
An identity in math is an equation that is always true. There are many useful algebraic identities which are equations that are true for all values of variables. Algebraic identities are helpful when dealing with polynomials (expressions with sums or differences of two or more terms) and binomials raised to a power (expressions with sums/differences of two terms). There are many algebraic identities, but some of the most commonly used identities in algebra are:
The Square of a Sum (Binomial): (a+b)2=a 2+2 a b+b 2
The Square of a Difference (Binomial): (a−b)2=a 2−2 a b+b 2
The Difference of Two Squares: a 2−b 2=(a+b)(a−b)
Binomial multiplication (with two like-variables): (x+a)(x+b)=x 2+(a+b)x+a b
The Square of a Sum (Trinomial): (a+b+c)2=a 2+b 2+c 2+2 a b+2 b c+2 a c
The Cube of a Sum (Binomial): (a+b)3=a 3+b 3+3 a b(a+b)
The Cube of a Difference (Binomial): (a−b)3=a 3−b 3−3 a b(a−b)
The Sum of Two Cubes: a 3+b 3=(a+b)(a 2−a b+b 2)
The Difference of Two Cubes: a 3−b 3=(a−b)(a 2+a b+b 2)
The FOIL Method: (a+b)(c+d)=a c+a d+b c+b d
These identities can be used to help solve for values, simplify polynomials, or re-write expressions in a different form. These identities are true for ALL values of variables and can be proven using various methods such as algebraic properties or geometrically. The key difference between any algebraic equation and an algebraic identity is the fact that these identities will remain true for any values of variables chosen. That is not necessarily the case for an equation that may be true for only specific values.
Frequently Asked Questions
What are some algebraic identities?
Some of the algebraic identities are the square of a sum, the square of a difference, the cube of a sum, or the cube of a difference. There is also the FOIL method identity that states (a+b)(c+d)=ac+ad+bc+bd.
How do you recognize an identity in algebra?
An algebraic identity is an equation that is true for any values of variables. This means that in order to identify an identity, it is important to ensure the equation is true for any value. To verify, you can manipulate one side of the equation so it is identical to the other side OR manipulate the equation until you have a true statement, such as a=a or 1=1.
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General Math Lessons
2 chapters | 80 lessons
Ch 1. General Mathematics Lessons
Large Numbers: Usage, Names & Order | What are Big Numbers?
Imperial System of Measurement | Overview, History & Units
Simplex Tableau Procedure & Examples | What is Simplex Method?
Algebraic Properties Definition, Equations & Examples
Basic Math Operations | Definition & Examples
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Geometry & Measurement | Definition, Basics & Examples
Statistical Graphs & Charts | Importance, Types & Uses
Angle Parts, Types & Applications in Real Life
Applications of Trigonometry in Real Life | Uses & Examples
Mathematical Terms | Definitions & Examples
Algebraic Identities | Definition & Examples
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Simpson's Paradox | Definition, History & Examples
Affine Geometry | Overview & Examples
Exact Differential Equations | Formula & Examples
Russell's Paradox | History, Principles & Application
Numerical Integration | Definition, Methods & Examples
Population Proportion | Formula, Symbol & Examples
What is Elementary Mathematics? | Topics & Examples
Proofs of Trigonometric Identities | Overview, Processes & Examples
Multiple Integral Overview, Formulas & Examples
What is an Integral? | Rules, Calculation & Examples
Pick's Theorem | Proof, Formula & Examples
Positional Notation Method & Definition
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Nth Root Definition & Formula
Solid of Revolution Definition, Formula & Examples
Absolute Convergence Overview, Proof & Examples
Euclidean Distance | Calculation, Formula & Examples
Test Statistic | Definition, Types & Formula
Parametric Statistics | Tests, Data & Distribution
Rates in Mathematics | Definition & Examples
Vigenre Cipher History, Example & Coding Variants
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Linearity in Algebra | Functions, Equations & Examples
Margin of Error Definition, Calculation & Formula
Partial Differential Equations | Definition, Types & Examples
Matrix Theory Definition, Basic Operations & Applications
Factorization Definition, Formulas & Methods
Affine Transformations | Definition, Properties & Examples
Vector Calculus | Overview, Identities & Formulas
Multivariable Calculus: Topics, Operations & Applications
Applied Statistics Overview & Principles
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Integration Techniques in Calculus
Geometric Representation | Theory and Examples
Differential Equations | Definition, Solutions & Examples
Differential Geometry | History, Types, & Applications
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190432 | https://www.axler.net/HFT.pdf | Harmonic Function Theory Second Edition Sheldon Axler Paul Bourdon Wade Ramey 17 July 2020 ©2001 Springer Contents Preface ix Acknowledgments xi Chapter 1 Basic Properties of Harmonic Functions 1 Definitions and Examples . . . . . . . . . . . . . . . . . . . . . . .
1 Invariance Properties . . . . . . . . . . . . . . . . . . . . . . . . . .
2 The Mean-Value Property . . . . . . . . . . . . . . . . . . . . . . . .
4 The Maximum Principle . . . . . . . . . . . . . . . . . . . . . . . . .
7 The Poisson Kernel for the Ball . . . . . . . . . . . . . . . . . . . .
9 The Dirichlet Problem for the Ball . . . . . . . . . . . . . . . . . .
12 Converse of the Mean-Value Property . . . . . . . . . . . . . . . .
17 Real Analyticity and Homogeneous Expansions . . . . . . . . . .
19 Origin of the Term “Harmonic” . . . . . . . . . . . . . . . . . . . .
25 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26 Chapter 2 Bounded Harmonic Functions 31 Liouville’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . .
31 Isolated Singularities . . . . . . . . . . . . . . . . . . . . . . . . . .
32 Cauchy’s Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . .
33 Normal Families . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35 Maximum Principles . . . . . . . . . . . . . . . . . . . . . . . . . . .
36 Limits Along Rays . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38 Bounded Harmonic Functions on the Ball . . . . . . . . . . . . . .
40 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42 v vi Contents Chapter 3 Positive Harmonic Functions 45 Liouville’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . .
45 Harnack’s Inequality and Harnack’s Principle . . . . . . . . . . .
47 Isolated Singularities . . . . . . . . . . . . . . . . . . . . . . . . . .
50 Positive Harmonic Functions on the Ball . . . . . . . . . . . . . .
55 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
56 Chapter 4 The Kelvin Transform 59 Inversion in the Unit Sphere . . . . . . . . . . . . . . . . . . . . . .
59 Motivation and Definition . . . . . . . . . . . . . . . . . . . . . . .
61 The Kelvin Transform Preserves Harmonic Functions . . . . . .
62 Harmonicity at Infinity . . . . . . . . . . . . . . . . . . . . . . . . .
63 The Exterior Dirichlet Problem . . . . . . . . . . . . . . . . . . . .
66 Symmetry and the Schwarz Reflection Principle . . . . . . . . . .
67 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
71 Chapter 5 Harmonic Polynomials 73 Polynomial Decompositions . . . . . . . . . . . . . . . . . . . . . .
74 Spherical Harmonic Decomposition of L2(S) . . . . . . . . . . .
78 Inner Product of Spherical Harmonics . . . . . . . . . . . . . . . .
82 Spherical Harmonics Via Differentiation . . . . . . . . . . . . . .
85 Explicit Bases of Hm(Rn) and Hm(S) . . . . . . . . . . . . . . .
92 Zonal Harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
94 The Poisson Kernel Revisited . . . . . . . . . . . . . . . . . . . . .
97 A Geometric Characterization of Zonal Harmonics . . . . . . . . 100 An Explicit Formula for Zonal Harmonics . . . . . . . . . . . . . 104 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 Chapter 6 Harmonic Hardy Spaces 111 Poisson Integrals of Measures . . . . . . . . . . . . . . . . . . . . . 111 Weak Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 The Spaces hp(B) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 The Hilbert Space h2(B) . . . . . . . . . . . . . . . . . . . . . . . . 121 The Schwarz Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . 123 The Fatou Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Contents vii Chapter 7 Harmonic Functions on Half-Spaces 143 The Poisson Kernel for the Upper Half-Space . . . . . . . . . . . 144 The Dirichlet Problem for the Upper Half-Space . . . . . . . . . . 146 The Harmonic Hardy Spaces hp(H) . . . . . . . . . . . . . . . . . 151 From the Ball to the Upper Half-Space, and Back . . . . . . . . . 153 Positive Harmonic Functions on the Upper Half-Space . . . . . . 156 Nontangential Limits . . . . . . . . . . . . . . . . . . . . . . . . . . 160 The Local Fatou Theorem . . . . . . . . . . . . . . . . . . . . . . . 161 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 Chapter 8 Harmonic Bergman Spaces 171 Reproducing Kernels . . . . . . . . . . . . . . . . . . . . . . . . . . 172 The Reproducing Kernel of the Ball . . . . . . . . . . . . . . . . . 176 Examples in bp(B) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 The Reproducing Kernel of the Upper Half-Space . . . . . . . . . 185 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 Chapter 9 The Decomposition Theorem 191 The Fundamental Solution of the Laplacian . . . . . . . . . . . . 191 Decomposition of Harmonic Functions . . . . . . . . . . . . . . . 193 Bôcher’s Theorem Revisited . . . . . . . . . . . . . . . . . . . . . . 197 Removable Sets for Bounded Harmonic Functions . . . . . . . . 200 The Logarithmic Conjugation Theorem . . . . . . . . . . . . . . . 203 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 Chapter 10 Annular Regions 209 Laurent Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 Isolated Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . 210 The Residue Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 213 The Poisson Kernel for Annular Regions . . . . . . . . . . . . . . 215 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 Chapter 11 The Dirichlet Problem and Boundary Behavior 223 The Dirichlet Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 223 Subharmonic Functions . . . . . . . . . . . . . . . . . . . . . . . . . 224 Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey viii Contents The Perron Construction . . . . . . . . . . . . . . . . . . . . . . . . 226 Barrier Functions and Geometric Criteria for Solvability . . . . 227 Nonextendability Results . . . . . . . . . . . . . . . . . . . . . . . . 233 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 Appendix A Volume, Surface Area, and Integration on Spheres 239 Volume of the Ball and Surface Area of the Sphere . . . . . . . . 239 Slice Integration on Spheres . . . . . . . . . . . . . . . . . . . . . . 241 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 Appendix B Harmonic Function Theory and Mathematica 247 References 249 Symbol Index 251 Index 255 Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Preface Harmonic functions—the solutions of Laplace’s equation—play a crucial role in many areas of mathematics, physics, and engineering.
But learning about them is not always easy. At times the authors have agreed with Lord Kelvin and Peter Tait, who wrote (, Preface) There can be but one opinion as to the beauty and utility of this analysis of Laplace; but the manner in which it has been hitherto presented has seemed repulsive to the ablest mathematicians, and difficult to ordinary mathematical students.
The quotation has been included mostly for the sake of amusement, but it does convey a sense of the difficulties the uninitiated sometimes encounter.
The main purpose of our text, then, is to make learning about har-monic functions easier. We start at the beginning of the subject, assum-ing only that our readers have a good foundation in real and complex analysis along with a knowledge of some basic results from functional analysis. The first fifteen chapters of , for example, provide suffi-cient preparation.
In several cases we simplify standard proofs. For example, we re-place the usual tedious calculations showing that the Kelvin transform of a harmonic function is harmonic with some straightforward obser-vations that we believe are more revealing.
Another example is our proof of Bôcher’s Theorem, which is more elementary than the classi-cal proofs.
We also present material not usually covered in standard treatments of harmonic functions (such as , , and ). The section on the Schwarz Lemma and the chapter on Bergman spaces are examples. For ix x Preface completeness, we include some topics in analysis that frequently slip through the cracks in a beginning graduate student’s curriculum, such as real-analytic functions.
We rarely attempt to trace the history of the ideas presented in this book. Thus the absence of a reference does not imply originality on our part.
For this second edition we have made several major changes. The key improvement is a new and considerably simplified treatment of spherical harmonics (Chapter 5). The book now includes a formula for the Laplacian of the Kelvin transform (Proposition 4.6). Another ad-dition is the proof that the Dirichlet problem for the half-space with continuous boundary data is solvable (Theorem 7.11), with no growth conditions required for the boundary function.
Yet another signifi-cant change is the inclusion of generalized versions of Liouville’s and Bôcher’s Theorems (Theorems 9.10 and 9.11), which are shown to be equivalent. We have also added many exercises and made numerous small improvements.
In addition to writing the text, the authors have developed a soft-ware package to manipulate many of the expressions that arise in har-monic function theory. Our software package, which uses many results from this book, can perform symbolic calculations that would take a prohibitive amount of time if done without a computer. For example, the Poisson integral of any polynomial can be computed exactly. Ap-pendix B explains how readers can obtain our software package free of charge.
The roots of this book lie in a graduate course at Michigan State University taught by one of the authors and attended by the other au-thors along with a number of graduate students. The topic of harmonic functions was presented with the intention of moving on to different material after introducing the basic concepts. We did not move on to different material. Instead, we began to ask natural questions about harmonic functions.
Lively and illuminating discussions ensued.
A freewheeling approach to the course developed; answers to questions someone had raised in class or in the hallway were worked out and then presented in class (or in the hallway). Discovering mathematics in this way was a thoroughly enjoyable experience. We will consider this book a success if some of that enjoyment shines through in these pages.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Acknowledgments Our book has been improved by our students and by readers of the first edition. We take this opportunity to thank them for catching errors and making useful suggestions.
Among the many mathematicians who have influenced our outlook on harmonic function theory, we give special thanks to Dan Luecking for helping us to better understand Bergman spaces, to Patrick Ahern who suggested the idea for the proof of Theorem 7.11, and to Elias Stein and Guido Weiss for their book , which contributed greatly to our knowledge of spherical harmonics.
We are grateful to Carrie Heeter for using her expertise to make old photographs look good.
At our publisher Springer we thank the mathematics editors Thomas von Foerster (first edition) and Ina Lindemann (second edition) for their support and encouragement, as well as Fred Bartlett for his valuable assistance with electronic production.
xi Chapter 1 Basic Properties of Harmonic Functions Definitions and Examples Harmonic functions, for us, live on open subsets of real Euclidean spaces. Throughout this book, n will denote a fixed positive integer greater than 1 and Ωwill denote an open, nonempty subset of Rn. A twice continuously differentiable, complex-valued function u defined on Ωis harmonic on Ωif ∆u ≡0, where ∆= D1 2+· · ·+Dn 2 and Dj 2 denotes the second partial derivative with respect to the jth coordinate variable. The operator ∆is called the Laplacian, and the equation ∆u ≡0 is called Laplace’s equation. We say that a function u defined on a (not necessarily open) set E ⊂Rn is harmonic on E if u can be extended to a function harmonic on an open set containing E.
We let x = (x1, . . . , xn) denote a typical point in Rn and let |x| = (x12 + · · · + xn2)1/2 denote the Euclidean norm of x.
The simplest nonconstant harmonic functions are the coordinate functions; for example, u(x) = x1. A slightly more complex example is the function on R3 defined by u(x) = x12 + x22 −2x32 + ix2.
As we will see later, the function 1 2 Chapter 1. Basic Properties of Harmonic Functions u(x) = |x|2−n is vital to harmonic function theory when n > 2; the reader should verify that this function is harmonic on Rn \ {0}.
We can obtain additional examples of harmonic functions by dif-ferentiation, noting that for smooth functions the Laplacian commutes with any partial derivative. In particular, differentiating the last exam-ple with respect to x1 shows that x1|x|−n is harmonic on Rn{0} when n > 2. (We will soon prove that every harmonic function is infinitely differentiable; thus every partial derivative of a harmonic function is harmonic.) The function x1|x|−n is harmonic on Rn{0} even when n = 2. This can be verified directly or by noting that x1|x|−2 is a partial derivative of log |x|, a harmonic function on R2 \ {0}. The function log |x| plays the same role when n = 2 that |x|2−n plays when n > 2. Notice that limx→∞log |x| = ∞, but limx→∞|x|2−n = 0; note also that log |x| is nei-ther bounded above nor below, but |x|2−n is always positive. These facts hint at the contrast between harmonic function theory in the plane and in higher dimensions. Another key difference arises from the close connection between holomorphic and harmonic functions in the plane—a real-valued function on Ω⊂R2 is harmonic if and only if it is locally the real part of a holomorphic function. No comparable result exists in higher dimensions.
Invariance Properties Throughout this book, all functions are assumed to be complex valued unless stated otherwise.
For k a positive integer, let Ck(Ω) denote the set of k times continuously differentiable functions on Ω; C∞(Ω) is the set of functions that belong to Ck(Ω) for every k. For E ⊂Rn, we let C(E) denote the set of continuous functions on E.
Because the Laplacian is linear on C2(Ω), sums and scalar multiples of harmonic functions are harmonic.
For y ∈Rn and u a function on Ω, the y-translate of u is the func-tion on Ω+ y whose value at x is u(x −y). Clearly, translations of harmonic functions are harmonic.
For a positive number r and u a function on Ω, the r-dilate of u, denoted ur , is the function Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Invariance Properties 3 (ur )(x) = u(rx) defined for x in (1/r)Ω= {(1/r)w : w ∈Ω}. If u ∈C2(Ω), then a simple computation shows that ∆(ur ) = r 2(∆u)r on (1/r)Ω. Hence dilates of harmonic functions are harmonic.
Note the formal similarity between the Laplacian ∆= D1 2+· · ·+Dn 2 and the function |x|2 = x12 + · · · + xn2, whose level sets are spheres centered at the origin. The connection between harmonic functions and spheres is central to harmonic function theory. The mean-value prop-erty, which we discuss in the next section, best illustrates this connec-tion. Another connection involves linear transformations on Rn that preserve the unit sphere; such transformations are called orthogonal.
A linear map T : Rn →Rn is orthogonal if and only if |Tx| = |x| for all x ∈Rn. Simple linear algebra shows that T is orthogonal if and only if the column vectors of the matrix of T (with respect to the standard basis of Rn) form an orthonormal set.
We now show that the Laplacian commutes with orthogonal trans-formations; more precisely, if T is orthogonal and u ∈C2(Ω), then ∆(u ◦T) = (∆u) ◦T on T −1(Ω). To prove this, let [tjk] denote the matrix of T relative to the standard basis of Rn. Then Dm(u ◦T) = n X j=1 tjm(Dju) ◦T, where Dm denotes the partial derivative with respect to the mth coordi-nate variable. Differentiating once more and summing over m yields ∆(u ◦T) = n X m=1 n X j,k=1 tkmtjm(DkDju) ◦T = n X j,k=1 n X m=1 tkmtjm (DkDju) ◦T = n X j=1 (DjDju) ◦T = (∆u) ◦T, Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 4 Chapter 1. Basic Properties of Harmonic Functions as desired. The function u ◦T is called a rotation of u. The preced-ing calculation shows that rotations of harmonic functions are har-monic.
The Mean-Value Property Many basic properties of harmonic functions follow from Green’s identity (which we will need mainly in the special case when Ωis a ball): 1.1 Z Ω (u∆v −v∆u) dV = Z ∂Ω (uDnv −vDnu) ds.
Here Ωis a bounded open subset of Rn with smooth boundary, and u and v are C2-functions on a neighborhood of Ω, the closure of Ω.
The measure V = Vn is Lebesgue volume measure on Rn, and s de-notes surface-area measure on ∂Ω(see Appendix A for a discussion of integration over balls and spheres). The symbol Dn denotes differen-tiation with respect to the outward unit normal n. Thus for ζ ∈∂Ω, (Dnu)(ζ) = (∇u)(ζ) · n(ζ), where ∇u = (D1u, . . . , Dnu) denotes the gradient of u and · denotes the usual Euclidean inner product.
Green’s identity (1.1) follows easily from the familiar divergence the-orem of advanced calculus: 1.2 Z Ω div w dV = Z ∂Ω w · n ds.
Here w = (w1, . . . , wn) is a smooth vector field (a Cn-valued function whose components are continuously differentiable) on a neighborhood of Ω, and div w, the divergence of w, is defined to be D1w1+· · ·+Dnwn.
To obtain Green’s identity from the divergence theorem, simply let w = u∇v −v∇u and compute.
The following useful form of Green’s identity occurs when u is har-monic and v ≡1: 1.3 Z ∂Ω Dnu ds = 0.
Green’s identity is the key to the proof of the mean-value property.
Before stating the mean-value property, we introduce some notation: B(a, r) = {x ∈Rn : |x −a| < r} is the open ball centered at a of Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey The Mean-Value Property 5 radius r; its closure is the closed ball B(a, r); the unit ball B(0, 1) is denoted by B and its closure by B. When the dimension is important we write Bn in place of B. The unit sphere, the boundary of B, is denoted by S; normalized surface-area measure on S is denoted by σ (so that σ(S) = 1). The measure σ is the unique Borel probability measure on S that is rotation invariant (meaning σ T(E) = σ(E) for every Borel set E ⊂S and every orthogonal transformation T).
1.4 Mean-Value Property: If u is harmonic on B(a, r), then u equals the average of u over ∂B(a, r). More precisely, u(a) = Z S u(a + rζ) dσ(ζ).
Proof: First assume that n > 2. Without loss of generality we may assume that B(a, r) = B. Fix ε ∈(0, 1). Apply Green’s identity (1.1) with Ω= {x ∈Rn : ε < |x| < 1} and v(x) = |x|2−n to obtain 0 = (2 −n) Z S u ds −(2 −n)ε1−n Z εS u ds − Z S Dnu ds −ε2−n Z εS Dnu ds.
By 1.3, the last two terms are 0, thus Z S u ds = ε1−n Z εS u ds, which is the same as Z S u dσ = Z S u(εζ) dσ(ζ).
Letting ε →0 and using the continuity of u at 0, we obtain the desired result.
The proof when n = 2 is the same, except that |x|2−n should be replaced by log |x|.
Harmonic functions also have a mean-value property with respect to volume measure. The polar coordinates formula for integration on Rn is indispensable here. The formula states that for a Borel measurable, integrable function f on Rn, Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 6 Chapter 1. Basic Properties of Harmonic Functions 1.5 1 nV(B) Z Rn f dV = Z ∞ 0 r n−1 Z S f (rζ) dσ(ζ) dr (see , Chapter 8, Exercise 6). The constant nV(B) arises from the normalization of σ (choosing f to be the characteristic function of B shows that nV(B) is the correct constant).
1.6 Mean-Value Property, Volume Version: If u is harmonic on B(a, r), then u(a) equals the average of u over B(a, r). More precisely, u(a) = 1 V B(a, r) Z B(a,r) u dV.
Proof: We can assume that B(a, r) = B. Apply the polar coordi-nates formula (1.5) with f equal to u times the characteristic function of B, and then use the spherical mean-value property (Theorem 1.4).
We will see later (1.24 and 1.25) that the mean-value property char-acterizes harmonic functions.
We conclude this section with an application of the mean value prop-erty. We have seen that a real-valued harmonic function may have an isolated (nonremovable) singularity; for example, |x|2−n has an isolated singularity at 0 if n > 2. However, a real-valued harmonic function u cannot have isolated zeros.
1.7 Corollary: The zeros of a real-valued harmonic function are never isolated.
Proof: Suppose u is harmonic and real valued on Ω, a ∈Ω, and u(a) = 0. Let r > 0 be such that B(a, r) ⊂Ω. Because the average of u over ∂B(a, r) equals 0, either u is identically 0 on ∂B(a, r) or u takes on both positive and negative values on ∂B(a, r). In the later case, the connectedness of ∂B(a, r) implies that u has a zero on ∂B(a, r).
Thus u has a zero on the boundary of every sufficiently small ball centered at a, proving that a is not an isolated zero of u.
The hypothesis that u is real valued is needed in the preceding corol-lary. This is no surprise when n = 2, because nonconstant holomorphic functions have isolated zeros. When n ≥2, the harmonic function Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey The Maximum Principle 7 (1 −n)x12 + n X k=2 xk2 + ix1 is an example; it vanishes only at the origin.
The Maximum Principle An important consequence of the mean-value property is the fol-lowing maximum principle for harmonic functions.
1.8 Maximum Principle: Suppose Ωis connected, u is real valued and harmonic on Ω, and u has a maximum or a minimum in Ω. Then u is constant.
Proof: Suppose u attains a maximum at a ∈Ω. Choose r > 0 such that B(a, r) ⊂Ω. If u were less than u(a) at some point of B(a, r), then the continuity of u would show that the average of u over B(a, r) is less than u(a), contradicting 1.6. Therefore u is constant on B(a, r), proving that the set where u attains its maximum is open in Ω. Because this set is also closed in Ω(again by the continuity of u), it must be all of Ω(by connectivity). Thus u is constant on Ω, as desired.
If u attains a minimum in Ω, we can apply this argument to −u.
The following corollary, whose proof immediately follows from the preceding theorem, is frequently useful. (Note that the connectivity of Ωis not needed here.) 1.9 Corollary: Suppose Ωis bounded and u is a continuous real-valued function on Ωthat is harmonic on Ω. Then u attains its maximum and minimum values over Ωon ∂Ω.
The corollary above implies that on a bounded domain a harmonic function is determined by its boundary values.
More precisely, for bounded Ω, if u and v are continuous functions on Ωthat are har-monic on Ω, and if u = v on ∂Ω, then u = v on Ω. Unfortunately this can fail on an unbounded domain. For example, the harmonic func-tions u(x) = 0 and v(x) = xn agree on the boundary of the half-space {x ∈Rn : xn > 0}.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 8 Chapter 1. Basic Properties of Harmonic Functions The next version of the maximum principle can be applied even when Ωis unbounded or when u is not continuous on Ω.
1.10 Corollary: Let u be a real-valued, harmonic function on Ω, and suppose lim sup k→∞ u(ak) ≤M for every sequence (ak) in Ωconverging either to a point in ∂Ωor to ∞.
Then u ≤M on Ω.
Remark: To say that (ak) converges to ∞means that |ak| →∞. The corollary is valid if “lim sup” is replaced by “lim inf” and the inequalities are reversed.
Proof of Corollary 1.10: Let M′ = sup{u(x) : x ∈Ω}, and choose a sequence (bk) in Ωsuch that u(bk) →M′.
If (bk) has a subsequence converging to some point b ∈Ω, then u(b) = M′, which implies u is constant on the component of Ωcon-taining b (by the maximum principle). Hence in this case there is a sequence (ak) in Ωconverging to a boundary point of Ωor to ∞on which u = M′, and so M′ ≤M.
If no subsequence of (bk) converges to a point in Ω, then (bk) has a subsequence (ak) converging either to a boundary point of Ωor to ∞.
Thus in in this case we also have M′ ≤M.
Theorem 1.8 and Corollaries 1.9 and 1.10 apply only to real-valued functions. The next corollary is a version of the maximum principle for complex-valued functions.
1.11 Corollary: Let Ωbe connected, and let u be harmonic on Ω. If |u| has a maximum in Ω, then u is constant.
Proof: Suppose |u| attains a maximum value of M at some point a ∈Ω. Choose λ ∈C such that |λ| = 1 and λu(a) = M. Then the real-valued harmonic function Re λu attains its maximum value M at a; thus by Theorem 1.8, Re λu ≡M on Ω. Because |λu| = |u| ≤M, we have Im λu ≡0 on Ω. Thus λu, and hence u, is constant on Ω.
Corollary 1.11 is the analogue of Theorem 1.8 for complex-valued harmonic functions; the corresponding analogues of Corollaries 1.9 Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey The Poisson Kernel for the Ball 9 and 1.10 are also valid. All these analogues, however, hold only for the maximum or lim sup of |u|. No minimum principle holds for |u| (consider u(x) = x1 on B).
We will be able to prove a local version of the maximum principle after we prove that harmonic functions are real analytic (see 1.29).
The Poisson Kernel for the Ball The mean-value property shows that if u is harmonic on B, then u(0) = Z S u(ζ) dσ(ζ).
We now show that for every x ∈B, u(x) is a weighted average of u over S. More precisely, we will show there exists a function P on B × S such that u(x) = Z S u(ζ)P(x, ζ) dσ(ζ) for every x ∈B and every u harmonic on B.
To discover what P might be, we start with the special case n = 2.
Suppose u is a real-valued harmonic function on the closed unit disk in R2. Then u = Re f for some function f holomorphic on a neigh-borhood of the closed disk (see Exercise 11 of this chapter). Because u = (f + f )/2, the Taylor series expansion of f implies that u has the form u(rζ) = ∞ X j=−∞ ajr |j|ζj, where 0 ≤r ≤1 and |ζ| = 1. In this formula, take r = 1, multiply both sides by ζ−k, then integrate over the unit circle to obtain ak = Z S u(ζ)ζ−k dσ(ζ).
Now let x be a point in the open unit disk, and write x = rη with r ∈[0, 1) and |η| = 1. Then Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 10 Chapter 1. Basic Properties of Harmonic Functions u(x) = u(rη) 1.12 = ∞ X j=−∞ Z S u(ζ)ζ−j dσ(ζ) r |j|ηj = Z S u(ζ) ∞ X j=−∞ r |j|(ηζ−1)j dσ(ζ).
Breaking the last sum into two geometric series, we see that u(x) = Z S u(ζ) 1 −r 2 |rη −ζ|2 dσ(ζ).
Thus, letting P(x, ζ) = (1 −|x|2)/|x −ζ|2, we obtain the desired for-mula for n = 2: u(x) = Z S u(ζ)P(x, ζ) dσ(ζ).
Unfortunately, nothing as simple as this works in higher dimen-sions. To find P(x, ζ) when n > 2, we start with a result we call the symmetry lemma, which will be useful in other contexts as well.
1.13 Symmetry Lemma: For all nonzero x and y in Rn, y |y| −|y|x = x |x| −|x|y .
Proof: Square both sides and expand using the inner product.
To find P for n > 2, we try the same approach used in proving the mean-value property. Suppose that u is harmonic on B. When proving that u(0) is the average of u over S, we applied Green’s identity with v(y) = |y|2−n; this function is harmonic on B \ {0}, has a singularity at 0, and is constant on S. Now fix a nonzero point x ∈B. To show that u(x) is a weighted average of u over S, it is natural this time to try v(y) = |y −x|2−n. This function is harmonic on B \ {x}, has a singularity at x, but unfortunately is not constant on S. However, the symmetry lemma (1.13) shows that for y ∈S, |y −x|2−n = |x|2−n y − x |x|2 2−n .
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey The Poisson Kernel for the Ball 11 x / |x | |y |x 0 |x |y y / |y | The symmetry lemma: the two bold segments have the same length.
Notice that the right side of this equation is harmonic (as a function of y) on B. Thus the difference of the left and right sides has all the properties we seek.
So set v(y) = L(y) −R(y), where L(y) = |y −x|2−n, R(y) = |x|2−n y − x |x|2 2−n , and choose ε small enough so that B(x, ε) ⊂B. Now apply Green’s identity (1.1) much as in the proof of the mean-value property (1.4), with Ω= B \ B(x, ε). We obtain 0 = Z S uDnv ds −(2 −n)s(S)u(x) − Z ∂B(x,ε) uDnR ds + Z ∂B(x,ε) RDnu ds (the mean-value property was used here). Because uDnR and RDnu are bounded on B, the last two terms approach 0 as ε →0. Hence u(x) = 1 2 −n Z S uDnv dσ.
Setting P(x, ζ) = (2 −n)−1(Dnv)(ζ), we have the desired formula: 1.14 u(x) = Z S u(ζ)P(x, ζ) dσ(ζ).
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 12 Chapter 1. Basic Properties of Harmonic Functions A computation of Dnv, which we recommend to the reader (the sym-metry lemma may be useful here), yields 1.15 P(x, ζ) = 1 −|x|2 |x −ζ|n .
The function P derived above is called the Poisson kernel for the ball; it plays a key role in the next section.
The Dirichlet Problem for the Ball We now come to a famous problem in harmonic function theory: given a continuous function f on S, does there exist a continuous func-tion u on B, with u harmonic on B, such that u = f on S? If so, how do we find u? This is the Dirichlet problem for the ball. Recall that by the maximum principle, if a solution exists, then it is unique.
We take our cue from the last section. If f happens to be the re-striction to S of a function u harmonic on B, then u(x) = Z S f (ζ)P(x, ζ) dσ(ζ) for all x ∈B. We solve the Dirichlet problem for B by changing our perspective. Starting with a continuous function f on S, we use the formula above to define an extension of f into B that we hope will have the desired properties.
The reader who wishes may regard the material in the last section as motivational. We now start anew, using 1.15 as the definition of P(x, ζ).
For arbitrary f ∈C(S), we define the Poisson integral of f, denoted P[f ], to be the function on B given by 1.16 Pf = Z S f (ζ)P(x, ζ) dσ(ζ).
The next theorem shows that the Poisson integral solves the Dirich-let problem for B.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey The Dirichlet Problem for the Ball 13 Johann Peter Gustav Lejeune Dirichlet (1805–1859), whose attempt to prove the stability of the solar system led to an investigation of harmonic functions.
1.17 Solution of the Dirichlet problem for the ball: Suppose f is continuous on S. Define u on B by u(x) = ( Pf if x ∈B f (x) if x ∈S.
Then u is continuous on B and harmonic on B.
The proof of 1.17 depends on harmonicity and approximate-identity properties of the Poisson kernel given in the following two proposi-tions.
1.18 Proposition: Let ζ ∈S. Then P(·, ζ) is harmonic on Rn \ {ζ}.
We let the reader prove this proposition. One way to do so is to write P(x, ζ) = (1 −|x|2)|x −ζ|−n and then compute the Laplacian of P(·, ζ) using the product rule 1.19 ∆(uv) = u∆v + 2∇u · ∇v + v∆u, which is valid for all real-valued twice continuously differentiable func-tions u and v.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 14 Chapter 1. Basic Properties of Harmonic Functions 1.20 Proposition: The Poisson kernel has the following properties: (a) P(x, ζ) > 0 for all x ∈B and all ζ ∈S; (b) Z S P(x, ζ) dσ(ζ) = 1 for all x ∈B; (c) for every η ∈S and every δ > 0, Z |ζ−η|>δ P(x, ζ) dσ(ζ) →0 as x →η.
Proof: Properties (a) and (c) follow immediately from the formula for the Poisson kernel (1.15).
Taking u to be identically 1 in 1.14 gives (b). To prove (b) with-out using the motivational material in the last section, note that for x ∈B \ {0}, we have Z S P(x, ζ) dσ(ζ) = Z S P |ζ|x, ζ |ζ| dσ(ζ) = Z S P |x|ζ, x |x| dσ(ζ), where the last equality follows from the symmetry lemma (1.13). Propo-sition 1.18 tells us that P |x|ζ, x |x| , as a function of ζ, is harmonic on B. Thus by the mean-value property we have Z S P(x, ζ) dσ(ζ) = P 0, x |x| = 1, as desired. Clearly (b) also holds for x = 0, completing the proof.
Proof of Theorem 1.17: The Laplacian of u can be computed by differentiating under the integral sign in 1.16; Proposition 1.18 then shows that u is harmonic on B.
To prove that u is continuous on B, fix η ∈S and ε > 0. Choose δ > 0 such that |f (ζ) −f (η)| < ε whenever |ζ −η| < δ (and ζ ∈S).
For x ∈B, (a) and (b) of Proposition 1.20 imply that Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey The Dirichlet Problem for the Ball 15 |u(x) −u(η)| = Z S f (ζ) −f (η) P(x, ζ) dσ(ζ) ≤ Z |ζ−η|≤δ |f (ζ) −f (η)| P(x, ζ) dσ(ζ) + Z |ζ−η|>δ |f (ζ) −f (η)| P(x, ζ) dσ(ζ) ≤ε + 2∥f∥∞ Z |ζ−η|>δ P(x, ζ) dσ(ζ), where ∥f ∥∞denotes the supremum of |f | on S. The last term above is less than ε for x sufficiently close to η (by Proposition 1.20(c)), proving that u is continuous at η.
We now prove a result stronger than that expressed in 1.14.
1.21 Theorem: If u is a continuous function on B that is harmonic on B, then u = P[u|S] on B.
Proof: By 1.17, u −P[u|S] is harmonic on B and extends continu-ously to be 0 on S. The maximum principle (Corollary 1.9) now implies that u −P[u|S] is 0 on B.
Because translations and dilations preserve harmonic functions, our results can be transferred easily to any ball B(a, r). Specifically, given a continuous function f on ∂B(a, r), there exists a unique continuous function u on B(a, r), with u harmonic on B(a, r), such that u = f on ∂B(a, r). In this case we say that u solves the Dirichlet problem for B(a, r) with boundary data f.
We now show that every harmonic function is infinitely differen-tiable. In dealing with differentiation in several variables the following notation is useful: a multi-index α is an n-tuple of nonnegative inte-gers (α1, . . . , αn); the partial differentiation operator Dα is defined to be D1 α1 . . . Dn αn (Dj 0 denotes the identity operator). For each ζ ∈S, the function P(·, ζ) is infinitely differentiable on B; we denote its αth partial derivative by DαP(·, ζ) (here ζ is held fixed).
If u is continuous on B and harmonic on B, then u(x) = Z S u(ζ)P(x, ζ) dσ(ζ) Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 16 Chapter 1. Basic Properties of Harmonic Functions for every x ∈B. Differentiating under the integral, we easily see that u ∈C∞(B); the formula 1.22 Dαu(x) = Z S u(ζ)DαP(x, ζ) dσ(ζ) holds for every x ∈B and every multi-index α.
The preceding argument applies to any ball after a translation and dilation. As a consequence, every harmonic function is infinitely dif-ferentiable.
The following theorem should remind the reader of the behavior of a uniformly convergent sequence of holomorphic functions.
1.23 Theorem: Suppose (um) is a sequence of harmonic functions on Ωsuch that um converges uniformly to a function u on each compact subset of Ω.
Then u is harmonic on Ω.
Moreover, for every multi-index α, Dαum converges uniformly to Dαu on each compact subset of Ω.
Proof: Given B(a, r) ⊂Ω, we need only show that u is harmonic on B(a, r) and that for every multi-index α, Dαum converges uniformly to Dαu on each compact subset of B(a, r). Without loss of generality, we assume B(a, r) = B.
We then know that um(x) = Z S um(ζ)P(x, ζ) dσ(ζ) for every x ∈B and every m. Taking the limit of both sides, we obtain u(x) = Z S u(ζ)P(x, ζ) dσ(ζ) for every x ∈B. Thus u is harmonic on B.
Let α be a multi-index and let x ∈B. Then Dαum(x) = Z S um(ζ)DαP(x, ζ) dσ(ζ) → Z S u(ζ)DαP(x, ζ) dσ(ζ) = Dαu(x).
If K is a compact subset of B, then DαP is uniformly bounded on K ×S, and so the convergence of Dαum to Dαu is uniform on K, as desired.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Converse of the Mean-Value Property 17 Converse of the Mean-Value Property We have seen that every harmonic function has the mean-value prop-erty. In this section, we use the solvability of the Dirichlet problem for the ball to prove that harmonic functions are the only continuous func-tions having the mean-value property. In fact, the following theorem shows that a continuous function satisfying a weak form of the mean-value property must be harmonic.
1.24 Theorem: Suppose u is a continuous function on Ω. If for each x ∈Ωthere is a sequence of positive numbers rj →0 such that u(x) = Z S u(x + rjζ) dσ(ζ) for all j, then u is harmonic on Ω.
Proof: Without loss of generality, we can assume that u is real valued. Suppose that B(a, R) ⊂Ω. Let v solve the Dirichlet problem for B(a, R) with boundary data u on ∂B(a, R). We will complete the proof by showing that v = u on B(a, R).
Suppose that v −u is positive at some point of B(a, R). Let E be the subset of B(a, R) where v −u attains its maximum. Because E is compact, E contains a point x farthest from a. Clearly x ∈B(a, R), so there exists a ball B(x, r) ⊂B(a, R) such that u(x) equals the average of u over ∂B(x, r).
Because v is harmonic, we have (v −u)(x) = Z S (v −u)(x + rζ) dσ(ζ).
But (v −u)(x + rζ) ≤(v −u)(x) for all ζ ∈S, with strict inequality on a nonempty open subset of S (because of how x was chosen), con-tradicting the equation above. Thus v −u ≤0 on B(a, R). Similarly, v −u ≥0 on B(a, R).
The proof above can be modified to show that if u is continuous on Ωand satisfies a local mean-value property with respect to volume measure, then u is harmonic on Ω; see Exercise 22 of this chapter.
The hypothesis of continuity is needed in Theorem 1.24. To see this, let Ω= Rn and define u by Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 18 Chapter 1. Basic Properties of Harmonic Functions u(x) = 1 if xn > 0 0 if xn = 0 −1 if xn < 0.
Then u(x) equals the average of u over every sphere centered at x if xn = 0, and u(x) equals the average of u over all sufficiently small spheres centered at x if xn ̸= 0. But u is not even continuous, much less harmonic, on Rn.
In the following theorem we replace the continuity assumption with the weaker condition of local integrability (a function is locally inte-grable on Ωif it is Lebesgue integrable on every compact subset of Ω).
However, we now require that the averaging property (with respect to volume measure) hold for every radius.
1.25 Theorem: If u is a locally integrable function on Ωsuch that u(a) = 1 V B(a, r) Z B(a,r) u dV whenever B(a, r) ⊂Ω, then u is harmonic on Ω.
Proof: By Exercise 22 of this chapter, we need only show that u is continuous on Ω. Fix a ∈Ωand let (aj) be a sequence in Ωconverg-ing to a. Let K be a compact subset of Ωwith a in the interior of K.
Then there exists an r > 0 such that B(aj, r) ⊂K for all sufficiently large j. Because u is integrable on K, the dominated convergence the-orem shows that u(aj) = 1 V B(a, r) Z B(aj,r) u dV = 1 V B(a, r) Z K uχB(aj,r) dV → 1 V B(a, r) Z K uχB(a,r) dV = u(a) (as usual, χE denotes the function that is 1 on E and 0 offE). Thus u is continuous on Ω, as desired.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Real Analyticity and Homogeneous Expansions 19 Real Analyticity and Homogeneous Expansions We saw in the section before last that harmonic functions are in-finitely differentiable. A much stronger property will be established in this section—harmonic functions are real analytic. Roughly speaking, a function is real analytic if it is locally expressible as a power series in the coordinate variables x1, x2, . . . , xn of Rn.
To make this more precise, we need to discuss what is meant by a series of complex numbers of the form P cα, where the summation is over all multi-indices α. (The full range of multi-indices will always be intended in a series unless indicated otherwise.) The problem is that there is no natural ordering of the set of all multi-indices when n > 1.
However, suppose we know that P cα is absolutely convergent, i.e., that sup X α∈F |cα| < ∞, where the supremum is taken over all finite subsets F of multi-indices.
All orderings α(1), α(2), . . . of multi-indices then yield the same value for P∞ j=1 cα(j); hence we may unambiguously write P cα for this value.
We will only be concerned with such absolutely convergent series.
The following notation will be convenient when dealing with multi-ple power series: for x ∈Rn and α = (α1, α2, . . . , αn) a multi-index, define xα = x1α1x2α2 . . . xnαn, α! = α1!α2! . . . αn!, |α| = α1 + α2 + · · · + αn.
A function f on Ωis real analytic on Ωif for every a ∈Ωthere exist complex numbers cα such that f (x) = X cα(x −a)α for all x in a neighborhood of a, the series converging absolutely in this neighborhood.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 20 Chapter 1. Basic Properties of Harmonic Functions Some basic properties of such series are contained in the next propo-sition. Here it will be convenient to center the power series at a = 0, and to define R(y) = {x ∈Rn : |xj| < |yj|, j = 1, 2, . . . , n} for y ∈Rn; R(y) is the n-dimensional open rectangle centered at 0 with “corner y”. To avoid trivialities we will assume that each compo-nent of y is nonzero.
1.26 Theorem: Suppose {cαyα} is a bounded set. Then: (a) For every multi-index β, the series X α Dβ(cαxα) converges absolutely on R(y) and uniformly on compact subsets of R(y).
(b) The function f defined by f (x) = P cαxα for x ∈R(y) is in-finitely differentiable on R(y). Moreover, Dβf (x) = X α Dβ(cαxα) for all x ∈R(y) and for every multi-index β.
Furthermore, cα = Dαf (0)/α! for every multi-index α.
Remarks: 1. To say the preceding series converges uniformly on a set means that every ordering of the series converges uniformly on this set in the usual sense.
2. The theorem shows that every derivative of a real-analytic func-tion is real analytic, and that if P aαxα = P bαxα for all x in a neigh-borhood of 0, then aα = bα for all α.
Proof of Theorem 1.26: We first observe that on the rectangle R (1, 1, . . . , 1) , we have X α Dβ(xα) = Dβ[(1 −x1)−1(1 −x2)−1 . . . (1 −xn)−1] for every multi-index β, as the reader should verify (start with β = 0).
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Real Analyticity and Homogeneous Expansions 21 Now assume that |cαyα| ≤M for every α. If K is a compact subset of R(y), then K ⊂R(ty) for some t ∈(0, 1). Thus for every x ∈K and every multi-index α, |cαxα| ≤t|α||cαyα| ≤Mt|α|.
By the preceding paragraph, P t|α| = (1 −t)−n < ∞, establishing the absolute and uniform convergence of P cαxα on K. Similar reasoning, with a little more bookkeeping, applies to P Dβ(cαxα). This completes the proof of (a).
Letting f (x) = P cαxα for x ∈R(y), the uniform convergence on compact subsets of R(y) of the series P Dβ(cαxα) for every β shows that f ∈C∞ R(y) , and that Dβf (x) = P Dβ(cαxα) in R(y) for ev-ery β. The formula for the Taylor coefficients cα follows from this by computing the derivatives of f at 0.
A word of caution: Theorem 1.26 does not assert that rectangles are the natural domains of convergence of multiple power series. For example, in two dimensions the domain of convergence of P∞ j=1(x1x2)j is {(x1, x2) ∈R2 : |x1x2| < 1}.
The next theorem shows that real-analytic functions enjoy certain properties not shared by all C∞-functions.
1.27 Theorem: Suppose Ωis connected, f is real analytic in Ω, and f = 0 on a nonempty open subset of Ω. Then f ≡0 in Ω.
Proof: Let ω denote the interior of {x ∈Ω: f (x) = 0}. Then ω is an open subset of Ω. If a ∈Ωis a limit point of ω, then all derivatives of f vanish at a by continuity, implying that the power series of f at a is identically zero; hence a ∈ω. Thus ω is closed in Ω. Because ω is nonempty by hypothesis, we must have ω = Ωby connectivity, giving f ≡0 in Ω.
1.28 Theorem: If u is harmonic on Ω, then u is real analytic in Ω.
Proof: It suffices to show that if u is harmonic on B, then u has a power series expansion converging to u in a neighborhood of 0.
The main idea here is the same as in one complex variable—we use the Poisson integral representation of u and expand the Poisson kernel Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 22 Chapter 1. Basic Properties of Harmonic Functions in a power series. Unfortunately the details are not as simple as in the case of the Cauchy integral formula.
Suppose that |x| < √ 2 −1 and ζ ∈S. Then 0 < |x −ζ|2 < 2, and thus P(x, ζ) = (1 −|x|2)(|x −ζ|2)−n/2 = (1 −|x|2) ∞ X m=0 cm(|x|2 −2x · ζ)m, where P∞ m=0 cm(t −1)m is the Taylor series of t−n/2 on the interval (0, 2), expanded about the midpoint 1.
After expanding the terms (|x|2 −2x · ζ)m and rearranging (permissible, since we have all of the absolute convergence one could ask for), the Poisson kernel takes the form P(x, ζ) = X α xαqα(ζ), for x ∈( √ 2−1)B and ζ ∈S, where each qα is a polynomial. This latter series converges uniformly on S for each x ∈( √ 2 −1)B.
Thus if u is harmonic on B, u(x) = Z S u(ζ)P(x, ζ) dσ(ζ) = X α Z S uqα dσ xα for all x ∈( √ 2 −1)B. This is the desired expansion of u near 0.
Unfortunately, the multiple power series at 0 of a function har-monic on B need not converge in all of B. For example, the function u(z) = 1/(1 −z) is holomorphic (hence harmonic) on the open unit disk of the complex plane. Writing z = x + iy = (x, y) ∈R2, we have u(z) = ∞ X m=0 (x + iy)m = ∞ X m=0 m X j=0 m j xj(iy)m−j for z ∈B2. As a multiple power series, the last sum above converges absolutely if and only if |x| + |y| < 1, and hence does not converge in all of B2. The reader should perhaps take a moment to meditate on the difference between the “real-analytic” and “holomorphic” power series of u.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Real Analyticity and Homogeneous Expansions 23 As mentioned earlier, the real analyticity of harmonic functions al-lows us to prove a local maximum principle.
1.29 Local Maximum Principle: Suppose Ωis connected, u is real valued and harmonic on Ω, and u has a local maximum in Ω. Then u is constant.
Proof: If u has a local maximum at a ∈Ω, then there exists a ball B(a, r) ⊂Ωsuch that u ≤u(a) in B(a, r). By Theorem 1.8, u is constant on B(a, r). Because u is real analytic on Ω, u ≡u(a) in Ωby Theorem 1.27.
Knowing that harmonic functions locally have power series expan-sions enables us to express them locally as sums of homogeneous har-monic polynomials. This has many interesting consequences, as we will see later. In the remainder of this section we develop a few basic re-sults, starting with a brief discussion of homogeneous polynomials.
A polynomial is by definition a finite linear combination of mono-mials xα. A polynomial p of the form p(x) = X |α|=m cαxα is said to be homogeneous of degree m; here we allow m to be any nonnegative integer. Equivalently, a polynomial p is homogeneous of degree m if p(tx) = tmp(x) for all t ∈R and all x ∈Rn. This last formulation shows that a homo-geneous polynomial is determined by its restriction to S: if p and q are homogeneous of degree m and p = q on S, then p = q on Rn. (This is not true of polynomials in general; for example, 1 −|x|2 ≡0 on S.) Note also that if p is a homogeneous polynomial of degree m, then so is p ◦T for every linear map T from Rn to Rn.
It is often useful to express functions as infinite sums of homo-geneous polynomials.
Here is a simple uniqueness result for such sums.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 24 Chapter 1. Basic Properties of Harmonic Functions 1.30 Proposition: Let r > 0. If pm and qm are homogeneous poly-nomials of degree m, m = 0, 1, . . . , and if ∞ X m=0 pm(x) = ∞ X m=0 qm(x) for all x ∈rB (both series converging pointwise in rB), then pm = qm for every m.
Proof: Fix ζ ∈S. Since the two series above converge and are equal at each point in rB, we have ∞ X m=0 pm(ζ)tm = ∞ X m=0 qm(ζ)tm for all t ∈(−r, r). By the uniqueness of coefficients of power series in one variable, pm(ζ) = qm(ζ) for every m. This is true for every ζ ∈S, and thus pm = qm on S for all m. By the preceding remarks, pm = qm on Rn for every m.
Suppose now that u is harmonic near 0. Letting pm(x) = X |α|=m Dαu(0) α!
xα, we see from Theorem 1.28 that u(x) = ∞ X m=0 pm(x) for x near 0. Because each pm is homogeneous of degree m, the latter series is called the homogeneous expansion of u at 0. Remarkably, the harmonicity of u implies that each pm is harmonic. To see this, observe that ∆u = P ∆pm ≡0 near 0, and that each ∆pm is homogeneous of degree m −2 for m ≥2 (and is 0 for m < 2). From 1.30 we conclude ∆pm ≡0 for every m. We have thus represented u near 0 as an infinite sum of homogeneous harmonic polynomials.
Translating this local result from 0 to any other point in the domain of u, we have the following theorem.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Origin of the Term “Harmonic” 25 1.31 Theorem: Suppose u is harmonic on Ωand a ∈Ω. Then there exist harmonic homogeneous polynomials pm of degree m such that 1.32 u(x) = ∞ X m=0 pm(x −a) for all x near a, the series converging absolutely and uniformly near a.
Homogeneous expansions are better behaved than multiple power series. As we will see later (5.34), if u is harmonic on Ωand B(a, r) ⊂Ω, then the homogeneous expansion 1.32 is valid for all x ∈B(a, r).
This is reminiscent of the standard power series result for holomor-phic functions of one complex variable. Indeed, if u is holomorphic on Ω⊂R2 = C, then by the uniqueness of homogeneous expansions, 1.32 is precisely the holomorphic power series of u on B(a, r).
Origin of the Term ‘‘Harmonic’’ The word “harmonic” is commonly used to describe a quality of sound. Harmonic functions derive their name from a roundabout con-nection they have with one source of sound—a vibrating string.
Physicists label the movement of a point on a vibrating string “har-monic motion”. Such motion may be described using sine and cosine functions, and in this context the sine and cosine functions are some-times called harmonics. In classical Fourier analysis, functions on the unit circle are expanded in terms of sines and cosines. Analogous ex-pansions exist on the sphere in Rn, n > 2, in terms of homogeneous harmonic polynomials (see Chapter 5). Because these polynomials play the same role on the sphere that the harmonics sine and cosine play on the circle, they are called spherical harmonics. The term “spherical har-monic” was apparently first used in this context by William Thomson (Lord Kelvin) and Peter Tait (see , Appendix B). By the early 1900s, the word “harmonic” was applied not only to homogeneous polynomi-als with zero Laplacian, but to any solution of Laplace’s equation.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 26 Chapter 1. Basic Properties of Harmonic Functions Exercises 1.
Show that if u and v are real-valued harmonic functions, then uv is harmonic if and only if ∇u · ∇v ≡0.
2.
Suppose Ωis connected and u is a real-valued harmonic function on Ωsuch that u2 is harmonic. Prove that u is constant. Is this still true without the hypothesis that u is real valued?
3.
Show that ∆(|x|t) = t(t + n −2)|x|t−2.
4.
Laplacian in polar coordinates: Suppose u is a twice continuously differentiable function of two real variables. Define a function U by U(r, θ) = u(r cos θ, r sin θ). Show that ∆u = 1 r ∂ ∂r r ∂U ∂r + 1 r 2 ∂2U ∂θ2 .
5.
Laplacian in spherical coordinates: Suppose u is a twice continu-ously differentiable function of three real variables. Define U by U(ρ, θ, ϕ) = u(ρ sin ϕ cos θ, ρ sin ϕ sin θ, ρ cos ϕ). Show that ∆u = 1 ρ2 ∂ ∂ρ ρ2 ∂U ∂ρ !
+ 1 ρ2 sin ϕ ∂ ∂ϕ sin ϕ ∂U ∂ϕ !
+ 1 ρ2 sin2ϕ ∂2U ∂θ2 .
6.
Suppose g is a real-valued function in C2(Rn) and f ∈C2(R).
Prove that ∆(f ◦g)(x) = f ′′ g(x) |∇g(x)|2 + f ′ g(x) ∆g(x).
7.
Show that if u is a positive function in C2(Ω) and t is a constant, then ∆(ut) = tut−1∆u + t(t −1)ut−2|∇u|2.
8.
Show that if u, v are functions in C2(Ω) with u positive, then ∆(uv) = vuv−1∆u + uv(log u)∆v + v(v −1)uv−2|∇u|2 + uv(log u)2|∇v|2 + 2uv−1(1 + v log u)∇u · ∇v.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Exercises 27 9.
Suppose A is an m-by-n matrix of real numbers. Think of each x ∈Rn as a column vector, so that Ax is then a column vector in Rm. Show that ∆(|Ax|) = |Ax|2|A|2 2 −|AtAx|2 |Ax|3 , where |A|2 2 is the sum of the squares of the entries of A and At denotes the transpose of A.
10.
Let u be harmonic on R2. Show that if f is holomorphic or con-jugate holomorphic on C, then u ◦f is harmonic.
11.
Suppose u is real valued and harmonic on B2. For (x, y) ∈B2, define v(x, y) = Z y 0 (D1u)(x, t) dt − Z x 0 (D2u)(t, 0) dt.
Show that u + iv is holomorphic on B2.
12.
Suppose u is a harmonic function on Ω. Prove that the function x 7→x · ∇u(x) is harmonic on Ω. (For a converse to this exercise, see Exercise 23 in Chapter 5.) 13.
Let T : Rn →Rn be a linear transformation such that u ◦T is harmonic on Rn whenever u is harmonic on Rn. Prove that T is a scalar multiple of an orthogonal transformation.
14.
Suppose Ωis a bounded open subset of Rn with smooth bound-ary and u is a smooth function on Ωsuch that ∆(∆u) = 0 on Ω and u = Dnu = 0 on ∂Ω. Prove that u = 0.
15.
Suppose that Ωis connected and that u is real valued and har-monic on Ω. Show that if u is nonconstant on Ω, then u(Ω) is open in R. (Thus u is an open mapping from Ωto R.) 16.
Suppose Ωis bounded and ∂Ωis connected. Show that if u is a real-valued continuous function on Ωthat is harmonic on Ω, then u(Ω) ⊂u(∂Ω). Is this true for complex-valued u?
17.
A function is called radial if its value at x depends only on |x|.
Prove that a radial harmonic function on B is constant.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 28 Chapter 1. Basic Properties of Harmonic Functions 18.
Give another proof that R S P(x, ζ) dσ(ζ) = 1 for every x ∈B by showing that the function x 7→ R S P(x, ζ) dσ(ζ) is harmonic and radial on B.
19.
Show that P[f ◦T] = P[f ] ◦T for every f ∈C(S) and every orthogonal transformation T.
20.
Find the Poisson kernel for the ball B(a, R).
21.
Use the mean-value property and its converse to give another proof that the uniform limit of a sequence of harmonic functions is harmonic.
22.
Suppose u is a continuous function on Ω, and that for each x ∈Ω there is a sequence of positive numbers rj →0 such that u(x) = 1 V B(x, rj) Z B(x,rj) u dV for each j. Prove that u is harmonic on Ω.
23.
One-Radius Theorem: Suppose u is continuous on B and that for every x ∈B, there exists a positive number r(x) ≤1 −|x| such that u(x) = Z S u(x + r(x)ζ) dσ(ζ).
Prove that u is harmonic on B.
24.
Show that the one-radius theorem fails if the assumption “u is continuous on B” is relaxed to “u is continuous on B”. (Hint sug-gested by Walter Rudin: When n = 2, set u(x) = am + bm log |x| on the annulus {1 −2−m ≤|x| ≤1 −2−m−1}, where the con-stants am, bm are chosen inductively. Proceed analogously when n > 2.) 25.
Hopf Lemma: Suppose that u is real valued, nonconstant, and harmonic on B. Show that if u attains its maximum value on B at ζ ∈S, then there is a positive constant c such that u(ζ) −u(rζ) ≥c(1 −r) for all r ∈(0, 1). Conclude that (Dnu)(ζ) > 0.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Exercises 29 26.
Show that the previous exercise can fail if instead of having a maximum at ζ, the restriction u|S has only a strict local maxi-mum at ζ. (Hint: Take n = 2 and u(x, y) = x2 −y2 −3x.) 27.
Prove that a harmonic function on B whose normal derivative vanishes identically on S is constant.
28.
Show that the previous result holds if the ball is replaced by a bounded smooth domain in Rn that has an internally tangent ball at each boundary point.
29.
Show that a polynomial p is homogeneous of degree m if and only if x · ∇p = mp.
30.
Prove that if p is a harmonic polynomial on Rn that is homoge-neous of degree m, then p/|x|2m+n−2 is harmonic on Rn \ {0}.
31.
Suppose that P cαxα converges in R(y). Prove that P cαxα is real analytic in R(y).
32.
A function u: Ω→Rm is called real analytic if each component of u is real analytic. Prove that the composition of real-analytic functions is real analytic. Deduce, as a corollary, that sums, prod-ucts, and quotients of real-analytic functions are real analytic.
33.
Let m be a positive integer. Characterize all real-analytic func-tions u on Rn such that u(tx) = tmu(x) for all x ∈Rn and all t ∈R.
34.
Show that the power series expansion of a function harmonic on Rn converges everywhere on Rn.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Chapter 2 Bounded Harmonic Functions Liouville’s Theorem Liouville’s Theorem in complex analysis states that a bounded holo-morphic function on C is constant. A similar result holds for harmonic functions on Rn. The simple proof given below is taken from Edward Nelson’s paper , which is one of the rare mathematics papers not containing a single mathematical symbol.
2.1 Liouville’s Theorem: A bounded harmonic function on Rn is constant.
Proof: Suppose u is a harmonic function on Rn, bounded by M.
Let x ∈Rn and let r > 0. By the volume version of the mean-value property (Theorem 1.6), |u(x) −u(0)| = 1 V B(0, r) Z B(x,r) u dV − Z B(0,r) u dV ≤M V(Dr ) V B(0, r) , where Dr denotes the symmetric difference of B(x, r) and B(0, r), so that Dr = [B(x, r) ∪B(0, r)] \ [B(x, r) ∩B(0, r)]. The last expression above tends to 0 as r →∞. Thus u(x) = u(0), and so u is constant.
Liouville’s Theorem leads to an easy proof of a uniqueness theorem for bounded harmonic functions on open half-spaces. The upper half-31 32 Chapter 2. Bounded Harmonic Functions space H = Hn is the open subset of Rn defined by H = {x ∈Rn : xn > 0}.
In this setting we often identify Rn with Rn−1×R, writing a typical point z ∈Rn as z = (x, y), where x ∈Rn−1 and y ∈R. We also identify ∂H with Rn−1.
A harmonic function on a compact set is determined by its restric-tion to the boundary (this follows from the maximum principle). How-ever, a harmonic function on a closed half-space is not determined by restriction to the boundary. For example, the harmonic function u on H defined by u(x, y) = y agrees on the boundary of the half-space with the harmonic function 0. The next result shows that this behavior can-not occur if we consider only harmonic functions that are bounded.
2.2 Corollary: Suppose u is a continuous bounded function on H that is harmonic on H. If u = 0 on ∂H, then u ≡0 on H.
Proof: For x ∈Rn−1 and y < 0, define u(x, y) = −u(x, −y), thereby extending u to a bounded continuous function defined on all of Rn. Clearly u satisfies the local mean-value property specified in Theorem 1.24, so u is harmonic on Rn. Liouville’s Theorem (2.1) now shows that u is constant on Rn.
In Chapter 7 we will study harmonic functions on half-spaces in detail.
Isolated Singularities Everyone knows that an isolated singularity of a bounded holomor-phic function is removable. We now show that the same is true for bounded harmonic functions.
We call a point a ∈Ωan isolated singularity of any function u de-fined on Ω{a}. When u is harmonic on Ω{a}, the isolated singularity a is said to be removable if u has a harmonic extension to Ω.
2.3 Theorem: An isolated singularity of a bounded harmonic func-tion is removable.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Cauchy’s Estimates 33 Proof: It suffices to show that if u is bounded and harmonic on B{0}, then u has a harmonic extension to B. Without loss of generality, we can assume that u is real valued. The only candidate for a harmonic extension of u to B is the Poisson integral P[u|S].
Assume first that n > 2. For ε > 0, define the harmonic function vε on B \ {0} by vε(x) = u(x) −Pu|S + ε(|x|2−n −1).
Observe that as |x| →1, we have vε(x) →0 (by 1.17), while the bound-edness of u shows that vε(x) →∞as x →0. By Corollary 1.10 (with “lim sup” replaced by “lim inf”), vε ≥0 in B \ {0}. Letting ε →0, we conclude that u −P[u|S] ≥0 on B \ {0}. Replacing u by −u, we also have u −P[u|S] ≤0, giving u = P[u|S] on B \ {0}. Thus P[u|S] is the desired harmonic extension of u to B.
The proof when n = 2 is the same, except that (|x|2−n −1) should be replaced by log 1/|x|.
Cauchy’s Estimates If f is holomorphic and bounded by M on a disk B(a, r) ⊂C, then |f (m)(a)| ≤m! M r m for every nonnegative integer m; these are Cauchy’s Estimates from complex analysis. The next theorem gives the comparable results for harmonic functions defined on balls in Rn.
2.4 Cauchy’s Estimates: Let α be a multi-index. Then there is a positive constant Cα such that |Dαu(a)| ≤CαM r |α| for all functions u harmonic and bounded by M on B(a, r).
Proof: We can assume that a = 0. If u is harmonic and bounded by M on B, then by 1.22 we have Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 34 Chapter 2. Bounded Harmonic Functions |Dαu(0)| = Z S u(ζ)DαP(0, ζ) dσ(ζ) ≤M Z S |DαP(0, ζ)| dσ(ζ) = CαM, where Cα = R S |DαP(0, ζ)| dσ(ζ).
If u is harmonic and bounded by M on B(0, r), then applying the result in the previous paragraph to the r-dilate ur shows that |Dαu(0)| ≤CαM r |α| .
Replacing r by r −ε and letting ε decrease to 0, we obtain the same conclusion if u is harmonic on the open ball B(0, r) and bounded by M there.
Augustin-Louis Cauchy (1789–1857), whose collected works consisting of 789 mathematics papers fill 27 volumes, made major contributions to the study of harmonic functions.
The following corollary shows that the derivatives of a bounded har-monic function on Ωcannot grow too fast near ∂Ω.
We let d(a, E) denote the distance from a point a to a set E.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Normal Families 35 2.5 Corollary: Let u be a bounded harmonic function on Ω, and let α be a multi-index. Then there is a constant C such that |Dαu(a)| ≤ C d(a, ∂Ω)|α| for all a ∈Ω.
Proof: For each a ∈Ω, apply Cauchy’s Estimates (Theorem 2.4) with r = d(a, ∂Ω).
Normal Families In complex analysis the term normal family refers to a collection of holomorphic functions with the property that every sequence in the collection contains a subsequence converging uniformly on compact subsets of the domain. The most useful result in this area (and the key tool in most proofs of the Riemann Mapping Theorem) states that a collection of holomorphic functions that is uniformly bounded on each compact subset of the domain is a normal family. We now prove the analogous result for harmonic functions.
2.6 Theorem: If (um) is a sequence of harmonic functions on Ωthat is uniformly bounded on each compact subset of Ω, then some subse-quence of (um) converges uniformly on each compact subset of Ω.
Proof: The key to the proof is the following observation: there exists a constant C < ∞such that for all u harmonic and bounded by M on any ball B(a, 2r), |u(x) −u(a)| ≤ sup B(a,r) |∇u| |x −a| ≤CM r |x −a| for all x ∈B(a, r). The first inequality is standard from advanced cal-culus; the second inequality follows from Cauchy’s Estimates (2.4).
Now suppose K ⊂Ωis compact, and let r = d(K, ∂Ω)/3. Because the set K2r = {x ∈Rn : d(x, K) ≤2r} is a compact subset of Ω, the sequence (um) is uniformly bounded by some M < ∞on K2r .
Let a, x ∈K and assume |x −a| < r. Then x ∈B(a, r) and |um| ≤M on Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 36 Chapter 2. Bounded Harmonic Functions B(a, 2r) ⊂K2r for all m, and so we conclude from the first paragraph that |um(x) −um(a)| ≤CM r |x −a| for all m. It follows that the sequence (um) is equicontinuous on K.
To finish, choose compact sets K1 ⊂K2 ⊂· · · ⊂Ω whose interiors cover Ω. Because (um) is equicontinuous on K1, the Arzela-Ascoli Theorem (, Theorem 11.28) implies (um) contains a subsequence that converges uniformly on K1. Applying Arzela-Ascoli again, there is a subsequence of this subsequence converging uniformly on K2, and so on. If we list these subsequences one after another in rows, then the subsequence obtained by traveling down the diagonal converges uniformly on each Kj, and hence on each compact subset of Ω.
Note that by Theorem 1.23, the convergent subsequence obtained above converges to a harmonic function; furthermore, every partial derivative of this subsequence converges uniformly on each compact subset of Ω.
Theorem 2.6 is often useful in showing that certain extrema exist.
For example, if a ∈Ω, then there exists a harmonic function v on Ω such that |v| < 1 on Ωand |∇v(a)| = sup{|∇u(a)| : u is harmonic on Ωand |u| < 1 on Ω}.
Maximum Principles Corollary 1.10 is the maximum principle in its most general form.
It states that if u is a real-valued harmonic function on Ωand u ≤M at the “boundary” of Ω, then u ≤M on Ω. The catch is that we need to consider ∞as a boundary point. (Again, the function u(x, y) = y on H shows why this is necessary.) Often it is possible to ignore the point at infinity when u is bounded; the next result shows that this is always possible in two dimensions.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Maximum Principles 37 2.7 Theorem: Suppose Ω⊂R2 and Ω̸= R2. If u is a real-valued, bounded harmonic function on Ωsatisfying 2.8 lim sup k→∞ u(ak) ≤M for every sequence (ak) in Ωconverging to a point in ∂Ω, then u ≤M on Ω.
Proof: Because Ω̸= R2, ∂Ωis not empty. Let ε > 0, and choose a sequence in Ωconverging to a point in ∂Ω. By hypothesis, u is less than M + ε on the tail end of this sequence. It follows that there is a closed ball B(a, r) ⊂Ωon which u < M + ε.
Define Ω′ = Ω\ B(a, r), and set v(z) = log z −a r for z ∈Ω′. Then v is positive and harmonic on Ω′, and v(z) →∞as z →∞within Ω′.
For t > 0, we now define the harmonic function wt on Ω′ by wt = u −M −ε −tv.
By 2.8 and the preceding remarks, lim supk→∞wt(ak) < 0 for every se-quence (ak) in Ω′ converging to a point in ∂Ω′, while the boundedness of u on Ω′ shows that wt(ak) →−∞for every sequence (ak) converg-ing to ∞within Ω′. By Corollary 1.10, wt < 0 on Ω′.
We now let t →0 to obtain u ≤M + ε on Ω′. Because u < M + ε on B(a, r), we have u ≤M + ε on all of Ω. Finally, since ε is arbitrary, u ≤M on Ω, as desired.
The higher-dimensional analogue of Theorem 2.7 fails. For an ex-ample, let Ω= {x ∈Rn : |x| > 1} and set u(x) = 1 −|x|2−n. If n > 2, then u is a bounded harmonic function on Ωthat vanishes on ∂Ωbut is not identically 0 on Ω. (In fact, u is never zero on Ω.) The proof of Theorem 2.7 carries over to higher dimensions except for one key point.
Specifically, when n = 2, there exists a positive harmonic function v on Rn \ B such that v(z) →∞as z →∞. When n > 2, there exists no such v; in fact, every positive harmonic function on Rn \ B has a finite limit at ∞when n > 2 (Theorem 4.10).
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 38 Chapter 2. Bounded Harmonic Functions The following maximum principle is nevertheless valid in all dimen-sions. Recall that Hn denotes the upper half-space of Rn.
2.9 Theorem: Suppose Ω⊂Hn.
If u is a real-valued, bounded harmonic function on Ωsatisfying lim sup k→∞ u(ak) ≤M for every sequence (ak) in Ωconverging to a point in ∂Ω, then u ≤M on Ω.
Proof: For (x, y) ∈Ω, define v(x, y) = n−1 X k=1 log xk2 + (y + 1)2 .
Then v is positive and harmonic on Ω, and v(z) →∞as z →∞within Hn. Having obtained v, we can use the ideas in the proof of Theorem 2.7 to finish the proof. The details are even easier here and we leave them to the reader.
Limits Along Rays We now apply some of the preceding results to study the boundary behavior of harmonic functions defined in the upper half-plane H2. We will need the notion of a nontangential limit, which for later purposes we define for functions on half-spaces of arbitrary dimension.
Given a ∈Rn−1 and α > 0, set Γα(a) = {(x, y) ∈Hn : |x −a| < αy}.
Geometrically, Γα(a) is a cone with vertex a and axis of symmetry par-allel to the y-axis.
We have Γα(a) ⊂Γβ(a) if α < β, and Hn is the union of the sets Γα(a) as α ranges over (0, ∞).
A function u defined on Hn is said to have a nontangential limit L at a ∈Rn−1 if for every α > 0, u(z) →L as z →a within Γα(a). The term “nontangential” is used because no curve in Γα(a) that approaches a Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Limits Along Rays 39 The cone Γα(a).
can be tangent to ∂Hn. Exercise 17 of this chapter shows that a bounded harmonic function on Hn can have a nontangential limit at a point of ∂Hn even though the ordinary limit does not exist at that point.
The following theorem for bounded harmonic functions on H2 as-serts that a nontangential limit can be deduced from a limit along a certain one-dimensional set.
2.10 Theorem: Suppose that u is bounded and harmonic on H2. If 0 < θ1 < θ2 < π and lim r→0 u(reiθ1) = L = lim r→0 u(reiθ2), then u has nontangential limit L at 0.
Proof: We may assume L = 0.
If the theorem is false, then for some α > 0, u(z) fails to have limit 0 as z →0 within Γα(0). This means that there exists an ε > 0 and a sequence (zj) tending to 0 within Γα(0) such that 2.11 |u(zj)| > ε for all j.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 40 Chapter 2. Bounded Harmonic Functions Define K = [−α, α] × {1}, and write zj = rjwj, where wj ∈K and rj > 0. Because zj →0, we have rj →0.
Setting uj(z) = u(rjz), note that (uj) is a uniformly bounded se-quence of harmonic functions on H2. By Theorem 2.6, there exists a subsequence of (uj) that converges uniformly on compact subsets of H2 to a bounded harmonic function v on H2; for simplicity we denote this subsequence by (uj) as well.
Examining the limit function v, we see that v(reiθ1) = lim j→∞uj(reiθ1) = lim j→∞u(rjreiθ1) = 0 for all r > 0. Similarly, v(reiθ2) = 0 for all r > 0. The reader may now be tempted to apply Theorem 2.7 to the region between the two rays; unfortunately we do not know that v(z) →0 as z →0 between the given rays. We avoid this problem by observing that the function z 7→v(ez) is bounded and harmonic on the strip Ω= {z = x + iy : θ1 < y < θ2}, and that v(ez) extends continuously to Ωwith v(ez) = 0 on ∂Ω. By Theorem 2.7, v(ez) ≡0 on Ω, and thus v ≡0 on H2.
The sequence (uj) therefore converges to 0 uniformly on compact subsets of H2. In particular, uj →0 uniformly on K. It follows that uj(wj) = u(zj) →0 as j →∞, contradicting 2.11.
Does a limit along one ray suffice to give a nontangential limit in Theorem 2.10? To see that the answer is no, consider the bounded harmonic function u on H2 defined by u(reiθ) = θ for θ ∈(0, π). This function has a limit along each ray in H2 emanating from the origin, but different rays yield different limits. (One ray will suffice for a bounded holomorphic function; see Exercise 22 of this chapter.) Bounded Harmonic Functions on the Ball In the last chapter we defined the Poisson integral P[f ] assuming that f is continuous on S. We can easily enlarge the class of functions f for which P[f ] is defined. For example, if f is a bounded (Borel) measurable function on S, then Pf = Z S f (ζ)P(x, ζ) dσ(ζ) Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Bounded Harmonic Functions on the Ball 41 defines a bounded harmonic function on B; we leave the verification to the reader.
Allowing bounded measurable boundary data gives us many more examples of bounded harmonic functions on B than could otherwise be obtained. For example, in Chapter 6 we will see that the extremal function in the Schwarz Lemma for harmonic functions is the Poisson integral of a bounded discontinuous function on S. In that chapter we will also prove a fundamental result (Theorem 6.13): given a bounded harmonic function u on B, there exists a bounded measurable f on S such that u = P[f ] on B.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 42 Chapter 2. Bounded Harmonic Functions Exercises 1.
Give an example of a bounded harmonic function on B that is not uniformly continuous on B.
2.
(a) Suppose u is a harmonic function on B \ {0} such that |x|n−2u(x) →0 as x →0.
Prove that u has a removable singularity at 0.
(b) Suppose u is a harmonic function on B2 \ {0} such that u(x)/ log |x| →0 as x →0.
Prove that u has a removable singularity at 0.
3.
Suppose that u is harmonic on Rn and that u(x, 0) = 0 for all x ∈Rn−1. Prove that u(x, −y) = −u(x, y) for all (x, y) ∈Rn.
4.
Under what circumstances can a function harmonic on Rn vanish on the union of two hyperplanes?
5.
Use Cauchy’s Estimates (Theorem 2.4) to give another proof of Liouville’s Theorem (Theorem 2.1).
6.
Let K be a compact subset of Ωand let α be a multi-index. Show that there is a constant C = C(Ω, K, α) such that |Dαu(a)| ≤C sup{|u(x)| : x ∈Ω} for every function u harmonic on Ωand every a ∈K.
7.
Suppose u is harmonic on Rn and |u(x)| ≤A(1 + |x|p) for all x ∈Rn, where A is a constant and p ≥0. Prove that u is a polynomial of degree at most p.
8.
Prove if (um) is a pointwise convergent sequence of harmonic functions on Ωthat is uniformly bounded on each compact sub-set of Ω, then (um) converges uniformly on each compact subset of Ω.
9.
Show that if u is the pointwise limit of a sequence of harmonic functions on Ω, then u is harmonic on a dense open subset of Ω.
(Hint: Baire’s Theorem.) Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Exercises 43 10.
Let u be a bounded harmonic function on B. Prove that sup x∈B (1 −|x|)|∇u(x)| < ∞.
11.
The set of harmonic functions u on B satisfying the inequality in Exercise 10 is called the harmonic Bloch space. Prove the har-monic Bloch space is a Banach space under the norm defined by ∥u∥= sup x∈B (1 −|x|)|∇u(x)| + |u(0)|.
12.
Give an example of an unbounded harmonic function in the har-monic Bloch space.
13.
Prove that if u is in the harmonic Bloch space and α is a multi-index with |α| > 0, then sup x∈B (1 −|x|)|α||Dαu(x)| < ∞.
14.
For a ∈B, let Ba denote the ball centered a with radius 1−|a| 2 .
Prove that if u is harmonic on B, then u is in the harmonic Bloch space if and only if sup a∈B 1 V(Ba) Z Ba |u −u(a)| dV < ∞.
15.
Let U denote the set of harmonic functions u on B such that u(0) = 0 and sup x∈B (1 −|x|)|∇u(x)| ≤1.
Prove that there exists a function v ∈U such that Z S |v(ζ/2)| dσ(ζ) = sup u∈U Z S |u(ζ/2)| dσ(ζ).
16.
Suppose Ω⊂Hn and that u is a continuous bounded function on Ωthat is harmonic on Ω. Prove that if u = 0 on ∂Ω, then u ≡0 on Ω.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 44 Chapter 2. Bounded Harmonic Functions 17.
Let f (z) = e−i/z. Show that f is a bounded holomorphic function on H2, that f has a nontangential limit at the origin, but that f does not have a limit along some curve in H2 terminating at the origin.
18.
Suppose 0 < θ1 < θ2 < π and L1, L2 ∈C. Show that there is a bounded harmonic function u on H2 such that u(reiθk) →Lk as r →0 for k = 1, 2.
19.
Suppose u is a bounded harmonic function on H2 with limits at 0 along two distinct rays. Specifically, suppose lim r→0 u(reiθk) = Lk for k = 1, 2, where 0 < θ1 < θ2 < π. Show that limr→0 u(reiθ) exists for every θ ∈(0, π), and evaluate this limit as a function of θ.
20.
Define f (z) = ei log z, where log z denotes the principal-valued logarithm. Show that f is a bounded holomorphic function on H2 whose real and imaginary parts fail to have a limit along every ray in H2 emanating from the origin.
21.
Let θ0 ∈(0, π).
Prove that there exists a bounded harmonic function u on H2 such that limr→0 u(reiθ) exists if and only if θ = θ0.
(Hint: Do this first for θ0 = π/2 by letting u(z) = Re ei log z and considering u(x, y) −u(−x, y).) 22.
Let f be a bounded holomorphic function on H2, and suppose limr→0 f (reiθ) exists for some θ ∈(0, π). Prove that f has a nontangential limit at 0.
23.
Let u be a bounded harmonic function on H3 that has the same limit along two distinct rays in H3 emanating from 0. Need u have a nontangential limit at 0?
24.
Prove that when n > 2 there does not exist a harmonic function v on Rn \ B such that v(z) →∞as z →∞.
25.
Let K denote a compact line segment contained in B3. Show that every bounded harmonic function on B3 \ K extends to be har-monic on B3.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Chapter 3 Positive Harmonic Functions This chapter focuses on the special properties of positive harmonic functions. We will describe the positive harmonic functions defined on all of Rn (Liouville’s Theorem), show that positive harmonic func-tions cannot oscillate wildly (Harnack’s Inequality), and characterize the behavior of positive harmonic functions near isolated singularities (Bôcher’s Theorem).
Liouville’s Theorem In Chapter 2 we proved that a bounded harmonic function on Rn is constant. We now improve that result. In Chapter 9 we will improve even the result below (see 9.10).
3.1 Liouville’s Theorem for Positive Harmonic Functions: A posi-tive harmonic function on Rn is constant.
Proof: The proof is a bit more delicate than that given for bounded harmonic functions.
Let u be a positive harmonic function defined on Rn. Fix x ∈Rn. Let r > |x|, and let Dr denote the symmetric difference of the balls B(x, r) and B(0, r). The volume version of the mean-value property (1.6) shows that u(x) −u(0) = 1 V B(0, r) hZ B(x,r) u dV − Z B(0,r) u dV i .
45 46 Chapter 3. Positive Harmonic Functions Because the integrals of u over B(x, r) ∩B(0, r) cancel (see 3.2), we have |u(x) −u(0)| ≤ 1 V B(0, r) Z Dr u dV ≤ 1 V B(0, r) Z B(0,r+|x|)\B(0,r−|x|) u dV = 1 V B(0, r) hZ B(0,r+|x|) u dV − Z B(0,r−|x|) u dV i = u(0)(r + |x|)n −(r −|x|)n r n .
Note that the positivity of u was used in the first inequality.
Now letting r →∞, we see that u(x) = u(0), proving that u is constant.
−r |x| − r 0 x r r + |x| 3.2 The symmetric difference Dr (shaded) of B(x, r) and B(0, r).
Liouville’s Theorem for positive harmonic functions leads to an easy proof that a positive harmonic function on R2 \ {0} is constant.
3.3 Corollary: A positive harmonic function on R2 \ {0} is constant.
Proof: If u is positive and harmonic on R2 \ {0}, then the function z 7→u(ez) is positive and harmonic on R2 (= C) and hence (by 3.1) is constant. This proves that u is constant.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Harnack’s Inequality and Harnack’s Principle 47 The higher-dimensional analogue of Corollary 3.3 fails; for example, the function |x|2−n is positive and harmonic on Rn \ {0} when n > 2.
We will classify the positive harmonic functions on Rn \ {0} for n > 2 after the proof of Bôcher’s Theorem; see Corollary 3.14.
Harnack’s Inequality and Harnack’s Principle Positive harmonic functions cannot oscillate too much on a com-pact set K ⊂Ωif Ωis connected; the precise statement is Harnack’s Inequality (3.6). We first consider the important special case where Ω is the open unit ball.
3.4 Harnack’s Inequality for the Ball: If u is positive and harmonic on B, then 1 −|x| (1 + |x|)n−1 u(0) ≤u(x) ≤ 1 + |x| (1 −|x|)n−1 u(0) for all x ∈B.
Proof: If u is positive and harmonic on the closed unit ball B, then u(x) = Pu|S = Z S u(ζ) 1 −|x|2 |x −ζ|n dσ(ζ) ≤ 1 −|x|2 (1 −|x|)n Z S u(ζ) dσ(ζ) = 1 + |x| (1 −|x|)n−1 u(0) for all x ∈B. If u is positive and harmonic on B, apply the estimate above to the dilates ur and take the limit as r →1.
This gives us the second inequality of the theorem. The first inequality is proved similarly.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 48 Chapter 3. Positive Harmonic Functions Define α(t) = (1−t)/(1+t)n−1 and β(t) = (1+t)/(1−t)n−1. After a translation and a dilation, 3.4 tells us that if u is positive and harmonic on B(a, R), and |x −a| ≤r < R, then 3.5 α(r/R)u(a) ≤u(x) ≤β(r/R)u(a).
3.6 Harnack’s Inequality: Suppose that Ωis connected and that K is a compact subset of Ω. Then there is a constant C ∈(1, ∞) such that 1 C ≤u(y) u(x) ≤C for all points x and y in K and all positive harmonic functions u on Ω.
Proof: We will prove that there is a constant C ∈(1, ∞) such that u(y)/u(x) ≤C for all x, y ∈K and all positive harmonic functions u on Ω. Because x and y play symmetric roles, the other inequality will also hold.
For (x, y) ∈Ω× Ω, define s(x, y) = sup{u(y)/u(x) : u is positive and harmonic on Ω}.
We first show that s < ∞on Ω× Ω.
Fix x ∈Ω, and define E = {y ∈Ω: s(x, y) < ∞}.
Because x ∈E, E is not empty. If y ∈E, we may choose r > 0 such that B(y, 2r) ⊂Ω. By 3.5, u ≤β(1/2)u(y) on B(y, r) for all positive harmonic functions u on Ω. We then have B(y, r) ⊂E, proving that E is open. If z ∈Ωis a limit point of E, there exists an r > 0 and a y ∈E such that z ∈B(y, r) ⊂B(y, 2r) ⊂Ω. By 3.5, u(z) ≤β(1/2)u(y) for all positive harmonic functions u on Ω. We then have z ∈E, proving that E is closed. The connectivity of Ωtherefore shows that E = Ω.
We now know that s is finite at every point of Ω× Ω. Let K ⊂Ωbe compact, and let (a, b) ∈K × K. Then by 3.5, u(y) u(x) ≤β(1/2)u(b) α(1/2)u(a) ≤β(1/2) α(1/2)s(a, b) Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Harnack’s Inequality and Harnack’s Principle 49 for all (x, y) in a neighborhood of (a, b), and for all positive harmonic functions u on Ω.
Because K × K is covered by finitely many such neighborhoods, s is bounded above on K × K, as desired.
Note that the constant C in 3.6 may depend upon Ωand K, but that C is independent of x, y, and u.
An intuitive way to remember Harnack’s Inequality is shown in 3.7.
Here we have covered K with a finite chain of overlapping balls (possi-ble, since Ωis connected); to compare the values of a positive harmonic function at any two points in K, we can think of a finite chain of inequal-ities of the kind expressed in 3.5.
3.7 K covered by overlapping balls.
Harnack’s Inequality leads to an important convergence theorem for harmonic functions known as Harnack’s Principle. Consider a mono-tone sequence of continuous functions on Ω. The pointwise limit of such a sequence need not behave well—it could be infinite at some points and finite at other points. Even if it is finite everywhere, there is no reason to expect that our sequence converges uniformly on every compact subset of Ω. Harnack’s Principle shows that none of this bad behavior can occur for a monotone sequence of harmonic functions.
3.8 Harnack’s Principle: Suppose Ωis connected and (um) is a point-wise increasing sequence of harmonic functions on Ω. Then either (um) converges uniformly on compact subsets of Ωto a function harmonic on Ωor um(x) →∞for every x ∈Ω.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 50 Chapter 3. Positive Harmonic Functions Proof: Replacing um by um −u1 + 1, we can assume that each um is positive on Ω. Set u(x) = limm→∞um(x) for each x ∈Ω.
First suppose u is finite everywhere on Ω. Let K be a compact subset of Ω. Fix x ∈K. Harnack’s Inequality (3.6) shows there is a constant C ∈(1, ∞) such that um(y) −uk(y) ≤C um(x) −uk(x) for all y ∈K, whenever m > k. This implies (um) is uniformly Cauchy on K, and thus um →u uniformly on K, as desired. Theorem 1.23 shows that the limit function u is harmonic on Ω.
Now suppose u(x) = ∞for some x ∈Ω. Let y ∈Ω. Then Harnack’s Inequality (3.6), applied to the compact set K = {x, y}, shows that there is a constant C ∈(1, ∞) such that um(x) ≤Cum(y) for every m.
Because um(x) →∞, we also have um(y) →∞, and so u(y) = ∞. This implies that u is identically ∞on Ω.
Isolated Singularities In this section we prove Bôcher’s Theorem, which characterizes the behavior of positive harmonic functions in the neighborhood of an iso-lated singularity. Recall that when n = 2, the function log(1/|x|) is positive and harmonic on B{0}, while when n > 2, the function |x|2−n is positive and harmonic on B \ {0}. Roughly speaking, Bôcher’s Theo-rem says that near an isolated singularity, a positive harmonic function must behave like one of these functions.
3.9 Bôcher’s Theorem: If u is positive and harmonic on B{0}, then there is a function v harmonic on B and a constant b ≥0 such that u(x) = ( v(x) + b log(1/|x|) if n = 2 v(x) + b|x|2−n if n > 2 for all x ∈B \ {0}.
The next three lemmas will be used in the proof of Bôcher’s The-orem (our proof of Bôcher’s Theorem is taken from , which also contains references to several other proofs of this result). The first lemma describes the spherical averages of a function harmonic on a Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Isolated Singularities 51 punctured ball. Given a continuous function u defined on B \ {0}, we define Au to be the average of u over the sphere of radius |x|: Au = Z S u(|x|ζ) dσ(ζ) for x ∈B \ {0}.
3.10 Lemma: Suppose u is harmonic on B \ {0}.
Then there exist constants a, b ∈C such that Au = ( a + b log(1/|x|) if n = 2 a + b|x|2−n if n > 2 for all x ∈B \ {0}. In particular, A[u] is harmonic on B \ {0}.
Proof: Let ds denote surface-area measure (unnormalized). Define f on (0, 1) by f (r) = Z S u(rζ) ds(ζ); so Au is a constant multiple of f (|x|). Because u is continuously differentiable on B \ {0}, we can compute f ′ by differentiating under the integral sign, obtaining f ′(r) = Z S ζ · (∇u)(rζ) ds(ζ) = r −n Z rS τ · (∇u)(τ) ds(τ).
Suppose 0 < r0 < r1 < 1 and Ω= {x ∈Rn : r0 < |x| < r1}. The divergence theorem, applied to ∇u, shows that Z ∂Ω n · ∇u ds = Z Ω ∆u dV; here n denotes the outward unit normal on Ω. Because u is harmonic on Ω, the right side of this equation is 0. Note also that ∂Ω= r0S ∪r1S and that n = −τ/r0 on r0S and n = τ/r1 on r1S. Thus the last equation above implies that 1 r0 Z r0S τ · (∇u)(τ) ds(τ) = 1 r1 Z r1S τ · (∇u)(τ) ds(τ), which means f ′(r) is a constant multiple of r 1−n (for 0 < r < 1). This proves f(r) is of the desired form.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 52 Chapter 3. Positive Harmonic Functions An immediate consequence of the lemma above is that every radial harmonic function on B \ {0} is of the form given by the conclusion of 3.10 (a function is radial if its value at x depends only on |x|). Proof: if u is radial, then u = A[u]. (For another proof, see Exercise 13 of this chapter.) The next lemma is a version of Harnack’s Inequality that allows x and y to range over a noncompact set provided |x| = |y|.
3.11 Lemma: There exists a constant c > 0 such that for every positive harmonic function u on B \ {0}, cu(y) < u(x) whenever 0 < |x| = |y| ≤1/2.
Proof: Harnack’s Inequality (3.6), with Ω= B \ {0} and K = (1/2)S, shows there is a constant c > 0 such that for all positive harmonic u on B \ {0}, we have cu(y) < u(x) whenever |x| = |y| = 1/2. Applying this result to the dilates ur , 0 < r < 1, gives the desired conclusion.
The following result characterizes the positive harmonic functions on B \ {0} that are identically zero on S. This is really the heart of our proof of Bôcher’s Theorem.
3.12 Lemma: Suppose u is positive and harmonic on B \ {0} and u(x) →0 as |x| →1. Then there exists a constant b > 0 such that u(x) = ( b log(1/|x|) if n = 2 b(|x|2−n −1) if n > 2 for all x ∈B \ {0}.
Proof: By Lemma 3.10, we need only show that u = A[u] on B{0}.
Suppose we could show that u ≥A[u] on B \ {0}. Then if there were a point x ∈B \ {0} such that u(x) > Au, we would have Au > A A[u] (x) = Au, a contradiction. Thus we need only prove that u ≥A[u] on B \ {0}, which we now do.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Isolated Singularities 53 Let c be the constant of Lemma 3.11. By Lemma 3.10, u −cA[u] is harmonic on B \ {0}. By Lemma 3.11, u(x) −cAu > 0 whenever 0 < |x| ≤1/2, and clearly u(x) −cAu →0 as |x| →1 by our hypothesis on u. The minimum principle for harmonic functions (1.10) thus shows that u −cA[u] > 0 on B \ {0}.
We wish to iterate this result. For this purpose, define g(t) = c + t(1 −c) for t ∈[0, 1]. Suppose we know that 3.13 w = u −tA[u] > 0 on B{0} for some t ∈[0, 1]. Since w(x) →0 as |x| →1, the preceding argument may be applied to w, yielding w −cA[w] = u −g(t)A[u] > 0 on B {0}. This process may be continued. Letting g(m) denote the mth iterate of g, we see that 3.13 implies u −g(m)(t)A[u] > 0 on B \ {0} for m = 1, 2, . . . .
But g(m)(t) →1 as m →∞for every t ∈[0, 1], so that 3.13 holding for some t ∈[0, 1] implies u−A[u] ≥0 on B{0}. Since 3.13 obviously holds when t = 0, we have u−A[u] ≥0 on B \ {0}, as desired.
Now we are ready to prove Bôcher’s Theorem (3.9).
Proof of Bôcher’s Theorem: We first assume that n > 2 and that u is positive and harmonic on B \ {0}. Define a harmonic function w on B \ {0} by w(x) = u(x) −Pu|S + |x|2−n −1.
As |x| →1, we have w(x) →0 (by 1.17), and as |x| →0, we have w(x) →∞(because u is positive and P[u|S] is bounded on B \ {0}).
By the minimum principle (1.10), we conclude that w is positive on B \ {0}.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 54 Chapter 3. Positive Harmonic Functions Lemma 3.12, applied to w, shows that u(x) = v(x) + b|x|2−n on B \ {0} for some v harmonic on B and some constant b. Now letting x →0, we see that the positivity of u implies that b ≥0; we have thus proved Bôcher’s Theorem in the case where u is positive and harmonic on B \ {0}.
For the general positive harmonic u on B \ {0}, we may apply the result above to the dilate u(x/2), so that u(x/2) = v(x) + b|x|2−n on B \ {0} for some v harmonic on B and some constant b ≥0. This implies that u(x) = v(2x) + b22−n|x|2−n on (1/2)B{0}, which shows that u(x)−b22−n|x|2−n extends harmon-ically to (1/2)B, and hence to B. Thus the proof of Bôcher’s Theorem is complete in the case where n > 2. The proof of the n = 2 case is the same, except that log(1/|x|) should be replaced by |x|2−n.
In Chapter 9, in the section Bôcher’s Theorem Revisted, we will see another approach to this result.
We conclude this section by characterizing the positive harmonic functions on Rn {0} for n > 2. (Recall that by 3.3, a positive harmonic function on R2 \ {0} is constant.) 3.14 Corollary: Suppose n > 2.
If u is positive and harmonic on Rn \ {0}, then there exist constants a, b ≥0 such that u(x) = a + b|x|2−n for all x ∈Rn \ {0}.
Proof: Suppose u is positive and harmonic on Rn \ {0}. Then on B \ {0} we may write u(x) = v(x) + b|x|2−n as in Bôcher’s Theorem (3.9). The function v extends harmonically to all of Rn by setting v(x) = u(x) −b|x|2−n for x ∈Rn \ B. Because lim infx→∞v(x) ≥0, the minimum principle (1.10) implies that v is nonnegative on Rn. By 3.1, v is constant, completing the proof.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Positive Harmonic Functions on the Ball 55 Positive Harmonic Functions on the Ball At the end of Chapter 2 we briefly discussed how it is possible to define P[f ] when f is not continuous, and indicated that it was neces-sary to do so in order to characterize the bounded harmonic functions on B. A similar idea works for positive harmonic functions on B—given a positive finite Borel measure µ on S, we can define Pµ = Z S P(x, ζ) dµ(ζ) for x ∈B.
The function so defined is positive and harmonic on B, as the reader can check by differentiating under the integral sign or by using the converse to the mean-value property. In Chapter 6 we will show (see 6.15) that every positive harmonic function on B is the Poisson integral of a measure as above. Many important consequences follow from this characterization, among them the result (see 6.44) that every positive harmonic function on B has boundary values almost everywhere on S, in a sense to be made precise in Chapter 6.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 56 Chapter 3. Positive Harmonic Functions Exercises 1.
Use 3.5 to give another proof of Liouville’s Theorem for positive harmonic functions (3.1).
2.
Can equality hold in either of the inequalities in Harnack’s In-equality for the ball (3.4)?
3.
Show that for every multi-index α there exists a constant Cα such that |Dαu(x)| ≤ Cαu(0) (1 −|x|)|α|+n−1 for every x ∈B and every positive harmonic u on B. Use this to give another proof of Liouville’s Theorem for positive harmonic functions.
4.
Let Ωbe an open square in R2. Prove that there exists a positive harmonic function u on Ωsuch that u(z)d(z, ∂Ω) is unbounded on Ω.
5.
Define s on Ω× Ωby s(x, y) = sup{u(y)/u(x) : u is positive and harmonic on Ω}.
Prove that s is continuous on Ω× Ω.
6.
Suppose u is positive and harmonic on the upper half-space H.
Prove that if z ∈H and u is bounded on the ray {rz : r > 0}, then u is bounded in the cone Γα(0) for every α > 0.
7.
Suppose u is positive and harmonic on the upper half-space H, z ∈H, and u(rz) →L as r →0, where L ∈[0, ∞]. Show that if L = ∞, then u has nontangential limit ∞at 0. Prove a similar result for the case L = 0. Show that u need not have a nontan-gential limit at 0 if L ∈(0, ∞).
8.
Prove the analogue of Theorem 2.10 for positive harmonic func-tions on H2 with a common limit along two distinct rays.
9.
Suppose u is positive and harmonic on H. Show that u has non-tangential limit L at 0 if and only if limr→0 u(rz) = L for every z ∈H.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Exercises 57 10.
Show that if pointwise divergence to ∞occurs in Harnack’s Prin-ciple (3.8), then the divergence is “uniform” on compact subsets of Ω.
11.
Prove that a pointwise convergent sequence of positive harmonic functions on Ωconverges uniformly on compact subsets of Ω.
12.
Suppose Ωis connected and (um) is a sequence of positive har-monic functions on Ω. Show that at least one of the following statements is true: (a) (um) contains a subsequence diverging to ∞pointwise on Ω; (b) (um) contains a subsequence converging uniformly on com-pact subsets of Ω.
13.
Suppose that u is a radial function in C2(B \ {0}). Let g be the function on (0, 1) defined by g(|x|) = u(x). Compute ∆u in terms of g and its derivatives. Use this to prove that a radial harmonic function on B \ {0} must be of the form given by the conclusion of 3.10.
14.
Prove that the constant b and the function v in the conclusion of Bôcher’s Theorem (3.9) are unique.
15.
Suppose n > 2. Assume a ∈Ωand u is harmonic on Ω\ {a}.
Show that if u is positive on some punctured ball centered at a, then there exists a nonnegative constant b and a harmonic func-tion v on Ωsuch that u(x) = b|x −a|2−n + v(x) on Ω\ {a}.
16.
(a) Suppose n > 2. Let u be harmonic on B \ {0}. Show that if lim inf x→0 u(x)|x|n−2 > −∞, then there exists a function v harmonic on B and a constant b such that u(x) = b|x|2−n + v(x) on B.
(b) Formulate and prove a similar result for n = 2.
17.
Let A = {a1, a2, . . . } denote a discrete subset of Rn. Characterize the positive harmonic functions on Rn \ A.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Chapter 4 The Kelvin Transform The Kelvin transform performs a role in harmonic function the-ory analogous to that played by the transformation f (z) 7→f (1/z) in holomorphic function theory. For example, it transforms a function harmonic inside the unit sphere into a function harmonic outside the sphere. In this chapter, we introduce the Kelvin transform and use it to solve the Dirichlet problem for the exterior of the unit sphere and to obtain a reflection principle for harmonic functions. Later, we will use the Kelvin transform in the study of isolated singularities of harmonic functions.
Inversion in the Unit Sphere When studying harmonic functions on unbounded open sets, we will often find it convenient to append the point ∞to Rn. We topologize Rn ∪{∞} in the natural way: a set ω ⊂Rn ∪{∞} is open if it is an open subset of Rn in the ordinary sense or if ω = {∞} ∪(Rn \ E), where E is a compact subset of Rn. The resulting topological space is compact and is called the one-point compactification of Rn. Via the usual stereographic projection, Rn ∪{∞} is homeomorphic to the unit sphere in Rn+1.
The map x 7→x∗, where x∗= x/|x|2 if x ̸= 0, ∞ 0 if x = ∞ ∞ if x = 0 59 60 Chapter 4. The Kelvin Transform is called the inversion of Rn ∪{∞} relative to the unit sphere. Note that if x ∉{0, ∞}, then x∗lies on the ray from the origin determined by x, with |x∗| = 1/|x|. The reader should verify that the inversion map is continuous, is its own inverse, is the identity on S, and takes a neighborhood of ∞onto a neighborhood of 0.
For any set E ⊂Rn ∪{∞}, we define E∗= {x∗: x ∈E}.
The inversion map preserves the family of spheres and hyperplanes in Rn (if we adopt the convention that the point ∞belongs to every hyperplane). To see this, observe that a set E ⊂Rn is a nondegenerate sphere or hyperplane if and only if 4.1 E = {x ∈Rn : a|x|2 + b · x + c = 0}, where b ∈Rn and a, c are real numbers satisfying |b|2 −4ac > 0. We easily see that if E has the form 4.1, then E∗has the same form (with the roles of a and c reversed); inversion therefore preserves the family of spheres and hyperplanes, as claimed.
Recall that a C1-map Ψ : Ω→Rn is said to be conformal if it pre-serves angles between intersecting curves; this happens if and only if the Jacobian Ψ ′(x) is a scalar multiple of an orthogonal transformation for each x ∈Ω.
4.2 Proposition: The inversion x 7→x∗is conformal on Rn \ {0}.
Proof: Set Ψ(x) = x∗= x/|x|2. Fix y ∈Rn \ {0}. Choose an or-thogonal transformation T of Rn such that Ty = (|y|, 0, . . . , 0). Clearly Ψ = T −1 ◦Ψ ◦T, so that Ψ ′(y) = T −1 ◦Ψ ′ T(y) ◦T.
Thus to complete the proof we need only show that Ψ ′ T(y) , which equals Ψ ′(|y|, 0, . . . , 0), is a scalar multiple of an orthogonal transfor-mation. However, a simple calculation, which we leave to the reader, shows that the matrix of Ψ ′(|y|, 0, . . . , 0) is diagonal, with −1/|y|2 in the first position and 1/|y|2 in the other diagonal positions. Hence the proof of the proposition is complete.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Motivation and Definition 61 Motivation and Definition Suppose E is a compact subset of Rn. If u is harmonic on Rn \E, we naturally regard ∞as an isolated singularity of u. When should we say that u has a removable singularity at ∞? There is an obvious answer when n = 2, because here the inversion x 7→x∗preserves harmonic functions: if Ω⊂R2 \ {0} and u is harmonic on Ω, then the function x 7→u(x∗) is harmonic on Ω∗. (Note that on R2 = C, inversion is the map z 7→1/ z.) When n = 2, then, we say that u is harmonic at ∞ provided the function x 7→u(x∗) has a removable singularity at 0.
Unfortunately, the inversion map does not preserve harmonic func-tions when n > 2 (consider, for example, u(x) = |x|2−n). Nevertheless, there is a transformation involving the inversion that does preserve har-monic functions for all n ≥2; it is called the Kelvin transformation in honor of Lord Kelvin who discovered it in the 1840s .
We can guess what this transformation is by applying the symme-try lemma to the Poisson kernel. Fixing ζ ∈S, recall that P(·, ζ) is harmonic on Rn \ {ζ} (1.18). By the symmetry lemma (1.13), we have |x −ζ| = |x|−1x −|x|ζ for all x ∈Rn \ {0}. Applying this to P(x, ζ) = (1 −|x|2)/|x −ζ|n, we easily compute that 4.3 P(x, ζ) = −|x|2−nP(x∗, ζ) for all x ∈Rn \ {0, ζ}. The significant fact here is that the right side of 4.3 is a harmonic function of x on Rn \ {0, ζ}. Except for the minus sign, the definition of the Kelvin transformation is staring us in the face.
Thus, given a function u defined on a set E ⊂Rn \ {0}, we define the function K[u] on E∗by Ku = |x|2−nu(x∗); the function K[u] is called the Kelvin transform of u. Note that when n = 2, Ku = u(x∗).
We easily see that K K[u] = u for all functions u as above; in other words, the Kelvin transform is its own inverse.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 62 Chapter 4. The Kelvin Transform The transform K is also linear—if u, v are functions on E and b, c are constants, then K[bu + cv] = bK[u] + cK[v] on E∗.
The Kelvin transform preserves uniform convergence on compact sets. Specifically, suppose E is a compact subset of Rn \ {0} and (um) is a sequence of functions on E. Then (um) converges uniformly on E if and only if (K[um]) converges uniformly on E∗.
The Kelvin Transform Preserves Harmonic Functions In this section we will see that the Kelvin transform of every har-monic function is harmonic. We begin with a simple computation.
4.4 Lemma: If p is a polynomial on Rn homogeneous of degree m, then ∆(|x|2−n−2mp) = |x|2−n−2m∆p.
Proof: Let t ∈R. Use the product rule for Laplacians (1.19) along with Exercise 3 in Chapter 1 to get ∆(|x|tp) = |x|t∆p + 2t|x|t−2x · ∇p + t(t + n −2)|x|t−2p.
If p is homogeneous of degree m, then x · ∇p = mp (see Exercise 29 in Chapter 1), so the equation above reduces to 4.5 ∆(|x|tp) = |x|t∆p + t(2m + t + n −2)|x|t−2p.
Taking t = 2 −n −2m now gives the conclusion of the lemma.
If p is homogeneous of degree m, then clearly K[p] = |x|2−n−2mp.
This observation is used twice in the proof of the next proposition, which shows that the Kelvin transform comes close to commuting with the Laplacian.
4.6 Proposition: If u is a C2 function on an open subset of Rn \ {0}, then ∆(K[u]) = K[|x|4∆u].
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Harmonicity at Infinity 63 Proof: First suppose that p is a polynomial on Rn homogeneous of degree m. Then ∆(K[p]) = ∆(|x|2−n−2mp) = |x|2−n−2m∆p = K[|x|4∆p], where the second equality follows from Lemma 4.4 and the third equal-ity holds because |x|4∆p is homogeneous of degree m + 2.
The paragraph above shows that the proposition holds for poly-nomials (by linearity).
Because polynomials are locally dense in the C2-norm, the result holds for arbitrary C2 functions u, as desired.
We come now to the the crucial property of the Kelvin transform.
4.7 Theorem: If Ω⊂Rn \ {0}, then u is harmonic on Ωif and only if K[u] is harmonic on Ω∗.
Proof: From the previous proposition, we see that ∆(K[u]) ≡0 if and only if ∆u ≡0.
Harmonicity at Infinity Because the Kelvin transform preserves harmonicity, we make the following definition: if E ⊂Rn is compact and u is harmonic on Rn \ E, then u is harmonic at ∞provided K[u] has a removable singularity at the origin. Notice that in the n = 2 case this definition is consistent with our previous discussion.
If u is harmonic at ∞, then K[u] has a finite limit L at 0; in other words lim x→0 |x|2−nu(x/|x|2) = L.
From this we see that if u is harmonic at ∞, then limx→∞u(x) = 0 when n > 2, while limx→∞u(x) = L when n = 2. This observation leads to characterizations of harmonicity at ∞. We begin with the n > 2 case.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 64 Chapter 4. The Kelvin Transform 4.8 Theorem: Assume n > 2. Suppose u is harmonic on Rn \ E, where E ⊂Rn is compact.
Then u is harmonic at ∞if and only if limx→∞u(x) = 0.
Proof: We have just noted above that if u is harmonic at ∞, then limx→∞u(x) = 0.
To prove the other direction, suppose that limx→∞u(x) = 0. Then |x|n−2Ku →0 as x →0. By Exercise 2(a) of Chapter 2, K[u] has a removable singularity at 0, which means u is harmonic at ∞.
Now we turn to the characterization of harmonicity at ∞in the n = 2 case.
4.9 Theorem: Suppose u is harmonic on R2 \ E, where E ⊂R2 is compact. Then the following are equivalent: (a) u is harmonic at ∞; (b) limx→∞u(x) = L for some complex number L; (c) u(x)/log |x| →0 as x →∞; (d) u is bounded on a deleted neighborhood of ∞.
Proof: We have already seen that (a) implies (b).
That (b) implies (c) is trivial.
Suppose now that (c) holds. Then Ku/log |x| →0 as x →0. By Exercise 2(b) of Chapter 2, K[u] has a removable singularity at 0. Thus u is harmonic at ∞, which implies (d).
Finally, suppose that (d) holds, so that u is bounded on a deleted neighborhood of ∞. Then Ku = u(x∗) is bounded on a deleted neighborhood of 0. Thus by Theorem 2.3, (a) holds, completing the proof.
Boundedness near ∞is thus equivalent to harmonicity at ∞when n = 2, but not when n > 2. We now take up the question of bounded-ness near ∞in higher dimensions.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Harmonicity at Infinity 65 4.10 Theorem: Suppose n > 2 and u is harmonic and real valued on Rn \ E, where E is compact. Then the following are equivalent: (a) u is bounded in a deleted neighborhood of ∞; (b) u is bounded above or below in a deleted neighborhood of ∞; (c) u −c is harmonic at ∞for some constant c; (d) u has a finite limit at ∞.
Proof: The implications (a) ⇒(b) and (d) ⇒(a) are trivial. If (c) holds then u has limit c at ∞by Theorem 4.8; hence (c) ⇒(d). We complete the proof by showing (b) ⇒(c).
Without loss of generality, we assume that u is positive in a deleted neighborhood of ∞. Thus the Kelvin transform K[u] is positive in a deleted neighborhood of 0. By Bôcher’s Theorem (3.9) there is a con-stant c such that Ku −c|x|2−n extends harmonically across 0.
Applying the Kelvin transform shows that u −c is harmonic at ∞.
Conditions (a), (c), and (d) of Theorem 4.10 are equivalent without the hypothesis that u is real valued.
Note that Theorem 4.10 provides a new proof of Liouville’s Theorem for positive harmonic functions (3.1). Specifically, if n > 2 and u is positive and harmonic on Rn, then by Theorem 4.10 u must have finite limit c at ∞. By the maximum/minimum principle, u ≡c. This new proof of Liouville’s Theorem amounts to the observation that—via the Kelvin transform—Bôcher’s Theorem implies Liouville’s Theorem, at least for n > 2.
The implication also holds when n = 2. If u is positive and harmonic on R2, then by Bôcher’s Theorem (3.9) there is a constant b ≥0 such that v(x) = Ku −b log |1/x| has harmonic extension across 0.
Thus v is an entire harmonic function. If b > 0, then we would have limx→∞v(x) = ∞, which contradicts the minimum principle.
Thus b = 0 and limx→∞v(x) = u(0), from which it follows that v ≡u(0).
Hence u ≡u(0), as desired.
In Chapter 9, we will see that Liouville’s Theorem implies Bôcher’s Theorem when n > 2, and we will present generalized versions of these theorems.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 66 Chapter 4. The Kelvin Transform The Exterior Dirichlet Problem In Chapter 1, we solved the Dirichlet problem for the interior of the unit sphere S—given any f ∈C(S), there is a unique function u harmonic on B and continuous on B such that u|S = f . To solve the corresponding problem for the exterior of the unit sphere, we define the exterior Poisson kernel, denoted Pe, by setting Pe(x, ζ) = |x|2 −1 |x −ζ|n for |x| > 1 and ζ ∈S. Given f ∈C(S), we define the exterior Poisson integral Pe[f ] by Pef = Z S f (ζ)Pe(x, ζ) dσ(ζ) for |x| > 1.
4.11 Theorem: Suppose f ∈C(S). Then there is a unique function u harmonic on B∗and continuous on B ∗such that u|S = f . Moreover, u = Pe[f ] on B∗\ {∞}.
Remark: For n > 2, we are not asserting that there exists a unique continuous u on B∗, with u harmonic on {x ∈Rn : |x| > 1}, such that u|S = f. For example, the functions 1 −|x|2−n and 0, which agree on S, are both harmonic on Rn \ {0}. The uniqueness assertion in the theorem above comes from the requirement that u be harmonic at ∞ (recall that ∞∈B∗).
Proof of Theorem 4.11: Let v ∈C(B) denote the solution of the Dirichlet problem for B with boundary data f on S, so that v|S = f and v(x) = Z S f (ζ)P(x, ζ) dσ(ζ) for x ∈B. The function u = K[v] is then harmonic on B∗(if we set u(∞) = limx→∞Kv), u is continuous on B ∗, and u|S = f.
We have u(x) = Z S f (ζ)|x|2−nP(x∗, ζ) dσ(ζ) for |x| > 1. By 4.3, this gives u = Pe[f ] on B∗\ {∞}, as desired.
The uniqueness of u follows from the maximum principle.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Symmetry and the Schwarz Reflection Principle 67 Symmetry and the Schwarz Reflection Principle Given a hyperplane E, we say that a pair of points are symmetric about E if E is the perpendicular bisector of the line segment joining these points. For each x ∈Rn, there exists a unique xE ∈Rn such that x and xE are symmetric about E; we call xE the reflection of x in E.
Clearly (xE)E = x for every x ∈Rn.
We say Ωis symmetric about the hyperplane E if ΩE = Ω, where ΩE = {xE : x ∈Ω}.
Ωis symmetric about E.
If T is a translation, dilation, or rotation, then T preserves symmetry about hyperplanes; in other words, if T is any of these maps and E is a hyperplane, then T(x) and T(xE) are symmetric about T(E) for all x ∈Rn.
Given a hyperplane E = {x ∈Rn : b · x = c}, where b is a nonzero vector in Rn and c is a real number, we set E+ = {x ∈Rn : b · x > c}; geometrically, E+ is an open half-space with ∂E+ = E.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 68 Chapter 4. The Kelvin Transform We now come to the Schwarz reflection principle for hyperplanes; the reader who has done Exercise 3 in Chapter 2 can probably guess the proof.
4.12 Theorem: Suppose Ωis symmetric about a hyperplane E. If u is continuous on Ω∩E+, u is harmonic on Ω∩E+, and u = 0 on Ω∩E, then u extends harmonically to Ω.
Proof: We may assume that E = {(x, y) ∈Rn : y = 0} and that E+ = {(x, y) ∈Rn : y > 0}. The function v(x, y) = ( u(x, y) if (x, y) ∈Ωand y ≥0 −u(x, −y) if (x, y) ∈Ωand y < 0 is continuous on Ωand satisfies the mean-value property. Hence, by Theorem 1.24, v is a harmonic extension of u to all of Ω.
We now extend the notions of symmetry and reflection to spheres.
If E = S, the unit sphere, then inversion is the natural choice for the reflection map x 7→xE. So here we set xE = x∗. More generally, if E = ∂B(a, r), we define 4.13 xE = a + r 2(x −a)∗, and we say that x and xE are symmetric about E. Note that the center of E and the point at infinity are symmetric about E. We say that Ωis symmetric about the sphere E if ΩE = Ω, where ΩE = {xE : x ∈Ω}; see 4.14.
We remark in passing that symmetry about a hyperplane can be viewed as a limiting case of symmetry about a sphere; see Exercise 10 of this chapter. (We adopt the convention that ∞E = ∞when E is a hyperplane.) Translations, dilations, and rotations obviously preserve symmetry about spheres.
The inversion map also preserves symmetry—about spheres as well as hyperplanes—although this is far from obvious. Let us look at a special case we need below. Suppose E is the sphere with center (0, . . . , 0, 1) and radius 1. Then E contains the origin, so that E∗ is a hyperplane; in fact, E∗= {(x, y) ∈Rn : y = 1/2} ∪{∞}, Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Symmetry and the Schwarz Reflection Principle 69 4.14 Ωis symmetric about E.
as the reader can easily check. Assume for the moment that n = 2; here we identify R2 with C, so that inversion is the map z 7→1/ z. Given z ∈C{0}, we need to show that z∗and (zE)∗are symmetric about E∗.
A moment’s reflection shows that to do this we need only verify that the conjugate of z∗−i/2 equals (zE)∗−i/2; this bit of algebra we leave to the reader. To go from R2 to Rn with n > 2, observe that inversion preserves every linear subspace of Rn. Given z ∈Rn{0}, then, we look at the two-dimensional plane determined by 0, z, and zE. Because the center of E is on the line determined by z and zE, this plane contains the (0, y)-axis. We can thus view this plane as C, with (0, . . . , 0, 1) playing the role of i. The proof for R2 therefore shows that z∗and (zE)∗are symmetric about E∗in Rn.
We can now prove the Schwarz reflection principle for regions sym-metric about spheres.
4.15 Theorem: Suppose Ωis a region symmetric about ∂B(a, r). If u is continuous on Ω∩B(a, r), u is harmonic on Ω∩B(a, r), and u = 0 on Ω∩∂B(a, r), then u extends harmonically to Ω.
Proof: We may assume a = (0, . . . , 0, 1) and r = 1; we are then dealing with the sphere E discussed above. Because Ωis symmetric about E, Ω∗is symmetric about the hyperplane E∗, as we just showed.
Our hypotheses on u now show that the Schwarz reflection principle for hyperplanes (4.12) can be applied to the Kelvin transform of u.
Accordingly, K[u] extends to a function v harmonic on Ω∗. Because Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 70 Chapter 4. The Kelvin Transform the Kelvin transform is its own inverse, K[v] extends u harmonically to Ω.
Let us explicitly identify the harmonic reflection of u across a sphere in the concrete case of S, the unit sphere.
4.16 Theorem: Suppose Ωis connected and symmetric about S. If u is continuous on Ω∩B, u is harmonic on Ω∩B, and u = 0 on Ω∩S, then the function v defined on Ωby v(x) = ( u(x) if x ∈Ω∩B −Ku if x ∈Ω∩(Rn \ B) is the unique harmonic extension of u to Ω.
Proof: Set a = (0, . . . , 0, 1) and define w(x) = v(x−a); the domain of the function w is then Ω+ a, which is symmetric about the sphere E of the previous proof. We will be done if we can show that K[w] has the appropriate reflection property about the hyperplane E∗. What we need to show, then, is that K[w] (x + a)∗ = −K[w] (x∗+ a)∗ for all x ∈Ω. This amounts to showing that |x + a|n−2v(x) = −|x∗+ a|n−2v(x∗) for all x ∈Ω. By Exercise 1 of this chapter, |x∗+a|/|x+a| = |x|−1. We therefore need only show that v = −K[v] on Ω. But this last identity follows easily from the definition of v.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Exercises 71 Exercises 1.
Show that if ζ ∈S and x ∈Rn \ {0}, then |x∗+ ζ| = |x + ζ| |x| .
2.
Show that at x ∈Rn {0}, the determinant of the Jacobian of the inversion map equals −1/|x|2n.
3.
Let f be a function of one complex variable that is holomorphic on the complement of some disk. We say that f is holomorphic at ∞provided f (1/z) has a removable singularity at 0. Show that the following are equivalent: (a) f is holomorphic at ∞; (b) f is bounded on a deleted neighborhood of ∞; (c) limz→∞f (z)/z = 0.
4.
Assume ω ⊂Rn ∪{∞} is open. Show that u is harmonic on ω if and only if K[u] is harmonic on ω∗.
5.
(a) Show that if n > 2, then the only harmonic function on Rn ∪{∞} is identically zero.
(b) Prove that all harmonic functions on R2 ∪{∞} are constant.
6.
Suppose that u is harmonic and positive on R2 \ E, where E is compact. Characterize the behavior of u near ∞.
7.
Prove that the solution to the exterior Dirichlet problem in The-orem 4.11 is unique.
8.
Suppose that f is continuous on ∂B(a, r) and that u solves the Dirichlet problem for B(a, r) with boundary data f . What is the solution (expressed in terms of u) of the Dirichlet problem for (Rn ∪{∞}) \ B(a, r) with boundary data f ?
9.
Let E denote the hyperplane {x ∈Rn : b · x = c}, where b is a nonzero vector in Rn and c is a real number. For x ∈Rn, show that the reflection xE is given by the formula xE = x −2 (x · b) −c b |b|2 .
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 72 Chapter 4. The Kelvin Transform 10.
Let E denote the hyperplane Rn−1 × {0}. Fix a point z = (0, y) in the upper half-space.
Show that the reflections of z about spheres of radius R centered at (0, R) converge to zE = (0, −y) as R →∞.
11.
Suppose E is a sphere of radius r centered at a, with 0 ∉E.
Show that the radius of E∗is r/ r 2 −|a|2 and the center of E∗ is a/(|a|2 −r 2).
12.
Show that the inversion map preserves symmetry about spheres and hyperplanes. In other words, if E is a sphere or hyperplane, then x∗and (xE)∗are symmetric about E∗for all x.
13.
Let E be a compact subset of S with nonempty interior relative to S. Prove that there exists a nonconstant bounded harmonic function on Rn \ E.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Chapter 5 Harmonic Polynomials Recall the Dirichlet problem for the ball in Rn: given f ∈C(S), find u ∈C(B) such that u is harmonic on B and u|S = f . We know from Chapter 1 that u(x) = Pf = Z S f (ζ) 1 −|x|2 |x −ζ|n dσ(ζ) for x ∈B. To prove that P[f ] is harmonic on B, we computed its Lapla-cian by differentiating under the integral sign in the equation above and noting that for each fixed ζ ∈S, the Poisson kernel (1 −|x|2)/|x −ζ|n is harmonic as a function of x.
Suppose now that f is a polynomial on Rn restricted to S. For fixed ζ ∈S, the Poisson kernel (1 −|x|2)/|x −ζ|n is not a polynomial in x, so nothing in the formula above suggests that P[f] should be a poly-nomial. Thus our first result in this chapter should be somewhat of a surprise: P[f ] is indeed a polynomial, and its degree is at most the degree of f .
Further indications of the importance of harmonic polynomials will come when we prove that every polynomial on Rn can be written as the sum of a harmonic polynomial and a polynomial multiple of |x|2.
This result will then be used to decompose the Hilbert space L2(S) into a direct sum of spaces of harmonic polynomials. As we will see, this decomposition is the higher-dimensional analogue of the Fourier series decomposition of a function on the unit circle in R2.
Our theory will lead to a fast algorithm for computing the Poisson integral of any polynomial. The algorithm involves differentiation, but not integration!
73 74 Chapter 5. Harmonic Polynomials Next, we will use the Kelvin transform to find an explicit basis for the space of harmonic polynomials that are homogeneous of degree m.
The chapter concludes with a study of zonal harmonics, which are used to decompose the Poisson kernel and to show that the homogeneous expansion of a harmonic function has nice convergence properties.
Polynomial Decompositions We begin with a crucial theorem showing that the Poisson integral of a polynomial is a polynomial of a special form. The proof uses, without comment, the result that the Poisson integral gives the unique solution to the Dirichlet problem.
Note that the theorem below implies that if p is a polynomial, then the degree of P[p|S] is less than or equal to the degree of p.
This inequality can be strict; for example, if p(x) = |x|2, then P[p|S] ≡1.
5.1 Theorem: If p is a polynomial on Rn of degree m, then P[p|S] = (1 −|x|2)q + p for some polynomial q of degree at most m −2.
Proof: Let p be a polynomial on Rn of degree m. If m = 0 or m = 1, then p is harmonic and hence P[p|S] = p, so the desired result follows by taking q = 0. Thus we can assume that m ≥2.
For any choice of q, the function (1−|x|2)q+p equals p on S. Thus to solve the Dirichlet problem for B with boundary data p|S, we need only find q such that (1 −|x|2)q + p is harmonic. In other words, to prove the theorem we need only show that there exists a polynomial q of degree at most m −2 such that 5.2 ∆ (1 −|x|2)q = −∆p.
To do this, let W denote the vector space of all polynomials on Rn of degree at most m −2, and define a linear map T : W →W by T(q) = ∆ (1 −|x|2)q .
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Polynomial Decompositions 75 If T(q) = 0, then (1 −|x|2)q is a harmonic function; this harmonic function equals 0 on S, and hence by the maximum principle equals 0 on B; this forces q to be 0. Thus T is injective.
We now use the magic of linear algebra (an injective linear map from a finite-dimensional vector space to itself is also surjective) to conclude that T is surjective. Hence there exists a polynomial q of degree at most m −2 such that 5.2 holds, and we are done.
The following corollary will be a key tool in our proof of the direct-sum decomposition of the polynomials (Proposition 5.5). Here “poly-nomial” means a polynomial on Rn, and “nonzero” means not identi-cally 0.
5.3 Corollary: No nonzero polynomial multiple of |x|2 is harmonic.
Proof: Suppose p is a nonzero polynomial on Rn of degree m and |x|2p is harmonic. Because p|S = (|x|2p)|S, the Poisson integral P[p|S] must equal the harmonic polynomial |x|2p, which has degree m + 2. This contradicts the previous theorem, which implies that the degree of P[p|S] is at most m.
Every polynomial p on Rn with degree m can be uniquely written in the form p = Pm j=0 pj, where each pj is a homogeneous polynomial on Rn of degree j. We call pj the homogeneous part of p of degree j.
Note that ∆p = Pm j=0 ∆pj, and thus p is harmonic if and only if each pj is harmonic (because a polynomial is identically 0 if and only if each homogeneous part of it is identically 0).
In the next section we will be working in L2(S). Two distinct poly-nomials of the same degree can have equal restrictions to S, but two homogeneous polynomials of the same degree that agree on S must agree everywhere.
Thus we will find it convenient to restrict atten-tion to homogeneous polynomials. Let us denote by Pm(Rn) the com-plex vector space of all homogeneous polynomials on Rn of degree m.
Let Hm(Rn) denote the subspace of Pm(Rn) consisting of all homoge-neous harmonic polynomials on Rn of degree m. For example, 5.4 p(x, y, z) = 8x5 −40x3y2 +15xy4 −40x3z2 +30xy2z2 +15xz4 is an element of H5(R3), as the reader can verify; we have used (x, y, z) in place of (x1, x2, x3) to denote a typical point in R3.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 76 Chapter 5. Harmonic Polynomials In the following proposition, we write Pm(Rn) as the algebraic di-rect sum of the two subspaces Hm(Rn) and |x|2Pm−2(Rn), meaning that every element of Pm(Rn) can be uniquely written as the sum of an element of Hm(Rn) and an element of |x|2Pm−2(Rn). In the next section we will see that this is an orthogonal decomposition when we restrict all functions to S and use the usual inner product that comes from surface-area measure.
5.5 Proposition: If m ≥2, then Pm(Rn) = Hm(Rn) L |x|2Pm−2(Rn).
Proof: Let p ∈Pm(Rn). Then p = P[p|S] + |x|2q −q for some polynomial q of degree at most m −2 (by Theorem 5.1). Take the homogeneous part of degree m of both sides of the equation above, getting 5.6 p = pm + |x|2qm−2, where pm is the homogeneous part of degree m of the harmonic func-tion P[p|S] (and hence pm ∈Hm(Rn)) and qm−2 is the homogeneous part of degree m −2 of q (and hence qm−2 ∈Pm−2(Rn)). Thus every element of Pm(Rn) can be written as the sum of an element of Hm(Rn) and an element of |x|2Pm−2(Rn).
To show that this decomposition is unique, suppose that pm + |x|2qm−2 = e pm + |x|2 e qm−2, where pm, e pm ∈Hm(Rn) and qm−2, e qm−2 ∈Pm−2(Rn). Then pm −e pm = |x|2(e qm−2 −qm−2).
The left side of the equation above is harmonic, and the right side is a polynomial multiple of |x|2. Thus Corollary 5.3 implies that pm = e pm and qm−2 = e qm−2, as desired.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Polynomial Decompositions 77 The map p 7→pm, where p ∈Pm(Rn) and pm ∈Hm(Rn) are as in 5.6, is called the canonical projection of Pm(Rn) onto Hm(Rn). Later we will find a formula for this projection (see Theorem 5.18).
We now come to the main result of this section. As usual, [t] denotes the largest integer less than or equal to t. Thus in the theorem below, the last index m−2k equals 0 or 1, depending upon whether m is even or odd.
5.7 Theorem: Every p ∈Pm(Rn) can be uniquely written in the form p = pm + |x|2pm−2 + · · · + |x|2kpm−2k, where k = [ m 2 ] and each pj ∈Hj(Rn).
Proof: The desired result obviously holds when m = 0 or m = 1, because Pm(Rn) = Hm(Rn) in those cases. Thus we can assume that m ≥2.
Suppose that p ∈Pm(Rn). By the previous proposition, p can be uniquely written in the form p = pm + |x|2q, where pm ∈Hm(Rn) and q ∈Pm−2(Rn). By induction, we can as-sume that the theorem holds when m is replaced by m −2. Taking the unique decomposition for q given by the theorem and plugging it into the equation above gives the desired decomposition of p. This decomposition is unique because pm is uniquely determined and the decomposition of q is also uniquely determined.
If p ∈Pm(Rn) and pm, pm−2, . . . , pm−2k are as in the theorem above, then the solution to the Dirichlet problem for B with boundary data p|S is pm + pm−2 + · · · + pm−2k.
To see this, observe that the function above is harmonic and that it agrees with p on S. Later in this chapter we will develop an algorithm for computing pm, pm−2, . . . , pm−2k (and thus P[p|S]) from p.
We finish this section by computing dim Hm(Rn), the dimension (over C) of the vector space Hm(Rn). Because H0(Rn) is the space of constant functions, dim H0(Rn) = 1. Because H1(Rn) is the space of Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 78 Chapter 5. Harmonic Polynomials linear functions on Rn, we have dim H1(Rn) = n. The next proposition takes care of higher values of m.
5.8 Proposition: If m ≥2, then dim Hm(Rn) = n + m −1 n −1 !
− n + m −3 n −1 !
.
Proof: We begin by finding dim Pm(Rn). Because the monomials {xα : |α| = m} form a basis of Pm(Rn), dim Pm(Rn) equals the number of distinct multi-indices α = (α1, . . . , αn) with |α| = m. Adding 1 to each αj, we see that dim Pm(Rn) equals the number of multi-indices α = (α1, . . . , αn), with each αj > 0, such that |α| = n + m.
Now consider removing n−1 integers from the interval (0, n + m) ⊂R. This partitions (0, n + m) into n disjoint open intervals. Letting α1, . . . , αn denote the lengths of these intervals, taken in order, we have n X j=1 αj = n + m.
Each choice of n −1 integers from (0, n + m) thus generates a multi-index α with |α| = n + m, and each multi-index of degree n + m arises from one and only one such choice. The number of such choices is, of course, n+m−1 n−1 . Thus dim Pm(Rn) = n + m −1 n −1 !
.
From Proposition 5.5 we have dim Hm(Rn) = dim Pm(Rn) −dim Pm−2(Rn).
Combining the last two equations gives the desired result.
Spherical Harmonic Decomposition of L2(S) In Proposition 5.5, we showed that the space of homogeneous poly-nomials of degree m decomposes as the direct sum of the space of Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Spherical Harmonic Decomposition of L2(S) 79 harmonic homogeneous polynomials of degree m and |x|2 times the homogeneous polynomials of degree m −2. Now we turn to ideas re-volving around orthogonal direct sums, which means that we need to introduce an inner product.
Because a homogeneous function on Rn is determined by its restric-tion to S, we follow the natural impulse to work in L2(S, dσ), which we denote simply by L2(S). In other words, L2(S) denotes the usual Hilbert space of Borel-measurable square-integrable functions on S with inner product defined by ⟨f , g⟩= Z S f g dσ.
Our main result in this section will be a natural orthogonal decompo-sition of L2(S).
Homogeneous polynomials on Rn of different degrees, when re-stricted to S, are not necessarily orthogonal in L2(S).
For example, x12 and x14 are not orthogonal in this space because their product is positive everywhere on S. However, the next proposition shows that if the homogeneous polynomial of higher degree is harmonic, then we indeed have orthogonality (because Hm(Rn) is closed under complex conjugation).
5.9 Proposition: If p, q are polynomials on Rn and q is harmonic and homogeneous with degree higher than the degree of p, then Z S pq dσ = 0.
Proof: The desired conclusion involves only the values of p and q on S. Hence by linearity and Theorem 5.7, it suffices to prove the proposition when p is replaced by a homogeneous harmonic polyno-mial. Thus we can assume that p ∈Hk(Rn) and that q ∈Hm(Rn), where k < m.
Green’s identity (1.1) implies that 5.10 Z S (pDnq −qDnp) dσ = 0.
But for ζ ∈S, (Dnp)(ζ) = d dr p(rζ)|r=1 = d dr r kp(ζ) |r=1 = kp(ζ).
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 80 Chapter 5. Harmonic Polynomials Similarly, Dnq = mq on S. Thus 5.10 implies that (m −k) Z S pq dσ = 0.
Because k < m, the last integral vanishes, as desired.
Obviously |x|2Pm−2(Rn) restricted to S is the same as Pm−2(Rn) restricted to S. Thus the last proposition shows that if we restrict all functions to S, then the decomposition given in Proposition 5.5 is an or-thogonal decomposition with respect to the inner product on L2(S).
The restriction of Hm(Rn) to S is sufficiently important to receive its own name and notation. A spherical harmonic of degree m is the restriction to S of an element of Hm(Rn). The collection of all spherical harmonics of degree m will be denoted by Hm(S); thus Hm(S) = {p|S : p ∈Hm(Rn)}.
The map p 7→p|S provides an identification of the complex vector space Hm(Rn) with the complex vector space Hm(S). We use the no-tation Hm(S) when we want to emphasize that we are considering the functions to be defined only on S.
For example, take n = 3 and consider the function 5.11 q(x, y, z) = 15x −70x3 + 63x5 defined for (x, y, z) ∈S. Is q an element of H5(S)? Although q ap-pears to be neither harmonic nor homogeneous of degree 5, note that on S we have q(x, y, z) = 15x(x2 + y2 + z2)2 −70x3(x2 + y2 + z2) + 63x5.
The right side of the equation above is a homogeneous polynomial on R3 of degree 5, and as the reader can check, it is harmonic. Thus q, as defined by 5.11, is indeed an element of H5(S). (To save a bit of work, note that the right side of the equation above equals the polyno-mial p ∈H5(R3) defined by 5.4, so q = p|S. Examples 5.4 and 5.11 were generated using the software described in Appendix B.) Restating some previous results in terms of spherical harmonics, we see that Proposition 5.9 implies that Hk(S) is orthogonal to Hm(S) in L2(S) whenever k ̸= m. Theorem 5.7 implies that if p is a polynomial Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Spherical Harmonic Decomposition of L2(S) 81 on Rn of degree m, then p|S can be written as a sum of spherical har-monics of degree at most m. In our next theorem, we will use these results to decompose L2(S) into an infinite direct sum of spaces of spherical harmonics.
We will need a bit of Hilbert space theory.
Recall that if H is a complex Hilbert space, then we write H = L∞ m=0 Hm when the following three conditions are satisfied: (a) Hm is a closed subspace of H for every m.
(b) Hk is orthogonal to Hm if k ̸= m.
(c) For every x ∈H, there exist xm ∈Hm such that x = x0 + x1 + · · · , the sum converging in the norm of H.
When (a), (b), and (c) hold, the Hilbert space H is said to be the direct sum of the spaces Hm. If this is the case, then the expansion in (c) is unique. Also, if (a) and (b) hold, then (c) holds if and only if the complex linear span of S∞ m=0 Hm is dense in H.
We can now easily prove the main result of this section.
5.12 Theorem: L2(S) = L∞ m=0 Hm(S).
Proof: Condition (a) above holds because each Hm(S) is finite di-mensional and hence is closed in L2(S).
We have already noted that condition (b) above follows from Propo-sition 5.9.
To verify condition (c), we need only show that the linear span of S∞ m=0 Hm(S) is dense in L2(S). As we have already noted, Theorem 5.7 implies that if p is a polynomial on Rn, then p|S can be written as a finite sum of elements of S∞ m=0 Hm(S). By the Stone-Weierstrass The-orem (see , Theorem 7.33), the set of restrictions p|S, as p ranges over all polynomials on Rn, is dense in C(S) with respect to the supre-mum norm. Because C(S) is dense in L2(S) and the L2-norm is less than or equal to the L∞-norm on S, this implies that the linear span of S∞ m=0 Hm(S) is dense in L2(S), as desired.
The theorem above reduces to a familiar result when n = 2. To see this, suppose p ∈Hm(R2) is real valued. Then p is the real part Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 82 Chapter 5. Harmonic Polynomials of an entire holomorphic function f whose imaginary part vanishes at the origin. The Cauchy-Riemann equations imply that all (complex) derivatives of f except the mth derivative vanish at the origin. Thus f = czm for some complex constant c, and so p = czm + czm.
This implies that Hm(R2) is the complex linear span of {zm, zm}. Thus Hm(S), as a space of functions of the variable eiθ, is the complex linear span of {eimθ, e−imθ} (or of {cos mθ, sin mθ}). Hence for f ∈L2(S), the decomposition promised by the theorem above takes the form f = ∞ X m=−∞ ameimθ, where the sum converges in L2(S). In other words, when n = 2 the decomposition given by the theorem above is just the standard Fourier series expansion of a function on the circle.
When n > 2, we can think of the theorem above as providing an expansion for functions f ∈L2(S) analogous to the Fourier series ex-pansion, with spherical harmonics playing the roles of the exponential functions eimθ (or of the trigonometric functions cos mθ, sin mθ).
Inner Product of Spherical Harmonics Suppose p = P α bαxα and q = P α cαxα are harmonic polynomials on Rn. In this section we focus on the question of computing the inner product of p and q in L2(S). We denote this inner product by ⟨p, q⟩, although technically ⟨p|S, q|S⟩would be more correct.
Each of p, q can be written as a sum of homogeneous harmonic poly-nomials, and we can expand the inner product ⟨p, q⟩accordingly. By Proposition 5.9, the inner product of terms coming from the homoge-neous parts of different degrees equals 0. In other words we could, if desired, assume that p and q are homogeneous harmonic polynomials of the same degree. Even then it appears that the best we could do would be to write ⟨p, q⟩= X α X β bαcβ Z S xα+β dσ(x).
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Inner Product of Spherical Harmonics 83 The integral over S of the monomial xα+β was explicitly calculated by Hermann Weyl in Section 3 of ; using that result would complete a formula for ⟨p, q⟩. We will take a different approach.
We have no right to expect the double-sum formula above to reduce to a single-sum formula of the form ⟨p, q⟩= X α bαcαwα, because distinct mononials of the same degree are not necessarily or-thogonal in L2(S).
For example, x12 and x22 are not orthogonal in this space because their product is positive everywhere on S. However, a single-sum formula as above is the main result of this section; see Theorem 5.14.
The single-sum formula that we will prove makes it appear that the monomials form an orthonormal set in L2(S), which, as noted above, is not true. But we are dealing here only with harmonic polynomials, and no monomial of degree above 1 is harmonic. In some mysterious fashion being harmonic forces enough cancellation in the double sum to collapse it into a single sum.
The following lemma will be a key tool in our proof of the single-sum formula.
5.13 Lemma: If m > 0 and p, q ∈Hm(Rn), then Z S pq dσ = 1 m(n + 2m −2) Z S ∇p · ∇q dσ.
Proof: Fix m > 0 and p, q ∈Hm(Rn).
Using the homogeneity of pq, we see, just as in the proof of Proposition 5.9, that pq equals (1/2m) times the normal derivative of pq on S. Thus Z S pq dσ = 1 2mnV(B) Z S ∇(pq) · n ds = 1 2mnV(B) Z B ∆(pq) dV, where the nV(B) term appears in the first equality because of the switch from normalized surface-area measure dσ to surface-area measure ds (see A.2 in Appendix A) and the second equality comes from the diver-gence theorem (1.2). Convert the last integral to polar coordinates (1.5), Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 84 Chapter 5. Harmonic Polynomials apply the product rule for the Laplacian (1.19), and use the homogene-ity of ∇p and ∇q to get Z S pq dσ = 1 m Z 1 0 r n+2m−3 Z S ∇p · ∇q dσ dr = 1 m(n + 2m −2) Z S ∇p · ∇q dσ, as desired.
Now we can prove the surprising single-sum formula for the inner product of two harmonic polynomials.
5.14 Theorem: If p = P α bαxα and q = P α cαxα are harmonic polynomials on Rn, then ⟨p, q⟩= X α bαcαwα, where wα = α!
n(n + 2) . . . (n + 2|α| −2).
Proof: Every harmonic polynomial can be written as a finite sum of homogeneous harmonic polynomials. We already know (from Proposi-tion 5.9) that homogeneous harmonic polynomials of different degrees are orthogonal in L2(S). Thus it suffices to prove the theorem under the assumption that p, q ∈Hm(Rn) for some nonnegative integer m.
Because Hm(Rn) is closed under complex conjugation, we can also assume, with no loss of generality, that each cα ∈R.
If m = 0, then p, q are constant and the desired result obviously holds (where the empty product in the denominator of the formula defining wα is interpreted, as usual, to equal 1).
So fix m > 0 and assume, by induction, that the theorem holds for smaller values of m. Let ej = (0, . . . , 0, 1, 0, . . . , 0), where the 1 appears in the jth slot. Now ∇p · ∇q is a sum of terms, each of which is a prod-uct of harmonic polynomials. Thus using our induction hypothesis we have Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Spherical Harmonics Via Differentiation 85 Z S ∇p · ∇q dσ = n X j=1 Z S X α bααjxα−ej X α cααjxα−ej dσ(x) = n X j=1 X α bαcααj2 (α −ej)!
n(n + 2) . . . (n + 2m −4) = X α bαcα n X j=1 αj α!
n(n + 2) . . . (n + 2m −4) = X α bαcα α! m n(n + 2) . . . (n + 2m −4).
The equation above, when combined with Lemma 5.13, gives the de-sired formula.
Spherical Harmonics Via Differentiation Compute a few partial derivatives of the function |x|2−n. You will find the answer is always of the same form—a polynomial divided by a power of |x|. For example, D1 2(|x|2−n) = (2 −n)(|x|2 −nx12) |x|n+2 .
Notice here that the polynomial in the numerator is harmonic. This is no accident—differentiating |x|2−n exactly k times will always leave us with a homogeneous harmonic polynomial of degree k divided by |x|n−2+2k, as we will see in Lemma 5.15. We will actually see much more than this, when we show (Theorem 5.18) that this procedure gives a for-mula for the canonical projection of Pm(Rn) onto Hm(Rn). This sec-tion concludes with the development of a fast algorithm for finding the Poission integral of a polynomial via differentiation (Theorem 5.21).
The Kelvin transform will play a key role here.
To see why, ob-serve that the Kelvin transform applied to the example in the paragraph above leaves us with the harmonic polynomial in the numerator. This indicates how we will obtain homogeneous harmonic polynomials—we first differentiate |x|2−n, and then we apply the Kelvin transform.
For p = P α cαxα a polynomial on Rn, we define p(D) to be the differential operator P α cαDα.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 86 Chapter 5. Harmonic Polynomials 5.15 Lemma: If p ∈Pm(Rn), then K[p(D)|x|2−n] ∈Hm(Rn).
Proof: First we will show that K[p(D)|x|2−n] ∈Pm(Rn). By lin-earity, we need only prove this in the special case when p is a monono-mial. To get started, note that the desired result obviously holds when m = 0. Now we will use induction, assuming that the result holds for some fixed m, and then showing that it also holds for m + 1.
Let α be a multi-index with |α| = m. By our induction hypothesis, there exists u ∈Hm(Rn) such that K[Dα|x|2−n] = u.
Take the Kelvin transform of both sides of the equation above, getting Dα|x|2−n = |x|2−n−2mu.
Fix an index j, and differentiate both sides of the equation above with respect to xj, getting DjDα|x|2−n = (2 −n −2m)xj|x|−n−2mu + |x|2−n−2mDju = |x|2−n−2(m+1) (2 −n −2m)xju + |x|2Dju 5.16 = |x|2−n−2(m+1)v, where v ∈Pm+1(Rn). Now take the Kelvin transform of both sides of the equation above, getting K[DjDα|x|2−n] = v.
Thus K[DjDα|x|2−n] ∈Pm+1(Rn). Because DjDα represents differ-entiation with respect to an arbitrary multi-index of order m + 1, this completes the induction argument.
All that remains is to prove that K[p(D)|x|2−n] is harmonic. But |x|2−n is harmonic and every partial derivative of any harmonic func-tion is harmonic, so p(D)|x|2−n is harmonic. The proof is completed by recalling that the Kelvin transform of every harmonic function is harmonic (Theorem 4.7).
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Spherical Harmonics Via Differentiation 87 Suppose p ∈Pm(Rn). Proposition 5.5 gives a unique decomposition p = pm + |x|2q, where pm ∈Hm(Rn) and q ∈Pm−2(Rn). The previous lemma states that K[p(D)|x|2−n] ∈Hm(Rn). So p determines two harmonic poly-nomials, pm and K[p(D)|x|2−n], leading to an investigation of the re-lationship between them. As we will see (Theorem 5.18), one of these harmonic polynomials is a constant multiple of the other, with the con-stant depending only on m and n. The key to proving this is the follow-ing lemma, which we can guess by looking at the proof of Lemma 5.15.
Specifically, note that in 5.16, u gets multiplied by xj, just as Dα is multiplied by Dj. An extra factor of 2 −n −m also appears. Thus 5.16 suggests the following lemma, where cm is the constant defined by cm = m−1 Y k=0 (2 −n −2k).
Although cm depends upon n as well as m, we are assuming that n > 2 is fixed. For n = 2, the definition of cm and the analogue of the follow-ing lemma are given in Exercise 14 of this chapter.
5.17 Lemma: If n > 2 and p ∈Pm(Rn), then K[p(D)|x|2−n] = cm(p −|x|2q) for some q ∈Pm−2(Rn).
Proof: The proof is a modification of the proof of the previous lemma. By linearity, we need only consider the case when p is a mono-mial. The desired result obviously holds when m = 0. Now we will use induction, assuming that the result holds for some fixed m, and then showing that it also holds for m + 1.
Let α be a multi-index with |α| = m. By our induction hypothesis, there exists q ∈Pm−2(Rn) such that K[Dα|x|2−n] = cm(xα −|x|2q).
Follow the proof of Lemma 5.15, setting u = cm(xα−|x|2q), taking the Kelvin transform of both sides of the equation above, and then applying Dj to both sides, getting (see 5.16) Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 88 Chapter 5. Harmonic Polynomials DjDα|x|2−n = |x|2−n−2(m+1)cm h (2 −n −2m)xj(xα −|x|2q) + |x|2Dju cm i = |x|2−n−2(m+1)cm+1(xjxα −|x|2v), where v ∈Pm−1(Rn). Now take the Kelvin transform of both sides of the equation above, getting K[DjDα|x|2−n] = cm+1(xjxα −|x|2v).
Because xjxα represents an arbitrary monomial of order m + 1, this completes the induction argument and the proof.
In the next theorem we will combine the last two lemmas. Recall that the canonical projection of Pm(Rn) onto Hm(Rn) comes from the decomposition given by Proposition 5.5. By the orthogonal projection onto Hm(S), we mean the usual orthogonal projection of the Hilbert space L2(S) onto the closed subspace Hm(S). In part (b) of the next the-orem, to be formally correct we should have written (p(D)|x|2−n)|S/cm instead of p(D)|x|2−n/cm.
5.18 Theorem: Suppose n > 2 and p ∈Pm(Rn). Then: (a) The canonical projection of p onto Hm(Rn) is K[p(D)|x|2−n]/cm.
(b) The orthogonal projection of p|S onto Hm(S) is p(D)|x|2−n/cm.
Proof: By Lemma 5.17, we can write 5.19 p = K[p(D)|x|2−n]/cm + |x|2q for some q ∈Pm−2(Rn). Lemma 5.15 shows that the first term on the right side of this equation is in Hm(Rn). Thus this equation is the unique decomposition of p promised by Proposition 5.5, and further-more K[p(D)|x|2−n]/cm is the canonical projection of p onto Hm(Rn), which proves (a).
To prove (b), restrict both sides of 5.19 to S, getting p|S = p(D)|x|2−n/cm + q|S.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Spherical Harmonics Via Differentiation 89 By Proposition 5.9, q|S is orthogonal to Hm(S). Thus taking the or-thogonal projection onto Hm(S) of both sides of the equation above gives (b).
See Exercise 14 of this chapter for the analogue of the preceding theorem for n = 2.
As an immediate corollary of the theorem above, we get the follow-ing unusual identity for homogeneous harmonic polynomials.
5.20 Corollary: If n > 2 and p ∈Hm(Rn), then p = K[p(D)|x|2−n]/cm.
Recall from Theorem 5.7 that for p ∈Pm(Rn), there is a unique decomposition of the form p = pm + |x|2pm−2 + · · · + |x|2kpm−2k, where k = [ m 2 ] and each pj ∈Hj(Rn). Recall also that the solution to the Dirichlet problem for B with boundary data p|S equals pm + pm−2 + · · · + pm−2k.
Part (a) of the previous theorem gives a fast algorithm for computing pm, pm−2, . . . , pm−2k and thus for computing the Poisson integral of any polynomial. Specifically, pm can be computed from the formula pm = K[p(D)|x|2−n]/cm. Use this to then solve for q ∈Pm−2(Rn) in the decomposition p = pm+|x|2q. To find pm−2, repeat this procedure with q in place of p and m −2 in place of m. Continue in this fashion, finding pm, pm−2, . . . , pm−2k.
The algorithm for computing the Poisson integral of a polynomial described in the paragraph above relies on differentiation rather than integration. We have found it typically to be several orders of magni-tude faster than algorithms involving integration. The next theorem gives another algorithm, also using only differentiation, for the exact computation of Poisson integrals of polynomials. We have found it to be even faster than the algorithm described in the paragraph above, typically by a factor of about 2.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 90 Chapter 5. Harmonic Polynomials The algorithm described by the next theorem is used by the soft-ware discussed in Appendix B. This software shows, for example, that if n = 5 then the Poisson integral of x15x2 equals x15x2 −10x13x2|x|2 13 + 15x1x2|x|4 143 + 10x13x2 13 −10x1x2|x|2 39 + 5x1x2 33 .
Note that in the solution above, the homogeneous part p6 of highest or-der (the first three terms above) consists of the original function x15x2 plus a polynomial multiple of |x|2. This is expected, as we know that p6 = x15x2 −|x|2q for some q ∈P4(R5).
Finally, we need one bit of notation. Define ∆0p = p, and then for i a positive integer inductively define ∆ip = ∆(∆i−1p).
In the theorem below, we could have obtained a formula for ci,j in closed form. However, in the inductive formulas given here are more efficient for computation. These formulas come from , which in turn partially based its derivation on ideas from .
5.21 Theorem: Suppose p ∈Pm(Rn) has the decomposition p = pm + |x|2pm−2 + · · · + |x|2kpm−2k, where k = [ m 2 ] and each pm−2j ∈Hm−2j(Rn). Then pm−2j = k X i=j ci,j|x|2(i−j)∆ip for j = 0, . . . , k, where c0,0 = 1 and cj,j = cj−1,j−1(2m + n −2j) 2j(2m + n + 2 −4j)(2m + n −4j) for j = 1, . . . , k and ci,j = ci−1,j 2(j −i)(2m + n −2 −2j −2i) for i = j + 1, . . . , k.
Proof: As a special case of 4.5, we have ∆(|x|2iq) = 2i(2m + n −2 −2i)|x|2i−2q Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Spherical Harmonics Via Differentiation 91 for q ∈Hm−2i(Rn). Repeated application of this equation shows that for every nonnegative integer j, the operator |x|2j∆j equals a constant times the identity operator on |x|2iHm−2i(Rn). Denoting this constant by bi,j, note that bi,j = 0 if and only if j > i. Furthermore, the reader should verify that 5.22 bj,j = 2jj!
j Y i=1 (2m + n −2j −2i).
For j = 0, . . . , k, apply the operator |x|2j∆j to both sides of the equation p = Pk i=0 |x|2ipm−2i, getting the lower-triangular system 5.23 |x|2j∆jp = k X i=j bi,j|x|2ipm−2i.
Let (ci,j) denote the matrix inverse of the (k + 1)-by-(k + 1) matrix (bi,j); thus (ci,j) is also a lower-triangular matrix.
View the system 5.23 as a matrix equation whose right side consists of the row matrix of unknowns |x|2ipm−2i times the matrix (bi,j). Now multiply (on the right) both sides of this matrix equation by the matrix (ci,j) to solve for |x|2ipm−2i, then divide by |x|2i and interchange i and j to obtain 5.24 pm−2j = k X i=j ci,j|x|2(i−j)∆ip, for j = 0, . . . , k, as desired.
The only remaining task is to prove the inductive formulas for cj,j and ci,j. The diagonal entries of the inverse of a lower-triangular matrix are easy to compute. Specifically, we have cj,j = 1/bj,j. The claimed inductive formula for cj,j now follows from 5.22.
To prove the inductive formula for ci,j, fix j and use 4.5 to take the Laplacian of both sides of 5.24, then multiply by |x|2j, getting 0 = k X i=j ci,j |x|2i∆i+1p + 2(i −j)(2m + n −2 −2j −2i)|x|2i−2∆ip = k X i=j+1 ci−1,j + 2(i −j)(2m + n −2 −2j −2i)ci,j |x|2i−2∆ip, Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 92 Chapter 5. Harmonic Polynomials where the second equality is obtained from the first by breaking the sum into two parts, replacing i by i −1 in the first part, and recombin-ing the two sums (after the change of summation, the first summation should go to k + 1, but the (k + 1)-term equals 0; similarly, the second sum should start at j, but the j-term equals 0).
The equality above must hold for all p ∈Pm(Rn) (the ci,j are inde-pendent of p). This can happen only if ci−1,j + 2(i −j)(2m + n −2 −2j −2i)ci,j = 0, which gives our desired inductive formula.
Explicit Bases of Hm(Rn) and Hm(S) Theorem 5.18 implies that {K[Dα|x|2−n] : |α| = m} spans Hm(Rn) and that {Dα|x|2−n : |α| = m} spans Hm(S). In the next theorem, we find an explicit subset of each of these spanning sets that is a basis.
5.25 Theorem: If n > 2 then the set {K[Dα|x|2−n] : |α| = m and α1 ≤1} is a vector space basis of Hm(Rn), and the set {Dα|x|2−n : |α| = m and α1 ≤1} is a vector space basis of Hm(S).
Proof: Let B = {K[Dα|x|2−n] : |α| = m and α1 ≤1}.
We will first show that B spans Hm(Rn).
For this we need only show that K[Dα|x|2−n] is in the span of B for every multi-index α of degree m (by Theorem 5.18). So suppose α is a multi-index of degree m. If α1 is 0 or 1, then K[Dα|x|2−n] is in B by definition. Now we use induction on α1. Suppose that α1 > 1 and that K[Dβ|x|2−n] is in the span of B for all multi-indices β of degree m whose first components are less than α1. Because ∆|x|2−n ≡0, we have Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Explicit Bases of Hm(Rn) and Hm(S) 93 K[Dα|x|2−n] = K[D1 α1−2D2 α2 . . . Dn αn(D1 2|x|2−n)] = −K D1 α1−2D2 α2 . . . Dn αn n X j=2 Dj 2|x|2−n = − n X j=2 K[D1 α1−2D2 α2 . . . Dn αn(Dj 2|x|2−n)].
By our induction hypothesis, each of the summands in the last line is in the span of B, and therefore K[Dα|x|2−n] is in the span of B. We conclude that B spans Hm(Rn).
To complete the proof that B is a basis of Hm(Rn), we show that the cardinality of B is at most the dimension of Hm(Rn). We have B = {K[Dα|x|2−n]}, where α ranges over multi-indices of length m that are not of the form (β1 +2, β2, . . . , βn) with |β| = m−2. Therefore the cardinality of B is at most #{α : |α| = m} −#{β : |β| = m −2}, where # denotes cardinality. But from Proposition 5.5, we know that this difference equals the dimension of Hm(Rn).
Having shown that B is a basis of Hm(Rn), we can restrict to S, obtaining the second assertion of this theorem.
The software described in Appendix B uses Theorem 5.25 to con-struct bases of Hm(Rn) and Hm(S). For example, this software pro-duces the following vector space basis of H4(R3): {3|x|4 −30|x|2x22 + 35x24, 3|x|2x2x3 −7x23x3, |x|4 −5|x|2x22 −5|x|2x32 + 35x22x32, 3|x|2x2x3 −7x2x33, 3|x|4 −30|x|2x32 + 35x34, 3|x|2x1x2 −7x1x23, |x|2x1x3 −7x1x22x3, |x|2x1x2 −7x1x2x32, 3|x|2x1x3 −7x1x33}.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 94 Chapter 5. Harmonic Polynomials Although the previous theorem is valid only when n > 2, the n = 2 case is easy—earlier in the chapter we saw that {zm, zm} is a basis of Hm(R2) and {eimθ, e−imθ} is a basis of Hm(S).
Zonal Harmonics We continue to view Hm(S) as an inner product space with the L2(S)-inner product. Fix a point η ∈S, and consider the linear map Λ: Hm(S) →C defined by Λ(p) = p(η).
Because Hm(S) is a finite-dimensional inner-product space, there ex-ists a unique function Zm(·, η) ∈Hm(S) such that p(η) = ⟨p, Zm(·, η)⟩= Z S p(ζ)Zm(ζ, η) dσ(ζ) for all p ∈Hm(S). The spherical harmonic Zm(·, η) is called the zonal harmonic of degree m with pole η. The terminology comes from geo-metric properties of Zm that will be explained shortly.
We easily compute Zm when n = 2. Clearly Z0 ≡1. For m > 0, Hm(S) is the two-dimensional space spanned by {eimθ, e−imθ}, as we saw earlier. Thus if we fix eiϕ ∈S, there are constants α, β ∈C such that Zm(eiθ, eiϕ) = αeimθ + βe−imθ. The reproducing property of the zonal harmonic then gives γeimϕ + δe−imϕ = Z 2π 0 (γeimθ + δe−imθ)(αe−imθ + βeimθ) dθ 2π = γα + δβ for every γ, δ ∈C. Thus α = e−imϕ and β = eimϕ. We conclude that 5.26 Zm(eiθ, eiϕ) = eim(θ−ϕ) + eim(ϕ−θ) = 2 cos m(θ −ϕ).
Later (5.38) we will find an explicit formula for zonal harmonics in higher dimensions.
We now return to the case of arbitrary n ≥2. The next proposition gives some basic properties of zonal harmonics. The proof of (c) below Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Zonal Harmonics 95 uses orthogonal transformations, which play an important role in our study of zonal harmonics. We let O(n) denote the group of orthogo-nal transformations on Rn. Observe that Hm(Rn) is O(n)-invariant, meaning that if p ∈Hm(Rn) and T ∈O(n), then p ◦T ∈Hm(Rn). It follows that Hm(S) is O(n)-invariant as well.
5.27 Proposition: Suppose ζ, η ∈S and m ≥0. Then: (a) Zm is real valued.
(b) Zm(ζ, η) = Zm(η, ζ).
(c) Zm ζ, T(η) = Zm T −1(ζ), η for all T ∈O(n).
(d) Zm(η, η) = dim Hm(Rn).
(e) |Zm(ζ, η)| ≤dim Hm(Rn).
Proof: To prove (a), suppose p ∈Hm(S) is real valued. Then 0 = Im p(η) = Im Z S p(ζ)Zm(ζ, η) dσ(ζ) = − Z S p(ζ) Im Zm(ζ, η) dσ(ζ).
Defining p by p(ζ) = Im Zm(ζ, η) yields Z S Im Zm(ζ, η) 2 dσ(ζ) = 0, which implies Im Zm ≡0, proving (a).
To prove (b), consider any orthonormal basis e1, . . . , ehm of Hm(S), where hm = dim Hm(S) = dim Hm(Rn) (see Proposition 5.8 for an explicit formula for hm). By standard Hilbert space theory, Zm(·, η) = hm X j=1 ⟨Zm(·, η), ej⟩ej = hm X j=1 ej(η)ej.
Thus 5.28 Zm(ζ, η) = hm X j=1 ej(η)ej(ζ).
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 96 Chapter 5. Harmonic Polynomials Because Zm is real valued, the equation above is unchanged after com-plex conjugation, which implies (b).
To prove (c), let T ∈O(n). For every p ∈Hm(S) we have p T(η) = (p ◦T)(η) = Z S p T(ζ) Zm(ζ, η) dσ(ζ) = Z S p(ζ)Zm T −1(ζ), η dσ(ζ), the last equality following from the rotation invariance of σ. By the uniqueness of the zonal harmonic, the equation above gives (c).
To prove (d), note that taking ζ = T(η) in (c) gives Zm T(η), T(η) = Zm(η, η).
Thus the function η 7→Zm(η, η) is constant on S.
To evaluate this constant, take ζ = η in 5.28, obtaining Zm(η, η) = hm X j=1 |ej(η)|2.
Now integrate both sides of the equation above over S, getting Zm(η, η) = Z S hm X j=1 |ej(η)|2 dσ(η) = hm = dim Hm(Rn), which gives (d).
To prove (e), note that ∥Zm(·, η)∥2 2 = ⟨Zm(·, η), Zm(·, η)⟩= Zm(η, η) = dim Hm(Rn), where ∥∥2 denotes the norm in L2(S). Now |Zm(ζ, η)| = |⟨Zm(·, ζ), Zm(·, η)⟩| ≤∥Zm(·, ζ)∥2∥Zm(·, η)∥2 = dim Hm(Rn), completing the proof.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey The Poisson Kernel Revisited 97 Exercise 19 of this chapter deals with the question of when the in-equality in part (e) of the proposition above is an equality.
Our previous decomposition L2(S) = L∞ m=0 Hm(S) has an elegant restatement in terms of zonal harmonics, as shown in the next theorem.
Note that this is just the Fourier series decomposition when n = 2.
5.29 Theorem: Suppose f ∈L2(S). Let pm(η) = ⟨f , Zm(·, η)⟩for m ≥0 and η ∈S . Then pm ∈Hm(S) and f = ∞ X m=0 pm in L2(S).
Proof: By Theorem 5.12, we can write f = P∞ m=0 qm for some choice of qm ∈Hm(S), where the infinite sum converges in L2(S). The proof is completed by noticing that pm(η) = ⟨f , Zm(·, η)⟩= ∞ X k=0 qk, Zm(·, η) = ⟨qm, Zm(·, η)⟩= qm(η), where the third equality comes from the orthogonality of spherical har-monics of different degrees (Proposition 5.9).
The Poisson Kernel Revisited Every element of Hm(S) has a unique extension to an element of Hm(Rn); given p ∈Hm(S), we will let p denote this extension as well.
In particular, the notation Zm(·, ζ) will now often refer to the extension of this zonal harmonic to an element of Hm(Rn).
Suppose x ∈Rn. If x ̸= 0 and p ∈Hm(Rn), then p(x) = |x|mp(x/|x|) 5.30 = |x|m Z S p(ζ)Zm(x/|x|, ζ) dσ(ζ) = Z S p(ζ)Zm(x, ζ) dσ(ζ).
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 98 Chapter 5. Harmonic Polynomials We easily check that the first and last terms above also agree when x = 0. Note that Zm(x, ·) is a spherical harmonic of degree m for each fixed x ∈Rn.
Our next result uses the equation above to expresses the Poisson integral of a polynomial in terms of zonal harmonics.
5.31 Proposition: Let p be a polynomial on Rn of degree m. Then Pp|S = m X k=0 Z S p(ζ)Zk(x, ζ) dσ(ζ) for every x ∈B.
Proof: By Theorem 5.1, P[p|S] is a polynomial of degree at most m and hence can be written in the form 5.32 P[p|S] = m X k=0 pk, where each pk ∈Hk(Rn). For each x ∈B and each k we have pk(x) = Z S pk(ζ)Zk(x, ζ) dσ(ζ) = Z S m X j=0 pj(ζ)Zk(x, ζ) dσ(ζ) = Z S p(ζ)Zk(x, ζ) dσ(ζ), where the first equality comes from 5.30, the second equality comes from the orthogonality of spherical harmonics of different degrees (see Proposition 5.9), and the third equality holds because p and its Poisson integral Pm j=0 pj agree on S.
Combining the last equation with 5.32 gives the desired result.
The proposition above leads us to the zonal harmonic expansion of the Poisson kernel.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey The Poisson Kernel Revisited 99 5.33 Theorem: For every n ≥2, P(x, ζ) = ∞ X m=0 Zm(x, ζ) for all x ∈B, ζ ∈S. The series converges absolutely and uniformly on K × S for every compact set K ⊂B.
Proof: For a fixed n, Proposition 5.27(e) and Exercise 10 of this chapter show that there exists a constant C such that |Zm(x, ζ)| ≤Cmn−2|x|m for all x ∈Rn, ζ ∈S. The series P∞ m=0 Zm(x, ζ) therefore has the desired convergence properties.
Fix x ∈B. From Propositions 5.31 and 5.9 we see that Z S f (ζ)P(x, ζ) dσ(ζ) = Z S f (ζ) ∞ X m=0 Zm(x, ζ) dσ(ζ) whenever f is the restriction of a polynomial to S. Because such func-tions are dense in L2(S), this implies that P(x, ζ) = P∞ m=0 Zm(x, ζ) for almost every ζ ∈S. But all the functions involved are continuous, so we actually have equality everywhere, as desired.
When n = 2, we can express the theorem above in a familiar form.
Recall that we used complex analysis (see 1.12) to show that the Poisson kernel for B2 takes the form P(reiθ, eiϕ) = ∞ X m=−∞ r |m|eim(θ−ϕ) = 1 + ∞ X m=1 r m2 cos m(θ −ϕ) for all r ∈[0, 1) and all θ, ϕ ∈[0, 2π]. By 5.26, this is exactly the expansion in the theorem above.
The preceding theorem enables us to prove that the homogeneous expansion of an arbitrary harmonic function has the stronger conver-gence property discussed after Theorem 1.31.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 100 Chapter 5. Harmonic Polynomials 5.34 Corollary: If u is a harmonic function on B(a, r), then there exist pm ∈Hm(Rn) such that u(x) = ∞ X m=0 pm(x −a) for all x ∈B(a, r), the series converging absolutely and uniformly on compact subsets of B(a, r).
Proof: We first assume that u is harmonic on B. For any x ∈B, Theorem 5.33 gives u(x) = Z S u(ζ)P(x, ζ) dσ(ζ) = ∞ X m=0 Z S u(ζ)Zm(x, ζ) dσ(ζ).
Letting pm(x) = R S u(ζ)Zm(x, ζ) dσ(ζ) for x ∈Rn, observe that pm ∈Hm(Rn). As in the proof of Theorem 5.33, |pm(x)| ≤Cmn−2|x|m Z S |u| dσ for all x ∈Rn, and thus the series P pm converges absolutely and uniformly to u on compact subsets of B.
After a translation and dilation, the preceding argument shows that if u is harmonic on B(a, r), then u has an expansion of the desired form in each B(a, s), 0 < s < r. By the uniqueness of homogeneous expansions, all of these expansions are the same, and thus u has the desired expansion on B(a, r).
A Geometric Characterization of Zonal Harmonics In this section we give a simple geometric characterization of zonal harmonics. Recall the definition of a “parallel” from cartography: if we identify the surface of the earth with S ⊂R3 so that the north pole is at (0, 0, 1), then a parallel is simply the intersection of S with any plane perpendicular to the z-axis. The notion of a parallel is easily extended to all dimensions. Specifically, given η ∈S, we define a parallel orthog-onal to η to be the intersection of S with any hyperplane perpendicular to η.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey A Geometric Characterization of Zonal Harmonics 101 η Parallels orthogonal to η.
We claim that the zonal harmonic Zm(·, η) is constant on each par-allel orthogonal to η. To prove this, observe that a function f on S is constant on parallels orthogonal to η if and only if f ◦T −1 = f for every T ∈O(n) with T(η) = η. Thus Proposition 5.27(c) proves our claim.
Our goal is this section is to show that scalar multiples of Zm(·, η) are the only members of Hm(S) that are constant on parallels orthog-onal to η (Theorem 5.37). This geometric property explains how zonal harmonics came to be named—the term “zonal” refers to the “zones” between parallels orthogonal to the “pole” η.
We will use two lemmas to prove our characterization of zonal har-monics.
The first lemma describes the power series expansion of a real-analytic radial function.
5.35 Lemma: If f is real analytic and radial on Rn, then there exist constants cm ∈C such that f (x) = ∞ X m=0 cm|x|2m for all x near 0.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 102 Chapter 5. Harmonic Polynomials Proof: Assume first that f ∈Pm(Rn) and that f is not identically 0.
Because f is radial, it has a constant value c ̸= 0 on S, which implies that f(x) = c|x|m for all x ∈Rn. Clearly m is even (otherwise f would not be a polynomial). Thus f has the desired form in this case.
Now suppose that f is real analytic and radial, and that P pm is the homogeneous expansion of f near 0. Let T ∈O(n). Because f is radial, f = f ◦T, which gives P pm = P pm ◦T near 0. Since pm is a homogeneous polynomial of degree m, the same is true of pm ◦T, so that pm = pm ◦T for every m by the uniqueness of the homogeneous expansion of f. This is true for every T ∈O(n), and therefore each pm is radial. The result in the previous paragraph now completes the proof.
The next lemma is the final tool we need for our characterization of zonal harmonics. Recall that we can identify Rn with Rn−1 × R, writing a typical point z ∈Rn as z = (x, y).
5.36 Lemma: Suppose that u is harmonic on Rn and that u(·, y) is radial on Rn−1 for each y ∈R. Suppose further that u(0, y) = 0 for all y ∈R. Then u ≡0.
Proof: Recall that the power series of a function harmonic on Rn converges everywhere on Rn (see Exercise 34 in Chapter 1). Because u is real analytic on Rn and each u(·, y) is radial on Rn−1, Lemma 5.35 implies that the expansion of u takes the form u(x, y) = ∞ X m=0 cm(y)|x|2m, where each cm is a real-analytic function of y. Because u is harmonic, we obtain 0 = ∆u(x, y) = ∞ X m=0 cm′′(y)|x|2m + ∞ X m=1 αmcm(y)|x|2(m−1) = ∞ X m=0 [cm′′(y) + αm+1cm+1(y)]|x|2m, Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey A Geometric Characterization of Zonal Harmonics 103 where αm = 2m(2m + n −3). Looking at the last series, we see that each term in brackets vanishes. Because c0(y) = u(0, y) = 0, we easily verify by induction that each cm is identically zero. Thus u ≡0, as desired.
Let N denote the north pole (0, . . . , 0, 1). We can now characterize the zonal harmonics geometrically.
5.37 Theorem: Let η ∈S. A spherical harmonic of degree m is con-stant on parallels orthogonal to η if and only if it is a constant multiple of Zm(·, η).
Proof: We have already seen that Zm(·, η) is constant on parallels orthogonal to η.
For the converse, we may assume m ≥1. For convenience we first treat the case η = N. So suppose p ∈Hm(Rn) is constant on parallels orthogonal to N. For T ∈O(n −1), we then have p(Tx, y) = p(x, y) for all (x, y) ∈S, and hence for all (x, y) ∈Rn. Because this holds for all T ∈O(n −1), we conclude that p(·, y) is radial on Rn−1 for each y ∈R. In particular, Zm (·, y), N , regarded as an element of Hm(Rn), is radial on Rn−1 for each y ∈R.
Now choose c such that p(N) = cZm(N, N), and define u = p −cZm(·, N).
Then u is harmonic on Rn, u(·, y) is radial on Rn−1 for each y ∈R, and u(0, y) = u(yN) = ymu(N) = 0 for every y ∈R. By Lemma 5.36, u ≡0. Thus p is a constant multiple of Zm(·, N), as desired.
For the general η ∈S, choose T ∈O(n) such that T(N) = η. If p ∈Hm(S) is constant on parallels orthogonal to η, then p ◦T is constant on parallels orthogonal to N. Hence p◦T is a constant multiple of Zm(·, N), which implies that p is a constant multiple of Zm(·, N)◦T −1, which, by Proposition 5.27(c), equals Zm(·, η).
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 104 Chapter 5. Harmonic Polynomials An Explicit Formula for Zonal Harmonics The expansion of the Poisson kernel given by Theorem 5.33 allows us to find an explicit formula for the zonal harmonics.
5.38 Theorem: Let x ∈Rn and let ζ ∈S. Then Zm(x, ζ) equals (n+2m−2) [m/2] X k=0 (−1)k n(n + 2) . . . (n + 2m −2k −4) 2kk! (m −2k)!
(x·ζ)m−2k|x|2k for each m > 0.
Proof: The function (1 −z)−n/2 is holomorphic on the unit disk in the complex plane, and so it has a power series expansion 5.39 (1 −z)−n/2 = ∞ X k=0 ckzk for |z| < 1. We easily compute that 5.40 ck = n 2 n 2 + 1 . . .
n 2 + k −1 k!
.
Fix ζ ∈S.
For |x| small, 5.39 and the binomial formula imply that P(x, ζ) = (1 −|x|2)(1 + |x|2 −2x · ζ)−n/2 = (1 −|x|2) ∞ X k=0 ck(2x · ζ −|x|2)k = (1 −|x|2) ∞ X k=0 ck k X j=0 (−1)j k j 2k−j(x · ζ)k−j|x|2j.
By Theorem 5.33, Zm(·, ζ) is equal to the sum of the terms of degree m in the power series representation of P(·, ζ). Thus the formula above implies that 5.41 Zm(x, ζ) = qm(x) −|x|2qm−2(x), Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey An Explicit Formula for Zonal Harmonics 105 where qm and qm−2 are the sums of terms of degree m and m −2, respectively, in the double series above. It is easy to see that qm(x) = X m/2≤k≤m ck(−1)m−k k m−k 22k−m(x · ζ)2k−m|x|2m−2k.
Replacing the index k by m −k in this sum shows that qm(x) = [m/2] X k=0 cm−k(−1)k m−k k 2m−2k(x · ζ)m−2k|x|2k.
Using 5.40, the last equation becomes qm(x) = [m/2] X k=0 (−1)k n(n + 2) . . . (n + 2m −2k −2) 2kk! (m −2k)!
(x · ζ)m−2k|x|2k.
By replacing m by m−2, we obtain a formula for qm−2. In that formula, replace the index k by k−1, and then combine terms in 5.41 to complete the proof.
Note that for x ∈S, the expansion in the theorem above shows that Zm(x, ζ) is a function of x · ζ. We could have predicted this by recalling from the last section that on S, the zonal harmonic Zm(·, ζ) is constant on parallels orthogonal to ζ.
The formula for zonal harmonics given by the theorem above may be combined with Proposition 5.31 and the formula for the integral over S of any monomomial (, Section 3) to calculate explicitly the Poisson integral of any polynomial. However, this procedure is typi-cally several orders of magnitude slower than the algorithm given by Theorem 5.21.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 106 Chapter 5. Harmonic Polynomials Exercises 1.
Suppose p is a polynomial on Rn such that p|S ≡0. Prove that there exists a polynomial q such that p = (1 −|x|2)q.
2.
Suppose p is a homogeneous polynomial on Rn and u = P[p|S].
Prove that u is a homogeneous polynomial with the same degree as p if and only if p is harmonic.
3.
Suppose p is a polynomial on Rn and u = P[p|S]. Prove that the degree of u is less than the degree of p if and only if the homo-genenous part of p of highest degree is a polynomial multiple of |x|2.
4.
Suppose that f is a homogeneous polynomial on Rn of even (re-spectively, odd) degree. Prove that P[f ] is a polynomial consist-ing only of terms of even (respectively, odd) degree.
5.
Suppose E is an open ellipsoid in Rn.
(a) Prove that if p is a polynomial on Rn of degree at most m, then there exists a harmonic polynomial q on Rn of degree at most m such that q|∂E = p|∂E.
(b) Use part (a) and the Stone-Weierstrass Theorem to show that if f ∈C(E), then there exists u ∈C(E) such that u|∂E = f and u is harmonic on E.
6.
Let f be a polynomial on Rn. Prove that Pe[f |S], the exterior Poisson integral of f |S (see Chapter 4), extends to a function that is harmonic on Rn \ {0}.
7.
Generalized Dirichlet Problem: Show that if f and g are polyno-mials on Rn, then there is a unique polynomial p with p|S = f |S and ∆p = g. (The software described in Appendix B can find p explicitly.) 8.
From Pascal’s triangle we know N+1 M = N M + N M−1 . Use this and Proposition 5.8 to show that dim Hm(Rn) = n + m −2 n −2 !
+ n + m −3 n −2 !
for m ≥1.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Exercises 107 9.
Prove that dim Hm(Rn) < dim Hm+1(Rn) when n > 2.
10.
Prove that for a fixed n, dim Hm(Rn) mn−2 → 2 (n −2)!
as m →∞.
11.
Prove that Z S |x12 −x22|2 dσ(x) = 4 n(n + 2).
12.
Suppose p, q ∈Hm(Rn). Prove that p(D)[q] = n(n + 2) . . . (n + 2m −2) Z S pq dσ.
(Note that the left side of the equation above, which appears to be a function, is actually a constant because p and q are both homogeneous polynomials of degree m.) 13.
Where in the proof of Lemma 5.17 was the hypothesis n > 2 used?
14.
For n = 2, let cm = (−2)m−1(m −1)!.
Suppose m > 0 and p ∈Pm(R2).
(a) Prove that K[p(D) log |x|] = cm(p −|x|2q) for some q ∈Pm−2(R2).
(b) Prove that the orthogonal projection of p onto Hm(R2) is K[p(D) log |x|]/cm.
15.
Given a polynomial f on Rn, how would you go about determin-ing whether or not f |S is a spherical harmonic?
16.
Prove that if p ∈Hm(Rn), then DjK[p] = K[|x|2Djp + (2 −n −2m)xjp] for 1 ≤j ≤n.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 108 Chapter 5. Harmonic Polynomials 17.
Prove that if p ∈Hm(Rn), where n > 2 and m > 0, then K[p] = 1 m(4 −n −2m) n X j=1 DjK[Djp].
18.
Let f ∈C(S). The Neumann problem for B with boundary data f is to find a function harmonic on B whose outward normal derivative on S equals f and whose value at the origin is 0.
(a) Show that the Neumann problem with boundary data f has at most one solution.
(b) Show if the Neumann problem with boundary data f has a solution, then R S f dσ = 0.
(c) Show that if p is a polynomial on Rn, then the Neumann problem with boundary data p|S has a solution if and only if R S p dσ = 0. Describe how you would calculate a solution to the Neumann problem with boundary data p|S from the solution to the Dirichlet problem with boundary data p|S, and vice versa.
19.
(a) For n = 2, find a necessary and sufficient condition for equality in Proposition 5.27(e).
(b) Prove that if n > 2 and m > 0, then the inequality in Propo-sition 5.27(e) is an equality if and only if ζ = η or ζ = −η.
20.
Define PM(x, ζ) = PM m=0 Zm(x, ζ). Show that for fixed ζ ∈S, inf x∈B PM(x, ζ) →−∞ as M →∞, even though for each fixed x ∈B, PM(x, ζ) →P(x, ζ) > 0 as M →∞(by Theorem 5.33).
21.
Fix x ∈B. For f = P(x, ·), what is the expansion given by The-orem 5.29? Show how this could be used to give an alternative proof of Theorem 5.33.
22.
Give an example of a real-analytic function on B whose homoge-neous expansion (about 0) does not converge in all of B. (Com-pare this with Corollary 5.34.) Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Exercises 109 23.
Suppose u ∈C1(B) is such that the function x 7→x · ∇u(x) is harmonic on B. Prove that u is harmonic on B.
24.
Suppose u is harmonic on Rn and u is constant on parallels or-thogonal to η ∈S. Show that there exist c0, c1, . . . ∈C such that u(x) = ∞ X m=0 cmZm(x, η) for all x ∈Rn.
25.
Suppose that u is harmonic on B, u is constant on parallels or-thogonal to η ∈S, and u(rη) = 0 for infinitely many r ∈[−1, 1].
Prove that u ≡0 on B.
26.
Show that u need not vanish identically in Exercise 25 if “har-monic on B” is replaced by “continuous on B and harmonic on B”.
(Suggestion: Set qm(x) = Zm(x, η)/(dim Hm(Rn)) and consider a sum of the form P∞ k=1(−1)kckqmk(x), where the coefficients ck are positive and summable and the integers mk are widely spaced.) 27.
Show that there exists a nonconstant harmonic function u on R2 that is constant on parallels orthogonal to eiθ as well as on par-allels orthogonal to eiϕ if and only if θ −ϕ is a rational multiple of π.
28.
Suppose n > 2 and ζ, η ∈S. Under what conditions can a func-tion on S be constant on parallels orthogonal to ζ as well as on parallels orthogonal to η?
29.
Fix a positive integer m. By Theorem 5.38, there is a polynomial q of one variable such that Zm(η, ζ) = q(η · ζ) for all η, ζ ∈S.
Prove that if n is even, then each coefficient of q is an integer.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Chapter 6 Harmonic Hardy Spaces Poisson Integrals of Measures In Chapter 1 we defined the Poisson integral of a function f ∈C(S) to be the function P[f ] on B given by 6.1 Pf = Z S f (ζ)P(x, ζ) dσ(ζ).
We now extend this definition: for µ a complex Borel measure on S, the Poisson integral of µ, denoted P[µ], is the function on B defined by 6.2 Pµ = Z S P(x, ζ) dµ(ζ).
Differentiating under the integral sign in 6.2, we see that P[µ] is har-monic on B.
The set of complex Borel measures on S will be denoted by M(S).
The total variation norm of µ ∈M(S) will be denoted by ∥µ∥. Recall that M(S) is a Banach space under the total variation norm. By the Riesz Representation Theorem, if we identify µ ∈M(S) with the linear functional Λµ on C(S) given by Λµ(f ) = Z S f dµ, then M(S) is isometrically isomorphic to the dual space of C(S). (A good source for these results is .) 111 112 Chapter 6. Harmonic Hardy Spaces We will also deal with the Banach spaces Lp(S), 1 ≤p ≤∞. When p ∈[1, ∞), Lp(S) consists of the Borel measurable functions f on S for which ∥f ∥p = Z S |f |p dσ 1/p < ∞; L∞(S) consists of the Borel measurable functions f on S for which ∥f ∥∞< ∞, where ∥f ∥∞denotes the essential supremum norm on S with respect to σ. The number q ∈[1, ∞] is said to be conjugate to p if 1/p + 1/q = 1. If 1 ≤p < ∞and q is conjugate to p, then Lq(S) is the dual space of Lp(S). Here we identify g ∈Lq(S) with the linear functional Λg on Lp(S) defined by Λg(f ) = Z S f g dσ.
Note that because σ is a finite measure on S, Lp(S) ⊂L1(S) for all p ∈[1, ∞]. Recall also that C(S) is dense in Lp(S) for 1 ≤p < ∞.
It is natural to identify each f ∈L1(S) with the measure µf ∈M(S) defined on Borel sets E ⊂S by 6.3 µf (E) = Z E f dσ.
Shorthand for 6.3 is the expression dµf = f dσ. The map f 7→µf is a linear isometry of L1(S) into M(S). We will often identify functions in L1(S) as measures in this manner without further comment.
For f ∈L1(S), we will write P[f ] in place of P[µf ]. Here one could also try to define P[f ] as in 6.1. Fortunately the two definitions agree, because if ϕ is a bounded Borel measurable function on S (in particular, if ϕ = P(x, ·)), then R S ϕ dµf = R S ϕf dσ. Our notation is thus consis-tent with that defined previously for continuous functions on S.
Throughout this chapter, when given a function u on B, the notation ur will refer to the function on S defined by ur (ζ) = u(rζ); here, of course, 0 ≤r < 1.
We know that if f ∈C(S), then P[f ] has a continuous extension to B. What can be said of the more general Poisson integrals defined above? We begin to answer this question in the next two theorems.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Poisson Integrals of Measures 113 6.4 Theorem: The following growth estimates apply to Poisson inte-grals: (a) If µ ∈M(S) and u = P[µ], then ∥ur ∥1 ≤∥µ∥for all r ∈[0, 1).
(b) If 1 ≤p ≤∞, f ∈Lp(S), and u = P[f ], then ∥ur ∥p ≤∥f ∥p for all r ∈[0, 1).
Proof: The identity 6.5 P(rη, ζ) = P(rζ, η), valid for all η, ζ ∈S and all r ∈[0, 1), will be used to prove both (a) and (b).
To prove (a), let µ ∈M(S) and set u = P[µ]. For η ∈S and r ∈[0, 1), |u(rη)| ≤ Z S P(rη, ζ) d|µ|(ζ), where |µ| denotes the total variation measure associated with µ. Fu-bini’s theorem and 6.5 then give ∥ur ∥1 = Z S |u(rη)| dσ(η) ≤ Z S Z S P(rη, ζ) d|µ|(ζ) dσ(η) = Z S Z S P(rζ, η) dσ(η) d|µ|(ζ) = ∥µ∥.
For (b), assume first that 1 ≤p < ∞. Let f ∈Lp(S) and set u = P[f ].
Then |u(rη)| ≤ Z S |f (ζ)| P(rη, ζ) dσ(ζ).
By Jensen’s inequality, |u(rη)|p ≤ Z S |f (ζ)|pP(rη, ζ) dσ(ζ).
Integrate this last expression over S and use an argument similar to that given for (a) to get ∥ur ∥p ≤∥f ∥p, as desired.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 114 Chapter 6. Harmonic Hardy Spaces The case f ∈L∞(S) is the easiest. With u = P[f ], we have |u(rη)| ≤ Z S |f (ζ)| P(rη, ζ) dσ(ζ) ≤∥f∥∞ Z S P(rη, ζ) dσ(ζ) = ∥f∥∞.
Our first consequence of the last theorem is that ∥ur ∥p is an in-creasing function of r for each harmonic function u. A necessary and sufficient condition for the inequality in the corollary below to be an equality is given in Exercise 4 of this chapter.
6.6 Corollary: If u is harmonic on B and 0 ≤r ≤s < 1, then ∥ur ∥p ≤∥us∥p for all p ∈[1, ∞].
Proof: Suppose u is harmonic on B and 0 ≤r ≤s < 1. The idea of the proof is to think of ur as a dilate of the Poisson integral of us; then the result follows from the previous theorem. More specifically, ∥ur ∥p = ∥P[us] r s ∥p ≤∥us∥p, where the equality follows from Theorem 1.21 and the inequality fol-lows from Theorem 6.4(b).
If f ∈C(S) and u = P[f ], we know that ur →f in C(S) as r →1.
This fact and Theorem 6.4 enable us to prove the following result on Lp-convergence.
6.7 Theorem: Suppose 1 ≤p < ∞. If f ∈Lp(S) and u = P[f ], then ∥ur −f∥p →0 as r →1.
Proof: Let p ∈[1, ∞), let f ∈Lp(S), and set u = P[f ]. Fix ε > 0, and choose g ∈C(S) with ∥f −g∥p < ε. Setting v = P[g], we have ∥ur −f ∥p ≤∥ur −vr ∥p + ∥vr −g∥p + ∥g −f∥p.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Weak Convergence 115 Now (ur −vr ) = (P[f −g])r , hence ∥ur −vr ∥p < ε by Theorem 6.4.
Note also that ∥vr −g∥p ≤∥vr −g∥∞. Thus ∥ur −f ∥p < ∥vr −g∥∞+ 2ε.
Because g ∈C(S), we have ∥vr −g∥∞→0 as r →1. It follows that lim supr→1 ∥ur −f∥p ≤2ε. Since ε is arbitrary, ∥ur −f ∥p →0, as desired.
Theorem 6.7 fails when p = ∞. In fact, for f ∈L∞(S) and u = P[f ], we have ∥ur −f ∥∞→0 as r →1 if and only if f ∈C(S), as the reader should verify.
In the case µ ∈M(S) and u = P[µ], one might ask if the L1-functions ur always converge to µ in M(S). Here as well the answer is negative.
Because L1(S) is a closed subspace of M(S), ur →µ in M(S) precisely when µ is absolutely continuous with respect to σ.
We will see in the next section that there is a weak sense in which convergence at the boundary occurs in the cases discussed in the two paragraphs above.
Weak Convergence A useful concept in analysis is the notion of weak convergence.
Suppose X is a normed linear space and X∗is the dual space of X. If Λ1, Λ2, . . . ∈X∗, then the sequence (Λk) is said to converge weak to Λ ∈X∗provided limk→∞Λk(x) = Λ(x) for every x ∈X.
In other words, Λk →Λ weak precisely when the sequence (Λk) converges pointwise on X to Λ. We will also deal with one-parameter families {Λr : r ∈[0, 1)} ⊂X∗; here we say that Λr →Λ weak if Λr (x) →Λ(x) as r →1 for each fixed x ∈X.
A simple observation we need later is that if Λk →Λ weak, then 6.8 ∥Λ∥≤lim inf k→∞ ∥Λk∥.
Here ∥Λ∥is the usual operator norm on the dual space X∗defined by ∥Λ∥= sup{|Λ(x)| : x ∈X, ∥x∥≤1}.
Convergence in norm implies weak convergence, but the converse is false.
A simple example is furnished by ℓ2, the space of square Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 116 Chapter 6. Harmonic Hardy Spaces summable sequences. Because ℓ2 is a Hilbert space, (ℓ2)∗= ℓ2. Let ek denote the element of ℓ2 that has 1 in the kth slot and 0’s elsewhere.
Then for each a = (a1, a2, . . . ) ∈ℓ2, ⟨a, ek⟩= ak for every k. Thus ek →0 weak in ℓ2 as k →∞, while ∥ek∥= 1 in the ℓ2-norm for every k.
This example also shows that inequality may occur in 6.8.
Our next result is the replacement for Theorem 6.7 in the cases mentioned at the end of the last section.
6.9 Theorem: Poisson integrals have the following weak conver-gence properties: (a) If µ ∈M(S) and u = P[µ], then ur →µ weak in M(S) as r →1.
(b) If f ∈L∞(S) and u = P[f ], then ur →f weak in L∞(S) as r →1.
Proof: Recall that C(S)∗= M(S). Suppose µ ∈M(S), u = P[µ], and g ∈C(S). To prove (a), we need to show that 6.10 Z S gur dσ → Z S g dµ as r →1.
Working with the left-hand side of 6.10, we have Z S gur dσ = Z S g(ζ) Z S P(rζ, η) dµ(η) dσ(ζ) = Z S Z S g(ζ)P(rη, ζ) dσ(ζ) dµ(η) = Z S Pg dµ(η), where we have used 6.5 again. Because g ∈C(S), Pg →g(η) uniformly on S as r →1. This proves 6.10 and completes the proof of (a).
The proof of (b) is similar. We first recall that L1(S)∗= L∞(S). With f ∈L∞(S) and u = P[f ], we thus need to show that 6.11 Z S gur dσ → Z S gf dσ Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey The Spaces hp(B) 117 as r →1, for each g ∈L1(S).
Using the same manipulations as above, we see that the left side of 6.11 equals R S(P[g])r f dσ. By The-orem 6.7, (P[g])r →g in L1(S) as r →1.
Because f ∈L∞(S), we have (P[g])r f →gf in L1(S). This proves 6.11, completing the proof of (b).
In Chapter 2 we told the reader that every bounded harmonic func-tion on B is the Poisson integral of a bounded measurable function on S. In Chapter 3, we claimed that each positive harmonic function on B is the Poisson integral of some positive measure on S. We will prove these results in the next section. The key to these proofs is the following fundamental theorem on weak convergence.
6.12 Theorem: If X is a separable normed linear space, then ev-ery norm-bounded sequence in X∗contains a weak convergent sub-sequence.
Proof: Assume (Λm) is a norm-bounded sequence in X∗.
Then (Λm) is both pointwise bounded and equicontinuous on X (equiconti-nuity follows from the linearity of the functionals Λm). By the Arzela-Ascoli Theorem for separable metric spaces (Theorem 11.28 in ), (Λm) contains a subsequence (Λmk) converging uniformly on compact subsets of X. In particular, (Λmk) converges pointwise on X, which implies that (Λmk) converges weak to some element of X∗.
In the next section we will apply the preceding theorem to the sep-arable Banach spaces C(S) and Lq(S), 1 ≤q < ∞.
The Spaces hp(B) The estimates obtained in Theorem 6.4 suggest the definition of some new function spaces. For 1 ≤p ≤∞, we define hp(B) to be the class of functions u harmonic on B for which ∥u∥hp = sup 0≤r<1 ∥ur ∥p < ∞.
Thus hp(B) consists of the harmonic functions on B whose Lp-norms on spheres centered at the origin are uniformly bounded.
Because Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 118 Chapter 6. Harmonic Hardy Spaces ∥ur ∥p is an increasing function of r for each harmonic function u (Corollary 6.6), we have ∥u∥hp = lim r→1 ∥ur ∥p for each u ∈hp(B).
Note that h∞(B) is the collection of bounded harmonic functions on B, and that ∥u∥h∞= sup x∈B |u(x)|.
We refer to the spaces hp(B) as “harmonic Hardy spaces”. The usual “Hardy spaces”, denoted by Hp(B2), consist of the functions in hp(B2) that are holomorphic on B2; they are named in honor of the mathemati-cian G. H. Hardy, who first studied them.
It is straightforward to verify that hp(B) is a normed linear space under the norm ∥∥hp. A consequence of Theorem 6.13 below is that hp(B) is a Banach space.
Here are some observations that can be elegantly stated in terms of the hp-spaces: (a) The map µ →P[µ] is a linear isometry of M(S) into h1(B).
(b) For 1 < p ≤∞, the map f →P[f ] is a linear isometry of Lp(S) into hp(B).
Let us verify these claims. First, the maps in question are clearly linear. Second, in the case of (a), we have ∥P[µ]∥h1 ≤∥µ∥ by Theorem 6.4. On the other hand, 6.8 and Theorem 6.9 show that ∥µ∥≤lim inf r→1 ∥(P[µ])r ∥1 = ∥P[µ]∥h1.
This proves (a). The proof of (b) when p = ∞is similar. The proof of (b) when 1 < p < ∞is even easier, following from Theorem 6.7.
We now prove the remarkable result that the maps in (a) and (b) above are onto.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey The Spaces hp(B) 119 6.13 Theorem: The Poisson integral induces the following surjective isometries: (a) The map µ 7→P[µ] is a linear isometry of M(S) onto h1(B).
(b) For 1 < p ≤∞, the map f 7→P[f ] is a linear isometry of Lp(S) onto hp(B).
Proof: All that remains to be verified in (a) is that the range of the map µ 7→P[µ] is all of h1(B). To prove this, suppose u ∈h1(B). By definition, this means that the family {ur : r ∈[0, 1)} is norm-bounded in L1(S), and hence in M(S) = C(S)∗. Theorem 6.12 thus implies there exists a sequence rj →1 such that the sequence urj converges weak to some µ ∈M(S). The proof of (a) will be completed by showing that u = P[µ].
Fix x ∈B. Because the functions y 7→u(rjy) are harmonic on B, we have 6.14 u(rjx) = Z S u(rjζ)P(x, ζ) dσ(ζ) for each j. Now let j →∞. Simply by continuity, the left side of 6.14 converges to u(x). On the other hand, because P(x, ·) ∈C(S), the right side of 6.14 converges to Pµ. Therefore u(x) = Pµ, and thus u = P[µ] on B, as desired.
The proof of (b) is similar. Fix p ∈(1, ∞], let u ∈hp(B), and let q be the number conjugate to p. Then the family {ur : r ∈[0, 1)} is norm-bounded in Lp(S) = Lq(S)∗. By Theorem 6.12, there exists a sequence rj →1 such that urj converges weak to some f ∈Lp(S). The argument given in the paragraph above may now be used, essentially verbatim, to show that u = P[f ]; the difference is that here we need to observe that P(x, ·) ∈Lq(S). We leave it to the reader to fill in the rest of the proof.
The theorem above contains the assertion made in Chapter 2 that if u is bounded and harmonic on B, then u = P[f ] for some f ∈L∞(S).
We next take up the claim made in Chapter 3.
6.15 Corollary: If u is positive and harmonic on B, then there is a unique positive measure µ ∈M(S) such that u = P[µ].
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 120 Chapter 6. Harmonic Hardy Spaces Proof: Suppose u is positive and harmonic on B. Then Z S |ur | dσ = Z S ur dσ = u(0) for every r ∈[0, 1), the last equality following from the mean-value property. Thus u ∈h1(B), which by Theorem 6.13 means that there is a unique µ ∈M(S) such that u = P[µ]. Being the weak limit of the positive measures ur (Theorem 6.9(a)), µ is itself positive.
Our next proposition gives a growth estimate for functions in hp(B).
For a slight improvement of this proposition, see Exercise 11 of this chapter.
6.16 Proposition: Suppose 1 ≤p < ∞. If u ∈hp(B), then |u(x)| ≤ 1 + |x| (1 −|x|)n−1 1/p ∥u∥hp for all x ∈B.
Proof: Suppose u ∈hp(B) and x ∈B.
First consider the case where 1 < p < ∞. By Theorem 6.13, there exists f ∈Lp(S) such that u = P[f ]; furthermore ∥u∥hp = ∥f ∥p. Let q be the number conjugate to p. Now |u(x)| = Z S f (ζ)P(x, ζ) dσ(ζ) ≤ Z S P(x, ζ)q dσ(ζ) 1/q ∥u∥hp.
6.17 Notice that Z S P(x, ζ)q dσ(ζ) ≤sup ζ∈S P(x, ζ)q−1 Z S P(x, ζ) dσ(ζ) = 1 + |x| (1 −|x|)n−1 q−1 6.18 Combining 6.17 and 6.18 gives the desired result.
The p = 1 case is similar and is left to the reader.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey The Hilbert Space h2(B) 121 We conclude this section with a result that will be useful in the next chapter.
6.19 Theorem: Let ζ ∈S. Suppose that u is positive and harmonic on B, and that u extends continuously to B \ {ζ} with u = 0 on S \ {ζ}.
Then there exists a positive constant c such that u = cP(·, ζ).
Proof: We have u = P[µ] for some positive µ ∈M(S) by The-orem 6.15, and we have ur →µ weak in M(S) as r →1 by Theo-rem 6.9. The hypotheses on u imply that the functions ur converge to 0 uniformly on compact subsets of S \ {ζ} as r →1.
Therefore R S ϕ dµ = 0 for any continuous ϕ on S that is zero near ζ. This im-plies that µ(S \ {ζ}) = 0, and thus that µ is a point mass at ζ. The conclusion of the theorem is immediate from this last statement.
The Hilbert Space h2(B) The map f 7→P[f ] is a linear isometry of L2(S) onto h2(B) (by Theo-rem 6.13). Because L2(S) is a Hilbert space, we can use this isometry to transfer a Hilbert space structure to h2(B). Specifically, we can define ⟨P[f ], P[g]⟩= ⟨f , g⟩= Z S f g dσ for f , g ∈L2(S), where we use ⟨, ⟩to denote the inner product on both h2(B) and L2(S), allowing the context to make clear which is in-tended.
Given u, v ∈h2(B), it would be nice to have an intrinsic formula for ⟨u, v⟩that does not involve finding f , g ∈L2(S) such that u = P[f ] and v = P[g]. Fortunately, Theorem 6.7 leads to such a formula. We have ur →f and vr →g in L2(S), and thus ⟨ur , vr ⟩→⟨f , g⟩. Hence ⟨u, v⟩= lim r→1 Z S u(rζ)v(rζ) dσ(ζ).
For f ∈L2(S) and x ∈B, we have 6.20 Pf = ⟨f , P(x, ·)⟩.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 122 Chapter 6. Harmonic Hardy Spaces To translate this to an intrinsic formula on h2(B), we need to find the Poisson integral of P(x, ·). In other words, we need to extend P(x, ·), which is currently defined on S, to a harmonic function on B. To do this, note that P(x, ζ) = 1 −|x|2 |x −ζ|n = 1 −|x|2 (1 −2x · ζ + |x|2)n/2 for ζ ∈S. We extend the domain of P by defining 6.21 P(x, y) = 1 −|x|2|y|2 (1 −2x · y + |x|2|y|2)n/2 for all x, y ∈Rn × Rn for which the denominator above is not 0. Note that this agrees with our previous definition when y ∈S.
Our extended Poisson kernel P has the pleasant properties that P(x, y) = P(y, x) and P(x, y) = P(|x|y, x/|x|). The last equation shows that for x fixed, P(x, ·) is a harmonic function (because it is a dilate of a harmonic function). In particular, for x ∈B, the func-tion P(x, ·) is harmonic on B and hence is the function in h2(B) that corresponds to the unextended Poisson kernel P(x, ·) ∈L2(S). The extended Poisson kernel will play a major role when we study Bergman spaces in Chapter 8.
Translating 6.20 to h2(B), we have the intrinsic formula 6.22 u(x) = ⟨u, P(x, ·)⟩ for all x ∈B and u ∈h2(B). The usefulness of this viewpoint is demon-strated by the next proposition, which gives a sharp growth estimate for functions in h2(B), slightly better than the p = 2 case of Proposi-tion 6.16.
6.23 Proposition: If u ∈h2(B), then |u(x)| ≤ s 1 + |x|2 (1 −|x|2)n−1 ∥u∥h2 for all x ∈B.
Proof: Suppose u ∈h2(B) and x ∈B. From the Cauchy-Schwarz inequality and 6.22, we have Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey The Schwarz Lemma 123 |u(x)| ≤∥P(x, ·)∥h2∥u∥h2.
Now ∥P(x, ·)∥2 h2 = ⟨P(x, ·), P(x, ·)⟩= P(x, x), where the last equality comes from 6.22. Use 6.21 to compute P(x, x) and complete the proof.
The Schwarz Lemma The Schwarz Lemma in complex analysis states that if h is holo-morphic on B2 with |h| < 1 and h(0) = 0, then |h(z)| ≤|z| for all z ∈B2; furthermore, if equality holds at any nonzero z ∈B2, then h(z) = λz for all z ∈B2, where λ is a complex number of modulus one.
In this section we take up the Schwarz Lemma for functions harmonic on Bn.
Hermann Amandus Schwarz (1843–1921), whose reflection principle we used in Chapter 4 and whose lemma we now extend to harmonic functions, is also noted for his discovery of a procedure for solving the Dirichlet problem.
Let S+ denote the northern hemisphere {ζ ∈S : ζn > 0} and let S−denote the southern hemisphere {ζ ∈S : ζn < 0}. We define a harmonic function U = Un on B by setting Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 124 Chapter 6. Harmonic Hardy Spaces U = P[χS+ −χS−].
In other words, U is the Poisson integral of the function that equals 1 on S+ and −1 on S−. Note that U is harmonic on B, with |U| < 1 and U(0) = 0.
The following theorem shows that U and its rotations are the ex-tremal functions for the Schwarz Lemma for harmonic functions. Re-call that N = (0, . . . , 0, 1) denotes the north pole of S.
6.24 Harmonic Schwarz Lemma: Suppose that u is harmonic on B, |u| < 1 on B, and u(0) = 0. Then |u(x)| ≤U(|x|N) for every x ∈B. Equality holds for some nonzero x ∈B if and only if u = λ(U ◦T), where λ is a complex constant of modulus 1 and T is an orthogonal transformation.
Proof: Fix x ∈B. After a rotation, we can assume that x lies on the radius from 0 to N, so that x = |x|N.
First we consider the case where u is real valued. By Theorem 6.13, there is a real-valued function f ∈L∞(S) such that u = P[f ] and ∥f ∥∞≤1.
We claim that u(x) ≤U(x). This inequality is equivalent to the inequality Z S− 1 + f (ζ) P(x, ζ) dσ ≤ Z S+ 1 −f (ζ) P(x, ζ) dσ.
Because x = |x|N, we have P(x, ζ) = (1 −|x|2)/(1 + |x|2 −2|x|ζn)n/2, so the inequality above is equivalent to Z S− 1 + f (ζ) (1 + |x|2 −2|x|ζn)n/2 dσ(ζ) 6.25 ≤ Z S+ 1 −f (ζ) (1 + |x|2 −2|x|ζn)n/2 dσ(ζ).
The condition u(0) = 0 implies that R S−f dσ = − R S+ f dσ. Thus, since ζn is negative on S−and positive on S+, we have Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey The Schwarz Lemma 125 Z S− 1 + f (ζ) (1 + |x|2 −2|x|ζn)n/2 dσ(ζ) ≤ Z S− 1 + f (ζ) (1 + |x|2)n/2 dσ(ζ) = Z S+ 1 −f (ζ) (1 + |x|2)n/2 dσ(ζ) ≤ Z S+ 1 −f (ζ) (1 + |x|2 −2|x|ζn)n/2 dσ(ζ).
Thus 6.25 holds, completing the proof that u(x) ≤U(x). Note that if x ̸= 0, then the last two inequalities are equalities if and only if f = 1 almost everywhere on S+ and f = −1 almost everywhere on S−. In other words, we have u(x) = U(x) if and only if u = U.
Now remove the restriction that u be real valued. Choose β ∈C such that |β| = 1 and βu(x) = |u(x)|. Apply the result just proved to the real part of βu, getting |u(x)| ≤U(x), with equality if and only if βu = U.
Note that while the extremal functions for the Schwarz Lemma for holomorphic functions are the entire functions z 7→λz (with |λ| = 1), the extremal functions for the Harmonic Schwarz Lemma are discontin-uous at the boundary of B. Later in this section we will give a concrete formula for U when n = 2; Exercise 24 of this chapter gives formulas for U(|x|N) when n = 3, 4. The software package described in Ap-pendix B can compute U(|x|N) for higher values of n.
The Schwarz Lemma for holomorphic functions has a second part that we did not mention earlier. Specifically, if h is holomorphic on B2 and |h| < 1 on B2, then |h′(0)| ≤1; equality holds if and only if h(z) = λz for some constant λ of modulus one. (Almost all complex analysis texts add the hypothesis that h(0) = 0, which is not needed for this part of the Schwarz Lemma.) The next theorem gives the cor-responding result for harmonic functions. Here the gradient takes the place of the holomorphic derivative.
6.26 Theorem: Suppose u is a real-valued harmonic function on Bn and |u| < 1 on Bn. Then |(∇u)(0)| ≤2V(Bn−1) V(Bn) .
Equality holds if and only if u = U ◦T for some orthogonal transforma-tion T.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 126 Chapter 6. Harmonic Hardy Spaces Proof: We begin by investigating the size of the partial derivative Dnu(0). By Theorem 6.13, there is a real-valued function f ∈L∞(S) such that u = P[f ] and ∥f∥∞≤1. Differentiating under the integral sign in the Poisson integral formula, we have Dnu(0) = Z S f (ζ)DnP(0, ζ) dσ(ζ) = n Z S f (ζ)ζn dσ(ζ) ≤n Z S |ζn| dσ(ζ).
Equality holds here if and only if f = 1 almost everywhere on S+ and f = −1 almost everywhere on S−, which is equivalent to saying that u equals U. The last integral can be easily evaluated using A.6 from Appendix A: n Z S |ζn| dσ(ζ) = 2 V(Bn) Z Bn−1 (1 −|x|2)−1/2(1 −|x|2)1/2 dVn−1(x) = 2V(Bn−1) V(Bn) .
Thus Dnu(0) ≤2V(Bn−1)/V(Bn), with equality if and only if u = U.
Applying this result to rotations of u, we see that every directional derivative of u at 0 is bounded above by 2V(Bn−1)/V(Bn); the length of ∇u(0) is therefore bounded by the same constant, with equality if and only if u is a rotation of U.
The bound given above on |(∇u)(0)| could not be improved if we added the hypothesis that u(0) = 0, because the extremal function already satisfies that condition.
When n = 2, the preceding theorem shows that |(∇u)(0)| ≤4/π.
Note that the optimal constant 4/π is larger than 1, which is the optimal constant for the Schwarz Lemma for holomorphic functions.
Theorem 6.26 fails for complex-valued harmonic functions (Exer-cise 23 of this chapter). The gradient, which points in the direction of maximal increase for a real-valued function, seems to have no natural geometric interpretation for complex-valued functions.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey The Schwarz Lemma 127 We now derive an explicit formula for the extremal function U when n = 2. Here the arctangent of a real number is always taken to lie in the interval (−π/2, π/2).
The graph of the harmonic function U2 along with the boundary of its domain. On the upper-half of unit circle in the xy-plane, this function equals 1; on the lower half of the circle it equals −1.
6.27 Proposition: Let z = (x, y) be a point in B2. Then U2(x, y) = 2 π arctan 2y 1 −x2 −y2 and U2(|z|N) = 4 π arctan |z|.
Proof: Think of z = x + iy as a complex variable. The conformal map z 7→(1+z)/(1−z) takes B2 onto the right half-plane. The function z 7→log[(1 + z)/(1 −z)], where log denotes the principal branch of the logarithm, is therefore holomorphic on B2. Multiplying the imaginary part of this function by 2/π, we see that the function u defined by u(x, y) = 2 π arctan 2y 1 −x2 −y2 is harmonic on B2.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 128 Chapter 6. Harmonic Hardy Spaces Because u is bounded on B2, Theorem 6.13 implies that u = P[f ] on B2 for some f ∈L∞(S). Theorem 6.9 shows that ur →f weak in L∞(S) as r →1. But note that u extends to be continuous on B2 ∪S+ ∪S−, with u = 1 on S+ and u = −1 on S−; thus ur →u|S weak in L∞(S) as r →1 (by the dominated convergence theorem). Hence f = u|S almost everywhere on S. Thus u = U2, completing the proof of the first part of the proposition.
The second assertion in the proposition now follows from standard double-angle identities from trigonometry.
The Fatou Theorem Recall the cones Γα(a) defined in the section Limits Along Rays of Chapter 2. We will use these cones to define nontangential approach regions in the ball. For α > 0 and ζ ∈S, we first translate and rotate Γα(0) to obtain a new cone with vertex ζ and axis of symmetry con-taining the origin. This new cone crashes through the sphere on the side opposite of ζ, making it unsuitable for a nontangential approach region in B. To fix this, consider the largest ball B 0, r(α) contained in the new cone (we do not need to know the exact value of r(α)). Taking the convex hull of B 0, r(α) and the point ζ, and then removing the point ζ, we obtain the open set Ωα(ζ) pictured here.
The region Ωα(ζ) has the properties we seek for a nontangential approach region in B with vertex ζ.
Specifically, Ωα(ζ) stays away from the sphere except near ζ, and near ζ it equals the translated and rotated version of Γα(0) with which we started.
We have Ωα(ζ) ⊂Ωβ(ζ) if α < β, and B is the union of the sets Ωα(ζ) as α ranges over (0, ∞).
Note that T Ωα(ζ) = Ωα T(ζ) for every orthogonal transformation T on Rn. This allows us to transfer statements about the geometry of, say, Ωα(N) to any Ωα(ζ).
A function u on B is said to have nontangential limit L at ζ ∈S if for each α > 0, we have u(x) →L as x →ζ within Ωα(ζ).
In this section we prove that if u ∈h1(B), then u has a nontangential limit at almost every ζ ∈S. (In this chapter, the term “almost every-where” will mean “almost everywhere with respect to σ”.) Theorems Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey The Fatou Theorem 129 The nontangential approach region Ωα(ζ).
asserting the almost everywhere existence of limits within approach regions are commonly referred to as “Fatou Theorems”. The first such result was proved by Fatou , who in 1906 showed that bounded har-monic functions in the open unit disk have nontangential limits almost everywhere on the unit circle.
We approach the Fatou theorem for h1(B) via several operators known as “maximal functions”. Given a function u on B and α > 0, the nontangential maximal function of u, denoted by Nα[u], is the function on S defined by Nαu = sup x∈Ωα(ζ) |u(x)|.
The radial maximal function of u, denoted by R[u], is the function on S defined by Ru = sup 0≤r<1 |u(rζ)|.
Clearly Ru ≤Nαu for every ζ ∈S and every α > 0. The following theorem shows that, up to a constant multiple, the reverse inequality holds for positive harmonic functions on B.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 130 Chapter 6. Harmonic Hardy Spaces 6.28 Theorem: For every α > 0, there exists a constant Cα < ∞such that Nαu ≤CαRu for all ζ ∈S and all positive harmonic functions u on B.
Proof: Let ζ ∈S. The theorem then follows immediately from the existence of a constant Cα such that 6.29 P(x, η) ≤CαP(|x|ζ, η) for all x ∈Ωα(ζ) and all η ∈S. To prove 6.29, apply the law of cosines to the triangle with vertices 0, x, and ζ in 6.30 to see that there is a constant Aα such that |x −ζ| < Aα(1 −|x|) for all x ∈Ωα(ζ).
6.30 |x −ζ| is comparable to (1 −|x|) for x ∈Ωα(ζ).
Thus Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey The Fatou Theorem 131 |x|ζ −η ≤ |x|ζ −x + |x −η| ≤|ζ −x| + |x −η| ≤(Aα + 1)|x −η| for all x ∈Ωα(ζ) and all η ∈S.
This shows that 6.29 holds with Cα = (1 + Aα)n.
We turn now to a key operator in analysis, the Hardy-Littlewood maximal function. For ζ ∈S and δ > 0, define κ(ζ, δ) = {η ∈S : |η −ζ| < δ}.
Thus κ(ζ, δ) is the open “spherical cap” on S with center ζ and radius δ.
(Note that κ(ζ, δ) = S when δ > 2.) The Hardy-Littlewood maximal function of µ ∈M(S), denoted by M[µ], is the function on S defined by Mµ = sup δ>0 |µ| κ(ζ, δ) σ κ(ζ, δ) .
Suppose µ ∈M(S) is positive and δ > 0 is fixed.
Let (ζj) be a sequence in S such that ζj →ζ. Because the characteristic functions of κ(ζj, δ) converge to 1 pointwise on κ(ζ, δ) as j →∞, Fatou’s Lemma shows that µ κ(ζ, δ) ≤lim inf j→∞ µ κ(ζj, δ) .
In other words, the function ζ 7→µ κ(ζ, δ) is lower-semicontinuous on S.
From the definition of M[µ], we conclude that M[µ] is the supremum of lower-semicontinuous functions on S, and thus M[µ] is lower-semicontinuous. In particular, M[µ]: S →[0, ∞] is Borel mea-surable.
In the next theorem we begin to see the connection between the Hardy-Littlewood maximal function and the Fatou Theorem.
6.31 Theorem: If µ ∈M(S) and u = P[µ], then Ru ≤Mµ for all ζ ∈S.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 132 Chapter 6. Harmonic Hardy Spaces Proof: Observe that if f is a continuous, positive, and increasing function on [−1, 1], then given ε > 0, there exists a step function ϕ = c0χ[−1,1] + m X j=1 cjχ(tj,1] such that f ≤ϕ ≤f + ε on [−1, 1]; here −1 < t1 < · · · < tm < 1 and c0, . . . , cm ∈[0, ∞).
We may assume µ is positive and that ζ = N. Fix r ∈0, 1). Then P(rN, η) = f (ηn), where f (t) = 1 −r 2 (1 −2rt + r 2)n/2 for t ∈[−1, 1]. Let ε > 0. Because f has the properties specified in the first paragraph, there exists a step function ϕ as above with P(rN, η) ≤ϕ(ηn) ≤P(rN, η) + ε for all η ∈S.
Now for any t ∈R, the function on S defined by η 7→χ(t,1 is the characteristic function of an open cap centered at N. We conclude that there are caps κ0, . . . , κm, centered at N, and nonnegative numbers c0, . . . , cm, such that 6.32 P(rN, η) ≤ m X j=0 cjχκj(η) ≤P(rN, η) + ε for all η ∈S.
Integrating the first inequality in 6.32 over S with respect to µ, we get u(rN) ≤ m X j=0 cjµ(κj) = m X j=0 cjσ(κj) µ(κj)/σ(κj) ≤Mµ m X j=0 cjσ(κj) ≤Mµ(1 + ε).
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey The Fatou Theorem 133 The last inequality follows by integrating the second inequality in 6.32 over S with respect to σ.
Because ε is arbitrary, we conclude that u(rN) ≤Mµ, and thus Ru ≤Mµ, as desired.
Theorem 6.37 below estimates the σ-measure of the set where M[µ] is large. The “covering lemma” that we prove next will be a crucial ingre-dient in its proof. We abuse notation slightly and adopt the convention that if κ = κ(ζ, δ), then 3κ denotes the cap κ(ζ, 3δ).
6.33 Covering Lemma: Given caps κj = κ(ζj, δj), j = 1, . . . , m, there exists a subset J ⊂{1, . . . , m} such that: (a) The collection {κj : j ∈J} is pairwise disjoint; (b) m [ j=1 κj ⊂ [ j∈J 3κj.
Proof: We describe an inductive procedure for selecting the desired subcollection.
Start by choosing a cap κj1 having the largest radius among the caps κ1, . . . , κm. If all caps intersect κj1, we stop. Otherwise, remove the caps intersecting κj1, and from those remaining, select one of largest radius and denote it by κj2. If all the remaining caps intersect κj2, we stop; otherwise we continue as above. This process gives us a finite subcollection {κj : j ∈J}, where J = {j1, j2, . . . }.
The subcollection {κj : j ∈J} clearly satisfies (a).
Given κ ∈{κ1, . . . , κm}, let κ′ denote the first cap in the sequence κj1, κj2, . . . such that κ ∩κ′ is nonempty. The way in which the caps in {κj : j ∈J} were chosen shows that the radius of κ′ is at least as large as that of κ. By the triangle inequality, κ ⊂3κ′, proving (b).
In proving the next theorem we will need the fact that there exist constants a > 0, A < ∞, depending only on the dimension n, such that 6.34 aδn−1 ≤σ κ(ζ, δ) ≤Aδn−1 for all ζ ∈S and all δ ∈(0, 2]. Intuitively, κ(ζ, δ) looks like an (n−1)-dimensional ball of radius δ for small δ > 0, indicating that 6.34 is correct. One may verify 6.34 rigorously by using formula A.3 in Ap-pendix A.
From 6.34 we see that Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 134 Chapter 6. Harmonic Hardy Spaces 6.35 σ(3κ) ≤3n−1(A/a)σ(κ) for all caps κ ⊂S.
To motivate our next result, note that if f ∈L1(S) and t > 0, then tσ({|f| > t}) ≤ Z {|f|>t} |f | dσ ≤∥f∥1, giving 6.36 σ({|f | > t}) ≤∥f∥1 t .
Here we have used the abbreviated notation {|f | > t} to denote the set {ζ ∈S : |f (ζ)| > t}. The next theorem states that for µ ∈M(S), the Hardy-Littlewood maximal function M[µ] is almost in L1(S), in the sense that it satisfies an inequality resembling 6.36.
6.37 Theorem: For every µ ∈M(S) and every t ∈(0, ∞), σ({M[µ] > t}) ≤C∥µ∥ t , where C = 3n−1(A/a).
Proof: Suppose t ∈(0, ∞). Let E ⊂{M[µ] > t} be compact. Then for each ζ ∈E, there is a cap κ centered at ζ with |µ|(κ)/σ(κ) > t.
Being compact, E is covered by finitely many such caps. From these we may choose a subcollection with the properties specified in 6.33.
Thus there are pairwise disjoint caps κ1, . . . , κN such that 3κ1, . . . , 3κN cover E, and such that |µ|(κj)/σ(κj) > t for j = 1, . . . , N. By 6.35 and the definition of C, we therefore have σ(E) ≤ N X j=1 σ(3κj) ≤C N X j=1 σ(κj) ≤C N X j=1 |µ|(κj) t ≤C∥µ∥ t ; the pairwise disjointness of the caps κ1, . . . , κN was used in the last inequality. Taking the supremum over all compact E ⊂{M[µ] > t} now gives the conclusion of the theorem.
Let us write M[f ] in place of M[µ] when dµ = f dσ for f ∈L1(S).
The conclusion of Theorem 6.37 for f ∈L1(S) is then Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey The Fatou Theorem 135 6.38 σ({M[f ] > t}) ≤C∥f ∥1 t .
We now prove the Fatou theorem for Poisson integrals of functions in L1(S).
6.39 Theorem: If f ∈L1(S), then P[f ] has nontangential limit f (ζ) at almost every ζ ∈S.
Proof: For f ∈L1(S) and α > 0, define the function Lα[f ] on S by Lαf = lim sup x→ζ x∈Ωα(ζ) |Pf −f (ζ)|.
We first show that Lα[f ] = 0 almost everywhere on S.
Note that 6.40 Lα[f ] ≤Nα P[|f|] + |f |, and that Lα[f1+f2] ≤Lα[f1]+Lα[f2] (both statements holding almost everywhere on S). Note also that Lα[f ] ≡0 for every f ∈C(S).
Now fix f ∈L1(S) and α > 0. Also fixing t ∈(0, ∞), our main goal is to show that σ({Lα[f ] > 2t}) = 0.
Given ε > 0, we may choose g ∈C(S) such that ∥f −g∥1 < ε. We then have Lα[f ] ≤Lα[f −g] + Lα[g] = Lα[f −g] ≤Nα P[|f −g|] + |f −g| ≤CαR P[|f −g|] + |f −g| ≤CαM[|f −g|] + |f −g|, this holding at almost every point of S. In this string of inequalities we have used 6.40, 6.28, and 6.31 in succession.
We thus have 6.41 {Lα[f ] > 2t} ⊂{CαM[|f −g|] > t} ∪{|f −g| > t}.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 136 Chapter 6. Harmonic Hardy Spaces By 6.38 and 6.36, the σ-measure of the right side of 6.41 is less than or equal to CCα∥f −g∥1 t + ∥f −g∥1 t .
Recalling that ∥f −g∥1 < ε and that ε is arbitrary, we have shown that the set {Lα[f ] > 2t} is contained in sets of arbitrarily small σ-measure, and therefore σ({Lα[f] > 2t}) = 0.
Because this is true for every t ∈(0, ∞), we have proved Lα[f ] = 0 almost everywhere on S.
To finish, let f ∈L1(S), and define Em = {Lm[f ] = 0} for m = 1, 2, . . . . We have shown that Em is a set of full measure on S for each m, and thus T Em is a set of full measure. At each ζ ∈ T Em, P[f ] has nontangential limit f (ζ), which is what we set out to prove.
Recall that µ ∈M(S) is said to be singular with respect to σ, writ-ten µ ⊥σ, if there exists a Borel set E ⊂S such that σ(E) = 0 and |µ|(E) = ∥µ∥. Recall also that each µ ∈M(S) has a unique decomposi-tion dµ = f dσ + dµs, where f ∈L1(S) and µs ⊥σ; this is called the Lebesgue decomposition of µ with respect to σ. The following result is the second half of the Fatou Theorem for h1(B).
6.42 Theorem: If µ ⊥σ, then P[µ] has nontangential limit 0 almost everywhere on S.
Proof: Much of the proof is similar to that of Theorem 6.39, and so we will be brief about certain details.
It suffices to prove the theorem for positive measures, so suppose µ ∈M(S) is positive and µ ⊥σ. For α > 0, define Lαµ = lim sup x→ζ x∈Ωα(ζ) Pµ for ζ ∈S. Fixing t ∈(0, ∞), the proof will be completed by showing that σ({Lα[µ] > 2t}) = 0.
Let ε > 0.
Because µ ⊥σ, the regularity of µ implies the exis-tence of a compact set K ⊂S such that σ(K) = 0 and µ(S \ K) < ε.
Writing µ = µ1 + µ2, with dµ1 = χK dµ and dµ2 = χS\K dµ, observe that Lα[µ1] = 0 on S \ K (see Exercise 2 of this chapter) and that ∥µ2∥= µ(S \ K) < ε.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey The Fatou Theorem 137 Because Lα[µ] ≤Lα[µ1] + Lα[µ2], we have {Lα[µ] > 2t} ⊂{Lα[µ1] > t} ∪{Lα[µ2] > t} 6.43 ⊂K ∪{CαM[µ2] > t}.
(The inequality Lα[µ2] ≤CαM[µ2] is obtained as in the proof of Theo-rem 6.39.) Recalling that σ(K) = 0, we see by Theorem 6.37 that the left side of 6.43 is contained in a set of σ-measure at most (CCα∥µ2∥)/t, which is less than (CCαε)/t.
Since ε is arbitrary, we conclude that σ({Lα[µ] > 2t}) = 0, as desired.
Theorems 6.39 and 6.42 immediately give the following result.
6.44 Corollary: Suppose µ ∈M(S) and dµ = f dσ + dµs is the Lebesgue decomposition of µ with respect to σ. Then P[µ] has non-tangential limit f (ζ) at almost every ζ ∈S.
If u ∈h1(B), then u = P[µ] for some µ ∈M(S) by Theorem 6.13.
Corollary 6.44 thus implies that u has nontangential limits almost ev-erywhere on S. Because hp(B) ⊂h1(B) for all p ∈[1, ∞], the same result holds for all u ∈hp(B).
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 138 Chapter 6. Harmonic Hardy Spaces Exercises 1.
Show that if f ∈L1(S) and ζ ∈S is a point of continuity of f , then P[f ] extends continuously to B ∪{ζ}.
2.
Suppose V ⊂S is open, µ ∈M(S), and |µ|(V) = 0. Show that if ζ ∈V, then Pµ →0 as x →ζ unrestrictedly in B.
3.
Suppose that µ ∈M(S) and ζ ∈S. Show that (1 −r)n−1Pµ →2µ({ζ}) as r →1.
4.
Suppose that u is harmonic function on B and 0 ≤r < s < 1.
(a) Prove that ∥ur ∥1 = ∥us∥1 if and only if there is a constant c such that cu|sB is positive.
(b) Suppose 1 < p ≤∞. Prove that ∥ur ∥p = ∥us∥p if and only u is constant.
5.
(a) Give an example of a normed linear space and a weak con-vergent sequence in its dual space that is not norm-bounded.
(b) Prove that in the dual space of a Banach space, every weak convergent sequence is norm-bounded. (Hint: Use the uni-form boundedness principle.) 6.
It is easy to see that if µj →µ in M(S), then P[µj] →P[µ] uni-formly on compact subsets of B. Prove that the conclusion is still valid if we assume only that µj →µ weak in M(S).
7.
Suppose that (µj) is a norm-bounded sequence in M(S) such that (P[µj]) converges pointwise on B. Prove that (µj) is weak convergent in M(S).
8.
Prove directly (that is, without the help of Theorem 6.13) that hp(B) is a Banach space for every p ∈[1, ∞].
9.
Prove that a real-valued function on B belongs to h1(B) if and only if it is the difference of two positive harmonic functions on B.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Exercises 139 10.
Let ζ ∈S. Show that P(·, ζ) ∈hp(B) for p = 1 but not for any p > 1.
11.
Suppose 1 < p < ∞and u ∈hp(B). Prove that (1 −|x|)(n−1)/pu(x) →0 as |x| →1.
12.
A family of functions F ⊂L1(S) is said to be uniformly integrable if for every ε > 0, there exists a δ > 0 such that R E |f| dσ < ε whenever f ∈F and σ(E) < δ. Show that a harmonic function u on B is the Poisson integral of a function in L1(S) if and only if the family {ur : r ∈[0, 1)} is uniformly integrable.
13.
Prove that there exists u ∈h1(B) such that u B ∩B(ζ, ε) = R for all ζ ∈S, ε > 0. (Hint: Let u = P[µ], where µ is a judiciously chosen sum of point masses.) 14.
Suppose that p ∈[1, ∞) and u is harmonic on B.
Show that u ∈hp(B) if and only if there exists a harmonic function v on B such that |u|p ≤v on B.
15.
Suppose n > 2.
Show that if u is positive and harmonic on {x ∈Rn : |x| > 1}, then there exists a unique positive measure µ ∈M(S) and a unique nonnegative constant c such that u(x) = Peµ + c(1 −|x|2−n) for |x| > 1. (Here Pe is the external Poisson kernel defined in Chapter 4.) State and prove an analogous result for the case n = 2.
16.
Let Ωdenote B3 minus the x3-axis. Show that every bounded harmonic function on Ωextends to be harmonic on B3.
17.
Suppose ζ ∈S and f is positive and continuous on S \ {ζ}.
Need there exist a positive harmonic function u on B that extends continuously to B \ {ζ} with u = f on S \ {ζ}?
18.
Suppose E ⊂S and σ(E) = 0. Prove that there exists a positive harmonic function u on B such that u has limit ∞at every point of E.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 140 Chapter 6. Harmonic Hardy Spaces 19.
Let F denote the family of all positive harmonic functions u on B such that u(0) = 1. Compute inf{u(N/2) : u ∈F} and sup{u(N/2) : u ∈F}. Do there exist functions in F that attain either of these extreme values at N/2? If so, are they unique?
20.
Find all extreme points of F, where F is the family defined in the previous exercise. (A function in F is called an extreme point of F if it cannot be written as the average of two distinct functions in F.) 21.
Show that Z S P(x, ζ)P(y, ζ) dσ(ζ) = 1 −|x|2|y|2 (1 −2x · y + |x|2|y|2)n/2 for all x, y ∈B.
22.
Prove that if u ∈h2(B), u(0) = 0, and ∥u∥h2 ≤1, then |u(x)| ≤ s 1 + |x|2 (1 −|x|2)n−1 −1 for all x ∈B.
23.
Show that the bound on |(∇u)(0)| given by Theorem 6.26 can fail if the requirement that u be real valued is dropped.
24.
Show that U3(|x|N) = 1 |x| " 1 −1 −|x|2 p 1 + |x|2 # and U4(|x|N) = 2 π (1 + |x|2)2 arctan |x| −|x|(1 −|x|2) |x|2(1 + |x|2) .
(Hint: Evaluate the Poisson integrals that define U3(|x|N) and U4(|x|N), using an appropriate result from Appendix A. Be pre-pared for some hard calculus.) 25.
Suppose u is harmonic on B and P∞ m=0 pm is the homogeneous expansion of u about 0. Prove that ∥u∥h2 = ∞ X m=0 Z S |pm|2dσ 1/2 .
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Exercises 141 26.
Schwarz Lemma for h2-functions: Prove that if u is harmonic on B and ∥u∥h2 ≤1, then |∇u(0)| ≤√n. Find all functions for which this inequality is an equality.
27.
For a smooth function u on B, we define the radial derivative DRu by setting DRu(x) = x · ∇u(x) for x ∈B. Show that there exist positive constants c and C, depending only on the dimension n, such that c∥u∥h2 ≤|u(0)| + Z B |DRu(x)|2(1 −|x|) dV(x) 1/2 ≤C∥u∥h2 for all u harmonic on B. (Hint: Use the homogeneous expansion of u, Exercise 29 in Chapter 1, and polar coordinates.) 28.
(a) Find a measure µ ∈M(S) with M[µ] ∉L1(S).
(b) Can the measure µ in part (a) be chosen to be absolutely continuous with respect to σ?
29.
Let µ ∈M(S). Show that if lim δ→0 µ κ(ζ, δ) σ κ(ζ, δ) = L ∈C, then limr→1 Pµ = L. (Suggestion: Without loss of general-ity, ζ = N. For η near N, approximate P(rN, η) as in the proof of Theorem 6.31.) 30.
Let f ∈L1(S). A point ζ ∈S is called a Lebesgue point of f if lim δ→0 1 σ κ(ζ, δ) Z κ(ζ,δ) |f −f (ζ)| dσ = 0.
Show that almost every ζ ∈S is a Lebesgue point of f . (Hint: Imitate the proof of Theorem 6.39.) 31.
For u a function on B, let u∗(ζ) denote the nontangential limit of u at ζ ∈S, provided this limit exists. Show that if 1 < p ≤∞ and u ∈hp(B), then u = P[u∗], while this need not hold if u ∈h1(B).
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 142 Chapter 6. Harmonic Hardy Spaces 32.
Let f (z) = e(1+z)/(1−z) for z ∈B2. Show that the holomorphic function f has a nontangential limit with absolute value 1 at almost every point of S, even though f is unbounded on B2. Ex-plain why this does not contradict hp-theory.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Chapter 7 Harmonic Functions on Half-Spaces In this chapter we study harmonic functions defined on the upper half-space H. Harmonic function theory on H has a distinctly differ-ent flavor from that on B. One advantage of H over B is the dilation-invariance of H. We have already put this to good use in the section Limits Along Rays in Chapter 2. Some disadvantages that we will need to work around: ∂H is not compact, and Lebesgue measure on ∂H is not finite.
Recall that we identify Rn with Rn−1 × R, writing a typical point z ∈Rn as z = (x, y), where x ∈Rn−1 and y ∈R. The upper half-space H = Hn is the set H = {(x, y) ∈Rn : y > 0}.
We identify Rn−1 with Rn−1 × {0}; with this convention we then have ∂H = Rn−1.
For u a function on H and y > 0, we let uy denote the function on Rn−1 defined by uy(x) = u(x, y).
The functions uy play the same role on the upper half-space that the dilations play on the ball.
143 144 Chapter 7. Harmonic Functions on Half-Spaces The Poisson Kernel for the Upper Half-Space We seek a function PH on H × Rn−1 analogous to the Poisson kernel for the ball. Thus, for each fixed t ∈Rn−1, we would like PH(·, t) to be a positive harmonic function on H having the appropriate approximate-identity properties (see 1.20).
Fix t = 0 temporarily; we will concentrate first on finding PH(·, 0).
Taking our cue from Theorem 6.19, we look for a positive harmonic function on H that extends continuously to H \ {0} with boundary val-ues 0 on Rn−1 \ {0}. One such function is u(x, y) = y, but obviously this is not what we want—u doesn’t “blow up” at 0 as we know PH(·, 0) should. On the other hand, u does blow up at ∞. Applying the Kelvin transform, we can move the singularity of u from ∞to 0 and arrive at the desired function.
Thus, with u(x, y) = y, let us define v = K[u] on H, where K is the Kelvin transform introduced in Chapter 4.
A simple computation shows that v(x, y) = y (|x|2 + y2)n/2 for all (x, y) ∈H.
Because the inversion map preserves the upper half-space and the Kelvin transform preserves harmonic functions, we know without any computation that v is a positive harmonic function on H.
The function v has the property that vy(x) = y−(n−1)v1(x/y) for all (x, y) ∈H. Therefore the change of variables x 7→yx′ shows that R Rn−1 vy(x)dx is the same for all y > 0. (Here dx denotes Lebesgue measure on Rn−1.) Because R Rn−1 v1(x) dx < ∞(verify using polar coordinates—see 1.5), there exists a positive constant cn such that cn Z Rn−1 vy(x) dx = cn Z Rn−1 y (|x|2 + y2)n/2 dx = 1 for all y > 0. We will show that cn = 2/ nV(Bn) at the end of this section.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey The Poisson Kernel for the Upper Half-Space 145 The function cnv has all the properties we sought for PH(·, 0). To obtain PH(·, t), we simply translate cnv by t. Thus we now make our official definition: for z = (x, y) ∈H and t ∈Rn−1, set PH(z, t) = cn y (|x −t|2 + y2)n/2 .
The function PH is called the Poisson kernel for the upper half-space.
Note that PH can be written as PH(z, t) = cn y |z −t|n .
In this form, PH reminds us of the Poisson kernel for the ball.
(If (x, y) ∈H, then y is the distance from (x, y) to ∂H; analogously, the numerator of the Poisson kernel for B is roughly the distance to ∂B.) The work above shows that PH(·, t) is positive and harmonic on H for each t ∈Rn−1. We have also seen that 7.1 Z Rn−1 PH(z, t) dt = 1 for each z ∈H.
The next result gives the remaining approximate-identity property that we need to solve the Dirichlet problem for H.
7.2 Proposition: For every a ∈Rn−1 and every δ > 0, Z |t−a|>δ PH(z, t) dt →0 as z →a.
We leave the proof of Proposition 7.2 to the reader; it follows without difficulty from the definition of PH.
Let us now evaluate the normalizing constant cn. We accomplish this with a slightly underhanded trick: 1 cn = 2 π Z ∞ 0 1 1 + y2 Z Rn−1 y (|x|2 + y2)n/2 dx dy = 2 π Z H y (1 + y2)(|x|2 + y2)n/2 dx dy = 2nV(B) π Z S+ Z ∞ 0 ζn 1 + (rζn)2 dr dσ(ζ) = nV(B)/2, Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 146 Chapter 7. Harmonic Functions on Half-Spaces where the third equality is obtained by switching to polar coordinates (see 1.5) and S+ denotes the upper half-sphere.
The Dirichlet Problem for the Upper Half-Space For µ a complex Borel measure on Rn−1, the Poisson integral of µ, denoted by PH[µ], is the function on H defined by PHµ = Z Rn−1 PH(z, t) dµ(t).
We can verify that PH[µ] is harmonic on H by differentiating under the integral sign, or by noting that PH[µ] satisfies the volume version of the mean-value property on H.
We let M(Rn−1) denote the set of complex Borel measures on Rn−1.
With the total variation norm ∥∥, the Banach space M(Rn−1) is the dual space of C0(Rn−1), the space of continuous functions f on Rn−1 that vanish at ∞(equipped with the supremum norm).
For 1 ≤p < ∞, Lp(Rn−1) denotes the space of Borel measurable functions f on Rn−1 for which ∥f ∥p = Z Rn−1 |f (x)|p dx 1/p < ∞; L∞(Rn−1) consists of the Borel measurable functions f on Rn−1 for which ∥f ∥∞< ∞, where ∥f ∥∞denotes the essential supremum norm on Rn−1 with respect to Lebesgue measure.
Recall that on S, if p > q then Lp(S) ⊂Lq(S). On Rn−1, if p ̸= q then neither of the spaces Lp(Rn−1), Lq(Rn−1) contains the other. The reader should keep this in mind as we develop Poisson-integral theory in this new setting.
The Poisson integral of f ∈Lp(Rn−1), for any p ∈[1, ∞], is the function PH[f ] on H defined by PHf = Z Rn−1 f (t)PH(z, t) dt.
Because PH(z, ·) belongs to Lq(Rn−1) for every q ∈[1, ∞], the inte-gral above is well-defined for every z ∈H (by Hölder’s inequality). An Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey The Dirichlet Problem for the Upper Half-Space 147 argument like the one given for PH[µ] shows that PH[f ] is harmonic on H.
We now prove a result that the reader has surely already guessed.
7.3 Solution of the Dirichlet Problem for H: Suppose f is continu-ous and bounded on Rn−1. Define u on H by u(z) = ( PHf if z ∈H f (z) if z ∈Rn−1.
Then u is continuous on H and harmonic on H. Moreover, |u| ≤∥f ∥∞ on H.
Proof: The estimate |u| ≤∥f∥∞on H is immediate from 7.1. We already know that u is harmonic on H.
The proof that u is continuous on H is like that of Theorem 1.17.
Specifically, let a ∈Rn−1 and δ > 0. Then |u(z) −f (a)| = Z Rn−1 f (t) −f (a) PH(z, t) dt ≤ Z |t−a|≤δ |f (t) −f (a)| PH(z, t) dt + 2∥f∥∞ Z |t−a|>δ PH(z, t) dt for all z ∈H. If δ is small, the integral over {|t −a| ≤δ} will be small by the continuity of f at a and 7.1. The integral over {|t −a| > δ} approaches 0 as z →a by Proposition 7.2.
In the special case where f is uniformly continuous on Rn−1, we can make a stronger assertion: 7.4 Theorem: If f is bounded and uniformly continuous on Rn−1 and u = PH[f ], then uy →f uniformly on Rn−1 as y →0.
Proof: The uniform continuity of f on Rn−1 shows that the esti-mates in the proof of Theorem 7.3 can be made uniformly in a.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 148 Chapter 7. Harmonic Functions on Half-Spaces See Exercise 4 of this chapter for a converse to Theorem 7.4.
The next result follows immediately from Corollary 2.2 and Theo-rem 7.3; we state it as a theorem because of its importance.
7.5 Theorem: Suppose u is a continuous bounded function on H that is harmonic on H. Then u is the Poisson integral of its boundary values.
More precisely, u = PH[u|Rn−1] on H.
We now take up the more general Poisson integrals defined earlier.
Certain statements and proofs closely parallel those in the last chapter; we will be brief about details in these cases.
7.6 Theorem: The following growth estimates apply to Poisson inte-grals: (a) If µ ∈M(Rn−1) and u = PH[µ], then ∥uy∥1 ≤∥µ∥for all y > 0.
(b) Suppose 1 ≤p ≤∞.
If f ∈Lp(Rn−1) and u = PH[f ], then ∥uy∥p ≤∥f∥p for all y > 0.
Proof: The identity 7.7 PH (x, y), t = PH (t, y), x , valid for all x, t ∈Rn−1 and y > 0, is the replacement for 6.5 in this context. The rest of the proof is the same as that of Theorem 6.4.
The next result is the upper half-space analogue of Theorem 6.7.
Here the noncompactness of ∂H = Rn−1 forces us to do a little extra work.
7.8 Theorem: Suppose that 1 ≤p < ∞.
If f ∈Lp(Rn−1) and u = PH[f ], then ∥uy −f ∥p →0 as y →0.
Proof: We first prove the theorem for f ∈Cc(Rn−1), the set of con-tinuous functions on Rn−1 with compact support. Because Cc(Rn−1) is dense in Lp(Rn−1) for 1 ≤p < ∞, the approximation argument used in proving Theorem 6.7 (together with Theorem 7.6) will finish the proof.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey The Dirichlet Problem for the Upper Half-Space 149 Let f ∈Cc(Rn−1), and set u = PH[f ]. Choose a ball B(0, R) that contains the support of f. Because f is uniformly continuous on Rn−1, Theorem 7.4 implies that uy →f uniformly on Rn−1 as y →0. Thus to show that ∥uy −f ∥p →0, we need only show that 7.9 Z |x|>2R |uy(x)|p dx →0 as y →0.
For |x| > 2R, we have |uy(x)|p ≤ Z |t|<R |f (t)|p cny (|x −t|2 + y2)n/2 dt ≤ Cy (|x| −R)n , where C = cn∥f ∥p ∞Vn−1 B(0, R) and the first inequality follows from Jensen’s inequality. It is now easy to verify 7.9 by integrating in polar coordinates (1.5).
As in the last chapter, weak convergence replaces norm conver-gence for Poisson integrals of measures and L∞-functions.
7.10 Theorem: Poisson integrals have the following weak conver-gence properties: (a) If µ ∈M(Rn−1) and u = PH[µ], then uy →µ weak in M(Rn−1) as y →0.
(b) If f ∈L∞(Rn−1) and u = PH[f ], then uy →f weak in L∞(Rn−1) as y →0.
Proof: The Banach spaces M(Rn−1) and L∞(Rn−1) are, respectively, the dual spaces of C0(Rn−1) and L1(Rn−1). Note that if g ∈C0(Rn−1), then g is uniformly continuous on Rn−1, and therefore (PH[g])y →g uniformly on Rn−1 as y →0 (by Theorem 7.4). The proof of Theo-rem 6.9 can thus be used here, essentially verbatim. Again, the identity 7.7 replaces 6.5.
If f is continuous and bounded on Rn−1, then there is a function on the closed half-space H that is harmonic on H and agrees with f on the boundary Rn−1; see 7.3. What happens if we drop the assumption that Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 150 Chapter 7. Harmonic Functions on Half-Spaces f is bounded? Without any growth estimate on f , we cannot expect to find a solution to the Dirichlet problem with boundary data f by integrating f against some kernel as in 7.3. Nevertheless, the following theorem asserts that a solution exists with just the assumption that f is continuous. We are not asserting any sort of uniqueness for the solution, because a multiple of y can be added to any solution to obtain another solution.
7.11 Theorem: Suppose f ∈C(Rn−1). Then there exists u ∈C(H) such that u is harmonic on H and u|Rn−1 = f .
Proof: We will construct a sequence of functions u0, u1, . . . in C(H) such that for each k the following hold: (a) uk is harmonic on H; (b) uk(x, 0) = f (x) for all x ∈Rn−1 with |x| ≤k; (c) |(uk+1 −uk)(x, y)| < 1 2k for all (x, y) ∈H with |(x, y)| ≤k/2.
This will prove the theorem, because (c) implies that the sequence (uk) converges uniformly on each compact subset of H to a function u ∈C(H); from (a) and Theorem 1.23 we have that u is harmonic on H; from (b) we have that u|Rn−1 = f.
We construct the sequence (uk) inductively, starting by taking u0 to be the constant function whose value is f (0). Now fix k and suppose that we have uk ∈C(H) satisfying (a) and (b) above. To construct uk+1, let w ∈C(H) be such that w is harmonic on H and w(x, 0) = f (x) for all x ∈Rn−1 with |x| ≤k + 1 (to see that such a w exists, extend f|(k+1)Bn−1 to a bounded continuous function on Rn−1 and then use 7.3).
Now (w −uk)(x, 0) = 0 for all x ∈Rn−1 with |x| ≤k. Thus by the Schwarz reflection principle (4.12), (w −uk)|kB∩H extends to a harmonic function v on kB. The proof of 4.12 shows that 7.12 v(x, y) = −v(x, −y) for all (x, y) ∈kB.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey The Harmonic Hardy Spaces hp(H) 151 The expansion of v into an infinite sum of homogeneous harmonic polynomials converges uniformly to v on (k/2)B (see 5.34). Taking an appropriate partial sum, we conclude that there is a harmonic polyno-mial p such that |(v −p)(x, y)| < 1 2k for all (x, y) ∈(k/2)B. Note that p(x, 0) = 0 for all x ∈Rn−1, because 7.12 implies that the power series expansion of v contains only odd powers of y.
Now let uk+1 = w −p|H. Then uk+1 ∈C(H) and uk+1 is harmonic on H. Furthermore, uk+1(x, 0) = f (x) for all x ∈Rn−1 with |x| ≤k+1.
Finally, if (x, y) ∈H with |(x, y)| ≤k/2, then |(uk+1 −uk)(x, y)| = |(w −p −uk)(x, y)| = |(v −p)(x, y)| < 1 2k and thus uk+1 has all the desired properties.
The Harmonic Hardy Spaces hp(H) For p ∈[1, ∞], we define the harmonic Hardy space hp(H) to be the normed vector space of functions u harmonic on H for which ∥u∥hp = sup y>0 ∥uy∥p < ∞.
Note that h∞(H) is simply the collection of bounded harmonic func-tions on H, and that ∥u∥h∞= sup z∈H |u(z)|.
We leave it to the reader to verify that hp(H) is a normed linear space under the norm ∥∥hp.
As the reader should suspect by considering what happens in the ball, if u ∈hp(H) then the norms ∥uy∥p increase as y →0 to ∥u∥hp.
To prove this, we need to do some extra work because of the noncom-pactness of ∂H = Rn−1. We begin with the following result.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 152 Chapter 7. Harmonic Functions on Half-Spaces 7.13 Theorem: Let p ∈[1, ∞). Then there exists a constant C, de-pending only on p and n, such that |u(x, y)| ≤C∥u∥hp y(n−1)/p for all u ∈hp(H) and all (x, y) ∈H. In particular, every u ∈hp(H) is bounded on H + (0, y) for each y > 0.
Proof: Let (x0, y0) ∈H, and let ω denote the open ball in Rn with center (x0, y0) and radius y0/2. The volume version of the mean-value property, together with Jensen’s inequality, shows that |u(x0, y0)|p ≤ 1 Vn(ω) Z ω |u|p dVn 7.14 = 2n Vn(B)y0n Z ω |u|p dVn.
Setting Ω= {(x, y) ∈H : y0/2 < y < 3y0/2}, we have Z ω |u|p dVn ≤ Z Ω |u|p dVn = Z 3y0/2 y0/2 Z Rn−1 |u(x, y)|p dx dy ≤y0(∥u∥hp)p.
This estimate and 7.14 give the conclusion of the theorem after taking pth roots.
Theorems 7.5 and 7.13 show that if p ∈[1, ∞] and u ∈hp(H), then for each y > 0 we have 7.15 u z + (0, y) = PHuy for all z ∈H.
The next corollary is not entirely analogous to Corollary 6.6 because the conclusion that ∥uy∥p increases as y decreases is not true for an arbitrary harmonic function on H. For example, if u(x, y) = y −1, then ∥u1∥p = 0 while ∥uy∥p = ∞for all y ̸= 1. Thus we have the hypothesis in the next corollary that u ∈hp(H).
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey From the Ball to the Upper Half-Space, and Back 153 7.16 Corollary: Suppose 1 ≤p ≤∞and u ∈hp(H). Then ∥uy2∥p ≤∥uy1∥p whenever 0 < y1 ≤y2. Furthermore, ∥u∥hp = lim y→0 ∥uy∥p.
Proof: The idea is the same as in the proof of Corollary 6.6. Specif-ically, if 0 < y1 ≤y2 then ∥uy2∥p = ∥P[uy1]y2−y1∥p ≤∥uy1∥p, where the equality follows from 7.15 and the inequality follows from Theorem 7.6(b).
The formula for ∥u∥hp now immediately follows from the definition of ∥u∥hp and the first part of the corollary.
The next theorem is the analogue for the half-space of Theorem 6.13 for the ball. The results we have proved so far in this chapter allow the proof from the ball to carry over directly to the half-space, as the reader should verify.
7.17 Theorem: The Poisson integral induces the following surjective isometries: (a) The map µ 7→PH[µ] is a linear isometry of M(Rn−1) onto h1(H).
(b) For 1 < p ≤∞, the map f 7→PH[f ] is a linear isometry of Lp(Rn−1) onto hp(H).
From the Ball to the Upper Half-Space, and Back Recall the inversion map x 7→x∗defined in Chapter 4. This map takes spheres containing 0 onto hyperplanes, and takes the interiors of such spheres onto open half-spaces. Composing the inversion map with appropriate translations and dilations will give us a one-to-one Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 154 Chapter 7. Harmonic Functions on Half-Spaces map of B onto H. There are many such maps; the one we choose below has the advantage of being its own inverse under composition.
Let N = (0, 1) and S = (0, −1) (here 0 denotes the origin in Rn−1); we can think of N and S as the north and south poles of the unit sphere S.
Now define Φ: Rn \ {S} →Rn \ {S} by Φ(z) = 2(z −S)∗+ S.
It is easy to see that Φ is a one-to-one map of Rn \ {S} onto itself. We can regard Φ as a homeomorphism of Rn ∪{∞} onto itself by defining Φ(S) = ∞and Φ(∞) = S. The reader may find it helpful to keep the following diagram in mind as we proceed.
N S N 0 0 ∞ Φ Φ maps B onto H and H onto B.
The next result summarizes the basic properties of Φ.
7.18 Proposition: The map Φ has the following properties: (a) Φ Φ(z) = z for all z ∈Rn ∪{∞}; (b) Φ is a conformal, one-to-one map of Rn \ {S} onto Rn \ {S}; (c) Φ maps B onto H and H onto B; (d) Φ maps S \ {S} onto Rn−1 and Rn−1 onto S \ {S}.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey From the Ball to the Upper Half-Space, and Back 155 Proof: The proof of (a) is a simple computation.
In (b), only conformality needs to be checked. Recalling that the inversion map is conformal (Proposition 4.2), we see that Φ is the com-position of conformal maps, and hence is itself conformal.
We prove (c) and (d) together. Noting that Φ(S) = ∞, we know that Φ maps S \ {S} onto some hyperplane. Because the inversion map pre-serves the (0, y)-axis, the same is true Φ. The conformality of Φ thus shows that Φ(S \ {S}) is a hyperplane perpendicular to the (0, y)-axis.
Since Φ(N) = 0, we must have Φ(S \ {S}) = Rn−1. It follows that Φ(B) is either the upper or lower half-space. Because Φ(0) = N, we have Φ(B) = H, as desired.
We now introduce a modified Kelvin transform K that will take har-monic functions on B to harmonic functions on H and vice-versa. Given any function u defined on a set E ⊂Rn \ {S}, we define the function K[u] on Φ(E) by Ku = 2(n−2)/2|z −S|2−nu Φ(z) .
Note that when n = 2, Ku = u Φ(z) .
The factor 2(n−2)/2 is included so that K will be its own inverse.
That is, we claim K K[u] = u for all u as above, a computation we leave to the reader.
The transform K is linear—if u, v are functions on E and b, c are constants, then K[bu + cv] = bK[u] + cK[v] on Φ(E).
Finally, K preserves harmonicity. The real work for the proof of this was done when we proved Theorem 4.7.
7.19 Proposition: If Ω⊂Rn \ {S}, then u is harmonic on Ωif and only if K[u] is harmonic on Φ(Ω).
Proof: Because K is its own inverse, it suffices to prove only one direction of the theorem. So suppose that u is harmonic on Ω. Define Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 156 Chapter 7. Harmonic Functions on Half-Spaces a harmonic function v on 1 2(Ω−S) by v(z) = u(2z + S). By Theo-rem 4.7, the Kelvin transform K[v] is harmonic on 2(Ω−S)∗, and thus Kv is harmonic on 2(Ω−S)∗+ S = Φ(Ω). But, as is easily checked, Kv = 2(2−n)/2Ku, so that K[u] is harmonic on Φ(Ω), as desired.
Positive Harmonic Functions on the Upper Half-Space Because the modified Kelvin transform K takes positive functions to positive functions, Proposition 7.19 shows that K preserves the class of positive harmonic functions. Thus u is positive and harmonic on H if and only if K[u] is positive and harmonic on B. This will allow us to transfer our knowledge about positive harmonic functions on the ball to the upper half-space. For example, we can now prove an analogue of Theorem 6.19 for the upper half-space.
7.20 Theorem: Let t ∈Rn−1. Suppose that u is positive and harmonic on H, and that u extends continuously to H {t} with boundary values 0 on Rn−1 \ {t}. Suppose further that 7.21 u(0, y) y →0 as y →∞.
Then u = cPH(·, t) for some positive constant c.
Proof: The function K[u] is positive and harmonic on B. Thus by 6.19, K[u] = P[µ] for some positive µ ∈M(S), where as usual P denotes the Poisson kernel for the ball. Our hypothesis on u implies that K[u] extends continuously to B \ {S, Φ(t)}, with boundary values 0 on S \ {S, Φ(t)}.
The argument used in proving Theorem 6.19 then shows that µ is the sum of point masses at S and Φ(t).
An easy computation gives Ku = 2(n−2)/2(1 −r)2−nu 0, 1 + r 1 −r Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Positive Harmonic Functions on the Upper Half-Space 157 for every r ∈[0, 1). Now from 7.21 we see that (1−r)n−1Ku →0 as r →1, and this implies µ({S}) = 0 (see Exercise 3 in Chapter 6).
Thus µ is a point mass at Φ(t), and therefore K[u] is a constant times P ·, Φ(t) . Because PH(·, t) also satisfies the hypotheses of The-orem 7.20, K[PH(·, t)] is a constant times P ·, Φ(t) as well. Thus K[u] = cK[PH(·, t)] for some positive constant c.
Applying K to both sides of the last equation, we see that the linearity of K gives the conclusion of the theorem.
We can think of the next result as the “t = ∞” case of Theorem 7.20.
7.22 Theorem: Suppose that u is positive and harmonic on H and that u extends continuously to H with boundary values 0 on Rn−1. Then there exists a positive constant c such that u(x, y) = cy for all (x, y) ∈H.
Proof: The function K[u] is positive and harmonic on B, extends continuously to B \ {S}, and has boundary values 0 on S \ {S}. By The-orem 6.19, K[u] is a constant times P(·, S). Because the same is true of K[v], where v(x, y) = y on H, K[u] is a constant times K[v]. As in the proof of the last theorem, this gives us the desired conclusion.
The modified Kelvin transform K allows us to derive the relation between P and PH, the Poisson kernels for B and H, with a minimum of computation.
7.23 Theorem: PH(z, t) = 2n−2cn(1 + |t|2)−n/2|z −S|2−nP Φ(z), Φ(t) for all z ∈H and t ∈Rn−1.
Proof: Fix t ∈Rn−1, and let u(z) denote the right side of the equa-tion above that we want to prove. Then u is positive and harmonic on H, and it is easy to check that u extends continuously to H \ {t} with boundary values 0 on Rn−1 \ {t}. We also see that u(0, y)/y →0 (with plenty of room to spare) as y →∞. Thus by Theorem 7.20, u is a con-stant multiple of PH(·, t). Evaluating at z = N now gives the desired result.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 158 Chapter 7. Harmonic Functions on Half-Spaces We turn now to the problem of characterizing the positive harmonic functions on H. We know that if µ is a finite positive Borel measure on Rn−1, then PH[µ] is a positive harmonic function on H. Unlike the case for the ball, however, not all positive harmonic functions on H arise in this manner. In the first place, PH[µ] defines a positive harmonic func-tion on H for some positive measures µ that are not finite—Lebesgue measure on Rn−1, for example. Secondly, the positive harmonic func-tion y is not the Poisson integral of anything that lives on Rn−1.
Let us note that if µ is any positive Borel measure on Rn−1, then 7.24 PHµ = Z Rn−1 PH(z, t) dµ(t) is well-defined as a number in [0, ∞] for every z ∈H. We claim that 7.24 defines a positive harmonic function on H precisely when 7.25 Z Rn−1 dµ(t) (1 + |t|2)n/2 < ∞.
To see this, note that if z ∈H is fixed, then PH(z, t), as a function of t, is bounded above and below by positive constant multiples of (1 + |t|2)−n/2. Thus if 7.24 is finite for some z ∈H, then it is finite for all z ∈H, and this happens exactly when 7.25 occurs. In this case PH[µ] is harmonic on H, as can be verified by checking the volume version of the mean-value property.
We now state the main result of this section.
7.26 Theorem: If u is positive and harmonic on H, then there exists a positive Borel measure µ on Rn−1 and a nonnegative constant c such that u(x, y) = PHµ + cy for all (x, y) ∈H.
The main idea in the proof of this result is the observation that if u is positive and harmonic on H, then K[u] is positive and harmonic on B, and hence is the Poisson integral of a positive measure on S. The restriction of this measure to S \ {S} gives rise to the measure µ, and the mass of this measure at S gives rise to the term cy.
Before coming to the proof of Theorem 7.26 proper, we need to understand how measures on S pull back, via the map Φ, to measures Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Positive Harmonic Functions on the Upper Half-Space 159 on Rn−1. For any positive ν ∈M(S), we can define a positive measure ν ◦Φ ∈M(Rn−1) by setting (ν ◦Φ)(E) = ν Φ(E) for every Borel set E ⊂Rn−1. We then have the following “change of variables formula”, valid for every positive Borel measurable function f on S \ {S}: 7.27 Z S{S} f dν = Z Rn−1(f ◦Φ) d(ν ◦Φ).
The last equation is easy to verify when f is a simple function on S{S}; the full result follows from this by the monotone convergence theo-rem.
Proof of Theorem 7.26: If u is positive and harmonic on H, then K[u] is positive and harmonic on B, and thus K[u] = P[λ] for some positive measure λ ∈M(S). Define ν ∈M(S) by dν = χS{S} dλ. We then have K[u] = P[ν] + λ({S})P(·, S).
By the linearity of K, u = K P[ν] + λ({S})K[P(·, S)].
From Theorem 7.22 it is easy to see that K[P(·, S)] is a constant multi-ple of y on H. The proof will be completed by showing that K P[ν] = PH[µ] for some positive Borel measure µ on Rn−1.
Because ν({S}) = 0, Pν = Z S{S} P(z, ζ) dν(ζ) for all z ∈B. Thus by 7.27, K P[ν] (z) = Z S{S} 2(n−2)/2|z −S|2−nP Φ(z), ζ dν(ζ) = Z Rn−1 2(n−2)/2|z −S|2−nP Φ(z), Φ(t) d(ν ◦Φ)(t) for every z ∈H. In the last integral we may multiply and divide by ψ(t), where ψ(t) = 2(n−2)/2cn(1+|t|2)−n/2. With dµ = (1/ψ) d(ν ◦Φ), we then have K P[ν] = PH[µ], as desired.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 160 Chapter 7. Harmonic Functions on Half-Spaces Nontangential Limits We now look briefly at the Fatou Theorem for the Poisson integrals discussed in this chapter. Rather than tediously verifying that the maxi-mal function arguments of the last chapter carry through to the present setting, we use the modified Kelvin transform K to transfer the Fatou Theorem from B to H.
The notion of a nontangential limit for a function on H was defined in Chapter 2; the analogous definition for a function on B was given in Chapter 6. We leave it to the reader to verify the following asser-tion, which follows from the conformality of the map Φ: a function u on H has a nontangential limit at t ∈Rn−1 if and only if K[u] has a nontangential limit (within B) at Φ(t).
Another observation that we leave to the reader is that the map Φ preserves sets of measure zero. More precisely, a Borel set E ⊂Rn−1 has Lebesgue measure 0 if and only if Φ(E) has σ-measure 0 on S; this follows easily from the smoothness of Φ.
In this chapter, the term “almost everywhere” will refer to Lebesgue measure on Rn−1. Putting the last two observations together, we see that a function u on H has nontangential limits almost everywhere on Rn−1 if and only if K[u] has nontangential limits σ-almost every-where on S.
The next result is the Fatou theorem for Poisson integrals of func-tions in Lp(Rn−1).
7.28 Theorem: Let p ∈[1, ∞]. If f ∈Lp(Rn−1), then PH[f ] has nontangential limit f (x) at almost every x ∈Rn−1.
Proof: Because every real-valued function in Lp(Rn−1) is the dif-ference of two positive functions in Lp(Rn−1), we may assume that f ≥0. The function u = PH[f ] is then positive and harmonic on H, and thus K[u] is positive and harmonic on B. By 6.15 and 6.44, K[u] has nontangential limits σ-almost everywhere on S. As observed ear-lier, this implies that u has a nontangential limit g(x) for almost every x ∈Rn−1.
We need to verify that f = g almost everywhere. For p < ∞, The-orem 7.8 asserts that ∥uy −f ∥p →0 as y →0; thus some subse-quence (uyk) converges to f pointwise almost everywhere on Rn−1, and hence f = g. For p = ∞, Theorem 7.10(b) shows that uy →f weak Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey The Local Fatou Theorem 161 in L∞(Rn−1) as y →0. But we also have uy →g weak in L∞(Rn−1) by the dominated convergence theorem, and so we conclude f = g.
The theorem above shows, by Theorem 7.17, that if u ∈hp(H) and p ∈(1, ∞], then u has nontangential limits almost everywhere on Rn−1.
The next theorem gives us the same result for h1(H) as a corollary.
7.29 Theorem: Suppose µ ∈M(Rn−1) is singular with respect to Lebesgue measure. Then PH[µ] has nontangential limit 0 almost ev-erywhere on Rn−1.
Proof: We may assume that µ is positive. By analogy with 7.27, we define µ◦Φ ∈M(S) by setting (µ◦Φ)(E) = µ Φ(E{S}) for every Borel set E ⊂S. We then have K P[µ ◦Φ] = PH[ν], where dν = (1/ψ) dµ and ψ is as in the proof of 7.26. Because µ is singular with respect to Lebesgue measure on Rn−1, µ ◦Φ is singular with respect to σ. By 6.42, P[µ ◦Φ] has nontangential limit 0 almost everywhere on S. The equation above tells us that PH[ν] has nontan-gential limit 0 almost everywhere on Rn−1. From this we easily deduce that PH[µ] has nontangential limit 0 almost everywhere on Rn−1.
The Local Fatou Theorem The Fatou Theorems obtained so far in this book apply to Poisson integrals of functions or measures. In this section we prove a different kind of Fatou theorem—one that applies to arbitrary harmonic func-tions on H satisfying a certain local boundedness condition.
We will need to consider truncations of the cones Γα(a) defined in Chapter 2. Thus, for any h > 0, we define Γ h α (a) = {(x, y) ∈H : |x −a| < αy and y < h}.
A function u on H is said to be nontangentially bounded at a ∈Rn−1 if u is bounded on some Γ h α (a). Note that if u is continuous on H, then u is nontangentially bounded at a if and only if u is bounded on Γ 1 α(a) for some α > 0. We can now state the main result of this section.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 162 Chapter 7. Harmonic Functions on Half-Spaces The truncated cone Γ h α (a).
7.30 Local Fatou Theorem: Suppose that u is harmonic on H and E ⊂Rn−1 is the set of points at which u is nontangentially bounded.
Then u has a nontangential limit at almost every point of E.
A remarkable feature of this theorem should be emphasized. For each a ∈E, we are only assuming that u is bounded in some Γ h α (a); in particular, α can depend on a. Nevertheless, the theorem asserts the existence of a set of full measure F ⊂E such that u has a limit in Γα(a) for every a ∈F and every α > 0.
The following lemma will be important in proving the Local Fatou Theorem. Figure 7.32 may be helpful in picturing the geometry of the region Ωmentioned in the next three lemmas.
7.31 Lemma: Suppose E ⊂Rn−1 is Borel measurable, α > 0, and Ω= [ a∈E Γ 1 α(a).
Then there exists a positive harmonic function v on H such that v ≥1 on (∂Ω) ∩H and such that v has nontangential limit 0 almost every-where on E.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey The Local Fatou Theorem 163 Proof: Define a positive harmonic function w on H by w(x, y) = PHχEc + y, where χEc denotes the characteristic function of Ec, the complement of E in Rn−1. By Theorem 7.28, w has nontangential limit 0 almost everywhere on E.
7.32 Ω= [ a∈E Γ 1 α(a).
We wish to show that w is bounded away from 0 on (∂Ω) ∩H. Be-cause w(x, 1) ≥1, we have w ≥1 on the “top” of ∂Ω. Next, observe that (x, y) belongs to Γα(a) if and only if a ∈B(x, αy) (where B(x, αy) de-notes the ball in Rn−1 with center x and radius αy). So if (x, y) ∈∂Ω and 0 < y < 1, then (x, y) ∉Γα(a) for all a ∈E (otherwise (x, y) ∈Ω), giving B(x, αy) ⊂Ec. Therefore PHχEc = Z Ec cny (|x −t|2 + y2)n/2 dt ≥ Z B(x,αy) cny (|x −t|2 + y2)n/2 dt = Z B(0,α) cn (|t|2 + 1)n/2 dt.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 164 Chapter 7. Harmonic Functions on Half-Spaces Denoting the last expression by cα (a constant less than 1 that depends only on α and n), we see that if v = w/cα, then v satisfies the conclu-sion of the lemma.
The crux of the proof of 7.30 is the following weaker version of the Local Fatou Theorem.
7.33 Lemma: Let E ⊂Rn−1 be Borel measurable, let α > 0, and let Ω= [ a∈E Γ 1 α(a).
Suppose u is harmonic on H and bounded on Ω. Then for almost every a ∈E, the limit of u(z) exists as z →a within Γα(a).
Proof: Because every Borel set can be written as a countable union of bounded Borel sets, we may assume E is bounded. We may also assume that u is real valued.
Because u is continuous on H and E is bounded, we may assume that |u| ≤1 on the open set Ω′ = [ a∈E Γ 2 α(a).
Choose a sequence (yk) in the interval (0, 1) such that yk →0, and set Ek = [Ω−(0, yk)] ∩Rn−1. Each Ek is an open subset of Rn−1 that contains E. (At this point we suggest the reader start drawing some pictures.) For x ∈Rn−1, define fk(x) = χEk(x)u(x, yk).
Because (x, yk) ∈Ωif and only if x ∈Ek, we have |fk| ≤1 on Rn−1 for every k. The sequence (fk), being norm-bounded in L∞(Rn−1), has a subsequence, which we still denote by (fk), that converges weak to some f ∈L∞(Rn−1).
Now each fk is continuous on Ek (because Ek is open), and thus PH[fk] extends continuously to H ∪Ek (see Exercise 17(a) of this chap-ter). The function uk given by uk(x, y) = PHfk −u(x, y + yk) Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey The Local Fatou Theorem 165 is thus harmonic on H and extends continuously to H ∪Ek, with uk = 0 on Ek. In particular, uk is continuous on Ωwith uk = 0 on E. Further-more, because Ω+ (0, yk) ⊂Ω′, we have |uk| ≤2 on Ω.
Now let v denote the function of Lemma 7.31 with respect to Ω.
Then lim infz→∂Ω(2v −uk)(z) ≥0. By the minimum principle (1.10), 2v −uk ≥0 on Ω. Letting k →∞, we then see that 2v−(PH[f ]−u) ≥0 on Ω. Because this argument applies as well to 2v + uk, we conclude that |PH[f ] −u| ≤2v on Ω.
By Theorem 7.28, PH[f ] has nontangential limits almost everywhere on Rn−1, while Lemma 7.31 asserts v has nontangential limits 0 almost everywhere on E. From this and the last inequality, the desired limits for u follow.
Recall that if E ⊂Rn−1 is Borel measurable, then a point a ∈E is said to be a point of density of E provided lim r→0 Vn−1 B(a, r) ∩E Vn−1 B(a, r) = 1.
By the Lebesgue Differentiation Theorem (, Theorem 7.7), almost every point of E is a point of density of E.
Points of density of E are where we can expect the cones defining Ω in Lemma 7.33 to “pile up”; this will allow us to pass from 7.33 to the stronger assertion in 7.30.
7.34 Lemma: Suppose E ⊂Rn−1 is Borel measurable, α > 0, and Ω= [ a∈E Γ 1 α(a).
Suppose u is continuous on H and bounded on Ω. If a is a point of density of E, then u is bounded in Γ 1 β(a) for every β > 0.
Proof: Let a be a point of density of E, and let β > 0. It suffices to show that Γ h β (a) ⊂Ωfor some h > 0.
Choose δ > 0 such that 7.35 Vn−1 B(a, r) ∩E Vn−1 B(a, r) > 1 − α α + β !n−1 whenever r < δ; we may assume δ/(α + β) < 1. Set h = δ/(α + β), and let (x, y) ∈Γ h β (a). Then B(x, αy) ⊂B(a, (α + β)y). This implies Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 166 Chapter 7. Harmonic Functions on Half-Spaces B(x, αy)∩E is nonempty; otherwise we violate 7.35 (take r = (α+β)y).
Choosing any b ∈B(x, αy) ∩E, we have (x, y) ∈Γ 1 α(b), and thus Γ h β (a) ⊂Ω, as desired.
Proof of Theorem 7.30: We are assuming u is harmonic on H and E is the set of points in Rn−1 at which u is nontangentially bounded.
For k = 1, 2, . . ., set Ek = {a ∈Rn−1 : |u| ≤k on Γ 1 1/k(a)}. Then each Ek is a closed subset of Rn−1, and E = S Ek (incidentally proving that the set E is Borel measurable). Applying Lemma 7.34 to each Ek, and recalling that the points of density of Ek form a set of full measure in Ek, we see that there is a set of full measure F ⊂E such that u is bounded on Γ 1 α(a) for every a ∈F and every α > 0. For each positive integer k, we can write F as F = S Fj, where Fj = {a ∈F : |u| ≤j on Γ 1 k (a)}.
Lemma 7.33, applied to Fj, now shows that u has nontangential limits almost everywhere on E, as desired.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Exercises 167 Exercises 1.
Assume n = 2. For each t ∈R, find a holomorphic function gt on H such that PH(·, t) = Re gt.
2.
In Chapter 1, we calculated P(x, ζ) as a normal derivative on ∂B of an appropriate modification of |x −ζ|2−n (n > 2). Using an appropriate modification of |z−t|2−n, find a function whose nor-mal derivative on ∂H is PH(z, t).
3.
Let µ ∈M(Rn−1) and let u = PH[µ]. Prove that Z Bn−1 uy(x) dx →µ(Bn−1) + µ(∂Bn−1) 2 as y →0.
4.
Let p ∈[1, ∞] and assume u ∈hp(H). Show that if the functions uy converge uniformly on Rn−1 as y →0, then u extends to a bounded uniformly continuous function on H.
5.
For ζ ∈S, show that Φ(ζ) = (ζ1, ζ2, . . . , ζn−1, 0) 1 + ζn .
6.
For (x, y) ∈Rn \ {S}, show that 1 −|Φ(x, y)|2 = 4y |x|2 + (y + 1)2 .
7.
Show that if n = 2, then Φ(z) = 1 −iz z −i for every z ∈C \ {−i}.
8.
Suppose ζ ∈S and f ∈C(S \ {ζ}).
Prove that there exists u ∈C(B \ {ζ}) such that u is harmonic on B and u|S{ζ} = f .
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 168 Chapter 7. Harmonic Functions on Half-Spaces 9.
Prove that Z S f dσ = cn2n−2 Z Rn−1 f Φ(t) (1 + |t|2)1−n dt for every positive Borel measurable function f on S. (Hint: Be-cause Φ: S{S} →Rn−1 is smooth, there exists a smooth function w on Rn−1 such that d(σ ◦Φ) = w dt. To find w, apply 7.27 with ν = σ.) 10.
Using the result of the last exercise, show that Z Rn−1 f (t) dt = 2 cn Z S f Φ(ζ) (ζn + 1)1−n dσ(ζ) for every positive Borel measurable function f on Rn−1.
11.
(a) Let µ be a positive Borel measure on Rn−1 that satisfies 7.25, and set u = PH[µ]. Show that limy→∞u(0, y)/y = 0.
(b) Let u be a positive harmonic function on H.
Show that lim infy→0 u(0, y)/y > 0.
12.
Show that if u is positive and harmonic on H, then the decompo-sition u(x, y) = PHµ + cy of Theorem 7.26 holds for a unique positive Borel measure µ on Rn−1 and a unique nonneg-ative constant c.
13.
Let µ be a positive Borel measure on Rn−1 that satisfies 7.25, and set u = PH[µ]. Prove that lim y→0 Z Rn−1 ϕ(t)uy(t) dt = Z Rn−1 ϕ(t) dµ(t) for every continuous function ϕ on Rn−1 with compact support.
14.
Prove that K[hp(H)] ⊂h1(B) for every p ∈[1, ∞]. (Hint: Exer-cise 9 in Chapter 6 may be helpful here.) 15.
Let p ∈[1, ∞] and let f ∈Lp(Rn−1). Show that K[f ] ∈L1(S).
16.
Let p ∈[1, ∞] and let f ∈Lp(Rn−1). Show that K h P K[f ] i (z) = PHf for every z ∈H.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Exercises 169 17.
Assume that f is measurable on Rn−1 and that Z Rn−1 |f (t)|(1 + |t|2)−n/2 dt < ∞.
(a) Show that if f is continuous at a, then PH[f ] →f (a) as z →a within H.
(b) Show that PH[f ] tends nontangentially to f almost every-where on Rn−1. (Hint: Let g denote f times the character-istic function of some large ball.
Apply Theorem 7.28 to PH[g]; apply part (a) to PH[f −g].) 18.
Let µ be a positive Borel measure on Rn−1 that satisfies 7.25.
Show that if µ is singular with respect to Lebesgue measure, then PH[µ] has nontangential limit 0 at almost every point of Rn−1.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Chapter 8 Harmonic Bergman Spaces Throughout this chapter, p denotes a number satisfying 1 ≤p < ∞.
The Bergman space bp(Ω) is the set of harmonic functions u on Ωsuch that ∥u∥bp = Z Ω |u|p dV 1/p < ∞.
We often view bp(Ω) as a subspace of Lp(Ω, dV). The spaces bp(Ω) are named in honor of Stefan Bergman, who studied analogous spaces of holomorphic functions.
Stefan Bergman (1895–1977), whose book popularized the study of spaces of holomorphic functions belonging to Lp with respect to volume measure.
171 172 Chapter 8. Harmonic Bergman Spaces Reproducing Kernels For fixed x ∈Ω, the map u 7→u(x) is a linear functional on bp(Ω); we refer to this map as point evaluation at x. The following proposition shows that point evaluation is continuous on bp(Ω).
8.1 Proposition: Suppose x ∈Ω. Then |u(x)| ≤ 1 V(B)1/pd(x, ∂Ω)n/p ∥u∥bp for every u ∈bp(Ω).
Proof: Let r be a positive number with r < d(x, ∂Ω), and apply the volume version of the mean-value property to u on B(x, r). After taking absolute values, Jensen’s inequality gives |u(x)|p ≤ 1 V B(x, r) Z B(x,r) |u|p dV ≤ 1 r nV(B)∥u∥p bp.
The desired inequality is now obtained by taking pth roots and letting r →d(x, ∂Ω).
The next result shows that point evaluation of every partial deriva-tive is also continuous of bp(Ω).
8.2 Corollary: For every multi-index α there exists a constant Cα such that |Dαu(x)| ≤ Cα d(x, ∂Ω)|α|+n/p ∥u∥bp for all x ∈Ωand every u ∈bp(Ω).
Proof: Apply 8.1 and Cauchy’s Estimates (2.4) to u on the ball of radius d(x, ∂Ω)/2 centered at x.
The next proposition implies that bp(Ω) is a Banach space.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Reproducing Kernels 173 8.3 Proposition: The Bergman space bp(Ω) is a closed subspace of Lp(Ω, dV).
Proof: Suppose uj →u in Lp(Ω, dV), where (uj) is a sequence in bp(Ω) and u ∈Lp(Ω, dV). We must show that, after appropriate modification on a set of measure zero, u is harmonic on Ω.
Let K ⊂Ωbe compact. By Proposition 8.1, there is a constant C < ∞ such that |uj(x) −uk(x)| ≤C∥uj −uk∥bp for all x ∈K and all j, k. Because (uj) is a Cauchy sequence in bp(Ω), the inequality above implies that (uj) is a Cauchy sequence in C(K).
Hence (uj) converges uniformly on K.
Thus (uj) converges uniformly on compact subsets of Ωto a func-tion v that is harmonic on Ω(Theorem 1.23).
Because uj →u in Lp(Ω, dV), some subsequence of (uj) converges to u pointwise almost everywhere on Ω. It follows that u = v almost everywhere on Ω, and thus u ∈bp(Ω), as desired.
Taking p = 2, we see that the last proposition shows that b2(Ω) is a Hilbert space with inner product ⟨u, v⟩= Z Ω uv dV.
For each x ∈Ω, the map u 7→u(x) is a bounded linear functional on the Hilbert space b2(Ω) (by Proposition 8.1). Thus there exists a unique function RΩ(x, ·) ∈b2(Ω) such that u(x) = Z Ω u(y)RΩ(x, y) dV(y) for every u ∈b2(Ω). The function RΩ, which can be viewed as a func-tion on Ω× Ω, is called the reproducing kernel of Ω.
The basic properties of RΩgiven below are analogous to properties of the zonal harmonics we studied in Chapter 5 (even the proofs are the same).
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 174 Chapter 8. Harmonic Bergman Spaces 8.4 Proposition: The reproducing kernel of Ωhas the following properties: (a) RΩis real valued.
(b) If (um) is an orthonormal basis of b2(Ω), then RΩ(x, y) = ∞ X m=1 um(x)um(y) for all x, y ∈Ω.
(c) RΩ(x, y) = RΩ(y, x) for all x, y ∈Ω.
(d) ∥RΩ(x, ·)∥b2 = p RΩ(x, x) for all x ∈Ω.
Proof: To prove (a), suppose that u ∈b2(Ω) is real valued and x ∈Ω. Then 0 = Im u(x) = Im Z Ω u(y)RΩ(x, y) dV(y) = − Z Ω u(y) Im RΩ(x, y) dV(y).
Take u = Im RΩ(x, ·), obtaining Z Ω Im RΩ(x, y) 2 dV(y) = 0, which implies Im RΩ≡0. We conclude that each RΩis real valued, as desired.
To prove (b), let (um) be any orthonormal basis of b2(Ω). (Recall that L2(Ω, dV), and hence b2(Ω), is separable.) By standard Hilbert space theory, RΩ(x, ·) = ∞ X m=1 ⟨RΩ(x, ·), um⟩um = ∞ X m=1 um(x)um Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Reproducing Kernels 175 for each x ∈Ω, where the infinite sums converge in norm in b2(Ω).
Since point evaluation is a continuous linear functional on b2(Ω), the equation above shows that the conclusion of (b) holds.
To prove (c), note that (b) shows that RΩ(x, y) = RΩ(y, x), while (a) shows that RΩ(x, y) = RΩ(x, y) for all x, y ∈Ω. Putting these two equations together gives (c).
To prove (d), let x ∈Ω. Then ∥RΩ(x, ·)∥2 b2 = ⟨RΩ(x, ·), RΩ(x, ·)⟩ = RΩ(x, x), where the second equality follows from the reproducing property of RΩ(x, ·). Taking square roots gives (d).
Because b2(Ω) is a closed subspace of the Hilbert space L2(Ω, dV), there is a unique orthogonal projection of L2(Ω, dV) onto b2(Ω). This self-adjoint projection is called the Bergman projection on Ω; we denote it by QΩ. The next proposition establishes the connection between the Bergman projection and the reproducing kernel.
8.5 Proposition: If x ∈Ω, then QΩu = Z Ω u(y)RΩ(x, y) dV(y) for all u ∈L2(Ω, dV).
Proof: Let x ∈Ωand u ∈L2(Ω, dV). Then QΩu = ⟨QΩ[u], RΩ(x, ·)⟩ = ⟨u, RΩ(x, ·)⟩ = Z Ω u(y)RΩ(x, y) dV(y), where the first equality above follows from the reproducing property of RΩ(x, ·), the second equality holds because QΩis a self-adjoint projec-tion onto a subspace containing RΩ(x, ·), and the third equality follows from the definition of the inner product and Proposition 8.4(a).
In the next section, we will find a formula for computing QB[p] when p is a polynomial; see 8.14 and 8.15.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 176 Chapter 8. Harmonic Bergman Spaces The Reproducing Kernel of the Ball In this section we will find an explicit formula for the reproducing kernel of the unit ball. We begin by looking at the space Hm(Rn), which consists of the harmonic polynomials on Rn that are homogeneous of degree m. Recall that the zonal harmonics introduced in Chapter 5 are reproducing kernels for Hm(Rn). Thus if p ∈Hm(Rn), then 8.6 p(x) = Z S p(ζ)Zm(x, ζ) dσ(ζ) for each x ∈Rn (by 5.30). By using polar coordinates, we will obtain an analogue of 8.6 involving integration over B instead of S.
First we extend the zonal harmonic Zm to a function on Rn × Rn.
We do this by making Zm homogeneous in the second variable as well as in the first; in other words, we set 8.7 Zm(x, y) = |x|m|y|mZm(x/|x|, y/|y|).
(If either x or y is 0, we define Zm(x, y) to be 0 when m > 0; when m = 0, we define Z0 to be identically 1.) With this extended definition, Zm(x, ·) ∈Hm(Rn) for each x ∈Rn; also, Zm(x, y) = Zm(y, x) for all x, y ∈Rn.
We now derive the analogue of 8.6 for integration over B. For every p ∈Hm(Rn), we have Z B p(y)Zm(x, y) dV(y) = nV(B) Z 1 0 r n−1 Z S p(rζ)Zm(x, rζ) dσ(ζ) dr = nV(B) Z 1 0 r n+2m−1 Z S p(ζ)Zm(x, ζ) dσ(ζ) dr = nV(B)p(x) Z 1 0 r n+2m−1 dr = nV(B) n + 2mp(x) for each x ∈Rn. In other words, p(x) equals the inner product of p with (n + 2m)Zm(x, ·)/ nV(B) for every p ∈Hm(Rn).
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey The Reproducing Kernel of the Ball 177 Now, Hk(Rn) is orthogonal to Hm(Rn) in b2(B) if k ̸= m, as can be verified using Proposition 5.9 and polar coordinates (1.5). Thus if p is a harmonic polynomial of degree M, then p(x) equals the inner product of p with PM m=0(n + 2m)Zm(x, ·)/ nV(B) . Taking M = ∞in the last sum gives us a good candidate for the reproducing kernel of the ball; Theorem 8.9 will show that this is the right guess. The following lemma will be useful in proving this theorem.
8.8 Lemma: The set of harmonic polynomials is dense in b2(B).
Proof: First note that if u ∈L2(B, dV), then ur →u in L2(B, dV) as r →1. (For u ∈C(B), use uniform continuity; the general result follows because C(B) is dense in L2(B, dV).) Thus any u ∈b2(B) can be approximated in b2(B) by functions harmonic on B. But by 5.34, every function harmonic on B can be approximated uniformly on B, and hence in L2(B, dV), by harmonic polynomials.
Now we can express the reproducing kernel of the ball as an infinite linear combination of zonal harmonics. We will use the theorem below to derive an explicit formula for RB.
8.9 Theorem: If x, y ∈B then 8.10 RB(x, y) = 1 nV(B) ∞ X m=0 (n + 2m)Zm(x, y).
The series converges absolutely and uniformly on K × B for every com-pact K ⊂B.
Proof: For x, y ∈B \ {0} we have |Zm(x, y)| = |x|m|y|m|Zm(x/|x|, y/|y|)| ≤|x|m|y|m dim Hm(Rn), where the inequality comes from Proposition 5.27(e). Now Exercise 10 in Chapter 5 shows that the infinite series in 8.10 has the convergence properties claimed in the theorem. Thus if F(x, y) denotes the right side of 8.10, then F(x, ·) is a bounded harmonic function on B for each x ∈B. In particular, F(x, ·) ∈b2(B) for each x ∈B.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 178 Chapter 8. Harmonic Bergman Spaces Now fix x ∈B. The discussion before the statement of Lemma 8.8 shows that p(x) = ⟨p, F(x, ·)⟩whenever p is a harmonic polynomial.
Because point evaluation is continuous on b2(B) and harmonic polyno-mials are dense in b2(B), we have u(x) = ⟨u, F(x, ·)⟩for all u ∈b2(B).
Hence F is the reproducing kernel of the ball.
Our next goal is to evaluate explicitly the infinite sum in 8.10. Be-fore doing so, note that a natural guess about how to find a formula for the reproducing kernel would be to find an orthonormal basis of b2(B) and then try to evaluate the infinite sum in Proposition 8.4(b).
This approach is feasible when n = 2 (see Exercise 14 of this chapter).
However, there appears to be no canonical choice for an orthonormal basis of b2(B) when n > 2.
Recall (see 6.21) that the extended Poisson kernel is defined by 8.11 P(x, y) = 1 −|x|2|y|2 (1 −2x · y + |x|2|y|2)n/2 for all x, y ∈Rn × Rn for which the denominator above is not 0. Fol-lowing 6.21, we noted some properties of the extended Poisson kernel: P(x, y) = P(y, x) = P(|x|y, x/|x|), and for x fixed, P(x, ·) is a har-monic function.
The key connection between the extended Poisson kernel and RB is the formula for the Poisson kernel given by Theorem 5.33, which states that P(x, ζ) = ∞ X m=0 Zm(x, ζ) for x ∈B and ζ ∈S. For x, y ∈B, this implies that ∞ X m=0 Zm(x, y) = ∞ X m=0 Zm(|y|x, y/|y|) = P(|y|x, y/|y|) = P(x, y).
Returning to 8.10, observe that Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey The Reproducing Kernel of the Ball 179 ∞ X m=0 2mZm(x, y) = ∞ X m=0 d dt t2mZm(x, y)|t=1 = d dt ∞ X m=0 t2mZm(x, y) |t=1 = d dt ∞ X m=0 Zm(tx, ty) |t=1 = d dt P(tx, ty)|t=1.
Thus 8.10 implies the beautiful equation 8.12 RB(x, y) = nP(x, y) + d dt P(tx, ty)|t=1 nV(B) .
This simple representation gives us a formula in closed form for the reproducing kernel RB.
8.13 Theorem: Let x, y ∈B. Then RB(x, y) = (n −4)|x|4|y|4 + (8x · y −2n −4)|x|2|y|2 + n nV(B)(1 −2x · y + |x|2|y|2)1+n/2 .
Proof: Compute using 8.12 and 8.11.
The next result gives a formula for the Bergman projection on the unit ball. It should be compared to Theorem 5.1 and Proposition 5.31.
8.14 Theorem: Let p be a polynomial on Rn of degree m. Then QB[p] is a polynomial of degree at most m. Moreover, QBp = 1 nV(B) m X k=0 (n + 2k) Z B p(y)Zk(x, y) dV(y) for every x ∈B.
Proof: Fix x ∈B. For each r ∈(0, 1), the function pr is a polyno-mial of degree m. Thus by Proposition 5.9 we have Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 180 Chapter 8. Harmonic Bergman Spaces Z S p(rζ)Zk(x, ζ) dσ(ζ) = 0 for all k > m. Hence Z B p(y)Zk(x, y) dV(y) nV(B) = Z 1 0 r n+k−1 Z S p(rζ)Zk(x, ζ) dσ(ζ) dr = 0 for all k > m. Combining this result with 8.10 and Proposition 8.5 gives the desired equation.
Recall that Pm(Rn) denotes the space of polynomials on Rn that are homogeneous of degree m.
The following corollary shows how to compute the Bergman projection of a polynomial from its Poisson integral.
8.15 Corollary: Suppose p ∈Pm(Rn) and that Pm k=0 pk is the solution to the Dirichlet problem for the ball with boundary data p|S, where each pk ∈Hk(Rn). Then QB[p] = m X k=0 n + 2k n + k + mpk.
Proof: For 0 ≤k ≤m and x ∈B, we have Z B p(y)Zk(x, y) dV(y) nV(B) = Z 1 0 r n−1 Z S p(rζ)Zk(x, rζ) dσ(ζ) dr = Z 1 0 r n+k+m−1 Z S p(ζ)Zk(x, ζ) dσ(ζ) dr = pk(x) n + k + m, where the last equality comes from Proposition 5.31. Combining the last equality with Theorem 8.14 now gives the desired result.
If p is a polynomial on Rn, then the software described in Ap-pendix B computes the Bergman projection QB[p] by first computing Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Examples in bp(B) 181 the Poisson integral P[p] (using Theorem 5.21) and then uses the corol-lary above to compute QB[p]. For example, if n = 6 and p(x) = x14x2, then this software computes that QBp = x2 + |x|4x2 40 −|x|2x12x2 2 + x14x2 −3|x|2x2 56 + 3x12x2 7 .
Let us make an observation in passing. We have seen that if p is a polynomial on Rn, then the Poisson integral P[p] and the Bergman projection QB[p] are both polynomials. Each can be thought of as the solution to a certain minimization problem. Specifically, P[p] mini-mizes ∥p −u∥L∞(S), while QB[p] minimizes ∥p −u∥L2(B,dV), where both minimums are taken over all functions u harmonic on B.
Curiously, if p is a homogeneous polynomial, then the two harmonic approximations P[p] and QB[p] agree only if p is harmonic (see Exer-cise 19 in this chapter).
Examples in bp(B) Because the Poisson integral is a linear isometry of Lp(S) onto hp(B) (p > 1) and of M(S) onto h1(B) (Theorem 6.13), we easily see that hp(B) ̸= hq(B) whenever p ̸= q. We now prove the analogous result for the Bergman spaces of B.
8.16 Proposition: If 1 ≤p < q < ∞, then bq(B) is a proper subset of bp(B).
Proof: Suppose 1 ≤p < q < ∞. Because B has finite volume mea-sure, clearly bq(B) ⊂bp(B). To prove that this inclusion is proper, consider the identity map from bq(B) into bp(B). This map is linear, one-to-one, and bounded (by Hölder’s inequality).
If this map were onto, then the inverse mapping would be continuous by the open map-ping theorem, and so there would exist a constant C < ∞such that Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 182 Chapter 8. Harmonic Bergman Spaces 8.17 ∥u∥bq ≤C∥u∥bp for all u ∈bp(B).
We will show that 8.17 fails. For m = 1, 2, . . ., choose a homoge-neous harmonic polynomial um of degree m, with um ̸≡0. Integrating in polar coordinates (1.5), we find that ∥um∥bp = Z S |um|p dσ 1/p nV(B) Z 1 0 r pm+n−1 dr 1/p , a similar result holding for ∥um∥bq. Because Lr -norms on S with re-spect to σ increase as r increases (Hölder’s inequality), we have ∥um∥bq ∥um∥bp ≥ nV(B)/(qm + n) 1/q nV(B)/(pm + n) 1/p .
As m →∞, the expression on the right of the last inequality tends to ∞. Therefore 8.17 fails, proving that the identity map from bq(B) into bp(B) is not onto. Thus bq(B) is properly contained in bp(B), as desired.
We turn now to some other properties of the Bergman spaces on the ball. First note that hp(B) ⊂bp(B) for all p ∈[1, ∞), as an easy integration in polar coordinates (1.5) shows. (In fact, hp(B) ⊂bq(B) for all q < pn/(n −1); see Exercise 21 of this chapter.) However, each of the spaces bp(B) contains functions not belonging to any hq(B), as we show below. In fact, we will construct a function in every bp(B) that at every point of the unit sphere fails to have a radial limit; such a function cannot belong to any hq(B) by Corollary 6.44. We begin with a lemma that will be useful in this construction.
8.18 Lemma: Let fm(ζ) = eimζ1 for ζ ∈S and m = 1, 2, . . . . Then P[fm] →0 uniformly on compact subsets of B as m →∞.
Proof: Let g ∈C(S). Using the slice integration formula (A.5 in Appendix A), we see that R S fmg dσ equals a constant (depending only on n) times Z 1 −1 (1 −t2) n−3 2 eimt Z Sn−1 g(t, p 1 −t2 ζ) dσn−1(ζ) dt, Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Examples in bp(B) 183 where Sn−1 denotes the unit sphere in Rn−1 and dσn−1 denotes nor-malized surface-area measure on Sn−1. The Riemann-Lebesgue Lemma then shows that R S fmg dσ →0 as m →∞.
In particular, taking g = P(x, ·) for x ∈B, we see that P[fm] →0 pointwise on B.
Because |fm| ≡1 on S, we have |P[fm]| ≤1 on B for each m.
Thus by Theorem 2.6, every subsequence of (P[fm]) contains a sub-sequence converging uniformly on compact subsets of B. Because we already know that P[fm] →0 pointwise on B, we must have P[fm] →0 uniformly on compact subsets of B.
The harmonic functions of Lemma 8.18 extend continuously to B with boundary values of modulus one everywhere on S, yet converge uniformly to zero on compact subsets of B. In this they resemble the harmonic functions zm in the unit disk of the complex plane.
8.19 Theorem: Let α: [0, 1) →[1, ∞) be an increasing function with α(r) →∞as r →1. Then there exists a harmonic function u on B such that (a) |u(rζ)| < α(r) for all r ∈[0, 1) and all ζ ∈S; (b) at every point of S, u fails to have a finite radial limit.
Proof: Choose an increasing sequence of numbers sm ∈[0, 1) such that α(sm) > m+1. From the sequence (P[fm]) of Lemma 8.18, choose a subsequence (vm) with |vm| < 2−m on smB. Suppose r ∈[sm, sm+1).
Because each vm is bounded by 1 on B, we have ∞ X k=1 |vk(rζ)| = m X k=1 |vk(rζ)| + ∞ X k=m+1 |vk(rζ)| < m + 2−(m+1) + 2−(m+2) + · · · < m + 1 < α(sm) ≤α(r).
Thus P |vm(rζ)| < α(r) for all r ∈(0, 1) and all ζ ∈S; furthermore, P |vm| converges uniformly on compact subsets of B.
From the sequence (vm) we inductively extract a subsequence (um) in the following manner. Set u1 = v1. Because u1 is continuous on B, Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 184 Chapter 8. Harmonic Bergman Spaces we may choose r1 ∈[0, 1) such that |u1(rζ) −u1(ζ)| < 1/4 for all r ∈[r1, 1] and all ζ ∈S. Suppose we have chosen u1, u2, . . . , um from v1, v2, . . . and that we have radii 0 < r1 < · · · < rm < 1 such that 8.20 m X k=1 |uk(rζ) −uk(sζ)| < 1/4 for all r, s ∈[rm, 1], ζ ∈S.
We then select um+1 such that |um+1| < 2−(m+1) on rmB. Now choose rm+1 ∈(rm, 1) so that 8.20 holds with m + 1 in place of m. The radius rm+1 can be chosen since each uk is continuous on B.
Having obtained the subsequence (um) from (vm) (as well as the accompanying sequence (rm) of radii), we define u = ∞ X m=1 um.
From the first paragraph of the proof we know that |u(rζ)| < α(r) for all r ∈[0, 1), and that P um converges uniformly on compact subsets of B, which implies that u is harmonic on B.
We now show that at each point of S, u fails to have a radial limit.
(Here is where we use the fact that |um| ≡1 on S for every m.) We have |u(rm+1ζ) −u(rmζ)| ≥|um+1(rm+1ζ) −um+1(rmζ)| − X k̸=m+1 |uk(rm+1ζ) −uk(rmζ)| ≥|um+1(ζ)| −|um+1(rm+1ζ) −um+1(ζ)| −|um+1(rmζ)| −1/4 −2 ∞ X m+2 2−k ≥1 −1/4 −2−(m+1) −1/4 −2 ∞ X m+2 2−k ≥1/2 −2 ∞ X m+1 2−k.
Thus for each ζ ∈S, the sequence u(rmζ) fails to have a finite limit as m →∞, which implies that u fails to have a finite radial limit at ζ.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey The Reproducing Kernel of the Upper Half-Space 185 8.21 Corollary: There is a function u belonging to T p<∞bp(B) such that at every point of S, u fails to have a radial limit.
Proof: Let α(r) = 1+log 1 1−r , and let u be the corresponding func-tion guaranteed by Theorem 8.19. Integrating in polar coordinates (1.5), we easily check that u belongs to bp(B) for every p ∈[1, ∞).
The Reproducing Kernel of the Upper Half-Space The goal of this section is to find an explicit formula for the re-producing kernel of the upper half-space. A well-motivated, although computationally tedious, method of deriving this formula is given in Exercise 24 of this chapter. We will present a slicker method relying on the magic of integration by parts.
As we did for B, we will derive the reproducing kernel of H in terms of the Poisson kernel. Recall that for z ∈H and t ∈Rn−1, the Poisson kernel for H is the function PH(z, t) = 2 nV(B) zn |z −t|n .
For w ∈Rn, define w = (w1, . . . , wn−1, −wn); note that w is the usual complex conjugate of w on R2 = C. We now extend the domain of PH by defining 8.22 PH(z, w) = 2 nV(B) zn + wn |z −w|n for z ̸= w. Note that PH(z, w) = PH(w, z) and PH z + (0, r), w = PH z, w + (0, r) for r ∈R whenever these expressions make sense.
Thus PH(z, w) = PH(w, z) = PH w + (0, zn), z −(0, zn) for all z, w ∈H. Thus PH(z, ·) is harmonic on {w ∈Rn : wn > −zn} for each z ∈H (being the translate of a harmonic function).
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 186 Chapter 8. Harmonic Bergman Spaces Before proving the main result of this section, Theorem 8.24, we prove an analogue of Lemma 8.8 for H.
8.23 Lemma: The set of functions that are harmonic and square inte-grable on a half-space larger than H is dense in b2(H).
Proof: Let u ∈b2(H). For δ > 0, the function z 7→u z + (0, δ) belongs to b2 {z ∈Rn : zn > −δ} . But the functions u z + (0, δ) converge to u(z) in L2(H, dV) as δ →0.
(This follows by uniform continuity if u is continuous and has compact support in H; the set of such functions is dense in L2(H, dV).) Now we can give an explicit formula for the reproducing kernel of the upper half-space.
8.24 Theorem: For all z, w ∈H, RH(z, w) = −2 ∂ ∂wn PH(z, w) = 4 nV(B) n(zn + wn)2 −|z −w|2 |z −w|n+2 .
Proof: The second equality follows, with some simple calculus, from 8.22. Note that this equality implies ∂PH(z, w)/∂wn belongs to b2(H) for each fixed z ∈H (see Exercise 1 in Appendix A). The re-mainder of the proof will be devoted to showing that the first equality holds.
Fix z ∈H. Suppose δ > 0 and u ∈b2({w ∈Rn : wn > −δ}).
Then Z H u(w) ∂ ∂wn PH(z, w) dV(w) 8.25 = Z Rn−1 Z ∞ 0 u(x, y) ∂ ∂y PH z, (x, y) dy dx.
Now, u is bounded and harmonic on H by 8.1. Thus, after integrating by parts in the inner integral, the right side of 8.25 becomes − Z Rn−1 u(x, 0)PH(z, x) dx − Z Rn−1 Z ∞ 0 " ∂ ∂y u(x, y) # PH z, (x, y) dx dy, Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey The Reproducing Kernel of the Upper Half-Space 187 which equals 8.26 −u(z) − Z ∞ 0 Z Rn−1 " ∂ ∂y u(x, y) # PH z, (x, y) dx dy.
Notice that we reversed the order of integration to arrive at 8.26. This is permissible if the integrand in 8.26 is integrable over H. To verify this, note that by Corollary 8.2 there exists a constant C < ∞such that | ∂ ∂y u(x, y)| ≤ C (y + δ)1+n/2 .
Note also that 8.27 PH z, (x, y) = PH z + (0, y), (x, 0) , which implies R Rn−1 PH z, (x, y) dx = 1 for each y > 0. The reader can now easily verify that the integrand in 8.26 is integrable over H.
For each y > 0, the term in brackets in 8.26 is the restriction to Rn−1 of the function w 7→Dnu w + (0, y) , which is bounded and harmonic on H. Thus by 8.27, the integral over Rn−1 in 8.26 equals (Dnu) z + (0, 2y) . Therefore 8.26 equals −u(z) − Z ∞ 0 (Dnu) z + (0, 2y) dy = −u(z)/2, where the last equality holds because u z + (0, 2y) →0 as y →∞ (by 8.1).
Let F(w) = −2∂PH(z, w)/∂wn. We have shown that u(z) = ⟨u, F⟩ whenever u is harmonic and square integrable on a half-space larger than H. The set of such functions u is dense in b2(H) (Lemma 8.23), and thus the proof is complete.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 188 Chapter 8. Harmonic Bergman Spaces Exercises 1.
Prove that bp(Rn) = {0}.
2.
Suppose that u ∈bp(Ω). Prove that d(x, ∂Ω)n/p|u(x)| →0 as x →∂Ω.
3.
Prove that if u ∈bp({x ∈Rn : |x| > 1}), then u is harmonic at ∞.
4.
Suppose u ∈bp(H) and y > 0.
Prove that u(x, y) →0 as |x| →∞in Rn−1.
5.
Prove that if u is a harmonic function on Rn such that Z Rn |u(x)|(1 + |x|)λ dV(x) < ∞ for some λ ∈R, then u is a polynomial.
6.
(a) Assume that n > 2 and p ≥n/(n −2). Prove that if u is in bp(B \ {0}), then u has a removable singularity at 0.
(b) Show that the constant n/(n −2) in part (a) is sharp.
(c) Show that there exists a function in T p<∞bp(B2 \ {0}) that fails to have a removable singularity at 0.
7.
Prove that bp(Rn \ {0}) = {0}.
8.
Prove that RrΩ+a(x, y) = r −nRΩ x −a r , y −a r for all r > 0, a ∈Rn.
9.
Prove that ∥RΩ(x, ·)∥b2 ≤ 1 p V(B)d(x, ∂Ω)n .
10.
Suppose Ω1 ⊂Ω2 ⊂· · · is an increasing sequence of open sub-sets of Rn and Ω= S∞ k=1 Ωk. Prove that RΩ(x, y) = lim k→∞RΩk(x, y) for all x, y ∈Ω.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Exercises 189 11.
Suppose a1, . . . , am are points in Ω. Let A be the m-by-m matrix whose entry in row j, column k, equals RΩ(aj, ak). Prove that A is positive semidefinite.
12.
Show that the harmonic Bloch space is properly contained in bp(B) for every p < ∞. (See Exercise 11, Chapter 2, for the defi-nition of the harmonic Bloch space.) 13.
Show that u(x) = Z B u(y)RB(x, y) dV(y) for all u ∈bp(B) and for all p ∈[1, ∞).
14.
Assume n = 2, and set uk(reiθ) = r |k|eikθ, k = 0, ±1, . . . . Find constants ck so that {ckuk} is an orthonormal basis of b2(B), and then use Proposition 8.4(b) to find a formula for the reproducing of kernel of B2.
15.
(a) Prove there are positive constants C1, C2 such that C1 (1 −|x|)n/2 ≤∥RB(x, ·)∥b2 ≤ C2 (1 −|x|)n/2 for all x ∈B.
(b) Find an estimate analogous to (a) for ∥RH(z, ·)∥b2.
16.
Show that RB(x, ·)/∥RB(x, ·)∥b2 converges to 0 weak in b2(B) as |x| →1.
17.
Show that QB[x12] = 1 n + 2 + x12 −∥x∥2 n .
18.
Prove that if p ∈Pm(Rn) and QB[p] = 0, then p = 0.
19.
Prove that if p ∈Pm(Rn) and P[p] = QB[p], then p is harmonic.
20.
Fix ζ ∈S. Show that P(·, ζ) ∈bp(B) for p < n/(n −1). Also show that P(·, ζ) ∉bn/(n−1)(B).
21.
Show that hp(B) ⊂bq(B) for q < pn/(n −1).
22.
Prove that every infinite-dimensional closed subspace of b2(B) contains a function not in h2(B).
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 190 Chapter 8. Harmonic Bergman Spaces 23.
Show that functions in bp(B) belong to appropriate Hardy spaces of balls internally tangent to B.
More precisely, suppose that 1 ≤q < (n −1)p/(2n) and u ∈bp(B). Prove that if a ∈B \ {0}, then the function x 7→u a + (1 −|a|)x is in hq(B).
24.
Derive the formula for the reproducing kernel RH (Theorem 8.24) by writing H = S∞ k=1 B(kN, k) and then using Exercise 10 of this chapter and 8.13.
25.
Show that every positive harmonic function on B is in b1(B). Are there any positive harmonic functions on H that are in b1(H)?
26.
Suppose δ > 0 and u ∈bp({z ∈Rn : zn > −δ}). Show that Dαu ∈bp(H) for every multi-index α.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Chapter 9 The Decomposition Theorem If K ⊂Ωis compact and u is harmonic on Ω\ K, then u might be badly behaved near both ∂K and ∂Ω; see, for example, Theorem 11.18.
In this chapter we will see that u is the sum of two harmonic functions, one extending harmonically across ∂K, the other extending harmoni-cally across ∂Ω. More precisely, u has a decomposition of the form u = v + w on Ω\ K, where v is harmonic on Ωand w is harmonic on Rn \ K.
Furthermore, there is a canonical choice for w that makes this decom-position unique.
This result, which we call the decomposition theorem, has many ap-plications. In this chapter we will use it to prove a generalization of Bôcher’s Theorem, to show that bounded harmonic functions extend harmonically across smooth sets of dimension n −2, and to prove the logarithmic conjugation theorem. In Chapter 10, we will use the decom-position theorem to obtain a “Laurent” series expansion for harmonic functions on annular domains in Rn.
The Fundamental Solution of the Laplacian We have already seen how important the functions |x|2−n (n > 2) and log |x| (n = 2) are to harmonic function theory. Another illustra-191 192 Chapter 9. The Decomposition Theorem tion of their importance is that they give rise to integral operators that invert the Laplacian; we will need these operators in the proof of the decomposition theorem.
The support of a function g on Rn, denoted supp g, is the closure of the set {x ∈Rn : g(x) ̸= 0}. We let Ck c = Ck c (Rn) denote the set of functions in Ck(Rn) that have compact support. We will frequently use the abbreviation dy for the usual volume measure dV(y).
We now show how g can be reconstructed from ∆g if g ∈C2 c .
9.1 Theorem (n > 2): If g ∈C2 c , then g(x) = 1 (2 −n)nV(B) Z Rn(∆g)(y)|x −y|2−n dy for every x ∈Rn.
9.2 Theorem (n = 2): If g ∈C2 c , then g(x) = 1 2π Z R2(∆g)(y) log |x −y| dy for every x ∈R2.
Proof: We present the proof for n > 2, leaving the minor modi-fications needed for n = 2 to the reader (Exercise 1 of this chapter).
Note first that the function |x|2−n is locally integrable on Rn (use polar coordinates 1.5).
Fix x ∈Rn. Choose r large enough so that B(0, r) contains both x and supp g. For small ε > 0, set Ωε = B(0, r) \ B(x, ε). Because g is supported in B(0, r), Z Rn(∆g)(y)|x −y|2−n dy = lim ε→0 Z Ωε (∆g)(y)|x −y|2−n dy.
Now apply Green’s identity (1.1) Z Ω (u∆g −g∆u) dV = Z ∂Ω (uDng −gDnu) ds, with u(y) = |x −y|2−n and Ω= Ωε. Since g = 0 near ∂B(0, r), only the surface integral over ∂B(x, ε) comes into play. Recalling that the Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Decomposition of Harmonic Functions 193 unnormalized surface area of S is nV(B) (see A.2 in Appendix A), we calculate that lim ε→0 Z Ωε (∆g)(y)|x −y|2−n dy = (2 −n)nV(B)g(x), which gives the desired conclusion.
For x ∈Rn \ {0}, set F(x) = ( [(2 −n)nV(B)]−1|x|2−n if n > 2 (2π)−1 log |x| if n = 2.
The function F is called the fundamental solution of the Laplacian; it serves as the kernel of an integral operator that inverts the Laplacian on C2 c . To see this, define 9.3 (Tg)(x) = Z Rn g(y)F(x −y) dy = Z Rn g(x −y)F(y) dy for g ∈Cc. Now suppose g ∈C2 c . Then T(∆g) = g by 9.1 or 9.2. On the other hand, differentiation under the integral sign on the right side of 9.3 shows that ∆(Tg) = T(∆g); applying 9.1 or 9.2 again, we see that ∆(Tg) = g. Thus T ◦∆= ∆◦T = I, the identity operator, on the space C2 c ; in other words, T = ∆−1 on C2 c .
We can now solve the inhomogeneous equation 9.4 ∆u = g for any g ∈C2 c ; we simply take u = Tg. Equation 9.4 is often referred to as Poisson’s equation.
Decomposition of Harmonic Functions The reader is already familiar with a result from complex analy-sis that can be interpreted as a decomposition theorem. Specifically, suppose 0 < r < R < ∞, K = B(0, r), and Ω= B(0, R). Assume f is holomorphic on the annulus Ω\K, and let P∞ −∞akzk be the Laurent ex-pansion of f on Ω\ K. Setting g(z) = P∞ 0 akzk and h(z) = P−1 −∞akzk, we see that f = g +h on Ω\K, that g extends to be holomorphic on Ω, Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 194 Chapter 9. The Decomposition Theorem and that h extends to be holomorphic on (C ∪{∞}) \ K. The Laurent series expansion therefore gives us a decomposition for holomorphic functions (in the special case of annular regions in C).
The decom-position theorem (9.6 and 9.7) is the analogous result for harmonic functions.
We will need a large supply of smooth functions in the proof of the decomposition theorem; the following lemma provides what we want.
9.5 Lemma: Suppose K ⊂Ωis compact. Then there exists a function ϕ ∈C∞ c (Rn) such that ϕ ≡1 on K, supp ϕ ⊂Ω, and 0 ≤ϕ ≤1 on Rn.
Proof: Define a C∞-function f on R by setting f (t) = ( e−1/t if t > 0 0 if t ≤0, and define a function ψ ∈C∞ c (Rn) by setting ψ(y) = cf (1 −2|y|2), where the constant c is chosen so that R Rn ψ(y) dy = 1. Note that supp ψ ⊂B.
For r > 0, let ψr (y) = r −nψ(y/r). Observe that supp ψr ⊂rB and that R Rn ψr (y) dy = 1. Now set r = d(K, ∂Ω)/3 and define ω = {x ∈Ω: d(x, K) < r}.
Finally, put ϕ(x) = Z ω ψr (x −y) dy for x ∈Rn. Differentiation under the integral sign above shows that ϕ ∈C∞. Clearly 0 ≤ϕ ≤1 on Rn. Because ψr (x −y) is supported in B(x, r), we have ϕ(x) = 1 whenever x ∈K and ϕ(x) = 0 whenever d(x, K) > 2r.
We now prove the decomposition theorem; the n > 2 case differs from the n = 2 case, so we state the two results separately.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Decomposition of Harmonic Functions 195 9.6 Decomposition Theorem (n > 2): Let K be a compact subset of Ω. If u is harmonic on Ω\ K, then u has a unique decomposition of the form u = v + w, where v is harmonic on Ωand w is a harmonic function on Rn \ K satisfying limx→∞w(x) = 0.
9.7 Decomposition Theorem (n = 2): Let K be a compact subset of Ω. If u is harmonic on Ω\ K, then u has a unique decomposition of the form u = v + w, where v is harmonic on Ωand w is a harmonic function on R2 \ K satisfying limx→∞w(x) −b log |x| = 0 for some constant b.
Proof: We present the proof for n > 2 (Theorem 9.6), leaving the changes needed for n = 2 to the reader (Exercise 3 of this chapter).
As a notation convenience, for E any subset of Rn and r > 0, let Er = {x ∈Rn : d(x, E) < r}.
Suppose first that Ωis a bounded open subset of Rn.
Choose r small enough so that Kr and (∂Ω)r are disjoint. By Lemma 9.5, there is a function ϕr ∈C∞ c (Rn) supported in Ω\ K such that ϕr ≡1 on Ω\ Kr ∪(∂Ω)r ; Figure 9.8 may be helpful.
For x ∈Ω\ Kr ∪(∂Ω)r , apply Theorem 9.1 to the function uϕr , which can be thought of as a function in C∞ c (Rn), to obtain u(x) = (uϕr )(x) = 1 (2 −n)nV(B) Z Rn ∆(uϕr )(y)|x −y|2−n dy = 1 (2 −n)nV(B) Z (∂Ω)r ∆(uϕr )(y)|x −y|2−n dy + 1 (2 −n)nV(B) Z Kr ∆(uϕr )(y)|x −y|2−n dy = vr (x) + wr (x), where vr (x) is [(2 −n)nV(B)]−1 times the integral over (∂Ω)r and wr (x) is [(2 −n)nV(B)]−1 times the integral over Kr . Differentiation Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 196 Chapter 9. The Decomposition Theorem 9.8 ϕr ≡1 on the shaded region.
under the integral sign shows that vr is harmonic on Ω(∂Ω)r and that wr is harmonic on Rn \ Kr . We also see that wr (x) →0 as x →∞.
Suppose now that s < r. Then, as in the previous paragraph, we obtain the decomposition u = vs +ws on Ω(Ks ∪∂Ωs). We claim that vr = vs on Ω\ (∂Ω)r and wr = ws on (Rn ∪{∞}) \ Kr . To see this, note that if x ∈Ω\ Kr ∪(∂Ω)r , then vr (x)+wr (x) = vs(x)+ws(x) (because both sides equal u(x)). Thus, wr −ws is a harmonic function on Rn \ Kr that extends to be harmonic on Rn (wr −ws agrees with vs−vr near Kr ). Because both wr and ws tend to 0 at infinity, Liouville’s Theorem (2.1) implies that wr −ws ≡0. Thus wr = ws and vr = vs on Ω\ Kr ∪(∂Ω)r , as claimed.
For x ∈Ω, we may thus set v(x) = vr (x) for all r small enough so that x ∈Ω\ (∂Ω)r . Similarly, for x ∈Rn \ K, we set w(x) = wr (x) for small r. We have arrived at the desired decomposition u = v + w.
Now suppose that Ωis unbounded and u is harmonic on Ω\ K.
Choose R large enough so that K ⊂B(0, R) and let ω = Ω∩B(0, R).
Observe that K is a compact subset of the bounded open set ω and that u is harmonic on ω \ K. Applying the result just proved for bounded open sets, we have u(x) = e v(x) + w(x) for x ∈ω \ K, where e v is harmonic on ω and w is a harmonic function on Rn \ K satisfying limx→∞w(x) = 0. Notice that the difference u −w is harmonic on Ω\ K and extends harmonically across K because it Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Bôcher’s Theorem Revisited 197 agrees with e v near K. Set v = u −w; the sum v + w is then the desired decomposition of u.
Finally, the proof of the uniqueness of the decomposition is similar to the proof given above that wr = ws and vr = vs on Ω\ Kr ∪(∂Ω)r .
Note that the function w of Theorem 9.6 is harmonic at ∞, by The-orem 4.8. Note also that if u is real valued, then the functions v and w appearing in the decompositon of u (Theorem 9.6 or Theorem 9.7) will also be real valued. This can be proved either by looking at the proofs of Theorem 9.6 and Theorem 9.7 or by taking the real parts of both sides of the decomposition u = v + w and using the uniqueness of the decomposition.
Bôcher’s Theorem Revisited The remainder of this chapter consists of applications of the decom-position theorem. We begin by using it to obtain Bôcher’s Theorem (3.9) as a consequence of Liouville’s Theorem in the n > 2 setting.
9.9 Bôcher’s Theorem (n > 2): Let a ∈Ω. If u is harmonic on Ω\ {a} and positive near a, then there is a harmonic function v on Ω and a constant b ≥0 such that u(x) = v(x) + b|x −a|2−n for all x ∈Ω\ {a}.
Proof: Without loss of generality we can assume that u is real val-ued and a = 0.
By the decomposition theorem (9.6), we can write u = v + w, where v is harmonic on Ω, w is harmonic on Rn \ {0}, and limx→∞w(x) = 0. We will complete the proof by showing that w(x) = b|x|2−n for some constant b ≥0.
Because u is positive near 0 and v is bounded near 0, w = u −v is bounded below near 0. Let ε > 0 and set h(x) = w(x) + ε|x|2−n.
Then limx→0 h(x) = ∞and limx→∞h(x) = 0, so the minimum principle (1.10) implies that h ≥0 on Rn \ {0}. Letting ε →0, we conclude that w ≥0 on Rn \ {0}.
Because w tends to zero at ∞, the Kelvin transform K[w] has a removable singularity at 0 (see Exercise 2(a) in Chapter 2). Thus K[w] Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 198 Chapter 9. The Decomposition Theorem extends to be nonnegative and harmonic on all of Rn. By Liouville’s Theorem for positive harmonic functions (3.1), K[w] = b for some constant b ≥0. Therefore w(x) = b|x|2−n, completing the proof.
The preceding argument does not yield a proof of Bôcher’s Theorem when n = 2. (One difficulty is that the function w provided by the decomposition theorem no longer vanishes at ∞.) We can, however, still use the decomposition theorem to prove Bôcher’s Theorem in the n = 2 case. We will actually obtain a generalized version of Bôcher’s Theorem. Our proof relies on the following improvement of Liouville’s Theorem for positive harmonic functions (3.1).
9.10 Generalized Liouville Theorem: Suppose that u is a real-valued harmonic function on Rn and lim inf x→∞ u(x) |x| ≥0.
Then u is constant on Rn.
Proof: Fix x ∈Rn, let ε > 0 be arbitrary, and choose r > |x| such that u(y)/|y| ≥−ε whenever |y| > r −|x|. By the volume version of the mean-value property, u(x) −u(0) = 1 V B(0, r) hZ B(x,r) u dV − Z B(0,r) u dV i .
Let Dr denote the symmetric difference of the balls B(x, r) and B(0, r) (see Figure 3.2) and let Ar denote the annulus B(0, r +|x|)\B(0, r −|x|).
Then |u(x) −u(0)| ≤ 1 V B(0, r) Z Dr |u| dV ≤ 1 V B(0, r) Z Ar |u| dV.
For every y in the annulus Ar over which the last integral is taken, we have |u(y)| ≤2ε|y| + u(y) ≤4εr + u(y), where the first inequality is trivial when u(y) ≥0 and follows from our choice of r when u(y) < 0. Combining the last two sets of inequalities, we have Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Bôcher’s Theorem Revisited 199 |u(x) −u(0)| ≤ 1 V B(0, r) h 4εrV(Ar ) + Z Ar u dV i = 4εr + u(0) (r + |x|)n −(r −|x|)n r n .
Take the limit as r →∞, getting |u(x) −u(0)| ≤8εn|x|.
Now take the limit as ε →0, getting u(x) = u(0), as desired.
Note that u satisfies the hypothesis of the Generalized Liouville The-orem if and only if lim inf x→0 |x|n−1Ku ≥0, which explains the hypothesis of the following result.
9.11 Generalized Bôcher Theorem: Let a ∈Ω. Suppose that u is a real-valued harmonic function on Ω\ {a} and lim inf x→a |x −a|n−1u(x) ≥0.
Then there is a harmonic function v on Ωand a constant b ∈R such that u(x) = ( v(x) + b log |x −a| if n = 2 v(x) + b|x −a|2−n if n > 2 for all x ∈Ω\ {a}.
Proof: We will assume that n = 2, leaving the easier n > 2 case as an exercise for the reader.
Without loss of generality, we may assume that a = 0. Because u is harmonic on Ω\ {0}, it has a decomposition u = v + w, where v is harmonic on Ωand w is a harmonic function on R2 {0} sat-isfying limx→0(w(x)−b log |x|) = 0 for some constant b ∈R. Because v is continuous at 0, our hypothesis on u implies that Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 200 Chapter 9. The Decomposition Theorem lim inf x→0 |x|w(x) ≥0.
Now set h(x) = w(x) −b log |x| for x ∈R2 \ {0}, and observe that the Kelvin transform K[h] has a removable singularity at 0. Moreover, lim inf x→∞ Kh |x| = lim inf x→∞ w(x/|x|2) −b log |1/x| |x| = lim inf x→0 |x|w(x) ≥0.
Thus, by the the Generalized Liouville Theorem (9.10), K[h] must be constant; in fact, it must be zero because its value at 0 is 0. Hence h = 0, which implies that w(x) = b log |x|. Thus u has the desired form.
The preceding proof shows how the Generalized Bôcher Theorem follows from the Generalized Liouville Theorem. It is even easier to show that the Generalized Liouville Theorem follows from the General-ized Bôcher Theorem (Exercise 8 of this chapter); hence, these results are equivalent. The authors first learned of these generalizations of Bôcher’s and Liouville’s Theorems in and .
Removable Sets for Bounded Harmonic Functions Let h∞(Ω) denote the collection of bounded harmonic functions on Ω. We say that a compact set K ⊂Ωis h∞-removable for Ωif every bounded harmonic function on Ω\ K extends to be harmonic on Ω.
The following theorem shows that if K is h∞-removable for some Ω containing K, then K is h∞-removable for every Ωcontaining K. Note that by Liouville’s Theorem, K is h∞-removable for Rn if and only if every bounded harmonic function on Rn \ K is constant.
9.12 Theorem: Let K be a compact subset of Ω.
Then K is h∞-removable for Ωif and only if K is h∞-removable for Rn.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Removable Sets for Bounded Harmonic Functions 201 Proof: If K is h∞-removable for Ω, then clearly K is h∞-removable for Rn.
To prove the converse, we use the decomposition theorem.
The n > 2 case is easy. Suppose that K is h∞-removable for Rn and that u is bounded and harmonic on Ω\K. Let u = v+w be the decomposition given by (9.6). Because n > 2, w(x) →0 as x →∞. The boundedness near K of w = u −v thus shows that w is bounded and harmonic on Rn \ K. By hypothesis, w extends to be harmonic on Rn, and thus w ≡0 by Liouville’s Theorem. Hence u = v, and thus u extends to be harmonic on Ω, as desired.
The n = 2 case is more difficult (a rare occurrence); this is because w need not have limit 0 at ∞. We will show that if there is a bounded harmonic function on Ω\ K that does not extend to be harmonic on Ω, then there is a nonconstant bounded harmonic function on R2 \ K.
We may assume that each connected component of K is a point, in other words, that K is totally disconnected. Otherwise some com-ponent of K consists of more than one point. The Riemann Mapping Theorem then implies the existence of a holomorphic map of the Rie-mann sphere minus that component onto B2, giving us a nonconstant bounded harmonic function on R2 \ K, as desired.
Let u be a bounded harmonic function on Ω\K that does not extend to be harmonic on Ω. Then there exist distinct points x and y in K such that u does not extend harmonically to any neighborhood of x nor to any neighborhood of y. (If only one such point in K existed, then we would have found a nonremovable isolated singularity of a bounded harmonic function, contradicting Theorem 2.3.) Having obtained x and y, observe that the total disconnectivity of K shows that there exist disjoint open sets Ωx and Ωy (open in R2), with x ∈Ωx and y ∈Ωy, such that K ⊂Ωx ∪Ωy.
Now u is harmonic on (Ω∩Ωx)(Ωx∩K), so by Theorem 9.7 we have the decomposition u = vx + wx, where vx is harmonic on Ω∩Ωx and wx is harmonic on R2 \ (Ωx ∩K), with limz→∞wx(z) −bx log |z| = 0 for some constant bx. We also have a similar decomposition of u on (Ω∩Ωy) \ (Ωy ∩K). Note that wx is not constant, otherwise u would extend harmonically to a neighborhood of x. Note also that if bx were 0, then wx would be a nonconstant bounded harmonic function on R2\K, and we would be done; we may thus assume that bx is nonzero.
Setting h = wy −(by/bx)wx, we claim h is the desired nonconstant bounded harmonic function on R2 \ K. To see this, note that both wx Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 202 Chapter 9. The Decomposition Theorem and wy are bounded near K, and limz→∞h(z) = 0; this proves h is bounded and harmonic on R2 \ K. If h were constant, then wy would extend harmonically to a neighborhood of y, which would mean that u would extend harmonically to a neighborhood of y, a contradiction.
As an aside, note that the analogue of Theorem 9.12 for positive harmonic functions fails when n = 2: the compact set {0} is removable for positive harmonic functions on R2 \ {0} (see 3.3), but {0} is not removable for positive harmonic functions on B2 \ {0}.
Recall that if K ⊂Ωis a single point, then K is h∞-removable for Ω (Theorem 2.3). Our next theorem improves that result, stating (roughly) that if the dimension of K is less than or equal to n −2, then K is h∞-removable.
9.13 Theorem: If 1 ≤k ≤n −2 and Ψ : Bk →Ωis a C1-map, then Ψ(Bk) is h∞-removable for Ω.
Proof: By Theorem 9.12, we need only show that if u is bounded and harmonic on Rn \Ψ(Bk), then u is constant. Without loss of gener-ality, we assume u is real valued. By Theorem 4.10, there is a constant L such that u has limit L at ∞. Let ε > 0 and set v(x) = u(x) + ε Z Bk |x −Ψ(y)|2−n dVk(y) for x ∈Rn \ Ψ(Bk). Note that v is harmonic on Rn \ Ψ(Bk) and that v has limit L at ∞. Suppose we know that 9.14 Z Bk |x −Ψ(y)|2−n dVk(y) →∞ as x →Ψ(Bk).
The boundedness of u then shows v(x) →∞as x →Ψ(Bk). By the minimum principle, v ≥L on Rn \ Ψ(Bk). Letting ε →0, we conclude that u ≥L on Rn \ Ψ(Bk). A similar argument then gives u ≤L on Rn\Ψ(Bk), so that u is constant, as desired. In other words, to complete the proof, we need only show that 9.14 holds.
To prove 9.14, first suppose that x ∈Ψ(Bk). Then x = Ψ(z) for some z ∈Bk. Because Ψ has a continuous derivative, there is a constant C ∈(0, ∞) such that |Ψ(z)−Ψ(y)| ≤C|z−y| for every y ∈Bk. Thus Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey The Logarithmic Conjugation Theorem 203 Z Bk |x −Ψ(y)|2−n dVk(y) = Z Bk |Ψ(z) −Ψ(y)|2−n dVk(y) ≥C2−n Z Bk |z −y|2−n dVk(y) = ∞, where the last equality comes from Exercise 10 of this chapter. The assertion in 9.14 now follows from Fatou’s Lemma.
Note that B1 is the interval [−1, 1]. Thus, any smooth compact arc in Ωis h∞-removable for Ωprovided n > 2. Exercise 13 in Chapter 4 and Exercise 12 of this chapter show that compact sets of dimension n −1 are not h∞-removable.
The Logarithmic Conjugation Theorem In this section, Ωwill denote a connected open subset of R2. We say that Ωis finitely connected if R2 \ Ωhas finitely many bounded components. Recall that Ωis simply connected if R2\Ωhas no bounded components.
If u is the real part of a holomorphic function f on Ω, then the imaginary part of f is called a harmonic conjugate of u. When Ωis simply connected, a real-valued harmonic function on Ωalways has a harmonic conjugate (, Chapter VIII, Theorem 2.2).
The following theorem has been called the logarithmic conjuga-tion theorem because it shows that a real-valued harmonic function on a finitely connected domain has a harmonic conjugate provided that some logarithmic terms are subtracted.
9.15 Logarithmic Conjugation Theorem: Let Ωbe a finitely con-nected domain. Let K1, . . . , Km be the bounded components of R2 \ Ω, and let aj ∈Kj for j = 1, . . . , m. If u is real valued and harmonic on Ω, then there exist f holomorphic on Ωand b1, . . . , bm ∈R such that u(z) = Re f (z) + b1 log |z −a1| + · · · + bm log |z −am| for all z ∈Ω.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 204 Chapter 9. The Decomposition Theorem Proof: We prove the theorem by induction on m, the number of bounded components in the complement of Ω. To get started, recall that if m = 0 then Ωis simply connected and u = Re f for some function f holomorphic on Ω.
Suppose now that m > 0, and that the theorem is true with m−1 in place of m. With Ωas in the statement of the theorem, set Ω′ = Ω∪Km, so that Ω′ is a finitely connected domain whose complement has m−1 bounded components. Because u is harmonic on Ω′ \ Km, 9.7 gives the decomposition u = v+w, where v is harmonic on Ω′ and w is harmonic on R2 \ Km, with limz→∞w(z) −b log |z| = 0 for some constant b.
Because v satisfies the induction hypothesis, we will be done if we can show that 9.16 w(z) = Re g(z) + b log |z −am| for some function g holomorphic on R2 \ Km.
To verify 9.16, set h(z) = w(z) −b log |z −am| for z ∈R2 \ Km. We easily calculate that h(z) →0 as z →∞; thus h extends to be harmonic on (C ∪{∞})\Km. Now, (C ∪{∞})\Km can be viewed as a simply connected region on the Riemann sphere. On such a region every real-valued harmonic function has a harmonic conjugate.
This gives 9.16, and thus completes the proof of the theorem.
As an application of the logarithmic conjugation theorem, we now give a series development for functions harmonic on annuli.
9.17 Theorem: If u is real valued and harmonic on the annulus A = {z ∈R2 : r0 < |z| < r1}, then u has a series development of the form 9.18 u(reiθ) = b log r + ∞ X k=−∞ (ckr k + c−kr −k)eikθ.
The series converges absolutely for each reiθ ∈A and uniformly on compact subsets of A.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey The Logarithmic Conjugation Theorem 205 Proof: Use the logarithmic conjugation theorem (9.15) with Ω= A, K1 = {z ∈R2 : |z| ≤r0}, and a1 = 0, to get u(z) = b log |z| + Re f (z) for some holomorphic function f on A. On A, f has a Laurent series expansion f (z) = ∞ X k=−∞ ckzk that converges absolutely and uniformly on compact subsets of A.
Now, u(z) = b log |z| + f (z) + f (z) 2 ; the series representation 9.18 for u is obtained by setting z = reiθ and replacing f with its Laurent series.
The series representation 9.18 gives another proof that the averages of u over circles of radius r satisfy the n = 2 part of 3.10.
In Chapter 10 we consider the problem of obtaining an analogous series representation for functions harmonic on annular domains in Rn.
There, as one might expect, the decomposition theorem (9.6, 9.7) will play an important role.
Additional applications of the logarithmic conjugation theorem may be found in .
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 206 Chapter 9. The Decomposition Theorem Exercises 1.
Prove Theorem 9.2.
2.
Suppose n > 2.
Given g ∈C2 c (Rn), show that there exists a unique u ∈C2(Rn) such that ∆u = g and u(x) →0 as x →∞.
What happens when n = 2?
3.
Prove the decomposition theorem in the n = 2 case (9.7).
4.
For Ω⊂R2, define the operator ∂on C1(Ω) by ∂= 1 2 ∂ ∂x + i ∂ ∂y .
Show that if f ∈C1(Ω), then f is holomorphic on Ωif and only if ∂f = 0.
5.
Show that if g ∈C1 c (R2), then g(w) = 1 π Z R2 (∂g)(z) w −z dV2(z) for all w ∈C.
(Hint: Imitate the proof of 9.1, using Green’s Theorem instead of Green’s identity.) 6.
Let Ω⊂C, let K ⊂Ωbe compact, and let f be holomorphic on Ω\K. Using the previous exercise and an argument similar to the proof of the decomposition theorem, prove that f has a unique decomposition of the form f = g + h, where g is holomorphic on Ωand h is holomorphic on C \ K, with limz→∞h(z) = 0.
7.
Prove the Generalized Bôcher Theorem (9.11) when n > 2.
8.
Show that the Generalized Liouville Theorem (9.10) is a conse-quence of the Generalized Bôcher Theorem (9.11).
9.
If u is a real-valued harmonic function on Rn such that u(x)/|x| is bounded below for x near ∞, must u be constant?
10.
Let x ∈Bn and let c ∈R. Prove that Z Bn |x −y|c dV(y) = ∞ if and only if c ≤−n.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Exercises 207 11.
Does the conclusion of Theorem 9.13 remain true if we merely assume that Ψ is continuous on Bk?
12.
Suppose E is a compact subset of Rn−1 ⊂Rn with positive (n−1)-dimensional Lebesgue measure. Show that u(y) = Z E |x −y|2−n dVn−1(x) defines a function on Rn that is continuous, bounded, and non-constant. Show that this function is harmonic on Rn \ E.
13.
Suppose Ψ : Bk →Bn is a C1-map, where 1 ≤k ≤n−2. Show that if Ψ(Bk) is closed in Bn, then every bounded harmonic function on Bn \ Ψ(Bk) extends to a bounded harmonic function on Bn.
(Note that Exercise 16 in Chapter 6 is a special case of this exer-cise.) 14.
Every polynomial p(x, y) on R2 extends to a polynomial p(z, w) on C2 by replacing x and y with the complex numbers z and w in the expansion of p.
Show that if p is a harmonic poly-nomial on R2 with real coefficients, then the imaginary part of 2p(z/2, −iz/2) is a harmonic conjugate of p.
15.
Let Ω⊂R2 be finitely connected, and let K1, K2, . . . , Km be the bounded components of R2 \ Ω. Let aj, a′ j ∈Kj. Suppose that u is real valued and harmonic on Ω. Prove that if f , g are holo-morphic functions on Ωand bj, b′ j ∈R satisfy u(z) = Re f (z) + b1 log |z −a1| + · · · + bm log |z −am| = Re g(z) + b′ 1 log |z −a′ 1| + · · · + b′ m log |z −a′ m|, then bj = b′ j. How are f and g related?
16.
Using the logarithmic conjugation theorem (9.15), give another proof of the n = 2 case of the Generalized Bôcher Theorem (9.11).
17.
Use the series representation 9.18 to show that the Dirichlet problem for an annulus in R2 is solvable. More precisely, show that if A is an annulus in R2 and f is continuous on ∂A, then there is a function u harmonic on A and continuous on A such that u|∂A = f .
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Chapter 10 Annular Regions An annular region is a set of the form {x ∈Rn : r0 < |x| < r1}; here r0 ∈[0, ∞) and r1 ∈(0, ∞]. Thus an annular region is the re-gion between two concentric spheres, or is a punctured ball, or is the complement of a closed ball, or is Rn \ {0}.
Laurent Series If u is harmonic on B, then 5.34 gives the expansion u(x) = ∞ X m=0 pm(x), where pm is a homogeneous harmonic polynomial of degree m and the series converges absolutely and uniformly on compact subsets of B.
This expansion is reminiscent of the power series expansion for holo-morphic functions. We now take up the analogous Laurent series de-velopment for harmonic functions on annular regions.
10.1 Laurent Series (n > 2): Suppose u is harmonic on an annular region A. Then there exist unique homogeneous harmonic polynomials pm and qm of degree m such that u(x) = ∞ X m=0 pm(x) + ∞ X m=0 qm(x) |x|2m+n−2 on A. The convergence is absolute and uniform on compact subsets of A.
209 210 Chapter 10. Annular Regions Proof: Suppose A has inner radius r0 ∈[0, ∞) and outer radius r1 ∈(0, ∞]. By the decomposition theorem (9.6) we have u = v + w, where v is harmonic on r1B and w is harmonic on (Rn ∪{∞})\r0B. Be-cause v is harmonic on the ball r1B, there are homogeneous harmonic polynomials pm such that 10.2 v(x) = ∞ X m=0 pm(x) on r1B. The Kelvin transform of K[w] is harmonic on the ball (1/r0)B, and so there are homogeneous harmonic polynomials qm such that Kw = ∞ X m=0 qm(x) on (1/r0)B. Applying the Kelvin transform to both sides of this equa-tion, we have 10.3 w(x) = ∞ X m=0 qm(x) |x|2m+n−2 on Rn \ r0B. Combining the series expansions 10.2 and 10.3, we obtain the desired expansion for u on A. The series 10.2 and 10.3 converge absolutely and uniformly on compact subsets of A, and hence so does the Laurent series expansion of u. Uniqueness of the expansion follows from the uniqueness of the decomposition u = v + w and of the series expansions 10.2 and 10.3.
The preceding proof does not quite work when n = 2 because the decomposition theorem takes a different form in that case (see 9.6, 9.7). Exercise 1 of this chapter develops the Laurent series expansion for harmonic functions when n = 2.
Isolated Singularities Suppose n > 2, a ∈Ω, and u is harmonic on Ω\ {a}. By Theo-rem 10.1, there are homogeneous harmonic polynomials pm and qm such that Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Isolated Singularities 211 u(x) = ∞ X m=0 pm(x −a) + ∞ X m=0 qm(x −a) |x −a|2m+n−2 for x in a deleted neighborhood of a. We call the function 10.4 ∞ X m=0 qm(x −a) |x −a|2m+n−2 the principal part of u at a and classify the singularity at a accord-ingly. Specifically, u has a removable singularity at a if each term in the principal part is zero; u has a pole at a if the principal part is a finite sum of nonzero terms; u has an essential singularity at a if the principal part has infinitely many nonzero terms.
If u has a pole at a, with principal part given by 10.4, and M is the largest integer such that qM ̸= 0, then we say that the pole has order M + n −2. For example, if α is a multi-index, then Dα|x|2−n has a pole of order |α| + n −2 at 0. Theorem 10.5(b) below shows why the order of a pole has been defined in this manner. We call a pole of order n −2 a fundamental pole (because the principal part is then a multiple of the fundamental solution defined in Chapter 9).
10.5 Theorem (n > 2): If u is harmonic with an isolated singularity at a, then u has (a) a removable singularity at a if and only if lim x→a |x −a|n−2|u(x)| = 0; (b) a pole at a of order M + n −2 if and only if 0 < lim sup x→a |x −a|M+n−2|u(x)| < ∞; (c) an essential singularity at a if and only if lim sup x→a |x −a|N|u(x)| = ∞ for every positive integer N.
Proof: The proof of (a) follows from Exercise 2(a) in Chapter 2.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 212 Chapter 10. Annular Regions For the remainder of the proof, we assume that u has principal part at a given by w(x) = P∞ m=0 qm(x −a)/|x −a|2m+n−2.
To prove (b), first suppose that u has a pole at a of order M + n −2.
Then the homogeneity of each qm implies that lim sup x→a |x −a|M+n−2|u(x)| = sup S |qM|.
The right side of this equation is positive and finite, and hence so is the left side, proving one direction of (b).
Conversely, suppose 0 < lim supx→a |x −a|M+n−2|u(x)| < ∞. Then there is a constant C < ∞such that |w(a + rζ)| ≤C/r M+n−2 for small r > 0 and ζ ∈S. Let j be an integer with j > M. Then Z S |qj(ζ)|2 dσ(ζ) r 2j+2n−4 ≤ ∞ X m=0 Z S |qm(ζ)|2 dσ(ζ) r 2m+2n−4 = Z S |w(a + rζ)|2 dσ(ζ) ≤ C2 r 2M+2n−4 for small r > 0; here we have used the orthogonality of spherical har-monics of different degree (Proposition 5.9).
Letting r →0, we get R S |qj|2 dσ = 0, so that qj ≡0. Thus u has a pole at a of order at most M +n−2. Because lim supx→a |x−a|M+n−2|u(x)| is positive, the order of the pole is at least M + n −2, completing the proof of (b).
To prove (c), first suppose that lim supx→a |x −a|N|u(x)| = ∞for every positive integer N. By (a) and (b), u can have neither a remov-able singularity nor a pole at a, and thus u has an essential singularity at a.
To prove the other direction of (c), suppose there is a positive integer N such that lim supx→a |x −a|N|u(x)| < ∞. By the argument used in proving (b), this implies that qj ≡0 for all sufficiently large j. Thus u does not have an essential singularity at a, completing the proof of (c).
The analogue of the theorem above for n = 2, along with the appro-priate definitions, is given in Exercise 2 of this chapter.
Recall that Picard’s Theorem states if f is a holomorphic function with essential singularity at a, then f assumes all complex values, with Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey The Residue Theorem 213 one possible exception, infinitely often on every deleted neighborhood of a. Picard’s Theorem has the following analogue for real-valued har-monic functions.
10.6 Theorem: Let u be a real-valued harmonic function with either an essential singularity or a pole of order greater than n −2 at a ∈Rn.
Then u assumes every real value infinitely often near a.
Proof: By Bôcher’s Theorem (3.9), u cannot be bounded above or below on any deleted neighborhood of a. Thus, for every small r > 0, the connected set u B(a, r) \ {a} must be all of R. This implies that u assumes every real value infinitely often near a.
There is no analogue of Theorem 10.6 for complex-valued harmonic functions.
The Residue Theorem Suppose u ∈C2(Ω). Then u is harmonic on Ωif and only if Z ∂B(a,r) Dnu ds = 0 for every closed ball B(a, r) ⊂Ω; as usual, Dn denotes the derivative with respect to the outward normal n and ds denotes (unnormalized) surface-area measure. Proof: apply Green’s identity (1.1) with v ≡1 to small closed balls contained in Ω. We can think of this result as an analogue of Morera’s Theorem for holomorphic functions.
Integrating the normal derivative over the boundary also yields a “residue theorem” of sorts. Suppose n > 2 and the harmonic function u has an isolated singularity at a, with Laurent series expansion at a given by u(x) = ∞ X m=0 pm(x −a) + ∞ X m=0 qm(x −a) |x −a|2m+n−2 .
We call the constant q0 the residue of u at a, and write Res(u, a) = q0.
The following proposition and theorem justify this terminology.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 214 Chapter 10. Annular Regions 10.7 Proposition (n > 2): If u is harmonic on B(a, r) \ {a}, then Res(u, a) = 1 (2 −n)nV(B) Z ∂B(a,r) Dnu ds.
Proof: Without loss of generality, assume a = 0. Suppose u(x) = ∞ X m=0 pm(x) + ∞ X m=0 qm(x) |x|2m+n−2 is the Laurent expansion of u about 0.
The first sum is harmonic on B(0, r); hence, the integral of its normal derivative over ∂B(0, r) is zero. The integral of the normal derivative of the second sum equals q0r 1−n(2 −n) Z ∂B(0,r) ds + ∞ X m=1 (2 −n −m)r 1−n−2m Z ∂B(0,r) qm ds.
The value of the first integral is q0nV(B)(2 −n); all other integrals vanish by the mean-value property.
10.8 Residue Theorem (n > 2): Suppose Ωis a bounded open set with smooth boundary. Let a1, . . . , ak be distinct points in Ω. If u is harmonic on Ω\ {a1, . . . , ak}, then Z ∂Ω Dnu ds = (2 −n)nV(B) k X j=1 Res(u, aj).
Proof: Choose r > 0 so that B(a1, r), . . . , B(ak, r) are pairwise dis-joint and all contained in Ω. Set ω = Ω\ Sk j=1 B(aj, r) . Then Z ∂ω Dnu ds = 0 by Green’s identity (1.3). Hence Z ∂Ω Dnu ds = − k X j=1 Z ∂B(aj,r) Dnu ds = (2 −n)nV(B) k X j=1 Res(u, aj) (note that n points toward aj on ∂B(aj, r)).
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey The Poisson Kernel for Annular Regions 215 See Exercise 8 of this chapter for statement of the residue theorem when n = 2.
The Poisson Kernel for Annular Regions Let A be a bounded annular region. If f is a continuous function on ∂A, does f have a continuous extension to A that is harmonic on A?
In this section we will see that this question, called the Dirichlet prob-lem for A, has an affirmative answer if the inner radius of A is positive.
In fact, we will find a Poisson-integral type formula for the solution. (In the next chapter, we show that the Dirichlet problem is solvable on a much wider class of domains, although in the more general context we will not have an explicit integral formula for the solution.) Fix r0 ∈(0, 1). Throughout this section, we assume that A is the annular region {x ∈Rn : r0 < |x| < 1}. This is no loss of generality because dilations preserve harmonic functions.
To discover the formula for solving the Dirichlet problem for A, we begin with a special case. Suppose g ∈Hm(S) for some m ≥0.
Consider the problem of finding a continuous function u on A that is harmonic on A, with u = g on S and u = 0 on r0S. We first extend g to a harmonic homogeneous polynomial of degree m (which we also denote by g). The Kelvin transform of g is then harmonic on Rn \ {0}; the homogeneity of g shows that Kg = g(x)/|x|2m+n−2. Thus the function u defined by u(x) = 1 −(r0/|x|)2m+n−2 1 −r02m+n−2 g(x) solves the Dirichlet problem in this special case.
Let us define 10.9 bm(x) = 1 −(r0/|x|)2m+n−2 1 −r02m+n−2 , so that u(x) = bm(x)g(x), where u is the function displayed above.
Recall that integration against the zonal harmonic Zm(x, ζ) reproduces the values of functions in Hm(Rn) (5.30). Thus we can rewrite our formula for the solution u as follows: u(x) = bm(x) Z S g(ζ)Zm(x, ζ) dσ(ζ).
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 216 Chapter 10. Annular Regions Going one step further, if g = PM m=0 gm, where gm ∈Hm(S), then adding the solutions obtained for each gm in the previous paragraph solves the Dirichlet problem for A with boundary data g on S and 0 on r0S. Explicitly, u(x) = Z S g(ζ) M X m=0 bm(x)Zm(x, ζ) dσ(ζ).
Note that for each ζ ∈S, the function x 7→bm(x)Zm(x, ζ) is harmonic on A (because it equals a constant times Zm(x, ζ) plus a constant times KZm(·, ζ)).
Any polynomial restricted to S is the sum of spherical harmonics (from Theorem 5.7). Furthermore, the set of polynomials is dense in C(S) by the Stone-Weierstrass Theorem (see , Theorem 7.33). Sup-pose, then, that g is an arbitrary continuous function on S. To find a continuous function u on A that is harmonic on A, with u = g on S and u = 0 on r0S, the results above suggest that we try u(x) = Z S g(ζ)PA(x, ζ) dσ(ζ), where 10.10 PA(x, ζ) = ∞ X m=0 bm(x)Zm(x, ζ).
Note that 0 < bm(x) < 1 for x ∈A. Thus the last series converges absolutely and uniformly on K × S for every compact K ⊂A, as in the proof of Theorem 5.33. In particular, for each ζ ∈S, the function PA(·, ζ) is harmonic on A.
We handle the Dirichlet problem for A with boundary data h on r0S and 0 on S in a similar manner. Thus a process like the one above suggests that the solution u is given by the formula u(x) = Z S h(r0ζ)PA(x, r0ζ) dσ(ζ), where 10.11 PA(x, r0ζ) = ∞ X m=0 cm(x)Zm(x, ζ) Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey The Poisson Kernel for Annular Regions 217 and 10.12 cm(x) = |x|−m(r0/|x|)m+n−2 1 −|x|2m+n−2 1 −r02m+n−2 .
Note that |cm(x)Zm(x, ζ)| < (r0/|x|)m+n−2|Zm(x/|x|, ζ)| for x ∈A, so the infinite sum in 10.11 converges absolutely and uniformly on K × S for every compact K ⊂A, as in the proof of Theorem 5.33. In particular, for each ζ ∈S, the function PA(·, r0ζ) is harmonic on A.
Having approached the Dirichlet problem for A “one sphere at a time”, we easily guess what to do for an arbitrary f ∈C(∂A)—we sim-ply add the two candidate solutions obtained above.
We now make the formal definitions. For n > 2, PA is the function on A × ∂A defined by 10.9–10.12. (For n = 2 and m = 0, the terms b0(x) and c0(x) must be replaced by appropriate modifications of log |x|; Exercise 10 of this chapter asks the reader to make the necessary ad-justments.) For f ∈C(∂A), the Poisson integral of f, denoted PA[f ], is the function on A defined by PAf = Z S f (ζ)PA(x, ζ) dσ(ζ) + Z S f (r0ζ)PA(x, r0ζ) dσ(ζ).
We have already done most of the work needed to show that PA[f ] solves the Dirichlet problem for f .
10.13 Theorem (n > 2): Suppose f is continuous on ∂A. Define u on A by u(x) = ( PAf if x ∈A f (x) if x ∈∂A.
Then u is continuous on A and harmonic on A.
Proof: The function PA[f ] is the sum of two harmonic functions, and hence is harmonic.
To complete the proof, we need only show that u is continuous on A.
The discussion above shows that u is continuous on A in the case where f|S and f (r0·)|S are both finite sums of spherical harmonics. By The-orem 5.7 and the Stone-Weierstrass Theorem (see , Theorem 7.33), such functions are dense in C(∂A). For the general f ∈C(∂A), we ap-proximate f uniformly on ∂A with functions f1, f2, . . . from this dense Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 218 Chapter 10. Annular Regions subspace; the corresponding solutions u1, u2, . . . then converge uni-formly to u on A by the maximum principle, proving that u is contin-uous on A.
The hypothesis that r0 be greater than 0 is needed to solve the Dirichlet problem for the annular region A. For example, there is no function u harmonic on the punctured ball B \ {0}, with u continuous on B, satisfying u = 1 on S and u(0) = 0: if there were such a function, then it would be bounded and harmonic on B {0}, hence would extend harmonically to B (by 2.3), contradicting the minimum principle.
The results in this section are used by the software described in Appendix B to solve the Dirichlet problem for annular regions.
For example, the software computes that the harmonic function on the an-nular region {x ∈R3 : 2 ≤|x| ≤3} that equals x12 when |x| = 2 and equals x1x2x3 when |x| = 3 is −8 3− 2592 211|x|3 + 8 |x| + 32|x|2 633 −32x12 211 + 7776x12 211|x|5 + 2187x1x2x3 2059 −279936x1x2x3 2059|x|7 .
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Exercises 219 Exercises 1.
Suppose u is harmonic on an annular region A in R2. Show that there exist pm, qm ∈Hm(R2) such that u(x) = ∞ X m=0 pm(x) + q0 log |x| + ∞ X m=1 qm(x) |x|2m on A. Show also that the series converges absolutely and uni-formly on compact subsets of A.
2.
Suppose u is a harmonic function with an isolated singularity at a ∈R2. The principal part of u at a is defined to be q0 log |x −a| + ∞ X m=1 qm(x −a) |x −a|2m , where u has been expanded about a as in Exercise 1. We say that u has a fundamental pole at a if the principal part is a nonzero multiple of log |x|. We say that u has a pole at a of order M if there is a largest positive integer M such that qM ̸= 0. We say that u has an essential singularity at a if the principal part has infinitely many nonzero terms. Prove that u has (a) a removable singularity at a if and only if lim x→a u(x) log |x −a| = 0; (b) a fundamental pole at a if and only if 0 < lim x→a u(x) log |x −a| < ∞; (c) a pole at a of order M if and only if 0 < lim sup x→a |x −a|M|u(x)| < ∞; (d) an essential singularity at a if and only if lim sup x→a |x −a|N|u(x)| = ∞ for every positive integer N.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 220 Chapter 10. Annular Regions 3.
Give an example of a harmonic function of n variables, n > 2, that has an essential singularity at 0.
4.
Let u be a real-valued harmonic function with an isolated singu-larity at a ∈Rn. Show that u has a fundamental pole at a if and only if lim x→a |u(x)| = ∞.
5.
Suppose n > 2 and u is a harmonic function with an isolated singularity at a. Prove that lim x→a |x −a|n−2u(x) exists (as a complex number) if and only if u has either a remov-able singularity or a fundamental pole at a.
6.
Singularities at ∞: Suppose u is harmonic on a deleted neigh-borhood of ∞. The singularity of u at ∞is classified using the Laurent expansion of the Kelvin transform K[u] at 0; for exam-ple, if the Laurent expansion of K[u] at 0 has vanishing principal part, then we say u has a removable singularity at ∞.
(a) Show that u has an essential singularity at ∞if and only if lim sup x→∞ |u(x)| |x|M = ∞ for every positive integer M.
(b) Find growth estimates that characterize the other types of singularities at ∞.
7.
(a) Identify those functions that are harmonic on Rn, n > 2, with fundamental pole at ∞.
(b) Identify those functions that are harmonic on R2 with fun-damental pole at ∞.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Exercises 221 8.
Suppose n = 2 and the harmonic function u has an isolated singularity at a ∈R2, with Laurent series expansion u(x) = ∞ X m=0 pm(x −a) + q0 log |x −a| + ∞ X m=1 qm(x −a) |x −a|2m .
We say that the constant q0 is the residue of u at a and write Res(u, a) = q0. Prove that if u is harmonic on B(a, r){a}, then Res(u, a) = 1 2π Z ∂B(a,r) Dnu ds.
Also prove an analogue of the residue theorem (10.8) for the case n = 2.
9.
Show how formulas 10.11 and 10.12 are derived.
10.
Find the correct replacements for 10.9 and 10.12 when n = 2 and m = 0, and use this to solve the Dirichlet problem for annular regions in the plane.
11.
Let 0 < r0 < 1 and let A = {x ∈Rn : r0 < |x| < 1}. Let p, q be polynomials on Rn, and let f be the function on ∂A that equals p on r0S and equals q on S. Prove that PA[f ] extends to a function that is harmonic on Rn \ {0}.
12.
Generalized Annular Dirichlet Problem: Suppose that A is the annulus {x ∈Rn : r0 < |x| < r1}, where 0 < r0 < r1 < ∞. Prove that if f, g, h are polynomials on Rn, then there is a function u ∈C(A) such that u = f on r0S, u = g on r1S, and ∆u = h on A. Show that if n > 2, then u is a finite sum of functions of the form p(x)/|x|m, where p is a polynomial on Rn and m is a nonnegative integer. (The software described in Appendix B can find u explicitly.) Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Chapter 11 The Dirichlet Problem and Boundary Behavior In this chapter we construct harmonic functions on Ωthat behave in a prescribed manner near ∂Ω. Here we are interested in general do-mains Ω⊂Rn; the techniques we developed for the special domains B and H will not be available. Most of this chapter will concern the Dirichlet problem. In the last section, however, we will study a differ-ent kind of boundary behavior problem—the construction of harmonic functions on Ωthat cannot be extended harmonically across any part of ∂Ω.
The Dirichlet Problem If f is a continuous function on ∂Ω, does f have a continuous ex-tension to Ωthat is harmonic on Ω? This is the Dirichlet problem for Ω with boundary data f. If the answer is affirmative for all continuous f on ∂Ω, we say that the Dirichlet problem for Ωis solvable. Recall that the Dirichlet problem is solvable for B (Theorem 1.17) and for the re-gion between two concentric spheres (Theorem 10.13), but not for the punctured ball B \ {0} (see the remark following the proof of 10.13).
The Dirichlet problem is sometimes referred to as “the first boundary value problem of potential theory”. The search for its solution led to the development of much of harmonic function theory.
We will obtain a necessary and sufficient condition for the Dirichlet problem to be solvable for bounded Ω. Although the condition is not 223 224 Chapter 11. The Dirichlet Problem and Boundary Behavior entirely satisfactory, it leads in many cases to easily verified geometric criteria that imply the Dirichlet problem is solvable. We will see, for example, that the Dirichlet problem is solvable for bounded Ωwhenever Ωis convex or whenever ∂Ωis “smooth”.
Note that when Ωis bounded, the maximum principle (1.9) implies that if a solution to the Dirichlet problem exists, then it is unique.
Subharmonic Functions We follow the so-called Perron approach in solving the Dirichlet problem. This ingenious method constructs a solution as the supre-mum of a family of subharmonic functions.
In this book, we call a real-valued function u subharmonic on Ωprovided u is continuous on Ωand u satisfies the submean-value property on Ω. The latter require-ment is that for each a ∈Ω, there exists a closed ball B(a, R) ⊂Ωsuch that 11.1 u(a) ≤ Z S u(a + rζ) dσ(ζ) whenever 0 < r ≤R. Note that we are not requiring 11.1 to hold for all r < d(a, ∂Ω). (But see Exercise 5 of this chapter.) Obviously every real-valued harmonic function on Ωis subharmonic on Ω. A finite sum of subharmonic functions is subharmonic, as is any positive scalar multiple of a subharmonic function. In Exercise 8 of this chapter we ask the reader to prove that a real-valued u ∈C2(Ω) is subharmonic on Ωif and only if ∆u ≥0 on Ω. Thus u(x) = |x|2 is a subharmonic function on Rn that is not harmonic. This example shows that subharmonic functions do not satisfy the minimum princi-ple. They do, however, satisfy the maximum principle.
11.2 Theorem: Suppose Ωis connected and u is subharmonic on Ω.
If u has a maximum in Ω, then u is constant.
Proof: Suppose u attains a maximum at a ∈Ω. Choose a closed ball B(a, R) ⊂Ωas in 11.1. We have u ≤u(a) on B(a, R). If u were less than u(a) at any point of B(a, R), then the continuity of u would show that 11.1 fails for some r < R. Thus u ≡u(a) on B(a, R). The set where u attains its maximum is therefore an open subset of Ω. Because Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Subharmonic Functions 225 this set is also closed in Ω, it must be all of Ωby the connectivity of Ω, proving that u is constant on Ω.
The following theorem indicates another sense in which subhar-monic functions are “sub”-harmonic.
11.3 Theorem: Let Ωbe bounded. Suppose u and v are continuous on Ω, u is subharmonic on Ω, and v is harmonic on Ω. If u ≤v on ∂Ω, then u ≤v on Ω.
Proof: Suppose u −v ≤0 on ∂Ω. Because u −v is subharmonic on Ω, the maximum principle (11.2) shows that u −v ≤0 through-out Ω.
The proof of the next result follows easily from the definition of subharmonic functions, and we leave it to the reader.
11.4 Proposition: If u1 and u2 are subharmonic functions on Ω, then max{u1, u2} is subharmonic on Ω.
Although Proposition 11.4 is easy, it indicates why subharmonic functions are useful—they can be “bent” in ways that harmonic func-tions simply would not tolerate. The next theorem is a more sophisti-cated bending result.
11.5 Theorem: Suppose that u is subharmonic on Ωand B(a, R) ⊂Ω.
Let w be the function that on Ω\ B(a, R) equals u|Ω\B(a,R) and that on B(a, R) equals the solution to the Dirichlet problem for B(a, R) with boundary data u|∂B(a,R). Then w is subharmonic on Ωand u ≤w.
Proof: The inequality u ≤w on Ωfollows from Theorem 11.3. The continuity of w on Ωis clear.
To verify that w satisfies the submean-value property on Ω, let b ∈Ω. If b ∈B(a, R), then the harmonicity of w near b implies that w(b) = R S w(b + rζ) dσ(ζ) for all sufficiently small r. If b ∉B(a, R), then u(b) = w(b). The subharmonicity of u, coupled with the inequal-ity u ≤w on Ω, then implies that w(b) ≤ R S w(b + rζ) dσ(ζ) for all sufficiently small r. Thus w is subharmonic on Ω.
We call the function w defined in Theorem 11.5 the Poisson modifi-cation of u for B(a, R).
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 226 Chapter 11. The Dirichlet Problem and Boundary Behavior The Perron Construction In this and the next section, Ωwill denote a bounded open subset of Rn and f will denote a continuous real-valued function on ∂Ω. (Note that the Dirichlet problem is no less general if one assumes the bound-ary data to be real.) We define Sf to be the family of real-valued, con-tinuous functions u on Ωthat are subharmonic on Ωand satisfy u ≤f on ∂Ω. The collection Sf is often called the Perron family for f . Note that Sf is never empty—it contains the constant function u(x) = m, where m is the minimum value of f on ∂Ω. (We are already using the boundedness of Ω.) The two bending processes mentioned in the last section preserve the family Sf . Specifically, if u1, u2 belong to Sf , then so does the function max{u1, u2}; if u ∈Sf and B(a, R) ⊂Ω, then the Poisson modification of u for B(a, R) belongs to Sf .
Perron’s candidate solution for the Dirichlet problem with boundary data f is the function defined on Ωby Pf = sup{u(x) : u ∈Sf }.
We call P[f ] the Perron function for f . Note that m ≤P[f ] ≤M on Ω, where m and M are the minimum and maximum values of f on ∂Ω.
Note also that P[f ] ≤f on ∂Ω.
To motivate the definition of P[f ], suppose that v solves the Dirich-let problem for Ωwith boundary data f . Then v ∈Sf , so that v ≤P[f ] on Ω. On the other hand, Theorem 11.3 shows that every function in Sf is bounded above on Ωby v, so that P[f ] ≤v on Ω. In other words, if there is a solution, it must be P[f ].
Remarkably, even though P[f ] may not be a solution, it is always harmonic.
11.6 Theorem: P[f ] is harmonic on Ω.
Proof: Let B(a, R) ⊂Ω. It suffices to show that P[f ] is harmonic on B(a, R).
Choose a sequence (uk) in Sf such that uk(a) →Pf . Replac-ing uk by the Poisson modification of max{u1, . . . , uk} for B(a, R), we obtain a sequence in Sf that increases on Ωand whose terms are har-monic on B(a, R). Denoting this new sequence by (uk) as well, we still Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Barrier Functions and Geometric Criteria for Solvability 227 have uk(a) →Pf < ∞. By Harnack’s Principle (3.8), (uk) con-verges uniformly on compact subsets of B(a, R) to a function u har-monic on B(a, R). The proof will be completed by showing P[f ] = u on B(a, R).
We clearly have u ≤P[f ] on B(a, R).
To prove the reverse in-equality, let v ∈Sf ; we need to show that v ≤u on B(a, R).
Let vk denote the Poisson modification of max{uk, v} for B(a, R). Because u(a) = Pf and vk ∈Sf , we have vk(a) ≤u(a) for all k. Further-more, vk is harmonic on B(a, R) and max{uk, v} ≤vk on B(a, R) by subharmonicity. Thus for positive r < R, u(a) ≥vk(a) = Z S vk(a + rζ) dσ(ζ) ≥ Z S max{uk, v} dσ(ζ).
Letting k →∞, we see that the mean-value property of u on B(a, R) gives Z S u(a + rζ) dσ(ζ) ≥ Z S max{u, v} dσ(ζ).
It follows that v ≤u on B(a, R), proving that P[f ] ≤u on B(a, R), as desired.
Barrier Functions and Geometric Criteria for Solvability Let ζ ∈∂Ω. We call a continuous real-valued function u on Ωa bar-rier function for Ωat ζ provided that u is subharmonic on Ω, u < 0 on Ω{ζ}, and u(ζ) = 0. When such a u exists, we say that Ωhas a barrier at ζ. (After Theorems 11.11 and 11.16, the reader may concede that the term “barrier” is apt. Poincaré introduced barrier functions into the study of the Dirichlet problem; Lebesgue coined the term “barrier” and generalized the notion.) Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 228 Chapter 11. The Dirichlet Problem and Boundary Behavior 11.7 Theorem: If Ωhas a barrier at ζ ∈∂Ω, then Pf →f (ζ) as x →ζ within Ω.
Proof: Suppose u is a barrier function for Ωat ζ ∈∂Ω. Let ε > 0.
By the continuity of f on ∂Ω, we may choose a ball B(ζ, r) such that f (ζ)−ε < f < f (ζ)+ε on ∂Ω∩B(ζ, r). Because u is negative and con-tinuous on the compact set Ω\ B(ζ, r), there exists a positive constant C such that 11.8 f (ζ) −ε + Cu < f < f (ζ) + ε −Cu on ∂Ω\ B(ζ, r). The nonpositivity of u then shows that 11.8 is valid everywhere on ∂Ω.
We claim that 11.9 f (ζ) −ε + Cu ≤P[f ] ≤f (ζ) + ε −Cu on Ω. The first inequality in 11.9 holds because f (ζ) −ε + Cu ∈Sf .
For the other inequality, let v ∈Sf . Then v ≤f on ∂Ω, and therefore v + Cu < f (ζ) + ε on ∂Ωby 11.8.
Theorem 11.3 then shows that v + Cu < f (ζ) + ε on Ω, from which the inequality on the right of 11.9 follows.
Because u(ζ) = 0 and ε is arbitrary, the continuity of u and 11.9 give us the desired convergence of Pf to f (ζ) as x →ζ within Ω.
11.10 Theorem: The Dirichlet problem for bounded Ωis solvable if and only if Ωhas a barrier at each ζ ∈∂Ω.
Proof: Suppose that the Dirichlet problem for Ωis solvable and ζ ∈∂Ω. The function f defined on ∂Ωby f (x) = −|x−ζ| is continuous on ∂Ω; the solution to the Dirichlet problem for Ωwith boundary data f is then a barrier function for Ωat ζ.
Conversely, suppose that each point of ∂Ωhas a barrier function.
Theorems 11.6 and 11.7 then show that P[f ] solves the Dirichlet prob-lem with boundary data f whenever f is continuous and real valued on ∂Ω.
Theorem 11.10 reduces the Dirichlet problem to a local boundary behavior question that we call the barrier problem.
Have we made Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Barrier Functions and Geometric Criteria for Solvability 229 progress, or have we merely traded one boundary behavior problem for another?
We will see that the barrier problem can be solved in many cases of interest. For example, the next result will enable us to prove that the Dirichlet problem is solvable for any bounded convex domain.
11.11 External Ball Condition: If ζ ∈∂Ωbelongs to a closed ball contained in the complement of Ω, then Ωhas a barrier at ζ.
Proof: Suppose B(a, r) is a closed ball in the complement of Ωsuch that ∂B(a, r) ∩∂Ω= {ζ}. Define u on Rn \ {a} by u(x) = ( log r −log |x −a| if n = 2 |x −a|2−n −r 2−n if n > 2.
Then u is a barrier for Ωat ζ.
The external ball condition.
Consider now the case where Ωis bounded and convex. Each ζ ∈∂Ω then belongs to a closed half-space contained in the complement of Ω, which implies that the external ball condition is satisfied at each ζ ∈∂Ω. By 11.11, Ωhas a barrier at every point in its boundary. The following corollary is thus a consequence of Theorem 11.10.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 230 Chapter 11. The Dirichlet Problem and Boundary Behavior 11.12 Corollary: If Ωis bounded and convex, then the Dirichlet prob-lem for Ωis solvable.
Theorem 11.11 also enables us to prove that the Dirichlet problem is solvable for bounded Ωwhen ∂Ωis sufficiently smooth. To make this more precise, let us say that Ωhas Ck-boundary if for every ζ ∈∂Ω there exists a neighborhood ω of ζ and a real-valued ϕ ∈Ck(ω) sat-isfying the following conditions: (a) Ω∩ω = {x ∈ω : ϕ(x) < 0}; (b) ∂Ω∩ω = {x ∈ω : ϕ(x) = 0}; (c) ∇ϕ(ζ) ̸= 0 for every ζ ∈∂Ω∩ω.
Here k is any positive integer. The function ϕ is called a local defining function for Ω.
Assume now that Ωhas C2-boundary. For simplicity, suppose that 0 ∈∂Ωand that the tangent space of ∂Ωat 0 is Rn−1 × {0}.
Then near 0, ∂Ωis the graph of a C2-function ψ, where ψ is defined near 0 ∈Rn−1 and ∇ψ(0) = 0; this follows easily from the implicit function theorem. Because ∇ψ(0) = 0, we have |ψ(x)| = O(|x|2) as x →0 (by Taylor’s Theorem). This implies (we leave the details to the reader) that Ωsatisfies the external ball condition at 0. The preceding argument, after a translation and rotation, applies to any boundary point of Ω.
From Theorems 11.10 and 11.11 we thus obtain the following result.
11.13 Corollary: If Ωis bounded and has C2-boundary, then the Dirichlet problem for Ωis solvable.
A domain with C1-boundary need not satisfy the external ball condi-tion (Exercise 16(a) of this chapter). We now take up a condition on ∂Ω that covers the C1-case as well as many “nonsmooth” cases. The proto-type for this more-general Ωis the domain B \Γ α(0), where Γα(0) is the cone defined in Chapter 2; see the following diagram, where B \ Γ α(0) and one of its dilates are pictured.
We will need the following maximum principle for B \ Γ α(0).
11.14 Lemma: Let Ω= B \ Γ α(0).
Suppose u is real valued and continuous on Ω\ {0}, u is bounded and harmonic on Ω, and u ≤M on ∂Ω\ {0}. Then u ≤M on Ω.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Barrier Functions and Geometric Criteria for Solvability 231 Proof: If n > 2, apply the maximum principle (Corollary 1.10) to the function u(x) −M −ε|x|2−n on Ωand let ε →0. When n = 2, the same argument applies with log 1/|x| in place of |x|2−n.
0 B \ Γ α(0) and one of its dilates.
With Ωas in Lemma 11.14, note that rΩ= (rB) ∩Ωfor every r ∈(0, 1). This fact will be crucial in proving the next result.
11.15 Lemma: Let Ω= B \ Γ α(0). On ∂Ω, set f (x) = |x|, and on Ω, set u = P[f ]. Then −u is a barrier for Ωat 0.
Proof: The function u is harmonic on Ω, with 0 ≤u ≤1 on Ω. The external ball condition holds at each point of ∂Ω\ {0}, so u is contin-uous and positive on Ω\ {0}. The proof will be completed by showing that lim supx→0 u(x) = 0.
Fix r ∈(0, 1). By the maximum principle, there exists a constant c < 1 such that u ≤c on ∂(rΩ) \ {0}. Now define v(x) = u(x) −max{r, c}u(x/r) for x ∈rΩ. It is easy to check that v ≤0 on ∂(rΩ) \ {0}. Applying Lemma 11.14, we obtain v ≤0 on rΩ. Thus Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 232 Chapter 11. The Dirichlet Problem and Boundary Behavior lim sup x→0 u(x) ≤max{r, c} lim sup x→0 u(x/r) = max{r, c} lim sup x→0 u(x).
Because max{r, c} < 1, this implies that lim supx→0 u(x) = 0, which completes the proof.
The next result shows that the Dirichlet problem is solvable for ev-ery bounded domain satisfying the “external cone condition” at each of its boundary points. By a cone we shall mean any set of the form a+T Γ h α (0) , where a ∈Rn, T is a rotation, and Γ h α (0) is the truncation of Γα(0) defined in Chapter 7. We refer to a as the vertex of such a cone.
11.16 External Cone Condition: If Ωis bounded and ζ ∈∂Ωis the vertex of a cone contained in the complement of Ω, then Ωhas a barrier at ζ.
Proof: Subharmonic functions are preserved by translations, dila-tions, and rotations, so without loss of generality we may assume that ζ = 0 and that Ω∩B ⊂B \ Γ α(0) for some α > 0. Consider now the function −u obtained in Lemma 11.15. This function is identically −1 on S \ Γ α(0). Thus if we extend this function by defining it to equal −1 on Rn \ B, the result is a barrier for Ωat 0.
An Ωsatisfying the external cone condition.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Nonextendability Results 233 Nonextendability Results We now turn from the Dirichlet problem to another kind of bound-ary behavior question. Recall that if Ω⊂R2 = C, then there exists a function holomorphic on Ωthat does not extend holomorphically to any larger set. This is usually proved by first showing that each discrete subset of Ωis the zero set of some function holomorphic on Ω. Because a discrete subset of Ωthat clusters at each point of ∂Ωcan always be chosen, some holomorphic function on Ωis nonextendable.
Remarkably, there exist domains in Cm, m > 1, for which every holomorphic function extends across the boundary. For example, every holomorphic function on C2 \ {0} extends to be entire on C2; see , pages 5–6, for details.
What about harmonic functions of more than two real variables?
The next result shows that given any Ω⊂Rn, n > 2, a nonextendable positive harmonic function on Ωcan always be produced.
11.17 Theorem: Suppose n > 2 and Ω⊂Rn. Then there exists a positive harmonic function u on Ωsuch that lim sup x→ζ u(x) = ∞ for every ζ ∈∂Ω.
Proof: Assume first that Ωis connected. Let {ζ1, ζ2, . . . } be a count-able dense subset of ∂Ω. Fixing a ∈Ω, we may choose positive con-stants cm such that cm|a −ζm|2−n < 2−m for m = 1, 2, . . . . For x ∈Ω, define u(x) = ∞ X m=1 cm|x −ζm|2−n.
Each term in this series is positive and harmonic on Ω, and the series converges at a ∈Ω. By Harnack’s Principle, u is harmonic on Ω, and we easily verify that u satisfies the conclusion of the theorem.
If Ωis not connected, we apply the preceding to each connected component of Ωto produce the desired function.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 234 Chapter 11. The Dirichlet Problem and Boundary Behavior If Ω⊂R2, it may not be possible to construct a positive harmonic function on Ωthat is unbounded near every point of ∂Ω. Recall, for example, that every positive harmonic function on R2 \ {0} is constant (Corollary 3.3). Our next result, however, is valid for all n ≥2.
11.18 Theorem: Let Ω⊂Rn. Then there exists a real-valued harmonic function u on Ωsuch that lim inf x→ζ u(x) = −∞, lim sup x→ζ u(x) = ∞ for every ζ ∈∂Ω.
Proof: In the proof we will assume that n > 2; we leave the n = 2 case as an exercise.
Let I denote the set of isolated points of ∂Ω. We assume that ∂Ω\ I is nonempty; the proof that follows will easily adapt to the case ∂Ω= I.
Select disjoint countable dense subsets D−and D+ of ∂Ω\ I, and write D−∪D+ ∪I = {ζ1, ζ2, . . . }.
Now choose pairwise disjoint compact sets E1, E2, . . . such that for each m, (a) Em ⊂Ω∪{ζm}; (b) ζm is a limit point of Em.
For ζm ∈I we insist that Em be a closed ball of positive radius centered at ζm. For ζm ∈D−∪D+ we will not be as fussy; for example, we can take Em to be the union of {ζm} with a sequence in Ωconverging to ζm.
The pairwise disjointness of {Em} is easy to arrange by induction.
Set v(x) = |x|2−n, w(x) = x1|x|−n, and for m = 1, 2, . . ., define um(x) = −v(x −ζm) if ζm ∈D− v(x −ζm) if ζm ∈D+ w(x −ζm) if ζm ∈I.
Note that um is harmonic on Rn \ {ζm}.
Choose compact sets K1, K2, . . . ⊂Ωsuch that K1 ⊂int K2 ⊂K2 ⊂· · · Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Nonextendability Results 235 and Ω= ∪Km. We may then choose positive constants cm such that cm|um| ≤2−m on Km ∪E1 ∪· · · ∪Em−1. (Let E0 be the empty set.) Finally, define u = ∞ X m=1 cmum.
Because this series converges uniformly on compact subsets of Ω, u is harmonic on Ω.
To check the boundary behavior of u, we first consider the case where ζ ∈∂Ω\ I. Let ε > 0. Then B(ζ, ε) contains some ζm ∈D−.
On the corresponding Em, the series P j̸=m cjuj converges uniformly to a function continuous on Em. Because cmum(x) →−∞as x →ζm within Em, the infimum of u over B(ζ, ε) ∩Ωis −∞.
Similarly, the supremum of u over B(ζ, ε) ∩Ωis ∞, giving us the desired conclu-sion.
Now suppose that ζ ∈I. If ε > 0, then B(ζ, ε) contains some ζm ∈I.
The series P j̸=m cjuj then converges uniformly to a continuous func-tion on the closed ball Em. Because cmum in this case maps any punc-tured ball B(ζm, r) \ {ζm} onto R, we are done.
Note that if Ωis locally connected near ∂Ω(for example, if Ωis convex or has C1-boundary), then the function u of Theorem 11.18 satisfies u B(ζ, ε) ∩Ω = R for every ζ ∈∂Ωand every ε > 0. (Also see Exercise 22 of this chapter.) Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 236 Chapter 11. The Dirichlet Problem and Boundary Behavior Exercises 1.
Suppose that Ωis a simply connected domain in the plane whose boundary is a Jordan curve. Explain how to use a suitable ver-sion of the Riemann mapping theorem to show that the Dirichlet problem for Ωis solvable.
2.
Show that every translation, dilation, and rotation of a subhar-monic function is subharmonic.
3.
Suppose that u is subharmonic on Ωand that ϕ is increasing and convex on an open interval containing u(Ω). Prove that ϕ ◦u is subharmonic on Ω.
4.
Let u be a real-valued continuous function on Ω. Show that u is subharmonic on Ωif and only if for every compact K ⊂Ωthe following holds: if u ≤v on ∂K, where v is continuous on K and harmonic on the interior of K, then u ≤v on K.
5.
Suppose that u is subharmonic on Ωand a ∈Ω. Show that the function r 7→ R S u(a + rζ) dσ(ζ) is increasing. Conclude that the submean-value inequality 11.1 is valid for all r < d(a, ∂Ω).
6.
Show that if a sequence of functions subharmonic on Ωcon-verges uniformly on compact subsets of Ω, then the limit func-tion is subharmonic on Ω.
7.
Show that |u|p is subharmonic on Ωwhenever u is harmonic on Ωand 1 ≤p < ∞. (This and Exercise 5 imply that ∥ur ∥p is an increasing function of r, giving an alternative proof of Corol-lary 6.6.) 8.
Suppose u ∈C2(Ω) is real valued. Use Taylor’s Theorem to show that u is subharmonic on Ωif and only if ∆u ≥0 on Ω. (Hint: To show ∆u ≥0 implies u is subharmonic, first assume ∆u > 0.
Then consider the functions u(x) + ε|x|2.) 9.
Show that |x|p is subharmonic on Rn \ {0} for every p > 2 −n.
Also show that |x|p is subharmonic on Rn for every p > 0.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Exercises 237 10.
(a) Show that a subharmonic function on R2 that is bounded above must be constant.
(b) Suppose n > 2. Find a nonconstant subharmonic function on Rn that is bounded above.
11.
Assume that f is holomorphic on Ω⊂R2 = C and that u is subharmonic on f (Ω). Prove that u ◦f is subharmonic on Ω.
12.
Give an easier proof of Theorem 11.16 for the case n = 2.
13.
With n = 3, let Ωdenote the open unit ball with the x3-axis removed. Show that the Dirichlet problem for Ωis not solvable.
14.
Show that when n = 2, the Dirichlet problem is solvable for bounded open sets satisfying an “external segment condition”.
(Hint: An appropriate conformal map of B2 \ {0} onto the plane minus a line segment may be useful.) 15.
Show that the Dirichlet problem is solvable for B3 \ (H2 × {0}).
16.
(a) Give an example of a bounded Ωwith C1-boundary such that the external ball condition fails for some ζ ∈∂Ω.
(b) Show that a domain with C1-boundary satisfies the external cone condition at each of its boundary points.
17.
Suppose Ωis bounded. Show that the Dirichlet problem for Ωis solvable if and only if P[−f ] = −P[f ] on Ωfor every real-valued continuous f on ∂Ω.
18.
Suppose Ωis bounded and a ∈Ω.
Show that there exists a unique positive Borel measure µa on ∂Ω, with µa(∂Ω) = 1, such that Pf = Z ∂Ω f dµa for every real-valued continuous f on ∂Ω. (The measure µa is called harmonic measure for Ωat a.) 19.
Prove Theorem 11.18 in the case n = 2.
20.
Show that if Ω⊂R2 is bounded, then there exists a positive harmonic function u on Ωsuch that lim supx→ζ u(x) = ∞for every ζ ∈∂Ω.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 238 Chapter 11. The Dirichlet Problem and Boundary Behavior 21.
Suppose Ω⊂C and {ζ1, ζ2, . . . } is a countable dense subset of ∂Ω. Construct a holomorphic function on Ωof the form ∞ X m=1 cm z −ζm that does not extend across any part of ∂Ω. (This should be easier than the proof of Theorem 11.18.) 22.
Given an arbitrary Ω⊂Rn, does there always exist a real-valued harmonic u on Ωsuch that u B(ζ, ε) ∩Ω = R for every ζ ∈∂Ω and every ε > 0?
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Appendix A Volume, Surface Area, and Integration on Spheres Volume of the Ball and Surface Area of the Sphere In this section we compute the volume of the unit ball and surface area of the unit sphere in Rn.
Recall that B = Bn denotes the unit ball in Rn and that V = Vn denotes volume measure in Rn. We begin by evaluating the constant V(B), which appears in several formulas throughout the book.
A.1 Proposition: The volume of the unit ball in Rn equals πn/2 (n/2)!
if n is even, 2(n+1)/2π(n−1)/2 1 · 3 · 5 · · · n if n is odd.
Proof: Assume n > 2, denote a typical point in Rn by (x, y), where x ∈R2 and y ∈Rn−2, and express the volume Vn(Bn) as an iterated integral: Vn(Bn) = Z Bn 1 dVn = Z B2 Z (1−|x|2)1/2Bn−2 1 dVn−2(y) dV2(x).
239 240 Appendix A. Volume, Surface Area, and Integration on Spheres The inner integral on the last line equals the (n−2)-dimensional volume of a ball in Rn−2 with radius (1 −|x|2) 1/2. Thus Vn(Bn) = Vn−2(Bn−2) Z B2 (1 −|x|2) (n−2)/2 dV2(x).
Switching to the usual polar coordinates in R2, we get Vn(Bn) = Vn−2(Bn−2) Z π −π Z 1 0 (1 −r 2)(n−2)/2r dr dθ = 2π n Vn−2(Bn−2).
The last formula can be easily used to prove the desired formula for Vn(Bn) by induction in steps of 2, starting with the well-known results V2(B2) = π and V3(B3) = 4π/3.
Readers familiar with the gamma function should be able to rewrite the formula given by A.1 as a single expression that holds whether n is even or odd (see Exercise 7 of this appendix).
Turning now to surface-area measure, we let Sn denote the unit sphere in Rn. Unnormalized surface-area measure on Sn will be de-noted by sn and normalized surface-area measure on Sn will be de-noted by σn. Some of the arguments we give in the remainder of this appendix will be more intuitive than rigorous; the reader should have no trouble filling in the missing details. We presume some familiarity with surface-area measure.
Let us now find the relationship between Vn(Bn) and sn(Sn). We do this with an old trick from calculus. For h ≈0 we have (1 + h)n −1 Vn(Bn) = Vn (1 + h)Bn −Vn(Bn) ≈sn(Sn)h.
Dividing by h and letting h →0, we obtain nVn(Bn) = sn(Sn). We record this result in the following proposition.
A.2 Proposition: The unnormalized surface area of the unit sphere in Rn equals nVn(Bn).
A more common notation is Sn−1, which emphasizes that the sphere has dimension n −1 as a manifold. We use Sn to emphasize that the sphere lives in Rn.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Slice Integration on Spheres 241 Slice Integration on Spheres The map ψ: Bn−1 →Sn defined by ψ(x) = x, q 1 −|x|2 parameterizes the upper hemisphere of Sn. The corresponding change of variables is given by the formula A.3 dsn ψ(x) = dVn−1(x) p 1 −|x|2 .
Equation A.3 is found in most calculus texts in the cases n = 2, 3.
Consider now the map Ψ : Bn−k × Sk →Sn defined by Ψ(x, ζ) = x, q 1 −|x|2 ζ .
Here 1 ≤k < n. The map Ψ is one-to-one, and the range of Ψ is Sn minus a set that has sn-measure 0 (namely, the set of points on Sn whose last k coordinates vanish). We wish to find the change of variables formula associated with Ψ.
Observe that Bn−k ×Sk is an (n−1)-dimensional submanifold of Rn whose element of surface area is d(Vn−k × sk). For fixed x, Ψ changes (k −1)-dimensional area on {x} × Sk by the factor (1 −|x|2)(k−1)/2.
For fixed ζ, A.3 shows that Ψ changes (n −k)-dimensional area on Bn−k ×{ζ} by the factor (1−|x|2)−1/2. Furthermore, the submanifolds Ψ({x}×Sk) and Ψ(Bn−k ×{ζ}) are perpendicular at their point of inter-section, as is easily checked. The last statement implies that Ψ changes (n−1)-dimensional measure on Bn−k ×Sk by the product of the factors above. In other words, dsn Ψ(x, ζ) = (1 −|x|2)(k−2)/2 dVn−k(x) dsk(ζ).
The last equation and A.2 lead to the useful formula given in the next theorem. This formula shows how the integral over a sphere can be calculated by iterating an integral over lower-dimensional spheri-cal slices. We state the formula in terms of normalized surface-area measure because that is what we have used most often.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 242 Appendix A. Volume, Surface Area, and Integration on Spheres A.4 Theorem: Let f be a Borel measurable, integrable function on Sn.
If 1 ≤k < n, then Z Sn f dσn = k n V(Bk) V(Bn) Z Bn−k (1 −|x|2) k−2 2 Z Sk f x, q 1 −|x|2 ζ dσk(ζ) dVn−k(x).
Some cases of Theorem A.4 deserve special mention. We begin by choosing k = n −1, which is the largest permissible value of k. This corresponds to decomposing Sn into spheres of one less dimension by intersecting Sn with the family of hyperplanes orthogonal to the first coordinate axis. The ball Bn−k is just the unit interval (−1, 1), and so for x ∈Bn−k we can write x2 instead of |x|2. Thus we obtain the following corollary of Theorem A.4.
A.5 Corollary: Let f be a Borel measurable, integrable function on Sn.
Then Z Sn f dσn = n −1 n V(Bn−1) V(Bn) Z 1 −1 (1 −x2) n−3 2 Z Sn−1 f x, p 1 −x2 ζ dσn−1(ζ) dx.
At the other extreme we can choose k = 1. This corresponds to decomposing Sn into pairs of points by intersecting Sn with the family of lines parallel to the nth coordinate axis. The sphere S1 is the two-point set {−1, 1}, and dσ1 is counting measure on this set, normalized so that each point has measure 1/2.
Thus we obtain the following corollary of Theorem A.4.
A.6 Corollary: Let f be a Borel measurable, integrable function on Sn.
Then Z Sn f dσn = 1 nV(Bn) Z Bn−1 f x, p 1 −|x|2 + f x, − p 1 −|x|2 p 1 −|x|2 dVn−1(x).
Let us now try k = 2 (assuming n > 2).
Thus in A.4 the term (1 −|x|2)(k−2)/2 disappears. The variable ζ in the formula given by Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Slice Integration on Spheres 243 Theorem A.4 now ranges over the unit circle in R2, so we can replace ζ by (cos θ, sin θ), which makes dσ2(ζ) equal to dθ/(2π). Thus we obtain the following corollary of A.4.
A.7 Corollary (n > 2): Let f be a Borel measurable, integrable func-tion on Sn. Then Z Sn f dσn = 1 nV(Bn) Z Bn−2 Z π −π f x, q 1 −|x|2 cos θ, q 1 −|x|2 sin θ dθ dVn−2(x).
An important special case of the last result occurs when n = 3.
In this case Bn−2 is just the interval (−1, 1), and we get the following corollary.
A.8 Corollary: Let f be a Borel measurable, integrable function on S3.
Then Z S3 f dσ3 = 1 4π Z 1 −1 Z π −π f x, p 1 −x2 cos θ, p 1 −x2 sin θ dθ dx.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 244 Appendix A. Volume, Surface Area, and Integration on Spheres Exercises 1.
Prove that Z Rn 1 (|x| + 1)p dV(x) < ∞ if and only if p > n.
2.
(a) Consider the region on the unit sphere in R3 lying between two parallel planes that intersect the sphere. Show that the area of this region depends only on the distance between the two planes. (This result was discovered by the ancient Greeks.) (b) Show that the result in part (a) does not hold in Rn if n ̸= 3 and “planes” are replaced by “hyperplanes”.
3.
Let f be a Borel measurable, integrable function on the unit sphere S4 in R4. Define a function Ψ mapping the rectangular box [−1, 1] × [−1, 1] × [−π, π] to S4 by setting Ψ(x, y, θ) equal to x, q 1 −x2 y, q 1 −x2 q 1 −y2 cos θ, q 1 −x2 q 1 −y2 sin θ .
Prove that Z S4 f dσ4 = 1 2π2 Z 1 −1 q 1 −x2 Z 1 −1 Z π −π f Ψ(x, y, θ) dθ dy dx.
4.
Without writing down anything or using a computer, evaluate Z S7 ζ1 2 dσ(ζ).
5.
Let m be a positive integer. Use A.5 to give an explicit formula for Z 1 −1 (1 −x2)m/2 dx.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Exercises 245 6.
Suppose that m is a positive integer.
(a) Prove that Z S3 ζ1 m dσ(ζ) = ( 1 m+1 if m is even 0 if m is odd.
(b) Find a formula for Z S4 ζ1 m dσ(ζ).
7.
For readers familiar with the gamma function Γ: prove that the volume of the unit ball in Rn equals πn/2 Γ n 2 + 1 .
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Appendix B Harmonic Function Theory and Mathematica Using the computational environment provided by Mathematica, the authors have written software to manipulate many of the expres-sions that arise in the study of harmonic functions. This software al-lows the user to make symbolic calculations that would take a pro-hibitive amount of time if done without a computer. For example, Pois-son integrals of polynomials can be computed exactly.
Our software for symbolic manipulation of harmonic functions is available over the internet without charge. It is distributed as a Mathe-matica package that will work on any computer that runs Mathematica.
This Mathematica package and the instructions for using it are avail-able at and in the standard electronic mathematical archives (search for files containing HFT or ComputingWithHarmonicFunctions in their name). Comments, sugges-tions, and bug reports should be sent to axler@sfsu.edu.
Here are some of the capabilities of our Mathematica package: • symbolic calculus in Rn, including integration on balls and spheres; • solution of the Dirichlet problem for balls, ellipsoids, annular regions, and exteriors of balls in Rn (exact solutions with polynomial data); • solution of the Neumann problem for balls, ellipsoids and exteriors of balls in Rn (exact solutions with polynomial data); Mathematica is a registered trademark of Wolfram Research, Inc.
247 248 Appendix B. Harmonic Function Theory and Mathematica • computation of bases for spaces of spherical harmonics in Rn; • computation of the Bergman projection for balls in Rn; • manipulations with the Kelvin transform K and the modified Kelvin transform K; • computation of the extremal function given by the Harmonic Schwarz Lemma (6.24) for balls in Rn; • computation of harmonic conjugates in R2.
New features are frequently added to this software.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey References Harmonic function theory has a rich history and a continuing high level of research activity. MathSciNet lists over one thousand papers published just in the first two decades of the twenty-first century for which the review contains the phrase “harmonic function”. Although we have drawn freely from this heritage, we list here only those works cited in the text.
V. Anandam and M. Damlakhi, Bôcher’s Theorem in R2 and Carathéodory’s Inequality, Real Analysis Exchange 19 (1993/94), 537–539.
Sheldon Axler, Harmonic functions from a complex analysis viewpoint, American Mathematical Monthly 93 (1986), 246–258.
Sheldon Axler, Paul Bourdon, and Wade Ramey, Bôcher’s Theorem, American Mathematical Monthly 99 (1992), 51–55.
Sheldon Axler and Wade Ramey, Harmonic polynomials and Dirichlet-type problems, Proceedings of the American Mathematical Society 123 (1995), 3765–3773.
Stefan Bergman, The Kernel Function and Conformal Mapping, American Mathematical Society, 1950.
Ronald R. Coifman and Guido Weiss, Analyse Harmonique Non-commutative sur Certains Espaces Homogenes, Lecture Notes in Mathematics, Springer, 1971.
John B. Conway, Functions of One Complex Variable, second edition, Graduate Texts in Mathematics, Springer, 1978.
249 250 References P. Fatou, Séries trigonométriques et séries de Taylor, Acta Mathematica 30 (1906), 335–400.
L. L. Helms, Introduction to Potential Theory, Wiley-Interscience, 1969.
Yujiro Ishikawa, Mitsuru Nakai, and Toshimasa Tada, A form of classical Picard Principle, Proceedings of the Japan Academy, Series A, Mathematical Sciences 72 (1996), 6–7.
Oliver Dimon Kellogg, Foundations of Potential Theory, Springer, 1929.
Steven G. Krantz, Function Theory of Several Complex Variables, John Wiley, 1982.
Edward Nelson, A proof of Liouville’s Theorem, Proceedings of the American Mathematical Society 12 (1961), 995.
Walter Rudin, Principles of Mathematical Analysis, third edition, McGraw-Hill, 1976.
Walter Rudin, Real and Complex Analysis, third edition, McGraw-Hill, 1987.
Elias M. Stein and Guido Weiss, Fourier Analysis on Euclidean Spaces, Princeton University Press, 1971.
William Thomson (Lord Kelvin), Extraits de deux lettres adressées à M. Liouville, Journal de Mathematiques Pures et Appliqués 12 (1847), 256–264.
William Thomson (Lord Kelvin) and Peter Guthrie Tait, Treatise on Natural Philosophy, Cambridge University Press, 1867.
John Wermer, Potential Theory, Lecture Notes in Mathematics, Springer, 1974.
Hermann Weyl, On the volume of tubes, American Journal of Mathematics 61 (1939), 461–472.
Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Symbol Index Symbols are sorted by ignoring everything except Latin and Greek letters. For sorting purposes, Greek letters are assumed to be spelled out in full with Latin letters. For example, ΩE is translated to “OmegaE”, which is then sorted with other entries beginning with “O” and before O(n), which translates to “On”. The symbols that contain no Latin or Greek letters appear first, sorted by page number.
| |, 1 ∇, 4 L , 76, 81 [ ], 77 ⟨, ⟩, 79 ⊥, 136 ∂, 206 A, 215 α, 15 α!, 19 |α|, 19 A[u], 51 B, 5 B, 5 B(a, r), 4 B(a, r), 5 bm, 215 Bn, 5 ∥∥bp, 171 bp(Ω), 171 Ck c , 192 Cc(Rn−1), 148 C(E), 2 χE, 18 Ck(Ω), 2 cm, 87, 217 cm for n = 2, 107 cn, 144 C∞(Ω), 2 C0(Rn−1), 146 d(a, E), 34 Dα, 15 ∆, 1 Dm, 3 Dn, 4 Dr , 31, 45 ds, 4 251 252 Symbol Index dσ, 5 dσn, 240 dsn, 240 dV, 4 dVn, 4 dx, 144 dy, 192 E∗, 60 Γ, 245 Γα, 38 Γ h α , 161 H, 32 hm, 95 Hm(Rn), 75 Hm(S), 80 Hn, 32 h∞(Ω), 200 ∥∥hp, 117, 151 hp(B), 117 hp(H), 151 K, 61 K, 155 3κ, 133 κ(ζ, δ), 131 Lp(Rn−1), 146 Lp(S), 112 L2(S), 79 M, 131 M(Rn−1), 146 M(S), 111 ∥µ∥, 111 µf , 112 n, 1 n, 4 N, 103 Nα[u], 129 Ω, 1 Ωα(ζ), 128 ΩE, 67 O(n), 95 ∥∥p, 112, 146 PA[f], 217 PA(x, ζ), 216 p(D), 85 Pe[f], 66 Pe(x, ζ), 66 P[f], 12, 112 P[f], 226 PH[f], 146 Φ, 154 PH[µ], 146 PH(z, t), 145 PH(z, w), 185 Pm(Rn), 75 P[µ], 111 P(x, y), 122 P(x, ζ), 12 QΩ, 175 R, 20 R, 129 Res, 213 Rn ∪{∞}, 59 RΩ, 173 s, 4 S, 5 S+, 123 S−, 123 S, 154 Sf , 226 σ, 5 σn, 240 Sn, 240 sn, 240 supp, 192 U, 123 Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Symbol Index 253 Un, 123 ur , 2, 112 uy, 143 V, 4 Vn, 4 w, 185 x∗, 59 X∗, 115 xα, 19 xE, 67, 68 Zm(x, y), 176 Zm(ζ, η), 94 Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Index annular region, 209 approximate identity, 13, 144 Arzela-Ascoli Theorem, 36 Baire’s Theorem, 42 balls internally tangent to B, 190 barrier, 227 barrier function, 227 barrier problem, 228 basis of Hm(Rn), Hm(S), 92 Bergman projection, 175 Bergman space, 171 Bergman, Stefan, 171 Bloch space, 43, 189 Bˆ ocher’s Theorem, 50, 57, 197, 199 boundary data, 15 bounded harmonic function, 31 bounded harmonic function on B, 40, 119 canonical projection of Pm(Rn) onto Hm(Rn), 77 Cauchy’s Estimates, 33 Cauchy, Augustin-Louis, 34 computer package, 80, 90, 93, 106, 125, 180, 218, 221, 247 cone, 232 conformal map, 60 conjugate index, 112 convex region, 230 covering lemma, 133 decomposition theorem, 195 decomposition theorem for holomorphic functions, 206 degree, 23, 80 dilate, 2 direct sum, 81 Dirichlet problem, 12, 223 Dirichlet problem for H, 146 Dirichlet problem for annular regions, 215 Dirichlet problem for annular regions (n = 2), 221 Dirichlet problem for convex regions, 230 Dirichlet problem for smooth regions, 230 Dirichlet, Johann Peter Gustav Lejeune, 13 255 256 Index divergence theorem, 4 dual space, 115 equicontinuity, 117 essential singularity, 211 essential singularity (n = 2), 219 essential singularity at ∞, 220 exterior Dirichlet problem, 66 exterior Poisson integral, 66 exterior Poisson kernel, 66 external ball condition, 229 external cone condition, 232 external segment condition, 237 extremal function, 124 extreme point, 140 Fatou Theorem, 128, 160 finitely connected, 203 Fourier series, 82, 97 fundamental pole, 211 fundamental pole (n = 2), 219 fundamental pole at ∞, 220 fundamental solution of the Laplacian, 193 gamma function, 245 generalized annular Dirichlet problem, 221 generalized Dirichlet problem, 106 Green’s identity, 4 Hardy, G. H., 118 Hardy-Littlewood maximal function, 131 harmonic, 1 harmonic at ∞, 63 harmonic Bergman space, 171 harmonic Bloch space, 43, 189 harmonic conjugate, 203 harmonic functions, limits of, 16, 49 harmonic Hardy space, 118, 151 harmonic measure, 237 harmonic motion, 25 harmonics, 25 Harnack’s Inequality, 48 Harnack’s Inequality for B, 47, 56 Harnack’s Principle, 49 holomorphic at ∞, 71 homogeneous expansion, 24, 99 homogeneous harmonic polynomial, 24, 75 homogeneous part, 75 homogeneous polynomial, 23, 75 Hopf Lemma, 28 inversion, 60 inversion map, 153 isolated singularity, 32, 210 isolated singularity at ∞, 61 isolated singularity of positive harmonic function, 50 isolated zero, 6 Kelvin transform, 59, 61, 155 Laplace’s equation, 1 Laplacian, 1 Laurent series, 193, 209 law of cosines, 130 Lebesgue decomposition, 136 Lebesgue Differentiation Theorem, 165 Lebesgue point, 141 limits along rays, 39 Liouville’s Theorem, 31 Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Index 257 Liouville’s Theorem for positive harmonic functions, 45, 56, 198 local defining function, 230 Local Fatou Theorem, 161 locally connected, 235 locally integrable, 18 logarithmic conjugation theorem, 203 Mathematica, 247 maximum principle, 7, 36 maximum principle for subharmonic functions, 224 maximum principle, local, 23 mean-value property, 5 mean-value property, converse of, 17 mean-value property, volume version, 6 minimum principle, 7 Morera’s Theorem, 213 multi-index, 15 Neumann problem, 108 nonextendability of harmonic functions, 233 nontangential approach region, 38, 128 nontangential limit, 38, 128, 160 nontangential maximal function, 129 nontangentially bounded, 161 normal family, 35 north pole, 103 one-point compactification, 59 one-radius theorem, 28 open mapping, 27 open mapping theorem, 181 operator norm, 115 order of a pole, 211 order of a pole (n = 2), 219 orthogonal transformation, 3, 95 orthonormal basis of b2(B2), 189 parallel orthogonal to η, 100 Perron family, 226 Perron function, 226 Picard’s Theorem, 212 point evaluation, 172 point of density, 165 Poisson integral, 12 Poisson integral for H, 146 Poisson integral for annular region, 217 Poisson integral of a measure, 111 Poisson integral of a polynomial, 74, 89, 98, 105 Poisson kernel, 9, 12, 99, 122, 157 Poisson kernel for H, 144, 157, 185 Poisson kernel for annular region, 215 Poisson kernel, expansion into zonal harmonics, 99 Poisson modification, 225 Poisson’s equation, 193 polar coordinates, integration in, 6 polar coordinates, Laplacian in, 26 pole, 211 pole (n = 2), 219 positive harmonic function, 45 positive harmonic function on B, 55, 119 positive harmonic function on H, 156 Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey 258 Index positive harmonic function on R2 \ {0}, 46 positive harmonic function on Rn \ {0}, 54 power series, 19 principal part, 211 principal part (n = 2), 219 product rule for Laplacian, 13 punctured ball, 218 radial derivative, 141 radial function, 27, 52, 57, 101 radial harmonic function, 52, 57 radial limit, 185 radial maximal function, 129 real analytic, 19 reflection about a hyperplane, 67 reflection about a sphere, 68 removable sets, 200 removable singularity, 32, 188, 211 removable singularity at ∞, 61, 220 reproducing kernel, 172 reproducing kernel of B, 176 reproducing kernel of H, 185 residue, 213 residue (n = 2), 221 residue theorem, 213 residue theorem (n = 2), 221 Riemann-Lebesgue Lemma, 183 Riesz Representation Theorem, 111 rotation, 4 Schwarz Lemma, 123 Schwarz Lemma for ∇u, 125 Schwarz Lemma for h2, 141 Schwarz reflection principle, 67 Schwarz, Hermann Amandus, 123 separable normed linear space, 117 simply connnected, 203 singular measure, 136 singularity at ∞, 220 slice integration, 241 smooth boundary, 230 software for harmonic functions, 80, 90, 93, 106, 125, 180, 218, 221, 247 spherical average, 50 spherical cap, 131 spherical coordinates, Laplacian in, 26 spherical harmonic, 25, 80 spherical harmonics via differentiation, 85 Stone-Weierstrass Theorem, 81, 106, 216, 217 subharmonic, 224 submean-value property, 224 support, 192 surface area of S, 240 symmetry about a hyperplane, 67 symmetry about a sphere, 68 symmetry lemma, 10 total variation norm, 111 translate, 2 uniform boundedness principle, 138 uniformly continuous function, 147, 167 uniformly integrable, 139 upper half-space, 143 volume of B, 239 weak convergence, 115, 149 Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Index 259 zonal harmonic, 173 zonal harmonic expansion of Poisson kernel, 99 zonal harmonic with pole η, 94 zonal harmonic, formula for, 104 zonal harmonic, geometric characterization, 100 Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey Back Cover The authors have taken unusual care to motivate concepts and sim-plify proofs in this book about harmonic functions in Euclidean space.
Readers with a background in real and complex analysis at the begin-ning graduate level will feel comfortable with the material presented here. Topics include basic properties of harmonic functions, Poisson integrals, the Kelvin transform, harmonic polynomials, spherical har-monics, harmonic Hardy spaces, harmonic Bergman spaces, the decom-position theorem, Laurent expansions, isolated singularities, and the Dirichlet problem.
This new edition contains a completely rewritten chapter on har-monic polynomials and spherical harmonics, as well as new material on Bôcher’s Theorem, norms for harmonic Hardy spaces, the Dirichlet problem for the half space, and the relationship between the Laplacian and the Kelvin transform. In addition, the authors have included new exercises and have made numerous minor improvements throughout the text.
The authors have developed a software package, available electroni-cally without charge, that uses results from this book to calculate many of the expressions that arise in harmonic function theory. For example, the Poisson integral of any polynomial can be computed exactly. |
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Roots of unity
The Roots of unity are a topic closely related to trigonometry. Roots of unity come up when we examine the complexroots of the polynomial.
Contents
[hide]
1 Solving the Equation
2 Geometry
3 Properties
4 Uses
5 See Also
Solving the Equation
First, we note that since we have an nth degree polynomial, there will be n complex roots.
Now, we can convert everything to polar form by letting , and noting that for , to get . The magnitude of the RHS is 1, making (magnitude is always expressed as a positive real number). This leaves us with .
Taking the natural logarithm of both sides gives us . Solving this gives . Additionally, we note that for each of we get a distinct value for , but once we get to , we start getting coterminal angles.
Thus, the solutions to are given by for . We could also express this in trigonometric form as
Geometry
All of the roots of unity lie on the unit circle in the complex plane. This can be seen by considering the magnitudes of both sides of the equation . If we let , we see that , since the magnitude of the RHS of is 1, and for two complex numbers to be equal, both their magnitudes and arguments must be equivalent.
Additionally, we can see that when the nth roots of unity are connected in order (more technically, we would call this their convex hull), they form a regular n-sided polygon. This becomes even more evident when we look at the arguments of the roots of unity.
Properties
Listed below is a quick summary of important properties of roots of unity.
They occupy the vertices of a regular n-gon in the complex plane.
For , the sum of the n th roots of unity is 0. More generally, if is a primitive n th root of unity (i.e. for ), then
This is an immediate result of Vieta's Formulas on the polynomial and Newton's sums.
If is a primitive n th root of unity, then the roots of unity can be expressed as .
Also, don't overlook the most obvious property of all! For each th root of unity, , we have that
Uses
Roots of unity show up in many surprising places. Here, we list a few:
Geometry
Factoring
Number theory
See Also
Complex numbers
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Definition
Complex numbers
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Decimal to Binary and more…
Why the digits aren’t always 0 to 9.
Suraj Naranatt
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Nov 16, 2020
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The digits 0 to 9 are one of the first things which people learn in school and for good reason. Being able to identify these ten digits in your environment, be it in school, on a clock, or even in the elevator is obviously useful. However, these aren’t numbers in the true sense. Here’s why.
Representation of Numbers
The ten digits [0, 1, 2, 3, 4, 5, 6, 7, 8, 9] are used to represent numbers. On a sheet of paper, or even a computer screen, these are just characters. In mathematics, numbers are not the same as their representations, which means that I can use any other set of characters for the representation of numbers.
For example, you may choose the character to represent the value 0. Similarly let 😀= 1, 😁= 2, 😅= 3, 🤣= 4, 😇= 5, 😱= 6, 😑= 7, 😮= 8 and 😋= 9.
To count up to small numbers, say upto 20, we can still come up with other characters for representing numbers. For example, in the Roman numeral system, the number 1000 is represented by the symbol M, while 1001 is represented by MI in the system. Smaller numbers such as 9 are represented as IX i.e., 1 less than 10 where X=10.
The problem with such systems is that for large numbers, a huge number of symbols need to be memorized. A ludicrous example is the representation of the number 3999 as MMMCMXCIX in Roman numerals. For this reason, number systems based on positional notation were introduced.
Positional Notation
The Arabic number system, which most people are familiar with uses positional notation to represent numbers. If I were to ask you to write the number sixty-five thousand five hundred and thirty-six onto a piece of paper, you would easily be able to do so. This is because a finite number of symbols are used in the Arabic system. The digits are the elements of the set
D = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}.
In positional systems, each digit is multiplied by a base number raised to an exponent. The exponent used for a digit depends on the position of the digit in the number. For example, the number 1234 has the value (4 × 10⁰ + 3 × 10¹ + 2 × 10² + 1 × 10³). The Arabic number system uses the number 10 as the base, hence it is called the decimal number system. From now on, we will use the attach 10 to the subscript of a number to show that the number is represented in the decimal number system i.e., 1234₁₀.
In general, for a number system using the base b, a number can be represented as
a ₙ-₁ a ₙ-₂ …a ₁ a ₀, where a ₀, a ₁,…a ₙ-₁ are digits of the number.
The value of the number is given by a ₀ × b₀ + a ₁ × b₁ + … + aₙ-₁ × bₙ-₁
If we use 8 as a base, then the number 1234₁₀ is represented as 2322₈, where the subscript denotes the base used. To verify that this is indeed the correct representation of the value of 1234₁₀, use the same rule of positional notation.
2322₈ = (2 × 8⁰)₁₀ + (2 × 8¹)₁₀ + (3 × 8²)₁₀ + (2 × 8³)₁₀ = (2 + 16 + 192 + 1024)₁₀ = 1234₁₀
Using 8 as the base restricts the number of symbols which can be used to represent numbers. This number system is called the octal number system. The allowed set of symbols is the set
O = {0, 1, 2, 3, 4, 5, 6, 7}.
In a similar way, if we use 16 as a base, then the number 1234₁₀ can be represented as 4D2₁₆. This representation is called the hexadecimal representation of the number 1234₁₀ and is called the hexadecimal number system. Numbers can be formed using the elements of the set
H = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F}.
Another common number system (at least for computers) is the binary number system. This uses the number 2 as base and has only 2 allowed symbols — 0 and 1. The number 1234₁₀ is represented as 10011010010₂.
Among the three systems, conversions between binary and decimal are frequently performed. However, I will also explain how to convert representations in the other two systems into decimal.
Decimal to Binary Conversion
Numbers represented in the decimal number system can be converted to the binary system by repeated division. An example is shown below.
MSB = Most Significant Bit; LSB = Least Significant Bit
Converting Decimal to Binary
The above image shows a decimal number being divided until the number becomes 1. The remainder in each stage of division is a bit in the binary representation of the number, starting with the least significant bit obtained by dividing the number for the first time.
Decimal to Octal Conversion
Converting from the Decimal system to the octal system is very similar to the conversion shown above, except that the base is now 8 instead of 2. This means that the number to be converted will be repeatedly divided by 8 and the remainders in each stage make up the digits of the octal number. The number 1234₁₀ in octal representation is shown below.
Converting Decimal to Octal
Decimal to Hexadecimal Conversion
The process is similar to that used for conversion to octal and binary representation. Since the base is 16, the decimal number to be converted is divided by 16 instead of 8 or 2. However, there is a small difference.
In case the remainder is greater than 9, then the characters A to F are used in place of the numbers 10 to 15 respectively. A = 10, B = 11, C = 12, D = 13, E = 14, F = 15.
Converting Decimal to Hexadecimal
Hex, Oct or Bin to Decimal
Converting Hexadecimal, Octal or Binary representations back to Decimal is quite easy. As the value of a number is given by the sum a₀ × b₀ + a₁ × b₁ + … + aₙ-₁ × bₙ-₁, setting the value of base to either 16, 8 or 2 will allow you to find the decimal representation of the number from hexadecimal, octal and binary representations respectively.
2322₈ = (2 × 8⁰)₁₀+ (2 × 8¹)₁₀ + (3 × 8²)₁₀ + (2 × 8³)₁₀ = (2 + 16 + 192 + 1024)₁₀ = 1234₁₀
4D2₁₆ = (2 × 16⁰)₁₀ + (13 × 16¹)₁₀ + (4 × 16²)₁₀ = (2 + 208 + 1024)₁₀ = 1234₁₀
100110100102 = (0 × 2⁰)₁₀ + (1 × 2¹)₁₀ + (0 × 2²)₁₀ + (0 × 2³)₁₀ + (1 × 2⁴)₁₀ + (0 × 2⁵)₁₀ + (1 × 2⁶)₁₀ + (1 × 2⁷)₁₀ + (0 × 2⁸)₁₀ + (0 × 2⁹)₁₀ + (1 × 2¹⁰)₁₀
= (0 + 2 + 0 + 0 + 16 + 0 + 64 + 128 + 0 + 0 + 1024)₁₀ = 1234₁₀
Binary to Hexadecimal
Numbers in their binary representations can be converted to hexadecimal numbers by dividing the binary number into collections of 4-bit numbers, starting with the LSB.
For example, 10011010010₂ can be divided into three 4-bit numbers.
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Converting Binary to Hexadecimal
As you may have noticed, the number has been padded with a single zero to create collections of 4-bit numbers. If there were just 6 bits in the binary representation of a number, then the two zeros will be padded with the MSB, to create a collection containing two 4-bit numbers.
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Conversion to hexadecimal representation takes place by converting these smaller 4-bit numbers into their hexadecimal equivalents.
For example, 1010₂ = 10₁₀ = A₁₆. Remembering the binary representations of each digit in hexadecimal is often helpful. The following chart can be used.
Use for Binary to Hexadecimal conversion
Thus the hexadecimal number is 4D2₁₆.
Binary to Octal
Instead of dividing the binary number into collections of 4-bit numbers, the binary number is divided into collections of 3-bit numbers.
Press enter or click to view image in full size
Conversion of Binary to Octal
Each of the 3-but numbers is converted into its octal equivalent representation and concatenated to produce the correct octal number. The following chart can be used.
Use for Binary to Octal conversion
Thus, the octal representation of the binary number is 2322₈.
Hexadecimal to Binary
Converting from hexadecimal representation to binary uses the table used to convert binary representation to hexadecimal. The process is followed in reverse order. The digits of the hexadecimal number are converted to 4-bit binary numbers and are appended back in the same order.
An example is shown below.
Press enter or click to view image in full size
Hexadecimal to Binary conversion
Each digit is converted to a 4-bit binary representation and then concatenated in the same order.
Octal to Binary
Converting from octal representation to binary uses the table used to convert binary representation to octal. The digits of the octal number are converted to 3-bit binary numbers and are appended back in the same order.
An example is shown below
Press enter or click to view image in full size
Octal to Binary conversion
Each digit is converted to a 3-bit binary representation and then concatenated in the same order.
Fractional Parts
Let’s take a new number to convert into binary 1234.56
In this case, the fractional part is separated from the integer part, thus the number 0.56. The integer part is converted into binary using the method shown before, but the conversion of the fractional part is different.
To convert the fractional part into its binary representation, it must be multiplied by the base (in this case, base=2).
Press enter or click to view image in full size
Converting numbers with fractional parts
The number of times the fractional part must be multiplied depends on the required precision of the conversion. Therefore the number in binary representation is 10011010010.10001₂.
The same process can be followed to convert the number into octal or hexadecimal representation by multiplying by 8 or 16 respectively.
To convert a binary/octal/hexadecimal representation to decimal, it must be noted that the value of digits after the decimal point is given by the sum of values of each digit, where the value of a digit is the product of base raised to the exponent and the digit itself. As you may have noticed in the representation a ₙ-₁aₙ-₂…a₁a₀, only the value of the integer part can be calculated. To calculate the value of the fractional part, the exponent used by the base becomes increasingly negative with each digit in the fraction. This is better explained like this: 0.10001₂ = 1 × 2^(–1) + 0 × 2^(–2) + 0 × 2^(–3) +
0 × 2^(-4) + 1 × 2^(–5) = 0.5 + 0 + 0 + 0 + 0.03125 = 0.53125 ≈ 0.5
where 2 is the base.
Note that this process is not very accurate at calculating the decimal representation, since accuracy depends on the number of binary digits used. For other bases, a similar process is used (change the base to 8 for octal and 16 to hexadecimal).
Enfin
The base-12 number system, also called the duodecimal system makes division slightly easier since the number 12 is divisible by 1, 2, 3, 4, 6, and 12. Hexadecimal representation of numbers is used to represent colors in computers. Similarly, binary numbers form the basis for the representation of data within a computer. Conversion between the systems is sometimes tedious when done by hand, and so writing a program to do the same is obviously better. The python programming language allows easy conversion between the three common representations.
Open a python interpreter. Let the number to convert be 1234 in decimal. Using the hex() built-in function, the number can be converted into its hexadecimal representation. Similarly, the oct() function and the bin() functions can be used to convert a decimal number to octal and binary representations.
hex(1234) returns ‘0x4d2’
bin(1234) returns ‘0b10011010010’
oct(1234) returns ‘0o2322’
However, the functions don’t work for floating-point decimal representations.
Hopefully, you will now be able to convert those binary numbers into something better.
Further Reading
Roman Numerals (Wikipedia)
What is Mathematics?: An Elementary Approach to Ideas and Methods (ch.1)
Binary System and More
Duodecimal Number System
Python built-in Functions
For other interesting posts, check out my posts on our blogTechprodezza.
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Welcome to our Rubber Durometer Scale Section
A durometer scale is a type of measurement for rubber material hardness. The rubber durometer chart below gives you an idea of the rubber hardness that you want for your application. Generally, most rubber materials fall under the rubber durometer scale of Shore A. Thus, if you need a rubber or O-ring durometer that feels like a running shoe sole, review our rubber hardness chart below, then pick Shore 70A. A rubber durometer of Shore 70A is the most commonly chosen material hardness for all applications.
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190436 | https://people.math.harvard.edu/~demarco/ConvexShapes.pdf | CONVEX SHAPES AND HARMONIC CAPS LAURA DEMARCO AND KATHRYN LINDSEY Abstract. Any planar shape P ⊂C can be embedded isometrically as part of the boundary surface S of a convex subset of R3 such that ∂P supports the positive curvature of S. The complement Q = S \P is the associated cap. We study the cap construction when the curvature is harmonic measure on the boundary of (ˆ C\P, ∞).
Of particular interest is the case when P is a filled polynomial Julia set and the curvature is proportional to the measure of maximal entropy.
1. Introduction A planar shape is a compact, connected subset of the Euclidean plane that con-tains at least two points and has connected complement. Given a probability measure µ supported on the boundary of a planar shape P, we investigate the existence of a conformal metric ρ = ρ(z)|dz| on the Riemann sphere ˆ C so that (i) P, with its Euclidean metric from R2, embeds locally-isometrically into (ˆ C, ρ); and (ii) the curvature distribution ωρ = −∆log ρ(z) on ˆ C is equal to the push-forward of 4πµ under the embedding.
If ρ exists, then it is uniquely determined up to isometry (c.f. [4, §3.5, Theorem 1]), and we will denote it by ρ(P, µ).
A. D. Alexandrov’s theorems on convex surfaces [3, 2, 4] assert that any abstract metrized sphere with non-negative curvature is isometric to the boundary surface of a convex body in R3 with its induced metric (unique up to rigid motions of R3). In particular, the metrized sphere (ˆ C, ρ(P, µ)) will have a unique convex 3D realization.
The convex body may be degenerate, meaning that it lies in a plane and the sphere is viewed as the double of a convex planar region. Conversely, the surface of any compact, convex body in R3 (not contained in a line) may be endowed with a com-plex structure and uniformized so that it is isometric to the Riemann sphere with a conformal metric of non-negative curvature; see, e.g. . Thus, the existence of ρ(P, µ) may be viewed as a problem of “folding” the shape P into R3 and taking its convex hull, in such a way that the curvature of the resulting convex body is given by 4πµ.
The complement of P in (ˆ C, ρ(P, µ)) will be called the cap of (P, µ) and denoted by ˆ Pµ. By construction, the metric on the cap is flat, so there is a locally isometric Date: November 30, 2016.
1 2 LAURA DEMARCO AND KATHRYN LINDSEY development map D : ˆ Pµ →(C, |dz|).
We say the cap is planar if the development D is injective.
Our first observation is that there always exists a probability measure µ supported on ∂P so that the metric ρ(P, µ) exists (see §2.1 for a simple but degenerate construc-tion). We also observe that not all caps are planar, and we give examples in Section 2.
The harmonic cap. We are especially interested in the case where P is a connected filled Julia set K(f) of a polynomial f : C →C and the prescribed measure µ is the measure of maximal entropy supported on the boundary of K(f); see details in §2.4. This metrized sphere was defined in [13, Section 12] for an arbitrary rational map f : P1 →P1 of degree > 1. Questions about the features of its 3-dimensional realization were first posed by C. McMullen and W. Thurston.
To this end, we examine arbitrary planar shapes P ⊂C, and we let µ be the harmonic measure for the domain ˆ C \ P relative to ∞. By definition, µ is the push-forward of the Lebesgue measure on the unit circle S1 (normalized to have total mass 1) under a conformal isomorphism Φ : C \ D →C \ P; the measure µ is well defined even if Φ is not everywhere defined on S1. In this setting, the metric ρ(P, µ) is simply an extension of the Euclidean metric |dz| on P; it can be expressed in terms of the Green function GP(z) = log |Φ−1(z)| for z ∈C \ P. Setting GP(z) = 0 for z ∈P, we have ρ(P, µ) = e−2GP (z)|dz|.
Observe that the metric ρ(P, µ) is continuous on all of ˆ C: GP is continuous on C (by solvability of the Dirichlet problem on simply-connected domains), and it grows as γ + log |z| + o(1) as z →∞for some γ ∈R. The cap ˆ Pµ is called the harmonic cap of P.
Theorem 1.1. Let P be any planar shape and let µ be the harmonic measure on ∂P, relative to ∞. Let Φ : ˆ C \ D →ˆ C \ P be a conformal isomorphism with Φ(∞) = ∞.
A Euclidean development of the harmonic cap ˆ Pµ is given by the locally univalent function g : D →C defined by g(z) = Z z 0 Φ′(1/x) dx.
Moreover, there exist planar shapes P for which the harmonic cap is not planar.
As an example, the harmonic cap of a closed interval is planar; its development is shown in Figure 2.3 for P = [−2, 2] where g(z) = z −z3/3. A non-planar example is described in §2.3.
CONVEX SHAPES AND HARMONIC CAPS 3 Theorem 1.1 allows one to appeal to the theory of univalent functions for conditions on P that guarantee planarity of the harmonic cap. If the harmonic cap is planar, then the construction can be iterated, to find the harmonic cap of the development of a harmonic cap. It would be interesting to understand the properties of this dynamical system on a class of planar shapes. (The closed unit disk is a fixed point of this operation; see Example 4.1.) Constructing a cap. Given the data of a conformal metric (ˆ C, ρ) with non-negative curvature distribution, it is a notoriously difficult problem to construct the 3D real-ization, even for polyhedral metrics (as we discuss below). But it turns out that a development of a cap ˆ Pµ in C can be easily produced on the computer.
For planar shapes that are Jordan domains with rectifiable boundaries, a cap ˆ Pµ will have boundary of the same length as ∂P. A perimeter gluing of P and ˆ Pµ is the boundary identification (by arclength) between ∂P and ∂ˆ Pµ that produces (ˆ C, ρ(P, µ)).
Theorem 1.2. Let P be a planar shape with a piecewise-differentiable Jordan curve boundary, and let µ be a nonnegative Borel probability measure supported on the boundary of P. Let s be a counterclockwise, unit-speed parametrization of ∂P, and write s(t) = R t 0 eiα(x) dx for a real-valued function α. If the cap ˆ Pµ exists, then the boundary of its Euclidean development is parameterized in the clockwise direction by ˆ s(t) = Z t 0 ei(α(x)−κ(x)) dx where κ(t) = 4πµ(s(0, t]), and the perimeter gluing is given by s(t) ∼ˆ s(t), .
Given an arbitrary planar shape P, we can approximate it by a shape P ′ with piecewise-differentiable Jordan curve boundary and approximate any given measure µ on ∂P with a probability measure supported on the boundary of P ′. In this way, Theorem 1.2 supplies a straightforward strategy to illustrate the caps. In practice, we use polygonal approximations to the planar shape P with discrete curvature supported on the vertices. See Figures 1.1 and 1.2. A theorem of Reshetnyak states that weak convergence of the curvature distributions as measures on ˆ C implies convergence of the metrics [24, Theorem 7.3.1], .
For polygonal planar shapes with arbitrary probability measures µ supported on their vertices, our cap-drawing algorithm (which follows the proof of Theorem 1.2) can be used to draw the parametrization ˆ s, independent of the existence of the metric extension ρ(P, µ). For many examples, the curve ˆ s fails to form a closed loop or has a shape that cannot be the boundary parametrization of any Euclidean development of a cap (e.g., it may have positive winding number around a point in the plane, while the boundary of a cap development, traversed in the clockwise direction, will wind non-positively around all points). For example, if P is a triangle, there is a unique measure µ supported on the vertices of P that gives rise to a cap: any associated cap 4 LAURA DEMARCO AND KATHRYN LINDSEY Figure 1.1. In blue, a square; in orange, its harmonic cap, with one at-taching point indicated in black. The perimeter gluing is by arclength. There is a unique realization of the glued shapes as the boundary of a convex body in R3. The harmonic measure on the boundary of the square was approx-imated by a discrete measure supported on 500 points, using the Riemann mapping function in Sage . Image generated with Mathematica.
is necessarily a triangle whose sidelengths are the same as those of P, implying the cap is a reflected copy of P, the convex shape is degenerate, and µ(v) = (π −θ)/(2π) where v is a vertex of P with internal angle θ. In general, the questions of when the metric ρ(P, µ) exists and when the cap ˆ Pµ is planar are quite delicate, even in the polygonal setting.
Problem 1.3. For polygons of N sides, with side lengths {ℓ1, . . . , ℓN} and internal angle θi at each of its vertices vi, give an explicit description of the discrete curvature distributions µ = {µi} supported on the vertices vi so that the metric ρ(P, µ) exists.
Provide conditions under which the polygonal cap ˆ Pµ is planar.
Problem 1.3 is related to the geometry of the space of polygons with fixed side lengths and no boundary crossings, which, to our knowledge, has never been de-scribed. See where it is proved that the space is connected and contractible.
The 3-dimensional realization. Recall, by Alexandrov’s theorems ([4, 2, 3], ), for nonnegative µ there is a unique way to fold the Euclidean development of P and ˆ Pµ to form the boundary surface of a convex shape in R3. We may view the output of the cap-drawing algorithm, as in Figures 1.1 and 1.2, as paper cut-outs to be creased and glued to form the desired shape. Unfortunately, the exact shape of the 3-dimensional realization is not at all clear from the development alone. Even the set of folding lines inside P and ˆ Pµ is a mystery in general. Quoting from Alexandrov in translation [4, CONVEX SHAPES AND HARMONIC CAPS 5 Figure 1.2. In blue, a polygonal approximation to the filled Julia set of f(z) = z2 −1 with 211 vertices, the preimages of z = 2.0 under f11. The discrete probability measure that assigns equal mass to each of its 211 vertices approximates the harmonic measure on the filled Julia set. In orange, the polygonal cap associated to this polygon with discrete curvature measure.
There is a unique realization of the glued shapes as the boundary of a convex body in R3. Image generated with Mathematica.
p.100], “To determine the structure of a polyhedron from a development, i.e., to indicate its genuine edges in the development, is a problem whose general solution seems hopeless.” But in the case of harmonic measure on a planar shape, especially when the shape is the filled Julia set of a polynomial, there may be specialized ways to attack the problem.
Not long ago, Bobenko and Izmestiev devised an illuminating and constructive proof of Alexandrov’s realization theorem for polyhedral metrics , implementing their algorithm and making it publicly available. Unfortunately, the algorithm was not practical for the polyhedra that closely approximate the metrics for polynomial Julia sets . Laurent Bartholdi modified their strategy to handle some dynamical examples, such as the filled Julia set of f(z) = z2 −1 shown in Figure 1.3.
Formally, the convex 3D realization of (ˆ C, ρ(P, µ)) determines a Euclidean lamina-tion on the interiors of P and ˆ Pµ, consisting of the geodesic line segments that must be folded to form the 3D shape. We call this the bending lamination of the pair (P, µ). If one also retains the data of the dihedral angles (the amount of the fold along each leaf of the lamination), we obtain a measured lamination, uniquely determined by the pair (P, µ). We leave the following as an open problem: Problem 1.4. Suppose µ is the harmonic measure relative to ∞on the boundary of a planar shape P. Describe the (measured) bending lamination of (P, µ).
6 LAURA DEMARCO AND KATHRYN LINDSEY Figure 1.3. Two views of Bartholdi’s polyhedral approximation to the 3D realization of the filled Julia set of f(z) = z2 −1 with its harmonic measure, computed with 211 vertices. An illustration of the filled Julia set is superimposed onto the image. Graphic created with glc player.
Other comments and acknowledgments. In the course of this project, we were introduced to the vast literature of the computational geometry community. Quite a bit of research has gone towards visualizing the 3D realizations of Alexandrov’s convex polyhedral metrics and related problems. Most notably, we mention that we learned much from the work of Demaine and O’Rourke and their co-authors; see, e.g.
[10, 11].
We would like to thank Curt McMullen and, posthumously, Bill Thurston, for introducing us to this problem and for many interesting conversations on the topic over the past 15 years. In particular, the idea of representing a Julia set and its cap as paper cut-outs is due to Thurston. Our perspective on caps and bending is also inspired by the theory of pleated surfaces and Thurston’s study of spaces of polyhedra [25, 27], and the geometry of filled Julia sets for homogeneous polynomial maps .
We are grateful to Laurent Bartholdi, Ilia Binder, Robert Connelly, David Dumas, and Amie Wilkinson for helpful discussions. Finally, we thank the anonymous referee for many thoughtful, useful suggestions.
Our research was supported by the National Science Foundation and the Simons Foundation.
2. Caps, spirals, and Julia sets In this section, we observe that for every planar shape P, there is a probability measure µ on its boundary so that the metric ρ(P, µ) on ˆ C exists, by simple con-structions in R2. We provide examples to illustrate the failure of planarity of a cap.
We conclude the section with examples of harmonic caps coming from polynomial dynamical systems f : C →C. Formal definitions and the proofs of our theorems will be given in Sections 3 and 4.
2.1. The naive cap. Let P be a planar shape that is not contained in a line. Let ¯ P be the convex hull of P in the plane. The naive cap ˆ P is the union of ¯ P and a copy of each connected component of ¯ P \ P (the flaps), glued along their boundaries CONVEX SHAPES AND HARMONIC CAPS 7 in ∂¯ P. Then P and ˆ P glue to determine a degenerate convex body, and the metrized sphere is a doubled copy of ¯ P. Its curvature is supported in the intersection of ∂P with ∂¯ P. Unfolding the flaps of the naive cap ˆ P determines a Euclidean develop-ment. We can appeal to the Uniformization Theorem or to Reshetnyak’s theorem on isothermal coordinates [24, Theorem 7.1.2] to conclude that this degenerate surface can be represented as a conformal metric on the Riemann sphere ˆ C.
If P is an interval, then we can produce a cap by bending P into an L-shape in the plane, introducing an angle at the midpoint of P, and then taking the convex hull of this new shape in R2. Viewing the resulting triangle as a degenerate convex body in R3, we produce a metrized sphere with 3 concentrated points of curvature, at the two endpoints of P and at its midpoint. As P = ∂P in this example, we have shown the existence of a probability measure µ supported in ∂P and giving rise to a metric ρ(P, µ) on ˆ C. The developed cap ˆ Pµ will be a rhombus. For example, if the angle is chosen to be π/3, then the triangle will be equilateral, and µ will assign equal mass to each of the three cone points.
2.2. The naive cap is not always planar. Start with a convex polygonal shape in the plane with an external angle of about π/16 at one vertex. Remove two very thin spiral channels from the polygon that begin on adjacent edges of the polygon and spiral around one another, as in the left image of Figure 2.1. If the spirals are sufficiently intertwined, then the spiral flaps on the developed naive cap will overlap.
The right side of Figure 2.1 shows the spirals reflected across the edges of the polygon.
Figure 2.1. Left: A piece of a convex polygon (lying above the red and green line segments) minus two narrow spiral channels (shown in orange and blue) that begin from adjacent edges of the polygon. Each channel cut from the polygon is so narrow that we depict it as a curve. Right: A piece of its naive cap (again, above the red and green segments) with the two spiral flaps reflected outward, illustrating a non-planar Euclidean development.
2.3. Non-planar example for harmonic measure. For the harmonic cap, it is possible to construct an example similar to that of §2.2. Indeed, very skinny channels removed from any planar shape will have negligible harmonic measure, and so we can arrange for overlapping spirals in the cap.
8 LAURA DEMARCO AND KATHRYN LINDSEY Figure 2.2. Left: Two narrow spiral channels (shown in orange and blue) cut from the interior of a square planar shape (a segment of which is shown in green). Each channel cut from the polygon is so narrow that we depict it as a curve. Right: The two spirals on the exterior of the clover-shaped harmonic cap of the square, illustrating a non-planar Euclidean development.
A complete and accurate picture of the harmonic cap of the square is shown in Figure 1.1.
More precisely, begin with a square planar shape and choose a tiny ε > 0. The harmonic cap for the square is shown in Figure 1.1. Now remove two very skinny spiral channels from the square, emanating from a single edge, as in the left image of Figure 2.2; the openings of each channel should have width smaller than ε. The openings of the two spiral channels can be placed at a specified distance apart from one another, so that the harmonic measure of the interval between them is approximately equal to 1/32 of the total mass. (The number 1/32 is chosen because it is 1/4π times the curvature of π/8 for the polygon vertex shown in Figure 2.1). We can choose ε > 0 as small we wish so that the harmonic measure along the spiral boundaries is almost 0. Indeed, as the width of the spiral channels shrinks to 0, the domains ˆ C \ P are converging in the Carath´ eodory sense to the complement of the square; see, e.g., [15, §3.1].
Recall that the boundary of the cap development is parameterized by the formula of Theorem 1.2. The parametrization of the spirals on the cap, which will lie outside the clover-like harmonic cap for the square, will be essentially equal to a reflection of their original parametrizations (because κ will be essentially constant along their boundaries, having chosen the harmonic measure of the spirals to be near 0). On the other hand, the non-trivial portion of harmonic measure on the boundary of the square between the spiral-channel openings will curve the boundary of the cap so the spirals overlap. The change in tangent direction of the clover cap between the two attaching points of the spirals will be π/8, by construction. See Figure 2.2.
CONVEX SHAPES AND HARMONIC CAPS 9 2.4. Polynomial Julia sets. Now assume that f : C →C is a complex polynomial of degree d ≥2. Its filled Julia set is K(f) = {z ∈C : sup n |f n(z)| < ∞}.
Assume that K(f) is connected, so it is a planar shape. A planar development of its cap is given by the formula of Theorem 1.1. We can parameterize the boundary of the cap’s development for smooth or polygonal approximations to K(f) using Theorem 1.2.
The Green function for K(f) can be computed dynamically, as Gf(z) = lim n→∞ 1 dn log+ |f n(z)|.
The harmonic measure µf = 1 2π∆Gf is the unique measure of maximal entropy for f, and its support is equal to the Julia set J(f) = ∂K(f) [8, 20, 17]. The metric on ˆ C is defined by ρf = e−2Gf(z)|dz| for z ∈C, with curvature distribution ωf = −∆log ρf(z) = 4πµf.
Example 2.1. Let f(z) = z2. Then K(f) is the closed unit disk and Gf(z) = log+ |z|.
The measure µf is the Lebesgue measure on the circle. By symmetry, the harmonic cap is also a closed disk of radius 1. It follows that the convex realization in R3 is a degenerate closed disk.
Example 2.2. Let f(z) = z2 −2. Then K(f) is the real interval [−2, 2], and the metric on the sphere and the Euclidean development of the harmonic cap can be computed explicitly. The Riemann map from the complement of the unit disk to the complement of K(f) is given by Φf(z) = z + 1 z.
Applying Theorem 1.1, the cap is the image of the holomorphic function g : D →C defined by g(z) = Z z 0 Φ′ f(1/x) dx = Z z 0 (1 −z2) dz = z −z3/3.
See Figure 2.3. The convex realization in R3 is degenerate.
Example 2.3. Let f(z) = z2 + 1/4. A polygonal approximation to its filled Julia set and the harmonic cap are shown in Figure 2.4. The convex realization in R3 is nondegenerate; indeed, if the filled Julia set were contained in a plane in R3, then its convex hull would also lie in the surface, and then the curvature could not be supported on all of J(f).
10 LAURA DEMARCO AND KATHRYN LINDSEY Figure 2.3. A Euclidean development of the cap for the real interval P = [−2, 2] equipped with its harmonic measure. The figure shown is the image of the unit circle under g(z) = z −z3/3, so the cusps lie at z = ±2/3. To form the metrized sphere, the cap is folded in half along the segment joining the cusp points, and the interval P forms the seam. The resulting convex body is degenerate. See Example 2.2.
Figure 2.4. In blue, a polygonal approximation to the filled Julia set of f(z) = z2 + 1/4 with 211 vertices, the preimages of z = 0.5 under f11. The approximation to harmonic measure puts equal weight on each of the 211 vertices. In orange, the polygonal cap for this discrete curvature distribution.
A single attaching point is shown in black. There is a unique realization of the glued shapes as the boundary of a convex body in R3. Image generated with Mathematica.
3. Metrics and curvature In this section, we formalize the notions of curvature and metric from the point of view of Euclidean geometry, and we prove Theorem 1.2. In Proposition 3.1, we present an asymptotic formula for curvature when the boundary of the planar shape is a smooth Jordan curve, in terms of the circumference of small circles.
CONVEX SHAPES AND HARMONIC CAPS 11 3.1. Polyhedra and cone angles. A convex polyhedron in R3 is the intersection of finitely many closed halfspaces. It is said to be degenerate if it lies in a plane. When the polyhedron is non-degenerate and bounded, its boundary surface is topologically a sphere, and the Euclidean metric from R3 induces an intrinsic path metric on the sphere. If the polyhedron is degenerate and bounded, but not contained in a line, we will still view its boundary as a topological sphere, doubling the planar polygon and gluing along the polygonal boundary.
Abstractly, a convex polyhedral metric on a 2-dimensional sphere is an intrinsic metric with non-negative curvature concentrated at finitely many points. In other words, in a small neighborhood of all but finitely many points, the surface is isometric to a region in R2. In a neighborhood of each of the finitely many cone points, the surface is isometric to the point of a cone. The curvature of a cone point is equal to the angle deficit at the point; that is, if the circumference of any small circle of radius r centered at the cone point is equal to C(r), then the curvature is equal to (2πr −C(r))/r. By the Gauss-Bonnet formula, the sum of the curvatures over all cone points on the sphere is equal to 4π.
In , A. D. Alexandrov examines the geometry of convex polyhedra in detail.
He presents his proof from that any abstract polyhedral metric on a sphere is isometric to the boundary of a (possibly degenerate) convex polyhedron. Furthermore, the polyhedron in R3 is unique, up to Euclidean isometries.
Given a polyhedral metric on the sphere, and a simply-connected subset U of the sphere minus its cone points, a Euclidean development of U is a local isometry U →R2.
Suppose we are given the image I ⊂R2 of a Euclidean development of a full-area, simply-connected subset U of the sphere. Then, as a consequence of Alexandrov’s theorem, the convex polyhedron in R3 is uniquely determined by I and the gluing along its boundary (that reconstructs the topological sphere). In particular, the planar development and the gluing information will uniquely determine the edges of the polyhedron and their dihedral angles in R3 – information that is not locally apparent.
3.2. More general metrics of non-negative curvature. In , Alexandrov presents the proof of a more general realization result; see Chapter 1 of for a summary.
Given any abstract intrinsic metric on the sphere of non-negative curvature, it is re-alizable as the boundary of a (possibly degenerate) convex body in R3. His argument relies on a convergence statement, first approximating the metric by polyhedral met-rics, realizing the convex polyhedra, and then showing that the polyhedra converge to the desired convex body in R3.
Curvature is carefully treated by Alexandrov.
It is defined by an additive set function ω as follows. The curvature of a point is, as for a polyhedron, 2π minus the 12 LAURA DEMARCO AND KATHRYN LINDSEY cone angle of the point. That is, (3.2.1) ω({x}) = lim r→0+ 2πr −C(x, r) r where C(x, r) is the circumference of the circle of radius r centered at the point x.
The curvature of a geodesic line segment will always be 0. The curvature of a (small) geodesic triangle is its internal angle surplus, defined as the sum of the internal angles of the triangle minus π. The curvature of a more general region is computed by triangulation. See [21, Chapter 1, page 18].
Y. G. Reshetnyak, who was a student of Alexandrov, reformulated Alexandrov’s theory of metrics and curvature on a surface in complex-analytic language, expressing curvature as a finite Borel measure . We exploit this useful point of view in Section 4.
3.3. Parametrization of the cap. Suppose that a planar shape P is the closure of a Jordan domain with a piecewise-differentiable boundary.
Fix a nonnegative Borel measure µ on the boundary of P. Let L be the length of ∂P. Let s be a piecewise-differentiable parametrization by arclength of the boundary of P, in the counterclockwise direction, and write s′(t) = eiα(t) for a piecewise-continuous function α : [0, L] →R. For t ∈[0, L], we define a curvature function κ : [0, L] →[0, 4π] by κ(0) = 0 and (3.3.1) κ(t) = 4πµ(s(0, t]) for all t ∈(0, L], so that κ is monotone increasing with κ(L) = 4π. Recall that Theorem 1.2 asserts that, if the cap ˆ Pµ exists, then its boundary can be parameterized in the clockwise direction by ˆ s(t) = Z t 0 ei(α(x)−κ(x)) dx.
Proof of Theorem 1.2. Suppose first that P is a polygon in the complex plane and µ is a discrete probability measure supported on the vertices of P. Denote the vertices of P by v0, v1, . . . , vN = v0, oriented counterclockwise, and set ℓj = |vj −vj−1| to be the length of the j-th edge. We may assume for simplicity that v0 = 0 and v1 = ℓ1 lies on the positive real axis. Let θj be the internal angle of P at vertex vj, so that N X j=1 (π −θj) = 2π CONVEX SHAPES AND HARMONIC CAPS 13 and α(t) = k−1 X j=1 (π −θj) for k−1 X j=1 ℓj ≤t < k X j=1 ℓj for each k = 1, . . . N. Thus P is parameterized by s(t) = Z t 0 eiα(x) dx.
If ˆ Pµ exists, then it has a polygonal boundary with the same edge lengths as P.
We label its vertices in the clockwise direction by ˆ v0, ˆ v1, . . . , ˆ vN = ˆ v0. We may assume for simplicity that ˆ v0 = v0 and ˆ v1 = v1. The curvature condition implies that the internal angle ˆ θj at vertex ˆ vj must satisfy 4πµ(vj) = 2π −θj −ˆ θj.
Therefore, the clockwise parametrization ˆ s of ˆ Pµ will satisfy ˆ s′(t) = eiˆ α(t) with ˆ α(t) = − k−1 X j=1 (π −ˆ θj) for k−1 X j=1 ℓj ≤t < k X j=1 ℓj = α(t) − k−1 X j=1 4πµ(vj) for k−1 X j=1 ℓj ≤t < k X j=1 ℓj = α(t) −κ(t) In other words, the parametrization of the boundary of ˆ Pµ is given in a clockwise orientation by ˆ s(t) = Z t 0 ei(α(x)−κ(x)) dx.
If P is an arbitrary planar shape with piecewise-differentiable boundary, and if µ is any probability measure supported on the boundary of P, then the pair (P, µ) can be approximated by a sequence of polygons (Pn, µn) so that the vertices of Pn lie in ∂P for all n, and µn is a discrete probability measure supported on the vertices of Pn.
We may construct the polygons Pn so that the arclength parametrizations sn of ∂Pn converge uniformly to s and that the angle functions ρn →ρ uniformly. Furthermore, by choosing the vertices of Pn carefully, we may assume that for every ε > 0, all atoms of mass at least ε for µ are vertices of Pn and atoms of µn for all n ≥n(ε) > 0.
In this way, we can also arrange that the curvature functions κn converge uniformly to the curvature function κ. These choices for (Pn, µn) imply that the integrals Z t 0 ei(ρn(x)−κn(x)) dx − → Z t 0 ei(ρ(x)−κ(x)) dx as n →∞for all t ∈[0, |∂P|]. In other words, if the cap ˆ Pµ exists, then the desired boundary parametrization will be uniformly approximated by the curves ˆ sn defined 14 LAURA DEMARCO AND KATHRYN LINDSEY by ˆ sn(t) = Z t 0 ei(ρn(x)−κn(x)) dx.
Note that the curves ˆ sn are not necessarily closed loops, as the approximating polyg-onal caps ˆ Pµn may not exist.
□ 3.4. Circumference and curvature. If the boundary of the planar domain P and the measure µ are smooth enough, then the curvature of §3.2 satisfies the following relation, as a consequence of Theorem 1.2.
Proposition 3.1. Let P be a planar shape with boundary parametrized by arclength by s : [0, L] →∂P such that s is twice continuously-differentiable, and let µ be a probability measure on ∂P which is absolutely continuous with respect to arclength with a continuous density function.
Suppose the metric ρ(P, µ) exists.
For each x ∈∂P, let C(x, r) denote the circumference of a circle in (ˆ C, ρ(P, µ)) centered at x of radius r > 0. Then lim r→0+ 2πr −C(s(t), r) r2 = δ(t), where s∗µ = δ(t) dt on the interval [0, L].
It is interesting to compare the statement of Proposition 3.1 to the formula (3.2.1) for the Alexandrov curvature of a point, ω({x}) = lim r→0+ 2πr −C(x, r) r , and to the Bertrand-Puiseux formula for the Gaussian curvature κ when the metric on a surface is smooth, κ(x) = lim r→0+ 3 2πr −C(x, r) πr3 [26, page 147].
In our setting, the curvature of the surface is supported on a 1-dimensional curve, so the circumference discrepancy is proportional to r2.
We begin with a simple geometric lemma.
Lemma 3.2. For real numbers R > r > 0, let A(R, r) be the arclength of the inter-section of a closed disk of radius R and a circle of radius r centered at a boundary point of the disk. Then lim r→0+ πr −A(R, r) r2 = 1 R.
Proof. Assume the center of the radius r circle is at the origin in R2, and the disk of radius R is tangent to the x-axis at the origin. These two circles are given by the equations x2 + (y −R)2 = R2 and x2 + y2 = r2. These two circles intersect in two CONVEX SHAPES AND HARMONIC CAPS 15 points: ± q r2 − r4 4R2, r2 2R . Hence, A(R, r) = r π −2 tan−1 r2 √ 4R2r2−r4 . Then lim r→0+ πr −A(R, r) r2 = lim r→0+ 2 tan−1 r2 √ 4R2r2−r4 r = 1 R.
□ Proof of Proposition 3.1. The curvature function of equation (3.3.1) is computed as κ(t) = µ(s(0, t]) = Z t 0 δ(x) dx.
For each t ∈[0, L] and each small r > 0, the circumference C(s(t), r) is the sum of the lengths of two circular arcs: the arc in P to the “left” of s(t) (relative to the counterclockwise orientation on ∂P), whose length we will denote by Cr(t), and the arc in ˆ Pµ to the “right” of ˆ s(t) (relative to the clockwise orientation on ∂ˆ Pµ), whose length we will denote by ˆ Cr(t). Classical plane geometry tells us that the radius of the osculating circle to the plane curve s at s(t) is 1/|s′′(t)| = 1/|α′(t)|, using the notation of Theorem 1.2. Likewise, from Theorem 1.2, the radius of the osculating circle to the plane curve ˆ s at ˆ s(t) equals 1/|ˆ s′′(t)| = 1/|α′(t) −κ′(t)|.
For α′(t) > 0, the osculating circle is to the left of s(t), so lim r→0 πr −Cr(t) r2 = |α′(t)| = α′(t) by Lemma 3.2. For α′(t) < 0, the osculating circle is to the right of s(t), so lim r→0 πr −Cr(t) r2 = lim r→0 πr − 2πr −A 1 |α′(t)|, r r2 = −|α′(t)| = α′(t) by Lemma 3.2. Thus limr→0 πr−Cr(t) r2 = α′(t), regardless of the sign of α′(t). Similarly, lim r→0 πr −ˆ Cr(t) r2 = −(α′(t) −κ′(t)) = δ(t) −α′(t) regardless of the sign of α′(t) −κ′(t). Hence, lim r→0 2πr −C(s(t), r) r2 = lim r→0 πr −Cr(t) r2 + lim r→0 πr −ˆ Cr(t) r2 = α′(t) + δ(t) −α′(t) = δ(t).
□ 16 LAURA DEMARCO AND KATHRYN LINDSEY 4. Harmonic measure and holomorphic 1-forms In this section, we present curvature in the setting of conformal metrics, allowing us to use tools from complex analysis to address our geometric questions. This per-spective was first formalized by Reshetnyak . We present the proof of Theorem 1.1 and derive an alternative proof of the parametrization of the harmonic cap from Theorem 1.2. Finally, we revisit the general problem of existence of the metric ρ(P, µ) in Proposition 4.2.
4.1. Complex-analytic point of view. A smooth conformal metric on a domain in C can be expressed as ρ(z)|dz| for a smooth and positive function ρ.
The metric has non-negative curvature if U(z) = −log ρ(z) is a subharmonic function. Working with a more general class of metrics, we will only require that U be subharmonic, not necessarily differentiable or everywhere finite. We will also require that all pairs of points have finite distance from one another. These requirements can be formulated in terms of the curvature of the metric, as we explain below.
Formally, a conformal metric ρ on ˆ C is a (singular) Hermitian metric on the tangent bundle T ˆ C ≃OP1(2), and the curvature form of the metric is the positive measure given in local coordinates by ωρ = −∆log ρ (with ∆= 2i∂¯ ∂taken in the sense of distributions), so that Z ˆ C ωρ = 4π.
In more classical terms, for a smooth metric ρ, the Gaussian curvature is computed locally as κρ = −∆log ρ ρ2 .
See, for example, [1, §1.5] or [19, §2.2].
That U = −log ρ is subharmonic guarantees that the curvature form ωρ ≥0 as a distribution. Finite diameter is guaranteed by the assumption that ωρ({z0}) < 2π for all z0 ∈ˆ C [24, p.100]. Recall from §3.4 that concentrated curvature, at points z0 ∈ˆ C where 0 < ωρ({z0}) < 2π, corresponds to cone points in the local geometry.
Also in this setting, a computation shows that the circumference C(z0, r) of a small circle around z0 of radius r > 0 will satisfy [24, Lemma 8.1.1] lim r→0+ 2πr −C(z0, r) r = ωρ({z0}).
Conversely, every probability measure µ on ˆ C with µ({z}) < 1/2 for all z gives rise to a conformal metric of finite diameter with curvature distribution 4πµ, unique up to scale. Indeed, there is a one-to-one correspondence between probability measures CONVEX SHAPES AND HARMONIC CAPS 17 µ on ˆ C and their potentials, up to an additive constant, which can be viewed as logarithmically-homogeneous, plurisubharmonic functions Gµ on the tautological line bundle C2 \ {(0, 0)} →P1; see, e.g., [16, Theorem 5.9] and [13, Section 12]. The function Gµ will satisfy (2π)−1∆Gµ(z, 1) = µ in local coordinates z on ˆ C, and the conformal metric is expressed as ρµ = e−2Gµ(z,1)|dz|.
The identification between measures and their potentials is continuous, taking the L1 loc topology on potentials and the weak topology on measures. Moreover, convergence of curvatures implies convergence of the metrics [24, Theorem 7.3.1].
4.2. Harmonic measure as curvature. Let P be a compact, connected set in C containing at least 2 points, so that P is a planar shape as defined in the Introduction.
Let GP : C →R be the Green function for P; it is the unique continuous function on C satisfying (1) GP ≡0 on P, (2) GP(z) = log |z| + O(1) for z near ∞, and (3) GP is harmonic on C \ P. Then define a metric on C by ρP = e−2GP (z)|dz|.
By elementary potential theory, the function GP satisfies GP(z) = log(z)+γ+o(1) for z near ∞for some real number γ, so the metric extends uniquely by continuity across z = ∞. Note that this metric is flat (with 0 curvature) away from the boundary ∂P.
Its curvature form ωP = 2∆GP is equal to (4π times) the harmonic measure on ∂P (more precisely, the harmonic measure for the domain ˆ C \ P, relative to the point ∞).
Example 4.1. Let P be the closed unit disk. Then GP(z) = log+ |z| = max{0, log |z|}, and the curvature form ωP is arclength measure on the unit circle, normalized to have total length 4π. By the symmetry of P, it is not hard to see that Alexandrov’s real-ization of (ˆ C, ρP) will be the degenerate doubled flat disk.
4.3. The harmonic cap. Let P be any planar shape. Let Φ be the Riemann map from the complement of the unit disk to the complement of P, sending infinity to infinity. Consider the holomorphic 1-form η = 1 (Φ−1(z))2 dz on the complement of P. Since the Green function satisfies GP(z) = log |Φ−1(z)| on ˆ C \ P, we see that |η| is precisely the conformal metric ρP defined above, when restricted to the complement of P. Recall that Theorem 1.1 asserts that a Euclidean 18 LAURA DEMARCO AND KATHRYN LINDSEY development of the harmonic cap of P is given by the locally univalent function g : D →C defined by g(z) = Z z 0 Φ′(1/x) dx.
It also asserts that there exist examples where the locally univalent g fails to be univalent.
Proof of Theorem 1.1. Define F : ˆ C \ P →C by F(z) = Z z ∞ η = Z z ∞ 1 (Φ−1(ζ))2 dζ.
By definition, we have η = dF = F ∗(dw), where dw is the standard holomorphic 1-form on the plane. Since |η| is the desired conformal metric, and since |η| = F ∗|dw|, we conclude that F is a Euclidean development of the harmonic cap parametrized by z in ˆ C \ P. Now set ι(x) = 1/x. Then, to parameterize the cap by z ∈D, we pull η back to D by Φ ◦ι, so that D(z) = Z z 0 ι∗Φ∗η = Z z 0 ι∗ Φ′(ζ) ζ2 dζ = − Z z 0 Φ′(1/x) dx.
The local invertibility of D is clear because D′(z) = −Φ′(1/x) ̸= 0 for all x ∈D. Our desired function is g(z) = −D(z), which is clearly an isometric presentation.
It remains to observe that there exist planar shapes P for which the development g fails to be injective. We constructed such an example in §2.3, where P is a square minus two thin spiral channels.
□ 4.4. Harmonic cap boundary parametrization. Here we present an alternative proof of the cap parametrization in Theorem 1.2, in the special setting of harmonic measure.
As in Theorem 1.2, assume that P has a piecewise-differentiable boundary which is a Jordan curve parameterized by arclength by s : [0, L] →C. Recall that s′(t) = eiα(t) for some piecewise continuous function α : [0, L] →R. Let Φ be a Riemann map from the complement of the unit disk to the complement of P, sending infinity to infinity.
Then Φ extends to a homeomorphism from the unit circle to the boundary of P.
Define the conformal angle θ : [0, L] →R by θ(t) := arg(Φ−1(s(t))).
Without loss of generality, we may assume θ(0) = 0 so that θ defines a homeomor-phism from [0, L] to [0, 2π]. It follows that the curvature function of (3.3.1) for the harmonic measure µ on ∂P is equal to κ(t) = 4πµ(s(0, t]) = 2θ(t).
CONVEX SHAPES AND HARMONIC CAPS 19 Therefore, from Theorem 1.2, we know that the parametrization of the boundary of the harmonic cap is given by (4.4.1) ˆ s(t) = Z t 0 ei(α(x)−2θ(x)) dx.
Theorem 1.1 grants an alternate proof of (4.4.1). Indeed, with the g : D →C of Theorem 1.1, a parametrization of the boundary of the harmonic cap is given by ˆ s(t) = −g(1/Φ−1(s(t))) = −g(e−iθ(t)).
Moreover, the derivative of g is g′(z) = Φ′(1/z), and therefore, ˆ s′(t) = −g′(1/Φ−1(s(t)))−(Φ−1)′(s(t)) s′(t) Φ−1(s(t))2 = −Φ′(Φ−1(s(t))) −Φ′(Φ−1(s(t))) s′(t) Φ−1(s(t))2 = ei(α(t)−2θ(t)).
4.5. Metric existence for general measures. We conclude by returning to our original problem about the existence of a metric ρ(P, µ), for the case where P is a planar shape with Jordan curve boundary and the probability measure µ is arbitrary.
Suppose that J is a Jordan curve in ˆ C, cutting the sphere into Jordan domains A and B. We may assume that 0 ∈A and ∞∈B. Suppose that ν is a probability measure supported on J, and let U(z) = Z C log |z −w| dν(w) be a potential function for ν with logarithmic singularity at ∞. The conformal metric e−2U(z)|dz| on C extends to ˆ C and has curvature distribution equal to 4πν. Since A is simply connected, there exists a non-vanishing analytic function φ : A →C so that U(z) = log |φ(z)|.
The function φ is determined uniquely, up to postcomposition by a rotation. Set fν(z) = Z z 0 dz φ(z)2 for z ∈A. Then fν : A →C is a locally-univalent Euclidean development of A into the plane. It extends continuously to the boundary curve J. This proves the following proposition.
Proposition 4.2. Let P be a planar shape with Jordan curve boundary, and let µ be a probability measure supported on ∂P. The metric ρ(P, µ) on ˆ C exists if and only if there is a pair (J, ν) of a Jordan curve bounding a region A in ˆ C and probability measure supported on J so that fν(A) = P and (fν)∗ν = µ.
20 LAURA DEMARCO AND KATHRYN LINDSEY When µ is the harmonic measure on ∂P, observe that we may take J = ∂P and ν = µ in the statement of Proposition 4.2. Indeed, the potential function for harmonic measure satisfies U ≡0 on P so that fν = Id.
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Department of Mathematics, Northwestern University, Evanston, IL 60208, U.S.A.
E-mail address: demarco@math.northwestern.edu Department of Mathematics, University of Chicago, Chicago, IL 60637, U.S.A.
E-mail address: klindsey@math.uchicago.edu |
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4: Statically Equivalent Systems
Mechanics Map (Moore et al.)
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Wed, 28 Jul 2021 15:08:20 GMT
4.5: Equivalent Point Load
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Contents
1. Home
2. Bookshelves
3. Mechanical Engineering
4. Mechanics Map (Moore et al.)
5. 4: Statically Equivalent Systems
6. 4.5: Equivalent Point Load
Expand/collapse global location
Mechanics Map (Moore et al.)
Front Matter
1: Basics of Newtonian Mechanics
2: Static Equilibrium in Concurrent Force Systems
3: Static Equilibrium in Rigid Body Systems
4: Statically Equivalent Systems
5: Engineering Structures
6: Friction and Friction Applications
7: Particle Kinematics
8: Newton's Second Law for Particles
9: Work and Energy in Particles
10: Impulse and Momentum in Particles
11: Rigid Body Kinematics
12: Newton's Second Law for Rigid Bodies
13: Work and Energy in Rigid Bodies
14: Impulse and Momentum in Rigid Bodies
15: Vibrations with One Degree of Freedom
16: Appendix 1 - Vector and Matrix Math
17: Appendix 2 - Moment Integrals
Back Matter
4.5: Equivalent Point Load
Last updated Jul 28, 2021
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4.4: Distributed Forces
4.6: Chapter 4 Homework Problems
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Jacob Moore & Contributors
Pennsylvania State University Mont Alto via Mechanics Map
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Table of contents
1. Finding the Equivalent Point Load
2. Using Integration in 2D Surface Force Problems:
3. Using the Area and Centroid in 2D Surface Force Problems:
4. Using Integration in 3D Surface Force Problems:
5. Using Volume and Center of Volume in 3D Surface Force Problems:
6. Using Integration in Body Force Problems:
An equivalent point load is a single point force that will have the same effect on a body as the original loading condition, which is usually a distributed force. The equivalent point load should always cause the same linear acceleration and angular acceleration as the original force it is equivalent to (or cause the same reaction forces if the body is constrained). Finding the equivalent point load for a distributed force often helps simplify the analysis of a system by removing the integrals from the equations of equilibrium or equations of motion in later analysis.
Figure 4.5.1: If the body is unconstrained as shown on the left, the equivalent point load (shown as a solid vector) will cause the same linear and angular acceleration as the original distributed load (shown with dashed vectors). If the body is constrained as on shown on the right, the equivalent point load (shown as a solid vector) will cause the same reaction forces as the original distributed force (shown with dashed vectors).
Finding the Equivalent Point Load
When finding the equivalent point load, we need to find the magnitude, direction, and point of application of a single force that is equivalent to the distributed force we are given. In this course we will only deal with distributed forces with a uniform direction, in which case the direction of the equivalent point load will match the uniform direction of the distributed force. This leaves the magnitude and the point of application to be found. There are two options available to find these values:
We can find the magnitude and the point of application of the equivalent point load via integration of the force functions.
We can use the area/volume and the centroid/center of volume of the area or volume under the force function.
The first method is more flexible, allowing us to find the equivalent point load for any force function that we can make a mathematical formula for (assuming we have the skill in calculus to integrate that function). The second method is usually faster, assuming that we can look up the values for the area or volume under the force curve and the values for the centroid or center of volume for the area under the curve.
Using Integration in 2D Surface Force Problems:
Figure 4.5.2: The block shown above has a distributed force acting on it. The force function relates the magnitude of the force to the x position along the top of the box.
Finding the equivalent point load via integration always begins by determining the mathematical formula that is the force function. The force function mathematically relates the magnitude of the force (F) to the position (x). In this case the force is acting along a single line, so the position can be entirely determined by knowing the x coordinate, but in later problems we may also need to relate the magnitude of the force to the y and z coordinates. In our example above, we can relate magnitude of the force to the position by stating that the magnitude of the force at any point in Newtons per meter is equal to the x position in meters plus one.
The magnitude of the equivalent point load will be equal to the area under the force function. This will be the integral of the force function over its entire length (in this case, from x=0 to x=2).
(4.5.1)F eq=∫x min x max F(x)d x
Now that we have the magnitude of the equivalent point load such that it matches the magnitude of the original force, we need to adjust the position (x eq) such that it would cause the same moment as the original distributed force. The moment of the distributed force will be the integral of the force function (F(x)) times the moment arm about the origin (x). The moment of the equivalent point load will be equal to the magnitude of the equivalent point load that we just found times the moment arm for the equivalent point load (x eq). If we set these two things equal to one another and then solve for the position of the equivalent point load (x eq) we are left with the following equation.
(4.5.2)x eq=∫x min x max(F(x)∗x)d x F eq
Now that we have the magnitude, direction, and position of the equivalent point load, we can draw the point load in our original diagram. This point force can be used in place of the distributed force in further analysis.
Figure 4.5.3: The values for F eq and x eq that we have solved for are the magnitude and position of the equivalent point load.
Using the Area and Centroid in 2D Surface Force Problems:
As an alternative to using integration, we can use the area under the force curve and the centroid of the area under the force curve to find the equivalent point load's magnitude and point of application respectively.
Figure 4.5.4: The magnitude of the equivalent point load is equal to the area under the force function. Also, the equivalent point load will travel through the centroid of the area under the force function.
The magnitude(F eq) of the equivalent point load will be equal to the area under the force function. We can find this area using calculus, but there are often easier geometry-based ways of finding the area under the force function.
The equivalent point load will also travel through centroid of the area under the force function. This allows us to find the value for x eq. The centroid for many common shapes can be looked up in tables, and the parallel axis theorem can be used to determine the centroid of more complex shapes (see the Appendix page on centroids for more details).
Using Integration in 3D Surface Force Problems:
Figure 4.5.5: The magnitude of the surface force in this example varies with both x and y.
With surface force in a 3D problem, the force is distributed over a surface, rather than along a single line. To find the magnitude of the equivalent point load we will again start by finding the mathematical equation for the force function. Because the force is distributed over an area rather than just a line, the magnitude of the force may be related to both the x and the y coordinate, rather than just the x coordinate as before.
The magnitude of the equivalent point load (F eq) will be equal to the volume under the force curve. To calculate this value we will integrate the force function over the area that the force is applied to. To integrate this function F(x,y) in terms of the area, we will need to break the integral down further, integrating over x and then integrating over y.
(4.5.3)F eq=∫F(x,y)d A=∫y min y max(∫x min x max F(x,y)d x)d y
Once we solve for the magnitude of the equivalent point load, we can then solve for the position of the equivalent point load. Since the force is spread over a surface, we will need to calculate both the x(x eq) and the y(y eq) coordinates for the position. The process for solving for these values is similar to what was done with only an x value, except we change the moment arm value to match the equivalent point load coordinate we are looking for.
(4.5.4)x eq=∫(F(x,y)∗x)d A F eq
(4.5.5)y eq=∫(F(x,y)∗x)d A F eq
In each of the equations above, we will need to expand out the area integral into x and y integrals (as we did for F eq) in order to be able to solve them.
Using Volume and Center of Volume in 3D Surface Force Problems:
Just as in the 2D problems, there are some available shortcuts to finding the equivalent point load in 3D surface force problems. For a force spread over an area, the magnitude(F eq) of the equivalent point load will be equal to the volume under the force function. The equivalent point load will also travel through the center of volume of the volume under the force function. This should allow you to determine both x eq and y eq.
The center of volume for a shape will be the same as the center of mass for a shape if the shape is assumed to have uniform density. It should be possible to look these values up for common shapes in a table. Again, the parallel axis theorem can be used to find the center of volume for more complex shapes (See the Center of Mass page in Appendix 2 for more details).
Using Integration in Body Force Problems:
When we jump to body forces, the magnitude of our force will vary with x, y, and z coordinates. This means that our force function can include all of these variables (F(x,y,z)). To find the magnitude of the equivalent point load we integrate over the volume, breaking the volume integral down into x, y, and then z integrals.
(4.5.6)F eq=∫F(x,y,z)d V(4.5.7)=∫z min z max(∫y min y max(∫x min x max F(x,y,z)d x)d y)d z
To find the point of application of the equivalent point load, we will need to find all three coordinate positions. To do this, we will expand out the equations we used with two coordinates to include the third coordinate (z eq).
(4.5.8)x eq=∫(F(x,y,z)∗x)d V F eq
(4.5.9)y eq=∫(F(x,y,z)∗y)d V F eq
(4.5.10)z eq=∫(F(x,y,z)∗z)d V F eq
Video lecture covering this section, delivered by Dr. Jacob Moore. YouTube source:
Example 4.5.1
Determine the magnitude and the point of application for the equivalent point load of the distributed force shown below.
Figure 4.5.6: problem diagram for Example 4.5.1; a bar attached to a wall experiences a distributed force whose magnitude varies linearly over part of its length.
Solution
Video 4.5.2: Worked solution to example problem 4.5.1, provided by Dr. Jacob Moore. YouTube source:
Example 4.5.2
Determine the magnitude and the point of application for the equivalent point load of the distributed force shown below.
Figure 4.5.7: problem diagram for Example 4.5.2; a bar attached to a wall experiences a distributed force whose magnitude varies quadratically over its length.
Solution
Video 4.5.3: Worked solution to example problem 4.5.2, provided by Dr. Jacob Moore. YouTube source:
Example 4.5.3
Determine the magnitude and the point of application for the equivalent point load of the distributed force shown below.
Figure 4.5.8: problem diagram for Example 4.5.3; a bar attached to a wall experiences a distributed force whose magnitude varies linearly over part of its length and remains constant for the remainder.
Solution
Video 4.5.4: Worked solution to example problem 4.5.3, provided by Dr. Jacob Moore. YouTube source:
Example 4.5.4
The wind has piled up sand into a corner on a building. The building supervisor is worried about the weight of the sand pushing against the roof of the basement below. The function describing force of the sand pushing down on the surface is given below. Find the magnitude, direction and point of application of the equivalent point load for the distributed force of the sand. Draw the equivalent point load in a diagram.
Figure 4.5.9: problem diagram for Example 4.5.4; sand accumulated in the corner of a building is assigned to a three-dimensional coordinate system and the distributed force it exerts on the floor beneath it is described with a force equation.
Solution
Video 4.5.5: Worked solution to example problem 4.5.4, provided by Dr. Jacob Moore. YouTube source:
This page titled 4.5: Equivalent Point Load is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Jacob Moore & Contributors (Mechanics Map) via source content that was edited to the style and standards of the LibreTexts platform.
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Chapter 8 – Capacity
Learning Objectives
After studying this chapter, you should be able to:
Explain the element of contractual capacity.
Identify when a contract is voidable or void due to lack of capacity.
Apply the sword and shield doctrine.
8.1 General Perspectives on Capacity
Contractual capacity refers to the legal ability of an individual or entity to enter into a binding contract and be held legally responsible for their actions and obligations under that contract. It is an important principle in contract law as it ensures that contracts are entered into voluntarily and that all parties involved are capable of understanding and fulfilling their contractual obligations. Capacity is necessary for a legally binding contract.
Most people who enter contracts have capacity. Because of that, and to avoid having to prove one’s capacity every single time they wish to enter a contract, capacity is presumed unless there is something about the person or the transaction that calls capacity into question. In instances where capacity is questioned, it is possible to have either limited capacity (also called limited competency), or be incapable of contracting at all due to a lack of capacity, or incompetency.
The rule of contractual capacity is important because it helps to protect individuals who may be vulnerable or not fully capable of making informed decisions. Minors, people with mental disabilities, and individuals who are under the influence of drugs or alcohol may not have the capacity to fully understand the terms of a contract or the consequences of entering into it. By requiring that parties have the legal capacity to enter into a contract, the law seeks to prevent unfair contracts where one party may take advantage of the other party’s lack of understanding of the legal consequences of an agreement. Further, requiring capacity to enter a contract helps to ensure that the terms of the contract are understood and agreed upon by both parties. Since a contract is a meeting of the minds, if someone lacks mental capacity to understand what they are agreeing to, it is unreasonable to hold that person to the consequences of the contract. At common law there are various classes of people who are presumed to lack the requisite capacity to fully understand the consequences of a contract. These include infants (minors), the mentally ill, and the intoxicated. This Chapter will review each of these in turn.
8.2 Minors (or “Infants”)
The General Rule
Minors – persons younger than eighteen years of age in most states – may only enter into voidable contracts. This is because minors do not have full legal capacity to enter into contracts. Minors can enter into most contracts, but they can avoid their contracts, up to and within a reasonable time after reaching majority, while the other contracting party with full contractual capacity cannot. The idea is that minors do not stand on an equal footing with adults, and it is unfair to require them to abide by contracts made when they have immature judgment.
For example, suppose a 16-year-old enters into a contract to purchase a new phone from a mobile phone company. Since the 16-year-old is a minor and not yet legally capable of entering into a contract, the contract is voidable. This means that the 16-year-old has the option to either affirm the contract or void it. If the 16-year-old decides to go ahead with the purchase, the contract will be binding on the mobile phone company. However, if the 16-year-old chooses to avoid the contract, they will be released from any obligations or liabilities that may have arisen from the contract, and they will be entitled to a refund if they have already made any payments. This is called disaffirming a contract. To disaffirm a contract means to repudiate or reject a contract that has been previously entered into.
Most people will recognize the 16-year-old in the above example as a minor. But legal term infant is also used to describe the 16-year-old contracting party above. The words minor and infant are mostly synonymous, but there are some differences between the terms. Historically speaking, in a state where the legal age to drink alcohol is twenty-one, a twenty-year-old would be a minor, but not an infant, because infancy is under eighteen. A seventeen-year-old may avoid contracts (usually), but an eighteen-year-old, while legally bound to his contracts, cannot legally drink alcohol. This is why some states use the term infant for one who may avoid his contracts even though, of course, in everyday terms we think of an infant as a baby.
In nearly all U.S. states, an 18-year-old may assent to a binding contract, but there may still be some exceptions associated with age. For those under twenty-one, there may also be legal impediments to holding certain kinds of jobs, and agreeing to certain kinds of contracts, marrying, leaving home, and drinking alcohol. There is no uniform set of rules since each state can make their own laws in these areas. Because of these ambiguities, contracting adults like landlords or creditors may still require parents to co-sign contracts for a young adult over the age of 18. This is lawful, but is often unnecessary.
The exact day of “majority” – the day on which the limitations on contracting as a minor or infant vanish – can also vary. The old common-law rule put it on the day before the twenty-first birthday. Many states have changed this rule so that majority commences on the day of the eighteenth birthday. For the most part, the remainder of this Chapter will use the word minor to describe a contracting party under the age of 18. This is consistent with the law in New Jersey, where by statute an individual under the age of 18 is defined as an infant, and therefore lacks the legal capacity to contract.
Even though a minor lacks the legal capacity to contract, the minor’s contract is not automatically invalid. Instead, the minor has the right to choose to either affirm the contract or void it which makes the minor’s contract voidable. This means that despite having limited competence minors can enter contracts, but most of those contracts can be avoided, or disaffirmed, without any reason for doing so other than the party is a minor. If a lawsuit about the contract was filed, the minor would claim the defense of infancy, and this defense would be sufficient unless there are other reasons why the contract could not be disaffirmed. Should the adult wish to get out of the contract, they cannot make the same claim. The adult cannot seek to disaffirm the contract; only the minor can.
When a minor disaffirms a contract, the minor has the legal right of a minor to walk away from a contract and be released from any obligations or liabilities arising from it. In addition, the minor is restored to their original position, meaning that they would be entitled to the return of any consideration, without the responsibility of being liable for damages under the contract. A minor need only return whatever part of the consideration in the contract that they still have. So, if the 16-year-old in our example above opts to disaffirm their purchase of the new phone one month after entering the contract, the minor may disaffirm. The minor would be entitled to the return of any down payment or other fees paid for the phone. The minor will return the phone but would not be liable for any damage done to the phone while it was in the minor’s possession, even if the phone was broken. This is called the general rule of law pertaining to contracts entered into by minors. This means that the majority of states, but not all, follow this rule.
Exceptions to the Minor’s Right to Disaffirm
There are several exceptions and limitations to the minor’s right to disaffirm. These are discussed below.
Duty to Return Consideration Received
As we explored under the general rule, when a minor disaffirms a contract their own obligation would be to return an item of consideration if that item is still in the minor’s possession. For instance, the minor would have to return the untouched groceries in a contract with a grocery store. Under the general rule, if some of the groceries were already eaten, there would be nothing to return and thus the minor would not be liable for the value of the groceries that were consumed. Some courts, however, have required more from the minor when necessary to avoid injustice to the adult. Suppose that our 16-year-old minor that purchased the cell phone decided to disaffirm their purchase after dropping and breaking the phone. There are some courts that would limit what the minor could receive in disaffirming the contract. For example, a court could determine that the minor should not receive full value back for the phone, but instead receive only that value associated with the (wrecked) phone at the time of the disaffirmance. Since this is contrary to the general rule, it is called the “minority rule” of law, where minority here means that less than half of the states follow this rule. On this area of law, New Jersey is in the minority, typically requiring that a minor return all of the consideration (or the value of that consideration) that was received in order to disaffirm.
Necessaries
A necessary is a basic need of life such as food, clothing, shelter and basic medical services. This has historically been the definition at common law. In recent years, however, the courts have expanded the concept so that in many states today, necessities include property and services that will enable a person to earn a living and to provide for those dependent on them.
When a contract with a minor is for an item that is a necessary, an exception to the general rule that minors can disaffirm their contracts is that minors are generally liable for the reasonable cost of necessities. The reason behind this change from the general rule that minors may disaffirm without further obligation is that denying minors a full right of contract for necessaries would actually harm minors, not protect them. The minor will not be obligated to perform the contract so technically the minor can still disaffirm. However, the minor will be responsible to the other party for the reasonable value of the contracted necessary. For example, suppose a minor enters into a contract with a grocery store to purchase food items where the food qualifies as a necessary. If the minor later disaffirms the contract, they will not be liable for the contract price of the food items if they have not been consumed or used by the minor. Those items can be returned, which is required under the general rule. However, if the minor has already consumed or used the food items, and those food items are necessaries, the minor is responsible for the reasonable cost of the food items. The requirement that the minor be liable for the reasonable value of the necessary is rooted in the theory of quasi contract, which was explored in an earlier Chapter.
So, how does this rule protect minors? Since the rule provides protection to the seller of the necessary, in this case the food, that seller will want to sell to minors that need food for nourishment, because the seller does not risk the harsh consequences of the general rule.
Nonvoidable Contracts
There are some contracts that are considered nonvoidable. The contracts that are nonvoidable can vary by state, and can include contracts such as insurance, education or medical care, bonding agreements, stocks, or bank accounts, and child support agreements. The rationale behind making these contract nonvoidable is this: if a contract is voidable it is a disincentive to an adult party to contract with a minor (or other party that lacks capacity) since there is a risk that the minor will disaffirm the contract in the future. Many sellers simply will not take that risk and instead refuse to contract with minors completely. There are certain types of contracts, though, that we want to be available to minors and so making those contracts nonvoidable encourages these types of contracts with minors. For example, it is beneficial both to a minor and to society that the minor receive an education. Education can be expensive, and so to help assure that education loans are available to minors, the right of the minor to disaffirm that loan in the future is limited or eliminated.
Misrepresentation of Age
A minor that misrepresents their age in order to enter a contract may be limited in their ability to later disaffirm that contract. Typically, an adult that simply thought that the minor was over the age of 18 could not raise that as a defense to the minor’s attempt to disaffirm the contract. This is true even if the adult’s belief that the minor was older was reasonable. Importantly, the minor must affirmatively misrepresent their age. Even so, this limitation on disaffirmance will depend on the state. A Michigan statute, for instance, prohibits an infant from disaffirming if he has signed a “separate instrument containing only the statement of age, date of signing and the signature.” And some states estop him from claiming to be an infant even if he less expressly falsely represented himself as an adult. Estoppel is a refusal by the courts on equitable grounds to allow a person to escape liability on an otherwise valid defense; unless the infant can return the consideration, the contract will be enforced. It is a question of fact how far a non-express (an implied) misrepresentation will be allowed to go before it is considered so clearly misleading as to stray into the prohibited area. Some states hold the infant liable for damages for the tort of misrepresentation, but others do not. As William Prosser, the noted torts scholar, said of cases paying no attention to an infant’s lying about his age, “The effect of the decisions refusing to recognize tort liability for misrepresentation is to create a privileged class of liars who are a great trouble to the business world.”
Tort Connected with a Contract
Although the lack of contractual capacity can protect a minor from a contract by permitting the minor to disaffirm, this same rule does not extend to other areas of the law. Recall that minors can be liable for their torts (e.g., assault, trespass, nuisance, negligence). So, if there is also a tort claim connected with a contract involving a minor, the minor can be liable for the tort unless the tort suit is only an indirect method of enforcing the contract. For example, Tandy, who is 16, agrees to mow the neighbor’s lawn for $50. The neighbor pays Tandy up front. Tandy can disaffirm this contract because Tandy is a minor. If Tandy doesn’t mow the lawn, the neighbor could not sue for fraud because that would be an indirect attempt to enforce the contract. However, let’s say that while mowing the lawn, Tandy accidentally damages the neighbor’s flower bed with the lawnmower. The neighbor could then sue Tandy for the tort of negligence, which is a civil wrong resulting from a failure to exercise reasonable care.
Ratification
When a minor becomes an adult (i.e., reaches the age of majority when the “disability” that rendered the party of limited contractual capacity has been lifted) there are two options for an existing contract. The party to the contract can disaffirm that contract up to a reasonable period of time after reaching the age of majority, where what is reasonable will depend on the circumstances. Or, the contract can be ratified. Ratification takes place upon accepting or confirming a contract that was originally voidable because a party lacked contractual capacity.
For example, suppose a minor signs a contract to buy a car, but the contract is voidable due to the minor’s age. Once the minor reaches the age of majority, they may ratify the contract, thereby making it legally binding. They may ratify the contract by sending a letter to the seller of the car letting them know they intend to ratify the contract. This is express ratification. Or, they may ratify the contract by doing nothing to disaffirm it while continuing to drive the car. This is implied ratification. Express ratification occurs when a party explicitly confirms or approves the previously voidable contract. Implied ratification, on the other hand, occurs when a party behaves in a manner consistent with accepting a contract, or in a manner inconsistent with disaffirming the contract. Both express and implied ratification have the effect of making the previously voidable contract legally binding.
Activity 8A
You be the Judge
In the case of Hojnowski v. Vans Skate Park, the New Jersey Supreme Court heard a dispute involving the enforceability of a liability waiver signed by a parent on behalf of their minor child. The plaintiffs were the parents of a 12-year-old child who was injured at Vans Skate Park that had signed an extensive liability waiver and exculpatory clause when enrolling their child in a skateboarding program. This waiver was signed prior to the Plaintiffs having any access to the Skate Park.
While skating at the park on a later date, the child suffered a broken femur allegedly due to the conduct of a much more aggressive skater, of whose conduct the parents had already complained. Under the waiver the parents agreed that neither they, nor the child, would hold Vans Skate Park liable in case of an injury. In addition, the parents agreed that if there were a dispute between the parties that it would be submitted to arbitration, and not heard through the court system. The child did not sign the waiver.
Parents sued the Skate Park on their own behalf and for their child, alleging various tort claims, and the Skate Park defended the lawsuit stating that the parental waiver was an agreement between the parties that (1) limited the Skate Park’s liability, and (2) required the arbitration of any disputes between the parties. The case made its way to the New Jersey Supreme Court which ruled in favor of the plaintiffs in part stating that parental waivers of a child’s future claims for injuries in recreational activities are unenforceable. The Court held that parents cannot bind their children to contractual provisions that waive the child’s rights to sue for injuries caused by the negligence of others. The decision in this case established an important precedent in New Jersey, indicating that liability waivers signed by parents on behalf of their minor children in recreational activities are generally unenforceable. However, the Court did find that the agreement to arbitrate claims was enforceable so that the Hojnowskis must use arbitration to pursue their claims.
Question: Would having the child sign the waiver themselves change the outcome of this case? Why or why not?
Question: Even though the Court found that the parental waiver in this case was unenforceable, it still found that the parent could bind the child to use arbitration for the resolution of the dispute. Why do you think these matters were treated differently?
Read about this case on Google Scholar.
8.3 Persons Who Are Mentally Incapacitated
Mental incapacity can result from mental illness, physical illness, or deficiency, and may have the effect of causing full incompetence, limited competence, or have no impact on contractual capacity at all. As an initial determination, the impact of the illness or deficiency on the ability of the party to understand the legal consequences of entering into a contract should be reviewed. A person who has an illness or deficiency that does not impact the ability to understand the legal consequences of a contract would not qualify for mental incapacity, and therefore could enter a valid contract.
When a person is suffering from an illness or deficiency that prevents them from understanding the legal consequences of entering a contract, a contract entered during this time period is voidable due to limited competence. The time period to avoid the contract would be during the contract or when capacity is restored. For this limitation on competency, it is more likely that avoiding a contract will occur once the deficiency is lifted, restoring mental understanding. A disaffirmance may also take place by a guardian, if a court appoints a guardian and a voidable contract exists at that time of appointment.
When a person that is suffering from an illness or deficiency is adjudicated to be fully incompetent in a court proceeding, any further efforts of that person to enter a contract will result in a void contract. In this situation, a guardian will be appointed to proceed with contracting for the incompetent party in the future.
For example, suppose a seller diagnosed with a mental illness agrees to sell their property to a buyer. Even though the seller has a diagnosis, this contract will be valid unless the mental illness has impacted the seller’s ability to understand the legal consequences of the contract. If the seller, due to their mental illness, cannot understand the legal consequences of the contract, the contract may be considered voidable. The seller would be able to disaffirm the contract. Similarly, the buyer may not be able to enforce the contract and may have to return any money or property received from the seller. If the seller has been adjudicated incompetent in a Court proceeding, the contract would be void. This means that the guardian responsible for making decisions on the seller’s behalf could avoid the contract.
As in the situation with a minor, if the contract was for a necessity, the other party may have a valid claim against the estate of the one who is mentally incapacitated in order to prevent unjust enrichment. In any case, when a contract is disaffirmed, the mentally incapacitated person must return any property in their possession to the other contracting party. If the contract was fair and the other party had no knowledge of the mental illness, the court has the power to order other relief.
Activity 8B
Case Debate: Protection or Oppression? #FreeBritney
Britney Spears, a well-known pop singer, was placed under a conservatorship (or guardianship) in 2008. This is a legal arrangement where a guardian (conservator) is appointed to manage the personal and financial affairs of an individual (conservatee) who is deemed to lack the contractual capacity to make such decisions and enter contracts.
The conservatorship was established following a series of highly publicized personal and legal challenges in Britney Spears’ life. The conservatorship was implemented due to concerns about Britney Spears’ contractual capacity. It brought into question whether she had the legal capacity to enter into contracts and fully understand the implications and consequences of her agreements. Under the conservatorship, her father, Jamie Spears, initially served as the conservator, with control over her finances, career decisions, and personal life. A professional conservator, Jodi Montgomery, was later appointed as a temporary replacement for Jamie Spears to handle Britney’s personal affairs.
Despite being under a conservatorship, Britney Spears had a successful show in residency in Las Vegas from December 2013 to December 2017. Widely regarded as a successful venture for Britney Spears both artistically and commercially, Britney demonstrated her talent, professionalism, and ability to deliver captivating performances on stage.
The conservatorship remained in place for many years, even while Britney Spears sought review of the conservatorship expressing her desire to end the conservatorship and regain control over her life and finances. At various times while the conservatorship was in place, Britney Spears’ fans and the media questioned the conservatorship and its impact on her life. In 2019, the “#FreeBritney” movement gained momentum, with supporters advocating for an end to the conservatorship and greater transparency about Spears’ well-being.
In June 2021, Britney Spears made a statement during a court hearing, expressing her desire to terminate the conservatorship. She revealed details about the restrictive nature of the arrangement, claiming that she had been subjected to abuse, forced medication, and denial of personal freedoms. Eventually the conservatorship was dissolved and currently Britney Spears has regained full contractual capacity.
Question: Contractual capacity is intended to provide legal protections to those that do not understand the legal consequences of their contracts and to safeguard individuals from the consequences of contracting with unscrupulous adults. With such a safeguard available for those with limited capacity, why isn’t this enough protection for incompetent adults. In other words, what is the purpose of having a separate action where a Court could find a party incompetent and completely remove their right to contract and appoint a guardian?
Question: One of the justifications for continuing a guardianship is that the guardianship is working. When this is the perception, it can be very difficult for the party adjudicated incompetent to show that they can handle their own affairs. So, a guardianship that is working perpetuates the guardianship itself. When should a guardianship or conservatorship end? When should it have ended in the Britney Spears case?
Question: Should there be limits on guardianships as they pertain to contractual rights and obligations? What is the best way to balance the right to contract with the well-being of the individual subject to limitations imposed by a Court?
Debate the Case: Find and review two resources dealing with the #FreeBritney movement. Do you think that the Britney Spears conservatorship was a necessary safeguard or an infringement on the right to contract?
8.4 Persons who are Intoxicated
People who are intoxicated at the time of entering a contract may have limited competency if the nature of their intoxication rendered the person unable to understand the nature and consequences of a contract. Intoxication could be the effect of alcohol, drugs, or other intoxicating substances, and could be voluntary or involuntary.
If a person is so drunk that he has little awareness of his acts, and if the other person knows this, any contract that results is voidable. Should the intoxicated person disaffirm the contract when regaining competency, he is obligated to refund the consideration to the other party unless he dissipated it during his drunkenness. If the other person is reasonably unaware of his intoxicated state, however, an offer or acceptance of fair terms would be upheld by most courts.
If a person is only partially inebriated and has some (but not a full) understanding of his actions, courts will review factors in determining whether the transaction would be voidable or valid, for example, whether the other party induced the drunkenness, the adequacy of the consideration exchanged, and whether the transaction is one which a reasonably competent person might have made.
A person who was intoxicated at the time he made the contract may nevertheless subsequently ratify it. Thus, where a party, several times involuntarily committed for alcoholism, executed a promissory note in an alcoholic stupor but later, while sober, paid the interest on the past-due note, he was denied the defense of intoxication; the court said he had ratified his contract. In any event, intoxication is a disfavored defense on public policy grounds.
Case 8.1
Cameron v. Power Co., 50 S.E. 695 (N.C. 1905)
WALKER, J.
This action was brought to recover damages for the breach of a contract whereby the plaintiff agreed to sell and the defendant to buy a Corliss engine. … The defendant in its
answer [to the complaint] admitted that its president had signed a contract, and pleaded specially that at the time of signing it he was so drunk that he did not have sufficient mental capacity to contract with the plaintiff for the engine. The court, without objection, submitted only one issue to the jury, which is as follows: “What damage, if any, is the plaintiff entitled to recover of the defendant?” The jury answered “Nothing.” Judgment was entered accordingly.
The question presented for our consideration arises upon an exception to the charge of the court regarding the drunkenness of the plaintiff’s agent and its sufficiency to avoid the contract.
…
We have examined the charge of the court with care and cannot find that his Honor said anything not in strict accordance with the law, as we now declare it to be. He charged the jury as follows: “The mere fact that the defendant’s president was drinking was not sufficient, but the jury must find that he was so intoxicated that he could not understand the nature and scope of what he was doing. If the jury find from the greater weight of the testimony that the agent was drinking, it would not be sufficient to invalidate the contract, but if the jury find that the defendant’s president, at the time he signed the contract or order for the engine, was so drunk as to be incapable of knowing the effect of what he was doing, then the contract or order would not be binding upon the defendant. Whether or not he was so intoxicated as to render him incompetent to contract is a question for the jury upon all the evidence.” We think this was a clear and sufficient exposition of the law applicable to the facts of the case. What the judge said in his reference to the nature of the transaction in which the agent was engaged and its importance or magnitude … was evidently intended to point what he had already said as to the true test of mental capacity, and to impress upon them, as an essential condition of the validity of the contract, that the agent of the defendant at the time he signed the paper must have been sober enough to understand the nature of the transaction and
the effect or consequence of his act, and not that he must have been able to act with wisdom or discretion.
No error.
Case Questions
The Court notes that the question of competency in this case is one for the jury to consider based on the evidence in the case. What type of evidence would you expect to see in a case where capacity is being challenged?
If the Court found that the Plaintiff’s agents were too intoxicated to understand the legal consequences of the transaction, what would happen to the underlying contract in the case? Would there be damages?
In this case, the intoxication was voluntary. Do you think that the case would have turned out differently if the intoxication was involuntary? Why or why not?
8.5 The Sword and Shield Doctrine
Throughout the Chapter, we’ve explored ways a person with limited competency can seek to avoid unwanted contracts. As stated previously, the purpose of allowing disaffirmance in these cases is to protect people with limited competency from being taken advantage of, much like a “shield.”
The sword and shield doctrine is a legal principle that pertains to the use of a contract’s terms as both a sword to enforce the contract and a shield to defend against claims under the contract. The “sword” aspect of the doctrine refers to the use of a contract’s terms by a party seeking to enforce the contract. This means that a party can use the terms of the contract to compel the other party to perform their obligations under the contract. The “shield” aspect of the doctrine, on the other hand, refers to the use of a contract’s terms as a defense against claims under the contract. This means that a party can use the terms of the contract to defend against claims made by the other party under the contract.
In cases of capacity, the sword and shield doctrine can be used to protect the interests of parties who lack contractual capacity, such as minors or individuals with mental disabilities.
The shield aspect of the doctrine can be used to defend against claims under a contract made by a party who lacks contractual capacity. For example, if a minor enters into a contract to purchase a car and the car is defective, the minor can use the shield aspect of the doctrine to defend against the seller’s claim for payment by arguing that they lacked contractual capacity to enter into the contract and are therefore not bound by its terms. This is possible even if the minor would seek to keep the car if it were not defective, thus minority is a shield.
The sword aspect of the doctrine can also be used by a party who lacks contractual capacity to enforce the terms of a contract that are beneficial to them. For example, if a minor enters into a contract to sell their artwork, and the buyer fails to make payment as required by the contract, the minor can use the sword aspect of the doctrine to compel the buyer to make payment as required by the contract.
At the same time, a sword has two sides, and some courts have reviewed attempts to disaffirm contracts through this doctrine, finding in some cases that the person with limited competency is using their right to disaffirm to achieve an unfair advantage over a competent adult in a contract. For example, suppose a 16-year-old minor signs a contract with an adult to buy a car. The minor then decides to disaffirm the contract having already driven the car for several months and causing damage to it. The adult may ask the court to review the circumstance of the disaffirmance to see if the minor is unfairly using this right to avoid paying for the damages they caused to the car while they were using it.
In such cases, the court may review the circumstances surrounding the disaffirmance to determine if the minor is attempting to take advantage of the adult (using disaffirmance as a sword) or if they have a legitimate reason to avoid the contract (a shield). If the court finds that the person with limited competence is unfairly using the ability to disaffirm, the court could rule in favor of the adult and hold the minor responsible for any damages they caused under the contract.
In general, the sword and shield doctrine is meant to balance the interests of parties in contractual dealings and ensure that both parties are held to the same standard of performance. However, in cases where one party lacks contractual capacity, the doctrine must be applied in a way that considers the particular circumstances of that party and their intentions in disaffirming a contract.
Activity 8C
Which is Which?
Activity 8D
What’s your Verdict?
Video games are popular, as are in-app purchases within them. Should minors be able to disaffirm in-app video game purchases? Does it matter if the funding for the purchase originated from the minor’s own money, or from the parent? Explain your position using the information discussed in this Chapter.
End of Chapter Exercises
Langstraat was seventeen when he purchased a motorcycle. When applying for insurance, he signed a “Notice of Rejection,” declining to purchase uninsured motorist coverage. He was involved in an accident with an uninsured motorist and sought to disaffirm his rejection of the uninsured motorist coverage on the basis of infancy. May he do so?
Ivar, an infant, bought a used car—not a necessity—for $9,500. Seller took advantage of Ivar’s infancy: the car was really worth only $5,500. Can Ivar keep the car but disclaim liability for the $4,000 difference?
If Ivar bought the car and it was a necessity, could he disclaim liability for the $4,000?
If Ivar bought the car, and subsequently drove the car while it was having mechanical difficulties, resulting in blowing out the engine, can Ivar disaffirm the car? If so, will he have any liability in doing so? Under what theory?
Alice Ace found her adult son’s Christmas stocking; Mrs. Ace herself had made it fifty years before. It was considerably deteriorated. Isabel, sixteen, handy with knitting, agreed to reknit it for $100, which Mrs. Ace paid in advance. Isabel, regrettably, lost the stocking. She returned the $100 to Mrs. Ace, who was very upset. May Mrs. Ace now sue Isabel for the loss of the stocking (conversion) and emotional distress?
References
Dodson v. Shrader, 824 S.W.2d 545 (Tenn. 1992).
First State Bank of Sinai v. Hyland, 399 N.W.2d 894 (S.D. 1987).
Gastonia Personnel Corp. v. Rogers, 172 S.E.2d 19 (N.C. 1970).
Hojnowski v. Vans Skate Park, 187 N.J. 323, 901 A.2d 381 (2006)
N.J.S.A. 9:17B-1
Albert, Clark, Guide to New Jersey Contract Law, NJICLE
William L. Prosser, Handbook of the Law of Torts, 4th ed. (St. Paul, MN: West, 1971), 999.
Restatement (Second) of Contracts, Section 13.
Restatement (Second) of Contracts, Section 16(b). |
190439 | https://mathcentral.quora.com/math-tau-n-text-is-an-odd-integer-if-and-only-if-n-is-a-perfect-square-math | [math]\tau (n)\,\,\text{is an odd integer if and only if n is a perfect square.}[/math]? - Math Central - Quora
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τ(n)is an odd integer if and only if n is a perfect square.τ(n)is an odd integer if and only if n is a perfect square.?
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Mohammad Afzaal Butt
B.Sc in Mathematics&Physics, Islamia College Gujranwala (Graduated 1977)
·4y
Let τ(n)is odd, then Let τ(n)is odd, then
τ(n)=(k 1+1)(k 2+1)(k 3+1)⋯(k r+1)is odd for τ(n)=(k 1+1)(k 2+1)(k 3+1)⋯(k r+1)is odd for
n=p k 1 1⋅P k 2 2⋅k 3 3⋯p k r r n=p 1 k 1⋅P 2 k 2⋅3 k 3⋯p r k r
∵τ(n)is odd, all k i must be even∵τ(n)is odd, all k i must be even
⟹n=p 2 s 1 1⋅P 2 s 2 2⋅2 s 3 3⋯p 2 s r r⟹n=p 1 2 s 1⋅P 2 2 s 2⋅3 2 s 3⋯p r 2 s r
⟹n=(p s 1 1⋅P s 2 2⋅s 3 3⋯p s r r)2⟹n=(p 1 s 1⋅P 2 s 2⋅3 s 3⋯p r s r)2
∴n is a perfect square∴n is a perfect square
Conversely let n is a perfect square Conversely let n is a perfect square
n=(p s 1 1⋅P s 2 2⋅s 3 3⋯p s r r)2 n=(p 1 s 1⋅P 2 s 2⋅3 s 3⋯p r s r)2
⟹n=p 2 s 1 1⋅P 2 s 2 2⋅2 s 3 3\cdo⟹n=p 1 2 s 1⋅P 2 2 s 2⋅3 2 s 3\cdo
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Let τ(n)is odd, then Let τ(n)is odd, then
τ(n)=(k 1+1)(k 2+1)(k 3+1)⋯(k r+1)is odd for τ(n)=(k 1+1)(k 2+1)(k 3+1)⋯(k r+1)is odd for
n=p k 1 1⋅P k 2 2⋅k 3 3⋯p k r r n=p 1 k 1⋅P 2 k 2⋅3 k 3⋯p r k r
∵τ(n)is odd, all k i must be even∵τ(n)is odd, all k i must be even
⟹n=p 2 s 1 1⋅P 2 s 2 2⋅2 s 3 3⋯p 2 s r r⟹n=p 1 2 s 1⋅P 2 2 s 2⋅3 2 s 3⋯p r 2 s r
⟹n=(p s 1 1⋅P s 2 2⋅s 3 3⋯p s r r)2⟹n=(p 1 s 1⋅P 2 s 2⋅3 s 3⋯p r s r)2
∴n is a perfect square∴n is a perfect square
Conversely let n is a perfect square Conversely let n is a perfect square
n=(p s 1 1⋅P s 2 2⋅s 3 3⋯p s r r)2 n=(p 1 s 1⋅P 2 s 2⋅3 s 3⋯p r s r)2
⟹n=p 2 s 1 1⋅P 2 s 2 2⋅2 s 3 3⋯p 2 s r r⟹n=p 1 2 s 1⋅P 2 2 s 2⋅3 2 s 3⋯p r 2 s r
τ(n)=(2 s 1+1)(2 s 2+1)(2 s 3+1)⋯+(2 s r+1)τ(n)=(2 s 1+1)(2 s 2+1)(2 s 3+1)⋯+(2 s r+1)
⟹τ(n)is odd⟹τ(n)is odd
Q.E.D Q.E.D
9 4
John Calligy
4y
Suppose d∣n.d∣n. Then also (n/d)∣n.(n/d)∣n. So the divisors of n n come in pairs {d,n/d},{d,n/d}, with one divisor less than √n n and the other greater than √n,n, and this is an even number. If and only if n n is a perfect square, there is one more divisor, √n.n.
9 2
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A simpler way to minimize the mean of squares, given the mean?
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Suppose I have N N real numbers and I already know their mean, x¯x¯:
x¯=1 N∑i=1 N x i x¯=1 N∑i=1 N x i
(but I don't know the individual values x 1,x 2,…,x N)x 1,x 2,…,x N).
I want to find the smallest possible value of the mean of the squares, y y:
y=1 N∑i=1 N x 2 i y=1 N∑i=1 N x i 2
I thought a bit about the case where N=2 N=2 and it seemed like the answer should be x 1=x 2=x¯x 1=x 2=x¯. For example, if x¯=5 x¯=5, then 5 2+5 2=50 5 2+5 2=50, but 4 2+6 2=52 4 2+6 2=52 and 3 2+7 2=58 3 2+7 2=58, and so on.
I feel like this should be a very simple thing to prove for any N N, but I had to dig into the back of an old (engineering) textbook to recall the method of Lagrange multipliers... which (unless I made a mistake) indeed gave me the answer x 1=x 2=x 3=⋯=x N=x¯x 1=x 2=x 3=⋯=x N=x¯. Or, using the vector notation of the textbook: x––=x¯1–x =x¯1 . I wrote my calculations below, just in case this result is wrong.
My question is: Is there a "simpler" way to prove this result (e.g. using basic high school mathematics)? Is it over-complicated to view this as a constrained optimization problem?
Many thanks for any help.
My Calculations:
Function to minimize (subject to constraint):
f(x––)=1 N 1–T x––2 f(x _)=1 N 1 _ T x _ 2
Constraint:
c(x––)=1 N 1–T x––−x¯=0 c(x )=1 N 1 _ T x −x¯=0
Unconstrained function to minimize:
h(x––)=f(x––)+λ c(x––)h(x )=f(x )+λ c(x _)
Minimize to get λ λ:
∂f∂x––+∂∂x––(λ c(x––))=0–∂f∂x +∂∂x (λ c(x _))=0 _
2 N x––+λ N 1–=0–2 N x +λ N 1 =0 _
2 N 1–T x––+λ N 1–T 1–=0 2 N 1 _ T x +λ N 1 _ T 1 =0
λ=−2 x¯λ=−2 x¯
Substitute λ λ back in to solve for x x:
2 N x––=2 x¯N 1–2 N x _=2 x¯N 1 _
x––=x¯1–x _=x¯1 _
calculus
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asked Jul 29, 2021 at 15:44
HarryHarry
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See the RMS-AM inequality (root mean square a.k.a. quadratic mean).dxiv –dxiv 2021-07-29 16:44:17 +00:00 Commented Jul 29, 2021 at 16:44
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As you have already noted, the mean of the square of a pair of numbers withe fixed average is attained when both are of equal magnitude. Therefore for any finite set of numbers the same is true, since if two numbers in the set are different, they can be replaced by a better pair. Given that the set is finite, a finite number of steps will suffice.
More direct approach - proof by contradiction: Assume ∑x 2 i∑x i 2 is minimum with x 2 i≠x 2 j x i 2≠x j 2 for some i≠j i≠j. Replace pair with terms of equal average magnitude. The sum will be lower, contradicting minimum assumption.
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edited Jul 30, 2021 at 17:53
answered Jul 29, 2021 at 16:06
herb steinbergherb steinberg
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Ah, that's a clever way to think about it. I still had some trouble just proving it for the N=2 N=2 case... but I think I'm right in saying the two values must be symmetrical about the mean, so x 1=x¯+α x 1=x¯+α and x 2=x¯−α x 2=x¯−α (for some α α). Then the sum of the squares is y=x 2 1+x 2 2=2(x¯2+α 2)y=x 1 2+x 2 2=2(x¯2+α 2) (hence the values 50, 52 and 58 in my example). Then this is clearly minimized by choosing α=0 α=0.Harry –Harry 2021-07-29 17:36:09 +00:00 Commented Jul 29, 2021 at 17:36
In fact, I think this generalizes easily. I can simply write x––=x¯1–+α––x =x¯1 +α , for some vector α––α _ which must sum to zero in order for x¯x¯ to remain as the mean (so 1–T α––=0 1 _ T α =0). Then I want to minimize N y=x––T x––=N x¯2+x¯1–T α––+x¯α––T 1–+α––T α––=N x¯2+α––T α––N y=x _ T x =N x¯2+x¯1 _ T α +x¯α _ T 1 +α _ T α =N x¯2+α _ T α , which is obviously minimized when α––=0–α =0 _.Harry –Harry 2021-07-29 18:16:04 +00:00 Commented Jul 29, 2021 at 18:16
@Harry I added a simpler proof to my original answer.herb steinberg –herb steinberg 2021-07-30 17:54:57 +00:00 Commented Jul 30, 2021 at 17:54
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Is this short enough?
Take your N N values as the population so E[X]=1 n∑x i=x¯E[X]=1 n∑x i=x¯.
1 n∑x 2 i=E[X 2]=(E[X])2+var(X)1 n∑x i 2=E[X 2]=(E[X])2+var(X).
var(X)≥0 var(X)≥0 with equality iff X=E[X]X=E[X].
So 1 n∑x 2 i=(1 n∑x i)2=x¯2 1 n∑x i 2=(1 n∑x i)2=x¯2 is the minimum given x¯x¯ the mean and occurs iff each x i=x¯x i=x¯.
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answered Jul 29, 2021 at 15:50
HenryHenry
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I'm a little confused by the notation. Is X X a random variable? I hadn't quite imagined that anything was random. Or does X X correspond to the set of x i x i values, for i=1,2,…,N i=1,2,…,N? (Like the vector x––x _ in the question).Harry –Harry 2021-07-29 16:39:50 +00:00 Commented Jul 29, 2021 at 16:39
@Harry I was treating X X as a single random sample from the x i x i s so its expectation is their mean and its variance is their variance etc. Then using a well known expression relation the second moment to the variance and saying the second moment is minimised when the variance is 0 0 Henry –Henry 2021-07-29 16:45:05 +00:00 Commented Jul 29, 2021 at 16:45
Aha... then I think this may be equivalent to the solution I stumbled upon in my response to herb steinberg. I think you are defining v a r(X)v a r(X) as something like ∥x––−x¯1–∥2‖x −x¯1 ‖2, which corresponds to α––T α––α _ T α _ in my solution.Harry –Harry 2021-07-29 18:42:55 +00:00 Commented Jul 29, 2021 at 18:42
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190441 | https://optimization-online.org/wp-content/uploads/2023/09/OSPBRevision2.pdf | Using orthogonally structured positive bases for constructing positive k-spanning sets with cosine measure guarantees Warren Hare ∗ Gabriel Jarry-Bolduc † S´ ebastien Kerleau‡ Cl´ ement W. Royer§¶ October 3, 2023 Abstract Positive spanning sets span a given vector space by nonnegative linear combinations of their elements. These have attracted significant attention in recent years, owing to their extensive use in derivative-free optimization. In this setting, the quality of a positive spanning set is assessed through its cosine measure, a geometric quantity that expresses how well such a set covers the space of interest. In this paper, we investigate the construction of positive k-spanning sets with geometrical guarantees. Our results build on recently identified positive spanning sets, called orthogonally structured positive bases. We first describe how to identify such sets and compute their cosine measures efficiently. We then focus our study on positive k-spanning sets, for which we provide a complete description, as well as a new notion of cosine measure that accounts for the resilient nature of such sets. By combining our results, we are able to use orthogonally structured positive bases to create positive k-spanning sets with guarantees on the value of their cosine measures.
Keywords Positive spanning sets; Positive k-spanning sets; Positive bases; Positive k-bases; Cosine measure; k-cosine measure; Derivative-free optimization.
2020 Mathematics Subject Classification 15A03; 15A21; 15B30; 15B99; 90C56.
∗Department of Mathematics, University of British Columbia, Kelowna, British Columbia, Canada. Hare’s research is partially funded by the Natural Sciences and Engineering Research Council of Canada (cette recherche est partiellement financ´ ee par le Conseil de recherches en sciences naturelles et en g´ enie du Canada), Discover Grant #2018-03865. ORCID 0000-0002-4240-3903 (warren.hare@ubc.ca).
†Mathematics and Statistics Department, Saint Francis Xavier Univeristy, Antigonish, Nova Scotia, Canada.
Jarry-Bolduc’s research is partially funded through the Natural Sciences and Engineering Research Council of Canada (cette recherche est partiellement financ´ ee par le Conseil de recherches en sciences naturelles et en g´ enie du Canada), Discover Grant #2018-03865. ORCID 0000-0002-1827-8508 (gabjarry@alumni.ubc.ca).
‡LAMSADE, CNRS, Universit´ e Paris Dauphine-PSL, Place du Mar´ echal de Lattre de Tassigny, 75016 Paris, France (sebastien.kerleau@lamsade.dauphine.fr).
§LAMSADE, CNRS, Universit´ e Paris Dauphine-PSL, Place du Mar´ echal de Lattre de Tassigny, 75016 Paris, France. Royer’s research is partially funded by Agence Nationale de la Recherche through program ANR-19-P3IA-0001 (PRAIRIE 3IA Institute). ORCID 0000-0003-2452-2172 (clement.royer@lamsade.dauphine.fr).
¶Research for this paper is also supported by France-Canada Research Funds 2022.
1 1 Introduction Positive spanning sets, or PSSs, are sets of vectors that span a space of interest through non-negative linear combinations. These sets were first investigated by Davis , with a particular emphasis on inclusion-wise minimal PSSs, also called positive bases [4, 15]. Such a restriction guarantees bounds on the minimal and maximal size of a positive basis [1, 4]. PSSs of this form are now well understood in the literature . Any positive basis is amenable to a decomposition into minimal positive bases over subspaces , a result that recently lead to the characterization of nicely structured positive bases , hereafter called orthogonally structured positive bases, or OSPBs.
Positive spanning sets and positive bases are now a standard tool to develop derivative-free optimization algorithms [5, 3]. The idea is to use directions from a PSS in replacement for derivative information. In that setting, it becomes critical to quantify how well directions from a PSS cover the space of interest, which is typically assessed through the cosine measure of this PSS [11, 10]. In general, no closed form expression is available for this cosine measure, yet various computing techniques have been proposed [6, 19]. In particular, a deterministic algorithm to compute the cosine measure of any PSS was recently proposed by Hare and Jarry-Bolduc .
The cosine measure of certain positive bases is known, along with upper bounds for positive bases of minimal and maximal sizes [16, 18]. The intermediate size case is less understood, even though recent progress was made in the case of orthogonally structured positive bases .
Meanwhile, another category of positive spanning sets introduced in the 1970s has received little attention in the literature. Those sets, called positive k-spanning sets (PkSSs), can be viewed as resilient PSSs, since at least k of their elements must be removed in order to make them lose their positively spanning property . Unlike standard PSSs, only partial results are known regarding inclusion-wise minimal PkSS. These results depart from those for PSSs as they rely on polytope theory [13, 22]. Moreover, the construction of PkSSs based on PSSs has not been fully explored. To the best of our knowledge, the cosine measure of PkSSs has not been investigated, which partly prevents those sets from being used in derivative-free algorithms.
In this paper, we investigate positive k-spanning sets through the lens of orthogonally struc-tured positive bases and cosine measure. To this end, we refine previous results on OSPBs so as to obtain an efficient cosine measure calculation technique. We then explain how the notion of cosine measure can be generalized to account for the positive k-spanning property, thereby introducing a quantity called the k-cosine measure. To the best of our knowledge, this definition is new in both the derivative-free optimization and the positive spanning set literature. By combining those elements, we are able to build positive k-spanning sets from OSPBs as well as to provide a bound on their k-cosine measures. Our results pave the way for using positive k-spanning sets in derivative-free algorithms.
The rest of this paper is organized as follows. Section 2 summarizes important properties of positive spanning sets, including their characterization through the cosine measure, with a particular focus on the subclass of orthogonally structured positive bases (OSPBs). Section 3 describes an efficient way to compute the cosine measure of an OSPB based on leveraging its decomposition. Section 4 formalizes the main properties associated with positive k-spanning sets, introduces the k-cosine measure to help studying these sets and uses OSPBs to design PkSSs with guaranteed k-cosine measure.
Section 5 summarizes our findings and provides several perspectives of our work.
2 Notations Throughout this paper, we will work in the Euclidean space Rn with n ≥2, or a linear subspace thereof, denoted by L ⊂Rn. More generally, blackboard bold letters will be used to designate infinite-valued sets, such as linear (sub)spaces and half-spaces. The Minkowski sum {a + b, (a, b) ∈A × B} of two spaces A and B will be denoted by A + B or by A ⊥B if the two spaces are orthogonal to one another. The orthogonal to a given subspace A will be denoted accordingly by A⊥.
In order to allow for repetition among their elements, positive spanning sets will be seen as families of vectors. We will use calligraphic letters such as D to denote finite families of vectors. Given a family D ⊂Rn, the linear span of this family (i.e. the set of linear combinations of its elements) will be denoted by span(D). Bold lowercase (resp.
uppercase) letters will be used to designate vectors (resp. matrices). The notations 0n and 1n will respectively be used to designate the null vector and the all-ones vector in Rn, while In := {e1, · · · , en} will denote the family formed by coordinate basis vectors in Rn. Given a family Da of vectors in Rn, we will use Da to denote a matrix with n rows and whose columns correspond to the elements in Da, in no particular order unless otherwise stated. We will use Da {d} to denote a family obtained from Da by removing exactly one instance of the vector d.
Similarly, the notation Da ∪{d} will designate the family obtained from Da by adding one copy of the vector d. For instance, {d, d, d′} \ {d} = {d, d′} and {d, d′} ∪{d} = {d, d, d′}. Finally, the notation [ [1, m] ] will refer to the set {1, . . . , m}.
2 Background on positive spanning sets In this section, we introduce the main concepts and results on positive spanning sets that will be used throughout the paper. Section 2.1 defines positive spanning sets and positive bases.
Section 2.2 introduces the concept of orthogonally structured positive bases, that were first studied under a different name . Section 2.3 provides the definition of the cosine measure of a positive spanning set, along with its value for several commonly used positive bases.
2.1 Positive spanning sets and positive bases In this section, we recall the definitions of positive spanning sets and positive bases, based on the seminal paper of Davis .
Definition 2.1 (Positive span and positive spanning set) Let L be a linear subspace of Rn and m ≥1. The positive span of a family of vectors D = {d1, d2, . . . , dm} in L, denoted pspan(D), is the set pspan(D) := {u ∈L : u = α1d1 + · · · + αmdm | ∀i ∈[ [1, m] ], αi ≥0}.
A positive spanning set (PSS) of L is a family of vectors D such that pspan(D) = L.
When L is not specified, a positive spanning set is understood as positively spanning Rn.
Throughout the paper, we only consider positive spanning sets formed of nonzero vectors. Note, however, that we do not force all elements of a positive spanning set to be distinct. The next two lemmas are well known and describe basic properties of positive spanning sets.
Lemma 2.1 [18, Theorem 2.5] Let D be a finite set of vectors in a subspace L of Rn. The following statements are equivalent: 3 (i) D is a PSS of L.
(ii) For every nonzero vector u ∈L, there exists an element d ∈D such that u⊤d > 0.
(iii) span(D) = L and 0n can be written as a positive linear combination of the elements of D.
Lemma 2.2 [4, Theorem 3.7] If D is a PSS of L, then for any d ∈D the set D{d} linearly spans L.
Note that Lemma 2.2 implies that a PSS must contain at least dim(L) + 1 vectors. There is no upper bound on the number of elements that a PSS may contain, but such a restriction holds for a subclass of positive spanning sets called positive bases.
Definition 2.2 (Positive basis) Let L be a linear subspace of Rn of dimension ℓ≥1.
A positive basis of L of size ℓ+ s, denoted DL,s, is a PSS of L with ℓ+ s elements satisfying: ∀d ∈DL,s, pspan (DL,s{d}) ̸= L.
When L = Rn, such a set will be denoted by Dn,s.
In other words, positive bases of L are inclusion-wise minimal positive spanning sets of L .
Positive bases can also be defined thanks to the notion of positive independence.
Definition 2.3 (Positive independence) Let L be a linear subspace of Rn of dimension ℓ≥ 1. A family of vectors D in L is positively independent if, for any d ∈D, there exists a vector u ∈L such that u⊤d > 0 and u⊤v ≤0 for any v ∈D \ {d}.
A positive basis of L is thus a positively independent PSS of L.In the case L = Rn, we simply say that Dn,s is a positive basis (of size n + s).
One can show that the size of a positive basis of a subspace L is at least dim(L) + 1 and at most 2 dim(L) [2, 4].
Definition 2.4 Let L be a linear subspace of Rn of dimension ℓ≥1 and 1 ≤s ≤ℓ. A positive basis DL,s of L is called minimal if s = 1, in which case we denote it by DL = DL,1. The positive basis is called maximal if s = ℓ. If 1 < s < ℓ, we say that the positive basis has intermediate size.
Maximal and minimal positive bases have a well-understood structure [2, 17]. The structure of an arbitrary positive basis can however be quite complicated to analyze. In the next section, we describe a decomposition formula for positive bases that identifies a favorable structure.
2.2 Orthogonally structured positive bases A complete characterization of positive bases can be provided through a subspace decomposi-tion . Such a decomposition is based upon the concept of critical vectors defined below.
Definition 2.5 (Critical vectors) Let L be a subspace of Rn and DL,s be a positive basis of L. A vector c ∈L is called a critical vector of DL,s if it cannot replace an element of DL,s to form a positive spanning set, i.e.
∀d ∈DL,s, pspan ((DL,s \ {d}) ∪{c}) ̸= L.
4 Note that the zero vector is a critical vector for every positive basis of L, and that it is the only critical vector for a maximal positive basis . Moreover, if DL is a minimal positive basis and dim(L) ≥2, the set of critical vectors (known as the complete critical set) is given by − [ i̸=j pspan (DL \ {di, dj}) .
Using critical vectors, one may decompose any positive basis as a union of minimal positive bases.
Theorem 2.1 gives the result for a positive basis in Rn, by adapting the original decomposition result due to Romanowicz [21, Theorem 1] (see also [7, Lemma 25]).
Theorem 2.1 (Structure of a positive basis) Suppose that n ≥2, s ≥1, and consider a positive basis Dn,s of Rn. Then, either s = 1 or there exist subspaces L1, . . . , Ls of Rn such that Rn = L1 + L2 + · · · + Ls and Li ∩Lj = {0} for any i ̸= j, and there exist s associated minimal positive bases DL1, . . . , DLs such that Dn,s = DL1 ∪(DL2 + c1) ∪· · · ∪(DLs + cs−1), (1) where for any j = 1, . . . , s −1, we let DLj+1 + cj := d + cj d ∈DLj+1 , and the vector cj ∈L1 + · · · + Lj is a critical vector for DL1 ∪ j S i=2 (DLi + ci−1).
The result of Theorem 2.1 is actually an equivalence, in that any set that admits the decom-position (1) is a non-minimal positive basis of Rn [21, Theorem 1]. The example below provides an illustration for both Definition 2.5 and Theorem 2.1. One can easily check that the stated vector is indeed critical, and that the proposed decomposition matches (1).
Example 2.1 Consider the following positive basis of R4: D4,2 = 1 0 0 0 , 0 1 0 0 , −1 −1 2 2 , 1 1 −4 −4 0 0 1 0 , 0 0 0 1 .
The first four vectors of this set form a minimal positive basis for {[x y z z]⊤, (x, y, z) ∈R3} that admits [0 0 0.5 0.5]⊤as a critical vector. Therefore, a decomposition of the form (1) for D4,2 is: D4,2 = 1 0 0 0 , 0 1 0 0 , −1 −1 2 2 , 1 1 −4 −4 [ 0 0 0.5 −0.5 , 0 0 −0.5 0.5 + 0 0 0.5 0.5 .
In general, however, the decomposition (1) is hard to compute, as it requires to determine the critical vectors and the subspaces Li, that need not be orthogonal to one another. These considerations have lead researchers to consider a subclass of positive bases with a nicer decom-position , leading to Definition 2.6 below.
5 Definition 2.6 (Orthogonally structured positive bases) Suppose that n ≥2, and con-sider a positive basis Dn,s of Rn. If there exist subspaces L1, . . . , Ls of Rn such that Rn = L1 ⊥ L2 ⊥· · · ⊥Ls and associated minimal positive bases DL1, . . . , DLs such that Dn,s = DL1 ∪DL2 ∪· · · ∪DLs, (2) then Dn,s is called an orthogonally structured positive basis, or OSPB.
By construction, note that for any 1 ≤i < j ≤s, any pair of elements (d, d′) ∈DLi × DLj satisfies d⊤d′ = 0.
The class of orthogonally structured positive bases includes common positive bases, such as minimal ones and certain maximal positive bases such as those formed by the coordinate vectors and their negatives in Rn. More OSPBs can be obtained via numerical procedures, even for intermediate sizes . As will be shown in Section 3, it is also possible to compute their cosine measure in an efficient manner.
2.3 The cosine measure To end this background section, we define the cosine measure, a metric associated with a given family of vectors.
Definition 2.7 (Cosine measure and cosine vector set) Let D be a nonempty family of nonzero vectors in Rn. The cosine measure of D is given by cm (D) = min ∥u∥=1 u∈Rn max d∈D u⊤d ∥d∥.
The cosine vector set associated with D is given by cV(D) = arg min ∥u∥=1 u∈Rn max d∈D u⊤d ∥d∥.
The cosine measure and cosine vector set provide insights on the geometry of the elements of D in the space. Such concepts are of particular interest in the case of positive spanning sets, as shown by the result of Proposition 2.1. Its proof can be found in key references in derivative-free optimization [2, 3], but note that our analysis in Section 4 provides a more general proof in the context of positive k-spanning sets.
Proposition 2.1 Let D be a nonempty, finite family of vectors in Rn. Then D is a positive spanning set in Rn if and only if cm (D) > 0.
Proposition 2.1 shows that the positive spanning property can be checked by computing the cosine measure. The value of the cosine measure further characterizes how well that set covers the space, which is a relevant information in derivative-free algorithms . Both observations motivate the search for efficient techniques to compute the cosine measure.
We end this section by providing several examples of cosine measures for orthogonally struc-tured positive bases, that form the core interest of this paper. We focus on building those positive 6 bases from the coordinate vectors. A classical choice for a maximal positive basis consists in selecting the coordinate vectors and their negatives. In that case, it is known that cm (In ∪−In) = cm ({e1, . . . , en, −e1, . . . , −en}) = 1 √n.
One can also consider a minimal positive basis formed by the coordinate vectors and the sum of their negatives. Then, we have cm In ∪ ( − n X i=1 ei )!
= cm ( e1, . . . , en, − n X i=1 ei )!
= 1 p n2 + 2(n −1)√n .
To the best of our knowledge, the latter formula has only been recently established in [9, 20], and no formal proof is available in the literature. In the next section, we will develop an efficient cosine measure calculation technique for orthogonally structured positive bases that will provide a proof of this result as a byproduct.
3 Computing OSPBs and their cosine measure In this section, we describe how orthogonally structured positive bases can be identified using the decomposition introduced in the previous section. We also leverage this decomposition to design algorithms for detecting the OSPB structure and computing the cosine measure of a given OSPB.
3.1 Structure and detection of OSPBs The favorable structure of an OSPB can be revealed through the properties of its matrix rep-resentations, and in particular the Gram matrices associated with an OSPB, in the sense of Definition 3.1 below.
Definition 3.1 (Gram matrix) Let S be a finite family of m vectors in Rn and S ∈Rn×m a matrix representation of this family. The Gram matrix of S associated with S is the matrix G(S) = S⊤S.
Given a Gram matrix G(S) of S, any Gram matrix of S has the form P⊤G(S)P where P is a permutation matrix. We will show that when S is an OSPB, there exists a Gram matrix with a block-diagonal structure that reveals the decomposition of the basis, and thus its OSPB nature.
Theorem 3.1 Let n ≥2, s ≥2 and Dn,s ⊂Rn be a positive basis. Then, Dn,s is orthogonally structured if and only if one of its Gram matrices G(Dn,s) can be written as a block diagonal matrix with s diagonal blocks.
Proof. Suppose first that Dn,s is an OSPB associated to the decomposition Dn,s = DL1 ∪DL2 ∪· · · ∪DLs.
Let Dn,s be a matrix representation of Dn,s such that Dn,s = DL1 · · · DLs , where DLi is a matrix representation of DLi for every i ∈[ [1, s] ].
By orthogonality of those matrices, 7 the Gram matrix G(Dn,s) = D⊤ n,sDn,s is then block diagonal with s blocks corresponding to G(DL1), . . . , G(DLs), and thus the desired result holds.
Conversely, suppose that there exists a matrix representation Dn,s of Dn,s such that G(Dn,s) is block diagonal with s blocks. Then, by considering a partitioning with respect to those diago-nal blocks, one can decompose Dn,s into S1 · · · Ss so that G(Dn,s) = diag (G(S1), . . . , G(Ss)).
By definition of the Gram matrix, this structure implies that the columns of Si and Sj are or-thogonal for any i ̸= j, thus we only need to show that each Si is a minimal positive basis for its linear span. To this end, we use the fact that Dn,s is positively spanning. By Lemma 2.1(ii), there exists a vector u ∈Rn+s with positive coefficients such that Dn,su = 0n. By decomposing the vector u into s vectors u1, . . . , us according to the decomposition S1 · · · Ss , we obtain that Dn,su = n X i=1 Siui = 0n.
(3) By orthogonality of the columns of the Si matrices, the property (3) is equivalent to Siui = 0n ∀i ∈[ [1, s] ].
Using again Lemma 2.1(ii), this latter property is equivalent to Si being a PSS for its linear span. Let ni be the dimension of this span, and mi the number of elements in Si. Since the Si are orthogonal and the columns of Dn,s span Rn, we must have Ps i=1 ni = n. In addition, we also have Ps i=1 mi = n + s = Ps i=1 ni + s. Since mi > ni for all i ∈[ [1, s] ], we conclude that mi = ni + 1, and thus every Si is a minimal positive basis.
□ Note that the characterization stated by Theorem 3.1 trivially holds for minimal positive bases.
This theorem thus provides a characterization of the OSPB property through Gram matrices. One can then use this result to determine whether a positive basis has an orthogonal structure and, in that event, to highlight the associated decomposition.
Corollary 3.1 Let n ≥2, s ≥1 and Dn,s be a positive basis in Rn. Let Dn,s be a matrix representation of Dn,s. If there exists a permutation matrix P ∈R(n+s)×(n+s) such that the Gram matrix G(Dn,sP) is block diagonal with s blocks, then Dn,s is an OSPB, and its decomposition (2) is given by the columns of Dn,sP in that order.
Corollary 3.1 provides a principled way of detecting the OSPB structure, by checking all possible permutations of the elements in the basis. Although this is not the focus of this work, we remark that it can be done in an efficient manner by considering a graph whose Laplacian matrix L has non-zero coordinates in the exact same rows and columns as G(Dn,s). Indeed, the problem of finding the permutation for matrix L reduces to that of finding the number of connected components in the graph and can be solved in polynomial time .
3.2 Cosine measure of an orthogonally structured positive basis As seen in the previous section, orthogonally structured positive bases admit a decomposition into minimal positive bases that are orthogonal to one another. In this section, we investigate how this particular structure allows for efficient computation of the cosine measure.
Our approach builds on the algorithm introduced by Hare and Jarry-Bolduc , that com-putes the cosine measure of any positive spanning set in finite time (though the computation 8 time may be exponential in the problem dimension). This procedure consists of a loop over all possible linear bases contained in the positive spanning set. More formally, given a positive spanning set D in Rn and a linear basis B ⊂D, we let D be a matrix representation of D, and B the associated representation of B. One then defines γB := 1 p 1⊤ n G(B)−11n and uB := γBB−⊤1n.
(4) The vector uB is the only unitary vector such that u⊤ B d ∥d∥= γB for all d ∈B [6, Lemmas 12 and 13]. Computing the quantities (4) for all linear bases contained in D then gives both the cosine measure and the cosine vector set for D [6, Theorem 19]. Indeed, we have cm (D) = min B ⊂D B basis of Rn max d∈D u⊤ B d ∥d∥, (5) while the cosine vector set is given by cV(D) = uB : max d∈D u⊤ B d ∥d∥= cm (D) .
The next proposition shows how, when dealing with OSPBs, formula (5) can be simplified to give a more direct link between the quantities γB and the cosine measure.
Proposition 3.1 Let Dn,s be an OSPB of Rn and DL1, . . . , DLs be a decomposition of Dn,s into minimal positive bases. Then, cm (Dn,s) = min Bn ⊂Dn,s Bn basis of Rn γBn and cV(Dn,s) = {uBn : γBn = cm (Dn,s)}, where γBn and uBn are computed according to (4).
Proof. Using the decomposition of Dn,s, any linear basis Bn of Rn contained in Dn,s can be decomposed as Bn = BL1 ∪· · ·∪BLs where BLi ⊂DLi is a linear basis of Li. By construction, the vector uBn makes a positive dot product with every element of Bn (and thus with every element of any BLi) equal to γBn. Consequently, the vector uBn lies in the positive span of Bn, and its projection on any subspace Li lies in the positive span of BLi.
Meanwhile, for any i ∈[ [1, s] ], there exists di ∈DLi such that BLi = DLi \ {di}, and this vector satisfies u⊤ Bndi < 0 per Lemma 2.1(ii), leading to max d∈Dn u⊤ Bnd ∥d∥= max i∈[ [1,s] ] ( max d∈DLi u⊤ Bnd ∥d∥ ) = max i∈[ [1,s] ] ( max d∈BLi u⊤ Bnd ∥d∥ ) = max d∈Bn u⊤ Bnd ∥d∥= γBn, where the second equality comes from u⊤ Bndi < 0 and max d∈Dn u⊤ Bnd > 0, and the last equality holds by definition of γBn. Recalling that cm (Dn) is given by (5) concludes the proof.
□ One drawback of the approach described so far is that it requires computing the quantities (4) for all linear bases including in the positive spanning set of interest. In the case of OSPBs, we show below that the number of linear bases to be considered can be greatly reduced by leveraging to their decomposition into minimal positive bases.
9 Theorem 3.2 Let Dn,s be an OSPB of Rn and DL1, . . . , DLs be a decomposition of Dn,s into minimal positive bases. Let Bn be a linear basis contained in Dn,s such that Bn = ∪s i=1BLi, where every BLi is a linear basis of the subspace corresponding to DLi. For each basis BLi, define γBLi = 1 q 1⊤ LiG(BLi)−11Li , where BLi is a matrix representation of BLi and 1Li denotes the vector of all ones in Rdim(Li).
Then, γBn = 1 qPs i=1 γ−2 BLi = 1 qPs i=1 1⊤ LiG(BLi)−11Li , (6) with γBn being defined as in (4).
As a result, the cosine measure and cosine vector set of Dn,s are given by cm (Dn,s) = 1 qPs i=1 maxdi∈DLi γ−2 DLi{di} (7) and cV(Dn,s) = uBn : Bn = ∪s i=1BLi, uBn = [BL1 · · · BLs]−⊤1n qPs i=1 γ−2 BLi (8) respectively.
Proof. Let Bn be a matrix representation of Bn such that Bn = BL1 · · · BLs . Then, the Gram matrix of Bn is G(Bn) = diag (G(BL1), . . . , G(BLs)), implying that γBn = 1 p 1⊤ n G(Bn)−11n = 1 q 1⊤ n diag (G(BL1), . . . , G(BLs))−1 1n = 1 p 1⊤ n diag (G(BL1)−1, . . . , G(BLs)−1) 1n = 1 qPs i=1 1⊤ LiG(BLi)−11Li , which proves (6).
Recall now that every linear basis BLi can be written as DLi \ {di} for some di ∈DLi.
Combining this property together with (6) and Proposition 3.1, we obtain cm (Dn,s) = min Bn⊂Dn,s Bn basis of Rn γBn = min BL1,...,BLs BLi=DLi{di} 1 qPs i=1 γ−2 BLi = min d1,...,ds di∈DLi 1 qPs i=1 γ−2 DLi{di} = 1 qPs i=1 maxdi∈DLi γ−2 DLi{di} , 10 proving (7).
Finally, combining (7) with the result of Proposition 3.1 on the cosine vector set gives cV(Dn,s) = {uBn : γBn = cm (Dn,s)} = uBn : Bn = ∪s i=1BLi, 1 qPs i=1 γ−2 BLi = cm (Dn,s) Using a matrix representation Bn = BL1 · · · BLs along with the formula (4) for uBn leads to cV(Dn,s) = n uBn : Bn = ∪s i=1BLi, uBn = γBn[BL1 · · · BLs]−⊤1n o = uBn : Bn = ∪s i=1BLi, uBn = [BL1 · · · BLs]−⊤1n qPs i=1 γ−2 BLi which is precisely (8).
□ On a broader level, Theorem 3.2 shows that any calculation for a linear basis contained in an OSPB reduces to calculations on bases contained in each of the minimal positive bases in its decomposition. This observation leads to a principled way of computing the cosine measure from the OSPB decomposition, as described in Algorithm 1.
Algorithm 1: Cosine measure of an orthogonally structured positive basis 1 Input: An orthogonally structured positive basis Dn,s of Rn with normalized vectors.
2 Step 1. Compute a decomposition Dn,s = DL1 ∪· · · ∪DLs of Dn,s into s minimal positive bases DLi on subspaces of Rn.
3 Step 2. For every i ∈[ [1, s] ], compute βLi = max d∈DLi γ−2 DLi{d} and Ai = arg max d∈DLi γ−2 DLi{d}.
For all d ∈Ai, note B(d) Li = DLi \ {d}.
4 Step 3. Return the cosine measure cm (Dn,s) = 1 pPs i=1 βLi (9) and the cosine vector set cV(Dn,s) = [ (d1,...,ds)∈A1×···×As u ∈Rn, u = h B(d1) L1 · · · B(ds) Ls i−⊤ 1n pPs i=1 βLi , where B(d) Li is a matrix representation of B(d) Li .
11 In essence, Algorithm 1 is quite similar to the generic method proposed for positive spanning sets . However, Algorithm 1 reduces the computation of the cosine measure to that over minimal positive bases, which leads to significantly cheaper calculations. Indeed, for any minimal positive basis DLi involved in the decomposition, the algorithm considers |DLi| positive bases, and computes the quantities of interest (4) for each of those (which amounts to inverting or solving a linear system involving the associated Gram matrix). Overall, Algorithm 1 thus checks Ps i=1 |DLi| = |Dn,s| = n + s bases to compute the cosine measure (9). This result represents a significant improvement over the n+s n possible bases that are potentially required when the OSPB decomposition is not exploited, as in the algorithm for generic PSSs . We also recall from Section 3.1 that Step 1 in Algorithm 1 can be performed in polynomial time.
To end this section, we illustrate how Algorithm 1 results in a straightforward calculation in the case of some minimal positive bases, by computing explicitly the cosine measure of the minimal positive basis In ∪{−1n}. In this case, the calculation becomes easy thanks to the orthogonality of the vectors within In∪{−1}. Although the value (10) was recently stated [9, 20] and checked numerically using the method of Hare and Jarry-Bolduc , to the best of our knowledge the formal proof below is new.
Lemma 3.1 The cosine measure of the OSPB In ∪{−1n} is given by cm (In ∪{−1n}) = 1 p n2 + 2(n −1)√n .
(10) Proof. For the sake of simplicity, we normalize the last vector in the set (which does not change the value of the cosine measure) and we let Dn = In ∪{−1 √n1n} in the rest of the proof. Since Dn is a positive basis, it follows that cm (Dn) = min d∈Dn γDn{d}, Two cases are to be considered. Suppose first that d = −1 √n1n, so that Dn \ {d} = In. Then, any matrix representation of that basis B is such that G(B) = In. Consequently, γDn{d} = 1 p 1⊤ n G(B)−11n = 1 p 1⊤ n 1n = 1 √n.
Suppose now that d = ei for some i ∈[ [1, n] ]. In that case, considering the matrix representation Bi = h e1 · · · ei−1 ei+1 · · · en −1 √n1n i , we obtain that G(Bi) = " In−1 −1 √n1n−1 −1 √n1⊤ n−1 1 # and G(Bi)−1 = In−1 + 1n−11⊤ n−1 √n1n−1 √n1⊤ n−1 n Since these formulas do not depend on i, we obtain that for all i ∈[ [1, n] ], we have γDn{ei} = 1 p 1⊤ n G(Bi)−11n = 1 qPn j=1 Pn ℓ=1[G(Bi)−1]jℓ .
Summing all coefficients of G(Bi)−1 gives n −1 + (n −1)2 + 2(n −1)√n + n = n2 + 2(n −1)√n, hence γDn{ei} = 1 √ n2+2(n−1)√n. Comparing this value with 1 √n yields the desired conclusion. □ Algorithm 1 allows for efficient calculation of cosine measures for specific positive spanning sets that originate from OSPBs, as we will establish in Section 4 in the case of positive k-spanning sets.
12 4 Positive k-spanning sets and k-cosine measure In this section, we are interested in positive spanning sets that retain their positively spanning ability when one or more of their elements are removed.
Those sets were originally termed positive k-spanning sets , and we adopt the same terminology. Section 4.1 recalls the key definitions for positive k-spanning sets and positive k-bases, while Section 4.2 introduces the k-cosine measure, a generalization of the cosine measure from Section 2.3. Finally, Section 4.3 illustrates how to construct sets with guaranteed k-cosine measure based on OSPBs.
4.1 Positive k-spanning property Our goal for this section is to provide a summary of results on positive k-spanning sets that mimic the standard ones for positive spanning sets. We start by defining the property of interest.
Definition 4.1 (Positive k-span and positive k-spanning set) Let L be a linear subspace of Rn and k ≥1. The positive k-span of a finite family of vectors D in L, denoted pspank(D), is the set pspank(D) := \ S⊂D |S|≥|D|−k+1 pspan(S).
A positive k-spanning set (PkSS) of L is a family of vectors D such that pspank(D) = L.
As for Definition 2.1, when L is not specified, a PkSS should be understood as a PkSS of Rn. By construction, any PkSS of L must contain a PSS of L, and therefore is a PSS of L itself.
Moreover, the notions of positive k-span and PkSS with k = 1 coincide with that of positive span and PSS from Definition 2.1. Similar to PSSs, we will omit the subspace of interest when L = Rn.
The notion of positive spanning set is inherently connected to that of spanning set, a standard concept in linear algebra. We provide below a counterpart notion associated with PkSSs.
Definition 4.2 (k-span and k-spanning set) Let D be a finite family of vectors in Rn. the k-span of D, denoted spank(D), is defined by spank(D) = \ S⊂D |S|≥|D|−k+1 span(S).
Given a subspace L of Rn, a k-spanning set of L is a family of vectors D such that spank(D) = L.
Using the two definitions above, we can generalize the results of Lemma 2.1 and Lemma 2.2 to positive k-spanning sets. The proof is omitted as it merely amounts to combining the definitions with these two lemmas.
Lemma 4.1 Let L be a subspace of Rn and D a finite family of vectors in L. Let k ∈N satisfy 1 ≤k ≤|D|. The following statements are equivalent: (i) D is a PkSS of L.
(ii) For any nonzero vector u ∈L, there exist k elements d1, . . . , dk of D such that u⊤di > 0 for all i ∈[ [1, k] ].
13 (iii) spank(D) = L and for any S ⊂D of cardinality |D| −k + 1, the vector 0n can be written as a positive linear combination of the elements of S.
Lemma 4.2 Let L be a subspace of Rn and let the finite set D be a PkSS for L. Then, for any d ∈D, the k-span of the family D \ {d} is L.
The equivalence between statements (i) and (ii) from Lemma 4.1 motivates the term “posi-tive k-spanning sets”. Indeed, given a PkSS and a vector in the subspace it positively k-spans, there exist k elements of the PkSS that make an acute angle with that vector. Note however that the equivalence between statements (i) and (iii), as well as Definition 4.1, both provide an alternate characterization, namely that a PkSS is a PSS that retains this property when removing k −1 of its elements. This latter characterization implies that a PkSS must contain at least n + k vectors, although this bound is only tight when k = 1. Indeed, in [13, Corollary 5] it is shown that the minimal size of a positive k-spanning set in a subspace of dimension ℓis 2k + ℓ−1.
Throughout Section 2, we highlighted positive bases as a special class of positive spanning sets. We now define the counterpart notion for positive k-spanning sets, that are termed positive k-bases.
Definition 4.3 (Positive k-basis) Let L be a linear subspace of Rn. A positive k-basis of L is a PkSS D of L satisfying ∀d ∈D, pspank (D{d}) ̸= L.
Positive k-bases can be thought as inclusion-wise minimal positive k-spanning sets (similarly to the characterization of positive bases from Section 2.1). We showed earlier how the notion of positive independence can be used to give an alternative definition for positive bases. Let us generalize this idea and introduce the concept of positive k-independence, used to characterize positive k-bases.
Definition 4.4 (Positive k-independence) Let L be a linear subspace of Rn of dimension ℓ≥1 and let k ≥1. A family of vectors D in L is positively k-independent if |D| ≥k and, for any d ∈D, there exists a vector u ∈L having a positive dot product with exactly k elements of D, including d.
Using this new concept, positive k-bases can alternatively be defined as positively k-independent PkSSs. We mention in passing that the upper bound on the size of a positive k-basis has yet to be determined, though it is known to exceed 2kℓfor an ℓ-dimensional subspace .
4.2 The k-cosine measure This section aims at generalizing the cosine measure from Section 2.3 to positive k-spanning sets so that it characterizes the positive k-spanning property. Our proposal, called the k-cosine measure, is described in Definition 4.5.
Definition 4.5 (k-cosine measure) Let D be a finite family of nonzero vectors in Rn and let k ∈N satisfy 1 ≤k ≤|D|. The k-cosine measure of D is given by cmk (D) = min ∥u∥=1 u∈Rn max S⊂D |S|=k min d∈S u⊤d ∥d∥.
14 The k-cosine vector set associated with D is given by cVk(D) = arg min ∥u∥=1 u∈Rn max S⊂D |S|=k min d∈S u⊤d ∥d∥.
Note that Definition 2.7 is a special case of Definition 4.5 corresponding to k = 1. In its general form, this definition expresses how well the vectors of the family of interest are spread in the space through subsets of k vectors, which is related to property (ii) in Lemma 4.1. Our next result shows that this quantity characterizes PkSSs.
Theorem 4.1 Let D be a finite family of vectors in Rn and let k ∈N satisfy 1 ≤k ≤|D|. Then D is a positive k-spanning set in Rn if and only if cmk (D) > 0.
Proof. Without loss of generality we assume that all the elements of D are unit vectors.
Suppose first that D is a PkSS. Then, by Lemma 4.1, for any unit vector u ∈Rn, there exist k vectors in D that make a positive dot product with u. Let Su be a subset of D consisting of these k vectors. By construction, we have min d∈Su u⊤d > 0 and thus max S⊂D |S|=k min d∈S u⊤d ≥min d∈Su u⊤d > 0.
Since the result holds for any unit vector u, we conclude that cmk (D) > 0.
Conversely, suppose that cmk (D) > 0. For any unit vector u, we have by assumption that max S⊂D |S|=k min d∈S u⊤d > 0.
(11) In other words, at least k elements of D have a positive dot product with u. This proves that D satisfies statement (ii) from Lemma 4.1 and thus D is a positive k-spanning set.
□ As announced in Section 2.3, the proof of Theorem 4.1 covers that of Proposition 2.1 as the special case k = 1.
To end this section, we provide an alternate definition of the k-cosine measure that connects this notion to the cosine measure.
Theorem 4.2 Let D be a finite family of nonzero vectors in Rn and let k ∈N satisfy 1 ≤k ≤ |D|. Then, cmk (D) = min S⊂D |S|=|D|−k+1 cm (S) .
(12) Proof. Without loss of generality, we assume that D = {d1, . . . , dm} where each di is a unit vector. In order to show that min ∥u∥=1 max S⊂D |S|=k min d∈S u⊤d = min S⊂D |S|=m−k+1 cm (S) , 15 we prove the equivalent statement min ∥u∥=1 max S⊂D |S|=k min d∈S u⊤d = min ∥u∥=1 min S⊂D |S|=m−k+1 max d∈S u⊤d.
(13) To this end, let u be a unit vector in Rn. We reorder the elements of D so that for all i ≤m−1, u⊤di ≥u⊤di+1 and we define S− k = {d1, . . . , dk} and S+ k = {dk, . . . , dm}. By construction, one has u⊤dk = min d∈S− k u⊤d = max d∈S+ k u⊤d.
(14) Moreover, the definitions of S− k and S+ k also imply that S− k ∈arg max S⊂D |S|=k min d∈S u⊤d and S+ k ∈arg min S⊂D |S|=m−k+1 max d∈S u⊤d.
(15) Combining (14) and (15) gives max S⊂D |S|=k min d∈S u⊤d = min S⊂D |S|=m−k+1 max d∈S u⊤d.
(16) Since (16) holds for any unit vector u ∈Rn, it follows that min ∥u∥=1 max S⊂D |S|=k min d∈S u⊤d = min ∥u∥=1 min S⊂D |S|=m−k+1 max d∈S u⊤d, which is precisely (13) and, as a result, proves (12).
□ The formula (12) is associated with another characterization of PkSSs, namely that provided by Definition 4.1. In essence, the k-cosine measure of a family D is the minimum among all subsets of D with cardinality |D| −k + 1. A useful corollary of that result is that for any k ≥2, one has cm (D) = cm1 (D) ≥cm2 (D) ≥· · · ≥cmk (D) .
In the next section, we will show how PkSSs built using OSPBs satisfy stronger properties associated with the k-cosine measure.
4.3 Building positive k-spanning sets using OSPBs In this section, we describe how OSPBs can be used to generate positive k-spanning sets or positive k-bases with guarantees on their k-cosine measure.
A first approach towards constructing positive k-spanning sets consists in duplicating the same PSS k times, so that every direction remains even after taking out k−1 elements. However, such a strategy creates redundancy among the elements of the set. The purpose of this section is to introduce a more generic approach based on rotation matrices that allows for all vectors to be distinct.
Proposition 4.1 Let Dn,s be an OSPB of Rn with s ∈[ [1, n] ]. Let R(1), . . . , R(k) be k rotation matrices in Rn×n, and define D(j) n,s as the family of vectors obtained by applying R(j) to each 16 vector in Dn,s for any j ∈[ [1, k] ]. Then, the set D(1:k) n,s = k S j=1 D(j) n,s is a positive k-spanning set with cmk D(1:k) n,s ≥cm (Dn,s) .
(17) Proof. First, note that applying a rotation to every vector in a family does not change its cosine measure since rotations preserve angles, hence cm D(j) n,s = cm (Dn,s) for every j ∈[ [1, k] ].
Consider a family S ⊂D(1:k) n,s with |S| = D(1:k) n,s −k + 1. By the pigeonhole principle, this set must contain one of the k positive bases obtained by rotation. Letting D(jS) n,s denote that positive basis, we have cm (S) ≥cm D(jS) n,s = cm (Dn,s). Using Theorem 4.2, we obtain that cmk D(1:k) n,s = min S⊂D(1:k) n,s |S|= D(1:k) n,s −k+1 cm (S) ≥cm (Dn,s) .
In particular, this proves cmk D(1:k) n,s > 0, hence this set being positively k-spanning now fol-lows from Theorem 4.1.
□ Several remarks are in order regarding Proposition 4.1. First, note that the result extends to any positive spanning set, but we focus on OSPBs since in that case (17) gives a lower bound on the k-cosine measure that can be computed in polynomial time. Moreover, when R1 = · · · = Rk = In, i.e. when k identical copies of Dn,s are used, the resulting family is a positive k-basis and relation (17) holds with equality. As a result, its k-cosine measure can be computed in polynomial time using Algorithm 1. In general, however, the family generated through Proposition 4.1 is not necessarily a positive k-basis. For instance, if n = 2, setting D2 = {I2, −1}, k = 2, R1 = I2 and R2 = 1/2 − √ 3/2 √ 3/2 1/2 yields a family that is a positive 2-spanning set but not a positive 2-basis.
We now present a strategy based on rotations tailored to OSPBs. Recall that OSPBs can be decomposed into minimal positive bases over orthogonal subspaces. Our proposal consists in applying separate rotations to each of those minimal positive bases. Our key result is described in Theorem 4.3, and shows that one can define a strategy for rotating a minimal positive basis to obtain a positive k-basis. The proof relies on two technical results, the first one being an intrinsic property of minimal positive bases.
Lemma 4.3 Let DL = {d1, . . . , dℓ+1} be a minimal positive basis of an ℓ-dimensional subspace L in Rn. Then, there exists v1, . . . , vℓ+1 ∈L satisfying ∀i ∈[ [1, ℓ+ 1] ], d⊤ i vi > 0 and ∀1 ≤j < i ≤ℓ+ 1, d⊤ i vj < 0.
(18) Proof. By Lemma 2.1(ii), the zero vector 0n can be obtained by a positive linear combination of all elements in DL. Without loss of generality, we rescale the elements of DL to ensure that ℓ+1 P i=1 di = 0n. Now, since DL is a minimal positive basis, the set DL{dℓ+1} is a linear basis of L. Suppose that we augment this set with n −ℓvectors to form a basis Bn of Rn, and let Bn 17 be a matrix representation of that basis such that the first ℓcolumns correspond to d1, . . . , dℓ.
Then, we have di = Bnei for any i ∈[ [1, ℓ] ], while dℓ+1 = −Bn Pℓ j=1 ej , where we recall that e1, . . . , eℓare the first ℓvectors of the canonical basis.
We now define the vectors ( vi = B−⊤ n −Pℓ j=1 ej + (ℓ+ 1)ei i ∈[ [1, ℓ] ] vℓ+1 = −B−⊤ n Pℓ j=1 ej.
Using orthogonality of the coordinate vectors in Rn, we have that d⊤ i vi = ℓ∥ei∥2 = ℓ> 0 for every i ∈[ [1, ℓ] ] and v⊤ ℓ+1dℓ+1 = Pℓ j=1 ej 2 > 0. As a result, the first part of (18) holds.
In addition, for any 1 ≤j < i ≤ℓ+ 1, we obtain that d⊤ i vj = −∥ei∥2 = −1 < 0 if i < ℓ+ 1 and d⊤ ℓ+1vj = ℓ X i=1 ei 2 −(ℓ+ 1)∥ej∥2 = ℓ X i=1 ∥ei∥2 −(ℓ+ 1)∥ej∥2 = ℓ−(ℓ+ 1) = −1 < 0, thus the second part of (18) also holds.
□ Note that Lemma 4.3 is not equivalent to Definition 2.3 as the inequalities in (18) are strict.
Our second technical result uses the result of Lemma 4.3 to define a quantity characteristic of the angles between vectors in a minimal positive basis.
Lemma 4.4 Suppose that n ≥ℓ≥2 and let DL = {d1, . . . , dℓ+1} be a minimal positive basis of a ℓ-dimensional subspace L in Rn. Let v1, . . . , vℓ+1 be vectors in L that satisfy (18). For any pair (i, i′) ∈[ [1, ℓ+ 1] ]2, let di,i′ be the orthogonal projection of di on L ∩{vi′}⊥. Then, the quantity ρDL := max i,i′≤ℓ+1 d⊤ i di,i′ ∥di∥∥di,i′∥lies in the interval [0, 1).
Proof. Since di,i′ is a projection of di for any pair (i, i′), we have that d⊤ i di,i′ ≥0, hence ρDL ≥0.
Moreover, for any pair 1 ≤i ≤i′ ≤ℓ+ 1, we have d⊤ i vi′ ̸= 0 by (18), hence di / ∈L ∩{vj}⊥and d⊤ i di,i′ ∥di∥∥di,i′∥< 1. As a result, ρDL < 1, completing the proof.
□ The quantity defined in Lemma 4.4 is instrumental in defining suitable rotations matrices to produce a positive k-basis. The construction is described in Theorem 4.3.
Theorem 4.3 Suppose that n ≥ℓ≥2 and let DL = {d1, . . . , dℓ+1} be a minimal positive basis of an ℓ-dimensional subspace L in Rn. Let ρDL be the quantity defined in Lemma 4.4 for some vectors v1, . . . , vℓ+1 satisfying (18). Finally, let R(1) L , . . . , R(k) L be k rotation matrices in Rn×n such that (i) R(j) L u = u, for any j ∈[ [1, k] ] and any vector u ∈L⊥, (ii) for any pair (j, j′) ∈[ [1, k] ]2 and any pair (i, i′) ∈[ [1, ℓ+ 1] ]2, h R(j) L di i⊤ R(j′) L di′ ∥R(j) L di∥∥R(j′) L di′∥ = 1 ⇔ j = j′ and i = i′, 18 (iii) h R(j) L di i⊤ di ∥R(j) L di∥∥di∥> ρDL for any pair (i, j) ∈[ [1, ℓ+ 1] ] × [ [1, k] ].
Denote by D(j) L the family obtained by applying R(j) L to each vector in DL. Then, the family D(1:k) L = k S j=1 D(j) L is a positive k-basis of L with no identical elements.
Proof. Owing to assumptions (i) and (ii), we know that D(1:k) L is a subset of L with pairwise distinct elements. Moreover, Proposition 4.1 guarantees that D(1:k) L is a PkSS for L. Therefore we only need to show the positive k-independence of this set.
To this end, we consider an index i ∈[ [1, ℓ+ 1] ] and show that the vector vi makes a positive dot product with exactly k elements of D(1:k) L , these elements being R(1) L di, . . . , R(k) L di. For any j ∈[ [1, k] ], we deduce from condition (iii) in the theorem’s statement that h R(j) L di i⊤ di ∥R(j) L di∥∥di∥ > ρDL ≥ d⊤ i di,j ∥di∥∥di,j∥, where di,j is defined as in Lemma 4.4. Letting Θ(u, u′) denote the angle (with values in [0, π]) be-tween two elements u and u′ in Rn, the above property translates to Θ(R(j) L di, di) < Θ(di, di,j).
Using now that Θ(u, u′) ≤Θ(u, u′′) + Θ(u′′, u′) for any vector triplet (u, u′, u′′) ∈(Rn)3, we obtain Θ(R(j) L di, vi) ≤Θ(R(j) L di, di) + Θ(di, vi) < Θ(di,i, di) + Θ(di, vi) = π 2 , where the last equality comes from the fact that di,i and vi are orthogonal vectors. We have thus established that v⊤ i R(j) L di > 0 for all j ∈[ [1, k] ].
It now remains to show that v⊤ i R(j) L di′ ≤0 for any i′ ̸= i. First, using the same argument as above yields Θ(vi, di′) = π 2 + Θ(di′,i, di′) as well as Θ(vi, R(j) L di′) ≥Θ(vi, di′) −Θ(di′, R(j) L di′).
Applying condition (iii), we find that Θ(di′, R(j) L di′) < Θ(di′, di′,i), which finally leads to Θ(vi, R(j) L di′) ≥Θ(vi, di′) −Θ(di′, R(j) L di′) > π 2 , hence v⊤ i R(j) L di′ ≤0. Overall, we have established that the vector vi makes a positive scalar product with exactly k vectors in D(1:k) L . Since any vector d ∈D(1:k) L is obtained by applying a rotation to some element di ∈D(1:k) L , we conclude that D(1:k) L is positively k-independent, proving the desired result.
□ Note that Theorem 4.3 only applies when ℓ≥2 (thus requiring n ≥2 as well). Indeed, when the subspace L has dimension 1, none of the three conditions (i), (ii) and (iii) can be fulfilled.
Remark 4.1 The first two conditions enforced on the rotation matrices in Theorem 4.3 are designed to produce distinct vectors in L without affecting the orthogonal complement of L in Rn.
Indeed, Condition (i) guarantees that the rotation leaves any vector orthogonal to L invariant.
Condition (ii) enforces that all vectors produced by applying the rotations are distinct since the angle between two different vectors has cosine less than 1. Finally, condition (iii) ensures that the vectors are positively k-independent. These conditions can be ensured through careful control of the eigenvalues of those rotation matrices, according to the angles between the vectors in DL.
19 Theorem 4.3 provides a principled way of building positive k-bases from minimal positive bases. This approach naturally extends to OSPBs through their decomposition into orthogonal minimal positive bases. Using the technique described in Theorem 4.3 one can define rotation matrices that act on a single subspace from the decomposition of the OSPB without affecting the remaining subspaces thanks to orthogonality. The next corollary states the result.
Corollary 4.1 Let Dn,s be an OSPB and let DL1, . . . , DLs be a decomposition (2) of Dn,s. As-sume that for every i ∈[ [1, s] ], dim(Li) > 1. Let k ≥1 and for every i ∈[ [1, s] ], let R(1) Li , . . . , R(k) Li be k rotation matrices satisfying the properties (i), (ii) and (iii) from Theorem 4.3 relative to DLi. Then, the set D(1:k) n,s = k [ j=1 D(j) n,s, with D(j) n,s = s [ i=1 D(j) Li ∀j ∈[ [1, k] ], is a positive k-basis with cmk D(1:k) n,s ≥cm (Dn,s).
Corollary 4.1 thus provides a useful strategy to design positive k-bases with guaranteed k-cosine measure, since a lower bound on that quantity is given by cm (Dn,s) and that measure can be efficiently computed through Algorithm 1. In particular, applying this strategy to a minimal positive basis yields a simple way of generating a positive k-basis with k(n+1) vectors, as shown by Theorem 4.3 above.
Similarly to the result of Theorem 4.3, the result of Corollary 4.1 does not apply to all OSPBs. In particular, the maximal OSPB {In, −In} decomposes into n minimal positive bases over one-dimensional subspaces, which precludes the application of Theorem 4.3. Note however that Proposition 4.1 still provides a way to compute PkSSs with cosine measure guarantees for such a maximal OSPB.
5 Conclusion In this paper, we have studied two classes of positive spanning sets.
On the one hand, we have investigated orthogonally structured positive bases, that possess a favorable structure in that they decompose into minimal positive bases over orthogonal subspaces. By exploiting this property, we described an algorithm that computes this structure as well as the value of the cosine measure in polynomial time, thereby improving an existing procedure for generic positive spanning sets. On the other hand, we have conducted a detailed study of positive k-spanning sets, a relatively understudied class of PSSs with resilient properties. We have provided several characterizations of these sets, including the generalization of the cosine measure through the k-cosine measure. We have also leveraged OSPBs to build positive k-spanning sets with guarantees on their k-cosine measure based on rotations.
Our results open the way for several promising areas of research.
We believe that the cosine measure calculation technique can be further improved for positive bases by leveraging their decomposition, although the presence of critical vectors poses a number of challenges to overcome for dealing with positive bases that are not orthogonally structured.
Such results would also improve the practical calculation of k-cosine measures. Finally, designing derivative-free optimization techniques based on positive k-spanning sets with guarantees on their k-cosine measure is a natural application of our study, and is the subject of ongoing work.
20 References C. Audet, A short proof on the cardinality of maximal positive bases, Optim. Letters, 5 (2011), pp. 191–194.
C. Audet and W. Hare, Derivative-Free and Blackbox Optimization, Springer Series in Operations Research and Financial Engineering, Springer International Publishing, 2018.
A. Conn, K. Scheinberg, and L. Vicente, Introduction to derivative-free optimization, vol. 8, Siam, 2009.
C. Davis, Theory of positive linear dependence, American Journal of Mathematics, 76 (1954), pp. 733–746.
W. Hare and G. Jarry-Bolduc, Calculus identities for generalized simplex gradients: Rules and applications, Submitted to SIAM Journal on Optimization, (2018).
, A deterministic algorithm to compute the cosine measure of a finite positive spanning set, Optim. Lett., 14 (2020), pp. 1305–1316.
W. Hare, G. Jarry-Bolduc, and C. Planiden, Nicely structured positive bases with maximal cosine measure, Optim. Lett., (2023).
J. Hopcroft and R. Tarjan, Algorithm 447: Efficient algorithms for graph manipula-tion, Commun. ACM, 16 (1973), p. 372–378.
G. Jarry-Bolduc, Structures in Derivative Free Optimization: Approximating Gradients/ Hessians, and Positive Bases, PhD thesis, University of British Columbia, 2023. https: //circle.ubc.ca/.
T. G. Kolda, R. M. Lewis, and V. Torczon, Optimization by direct search: New perspectives on some classical and modern methods, SIAM Rev., 45 (2003), pp. 385–482.
R. M. Lewis and V. Torczon, Rank ordering and positive bases in pattern search algo-rithms., tech. rep., Institute for computer applications in science and engineering, Hampton VA, 1996.
D. A. Marcus, Minimal positive 2-spanning sets of vectors, Proceedings of the AMS, 82 (1981), pp. 165–172.
D. A. Marcus, Gale diagrams of convex polytopes and positive spanning sets of vectors, Discrete Applied Mathematics, 9 (1984), pp. 47–67.
A. Marsden, Eigenvalues of the Laplacian and their relationship to the connectedness of a graph, University of Chicago, REU, (2013).
R. McKinney, Positive bases for linear spaces, Transactions of the American Mathematical Society, 103 (1962), pp. 131–148.
G. Naevdal, Positive bases with maximal cosine measure, Optimization Letters, 13 (2019), pp. 1381–1388.
21 R. Regis, The calculus of simplex gradients, Optimization Letters, 9 (2015), pp. 845–865.
R. Regis, On the properties of positive spanning sets and positive bases, Optimization and Engineering, 17 (2016), pp. 229–262.
R. Regis, On the properties of the cosine measure and the uniform angle subspace, Comput.
Optim. Appl., 78 (2021), pp. 915–952.
L. Roberts and C. W. Royer, Direct search based on probabilistic descent in reduced spaces. arXiv:2204.01275v2, 2023.
Z. Romanowicz, Geometric structure of positive bases in linear spaces, Applicationes Mathematicae, 19 (1987), pp. 557–567.
R. Wotzlaw, Incidence graphs and unneighborly polytopes, PhD thesis, Technischen Uni-versit¨ at Berlin, 2009.
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Metric Prefixes and SI Units
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Introduction
Metric Prefixes are incredibly useful for describing quantities of the International System of Units (SI) in a more succinct manner.
When exploring the world of electronics, these units of measurement are very important and allow people from all over the world to communicate and share their work and discoveries. Some common units used in electronics include voltage for electrical potential difference, ampere for electrical current, watts for power, farad for capacitance, henry's for inductance, and ohms for resistance.
This tutorial will not only go over some of the most commonly used units in electronics but will also teach you the metric prefixes that help describe all of these base units in quantities ranging from the insanely large to the incredibly small.
Suggested Reading
If you would like to know more about the components that use the units and prefixes described in this tutorial, check out some of these related tutorials.
### Voltage, Current, Resistance, and Ohm's Law
Learn about Ohm's Law, one of the most fundamental equations in all electrical engineering.
### Resistors
A tutorial on all things resistors. What is a resistor, how do they behave in parallel/series, decoding the resistor color codes, and resistor applications.
### Capacitors
Learn about all things capacitors. How they're made. How they work. How they look. Types of capacitors. Series/parallel capacitors. Capacitor applications.
You should also be familiar with binary in order to help you understand binary prefixes.
### Binary
Binary is the numeral system of electronics and programming...so it must be important to learn. But, what is binary? How does it translate to other numeral systems like decimal?
SI Units
We've been measuring stuff for millennia, and our units used for those measures have been evolving since then. There are now dozens of units to describe physical quantities. For example, length can be measured by the foot, meter, fathom, chain, parsec, league, and so on. In order to better communicate measurements, we needed a standardized system of units, which every scientist and measurer could use to share their findings. This standardized system has come to be called the \ International System of Units \, abbreviated SI.
Physical SI Units
| Quantity | SI Unit | Unit Abbreviation |
---
| Time | second | s |
| Length | meter | m |
| Mass | gram | g |
| Temperature | kelvin | K |
| Force | newton | N |
While we can still use units like feet or miles for distance (instead of meters), liters to describe volume (instead of m3), and Fahrenheit or Celsius to describe temperature (instead of °K), the units above are a standardized way for every scientist to share their measurements. Using the units above means everyone is speaking the same language.
Common Electronics Units
In dealing with electronics, there are a handful of units we'll be encountering more often than others. These include:
| Quantity | SI Unit | Unit Abbreviation |
---
| Electric Potential Difference (Voltage) | volts | V |
| Electric Current | ampere | A |
| Power | watt | W |
| Energy/Work/Heat | joule | J |
| Electric Charge | coulomb | C |
| Resistance | ohm | Ω |
| Capacitance | farad | F |
| Inductance | henry | H |
| Frequency | hertz | Hz |
Now that we know the units, let's look at how they can be augmented with prefixes to make them even more usable!
The Prefixes
When first learning about metric prefixes, chances are you were taught these six prefixes first:
| | | |
---
| Prefix (Symbol) | Power | Numeric Representation |
| kilo (k) | 103 | 1,000 |
| hecto (h) | 102 | 100 |
| deka (da) | 101 | 10 |
| no prefix | 100 | 1 unit |
| deci (d) | 10-1 | 0.1 |
| centi (c) | 10-2 | 0.01 |
| milli (m) | 10-3 | 0.001 |
These are what we'll consider the standard six prefixes taught in most High School science courses. You may have even learned a fun mnemonic to go along with these such as Kangaroos Have Dirty Underwear During Cold Months. However, as you'll soon see, when learning about electronics and computer science, the range of prefixes well exceeds the standard six. While these prefixes cover a rang of 10-3 to 103, many electronic values can have a much larger range.
Describing the Large
| | | |
---
| Prefix (Symbol) | Power | Numeric Representation |
| yotta (Y) | 1024 | 1 septillion |
| zetta (Z) | 1021 | 1 sextillion |
| exa (E) | 1018 | 1 quintillion |
| peta (P) | 1015 | 1 quadrillion |
| tera (T) | 1012 | 1 trillion |
| giga (G) | 109 | 1 billion |
| mega (M) | 106 | 1 million |
| kilo (k) | 103 | 1 thousand |
| no prefix | 100 | 1 unit |
These above prefixes dramatically help describe quanities of units in large amounts. Instead of saying 3,200,000,000 Hertz, you can say 3.2 GigaHertz, or 3.2 GHz for shorthand written notation. This allows us to describe incredibly large numbers of units succinctly. There are also prefixes for helping communicate tiny numbers as well.
Describing the Small
| | | |
---
| Prefix (Symbol) | Power | Numeric Representation |
| no prefix | 100 | 1 unit |
| milli (m) | 10-3 | 1 thousandth |
| micro (µ) | 10-6 | 1 millionth |
| nano (n) | 10-9 | 1 billionth |
| pico (p) | 10-12 | 1 trillionth |
| femto (f) | 10-15 | 1 quadrillionth |
| atto (a) | 10-18 | 1 quintillionth |
| zepto (z) | 10-21 | 1 sextillionth |
| yocto (y) | 10-24 | 1 septillionth |
Now, instead one trillionth of a second, it can be referred to as a picosecond. One thing to notice about the prefixes for small values, is that their shorthand notations are all lower case while the large number prefixes are upper case (with the exception of kilo-, hecto- and deca-). This allows you to distinguish between the two when they use the same letter. As an example, one mW (milliwatt) does not equal one MW (megawatt).
Note: Since the upper case 'K' was already used to describe Kelvins, a lower case 'k' was chosen to represent the prefix kilo-. As you'll see in the Bits and Bytes section, there is also some confusion with k and K when dealing with the binary (base 2) prefixes.
Conversion
The beautiful thing about these metric prefixes is that, once you get the hang of conversion between a few of them, translating that ability to all the other prefixes is easy.
As a first simple example, lets translate 1 Ampere (A) into smaller values. A milliampere is 1 thousandth of the unit Ampere hence 1 Ampere is equal to 1000 milliamperes. Going further, 1 milliampere is equivalent to 1000 microamperes and so on. Going in the opposite direction, 1 Ampere is .001 Kiloampere, or 1000 Amperes is 1 Kiloampere. Now that's a lot of current!
As you may have noticed, switching between prefixes is the same as moving the decimal point over by 3 places. This is also the same as multiplying or dividing by 1000. When you're going up to a larger prefix, from Kilo to Mega for example, the decimal place is moved three places to the left. 100,000 Kilowatts equals 100 Megawatts. 10 Kilowatts equals .01 Megawatts. Mega is the prefix right above Kilo so regardless of whether we are talking about Watts, Amperes, Farads, or whatever unit, the movement of the decimal place by three positions to the left still works when moving up a prefix.
When moving down a prefix, let's say from nano- to pico-, the decimal place is moved three places to the right. 1 nanoFarad equals 1000 picoFarads. .5 nanoFarad equals 500 picoFarads. Here's a short list so you can see the pattern:
1 Giga- = 1000 Mega-
1 Mega- = 1000 Kilo-
1 Kilo- = 1000 units
1 unit = 1000 milli-
1 milli- = 1000 micro-
See the trend? Each prefix is a thousand times larger than the previous. While a little overwhelming at first, translation from one prefix to another eventually becomes second nature.
Bits and Bytes
Working with bits and bytes can cause a bit confusion (pun intended). Since computers work with base 2 numbers instead of base 10, it is often unclear which number base one is referring to when using the metric prefixes. For example, 1 Kilobyte is often used to mean 1000 bytes (base 10), or it can be used to represent 1024 bytes (base 2), resulting in misunderstandings.
To eliminate these mix-ups, the International Electrotechnial Commision came up with some new prefixes for the base 2 bits and bytes. These are referred to as binary prefixes.
| | | |
---
| Prefix (Symbol) | Power | Numeric Representation |
| exbi- (Ei-) | 260 | 1,152,921,504,606,846,976 |
| pebi- (Pi-) | 250 | 1,125,899,906,842,624 |
| tebi- (Ti-) | 240 | 1,099,511,627,776 |
| gibi (Gi-) | 230 | 1,073,741,824 |
| mebi- (Mi-) | 220 | 1,048,576 |
| kibi- (Ki-) | 210 | 1,024 |
| no prefix | 20 | 1 bit or byte |
Adopting this would mean 1 Megabyte = 1000 Kilobytes while 1 Mebibyte equals 1024 Kibibytes. Essentially for bits and bytes, each jump in prefix would be a multiple of 1024 (2^10) instead of 1000 (10^3). Unfortunately, this system is not widely used in practice, so anytime you hear a number of bytes or bits, you have to wonder if they are talking about them in base 2 or base 10.
Hard drive companies and others typically sell products in base 10 as it makes it sound larger. A 1 Terabyte hard drive will turn out to actually be about 931.3 Gibibytes.
This is where we run into the upper case and lower case 'k' situation. The proper prefix for kibi if 'Ki'. However, it will sometimes appear as just and upper case 'K', which, again, represents temperature in Kelvins. So, any time you hear the word Kilobyte, you still have to wonder if it signifies 1000 bytes (base 10) or 1024 bytes (base 2). On the other hand, if you see the term kibibyte, you know for sure it's talking about the base 2 version interpretation of digital storage (1024 bytes).
Converting Bits to Bytes and Bytes to Bits
We've covered converting bits and bytes to larger or smaller numbers of each, but there is also the matter of converting bits to bytes and vise versa. Remember that 1 Byte is equal to 8 bits (a majority of the time), and one bit is equal to 0.125 bytes (or 1/8). Granted, there are many orders of magnitude pertaining to bits, but byte is typically used most frequently. The practice of converting between one and the other is not all that common, but it is still useful information when dealing with electronics, especially when it comes to memory. For example, you could be writing code that stores individual bits, but your memory is defined as bytes.
Practice
Now for some practice exercises. We'll use standard abreviations for each unit type we'll convert:
A for Amperes
V for Volts
W for Watts
Hz for Hertz
F for Farads
H for Henry's
Ω for Ohms
s for Seconds
B for Bytes
b for bits
Conversion Example:
Convert: 400 mA to A
Answer: 400 mA = .4 A
Convert:
50 mA to A
10 nF to pF
500 kW to W
.01 mV to µV
20,000 kΩ to MΩ
4680 MHz to GHz
4 TiB to GiB
200 Mb to kb
.00007 s to µs
1450 nH to µH
Practice Answers
.05 A
10,000 pF
500,000 W
10 µV
20 MΩ
4.68 GHz
4096 GiB
200,000 kb
70 µs
1.45 µH
Soon, switching between prefixes when needed becomes very quick.
Resources and Going Further
Being able to convert numbers to the best prefix depending on the size of the number is an important skill to have. It allows you to avoid really long and messy numbers like 5,600,000 or .000000002. Using 5.6M or 2n allows you to convey the information faster and in a much tidier and easier to read format.
Interested in learning more foundational topics?
See our Engineering Essentials page for a full list of cornerstone topics surrounding electrical engineering.
Take me there!
Now that you are familiar with the metric prefixes, consider taking a look at our How to Use a Multimeter tutorial. Using a multimeter requires a good understanding of all the prefixes since your measurements will often show up as such.
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190443 | https://www.quora.com/If-the-first-term-of-the-arithmetic-progression-is-1-and-the-terms-are-odd-numbers-the-sum-of-its-odd-numbers-terms-is-175-thesum-of-its-even-number-terms-is-150-what-is-its-sequence | If the first term of the arithmetic progression is 1, and the terms are odd numbers, the sum of its odd numbers terms is 175, thesum of its even number terms is 150, what is its sequence? - Quora
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If the first term of the arithmetic progression is 1, and the terms are odd numbers, the sum of its odd numbers terms is 175, thesum of its even number terms is 150, what is its sequence?
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David Vanderschel
PhD in Mathematics&Physics, Rice (Houston neighborhood) (Graduated 1970) · Author has 37.6K answers and 50.1M answer views
·4y
Let A A be the average of all the terms and let n n be number of even terms.
Then n A=150 n A=150 and (n+1)A=175(n+1)A=175 implies A=25 A=25 and n=6.n=6.
The middle term, A=25,A=25, is the fourth odd term or the seventh term overall.
So the difference is (25–1)/6=4.(25–1)/6=4.
The k k th term is 4 k−3 4 k−3 for k k in [1,13].[1,13].
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An arithmetic progression has 20 terms, the sum of all even numbers is 350. Given that the first term is 5. What is the common difference and the eleventh term?
How many arithmetic progressions consisting of natural numbers exist, having an even first term, where the sum of odd-valued terms is 33 and the sum of even-valued terms is 44?
What is the next number in an arithmetic progression of three terms with a common difference of one where the first number is nine?
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Doug Dillon
Ph.D. Mathematics · Author has 12.4K answers and 11.4M answer views
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When one “finds” an AP, one finds a, d and n, and we know a.
Since the sums are greater than 1, the sequence is increasing. Since the sum of the even-numbered terms is less than the sum of the odd-numbers terms, there is one more odd than even-numbered terms and the last odd-numbered term is 25.
Suppose there are n odd-numbered terms and n-1 even-numbered terms.
The sum of the odd-numbered terms is[math] [/math]
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When one “finds” an AP, one finds a, d and n, and we know a.
Since the sums are greater than 1, the sequence is increasing. Since the sum of the even-numbered terms is less than the sum of the odd-numbers terms, there is one more odd than even-numbered terms and the last odd-numbered term is 25.
Suppose there are n odd-numbered terms and n-1 even-numbered terms.
The sum of the odd-numbered terms is n 2[2+2(n−1)d]=175 n 2[2+2(n−1)d]=175 and the sum of the even-numbered terms is n−1 2[2(1+d)+2(n−2)d]=150 n−1 2[2(1+d)+2(n−2)d]=150. Wr...
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To solve this problem, we need to define the arithmetic progression (AP) and analyze the given conditions.
Definitions
First term (a): 1
Common difference (d): Since the terms are odd numbers, d d must be an even number.
Sum of odd-numbered terms: 175
Sum of even-numbered terms: 150
General Term of the AP
The n n-th term of the arithmetic progression can be expressed as:
a n=a+(n−1)d=1+(n−1)d a n=a+(n−1)d=1+(n−1)d
Odd-Numbered Terms
The odd-numbered terms occur at the 1st, 3rd, 5th, etc., positions. The n n-th odd term can be expressed as:
a 2 n−1=1+(2 n−2)d=1+(2 n−2)d a 2 n−1=1+(2 n−2)d=1+(2 n−2)d
The sum of the first n n odd-numbered terms is
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To solve this problem, we need to define the arithmetic progression (AP) and analyze the given conditions.
Definitions
First term (a): 1
Common difference (d): Since the terms are odd numbers, d d must be an even number.
Sum of odd-numbered terms: 175
Sum of even-numbered terms: 150
General Term of the AP
The n n-th term of the arithmetic progression can be expressed as:
a n=a+(n−1)d=1+(n−1)d a n=a+(n−1)d=1+(n−1)d
Odd-Numbered Terms
The odd-numbered terms occur at the 1st, 3rd, 5th, etc., positions. The n n-th odd term can be expressed as:
a 2 n−1=1+(2 n−2)d=1+(2 n−2)d a 2 n−1=1+(2 n−2)d=1+(2 n−2)d
The sum of the first n n odd-numbered terms is:
S odd=∑k n=1 a 2 n−1=∑k n=1(1+(2 n−2)d)S odd=∑n=1 k a 2 n−1=∑n=1 k(1+(2 n−2)d)
This simplifies to:
S odd=k+(2 d)(k(k−1)2)=k+d k(k−1)S odd=k+(2 d)(k(k−1)2)=k+d k(k−1)
Setting this equal to 175, we have:
k+d k(k−1)=175(1)k+d k(k−1)=175(1)
Even-Numbered Terms
The even-numbered terms occur at the 2nd, 4th, 6th, etc., positions. The n n-th even term can be expressed as:
a 2 n=1+(2 n−1)d a 2 n=1+(2 n−1)d
The sum of the first n n even-numbered terms is:
S even=∑k n=1 a 2 n=∑k n=1(1+(2 n−1)d)=k+d∑k n=1(2 n−1)S even=∑n=1 k a 2 n=∑n=1 k(1+(2 n−1)d)=k+d∑n=1 k(2 n−1)
The sum of the first k k odd numbers is k 2 k 2, thus:
S even=k+d(k 2)=k+d k 2 S even=k+d(k 2)=k+d k 2
Setting this equal to 150, we have:
k+d k 2=150(2)k+d k 2=150(2)
Solving the System of Equations
We now have a system of two equations:
k+d k(k−1)=175 k+d k(k−1)=175
k+d k 2=150 k+d k 2=150
From equation (2), we can express d d:
d k 2=150−k⟹d=150−k k 2(3)d k 2=150−k⟹d=150−k k 2(3)
Substituting equation (3) into equation (1):
k+150−k k 2 k(k−1)=175 k+150−k k 2 k(k−1)=175
This simplifies to:
k+(150−k)(k−1)=175 k 2 k+(150−k)(k−1)=175 k 2
Expanding this gives:
k+150 k−150−k 2+k=175 k 2 k+150 k−150−k 2+k=175 k 2
Combining like terms:
152 k−150=175 k 2+k 2 152 k−150=175 k 2+k 2
This simplifies to:
176 k 2−152 k+150=0 176 k 2−152 k+150=0
Using the Quadratic Formula
Using the quadratic formula k=−b±√b 2−4 a c 2 a k=−b±b 2−4 a c 2 a:
a=176 a=176
b=−152 b=−152
c=150 c=150
Calculating the discriminant:
b 2−4 a c=(−152)2−4⋅176⋅150=23104−105600=−8256 b 2−4 a c=(−152)2−4⋅176⋅150=23104−105600=−8256
Since the discriminant is negative, it indicates that there are no real solutions under our assumption.
Conclusion
Given that the conditions of the problem might have inconsistencies or require specific values for d d and k k, it would be beneficial to re-evaluate the setup or provide further constraints to reach a valid arithmetic sequence.
If you have additional constraints or specific values for d d and k k, please share them for further analysis!
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Calvin Campbell
B.Sc. in Computer Science&Mathematics, University of the West Indies (Graduated 1984) · Author has 3.9K answers and 4.1M answer views
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We need to find the number of terms, n and the common difference, d of the A.P. series of odd numbers: 1 + (1 + (2-1)d ) + (1 + (3–1)d ) + (1 + (4–1)d) + …. + (1 + (n-1)d, since the nth term, tn is given by:
tn = t1 + (n-1)d = 1 + (n-1)d, and t1 = 1.
Since the sum of the odd numbered terms is 175, and the sum of the even numbered terms is 150, then the sum of the entire series of odd numbers is 175
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We need to find the number of terms, n and the common difference, d of the A.P. series of odd numbers: 1 + (1 + (2-1)d ) + (1 + (3–1)d ) + (1 + (4–1)d) + …. + (1 + (n-1)d, since the nth term, tn is given by:
tn = t1 + (n-1)d = 1 + (n-1)d, and t1 = 1.
Since the sum of the odd numbered terms is 175, and the sum of the even numbered terms is 150, then the sum of the entire series of odd numbers is 175 + 150 = 325.
Since the sum of the series is odd and all the terms are odd, then the number of terms must also be odd, because the sum of an odd number of odd numbers is odd. Example: The sum of 3 odd numbers, 1 + 3 + 5 = 9, which is an odd number.
Therefore, the number of odd numbered terms should be one more than the number of even numbered terms. If the number of terms of the original series is n, then the number of the even numbered terms is (n -1)/2, and the number of odd numbered terms is (n-1)/2 + 1 = (n + 1)/2.
Since the terms are odd, the common difference, d in the original series must be even, and the common differences of the series of the odd numbered terms and that of the even numbered terms must be 2 times d (2d).
The sum, S of an A.P. series is given by formula:
S = number of terms/2[2 first term + (number of terms - 1) common difference]
Therefore,
The sum of the odd numbered terms, SO:
SO = ((n+1)/2)/2[21 + ((n+1)/2 - 1) 2d] = 175
=> (n+1)[2 + (n+1)d - 2d] = 4 175 = 700 — Eqn. I
The sum of the even numbered terms, SE:
SE = ((n-1)/2)/2 [ 2(1+d) + ((n-1)/2 - 1)2d] = 150
=> (n-1)[2 + 2d + (n-1)d - 2d = 150 4 = 600 — Eqn. II
If you solve Eqn. I & Eqn. II for n and d simultaneously, yo...
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What is the sum of even and odd numbers from 1 to 100?
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Can an odd number and an even number make 100?
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Darryl Nester
25+ years teaching college math · Author has 791 answers and 2.1M answer views
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In arithmetic sequence and series, how do you solve for both initial term and number of terms (only the sum of terms, difference and nth term is given)?
An arithmetic series with initial term a, common difference d, and n terms
a+(a+d)+(a+2 d)+(a+3 d)+⋯+(a+(n−1)d)a+(a+d)+(a+2 d)+(a+3 d)+⋯+(a+(n−1)d)
can be rearranged as
n termsa+a+⋯+a+d(0+1+2+⋯+(n−1))=n a+1 2 d n(n−1)a+a+⋯+a⏞n terms+d(0+1+2+⋯+(n−1))=n a+1 2 d n(n−1)
(This is the standard formula for the sum of a finite arithmetic series.) So, suppose you have been given
the difference:d 1 2 the sum:S=n a+1 2 d n(n−1)the n th term:F=a+d(n−1)the difference:d 1 2 the sum:S=n a+1 2 d n(n−1)the n th term:F=a+d(n−1)
We must solve for n and a. We can eliminate a by noting that
2 S=2 n a+d n(2 S=2 n a+d n(
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An arithmetic series with initial term a, common difference d, and n terms
a+(a+d)+(a+2 d)+(a+3 d)+⋯+(a+(n−1)d)a+(a+d)+(a+2 d)+(a+3 d)+⋯+(a+(n−1)d)
can be rearranged as
n termsa+a+⋯+a+d(0+1+2+⋯+(n−1))=n a+1 2 d n(n−1)a+a+⋯+a⏞n terms+d(0+1+2+⋯+(n−1))=n a+1 2 d n(n−1)
(This is the standard formula for the sum of a finite arithmetic series.) So, suppose you have been given
the difference:d 1 2 the sum:S=n a+1 2 d n(n−1)the n th term:F=a+d(n−1)the difference:d 1 2 the sum:S=n a+1 2 d n(n−1)the n th term:F=a+d(n−1)
We must solve for n and a. We can eliminate a by noting that
2 S=2 n a+d n(n−1)=2 n F−d n(n−1))2 S=2 n a+d n(n−1)=2 n F−d n(n−1))
This is a quadratic equation in n (the only unknown value), so we can solve it:
d n 2−(2 F+d)n+2 S=0⟹n=2 F+d±√(2 F+d)2−8 d S 2 d d n 2−(2 F+d)n+2 S=0⟹n=2 F+d±(2 F+d)2−8 d S 2 d
Of course, we want a positive integer solution, if one exists. Once we have a value for n, the initial term a is F−d(n−1)F−d(n−1). For example:
d S F k=2 F+d√k 2−8 d S n 1 a 1 n 2 a 2 3 390 47 97 7 15 5 17.¯¯¯3−2 2 800 56 114 14 25 8 32−6 5 1525 121 247 3 24.4 4 25 1−1 5−1−3 7 5 3−2−4 0.5 16.5 6 12.5 9.5 3 5 22−4.5 d S F k=2 F+d k 2−8 d S n 1 a 1 n 2 a 2 3 390 47 97 7 15 5 17.3¯−2 2 800 56 114 14 25 8 32−6 5 1525 121 247 3 24.4 4 25 1−1 5−1−3 7 5 3−2−4 0.5 16.5 6 12.5 9.5 3 5 22−4.5
In the above, n 1 n 1 is the solution which corresponds to “–” in the numerator of the quadratic formula. We can see that sometimes there is one positive integer value for n, and sometimes there are two (that is, we cannot always determine a and n with certainty from the given information).
Note that sometimes there are no integer solutions for n, meaning that the given information is not consistent with an arithmetic series. For example, with d=1,S=4,F=3 d=1,S=4,F=3, we get no useful solutions … and it is pretty easy to recognize that if we end with 3, and construct previous terms by counting backwards by 1, there is no number of terms we could use to have the series add up to 4.
Finally, note that we could take the approach of treating the n th term as the first term, and negate the difference, so that instead of looking at the series as
a+(a+d)+(a+2 d)+⋯+F a+(a+d)+(a+2 d)+⋯+F
we would reverse the order and see it as
F+(F−d)+(F−2 d)+⋯+(F−d(n−1))F+(F−d)+(F−2 d)+⋯+(F−d(n−1))
This is not really different (or easier) than the approach described above, but might feel simpler. (In particular, thinking in this way makes it easier to recognize the “brute force” approach to finding a and n: start writing down terms until you get the desired sum. However, that’s not a practical approach in cases where n is large, and care must be taken to note the cases where there are two solutions.)
In arithmetic sequence and series, how do you solve for both initial term and number of terms (only the sum of terms, difference and nth term is given)?
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Enrico Gregorio
Associate professor in Algebra · Author has 18.4K answers and 16M answer views
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The sum of the first 20 terms of an arithmetic sequence is 760. What is the first term if the last term is 95? What is its common difference?
The sum of the first 20 terms of an arithmetic sequence is 760. What is the first term if the last term is 95? What is its common difference?
As few formulas as possible. Call a a your first term. Then you know that the terms are
a,a+d,…,a+19 d a,a+d,…,a+19 d
where d d is the common difference. Then we see that
a+19 d=95 a+19 d=95
20 a+(0+1+2+⋯+19)d=760 20 a+(0+1+2+⋯+19)d=760
You surely know how to compute the sum in parentheses, don’t you? It’s
19⋅20 2=190 19⋅20 2=190
Now the second equation simplifies to
2 a+19 d=76 2 a+19 d=76
and subtracting yields
a=76−95=−19 a=76−95=−19
and therefore
d=95−a 19=6 d=95−a 19=6
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Devika
Author has 107 answers and 132.7K answer views
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The sum of three terms of an arithmetic progression is 30. The sum of the squares of the terms is 318. What are the three terms?
Let the three terms be a-d, a, a+d
sum of the three terms = 30
a-d + a + a+d = 30
3a = 30
a = 10 ———-(1)
sum of the squares of the terms = 318
(a-d)^2 + a^2 + (a+d)^2 = 318
a^2 - 2ad + d^2 + a^2 + a^2 + 2ad + d^2 =318
3a^2 + 2d^2 = 318 ————-(2)
substituting (1) in (2),
3(10)^2 + 2d^2 = 318
3100 + 2d^2 = 318
300 + 2d^2 = 318
2d^2 = 318–300
2d^2 = 18
d^2 = 18/2
d^2 = 9
d = square root of 9
d = 3
a = 10
a-d = 10 - 3 = 7
a+d = 10 + 3 =13
Thus the terms are, 7 , 10 and 13
cross-checking,
sum of the three terms should be 30
10 + 7 + 13 = 30 (true)
sum of the squares of the three terms should be equal to 318
(7)^2 + (10)^2 +
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Let the three terms be a-d, a, a+d
sum of the three terms = 30
a-d + a + a+d = 30
3a = 30
a = 10 ———-(1)
sum of the squares of the terms = 318
(a-d)^2 + a^2 + (a+d)^2 = 318
a^2 - 2ad + d^2 + a^2 + a^2 + 2ad + d^2 =318
3a^2 + 2d^2 = 318 ————-(2)
substituting (1) in (2),
3(10)^2 + 2d^2 = 318
3100 + 2d^2 = 318
300 + 2d^2 = 318
2d^2 = 318–300
2d^2 = 18
d^2 = 18/2
d^2 = 9
d = square root of 9
d = 3
a = 10
a-d = 10 - 3 = 7
a+d = 10 + 3 =13
Thus the terms are, 7 , 10 and 13
cross-checking,
sum of the three terms should be 30
10 + 7 + 13 = 30 (true)
sum of the squares of the three terms should be equal to 318
(7)^2 + (10)^2 + ( 13)^2 = 49 + 100 + 169 = 318 (true)
hence the terms are 7 , 10 and 13
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I usually look at three things when I’m buying work shoes: comfort, cushioning and arch support; how sturdy the sole is; and whether I can actually afford to get more than one pair if I want them in different colours.
Ballerinas by Bata though, are what I wear the most. I didn’t know about them until recently, when a coworker recommended them to me, also spotted my favorite creator Siddhi Karwa styling them across Europe and I have been absolutely loving them.
They’re professional enough for work wear but don’t feel heavy and keep me comfortable throughout the day, even when I’m commuting to the office. I got mine for around ₹999 from Bata, which felt like a steal compared to some other brands I looked at. They’ve held up really well, and I can easily pair them with trousers, skirts for my work outfits. If you’re on a budget but still want something that is comfortable and follows fashion trends, Ballerinas by Bata are the perfect choice. I picked up mine from a Bata store near me, you can grab yours too.
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Sandy Norman
Assoc. Prof. Emeritus (Mathematics) · Author has 100 answers and 186.6K answer views
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How many consecutive odd numbers are there in an arithmetic progression with a given common difference?
This is a QGP prompt question and, like many of them, needs some clarification. A lot.
Is the progression infinite? Or finite? And if the latter, how many terms?
Does “How many consecutive odd numbers…” mean how many terms include something like 5, 7, 9, …. in succession? Or simply how many consecutive terms are odd?
What are allowable values for the first term of the progression? And the common differences?
Regardless, here are some possibilities:
We see above an example in which the sequence represents precisely the set of consecutive positive odd numbers; another in which consecutive terms are o
Continue Reading
This is a QGP prompt question and, like many of them, needs some clarification. A lot.
Is the progression infinite? Or finite? And if the latter, how many terms?
Does “How many consecutive odd numbers…” mean how many terms include something like 5, 7, 9, …. in succession? Or simply how many consecutive terms are odd?
What are allowable values for the first term of the progression? And the common differences?
Regardless, here are some possibilities:
We see above an example in which the sequence represents precisely the set of consecutive positive odd numbers; another in which consecutive terms are odd; and two progressions with no odds at all.
There is not much definitive we can say except that for infinite arithmetic progressions you might have no consecutive odds (or no consecutive terms that are odd) or you might have an infinite number.
Now, are these the only possibilities? Either 0 or an infinite number of them (consecutive odds or successive terms that are odd)?
I won’t give away the answer.
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Matheus Rocha
B.S. Electrical Engineering
·7y
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In arithmetic sequence and series, how do you solve for both initial term and number of terms (only the sum of terms, difference and nth term is given)?
The n n-th term of the sequence is
a n=a 1+(n−1)d,a n=a 1+(n−1)d,
where d d is the difference between consecutive terms. The sum of the first n n terms of the sequence is
S n=n a 1+a n 2.S n=n a 1+a n 2.
Solving this last equation for a 1 a 1,
a 1=2 S n n−a n.a 1=2 S n n−a n.
Replacing it in the expression for the n n-th term,
a n=2 S n n−a n+(n−1)d.a n=2 S n n−a n+(n−1)d.
Multiplying this last equation for n n and rearranging,
d n 2−(2 a n+d)n+2 S n=0.d n 2−(2 a n+d)n+2 S n=0.
If d=0 d=0, this implies
n=S n a n.n=S n a n.
If d≠0 d≠0, the equation will be a quadratic equation in n n, with the solutions
n=2 a n+d±√(2 a n+d)2−8 d S n 2 d.n=2 a n+d±(2 a n+d)2−8 d S n 2 d.
Notice that not always this expression will provide an in
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The n n-th term of the sequence is
a n=a 1+(n−1)d,a n=a 1+(n−1)d,
where d d is the difference between consecutive terms. The sum of the first n n terms of the sequence is
S n=n a 1+a n 2.S n=n a 1+a n 2.
Solving this last equation for a 1 a 1,
a 1=2 S n n−a n.a 1=2 S n n−a n.
Replacing it in the expression for the n n-th term,
a n=2 S n n−a n+(n−1)d.a n=2 S n n−a n+(n−1)d.
Multiplying this last equation for n n and rearranging,
d n 2−(2 a n+d)n+2 S n=0.d n 2−(2 a n+d)n+2 S n=0.
If d=0 d=0, this implies
n=S n a n.n=S n a n.
If d≠0 d≠0, the equation will be a quadratic equation in n n, with the solutions
n=2 a n+d±√(2 a n+d)2−8 d S n 2 d.n=2 a n+d±(2 a n+d)2−8 d S n 2 d.
Notice that not always this expression will provide an integer positive solution for n n, meaning that not always exists an arithmetic sequence with certain S n S n, d d and a n a n. We can verify this fact with one example. If a n a n is negative and d d is positive, all the sequence terms have to be negatives; thus, a positive value for S n S n is impossible.
Summarizing, if you know S n S n, d d and a n a n, being d≠0 d≠0, then you can determine the possible values for n n by the expression
n=2 a n+d±√(2 a n+d)2−8 d S n 2 d,n=2 a n+d±(2 a n+d)2−8 d S n 2 d,
and, after it, determine the corresponding value of a 1 a 1 for each possible value of n n by the expression
a 1=2 S n n−a n a 1=2 S n n−a n.
(You can also calculate a 1 a 1 first, replacing the expression for n n in the expression for a 1 a 1, but it will be more laborious).
If d=0 d=0, we have
n=S n a n n=S n a n
and
a 1=a n.a 1=a n.
Indeed, for d=0 d=0, all the terms of the sequence have to be equal each other.
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Sandesh Ghanta
BTECH in Computer Science, Amrita School of Engineering Amritapuri (Graduated 2020)
·7y
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What is the sum of even numbers in an arithmetic sequence?
Let us say the first term in the sequence is ‘a’ and the common difference is ‘d’.
So we can classify this into 4 cases
a is odd and d is odd
a is odd and d is even
a is even and d is odd
a is even and d is even
Case 1
The sequence is a, a+d, a+2d, a+3d…. now here alternate numbers are even starting from a+d. So the sum of the even numbers would be the sum of the sequence a+d, a+3d, a+5d… This is another arithmetic series with first term a+d and common difference 2d
Case 2
The sequence is a, a+d, a+2d, a+3d…. now we know that odd + even is always odd. So every term in this sequence is an odd number. So
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Let us say the first term in the sequence is ‘a’ and the common difference is ‘d’.
So we can classify this into 4 cases
a is odd and d is odd
a is odd and d is even
a is even and d is odd
a is even and d is even
Case 1
The sequence is a, a+d, a+2d, a+3d…. now here alternate numbers are even starting from a+d. So the sum of the even numbers would be the sum of the sequence a+d, a+3d, a+5d… This is another arithmetic series with first term a+d and common difference 2d
Case 2
The sequence is a, a+d, a+2d, a+3d…. now we know that odd + even is always odd. So every term in this sequence is an odd number. So the answer is 0
Case 3
The sequence is a, a+d, a+2d, a+3d…. now here alternate numbers are even starting from a. So the sum of the even numbers would be the sum of the sequence a, a+2d, a+4d… This is another arithmetic series with first term a and common difference 2d
Case 4
The sequence is a, a+d, a+2d, a+3d…. now we know that even + even is always even. So every term in this sequence is even. So the answer is nothing but the sum of the sequence itself.
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Bruce Stevens
Author has 2.8K answers and 1.5M answer views
·11mo
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The 16th term of an arithmetic progression is 40, and the sum of the first 5 terms is 5. What is the sum of the first 50 terms?
Set the question into AS equations:
t(16) = 40 = ( a + 15d ),
Number of diffs = ((n (n — 1)) / 2)
where n = 5,
Number of 1st terms = 5,
Sum(5) = 5 = ( 5a + 10d ).
Solve for the value, of the constant difference, d, between consecutive terms:
t(6) = 40 = ( a + 15d ), → a = ( 40 — 15d ),
Sum(5) = 5 = ( 5a + 10d ),
a = ( 1 — 2d ),
So,
( 40 — 15d ) = ( 1 — 2d ),
Group like terms:
( 15d — 2d ) = ( 40 — 1 ), → 13d = 39,
d = (39 / 13), → d = 3.
Solve for the value, of the 1st term, a:
a = ( 1 — 2d ), → a = ( 1 — 2(3) ),
a = ( 1 — 6 ), → a = ( — 5).
Solve for the value, of the sum of the first fifty terms, Sum(50);
Nu
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Set the question into AS equations:
t(16) = 40 = ( a + 15d ),
Number of diffs = ((n (n — 1)) / 2)
where n = 5,
Number of 1st terms = 5,
Sum(5) = 5 = ( 5a + 10d ).
Solve for the value, of the constant difference, d, between consecutive terms:
t(6) = 40 = ( a + 15d ), → a = ( 40 — 15d ),
Sum(5) = 5 = ( 5a + 10d ),
a = ( 1 — 2d ),
So,
( 40 — 15d ) = ( 1 — 2d ),
Group like terms:
( 15d — 2d ) = ( 40 — 1 ), → 13d = 39,
d = (39 / 13), → d = 3.
Solve for the value, of the 1st term, a:
a = ( 1 — 2d ), → a = ( 1 — 2(3) ),
a = ( 1 — 6 ), → a = ( — 5).
Solve for the value, of the sum of the first fifty terms, Sum(50);
Number of diffs = ((n (n — 1)) / 2 )
where n = 50,
Number of 1st terms = 50,
Sum(50) = ( 50a + 1225d ),
Sum(50) = ( 50( — 5) + 1225(3) ),
Sum(50) = ( — 250 + 3675 ),
Sum(50) = 3425.
P.S.
The arithmetic sequence is:
( — 5), ( — 2), 1, 4, 7, 10, 13, 16, 19, 22, …
The general term rule, of this arithmetic sequence, when n = 1, 2, 3, …, is:
t(n) = ( ( — 5) + ((n — 1) 3) )
OR
t(n) = ( 3n — 8 )
OR
t(n) = ( 3(n — 3) + 1 ).
The general summation rule, of this arithmetic sequence, when n =1, 2, 3, …, is:
Sum(n) = ( ( n(3n — 13) ) / 2 ).
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Manjunath Subramanya Iyer
I am a retired bank officer teaching maths · Author has 7.2K answers and 10.4M answer views
·5y
Related
An arithmetic progression has 20 terms, the sum of all even numbers is 350. Given that the first term is 5. What is the common difference and the eleventh term?
Suppose the AP is a, a+d, a+2d,………..a+19d
Given that the first term is 5
So the AP can be taken as 5, 5+d, 5+2d,…….5+19d
The even terms are 5+d, 5+3d,………5+19d.
Their sum is given to be 350.
Hence (10/2)(5+d+5+19d) = 350
Solving we get d= 3
11th term = a+10d = 5+30 = 35.
Upvote ·
9 5
Eugene Lin
JD, Independent Contractor, Real Estate Broker at Real Estate, Property Management (1999–present) · Author has 1.7K answers and 591.9K answer views
·4y
The “odd” series is 1,3,5,7,……, N, The sum of That is 175; the “even” series is 2,4,6,…..( N-1) The sum of that is 150, Add up these two series will created natural number series 1,2,3,…., (N-1), N, their total sum is (1+N)N/2=325，solve this equation to find what N is. This is simple algebra operation, expand the formula to N+ N^2=2x325=650, or (N-25)(N+26)=0, N=25， The sequence is 1,2,3,….,25 (all are natural numbers). QED
Upvote ·
9 5
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190445 | https://mathoverflow.net/questions/127250/a-graduate-course-on-sturm-liouville-theory | fa.functional analysis - A graduate course on Sturm Liouville theory? - MathOverflow
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A graduate course on Sturm Liouville theory?
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I have some general questions on Sturm-Liouville theory. We are planning to introduce a graduate course on Sturm-Liouville theory and every one has been asked to propose topics which might be suitable for the course.
I would like to know the following.
Is it worth to have a course exclusively on just Sturm-Liouville theory?
If we have a course like that what are the topics that could be introduced in the course?
Would it be possible for an exclusive course on sturm-liouville theory without much background in functional analysis.
Any suggestions would be really appreciated.
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edited Mar 5, 2016 at 6:49
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A more advanced/comprehensive course can be based on Atkinson's book Discrete and continuous boundary problems. No functional analysis is required, neither for Hilbert-Courant nor for Atkinson. (When Courant wrote the first volume of HC, functional analysis did not exist yet:-)
Another book which studies some of these questions in depth (and without any functional analysis) is Gantmakher-Krein, Oscillation matrices and kernels. (Gantmaher, Gantmacher...)
Is it worth offering such a course? The answer depends on whether you can find enough students who will enroll. This depends on your local conditions. Of course this is a beautiful and useful subject and worth learning.
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answered Apr 11, 2013 at 19:08
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Alexandre Eremenko
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Thanks for your comments and suggestions. I do have one more question. I think the students who might be taking this course are more oriented toward algebra. Is it possible to have a little bit of an algebraic flavor to this course? user8974 –user8974 2013-04-12 03:00:42 +00:00 Commented Apr 12, 2013 at 3:00
Yes, and Atkinson's book is good for this. It has more algebraic flavor on my opinion, because he considers discrete and continuous problems together. Same applies to Krein-Gantmakher book.Alexandre Eremenko –Alexandre Eremenko 2013-04-12 12:38:28 +00:00 Commented Apr 12, 2013 at 12:38
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The classic "Methods of Mathematical Physics" by Courant and Hilbert has a wealth of material on the Sturm-Liouville problem and its connections to various themes of mathematical physics. In fact, one could almost say that the treatise is constructed around this topic. Little or no knowledge of Functional Analysis is required. It is, of course, not the most up to date version but can be recommended as a first introduction--- it contains a multitude of explicit examples (Green functions, eigenfunction expansions, connections with the classical pde's of mathematical physics via separation of variables, special functions etc.) and is a joy to read.
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answered Apr 11, 2013 at 16:42
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I know that the students who will be taking this class are more oriented toward algebra. Is there any possibility of adding a bit of algebraic flavor to the course? user8974 –user8974 2013-04-11 18:24:10 +00:00 Commented Apr 11, 2013 at 18:24
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A full course on Sturm-Liouville is worthy of being given in my humble opinion. As already mentioned, the Sturm-Liouville problem is connected with many problems from pure and applied mathematics. For examples,
PDE, boundary problems, notion of Green function (as mentioned above)
some special functions and orthogonal polynomials
problems with ordinary differential equations (see the relevant chapters in Coddington-Levinson or in Whittaker): existence, uniqueness, explosion
problems with eigenvalues, their asymptotics, introduction to the theory of self-adjoint operators, WKB approximations, ...
the Weyl-Titchmarsh theory and its relation with complex analysis and Green's functions
numerical computations of the eigenvalue problems or of the solution (shooting methods, ...), as a way to introduce numerical analysis.
parabolic equations and links between semi-groups and resolvent (see e.g. ``The abstract Cauchy problem and Cauchy's problem for parabolic differential equations'' by E. Hille)
Feller theory of one-dimensional stochastic processes is also strongly related to SL problems
the Krein theory of the string (already mentioned), harmonic analysis, ...
and of course the many applications of the SL equation in physics and other... (on the history, see ``Sturm and Liouville's work on ordinary linear differential equations. The emergence of Sturm-Liouville theory'' by Lützen).
Actually, problems related to the SL equations are simple to write, because Sturm-Liouville operators only involve differential operators), yet their resolutions involve a great deal of ideas that have been generalized and extended in many directions. For me, dealing with SL problems could be a "smooth way" to learn and understand key ideas without relying on abstract settings. For example, I was recently very happy to read the elegant proof of Weyl on the limit circle and point circle, and to have a glimpse of its relations with the "resolution of identity", as it really enlightens it.
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edited Mar 6, 2016 at 10:37
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You could also include the symplectic interpretation of Sturm–Liouville theory (via of the Maslov index) given in Arnold's paper The Sturm theorems and symplectic geometry. This may be of interest to students who are more geometrically/topologically inclined.
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answered May 10, 2016 at 21:13
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I would recommend the book of E-C. Titchmarsh "Eigenfunction Expansions Associated with Second-order Differential Equations".
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answered Mar 5, 2016 at 5:34
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Factoring a Quadratic with a Negative Leading Coefficient
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Alex Hudyma, Amy McKenney
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Alex has been teaching Core Math, Statistics, Pre-calculus, and Calculus at the university level since 2018. She graduated from Western University with a Master's in Mathematics after completing her Honors Bachelor of Science in Mathematics at Lakehead University.
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Amy McKenney
Amy has taught high school mathematics for over 14 years. She has a master's degree in education from Plymouth State University and her undergraduate degree in mathematics. She is certified to teach grades 7-12 mathematics.
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Example SolutionsPractice Questions
Suppose we have a quadratic equation that has a negative coefficient in front of it, and we'd like to factor. This is no problem - we can actually factor the negative out to the front of the equation, and then work on factoring as normal. Let's go through a few different examples here.
Example Problem 1: Factoring a Negative Quadratic
Let's say we're asked to factor the equation y=−x 2−2 x+3.
We can see that if the negative weren't there, this would be a quadratic with a leading coefficient of 1 and we might attempt to factor by the sum-product method. Luckily, we can re-write this equation by factoring the negative sign out to the front. Doing this flips the sign of each term, and we get:
y=−(x 2+2 x−3).
Notice that we haven't changed the value of the equation - if we multiply the negative back in, we get back our original equation. Now we're able to focus on simply factoring the equation that is inside the brackets, and the negative sign comes along for the ride.
Examining the equation x 2+2 x−3 on its own, we could factor using the sum-product method by finding two numbers that multiply to −3 and add to 2. Working through the different possible factors of −3, we should find that −1 and 3 work. So we can factor x 2+2 x−3 into (x−1)(x+3). This means we can re-write our original equation as follows:
y=−x 2−2 x+3
=−(x 2+2 x−3)
=−((x−1)(x+3))
So we've done it. And, if we wish, we can remove the extra set of brackets. So by factoring the negative sign out front, we get our quadratic in its factored form of y=−(x−1)(x+3).
Example Problem 2: Factoring a Negative Quadratic
Let's do another one. Suppose we're given y=−x 2+9.
Noticing the negative leading coefficient, let's factor it out right away and focus on the resulting equation:
y=−(x 2−9).
Inside the brackets appears to be a difference of squares. So we can focus on factoring that difference of squares individually, leaving the negative sign out front:
y=−(x 2−9)
y=−(x 2−3 2)
=−((x−3)(x+3))
=−(x−3)(x+3).
And there we have our factored quadratic: y=−(x−3)(x+3).
Example Problem 3: Factoring a Negative Quadratic
We'll try one more. Suppose we'd like to factor the equation y=−6 x 2+3 x+3. This quadratic is a factor-by-grouping case, and can actually be solved by leaving the negative coefficient as it is. Alternately, we can factor the negative out front if we prefer, and we'll get the same answer. Let's try solving by factoring it out front. This gives us
y=−(6 x 2−3 x−3).
Now we're trying to factor the equation inside of the brackets by grouping. To do this, we need to find two numbers that multiply to 6⋅(−3), or −18, and add to −3. Upon checking the factors of −18, we find that the combination of −6 and 3 satisfy these requirements. So we factor by grouping inside of the brackets, keeping the negative out front:
y=−(6 x 2−3 x−3)
−(6 x 2−6 x+3 x−3)
−(6 x(x−1)+3(x−1))
−((x−1)(6 x+3))
−(x−1)(6 x+3).
Thus, our factored equation is y=−(x−1)(6 x+3).
Remember that not every quadratic equation can be factored, regardless of whether it has a negative leading coefficient or not. If the above methods don't work, we can always use the quadratic formula to find factors of a quadratic (provided they exist).
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Table of Contents
Example Problem 1
Example Problem 2
Example Problem 3
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Factoring a Quadratic with a Negative Leading Coefficient
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190449 | https://math.stackexchange.com/questions/1541739/counting-the-number-of-4-letter-strings-containing-a-and-b-and-having-no-repeate | combinatorics - Counting the number of 4-letter strings containing A and B and having no repeated letters - Mathematics Stack Exchange
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Counting the number of 4-letter strings containing A and B and having no repeated letters
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Asked 9 years, 10 months ago
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\begingroup
The letters ABCDEFGH are to be used to form strings of length 4. How many strings contain the letters A and B if repetitions are not allowed?
Here is my answer:
The total number of possible combinations are 8×7×6×5
The total number of combinations if A and B are not allowed is 6×5×4×3
So the answer is the difference of the two, which is (8×7×6×5) - (6×5×4×3) = 1320
Please tell me if I am correct and if not, point me in the right direction.
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edited Jan 2, 2017 at 0:01
Alex Riley
601 8 8 silver badges 21 21 bronze badges
asked Nov 22, 2015 at 21:10
satjavsatjav
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\endgroup
1
1 \begingroup You counted the strings that have A or B or both.\endgroup André Nicolas –André Nicolas 2015-11-22 21:14:38 +00:00 Commented Nov 22, 2015 at 21:14
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If the four-letter string must contain A and B, then there are \left(6 \atop 2 \right) = 15 pairs of letters you could choose from the remaining six letters C, D, E, F, G and H.
You now have 15 possible sets of four letters. Each set of four letters can be arranged in 4! = 24 different ways, giving you 15 \cdot 24 = 360 possible strings.
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edited May 15, 2017 at 10:27
answered Nov 22, 2015 at 21:21
Alex RileyAlex Riley
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If you choose the positions of A and B in that 4-letter string, then you get 6.5 possible strings. But then you can choose the positions of A and B in how many ways: 4.3, right? So in total you get 4.3.6.5 such strings. So my answer is 360.
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edited Nov 22, 2015 at 21:20
answered Nov 22, 2015 at 21:14
peter.petrovpeter.petrov
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\begingroup Um, when A and B are taken, aren't there only 6 more letters to choose from? So it should be 4.3.6.5 ?\endgroup satjav –satjav 2015-11-22 21:20:07 +00:00 Commented Nov 22, 2015 at 21:20
\begingroup@satjav Right, thanks.\endgroup peter.petrov –peter.petrov 2015-11-22 21:21:16 +00:00 Commented Nov 22, 2015 at 21:21
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190450 | https://library.achievingthedream.org/sanjaccollegealgebra/chapter/use-like-bases-to-solve-exponential-equations/ | Skip to content
Want to create or adapt books like this? Learn more about how Pressbooks supports open publishing practices.
169 Use like bases to solve exponential equations
The first technique involves two functions with like bases. Recall that the one-to-one property of exponential functions tells us that, for any real numbers b, S, and T, where , if and only if S= T.
In other words, when an exponential equation has the same base on each side, the exponents must be equal. This also applies when the exponents are algebraic expressions. Therefore, we can solve many exponential equations by using the rules of exponents to rewrite each side as a power with the same base. Then, we use the fact that exponential functions are one-to-one to set the exponents equal to one another, and solve for the unknown.
For example, consider the equation . To solve for x, we use the division property of exponents to rewrite the right side so that both sides have the common base, 3. Then we apply the one-to-one property of exponents by setting the exponents equal to one another and solving for x:
A General Note: Using the One-to-One Property of Exponential Functions to Solve Exponential Equations
For any algebraic expressions S and T, and any positive real number ,
How To: Given an exponential equation with the form , where S and T are algebraic expressions with an unknown, solve for the unknown.
Use the rules of exponents to simplify, if necessary, so that the resulting equation has the form .
Use the one-to-one property to set the exponents equal.
Solve the resulting equation, S= T, for the unknown.
Example 1: Solving an Exponential Equation with a Common Base
Solve .
Solution
Try It 1
Solve .
Rewriting Equations So All Powers Have the Same Base
Sometimes the common base for an exponential equation is not explicitly shown. In these cases, we simply rewrite the terms in the equation as powers with a common base, and solve using the one-to-one property.
For example, consider the equation . We can rewrite both sides of this equation as a power of 2. Then we apply the rules of exponents, along with the one-to-one property, to solve for x:
How To: Given an exponential equation with unlike bases, use the one-to-one property to solve it.
Rewrite each side in the equation as a power with a common base.
Use the rules of exponents to simplify, if necessary, so that the resulting equation has the form .
Use the one-to-one property to set the exponents equal.
Solve the resulting equation, S= T, for the unknown.
Example 2: Solving Equations by Rewriting Them to Have a Common Base
Solve .
Solution
Try It 2
Solve .
Solution
Example 3: Solving Equations by Rewriting Roots with Fractional Exponents to Have a Common Base
Solve .
Solution
Try It 3
Solve .
Solution
Q & A
Do all exponential equations have a solution? If not, how can we tell if there is a solution during the problem-solving process?
No. Recall that the range of an exponential function is always positive. While solving the equation, we may obtain an expression that is undefined.
Example 4: Solving an Equation with Positive and Negative Powers
Solve .
Solution
This equation has no solution. There is no real value of x that will make the equation a true statement because any power of a positive number is positive.
Analysis of the Solution
The figure below shows that the two graphs do not cross so the left side is never equal to the right side. Thus the equation has no solution.
Try It 4
Solve .
Solution |
190451 | https://intro.chem.okstate.edu/1515SP02/Lecture/Chapter12/Water4.html | The Lewis structure for water;
Water is polar. It has two lone-pairs of electrons on the central atom.
Because the oxygen atom is more electronnegative compared to the hydrogen atom it has a greater attraction to the electrons in the O-H bond. Additionally the lone-pairs of electrons on oxygen contribute to locating the partial negative charge on the oxygen atom in the moelcule. Therefore the hydrogen atoms carry a partical positive charge.
So what happens when several water molecules are located in close proximity? Can you draw a picture using the Lewis structures depicting the orientation of three water molecules to each other based on the partial charge carried on the oxygen and the hydrogen atoms?
Answer.
The adjacent water molecules align themselves so that the partial charges on adjacen molecules can form an attraction. We refer to the attraction between the lone-pair of electrons (where there is partial negative charge) on the oxygen atom of a water molecule and the partial positive charge on the hydrogen atom as a hydrogen-bond.
It is important to associate the hydrogen-bond with the intermolecular attraction between two water molecules, not the O-H covalent bond within (intra) the water molecule. |
190452 | https://sigmodrecord.org/publications/sigmodRecord/1612/pdfs/05_vision_Kamat.pdf | A Closer Look at Variance Implementations In Modern Database Systems Niranjan Kamat Arnab Nandi Computer Science & Engineering The Ohio State University {kamatn,arnab}@cse.osu.edu ABSTRACT Variance is a popular and often necessary component of aggregation queries. It is typically used as a secondary measure to ascertain statistical properties of the result such as its error. Yet, it is more expensive to compute than primary measures such as SUM, MEAN, and COUNT.
There exist numerous techniques to compute variance.
While the definition of variance implies two passes over the data, other mathematical formulations lead to a single-pass computation. Some single-pass formulations, how-ever, can suffer from severe precision loss, especially for large datasets.
In this paper, we study variance implementations in various real-world systems and find that major database systems such as PostgreSQL and most likely System X, a major commercial closed-source database, use a repre-sentation that is efficient, but suffers from floating point precision loss resulting from catastrophic cancellation.
We review literature over the past five decades on vari-ance calculation in both the statistics and database com-munities, and summarize recommendations on imple-menting variance functions in various settings, such as approximate query processing and large-scale distributed aggregation. Interestingly, we recommend using the math-ematical formula for computing variance if two passes over the data are acceptable due to its precision, paral-lelizability, and surprisingly computation speed.
1.
INTRODUCTION New large-scale distributed data management and analytics systems are being developed at a rapid pace, with the scalability aspect of computation be-ing their predominant development focus (except-ing ). Comparatively lesser e↵orts have been ex-pended on ensuring numerical correctness and sta-bility of algorithms. While such an approach can result in the queries being answered more quickly, it can also cause the computation to have a higher level of numerical imprecision.
The concern of achieving numerical stability and precision is pertinent in numerous computational Copied from above Mean 1 2 3 4 5 6 Postgres 10.48359 100.4836 1000.484 10000.48 100000.5 1000000 10 SQL Server 10.52669 100.5267 1000.527 10000.53 100000.5 1000001 10 Oracle 10.534 100.534 1000.534 10000.53 100000.5 1000001 10 StdDev1.96/sqrt(n) Postgres 0.055101 0.055101 0.055101 0.055101 0.055098 0.054875 0 SQL Server 0.054414 0.054414 0.054414 0.054414 0.054409 0.054779 0 System Y 0.060335 0.060335 0.060335 0.060335 0.060335 0.060335 0 Log Mean 1 2 3 4 5 6 PostgreSQL 9.3 1.02051019 2.00209516 3.00020997 4.000021 5.0000021 6.00000021 7.
System X 1.02229176 2.00228138 3.00022868 4.00002287 5.00000229 6.00000023 7.
System Y 1.02259317 2.00231295 3.00023185 4.00002319 5.00000232 6.00000023 7.
StdDev1.96/sqrt(n) Postgres -1.2588378 -1.2588378 -1.2588378 -1.2588378 -1.2588621 -1.260625 SQL Server -1.264292 -1.264292 -1.264292 -1.2642919 -1.2643257 -1.261388 -1 System Y -1.2194275 -1.2194275 -1.2194275 -1.2194275 -1.2194275 -1.2194275 -1 The previous figure had gaussian distribution this has uniform Random 1 2 3 4 5 6 Common 14.9 15.02 14.76 13.79 12.98 12.07 Textbook1pass 11.46 9.57 7.58 5.39 3.49 1.69 Updating 14.88 14.31 13.46 12.23 11.37 10.49 Updating pair 14.88 14.98 14.42 13.1 12.41 11.43 Two pass 15.05 14.94 14.85 14.81 14.99 14.8 Shifted 2 pass 12.72 11.97 10.95 9.91 9.02 7.97 -5 0 5 10 15 20 25 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Log(Mean, Confidence Interval) Shift Exponent PostgreSQL 9.3 System X System Y 0 2 4 6 8 10 12 14 16 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Correct Decimal Places Shift Exponent Total Variance Textbook1Pass Updating Updating pairwise Two Pass Shifted One Pass Figure 1: E↵ect of Variance Error on T-Test Con-fidence Intervals: As the magnitude of values in-creases (x-axis, true margin of error is kept con-sistent for each dataset), mean is expected to in-crease, and size of error bars is expected to stay the same. However, PostgreSQL and System X error bars (↵“ 0.05) vary widely, while System Y has correct error bars (details in Section 1.1).
scenarios; it is especially important in variance cal-culation, which has an ubiquitous presence in large-scale analytics and is known to su↵er from precision issues . Variance is an important aggregate func-tion and an essential tool in sampling-based aggre-gation queries. Typically used as a secondary mea-sure, it augments measures such as AVERAGE and provides an insight into data distribution beyond the primary measure.
Computation of variance, however, is susceptible to precision loss when the variance is much smaller than the mean .
There exist several techniques to compute vari-ance.
The standard formula uses two passes and provides an accurate estimate (Two Pass). Due to its perception of being more expensive, other tech-niques have been developed that require a single pass over the data. One such formula, while fast, is known to su↵er from precision loss (Textbook One Pass) due to catastrophic cancellation , an un-desirable e↵ect of a floating point operation that causes relative error to far exceed absolute error.
Figure 1 demonstrates this problem. As a side note, this problem a↵ects calculators as well .
28 SIGMOD Record, December 2016 (Vol. 45, No. 4) Another formula (Updating), as recommended by Knuth , has found a strong foothold in the database community, with numerous implementations citing him. However, this formula is constrained by the fact that it can only incorporate a single value into the current running estimates. It is unable to com-bine the estimates from di↵erent subsets of data.
Given the rise of large-scale data processing, mas-sive multi-core support and availability of GPUs, it is prudent to consider representations such as Pair-wise Updating, that can combine partial results at a larger scale instead of incrementally incorporat-ing a single data point. Further, Pairwise Updating is also known to provide better precision for both single ( ) and double precision input (Section 4).
Contributions & Outline: ‚ We catalog usage of di↵erent variance formulas in various open source database systems (Table 2).
‚ We experiment with di↵erent closed source and open source databases to investigate precision loss issues. We find that precision of PostgreSQL and System X deteriorates the most. After looking at the PostgreSQL source code, we can verify that it uses Textbook One Pass, and hypothesize that Sys-tem X does so as well.
‚ We empirically study the accuracy of the dif-ferent representations under varying additive shifts and dataset sizes including a hitherto unstudied one, which we call Total Variance.
‚ We recommend using Two Pass if performing two passes over the data is acceptable (Section 5), which seems counter-intuitive, but works due to its computational simplicity.
In the next subsection, we look at the adverse e↵ects of imprecise variance calculation. Section 2 presents di↵erent variance representations and their properties. We then note the representations used by modern databases in Section 3. Section 4 lists our analysis of the behavior of the di↵erent formu-las (using double precision input compared with sin-gle precision in Chan et al. ). We conclude with our recommendations for choosing an appropriate variance representation in current environments.
1.1 Impact of Variance Calculations Due to the pervasive use of variance, a loss of pre-cision can have an impact in a variety of di↵erent domains. In the following paragraphs, we look at some use cases where the lack of precision in vari-ance calculation can have adverse consequences.
Incorrect Output: It is possible to experimen-tally observe the loss of precision as incorrect out-put. To illustrate the pitfalls in using Textbook One Pass, 100 values were generated from a Uniformp0, 1q distribution and shifted by 10Shift Exponent for Shift Exponent varying from 1 to 15.
The variance as a result of data being shifted should be similar to the one without any shift. We verify this by adding and subtracting the shift exponent and note that the variance of the resultant dataset was close to the true sample variance. However, Figure 1, which depicts the sample mean and confidence interval, shows that PostgreSQL and System X su↵er from variance calculations being susceptible to precision loss due to the shift. We know that PostgreSQL uses Textbook One Pass and the pattern of the erro-neous calculations displayed by both hints towards System X using it as well. In contrast, other database systems su↵ered minor precision loss as expected (these results are not shown since they do not add any additional information to the figure). It should be noted that System Y was found to be highly im-mune to precision loss.
Visualization: Erroneous variance calculation can have a notable impact on visualizations as shown by Figure 1. While error bars should be similar, they instead vary widely and inaccurately for higher shift values for PostgreSQL and System X. We also found Datavore, which powers the Profiler visualization system , to use Two Pass.
Negative Variance: It is possible for variance to be negative while using Textbook One Pass – a theoretically impossible result.
We observed in the PostgreSQL source code that variance is set to zero, if negative. Figure 1 shows numerous values of 0 (missing error bars) for PostgreSQL (shift ex-ponent 8, 9, and 12) and System X (shift exponents 10 and 11), providing evidence of System X employ-ing a similar strategy for handling negative variance values and using Textbook One Pass.
Decision Support Systems: As a building block in popular algorithms, flaws in variance implemen-tations can have far-reaching impacts, e.g., in hy-pothesis testing, which is an integral part of deci-sion support systems. Having imprecise or incorrect variance estimates can greatly change the result of hypothesis testing.
Data Mining: Variance is an important tool in statistical analysis and machine learning algorithms such as Gaussian Naive Bayes, or Mixture of Gaus-sians based algorithms such as background model-ing, clustering, or topic modeling. For example, we found usage of Textbook One Pass within a graph-ics library of the R language. Similarly, MADlib was also found to have a call to the PostgreSQL variance function: thus, an erroneous calculation of variance can extend from the underlying databases to the systems built on top of them.
SIGMOD Record, December 2016 (Vol. 45, No. 4) 29 Name Formula Accuracy Passes Storage Parallel Two Pass S “ ∞N i“1pxi ´ ¯ xq2, ¯ x “ ∞N x“1 xi N 3 2 O(1) 3 Textbook 1 Pass S “ ∞N i“1 x2 i -1 N p∞N i“1 xiq2 7 1 O(1) 3 Shifted 1 Pass S “ ∞N i“1pxi ´ ¯ xq2 ´ p∞N i“1pxi ´ ¯ xqq2{N Varies 1 O(1) 3 Pairwise T1,mn “ T1,m Tm1,mn, S1,mn “ S1,m 3 1 O(ln(N)) 3 UpdatingSm1,mn m npmnqp n mT1,m ´ Tm1,mnq2 Updating-YC T1,j “ T1,j´1 xj 3 1 O(1) 7 S1,j “ S1,j´1 1 jpj´1qpjxj ´ T1,jq2 Updating-WWH M1,j “ M1,j´1 xj´M1,j´1 j , S1,j “ S1,j´1 3 1 O(1) 7 (Updating)pj ´ 1q ˆ pxj ´ M1,j´1q ˆ p xj´M1,j´1 j q Total Variance S “ ∞groups i“1 nipmi ´ ¯ xq2 pni ´ 1qvi ˘ 3 3 Varies Varies Table 1: Commonly used Formulas for Variance.
2.
VARIOUS VARIANCE FORMULATIONS Table 1 presents the common variance represen-tations . We use a similar naming convention to that used by Chan et al. . S represents the sum of squares. The sample variance can be given by S N´1, where N is the sample size. xi is the ith data point.
¯ x is the sample mean.
Mm,n is the mean of the data points from indexes m to n (both inclusive).
Tm,n is the total of the data points from indexes m to n (both inclusive). We have also provided To-tal Variance (derivation in the technical report ).
In its formula, ni, mi, and vi represent the count, mean, and variance respectively, of the ith group.
Textbook One Pass can be computationally dan-gerous as the quantities ∞N i“1 x2 i and 1 N p∞N i“1 xiq2 can nearly cancel each other out. The Pairwise Up-dating formula hierarchically combines pairs of vari-ance values and uses OplogpNqq storage while reduc-ing the relative errors from OpNq to OplogpNqq .
Updating-YC represents Youngs and Cramer for-mula and is essentially identical to Updating Pairwise when m “ 1 or n “ 1.
The Updating-WWH formula refers to the nearly identical for-mulas used by Welford et al. , West et al. , and Hanson et al. and has similar precision as Updating-YC. We have used the Updating-WWH representation for updates using a single data point, and denote it by Updating. Shifting the data by an exact or approximate value of ¯ x (Shifted One Pass) can also result in substantial accuracy gains .
2.1 Total Variance Since this is the first paper to introduce the To-tal Variance representation, we explain its steps in more details below. In the first pass, which is over the tuples, the variance (using one of the other for-mulas), mean, and count, of individual groups are computed. The second pass, over the groups thus formed, finds the overall mean of the data. In the third pass, over the groups, the overall variance is then found. Since the second and third passes are over the groups obtained as a result of the first pass, and di↵erent formulas can be used to compute vari-ance of individual groups in the first pass, complex-ity of the overall algorithm can vary widely. While second and third passes are highly parallelizable, its overall parallelizability is dependent upon the for-mula used to find variance of the groups. It is de-signed for combining variances of di↵erent groups and is agnostic to the representation used in the first pass – our implementation uses Updating.
Computing mean of individual groups is a well-researched subject with Tian et al. providing a good overview. We use a single pass algorithm to compute mean of individual groups and to combine means of groups as well. To handle a large num-ber of groups, one can look into using an aggrega-tion tree to combine means. The usual technique of mean estimation can be used in case the number of groups is large, at the cost of decreased precision.
There does not appear to be a theoretically ideal group size for Total Variance, and we could not determine one experimentally either .
In dis-tributed execution, one natural way is to consider data across di↵erent nodes as groups. Further, data within a node can be equally partitioned, so that each core works on a single subgroup.
2.2 Properties of Different Representations While Chan et al. provide an overview of the accuracy, passes, and storage required for most of the formulas given in Table 1 (other than Total Variance), their classification as being distributive, and thus the ability to be parallelized, has not been explicitly listed before, which we do. In Table 1, the Storage column depicts the extra space needed for computing variance, which is above and beyond that needed to store the data itself.
30 SIGMOD Record, December 2016 (Vol. 45, No. 4) The accuracy of Shifted One Pass depends on that of the mean estimate. Pairwise Updating is the only representation providing accurate results while being highly parallelizable and requiring a single pass. Additionally, as we will see in Section 4, the precision of Total Variance is slightly better than that of Updating Pairwise, which has the best pre-cision amongst all single pass algorithms. As a side note, Two Pass, Total Variance and Textbook One Pass are the only representations that can be repre-sented using a standard SQL query. Note that Ta-ble 2.1 of succinctly enumerates the error bounds of di↵erent formulations. Further, Kahan summa-tion [5,10] can help improve their precision.
2.3 Data Conditioning Data shifting and scaling are immensely useful in improving accuracy of algorithms .
For exam-ple, shifting the data by its mean is the basis for Shifted One Pass. Indeed, Chan et al. demon-strate the usefulness of shifting by an approximate mean computed using a sample of the data by prov-ing that it reduces the bounds of the condition num-ber.
Further, techniques such as dividing by the mean or using the log function can be helpful in improving the accuracy.
However, along with requiring additional computational resources these techniques can also worsen the accuracy under mali-cious datasets , and need careful user supervision.
2.4 Hybrid Formulae It is clear that di↵erent implementations can be used to find variance of di↵erent groups, and com-bine partial results.
Indeed, it has been brought to our attention that a commercial system uses the Updating-YC formula to compute variance at in-dividual nodes, and combines them using Pairwise Updating formula. Total Variance is a hybrid for-mula as well. This provokes an interesting piece of future work – choosing di↵erent representations at di↵erent computation steps, based on factors such as numerical precision, data partitioning, time for first result, number of passes permissible (Section 5).
This idea is elaborated upon in Section 5.
2.5 Current Recommendation Guidelines Chan et al. provide detailed recommendation guidelines for di↵erent variance formulas. They rec-ommend usage of Pairwise Updating for combining variances across multiple processors since it reduces the errors and is massively parallelizable if extra OplogpNqq space is available. Further, it is also the safest (least precision loss) algorithm to use within each processor, under the constraint of a single pass.
2.6 Extensibility to Other Measures Standard deviation, standard error, and coeffi-cient of variation are important statistical measures, and perform variance computation. Thereby, they are also a↵ected by the properties of the underlying representation.
Similarly, the properties will also extend to any user-defined measure whose variance can be expressed in a closed form as a function of the variance of one of the measure dimensions. For example, for a user-defined measure given by a ˚ AVGpaggq ` b, where a and b are constants and agg is a measure dimension, the variance of the measure can be given in closed form as a2˚VARIANCEpaggq.
3.
VARIANCE IMPLEMENTATIONS IN MODERN DATABASE SYSTEMS We looked at the code of multiple open source databases to find their variance representations. We also conjecture about two closed source ones through our experiments.
Database Formula PostgreSQL 9.4.4 Textbook One Pass MySQL 5.7 Updating Impala 2.1.5 Updating Pairwise Hive 1.2.1 Updating Pairwise Spark 1.4.1 Updating Pairwise SQLite No Variance Support System X Textbook One-pass (Conjecture) System Y Cannot Conjecture Table 2: Variance Implementations in Databases.
PostgreSQL uses Textbook One Pass and is thus susceptible to precision loss. MySQL uses Knuth’s modification of Welford’s updating formula. There-fore, it can only process a single additional data point, and cannot avail of the possible paralleliza-tion.
Spark 1.4.1 and Impala 2.1.5, on the other hand, use a modified version of Updating Pairwise.
Although the source code for System X is not available, we conjecture that it uses Textbook One Pass as its precision behavior was similar to that of PostgreSQL. System Y was found to have the best precision. We hypothesize that it uses higher precision variables, but cannot make any conjecture about the exact representation.
4.
EXPERIMENTAL ANALYSIS Chan et al. have looked at the precision of dif-ferent algorithms using single precision input. We present the precision results using double precision input. We also evaluate the precision of Total Vari-ance. We look at the precision in the variance cal-SIGMOD Record, December 2016 (Vol. 45, No. 4) 31 Gaussian 1 2 3 4 5 6 7 8 Common 13.94 13.94 13.85 13.21 12.25 11.47 10.29 9.45 Textbook1pass 11.54 9.64 7.63 5.61 3.63 1.42 0.07 0.16 Updating 13.82 13.65 12.92 11.81 11.01 9.97 8.76 7.95 Updating pair 13.95 13.97 13.65 12.71 11.79 10.98 9.84 8.81 Two pass 14.21 13.91 13.72 13.7 13.8 13.78 13.84 12.6 Shifted 2 pass 12.75 12.04 11 9.9 9.02 7.92 6.96 6 yes HALF CONFIDENCE INTERVAL EXPT Mean 0 1 2 3 4 5 6 7 Postgres 0.483594 10.48359 100.4836 1000.484 10000.48 100000.5 1000000 10000000 SQL Server 0.526688 10.52669 100.5267 1000.527 10000.53 100000.5 1000001 10000001 StdDev1.96/sqrt(n) Postgres 0.055101 0.055101 0.055101 0.055101 0.055101 0.055098 0.054875 0.031357 SQL Server 0.054414 0.054414 0.054414 0.054414 0.054414 0.054409 0.054779 0.048006 0.001 0.01 0.1 1 10 100 1000 10000 DECIMAL PLACES POSTGRES SQL SERVER Shift 1 2 3 4 5 6 7 8 Postgres 14 11 8 8 4 3 0 0 SQL Server 12 10 9 7 5 2 1 0 Oracle 17 17 17 17 17 17 17 17 MySQL 5.6 17 15 14 12 12 10 9 9 impala 15 14 12 12 12 11 9 9 Average Decimal Places 1 2 3 4 5 6 7 8 Postgres 12.8 10.7 9 6.9 4.9 2.5 0.7 0 SQL Server 12.5 10.7 8 6.7 4.4 2.2 0.6 0 Oracle 17 17 17 17 17 17 17 17 MySQL 5.6 15.3 14.4 13.5 11.9 11.4 10.8 9 8.5 Impala 14.9 13.8 12.9 11.8 11.3 10.1 9.1 8 0 5 10 15 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Correct Decimal Places Shift Exponent 0 50000 100000 150000 200000 250000 300000 1 2 3 4 Margin Of Error 0 2 4 6 8 10 12 14 16 18 1 2 3 4 5 6 7 8 Correct Decimal Places Shift Exponent PostgreSQL9.3 SQL Server 2014 Oracle MySQL 5.6 Impala 2.1.0 (a) With increasing shift exponent, all representations experience precision loss, though some more severely than others.
This figure deals with changing data size Shift 10^5 1 2 3 4 5 6 7 8 Common 16.91 11.99 12.59 12.98 13.5 13.75 13.46 13.03 Textbook 1 pass 5.06 4.36 4.12 3.49 3.12 2.59 2.2 1.41 Updating 11.4 11.46 11.57 11.37 11.33 11.36 11.47 11.32 Updating pair 11.33 11.51 11.93 12.41 12.89 13.26 13.34 13.1 Two pass 16.69 16.09 14.44 14.99 14.4 13.84 13.33 12.9 Shifted 2 pass 10.68 9.89 9.43 9.02 8.35 7.96 7.56 6.9 yes 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Shift Exponent 0 5 10 15 20 1 2 3 4 5 6 7 8 Correct Decimal Places Data Size Exponent Two Pass Total Variance Updating pairwise Updating Shifted One Pass Textbook One Pass (b) Precision generally decreases with in-creasing dataset size.
0.01 0.1 1 10 100 1000 10000 4 5 6 7 Execution Time (ms) Data Size Exponent (c) Two Pass provides results faster than others, excepting Textbook One Pass, which has the least numerical precision.
Figure 2: Two Pass not only has the highest precision, but also requires second lowest execution time.
culation o↵ered by the di↵erent databases. We also present the execution times of di↵erent algorithms on data sizes up to 100 million tuples. The results are the average over 100 runs. Experiments were performed using Ubuntu 14.04.05 LTS with a 4 core, 2.4 GHz Intel CPU, with 16 GB RAM, and 256 GB SSD storage, using a single execution thread.
Dataset: Although numerous benchmarks exist to evaluate the accuracy of numerical algorithms, they are constrained by their dataset size. For example, the biggest dataset in the NIST StRD bench-mark consists of 5000 points. Furthermore, for this dataset, the mean is not significantly larger than the standard deviation (µ “ 4.5348, σ “ 2.8673).
Therefore, in a similar vein as Tian et al. , we created synthetic datasets of di↵erent sizes using Uniformp0, 1q (variance being 1 12). They were shifted by adding values ranging from 101 to 1015.
4.1 Impact of Shift Numerical precision was evaluated using varying additive shifts, over a dataset of size 10000. Group size was set at 10 for Total Variance. We present our findings in Figure 2a, where Y-axis represents the number of correct decimal digits (non-fractional part of the result was 0). The results were as ex-pected , with Two Pass having the best precision, and Textbook One Pass the worst.
4.2 Impact of Data Size Since precision errors typically accumulate, we tried datasets of sizes from 10 to 100 million. The shift was set at 105. Figure 2b shows that precision generally worsens with increasing data size. Two Pass again outperforms other algorithms. Textbook One Pass consistently exhibits the worst precision.
Counter-intuitively, the precision of Total Vari-ance and Updating Pairwise was found to increase for the data size exponents from 2 to 6. We are un-able to conjecture the reason behind this behavior.
The precision error for Updating Pairwise increases as Oplogpnqq, while that for others (except Total Variance) increases as at least Opnq , where n is the data size.
Therefore, while we can expect the error in Updating Pairwise to not increase at the same rate as others, the error decrease is unex-pected. In the absence of theoretical error bounds for Total Variance, we cannot hypothesize about the possible cause. To ensure there were no irregu-larities, the experiment was repeated multiple times with similar results.
4.3 Single-Threaded Execution Speed We also looked at the execution time of di↵erent algorithms with increasing data size (Figure 2c).
Results with lower data sizes have not been pre-sented due to the computation taking minimal time.
Surprisingly, there was no discernible di↵erence in execution time between Two Pass and Shifted One Pass.
Only Textbook One Pass took lesser time than Two Pass.
We attribute the low execution time of Two Pass to simplicity of its computation.
4.4 Impact of Shift on Different Databases 0 5 10 15 20 1 2 3 4 5 6 7 8 Correct Decimal Places Shift Exponent PostgreSQL 9.3 System X System Y MySQL 5.6 Impala 2.1.0 Figure 3: Databases follow ex-pected precision patterns.
We look at variance precision for the di↵erent databases under varying additive shifts.
We took e↵orts to ensure di↵erent systems have similar data types. 100 points were chosen from a Uniformp0, 1q distribution.
Figure 3 shows that precision loss follows a simi-lar pattern in System X and PostgreSQL. Impala and MySQL have a similar error profile as well.
4.5 Miscellaneous Experiments In one of the other experiments, details in , we noted that changing group size in Total Variance did not have a significant e↵ect on the precision. In another experiment, multi-threaded execution gave 32 SIGMOD Record, December 2016 (Vol. 45, No. 4) us expected speedups for the parallelizable formula-tions. Finally, we inspected the mantissa of the two terms that compose Textbook One Pass to demon-strate the cause of catastrophic cancellation.
5.
CONCLUSION & RECOMMENDATIONS Precision issues associated with Textbook One Pass have been well documented. However, we have seen that databases such as PostgeSQL and likely Sys-tem X still use it. We recommend from the per-spective of safety to discontinue its usage. Though there might be arguments for its continued usage after warning the users in certain scenarios, the ar-guments against it far outweigh the speedup bene-fit and its ease of implementation. Although error inherently exists in approximate query processing, numerical precision errors are easy to eliminate and hard to apportion and therefore should be avoided whenever possible.
Hence, we recommend to the designers of databases, and statistics and analytics packages, to discontinue its usage. Further, it would be wise for users to perform a sanity check using ex-periments similar to those given in Section 4.1.
Previous work has recommended Pairwise Updat-ing from the perspective of precision, speed, and parallelizability .
However, we have seen from our experiments of up to 100 million data points, that the most accurate algorithm, Two Pass, takes lesser time than Updating, Updating Pairwise, and Total Variance. Further, it takes around the same amount of time as Shifted One Pass, which relies on mean estimation. Two Pass is also easy to im-plement and parallelize. Therefore, in the case that performing two passes over the data is ac-ceptable, Two Pass should be the preferred algorithm.
Determining whether two passes are acceptable, however, is a nuanced decision. When the data fits in memory, performing two passes over the data is clearly acceptable as all representations will incur the identical data read I/O cost. When the data cannot fit in memory, summing up the es-timated I/O and computation times can help deter-mine whether Two Pass will need the least amount of time, in which case it should be chosen.
In other cases, i.e., whenever Two Pass is not estimated to require the least execution time, there does not exist a clear winner, due to di↵erent algorithms having di↵erent strengths and weaknesses.
Updating provides faster results at lower precision, compared with Updating Pair-wise, without needing additional memory. Updating Pairwise is parallelizable, whereas Updating is not.
While Shifted One Pass provides quick results, its accuracy is dependent on correctness of the mean estimate.
Total Variance has good accuracy, al-though it takes longer to execute, and is dependent on the algorithm used to compute group statistics, while also needing multiple passes.
Hence, there does not exist any algorithm that dominates every other algorithm, resulting in there not being a clear choice. We can see that a query planner that devises hybrid formulas, while taking the data distribution, estimated I/O and computation costs, and the over-all strengths and weaknesses of di↵erent algorithms into consideration, appears to be an important and ideal piece of future work.
6.
ACKNOWLEDGMENT We acknowledge the generous support of U.S. NSF under awards IIS-1422977 and CAREER IIS-1453582.
We would also like to thank the reviewer for their highly insightful and helpful comments.
7.
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R. J. Hanson. Stably Updating Mean and Standard Deviation of Data. ACM, 1975.
J. M. Hellerstein, C. R´ e, F. Schoppmann, et al. The MADlib Analytics Library: or MAD Skills, the SQL. VLDB, 2012.
N. J. Higham. Accuracy and Stability of Numerical Algorithms. SIAM, 2002.
W. Kahan. Further Remarks on Reducing Truncation Errors. ACM, 8(1):40, 1965.
N. Kamat and A. Nandi. A Closer Look at Variance Implementations In Modern Database Systems. Arxiv TR, 2016.
S. Kandel et al. Profiler: Integrated Statistical Analysis and Visualization for Data Quality Assessment. AVI, 2012.
D. E. Knuth. Art of Computer Programming, Volume 2: Seminumerical Algorithms. 2014.
J. Rogers, J. Filliben, et al. StRD: Statistical Reference Datasets for Testing the Numerical Accuracy of Statistical Software, 1998.
Y. Tian, S. Tatikonda, and B. Reinwald.
Scalable and Numerically Stable Descriptive Statistics in SystemML. ICDE, 2012.
B. Welford. Note on a Method for Calculating Corrected Sums of Squares and Products.
Technometrics, 1962.
D. West. Updating Mean and Variance Estimates: An Improved Method. 1979.
E. A. Youngs et al. Some Results Relevant to Choice of Sum and Sum-of-Product Algorithms. Technometrics, 1971.
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190453 | https://www.paecon.net/PAEReview/issue105/Armstrong105.pdf | real-world economics review, issue no. 105 subscribe for free 57 History and origin of money in MMT and Austrian Economics: The difference methodology makes? Phil Armstrong1 [Gower Initiative for Modern Money Studies] Copyright: Phil Armstrong, 2023 You may post comments on this paper at 1. Introduction 7KLV DUWLFOH ZDV SURPSWHG E\ 3HU B\OXQG¶V UHFHQW FULWLTXH RI 5DQGDOO :UD\¶V DUWLFOH, ³7D[HV DUH IRU 5HGHPSWLRQ QRW 6SHQGLQJ´ (VHH B\OXQG 2022; :UD\ 2016). IW LV, KRZHYHU, RI JHQHUDO LQWHUHVW LQVRIDU DV it provides an opportunity to address some of the typical misunderstandings of Modern Monetary Theory (007) DQG WR GR VR EDVHG RQ D FRQWUDVW ZLWK DQ AXVWULDQ 6FKRRO DSSURDFK. 7KH PDMRULW\ RI ³LQIRUPHG FULWLTXH´ KDV WHQGHG WR RULJLQDWH IURP SRVW-Keynesians and, to a lesser extent, Marxism (for discussion see AUPVWURQJ 2023). B\OXQG¶V FULWLTXH HQFRPSDVVHV WKH VWDWH WKHRU\ RI PRQH\ ZKLFK XQGHUSLQV 007¶V DSSURDFK WR WKH RULJLQ RI PRQH\ (LWV ³PRQH\ VWRU\´ URRWHG LQ CKDUWDOLVP). IPSRUWDQWO\, PXFK RI WKH GLIIHUHQFH EHWZHHQ B\OXQG¶V AXVWULDQ DSSURDFK DQG 007 EHJLQV ZLth a contrasting methodology. Austrian economics is deductivist and focuses on the implications of the agency of the individual. Focusing on deductivism places less emphasis on history and more on building an axiomatic case. In combination with a focus on individual agency in market exchange it speaks to an origin of money in barter (so money is a spontaneous solution to the problem of barter and arises as a medium of exchange in market contexts). MMT follows the state theory of money and Chartalism and begins from what history, anthropology and archaeology tell us about the origin of money. As such, its focus is the emergence of debt, of a unit of account and of the role of the state in creating the conditions in which social relations of money can emerge, not least the role of state issuance of money as a means to appropriate resources, which in turn encourages market activity in order to acquire money tokens to pay taxes (so money presupposes the development of a unit of account, takes the form of a credit-debt, and becomes a general means of payment within market exchange in response to the activity of its originators). Arguably this latter approach makes MMT a form of retroduction within an open systems ontology. This is quite a different starting point to that presupposed by Bylund. I begin in section 2 with a brief summary of the methodological commitments of the Austrian school contrasted with my reading of MMT (which to be clear has not been explicitly acknowledged or discussed by all proponents of MMT). In section 3 I turn to the Austrian approach to the origin of money and in section 4 to that of state theory and MMT. Against the backdrop of the cumulative argument I WXUQ LQ VHFWLRQ 5 WR B\OXQG¶V VSHFLILF FDVH DQG LQ VHFWLRQ 6 I FRQFOXGH ZLWK D EULHI UHSULVH RI NH\ SRLQWV. 1 Phil Armstrong has been an economics teacher for more than forty years. He is an Associate at the Gower Initiative for Modern Money Studies. PArmstrong@yorkcollege.ac.uk real-world economics review, issue no. 105 subscribe for free 58 2. The methodology of the Austrian School contrasted with that employed by Modern Monetary Theorists Over the course of several years I interviewed many of the best known and influential heterodox economists, published as Can Heterodox Economics make a Difference: Conversations with Key Thinkers (Armstrong 2020a).2 Conducting these interviews confirmed that methodological perspective KDV D SURIRXQG LPSDFW RQ DQ HFRQRPLVW¶V ZRUN.3 According to Murray Rothbard at the Mises Institute, SUD[HRORJ\ LV WKH ³GLVWLQFWLYH PHWKRGRORJ\ RI AXVWULDQ HFRQRPLFV´ DQG LW: rests on the fundamental axiom that individual human beings act, that is, on the primordial fact that individuals engage in conscious actions toward chosen goals. The praxeological method spins out by verbal deduction the logical implications of that primordial fact. In short, praxeological economics is the structure of logical implications of the fact that individuals act. This structure is built on the fundamental axiom of action (Rothbard 2019 [and 1976/2011] emphasis in the original). Not only is an Austrian approach deductivist: since praxeology begins with a true axiom, A, all the propositions that can be deduced from this axiom must also be true. For if A implies B, and A is true, then B must also EH WUXH« >FXUWKHUPRUH@ DOO DFWLRQ LQ WKH UHDO ZRUOG, IXUWKHUPRUH, PXVW WDNH SODFH through time; all action takes place in some present and is directed toward the future (immediate or remote) attainment of an end. (Rothbard 2019 [and 1976/2011]). As Rothbard makes clear deduction has important implications for the role of history in economic theory: We arrived at [the implications of the axiom of action] by deducing the logical implications of the existing fact of human action, and hence deduced true conclusions IURP D WUXH D[LRP. ASDUW IURP WKH IDFW WKDW WKHVH FRQFOXVLRQV FDQQRW EH µWHVWHG¶ E\ historical or statistical means, there is no need to test them since their truth has already been established. Historical fact enters into these conclusions only by determining ZKLFK EUDQFK RI WKH WKHRU\ LV DSSOLFDEOH LQ DQ\ SDUWLFXODU FDVH« 0LVHV LQGHHG KHOG QRt RQO\ WKDW HFRQRPLF WKHRU\ GRHV QRW QHHG WR EH µWHVWHG¶ E\ KLVWRULFDO IDFW EXW DOVR WKDW it cannot EH VR WHVWHG« >6R HFRQRPLF WKHRU\ LV@ QRW D VWDWHPHQW RI ZKDW XVXDOO\ happens, but of what necessarily must happen. (Rothbard 2019 [and 1976/2011] emphasis added). Readers are no doubt aware that not all Austrian economists agree on first principles and there is a notable strand who are critical of mainstream economics understanding of equilibrium, use of mathematics, and pursuit of regularity which presupposes closed systems. Catallexy can, for example, be construed as an open systems concept and the coordination function of markets as a continual 2 See also Armstrong (2018, 2020b). 3 7KLV UHDVRQLQJ LV LQ OLQH ZLWK 6PLWKLQ¶V (2010) DSSURDFK. HH KLJKOLJKWV WKH LPSRUWDQFH RI WKH GHHSO\ KHOG SROLWLFDO views of economists to their mode of theorising and that the idea of taking an ethical stance based upon individualism (which characterises the Austrian School) as a starting point for analysis has great appeal, especially for those who consider social classes as an illegitimate starting point for analysis, having no independent existence apart from their constituent parts. real-world economics review, issue no. 105 subscribe for free 59 evolution. However, there is an obvious tension between praxeology and the role history plays in informing economic theory.4 MMT, in contrast, begins from observation of how a money system works (in order to make claims regarding how it could work if properly understood ± an obvious point of contention) and with due attention to history. Though not all MMT advocates would necessarily endorse this, it can be interpreted as a form of realist social science.5 For example, critical realism argues that all theory implies an ontology and this may be explicit or implicit and the world itself cannot be reduced to and so should not EH FRQIXVHG ZLWK WKH WKHRULHV ZH KROG RI LW (WKLV LV DQ ³HSLVWHPLF IDOODF\´). 7KH PRVt influential version RI WKLV LQ HFRQRPLFV LV 7RQ\ LDZVRQ¶V ZRUN, DQG ZKLOH LDZVRQ KDV KLV GLIIHUHQFHV ZLWK 007 RYHU WKHRU\ of money, the basic points about ontology still apply.6 Reality is stratified (so some parts build upon and presuppose others ± physical, chemical, biological, social etc.), emergent (the organisation of parts produces new entities with new powers), and is continually developing through time.7 Social reality is a combination of relative stability (since as conscious beings we plan, organise and determine our conditions of social existence for the purposes of reproduction, stability and security) and change (we organise to do things differently, we invent and innovate and evolution and unintended consequences apply to action). As such what we observe around us is the interplay of agency and structure and is RQO\ HYHU ³GHPL-UHJXODU´ DQG KLVWRU\ PDWWHUV LQ RSHQ V\VWHP SURFHVVHV. 7R PDNH VHQVH RI WKis the main tool of enquiry is retroduction rather than deduction or induction (though neither of these is irrelevant). Retroduction theorises and seeks evidence for possible underlying causal mechanisms that can account for relative degrees of regularity of outcomes and employs different ways to test out the role of such causal mechanisms (it is in various ways similar to abduction). While some of the argument is specific to critical realism, a commitment to open systems is common to heterodoxy and in any case, critical realism merely offers an under-labouring service. It is philosophy of social science or social theory with methodological implications. It is not economic theory and there are no exclusively critical realist methods. I would argue MMT retroduces real social mechanisms. MMT contains an explicit recognition of how institutional change impacts on the real mechanisms present in an economy. For example, MMT stresses that the social structures and institutions extant under the Gold Standard determined the actual behaviour of the authorities observed by economists as policy RXWFRPHV RU ³HYHQWV´. 007 KLJKOLJKWV WKH FRQWUDVW EHWZHen these Gold Standard institutions and the nature of contemporary institutions and mechanisms at work in monetary systems when a nation issues its own non-convertible currency where state and central bank must work hand-in-hand on a daily basis.8 4 See also Caldwell (1984). 5 See Armstrong and Morgan (2023). 6 See Lawson (1997, 2003, 2022); Mingers (2014). 7 See Bhaskar (2008 , 2015 [1978); Collier (1994). 8 FRU H[DPSOH, µAGYRFDWHV RI 007 FRQWHQG WKDW, XQGHU WKH JROG VWDQGDUG, JRYHUQPHQWV ZHUH FRQVWUDLQHG LQ WKHLU spending by their ability to tax and borrow. If a fiscal deficit existed there would be untaxed spending in the system which could be converted intR JROG DW D IL[HG UDWH. IQ WKLV FDVH WKH VWDWH ZRXOG QHHG WR RIIHU µPDUNHW-GHWHUPLQHG¶ rates to induce holders to buy non-convertible government debt rather than convert into gold (Mosler, 2012, p. 22) From an MMT perspective, social realities fundamentally changed in 1971 (when Nixon closed the gold window) and new structures, mechanisms and rules now apply for nations with their own sovereign currencies operating XQGHU IORDWLQJ H[FKDQJH UDWHV¶ (AUPVWURQJ, 2018, S. 21). real-world economics review, issue no. 105 subscribe for free 60 There are then, clear differences in terms of underlying perspective between an Austrian approach and MMT and this has consequences for the relative significance of history and thus of origin stories of money. 3. The Austrian Theory of Money There is no role for the state in the genesis of money in the orthodox ± and Austrian ± money narrative. IQJKDP (2004, S. 19, HPSKDVLV DGGHG) FRQVLGHUV WKDW ³DOO RUWKRGR[ HFRQRPLF DFFRXQWV RI PRQH\ DUH FRPPRGLW\ H[FKDQJH WKHRULHV. BRWK PRQH\¶V KLVWRULFDO Rrigins and logical conditions of existence are explained as the outcome of economic exchange in the market that evolves as a result of individual utility maximisation´. When Carl Menger (1892) articulated a story of money, his ontology was deeply rooted in the SUHVXSSRVLWLRQ ³WKDW WKH LQGLYLGXDO HQWHUV WKH ZRUOG HTXLSSHG ZLWK ULJKWV WR WKH IUHH GLVSRVDO RI KLV property and the pursuit of his economic self-interest, and that these rights are anterior to, and LQGHSHQGHQW RI, DQ\ VHUYLFH WKDW KH PD\ UHQGHU´ (7DZQH\, 1920, S.23). 0HQJHU¶V WKHRULVLQJ ZDV EDVHG RQ ³WKH VXEMHFWLYH JRDO-directed actions of individual agents- a view that continues to characterise the Austrian approach to economic WKHRU\´ (HDQGV, 2001, S. 39) DQG ³DQWLHPSLULFLVW GHGXFWLYLVP´ (LELG, S. 39). 7KH VWRU\ VWDQGV SXUHO\ RQ WKH ³D SULRUL WUXWK´ RI KLV SUHVXSSRVLWLRQV DQG KLV ORJLFDO GHGXFWLYH reasoning. IQGHHG, 0HQJHU¶V VHPLQDO DUWLFOH (1892) VHW RXW WKH AXVWULDQ SHUVSHFWLYH, ³0HQ KDYH EHHQ OHG, ZLWK increasing knowledge of their individual interests, each by his own economic interests, without convention, without legal compulsion, nay, even without any regard to the common interest, to H[FKDQJH JRRGV GHVWLQHG IRU H[FKDQJH (WKHLU µZDUHV¶) IRU RWKHU JRRGV HTXDOO\ GHVWLQHG IRU H[FKDQJH, EXW PRUH VDOHDEOH´ (0HQJHU, S. 244). HH GHYHORSV KLV DUJXPHQW IXUWKHU ZLWK, ³3XWWLQJ DVLGH DVVXPSWLRQV which are historically unsound, we can only come fully to understand the origin of money by learning to view the establishment of the social procedure, with which we are dealing, as the spontaneous outcome, the unpremeditated resultant, of particular, individual efforts of the members of a society, who have little E\ OLWWOH ZRUNHG WKHLU ZD\ WR D GLVFULPLQDWLRQ RI WKH GLIIHUHQW GHJUHHV RI VDOHDEOHQHVV LQ FRPPRGLWLHV´9 (Menger, p. 245). IW LV LPSRUWDQW WR VWUHVV WKDW 0HQJHU¶V DUWLFOH GRHV QRW LQFOXGH DQ\ UHDO-world evidence, indeed given his DGYRFDF\ RI ORJLFDO GHGXFWLYH UHDVRQLQJ, HPSLULFDO WHVWLQJ ZRXOG KDYH EHHQ VXSHUIOXRXV. KHYLQ DRZG¶V SKUDVH ³FRQMHFWXUDO KLVWRU\´10 (DRZG, 2000, S. 139) LV SHUWLQHQW KHUH. HH SRLQWV RXW WKDW, µ¶A FRQMHFWXUDO history provides a benchmark to assess the world we live in, but it is important to appreciate that it is 9 Likewise, Rothbard sees the development of money as the result of individual purposeful action within a market, ³IQ WKH SXUHO\ IUHH PDUNHW, QR RQH SHUVRQ RU JURXS FDQ KDYH FRQWURO RYHU PRQH. 0RQH\ DULVHV, RQ WKH IUHH PDUNHW, when one or more commodities, in particularly intense demand and possessing such other qualities as durability, portability, and divisibility, are chosen by individuals to serve as media of exchange. Once a commodity begins to be used as a medium, the process accelerates as this makes the good all the more valuable, until it finally comes to be used as a general medium for exchanges²DV D PRQH\´ (5RWKEDUG, 2011, S.709). 10 A W\SLFDO FRQMHFWXUDO KLVWRU\ ZRXOG SURFHHG DORQJ WKHVH OLQHV, LQ ³SULPLWLYH´ HFRQRPLHV H[FKDQJH ZDV EDVHG RQ barter but as societies developed, efficiency was improved by the introduction of one commodity as a means of exchange and a unit of value. A wide range of different commodities have been used in different societies at different times, but in the end precious metals emerged as the most efficient variant and a fixed quantity of a metal (typically gold or silver) of known purity became a standard. Eventually credit was introduced as a substitute for gold, requiring less direct use of metal and improving efficiency (Armstrong and Siddiqui 2019, p. 99). real-world economics review, issue no. 105 subscribe for free 61 not meant to provide an accurate description of how the world actually evolved [emphasis in original]. The conjectural history is a useful myth, and it is no criticism of a conjectural history to say that the world failed to evolve in the way it postulates´. 4. Heterodox Approaches: Credit and State Theories of Money AXVWULDQ HFRQRPLFV¶ SODFH LQ KHWHURGR[\ LV D PDWWHU RI VRPH FRQWURYHUV. 3XWWLQJ WKDW DVLGH, PDQ\ heterodox economists, including Modern Monetary Theorists (Armstrong, 2015) support some version of credit theory (Innes 1913, 1914) and state theory of money (Knapp, 1924).11 It is argued here that a consideration of the ontology of money ± or what money is ± should be the starting point. Modern Monetary Theory is entirely consistent with the view that money is credit and nothing but credit (Innes, 1913, 1914; Wray, 1998, 2004)12. 7KURXJKRXW KLVWRU\, FRPPRGLWLHV KDYH EHHQ XVHG DV PRQH\ ³WKLQJV´ (KH\QHV (1930, 9RO. 1, S. 14) RU PRQH\ ³VLJQLILHUV´ EXW FRPPRGLWLHV (L.H., FUHGLW WRNHQV WR WKH KROGHU DQG symbols of indebtedness to the issuer) have never been money itself and the conflation of money ³WKLQJV´ ZLWK PRQH\ LWVHOI, LQ WKLV ZD\ FRQVWLWXWHV DQ RQWRORJLFDO RU FDWHJRU\ HUURU (AUPVWURQJ DQG Siddiqui, 2019). Modern Monetary Theorists reject the conjectural history favoured by the Austrian School ± or the attempt to deduce a history of money without the state - and stress the role of money as providing a unit of account, an approach which tends to be compatible with a focus on the importance of the role of a central authority in the genesis of money, as opposed to market forces. Ingham (2004, p. 181) contends that discrete truck and barter would lead to the production of a vast array of bilateral exchange ratios, rather than the enduring unit of account required for the measurement of relative prices critical to the operation of the market. Rather than arising from a spontaneous process, a stable unit of account is required before D PDUNHW FDQ IXQFWLRQ; IRU IQJKDP, ³PRQH\ LV ORJLFDOO\ DQWHULRU DQG KLVWRULFDOO\ SULRU WR WKH PDUNHW´. Armstrong and Siddiqui (2019, p. 101) point out that the use of quantities of grain as a unit of account is well documented (Wray, 1998, pp. 47-8) but from a heterodox perspective, drawing directly from KH\QHV¶V ZRUN, WKLV XVH LV founded on state action rather than being a market outcome. The units of 11 Armstrong and Siddiqui (2019, p. 108) suggest a relationship between the credit theory of Innes and the state theory developed by Knapp. This follows from Smithin (2018, pp. 194-95) ZKR DUJXHV WKDW ³WKH VWXG\ RI PRQH\ DQG monetary issues should follow a fRXU VWDJH µVFKHPD¶ EHJLQQLQJ ZLWK D UHDOLVW VRFLDO RQWRORJ\, IROORZHG E\ HFRQRPLF VRFLRORJ\, PRQHWDU\ PDFURHFRQRPLFV DQG, ILQDOO\, SROLWLFDO HFRQRP\´. B\ XWLOLVLQJ WKLV VWUXFWXUH, FUHGLW WKHRU\ LV foundational and explains the ontology of money. The economic sociology of money, described by the state theory in the second stratum, explains how the particular form of credit we use as money was introduced and became embedded in society. 12 Anthropological studies of pre-modern societies have revealed the widespread existence of gift exchange, inter-community barter and the use of specific commodities to settle obligations under particular circumstances within societies (Polanyi, 1968; Neale, 1976). In the latter case the commodities (at least partially) possess the function RI D ³PHGLXP RI H[FKDQJH´ DQG IRU WKLV UHDVRQ PLJKW UHDVRQDEO\ GHVFULEHG DV ³PRQLHV´ E\ DQWKURSRORJLVWV (VHH Neale, 1976, pp. 31-45). However, from the standpoint of credit theory, the presence of commodities functioning LQ VXFK D ZD\ ZRXOG QRW EH VXIILFLHQW IRU D VRFLHW\ WR EH UHJDUGHG DV ³PRQHWL]HG´. I DJUHH ZLWK KH\QHV¶ GLVWLQFWLRQ EHWZHHQ D ³FRPPRGLW\ ZKLFK LV GLVFRQQHFWHG IURP D XQLW RI DFFRXQW DQG PHUHO\ XVHG LQ D ZD\ WR LPSURYH VSRW WUDQVDFWLRQV DQG D PRQH\ µWKLQJ¶ ZKLFK E\ YLUWXH RI LWV UHODWLRQVKLS WR D VWDQGDUG RU PRQH\ RI DFFRXQW EHFRPHV µPRQH\ SURSHU¶´. KH\QHV DGGV, ³VRPHWKLQJ ZKLFK LV PHUHO\ XVHG DV D FRQYHQLHQW PHGLXP RI H[FKDQJH RQ WKH VSRW may approach to being Money, since it may represent a means of holding General Purchasing Power. But if this is all that is involved, we have scarcely emerged from the stage of Barter. Money-Proper in the full sense of the term FDQ RQO\ H[LVW LQ UHODWLRQ WR D µ0RQH-of-AFFRXQW¶´ (KH\QHV, 1930 9RO. 1, S. 3). real-world economics review, issue no. 105 subscribe for free 62 account used in early empires were almost without exception based on grain quantities and led to the establishment of precious metal standards (Keynes, 1982, pp. 236-7). II ZH UHIHU WR ³D PLQD, VKHNHO RU pound, all the early money units were weight units based on either wheat or barley grains, with the nominal value of gold usually measured in wheat units, and the nominal value of silver measured in EDUOH\ XQLWV´ (:UD\, 1998, S. 48). :UD\ QRWHV WKDW D UXOHU ZRXOG EH DEOH HVWDEOLVK D PRQHWDU\ XQLW E\ setting it equal to a particular quantity of grains of gold, but the relative value of gold represented by its market price could change without the need to change the standard (ibid, emphasis added). Thus, the value of, for example, a shekel weight of gold could rise or (less frequently) fall against the abstract standard of the shekel. Modern Monetary Theorists consider a study of the historical development of money and the monetisation of economies to be very significant13, for example, an examination of the use of cowry is HQOLJKWHQLQJ DQG UXQV FRXQWHU WR LGHD WKDW WKH XVH RI ³SULPLWLYH PRQH\´ VSULQJV IURP D VSRQWDQHRXV SURFHVV. ³CRZU\ ZDV XVHG DV PRQH\ LQ DDKRPH\ GHVSLWH WKH IDFW LW ZDV QRW SURGXFHG GRPHVWLFDOO. IW needed to be imported and was then issued by the monarch. Without this state-directed process it could QRW KDYH EHHQ XVHG DV FXUUHQF\´ (3RODQ\L, 1968, SS. 280-305). Rather than being an aspect of a market-based evolutionary process it was an aspect of state actLYLW. ³CRZULH «JDLQHG WKH VWDWXV RI D FXUUHQF\ by virtue of state policy, which regulated its use and guarded against its proliferation by preventing VKLSORDGV IURP EHLQJ IUHHO\ LPSRUWHG´ (LELG, S. 299)´. Armstrong and Siddiqui (2019) point out that anthropological study (Humphrey and Hugh-Jones, 1992; Graeber, 2011) supports the contention that barter had no role in the development of money. Indeed, GHVSLWH H[WHQVLYH VWXG\, DQG EDUWHU¶V ZLGHVSUHDG H[LVWHQFe, no society founded on the use of barter has yet been found14, let alone a barter economy which spontaneously turned into a monetary one WKURXJK LQGLYLGXDO DFWLRQ. ³1R H[DPSOH RI D EDUWHU HFRQRP\, SXUH DQG VLPSOH, KDV HYHU EHHQ GHVFULEHG, let alone the emergence from it of money; all available ethnography suggests that there never has been VXFK D WKLQJ´ (HXPSKUH\, 1992, TXRWHG LQ GUDHEHU, 2011, S. 29). The nature and history of barter are separate from the nature and history of money; barter trades and monetary transactions apply in different situations. The key element that distinguishes the nature of barter from that of money is that barter involves only two parties in the exchange whereas a monetary transaction, in contrast, involves three. When a purchase is made the buyer provides the seller with a credit on a third party. This credit is money. There is no money in direct exchange; barter cannot provide the origins of money although it seems that barter exists alongside money (Armstrong and Siddiqui, 2019, p.111). Credit Theory of Money IQQHV (1913) GHILQHV PRQH\ DV FUHGLW, ³CUHGLW LV WKH SXUFKDVLQJ SRZHU VR RIWHQ PHQWLRQHG LQ HFRQRPLF ZRUNV DV EHLQJ RQH RI WKH SULQFLSDO DWWULEXWHV RI PRQH\, DQG« FUHGLW DQG FUHGLW DORQH LV PRQH\´. HH explains the relationship between credit and debt and in so doing describes the nature of money, ³:KHWKHU«WKH ZRUG FUHGLW RU GHEW LV XVHG, WKH WKLQJ VSRNHQ RI LV SUHFLVHO\ WKH VDPH LQ ERWK FDVHV, WKH 13 See Wray (2004); Henry (2004); Hudson (2004). 14 ³:KHWKHU ZH WXUQ WR WKH HYLGHQFH IURP KLVWRU\ RU WR WKH HYLGHQFH LQ DFFRXQWV E\ DQWKURSRORJLVWV, ZH GR QRW ILQG economic systems in which people depend upon bartering their labour or produce for the produce of others in order to get the necessities of daiO\ OLIH´ (1HDOH, 1976, S. 23). real-world economics review, issue no. 105 subscribe for free 63 one or the other word being used according as the situation is being looked at from the point of view of WKH FUHGLWRU RU RI WKH GHEWRU´. ³0RQH\, WKHQ, LV FUHGLW DQG QRWKLQJ EXW FUHGLW. A¶V PRQH\ LV B¶V GHEW WR KLP, DQG ZKHQ B SD\V KLV GHEW, A¶V PRQH\ GLVDSSHDUV. 7KLV LV WKH ZKROH WKHRU\ RI PRQH\´ (LELG, 1913). IQQHV GHILQHG VWDWH PRQH\ DV D IRUP RI FUHGLW, ³EYHU\ WLPH D FRLQ RU FHUWLILFDWH LV LVVXHG« A FUHGLW RQ WKH SXEOLF WUHDVXU\ LV RSHQHG, D SXEOLF GHEW LQFXUUHG´ (IQQHV, 1914). IQQHV UHFRJQLVHG WKDW D GHEW WR WKH state or tax liability can be paid by the retuUQ RI WKH JRYHUQPHQW¶V RZQ GHEW LQVWUXPHQW; LQ RWKHU ZRUGV, WKHUH H[LVWV ³WKH ULJKW RI WKH KROGHU RI WKH FUHGLW (WKH FUHGLWRU) WR KDQG EDFN WR WKH LVVXHU RI WKH GHEW (WKH GHEWRU) WKH ODWWHU¶V DFNQRZOHGJHPHQW RU REOLJDWLRQ, ZKHQ WKH IRUPHU EHFRPHV GHEWRU DQd the latter FUHGLWRU´ (IQQHV, 1914). IQQHV¶V ZRUN LV VLJQLILFDQW VLQFH LW SURYLGHV D SRZHUIXO FULWLTXH RI RUWKRGR[ WKHRU\ concerning the ontology of money. It highlights the weaknesses in the latter approach and provides a persuasive alternative perspective, namely money is credit in its essential nature. If we accept that money is credit15 and the monetary system is best characterised as simply a ledger of credits and debits, we are faced with a second question, namely how should we understand the history and sociology of money? Simply put, how did economies become monetised? State theory of Money Ingham (2004, p. 47), considers the Methodenstreit and the division of opinion between the German Historical School and the Austrian School, noting that the former group saw money as a means of accounting for and settling debts and regarded an approach to analysing money without a foundational role for the state as absurd. Consistent with this view, in the State Theory of Money (1924), Knapp DUJXHV WKDW LW LV WKH VWDWH WKDW GHFLGHV RQ WKH XQLW RI DFFRXQW DQG WKH ³PRQH\ WKLQJV´ WKDW DUH WR EH XVHG in settlement of debts denominated in this unit. Initially, the unit of account may be a weight of precious metal of given fineness. However, the state may choose to change the unit to a different metal by decree. Thus, the choice of unit is in the hands of the state rather than springing from a process involving individuals searching for the most efficient way of reducing the costs of barter. The state has the power WR FKRRVH WKH ³PRQH\ WKLQJV´ L.H., ZKDW PD\ EH XVHG WR VHWWOH GHEWV LQ WKH GHVLJQDWHG XQLW RI DFFRXQW (KQDSS, 1924, S. 15). ³IQ PRGHUQ PRQHWDU\ V\VWHPV SURFODPDWLRQ LV DOZD\V VXSUHPH´ (LELG, S. 31). 7KH role of the state is dominant in both the development of a unit account and in the monetisation of a society, rather than it being generated spontaneously by individuals maximising expected utility. MMT follows a Chartalist perspective arguing that, logically and practically, the emission of state money LV DQWHULRU WR LWV FROOHFWLRQ. FURP WKLV SHUVSHFWLYH, IROORZLQJ WKH ORJLF RI KQDSS¶V DSSURDFK, WD[DWLRQ serves, not to fund spending but to allow the state to provision itself by the transfer of resources from the private sector to itself. The importance of sequence is stressed in the MMT money story. It begins with a powerful stakeholder, more commonly the state, desiring to provision itself by transferring resources from the private sector to itself (Mosler, 2020). The government first levies a tax liability on its population and determines the means by which that liability can be satisfied, for example in a modern context, US dollars or UK pounds. The existence of the tax obligation creates willing private sector sellers of goods and services who require the state currency to pay their tax bill. The state can spend its currency to buy the goods and services available for sale. The state always spends by the issue of new money and is conceptualised as a currency-issuer. Once the non-government sector has acquired state money it can pay its taxes and, in addition, it may well be the case that the private sector wishes 15 Ingham (2004) points out that not all credit is money, but all money is credit. real-world economics review, issue no. 105 subscribe for free 64 to save state currency and so will offer sufficient goods and services for sale to the state in order to satisfy this demand. From this perspective, government deficit spending, or spending in excess of tax obligations, simply provides the state money which the non-government sector wishes to save (see below) Consistent with the credit theory of money, MMT conceptualises the state money held as saving by the non-government as a tax credit (Mosler, 2020). It will remain as saving until used to pay taxes. Alternatively, the state may offer the non-government sector the opportunity to buy interest-bearing state debt (ibid). 5. 5HVSRQGLQJ WR B\OXQG¶V &ULWLTXH 3HU B\OXQG¶V (2022) FULWLTXH LV D UHODWLYHO\ XQXVXDO HQJDJHPHQW ZLWK 007 DQG P\ UHVSRQVH ZLOO, I KRSH, give credit if credit is due, as well as providing clarification and articulating counterarguments as appropriate. In writing a reply, I will draw upon the text above when required. I argue here that the fundamental difference between MMT and the Austrian School lies at the level of methodology (as described above) DQG I KRSH WR IROORZ ³DRZ¶V KHXULVWLFV´ LQ WKLV UHVSRQVH. DRZ SRLQWV WR WKH SUDFWLFDO LPSOLFDWLRQV RI accepting methodological pluralism for the behaviour of economists, describing them in the form of heuristics- both positive and negative. I focus here on the former which consist of the following instructions for methodological pluralists, ³>U@HVSHFW WKH OHJLWLPDF\ RI DOWHUQDWLYH DSSURDFKHV DQG KDYH an understanding of them. Be prepared to justify your own approach relative to others, [b]e prepared to adapt your approach as events unfold and as a result of debate, [b]e open to drawing on other DSSURDFKHV IRU LGHDV, HYHQ LI WKH\ WXUQ LQWR VRPHWKLQJ HOVH LQ \RXU DSSURDFK´ (DRZ, 2017, S. 10, parentheses added). I EHJLQ ZLWK D FRPPHQW RQ B\OXQG¶V KDOI-PLVWDNHQ FRQWHQWLRQ (2022, S. 148), ³IQ WKH VFKRODUO\ OLWHUDWXUH, LQWHUHVW LQ 007 LV OLPLWHG, DQG ZKDW DWWHQWLRQ WKH DSSURDFK KDV JRWWHQ VR IDU KDV EHHQ SULPDULO\ FULWLFDO« One reason for this is likely that MMT focuses on policy prescriptions rather than explanations which PDNHV LW XQVXLWDEOH IRU UHVHDUFK´. 1RZ ZKLOH LW LV WUXH WKDW, ³IQ WKH VFKRODUO\ OLWHUDWXUH, LQWHUHVW LQ 007 LV OLPLWHG, DQG ZKDW DWWHQWLRQ WKH DSSURDFK KDV JRWWHQ VR IDU KDV EHHQ SULPDULO\ FULWLFDO´, LW LV certainly not the case that MMT focuses on policy prescriptions. Modern Monetary Theorists have produced a well-established body of theoretical work and its policy prescriptions follow from that theory (Mosler 2012, 2020; Wray 1998; Armstrong, forthcoming). MMT seeks to provide explanations of observed events which should be the case with all economic theory (as I argue above)16. 7KH PLVWDNHQ VXJJHVWLRQ WKDW ³007 IRFXVHV RQ SROLF\ SUHVFULSWLRQV´ LV FRPPRQO\ PDGH DQG LV WKH UHVXOW RI FULWLFV¶ IDLOXUH WR WDNH WKH WLPH WR HVWDEOLVK ZKDW MMT is really saying rather than accepting how it is reported in mainstream economic media and literature (Armstrong, 2023, forthcoming). Mainstream critiques, such as Mankiw (2019), fail to take the QHFHVVDU\ WLPH WR HQJDJH LQ D VFKRODUO\ PDQQHU. IQGHHG, 0DQNLZ¶V VKRUW DUWLFOH ZDV HDVLO\ GLVPLVVHG by Mitchell (2019a, 2019b). Bylund (pp. 148-150, SDUHQWKHVHV LQ WKH RULJLQDO) JLYHV D IDLU GHVFULSWLRQ RI CKDUWDOLVP DQG 007¶V FRQVLVWHQF\ ZLWK LW. HH WKHQ VXJJHVWV D SRWHQWLDO ZHDNQHVV LQ WKH 007 DUJXPHQW, ³II WKH FXUUHQF\ LV valued because (and only because) it is needed to pay the taxes owed to the government, then this 16 Mainstream (New Keynesian) theory has clearly failed to provide powerful explanation of real-world events (Armstrong 2018, 2020b). real-world economics review, issue no. 105 subscribe for free 65 GRHV QRW DOVR H[SODLQ ZK\ DFWRUV ZRXOG YDOXH LW PXFK EH\RQG WKHLU WD[ OLDELOLWLHV´. IQ RWKHU ZRUGV, ZK\ would the non-government sector want to net save government currency? Why not just acquire as much as they need, pay their tax bill, and carry on life as before? He also asks why non-government actors might need to acquire government money before it was necessary, lose flexibility and run the risk of it losing value over time? Why net save state money? To understand and answer this question we must reflect upon the Austrian PHWKRG DQG PRQH\ VWRU. AV QRWHG HDUOLHU, IRU WKH AXVWULDQ VFKRRO, ³PRQH\´ LV VLPSO\ D PHGLXP RI exchange which develops as a cost saving development of barter. It is a private sector invention flowing IURP ³SXUSRVHIXO KXPDQ DFWLRQ´ DV LQGLYLGXDOV PD[LPLVH VHOI-interest. As Mises (1998, p. 774, emphasis DGGHG) SXWV LW, ³A WKLQJ becomes money only by virtue of the fact that those exchanging commodities and serYLFHV FRPPRQO\ XVH LW DV D PHGLXP RI H[FKDQJH.´ Importantly, from this perspective, private sector money predates state involvement; private individuals DUH DOUHDG\ XVLQJ PRQH\ EHIRUH WKH VWDWH DWWHPSWV WR ³SLUDWH WKH V\VWHP´. 6R, IRU WKH AXVWULDQ 6FKRRO, it makes sense to question the idea of net saving of state money. The introduction of coercive taxation is seen as an unwelcome and inefficient disruption to pre-existing private markets and thus, a rational self-interested individual might reasonably be expected to simply access the state money required to settle tax liabilities and then continue to trade using the more trustworthy and familiar private money. Bylund (2022, p. 153-56, parentheses in the original) returns to the same point regarding the logic EHKLQG QHW VDYLQJ RI VWDWH PRQH\ ZKHQ FULWLFLVLQJ :UD\¶V FORDNURRP WLFNHW DQDORJ\17, which illustrates KRZ D GHEW LV UHGHHPHG E\ WKH UHWXUQ RI WKH LVVXHU¶V RZQ OLDELOLW\, ZLWK WKH IXUWKHU TXHVWLRQ, ³:K\ ZRXOG a guest acquire more than one token? (And why would you acquire tokens before you are ready to OHDYH?)´ However, as we noted above, this thinking is highly problematic and ably summed up by Neale (1976, pp. 8-9), ³DHVSLWH WKH IDFW PDQ\ D WH[W RQ PRQH\ VD\V WKDW PRQH\ RULJLQDWHG LQ WKH LQFRQYHQLHQFHV RI barter, that it was invented as a medium of exchange, or that a good commonly used in trade gradually evolved into a medium of exchange ± despite such statements, neither historical evidence nor by DUJXPHQW E\ DQDORJ\ IURP FRQWHPSRUDU\ QRQOLWHUDWH VRFLHWLHV OHQGV VXSSRUW WR WKLV VSHFXODWLYH KLVWRU\´. Simply put, the anthropological and historical evidence suggests that money is not a private invention ± the state is there at the start for good or ill (Armstrong, 2015). FURP WKH SHUVSHFWLYH DUJXHG IRU KHUH, VWDWH PRQH\¶V LQWURGXFWLRQ monetises a society, rather than competing with a pre-H[LVWLQJ SULYDWH ³PRQH\´ (RU PHGLXP RI H[FKDQJH). 6WLOO, WKH TXHVWLRQ UHPDLQV DV to why agents in a newly monetised society might have net saving desires for state money. Forstater and Mosler (1999) model the introduction of money into a society and note that taxpayers who do not ZLVK (RU GRQ¶W TXDOLI) WR ZRUN IRU WKH VWDWH PXVW VHHN RWKHU ZD\V RI REWDLQLQJ VWDWH FXUUHQF\, ³IQ WKH simplest case, individuals offer goods and services to those employed by the state in return for some of the currency originally earned from the State. Non taxpayers, too, are apt to become monetized, as when they see goods and services for sale they, too, desire units of the State currency of denomination. 17 :UD\ (2016, S.3) ILUVW HPSOR\V D ³FORDNURRP WLFNHW´ DQDORJ\, ³IQ GLVFXVVLQJ PRQH\, G.F. KQDSS (RQH RI WKH developers of the State Money Approach, adopted by Keynes and today by Modern Money Theory) made a useful analogy with the cloakroom token. When you drop off your coat at the cloakroom, the attendant offers you a token, usually with an identification number. The token is evidence of the debt of the cloakroom, which owes you a coat. Some hours later you return with the token. The attendant returns your coat. By accepting the token and meeting WKH REOLJDWLRQ WR UHWXUQ \RXU FRDW, WKH DWWHQGDQW KDV ³UHGHHPHG´ KHUVHOI RU KLPVHOI. 7KH VODWH LV ZLSHG FOHDQ; WKH GHEW LV GHVWUR\HG´. HH WKHQ WDONV DERXW WDOOLHV DQG SDSHU PRQH\ UHGHPSWLRQ DV LOOXVWUDWLRQV RI WKH Vtate theory (ibid). real-world economics review, issue no. 105 subscribe for free 66 They may, for example, sell their labor to those employed by the State, and then, with the currency XQLWV WKXV REWDLQHG, PDNH SXUFKDVHV IURP WD[SD\HUV QRW HPSOR\HG E\ WKH 6WDWH´ (FRUVWDWHU DQG 0RVOHU 1999, emphasis added). Here we see a pervasive logic behind the desire to net save state money. The whole population will REVHUYH JRRGV DQG VHUYLFHV EHLQJ PDGH DYDLODEOH IRU VDOH LQ WKH VWDWH¶V FXUUHQF. II QR PHPEHU RI society desires the goods and services available to buy using state money, we might reasonably expect net saving desires of state money by non-state agents to be zero. Net savings desire above zero reflects a positive preference to acquire such goods and services and will require both taxpayers and non-taxpayers alike to acquire more state currency than that required to pay their taxes. B\OXQG (2022, S. 150) IXUWKHU DUJXHV, ³« VHOOLQJ UHVRXUFHV WR WKH JRYHUQPHQW LQ H[FKDQJH IRU FXUUHQF\ QHHGHG WR SD\ WD[HV « PRQWKV RU HYHQ D \HDU ODWHU ZRXOG OLPLW WKH HFRQRPLF IOH[LELOLW\ RI WKH DFWRU DV resources were bound up in tax-paying tokens. This is a cost on actors accepting government currency before taxes are due. Further, if and to the extent the currency is (or is expected to be) inflationary, meaning it loses purchasing power over time, anyone acquiring currency earlier than necessary would sXIIHU ORVVHV. AFWRUV ZRXOG EH EHWWHU RII DFFHSWLQJ WKH JRYHUQPHQW FXUUHQF\ DW D ODWHU GDWH´. Again, this point deserves attention. The desire to acquire state currency ahead of the need to pay taxes reflects an aversion to risk. The possibility of being unable to acquire sufficient state currency to pay taxes ± and indeed, buy goods and services available for sale in state currency ± in the future will manifest in a positive net savings desire in the present. Once a society is monetised and uses state money to settle debts to the state and non-state agents18, it also likely that holding state money will add to flexibility rather than reduce it. Additionally, although all commodities and currencies can suffer unpredictable shifts in value in the future, it would be unreasonable to assume that agents would generally be less confident in (most) state money than commodity alternatives. Of course, lack of FRQILGHQFH LQ WKH VWDWH¶V DELOLW\ (RU ZLOOLQJQHVV) WR PDLQWDLQ WKH YDOXH RI LWV FXUUHQF\ ZLOO UHGXFH GHVLUH to hold it but, importantly, it will not eliminate it. Indeed, MMT accepts that inflation reduces net savings desires and very high inflation can reduce it significantly (Wray, 1998, p. 85). We now turn to a case study which examines the significance of the conflicting cultures and attitudes to money of the Bantu and the so-FDOOHG ³3LRQHHUV´, RU FRQTXHULQJ BULWLVK VHWWOHUV, LQ VRXWKHUQ AIULFD LQ the nineteenth century (Neale, 1976, pp. 77-81; see also Wray, 1998, pp. 57-61)19. Neale describes a situation where a society unaccustomed to the use of money was conquered by an outside monetised society and was then faced with offers of work, paid in money, by the conquerors (from Great Britain). 18 MMT recognises that banks are agents of the central bank (Mosler and Armstrong, 2019), granted the privilege of creating money in the form of bank deposits, denominated in the state unit of account, subject to strict regulatory UHTXLUHPHQWV. 6XFK ³EDQN PRQH\´ FDQ EH XVHG WR VHWWOH GHEWV EHWZHHQ QRQ-government sector agents but cannot directly settle tax debts to the state. A taxpayer might use a credit on a bank (a deposit) as a payment to the bank (LWV DJHQW), EXW WKH ILQDO VHWWOHPHQW RI D WD[ GHEW UHTXLUHV D UHVHUYH GUDLQ IURP D EDQN¶V UHVHUYH DFFRXQW WR WKH Treasury account at the central bank (Armstrong, 2015). 19 :UD\ (1998, S.59) DUJXHV WKDW WKLV H[SHULHQFH RI PRQHWL]DWLRQ ZDV D ³QHDUO\ XQLYHUVDO H[SHULHQFH WKURXJKRXW AIULFD´, QHYHUWKHOHVV, I GR QRW VXJJHVW WKDW because it happened that way it must happen that way everywhere. 5DWKHU I HPSOR\ DQDO\WLF JHQHUDOLVDWLRQ. HDYLQJ SXW IRUZDUG P\ K\SRWKHVLV EDVHG XSRQ FRUVWDWHU DQG 0RVOHU¶V (1999) PRGHO, WKLV FDVH VWXG\ WDNHV WKH IRUP RI DQ ³H[SHULPHQW´. 5REHUW LQ favour of the Chartalist sequence of spending and taxation] is limited to whether and to the extent that the government destroys the currency. If the government reuses the currency, then the currency is no ORQJHU D WRNHQ WKDW LV µUHGHHPHG¶´. HRZHYHU, WKLV DUJXPHQW UHIOHFWV WKH VDme conflation of money (i.e., WKH JRYHUQPHQW¶V GHEW RU WD[ FUHGLW WR WKH KROGHU) DQG WKH VLJQLILHU (RU WRNHQ) RI WKH GHEW. 2QFH WKH nature of money as credit is understood it becomes clear that the government never reuses revenue nor can it; tax revenue is merely the return if its own IOUs. Clearly the issuer of an IOU never needs to reuse it! It may reuse the tokens of indebtedness, but such action is of no consequence, for example, history shows us that a ruler using coin might choose to melt down all the returned coins and issue new coins, spend the coins again as signifiers of new debt (especially if they contain precious metal) or issue an entirely new token of debt for a range of reasons (Desan, 2014). It seems that, from a heterodox standpoint, the Austrian ontological error of confusing a money token, or signifier, with the money itself is again at the root of this misunderstanding. B\OXQG (2022, S. 156) DOVR FULWLFLVHV :UD\¶V XVH RI H[DPSOHV WR VXSSRUW KLV DUJXPHQW (:UD\, 2016, pp.3-10), ³FRU RQH, WKDW WKHUH DUH H[DPSOHV LOOXVWUDWLQJ KLV SRLQW GRHV QRW PHDQ WKDW DOO RU HYHQ PRVW KLVWRULFDO H[DPSOHV VXSSRUW KLV DUJXPHQW´. :KLOH WKLV Vtatement, taken in isolation, is clearly true, Bylund weakens his own case by failing to provide a single counterexample to illustrate the conjectural real-world economics review, issue no. 105 subscribe for free 69 history of money arising from barter. This should come as no surprise, since at time of writing, despite extensive historical and anthropological study, no such example has yet been found. B\OXQG¶V (2022, S.162) FRQWLQXHV ZLWK D IXUWKHU SRLQW, ³:UD\ DOVR RYHUORRNV WKH LPSRUWDQW IDFW WKDW WKH government currency has a legacy of being real money. Paper notes, whether issued by private banks or the central bank, used to be accepted because they ZHUH UHGHHPDEOH LQ SUHFLRXV PHWDO´. IQWHUHVWLQJO\, B\OXQG DJDLQ XVHV WKH WHUP, ³UHDO PRQH\´- consistent with the category error of conflating money itself with its signifier but, putting that aside (see above), the idea that redeemability in precious metal was the key to acceptability is itself open to challenge. Desan (2014, p. 319) notes ± with UHIHUHQFH WR WKH BDQN RI EQJODQG¶V LVVXH RI QRWHV UHGHHPDEOH LQ VSHFLH LQ LWV HDUO\ \HDUV ± that redeemability may well have appeared to be the lynchpin of the s\VWHP EXW LQ UHDOLW\, ³« WKH LPDJH offered of gold or silver in the vault gave the sense that an anchor existed ± even if the anchor was DFWXDOO\ HOVHZKHUH, LQ WKH VRXQG IXQFWLRQ RI WKH ILVFDO V\VWHP´. 5DWKHU WKDQ FRQYHUWLELOLW\ LQWR SUHFLRXV metals or other assets, acceptability of state money fundamentally depends on the robustness and effectiveness of the tax system. When the latter fails acceptability is necessarily adversely affected (Wray 1998, p. 85). Importantly, while it is true that government currency (i.e., the tokens of its indebtedness) has a legacy of precious metals, especially with reference to the gold standard, this was a choice made by states WKHPVHOYHV. 0RQHWDU\ V\VWHPV KDYH XWLOLVHG WRNHQV RU ³PRQH\ WKLQJV´ VXFK DV FRLQV, WDOOLHV RU banknotes to symbolise the debt20. A seller receives a physical token to show that they hold credit on WKH GHEWRU (WKH VWDWH RU RQ D SULYDWH LQGLYLGXDO RU LQVWLWXWLRQ). GROG LV QRW ³PRQH\´ EHFDXVH LW DURVH DV medium of exchange through private action. Rather, gold is monetised by the actions of the state under a gold standard. If the state stands by to purchase a given amount of gold for a fixed price in the unit of DFFRXQW WKH JROG LV WKXV LQWHUFKDQJHDEOH ZLWK WKH VWDWH¶V PRQH. 20 AUPVWURQJ DQG 6LGGLTXL (2019, S. 114) QRWH WKDW, ³FURP D PRGHUQ VWDQGSRLQW LW PLJKW VHHP ZDVWHIXO WR manufacture tokens or money things from precious metals with high intrinsic value and multiple uses instead of something with zero or close to zero intrinsic value. Why use precious metal? Minsky gives a clue when he notes, ³DQ\RQH FDQ FUHDWH PRQH\, WKH WULFN LV JHWWLQJ LW DFFHSWHG´ (0LQVN\, 1986, S. 228). :H VXJJHVW WKDW LQ D ZRUOG RI uncertainty about the future, issuing debt by using precious metal tokens would have had several advantages. First, it would raise the prestige of the issuer. Any state that can access gold or silver and use it to manufacture money tokens should be worthy of at least some respect. Second, the scarcity of precious metals would give the WRNHQV D ³IORRU YDOXH´. II WKH FXUUHQW PRQHWDU\ V\VWHP EURNH GRZQ DQG WKH WRNHQV ZHUH QR ORQJHU DFFHSWDEOH LQ payment of taxes then at least they would have some residual value. Third, this scarcity would add to the acceptability of the tokens from those who might fear that the possibility of irresponsible issue of tokens by the state in the future was a real threat and might lead, in turn to a reduced value of their monetary wealth. Lack of availability of precious metal would constrain the state from such actions. Fourth, fraudsters would find it hard to find precious metal relative to, say, a common material which would reduce (although not eliminate) the chance of counterfeiting. In principle, though, materials with little or no intrinsic value could have been (and indeed, were) chosen as money tokens, notably hazel wood tallies (Wray, 1998; Desan, 2014). However, the common choice of precious metal tokens has been the source of a great deal of confusion as category errors have proliferated in economics. Unfortunately, economists have committed an ontological error (or category error) when considering the actual QDWXUH RI PRQH\ DQG KDYH FRQIXVHG µPRQH\ WKLQJV¶ RU µVLJQLILHUV¶ (PRUH generally, tokens) which are producible FRPPRGLWLHV ZLWK WKH PRQH\ LWVHOI, ZKLFK LV QRW D SURGXFHG FRPPRGLW\ (IQJKDP, 2001).´ real-world economics review, issue no. 105 subscribe for free 70 6. Conclusion It seems that a methodological approach founded on initial axioms and deductive logic has come to dominate the economics academy following the Methodenstreit. Advocates of alternative approaches are concentrated in heterodox economics and other social sciences. With specific reference to money, Ingham (2004, p. 197) points out that the insights of the Historical School have largely disappeared from orthodox eFRQRPLFV, DQG LW KDV EHFRPH ³JHQHUDOO\ DFFHSWHG WKDW WKDW WKH RQWRORJ\ RI PRQH\ ZDV adequately dealW ZLWK E\ WKH YHQHUDEOH WKHRU\ LQ ZKLFK PRQH\¶V IXQFWLRQV ZHUH GHGXFHG IURP LWV VWDWXV DV D FRPPRGLW\´. 7KLV DUWLFOH VXSSRUWV IQJKDP¶V YLHZ WKDW WKLV, ³HQWDLOHG D VHULRXV ORJLFDO FDWHJRU\ HUURU. Such functions cannot be established in this manner; rather they are institutional facts that can only be DVVLJQHG LQ WKH FRQVWUXFWLRQ RI UHDOLW\´. Regardless of its form and substance money is always an abstract claim or credit ZKRVH µPRQH\QHVV¶ LV FRQIHUUHG E\ D PRQH\ RI DFFRXQW« PRQH\ LV QRW PHUHO\ VRFLDOO\ SURGXFHG« LW LV DOVR VRFLDOO\ constituted by the social relation of credit-debt. All money is debt in so far as issuers promise to accept their own money for any debt payment by any bearer of the money. The credibility of the promises forms a hierarchy of moneys WKDW KDYH GHJUHHV RI DFFHSWDELOLW. 7KH VWDWH¶V VRYHUHLJQ LVVXH RI OLDELOLWLHV XVXDOOy occupies the top place, as these are accepted in payment of taxes (Ingham 2004, p.198, emphasis in the original). It is also apparent that MMT and the Austrian School face a barrier to communication which we might UHDVRQDEO\ FDOO ³LQFRPPHQVXUDELOLW\ RI SDUDGLJPV´ (KXKQ 1962, AUPVWURQJ 2020D). 7KLV PDNHV IUXLWIXO dialogue difficult as both schools conceptualise the world differently, the former through a realist social ontological lens, the latter via axiomatic deductivism (Armstrong, 2020c). Specifically, I argue here that WKDW 007 DQG WKH AXVWULDQ 6FKRRO IDFH ³µmethodological incommensurability¶, DFFRUGLQJ WR ZKLFK WKere is no common measure between successive scientific theories, in the sense that theory comparison is sometimes a matter of weighing historically developing values, not following fixed, definitive rules (Sankey and Hoyningen-Huene 2001, vii-[Y)´. Thus, we might legitimately ask if anything can be gained from interaction between the Austrian School and MMT? As both an optimist and a pluralist, I believe so (see Dowd, interviewed in Armstrong 2020a), SURYLGHG µDRZ¶V KHXULVWLFV¶ DUH IROORZHG. IW LV VXrely beneficial to be encouraged to think about legitimate scholarly criticism and to produce a meaningful response to it. In his critique, Bylund (2022) stays firmly ³LQ SDUDGLJP´, IDLOV WR DSSUHFLDWH ³PHWKRGRORJLFDO SOXUDOLVP´ DQG ILQGV WKH LQVLJKWV RI 0MT beyond his UHDFK. IQ FRQFOXVLRQ, LW LV LPSRUWDQW WR VWUHVV WKDW D IXOO DSSUHFLDWLRQ RI WKLV DUWLFOH¶V GHIHQFH RI :UD\ (2016) requires a scholar to look beyond the reach of praxeology to a consideration of an alternative realist methodology and also to recognise the importance, not only of logic and theory but, importantly, of history and anthropology. Acknowledgements The author would like to thank L. Randall Wray for his support and Steve Laughton for his valuable comments on a draft of this paper. All errors remain entirely my responsibility. real-world economics review, issue no. 105 subscribe for free 71 References AUPVWURQJ, 3. (2015), ³HHWHURGR[ 9LHZV RI 0RQH\ DQG 0RGHUQ 0RQHWDU\ 7KHRU\´, moslereconomics.com, Accessed 28/08/2023. AUPVWURQJ, 3. (2018), ³0RGHUQ 0RQHWDU\ WKHRU\ DQG D HHWHURGR[ AOWHUQDWLYH 3DUDGLJP´, The Gower Initiative of Modern Money Studies, Dec 26, Modern Monetary Theory and a Heterodox Alternative Paradigm - The Gower Initiative for Modern Money Studies (gimms.org.uk)´, Accessed 28/08/2023. Armstrong, P. (2020a), Can Heterodox Economics make a Difference: Conversations with Key Thinkers, Cheltenham: Edward Elgar (2020). AUPVWURQJ, 3. (2020E), ³0RGHUQ 0RQHWDU\ 7KHRU\ DQG LWV 5HODWLRQVKLS WR HHWHURGR[ EFRQRPLFV´, Solent University, Accessed 28/08/2023. AUPVWURQJ, 3. (2020F), ³1RUZRRG HDQVRQ, 3DXO KUXJPDQ DQG 007´, Gower Initiative for Modern Money Studies, March 29, Accessed 28/08/2023. AUPVWURQJ, 3. (2022), ³A 0RGHUQ 0RQHWDU\ 7KHRU\ DGYRFDWH¶V UHVSRQVH WR µ0RGHUQ 0RQHWDU\ 7KHRU\ RQ PRQH\, VRYHUHLJQW\, DQG SROLF\: A 0DU[LVW FULWLTXH ZLWK UHIHUHQFH WR WKH EXUR]RQH DQG GUHHFH¶ E\ CRVWDV LDSDYLWVDV DQG 1LFROiV AJXLOD (2020)´, The Japanese Political Economy, Published online 18/03/2022. A modern PRQHWDU\ WKHRU\ DGYRFDWH¶V UHVSRQVH WR µPRGHUQ PRQHWDU\ WKHRU\ RQ PRQH\, VRYHUHLJQW\, DQG SROLF\: A PDU[LVW FULWLTXH ZLWK UHIHUHQFH WR WKH EXUR]RQH DQG GUHHFH¶ E\ CRVWDV LDSDYLWVDV DQG 1LFROiV AJXLOD (2020): The Japanese Political Economy: Vol 0, No 0 (tandfonline.com), Accessed 28/08/2023. AUPVWURQJ, 3. (2023), ³µ7RPEVWRQH IRU D 7RPEVWRQH¶: DHDOLQJ ZLWK WKH µEDG VFLHQFH¶ EHKLQG PDLQVWUHDP FULWLFLVP RI 007´, The Gower Initiative of Modern Money Studies, Jan. 28, µ7RPEVWRQH IRU D 7RPEVWRQH¶: DHDOLQJ ZLWK WKH µEDG VFLHQFH¶ EHKLQG PDLQVWUHDP FULWLFLVP RI 007 - The Gower Initiative for Modern Money Studies (gimms.org.uk), Accessed 28/08/2023. Armstrong, P. (forthcoming), The Contribution of Modern Monetary Theory to Heterodox Economics, title tbc, in Post Keynesian Economics: Key Debates and Contemporary Perspectives, John King and Therese Jefferson (eds.), Cheltenham: Edward Elgar. AUPVWURQJ, 3. DQG 0RUJDQ, J. (2023), ³007 DV SRVW-neoliberal economics: The role of methodology-SKLORVRSK\´, in Modern Monetary Theory: Key Insights, Leading thinkers. In L. Randall Wray, Phil Armstrong, Sara Holland, Claire Jackson-Prior, Prue Plumridge and Neil Wilson (eds.), Cheltenham: Edward Elgar. AUPVWURQJ, 3. DQG 6LGGLTXL, K. (2019), µ7KH FDVH IRU WKH 2QWRORJ\ RI 0RQH\ DV CUHGLW: 0RQH\ DV EHDUHU RU EDVLV RI ³YDOXH´, Real World Economics Review, Issue 90, December. Bhaskar, R. (2008/1975), A Realist View of Science, London: Routledge. Bhaskar, R. (2015/1978), The Possibility of Naturalism, London: Routledge. B\OXQG, 3. (2022), ³IV LW 0RQH\ EHFDXVH LW LV 5HGHHPHG LQ 7D[ 3D\PHQWV? A 5HVSRQVH WR KHOWRQ DQG :UD\´, Quarterly Journal of Austrian Economics, Vol. 25, No.4, Winter 2022, Is It Money Because It Is Redeemed in Tax Payments? A Response to Kelton and Wray | Published in Quarterly Journal of Austrian Economics (scholasticahq.com), Accessed 28/08/23. CDOGZHOO, B. (1984), ³3UD[HRORJ\ DQG LWV FULWLFV: DQ DSSUDLVDO´, History of Political Economy, 16:3, Duke University Press, Praxeology and Its Critics.pdf (duke.edu), Accessed 28/08/23. Collier, A. (1994) Critical Realism. London: Verso. Desan, C. (2014), Making Money: Coin, Currency, and the Coming of Capitalism, Oxford: Oxford University Press. DRZ, 6. (2018), ³3OXUDOLVW HFRQRPLFV: LV LW VFLHQWLILF?´ IQ DHFNHU 6, EOVQHU : & FOHFKWQHU 6 (HGV.) Advancing Pluralism in Teaching Economics: international perspectives on a textbook science, London: Routledge, 13-30. Dowd, K. interviewed in Armstrong, (2020a), Can Heterodox Economics make a Difference: Conversations with Key Thinkers, Cheltenham: Edward Elgar, pp. 60-78. real-world economics review, issue no. 105 subscribe for free 72 Graeber, D. (2011), Debt: The First 5000 Years, New York: Melville House. Grierson, P. (1977), The Origins of Money, London: Athlone Press. HDKQ, F. (1987) ³FRXQGDWLRQV RI 0RQHWDU\ 7KHRU.´ IQ 0. DHCHFFR DQG J. FLWRXVVL (HGV.) Monetary Theory and Institutions. London: MacMillan. Hands, D.W. (2001), Reflection without Rules. Cambridge: Cambridge University Press. HHQU\, J. (2004), ³7KH 6RFLDO 2ULJLQV RI 0RQH.´ IQ Credit and State Theories of Money, L.R. Wray (ed.), Cheltenham: Edward Elgar. HXGVRQ, 0. (2004), ³7KH AUFKDHRORJ\ RI 0RQH\: DHEW YHUVXV EDUWHU 7KHRULHV RI 0RQH\¶V 2ULJLQV.´ IQ Credit and State Theories of Money, L.R. Wray (ed.), Cheltenham: Edward Elgar. Humphrey, C. and Hugh-Jones, S. (eds.) (1992), Barter, Exchange and Value: An Anthropological Approach. Cambridge: Cambridge University Press. IQJKDP G. (2001), ³FXQGDPHQWDOV RI D 7KHRU\ RI 0RQH\: 8QWDQJOLQJ FLQH, LDSDYLWVDV DQG =HOL]HU.´ Economy and Society, 30 (3), pp. 304-323. Ingham, G. (2004), The Nature of Money. Oxford: Polity/Blackwell. IQQHV, A. 0. (1913), ³:KDW LV 0RQH\?´ Banking Law Journal, May, pp. 377-408. IQQHV, A. 0. (1914), ³7KH CUHGLW 7KHRU\ RI 0RQH.´ Banking Law Journal, January, pp. 151-168. Keynes, J. M. (1930), A Treatise on Money, 2 vols. New York: Harcourt and Brace. Keynes, J. M. (1982) Collected Writings Volume XXVIII, Cambridge: Cambridge University Press. Knapp, G. F. (1924), The State Theory of Money, New York: Augustus M. Kelley (1973). Kuhn, T. (2012/1962), The Structure of Scientific Revolutions, Chicago: University of Chicago Press. Lawson, T. (1997), Economics and Reality, London: Routledge. Lawson, T. (2003), Reorienting Economics, London: Routledge. LDZVRQ, 7. (2022), ³7ZR FRQFHSWLRQV RI WKH QDWXUH RI PRQH\: FODULI\LQJ GLIIHUHQFHV EHWZHHQ 007 DQG PRQH\ WKHRULHV VSRQVRUHG E\ VRFLDO SRVLWLRQLQJ WKHRU.´ Real-World Economics Review, 101: 2-19. 0DQNLZ, G. (2019), ³A 6NHSWLFV GXLGH WR 0RGHUQ 0RQHWDU\ 7KHRU\´, Harvard University, 12 December, Microsoft Word - MMT - Mankiw (harvard.edu), Accessed 28/08/23. 0HQJHU, C. (1892), ³2Q WKH 2ULJLQ RI 0RQH\´, Economic Journal volume 2, 239-55 (trans. Caroline A. Foley). Mingers, J. (2014), Systems Thinking, Critical Realism and Philosophy: A Confluence of Ideas, Abingdon: Routledge. Minsky, H. (1986), Stabilizing an Unstable Economy, New York: McGraw-Hill Professional. 0LWFKHOO, :. (2019D), µA 5HVSRQVH WR GUHJ 0DQNLZ ± 3DUW 1¶, DHFHPEHU 23, , Accessed 28/08/23. 0LWFKHOO, :. (2019E), µA 5HVSRQVH WR GUHJ 0DQNLZ ± 3DUW 2¶, DHFHPEHU 24, Accessed 28/08/23. Mosler, W. (2012), Soft Currency Economics II, US Virgin Islands: Valance. 0RVOHU, :. (2020), ³007 :KLWH 3DSHU´, MMT White Paper - Mosler Economics / Modern Monetary Theory, Accessed 28/08/23. 0RVOHU, Z. DQG AUPVWURQJ, 3. (2019), ³A DLVFXVVLRQ RI CHQWUDO BDQN 2SHUDWLRQV DQG IQWHUHVW 5DWH 3ROLF\´, The Gower Initiative for Modern Money Studies, February 24 Accessed 28/08/23. Central-Bank-Interest-Rate-Policy-Mosler-Armstrong.pdf (gimms.org.uk), Accessed 28/08/23. Neale, W. (1976), Monies in Societies, San Francisco: Chandlers and Sharp Publishers. real-world economics review, issue no. 105 subscribe for free 73 Polanyi, K. (1968), Primitive, Archaic and Modern Economies, George Dalton (ed.), New York: Anchor Books. Rothbard, M.N. (2011), Economic Controversies, First published in The Foundations of Modern Austrian Economics (1976). Economic Controversies (mises.org), Accessed 28/08/23. Sankey, H. and Hoyningen-Huene, P. (eds.) (2001), Incommensurability and Related Matters, Dordrecht: Kluwer: vii-xxxiv, In Oberheim (2018), The Stanford Encyclopaedia of Philosophy, The Incommensurability of Scientific Theories (Stanford Encyclopedia of Philosophy), Accessed 28/08/23. Smithin, J. (2010), Money, Enterprise and Income Distribution, London and New York: Routledge. Tawney, R.H. (1921), The Acquisitive Society, London: Bell and Sons. Wray, L. R. (1998), Understanding Modern Money, Cheltenham: Edward Elgar. Wray, L. R. (2004), Credit and State Theories of Money, Cheltenham: Edward Elgar. :UD\, L. 5. (2016), ³7D[HV DUH IRU 5HGHPSWLRQ 1RW 6SHQGLQJ´, World Economic Review, 7, pp.3-11, WEA-WER-7-Wray.pdf (worldeconomicsassociation.org), Accessed 28/08/23. Yin, R. (2003), Case Study Research: Design and Methods, 3rd edition. Thousand Oaks, CA: Sage Publications. Author contact: parmstrong@yorkcollege.ac.uk _________ SUGGESTED CITATION: Phil Armstrong, ³History and origin of money in MMT and Austrian Economics: The difference methodology makes?´, real-world economics review, issue no. 105, October 2023, pp. 57±73, You may post and read comments on this paper at |
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How do I draw moments on hinged gates?
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Hello I had a quick question regarding moments on hinged gates on submerged curves. Is there a general rule that we follow when we draw moments on hinges, do we draw the hinges in the direction the gate would open if enough force acts on it? Do we draw the moments in the direction the gate is current pushing against water? Please help me, I included an image as an example, where would I draw my moments on this hinge C? If its the former would this be the correct direction of moments?
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When the gate is drawn fully closed, it is understood that the system is in structural equilibrium with the condition $\sum M_C = 0$ is true, so there is no moment to show on the diagram.
If a mechanical device is required to assist the hinge to resist the rotation due to the hydrostatic pressure, then its presence must be indicated on the hinge point "C" with the direction in agreement with its purpose. In this case, the same direction as rotation produced by the force "P".
On the diagram above, it is understood that the rotational equilibrium is maintained thru $M_P + M_R - M_W = 0$. ($M_P$ & $M_W$ need not shown though.)
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190455 | https://www.youtube.com/watch?v=hUQXfU2lEAw | Art of Problem Solving: Completing the Square Part 2
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190456 | https://www.gauthmath.com/solution/1832185631531042/Simplify-256-3-4 | Solved: Simplify. 256^(frac 3)4 [Math]
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Simplify. 256^(frac 3)4
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100%(1 rated)
Answer
The answer is 64
Explanation
Rewrite the expression using the property $$a^{\frac{m}{n}} = \sqrt[n]{a^m}$$a n m=n a m
$$256^{\frac{3}{4}} = \sqrt{256^{3}}$$25 6 4 3=4 25 6 3
Express 256 as a power of 4
$$256 = 4^{4}$$256=4 4
Substitute $$4^{4}$$4 4 for 256 in the expression
$$\sqrt{256^{3}} = \sqrt{(4^{4})^{3}}$$4 25 6 3=4(4 4)3
Simplify the expression using the power of a power rule $$(a^m)^n = a^{mn}$$(a m)n=a mn
$$\sqrt{(4^{4})^{3}} = \sqrt{4^{12}}$$4(4 4)3=4 4 12
Rewrite the radical expression as a power
$$\sqrt{4^{12}} = (4^{12})^{\frac{1}{4}}$$4 4 12=(4 12)4 1
Simplify the expression using the power of a power rule $$(a^m)^n = a^{mn}$$(a m)n=a mn
$$(4^{12})^{\frac{1}{4}} = 4^{\frac{12}{4}} = 4^{3}$$(4 12)4 1=4 4 12=4 3
Calculate $$4^{3}$$4 3
$$4^{3} = 4 \times 4 \times 4 = 64$$4 3=4×4×4=64
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190457 | https://pmc.ncbi.nlm.nih.gov/articles/PMC8582750/ | Development and Validation of Sentences Without Semantic Context to Complement the Basic English Lexicon Sentences - PMC
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J Speech Lang Hear Res
. 2020 Oct 13;63(11):3847–3854. doi: 10.1044/2020_JSLHR-20-00174
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Development and Validation of Sentences Without Semantic Context to Complement the Basic English Lexicon Sentences
Erin R O'Neill
Erin R O'Neill
a Department of Psychology, University of Minnesota, Minneapolis
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a,✉, Morgan N Parke
Morgan N Parke
a Department of Psychology, University of Minnesota, Minneapolis
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a, Heather A Kreft
Heather A Kreft
a Department of Psychology, University of Minnesota, Minneapolis
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a, Andrew J Oxenham
Andrew J Oxenham
a Department of Psychology, University of Minnesota, Minneapolis
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a
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a Department of Psychology, University of Minnesota, Minneapolis
Disclosure: The authors have declared that no competing interests existed at the time of publication.
✉
Correspondence to Erin R. O'Neill: oneil554@umn.edu
Editor-in-Chief: Frederick (Erick) Gallun
Editor: Christian E. Stilp
✉
Corresponding author.
Received 2020 Apr 13; Revised 2020 Jul 8; Accepted 2020 Aug 2; Issue date 2020 Nov.
Copyright © 2020 The Authors
This work is licensed under a Creative Commons Attribution 4.0 International License.
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PMCID: PMC8582750 PMID: 33049146
Abstract
Purpose
The goal of this study was to develop and validate a new corpus of sentences without semantic context to facilitate research aimed at isolating the effects of semantic context in speech perception.
Method
The newly developed corpus contains nonsensical sentences but is matched in vocabulary and syntactic structure to the existing Basic English Lexicon (BEL) corpus. It consists of 20 lists, with each list containing 25 sentences and each sentence having four keywords. Each new list contains the same keywords as the respective list in the original BEL corpus, but the keywords within each list are scrambled across sentences to eliminate semantic context within each sentence, while maintaining the original syntactic structure. All sentences in the original and nonsense BEL corpora were recorded by the same two male and two female talkers.
Results
Mean intelligibility scores for each list were estimated by calculating the mean proportion of correct keywords achieved by 40 normal-hearing listeners for one male and one female talker. Although small but significant differences were found between some pairs of lists, mean performance for all 20 lists fell within the 95% confidence intervals of the mean.
Conclusions
Lists in the newly developed nonsense corpus are reasonably well equated for difficulty and can be used interchangeably in a randomized experimental design. Both the original and nonsense BEL sentences, all recorded by the same four talkers, are publicly available.
Supplemental Material
A number of different materials have been used in clinical and research settings to assess listeners' ability to understand speech. These include sentence lists, such as the Hearing in Noise Test (Nilsson et al., 1994), AzBio sentences (Spahr et al., 2012), the Bamford–Kowal–Bench Speech-in-Noise Test (Bench et al., 1979), and the Quick Speech-in-Noise Test (Killion et al., 2004), as well as isolated words, such as the consonant-nucleus-consonant (CNC) word list (Nilsson et al., 1996) and spondees (e.g., Harris, 1991; Turner et al., 2004). Sentences have some advantages over isolated words, in that they incorporate the co-articulation between words that occurs in natural communication. They also include varying degrees of semantic context that makes many words in sentences predictable to some extent, based on the preceding or following words. Although the inclusion of semantic context has ecological validity, it also imposes a degree of uncertainty regarding what was actually heard, as opposed to simply inferred, by the participants. This process of inferring, or “filling in,” may inflate the estimated audibility of speech and may potentially discount any additional listening effort exerted to achieve a given level of speech understanding (e.g., Sarampalis et al., 2009; Winn, 2016).
Interest in listening effort has grown in recent years in response to individuals with hearing loss and cochlear implants (CIs) reporting high levels of mental fatigue associated with daily listening (Hughes et al., 2018). Increased listening effort has also been inferred from laboratory studies using measures such as pupil dilation (e.g., Beatty, 1982; Kahneman & Beatty, 1966), which may result at least in part from increased semantic inference or “filling in” on the part of CI users (Winn, 2016). The Framework for Understanding Effortful Listening, developed by Pichora-Fuller et al. (2016), highlights the interaction between the semantic context and vocabulary of a speech passage and the differential cognitive and listening effort required to decode such passages. Other studies have also explored the difference in the use of semantic context between children and adults, with some indicating that semantic context is only maximally leveraged in adulthood and, as a result, children need higher levels of audibility to achieve the same level of speech understanding (Nittrouer & Boothroyd, 1990; Stelmachowicz et al., 2000).
Despite the growing interest in the connections between the use of semantic context, listening effort, development, and hearing loss, the speech materials currently available to explore this issue in adults remain limited. Most commonly, a set of sentences is used for which the final word in each sentence is either predictable or not, based on the preceding words in the sentence (Bilger et al., 1984; Lash et al., 2013). This type of sentence provides a controlled method for analyzing the effect of semantic context, but places artificial importance on the final word of each sentence that is not directly comparable to the ways in which semantic context operates in conversational speech typical of everyday environments. For example, listeners may use context occurring later in a sentence to resolve earlier ambiguities. In addition, the resulting measure of speech perception is limited to just one word per sentence, making it a relatively inefficient method to estimate overall intelligibility.
Some studies have tackled this problem from a different angle, using stimuli that eliminate semantic context by scrambling all the words in a sentence, but violate typical grammatical structure in the process (Boothroyd & Nittrouer, 1988). Other studies have maintained correct grammatical structure but have removed semantic context by replacing nouns and verbs with novel tokens, such as pseudowords (Carroll, 1883; Yamada & Neville, 2007). Many studies also utilize matrix sentences that maintain a consistent sentence structure (e.g., proper name + verb + number + color + noun; e.g., Bolia et al., 2000; Hagerman, 1982; Kollmeier et al., 2015), but the closed-set nature of the measure and limited number of choices for each word category make the results hard to interpret in terms of everyday speech perception outcomes. A perhaps more ecologically valid nonsense sentence corpus has been used to examine audio-visual cues on speech perception (Helfer, 1997) and to study the intelligibility of whispered speech (Freyman et al., 2012; Ruggles et al., 2014). It contains sentences with English words, typical grammar, syntax, and prosody, but no semantic context. These sentences thus sound correct, but do not make any logical sense. These “nonsense sentences” provide more information than sentences with only one target word and arguably provide more ecological validity than sentences with invalid syntax or sentences selected from a closed and known set of words in each position.
Although a nonsense sentence corpus offers the opportunity to study sentence perception in the absence of semantic context, it is difficult to quantify the effect of such semantic context because there is currently no equated sentence corpus containing semantic context for comparison. Specifically, if performance on these nonsense sentences is compared to performance on another existing corpus with semantic context, such as the AzBio (Spahr et al., 2012) or IEEE (1969) sentences, the influence of different talkers, vocabulary, number of keywords, and sentence length inherent to the different corpora cannot be eliminated as confounds (O'Neill et al., 2019). Although Boothroyd and Nittrouer (1988) and Stelmachowicz et al. (2000) constructed sentences with low and high semantic context, the vocabularly was targeted at young children, and the number of sentences was small compared to those of other commonly used corpora.
The purpose of this study was to develop, record, and validate a new nonsense sentence corpus, matched to an existing sentence corpus in terms of talkers, vocabulary, number of keywords, and sentence length. The Basic English Lexicon (BEL) sentence corpus (Calandruccio & Smiljanic, 2012) was employed due to its simple vocabulary and sentence structure, as well as its high degree of semantic predictability. The BEL sentences were originally designed to probe the hearing status of nonnative English speakers who may have limited knowledge of complex English vocabulary. The keywords in the corpus were derived from interviews with 100 nonnative English speakers, which focused on 20 topic categories, such as cooking, sports, extended family, work, and others. The keywords were then distributed across 20 lists of 25 sentences each, balancing the rate of occurrence of each keyword, syntactic structure of sentences, syllable counts, and high-frequency speech information (i.e., fricatives, affricatives). Native English speakers then listened to the sentences in background noise to ensure equal perceptual difficulty across lists. The new nonsense sentence corpus was developed in such a way that the keywords within each list were maintained from the original corpus, along with the grammatical structure of each sentence. Keywords within each list were scrambled across sentences, so the final lists consisted of sentences with typical grammatical structure but without semantic context. The resulting sentences were therefore syntactically correct and fully formed English sentences, but were also extremely unlikely and unpredictable contextually. Both the original and new nonsense BEL sentences were then recorded by the same two male and two female talkers (one younger and one older in each pair). Finally, normal-hearing participants were tested on the nonsense sentences spoken by one of the female and one of the male talkers to ensure that the keywords were not predictable and that the lists were balanced for intelligibility by normal-hearing listeners in the presence of background noise.
Method
Sentence Development
To maintain lexical and grammatical consistency with the original BEL sentence corpus, all lists in the nonsense BEL corpus contained the same vocabulary, keywords, and sentence structures as the original 20 lists of 25 sentences each. The original BEL sentence corpus utilizes different variations of a basic syntactic framework that consists of combinations of the following word categories: determiner (D), adjective (A), noun (N), pronoun (Pro), adverb (Adv), verb (V), and preposition (P). Although a number of different combinations of these basic word categories were used to create various syntactic structures, 12 variants account for 70% of all syntactic frameworks, with the remaining 30% of variants being slightly less common but still basic in the sense that they lacked syntactic complexity, such as embedded or proposed dependent clauses. An example of one of the common syntactic structures used is DANVA, with a corresponding sentence being, “The boiled fish smells bad.” To ensure that all sentence structures in the nonsense corpus matched those used in the original BEL corpus, the original syntactic structure for every sentence was identified, and each word was sorted into its appropriate word category. Once all of the words from sentences within a list were sorted into word categories, these words were randomized by computer within each word category and added back into the original sentence structures to create novel nonsense sentences. The randomly assigned combination of words from appropriate word categories (as determined by the necessary syntactic structure of the sentence) were then checked manually for verb tense and noun plurality agreement, and if any disagreement was found, the word causing the grammatical violation was replaced with another randomly selected word from that word category, until the grammatical features were in agreement. The resulting 20 novel lists contained the same 25 sentence structures and 100 keywords as the original BEL sentence list from which it was derived, but was devoid of semantic context. An example of one such nonsense list is shown in Table 1. The complete list of all 500 nonsense sentences can be found in the Supplemental Material S1.
Table 1.
List 11 of the original Basic English Lexicon corpus and Basic English Lexicon nonsense corpus.
| Syntactic structure | Original sentence using described structure | Nonsense sentence using same structure |
:---:
| DNVPAN | The MEETING STARTS in TWENTY MINUTES. | The WINDOWS LEARNED in BROWN SECRETARY. |
| DNVAN | The CUSTOMERS HATE BLACK TEA. | The MEAL PLANNED TWENTY KIDS. |
| DANVA | The SICK PERSON FEELS BETTER. | That COOL ROOM DRINKS HERE. |
| DANVAdvA | That BROWN BIRD is ALWAYS HERE. | A DANGEROUS BIRD was REALLY ORANGE. |
| DANVProAN | The THREE COUSINS did their MATH HOMEWORK. | The TWO GLOVES had their ENGLISH HORSE. |
| DANVDN | The DARK CLOUD COVERED the SKY. | The CHICKEN MOVIE CLIMBED the SON. |
| DANVN | The GROCERY STORE SELLS FOOD. | The GROCERY PERSON NEEDS FARM. |
| DNVPDAN | The MOVIE STARTED in the SMALL ROOM. | The CAKE BIT in the BETTER SOUP. |
| DANVDAN | The CHICKEN SOUP was a TASTY MEAL. | The DIFFICULT JUICE was the BIRTHDAY TEST. |
| DANVAandA | The COOL NIGHT was COMFORTABLE and RELAXING. | The TASTY NIGHT was THREE and DARK. |
| DANVVAdv | The BIRTHDAY CARD was SENT LATE. | The TROUBLED GRADE is SENT EASILY. |
| DNVNAdv | The SECRETARY LEARNED SPANISH EASILY. | A RADIO FEELS PROFESSOR LOUDLY. |
| DANVPDN | The WHITE HORSE LIVES on a FARM. | The SMALL TEA SELLS over an APARTMENT. |
| ProNVAN | Our APARTMENT NEEDS MORE WINDOWS. | Our HOMEWORK BUYS MORE SKY. |
| ProNVAN | Our MOTHER DRINKS ORANGE JUICE. | Our RABBIT STARTED SICK CUSTOMERS. |
| DANVDN | The DANGEROUS SNAKE BIT the RABBIT. | A COMFORTABLE BOYFRIEND COVERED theCOUSINS. |
| DNVDAN | The PROFESSOR GAVE an UNFAIR GRADE. | The SNAKE PLAYED the UNFAIR WEDDING. |
| ProVANPDN | They PLAYED FAST MUSIC on the RADIO. | They HATE GREEN MINUTES on the CARD. |
| DANVAdvA | That ENGLISH TEST was REALLY DIFFICULT. | The MATH STORE was ALWAYS RELAXING. |
| DNandNVProN | The BOYFRIEND and GIRLFRIEND PLANNED their WEDDING. | The SISTER and CLOUD SCREAMED their MOTHER. |
| DNVAdvPDN | The KIDS SCREAMED LOUDLY in the PARK. | The MEETING LIVES LATE on the FENCE. |
| DANVN | The TROUBLED SON STOLE MONEY. | The WHITE MONEY GAVE KITTEN. |
| ProNVNAdv | His SISTER BUYS CAKE DAILY. | His SNOWMAN STOLE SPANISH DAILY. |
| DANVPDN | A LITTLE KITTEN CLIMBED over the FENCE. | The LITTLE GIRLFRIEND STARTS in that MUSIC. |
| DNVAAN | The SNOWMAN had TWO GREEN GLOVES. | The PARK did FAST BLACK FOOD. |
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Note. The syntactic structure for each sentence is shown in the far left column with word categories as follows: D = determiner, A = adjective, N = noun, V = verb, P = preposition, Adv = adverb, Pro = pronoun. Keywords are in uppercase and the four bold words indicate an example of how words were redistributed from one original sentence across the nonsense sentences.
Sentence Recordings
Four native speakers of American English, two women (aged 20 and 62 years) and two men (aged 26 and 63 years), recorded all 500 original BEL sentences, as well as all 500 newly created nonsense BEL sentences. All sentences were digitally recorded in a single-walled, sound-attenuating booth located in a quiet room, at a 22050-Hz sampling rate with 16-bit resolution, using a PMD670 solid state recorder (Marantz). Talkers were seated approximately 12 in. from an ME64 stationary microphone (Sennheiser) and instructed to keep their backs against the back of the chair to maintain a roughly constant distance from the microphone. Talkers were also instructed to maintain a stable level of speech and to read each sentence in a natural conversational manner. Each list of 25 sentences was printed on a separate sheet of paper, and talkers were instructed to pause in between lists, to avoid any sound contamination of the sentences due to page turns. Talkers were also instructed to take slightly longer than natural pauses in between each sentence, to aid with the subsequent splicing process. Finally, if the talkers hesitated or made a noticeable error, they were instructed to pause and repeat the sentence.
After the initial recording session by each talker, the sound files were digitally edited using Audacity software (free audio editor, Version 2.3.3, 2019) and each sentence was spliced and saved as an individual sound file. Each audio file was then cross-checked with the text of each sentence to ensure word accuracy, and was also checked for any audible distortions, microphone pops, clipping, or ambient sound. After the initial editing, each talker rerecorded any flagged sentences and the editing process was repeated until all sentence recordings were deemed adequate. A two-way repeated-measures analysis of variance (with a Huyhn–Feldt correction for lack of sphericity), with average sentence duration per list as a dependent variable and talker and corpus as factors, showed a significant effect of corpus (original vs. nonsense), F(1, 19) = 196.8, p< .001, η p 2 = .912, and talker, F(2.5, 47.9) = 234.9, p< .001, η p 2 = .925, and a significant interaction between corpus and talker, F(2, 38.9) = 7.8, p = .001, η p 2 = .291. On average, sentence durations were longer in the nonsense corpus (2.47 s) than in the original corpus (2.27 s), but each talker also spoke at different rates, which impacted overall durations for each corpus differently. The fastest talker was the older male (mean sentence duration = 2.16 s), followed by the younger female (2.27 s) and the younger male (2.37 s), with the older female having the slowest average speaking rate (2.67 s). Recordings from all four talkers for both the original BEL corpus and the nonsense BEL corpus are available for download from the Data Repository for the University of Minnesota (DRUM) at
Evaluation of Predictability
To ensure the original BEL corpus and nonsense BEL corpus did indeed differ in predictability due to semantic context, a “fill-in-the-blank” test was administered to 20 native speakers of American English. The participants (14 female, six male) were mostly undergraduate students and ranged in age from 19 to 30 years (M = 20.9 years, SD = 2.7). All experimental protocols were approved by the institutional review board of the University of Minnesota, and all participants provided informed written consent prior to enrollment. For each sentence in the original and nonsense BEL corpora, one keyword was randomly omitted and replaced by a blank. Sentences from two original lists and two nonsense lists were then combined in random order to create a 100-sentence fill-in-the-blank test for each participant. This was done for all lists of sentences so that each participant completed four lists and each of the 40 lists was completed by two participants. An excerpt from one of the fill-in-the-blank tests is shown in Table 2.
Table 2.
Example sentences from the fill in the blank test.
| Sentences shown to participants | Missing keyword | Sentence type |
:---:
| 1. The hungry teenagers eat _. | snacks | Original |
| 2. A shy _ traveled the people. | cousin | Nonsense |
| 3. The woman was weird in _ problems. | many | Nonsense |
| 4. The _ band played in a concert. | popular | Original |
| 5. The milk and cheese smelled _. | horrible | Original |
| 6. The thirsty _ was excited and black. | dish | Nonsense |
| 7. The people write after her _. | salad | Nonsense |
| 8. Her neighbors are _ and not silver. | bright | Nonsense |
| 9. The bears eat brown _. | performer | Nonsense |
| 10. The class broke _ twins. | scary | Nonsense |
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Note. The left column shows how the sentences appeared on the fill in the blank test. The middle column shows the missing word from each sentence, and the far right column denoted the corpus to which each sentence belongs.
The tests were administered on a computer in a quiet room, with participants typing their answers into a PDF document with an empty cell corresponding to each blank. Participants were instructed to fill in the blank in each sentence with the word they thought would fit best. Participants were told to type only one word per blank and to guess when they were unsure of the correct answer. Responses were scored in three ways. For the first scoring method, a response was only considered to be correct if it exactly matched the actual missing keyword. Since the correct verb tense was often ambiguous, the second scoring method also considered responses to be correct if the verb was correctly identified but the tense was incorrect. For example, if the missing keyword was “drinks,” but a participant answered “drank,” that response would be considered correct. In the final scoring method, any response that was a synonym, an antonym, or in the same semantic category as the actual missing keyword was considered correct. For example, if the missing keyword was “store,” but a participant responded with the word “shop,” the response would be marked as correct. A list of the original words along with the words that were accepted as having a similar semantic meaning is provided in the Supplemental Material S2. Taken together, these three scoring methods provided a more nuanced interpretation of the data, especially when considering how ambiguities may have been resolved if the sentences had been spoken, rather than read.
List Validation
To confirm that the 20 newly created lists of nonsense BEL sentences were equated for difficulty and could be used interchangeably in experimental design, 40 young, normal-hearing adults listened to all 20 lists of nonsense sentences, in background noise. Only three had previously participated in the written validation test described above, which had taken place 6 months prior. Twenty participants (19 women, one man) ranging in age from 18 to 22 years (M = 19.5 years, SD = 2.2) listened to sentences recorded by the older female talker, and 20 participants (15 women, five men) ranging in age from 18 to 24 years (M = 20.9 years, SD = 2.7) listened to sentences recorded by the younger male talker. Though traditionally validation studies have been conducted using only one talker (e.g., Calandruccio & Smiljanic, 2012; Nilsson et al., 1994), we chose to include two of the four recorded talkers for a more robust, gender- and age-balanced analysis of possible list-level differences in performance, while staying within reasonable time constraints for data collection. Normal hearing was defined as having pure-tone audiometric thresholds less than 20 dB HL at all octave frequencies between 250 and 8000 Hz with no reported history of hearing disorders. All experimental protocols were approved by the institutional review board of the University of Minnesota, and all participants provided informed written consent prior to enrollment.
All sentences were presented in Gaussian noise, spectrally shaped to match the long-term spectrum of the nonsense BEL corpus as recorded by either the older female talker or the younger male talker. Thus, the speech-shaped noise differed for the two groups of participants, but was mixed with the speech at the same signal-to-noise ratio (SNR) of −4 dB. This SNR was selected based on pilot testing to avoid floor and ceiling effects and to facilitate direct comparisons between lists and talkers. The speech was presented at a root-mean-square level of 65 dB SPL, as measured at the position corresponding to the participant's head, and the noise level was adjusted to produce the desired SNR. The noise was gated on 1 s before the beginning of each sentence and gated off 1 s after the end of each sentence. The stimuli were generated using MATLAB and converted via an E22 24-bit digital-to-analog converter (Lynx Studio Technology) at a sampling rate of 22050 Hz. The sounds were presented in a single-walled, sound-attenuating booth located in a quiet room via an amplifier and a single loudspeaker, placed approximately 1 m from the listener at 0° azimuth.
Participants responded to sentences by typing what they heard on a computer keyboard. They were encouraged to guess individual words, even if they had not heard or understood the entire sentence. Instructions were given orally, and participants were asked if they had any questions about procedures before beginning the task. Sentences were scored for keywords correct as a proportion of the total number of keywords presented. Initial scoring was automatic, with each error then checked manually for potential spelling errors or homophones (e.g., wait and weight), which were marked as correct. The 20 lists of nonsense BEL sentences were ordered randomly and split into four blocks (each containing five lists) for each participant. All testing was completed in one 2-hr session per participant with a short break after completion of the first two blocks.
Results and Discussion
Predictability of Nonsense and Original BEL Sentences
The mean proportions of correctly “filled in” words for each list of original and nonsense BEL sentences, scored using the three different scoring methods described above, are shown in Figure 1. As expected, participants were able to correctly guess the missing keywords at much higher rates for the original BEL sentences (M = 21%) than for the nonsense BEL sentences (M< 1%), which was also true when allowing tense or plurality errors (M = 22% vs. < 1%). Because of the dichotomous nature of the response variable, a binomial generalized linear mixed-effects model was used to analyze results with the number of correctly guessed keywords (out of the total number of keywords) as the response variable. Fixed effects included material type (original vs. nonsense) and scoring method, while random effects included an intercept for each participant. The model was implemented using the R programming language and the lme4 package (Bates et al., 2015). The significance of the fixed effects was tested by Wald χ 2 tests in a Type III analysis of deviance. All p values were corrected using the Holm correction and compared against a criterion of θ = 0.05 to assess statistical significance. The analysis of deviance confirmed that performance for original versus nonsense sentences was significantly different from each other across all three scoring methods, χ 2(1) = 91.7, p< .001. Post hoc contrast tests showed that this difference was significant when scoring for exact keywords, estimated odds ratio = 2.64, χ 2(1) = 91.7, p< .001, and when allowing tense or plurality errors, estimated odds ratio = 2.65, χ 2(1) = 101.3, p< .001. A significant difference was also found for the rates that counted semantically similar words as correct, 43% and 2% for original and nonsense BEL sentences, respectively; estimated odds ratio = 2.64, χ 2(1) = 240.6, p< .001. These results, showing correct response rates of 2% or less in all cases, confirm that the new nonsense BEL sentences provide minimal semantic context.
Figure 1.
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Mean proportion of correctly “filled in” words, averaged across all 20 lists of original Basic English Lexicon (BEL) sentences and nonsense BEL sentences, scored using three methods. The first set of bars represent the mean proportion of reported keywords that were exactly correct (Exact), the second set of bars show the proportion of keywords that were either exactly correctly or contained the right word but wrong tense or plurality (Exact + tense), and the third set of bars show the proportion of reported keywords that were either exactly correct, correct except for an error in tense or plurality, or had the same semantic meaning as the missing keyword (Exact + tense + meaning). Results for the original BEL sentences are shown by black bars and those for the nonsense BEL sentences are shown by white bars. Error bars represent 1 standard error of the mean between listeners.
Speech Intelligibility in Noise of Nonsense BEL Sentences
Speech perception results for all 20 lists of nonsense BEL sentences, as recorded by an older female talker and a younger male talker, are shown in Figure 2. Because of the dichotomous nature of the response variable, the same generalized linear mixed-effects model used to analyze results from the fill-in-the-blank experiment was also used to analyze the intelligibility results. Fixed effects included talker, list, and the interaction between the two, while random effects included an intercept for each participant. An analysis of deviance revealed a significant effect of talker, χ 2(1) = 44.7, p< .001; a significant effect of list, χ 2(19) = 343.1, p< .001; and a significant interaction between talker and list, χ 2(19) = 149.9, p< .001. The main effect of talker confirmed the higher scores achieved with the older female talker (M = 73%, SD = 5.0, range: 54%–89%) than the younger male talker (M = 60%, SD = 6.2, range: 39%–77%). Post hoc contrast tests also confirmed significant differences in performance across lists for both the older female talker, χ 2(19) = 343.1, p< .001, and the younger male talker, χ 2(19) = 218.5, p< .001.
Figure 2.
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Speech perception for nonsense Basic English Lexicon sentences recorded by an older female talker and a younger male talker. Red circles and blue squares show individual data for the male and female talker, respectively. The red and blue bars indicate mean performance for each list for the male and female talkers, and black bars show list averages across talker. The horizontal red, blue, and black dotted lines show the overall performance average across lists for the male talker, the female talker, and across talkers, respectively.
To illustrate performance differences between lists, independent of talker differences, scores are replotted in Figure 3, relative to each participant's mean score across sentence lists, with the horizontal bars representing the percentage point deviation from the mean for each talker (blue or red for female or male, respectively) or the mean across talkers (black). As shown in Figure 3, some lists produced consistently better-than-average (e.g., List 1) or worse-than-average (e.g., List 18) performance for both talkers, whereas others (e.g., List 16) produced different results depending on the talker. When averaged across talkers, no sentence list produced performance that was more than 7 percentage points away from the mean, and no sentence list was outside the 95% confidence intervals. The only lists for which mean performance deviated from overall average performance by more than 5 percentage points for both talkers were Lists 1 and 18. Interestingly, in a validation study of the original BEL sentences (Rimikis et al., 2013), performance for nonnative English speakers was also better than average for List 1 and poorest for List 18. Therefore, the differences observed in nonsense Lists 1 and 18 may be due to differences in vocabulary specific to these lists, rather than any effects of word scrambling. Figure 3, and the full dataset provided via the DRUM link ( can also be used to select subsets of lists that are more closely equated for performance for a given talker, or across talkers. For example, Lists 1–5 are very well matched, as are Lists 6–20, when excluding Lists 12 and 18.
Figure 3.
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Scores for nonsense Basic English Lexicon sentences recorded by an older female talker (blue) and a young male talker (red), plotted relative to each participant's mean across lists. The red and blue bars indicate mean relative scores for each list for the male and female talkers, respectively, and black bars show averages across the two talkers. The horizontal dotted lines show the 95% confidence interval for performance on all lists across talkers.
Conclusions
A corpus of 500 syntactically correct but semantically incongruent nonsense sentences, matched for vocabulary, number of keywords, sentence length, and talker to the existing BEL corpus, was developed and validated for list equivalency in young normal-hearing listeners. In conjunction with the original BEL corpus, this new nonsense BEL corpus can be used in research related to the use of semantic context in hearing-impaired and other populations and its association with speech understanding and listening effort. With further validation in listeners with hearing loss or CIs, the lists could also be used for clinical testing. The text of the original and nonsense BEL sentences , along with the recordings from all four talkers, are available for download at
Supplementary Material
Supplemental Material S1. Datasheet overview.
Click here for additional data file. (351KB, pdf)
Supplemental Material S2. Datasheet overview.
Click here for additional data file. (158.2KB, pdf)
Acknowledgments
This work was supported by National Institute on Deafness and Other Communication Disorders Grant R01 DC012262, awarded to Andrew J. Oxenham. We thank Benjamin Munson for his advice during the development of this new corpus and for his helpful comments on an earlier version of this article.
Funding Statement
This work was supported by National Institute on Deafness and Other Communication Disorders Grant R01 DC012262, awarded to Andrew J. Oxenham.
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Supplementary Materials
Supplemental Material S1. Datasheet overview.
Click here for additional data file. (351KB, pdf)
Supplemental Material S2. Datasheet overview.
Click here for additional data file. (158.2KB, pdf)
Articles from Journal of Speech, Language, and Hearing Research : JSLHR are provided here courtesy of American Speech-Language-Hearing Association
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About Polyarteritis Nodosa (PAN)
Last Updated on February 5, 2024
Polyarteritis nodosa (PAN) is a form of vasculitis that can occur in two distinct forms: cutaneous PAN (where inflammation involves primarily the vessels of the skin) and systemic PAN (where the entire bed of small to medium arteries may be involved). Some people believe that these two forms represent the two ends of the spectrum of PAN. In both cases, inflammation (-itis) of primarily medium and occasionally smaller arteries occurs, leading to restricted blood flow to the organs and tissues that these arteries supply, including the skin, joints, kidneys, gastrointestinal tract, heart, lungs and brain, among other organs.
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Brantley, Living with PAN, Diagnosed at 6-months old
What is Polyarteritis Nodosa (PAN)?
Polyarteritis nodosa (PAN) is a form of vasculitis that can occur in two distinct forms: cutaneous PAN (where inflammation involves primarily the vessels of the skin) and systemic PAN (where the entire bed of small to medium arteries may be involved). Some people believe that these two forms represent the two ends of the spectrum of PAN. In both cases, inflammation (-itis) of primarily medium and occasionally smaller arteries occurs, leading to restricted blood flow to the organs and tissues that these arteries supply, including the skin, joints, kidneys, gastrointestinal tract, heart, lungs and brain, among other organs. Cutaneous PAN involves primarily the skin and joints, and may follow a relatively mild course. Systemic PAN affects the body more extensively, and can cause symptoms such as fever, fatigue, weakness, weight loss or poor growth, muscle and joint aches, skin abnormalities, abdominal pain, and peripheral nerve injury which could cause areas of numbness in the legs and arms. When the kidney is involved, as it often is, high blood pressure can result. In some people, the vessel damage leads to weakened artery walls, which may then balloon out, forming aneurysms.
Prompt diagnosis and treatment are essential to avoid serious complications of PAN. Initially, corticosteroids are often used in high doses along with other medications to suppress inflammation and immune-mediated attack on the vessels. After an initial response to treatment, PAN may remain quiet in some individuals, or follow a course in others that may have periods of relapse and remission. Therefore, close ongoing medical observation is important over long periods of time. In cutaneous PAN, where just skin and perhaps joints are involved, treatment usually leads to a lasting remission.
Causes
Most children who develop PAN have been previously healthy. It is not known what triggers the immune system to go awry and attack one’s own blood vessels. In some cases, a recent infection appears to trigger PAN. Streptococcus has been associated with the development of PAN, especially the cutaneous form. Hepatitis B or other viral infections may trigger PAN in some individuals. In others, it might be a reaction to medication or unknown immune triggers. People of Middle Eastern origin with a mutation in the gene encoding an inflammatory protein called pyrin have a higher rate of PAN and some other vasculitis conditions than people without these mutations. Another genetic risk factor involves a gene that encodes for adenosine deaminase 2 (ADA2). People with a mutation of this gene may present with a clinical disease that looks very much like PAN. The onset of disease in individuals with a deficiency of ADA2 (DADA2) almost always occurs during childhood.
In most cases, though, we do not know what causes PAN.
Who Gets PAN?
Polyarteritis nodosa is a rare disease with an estimated annual incidence of 3 to 4.5 cases per 100,000 persons in the United States. A study from southern Sweden reported the annual incidence for PAN was 7/100,000 children. Asian populations appear to have a higher incidence. It may be more common still in Turkey and Israel, likely because of the high rate of carriers for mutations in the pyrin gene. It is most common in adults between the ages of 45 to 65 years, but PAN has been reported in early infancy as well. Boys are more commonly affected than girls.
Symptoms
PAN can affect almost any organ system, and the symptoms can be very different from one person to another. Often, the first symptoms are non-specific, with patients reporting fever, night sweats, loss of appetite and poor weight gain or even weight loss, rash, and severe muscle and joint pains evolving over a period of weeks to months.
Common signs and symptoms
Fatigue
Loss of appetite
Numbness or tingling in hands or feet
High blood pressure
Blood in stool
Chest pain
Trouble breathing
Fever
Lacy rash, skin nodules or ulcerations
Swollen lymph nodes
Other reported signs and symptoms
Testicular pain in boys
Abdominal pain
Inflamed mucous membranes
Sudden loss of strength of hands or feet
Spleen swelling and tenderness
Strokes and encephalopathy with brain malfunction and seizures
Poor blood flow/perfusion of fingers and toes
Nodules in the subcutaneous fat layer
Making The Diagnosis
Because most patients’ symptoms include fever and malaise, extensive testing is needed to be sure infection is not the cause. Unfortunately, there is not a single lab study that makes the diagnosis of PAN.
Laboratory studies: Blood and urine cultures are often performed to rule out infection as a cause of the fever and other symptoms. A complete blood count (CBC) often shows anemia. Non-specific tests for inflammation, such as the sedimentation rate (ESR) and c-reactive protein (CRP) are often elevated in active PAN. A full chemistry panel is performed to assess kidney and liver function. A urinalysis is performed to check for blood or protein in the urine. Sometimes, a gene study to look for the presence of DADA2 will be performed in children presenting with PAN symptoms.
Imaging studies: X-rays or other specialized imaging studies, such as ultrasound, angiograms and magnetic resonance angiography (MRA), are needed to visualize the affected vessels.
Biopsy: If uncertainty remains about the cause of symptoms, a biopsy of an affected vessel or tissue usually pins down the diagnosis.
Treatment
Treatment for PAN begins with corticosteroids (either oral prednisone or prednisolone, or IV methylprednisolone) to control the inflammation, but, in severe multi-organ involvement, corticosteroids are usually combined with an immune-suppressing drug called cyclophosphamide which will lower the number of inflammatory cells that can do damage to the arteries. Other immunosuppressants, such as methotrexate, azathioprine or mycophenolate mofetil, may be used to induce or to help maintain immunosuppression. Intravenous immunoglobulin (IVIG), an infusion of pooled antibodies may be beneficial in some cases. In refractory cases, i.e., those not responding to the treatments above, medications such as rituximab or other biological medications may be given. Complications, such as hypertension, are also treated with medication. If the PAN is related to hepatitis B, antiviral medications and sometimes plasma exchange are employed. Children with cutaneous PAN usually respond well just to corticosteroids. If a preceding streptococcal infection was implicated as a possible trigger of PAN, then antibiotic (often penicillin) prophylaxis may be considered.
In 2021 the American College of Rheumatology (ACR) published guidelines for the management of certain vasculitides, that were also endorsed by the Vasculitis Foundation (VF). Clinical practice guidelines are developed to reduce inappropriate care, minimize geographic variations in practice patterns, and enable effective use of health care resources. Guidelines and recommendations developed and/or endorsed by the ACR are intended to provide guidance for particular patterns of practice and not to dictate the care of a particular patient. The application of these guidelines should be made by the physician in light of each patient’s individual circumstances. Guidelines and recommendations are subject to periodic revision as warranted by the evolution of medical knowledge, technology, and practice.
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Possible Medication Side Effects
The strong immunosuppressants used to treat the inflammatory reaction in PAN have possible side effects, including lowering your body’s ability to fight infection, weakening of the bones (osteoporosis), high blood sugar, altered fat metabolism, acne and others. It is important for your doctor to closely monitor your progress and adjust your medication to the lowest effective dose to control the disease. Prevention of infection is also very important. Talk with your doctor about ways to do this includingimmunizations, avoiding ill contacts, and sanitizing surfaces that put you at high risk for infection.
Complications
A variety of complications may occur, depending on the organ systems involved. Aneurysms in the arteries that lead to the kidneys, liver, or intestines can cause impaired kidney function, abdominal pain, nausea, vomiting or GI bleeding. Affected arteries could develop blood clots, leading to organ injury such as myocardial infarction (heart attack), or loss of skin and subcutaneous tissue. Interestingly, the lungs are often spared in PAN.
Relapse
Even with effective treatment, the disease may be characterized by fluctuations in disease activity, with periods of flares and remission. If initial symptoms return or worsen, or if new symptoms develop, report this to your physician as soon as possible. Regular check-ups, ongoing lab tests and imaging are important to detect relapses or new organ involvement.
How is PAN in children different from PAN in adults?
Compared to adults with PAN, children usually start with healthy blood vessels, so the vessel inflammation and possible resulting vessel injury is often better tolerated in children. Children have a wonderful ability to heal and to rebound from even very severe illness. In small studies comparing children with adults, children are more likely to have the milder cutaneous PAN form than adults and were less likely to have kidney and brain involvement than the adult patients. The mortality rate in children was very low (0%), but it was 13.6% in adults in one study of 22 adults and 15 children. [Erden A, Batu E, Sonmez HE, et al. Int J Rheum Dis. 2017 PMID: 28626961; DOI: 10.1111/1756-185X.13120]. In a larger report of 69 children with PAN, the relapse rate was reported to be 35%, with a mortality rate below 5%. [Eleftheriou D, Dillon MJ, Tullus K, et al. Arthritis & Rheumatism. 2013, 65:2476-2485.
Your Medical Team
Effective treatment of PAN will require the coordinated care of a team of medical providers and specialists. In addition to your primary care provider, you will need to be followed by a pediatric rheumatologist. You may also need to see other specialists: dermatologist (skin), nephrologist (kidneys), cardiologist (heart specialist), or others as needed, depending on the organs involved. You might also benefit from a physical or occupational therapist at points in your course. Chronic illness and medication side effects can put a lot of stress on a child or teen, so your needs might include an educator or a psychotherapist to assure the best outcomes.
It may be helpful to keep a health care journal to help remember your team members. In it, you can track medications and doses, your symptoms, test results, and file notes from doctor appointments all in one place. To get the most out of your doctor visits, make a list of questions beforehand and bring a supportive friend or family member to provide a second set of ears and to take notes.
Remember, it’s up to you to be your own advocate. If you have concerns with your treatment plan, speak up. Your doctor may be able to adjust your dosage or offer different treatment options. If a medication makes you feel ill, discuss it with your doctor. Also, it is always your right to seek a second opinion.
Living with Polyarteritis Nodosa
Forty years ago, PAN was a fatal disease in many cases, especially systemic PAN. Now, it is a chronic condition that can usually be well-controlled with immunosuppressant medication, but that means you must take more precautions than other children and teenagers who do not have a chronic disease. This can frustrate and discourage many people, but the upside is that, in most situations, a full, productive life is possible with the help of your medical team. Sharing your experience with family and friends, connecting with others through a support group, or talking with a mental health professional can help.
Outlook
There is a lot of research and progress world-wide to help understand what causes PAN, the genetics that might pre-dispose someone to getting the disease, and the mechanisms of the vasculitis. This has led to improvement in treatment, and therefore survival rate and improved quality of life. Still, much remains to be learned about PAN, but with the research underway leading to clinical trials of newer medications, more progress is just around the corner.
Additional Resources
Order a free pediatric vasculitis empowerment kit which contains a guide for families and a guide for teachers.
Information on current clinical trials is posted on the Internet at www.clinicaltrials.gov. All studies receiving U.S. government funding, and some supported by private industry, are posted on this government website. For information about clinical trials being conducted at the National Institutes of Health (NIH) in Bethesda, MD, contact the NIH Patient Recruitment Office:
Tollfree: (800) 411-1222
TTY: (866) 411-1010
Email:prpl@cc.nih.gov
For information about clinical trials sponsored by private sources, contact: www.centerwatc.com
Learn more about participating in research.
PRINTO (Paediatric Rheumatology International Trials Organisation) has a website with information in many languages about most pediatric rheumatic and autoimmune diseases. To choose a language other than English, go to PRINTO.it, scroll down to select the country where that language is primary (for English, choose UK), and choose “information on paediatric rheumatic diseases.” At the bottom of the page, choose “first section.” From there, click on “rare juvenile primary systemic vasculitis.” Alternatively, copy and paste the following address into your browser:
PAN Videos
0:00 0:30 Introducing Dr. Kathy McKinnon. 1:26 Pop Quiz on PAN. 6:21 Blood vessel types and functions. 6:54 What is... 24 9 YouTube Video UExyNkJ3YXU2dVNtdnRoc3hEckpGTFQ5M0tyNHBwWW5seC4xMkVGQjNCMUM1N0RFNEUx
Overview of Polyarteritis Nodosa (PAN)
Introductions 0:15 What is PAN? 2:21 Why did I get/have PAN? 4:09 How does PAN make me sick? 6:33 Artery changes in PAN. 7:37 Are... 60 25 YouTube Video UExyNkJ3YXU2dVNtdnRoc3hEckpGTFQ5M0tyNHBwWW5seC4wOTA3OTZBNzVEMTUzOTMy
VF Disease Insights Polyarteritis Nodosa
11 1 YouTube Video UExyNkJ3YXU2dVNtdnRoc3hEckpGTFQ5M0tyNHBwWW5seC41MjE1MkI0OTQ2QzJGNzNG
ACRVF Guidelines Conf PAN Breakout
The VF’s Community Heroes program uplifts people living with or loving someone with vasculitis. The 2024 heroes embody resilience, courage and... 13 7 YouTube Video UExyNkJ3YXU2dVNtdnRoc3hEckpGTFQ5M0tyNHBwWW5seC41MzJCQjBCNDIyRkJDN0VD
Nina - Vasculitis Community Hero 2024
0:00 1:26 Case presentation. 2:19 Put out the fire and treatment strategies. 3:27 Cyclophosphamide (CYC). 5:24 Cyclophosphamide... 15 2 YouTube Video UExyNkJ3YXU2dVNtdnRoc3hEckpGTFQ5M0tyNHBwWW5seC5DQUNERDQ2NkIzRUQxNTY1
Alphabet Soup of Medications
A panel of people living with vasculitis share their experiences with how their sense of self shifted after their vasculitis diagnosis. While the... 38 1 YouTube Video UExyNkJ3YXU2dVNtdnRoc3hEckpGTFQ5M0tyNHBwWW5seC45NDk1REZENzhEMzU5MDQz
Finding New Meaning After Diagnosis
0:00 3:36 Understanding the biology of steroids. 9:34 Historical background of using glucocorticoids to treat human diseases. 23:39 Side... 26 4 YouTube Video UExyNkJ3YXU2dVNtdnRoc3hEckpGTFQ5M0tyNHBwWW5seC5GNjNDRDREMDQxOThCMDQ2
How do glucocorticoids (‘steroids’) work? 2025
0:00 0:37 Introductions 2:38 Charlie Granger introduction. 5:01 Greg Patterson introduction. 9:34 The mental and emotional impact of... 19 6 YouTube Video UExyNkJ3YXU2dVNtdnRoc3hEckpGTFQ5M0tyNHBwWW5seC40NzZCMERDMjVEN0RFRThB
VF Patient Roundtable: Chronic Pain and Vasculitis 2024
0:00 6:18 What happens to the body when taking corticosteroids? 8:57 What is adrenal insufficiency? 10:34 ... 46 4 YouTube Video UExyNkJ3YXU2dVNtdnRoc3hEckpGTFQ5M0tyNHBwWW5seC5EMEEwRUY5M0RDRTU3NDJC
What You Need to Know About Adrenal Insufficiency 2024
0:00 2:40 Welcome to the new and improved VF website. 3:29 Find information about each type of vasculitis. 5:47 Exploring the... 5 0 YouTube Video UExyNkJ3YXU2dVNtdnRoc3hEckpGTFQ5M0tyNHBwWW5seC45ODRDNTg0QjA4NkFBNkQy
Researching Online: Tour of the new and improved VF Website
0:00 0:42 Introductions 2:40 Poll: How would you describe your relationship with your provider? 4:11 Why shared decision-making is... 12 4 YouTube Video UExyNkJ3YXU2dVNtdnRoc3hEckpGTFQ5M0tyNHBwWW5seC4zMDg5MkQ5MEVDMEM1NTg2
Shared Decision Making: The Doctor - Patient Relationship 2024
0:00 1:55 Introductions 3:42 Cringe Questions: “ How can you be sick when you look so good? / How bad can vasculitis... 43 13 YouTube Video UExyNkJ3YXU2dVNtdnRoc3hEckpGTFQ5M0tyNHBwWW5seC41Mzk2QTAxMTkzNDk4MDhF
VF Patient Roundtable: Dealing with Vasculitis Cringe Questions
New Medications Presenter: Dr. Nicole Orzechowski Chapel Hill 2024 Regional Conference 0:00 1:26 Avocopan 2:33 ... 7 2 YouTube Video UExyNkJ3YXU2dVNtdnRoc3hEckpGTFQ5M0tyNHBwWW5seC5EQUE1NTFDRjcwMDg0NEMz
New Medications
3:00The 2024 Chapel Hill VF Conference Vasculitis Research: What’s New, What’s on the Horizon Presenter: Vimal Derebail, MD ... 13 0 YouTube Video UExyNkJ3YXU2dVNtdnRoc3hEckpGTFQ5M0tyNHBwWW5seC41QTY1Q0UxMTVCODczNThE
Vasculitis Research: What is on the horizon?
0:00 5:47 Lab testing. 7:11 Complete Blood Count (CBC) 9:58 Complete Blood Count in Vasculitis 11:52 CBC and treatment 12:49... 54 10 YouTube Video UExyNkJ3YXU2dVNtdnRoc3hEckpGTFQ5M0tyNHBwWW5seC4yMUQyQTQzMjRDNzMyQTMy
Understanding Your Lab Results 2024
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#### VF Welcomes Jocelyn Ashford to the Board of Directors
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#### A Family’s Personal Journey Through Vasculitis
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#### Leaving a Legacy Matters
#### Two Sides of a Kidney
#### Mom’s Determined to Take Back Control Over Daughter’s Vasculitis
#### “I Had No Idea If I was Going to Live or Die” – Alex’s Teenage Journey with Vasculitis
#### Top Vasculitis-Friendly Travel Tips for the Holidays
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#### VPPRN 6-Month Check-in Forms – Community Dashboard
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#### She Came. She Saw. She Loved. Tatum Hopper’s Family Celebrates Her Life
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#### David Massicotte-Azarniouch, MD, Awarded Two-Year Grant Through VF’s Young Investigator’s Award Program
#### ALERT: Changes Coming to Medicaid and Children’s Health Insurance Program Coverage
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#### MICHAEL AGATHIS | VF Patient Hero 2022
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#### Many Outstanding Questions Still Exist for Vasculitis Patients: The Massachusetts General Hospital Rheumatology Vasculitis Program Hopes to Find Some of the Answers
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#### 2022 V-RED Winners
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#### Introducing Our 2022-2023 VCRC-VF Fellow: Mohanad M. Elfishawi, MBBCh, MS
#### Introducing Our 2020-2021 VCRC-VF Fellow: Kinanah Yaseen, MD
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#### The Vasculitis Foundation Announces Exciting New Initiative: Vasculitis-Building Outcomes, Leading Discoveries
#### Twin Siblings Specializing in Vasculitis Are Working Together at KUMC
#### Rheumatology Fellow’s Current Research Offers Improved Guidance to Women with Vasculitis in all Areas of Reproduction
#### Co-Directors of Vanderbilt’s Vasculitis Clinic say it’s Critical to Have a Multidisciplinary Approach to Care
#### 2019 VCRC/VF Fellow Joins UPMC as Vasculitis Center Director
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#### 2021 V-RED Winners
#### Arkansas Endocrinologist Wins First Place in the Vasculitis Foundation’s 2021 V-RED Award Program
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#### Journey to Diagnosis, April 2021
#### The VF Welcomes Jocelyn Ashford to the Board of Directors
#### VF Welcomes Jocelyn Ashford to the Board of Directors
#### Pinochle Night Brings Friends Together During Pandemic
#### A Family’s Personal Journey Through Vasculitis
#### A New Blueprint for Vasculitis Care: Insights from a Landmark Summit
#### Leaving a Legacy Matters
#### Two Sides of a Kidney
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190459 | https://artofproblemsolving.com/wiki/index.php/Combination?srsltid=AfmBOorLS_8eSwAL9mhBAbnNp2NzFPvXut07LZ7_SOxxjXLLx65avY8r | Art of Problem Solving
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Combination
A combination, sometimes called a binomial coefficient, is a way of choosing objects from a set of where the order in which the objects are chosen is irrelevant. We are generally concerned with finding the number of combinations of size from an original set of size
Contents
[hide]
1 Video
2 Notation
3 Formula
3.1 Derivation
4 Formulas/Identities
5 Examples
6 See also
Video
This video goes over what Permutations & Combinations are, their various types, and how to calculate each type! It serves as a great introductory video to combinations, permutations, and counting problems in general! Permutations & Combinations Video
This video is a great introduction to permutations, combinations, and constructive counting:
Notation
The common forms of denoting the number of combinations of objects from a set of objects is:
Formula
Derivation
Consider the set of letters A, B, and C. There are different permutations of those letters. Since order doesn't matter with combinations, there is only one combination of those three.( are all equivalent in combination but different in permutation.)
In general, since for every permutation of objects from elements which is .Now since in permutation the order of arrangement matters( is not same as ) but in combinations the order of arrangement does not matter( is equivalent to ).For its derivation see this video.You need combinations when order doesn't matter, and you need permutations when order does matter. Combinations and permutations are useful tools in counting.
Suppose elements are selected out.For permutation elements can be arranged in ways.We have overcounted the number of combinations by times.So We have , or .
Formulas/Identities
One of the many proofs is by first inserting into the binomial theorem. Because the combinations are the coefficients of , and a and b disappear because they are 1, the sum is .
We can prove this by putting the combinations in their algebraic form. . As we can see, . By the commutative property, . Because , by the transitive property, we can conclude that this is true for all non-negative integers n and r where n is greater than or equal to r. Another reason this is true is because that choosing what you don't want is the same as choosing what you do want, because by choosing what you don't want, you imply that you choose the rest. This identity is also the reason why Pascal's Triangle is symmetrical.
Examples
2005 AIME II Problem 1
See also
Combinatorics
Combinatorial identity
Permutations
Pascal's Triangle
Generating function
Retrieved from "
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Combinatorics
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190460 | https://www.youtube.com/watch?v=LfV0lysyTss | ALEKS: Deducing the block of an element from an electron configuration
Roxi Hulet
32400 subscribers
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Posted: 26 Jul 2023
5 comments
Transcript:
in this video I'll show you how to solve the Alex problem called deducing the block of an element from an electron configuration to solve this problem you're definitely going to want to have a periodic table handy and I really like Alex's periodic table for this problem because part of this problem is asking us to classify elements as metals nonmetals or metalloids and Alex has got some good color coding for that one of the things that we're going to be doing on this problem is identifying the block location of elements and so I want to start by marking up this periodic table to identify the different block location so this is a terminology that we use when we're talking about electron configuration the elements that are in the first two columns on the periodic table are sometimes referred to as belonging to the S block and so when we're looking at a question like this it's asking you what block is as in what block is the element located all of the elements in these these two columns these these elements here are said to be located in the S block all of the elements over here all of these guys this is the P block so any element in this area is a p-block element the transition metals are in the D Block and then last but not least the lanthanides and the actinides are all in the F block now solving this problem is going to be a lot easier for you if you know how to use the periodic table to count off or or to identify or classify an electron configuration so for example if I said use the periodic table to write the electron configuration of fluorine and you said no problem I know how to do that that's one s two two s two two P five if you understand that type of notation that's going to make this problem a lot easier because that's really the main concept here if you don't know how to do that using the periodic table to classify electron configuration I think it's probably worth your time to go back and take a look at how that is done it'll only take you a couple minutes and it's going to make this problem a lot easier so what we have over here are some generic electron configurations Alex is using some letters to just sort of represent it could be anything this g means it could be any noble gas although Alex does say lighter than radon so anything not radon um and then it's using n's and M's to represent those principal quantum numbers like 1s or or 2p or 3p or 3s or whatever it's just like just using these to kind of stand in place of the principal quantum number I think that this problem is a little bit easier to look at if we just kind of ignore those letters so we really don't need to think about them and what I'm going to do is just kind of jump in with the first example and I want to do this one first because obviously that's the easiest one to do because it's a lot smaller than the others and I think honestly that is easier if we just ignore that n pretend like it's not there so what this problem is saying is hey go to the periodic table and find every single element that has an electron configuration that has this General pattern meaning some sort of noble gas followed by some principal quantum number and then S1 anything that has that type of type of electron configuration so it could be helium to S1 it could be neon 3s1 argon 4s1 Krypton 5s1 Xenon 6s1 the problem says though not radon don't include that so um this is a generic like electron configuration for all of these elements right here helium 2s1 neon 3s1 argon 4s1 I'm not going to read through them all again it was all how far down did I go all of these elements right here have that same general electron configuration the first question that we're looking at is what block are all of those elements in these five elements right here they're all in the S block and just so that you know there's never going to be more than one option for this like all of the elements that you find when you're looking at these electron configurations they're all going to be in the same column all of them will be in the same column and the elements that are in the same column are always going to be in the same block that's just how it works so there's always only going to be one to choose in this spot um these five elements right here based on the color coding for Alex these are all metals now there are going to be situations where you may choose more than one of these options so for example possibly um you choose these three elements as matching up with an electron configuration these would be p-block elements and they could be either metalloids or Metals you could choose or maybe you would choose these three elements again P block non-metal metal and metal uh non-metal metalloid and metal so it is possible that you would choose here one two or all three of those options just based on you know where you're located on the periodic table all right so let's do let's do this one next let's just work our way from the bottom to the top again I do think it's visually easier if you ignore the ends and the M's so what are we looking at we're looking for elements that have noble gas followed by s2d10 P1 so starting with neon noble gas followed by S2 uh oh here we have a problem because we need S2 followed by d10 3s2 is not followed by D anything so we can't start with neon for this particular one let's go to argon for Argon it would be 4s2 3D 10 P1 so that would give us gallium I'm just going to kind of Mark that we could try Krypton that would give us 5s2 4D 10 5 P1 so it would give us this guy right here and that's really like we don't need to keep going like that's that's enough information we're just going to keep going down this column these are all blue so they're all metals and they are all located in the P block not so bad all right let's try our last one my last one again I'm going to ignore the n and the M that is S2 F 14. starting at uh starting at we'll start with helium 2s2 we're not getting into the F's yet so we're not starting with helium Neon 3s2 we've got to go straight from the s to the F so this is also not correct argon goes for S2 3d10 so that also is not correct because our configuration doesn't have the D electrons in it Krypton 5s2 4010 nope Xenon 6s2 4f 14. so this one for this particular configuration there's only one element that works because we can't start with radon and that is a metal and it is in the F block |
190461 | https://books.google.com/books/about/Microbiology.html?id=wUkqAQAAMAAJ | Microbiology: An Introduction - Gerard J. Tortora, Berdell R. Funke, Christine L. Case - Google Books
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Microbiology: An Introduction
Gerard J. Tortora, Berdell R. Funke, Christine L. Case
Pearson Benjamin Cummings, 2010 - Science - 812 pages
This #1 selling non-majors microbiology book is praised for its straightforward presentation of complex topics, careful balance of concepts and applications, and proven art that teaches. In its Tenth Edition, Tortora/Funke/Case responds to the #1 challenge of the microbiology course: teaching a wide range of reader levels, while still addressing reader under-preparedness. The Tenth Edition meets readers at their respective skill levels.
First, the book signals core microbiology content to readers with the new and highly visual Foundation Figures that readers need to understand before moving forward in a chapter. Second, the book gives readers frequent opportunities for self-assessment with the new Check Your Understanding questions that correspond by number to the chapter Learning Objectives. Then, a new "visual learning" orientation includes: an increased number of the popular Diseases in Focus boxes, newly illustrated end-of-chapter Study Outlines that provide students with visual cues to remind them of chapter content, and new end-of-chapter Draw It questions. The all-new art program is contemporary without compromising Tortora/Funke/Case's hallmark reputation for precision and clarity. Content revisions include substantially revised immunity chapters and an increased emphasis on antimicrobial resistance, bioterrorism, and biofilms. The new Get Ready for Microbiology workbook and online practice and assessment materials help readers prepare for the course.
The Microbial World and You, Chemical Principles, Observing Microorganisms Through a Microscope, Functional Anatomy of Prokaryotic and Eukaryotic Cells, Microbial Metabolism, Microbial Growth, The Control of Microbial Growth, Microbial Genetics, Biotechnology and Recombinant DNA, Classification of Microorganisms, The Prokaryotes: Domains Bacteria and Archaea, The Eukaryotes: Fungi, Algae, Protozoa, and Helminths, Viruses, Viroids, and Prions, Principles of Disease and Epidemiology, Microbial Mechanisms of Pathogenicity, Innate Immunity: Nonspecific Defenses of the Host, Adaptive Immunity: Specific Defenses of the Host, Practical Applications of Immunology, Disorders Associated with the Immune System, Antimicrobial Drugs, Microbial Diseases of the Skin and Eyes, Microbial Diseases of the Nervous System, Microbial Diseases of the Cardiovascular and Lymphatic Systems, Microbial Diseases of the Respiratory System, Microbial Diseases of the Digestive System, Microbial Diseases of the Urinary and Reproductive Systems, Environmental Microbiology, Applied and Industrial Microbiology . Intended for those interested in learning the basics of microbiology.
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Contents
Quick Reference Table of Contents 1
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Microbiology: An Introduction
Gerard J. Tortora,Berdell R. Funke,Christine L. Case
No preview available - 2010
Microbiology: An Introduction
Gerard J. Tortora,Berdell R. Funke,Christine L. Case
No preview available - 2009
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activityagentalgaeamino acidsanaerobicanimalsantibioticsantibodiesantigenantimicrobialarchaeaatomsbacteriabacterial cellbacteriumbiofilmsblood cellsbodycalledcancercarboncausecell wallcellularChapterCHECK YOUR UNDERSTANDINGchemicalchromosomecolicommoncontainculturecyclecytoplasmdevelopeddiseasedisinfectantdrugseffectiveelectronendosporesenergyenzymeseukaryoticexamplefeverFigureflagellafungigenegeneticgenusglucosegram-negativegram-positivegrowthhost cellhumanhydrogenidentifyimmune systeminfectioninfluenzaingestedinhibitLEARNING OBJECTIVESlipidsmacrophagesmetabolicmicrobesmicrobiologymicrobiotamicroorganismsmicroscopemoleculesmRNAnucleotidesnutrientsoccurorganismsoxidationoxygenparasitespathogenspatientspenicillinpeptidoglycanphagephagocytesphagocytosisplasma membraneplasmidpneumoniaproduceprokaryoticproteinsprotozoareactionsrecombinantreleasedreplicationresistancerespiratoryresultribosomesskinspeciesstainstrandStreptococcusstructuresurfacesymptomssynthesistemperaturetiontissuetoxintransmissiontransmittedtreatmentusuallyvaccineviralvirusviruseswww.microbiologyplace.com
Bibliographic information
Title Microbiology: An Introduction
AuthorsGerard J. Tortora, Berdell R. Funke, Christine L. Case
Edition 10, illustrated
Publisher Pearson Benjamin Cummings, 2010
Original from the University of Michigan
Digitized Sep 12, 2011
ISBN 0321550072, 9780321550071
Length 812 pages
SubjectsScience
›
Life Sciences
›
Biology
Science / Life Sciences / Biology
Science / Life Sciences / Microbiology
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190462 | https://www.quora.com/How-do-you-simplify-the-following-Boolean-functions-to-a-minimum-number-of-literals-by-using-Boolean-algebra-F-ABC-A-AB-ABC | How to simplify the following Boolean functions to a minimum number of literals by using Boolean algebra: F= A'BC+ A + AB + ABC - Quora
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Vijay Mankar
HoD (Electronics) at Government Polytechnic Nagpur · Author has 7K answers and 11.2M answer views
·4y
F=A′B C+A+A B+A B C F=A′B C+A+A B+A B C
F=A′B C+A(1+B+B C)F=A′B C+A(1+B+B C)
F=A′B C+A(1)F=A′B C+A(1)
F=A′B C+A F=A′B C+A
F=(A+A′)(A+B C)F=(A+A′)(A+B C)
F=(1)(A+B C)F=(1)(A+B C)
F=A+B C F=A+B C
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How do I reduce this Boolean function with the minimum number of literals: ABC+AB'+ABC?
How do you simplify the following Boolean function to a minimum number of literals F (ABCD) =AB+CD+ABC+AB', C+ABCD?
What is the Boolean function to require literals BC+AC'+AB+BCD?
How do I reduce the Boolean function to minimize the number of literals A'BC + ABC' + ABC + A'BC'?
How do you simplify the following Boolean functions to a minimum number of literals: predicate(a) xy + xy′?
Rick McGeer
Inventor of the MBS algorithm for logic minimization. · Author has 5.5K answers and 12.1M answer views
·5y
Originally Answered: How can I simplify the following boolean function f=a'bc+a+ab'+abc' to a minimum numberof literals? ·
Find the primes using a Karnaugh Map:
The primes are a,b c a,b c. Both of these primes are essential (b c b c is required to cover ¯a b c a¯b c, a a is required to cover a¯b¯c,a¯b c,a b¯c a b¯c¯,a b¯c,a b c¯. ). So the minimum sum-of-products form is a+b c a+b c.
Incidentally, note that this is a unate function (in simplified form, each variable appears uncomplemented), and there is a longstanding theorem than every prime implicant of a unate function is essential.
Continue Reading
Find the primes using a Karnaugh Map:
The primes are a,b c a,b c. Both of these primes are essential (b c b c is required to cover ¯a b c a¯b c, a a is required to cover a¯b¯c,a¯b c,a b¯c a b¯c¯,a b¯c,a b c¯. ). So the minimum sum-of-products form is a+b c a+b c.
Incidentally, note that this is a unate function (in simplified form, each variable appears uncomplemented), and there is a longstanding theorem than every prime implicant of a unate function is essential.
Upvote ·
Deepti Mukherjee
M.Sc.(Math) in Science&Science, Guptipara, India (Graduated 1992) · Author has 4.2K answers and 408.8K answer views
·1y
F = A'BC + A +AB + ABC
=(A'BC + ABC) + A+AB
=BC(A’+A) + A +AB
By complement law, A’+A = 1
Therefore, F = BC.1 + A + AB
= BC +A+AB [by Identity law BC.1=BC]
= BC+A(1+B)
=BC + A [ by boundness law, 1+B=1]
=A+BC [by commutative law]
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Gil Byeong Chan
Studied at Kookmin University
·6y
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How do I reduce this Boolean function with the minimum number of literals: ABC+AB'+ABC?
ABC+AB'+ABC
First, let’s remove the duplicate term( + ABC)
(It’s obvious rather than a formula, isn’t it?).
ABC+AB'
This is same as below, which is just as obvious as the above.
A(BC+B')
At this point, you have two ways to solve it.
The first is to use formula, as like the another answer.
In the case you don’t remember it, just think about the condition naturally, by using your pure brain power.
What does (BC + B') actually means?
Yes, it’s “Every cases minus BC' ”
So, it is (BC')', which is same as (B' + C).
How is (BC')' = (B' + C) possible? Just look at the image.
The marked area is literally B
Continue Reading
ABC+AB'+ABC
First, let’s remove the duplicate term( + ABC)
(It’s obvious rather than a formula, isn’t it?).
ABC+AB'
This is same as below, which is just as obvious as the above.
A(BC+B')
At this point, you have two ways to solve it.
The first is to use formula, as like the another answer.
In the case you don’t remember it, just think about the condition naturally, by using your pure brain power.
What does (BC + B') actually means?
Yes, it’s “Every cases minus BC' ”
So, it is (BC')', which is same as (B' + C).
How is (BC')' = (B' + C) possible? Just look at the image.
The marked area is literally B' or C.
Visually drawing an image in mind or on paper can give you an intuition.
Therefore,
∴A(B C′)′o r A(B′+C)∴A(B C′)′o r A(B′+C)
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How do you minimize the following Boolean expression using Boolean identities: F (A, B, C) = A‟B+ BC‟+ BC+ AB‟C‟?
How can I simplify the Boolean expression (ABC + C) using Boolean algebra?
How can I simplify this Boolean algebra expression: (A'BC + AB'C + ABC' + ABC)?
How do I simplify this Boolean expression: (AB) (AB)?
How do you simplify the following Boolean expression: F (A,B, C) =A′B+BC′+BC+AB′C′?
Kevin Green
Have enjoyed building things with code for years. · Author has 1K answers and 1.3M answer views
·4y
How do you simplify the following Boolean functions to a minimum number of literals by using Boolean algebra: F= A'BC+ A + AB + ABC?
rule 1: A + AB = A
rule 2: A + A’B = A + B
looking at last 3 terms
A + AB + ABC
AB + ABC = AB
so have A + AB which reduces to A
leaving A’BC + A = A + A’BC = A + BC
Upvote ·
9 1
Vijay Mankar
HoD (Electronics) at Government Polytechnic Nagpur · Author has 7K answers and 11.2M answer views
·5y
Originally Answered: How can I simplify the following boolean function f=a'bc+a+ab'+abc' to a minimum numberof literals? ·
f=a′b c+a+a b′+a b c′f=a′b c+a+a b′+a b c′
f=a′b c+a(1++b′+b c′)1+X=1 f=a′b c+a(1++b′+b c′)⏟1+X=1
f=a′b c+a f=a′b c+a
f=(a+a′)(a+b c)f=(a+a′)(a+b c)
f=(1)(a+b c)f=(1)(a+b c)
f=a+b c■f=a+b c◼
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Vance Faber
Studied Mathematics · Author has 2.9K answers and 1.8M answer views
·5y
Originally Answered: How can I simplify the following boolean function f=a'bc+a+ab'+abc' to a minimum numberof literals? ·
This question comes up over and over. I am going to assume + is exclusive or. Then x'=1+x, x^2=x and 2x=0, so we have
f=bc+abc+a+a+ab+abc+ab=bc.
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Sayeed Sajjad Razin
Studied Biomedical Engineering at Bangladesh University of Engineering and Technology (Graduated 2024) · Author has 80 answers and 312.8K answer views
·Updated 2y
Related
How do I reduce this Boolean function with the minimum number of literals: ABC+AB'+ABC?
ABC + AB’ + ABC
=ABC + AB’ [A + A = A]
=A(B’ + BC)
=A(B’+ B)(B’+ C) [A+BC = (A+B)(A+C)]
=A(B’ + C) [A’ + A = 1]
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Pavel Juranek
Technician, Programmer, Analyst, Consultant · Author has 3.7K answers and 1.3M answer views
·3y
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How do I reduce the Boolean function to minimize the number of literals A'BC + ABC' + ABC + A'BC'?
A'BC + ABC' + ABC + A'BC' = ……………….. commutation
(A'BC + A'BC') + (ABC' + ABC) = …………… assotiation
A'B(C + C') + AB(C' + C) = A'B + AB = …… distribution
(A' + A)B = B …………………………………………… distribution
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9 9
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Roger Larson
Author has 5K answers and 4.7M answer views
·1y
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How do you simplify the following Boolean function F (x,y, z) = x’y’z’ + x”y + xyz” +xz to a minimum number of literals using algebraic manipulation and implement?
Do you really mean
F( x,y,z) = x’y’z’ + x”y + xyz” + xz
this is
F( x,y,z) = x’y’z’ + xy + xyz + xz
since y” = y likewise z” = z
F( x,y,z) = x’y’z’ + xy + xyz + xz = x’y’z’ + xy(1 +z) + xz = x’y’z’ + xy(1) + xz = x’y’z + xy + xz
F( x,y,z) = x’y’z’ + xy + xz
this is as much as it can be reduced
Or do mean
F( x,y,z) = x’y’z’ + x’y + z’ xy + xz
the x’y term can be with
x’y = x’y(1) = x’y( z + z’j = x’yz + x’yz’
the xz term can be written
xz = xz(1) = xz(y + y’) = xzy + xzy’
F(x,y,z) = x’y’z’ + x’(y)z’ + x’yz + (z’)xy + xzy + xzy’
F(x,y,z) = x’z’(y’ + y) + y(x’z’ + x’z + z’x + xz) + xz(y + y’)
F( x, y,z) = x’z’ +
Continue Reading
Do you really mean
F( x,y,z) = x’y’z’ + x”y + xyz” + xz
this is
F( x,y,z) = x’y’z’ + xy + xyz + xz
since y” = y likewise z” = z
F( x,y,z) = x’y’z’ + xy + xyz + xz = x’y’z’ + xy(1 +z) + xz = x’y’z’ + xy(1) + xz = x’y’z + xy + xz
F( x,y,z) = x’y’z’ + xy + xz
this is as much as it can be reduced
Or do mean
F( x,y,z) = x’y’z’ + x’y + z’ xy + xz
the x’y term can be with
x’y = x’y(1) = x’y( z + z’j = x’yz + x’yz’
the xz term can be written
xz = xz(1) = xz(y + y’) = xzy + xzy’
F(x,y,z) = x’y’z’ + x’(y)z’ + x’yz + (z’)xy + xzy + xzy’
F(x,y,z) = x’z’(y’ + y) + y(x’z’ + x’z + z’x + xz) + xz(y + y’)
F( x, y,z) = x’z’ + y + xz
Upvote ·
Vijay Mankar
Author has 7K answers and 11.2M answer views
·5y
Related
How do I simplify this Boolean expression: A'BC+AB'C+A'B'C' + AB'C + AB?
A′B C+A B′C+A′B′C′+A B′C+A B A′B C+A B′C+A′B′C′+A B′C+A B
=A′B C+A′B′C′+A B′C+A B′CA B′C+A B=A′B C+A′B′C′+A B′C+A B′C⏟A B′C+A B
=A′B C+A′B′C′+A B′C+A B=A′B C+A′B′C′+A B′C+A B
=A′B′C′+A′B C+A(B+B′C)=A′B′C′+A′B C+A(B+B′C)
=A′B′C′+A′B C+A(B+B′)(B+C)=A′B′C′+A′B C+A(B+B′)(B+C)
=A′B′C′+A′B C+A(1)(B+C)=A′B′C′+A′B C+A(1)(B+C)
=A′B′C′+A′B C+A B+A C=A′B′C′+A′B C+A B+A C
=A′B′C′+C(A′B+A(A′+A)(B+A)=(B+A)+A B=A′B′C′+C(A′B+A⏟(A′+A)(B+A)=(B+A)+A B
=A′B′C′+C(B+A)+A B=A′B′C′+C(B+A)+A B
=A′B′C′+B C+C A+A B=A′B′C′+B C+C A+A B
Continue Reading
A′B C+A B′C+A′B′C′+A B′C+A B A′B C+A B′C+A′B′C′+A B′C+A B
=A′B C+A′B′C′+A B′C+A B′CA B′C+A B=A′B C+A′B′C′+A B′C+A B′C⏟A B′C+A B
=A′B C+A′B′C′+A B′C+A B=A′B C+A′B′C′+A B′C+A B
=A′B′C′+A′B C+A(B+B′C)=A′B′C′+A′B C+A(B+B′C)
=A′B′C′+A′B C+A(B+B′)(B+C)=A′B′C′+A′B C+A(B+B′)(B+C)
=A′B′C′+A′B C+A(1)(B+C)=A′B′C′+A′B C+A(1)(B+C)
=A′B′C′+A′B C+A B+A C=A′B′C′+A′B C+A B+A C
=A′B′C′+C(A′B+A(A′+A)(B+A)=(B+A)+A B=A′B′C′+C(A′B+A⏟(A′+A)(B+A)=(B+A)+A B
=A′B′C′+C(B+A)+A B=A′B′C′+C(B+A)+A B
=A′B′C′+B C+C A+A B=A′B′C′+B C+C A+A B
Upvote ·
9 3
9 1
John R. Clymer
PhD in Electrical Engineering&Mathematics, Arizona State University (Graduated 1971) · Author has 1.2K answers and 490.6K answer views
·3y
Related
How do I reduce the Boolean function to minimize the number of literals A'BC + ABC' + ABC + A'BC'?
ABC + ABC’ = AB
ABC + A’BC = BC
A’BC’ + A’BC + ABC’ = BC’ + A’B
AB + A’B = B
BC + BC’ = B
F = B
Checked by KMap
Upvote ·
9 1
Samhitha Sree
B. Tech in Electrical and Electronics Engineering&C (programming language), Bhojreddy Engineering College (Graduated 2021) · Author has 62 answers and 367.1K answer views
·6y
Related
How do I reduce this Boolean function with the minimum number of literals: ABC+AB'+ABC?
Firstly by using idempotence law ABC+ABC=ABC
the expression would be reduced to
ABC+AB'
A(BC+B')
A(B'+C) using redundant literal rule(A'+AB=A'+B )
Upvote ·
9 2
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How do I reduce this Boolean function with the minimum number of literals: ABC+AB'+ABC?
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Back To Equations of FunctionsBack
7.4
Function Notation and Linear Functions
Written by:Andrew Gloag | Melissa Kramer |
Fact-checked by:The CK-12 Editorial Team
Last Modified: Sep 01, 2025
Suppose you just purchased a used car, and the number of miles on the odometer can be represented by the equation @$\begin{align}y = x + 30000\end{align}@$, where @$\begin{align}y\end{align}@$ is the number of miles on the odometer, and @$\begin{align}x\end{align}@$ is the number of miles you have driven it. Could you convert this equation to function notation? How many miles will be on the odometer if you drive the car 700 miles?
Function Notation
So far, the term function has been used to describe many of the equations we have been graphing. The concept of a function is extremely important in mathematics. Not all equations are functions. To be a function, for each value of @$\begin{align}x\end{align}@$ there is one and only one value for @$\begin{align}y\end{align}@$.
A function is a relationship between two variables such that the input value has ONLY one unique output value.
Recall that a function rule replaces the variable @$\begin{align}y\end{align}@$ with its function name, usually @$\begin{align}f(x)\end{align}@$. Remember that these parentheses do not mean multiplication. They separate the function name from the independent variable, @$\begin{align}x\end{align}@$.
@$$\begin{align}& \quad \ input\
& \quad \ \ \ \downarrow\
& \quad \underbrace{f(x)}= y \leftarrow output\
& \ function\
& \quad \ \ box\end{align}@$$
@$\begin{align}f(x)\end{align}@$ is read “the function @$\begin{align}f\end{align}@$ of @$\begin{align}x\end{align}@$” or simply “@$\begin{align}f\end{align}@$ of @$\begin{align}x\end{align}@$.”
If the function looks like this: @$\begin{align}h(x)=3x-1\end{align}@$, it would be read @$\begin{align}h\end{align}@$ of @$\begin{align}x\end{align}@$ equals 3 times @$\begin{align}x\end{align}@$ minus 1.
Function notation allows you to easily see the input value for the independent variable inside the parentheses.
You can think of a function as a machine. You start with an input (some value), the machine performs the operations (it does the work), and your output is the answer. For example, @$\begin{align}f(x)=3x+2\end{align}@$ takes some number, @$\begin{align}x\end{align}@$, multiplies it by 3 and adds 2. As a machine, it would look like this:
When you use the function machine to evaluate @$\begin{align}f(2)\end{align}@$, the solution is @$\begin{align}f(2)=8\end{align}@$.
Let's use function notation to complete the following problems:
Consider the function @$\begin{align}f(x)=-\frac{1}{2} x^2\end{align}@$. Evaluate @$\begin{align}f(4)\end{align}@$.
The value inside the parentheses is the value of the variable @$\begin{align}x\end{align}@$. Use the Substitution Property to evaluate the function for @$\begin{align}x=4\end{align}@$.
@$$\begin{align}f(4)& =-\frac{1}{2}(4^2)\
f(4)& = -\frac{1}{2} \cdot 16\
f(4)& =-8\end{align}@$$
To use function notation, the equation must be written in terms of @$\begin{align}x\end{align}@$. This means that the @$\begin{align}y-\end{align}@$variable must be isolated on one side of the equal sign.
Rewrite @$\begin{align}9x+3y=6\end{align}@$ using function notation.
The goal is to rearrange this equation so the equation looks like @$\begin{align}y=\end{align}@$. Then replace @$\begin{align}y=\end{align}@$ with @$\begin{align}f(x)=\end{align}@$.
@$$\begin{align}9x+3y& =6 && \text{Subtract} \ 9x \ \text{from both sides}.\
3y& =6-9x && \text{Divide by} \ 3.\
y& =\frac{6-9x}{3}=2-3x\
f(x)& =2-3x\end{align}@$$
A function is defined as @$\begin{align}f(x)=6x-36\end{align}@$. Determine @$\begin{align}f(2)\end{align}@$ and @$\begin{align}f(p)\end{align}@$
Substitute @$\begin{align}x = 2\end{align}@$ into the function @$\begin{align}f(x): \ f(2)=6 \cdot 2 - 36 = 12-36=-24\end{align}@$.
Substitute @$\begin{align}x = p\end{align}@$ into the function @$\begin{align}f(x): \ f(p)=6p-36\end{align}@$.
Examples
Example 1
Earlier, you were told that the number of miles on your used car can be represented by the equation @$\begin{align}y = x + 30000\end{align}@$ where @$\begin{align}y\end{align}@$ is the number of miles on the odometer and @$\begin{align}x\end{align}@$ is the number of miles you have driven it. What is this equation converted into function notation? How many miles will be on the odometer if you drive the car 700 miles?
To write an equation in function notation, replace the @$\begin{align}y\end{align}@$ variable with @$\begin{align}f(x)\end{align}@$. The equation written in function notation would be: @$\begin{align}f(x)= x + 30000\end{align}@$ where @$\begin{align}f(x)\end{align}@$ is the number of miles on the odometer.
To figure out how many miles will be on the odometer if you drive the car 700 miles, we want to find @$\begin{align}f(700)\end{align}@$.
@$$\begin{align}f(700) = 700 + 30000\
f(700)=30700\end{align}@$$
There will be 30,700 miles on the odometer if you drive the car 700 miles.
Example 2
Rewrite the equation @$\begin{align}2y-4x=10\end{align}@$ in function notation where @$\begin{align}f(x)=y\end{align}@$, and then evaluate @$\begin{align}f(-1), f(2), f(0)\end{align}@$, and @$\begin{align}f(z)\end{align}@$.
First we need to solve for @$\begin{align}y\end{align}@$.
Adding @$\begin{align}4x\end{align}@$ to both sides gives @$\begin{align}2y=4x+10\end{align}@$, and dividing by 2 gives @$\begin{align}y=2x+5.\end{align}@$
Now we just replace the @$\begin{align}y\end{align}@$ with @$\begin{align}f(x)\end{align}@$ to get @$\begin{align}f(x)=2x+5.\end{align}@$.
Now we can evaluate @$\begin{align}f(x)=y=2x+5\end{align}@$ for @$\begin{align}f(-1), f(2), f(0)\end{align}@$, and @$\begin{align}f(z)\end{align}@$:
@$\begin{align}f(-1)=2(-1)+5=-2+5=3\end{align}@$
@$\begin{align}f(2)=2(2)+5=4+5=9\end{align}@$
@$\begin{align}f(0)=2(0)+5=5\end{align}@$
@$\begin{align}f(z)=2z+5\end{align}@$
Review
How is @$\begin{align}f(x)\end{align}@$ read?
What does function notation allow you to do? Why is this helpful?
Define function. How can you tell if a graph is a function?
In 4 – 7, tell whether the graph is a function. Explain your reasoning.
In 8-14, rewrite each equation using function notation.
@$\begin{align}y=7x-21\end{align}@$
@$\begin{align}6x+8y=36\end{align}@$
@$\begin{align}x=9y+3\end{align}@$
@$\begin{align}y=6\end{align}@$
@$\begin{align}d=65t+100\end{align}@$
@$\begin{align}F=1.8C+32\end{align}@$
@$\begin{align}s=0.10(m)+25,000\end{align}@$
In 15 – 19, evaluate @$\begin{align}f(-3), f(7), f(0)\end{align}@$, and @$\begin{align}f(z)\end{align}@$.
@$\begin{align}f(x)=-2x+3\end{align}@$
@$\begin{align}f(x)=0.7x+3.2\end{align}@$
@$\begin{align}f(x)=\frac{5(2-x)}{11}\end{align}@$
@$\begin{align}f(t)=\frac{1}{2} t^2+4\end{align}@$
@$\begin{align}f(x)=3-\frac{1}{2} x\end{align}@$
Review (Answers)
Click HERE to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option.
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190464 | https://www.youtube.com/watch?v=7D7khBC4gZE | Example: Inverse of 3x3 Matrix using Gaussian Elimination
The Ryder Project
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so here were given our matrix a and while asks three questions about it so the first one is to find the determinant the second is to state why it has an inverse and the third is to actually find that inverse using the Gaussian elimination technique so let's start by doing the determinant of our matrix a so in order to do this we need to pick either a row or a column to expand around and usually you want to pick one that's got a lot of zeros in it to make your life a bit easier so that means if I look at this matrix the only one that has zeros in it is the second row so that's what I'm going to pick to do my expansion about so let me just write out our matrix cool so if I'm going to expand around our second row here we know that the top left element is treated as like a positive one we do the determinant and then it alternates as we move away from it so that means that this one here would be a negative we all say it again as we move across so negative this would be positive that one and then the zero would be back to negative all right so the first thing that we pick out if I using the second row is our zero element and it would be like negative zero and we would multiply by the determinant of what's left over but since it's zero there's not much point so let's move on to the next element which is going to be positive and it's a 1 and we need to multiply by the determinant of what's left over when we cover over the row in the column selphie ignore these for a moment oK we've covered over them all that were left with is 1 negative 1 negative 1 2 okay and then the final element that we have is 0 and it would be negative again since it's 0 there's not much point writing in the rest of the determinant calculation because it's just going to drop to 0 anyway all right so now we just need to expand out the determinant of a 2 by 2 matrix and that's really easy because it's that a - BC equation so one stays out the front ad is going to be 1 times 2 minus BD so that's going to be the negative 1 times negative 1 so in here we're going to get 2 and then we've got negative 1 times negative 1 which is positive 1 but then it's minus Dawei so minus positive 1 so we end up with an answer of 1 for the determinant of our matrix so that answers the first part here so next up we need to state why our matrix a has an inverse and there's two kind of requirements for an inverse so an inverse exists if it's got two things the first one is you have a square matrix okay so if we look at our matrix a it is indeed a square matrix it's a 3x3 square ok so we've definitely got that one and the second requirement is that the determinant of our matrix so put in a here does not equal zero okay anything other than zero and it's good so we've got one here so that means that we've addressed that requirement as well so since we've met both of these criteria then our matrix a has an inverse so maybe I would conclude this with criteria met for a cool all right so the last little bit is where and probably the bulk of the working is so we need to find the inverse of a using Gaussian elimination so in order to do that we need to set up our matrix that we're going to perform operations on so for Gaussian elimination we set it up with this kind of look so we have the matrix a on the left and I is the identity matrix on the right so let's pop it in so 1 2 negative 1 0 1 0 and negative 1 1 2 and our identity matrix remember that's why we have the ones on the main diagonal and zeros everywhere else to fill it out cool so that's what I've got there so now all we need to do is perform our row operations to get the left-hand side looking like this identity matrix and then whatever falls out on the right-hand side becomes our inverse now I'm gonna follow the pattern where I move around in this shape in order to solve for my inverse okay there are other ways to do it but the way that I'm gonna do it now is one that is always going to work so sometimes you might be able to shortcut it using a different method but yeah I'm just gonna go with this generic kind of one so if we're working in this pattern the first one that we look at is top left and what we want this to be is a 1 remember we're going for our identity matrix that looks like this okay so top left needs to be a 1 and we've got that the second one here that I'm working on is a 0 and again it's already looking like what it means to be so we don't need to do any operation on it so if we jump down to the next one though at the bottom here it's a negative 1 and we want it to be a 0 okay it's a bit messy but yeah we want it to be a 0 so we're going to need to perform an operation on it so what I would suggest is operating on Row 3 if I add on Row 1 to Row 3 I shall get negative 1 plus 1 and that goes to 0 which is what I'm looking for so let's do that all right so that means that the first row sorry isn't changing and neither is the second row so just copied those out but the third row is changing now so negative 1 that plus 1 gives us 0 1 plus 2 gives us 3 2 this is plus negative 1 it's going to take us to 1 0 plus 1 is 1 0 plus 0 is 0 and one plus zero is one all right so this one's fixed up so I'm going to move on to this one and we want it to be a zero okay so we have kind of two options on how we can get this to be a zero we can either add some multiple of Row two or we could add some multiple of Row one now I would suggest using row to here that's because we're going to end up with zero plus anything times zero is zero so this cell that we just fixed up isn't going to change however if you use Row one in your operation you're going to get zero plus or minus something times the 1 so it is going to change and then you're gonna have to go back and fix it again okay so based on that I'm going to use Row 2 in our operation that's meant to say R 3 so using our two if I do my bro 3 and I take away three of our two that should get me three minus three times one which goes to zero so that's what I'm looking for so again the first two rows and not changing so I'm just going to copy them out and now this third row by going to change so we've got zero minus three times zero which leaves us with zero three minus three times one which takes us to zero we've got one minus three times zero which would leave us with one we've got one minus three times zero again which should leave us with one with what zero minus one times three that should take us to negative three and we've got one minus three times zero which to just leave us with one awesome so now this one is fixed so moving on to our next one we have a 1 in here now that's actually what we were looking for if we go back and look at our reference we want to rock one in the bottom-right corner so that one's good don't need to do anything so next if I move up to this one here we've got a zero again that's what I'm already looking for in that so so I don't need to do anything with it so jumping up again get this top right we want this to be a zero so we are going to have to perform some operation to get there so what I would suggest this time is if I do this top row plus the bottom o negative one plus one takes me to zero so I should be able to get that back to where I want it so we're operating on Row one we want Row 1 plus Row 3 so let's go down and perform that so 1 plus 0 gives us 1 2 plus 0 gives us 2 negative 1 plus 1 gives us 0 I've got 1 plus 1 which is 2 0 plus negative 3 which is negative 3 and 0 plus 1 which is 1 so everything else remains the same because we haven't operated on it it's a bit neater cool all right so moving back here we have a 2 so we want to get this to to be a 0 that's the top middle here so what I would suggest for this one we're going to operate on Row 1 and if we do Row 1 minus 2 Row 2 we should be able to get this to go to 0 all right so Row 1 minus 2 times 0 so we're going to get um this just ends up being 1 again so 2 minus 2 times 1 that takes us to 0 0 minus 2 times 0 is 0 2 minus 2 times 0 is 2 negative 3 minus 2 times 1 that's going to take us to negative 5 1 minus 2 times 0 that's going to leave us with 1 again everything else just remains the same because we didn't operate on it cool alright so now looking at our last element we can see that this is ended up becoming a 1 and of course that is what we were looking for for it so now the entire left-hand side here we can say that that is the identity matrix and that means that the right-hand side is now the inverse of a matrix so we can conclude our final answer which was what is a inverse and we just need to copy this out so 2 negative 5 1 0 1 0 1 negative 3 1 that there is the end of the question so that's all there is for this video |
190465 | https://www.bbc.co.uk/bitesize/guides/z3cfxfr/revision/4 | The force-extension graph - How does the particle model relate to material under stress? - GCSE Combined Science Revision - OCR 21st Century - BBC Bitesize
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How does the particle model relate to material under stress?The force-extension graph
Forces are responsible for changing the motion of objects. If more than one force is present, the shape of an object can also be changed.
Part ofCombined ScienceMatter - Models and explanations
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Deformation
Extension
Springs
The force-extension graph
Practical - the effect of forces on springs
The force-extension graph
A force-extension graph can be used to calculate the work done in joules when stretching a spring.
Force should be plotted on the vertical (y) axis
Extension should be plotted on the horizontal (x) axis
As long as the limit of proportionality is not exceeded this should produce a straight line graph that passes through the origin as shown below.
In a force-extension graph:
the gradient is the spring constant
the area under the line is the work done in stretching the spring
Example:
Question
Using the graph, calculate the work done to extend the spring from 0 m to 0.10 m.
Show answer Hide answer
The area under the line is a triangle:
Areas = 1 2 × base × height
= 1 2 × 0.10 × 5 = 0.25 J
Next page Practical - the effect of forces on springs
Previous page Springs
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How does energy transform matter? - OCR 21st Century
How does the particle model explain the effects of heating?
States of matter: interactive activity - OCR 21st Century
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190466 | https://www.sciencedirect.com/topics/biochemistry-genetics-and-molecular-biology/ectoderm | Skip to Main content
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Chapters and Articles
You might find these chapters and articles relevant to this topic.
Review article
Mechanical modulation of osteochondroprogenitor cell fate
2008, The International Journal of Biochemistry & Cell BiologyMelissa L. Knothe Tate, ... Ulf R. Knothe
The ectoderm defines the outer edges or boundaries of the developing limb. As a tightly organized band of tissue that envelops the mesoderm, the ectoderm influences the balance of forces within the mesoderm. This tissue level force balance further influences force balances within individual cells, and ultimately transduces mechanical forces to the nucleus, where adaptation is manifested through modulation of gene activity. Interestingly, aggregation or lack of dissipation of cells around a common locus would likely result in compressive stresses within the condensation that would be equilibrated by the tensile stresses of the enveloping ectoderm (Fig. 2A). A mesenchymal condensation associated with increased cell division would result in a net expansion of tissue, creating a tensile force on the ectoderm, due to hoop stresses pushing the bounding envelope outwards (Fig. 2B). The differing effects of aggregation versus expansion may influence whether an osteogenic versus a chondrogenic condensation forms per se. Taking this a step further, the effect of aggregating cells versus expanding cell mass likely exert further influence on the subsequent persistence of epithelial mesenchymal transitions, perhaps influencing stem cell recruitment from the epithelium as well as definition of apoptotic zones during patterning and further morphogenetic events. Loci of condensation are bounded by zones of apopototic noncondensing mesenchyme, which delineate the individual elements of multi-element skeletal organs, e.g. the multiple bones and cartilages making up the hand and the vertebral column. In addition, areas of loose, noncondensing mesenchyme are areas observed to show increased vascularity (Ekblom et al., 1994), perhaps due to diminished cell viability in these areas or because these areas of less dense tissue provide less resistance to vascular invasion than the more dense tissue of the condensed mesenchyme.
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Journal2008, The International Journal of Biochemistry & Cell BiologyMelissa L. Knothe Tate, ... Ulf R. Knothe
Review article
CDH2 and CDH11 act as regulators of stem cell fate decisions
2015, Stem Cell ResearchStella Alimperti, Stelios T. Andreadis
Ectodermal lineage
The ectoderm is the first germ layer to emerge during gastrulation. In vertebrates, the ectoderm is responsible for the formation of the nervous system and spinal cord. The nervous system is formed during neurulation, when the neural tube is transformed into a primitive structure and eventually into the central nervous system. Early in neural tube development, the notochord and the dorsal aorta do not express CDH11, which is expressed during the later stages of neural tube formation and is important for brain and spinal cord development (Suzuki et al., 1997; Marthiens et al., 2002a). CDH11 is expressed in the limbic system of the brain, particularly in the hippocampus where it is thought to participate in the organization and stabilization of synaptic connections (Manabe et al., 2000). It is also expressed in the peripheral nervous system and, in particular, in motor and sensory axons during the period of active nerve elongation and path finding (Marthiens et al., 2002b; Padilla et al., 1998). CDH2 is present during neuroectoderm formation and is important for nervous system development (Kadowaki et al., 2007; Redies, 2000). CDH2 knockout mice die on day 10 of gestation due to heart defects and malformed neural tubes, although tissue development appears normal up to this stage (Radice et al., 1997). Others reported that CDH2 is involved in neuronal circuit maturation by contributing to axonal extension (Kimura et al., 1995). Finally, both CDH2 and CDH11 were shown to regulate neurite outgrowth through FGFR (Boscher and Mege, 2008),phosphoinositide phospholipase C (PLC) and CAM kinase pathways (Bixby et al., 1994; Riehl et al., 1996).
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Journal2015, Stem Cell ResearchStella Alimperti, Stelios T. Andreadis
Review article
Why the embryo still matters: CSF and the neuroepithelium as interdependent regulators of embryonic brain growth, morphogenesis and histiogenesis
2009, Developmental BiologyAngel Gato, Mary E. Desmond
There are three embryological tissues, ectoderm, mesoderm and endoderm that form the major tissues and organs of the vertebrate body. Ectoderm, the outer layer of early embryo differentiates into neural ectoderm, neural crest and skin ectoderm.
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Journal2009, Developmental BiologyAngel Gato, Mary E. Desmond
Review article
Application of hiPSCs in tooth regeneration via cellular modulation
2021, Journal of Oral BiosciencesHan Ngoc Mai, ... Han-Sung Jung
3.3 Ectodermal tissue regeneration
Ectoderm is the outermost layer of the three germ layers, which gives rise to numerous outer layers of the body, including the epidermis, hair, nails, oral epithelium, cornea, and olfactory epithelium. The central and peripheral nervous systems are derived from ectoderm .
The differentiation of iPSCs into embryonic ectoderm and its derivatives has been shown to exhibit immense potential in wound healing and tissue regeneration for surface ectodermal appendages, such as hair follicles, mammary glands, salivary glands, teeth, and neural ectoderm . Based on our understanding of embryonic development, we can determine the type of growth signaling molecules that must be employed to generate a specific organ. Some well-known signaling pathways responsible for further differentiation of the surface ectoderm include BMP2, WNT/β-catenin, ectodysplasin (Eda)/NF-κB, FGF, SHH, and TGF-β signaling pathways . A combination of iPSC ectodermal precursor cells in keratinocyte culture medium with RA, BMP, and WNT3a has been shown to transform cells into new dermal papilla cells . In vitro direct differentiation of hiPSCs into epithelial stem cells has been achieved previously using precise temporal regulation of the epidermal growth factor (EGF), RA, and BMP signals. Accordingly, it was re-established in vivo using the epithelium of the hair follicle and the interfollicular epidermis .
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Journal2021, Journal of Oral BiosciencesHan Ngoc Mai, ... Han-Sung Jung
Chapter
Scientific Fundamentals of Biotechnology
2011, Comprehensive Biotechnology (Third Edition)S.K.W. Oh, A.B.H. Choo
Glossary
Ectoderm
: The outer layer of cells in a developing embryo, which comprises mainly of the neural and epithelial lineages.
: The inner layer of cells that comprises the endocrine organs such as liver, pancreas, kidney, and lung.
: The middle layer of cells in the developing embryo that comprises muscle, bone, cartilage, heart, and blood lineages.
Multipotent
: Cells capable of differentiation to a more limited range of cell types (e.g., either the ectoderm, mesoderm, or endoderm layers).
Niche
: The anatomical location in adult tissues where stem cells reside.
Pluripotent
: Cells capable of differentiation into all three germ layers and all tissue types of the body.
Stem cells
: Cells capable of both unlimited self-renewal and differentiation to other cell types of the body.
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Reference work2011, Comprehensive Biotechnology (Third Edition)S.K.W. Oh, A.B.H. Choo
Review article
genes, neural crest cells and branchial arch patterning
2001, Current Opinion in Cell BiologyPaul A Trainor, Robb Krumlauf
Similar to the neuroepithelium, it has been suggested that the ectoderm is regionalized into territories, called ectomeres, that contribute to specific regions of the branchial arches . Currently, there is no evidence to support the idea that each ectomere represents a functional developmental unit. In contrast, however, there is evidence that the surface ectoderm plays a major role in the induction of odontogenesis during branchial arch development . The oral ectoderm of the first branchial arch directly regulates the patterning of the underlying neural crest mesenchyme into teeth and the ability to respond to these instructive or inducing signals is not confined to first arch neural crest cells . Fgf8, which is expressed in the anterior surface ectoderm of the first arch, is essential for determining the polarity of the branchial arch . Hence, the surface ectoderm plays an important role in patterning the branchial arch derivatives (Fig. 3).
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Journal2001, Current Opinion in Cell BiologyPaul A Trainor, Robb Krumlauf
Chapter
The Limb Field and the AER
2005, Bones and CartilageBrian K. Hall
It is most unfortunate that Saunders named the AER an ectodermal and not an epithelial ridge. The stage in development when the AER appears is way beyond any stage when ectoderm is still present, ectoderm being the name of a germ layer, epithelium (epithelia) being a term for the type of cellular organization in which cells exist as a layer(s) of connected cells on a basement membrane. Similarly, limb mesenchyme is often, perhaps usually, referred to as limb mesoderm. However, the embryonic stages when limb buds arise are well beyond the germ-layer stage, and mesoderm is a germ layer.
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Book2005, Bones and CartilageBrian K. Hall
Chapter
The Limb Field and the AER
2005, Bones and CartilageBrian K. Hall
Mesoderm, ectoderm and endoderm are three germ layers; epithelia and mesenchyme are two types of cellular organization. Mesoderm ≠ mesenchyme, nor does ectoderm ≠ epithelium.
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Book2005, Bones and CartilageBrian K. Hall
Review article
The Neural Crest: 150 years after His' discovery
2018, Developmental BiologyPatrick Pla, Anne H. Monsoro-Burq
1 Introduction
During animal development, the ectoderm, i.e. the outer embryonic germ layer, progressively undergoes a refined regional patterning essential for the development of many cells and organs. In the dorsalmost aspect of the gastrula and neurula-stage vertebrate embryo, the ectoderm forms the neural plate, which generates the neural tube and ultimately the central nervous system. More laterally and ventrally, the nonneural ectoderm forms the epidermis and the associated glands. At an early stage of ectoderm patterning, the frontier between the neural plate ectoderm (NP) and the nonneural ectoderm (NNE) is not a sharp line but rather a broad area called the neural (plate) border (NB). Definitive cell fates are not yet segregated in this area and the NB domain contributes to the dorsal neural tube, the neural crest (NC), the NNE and, anteriorly, the placode progenitors. This region has been defined by lineage tracing experiments and by the expression of a combination of marker genes.
Many vertebrate-specific anatomical features derive from neural border progenitors through the development of neural crest cells and placodes. The NC gives rise to most neurons and glia of the peripheral nervous system, to a large part of the craniofacial skeleton and mesenchyme, to the melanocytes and other pigment cells, to the outflow tract of the heart, and to adrenal medulla cells. Placodes form the olfactory epithelium, the inner ear, some neurons of the cranial ganglia, and the lens of the eye. Therefore, during their early development, the NC and placode cells share the ability to migrate and generate multiple cell types including sensory neurons and glial cells (for an extensive comparison between these two populations see Baker and Bronner-Fraser, 2001; Schlosser, 2008). Understanding the multi-step Gene Regulatory Network (GRN) that governs NB and early NC development is critical for understanding the molecular mechanisms underlying human diseases, notably affecting pigmentation, craniofacial development, peripheral nervous system and heart development. This review presents the current understanding on NB development, including NB induction and specification during early development, and the induction and segregation of the NC cell fate.
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Journal2018, Developmental BiologyPatrick Pla, Anne H. Monsoro-Burq
Review article
Invited Review - Odontology
2014, Experimental Cell ResearchAdrien Naveau, ... Ophir D. Klein
Abstract
The vertebrate ectoderm gives rise to organs that produce mineralized or keratinized substances, including teeth, hair, and claws. Most of these ectodermal derivatives grow continuously throughout the animal׳s life and have active pools of adult stem cells that generate all the necessary cell types. These organs provide powerful systems for understanding the mechanisms that enable stem cells to regenerate or renew ectodermally derived tissues, and remarkable progress in our understanding of these systems has been made in recent years using mouse models. We briefly compare what is known about stem cells and their niches in incisors, hair follicles, and claws, and we examine expression of Gli1 as a potential example of a shared stem cell marker. We summarize some of the features, structures, and functions of the stem cell niches in these ectodermal derivatives; definition of the basic elements of the stem cell niches in these organs will provide guiding principles for identification and characterization of the niche in similar systems.
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Journal2014, Experimental Cell ResearchAdrien Naveau, ... Ophir D. Klein
Related terms:
Stem Cell
Mesoderm
Germ Layer
Neural Plate
Neural Crest
Epiblast
Gastrulation
Embryonic Stem Cell
Mouse
Endoderm
View all Topics
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190467 | https://medium.com/@anonymsdc123/binomial-coefficient-744f152d4919 | Sitemap
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Binomial coefficient
Vikesh Laharpure
10 min readJun 8, 2022
“nCk” redirects here. For other uses, see NCK (disambiguation).
The binomial coefficients can be arranged to form Pascal’s triangle, in which each entry is the sum of the two immediately above.
Visualisation of binomial expansion up to the 4th power
In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers n ≥ k ≥ 0 and is written {\displaystyle {\tbinom {n}{k}}.}
It is the coefficient of the xk term in the polynomial expansion of the binomial power (1 + x)n, and is given by the formula
For example, the fourth power of 1 + x is and the binomial coefficient is the coefficient of the x2 term in successive rows. it gives a triangular array called Pascal’s triangle, satisfying the recurrence relation
The binomial coefficients occur in many areas of mathematics, and especially in combinatorics. The symbol is usually read as “n choose k” because there are ways to choose an (unordered) subset of k elements from a fixed set of n elements. For example, there are ways to choose 2 elements from the binomial coefficients can be generalized for any complex number z and integer k ≥ 0, and many of their properties continue to hold in this more general form.
History and notation
Andreas von Ettingshausen introduced the notation in 1826, although the numbers were known centuries earlier (see Pascal’s triangle). The earliest known detailed discussion of binomial coefficients is in a tenth-century commentary, by Halayudha, on an ancient Sanskrit text, Pingala’s Chandaḥśāstra. The second earliest description of binomial coefficients is given by Al-Karaji. In about 1150, the Indian mathematician Bhaskaracharya gave an exposition of binomial coefficients in his book Līlāvatī.
Alternative notations include C(n, k), nCk, nCk, Ckn, Cnk, and Cn,k in all of which the C stands for combinations or choices. Many calculators use variants of the C notation because they can represent it on a single-line display. In this form the binomial coefficients are easily compared to k-permutations of n, written as P(n, k), etc.
Definition and interpretations
The first few binomial coefficients on a left-aligned Pascal’s triangle.
For natural numbers (taken to include 0) n and k, the binomial coefficient can be defined as the coefficient of the monomial Xk in the expansion of (1 + X)n. The same coefficient also occurs (if k ≤ n) in the binomial formula(valid for any elements x, y of a commutative ring), which explains the name “binomial coefficient”.
Another occurrence of this number is in combinatorics, where it gives the number of ways, disregarding order, that k objects can be chosen from among n objects; more formally, the number of k-element subsets (or k-combinations) of an n-element set. This number can be seen as equal to the one of the first definition, independently of any of the formulas below to compute it: if in each of the n factors of the power (1 + X)n one temporarily labels the term X with an index i (running from 1 to n), then each subset of k indices gives after expansion a contribution Xk, and the coefficient of that monomial in the result will be the number of such subsets. This shows in particular that is a natural number for any natural numbers n and k. There are many other combinatorial interpretations of binomial coefficients (counting problems for which the answer is given by a binomial coefficient expression), for instance the number of words formed of n bits (digits 0 or 1) whose sum is k is given by , while the number of ways to write where every ai is a nonnegative integer is given by most of these interpretations are easily seen to be equivalent to counting k-combinations.
Integer-valued polynomials[edit]
Main article: Integer-valued polynomial
Each polynomial is integer-valued: it has an integer value at all integer inputs. (One way to prove this is by induction on k, using Pascal’s identity.) Therefore, any integer linear combination of binomial coefficient polynomials is integer-valued too. Conversely, (4) shows that any integer-valued polynomial is an integer linear combination of these binomial coefficient polynomials. More generally, for any subring R of a characteristic 0 field K, a polynomial in K[t] takes values in R at all integers if and only if it is an R-linear combination of binomial coefficient polynomials.
Sums of the binomial coefficients[edit]
The formula says the elements in the nth row of Pascal’s triangle always add up to 2 raised to the nth power. This is obtained from the binomial theorem by setting x = 1 and y = 1. The formula also has a natural combinatorial interpretation: the left side sums the number of subsets of {1, …, n} of sizes k = 0, 1, …, n, giving the total number of subsets. (That is, the left side counts the power set of {1, …, n}.) However, these subsets can also be generated by successively choosing or excluding each element 1, …, n; the n independent binary choices (bit-strings) allow a total of choices. The left and right sides are two ways to count the same collection of subsets, so they are equal.
In programming languages[edit]
The notation is convenient in handwriting but inconvenient for typewriters and computer terminals. Many programming languages do not offer a standard subroutine for computing the binomial coefficient, but for example both the APL programming language and the (related) J programming language use the exclamation mark: k ! n. The binomial coefficient is implemented in SciPy as scipy.special.comb.
from math import factorialdef binomial_coefficient(n: int, k: int) -> int: return factorial(n) // (factorial(k) factorial(n - k))
are very slow and are useless for calculating factorials of very high numbers (in languages such as C or Java they suffer from overflow errors because of this reason). A direct implementation of the multiplicative formula works well:
def binomial_coefficient(n: int, k: int) -> int: if k < 0 or k > n: return 0 if k == 0 or k == n: return 1 k = min(k, n - k) # Take advantage of symmetry c = 1 for i in range(k): c = c (n - i) / (i + 1) return c
(In Python, range(k) produces a list from 0 to k−1.)
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Pascal’s rule provides a recursive definition which can also be implemented in Python, although it is less efficient:
def binomial_coefficient(n: int, k: int) -> int: if k < 0 or k > n: return 0 if k > n - k: # Take advantage of symmetry k = n - k if k == 0 or n <= 1: return 1 return binomial_coefficient(n - 1, k) + binomial_coefficient(n - 1, k - 1)
The following implementation uses all these ideas
(define (binomial n k);; Helper function to compute C(n,k) via forward recursion (define (binomial-iter n k i prev) (if (>= i k) prev (binomial-iter n k (+ i 1) (/ ( (- n i) prev) (+ i 1)))));; Use symmetry property C(n,k)=C(n, n-k) (if (< k (- n k)) (binomial-iter n k 0 1) (binomial-iter n (- n k) 0 1)))
Implementation in the C language:
```
include unsigned long binomial(unsigned long n, unsigned long k) { unsigned long c = 1, i; if (k > n-k) // take advantage of symmetry k = n-k; for (i = 1; i <= k; i++, n--) { if (c/i >= ULONG_MAX/n) // return 0 on potential overflow return 0; c = c / i n + c % i n / i; // split c n / i into (c / i i + c % i) n / i } return c;}
```
The binomial coefficient is the number of ways of picking
unordered outcomes from possibilities, also known as a combination or combinatorial number. The symbols and are used to denote a binomial coefficient, and are sometimes read as “ choose .” therefore gives the number of k-subsets possible out of a set of distinct items. For example, The 2-subsets of are the six pairs and, so
. The number of lattice paths from the origin
to a point) is the binomial coefficient (Hilton and Pedersen 1991). The value of the binomial coefficient for nonnegative integers and is given by where denotes a factorial, corresponding to the values in Pascal’s triangle.
Writing the factorial as a gamma function allows the binomial coefficient to be generalized to noninteger arguments (including complex and) as Binomial coefficients for nonnegative integer therefore give a polynomial in where is a Pochhammer symbol. These rational coefficients are sometimes known as “generalized binomial coefficients.”
Using the gamma function symmetry formula for integer
and complex , this definition can be extended to negative integer arguments, making it continuous at all integer arguments as well as continuous for all complex arguments except for negative integer and noninteger, in which case it is infinite (Kronenburg 2011). This definition, given by (6) for negative integer and integer is in agreement with the binomial theorem, and with combinatorial identities with a few special exceptions (Kronenburg 2011).
The binomial coefficient is implemented in the Wolfram Language as Binomial[n, k], which follows the above convention starting in Version 8.
Plotting the binomial coefficient in the -plane (Fowler 1996) gives the beautiful plot shown above, which has a very complicated graph for negative and is therefore difficult to render using standard plotting programs. For a positive integer, the binomial theorem gives The finite difference analog of this identity is known as the Chu-Vandermonde identity. A similar formula holds for negative integers,
There are a number of elegant binomial sums. The binomial coefficients satisfy the identities The product of binomial coefficients is given by where is a hyperfactorial and is a factorial. As shown by Kummer in 1852, if is the largest power of a prime that divides, where and are nonnegative integers, then is the number of carries that occur when is added to in base
(Graham et al. 1989, Exercise 5.36, p. 245; Ribenboim 1989; Vardi 1991, p. 68). Kummer’s result can also be stated in the form that the exponent of a prime dividing is given by the number of integers for which where denotes the fractional part of . This inequality may be reduced to the study of the exponential sums , where is the Mangoldt function. Estimates of these sums are given by Jutila (1973, 1974), but recent improvements have been made by Granville and Ramare (1996).
R. W. Gosper showed that for all primes, and conjectured that it holds only for primes. This was disproved when Skiena (1990) found it also holds for the composite number . Vardi (1991, p. 63) subsequently showed that is a solution whenever is a Wieferich prime and that if with is a solution, then so is . This allowed him to show that the only solutions for composite are 5907, and, where 1093 and 3511 are Wieferich primes. Consider the binomial coefficients, the first few of which are 1, 3, 10, 35, 126, … (OEIS A001700). The generating function is These numbers are squarefree only for , 3, 4, 6, 9, 10, 12, 36, … (OEIS A046097), with no others known. It turns out that is divisible by 4 unless. belongs to a 2-automatic set , which happens to be the set of numbers whose binary representations contain at most two 1s: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 16, 17, 18, … (OEIS A048645). Similarly, is divisible by 9 unless belongs to a 3-automatic set, consisting of numbers for which the representation of in ternary consists entirely of 0s and 2s (except possibly for a pair of adjacent 1s). The initial elements of are 1, 2, 3, 4, 6, 7, 9, 10, 11, 12, 13, 18, 19, 21, 22, 27, … (OEIS A051382). If is squarefree, then must belong to . It is very probable that is finite, but no proof is known. Now, squares larger than 4 and 9 might also divide, but by eliminating these two alone, the only possible for are 1, 2, 3, 4, 6, 9, 10, 12, 18, 33, 34, 36, 40, 64, 66, 192, 256, 264, 272, 513, 514, 516, 576 768, 1026, 1056, 2304, 16392, 65664, 81920, 532480, and 545259520. All of these but the last have been checked, establishing that there are no other such that is squarefree for Erdős showed that the binomial coefficient with is a power of an integer for the single case (Le Lionnais 1983, p. 48). Binomial coefficients are squares when is a triangular number, which occur for 6, 35, 204, 1189, 6930, … (OEIS A001109). These values of have the corresponding values 9, 50, 289, 1682, 9801, … (OEIS A052436). The binomial coefficients are called central binomial coefficients, where is the floor function, although the subset of coefficients is sometimes also given this name. Erdős and Graham (1980, p. 71) conjectured that the central binomial coefficient is never squarefree for , and this is sometimes known as the Erdős squarefree conjecture. Sárkőzy’s theorem (Sárkőzy 1985) provides a partial solution which states that the binomial coefficient is never squarefree for all sufficiently large (Vardi 1991). Granville and Ramare (1996) proved that the only squarefree values are and 4. Sander (1992) subsequently showed that are also never squarefree for sufficiently large as long as is not “too big.”
For and distinct primes, then the function (◇) satisfies (Vardi 1991, p. 66). Most binomial coefficients with have a prime factor , and Lacampagne et al. (1993) conjecture that this inequality is true for all , or more strongly that any such binomial coefficient has least prime factor or with the exception for which , 19, 23, 29 (Guy 1994, p. 84). The binomial coefficient (mod 2) can be computed using the XOR operation XOR, making Pascal’s triangle mod 2 very easy to construct. Sondow (2005) and Sondow and Zudilin (2006) noted the inequality for a positive integer and a real number.
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190468 | https://www.broward.edu/students/asc/central-asc/_docs/functioncomposition.pdf | Composite Functions Given two functions, combine them in a way such that the outputs of one function become the inputs for the other, making it a composite function. (𝑓 ∘ 𝑔)(𝑥) = 𝑓(𝑔(𝑥)) OR (𝑓 ∘ 𝑔)(𝑥) = 𝑓 of g of x Evaluating Composite Functions Evaluate the function on the right side, and then substitute that result into the other function to find the answer. Example: Given 𝑓(𝑥) = 5𝑥 +2 and 𝑔(𝑥) = 1+𝑥2, find (𝑓 ∘ 𝑔)(3). Solutions: Step 1: Set up the equation and start from the right side. (𝑓 ∘ 𝑔)(3) = 𝑓(𝑔(3)) Notice 𝑔(3) is the input for 𝑓(𝑥), so start by solving for g(3). Given 𝑔(𝑥) = 1+𝑥2: 𝑔(3) = 1+(3)2 𝑔(3) = 10 Step 2: Now substitute the answer for 𝑔(3) into 𝑓(𝑥). 𝑓(𝑔(3)) = 𝑓(10) Given 𝑓(𝑥) = 5𝑥 + 2: 𝑓(9) = 5(10) + 2 𝑓(9) = 52 so (𝒇 ∘ 𝒈)(𝟑) = 52 3 52 𝑓 𝑔 9 𝑔(3) → 𝑓(10) → 52 Finding the Composite Function To compose two functions, redefine the composition by using the definition to find 𝑓(𝑔(𝑥)) or 𝑔(𝑓(𝑥)). Example: Given 𝑓(𝑥) = 𝑥2 + 4 and 𝑔(𝑥) = 1 𝑥, find (𝑔∘𝑓)(𝑥). Solution: Step 1: Set up the function using the definition. (𝑔∘𝑓)(𝑥) = 𝑔(𝑓(𝑥)) Notice 𝑓(𝑥) is the input for 𝑔(𝑥), so start with 𝑓(𝑥). Given 𝒇(𝒙) = 𝒙𝟐+ 𝟒: 𝑔(𝒇(𝒙)) = 𝑔(𝒙𝟐+ 𝟒) Step 2: Now substitute 𝑥2 + 4 into 𝑔(𝑥) for every 𝑥. Simplify as needed. Given 𝑔(𝒙) = 1 𝒙: 𝑔(𝒙𝟐+ 𝟒) = 1 (𝒙𝟐+ 𝟒) so 𝒈(𝒇(𝒙)) = 𝟏 𝒙𝟐+ 𝟒 Example: Given 𝑓(𝑥) = 𝑥2 + 2𝑥−3 and 𝑔(𝑥) = 𝑥+ 1 find 𝑓(𝑔(𝑥)). Solution: Since 𝑓(𝑔(𝑥)) uses 𝑔(𝑥) as the input for 𝑓, substitute 𝑥+ 1 for 𝑔(𝑥) and simplify. Step 1: Substitute. 𝑓(𝑔(𝑥)) = 𝑓(𝑥+ 1) 𝑓(𝑥+ 1) = (𝑥+ 1)2 + 2(𝑥+ 1) −3 Step 2: Simplify. 𝑓(𝑥+ 1) = (𝑥2 + 2𝑥+ 1) + 2𝑥+ 2 −3 𝒇(𝒙+ 𝟏) = 𝒙𝟐+ 𝟒𝒙 Practice Exercises: 1. Given 𝑓(𝑥) = 2𝑥−6 and 𝑔(𝑥) = 𝑥2 + 3, find 𝑔(𝑓(𝑥)).
2. Given 𝑓(𝑥) = 4 −𝑥 and 𝑔(𝑥) = 𝑥3 −1, find (𝑓∘𝑔)(𝑥).
3. Given 𝑓(𝑥) = 3𝑥+ 4 and 𝑔(𝑥) = 2𝑥, find (𝑓∘𝑔)(5).
4. Given 𝑓(𝑥) = 𝑥+ 7 and 𝑔(𝑥) = 1 𝑥2−1 find 𝑔(𝑓(2)). Answers: 1. 𝑔(𝑓(𝑥)) = 4𝑥2 −24𝑥+ 39 2. 𝑓(𝑔(𝑥)) = 5 −𝑥3 3. 𝑓(𝑔(5)) = 34 4. 𝑔(𝑓(2)) = 1 80 |
190469 | https://www.canyonchasers.net/2024/03/how-do-motorcycles-turn/ | Countersteering is probably the most misunderstood concepts in motorcycling. When I was learning to ride, it was widely dismissed. Then when I became a riding instructor, it was believed that it only worked above a certain speed. Now we know that it works at every speed. So it’s often cited as the one perfect solution to every cornering problem. Running wide in a corner? Countersteer. And if that isn’t working, countersteer more.
But I tend to be suspicious of overly simple or absolute answers, and so should you. The answer to almost every motorcycle question actually starts with it depends.
We all know that motorcycles turn by leaning, right? We press the handlebar in the direction of the turn. The bike leans over, and then we are told that motorcycles change direction because the tires are shaped like cones. When leaned over, it moves in a circular path because the force applied is directed at an angle. This is Newton’s principle of circular motion. Problem solved. Except motorcycles have two wheels.
This got me wondering what’s happening when we lean into a corner. Why are motorcycles stable
and how do they turn? It all kind of seems to defy logic, doesn’t it? What I discovered is that statement is more true than you may realize. We can all agree that motorcycles lean and then turn. But what’s happening?
Imagine that balancing a in the palm of our hand is is our motorcycle. And if we just start trying
to move, it’s going to fall in the opposite direction. That means if I want to go one way, I first have to move my hand the opposite direction. It’ll tip in the direction I want to go and I can follow it. I’m moving my hand, my contact patch in the opposite direction that I want to go.
It works the same way on a motorcycle. If I want the bike to lean to the left, I need to move the contact
patch to the right. We control this with the handlebars. Rotate the bars and it moves the contact patch sideways in the opposite direction. We have to steer into the direction of the undesired fall, and this is what we’re doing At slower speeds. The motorcycle starts to fall to the left. We steer to the left and it comes back up right, because we’re bringing the tires back underneath the center of mass. And this explains why it feels like we aren’t countersteering at slow speeds.
When the pace comes up. We use this to our advantage to get the motorcycle to lean into the direction
we want to go. To get the bike to lean left, we push forward on the handlebar in the direction we want to lean. We steer away from where we want to turn. I do this every time I turn and so do you.
But what happens next?
Take a bicycle wheel and we spin it, and it’s stable. This is called gyroscopic precession, seemingly
defying gravity. Right? According to Newton, spinning wheels are illustrating the principle of angular momentum. It maintains its orientation relative to the Earth’s axis or resists changes in its orientation.
This has been well-established thought since around 1910, when an entire volume of a three book series was dedicated specifically to the stability of a bicycle. Gyroscopes. But if we try to isolate this; take away steering and a moving bicycle falls over just as quickly as a stationary one.
If bikes are stable because of gyroscopic precession, why does the one with spinning gyroscopic wheels fall over just as fast as the one that’s just sitting there?
Maybe it’s because bikes are cleverly designed to steer themselves. In other videos, I often talk about the importance of trail. And if we draw a straight line down from the steering stem, not the forks, the steering axis intersects the ground in front of the contact patch. This is called trail. The tires contact patch trails the steering axis, and on bicycles and motorcycles it’s roughly 100mm. That floored me. Bicycles and motorcycles travel at vastly different speeds, yet the trail number between the two is more or less the same. Trail is like caster on a shopping cart. The the wheel follows where it steered.
Let’s put these two ideas together. Use a toy gyroscope. When we spin it forward the same direction as our wheels on a motorcycle. And then lean it to the left. Watch as it turns to the left, into the turn. So then gyroscopes don’t keep the bike upright, they turn the handlebars? It turns into the turn, not away from the turn, which will push the bike upright again.
So let’s get this straight. Two wheeled vehicles are stable because of gyroscopic precession and caster
work together right? Or maybe not.
A bunch of really smart researchers wanted to test this. So they built a bicycle that had an extra set of wheels that spun the opposite direction and eliminated all the angular momentum. They basically counteracted all those gyroscopic forces. Going all in, they also removed all trail. The contact patch was
in front of the steering axis, and they found the bicycle was still stable. Even when its path of travel
is disturbed, it regains stability and keeps going.
So if it’s not gyroscopes and it’s not trail, then what is it?
What they figured was stability was a result of mass distribution. The force of gravity also works to steer the bike into the lean. We can illustrate what this means. If we take a bicycle and we lift it up and tip it forward so there’s effectively no trail.
Watch what happens as we lean it to the side. The front tire still steers into the corner because gravity. And if I was strong enough to lift up a motorcycle, it would do the exact same thing.
What really surprised me with all of this is that bikes are still an active area of study. Scientists don’t currently know what it is about these combinations of variables that work together to give a bike stability, gyroscopes, trail or mass distribution. We know some combinations work while others don’t. We can fly drones on Mars, but scientists still don’t fully understand two wheeled vehicles.
The fact that the humble bike still stumps the world’s leading mathematicians and physicists can best
be summed up with a quote by leading quantum physicist Michael Brooks.
Forget mysterious dark matter and the inexplicable accelerating expansion of the universe. The bicycle represents a far more embarrassing hole in the accomplishments of physics.
Michael Brooks
What I eventually resolved was it’s just like caster in the steering geometry of our bikes, where the wheel goes, where it steered. Science is trying to explain or following our observations. In other words, physicists didn’t come out and say, hey, to turn left, press forward on the left handlebar. Riders observed this was happening, and people much smarter than me are just trying to understand why.
So that’s all fun and neat and all, but how can we apply any of this to our riding?
We’ve learned that in order for the motorcycle to actually change direction, the handlebars need to be able to be turned into the corner. That means that counter steering isn’t really so much about steering as it is about initiating and controlling lean so that the tires can be underneath the bike’s center of mass.
We’ve all experienced how counter steering is easy to feel when we try to change direction while under power. In fact, countersteering is pretty much the only way to get the tire to move out from underneath the bike and initiate a change in direction when we need to turn while on the throttle.
The advice that says that if we are running wide in a corner, we need to stay on the gas and just counter
steer to lean more. It’s predicated on the staying on the gas part, and indeed, there are a few situations
where we need to turn while on the gas, like a swerve or a chicane. It also works great in places
like fast, open sweeping corners, freeway on ramps, or big empty parking lots.
But we’re not freeway chasers and We’re not parking lot chasers. and we’re not John F Kennedy drive chasers. We’re canyon chasers. We want to go ride those scenic byways and back ways where the roads are wiggly and less predictable. The corners are blind more often than not, and the asphalt
can be wildly variable, where we have to be adaptable and adjustable riders if we want to survive. Where staying on the throttle and pushing the contact patch out from underneath us and throwing lean angle at a problem may not always work out.
Here’s the thing. Lean angle is not infinite. Not every bike is going to be capable of just leaning more. Not every rider is going to be capable of just leaning more. Not every tire is going to be capable of just leaning more. Sure, oftentimes we can lean more than we think, but there’s a point where we will run out of lean. Can you think of any real world situations where we may not want to push the contact patch out from underneath us? Have you ever found yourself in a situation where you maybe didn’t want to turn while accelerating? You know, maybe something like a downhill corner? Even something as common as a freeway off ramp?
What can an adaptable rider do in situations where turning while accelerating shouldn’t be our first choice or our best option? To find the answer, we need to venture out into space. The final frontier.
Let’s go take a look at actual rocket science. Orbital mechanics to be more specific. When we put something in orbit, it has to be going fast enough to stay in that orbit or that radius. What happens as it slows down? The orbit starts to decay. The radius gets smaller. When this starts to happen,
how did the fine folks at NASA get the International Space Station back into the correct orbit? They strap a rocket to it and accelerate. In other words, speed equals radius.
See, all those episodes of Star Trek are helping us become better riders.
Speed equals radius is an easy way to explain centripetal force. As riders that means for a given lean angle, the radius of a corner is determined by how fast we are going. We go fast in fast corners. We go slow through slow corners. Champ School has a great demonstration of this. For a given lean angle
at a steady speed, the radius of a corner stays the same. When the rider accelerates the corner gets bigger and when the rider slows down the radius of the corner gets smaller.
There are some other advantages from entering a corner with your brake light on. Slower speed requires less lean angle. Less lean angle means less risk because we have more available grip for lots of different reasons. We can also add and subtract small amounts of brake pressure for even more precision. And if we need to slow more. No big deal because we are already slowing.
Here’s the deal. Physics help us to explain a lot of things. It helps us to better understand why bikes are stable, why tires need to be cone shaped. Why moving the contact patch underneath the center of mass changes how much the bike leans and all that other cool stuff. But what ultimately matters more than the whys of the physics is our inputs. In other words, if the only tool we have is a hammer, every problem starts to look like a nail. Have more than one tool. Sometimes we need to turn while on the throttle with counter steering. Sometimes it’s turning while using some front brake pressure to control our speed to control our radius.
Efficiency is using the right tool at the right time become an adaptable rider. Use different tools
for different jobs and understand how each works.
So when we get into a corner too fast, should we just countersteer more?
Well it depends.
Sources:
Most People Don’t Know How Bikes Work
Bicycle Research Project at Cornell
Why Bicycles Do Not Fall
How do Bikes Stay Up?
How Bicycles Balance Themselves
Gyroscopic Precession
Spinning
Neil deGrasse Tyson Explains
Bicycle Mechanics and Dynamics
Bicycle Dynamics
Explaining the Gyro Effect
Centripetal Force
Centrifigul Force
Motorcycle Dynamics
Sport Riding Techniques
Bicycle and Motorcycle Dynamics
Circular Motion
Angular Momentum
Precession
Centrifugal Force
Centripetal Force
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190470 | https://www.taxgirl.com/2018/12/15/ask-the-taxgirl-splitting-household-expenses/ | Ask The Taxgirl: Splitting Up Household Expenses
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Ask The Taxgirl: Splitting Up Household Expenses
Ask The Taxgirl: Splitting Up Household Expenses
Kelly Phillips ErbDecember 15, 2018 May 21, 2020
Taxpayer asks:
Dear Taxgirl:
I am married and due to my husband’s tax debt, we file as married filing separately. I bought a house recently in my name only that we live in together, and he pays 1/2 of all house expenses. Is this considered income to me? The state we live in is Illinois.
Taxgirl says:
If you file as married filing separately, that means that you are reporting only your income and claiming only your deductions; your spouse’s income and expenses are reported separately.
Splitting household expenses happens all of the time and for all kinds of reasons (we do it in my family, too). Typically, since household expenses are personal in nature and are not deductible, there’s no corresponding income. So, from a federal income tax perspective, your spouse’s payments to you to help cover household expenses are tax neutral – in other words, no harm, no foul.
One quick caveat: You didn’t specifically reference a mortgage but if you do have a mortgage, the ownership and payment rules still apply for purposes of any home mortgage interest deduction. Also, remember that when you file separate returns, you and your spouse must both claim the standard deduction or both of you must itemize your deductions (you can’t itemize while your spouse claims a standard deduction).
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Kelly Phillips Erb
Kelly Phillips Erb is a tax attorney, tax writer, and podcaster.
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190471 | https://www.reddit.com/r/askmath/comments/1lqy3vj/i_did_this_problem_and_found_infinite_solutions/ | I did this problem and found Infinite solutions, but the comments say only 20 degrees work, did I do this right? : r/askmath
Skip to main contentI did this problem and found Infinite solutions, but the comments say only 20 degrees work, did I do this right? : r/askmath
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•3 mo. ago
nooble36
I did this problem and found Infinite solutions, but the comments say only 20 degrees work, did I do this right?
Geometry
I’ve tried 20, 25, 70, and 110 degrees and they all seem to work
I think this is infinite solutions, here’s my work: ACB = 180 - CAB - ABC = 20 AFB (F being center point) = 180 - FAB - ABF = 50 ADB = 180 - DAB - ABD = 40 AEB = 180 - EAB - EBA = 30 DFE = AFB = 50
Then from here: CDB = 180 - ADB = 140 CEA = 180 - AEB = 150 CDE + CED = 180 - ACB = 160 EDB + DEA= 180 - DFE = 130 CDE + EDB = CDB =140 CED + DEA = CEA = 150
Then, Since CDE + CED = 160 and CDE + EBA = 140 then CED - EBA = 20 CED + CDE = 160 and CED + DEA = 150 then CDE - DEA = 10
And as such CDE = DEA + 10, CED = 180 - CDE, and EBA = CED - 20
I think this proves infinite solutions, honestly I don’t know much more then a high school’s worth of math so I don’t know if that’s all I need, but it seems that every number that I put into that formula works and I don’t see any reason it wouldn’t be infinite solutions
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190472 | https://www.math.miami.edu/~wachs/papers/eulerian.pdf | q-EULERIAN POLYNOMIALS: EXCEDANCE NUMBER AND MAJOR INDEX JOHN SHARESHIAN1 AND MICHELLE L. WACHS2 Abstract. In this research announcement we present a new q-analog of a classical formula for the exponential generating function of the Eulerian poly-nomials. The Eulerian polynomials enumerate permutations according to their number of descents or their number of excedances. Our q-Eulerian polynomials are the enumerators for the joint distribution of the excedance statistic and the major index.
There is a vast literature on q-Eulerian polynomials that involves other combinations of Eulerian and Mahonian permutation statistics, but this is the first result to address the combination of excedance number and major index. We use symmetric function theory to prove our formula.
In particular, we prove a symmetric function version of our formula, which involves an intriguing new class of symmetric functions. We also discuss con-nections with (1) the representation of the symmetric group on the homology of a poset introduced by Bj¨ orner and Welker, (2) the representation of the symmetric group on the cohomology of the toric variety associated with the Coxeter complex of the symmetric group, studied by Procesi, Stanley, Stem-bridge, Dolgachev and Lunts, (3) the enumeration of words with no adjacent repeats studied by Carlitz, Scoville and Vaughan and by Dollhopf, Goulden and Greene, and (4) Stanley’s chromatic symmetric functions.
1. Introduction The subject of permutation statistics originated in the early 20th century work of Major Percy MacMahon [22, Vol.
I, pp. 135, 186; Vol.
II, p. viii], and has developed into an active and important area of enumerative combinatorics over the last four decades. It deals with the enumeration of permutations according to natural statistics. A permutation statistic is simply a function from the symmetric group Sn to the set of nonnegative integers. MacMahon studied four fundamental permutation statistics, the inversion index, the major index, the descent number and the excedance number, which we define below.
Let [n] denote the set {1, 2, . . . , n}. For each σ ∈Sn, the descent set of σ is defined to be DES(σ) := {i ∈[n −1] : σ(i) > σ(i + 1)}, and the excedance set is defined to be EXC(σ) := {i ∈[n −1] : σ(i) > i}.
Date: October 16, 2006; revised February 20, 2007.
2000 Mathematics Subject Classification. 05A30, 05E05, 05E25.
1Supported in part by NSF Grants DMS 0300483 and DMS 0604233, and the Mittag-Leffler Institute.
2Supported in part by NSF Grants DMS 0302310 and DMS 0604562, and the Mittag-Leffler Institute.
1 2 SHARESHIAN AND WACHS The descent number and excedance number are defined respectively by des(σ) := |DES(σ)| and exc(σ) := |EXC(σ)|.
For example, if σ = 32541, written in one line notation, then DES(σ) = {1, 3, 4} and EXC(σ) = {1, 3}; hence des(σ) = 3 and exc(σ) = 2. If i ∈DES(σ) we say that σ has a descent at i.
If i ∈EXC(σ) we say that σ(i) is an excedance of σ.
MacMahon [22, Vol. I, p. 186] observed that the descent number and excedance number are equidistributed, that is, the number of permutations in Sn with j descents equals the number of permutations with j excedances for all j. (There is a well-known combinatorial proof of this fact due to Foata [11, 14].) These numbers were first studied by Euler and have come to be known as the Eulerian numbers.
They are the coefficients of the Eulerian polynomials1 An(t) := X σ∈Sn tdes(σ) = X σ∈Sn texc(σ).
Any permutation statistic that is equidistributed with des and exc is said to be an Eulerian statistic.
The Eulerian numbers and the Eulerian polynomials have been extensively stud-ied in many different contexts in the mathematics and computer science literature.
For excellent treatments of this subject, see the classic lecture notes of Foata and Sch¨ utzenberger and Section 5.1 of Knuth’s classic book series “The Art of Computer Programming” . The exponential generating function formula, (1.1) X n≥0 An(t)zn n! = 1 −t ez(t−1) −t where A0(t) = 1, is attributed to Euler in [21, p. 39].
The major index of a permutation σ ∈Sn is defined by maj(σ) := X i∈DES(σ) i, and the inversion statistic is defined by inv(σ) := |{(i, j) : 1 ≤i < j ≤n & σ(i) > σ(j)}|.
MacMahon showed that the major index is equidistributed with the inversion statistic by establishing the first equality in (1.2) X σ∈Sn qmaj(σ) = [n]q! = X σ∈Sn qinv(σ), where [n]q := 1 + q + · · · + qn−1 and [n]q! := [n]q[n −1]q · · · q.
Rodrigues had earlier obtained the second equality. (An elegant combinatorial proof of the equidistribution of maj and inv was obtained by Foata [12, 14].) Any permutation statistic that is equidistributed with the major index and inversion index is said to be a Mahonian statistic.
1It is more common to define the Eulerian polynomials as P σ∈Sn tdes(σ)+1 q-EULERIAN POLYNOMIALS: EXCEDANCE NUMBER AND MAJOR INDEX 3 Note that by setting q = 1 in (1.2), one gets the formula n! for the number of permutations. Equation (1.2) is a beautiful “q-analog” of this formula and is the fundamental example of the subject of permutation statistics and q-analogs, in which one seeks to obtain nice q-analogs of enumeration formulas.
One can look for nice q-analogs of the Eulerian polynomials by considering the joint distributions of the Mahonian and Eulerian statistics given above. Consider the four possibilities, Ainv,des n (q, t) := X σ∈Sn qinv(σ)tdes(σ) Amaj,des n (q, t) := X σ∈Sn qmaj(σ)tdes(σ) Ainv,exc n (q, t) := X σ∈Sn qinv(σ)texc(σ) Amaj,exc n (q, t) := X σ∈Sn qmaj(σ)texc(σ).
There are many interesting results on the first three q-Eulerian polynomials and on multivariate distributions of all sorts of combinations of Eulerian and Mahonian statistics (for a sample see [1, 2, 5, 7, 13, 16, 17, 18, 19, 20, 26, 27, 29, 30, 31, 38]).
These include Stanley’s q-analog of (1.1) given by, X n≥0 Ainv,des n (q, t) zn [n]q! = 1 −t Expq(z(t −1)) −t where Ainv,des 0 (q, t) = 1 and Expq(z) := X n≥0 q( n 2) zn [n]q!.
Surprisingly, we have found no mention of the fourth q-Eulerian polynomial Amaj,exc n (q, t) anywhere in the literature. Here we announce the following remark-able q-analog of (1.1).
Theorem 1.1. The q-exponential generating function for Amaj,exc n (q, t) is given by (1.3) X n≥0 Amaj,exc n (q, t) zn [n]q! = (1 −tq) expq(z) expq(ztq) −tq expq(z) , where Amaj,exc 0 (q, t) = 1 and expq(z) := X n≥0 zn [n]q!.
An alternative formulation of (1.3) more closely analogous to (1.1) is given by (1.4) X n≥0 Amaj,exc n (q, t) zn [n]q! = 1 −tq expq(ztq)Expq(−z) −tq .
In fact, we prove the following more general result, which reduces to Theorem 1.1 when r = 1.
4 SHARESHIAN AND WACHS Theorem 1.2. Let Amaj,exc,fix n (q, t, r) := X σ∈Sn qmaj(σ)texc(σ)rfix(σ), where fix(σ) denotes the number of fixed points of σ ∈Sn, i.e., the number of i ∈[n] such that σ(i) = i. Then (1.5) X n≥0 Amaj,exc,fix n (q, t, r) zn [n]q! = (1 −tq) expq(rz) expq(ztq) −tq expq(z).
By setting t = 1 in (1.5) one obtains a formula of Gessel and Reutenauer , and by setting q = 1 one obtains a formula that is equivalent to (1.1). By setting r = 0, we obtain the new result, X n≥0 X σ∈Dn qmaj(σ)texc(σ) zn [n]q! = 1 −tq expq(ztq) −tq expq(z), where Dn is the set of derangements in Sn.
Alternative formulations of (1.5) are given by the recurrence relation Amaj,exc,fix n (q, t, r) = rn + n−2 X k=0 n k q Amaj,exc,fix k (q, t, r) tq[n −k −1]tq, and the formula Amaj,exc,fix n (q, t, r) = ⌊n 2 ⌋ X m=0 (tq)m X k0 ≥0 k1, . . . , km ≥2 P ki = n n k0, . . . , km q rk0 m Y i=1 [ki −1]tq, where n k q = [n]q!
[k]q![n −k]q! and n k0, . . . , km q = [n]q!
[k0]q![k1]q! · · · [km]q!.
In the next section we describe the techniques that were used to prove Theo-rem 1.2. They involve an interesting class of symmetric functions and a symmetric function identity (Theorem 2.1), which generalizes Theorem 1.2.
We prove the symmetric function identity by modifying a bijection of Gessel and Reutenauer and generalizing a bijection of Stembridge . After a preliminary version of this paper was circulated, Foata and Han extended Theorem 1.2, finding the generating function for the joint distribution of maj,exc,fix and des. Their result can also be obtained by specializing our symmetric function identity.
In Section 3 we discuss a connection with two graded representations of the symmetric group, which turn out to be isomorphic. We show that a specialization of the Frobenius characteristic of these representations yields Amaj,exc(q, t). One of the representations is the representation of the symmetric group on the cohomology of the toric variety associated with the Coxeter complex of the symmetric group.
This representation was studied by Procesi , Stanley , Stembridge , , and Dolgachev and Lunts . The other representation is the representation of the symmetric group on the homology of maximal intervals of a certain intriguing poset introduced by Bj¨ orner and Welker in their study of connections between poset topology and commutative algebra. In fact, our study of the latter representation q-EULERIAN POLYNOMIALS: EXCEDANCE NUMBER AND MAJOR INDEX 5 is what led us to discover formula (1.1) and its symmetric function generalization, in the first place.
Various authors have studied Mahonian (resp. Eulerian) partners to Eulerian (resp.
Mahonian) statistics whose joint distribution is equal to a known Euler-Mahonian distribution. We mention, for example, Foata , Foata and Zeilberger , Clarke, Steingr´ ımisson and Zeng , Haglund , Babson and Steingr´ ımsson , and Skandera . In Section 4 we define a new Mahonian statistic to serve as a partner for des in the (maj, exc) distribution. We do not have a simple proof of the equidistribution. We have a highly nontrivial proof which uses tools from poset topology and the symmetric function results announced in Sections 2 and 3.
Details of the proofs discussed in this announcement, as well as further conse-quences and open problems, will appear in a forthcoming paper.
2. Symmetric function generalization In this section we present a symmetric function generalization of Theorem 1.2.
Let H(z) = H(x, z) := X n≥0 hn(x)zn, where hn(x) denotes the complete homogeneous symmetric function in the indeter-minates x = (x1, x2, . . . ), that is hn(x) := X 1≤i1≤i2≤···≤in xi1xi2 . . . xin for n ≥1, and h0 = 1. By setting xi := qi−1, for all i, and z := z(1 −q) in H(x, z), one obtains expq(z), see . It follows that (2.1) (1 −t)H(x, zr) H(x, zt) −tH(x, z) xi := qi−1 z := z(1 −q) = (1 −t) expq(zr) expq(zt) −t expq(z).
We will construct for each n, j, k ≥0, a quasisymmetric function Qn,j,k(x) whose generating function P n,j,k≥0 Qn,j,k(x)tjrkzn specializes to X n≥0 X σ∈Sn qmaj(σ)−exc(σ)texc(σ)rfix(σ) zn [n]q!
when we set xi := qi−1 and z := z(1 −q). Thus by taking specializations of both sides of (2.4) below and setting t := tq, we obtain (1.5).
For σ ∈Sn, let ¯ σ be the barred word obtained from σ by placing a bar above each excedance. For example, if σ = 531462 then ¯ σ = ¯ 5¯ 314¯ 62. View ¯ σ as a word over the ordered alphabet {¯ 1 < ¯ 2 < · · · < ¯ n < 1 < 2 < · · · < n}.
We extend the definition of descent set from permutations to words w of length n over an ordered alphabet by letting DES(w) := {i ∈[n −1] : wi > wi+1}, where wi is the ith letter of w. Now define the excedance-descent set of a permu-tation σ ∈Sn to be EXD(σ) := DES(¯ σ).
6 SHARESHIAN AND WACHS For example, EXD(531462) = DES(¯ 5¯ 314¯ 62) = {1, 4}. The interesting thing about EXD is that for all σ ∈Sn, (2.2) X i∈EXD(σ) i = maj(σ) −exc(σ).
For S ⊆[n −1] and n ≥1, define the quasisymmetric function FS,n(x1, x2, . . . ) := X i1 ≥· · · ≥in j ∈S ⇒ij > ij+1 xi1 . . . xin, and let F∅,0 = 1.
A basic result in Gessel’s theory of quasisymmetric functions (see [33, Lemma 7.19.10]) is that FS,n(1, q, q2, . . . ) = q P s∈S s (1 −q)(1 −q2) . . . (1 −qn).
Hence it follows from (2.2) that for all σ ∈Sn, FEXD(σ),n(1, q, q2, . . . ) = qmaj(σ)−exc(σ) (1 −q)(1 −q2) . . . (1 −qn).
For any n, j, k ≥0, let Qn,j,k = Qn,j,k(x) := X σ ∈Sn exc(σ) = j fix(σ) = k FEXD(σ),n(x).
By taking the specialization of the generating function we get, (2.3) X n,j,k≥0 Qn,j,k(x)tjrkzn xi := qi−1 z := z(1 −q) = X n≥0 X σ∈Sn qmaj(σ)−exc(σ)texc(σ)rfix(σ) zn [n]q!.
It follows from (2.1) and (2.3) that by setting xi := qi−1, z := z(1−q) and t := tq in the following result we obtain Theorem 1.2.
Theorem 2.1.
(2.4) X n,j,k≥0 Qn,j,ktjrkzn = (1 −t)H(zr) H(zt) −tH(z).
The proof of this theorem requires an alternative characterization of Qn,j,k which is established by adapting a bijection that Gessel and Reutenauer introduced to enumerate permutations with a fixed descent set and a fixed cycle type. Gessel and Reutenauer deal with circular words over the alphabet of positive integers. We consider circular words over the alphabet of barred and unbarred positive integers.
For each such circular word and each starting position, one gets a linear word by reading the circular word in a clockwise direction.
If one gets a distinct linear word for each starting position, then the circular word is said to be primitive.
For example (¯ 1, 1, 1) is primitive while (¯ 1, 2, ¯ 1, 2) is not. The absolute value of a letter is the letter obtained by erasing the bar if there is one. We will say that a primitive circular word is a necklace if each letter that is followed (clockwise) by a letter greater in absolute value is barred and each letter that is followed by a letter smaller in absolute value is unbarred. Letters that are followed by letters equal in q-EULERIAN POLYNOMIALS: EXCEDANCE NUMBER AND MAJOR INDEX 7 absolute value have the option of being barred or not. A circular word consisting of one barred letter is not a necklace. For example the following circular words are necklaces: (¯ 1, 3, 1, ¯ 1, 2, 2), (¯ 1, 3, ¯ 1, ¯ 1, 2, 2), (¯ 1, 3, 1, ¯ 1, ¯ 2, 2), (¯ 1, 3, ¯ 1, ¯ 1, ¯ 2, 2), (3), while (¯ 1, ¯ 3, 1, 1, 2, ¯ 2) and (¯ 3) are not.
An ornament is a multiset of necklaces. The type λ(R) of an ornament R is the partition whose parts are the sizes of the necklaces in R. The weight w(R) of an ornament R is the product of the weights of the letters of R, where the weight of the letter a is the indeterminate x|a|, where |a| denotes the absolute value of a. For example λ((¯ 1, 2, 2), (¯ 1, ¯ 2, 3, 3, 2)) = (5, 3) and w((¯ 1, 2, 2), (¯ 1, ¯ 2, 3, 3, 2)) = x2 1x4 2x2 3.
For each partition λ and nonnegative integer j, let Rλ,j be the set of ornaments of type λ with j bars.
Theorem 2.2. For all λ ⊢n and j = 0, 1, . . . , n −1, let Qλ,j := X σ FEXD(σ),n summed over all permutations of cycle type λ with j excedances. Then Qλ,j = X R∈Rλ,j w(R).
This theorem is proved via a bijection between ornaments of type λ with j bars and permutations of cycle type λ with j excedances paired with “compatible” weakly decreasing sequences of positive integers. Our bijection is an adaptation of the Gessel-Reutenauer bijection which sends multisets of primitive circular words over an ordered alphabet to permutations paired with compatible weakly decreasing sequences over the same alphabet. The Gessel-Reutenauer bijection, which is also described in , can be viewed as a necklace analog of the bijection in Stanley’s theory of P-partitions . Here we need to order our alphabet by 1 < ¯ 1 < 2 < ¯ 2 < . . . .
Our map is obtained by first applying the Gessel-Reutenauer map to our orna-ments and then removing the bars from the barred letters in the compatible weakly decreasing sequence to obtain a weakly decreasing sequence of positive integers.
Theorem 2.2 has several interesting consequences. For one thing, it can be used to prove that the quasisymmetric functions Qλ,j and Qn,j,k are actually symmetric.
It also has the following useful consequence.
Corollary 2.3. For all n, j, k, Qn,j,k = hkQn−k,j,0.
It follows from Corollary 2.3 that Theorem 2.1 is equivalent to (2.5) X n,j≥0 Qn,j,0tjzn = 1 −t H(zt) −tH(z), 8 SHARESHIAN AND WACHS which in turn, is equivalent to the recurrence relation (2.6) Qn,j,0 = X 0 ≤m ≤n −2 j + m −n < i < j Qm,i,0hn−m.
We establish this recurrence relation by introducing another type of configuration, closely related to ornaments.
Define a banner B to be a word over the alphabet of barred and unbarred positive integers, where B(i) is barred if |B(i)| < |B(i + 1)| and B(i) is unbarred if |B(i)| > |B(i + 1)| or i = length(B). All other letters have the option of being barred. The weight of a banner is the product of the weights of its letters.
A Lyndon word over an ordered alphabet is a word that is strictly lexicographi-cally smaller than all its circular rearrangements. A Lyndon factorization of a word over an ordered alphabet is a factorization into a weakly lexicographically decreas-ing sequence of Lyndon words. It is a result of Lyndon (see [24, Theorem 5.1.5]) that every word has a unique Lyndon factorization. The Lyndon type of a word is the partition whose parts are the lengths of the words in its Lyndon factorization.
For each partition λ and positive integer j, let Bλ,j be the set of banners with j bars whose Lyndon type is λ.
By turning the Lyndon words in the Lyndon factorization of a banner into cir-cular words, we obtain an ornament. This map from banners to ornaments is the bijection whose existence is asserted in the following proposition.
Proposition 2.4. For any partition λ and nonnegative integer j, there is a weight-preserving bijection from Bλ,j to Rλ,j.
Corollary 2.5. Let Bn,j be the set of banners of length n with j bars whose Lyndon type has no parts of size 1. Then Qn,j,0 = X B∈Bn,j w(B) Define a marked sequence (α, i) to be a weakly increasing finite sequence α of positive integers together with an integer i such that 1 ≤i ≤length(α) −1. Let Mn be the set of marked sequences of length n and let Bn be the set of banners of length n whose Lyndon type has no parts of size 1.
Theorem 2.6. For all n ≥2, there is a bijection γ : Bn → [ 0≤m≤n−2 Bm × Mn−m, such that if γ(B) = (B′, (α, i)) then w(B) = w(B′)w(α) and bar(B) = bar(B′) + i, where bar(B) denotes the number of bars of B.
We will not describe the bijection here except to say that, when restricted to banners with distinct letters, it reduces to a bijection from permutations to marked words that Stembridge constructed to study the representation of the symmet-ric group on the cohomology of the toric variety assoiciated with the type A Coxeter complex. (We discuss this representation in Section 3.) Banners in Bn admit a q-EULERIAN POLYNOMIALS: EXCEDANCE NUMBER AND MAJOR INDEX 9 certain kind of decomposition, called a decreasing decomposition in . The de-creasing decomposition plays the role in our bijection that the cycle decomposition of permutations plays in Stembridge’s bijection.
Corollary 2.5 and Theorem 2.6 are all that is needed to establish the recurrence relation (2.6), which yields our main result, Theorem 2.1.
Remark.
If one applies the standard involution ω to the symmetric function appearing on the right hand side of (2.4), one gets a refinement of a symmetric function that has been studied by Carlitz, Scoville and Vaughan and Dollhopf, Goulden and Greene in connection with the enumeration of words with no adjacent repeats. It was pointed out to us by Richard Stanley that these words and our banners can be viewed as “dual” graph colorings in the sense of [34, Theorem 4.2].
3. Some Representation Theoretic Consequences The Frobenius characteristic ch is a fundamental homomorphism from the ring of representations of symmetric groups to the ring of symmetric functions. In this section we present two representations whose Frobenius characteristic is Qn,j := Pn k=0 Qn,j,k.
The first representation involves the toric variety associated with the Coxeter complex of a Weyl group. Let Xn be the toric variety associated with the Coxeter complex of Sn. The action of Sn on Xn induces a representation of Sn on the co-homology H2j(Xn) for each j = 0, . . . , n−1. (Cohomology in odd degree vanishes.) Stanley , using a formula of Processi , proves that X n≥0 n−1 X j=0 chH2j(Xn) tjzn = (1 −t)H(z) H(zt) −tH(z).
Combining this with Theorem 2.1 yields the following conclusion.
Theorem 3.1. For all j = 0, 1, . . . , n −1, chH2j(Xn) = Qn,j.
The second representation involves poset topology, a subject in which topological properties of a simplicial complex associated with a poset are studied, see .
The faces of the simplicial complex, called the order complex of the poset, are the chains of the poset. Here we consider the homology of the order complex of the Rees product of two simple posets. The Rees product is a poset construction recently introduced by Bj¨ orner and Welker in their study of relations between poset topology and commutative algebra.
Definition 3.2. Let P and Q be pure (ranked) posets with respective rank func-tions rP and rQ. The Rees product P ∗Q of P and Q is defined as follows: P ∗Q := {(p, q) ∈P × Q : rP (p) ≥rQ(q)} with order relation given by (p1, q1) ≤(p2, q2) if the following holds • p1 ≤P p2 • q1 ≤Q q2 • rP (p2) −rP (p1) ≥rQ(q2) −rQ(q1).
10 SHARESHIAN AND WACHS Let Bn be the Boolean algebra (ie., the lattice of subsets of [n] ordered by inclusion) and let Cn be the chain 1 < 2 < · · · < n. The maximum elements of (Bn \ {∅}) ∗Cn are of the form ([n], j), where j = 1, . . . , n. Let In,j be the set of elements of (Bn \ {∅}) ∗Cn that are smaller than ([n], j) and let ˜ Hi(In,j) be the reduced simplicial (complex) homology of the order complex of In,j. It follows from results of Bj¨ orner and Welker that homology vanishes below the top dimension n −2. The symmetric group Sn acts on In,j in an obvious way and this induces a representation on ˜ Hn−2(In,j). We prove the following result using techniques from poset topology.
Theorem 3.3.
1 + X n≥1 n X j=1 ch( ˜ Hn−2(In,j) ⊗sgn) tj−1zn = (1 −t)H(z) H(zt) −tH(z), where sgn denotes the sign representation. Consequently for all n, j, ch( ˜ Hn−2(In,j) ⊗sgn) = Qn,j−1 and as Sn-modules ˜ Hn−2(In,j) ⊗sgn ∼ = H2j−2(Xn).
We conjecture that for all λ and j, the symmetric function Qλ,j is also the Frobenius characteristic of some representation. One consequence of Theorem 2.2 is that Qλ,j can be described as a product of plethysms of symmetric functions of the form Q(n),i, where (n) denotes a partition with a single part. Hence if the conjecture holds for all Q(n),i then it holds in general.
We use ornaments and banners to show that if the conjecture does hold then the restriction to Sn−1 of the representation whose Frobenius characteristic is Q(n),i, has Frobenius characteristic Qn−1,i−1.
4. A new Mahonian statistic In this section we describe a new Mahonian statistic whose joint distribution with des is the same as the joint distribution of maj and exc.
An admissible inversion of σ ∈Sn is a pair (σ(i), σ(j)) such that the following conditions hold: • i < j • σ(i) > σ(j) • either ◦σ(j) < σ(j + 1) or ◦∃k such that i < k < j and σ(k) < σ(j).
Let ai(σ) := # admissible inversions of σ. Define the statistic aid(σ) := ai(σ) + des(σ).
For example, the admissible inversions of 24153 are (2, 1), (4, 1) and (4, 3). So aid(24153) = 3 + 2.
Theorem 4.1. For all n ≥1, X σ∈Sn qaid(σ)tdes(σ) = X σ∈Sn qmaj(σ)texc(σ).
q-EULERIAN POLYNOMIALS: EXCEDANCE NUMBER AND MAJOR INDEX 11 We do not have a direct proof of this simple identity except when t or q is 1.
Our proof relies on Theorem 1.1, a q-analog of Theorem 3.3, and techniques from poset topology. We consider the Rees product (Bn(q) \ {(0)}) ∗Cn, where Bn(q) is the lattice of subspaces of the vector space Fn q . Let In,j(q) be the set of elements in (Bn(q) \ {(0)}) ∗Cn that are less than the maximal element (Fn q , j). We first use a well-known tool from poset topology, called lexicographic shellability [3, 37], to prove that (4.1) dim ˜ Hn−2(In,j(q)) = X σ ∈Sn des(σ) = j −1 qai(σ).
We then use other tools from poset topology to prove a theorem analogous to Theorem 3.3, which states that (4.2) X n≥0 n X j=1 dim ˜ Hn−2(In,j(q))tj−1 zn [n]q! = (1 −t) expq(z) expq(zt) −t expq(z).
Theorem 4.1 now follows from Theorem 1.1 and equation (4.1).
5. Acknowledgements The research presented here began while both authors were visiting the Mittag-Leffler Institute as participants in a combinatorics program organized by Anders Bj¨ orner and Richard Stanley. We thank the Institute for its hospitality and support.
We are also grateful to Ira Gessel for some very useful discussions and to Dominique Foata for valuable comments, which included (1.4) and an improved formulation of Theorem 1.2.
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q-EULERIAN POLYNOMIALS: EXCEDANCE NUMBER AND MAJOR INDEX 13 Department of Mathematics, Washington University, St. Louis, MO 63130 E-mail address: shareshi@math.wustl.edu Department of Mathematics, University of Miami, Coral Gables, FL 33124 E-mail address: wachs@math.miami.edu |
190473 | https://telrp.springeropen.com/articles/10.1186/s41039-018-0082-z | Automatic distractor generation for multiple-choice English vocabulary questions | Research and Practice in Technology Enhanced Learning | Full Text
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Automatic distractor generation for multiple-choice English vocabulary questions
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Published: 01 October 2018
Automatic distractor generation for multiple-choice English vocabulary questions
Yuni Susanti1,
Takenobu Tokunaga1,
Hitoshi Nishikawa1&
…
Hiroyuki Obari2
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Research and Practice in Technology Enhanced Learningvolume 13, Article number:15 (2018) Cite this article
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Abstract
The use of automated systems in second-language learning could substantially reduce the workload of human teachers and test creators. This study proposes a novel method for automatically generating distractors for multiple-choice English vocabulary questions. The proposed method introduces new sources for collecting distractor candidates and utilises semantic similarity and collocation information when ranking the collected candidates. We evaluated the proposed method by administering the questions to real English learners. We further asked an expert to judge the quality of the distractors generated by the proposed method, a baseline method and humans. The results show that the proposed method produces fewer problematic distractors than the baseline method. Furthermore, the generated distractors have a quality that is comparable with that of human-made distractors.
Introduction
Recent advances in natural language processing (NLP) have enabled us to build more advanced applications in the educational field, especially in learning and testing. They include the utilisation of NLP technologies and language resources for automating student assessment, instruction and curriculum design. Among others, applying NLP to second-language learning has attracted extensive attention, including automated essay scoring (Shermis and Burstein 2003) and automatic question generation (Araki et al. 20167
, (pp. 1125–1136)."); Hoshino and Nakagawa 2005. A real-time multiple-choice question generation for language testing -a preliminary study. In Proceedings of the Second Workshop on Building Educational Applications Using NLP. Association for Computational Linguistics, Ann Arbor, (pp. 17–20)."); Pino and Eskenazi 2009. Semi-automatic generation of cloze question distractors effect of students’ l1. In Proceedings of the SLaTE 2009 - 2009 ISCA Workshop on Speech and Language Technology in Education. ISCA (International Speech Communication Association), Warwickshire."); Satria and Tokunaga 2017. Automatic generation of english reference question by utilising nonrestrictive relative clause. In Proceedings of the 9th International Conference on Computer Supported Education, (pp. 379–386)."); Sumita et al. 2005. Measuring non-native speakers’ proficiency of English by using a test with automatically-generated fill-in-the-blank questions. In Proceedings of the Second Workshop on Building Educational Applications Using NLP. Association for Computational Linguistics, Ann Arbor, (pp. 61–68)."); Susanti et al. 2015. Automatic generation of english vocabulary tests. In Proceedings of the 7th International Conference on Computer Supported Education. INSTICC, Setubal, (pp. 77–87).")). The use of fully automated systems in second-language learning could significantly reduce the burden on human experts to teach students, create tests and evaluate the development of student’s abilities. At the very least, they could assist the human experts with these teaching activities. Automated systems are also useful for the self-study for students. For instance, students would be able to work on questions automatically generated by the system according to their current ability.
Standardised English language tests such as Test of English as a Foreign Language (TOEFL), Test of English for International Communication (TOEIC) and International English Language Testing System (IELTS) often use a multiple-choice format (as depicted in Fig.1) because of its efficiency in scoring. However, the difficulty and cost of developing multiple-choice questions have been the primary challenge of this format (Downing 2006). Because of this, research on automatic question generation, especially on multiple-choice question generation, has attracted much attention recently. Various resources and technologies in NLP can potentially contribute to the automatic generation of multiple-choice questions, as has been done by Mitkov and Ha (2003)7
, (pp. 17–22).") who used the WordNet (Fellbaum 1998. WordNet: An Electronic Lexical Database. Cambridge: MIT Press.")) as a lexical dictionary resource. More recently, Susanti et al. (2015). Automatic generation of english vocabulary tests. In Proceedings of the 7th International Conference on Computer Supported Education. INSTICC, Setubal, (pp. 77–87).") also used WordNet to generate multiple-choice English vocabulary questions. Another study was performed by Araki et al. (2016). Generating questions and multiple-choice answers using semantic analysis of texts. In Proceedings of COLING 2016, the 26th International Conference on Computational Linguistics: Technical Papers, Osaka, Japan.
, (pp. 1125–1136)."), who applied semantic analysis to texts for generating multiple-choice questions for English language learners.
Fig. 1
Closest-in-meaning vocabulary question (source: a TOEFL iBT question from past test, taken from the official website, www.ets.org.)
Full size image
Rodriguez (2005x
.")) stated that the quality of multiple-choice questions relies heavily on the quality of their options. His claim is supported by Hoshino (2013). Relationship between types of distractor and difficulty of multiple-choice vocabulary tests in sentential context. Language Testing in Asia, 3(1), 16.
."), who noted that test takers tended to employ a choice-oriented strategy when working on multiple-choice questions. Therefore, the quality of the question options, especially the distractors (wrong options), affects the quality of the question, as inappropriate distractors enable the test takers to guess the answer easily (Moser et al. 2012. Refined distractor generation with lsa and stylometry for automated multiple choice question generation. In: Thielscher, M., & Zhang, D. (Eds.) In AI 2012: Advances in Articial Intelligence. Springer, Berlin, (pp. 95–106).")) or cause them to unnecessary confusion.
Nevertheless, few studies on automatic question generation have focused on distractor generation. Haladyna (2004) pointed out that generating distractors was the most difficult part of multiple-choice question generation. As in the manual writing of questions, developing appropriate distractors remains a difficult task in automatic question generation. Some studies have generated distractors for fill-in-the-blank language questions using simple techniques such as random selection from words in the same document (Hoshino and Nakagawa 2005), employing a thesaurus (Sumita et al. 2005) and collecting similar candidates of the target word in terms of their frequency and dictionary-based collocation (Liu et al. 20050
, (pp. 1–8).")).
Other studies have employed more advanced techniques and resources for distractor generation, mostly for the fill-in-the-blank English vocabulary questions. For instance, Pino and Eskenazi (2009) and Correia et al. (2010) used graphemic (morphological and orthographic) and phonetic variants of the target word as distractor candidates. Correia et al. (2010) employed lexical resources to filter distractor candidates considering the target word’s synonym, hyponym and hypernym. Sakaguchi et al. (2013) utilised common learner errors that were constructed from error-correction pairs on a language learning site, Lang-8Footnote 1. In each pair of corrections, the error was a candidate distractor for the target word. Zesch and Melamud (2014) applied context-sensitive lexical inference rules to generate verb distractors that are not semantically similar to the target in the fill-in-the-blank context but might be similar in another context. More recently, Jiang and Lee (2017) proposed the use of a semantic similarity measure based on the word2vec model (Mikolov et al. 20131
.")) for generating plausible distractors of Chinese fill-in-the-blank vocabulary questions.
The present study focuses on generating distractors for English vocabulary questions as used in TOEFL. Figure1 illustrates an example of the type of question we discuss in this study. The vocabulary questions in TOEFL have a distinct characteristic; instead of asking for the best word to fill the gap as in the fill-in-the-blank type questions, they ask for the closest-in-meaning word of a target word (the vocabulary being asked in a question) used in a reading passage. However, the result of this study can contribute to the fill-in-the-blank questions as well because both types of questions share the same requirements for their distractors.
We conducted two evaluations: (1) a test taker-based evaluation and (2) an expert-based evaluation. In the first evaluation, we asked English learners to complete a set of question that differed only with respect to the distractor set. Otherwise, they shared the same reading passage and correct answer taken from a human-made question item. Each distractor set contains three distractors that are either human-made, generated by the baseline method or generated by the proposed method. We evaluated the quality of the distractors from the test takers’ responses by applying Neural Test Theory (Shojima et al. 2008), which is a test theory for analysing test results that grades the test takers into several ranks. According to Susanti et al. (2017), this test is useful for evaluating the statistical characteristics of options in multiple-choice questions. In the expert-based evaluation, we asked a professional item writer to evaluate the same sets of distractors as in the test taker-based evaluation. We also discuss how the two evaluation results relate.
The contributions of the present study are as follows:
1)
We proposed a method of distractor generation for multiple-choice vocabulary questions that is superior to the state-of-the-art method.
2.
We thoroughly analysed the evaluation results from two different perspectives, i.e. the test taker-based and the expert-based evaluations.
The next section (“Methods” section) describes the distractor generation including the proposed method and the state-of-the-art method used as the baseline. The evaluation design is presented in the “Evaluation design” section. The “Results and discussion” section discusses the result and analysis of the evaluation and is followed by the conclusion and future work directions in the “Conclusion” section.
Methods
In this study, we implemented a distractor generation method introduced by Jiang and Lee (2017) as a baseline because their work is the latest state-of-the-art method that targets the most similar task to the current study. Although their method generates Chinese fill-in-the-blank vocabulary questions, the method is independent of the language because it takes a corpus-based approach. We can hence adapt the method for English by replacing the corpus. Another difference is the question type to generate, i.e. fill-in-the-blank questions versus closest-in-meaning questions. These questions differ in whether the target word is present in the options as a correct answer (fill-in-the-blank) or present in the reading passage (closest-in-meaning). There is no difference in the characteristics of the distractors in both types of vocabulary questions.
In the following, we describe the baseline in detail, followed by our proposed method. We then compare the two methods as summarised in Table1. For all methods, distractor generation consists of three steps: (1) distractor candidate collection, (2) distractor candidate filtering and (3) distractor candidate ranking.
Table 1 Methods to be compared
Full size table
Baseline method
Distractor candidate collection and ranking
To collect distractor candidates, Jiang and Lee (2017) extracted all the words in the Chinese Wiki corpus and ranked them on the basis of their various similarity criteria to the target word. The similarity criteria consist of the word difficulty level (frequency-based) similarity, spelling similarity, PMI-based word co-occurrence with the target word and word2vec-based word similarity. They ranked the candidates according to each criterion and evaluated the results. Their evaluation showed that the word2vec-based criterion outperformed the others; thus, in this study, we implemented this criterion for collecting the distractor candidates.
Jiang and Lee (2017) trained a word2vec model on the Chinese Wiki corpusFootnote 2. Because we adapted their method for English vocabulary questions, we used a word2vec model pre-trained on English WikipediaFootnote 3.
Distractor candidate filtering
Jiang and Lee (2017) filtered the ranked distractor candidates to remove candidates that are also considered to be an acceptable answer. They examined whether the distractor candidates collocate with the words in the rest of the carrier sentenceFootnote 4, by filtering based on the trigram and dependency relations.
Trigram filtering: the trigram is formed from the distractor candidate and its two adjacent words (the previous and following words) in the carrier sentence. We implemented this filtering without modification in our implementation for English vocabulary questions.
Dependency relation filtering: the implementation by Jiang and Lee (2017) considers all the dependency relations with the distractor as a head or child. We implemented this filtering with a small corpusFootnote 5, but the filtering did not remove any candidates. Hence, we decided not to implement this filtering.
The three highest ranked candidates after the filtering were chosen as the final distractors for the question.
Proposed method
Distractor candidate collection
We collected the distractor candidates from two main sources that reflect two different relations with the target word. The first source is synonyms of the words in the reading passage that have the same part of speech and tense as the target word, with an assumption that those words share the same topic of the reading passage. The second source is siblings of the target word in the WordNet taxonomy. Because siblings share the same hypernym, the siblings of the target word should share a similar meaning but also have a certain difference in meaning.
In addition to these two sources of distractor candidates, we utilise the JACET8000 word list (Ishikawa et al. 2003) as the third source of distractor candidates. JACET8000 is a word list designed for Japanese English learners. It ranks 8000 basic English words according to their frequency in the British National CorpusFootnote 6 supplemented with six million tokens of texts targeted at the needs of Japanese students. The 8000 words are divided into eight groups of 1000 words; each group corresponds to their level of word difficulty with level 1 being the easiest.
We consider JACET8000 suitable for generating English vocabulary questions because it has been compiled for the purpose of English learning. Our observations tell us that most distractors in human-made vocabulary questions have the same or almost the same level of difficulty as the correct answer. Thus, as the distractor candidates, the present study utilises the words in the JACET8000 word list for which the level differs at most by two levels from that of the correct answer. For example, if the correct answer is level 4, the distractor candidates are collected from the words of levels 2–6.
Furthermore, to top up insufficient distractor candidates from WordNet, we also add the synonyms of synonyms and words related to the target word according to the Merriam-Webster Dictionary.
Distractor candidate filtering
The collected distractor candidates are further filtered following English vocabulary questions writing guidelines (Heaton 1989), which are summarised below.
1)
Question options should have the same part of speech as the target word.
2)
Distractors should have a word difficulty level that is similar to that of the correct answer.
3)
Question options should have approximately the same length.
4)
A pair of synonyms in the question options should be avoided.
5)
Antonyms of the correct answer should be avoided as distractors.
6)
Distractors should be related to the correct answer, or come from the same general topic.
Vocabulary questions in the present study ask for the word closest-in-meaning to the target word. Thus, the distractors must not have the same or a very similar meaning to either the target word or the correct answer. To guarantee that the distractors are not synonyms of the target word, we filter out synonymous candidates using the synonym list from WordNet and the Merriam-Webster Dictionary in addition to the criteria specified by Heaton (1989).
Distractor candidate ranking
Although the distractors must have a different meaning from both the target word and the correct answer, they must also be able to distract the test takers from the correct answer. Because the present study focuses on the closest-in-meaning vocabulary question, distracting distractors should be similar to the target word or correct answer in some respects. Unlike fill-in-the-blank questions, where the target word and correct answer are the same, in the closest-in-meaning questions, we can utilise both the target word and correct answer to generate distractors so that the distractors are semantically close to the target word but far from the correct answer. To rank distractor candidates, the baseline adopts a word embedding-based semantic similarity measure. In contrast, the present study introduces a new ranking metric r(c) that aggregates word embedding-based semantic similarity and word collocation information for ranking the distractor candidates c with respect to the target word (tw), reading passage (rp) and correct answer (ca), which is given by:
r(c)=rank(sim(c,t w))+rank(col(c,r p))−rank(sim(c,c a))
(1)
where sim(w i,w j) is the semantic similarity between words w i and w j; col(w i,context) is a collocation measure of word w i and its adjacent two words in the given context, and rank(f(·)) returns the rank of the value of f(·) in descending order. We use ranks instead of their raw scores because they are easier to integrate into a single score.
To calculate sim(w i,w j), the present study employs the cosine similarity of the word vectors derived by the word embedding GloVe algorithm rather than word2vec because it is more efficient (Pennington et al. 20142
, (pp. 1532–1543).")). We used the pre-trained GloVe word vectorsFootnote 7. We calculate the collocation measure col(_w_ _i_,context) on the basis of the frequencies of two bigrams: (_w_ _i_−1,_w_ _i_) and (_w_ _i_,_w_ _i_+1) in the context. The bigram statistics were generated using the module provided by the NLTK Python PackageFootnote 8 and the English Text corpora in the same packageFootnote 9.
The idea behind Eq. (1) is that we want to obtain a distractor candidate c that is similar to target word tw (a large sim(c,t w), i.e. has a high rank(sim(c,t w))), and frequently collocates with the adjacent words in the reading passage rp (a large col(c,r p), i.e. has a high rank(col(c,r p))), but is not similar to the correct answer ca (a small sim(c,c a), i.e. has a low rank(sim(c,c a))). Thus, we prefer distractor candidates with a smaller value of r(c).
Evaluation design
Question data
We selected 45 target words (TW 1–45) from real closest-in-meaning vocabulary questions collected from the ETS official siteFootnote 10 and preparation books of TOEFL iBT, which are published by the official TOEFL organisationFootnote 11. The selection was made such that the part of speech categories of the target words were balanced.
For each question, we used the three human-made distractors in the original TOEFL question as a reference distractor set. We then determined two additional sets of three distractors using the baseline and proposed methods. For each automatically generated method, the set of three distractors was made by selecting from the top three candidates in the ranked candidate list of that method. The original reading passage and correct answer were used to automatically generate the distractors. In total, we prepared 135 question items with 45 items each set. The order of the distractors was randomised in each question.
We conducted two evaluations, test taker-based and expert-based evaluations; they are explained in the following sections.
Test taker-based evaluation
The aim of this evaluation is to evaluate the validity of the distractor candidates when they are used in a real test setting. We administered the question set described above to English learners and evaluated the quality of the distractors based on their responses. We used a Latin square design to design the question sets, as shown in Table2. For instance, in question set QS.A, the distractor sets for target words (TWs) 1 to 15 are generated by the baseline method and TWs 16 to 30 by the proposed method and TWs 31 to 45 are the original TOEFL distractors created by humans.
Table 2 Configuration of the question sets
Full size table
Participants
A total of 80 Japanese university undergraduate students participated in the experiment. We divided them into three student groups, G1, G2 and G3, according to their school class and administered a different question set to each student group. Table2 shows the assignment of the question sets to the student groups.
Experimental procedure
The experiment was conducted in the form of an online test. The participants completed the test using their own computer, but each group worked on the question set together in the same classroom. The experiment comprised three sessions. In each session, one of the three groups worked on their assigned question set. A session lasted roughly 30–40 min.
Expert-based evaluation
The aim of this evaluation is to evaluate the quality of the automatically generated distractors using a human expert. Because of limited resources, we asked one human expert to evaluate the questions. However, we believe his judgement is reliable because he is an experienced professional writer of these questions.
We provided the expert with an evaluation guideline that includes the question writing guidelines presented in the “Distractor candidate filtering” section. Given a target word and its corresponding reading passage, the expert evaluated each of the three distractor sets used in the test taker-based evaluation by giving it a score of 1–5, where 1 indicates very low quality and 5 indicates very high quality. We also provided an optional “comment” field where he could write any possible reasons for giving a low score to a set of distractors, or explain why distractors were problematic, if any existed in the set.
Results and discussion
Test taker-based evaluation
Correlation with test takers’ proficiency scores
We calculate the correlation between test takers’ scores on the questions and their TOEIC scores, which we treated as the ground truth proficiency scores. The idea is that if the test takers’ scores on the machine-made questions show a strong correlation with their TOEIC scores, then the machine-made questions are able to measure the test taker’s proficiency. Table3 presents the Pearson correlation coefficients between the scores.
Table 3 Pearson correlation coefficients between test scores (averaged for all groups)
Full size table
The scores on the questions generated by both automatic methods show a lower correlation with their TOEIC scores than those of the original TOEFL questions. However, all methods indicate a low correlation in absolute terms. This is because the TOEIC score reflects various kinds of English proficiency of the test takers, whereas the generated questions concern only their vocabulary.
Focusing on the vocabulary ability, we also calculated the correlation coefficients between test takers’ scores on the machine-generated questions and those of the original TOEFL questions. This yielded positive correlations with coefficients of 0.425 (t = 5.039, df = 38, p value < 0.05) for the proposed method and 0.302 (t = 6.08, df = 37, p value < 0.05) for the baseline. (Evans 1996) categorised a correlation coefficient of 0.425 as “moderate” and 0.302 as “weak” correlation. This result is encouraging because it indicates that questions using the proposed method are more successful than those created using the baseline at measuring the test takers’ proficiency with respect to the original TOEFL questions.
Neural Test Theory analysis
Neural Test Theory (NTT) (Shojima 2007) is a test theory for analysing test data that grades the test takers into several ranks (on an ordinal scale). The idea behind this theory is that a test cannot distinguish test takers who have nearly equal abilities; the most that a test can do is to grade them into ranks. Susanti et al. (2017) applied the nominal neural test (NNT) model (Shojima et al. 2008), which is a variant of NTT for nominal-polytomous data, which is suitable for our vocabulary multiple-choice questions. NTT is useful for evaluating the statistical characteristics of options in multiple-choice questions. The item category reference profile (Shojima et al. 2008), ICRP for short, is a feature of NNT representing the probability that the test taker in a certain rank selects a certain question option in their responses to a certain question. Susanti et al. (2017) claimed that ICRP can be used to clarify the validity of the question options because it shows how test takers at each rank behave against each option of the question. For instance, ICRP can be used to clarify whether a distractor correctly deceives the low-ranked test takers compared with the high-ranked test takers.
Susanti et al. (2017) further categorised the ICRP into six categories based on the magnitude of the relations between the probability that the option is selected by test takers in the corresponding student rank, as shown in Fig.2. According to Susanti et al. (2017), the MD options are most favourable for distractors because the role of a distractor is to deceive a test taker into selecting it instead of the correct answer. Hence, the options that tend to be more selected by the lower-ranked test takers are good distractors. Such options should show a decreasing curve similar to the MD options in Fig.2. They further claimed that the MD options are the best for distractors, followed by the CU1 and CD1 options, then the CU2 and CD2 options. The MI options are the worst options for distractors.
Fig. 2
ICRP categories (image source: Susanti et al. (2017))
Full size image
In the present study, we applied the NNT model to our student response data, following the settings of Susanti et al. (2017), and counted the number of distractors in each ICRP category for each method as shown in Table4.
Table 4 ICRPs of the distractors for each method
Full size table
The three methods produced more or less a similar number of the favourable MD distractors. However, as shown in the first row of Table4, the proposed method produces fewer MI distractors (least favourable category for a distractor) than the baseline. The original TOEFL questions, as expected, produced the smallest number of MI distractors. This result is encouraging because it shows that the proposed method succeeded in removing the problematic distractor candidates better than the baseline.
We further analysed the MI distractors to find the reasons these distractors were categorised as MI distractors. The probability of choosing the MI distractors increases as the the test taker’s rank increases. This indicates that more high-proficiency test takers are deceived by this distractor than low-proficiency test takers. Knowing the reasons helps us to understand the behaviour of each method when producing those distractors. We found that MI distractors could be classified into the following four categories. SYN The distractor is a synonym of the target word or correct answer, e.g. the distractor “support” for the target word “assistance”, where the correct answer is “help”. We looked up two dictionariesFootnote 12, and if the distractors are listed as a synonym in one of the dictionaries, we classified them in this category. This type of distractors is not appropriate for use in tests. CON This distractor can be replaced in the given context, e.g. the distractor “move” for the target word “cope” when the correct answer is “adapt” in the following context “…dinosaurs were left too crippled to cope, especially if, as some scientists believe …”. In this example, the distractor “move” is neither similar to the target word nor the correct answer, but it fits in the context even though it results in a different sentence meaning. We checked the collocation of these words by querying Google search with a distractor and the word it is adjacent to in the reading passage as the query. This kind of distractor is reasonable because the test takers sometimes try to select the option that best replaces the target word in the reading passage. REL This distractor is defined as a word related to the target word or correct answer in a dictionaryFootnote 13, e.g. the distractor “storm” is defined as a related word of the target word “bombard” in the Merriam-Webster Dictionary. This kind of distractor is also reasonable. UNK This type of distractor has an MI curve in Fig.2 without any convincing explanation such as one of the above three categories. These distractors can be safely used as a distractor although they are not very distracting.
Table5 presents the number of the MI distractors categorised according to the above reasons. The results in Table5 suggest the following conclusions.
Table 5 Categorisation of MI distractors
Full size table
CON is the principal reason for the MI distractors across all methods.
On the basis of the above categorisation, the SYN candidates should be rejected as distractors because they are potentially dangerous. None of the MI distractors from the original TOEFL questions and proposed method belong to this category, whereas 11 out of 40 MI distractors of the baseline do and should be rejected. These results indicate that the proposed method succeeded in filtering the problematic candidates in this SYN category.
The CON and REL distractors are considered to be reasonable distractors, even though they are MI distractors. According to Table5, 23 out of 24 original TOEFL distractors belong to this category. The proposed and baseline methods respectively made 14 out of 26 (54%) and 13 out of 40 (33%) reasonable distractors in the CON and REL categories. This result is encouraging because more than half of the MI distractors generated by the proposed method are distracting distractors.
Expert-based evaluation
We calculate the average judgement scores for all 45 test questions, and the result is as follows: 2.867 (SD 1.471) for the baseline, 4.333 (SD 0.977) for the proposed method and 4.444 (SD 0.840) for the original TOEFL distractors. These average scores indicate that the distractors generated by the proposed method have better quality than those generated by the baseline and comparable quality with respect to the original TOEFL distractors.
The human expert also wrote in a total of 135 comments for all questions. As explained in the “Expert-based evaluation” section, the comments were specifically given to low-scored distractors; in other words, the distractors with comments were the problematic distractors according to the human expert. In total, 71 distractors from the baseline had comments, followed by 39 distractors from the proposed method, and 25 distractors from the original TOEFL distractors. This result is encouraging because the proposed method produced fewer of problematic distractors than the baseline. We grouped the comments into the seven categories presented in Table6 along with the number of distractors in each category for each method. Note that a distractor can belong to more than one category, so the row “total number of problematic distractors” is not necessarily the sum of the distractors in all categories. A description of the seven categories follows.
Table 6 Categorisation of problematic distractors by human expert
Full size table
1)
Too similar to the correct answer or the target word. Comments explicitly state that a distractor is too similar to the correct answer, e.g. “the distractor ‘overcome’ is too close to the correct answer or ‘refined’ is too similar to the correct answer”
2)
Different word class. Comments concern the difference in the word classes of the correct answer/target word and distractors, e.g. “the distractor ‘arise’ is an intransitive verb, ‘digging out’ is a verb phrase while ‘extending’ and ‘destroying’ are not.”
3)
No relation to the correct answer. Comments concern criterion 6 in the “Distractor candidate filtering” section, e.g. “the distractor ‘battlefield’ is not related to the correct answer.”
4)
Different word difficulty. Comments concern criterion 2 in the “Distractor candidate filtering” section, e.g. “all the distractors are much more difficult than the correct answer.”
5)
Antonym of the correct answer. Comments concern criterion 5 in the “Distractor candidate filtering” section, e.g. “the distractor ‘separate’ is an antonym of the correct answer.”
6)
Synonym pair. Comments concern criterion 4 in the “Distractor candidate filtering” section, e.g. “the distractor ‘repel’ and ‘repulse’ are synonyms.”
7)
Others. Comments that are not classified into the above categories, e.g. “the distractor ‘financially rewarding’ should be changed, because it involves the same word as the correct answer.”
Comment category 1 is the severest category because if a distractor is too similar to either the correct answer or the target word, it makes the question invalid because it has more than one correct answer. In this respect, our result is encouraging because the proposed method generated fewer invalid questions in this category. The other comment categories are considered not to be as severe because they do not affect the validity of the question.
We calculated the correlation of the expert scores of the distractor sets. The correlation coefficients are 0.313 (statistically significant at p value ≤ 0.05) for the proposed method and human pair and 0.012 (not statistically significant) for the baseline and human pair. This indicates that the expert tends to give similar scores to the proposed method’s distractors and the original distractors. Hence, the distractors generated by the proposed method look more similar to the human-made distractors than those generated by the baseline method from the expert’s point of view.
Comparison of the expert and test taker-based results
As previously stated, the distractors with comments from the human expert are potentially problematic. We further analysed the behaviour of the distractors in each comment category when they were used in the real test, i.e. in the test taker-based evaluation in the “Test taker-based evaluation” section. We analysed only the responses of the high-proficiency test takers because it is important to determine why high-proficiency test takers were deceived by the problematic distractors. We summarise the results in the following.
1)
Too similar to the correct answer or the target word. In six out of 30 distractors, no test takers selected the distractor in this category, whereas an average 30% of the high-proficiency test takers selected the other 24 distractors. The distractors in this category must be verified by human experts before they are used in a real test because there is a chance that they are actually the correct answers. One example is the distractor “notion” in a question with the target word “concept” and the correct answer “idea”. In this example, “notion” is a synonym of both the target word and correct answer. Out of 19 test takers, 8 test takers chose the distractor “notion”, whereas only 2 test takers chose the correct answer “idea”. Those 8 test takers did not necessarily choose the wrong answer because the distractor “notion” was indeed correct. This supports the claim that a question should not have distractors with a meaning that is too similar to either the target word or the correct answer.
2)
Different word class. Less than 23% of the high-proficiency test takers selected 29 out of 32 distractors in this category. Although these distractors are not necessarily problematic, they are not very distracting.
3)
No relation to the correct answer. Less than 30% of the high-proficiency test takers selected these distractors. As above, although these distractors are not necessarily problematic, they are not very distracting.
4)
Different word difficulty. The distractors that are easier or more difficult than the other options will stand out and might not be selected by the test takers because their difficulty looks salient. This is supported by the fact that 59 out of 67 distractors in this category were selected by less than 30% of the high-proficiency test takers. Hence, the distractors in this category are not distracting distractors.
5)
Antonym of the correct answer. More than 50% of the high-proficiency test takers did not select three out of four distractors in this category. If the distractor and correct answer are antonym pair, this suggests that one of them is wrong. This kind of distractor is not distracting.
6)
Synonym pair. No high-proficiency test takers selected the distractors in this category in four out of ten questions. This is most likely because they found out that a synonym pair in the options could not be a correct answer because a question has only a single correct answer. The distractors in this category should be verified by a human expert before they are used in a real test. One example is the distractors “life-sized” and “lifelike” for a question with the target word “miniature” and the correct answer “small”. No test taker out of 23 chose neither “life-sized” nor “lifelike”. The test takers probably figured out that they were synonym pair, so both could not be the correct answer. This gives evidence that there should not be a synonym pair in the options because the test taker can easily rule them out as a correct answer.
We are also interested in how the problematic distractors from the test taker-based evaluation (the MI distractors) were evaluated by the human expert. The MI distractors are considered problematic because the probability of choosing this distractor increases as the proficiency of the test takers increases. This indicates that more high-proficiency test takers are deceived by this distractor than low-proficiency test takers. Table7 shows the intersection of the MI distractors in the test taker-based evaluation and the commented distractors by the human expert, which is categorised according to Table6. Again, because a distractor can belong to more than one category, the sum of distractors in all categories can be larger than the “intersection” row.
Table 7 Categorisation of problematic distractors (combined)
Full size table
Table7 shows that 60% (24 out of 40) of the MI distractors in the baseline were also considered problematic by the human expert. This indicates that the baseline distractors that were judged as problematic by the human expert also behaved inappropriately in the real test. However, the same conclusion could not be drawn from the MI distractors generated by the proposed method and from the original TOEFL questions. Only 35% (9 out of 26) and 21% (5 out of 24) distractors generated by the proposed method and those from the original TOEFL questions, respectively, were judged as problematic by the human expert. This is an encouraging result because those distractors, despite their low score given by the human expert, may still be used in a real test, i.e. the problem is not severe.
Conclusion
We have presented a novel method for generating distractors for multiple-choice English vocabulary questions. The quality of distractors directly influences the quality of the question because inappropriate distractors allow the test takers to either guess the correct answer easily or unnecessarily confuse them.
The proposed method extends the state-of-the-art method by introducing a new metric for ranking distractor candidates. The new metric aggregates both semantic similarity and word collocation information. The idea is to find distractors that are (1) close to the target word but far from the correct answer in their meaning and (2) collocated with the adjacent words in the given context.
We conducted two evaluations for assessing the quality of the generated distractors: (1) test taker-based evaluation and (2) expert-based evaluation. We prepared 45 questions where the original reading passages and correct answers were borrowed from the TOEFL vocabulary questions. We prepared three sets of distractors for each question: one generated by the baseline, one generated by the proposed method and one from the original TOEFL question.
In the test taker-based evaluation, we administered the generated questions to 80 English learners and analysed the quality of the distractors based on their responses. We calculated the correlation between the test taker’s scores on the automatically generated questions and their scores on the original TOEFL questions as well as the correlation with their TOEIC scores. As a result, the scores on the questions prepared using the proposed method correlate better with those of the original TOEFL questions than those using the baseline. However, there is no difference between them with respect to the correlation with the TOEIC scores. The scores on the original TOEFL questions showed the highest correlation with those of the TOEIC scores.
We further analysed the characteristics of the distractors using Neural Test Theory. The result showed that the proposed method produced fewer problematic MI distractors than the baseline.
The original TOEFL questions produced the least number of MI distractors. This is encouraging because the proposed method succeeded in removing problematic distractor candidates during their generation better than the baseline.
In the expert-based evaluation, we asked a human expert to judge the quality of the three sets of distractors on a scale from 1 (low quality) to 5 (high quality). The average scores indicate that the distractors generated by the proposed method are better in quality than those generated by the baseline and are comparable in quality with the original distractors.
Further analysis showed that 60% of the baseline problematic distractors from the test taker-based evaluation were also considered problematic by the human expert. In contrast, only 35% and 21% problematic distractors from the proposed method and the original TOEFL questions, respectively, were judged as problematic by the human expert. This is an encouraging result because these distractors can be used for a real test despite their low score from the human expert.
Although the proposed method removes synonyms of the target word and correct answer from the distractor candidates, the expert-based evaluation showed that it still produces problematic distractors that are too similar to the correct answer. The distractors of this category can make a generated question invalid because it appears to have multiple correct answers. The questions generated by the proposed method still need human validation before using for a real test.
Future work for this research includes improving the automatically generated distractors so that they are as close as possible in quality to the human-made ones, especially with respect to their distractive power and plausibility.
Notes
They used about 14 million sentences from Chinese Wikipedia.
Available at the model consists of 1000 dimensions, 10 skipgrams and no stemming.
In their paper, they use the term “carrier sentence” because the text is usually only a sentence, not necessarily a reading passage as in the present study.
Available as a package in the Natural Language Toolkit (NLTK).
The word vectors were trained on Wikipedia 2014+ Gigaword 5, which consists of 6B tokens 400K vocabulary uncased words and provides 100 dimensional vectors. This resource is available at
Corpora: Brown, ABC, Genesis, Web Text, Inaugural, Gutenberg, Treebank and Movie Reviews, available at www.nltk.org/nltk_data/
www.ets.org
The Official Guide to the New TOEFL iBT, 2007, published by McGraw-Hill, New York.
Oxford Dictionary of English (www.oxforddictionaries.com) and the Merriam-Webster Dictionary (www.dictionaryapi.com).
The Merriam-Webster Dictionary’s related-word feature.
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Department of Computer Science (W8E-6F), Tokyo Institute of Technology 2-12-1 Oookayama, Meguro-ku, Tokyo, 152-8552, Japan
Yuni Susanti,Takenobu Tokunaga&Hitoshi Nishikawa
College of Economics, Aoyama Gakuin University, Tokyo, Japan
Hiroyuki Obari
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Susanti, Y., Tokunaga, T., Nishikawa, H. et al. Automatic distractor generation for multiple-choice English vocabulary questions. RPTEL13, 15 (2018).
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Received: 01 March 2018
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Keywords
Distractor
English vocabulary question
Automatic distractor generation
Multiple-choice question
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Araki, J., Rajagopal, D., Sankaranarayanan, S., Holm, S., Yamakawa, Y., Mitamura, T. (2016). Generating questions and multiple-choice answers using semantic analysis of texts. In Proceedings of COLING 2016, the 26th International Conference on Computational Linguistics: Technical Papers, Osaka, Japan. (pp. 1125–1136).
Correia, R., Baptista, J., Mamede, N., Trancoso, I., Eskenazi, M. (2010). Automatic generation of cloze question distractors. In Proceedings of the Interspeech 2010 Satellite Workshop on Second Language Studies: Acquisition, Learning, Education and Technology, Tokyo.
Downing, S.M. (2006). Selected-response item formats in test development. In Handbook of Test Development. Lawrence Erlbaum Associates, Inc., Mahwah, (pp. 287–301).
Google Scholar
Evans, J.D. (1996). Straightforward statistics for the behavioral sciences. Pacific Grove, CA: Brooks/Cole Publishing.
Google Scholar
Fellbaum, C. (1998). WordNet: An Electronic Lexical Database. Cambridge: MIT Press.
Google Scholar
Haladyna, T.M. (2004). Developing and validating multiple-choice test items, 3rd edition. Mahwah: Lawrence Erlbaum Associates.
Google Scholar
Heaton, J.B. (1989). Writing English language tests. Hongkong: Longman Pub Group.
Google Scholar
Hoshino, A., & Nakagawa, H. (2005). A real-time multiple-choice question generation for language testing -a preliminary study. In Proceedings of the Second Workshop on Building Educational Applications Using NLP. Association for Computational Linguistics, Ann Arbor, (pp. 17–20).
ChapterGoogle Scholar
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190474 | https://goldbook.iupac.org/terms/view/O04363/plain | Title: oxidation number Long Title: IUPAC Gold Book - oxidation number DOI: 10.1351/goldbook.O04363 Status: current Definition In English is largely synonymous with oxidation state, and may be preferred when the value represents a mere parameter or number rather than being related to chemical systematics or a state of the atom in a compound. Etymologically, it stems from the no-longer-used term 'Stock number' (oxidation number of a central atom; the charge it would bear if all the ligands were removed along with the electron pairs that were shared with the central atom) and the likewise obsolete term Ewens–Bassett number (ion charge). Related Terms - Ewens–Bassett number: - central atom: - oxidation state: Sources - PAC, 2014, 86, 1017. 'Toward a Comprehensive Definition of Oxidation State' on page 1020-1021 ( - PAC, 2016, 88, 831. 'Comprehensive definition of oxidation state' on page 832 ( Other Outputs - html: - json: - xml: Citation: Citation: 'oxidation number' in IUPAC Compendium of Chemical Terminology, 5th ed. International Union of Pure and Applied Chemistry; 2025. Online version 5.0.0, 2025. 10.1351/goldbook.O04363 License: The IUPAC Gold Book is licensed under Creative Commons Attribution-ShareAlike CC BY-SA 4.0 International ( for individual terms. Collection: If you are interested in licensing the Gold Book for commercial use, please contact the IUPAC Executive Director at executivedirector@iupac.org . Disclaimer: The International Union of Pure and Applied Chemistry (IUPAC) is continuously reviewing and, where needed, updating terms in the Compendium of Chemical Terminology (the IUPAC Gold Book). Users of these terms are encouraged to include the version of a term with its use and to check regularly for updates to term definitions that you are using. Accessed: 2025-09-28T23:58:14+00:00 |
190475 | https://onlinemschool.com/math/formula/square/ | Study of mathematics online.
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Square. Formulas and Properties of a Square
Page Navigation: Definition of a square The basic properties of a square Diagonal of a square The perimeter of a square The area of a square The circumscribed circle of a square (circumcircle) The inscribed circle of a square (incircle)
Definition.
Square is a regular quadrilateral in which all four sides and angles are equal. Squares differ only in sides length but all four angles is right angles.
|
| |
| Fig.1 | Fig.2 |
The basic properties of a square
Squares can also be a parallelogram, rhombus or a rectangle if they have the same length of diagonals, sides and right angles.
All four sides of a square are same length, they are equal:AB = BC = CD = AD:AB = BC = CD = AD
Opposite side of a square are parallel:AB||CD, BC||AD
All four angles of a square are right angles:∠ABC = ∠BCD = ∠CDA = ∠DAB = 90°
Sum of the angles of a square are equal to 360 degrees:∠ABC + ∠BCD + ∠CDA + ∠DAB = 360°
Diagonal of a square are same length:AC = BD
Each diagonal of a square divides its into two equal symmetrical area
Diagonals of a square intersect its right angles, and share each other half:
| | | | |
--- --- |
| AC┴BD | | AO = BO = CO = DO = | d |
| 2 |
Intersection point of the diagonals is called the center of the square and also the circumcenter of the inscribed circle and circumscribed circle
Each diagonal divides the angle of the square in half, meaning they are bisectors of the angles of the square:ΔABC = ΔADC = ΔBAD = ΔBCD∠ACB = ∠ACD = ∠BDC = ∠BDA = ∠CAB = ∠CAD = ∠DBC = ∠DBA = 45°
Both diagonals divide the square into four equal triangle besides these triangles are both isosceles and rectangular:ΔAOB = ΔBOC = ΔCOD = ΔDOA
Diagonal of a square
Definition.
Diagonal of the square is any segment that connects two vertices opposite angles of the square.
Diagonal of any square is always greater than its side √2 times.
Diagonal of a square formulas
Formula of the square diagonal in terms of the square side:d = a·√2
Formula of the square diagonal in terms of the square area:d = √2A
Formula of the square diagonal in terms of the square perimeter:
| | |
--- |
| d = | P |
| 2√2 |
formula of the square diagonal in terms of the circumradius:d = 2R
formula of the square diagonal in terms of the diameter of the circumcircle:d = Dc
formula of the square diagonal in terms of the inradius:d = 2r√2
formula of the square diagonal in terms of the diameter of the incircle:d = Di√2
formula of the square diagonal in terms of the length of the segment l (Fig.2):
| | |
--- |
| d = l | 2√10 |
| 5 |
The perimeter of a square
Definition.
The perimeter of a square called the sum of the lengths of all sides of the square.
Perimeter of a square formulas
Formula of the square perimeter in terms of the square side: P = 4a
Formula of the square perimeter in terms of the square area: P = 4√A
Formula of the square perimeter in terms of the square diagonal: P = 2d√2
Formula of the square perimeter in terms of the circumradius: P = 4R√2
Formula of the square perimeter in terms of the diameter of the circumcircle: P = 2Dc√2
Formula of the square perimeter in terms of the inradius: P = 8r
Formula of the square perimeter in terms of the diameter of the incircle: P = 4Di
Formula of the square perimeter in terms of the length of the segment l (Fig.2):
| | |
--- |
| P = l | 8 |
| √5 |
The area of a square
Definition.
Area of the square is called the space is limited to the square of the parties, that is, within the perimeter of a square.
The area of a square larger than area of any quadrilateral with the same perimeter.
Area of a square formulas
Formula of the square area in terms of the square side:A = a2
Formula of the square area in terms of the square perimeter:
| | |
--- |
| A = | P2 |
| 16 |
Formula of the square area in terms of the square diagonal:
| | |
--- |
| A = | d2 |
| 2 |
Formula of the square area in terms of the circumradius:A = 2R2
Formula of the square area in terms of the diameter of the circumcircle:
| | |
--- |
| A = | Dc2 |
| 2 |
Formula of the square area in terms of the inradius: A = 4r2
Formula of the square area in terms of the diameter of the incircle: A = Di2
Formula of the square area in terms of the length of the segment l (Fig.2):
| | |
--- |
| A = l 2 | 16 |
| √5 |
The circumscribed circle of a square
Definition.
The circumscribed circle of a square (circumcircle) called circle which passes only four top corners of the square and has a center at the intersection of the diagonals of the square.
The circumradius lager then inradius fo √2 times.
The circumradius equal to half the diagonal.
The area of the circumcircle larger then area of the same square at is π/2 times.
Circumradius of a square formulas
Formula of the square circumradius in terms of the square side:
| | |
--- |
| R = a | √2 |
| 2 |
Formula of the square circumradius in terms of the square perimeter:
| | |
--- |
| R = | P |
| 4√2 |
Formula of the square circumradius in terms of the square area:
| | |
--- |
| R = | √2A |
| 2 |
Formula of the square circumradius in terms of the square diagonal:
| | |
--- |
| R = | d |
| 2 |
Formula of the square circumradius in terms of the diameter of the circumcircle:
| | |
--- |
| R = | Dc |
| 2 |
Formula of the square circumradius in terms of the inradius: R = r √2
Formula of the square circumradius in terms of the diameter of the incircle:
| | |
--- |
| R = Di | √2 |
| 2 |
Formula of the square circumcircle in terms of the length of the segmentl (Fig.2):
| | |
--- |
| R = l | √10 |
| 5 |
The inscribed circle of a square
Definition.
The inscribed circle of a square (incircle) called the circle is tangent to the middle of the square sides and a circumcenter at the intersection of the diagonals of the square.
The inradius equal to half a square side.
The area of a incircle smaller than area of the square is 4/π times.
Inradius of a square formulas
Formula the square inradius in terms of the square side:
| | |
--- |
| r = | a |
| 2 |
Formula the square inradius in terms of the square diagonal:
| | |
--- |
| r = | d |
| 2√2 |
Formula the square inradius in terms of the square perimeter:
| | |
--- |
| r = | P |
| 8 |
Formula the square inradius in terms of the square area:
| | |
--- |
| r = | √A |
| 2 |
Formula the square inradius in terms of the circumradius:
| | |
--- |
| r = | R |
| √2 |
Formula the square inradius in terms of the diameter of the circumcircle:
| | |
--- |
| r = | Dc |
| 2√2 |
7 Formula the square inradius in terms of the diameter of the incircle:
| | |
--- |
| r = | Di |
| 2 |
Formula the square inradius in terms of the length of the segmentl (Fig.2):
| | |
--- |
| r = | l |
| √5 |
Geometry formulas Square. Formulas and Properties of a Square Rectangle. Formulas and Properties of a Rectangle Parallelogram. Formulas and Properties of a Parallelogram Rhombus. Formulas and Properties of a Rhombus Circle, disk, segment, sector. Formulas and properties Ellipse. Formulas and properties of ellipse Cylinder. Formulas and properties of a cylinder Cone. Formulas, characterizations and properties of a cone Area. Formulas of area Perimeter. Formulas of perimeter Volume. Formulas of volume Surface Area Formulas
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190476 | https://goldbook.iupac.org/terms/view/C00876/plain | Title: catalyst Long Title: IUPAC Gold Book - catalyst DOI: 10.1351/goldbook.C00876 Status: current Definition A substance that increases the rate of a reaction without modifying the overall standard Gibbs energy change in the reaction; the process is called catalysis. The catalyst is both a reactant and product of the reaction. The words catalyst and catalysis should not be used when the added substance reduces the rate of reaction (see inhibitor). Catalysis can be classified as 'homogeneous catalysis', in which only one phase is involved, and 'heterogeneous catalysis', in which the reaction occurs at or near an interface between phases. Catalysis brought about by one of the products of a reaction is called autocatalysis. Catalysis brought about by a group on a reactant molecule itself is called intramolecular catalysis. The term catalysis is also often used when the substance is consumed in the reaction (for example: base-catalysed hydrolysis of esters). Strictly, such a substance should be called an activator. Related Terms - Catalysis: - Michaelis–Menten kinetics: - activator: - autocatalytic reaction: - bifunctional catalysis: - catalytic coefficient: - electron-transfer catalysis: - esters: - general acid catalysis: - general base catalysis: - hydrolysis: - inhibitor: - interface: - intramolecular catalysis: - micellar catalysis: - phase-transfer catalysis: - product: - pseudo-catalysis: - rate of reaction: - reactant: - specific catalysis: Source - PAC, 1996, 68, 149. 'A glossary of terms used in chemical kinetics, including reaction dynamics (IUPAC Recommendations 1996)' on page 155 ( Related References - PAC, 1990, 62, 2167. 'Glossary of atmospheric chemistry terms (Recommendations 1990)' on page 2178 ( - PAC, 1993, 65, 2291. 'Nomenclature of kinetic methods of analysis (IUPAC Recommendations 1993)' on page 2293 ( - PAC, 1994, 66, 1077. 'Glossary of terms used in physical organic chemistry (IUPAC Recommendations 1994)' on page 1093 ( Other Outputs - html: - json: - xml: Citation: Citation: 'catalyst' in IUPAC Compendium of Chemical Terminology, 5th ed. International Union of Pure and Applied Chemistry; 2025. Online version 5.0.0, 2025. 10.1351/goldbook.C00876 License: The IUPAC Gold Book is licensed under Creative Commons Attribution-ShareAlike CC BY-SA 4.0 International ( for individual terms. Collection: If you are interested in licensing the Gold Book for commercial use, please contact the IUPAC Executive Director at executivedirector@iupac.org . Disclaimer: The International Union of Pure and Applied Chemistry (IUPAC) is continuously reviewing and, where needed, updating terms in the Compendium of Chemical Terminology (the IUPAC Gold Book). Users of these terms are encouraged to include the version of a term with its use and to check regularly for updates to term definitions that you are using. Accessed: 2025-09-29T01:00:23+00:00 |
190477 | https://timberlinefinancial.com/blog/understanding-thresholds | Understanding Spending Thresholds and How to Make Them Work for You - Timberline Financial
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October 14, 2015
Understanding Spending Thresholds and How to Make Them Work for You
Spending thresholds help establish value in spending decisions. Often this mental exercise is done subconsciously. Think about it. When you find an item you like, you determine how much it costs. Then you either quickly decide to buy it, “because it’s only five bucks” or a consciously evaluate the item. At that point you either talk yourself into or out of the purchase depending on your mood, how much money is in your pocket, and how much you feel you need the item.
Let’s say there’s a barber who charges $30 a haircut. A person, for example, may think, it’s only a haircut, I’ll get it. That’s their subconscious spending threshold, $30 dollars. Everything below that amount is decided without much thought or energy. Everything over that amount needs some level of consideration.
Consumers have different spending thresholds that are largely determined by the financial situation and income. The higher the income the higher the threshold is likely to be. College students living on a part time income may think $5 in spending needs some thought. For others the amount may be as high as $50 or $100, before stopping to think about the spending decision.
Why Thresholds Matter
Most of us do not have unlimited funds and your spending threshold can eat into your budget. This is spending you for which you can’t account. eally Things you buy might include a daily paper, daily coffee, an appetizer at lunch, or $20 bucks to the Boy Scout fundraiser. If this occurs frequently it can amount to hundreds of dollars each month in extra expenses that have very little value.
Let’s say you’re spending threshold is $20 and you spend at that limit 4 times a week. This totals $320 a month or $3,840 per year. Add a partner’s spending and that can drain $7,680 out of the family budget in a year’s time. It’s easy to see how leaky spending could pay for a vacation, pay off a credit card, or add to your retirement fund.
How to Manage Threshold Spending Effectively
Realistically no one wants to track every penny that is spent. Many cannot track spending effectively for 30 days, let alone long enough to get thousands in debt paid off and money for retirement.
Strategies that actually work:
1)Consciously lower your threshold. Decide on a lower amount you will adhere to. For example, anything over $5, you will make a conscious decision about. Anything over $100 is discussed with your partner. This creates the habit of really thinking about your purchases and what value it ads to your life. It can also open communication about how money is being spent.
2)Set a spending range. Give yourself a set amount that you can spend however you want each week, month or paycheck. You can spend it on lottery tickets and lattes or save it for a larger purchase but the “allowance” is in the budget.
3)For recurring expenses you really enjoy, find a less expensive way to enjoy it. For example, instead of buying a daily paper, subscribe to it. Instead of the daily cup of coffee from a local coffee shop, buy a nice coffee maker to make your own. This fulfills the habits that offer value, without the higher expense of spontaneous purchases.
These solutions will grant you more control over spending and provide some structure that can be worked into the budget to meet your needs and wants. It will also increase the chances of successfully sticking with an established budget without creating a sense of deprivation.
If you are burdened with high amounts of credit card debt and are struggling to make your payments, or you’re just not seeing your balances go down, call Timberline Financial today for a free financial analysis.
Our team of highly skilled professionals will evaluate your current situation to see if you may qualify for one of our debt relief programs. You don’t have to struggle with high-interest credit card debt any longer.
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190478 | https://www.youtube.com/watch?v=zq3p0qUOvE0 | ADDING FRACTIONS and MIXED NUMBERS using a BAR MODEL
Clelland Maths
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Posted: 9 Mar 2021
This video will teach you how to use a bar model for adding fraction and mixed numbers. We will look at adding fractions with the same denominator using a bar model. We will then look at adding fractions with a different denominator using a bar model. This will extend to adding mixed numbers using a bar model. We will then look at subtracting fractions using a bar model and subtracting mixed numbers using a bar model. Finally, we will look at subtracting mixed numbers where the answer is a negative fraction and examples without using a bar model.
CHAPTERS
0:00 - Introduction
0:15 - Example 1 - Adding fractions with the same denominator using a bar model
1:08 - Example 2 - Adding fractions with different denominators using a bar model
2:56 - Example 3 - Adding fractions and mixed number using a bar model
5:09 - Example 4 - Adding mixed numbers using a bar model
8:31 - Example 5 - Subtracting fractions with different denominators using a bar model
9:36 - Example 6 - Subtracting mixed numbers using a bar model
11:04 - Example 7 - Subtracting mixed numbers with a negative fraction in the answer
13:09 - Example 8 - Further example Subtracting mixed numbers with a negative fraction in the answer
I hope you enjoyed this video on adding and subtracting fractions and mixed numbers using a bar model. This will be the first in a series of videos on operations with fractions using a bar model. My next video will look at multiplying fractions and mixed numbers using a bar model.
National 5 Playlist -
This video is suitable for pupils in Scotland studying National 5 Mathematics. In England, is would be GCSE Mathematics.
maths #clellandmaths #fractions Copyright © Scottish Qualifications Authority.
Solutions all my own and not endorsed or produced by SQA.
Transcript:
Introduction that's the clown here from cloud maps today we're going to be looking at how to add and subtract fractions and we're going to do that by using the bar model then hopefully by the end of the lesson we'll get away from the bar model and be able to just do it numerically so let's start with just a simple example we see what we're doing Example 1 - Adding fractions with the same denominator using a bar model example one so i want to do a third plus a third a very simple thing to do but common mistakes here is some people think that might be two sets and i'm going to show why not so we can model this by drawing a bar and splitting that bar up into three equal parts so each part would be a third now if i did that i would get a picture of it look like this and notice i've shaded in two of those parts because i need one third for the first third so that's the third and i need another third for the next four the third plus a third and just counting how many i've got i've got one third here another third here one plus one i've got two of them so i've got two thirds so the answer every problem is simply two thirds hopefully that makes sense let's look at a slightly more complicated example for when the denominators are different and what we need to do is adjust and fix that so example two says what's a Example 2 - Adding fractions with different denominators using a bar model third plus a quarter now if i was to draw a bar and put it in free but i'm not going to disappoint before if i draw a plus 24 now it's about free so i need i need a number which i can split so that when i split the bar up i can split into thirds but i can also split it into quarters well 3 and 4 go into 12 so i'm going to get a bar that is 12. so i've done that here split this into 12 pieces if you can count them 1 2 3 4 5 6 7 eight nine ten eleven twelve right now my first fraction is a third so a third of twelve that's four parts so i want to shade in four of these parts there's one part shaded there's two there's three there's four so four of those parts make a fart because i could shade then another four here to make another third and another four here to make an effort so i now know that this part here is one third and i can do the same sort of thing for the quarter so if i take the quarter and try and shade that i'm going to need 12 divided by 4 is free one two three parts so i know that that is a quarter of the bar because the bar's got 12 parts in it the quarter of 12 is three so that's a quarter so now i can just count how many pieces i've got out of 12. i've got one two three four five six seven so a third plus a quarter is seven out of twelve example three three and a quarter Example 3 - Adding fractions and mixed number using a bar model plus a fifth now three in a quarter freeze a whole number so i've got three holes where i've got a quarter plus a fifth so if i was to write this out longer way that's the same as writing three plus a quarter plus a fifth so the only part i need to worry about is doing this quarter plus a fifth and then just having for you on at the end so i can model the quarter plus the fifth part with a bar as normal this time my bar is split into 20 pieces because the smallest number that four and five going to is 20. so if i want to switch my above the quarters and fifths i'm going to need to cover bar which is 20 pieces so i'll start off with my quarter i'll take a little notice this time a quarter of 20 equals five so i'm going to need five pieces here so let's pick a color one two three four and five so those five pieces represent simply the fraction one quarter let's do the same thing for the fifth so if i take a little note at the side a fifth of 20 pieces equals four so i need to shade in four pieces one two three and four pieces so those four pieces is a third so now we can do the same as we did before just count our pieces one two three four five six seven eight nine we've got nine out of twenty so the answer our sum is three plus nine out of twenty well that's just three and nine twentieths and we're done next example so example Example 4 - Adding mixed numbers using a bar model four says four and two-thirds add three and a half okay so let's split this up a little bit we've got four plus two-thirds plus three plus a half let's rearrange that so the whole numbers are together and the fractions are together so we've got four add three plus two thirds plus a half whole numbers can just be added together as normal four plus four is seven so i need to do seven add two thirds add a half so the big thing i need to do is to first plus a half and work out what that is and add it on to seven so let's do that in the next slide by using the bar model this time you should be able to see the bottom numbers three and two of fractions so i'm going to have to use six as my model split a barn is six so that i can split up the thirds and halfs so four and two thirds plus three and a half i've got my bar ready to go so first of all i'm going to shade in two thirds so one third if i take another side a third of six equals two so there's one third shady then another two of them won't give me two votes just tidy that up a little bit okay i've shaded them two thirds of the bar so let's move on to our half so if we go back a half of six equals three oh i need to shade in three parts but i've not got three parts left to shade in that's a little bit of an issue but it's easily solved we could just model this by having another bar model out of six and shading in how many we need so let's take another bar model so this first bar model represents our two thirds in the second bar model we're just going to do what we did is before shading half of it so there's one piece shading two pieces and three pieces to give me a half of a bar shooter so that represents a half so we've got four and two-fourths plus three and a half we just need to count our pieces now we've got one two three four five six seven six now seven six if i think of about a hole a whole bar if i just move these about you where we see this if i move this to here you should be able to see i've got one whole bar plus this extra piece down here left over so i've got four five six seven a whole bar makes eight and then an extra six left over one out of six again i've got a whole bar so that's four five six seven plus an extra one makes eight because this is one and then i've got one out of six left over one sixth so the answer of the sum is 7.8 and the sixth Example 5 - Subtracting fractions with different denominators using a bar model example five take away this time we've got two thirds and minus a half now we could model this exactly same way in a bar but instead of adding bars we can think of taking bars off okay so let's try that we've got a bar which has been split into six pieces now if i shade in two thirds well one third would be two pieces so two first i need to shade in four pieces this time so let's take a color miss one two three four so these bars represent two-fourths and then you think about taking away half of the whole bar now the whole bar is six so half of them is free so if i take away three bits that should give them the answer so one two three i'm left with one piece out of six so the answer is simply one-sixth this time with mixed numbers Example 6 - Subtracting mixed numbers using a bar model five and a third minus three and a quarter still no problem there just like when we added we can do the whole parts and the fraction parts separately we can do the same with taken away so this is simply the same as doing five minus three plus third minus a quarter just keeping it as separate pieces so i can do a five minus three t straight away to get two and then i'm going to have to add on the answer to this sum a fourth minus a quarter so let's do a third minus a quarter on the next page so i've took a bar as we did before and i've split it with 12 because three and four going to 12 so let's start building up our model taking a third of 12. i'm going to have to shade in four pieces so let's do that now so shading in one font we'll need to shade in four pieces so there's one two three four four of that is a third the way a quarter now quarter would be three pieces so if i take away three pieces one two three even what i'm left with is just one piece out of twelve so i prefer to take away a quarter as a twelve so the answer of the sum is five take away three is two i first take away a quarter is a 12. so it's 2 and 1 12 left over this time we've got six and a Example 7 - Subtracting mixed numbers with a negative fraction in the answer quarter minus two in the third let's get away from using a bar and start doing this just numerically so the sum we're going to try to do is six take away two plus the answer to one quarter minus the third well six take away two is four all right let's do a quarter minus a third on a new page so we've got one quarter minus a third so if we're using a bar we would be needing to split into 12. so in other words we're going to make it silver 12 as our common denominator right so if i took a quarter of 12 i would have three pieces and if i took a third of 12 i would have four pieces for you take away four problem is i can't do three take away four using a picture i can't start off with three things and take away four but we can with maths we can do free take away four three take away four is negative one so the answer if we caught twelves minus 4 12 is minus a 12. now that seems a little bit funky a little bit strange how can we have minus of 12 is our answer let's go back to our question we've got 4 and we're going to have to now add minus a 12. we'd have to take away our 12 from four well that's okay that's okay because we can split our four into however many parts we want so let's take our four down to three if i had three then i could have another whole minus a 12. but a whole could just be 12 out of 12. so taking that step further that'll just be three plus a whole 12 out of 12 minus one over twelve so nah now we can do it right twelve out of twelve minus one out of twelve twelve minus one eleven at a twelve so the answer is simply three and eleven twelfths example eight let's try that again okay Example 8 - Further example Subtracting mixed numbers with a negative fraction in the answer so we've got fourteen and three eighths minus three and four nine so let's deal with our whole part first let's do our 14 minus 3 here equals 11. nice and we're going to have to do 3 8 minus 4 9. well let's take a separate page for that and see how we get on with that one so remember what fractions mean if you think of the first one for the apes it means take a bar split into eight pieces and shade in three of them right and then the next one says take a bar so put it in nine pieces and shade in four of them but we want to take a bound so we can split it into a and nine so we're gonna have to think of a number that a and nine go into well eight times nine is seventy two so if we used 72 that's going to work for us so 72 is going to be our common denominator so we split in a 72 for both of them now 8 goes into 72 nine times so for each eighth i need to shade in nine but i've got three of them nine threes so nine times three 27 similarly for the one ninth and nine goes into 78 times so for each ninth i need to put shade in eight pieces eight times four is 32 so i'll get 32 on top showing that a little bit like mathematically if i had three eights and you wanted a little method for this minus four knives common denominator seventy two half times this one by nine so i need to times the top by nine to get twenty seven enough times this one by eight so i need to times the total pi eight to get thirty two so that's another way to think of it twenty seven minus thirty two is negative 5 372. so going back to our question that we had we now know that this is negative 5 out of 72. so putting it together you've got 11 right answer first minus 5 over 72 so that means i've got 10 plus an extra one which we'll call 7272 minus a 5 out of 72. remember that means 10 plus 1 is 11. and i'll start with take away 572's so that gives me with 10 72 take away 5 is 67 and 6 7 is 72 and there's our answer this has been mr clem from cleveland mass hopefully you found this helpful and useful a simple starter and adding some track reactions with mixed numbers as well stay safe take care and good luck you |
190479 | https://journalofinequalitiesandapplications.springeropen.com/articles/10.1186/s13660-024-03107-3 | Advertisement
The best constant for inequality involving the sum of the reciprocals and product of positive numbers with unit sum
Journal of Inequalities and Applications
volume 2024, Article number: 29 (2024)
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Abstract
In this paper, we study a special algebraic inequality containing a parameter, the sum of reciprocals and the product of positive real numbers whose sum is 1. Using a new optimization argument the best values of the parameter are determined. In the case of three numbers the algebraic inequality has some interesting geometric applications involving a generalization of Euler’s inequality about the ratio of radii of circumscribed and inscribed circles of a triangle.
1 Introduction
Inequalities with sharp constants, or at least when good estimates can be given of the sharp constants, are of special interest both in themselves and when they are used for various applications. Just as one example, we mention the recent paper [201
")] in this Journal. Concerning the importance for various applications we refer to the recent books [21
")] and [19
")], and the references therein.
Consider the inequality
where (x_{1},x_{2},\ldots ,x_{n}>0); (\sum_{i = 1}^{n}{x_{i}}=1), for (n\geq 2). Here, (\lambda >0) is a real number and we are asked to find the best (maximal possible) λ for each n (see [7f
")]). If such a λ exists, then we will denote it by \(\lambda \_{n}\). Note that the right-hand side of the inequality (1)
where (x_{1},x_{2},\ldots ,x_{n}>0); (\sum_{i = 1}^{n}{x_{i}}=1), is a nondecreasing function of (\lambda >0). Hence, if (1) is true for a certain (\lambda =\lambda _{n}), then it is also true for all (0<\lambda \leq \lambda _{n}).
By the Cauchy–Schwarz inequality (\sum_{i = 1}^{n}{\frac{1}{x_{i}}}\geq n^{2} =f(n^{2})). Since the inequality holds true for (\lambda = n^{2}), it also holds true for all (0<\lambda \leq n^{2}). Hence, the best constant (\lambda =\lambda _{n}), if it exists, satisfies (\lambda _{n} \geq n^{2}).
Case (n=2). For the case (n=2) there is no best constant. If (n=2), then we obtain the inequality
where (x_{1},x_{2}>0); (x_{1}+x_{2}=1). This inequality is true for any (\lambda >0). Indeed, if we multiply both sides by ((1+(\lambda -4)x_{1}x_{2})), then we obtain
Since (x_{1}+x_{2}=1), the parameter λ cancels out, and we obtain
which is always true.
Case (n=3). For case (n=3) the best constant is (\lambda _{3}=25). We obtain the inequality
where (x_{1},x_{2},x_{3}>0); (x_{1}+x_{2}+x_{3}=1). This inequality is true only for (0<\lambda \leq 25). We can show this by substituting (x_{1}=x_{2}=\frac{1}{4}), (x_{3}=\frac{1}{2}) in this inequality. On the other hand, we can prove that
holds true. Hence, (\lambda =25) is the maximum possible value for this inequality (see ). In the solution to problem it was noted by D.B. Leep that the case (\lambda =25) is equivalent to a more general inequality (s_{1}^{3}s_{2}+48s_{2}s_{3}-25s_{1}^{2}s_{3}\ge 0) for symmetric polynomials (s_{1}=x_{1}+x_{2}+x_{3}), (s_{2}=x_{1}x_{2}+x_{2}x_{3}+x_{3}x_{1}), (s_{3}=x_{1}x_{2}x_{3}), which can also be written as
making case (n=3) almost trivial. Inequality (1) can also be written using symmetric polynomials, but as the results for cases (n=4) and (n=5) below suggest, there is no simple solution for (n>3). Let
If (\lambda >0), then (1) is equivalent to the inequality
which is homogeneous with respect to its variables (x_{1},\ldots , x_{n}).
There are some geometric applications of case (n=3). The inequality
where R and r are, respectively, the circumradius and inradius, and a, b, c are the sides of a triangle, holds true if (\mu \leq 8). Indeed, substituting (a=b=3), (c=2), and the corresponding values of (R=\frac{9}{4\sqrt{2}}) and (r=\frac{\sqrt{2}}{2}) in (2) we obtain (\mu \leq 8). Hence, again, if we can prove (2) for (\mu =8), then (\mu =8) will the best constant for the inequality (2). For (\mu =8) we obtain
which is a refinement of Euler’s inequality (\frac{R}{r}\geq 2) and follows directly from the case (n=3) (see [4, 5]).
Another geometric application is the following inequality about the sides a, b, c of a triangle that follows directly from the case (n=3) (see ):
This inequality can also be written as a quintic inequality of symmetric polynomials
which is a special case ((v=3)) of the following inequality mentioned in (see p. 244, where (v=u+1))
This general inequality is also easily proved if we put (a=x_{2}+x_{3}), (b=x_{1}+x_{3}), (c=x_{1}+x_{2}), and simplify to obtain
Similar quartic and sextic inequalities were studied in 88
"), [23.
")], and their references (see also [16, vol. 28. Kluwer Academic, Dordrecht (1989)
")], Chap. 3).
One more geometric application of case (n=3) is about the areas of triangles and needs the introduction of some notations. Let M be a point in a triangle (ABC). Extend lines AM, BM, and CM to intersect the sides of triangle (ABC) at (A_{0}), (B_{0}), and (C_{0}), respectively (see Fig. 1). Next, construct the parallel to (A_{0}C_{0}) through M, which intersects BA and BC at (C_{1}) and (A_{2}), respectively. Analogously, draw the parallel through M to (B_{0}A_{0}) (and to (B_{0}C_{0})) to find (A_{1}) and (B_{2}) (and (B_{1}) and (C_{2})). Denote
where the square brackets stand for the area of the triangles (see [3, 6]). Then,
Geometric application of case (n=3)
Case (n=4). For the case (n=4) the best constant is (\lambda _{4}=\frac{582\sqrt{97}-2054}{121}\approx 30.4) (see ). In this case, we obtain
where (x_{1},x_{2},x_{3},x_{4}>0); (x_{1}+x_{2}+x_{3}+x_{4}=1). Again, this inequality is true only for (\lambda \leq \frac{582\sqrt{97}-2054}{121}). Indeed, substituting in this inequality (x_{1}=x_{2}=x_{3}=\frac{5+\sqrt{97}}{72}), (x_{4}=\frac{19-\sqrt{97}}{24}), we obtain (0<\lambda \leq \frac{582\sqrt{97}-2054}{121}). On the other hand, we can prove that the inequality holds true for (\lambda = \frac{582\sqrt{97}-2054}{121}). Hence, (\lambda = \frac{582\sqrt{97}-2054}{121}) is the maximum possible value for this inequality.
Case (n=5). For the case (n=5) we obtain the inequality
where (x_{1},x_{2},x_{3},x_{4},x_{5}>0); (x_{1}+x_{2}+x_{3}+x_{4}+x_{5}=1), and it was conjectured in that the best constant is
where (\alpha =\sqrt{8119+48\sqrt{22{,}535}}). This conjecture for (\lambda _{5}) will be proved in the current paper. Also, it will be proved that the equality cases in this inequality occur when (x_{1}=x_{2}=x_{3}=x_{4}=x_{5}= \frac{1}{5}) and when, for example, (x_{1}=x_{2}=x_{3}=x_{4}=x=\frac{\alpha}{240} + \frac{241}{240\alpha} + \frac{7}{240}\approx 0.173), (x_{5}=1-4x\approx 0.308).
Case (n=6). This case was not studied before. Using Maple, the exact value of (\lambda _{6}) is calculated. Case (n=6) is possibly the last case for which these calculations of the exact value are possible.
Case (n\ge 7). In view of the fact that quintic and higher-order equations are, in general, not solvable in radicals, it is unlikely that there is a precise formula for the best constant in the cases (n\ge 7). Therefore, for the greater values of n ((n\ge 7)), instead of the exact value, it is reasonable to find some bounds or approximations for (\lambda _{n}). In the current paper, it is proved that
Some possible improvements for this symmetric double inequality are also discussed.
It is interesting to compare the results of the current paper with the results for the following similar inequality
where (x_{1},x_{2},\ldots ,x_{i}>0); (\sum_{i = 1}^{n} {x_{i}}=1). The best constant (\nu _{n}) for this inequality is known for all (n>1). See Corollary 2.13 in , where it is proved that (\nu \le \nu _{n} =n^{2}-\frac{n^{n}}{(n-1)^{n-1}}). In particular, if (\nu =0), then we obtain
with the equality case possible only when (x_{1}=\cdots =x_{n}=\frac{1}{n}). Inequality (5) also follows from the following inequality for (E_{i}=\frac{1}{{\binom{n}{i}}}s_{i}) (averages of (s_{i})),
which holds if and only if
for each (1\le m\le n) (see , Theorem 1.1; , p. 94, item 77). Indeed, it is sufficient to note that inequality (5) can be written as (E_{n-1}\le E_{1}^{n-1}). This means that the above conditions for (\alpha _{i}), (\beta _{i}) ((i=1,\ldots ,n)) are satisfied as
Since (5) will be essential in the following text, an independent proof of (5) and some generalizations will be given in the Appendix. Note also that
Using this and by comparing (1) and (5), we obtain that if (n>2), then (\lambda _{n} <+\infty ).
Special cases (n=3) and (n=4) of inequality (4) are also of interest for comparison with the corresponding cases of inequality (1). If (n=3), then the best constant inequality is (\frac{1}{x_{1}}+\frac{1}{x_{2}}+\frac{1}{x_{3}}\leq \frac{9}{4}+ \frac{1}{4x_{1}x_{2}x_{3}}), where (x_{1},x_{2},x_{3}>0); (x_{1}+x_{2}+x_{3}=1). Surprisingly, this inequality is also equivalent to a geometric inequality. One can show that it simplifies to (p^{2}\geq 16Rr-5r^{2}), where p is the semiperimeter of a triangle. The last geometric inequality also follows from the formula for the distance between the incenter I and the centroid G of a triangle: (\vert IG \vert ^{2}=\frac{1}{9}(p^{2}+5r^{2}-16Rr)) (see ). If (n=4), then the best constant inequality is (\frac{1}{x_{1}}+\frac{1}{x_{2}}+\frac{1}{x_{3}}+\frac{1}{x_{4}}\leq \frac{176}{27}+\frac{1}{27x_{1}x_{2}x_{3}x_{4}}), where (x_{1},x_{2},x_{3},x_{4}>0); (x_{1}+x_{2}+x_{3}+x_{4}=1) (see , Example 3).
The literature about symmetric polynomial inequalities is extensive 13f
")–[15, 17–19 (1968).
"), 17, 17 (2000).
"), 18, 67–83 (2004).
"), 24, 815–819 (1989).
"), 26. Preprint, Arxiv
")]. Some of the results of the current paper were presented at the Maple Conference 2021 .
2 Main results
Let us consider all cases for (n\ge 3) in a unified way. Assume first that ((x_{1},x_{2},\ldots ,x_{n} )\ne (\frac{1}{n}, \frac{1}{n},\ldots ,\frac{1}{n} )). Then, by using (5), inequality (1) can be written as
where (x_{1},x_{2},\ldots ,x_{n}>0); (\sum_{i = 1}^{n}{x_{i}}=1), for (n\geq 3). Let us denote the left-hand side of (6) by (g(x_{1},\ldots ,x_{n})), which is defined for all points of the bounded set
except for point (P_{0} (\frac{1}{n},\frac{1}{n},\ldots ,\frac{1}{n} )). For the points of boundary
function g is undefined and, obviously, for each (i=1,\ldots ,n),
Lemma 1
If (x>0), then
Proof
The limit can be interpreted as a single variable limit if we take
where not all constants (\gamma _{i}) are equal and (t\rightarrow 0). Hence, we calculate
where we used L’Hôpital’s rule twice and the fact that (n\sum_{i = 1}^{n}{\gamma _{i}^{2}}> (\sum_{i = 1}^{n}{\gamma _{i}} )^{2}) (the Cauchy–Schwarz inequality, the equality case is not possible as not all (\gamma _{i}) are equal). The proof is completed. □
In particular, if (\sum_{i = 1}^{n}{x_{i}}=1), then (x=\frac{1}{n}), and therefore, by Lemma 1,
As an immediate consequence of this and (6), we obtain an upper bound for the best constant
We want to use a well-known result in the analysis, which states that a continuous function over a compact set achieves its minimum (and maximum) values at certain points. For this purpose, let us change function (g(x_{1},\ldots ,x_{n})), to a new function (g_{1}) so that (g_{1}) is defined also at point (P_{0} (\frac{1}{n},\frac{1}{n},\ldots ,\frac{1}{n} )) and points of ∂C, and (g_{1}) is continuous in the compact set (\overline{C}=C\cup \partial C):
Since (g_{1}) is a continuous function in compact C̅, (g_{1}) reaches its extreme values somewhere in C̅. Obviously, (g_{1}) reaches its maximum value (\frac{\pi}{2}) at the boundary points ∂C where (\prod_{i=1}^{n}{x_{i}}=0), and the minimum value at a point of C. The minimum of g is achieved at the same point of C if the minimum point is different from (P_{0} (\frac{1}{n},\frac{1}{n},\ldots ,\frac{1}{n} )). In any case, (\inf_{\textbf{x}\in C}{g}=\tan (\min_{ \textbf{x}\in \overline{C}}{g_{1}} )). We use an optimization argument similar to [12, 25] but with 3 variables, to determine where these points must lie. This method can also be used for other inequalities involving only symmetric polynomials (s_{1}), (s_{n-1}), and (s_{n}).
Let (P(x_{1},x_{2},\ldots ,x_{n})) be a minimum point of (g_{1}). Select any 3 of the coordinates of ((x_{1},x_{2},\ldots ,x_{n})), say (x_{1}), (x_{2}), and (x_{3}). Let us assume that (x_{1}x_{2}x_{3}=\alpha ) and (x_{1}+x_{2}+x_{3}=\beta ). Since (P\in C), (\alpha ,\beta >0). Also, by the AM-GM inequality (\beta ^{3}\ge 27\alpha ) and it is known that if (\beta ^{3}= 27\alpha ), then (x_{1}=x_{2}=x_{3}). Hence, suppose that (\beta ^{3}>27\alpha ). Let us now take arbitrary positive numbers x, y, z such that (xyz=\alpha ) and (x+y+z=\beta ). Without loss of generality we can assume that (x\le y\le z). Since (x+z=\beta -y) and (xz=\frac{\alpha}{y}), the numbers x and z are the solutions of the quadratic equation (\delta ^{2}+(y-\beta )\delta +\frac{\alpha}{y}=0). If we take (y=t), then we obtain parametrization of the curve obtained by intersection of the plane (x+y+z=\beta ) and the surface (xyz=\alpha ):
Parameter t changes in the interval ([t_{1},t_{2}]), where (t_{1}) and (t_{2}) are the zeros of the cubic (\kappa (t)=t(t-\beta )^{2}-4\alpha ) in intervals ((0,\frac{\beta}{3} )) and ((\frac{\beta}{3},\beta )), respectively. The third zero (t_{3}) of (\kappa (t)) satisfies (t_{3}>\beta ) and therefore (t_{3}\notin [t_{1},t_{2}]). Since we are interested only with the case (x\le y\le z), we will take one half of this curve (see Fig. 2)
and in a smaller interval ([t^{}_{1},t^{}_{2}]), where (t^{}_{1}) and (t^{}_{2}) are the zeros of the cubic (\kappa ^{}(t)=\kappa (t)-t(3t-\beta )^{2}) in intervals ((t_{1},\frac{\beta}{3} )) and ((\frac{\beta}{3},t_{2} )), respectively. Again, since the third zero (t^{}_{3}) of (\kappa ^{}(t)) satisfies (t^{}_{3}>\beta ), (t^{}_{3}\notin [t_{1},t_{2}]). Note that if (t=t^{}_{1}), then (x=y), and if (t=t^{}_{2}), then (y=z). Consider the sum (\frac{1}{x}+\frac{1}{y}+\frac{1}{z}) and note that
Denote (h(t)=\frac{1}{t}+\frac{\beta t-t^{2}}{\alpha}), where (t\in [t^{}_{1},t^{}_{2}]). Hence, if (t\in (t^{}_{1},t^{}_{2})), then (h'(t)=-\frac{1}{t^{2}}+\frac{\beta}{\alpha}-\frac{2t}{\alpha}= \frac{(z-t)(t-x)}{xy^{2}z}> 0), and (h'(t^{}_{1})=h'(t^{}_{2})=0). Consequently, (h(t)) attains its minimum and maximum in the interval ([t^{}_{1},t^{}_{2}]) at the endpoints (t^{}_{1}), (t^{}_{2}), respectively. We are interested in making h smaller, which happens when the sum (\frac{1}{x}+\frac{1}{y}+\frac{1}{z}) is smaller. Hence, the minimum of (\frac{1}{x_{1}}+\frac{1}{x_{2}}+\frac{1}{x_{3}}) is reached when (x_{1}=x_{2}\le x_{3}). Since the coordinates (x_{1}), (x_{2}), and (x_{3}) were chosen arbitrarily, these results hold for any trio of coordinates. Therefore, the left-hand side of (6) is minimal only when there are at most 2 distinct numbers in the set ({x_{1},x_{2},\ldots ,x_{n}}) Furthermore, if the two numbers are distinct, then the smaller one is repeated (n-1) times in ({x_{1},x_{2},\ldots ,x_{n}}) i.e., (x_{1}=x_{2}=\cdots =x_{n-1}\le x_{n}). Consequently, in (6) we can restrict ourselves only to the case where (x_{1}=x_{2}=\cdots =x_{n-1}=x), (x_{n}=1-(n-1)x), where (0< x\le \frac{1}{n}). By substituting these in (6) and simplifying, we obtain
We will study the part of the denominator that is dependent on x, and for simplicity put (t=nx). Hence,
Denote the polynomial in the denominator by
where (0\le t\le 1). By taking the derivative and simplifying, we obtain
Since (p_{n}^{\prime }(0)=n>2) and (p_{n}^{\prime }(1)=-\frac{n(n-1)(n-2)}{6}<0), there is at least 1 zero of the polynomial (p_{n}^{\prime }(t) ) in the interval ((0,1)). On the other hand, by Descartes’ rule of signs (see p. 247 in [11p
")], or p. 28 in [22
")]) the number of positive zeros of \(p\_{n}^{\prime }(t)\) does not exceed the number of sign changes in the sequence of coefficients of \(p\_{n}^{\prime }(t)\), which is 1. Hence, \(p\_{n}^{\prime }(t)\) has exactly one zero \(t\_{n}\) in \([0,1]\), which is also the maximum point of \(p\_{n}(t)\). This means that there is exactly one point \(x=\frac{t\_{n}}{n}\) in \((0,\frac{1}{n} )\), such that \(x\_{1}=x\_{2}=\cdots =x\_{n-1}=x\), \(x\_{n}=1-(n-1)x\) makes the left-hand side of (6) minimal. This minimal value is also the best constant for (1):
For (n=3,4,5), and 6 it is possible to find the exact values of (t_{n}) and the corresponding (\lambda _{n}).
if (n=3), then (p_{3}(t)=(2-t)(1+2t) ), and (p_{3}^{\prime }(t)=3-4t ). Therefore, (t_{3}=\frac{3}{4}). By (8), the best constant is (\lambda _{3}=\frac{3^{2}}{1-\frac{3-1}{p_{3}(t_{3})}}=25) (see ).
if (n=4), then (p_{4}(t)=(3-2t)(1+2t+3t^{2}) ) and (p_{4}^{\prime }(t)=4+10t-18t^{2} ). Therefore, (t_{4}=\frac{5+\sqrt{97}}{18}). By (8), the best constant is (\lambda _{4}=\frac{4^{2}}{1-\frac{4-1}{p_{4}(t_{4})}}= \frac{582\sqrt{97}-2054}{121}\approx 30.423077) (see ).
if (n=5), then (p_{5}(t)=(4-3t)(1+2t+3t^{2}+4t^{3}) ) and (p_{5}^{\prime }(t)=5+12t+21t^{2}-48t^{3} ). Using Cardano’s formula and Maple, we find that (t_{5}=\frac{\theta +7+241\theta ^{-1}}{48}), where (\theta = (8119+48 \sqrt{22{,}535} )^{\frac{1}{3}}). By (8), the best constant is (\lambda _{5}=\frac{5^{2}}{1-\frac{5-1}{p_{5}(t_{5})}}\approx 40.090307), which coincides with the value of (\lambda _{5}) conjectured in .
if (n=6), then
Using Ferrari’s method and Maple, we find that
where
By (8), the best constant is (\lambda _{6}=\frac{6^{2}}{1-\frac{6-1}{p_{6}(t_{6})}}\approx 52.358913).
For larger values of n, we can give some bounds for (\lambda _{n}). We already found an upper bound (7). We will now focus on a similar lower bound.
Parametric space curve (blue and green) representing intersection of the plane (x+y+z=\beta ) (not shown) and the surface (xyz=\alpha ) (not shown)
By the AM-GM inequality,
where (x_{1},x_{2},\ldots ,x_{n}>0); (\sum_{i = 1}^{n}{x_{i}}=1), (G_{n}=\sqrt[n]{\prod_{i=1}^{n}{x_{i}}}), and (n\geq 2). Let us show that if (\lambda =\frac{n^{3}}{n-1}), then
where ((x_{1},x_{2},\ldots ,x_{n})\ne (\frac{1}{n},\frac{1}{n}, \ldots ,\frac{1}{n} )) and therefore (G_{n}=\sqrt[n]{\prod_{i=1}^{n}{x_{i}}}<\frac{1}{n}). Indeed, we can simplify (10) to
where (s=nG_{n}<1). This is easily proved, as we can write it in the following form
where noting (s<1) completes the proof. From (1), (9), and (10) it follows that if (\lambda \le \frac{n^{3}}{n-1}), then (1) holds true. This means that we have now a lower bound for the best constant:
Combining (7) and (11) we obtain the following symmetric double inequality.
Theorem 1
If (n>2), then
It is possible to improve these estimates in exchange for a less-elegant formula. For example, if we put (x_{1}=x_{2}=\cdots =x_{n-1}=\frac{1}{n+1}), (x_{n}=\frac{2}{n+1}), then we obtain from (6) a new upper bound for the best constant:
One can check that (13) is sharper than (6) for all (n>3). We can also prove that (\frac{n}{n+1}\le t_{n}) or equivalently, (p^{\prime}_{n} (\frac{n}{n+1} )\ge 0) for all (n\ge 3). Indeed,
For (n=3,4,\ldots ,25) one can check directly that ((1+\frac{1}{n} )^{n}-\frac{8}{3}+\frac{1}{n}- \frac{1}{3 n^{2}}\ge 0). For (n>25), one can use the fact that ((1+\frac{1}{n} )^{n}>\frac{8}{3}) and (\frac{1}{n}>\frac{1}{3 n^{2}}).
Data Availability
No datasets were generated or analysed during the current study.
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Acknowledgements
This work was completed with the support of ADA University Faculty Research and Development Fund.
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Appendix
Appendix
We will give a proof of (5) here. We can use the optimization argument given after Lemma 1 of the current paper, to maximize the left-hand side of (5), while keeping the right-hand side of (5) fixed. This is achieved when for any 3 of the coordinates, say (x_{1}), (x_{2}), and (x_{3}), of ((x_{1},x_{2},\ldots ,x_{n})), (x_{1}\le x_{2}=x_{3}). Hence, we can restrict ourselves only to the case where (x_{1}=x) and (x_{2}=\cdots =x_{n-1}=x_{n}=\frac{1-x}{n-1}), where (0< x\le \frac{1}{n}). For this particular case (5) is transformed into
which can be simplified to the correct inequality
The equality case is possible only when (nx=1).
Inequality (5) can also be written as the homogeneous inequality (A_{n}^{n-1}H_{n}\ge G_{n}^{n}), where (A_{n}), (H_{n}), and (G_{n}) are, respectively, the arithmetic, harmonic, and geometric means of arbitrary positive numbers (x_{1},\ldots ,x_{n}):
Since (A_{n}\ge G_{n}), automatically (A_{n}^{l}H_{n}\ge G_{n}^{l+1}) for any real number (l\ge n-1). It would be natural to ask whether the general inequality (A_{n}^{l}H_{n}\ge G_{n}^{l+1}) can hold true also for some real number (l< n-1). The answer to this question is negative. A counterexample is found if one takes (x_{1}=x_{2}=\cdots =x_{n-1}=1) and (x_{n}=x), where (x\rightarrow 0^{+}). Indeed, if (l< n-1), then (A_{n}^{l}H_{n}=O(x)) and (G_{n}^{l+1}=O(x^{\frac{l+1}{n}})\gg O(x)).
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Aliyev, Y.N. The best constant for inequality involving the sum of the reciprocals and product of positive numbers with unit sum.
J Inequal Appl 2024, 29 (2024).
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190480 | https://lifeiscomputation.com/it-is-not-intelligent-if-it-always-halts/ | It's Not Intelligent If It Always Halts: A Critical Perspective on Current Approaches to AGI - Life Is Computation
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It’s Not Intelligent If It Always Halts: A Critical Perspective on Current Approaches to AGI
Posted on April 6, 2023 September 5, 2023
Imagine a conversation with one of these newly released AI chatbots. You ask it it to solve a tricky math problem. It responds with “That seems kind of hard. Give me some time to think.”. After a few minutes it comes back with “I haven’t solve it yet. And I am not sure I can. Would you like me to continue working on it?”. Another few minutes pass and then it comes back with “Aha! I figured it out!” and it proceeds to explain a neat and creative solution.
This scenario can never occur with PaLM, BARD, GPT-4, or any of the other transformer-based large language models that are thought to be on the path to general intelligence. In all of these models, each word in the machine’s response is produced in a fixed amount of time. The model cannot go away and “think” for a while. This is one of the reasons why I believe a solely transformer-based model can never be “intelligent”. (If you disagree with my characterization of transformers here, see section 4 and also this post).
Summary: I argue here, that intelligence requires the ability to explore “trains of thought” that are potentially never-ending. One cannot know a priori if a certain train of thought will lead to a solution or if it is futile. The only way to find out is to actually explore. And this type of exploration comes with the risk of never knowing if you are on the path to a solution or if your current path will go on forever. Intelligence involves problem-solving, and problem-solving requires arbitrary amounts of time. If a computer program is bound to finish quickly by virtue of its architecture, it cannot possibly be capable of general problem-solving.
In the summary paragraph above, I appealed to a number of intuitive notions (e.g. “train of thought”, or “exploration”, or “problem-solving”). In order to make my argument rigorous, I have to first introduce a few concepts rooted in classical theory of computation. In section 1, I will introduce three types of computer programs. In section 2, I describe what an unintelligent problem-solver can look like. In section 3, I describe what is needed to make the unintelligent problem-solver intelligent. In section 4, I explain why transformers can never be general problem-solvers. In section 5, I briefly discuss what I think needs to be done to address this problem.
1. There are Three Types of Computer Programs
First, let us establish that there are three types of computer programs:
_(A) Programs that eventually finish executing (or “halt”)
(B) Programs that provably never halt
(C) Programs that never halt but for which there is no proof_
For our purposes, we are talking about computer program that take no inputs but may produce an output. Let’s call these types of programs “standalone programs”. For instance, take the standalone program below that tries to find a Pythagorean triple:
```
Program I
1: start with S = 3
2: for all positive integers a, b, and c where a+b+c=S
3: if a2 + b2 = c2
4: return tuple (a, b, c) and halt
5: end for loop
5: increment S by 1
6: go to line 2
```
We know this program will eventually output the tuple (3, 4, 5) because 3 2 + 4 2 = 5 2 and these three numbers have the smallest sum among all Pythagorean triples. So this program is in category (A). It’s a program that eventually halts.
What about category B? What is an example of a program that we know never halts? Here is a simple one:
```
Program II
1: start with x = 1
2: while x is not divisible by 3
3: if x is odd
4: increment x by 1
5: otherwise if x is even
6: decrement x by 1
7: end while loop
8: return x and halt
```
We can prove that this program never stops. It revisits the same state and falls in an infinite loop. The variable x starts with 1, then becomes 2, then 1, then 2, and this goes on forever.
The program above revisits repetitive states. But not all programs in category B have to repeat themselves. For example, take the following program that attempts to find a solution that fulfills a n+b n=c n for some n ≥ 3.
```
Program III
1: start with S = 6
2: start with n = 3
3: while n < S
4: for all positive integers a, b, and c where a+b+c+n=S
5: if an + bn = cn
6: return tuple (a, b, c, n) and exit
7: end for loop
8; increment n by 1
9: end while loop
10: increment S by 1
11: go to line 2
```
This program never revisits the same state. It keeps on trying new values for a, b, c, and n, trying larger and larger numbers as it progresses in time. But it also never halts. We are now certain it never halts because someone proved Fermat’s last theorem less than 30 years ago. So this program is also in category B.
So far we have examples for categories A and B. Now here is the strange thing. It is logically impossible for me to give you an example of a program that is in category C. Because in order for us to show that a program is in category C, we must somehow know that it never halts. But if I can prove that a program never halts then it falls in category B, not C. Category C consists of programs that cannot be proved to halt.
Butcategory C is not empty. We know that there are programs that fall in category C. In fact we can prove it! We just can’t give a specific example. I won’t explain how we know this, but suffice it to say that we have known this for close to a hundred years from the work of Turing, Church, Godel, and the other logicians/mathematicians of the 1930s.
Consider the following program:
```
Program IV
1: start with n = 7
2: while n is a new number never before encountered
3: if n is even
4: replace n with n/2
5: otherwise
6: replace n with 5n+1
7: return n
```
This program produces the Collatz 5n+1 sequence which goes like 7, 36, 18, 9, 46, 23, 116, 58, 29, … Currently, no one knows its fate. It might end by finally reaching a repetitive number (making it a category A program) or it may go on forever, never repeating the same number twice. If it does go on forever, it can be a category B program (meaning that we can somehow prove it goes on forever), or it might be unprovably never ending, in which case it will fall in category C.
If Collatz 5n+1 is a category A or B program, there is some hope in finding that out. But if it is a category C program we will never know if it is in A, B, or C! The fate of category C programs will always remain a mystery to humanity. Executing a category C program will forever generate novel patterns and relationships that we could not have predicted before running it. (See “Novelty generator” and “Collatz 5n+1 Simulator” on LifeWiki if you found this intriguing).
At every moment in time, we have programs that we know eventually halt (category A), programs that we know go on forever (category B), and programs that we have not yet categorized (which can be in category A, B, or C). Over time as we make progress, programs that had an unknown category fall into A or B (but never C). For example the third program I introduced above had an unknown category 30 years ago. Until Andrew Wiles came up with an ingenious proof and we discovered it was a category B program. This process can go on forever. We will never be done categorizing unknown programs, otherwise we could say “all that remains are category C programs” and we would then have found examples of category C programs, which we established is impossible. (This is why mathematics is a never-ending endeavor. We will never be done proving all provable theorems).
2. Let’s Build an Unintelligent General Problem Solver
Determining whether a program halts is such a general problem that you can take any other well-defined problem and reformulate it in terms of halting[and finding the final the state or the output of the program if it halts]. For example, I can rewrite a mathematical statement as a program such that halting is equivalent to proving that statement true. The program can iterate through all possible strings of text, in order of length, and checking whether that text is a valid proof for the statement in question. If the program finds a valid proof it halts. Otherwise it goes on forever.
But the halting problem is unsolvable. If it was solvable then there would be no category C programs. So there is no program that can accurately solve the generic problem of “does this program halt”.
Now we are going to do something that may first sound ambitious. We will sketch out a program that can take any standalone program as input and try to determine whether it halts or not. I will call this program “BruteBot“. Remember, that the only acceptable inputs to BruteBot are standalone programs, i.e. programs that do not take an input. BruteBot is not a standalone program itself because it takes inputs.
BruteBot is named such because it uses the most unintelligent brute force strategy possible. Given program X, it searches through all possible texts and for each text checks whether that is a valid proof that program X does or does not halt. In theory BruteBot can solve any solvable halting problem. If X is a program that eventually halts, BruteBot will eventually run into a text describing the progression of program X all the way until its end, which is a valid proof that X halts. (Note that BruteBot can also find the output of any category A program). If X is a category B program, that means there is a proof (expressible in text form) that X never halts. BruteBot will eventually run into it, given enough time and resources.
Image Generated Using DALL·E 2
The obvious trouble here is that BruteBot is practically useless. It is like the monkey typewriter that hits random keys and will eventually type up Hamlet. It needs years, perhaps millennia, to stumble upon the shortest texts that constitute valid proofs. And for difficult problems, like proving Fermat’s last theorem (or equivalently proving Program III never halts), there is no hope that it will ever stumble upon a valid proof given our time and resource constraints.
But there’s something very important to understand about BruteBot. When given a program that is in category A or category B, BruteBot eventually halts. But when BruteBot is given a category C program, it cannot possibly ever halt, because there is no valid proof for it to discover. So BruteBot(X) halts if and only if X is a category A or B program. If the input X is a category C program BruteBot(X) is also a category C program. I next argue that this feature is not limited to BruteBot. Any general problem-solver, intelligent or not, must have this feature. If it is guaranteed to distinguish between category A and category B programs it can only do so if it never halts when given a category C program.
3. General Problem-Solving Requires Unbounded Exploration with the Risk of Never Ending
Now let us imagine another general problem solver: GeniusBot. Like BruteBot, GenuisBot takes a standalone program as input and determines whether it halts. If it halts, it also returns the output of that program. We don’t know how this program works. Maybe it’s based on neural nets. Maybe it’s using a strange algorithm that its creators do not want to disclose. All we know is that it runs on a computer and it is strictly better than BruteBot. That means that GenuisBot not only eventually solves any solvable problem, it is also fast and practical. When we give it Program III (above), it solves Fermat’s Last Theorem on its own, never having seen Wiles’s proof.
Without knowing the inner workings of GeniusBot we are going to prove something about it: that it requires unbounded time to solve solvable problems. Assume the contrary: let’s say GeniusBot is able to solve any solvable problem within a certain amount of time, say, less than a minute. That means for every category A or B input, it answers within a minute. But what happens when we provide it with a category C input? Category C programs are unsolvable. But GeniusBot cannot report back that the input program is unsolvable, because otherwise it would have given us an example of a category C program! And that is impossible! Alternatively, if it gets stuck for category C programs, that is also self-contradictory! Because we could then use GeniusBot to determine if a program is in category C. All we need to do is wait for a minute (or whatever the time duration within which it can solve all solvable problems) and if GeniusBot has not responded by then, we know it was given a category C program. Remember we cannot ever determine the fate of a category C program (See part 1). So our initial assumption that GeniusBot solves all solvable problems within a fixed amount of time cannot be true.
What this means is that 1) there is no cap on how long GeniusBot needs to solve problems, and 2) there are some problems for which GeniusBot will go on forever and we cannot possibly know whether it will never end or if it just needs a little more time. (This latter conclusion means that if input X is a category C program, GeniusBot(X) is also a category C program).
Taking time to think isn’t just something humans do. Any intelligent problem-solver, biological or artificial, needs time to “think”, because problem-solving fundamentally takes time and it is similar to exploring an unknown territory. We proved this with minimal assumptions about how our general problem-solver, GeniusBot, works. Our only assumptions are that it can be run on a computer, it can solve any solvable problem provided sufficient resources (just like BruteBot can), and that it doesn’t give us incorrect answers.
People have come up with all sorts of definitions for intelligence (see this paper for a comprehensive review). I won’t add another formal definition to the list here. But for me intelligence is all about problem-solving. In my view, the quest for formalizing intelligence is about transforming something like BruteBot into something like GeniusBot: a program that is smart and efficient in how it searches for solutions for general problems. We defined our problems in terms of halting but AI doesn’t have to be explicitly about the halting problem. If you define “general problem-solving” broadly enough, it will inevitably include unprovably unsolvable problems (like category C programs). With a sufficiently general definition any problem will be reducible to the halting problem and, vice versa, the halting problem can be reduced to any type of problem. For example, any mathematical problem can be reduced to the halting problem, and likewise the halting problem for any program can be reduced to a problem in mathematics.
When I say intelligent problem-solving risks the possibility of never ending, that doesn’t mean it has to get stuck working on a single problem forever. That wouldn’t be very useful. Rather, the program can temporarily take a break from working on a certain problem and decide to go back to it later, perhaps once it has made insight on another related problem. It might end up never returning to that problem. Ideally, AI can get better with experience, drawing on acquired knowledge and using search heuristics that have practically worked out for it in the past. But no matter what it does, it cannot escape the fact that it needs unbounded time to search for solutions and that it will never know which problems are unprovably unsolvable. (This is not too different from how mathematicians do math, as individuals or even a whole community).
4. Why Transformers Alone Cannot Solve General Intelligence
Transformers are a feed-forward neural network architecture that has become one of the state-of-the-art methods in contemporary AI. They have led to an outburst of impressive language models that have stirred up lots of excitement, controversy, debate, and in some cases fear.
There is a prevailing belief that transformers are sufficient for performing any computational task, or even achieving general intelligence. A number of papers have claimed that transformers are Turing-complete, meaning that they can simulate any computer program (including category C programs). Those claims are demonstrably false. I explain why in a separate post. Apart from these papers, people often cite the universal approximation theorem (UAT) to support the idea that transformers are capable of computing any [computable] function. UAT has a misleading name and it does not imply the ability to solve all solvable functions. (I also wrote about this in a another post you can read here.)
Let us think about transformers in the context of all that we have learned up to here. Can a transformer be generally intelligent? We established that a generally intelligent program (like GeniusBot of the previous section) must take arbitrary amounts of time to solve problems. And for some inputs it will try to keep running forever. How can a transformer work like that? Transformers are feed-forward networks so they produce each token within a fixed amount of time. If we require it to only respond with the final answer, that certainly won’t work because that would mean it attempts to solve problems within a capped time limit, which we showed is impossible (section 3).
Another strategy is to allow it to “think out-loud” through chain-of-thought (CoT) prompting (see this paper and this other paper). With this method the transformer does not have to respond with the final answer immediately. It can produce a series of tokens that move toward the answer step by step. The architecture now includes a feed-back loop and is not strictly feed-forward anymore. It’s recurrent. We can even modify it to hide the intermediate tokens and keep its thought process to itself. At every step the transformer can use any of the previously produced tokens as input creating something like a memory scratchpad. This allows it to run for arbitrary amounts of time even allowing it to theoretically fall into an infinite loop! Have we solved the problem of unbounded time then? No, we have not.
Even a recurrent language model with CoT prompting does not fulfill the requirements outlined above, despite the fact that it can theoretically run forever for some inputs. A generally intelligent problem-solver must have the following property: for some inputs it must become a category C program (a program that never halts but it cannot be proved that it never halts). But the input size is fixed for a given transformer model. If the size of the input is fixed and there is a limit to the number of possible tokens (words, or even numbers with bounded precision) then the state of the transformer at any given moment is describable with a string of fixed length. So the only way for a transformer to go on forever is for it to revisit the same state. And that will make it a type B program; we can prove that it never halts. So there is no way for it to become a category C program. Computer programs, in general, can be category C programs because their states may need to be described by strings of unbounded length. Some programs, unlike transformers, do not have an inherent limit on the number of reachable states. Memory usage may depend on the input and be unbounded otherwise. (A program’s memory usage is the proportionally equivalent to the length of the string that describes that program’s state).
Ultimately, artificial intelligence must be formalized as a computer program. It’s fine if that program uses transformers to solve problems. That program can improve itself over time and its problem-solving strategies don’t have to stay constant. It can be highly dynamic and interactive, using feedback from its environment. But however it works, its state must not be describable with a string of fixed length. If the number of reachable states for a program is limited, independent of the input, then that program cannot become a category C program for any input. And it therefore cannot engage in general problem-solving. The AI program must be able to encounter paths of state changes (analogous to “trains of thought”) with no limit to the number of states in them. And that comes with the risk of being on a never ending path without being able to know it. This is not something that transformers, with their fixed size, can do.
One might argue that our brains are describable with a string of fixed length so my perspective on intelligence precludes humans. But that would be mistaking computer programs for the physical devices that run them. Dijskstra once said: “Computer Science is no more about computers than astronomy is about telescopes” and the same can be said about thought and the brain. My perspective on intelligence does not exclude humans. We are intelligent and we are capable of general problem-solving. I believe our physically limited brains execute programs whose states cannot be described with a string of fixed length. (Remember “fixed” or “bounded” is not the same as “finite”. You cannot describe any arbitrary integer with a fixed number of digits, but any integer can be described with a finite number of digits). When our problem-solving programs hit biological limits of memory, we somehow recognize it, and we then recruit resources outside of our brains for memory (e.g. fingers, or pebbles, or more commonly writing on paper). And before our problem-solving programs hit biological limits of time (e.g. limited life-spans of individuals) we communicate information about the state of our programs for others to be able to continue from where we left off. That is how mathematics, biology, physics, computer science, and many other problem-solving endeavors work. I do not see this as fundamentally different than taking a computer program from a resource constrained computer and continuing its execution on another computer, or having a computer program use cloud storage if it runs out of space on a local hard drive. These are all ways for dealing with resources constraints, independent of the computational problem that hits the constraints.
5. What needs to be done?
I hope to have convinced you that an intelligent general problem-solving program must have to property that it takes unbounded amounts of time to run. And that for some problems it will never halt but we would not be able to tell if it will eventually end (category C programs). But transformers currently lack this property. It may be tempting to try to solve this problem by hooking up a model like GPT-4 to an interface that allows it to run programs for unbounded lengths of time. I am skeptical that this approach can take us toward general intelligence.
The non-halting property is a necessary but not sufficient condition for intelligence. The unintelligent problem-solver (BruteBot) that we imagined above fulfills this necessary condition, yet is unintelligent. You can address the criticism I raised here by augmenting a transformer with something like an unbounded stack or memory pad that lets it run arbitrarily long programs. That would fulfill this condition but it will not necessarily make it intelligent. It would be like if you point out that cars cannot fly because flying requires an upward lift force, and someone responds by installing a powerful fan on the roof of a car. They may be able to lift the car but flying requires more than just getting off the ground. It may be wiser to forget about modifying the car and start over with something else.
The problem raised here is not just an abstract one. Transformer-based models are failing at solving intelligence benchmarks that that other approaches are succeeding in, like the Abstraction and Reasoning Corpus (ARC). I personally think program synthesis is the way to go and I am currently working on developing an approach to ARC. But I also think we should continue using large language models as tools. In the car metaphor, it would be unwise to design an airplane by modifying cars, but it would also be unwise to not use cars as transportation vehicles that can help us get to the goal of building an airplane.
The three images in this article were all created with the help of AI models. From top to bottom: Midjourney, DALL·E 2, and DreamStudio.
Posted in UncategorizedTagged Abstraction and Reasoning, AI, artificial intelligence, creativity, halting, halting problem, novelty, problem solving, thought, Turing machines
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190481 | https://chem.libretexts.org/Courses/Purdue/Chem_26505%3A_Organic_Chemistry_I_(Lipton)/Chapter_1._Electronic_Structure_and_Chemical_Bonding/1.08_Hybridization | Skip to main content
1.8: Hybridization
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: Jun 5, 2019
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1.7: Atomic Orbitals and Covalent Bonding
1.9: Representation of Molecular Structure
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( \newcommand{\kernel}{\mathrm{null}\,})
Hybridization was introduced to explain molecular structure when the valence bond theory failed to correctly predict them. It is experimentally observed that bond angles in organic compounds are close to 109o, 120o, or 180o. According to Valence Shell Electron Pair Repulsion (VSEPR) theory, electron pairs repel each other and the bonds and lone pairs around a central atom are generally separated by the largest possible angles.
Introduction
Carbon is a perfect example showing the value of hybrid orbitals. Carbon's ground state configuration is:
According to Valence Bond Theory, carbon should form two covalent bonds, resulting in a CH2, because it has two unpaired electrons in its electronic configuration.However, experiments have shown that CH2 is highly reactive and cannot exist outside of a reaction. Therefore, this does not explain how CH4 can exist. To form four bonds the configuration of carbon must have four unpaired electrons.
One way CH4 can be explained is, the 2s and the 3 2p orbitals combine to make four, equal energy sp3 hybrid orbitals. That would give us the following configuration:
Now that carbon has four unpaired electrons it can have four equal energy bonds. The hybridization of orbitals is favored because hybridized orbitals are more directional which leads to greater overlap when forming bonds, therefore the bonds formed are stronger. This results in more stable compounds when hybridization occurs.
The next section will explain the various types of hybridization and how each type helps explain the structure of certain molecules.
sp3 hybridization
sp3 hybridization can explain the tetrahedral structure of molecules. In it, the 2s orbitals and all three of the 2p orbitals hybridize to form four sp3 orbitals, each consisting of 75% p character and 25% s character. The frontal lobes align themselves in the manner shown below. In this structure, electron repulsion is minimized.
Energy changes occurring in hybridization
Hybridization of an s orbital with all three p orbitals (px , py, and pz) results in four sp3 hybrid orbitals. sp3 hybrid orbitals are oriented at bond angle of 109.5o from each other. This 109.5o arrangement gives tetrahedral geometry (Figure 4).
| Example: sp3 Hybridization in Methane |
| Because carbon plays such a significant role in organic chemistry, we will be using it as an example here. Carbon's 2s and all three of its 2p orbitals hybridize to form four sp3 orbitals. These orbitals then bond with four hydrogen atoms through sp3-s orbital overlap, creating methane. The resulting shape is tetrahedral, since that minimizes electron repulsion. Hybridization Lone Pairs: Remember to take into account lone pairs of electrons. These lone pairs cannot double bond so they are placed in their own hybrid orbital. This is why H2O is tetrahedral. We can also build sp3d and sp3d2 hybrid orbitals if we go beyond s and p subshells. |
sp2 hybridization
sp2 hybridization can explain the trigonal planar structure of molecules. In it, the 2s orbitals and two of the 2p orbitals hybridize to form three sp orbitals, each consisting of 67% p and 33% s character. The frontal lobes align themselves in the trigonal planar structure, pointing to the corners of a triangle in order to minimize electron repulsion and to improve overlap. The remaining p orbital remains unchanged and is perpendicular to the plane of the three sp2 orbitals.
Energy changes occurring in hybridization
Hybridization of an s orbital with two p orbitals (px and py) results in three sp2 hybrid orbitals that are oriented at 120o angle to each other (Figure 3). Sp2 hybridization results in trigonal geometry.
| Example: sp2 Hybridization in Aluminum Trihydride |
| In aluminum trihydride, one 2s orbital and two 2p orbitals hybridize to form three sp2 orbitals that align themselves in the trigonal planar structure. The three Al sp2 orbitals bond with with 1s orbitals from the three hydrogens through sp2-s orbital overlap. |
| Example: sp2 Hybridization in Ethene |
| Similar hybridization occurs in each carbon of ethene. For each carbon, one 2s orbital and two 2p orbitals hybridize to form three sp2 orbitals. These hybridized orbitals align themselves in the trigonal planar structure. For each carbon, two of these sp orbitals bond with two 1s hydrogen orbitals through s-sp orbital overlap. The remaining sp2 orbitals on each carbon are bonded with each other, forming a bond between each carbon through sp2-sp2 orbital overlap. This leaves us with the two p orbitals on each carbon that have a single carbon in them. These orbitals form a ? bonds through p-p orbital overlap, creating a double bond between the two carbons. Because a double bond was created, the overall structure of the ethene compound is linear. However, the structure of each molecule in ethene, the two carbons, is still trigonal planar. |
sp Hybridization
sp Hybridization can explain the linear structure in molecules. In it, the 2s orbital and one of the 2p orbitals hybridize to form two sp orbitals, each consisting of 50% s and 50% p character. The front lobes face away from each other and form a straight line leaving a 180° angle between the two orbitals. This formation minimizes electron repulsion. Because only one p orbital was used, we are left with two unaltered 2p orbitals that the atom can use. These p orbitals are at right angles to one another and to the line formed by the two sp orbitals.
Energy changes occurring in hybridization
Figure 1: Notice how the energy of the electrons lowers when hybridized.
These p orbitals come into play in compounds such as ethyne where they form two addition? bonds, resulting in in a triple bond. This only happens when two atoms, such as two carbons, both have two p orbitals that each contain an electron. An sp hybrid orbital results when an s orbital is combined with p orbital (Figure 2). We will get two sp hybrid orbitals since we started with two orbitals (s and p). sp hybridization results in a pair of directional sp hybrid orbitals pointed in opposite directions. These hybridized orbitals result in higher electron density in the bonding region for a sigma bond toward the left of the atom and for another sigma bond toward the right. In addition, sp hybridization provides linear geometry with a bond angle of 180o.
| Example: sp Hybridization in Magnesium Hydride |
| In magnesium hydride, the 3s orbital and one of the 3p orbitals from magnesium hybridize to form two sp orbitals. The two frontal lobes of the sp orbitals face away from each other forming a straight line leading to a linear structure. These two sp orbitals bond with the two 1s orbitals of the two hydrogen atoms through sp-s orbital overlap. Hybridization |
| Example: sp Hybridization in Ethyne |
| The hybridization in ethyne is similar to the hybridization in magnesium hydride. For each carbon, the 2s orbital hybridizes with one of the 2p orbitals to form two sp hybridized orbitals. The frontal lobes of these orbitals face away from each other forming a straight line. The first bond consists of sp-sp orbital overlap between the two carbons. Another two bonds consist of s-sp orbital overlap between the sp hybridized orbitals of the carbons and the 1s orbitals of the hydrogens. This leaves us with two p orbitals on each carbon that have a single carbon in them. This allows for the formation of two ? bonds through p-p orbital overlap. The linear shape, or 180° angle, is formed because electron repulsion is minimized the greatest in this position. Hybridization |
References
John Olmsted, Gregory M. Williams Chemistry: The Molecular Science Jones & Bartlett Publishers 1996. 366-371
Francis A. Carey Advanced Organic Chemistry Springer 2001. 4-6
L. G. Wade, Jr. Whitman College Organic Chemistry Fifth Edition 2003
Problems
Using the Lewis Structures, try to figure out the hybridization (sp, sp2, sp3) of the indicated atom and indicate the atom's shape.
The carbon.
The oxygen.
The carbon on the right.
Answers
sp2- Trigonal Planar
The carbon has no lone pairs and is bonded to three hydrogens so we just need three hybrid orbitals, aka sp2.
sp3 - Tetrahedral
Don't forget to take into account all the lone pairs. Every lone pair needs it own hybrid orbital. That makes three hybrid orbitals for lone pairs and the oxygen is bonded to one hydrogen which requires another sp3 orbital. That makes 4 orbitals, aka sp3.
sp - Linear
The carbon is bonded to two other atoms, that means it needs two hybrid orbitals, aka sp.
An easy way to figure out what hybridization an atom has is to just count the number of atoms bonded to it and the number of lone pairs. Double and triple bonds still count as being only bonded to one atom. Use this method to go over the above problems again and make sure you understand it. It's a lot easier to figure out the hybridization this way.
Contributors
Harpreet Chima (UCD), Farah Yasmeen
Further Reading
Carey 4th Edition On-Line Activity
Hybridization
Web Pages
Hybrid Orbitals
Hybrid Orbitals
Valence Bond Theory and Hybrid Orbitals
Hybridization of Carbon
Hybrid Orbitals
Hybridization
Hybridization
1.7: Atomic Orbitals and Covalent Bonding
1.9: Representation of Molecular Structure |
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2011 AMC 10B Problems/Problem 19
Contents
1 Problem
2 Solution 1
3 Solution 2
4 Solution 3
5 Solution 4
6 Video Solution
7 See Also
Problem
What is the product of all the roots of the equation
Solution 1
First, square both sides, and isolate the absolute value. Solve for the absolute value and factor.
Case 1:
Multiplying both sides by gives us Rearranging and factoring, we have
Case 2:
As above, we multiply both sides by to find Rearranging and factoring gives us
Combining these cases, we have . Because our first step of squaring is not reversible, however, we need to check for extraneous solutions. Plug each solution for back into the original equation to ensure it works. Whether the number is positive or negative does not matter since the absolute value or square will cancel it out anyways. Trying , we have Therefore, and are extraneous.
Checking , we have
The roots of our original equation are and and product is .
Solution 2
Square both sides, to get . Rearrange to get . Seeing that , substitute to get . We see that this is a quadratic in . Factoring, we get , so . Since the radicand of the equation can't be negative, the sole solution is . Therefore, can be or . The product is then .
Solution 3
First we note that ![Image 45: $x \in (-\infty,-4] \cup [4,\infty]$]( This will help us later with finding extraneous solutions. Next, we have two cases: . We note that is not in the range of possible 's and thus is not a solution.
. We again not that is an extraneous solution.
Thus, we have the two solutions and . Therefore product is .
-ConfidentKoala4
Solution 4
To make this problem easier to comprehend, we can define variable , with the condition that is always nonnegative. Also, since any number squared is always nonnegative, we can define . Then we can square both sides and substitute: Bringing the equation over to one side, we get: Solving for a by factoring, we get: so or . But must be nonnegative, so the only value that works is . If , then and , so can equal . Multiplying the two, we get .
~BeepTheSheep954
Video Solution
~savannahsolver
See Also
2011 AMC 10B (Problems • Answer Key • Resources)
Preceded by
Problem 18Followed by
Problem 20
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190483 | https://www.avoiderrors.com/how-to-use-advance-mode-calculator-windows-10/ | How to Use Advance Mode Calculator - Windows 10 - AvoidErrors
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How to Use Advance Mode Calculator – Windows 10
by Muhammad Imran Habib · September 26, 2018
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The Calculator app for Windows 10 is a friendly version of the desktop calculator which works on both mobile and desktop devices. You can open multiple instances of calculators at the same time on the desktop and can switch between the Standard, Scientific, Programmer, Date calculation, and Converter modes of the app.
To start, select the Start button, and then select Calculator from the list of apps.
Standard Mode:
The Standard mode, gives the ability to perform basic math functions like addition, subtraction, multiplication, and division.
There is the option to calculate square root, store numbers in the memory, and percentages. This mode is great for figuring out day-to-day math problems or tallying numbers.
Scientific Mode:
The Scientific mode gives you the functions of a standard scientific calculator. To switch to scientific mode click on View > Scientific Calculator or press Alt + 2.
In this mode you will find functions like Sin, Cos, Tan, Degrees, Radians, Grads etc. to perform the scientific calculations that you want.
Programmer mode:
The programmer calculator is for computer science persons. To get to this mode, click on View > Programmer Calculator or press Alt + 3. In this mode, you can switch between Hexadecimal, Decimal, Octal, and Binary number systems. You’ll see things like RoR, And, Rsh which are functions like logic gates and bit shifting.
Statistical mode:
The statistics calculator will be useful for statistical analysis. You find the average function, an average of the square of values, the sum of the values, sum of the square of the values, standard deviation, and standard deviation of the population. All these tools are basic to statistical mathematics.
Math with Dates:
For planning events or anything date-related, you can use date calculation function. You can calculate the difference between two dates, add or subtract days from a specific date etc.
Conversion of Measurement Units:
The unit conversion mode can compute angle, area, energy, length, power, pressure, temperature, time, velocity, volume, and weight and mass conversions. It can convert several different units of measurement within each category.
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Muhammad Imran Habib
Imran is a technology evangelist with 8 years of experience working with some of the Industry leading companies. Imran's expertise includes On-Prem/Virtual Infrastructure deployments, IT Solutions for SMEs, End User Computing Support.
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190484 | https://files.ele-math.com/articles/mia-22-76.pdf | Mathematical Inequalities & Applications Volume 22, Number 4 (2019), 1099–1122 doi:10.7153/mia-2019-22-76 ON A FUNCTIONAL EQUATION RELATED TO TWO–VARIABLE CAUCHY MEANS TIBOR KISS AND ZSOLT P ´ ALES Dedicated to Professor Josip Peˇ cari´ c on the occasion of his 70th birthday (Communicated by S. Varoˇ sanec) Abstract. In this paper, we are dealing with the solution of the functional equation ϕ x+y 2 ( f(x)−f(y)) = F(x)−F(y), concerning the unknown functions ϕ, f and F defined on a same open subinterval of the reals.
Improving the previous results related to this topic, we describe the solution triplets (ϕ, f,F) assuming only the continuity of ϕ .
As an application, under natural conditions, we also solve the equality problem of two-variable Cauchy means and two-variable quasi-arithmetic means.
1. Introduction Let J be a nonempty open real interval. We say that a two-place function M : J ×J →R is a two-variable mean on J if it possesses the Mean Value Property, that is, if min(x,y) ⩽M(x,y) ⩽max(x,y), (x,y ∈J).
If both of the above inequalities are sharp whenever x ̸= y, then M is called a strict mean. To formulate the problem what we would like to deal with, we need the following two special classes of means.
The Class of Cauchy Means. The Cauchy Mean Value Theorem states that if G,H : J →R are differentiable functions then, for all distinct elements x,y ∈J, there exists a point u in the open interval determined by the points x and y such that the equality G′(u) H(x)−H(y) = H′(u) G(x)−G(y) Mathematics subject classification (2010): 39B52, 46C99.
Keywords and phrases: Cauchy mean, quasi-arithmetic mean, functional equations involving means, equality problem of means.
The research of the first author was supported in part by the Hungarian Academy of Sciences and in part by the ´ UNKP-17-Doctorand New National Excellence Program of the Ministry of Human Capacities.
The research of the second author was supported by the Hungarian Scientific Research Fund (OTKA) Grant K111651.
c ⃝ , Zagreb Paper MIA-22-76 1099 1100 T. KISS AND Z. P ´ ALES holds. If H′ does not vanish on J, then, by the Rolle Mean Value Theorem, the function H must be injective on J, hence, in this case, the above equality is equivalent to G′ H′ (u) = G(x)−G(y) H(x)−H(y).
Now, one can see easily that u has to be unique if, in addition, the ratio G′/H′ is invert-ible on J. We note that, under the mentioned conditions, the invertibility of G′/H′ is equivalent to that it is continuous and strictly monotone on J . The precise formulation and the proof of this statement can be found as Remark 1 in the paper . To prove the essential part, the author uses the Darboux property of the derivative functions.
Motivated by the above observation, one can introduce the notion of Cauchy means as follows. We say that a mean M : J × J →R is a two-variable Cauchy mean if there exist differentiable functions G,H : J →R with 0 / ∈H′(J) such that the function G′/H′ is invertible on J and that, for all x,y ∈J, we have M(x,y) = G′ H′ −1 G(x)−G(y) H(x)−H(y) if x ̸= y and M(x,y) = x if x = y.
(1) In this case, we denote M by CG,H , where G and H will be called the generator functions of the Cauchy mean. There are many papers dealing with this class of means.
In particular, the equality and homogeneity problem of Cauchy means was completely solved by Losonczi in [20, 23] and , respectively. For the solution of the equality problem in , 7th-order differentiability was assumed. This regularity condition was reduced to first-order differentiability by Matkowski in . The comparison problem of Cauchy means was also studied by Losonczi . The so-called invariance equation for Cauchy means was solved by Berrone in . Characterization type results were obtained by Berrone and by Berrone–Moro .
The Class of Quasi-Arithmetic Means. We say that a mean M : J×J →R is a two-variable quasi-arithmetic mean if one can find a continuous strictly monotone function Φ : J →R such that, for all x,y ∈J, we have M(x,y) = Φ−1Φ(x)+ Φ(y) 2 =: AΦ(x,y).
The continuity and the strict monotonicity of Φ provides that Φ is invertible and that Φ(J) is an open subinterval of R. Hence the above expression is well-formulated and it indeed defines a strict mean on J. Similarly to the previous part, Φ will be called the generator function of the quasi-arithmetic mean.
In this paper we are going to focus on the equality problem of two-variable Cauchy means and two-variable quasi-arithmetic means. It is easy to check that the equality problem CG,H = AΦ on J can be rewritten as ϕ u + v 2 f(u)−f(v) = F(u)−F(v), (u,v ∈Φ(J)), (2) where we define ϕ := G′ H′ ◦Φ−1, f := H ◦Φ−1, and F := G◦Φ−1.
(3) ON A FUNCTIONAL EQUATION 1101 Motivated by this reformulation, to solve the equality problem, it is enough to describe the solution triplets (ϕ, f,F) of the functional equation ϕ x+ y 2 f(x)−f(y) = F(x)−F(y), (x,y ∈I), (4) where I stands for a nonempty open subinterval of R.
The functional equations having the form (4) have a rich literature and it was investigated by several authors. The firs remarkable result, from the year 1985, is due to J. Acz´ el , who solved (4) assuming that f is the identity function. This particular case was also dealt with by Volkmann . Without any regularity assumptions for the unknown functions, he proved that the pair (ϕ,F) solves the equation if and only if F is a polynomial of degree at most two and ϕ is its derivative. Independently, in 1979, Sh. Haruki investigated a pexiderized version of the equation of Acz´ el, where f was still the identity, and he proved the same result, again without any regularity conditions. The details of Haruki’s work can be found in . The solutions of the equation in (4) was first described in 2016, by Z. M. Balogh, O. O. Ibrogimov and B. S.
Mityagin in the paper . In their approach, the functions f and g was supposed to be three-times differentiable. Two years later, R. Łukasik improved this result in , by showing that the continuous differentiability of f and g is sufficient to obtain the same solutions. Now we present a different approach, which allows us to treat and solve (4) solely under the continuity of ϕ .
The functional equation (4) seems to be a particular case of the functional equation ϕ(x+ y) = F(x)G(y)+ H(x)L(y) m(x)+ n(y) which was studied by Lundberg in a series of papers , . However, the solutions therein have indirect forms and we could not find a way to deduce our results from those of and . Another problem, which was formulated by P. K. Sahoo and T. Riedel in their book [32, Section 2.7], remains still open: without any regularity conditions concerning the members of (ϕ, f,Ψ,F), solve the functional equation ϕ x+ y 2 f(x)−f(y) = Ψ x+ y 2 F(x)−F(y) , (x,y ∈R).
One of our main observation (which will turn out in the next section) is that equa-tion (4) has a strong connection to the functional equation ϕ x+ y 2 (f(x)+ f(y)) = F(x)+ F(y), (x,y ∈I), (5) which has been studied and solved by the authors in the paper under minimal regularity assumption in .
2. Connection among the functional equations (4) and (5) We say that a subinterval J ⊆I is proper if it has at least two elements and is different from I . A function g : I →R will be called locally non-constant on I if there 1102 T. KISS AND Z. P ´ ALES is no proper subinterval of I , where g is constant. In other words, g is locally non-constant on I if and only if, for all λ ∈R, the set g−1({λ}) has an empty interior in the domain I.
First we refer to the result about the solutions of equation (5). Observe, that the substitution x = y in the equation (5) immediately yields that F = ϕ · f holds on I.
Therefore, in view of [12, Theorem 11], we have the following statement.
THEOREM 1. Let (ϕ, f,F) : I →R3 such that ϕ and f are continuous on I.
Then the triplet (ϕ, f,F) solves functional equation (5) if and only if either (i) there exists an interval J ⊆I such that f(x) = 0 for all x ∈I \ J, the function ϕ is constant on 1 2(J + I), and F = ϕ · f , or (ii) f is nowhere zero on I and there exist constants a,b,c,d,γ ∈R with ad ̸= bc such that, for all x ∈I , we have F(x) = csin(√−γx)+ d cos(√−γx), f(x) = asin(√−γx)+ bcos(√−γx), ϕ(x) = csin(√−γx)+ d cos(√−γx) asin(√−γx)+ bcos(√−γx), (6) or F(x) = cx+ d, f(x) = ax+ b, ϕ(x) = cx+ d ax+ b, (7) or F(x) = csinh(√γx)+ d cosh(√γx), f(x) = asinh(√γx)+ bcosh(√γx), ϕ(x) = csinh(√γx)+ d cosh(√γx) asinh(√γx)+ bcosh(√γx), (8) whenever γ < 0 or γ = 0 or γ > 0, respectively.
It is worth noticing that the condition (ii) of the above theorem could be formulated in the following equivalent way: (ii)’ f is nowhere zero on I, there exists a constant γ ∈R such that f and F are lin-early independent solutions of the second-order linear homogeneous differential equation Y ′′ = γY , and ϕ = F/ f on I.
In fact, in the paper [12, Theorem 11], instead of the continuity of f , a weaker regularity assumption was made. However, the following consequence of Theorem 1, in which the regularity assumptions for f are completely eliminated, will be sufficient for our purposes.
ON A FUNCTIONAL EQUATION 1103 COROLLARY 1. Let (ϕ, f,F) : I →R3 such that ϕ is continuous and locally non-constant on I. Then the triplet (ϕ, f,F) solves functional equation (5) if and only if F = ϕ · f and either f = 0 and ϕ is arbitrary on I, or the alternative (ii) of Theorem 1 holds. Consequently, f is an infinitely many times differentiable function.
Proof. First we show that the assumptions made on ϕ and (5) imply that f is continuous on I . To do this, let x0 ∈I be arbitrarily fixed. Then there exists r > 0 such that [x0−2r,x0+2r] ⊆I. The function ϕ is non-constant on the interval [x0−r,x0 +r], consequently, there exists u0 ∈[x0 −r,x0 + r] such that ϕ(x0) ̸= ϕ(u0). Let y0 := 2u0 −x0. Then y0 ∈[x0 −2r,x0 + 2r], the function x →ϕ( x+y0 2 )−ϕ(x) is continuous on I, and, by the choice of u0, it is different from zero at the point x := x0. Therefore, there exists δ > 0 such that this function is different from zero also on the entire interval ]x0 −δ,x0 +δ[⊆I. Using this, and the equality F = ϕ · f , equation (5) directly implies that f(x) = f(y0)ϕ(y0)−ϕ( x+y0 2 ) ϕ( x+y0 2 )−ϕ(x) , (x ∈]x0 −δ,x0 + δ[).
Thus f coincides with a continuous function on the neighborhood ]x0 −δ,x0 + δ[ of x0, which means that f has to be continuous at x0. Because x0 was arbitrarily chosen, it follows that f is continuous on I.
Thus, by Theorem 1, we have the alternatives (i) and (ii) for the solutions of (5).
The function f is obviously infinitely many times differentiable, provided that we have the case (ii). On the other hand, having (i) and using that ϕ is locally non-constant, it follows that the subinterval J in alternative (i) must be empty. Hence, in this case, f is identically zero on I, which finishes the proof.
□ In order to describe the connection between functional equations (4) and (5), let us introduce the following notations. For a positive number h and a function f : I →R, define the subinterval interval Ih of I and the function δh f : Ih →R by Ih := (I −h)∩(I + h) and δh f(x) := f(x+ h)−f(x−h), (x ∈Ih).
With these notations, we have the following basic result, which derives a functional equation from (4) for ϕ and δh f , furthermore eliminates the function F .
THEOREM 2. Assume that the triplet of functions (ϕ, f,F) : I →R3 solves func-tional equation (4). Then, for all h > 0, we have ϕ x+ y 2 (δh f(x)+ δh f(y)) = ϕ(x)δh f(x)+ ϕ(y)δh f(y), (x,y ∈Ih), (9) that is, the triplet (ϕ,δh f,ϕ ·δh f) solves functional equation (5) on Ih. Furthermore, if ϕ is continuous and locally non-constant on I, then f is infinitely many times differ-entiable on I and ϕ x+ y 2 (f ′(x)+ f ′(y)) = ϕ(x)f ′(x)+ ϕ(y)f ′(y), (x,y ∈I), (10) that is, the triplet (ϕ, f ′,ϕ · f ′) satisfies functional equation (5) on I.
1104 T. KISS AND Z. P ´ ALES Proof. Assume that equation (4) holds. Let h > 0 and x,y ∈Ih. Then x ± h and y±h belong to I . Substituting the pairs (x−h,x+h), (x+h,y−h), (y−h,y+h) and (y+ h,x−h) into (4), we get the following four equalities: ϕ(x)(f(x−h)−f(x+ h)) = F(x−h)−F(x+ h), ϕ x+ y 2 (f(x+ h)−f(y−h)) = F(x+ h)−F(y−h), ϕ(y)(f(y−h)−f(y+ h)) = F(y−h)−F(y+ h), ϕ x+ y 2 (f(y+ h)−f(x−h)) = F(y+ h)−F(x−h).
Adding up the above equations side by side, the equality in (9) follows immediately.
Assume now that ϕ is continuous and locally non-constant on I. Applying Corol-lary 1 for the function δh f (instead of f ), it follows, for all positive real number h, that the function δh f is infinitely many times differentiable on the subinterval Ih ⊆I. Thus, based on the celebrated result of N. G. de Bruijn [7, 8], f can be written as f0 + g, where f0 : I →R is infinitely many times differentiable and g : R →R is additive.
Without loss of generality, we may assume that g vanishes on the set of the rationals or, equivalently, that f = f0 on the set I ∩Q. This immediately implies that (ϕ, f0,F) solves the equation (4) on the intersection I ∩Q.
Now, indirectly, assume that g is not identically zero. Then there exists x0 ∈I \Q such that g(x0) ̸= 0. Let (xn) ⊆I ∩Q be any sequence such that xn →x0 as n →∞.
Applying the equation (4) for the triplet (ϕ, f0,F), for any member of (xn), and for any point y ∈I ∩Q, we get that F(xn) = ϕ xn + y 2 (f0(xn)−f0(y))+ F(y), (n ∈N).
The continuity of ϕ and f0 provides that the limit of the left hand side exists as n →∞.
Denoting limn→∞F(xn) by λ , using the decomposition of f , and the equality f = f0 on I ∩Q, we get that λ = ϕ x0 + y 2 (f0(x0)−f0(y))+ F(y) = ϕ x0 + y 2 (f(x0)−f(y))+ F(y)−ϕ x0 + y 2 g(x0).
On the very right hand side, we can apply that (ϕ, f,F) solves equation (4) on I, hence λ = F(x0)−ϕ x0 + y 2 g(x0), (y ∈I ∩Q).
By our assumption, g(x0) ̸= 0, thus this equation is equivalent to ϕ x0 + y 2 = F(x0)−λ g(x0) , (y ∈I ∩Q), which means that the function ϕ is constant on the dense subset 1 2(x0 +(I ∩Q)) of the interval 1 2(x0+I). Thus, by its continuity, ϕ must be constant on the subinterval 1 2(x0+ ON A FUNCTIONAL EQUATION 1105 I) ⊆I, which contradicts the fact that it is locally non-constant on I. Consequently, g must vanish on I , and hence also on R. Therefore f = f0 on I, where f0 was an infinitely many times differentiable function.
Finally, we prove that (10) is also valid. Let x,y ∈I be arbitrary. Then, for small positive h, we have that x,y ∈Ih and hence (9) holds for x,y and for small positive h. Dividing equation (9) by 2h and taking the limit h →0+, we get that (10) is satis-fied.
□ In order to be able to apply Theorem 2 to establish the solutions of (4), we have to distinguish two main cases concerning ϕ . First we will deal with the case when ϕ is constant on a proper subinterval of I and then with the case when ϕ is locally non-constant. The second part of the result of Theorem 2 applies in the latter case. The investigation of the first case requires a detailed analysis, thus, firstly, we turn to this part.
3. Preliminary results In order to treat (4), first we investigate a special case, more precisely, we solve the much simple functional equation ϕ x+ y 2 (f(x)−f(y)) = 0, (x,y ∈I), (11) which contains the two unknown functions ϕ : I →R and f : I →R. We note that a functional equation having the form as in (11) can be derived from (4) assuming that F is an affine transformation of the function f , that is, there exists A,B ∈R such that F = Af + B on I.
For the brevity, we will frequently use the notations α := infI and β := supI.
If S ⊆I, then the complementary set I \ S will be simply denoted by Sc. For a given function g : I →R, denote the set g−1({0}) by Zg . For two subsets S,P ⊆R, let us denote the set (2P −S) ∩I by (S|P). That is, (S|P) consists of those elements of I that are reflections of an element of S with respect to some element of P.
PROPOSITION 1. Let P and S be arbitrary subsets of R.
(a) If P 1 ⊆P 2 ⊆R and S1 ⊆S2 ⊆R, then (S|P 1) ⊆(S|P 2) and (S1|P) ⊆(S2|P).
(b) If at least one of the sets P or S is open, then (S|P) is also open.
(c) If P,S ⊆R are intervals, then (S|P) is also an interval, furthermore we have inf(S|P) = max(α,2infP−supS) and sup(S|P) = min(2supP−infS,β). (12) (d) Consequently, for a given point p ∈I, the set (I| p) is the maximal open subinterval contained in I, which is symmetric with respect to p. Furthermore, by the equality (12), it follows that (I| p) is bounded unless I = R.
Proof. The statements (a), (b), (c), and (d) are direct consequences of the defini-tion.
□ 1106 T. KISS AND Z. P ´ ALES LEMMA 1. Let (ϕ, f) be a solution of equation (11) and p ∈I . Then f(x) = f(p) holds for all x ∈(p|Zϕ c).
Proof. If (p|Zϕc) is empty, then the statement is obvious. Therefore we may assume that (p|Zϕc) is nonempty and let x ∈(p|Zϕ c) be arbitrary. Then there exists u ∈Zϕ c such that x+p 2 = u holds. Applying equation (11) for x and p and using that ϕ(u) ̸= 0, we get f(x) = f(p).
□ PROPOSITION 2. Let (ϕ, f) be a solution of (11) such that Zϕ is closed in I and has an empty interior. Then f is constant on I.
Proof. The assumptions concerning Zϕ provide that Zϕc is a nonempty open subset of I. Therefore it can be written as a union of its components. Let K ⊆Zϕ c be any component and x,y ∈K be arbitrary. Then x+y 2 ∈K , that is ϕ( x+y 2 ) ̸= 0, hence, based on equation (11), we get that f(x) = f(y).
In the next step we show that f takes the same value on any two components of Zϕc . There is nothing to prove if Zϕ c has only one component. Thus we may assume that this is not the case and let K and L be different components of Zϕ c . By our assumption, the interior of Zϕ is empty, hence the intersection 1 2(K + L)∩(Zϕ c) cannot be empty. Let u be any point of the intersection. Then ϕ(u) ̸= 0, furthermore, there exist x ∈K and y ∈L such that x+y 2 = u. Using equation (11) for the points x and y, we obtain that f(x) = f(y). Consequently, there exists λ ∈R such that f(x) = λ whenever x ∈Zϕc .
Finally we show that the restriction f|Zϕ is also identically λ . If Zϕ is empty, then we are done. If not then let p ∈Zϕ be arbitrarily fixed. The assumptions concern-ing Zϕ yield that there is an open interval U ⊆Zϕ c such that J := (p|U) ⊆(p|Zϕc) is contained in I. Then, by Lemma 1 we have f|J ≡f(p). On the other hand, J is a nonempty interval in I , which implies that the intersection J ∩Zϕ c cannot be empty.
Consequently, f(p) = λ must hold.
□ In the following proposition, we establish the solutions of equation (11) supposing that the interior of Zϕ is nonempty. In this case, the interior of Zϕ is the union if its components which are maximal open subintervals of the interior of Zϕ . Provided that Zϕ is closed, the closure of such intervals are maximal closed subintervals of Zϕ .
PROPOSITION 3. Let (ϕ, f) be a solution of (11) such that Zϕ is a closed subset of I with a nonempty interior. Then, for any maximal closed subinterval J ⊆Zϕ , there exist constants λ,μ ∈R such that (I| infJ) ⊆Zf−λ and (I| supJ) ⊆Zf−μ.
Proof. Let J be any nonempty closed subinterval of Zϕ . If J = I, then ϕ is identically zero on I and, by (I| infJ) = (I| supJ) = / 0, the inclusions in the statement above hold for any constants λ and μ . Thus, we may assume that at least one of the endpoints of J belongs to I, say p := supJ ∈I. Then (I| p) is nonempty. Let a := inf(I| p) and b := sup(I| p).
ON A FUNCTIONAL EQUATION 1107 In the first step we are going to show that f is constant on ]a, p]. Let x < y in ]a, p] be arbitrary. Then p < x+β 2 and, due to the maximality of J and the openness of Zϕ c, one can find a nonempty open and bounded subinterval U contained in Zϕ c∩ p, x+β 2 .
Let r := 2|U| and consider first the case y −x < r. Then there is u < v in U such that v−u = 1 2(y−x), that is 2v−y = 2u−x =: w. A short calculation yields that w ∈I, which means that the intervals (y|U) and (x|U) have a common point in I. By In view of Lemma 1, f(x) = f(y) follows.
If r ⩽y−x, then there exist n ∈N and t0 < ··· < tn in [x,y] such that t0 = x, tn = y and ti −ti−1 < r for all i ∈{1,...,n}. Based on the previous argument, f(ti−1) = f(ti) follows for all i ∈{1,...,n}, particularly, we have again f(x) = f(y). Thus, there exists μ ∈R such that f(x) = μ if x ∈]a, p].
Now we show that f takes μ also on ]p,b[. Let x ∈]p,b[ be any point. Then, by the maximality of J, there exists u ∈Zϕ c with p < u < x. Then 2u −x belongs to ]a, p]. Applying equation (11) and using ϕ(u) ̸= 0, we immediately get that f(x) = f(2u −x) = μ .
□ To formulate our next theorem, for a given set S ⊆I, we introduce the notations S∗:= {x ∈I | x < infS} and S∗:= {x ∈I | supS < x}.
We note that, regardless of the topological properties of S, the sets S∗and S∗are always open in I and that S∗= S∗= I if and only if S is empty.
LEMMA 2. Let S ⊆I be nonempty, s∗:= inf 1 2(S + I) and s∗:= sup 1 2(S + I).
Then we have S∗= (I|s∗) or S∗= (I|s∗), (13) provided that −∞< s∗or s∗< +∞, respectively.
Proof. We prove only the first identity in (13), the second one can be shown simi-larly. Assume that −∞< s∗holds. Then, s∗is finite and by its definition, s∗= infS + α 2 < β + α 2 .
(14) Then α is also finite and S∗=]α,infS[=]α,2s∗−α[.
On the other hand, by formula (12) and the inequality in (14), (I|s∗) =]max(α,2s∗−β),min(2s∗−α,β)[=]α,2s∗−α[.
which finishes the proof.
□ THEOREM 3. Let ϕ, f : I →R such that Zϕ is closed. Then the pair (ϕ, f) is a solution of equation (11) if and only if either f is constant on I and ϕ : I →R is 1108 T. KISS AND Z. P ´ ALES any function or there exist a nonempty closed interval K ⊆I and constants λ∗,λ ∗∈R such that the inclusions K∗⊆Zf−λ∗, K∗⊆Zf−λ ∗ and 1 2(I + K) ⊆Zϕ, (15) are satisfied.
Proof. If f is constant on I, then, obviously, for any function ϕ : I →R, the pair (ϕ, f) is a solution of (11). Let K ⊆I be a nonempty closed interval, λ∗,λ ∗∈R such that (15) holds, finally let x,y ∈I be arbitrarily fixed. If either x,y ∈K∗or x,y ∈ K∗, then either f(x) = f(y) = λ∗or f(x) = f(y) = λ ∗, respectively, which imply the equality in (11). In the remaining case, we have that infK ⩽max(x,y) and min(x,y) ⩽ supK , therefore inf(K + I) = infK + α < max(x,y)+ min(x,y) < β + supK = sup(K + I), where, obviously, max(x,y) + min(x,y) = x + y. These mean that x+y 2 ∈1 2(K + I) ⊆ Zϕ , that is, ϕ x+y 2 = 0, whence the equality in (11) follows again.
Conversely, assume that (ϕ, f) is a solution of equation (11) such that f is non-constant on I. If Zϕ = I then the statement of the theorem holds with K = I. (Then K∗and K∗are empty and λ∗, λ ∗can be arbitrary.) Thus, in the rest of the proof, we may assume that Zϕ is a proper subset of I .
Let J :=]α,a[ and L :=]b,β[ , where a := sup{t ⩾α | f is constant on ]α,t[} and b := inf{t ⩽β | f is constant on ]t,β[}.
Then J and L are maximal intervals with respect to the property that f is constant on them. Let K := (J ∪L)c . By Proposition 2, the interior of Zϕ cannot be empty. Thus, Zϕ contains a maximal closed subinterval J0 which cannot be equal to I because Zϕ is a proper subset of I. Thus, one of the endpoints of J0, say p is in I. In view of Proposition 3, it follows that f is constant on (I|p), which is a nonempty interval contained either in J or in L. Therefore at least one of the intervals J or L is nonempty, consequently, K is different from I . By our assumption concerning f , the intervals J and L are disjoint and different from I, which implies that K is nonempty. We have obtained that K is a nonempty closed proper subinterval of I. Furthermore, we also have K∗= J and K∗= L, thus there exist λ∗,λ ∗∈R such that K∗⊆Zf−λ∗and K∗⊆Zf−λ ∗. Thus, the first two inclusions in (15) hold and we only need to prove that ϕ is identically zero on G := 1 2(K + I).
By K ̸= I, the interval G strictly contains K , thus G and J ∪L have common points. This yields that f cannot be constant on G. Restricting the equation (11) to G, by Proposition 2, we get that the interior of G ∩Zϕ is nonempty. Thus there exists a maximal nonempty open subinterval U0 ⊆G such that ϕ is identically zero on U0.
To complete the proof, it suffices to show that U0 = G. Assume, to the contrary, that this is not the case. Then we may also assume that s∗:= infG < p := infU0 holds.
ON A FUNCTIONAL EQUATION 1109 Let U ⊆I be a maximal closed subinterval such that U0 ⊆U and ϕ is identically zero on U . Then infU must be also equal to p. Applying Proposition 3 for the maximal closed subinterval U ⊆Zϕ , the function f is constant on (I| p). Then exactly one of the identities inf(I| p) = α or sup(I| p) = β can hold. By the maximality property of J and L, we have either (I| p) ⊆J or (I| p) ⊆L, respectively. Because of the symmetry, we may assume that (I| p) ⊆J. Then J is nonempty, which implies that α < a must hold. To finish the proof, we distinguish two cases.
Case 1. If α = s∗then, having the relation α < a, it follows that I is not bounded from below, that is, α = s∗= −∞= inf(I| p). Then there exists x ∈(I| p) such that x < 2p −a, which yields that a < 2p −x ∈(I| p) ⊆J. This contradicts that supJ = a.
Case 2. Assume now that α < s∗. Then, using Lemma 2 and that inf(I|s∗) = inf(I| p) = α , s∗< p, we obtain that J = K∗= (I|s∗) ⊊(I| p) ⊆K∗= J, which is a contradiction again.
Consequently, 1 2(K + I) = G = U0 ⊆Zϕ .
□ REMARK 1. One can easily see that the closedness of the set Zϕ was only used to establish the necessity of the statement of Theorem 3. The sufficiency holds without any assumption on ϕ .
4. Solutions of (4) assuming that ϕ is constant on a subinterval of I In this section we are going to describe the solutions of the functional equation (4) assuming that the function ϕ is constant on a proper subinterval of I. The next proposition clarifies the main connection between the equations (4) and (11).
PROPOSITION 4. Let (ϕ, f,F) be a solution of equation (4) on the interval I, A ∈R, and J ⊆I be a subinterval. Then the pair (ϕA, f) solves the equation (11) on J if and only if there exists B ∈R such that F = Af + B on I .
Proof. Assume that (ϕA, f) solves (11) on the interval J, that is, we have ϕA x+ y 2 (f(x)−f(y)) = ϕ x+ y 2 −A (f(x)−f(y)) = 0, (x,y ∈J).
On the other hand, particularly, equation (4) holds on J. Subtracting the above equation from (4) side by side, we get that F(x)−Af(x) = F(y)−Af(y), (x,y ∈J).
Consequently, there exists B ∈R such that F(x)−Af(x) = B for all x ∈J, thus F = Af + B holds on J.
1110 T. KISS AND Z. P ´ ALES Conversely, assume that there are constants A,B ∈R such that F = Af +B on the interval J. Using this and that (ϕ, f,F) is a solution of equation (4) on J, we obtain that ϕ x+ y 2 (f(x)−f(y)) = F(x)−F(y) = Af(x)+ B−Af(y)−B = A(f(x)−f(y)) for all x,y ∈J. This means that (ϕA, f) solves (11) on J, which finishes the proof.
□ The following consequence of Proposition 4 will play a key role in the proof of Theorem 4 below.
COROLLARY 2. Let (ϕ, f,F) be a solution of equation (4) on the interval I , A ∈ R, and J,K be subintervals of I such that f(J) = f(K). If (ϕA, f) solves equation (11) on J, then it solves equation (11) also on K .
Proof. Assume that (ϕA, f) solves equation (11) on J. Then, by Proposition 4, there exists B ∈R, such that F = Af +B on J. By f(J) = f(K), we get that the same formula holds on K , which, in view of Proposition 4 again, yields that (ϕA, f) solves equation (11) on K .
□ For a given subinterval J ⊆I, define the sequence of intervals (Jn) by the recur-sion J0 := J and Jn := (Jn−1|J), (n ∈N).
(16) By the definition (16), it is easy to see that Jn = J for all n ∈N provided that J is either empty or is a singleton or equals I. If J is nonempty then, for the brevity, denote infJn and supJn by an and bn whenever n ∈N∪{0}, respectively.
LEMMA 3. Let J be a subinterval of I. Then (Jn) is a nondecreasing sequence of intervals contained in I and, provided that J is nonempty, for all n ∈N, we have an = max(α,2a0 −bn−1) and bn = min(2b0 −an−1,β).
(17) Consequently, if α < an and bn < β for some n ∈N∪{0} then bk −ak = (2k + 1)(b0 −a0), (k ∈{0,1,...,n}).
(18) Finally, if J ̸= I is a proper subinterval, then J = J0 ̸= J1, furthermore, if α = an or bn = β for some n ∈N∪{0}, then Jn+k = Jn+k+1 holds for all k ∈N.
Proof. The monotonicity of the sequence (Jn) can be easily shown by induction with respect to n. Obviously, J0 = J ⊆(J|J) = J1 holds. Assume that Jn−1 ⊆Jn for some n ∈N. Then, by the property (a) of Proposition 1, we have Jn = (Jn−1|J) ⊂(Jn|J) = Jn+1.
These imply that Jn−1 ⊆Jn for all n ∈N.
Assuming that J is nonempty, the identities in (17) are easy consequences of (12) in Proposition 1. Now suppose that {an,bn} ⊆I for some n ∈N ∪{0}. Then, by the ON A FUNCTIONAL EQUATION 1111 strict monotonicity of (Jn), we have {ak,bk} ⊆I for all k ∈{0,1,...,n}. Using this, we prove (18) by induction with respect to k. If n = 0 then k = 0 and the statement is trivial. Assume that n ∈N. If k = 0 then the equality in (18) is trivial. Assume that (18) holds for some non-negative integer k ⩽n −1, that is, we have bk −ak = (2k + 1)(b0 −a0). Then, by (17), we obtain that bk+1 −ak+1 = 2b0 −ak −(2a0 −bk) = (bk −ak)+ 2(b0 −a0) = (2k + 1)(b0 −a0)+ 2(b0 −a0) = 2(k + 1)+ 1 (b0 −a0).
If J ̸= I is a proper subinterval, then a0 < b0 and one of the strict inequalities α < a0 or b0 < β holds. In the first case, a1 < a0. Indeed, if this were not true, then by (17), a1 = max(α,2a0 −b0) = a0. This implies that 2a0 −b0 = a0, hence a0 = b0, a contradiction. In the case b0 < β , the inequality b0 < b1 follows similarly.
Assume, finally, that there exists n ∈N such that an = α . Then, by the increas-ingness on (Jn), for all k ∈N, we have α ⩽an+k ⩽an = α , that is an+k = α . Further-more, by this and by the second identity in (17), it follows that bn+k = min(2b0 −α,β) whenever k ∈N. Analogously, if bn = β for some n ∈N, then bn+k = β and an+k = max(α,2a0 −β) whenever k ∈N.
□ COROLLARY 3. Using the notations of Lemma 3, we have J∞:= n∈N Jn = J if J is a singleton, (I|J) otherwise.
Proof. If J is either empty or a singleton or equals I, then the statement is trivial.
Let assume that J is a proper nonempty subinterval of I with a0 < b0. First we show that J∞symmetric with respect to J, that is, (J∞|J) = J∞. Indeed, (J∞|J) = n∈N Jn J = n∈N (Jn|J) = n∈N Jn+1 = J∞.
This immediatley implies that J∞⊆(I|J). To prove the reversed inclusion, let x ∈ (I|J) \ J∞be any point. Then there exist u ∈J and y ∈I such that x = 2u −y. We may assume that x < y. Obviously, in this case we must have y / ∈J∞. Indeed, if y were an element of J∞, then there would exist n ∈N such that y ∈Jn. Therefore x ∈(Jn|J) = Jn+1, which contradicts that x / ∈J∞. On the other hand, u ∈J∩]x,y[⊆ J∞∩]x,y[. Consequently, we must have J∞⊆]x,y[ . Using Lemma 3, we obtain that y−x ⩾diam(Jn) = bn −an = (2n + 1)(b0 −a0) > 0, (n ∈N), which is impossible. Therefore J∞= (I|J).
□ COROLLARY 4. Let (ϕ, f,F) be a solution of equation (4) on the interval I , A ∈ ϕ(I) and J ⊆ZϕA be a subinterval. Then there exists B ∈R such that F = Af + B holds on J∞.
1112 T. KISS AND Z. P ´ ALES Proof. If ϕ is identically A on J, then (ϕA, f) trivially solves equation (11) on J, thus, in view of Proposition 4, F is of the form Af + B on J = J0 for some B ∈R.
Assume now that this is valid on Jn for some n ∈N ∪{0} and let x ∈Jn+1 be any point. Then there exist u ∈J and y ∈Jn such that u = x+y 2 . Then ϕ(u) = A and, by the inductive hypothesis, F(y) = Af(y)+ B. Thus, applying equation (4) for x and y, we get that A(f(x)−f(y)) = F(x)− Af(y)+ B , which reduces to F(x) = Af(x)+B. This means that we have F = Af +B on Jn for ar-bitrary n ∈N∪{0}. Now our statement is an obvious consequence of the monotonicity of the sequence (Jn) and of the definition of the interval J∞.
□ Having these results, we introduce some special subintervals of I, which will be useful in the proof of our main theorem.
Suppose that (ϕ, f,F) is a solution of (4) on I. For a given A ∈ϕ(I), let I A := ]α,βA[ (where βA ∈[α,β]) be the maximal open subinterval I A ⊆I such that (ϕA, f) solves (11) on I A . Similarly, let IA :=]αA,β[ (where αA ∈[α,β]) denote the maximal open subinterval IA ⊆I such that (ϕA, f) solves (11) on IA . Finally, define I and I by I := A∈ϕ(I) I A and I := A∈ϕ(I) IA.
Obviously, the families {I A | A ∈ϕ(I)} and {IA | A ∈ϕ(I)} are chains with respect to the inclusion, therefore, I and I are (not necessarily nonempty) open subintervals contained in I. Then, for some uniquely determined elements α,β ∈[α,β], we have that I :=]α,β[ and I :=]α,β[.
COROLLARY 5. Let (ϕ, f,F) be a solution of (4). Then, for all A ∈ϕ(I) and for all subinterval J ⊆ZϕA , we have J∞⊆I or J∞⊆I.
Proof. Let A ∈ϕ(I) and J ⊆ZϕA be any subinterval. Then, by Corollary 4, there exists B ∈R, such that F = Af +B on J∞. This, in view of Proposition 4, is equivalent to that the pair (ϕA, f) solves the equation (11) on the interval J∞. Hence at least one of the inclusions J∞⊆I or J∞⊆I must hold.
□ The following lemmas are about some useful connections between the structure of the sets I and I, and the behavior of the members of the solutions triplets of equation (4).
LEMMA 4. Let (ϕ, f,F) be a solution of (4), A and A′ be distinct points in ϕ(I), furthermore J and K be intervals such that (ϕA, f) and (ϕA′, f) solves (11) on J and K , respectively. Then f is constant on J ∩K .
Proof. If J ∩K is empty, then the statement is trivial. If not, then we have ϕ x+ y 2 −A (f(x)−f(y)) = 0 and ϕ x+ y 2 −A′ (f(x)−f(y)) = 0 for all x,y ∈J ∩K . Subtracting these equations side by side, we get (A −A′)(f(x) − f(y)) = 0, which, by our assumption A ̸= A′, implies that f(x) = f(y).
□ ON A FUNCTIONAL EQUATION 1113 LEMMA 5. Let (ϕ, f,F) be a solution of (4). Then f is constant on I (resp. on I) or there uniquely exists A ∈ϕ(I) such that I = IA (resp. I = IA ).
Proof. We prove the statement only for I. To avoid the trivial case, we may as-sume that I is nonempty. Assume that there is no A ∈ϕ(I) such that I = IA . Then it is enough to show that f is constant on any member of the chain-union in the def-inition of I. To do this, let A ∈ϕ(I) be arbitrary and x ∈I \ IA . Then, there exists A′ ∈ϕ(I) different from A such that x ∈IA′ . Hence IA ⊆IA′ and, in view of Lemma 4, the function f is constant on IA ∩IA′ = IA .
□ LEMMA 6. Let (ϕ, f,F) be a solution of (4). Then, for all constants λ,μ ∈R with λ ̸= μ , the function ϕ is constant on the set 1 2(Zf−λ + Zf−μ).
Proof. If at least one of the sets Zf−λ and Zf−μ is empty, then 1 2(Zf−λ +Zf−μ) is also empty, thus, in this case, the statement is trivially true. Assume therefore that Zf−λ and Zf−μ are nonempty. Equation (4) implies that F also must be constant both on the sets Zf−λ and Zf−μ . Thus there exists not necessarily different numbers Cλ,Cμ ∈R such that F(u) = Cλ and F(v) = Cμ whenever u ∈Zf−λ and v ∈Zf−μ .
Now, applying equation (4) for any points x ∈Zf−λ and y ∈Zf−μ , we obtain that ϕ x+ y 2 = F(x)−F(y) f(x)−f(y) = Cλ −Cμ λ −μ .
Thus ϕ is indeed constant on 1 2(Zf−λ + Zf−μ).
□ LEMMA 7. Let (ϕ, f,F) be a solution of (4), furthermore assume that the closed subinterval S := I (I ∪I) has a nonempty interior and that it is different from I . Then the following statements hold.
(i) The function ϕ is locally non-constant on S.
(ii) If there exists γ ∈[α,β[ such that f is constant on the interval ]γ,β[ , then f must be constant on ]γ,α[. Similarly, if there is γ ∈]α,β] such that f is constant on ]α,γ[ , then f must be constant on ]β,γ[.
Proof. To prove (i), indirectly, assume that there exists A ∈ϕ(I) such that the interior of the intersection S ∩ZϕA is nonempty. If J ⊆S ∩ZϕA is any nonempty open subinterval, then, based on Corollary 3, J ⊆J∞holds. Furthermore, by Corollary 5, we obtain that J ⊆S ∩J∞⊆S ∩(I ∪I), where the last intersection is empty, contradicting that J was a proper open subinterval.
Now we prove the first assertion of statement (ii). Assume that there exists α ⩽ γ < β and λ ∈R such that f is identically λ on the interval ]γ,β[ . Let y be any point of the interior of S. Then there exists x ∈]γ,β[ such that x+y 2 belongs also to the interior of S. If f(y) ̸= λ , then, based on Lemma 6, ϕ must be constant on 1 2({y}+[x,β[) ⊆S, which, in view of the statement (i), is impossible. This shows that f(y) must equal λ .
1114 T. KISS AND Z. P ´ ALES A similar argument yields that f(β) = λ also holds. Otherwise, again by Lemma 6, ϕ would be constant on 1 2(β+]β,α[) ⊆S, which contradicts (i) again. Thus, f is indeed constant on ]γ,α[ .
The proof of the second assertion of (ii) can be treated similarly.
□ THEOREM 4. Let ϕ : I →R be a function with closed level sets in I and assume that there exists a level set of ϕ whose interior is nonempty. Then the triplet of functions (ϕ, f,F) solves functional equation (4) if and only if there exist constants A ∈ϕ(I) and B ∈R such that (ϕA, f) solves (11) on I and F = Af + B on I.
Proof. The sufficiency is trivial. Indeed, assuming there exist A ∈ϕ(I) and B ∈R such that (ϕA, f) solves (11) on I and F = Af + B on I and calculating both sides of the equation (4), one can conclude that (ϕ, f,F) solves functional equation (4). We note that the argument in this direction does not need any condition concerning the level sets of ϕ .
Now we turn to the necessity, that is, assume that (ϕ, f,F) solves functional equa-tion (4). In view of Proposition 4, it is enough to show that there exists A ∈ϕ(I), such that (ϕA, f) solves equation (11) on the interval I. This is trivially fulfilled provided that at least one of the functions f or ϕ is constant on I. Therefore, in the sequel, we suppose that f and ϕ are non-constant functions.
By our assumption, ϕ has at least one level set with a nonempty interior, which, based on Corollary 5, means that at least one of the sets I and I is nonempty. Without loss of the generality, we may assume that I ̸= / 0. If I = I, then we are done. Hence, in the rest of the proof we may also assume that I is a proper subinterval of I.
For the brevity, denote the interval I \ (I ∪I) by S. We are going to prove that S must be empty. More precisely, first we show that its interior is empty, then, secondly, that it cannot be a singleton.
Part 1. Indirectly assume that the interior of S is nonempty. Observe, that, by the statement (ii) of Lemma 7, the function f cannot be constant on I. Indeed, if f were constant on I =]α,β[ , then it would also be constant on ]α,α[= I ∪(S \ {α}). Then, for any A, the pair (ϕA, f) satisfies (11) on ]α,α[, which means that I would be strictly expandable, contradicting its definition. Therefore, based on Lemma 5, there uniquely exists A ∈ϕ(I) such that I = IA , that is, the pair (ϕA, f) solves equation (11) on the subinterval I. Applying Theorem 3 for the interval I , there exists a closed subinterval K ⊆I such that J := 1 2(I + K) ⊆ZϕA holds.
Using the notations of Theorem 3, we show that K∗∩I must be empty. If not, then, in view of the statement (ii) of Lemma 7, we have that f is constant on K∗∩]α,α[ .
This implies that f(I) = f(]α,α[). Using that the pair (ϕA, f) solves equation (11) on I, by Corollary 2, it follows that (ϕA, f) also solves equation (11) on ]α,α[. This contradicts the maximality property of I again.
Therefore, K∗∩I is empty, or equivalently, we have that supJ := β . By Lemma 3, the interval J1 and hence J∞contains the point β in its interior. On the other hand, Corollary 5 implies that J∞⊆I or J∞⊆I. Both inclusions are obviously impossible.
This final contradiction means that the interior of S must be indeed empty or, equiva-lently, the closed interval S is at most a singleton.
ON A FUNCTIONAL EQUATION 1115 Part 2. In this part we are going to show that S is neither a singleton. The proof of this part also goes indirectly, that is, let us assume that S = {p} ⊆I. Then both of the open intervals I and I are nonempty. Furthermore, there is no one-sided open neighborhood of p contained in I, where ϕ were constant. Otherwise, if there exists r > 0 such that ϕ is constant on the left neighborhood J :=]p −r, p] ⊆I of p. Then the interval J∞contains p in its interior, which, because of Corollary 5, is impossible.
A similar argument shows that ϕ cannot be constant on any right neighborhood of p.
Therefore, in view of Lemma 5 and Theorem 3, there exist nonempty open subin-tervals L,R ⊆I with supL = infR = p and real constants λL,λR ∈R such that f|L ≡λL and f|R ≡λR. We claim that λL and λR are necessarily equal to f(p). If not, then, by Lemma 6, we get that ϕ is constant on the interval 1 2(L+ p) and 1 2(p+R), respec-tively. These sets are left and right neighborhoods of p, which, in view of the previous argument, yields a contradiction.
Thus the restriction of f to the interval L ∪{p} ∪R is a constant function. Then we have f(I) = f(I ∪R). Since (ϕA, f) solves equation (11) on I, by Corollary 2, it follows that (ϕA, f) also solves equation (11) on I ∪R. This again contradicts the maximality property of I.
Consequently, S is empty or, equivalently, the open interval I ∩I is nonempty.
Particularly, we get that I is also nonempty. To avoid the trivial case, it can be supposed that I is different from I. The function f is non-constant on I, therefore we may assume that f is non-constant on I . Then, by Lemma 5, there uniquely exists A ∈ϕ(I) such that I = IA . If f is constant on I, then f(I) = f(I ∪I) = f(I). Thus, in view of Corollary 2, the pair (ϕA, f) solves equation (11) on the entire domain I, hence in this case we are done. Therefore in the sequel we may assume also that f is non-constant on I. By Lemma 5, there uniquely exists A′ ∈ϕ(I) such that I = IA′ . Now, obviously, we have two possibilities, namely either A ̸= A′ or A = A′.
Assume that A ̸= A′ holds. Then, by Lemma 4, there exists a real constant λ such that f is identically λ on the intersection M := I ∩I. Then, because of the maximality property of I and I, there is no r > 0 such that f takes λ also on the open subinterval (M +] −r,r[) ∩I. Consequently, for all n ∈N, there exists yn ∈]β,β + 1 n[∩I such that f(yn) ̸= λ . Then, by Lemma 6, the function ϕ must be constant on the interval 1 2 (I ∩I)+ yn for all n ∈N. The intersection of the family 1 2 (I ∩I)+ yn | n ∈N is nonempty, which implies that ϕ is constant on the union n∈N 1 2 (I ∩I) + yn ⊇ α+β 2 ,β . A similar argument shows that ϕ must be also constant on the interval α, α+β 2 . The level sets of ϕ are closed, therefore ϕ takes the same value on J := ]α,β[ . Then J∞contains the points α and β in its interior, which contradicts the statement of Corollary 5.
Finally, we obtained that A = A′ must hold. Then the pair (ϕA, f) trivially solves (11) on the sets I and I. By Proposition 4, this means that there exist B,B′ ∈R such that F = Af + B on I and F = Af + B′ on I . The nonemptyness of the intersection IA ∩IA yields that B = B′. Consequently, F = Af + B on I . This, based again on Proposition 4, equivalent to that (ϕA, f) solves (11) on the entire interval I, which finishes the proof.
□ 1116 T. KISS AND Z. P ´ ALES COROLLARY 6. Let ϕ : I →R be a function with closed level sets in I and assume that there exists a level set of ϕ whose interior is nonempty. Then the triplet of functions (ϕ, f,F) solves functional equation (4) if and only if either f is constant on I and ϕ,F : I →R are arbitrary functions or there exist A ∈ϕ(I), B ∈R, a nonempty closed interval K ⊆I , and constants λ∗,λ ∗∈R such that the inclusions K∗⊆Zf−λ∗, K∗⊆Zf−λ ∗ and 1 2(I + K) ⊆Zϕ−A (19) are satisfied, furthermore we have F = Af + B on I .
5. Solutions of (4) assuming that ϕ is continuous The following statement summarizes what we have established in the previous sections and establishes the main result of our paper.
THEOREM 5. Let ϕ : I →R be a continuous function. Then the triplet of functions (ϕ, f,F) : I →R3 solves equation (4) on I if and only if either (i) f and F are constant functions and ϕ is an arbitrary function on I, or (ii) there exist A ∈ϕ(I), a nonempty closed interval K ⊆I and constants λ∗,λ ∗,μ ∈ R such that the inclusions of (19) hold and F = Af + μ on I , or (iii) there exist constants A,B,C,D ∈R with AD ̸= BC and γ,μ,λ ∈R such that, for all x ∈I, one of the possibilities F(x) = −Ccos(√−γx)+ Dsin(√−γx)+ μ, f(x) = −Acos(√−γx)+ Bsin(√−γx)+ λ, (20) ϕ(x) = Csin(√−γx)+ Dcos(√−γx) Asin(√−γx)+ Bcos(√−γx) , or F(x) = 1 2Cx2 + Dx+ μ, f(x) = 1 2Ax2 + Bx+ λ, (21) ϕ(x) = Cx+ D Ax+ B , or F(x) = Ccosh(√γx)+ Dsinh(√γx)+ μ, f(x) = Acosh(√γx)+ Bsinh(√γx)+ λ, (22) ϕ(x) = Csinh(√γx)+ Dcosh(√γx) Asinh(√γx)+ Bcosh(√γx) ON A FUNCTIONAL EQUATION 1117 hold whenever γ < 0 or γ = 0 or γ > 0, respectively.
It is worth noticing that the condition (iii) of the theorem could be formulated in the following equivalent way: (iii)’ f and F are differentiable functions, f ′ is nowhere zero on I and there exists a constant γ ∈R such that f ′ and F′ are linearly independent solutions of the second-order linear homogeneous differential equation Y ′′ = γY and ϕ = F′/ f ′ on I .
Proof. The continuity of ϕ implies that, for all E ∈R, the level set Zϕ−E is closed in I.
It is obvious that if f and F are constants and ϕ is an arbitrary function on I, then (ϕ, f,F) : I →R3 solves equation (4) on I. If F is constant on I, then (ϕ, f) solves (11). Therefore, by Theorem 3, either f is constant on I or there exist a nonempty closed interval K ⊆I and constants λ∗,λ ∗∈R such that the inclusions in (15) hold.
Thus the alternative (ii) of our theorem is valid with A = 0. Therefore, in the sequel, we may assume that F is non-constant on I . Then f is also non-constant.
If there exists a level set of ϕ whose interior is nonempty, then, in view of Corol-lary 6, there exist A ∈ϕ(I), a nonempty closed interval K ⊆I and constants λ∗,λ ∗,μ ∈ R such that the inclusions (19) hold, furthermore F = Af + μ on I. This shows, that the alternative (ii) is valid in this case.
Finally, we consider the case when ϕ is locally non-constant and the functions f and F are non-constant on I. We are going to show that in this setting the alternative (iii) is valid. First we deal with the necessity of (iii). Assume that the triplet of functions (ϕ, f,F) : I →R3 solves equation (4) on I. Then, in view of Theorem 2, the function f is infinitely many times differentiable and (ϕ, f ′) satisfies the functional equation (10) on I. Consequently, one of the possibilities (i) or (ii) of Theorem 1 holds concerning the pair (ϕ, f ′).
If the alternative (i) of Theorem 1 were valid, then there would exist a subinterval J ⊆I such that f ′ is zero on I\J and ϕ is constant on 1 2(J+I). Using that ϕ is locally non-constant on I, we could conclude that J must be empty, which would yield that f , and hence F , is constant on I contradicting our assumptions. Therefore, we must have the alternative (ii) of Theorem 1. Then there exist real constants a,b,c,d,γ ∈R with the property ad ̸= bc such that, depending on the sign of γ , the functions ϕ and f ′ are of the forms (6), (7) or (8) listed in Theorem 1. To compute f and F , we distinguish three main cases.
Case γ < 0. Then, in view of (3) of Theorem 1, we have ϕ(x) = csin(√−γx)+ d cos(√−γx) asin(√−γx)+ bcos(√−γx) and f ′(x) = asin(√−γx)+bcos(√−γx) for all x ∈I. Therefore, there exists λ ∈R such that f(x) = 1 √−γ (−acos(√−γx)+ bsin(√−γx))+ λ, (x ∈I).
1118 T. KISS AND Z. P ´ ALES Replacing a,b,c, and d by A√−γ , B√−γ , C√−γ , and D√−γ , respectively, we can see that f and ϕ are of the form stated in (20). To conclude the proof, we compute ϕ x+y 2 (f(x)−f(y)) for x,y ∈I . To simplify the calculation, denote the points √−γx and √−γy by u and v, respectively. Then f(x)−f(y) = (−Acos(u)+ Bsin(u))−(−Acos(v)+ Bsin(v)) = 2sin u−v 2 Asin u+v 2 + Bcos u+v 2 .
Therefore, ϕ x+ y 2 (f(x)−f(y)) =Csin u+v 2 + Dcos u+v 2 Asin u+v 2 + Bcos u+v 2 ·2sin u−v 2 Asin u+v 2 + Bcos u+v 2 =2sin u−v 2 Csin u+v 2 + Dcos u+v 2 (23) =(−Ccos(u)+ Dsin(u))−(−Ccos(v)+ Dsin(v)) =(−Ccos(√−γx)+ Dsin(√−γx))−(−Ccos(√−γy)+ Dsin(√−γy)).
Using that (ϕ, f,F) solves (4), the above identity yields, for all x,y ∈I , that F(x)−F(y) = (−Ccos(√−γx)+ Dsin(√−γx))−(−Ccos(√−γy)+ Dsin(√−γy)), hence the mapping x →F(x) +Ccos(√−γx) −Dsin(√−γx) is constant on I. This shows that F is of the form (20) for some real number μ .
Case γ = 0. Then, by (2) of Theorem 1, for all x ∈I , we have ϕ(x) = cx+ d ax+ b and f ′(x) = ax+ b.
As before, we immediately get that there exists λ ∈R such that f(x) = a 2x2 + bx+ λ, (x ∈I).
Now, replacing a,b,c, and d by A,B,C, and D, respectively, we get, for all x,y ∈I, that ϕ x+ y 2 (f(x)−f(y)) =C x+y 2 + D A x+y 2 + B 1 2Ax2 + Bx−1 2Ay2 −By =C x+y 2 + D A x+y 2 + B Ax+ y 2 + B (x−y) = Cx+ y 2 + D (x−y) (24) =1 2Cx2 + Dx− 1 2Cy2 + Dy .
ON A FUNCTIONAL EQUATION 1119 Using that (ϕ, f,F) solves (4), the above identity yields, for all x,y ∈I, that we must have F(x)−F(y) = 1 2Cx2 + Dx− 1 2Cy2 + Dy , hence the mapping x →F(x) −1 2Cx2 −Dx is constant on I . This shows that F is of the form (21) for some real number μ .
Case γ > 0. Based on (3) of Theorem 1, in this case we have ϕ(x) = csinh(√γx)+ d cosh(√γx) asinh(√γx)+ bcosh(√γx) and f ′(x) = asinh(√γx)+ bcosh(√γx) for all x ∈I, thus there exists λ ∈R such that f(x) = 1 √γ (acosh(√γx)+ bsinh(√γx))+ λ, (x ∈I).
Substituting A√γ , B√γ , C√γ , and D√γ instead of the constants a,b,c, and d , re-spectively, we can see that f and ϕ are of the form stated in (22). Now we compute ϕ x+y 2 (f(x) −f(y)) for x,y ∈I. For the brevity, denote the elements √γx and √γy by u and v, respectively. Then, f(x)−f(y) = (Acosh(u)+ Bsinh(u))−(Acosh(v)+ Bsinh(v)) = 2sinh u−v 2 Asinh u+v 2 + Bcosh u+v 2 .
Therefore, for all x,y ∈I, we have ϕ x+ y 2 (f(x)−f(y)) =Csinh u+v 2 + Dcosh u+v 2 Asinh u+v 2 + Bcosh u+v 2 ·2sinh u−v 2 Asinh u+v 2 + Bcosh u+v 2 =2sinh u−v 2 Csinh u+v 2 + Dcosh u+v 2 (25) =(Ccosh(u)+ Dsinh(u))−(Ccosh(v)+ Dsinh(v)) =(Ccosh(√γx)+ Dsinh(√γx))−(Ccosh(√γy)+ Dsinh(√γy)).
Using that (ϕ, f,F) solves (4), the above identity yields, for all x,y ∈I , that F(x)−F(y) = (Ccosh(√γx)+ Dsinh(√γx))−(Ccosh(√γy)+ Dsinh(√γy)), hence the function x →F(x) −Ccosh(√γx) −Dsinh(√γx) is constant on I. This verifies that F is of the form (22) for some μ ∈R.
To show the sufficiency of alternative (iii), observe that in each of the above cases, the identities (23), (24), and (25) are satisfied, respectively. Therefore, the triplet (ϕ, f,F) satisfies functional equation (4).
□ REMARK 2. One can easily see that the triplets (ϕ, f,F) described in alternatives (i) and (ii) are solutions of (4) without assuming the continuity of ϕ . It remains an open problem if the continuity assumption about ϕ can be omitted from the formulation of Theorem 5.
1120 T. KISS AND Z. P ´ ALES 6. An application In this section, using our main result, we solve the equality problem of two-variable Cauchy means and two-variable quasi-arithmetic means solely under the con-ditions needed for their definitions.
THEOREM 6. Let J ⊆R be an open subinterval. Assume that G,H : J →R are differentiable functions such that 0 / ∈H′(J) and G′ H′ is invertible on J, and that Φ : J → R is continuous and strictly monotone. Then the functional equation CG,H(x,y) = AΦ(x,y), (x,y ∈J) (26) holds if and only if Φ is differentiable with a nonvanishing first derivative and there exist constants A,B,C,D ∈R with AD ̸= BC and μ,λ,γ ∈R such that G H = A B C D ψ1 ψ2 + μ λ (27) holds on J, where (ψ1(x),ψ2(x)) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ cos(√−γΦ(x)),sin(√−γΦ(x)) if γ < 0, Φ2(x),Φ(x) if γ = 0, cosh(√γΦ(x)),sinh(√γΦ(x)) if γ > 0, (x ∈J).
Proof. A simple calculation yields that, the triplet (G,H,Φ) solves functional equation (26) on J if and only if the triplet of functions (ϕ, f,F) := G′ H′ ◦Φ−1,H ◦ Φ−1,G◦Φ−1 solves the functional equation (4) on I := Φ(J). In view of this connec-tion, the sufficiency part of the statement is obvious. Thus we can turn to the proof of the necessity part, that is, assume that (26) is satisfied.
Due to the properties of Φ, the set Φ(J) is an open subinterval of R. By our assumption, the function G′ H′ is invertible on J , which, in view of Remark 1 of the paper , implies that it is continuous and strictly monotone. This and the assumptions concerning Φ imply that ϕ = G′ H′ ◦Φ−1 must be also continuous, hence the conditions of Theorem 5 are satisfied. Consequently, for the members of the triplet (ϕ, f,F), we have one the possibilities (i), (ii) or (iii) listed in Theorem 5.
Assume that there exists a nonempty open subinterval of I, where f is constant.
Then, by the continuity and strict monotonicity of Φ, the preimage of this interval is an open interval again, contained in J, where H is turned to be also constant. This contradicts the assumption 0 / ∈H′(J). This means that f must be locally non-constant on I, which means that the case (i) is excluded.
A similar argument shows that the case (ii) of Theorem 5 is also impossible. In-deed, assume indirectly that we have (ii). According to the previous part, the set K must coincide with I, that is, ϕ must be constant on 1 2(I + K) = I . Due to the prop-erties of Φ, this yields that there exists a nonempty open subinterval of J, where the ratio G′ H′ is constant, which contradicts that it is invertible.
ON A FUNCTIONAL EQUATION 1121 Consequently, there exist real constants A,B,C, and D with the property AD ̸= BC and γ ∈R such that one of the alternatives (20), (21) or (22) listed in the case (iii) of Theorem 5 must hold.
Assume that γ < 0 holds. Then, using the definitions of the functions ϕ, f and F , for all u ∈I = Φ(J), we have F(u) = G Φ−1(u) = −Acos(√−γu)+ Bsin(√−γu)+ μ and f(u) = H Φ−1(u) = −Ccos(√−γu)+ Dsin(√−γu)+ λ.
Writing x instead of Φ−1(u), defining ψ1(x) := cos(√−γΦ(x)) and ψ2(x) := sin(√−γΦ(x)) for x ∈J, and, finally, writing A and C instead of −A and −C, we get that the pair (G,H) is of the form written in (27). The proof in the cases γ = 0 and γ > 0 goes analogously.
Now we are able to prove that Φ is differentiable with a nonvanishing first deriva-tive. By the Rolle Mean Value Theorem, H has to be invertible on I, hence we have Φ−1 = H−1 ◦f . In view of Theorem 5, the function f is (continuously) differen-tiable and, a short calculation yields that its derivative is either positive or negative on its domain. Hence, taking the inverse of both sides of this last identity, we get that Φ = f −1 ◦H , which shows that Φ is differentiable. Finally, the assumption 0 / ∈H′(J) provides that Φ′ does not vanish on J.
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(Received August 16, 2018) Tibor Kiss Institute of Mathematics, MTA-DE Research Group “Equations, Functions and Curves” Hungarian Academy of Sciences and University of Debrecen P. O. Box 12, 4010 Debrecen, Hungary e-mail: kiss.tibor@science.unideb.hu Zsolt P´ ales Institute of Mathematics University of Debrecen H-4032 Debrecen, Egyetem t´ er 1, Hungary e-mail: pales@science.unideb.hu Mathematical Inequalities & Applications www.ele-math.com mia@ele-math.com |
190485 | https://www.chegg.com/homework-help/questions-and-answers/1-fig-9-28-water-flows-36-pipe-rate-223-mgd-determine-horsepower-pump-x-785-efficiency-pro-q85642569 | Your solution’s ready to go!
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Question: 1. In Fig. 9-28, water flows through the 36" pipe at the rate of 22.3 mgd. Determine the horsepower of the pump X A (78.5% efficiency) that will produce the flows and elevations for the system if the pressure head at X is zero ft. (Draw grade lines). Ans. 250 hp 2. How much water must the pump supply when the flow through the 900-mm pipe is 1.31 m3/s, and
In Fig. 9-28, water flows through the 36" pipe at the rate of
22.3 mgd. Determine the horsepower of the pump X
A (78.5% efficiency) that will produce the flows and
elevations for the system if the pressure
head at X is zero ft. (Draw grade
lines). Ans. 250 hp
How much water must the pump supply when the flow through the
900-mm pipe is 1.31 m3/s, and what is the pressure head
at A in Fig. 9-29? Ans. 1.10
m3/s, 58 m
The pressure head at A in pump A
B is 120 ft when the energy change in the system (see
Fig. 9-30) due to the pump is 153 hp. The lost head through valve
Z is 10 ft. Find all the flows and the elevation of
reservoir T. Sketch the hydraulic grade
lines. Ans. 87.4 ft
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190486 | https://medwinpublishers.com/IJFSC/carbon-monoxide-poisoning-death-in-the-wild-environment-in-tropical-areas-a-case-report.pdf | International Journal of Forensic Sciences ISSN: 2573-1734 MEDWIN PUBLISHERS Committed to Create Value for Researchers Carbon Monoxide Poisoning Death in the Wild Environment in Tropical Areas: a Case Report Int J Forens Sci Carbon Monoxide Poisoning Death in the Wild Environment in Tropical Areas: a Case Report WANG B1,XU YZ1,QIN XS1, ZHOU YC1, HA S1, QU YH1, CHEN JH1, CONG Bin1,2, DENG Jian-qiang1 1Department of Forensic Medicine of Hainan Medical University, Hainan Provincial Academician Workstation, tropical forensic medicine, Hainan Provincial Tropical Forensic Engineering Research Center, Hainan Haikou 571199;China 2Department of Forensic Medicine, Hebei Medical University. Shijiazhuang 050011;China Corresponding author: Deng Jian-qiang, Department of Forensic Medicine of Hainan Medical University, Hainan Provincial Academician Workstation(tropical forensic medicine), Hainan Provincial Tropical Forensic Engineering Research Center, Hainan Haikou 571199; China Chen Jian-hua, Department of Forensic Medicine, Hebei Medical University, Shijiazhuang 050011, China Case Report Volume 8 Issue 2 Received Date: February 28, 2023 Published Date: April 03, 2023 DOI: 10.23880/ijfsc-16000294 Abstract Carbon monoxide (CO) poisoning is a major public health concern and a common cause of death. Deaths caused by carbon monoxide poisoning in the wild of tropics are rare and special. In this case, the measurement of percentage carboxyhemoglobin in the heart blood of the deceased was only 17.5%. After eliminating the causes of death such as self disease, mechanical injury, mechanical asphyxia and other toxicosis, the body was considered to have died from CO poisoning. The particularity of this case can prompt the identification thinking and broaden the judgment method for forensic scientists suspected of CO poisoning death. Keywords: Carbon Monoxide Poisoning; Carboxyhemoglobin; Cause of Death Abbreviations: CO: Carbon Monoxide; COHb: Carboxyhemoglobin.
Introduction Carbon monoxide (CO) is a colorless, tasteless, non-irritating toxic gas. Exogenous carbon monoxide is usually produced by incomplete combustion of carbon-containing compounds, such as car exhaust leakage, fire and unsafe heating systems [1-3]. Carbon monoxide poisoning is common in forensic practice under hermetic environment in cold regions due to keep warm needs, but rare in tropical areas, especially in the field environment [4,5]. After entering the blood, carbonmonoxide can combine with hemoglobin (Hb) to form carboxyhemoglobin (COHb). The affinity of CO and Hb is 200 to 250 times greater than the affinity of oxygen and Hb, and the dissociation rate of COHb is 3600 times slower than that of oxygenated hemoglobin, thus reducing the oxygen carrying capacity of hemoglobin [6,7]. Finally, it will lead to hypoxia of tissues and organs, so that make blood vessel compensatory dilation and extravasated. ATP is rapidly consumed in oxygen-free environment, resulting in the accumulation of sodium ions in cells, which cause tissue edema. When the body suffers from severe hypoxia, a series of pathophysiological changes will follow, and CO poisoning can cause death at last . The measurement of percentage carboxyhemoglobin in the corpse (%COHb) is an important reference index for judging the death International Journal of Forensic Sciences 2 Wang B, et al. Carbon Monoxide Poisoning Death in the Wild Environment in Tropical Areas: a Case Report. Int J Forens Sci 2023, 8(2): 000294.
Copyright© Wang B, et al.
of CO poisoning. However, due to the influence of many factors such as postmortem time, temperature, corruption, pathological state of the body and individual differences, the identification of the cause of death in CO poisoning cases has increased difficulty . This article reports a case suspected of CO poisoning and incisive wound with hesitation mark in the wild in tropical areas, which provides a reference for the identification of similar forensic cases.
Case Story and Scene Investigation In October 2022, located in a natural scenic spot in northern Hainan Island, China, which in the tropical region, a woman’s body was found in the grass beside a remote forest. The body is in a prone position, with the face tilted to the left, the upper limbs flexed and stretched forward, and the hands placed under the face. Beside the body, it is a single camouflage waterproof tent with both lower limbs placed in the tent. The site investigation found that there was burnt charcoal left in a stainless steel bowl in the tent, as well as a number of unused strips of charcoal. At the same time, a carbon monoxide alarm was also found, which required three batteries to work properly, and one of the batteries fell off. All items on the scene are placed in order. The possibility of criminal cases is excluded after comprehensive investigation.
According to the ID card of the deceased found at the scene, the police investigated the track of the deceased’s life activities before her death. After reviewing the camera image data at the entrance and exit of the scenic spot, it was determined that the woman entered the scenic spot alone with the items found at the scene two days before the body was found. She expressed suicidal tendencies when contacting with her family before her death. The local climate conditions at the time of the discovery of the body were cloudy, with temperatures ranging from 21°C to 30°C, and northwesterly winds 2-4 (Figures 1-3).
Figure 1: The scene, the body was located near a forest in the wild.
Figure 2: Burnt carbon ash in the tent.
International Journal of Forensic Sciences 3 Wang B, et al. Carbon Monoxide Poisoning Death in the Wild Environment in Tropical Areas: a Case Report. Int J Forens Sci 2023, 8(2): 000294.
Copyright© Wang B, et al.
One of the batteries fell off and could not work normally.
Figure 3: Carbon Monoxide Alarm Found on Site. Corpse External Examination The postmortem surface examination was carried out immediately after the site inspection. The main findings were as follows: The lividity was slightly bright red and located at the un- pressed part of the ventral side, which could fade under strong pressure. Rigor mortis existed at all joints of the body, strong and hard. Bilateral eyelid conjunctiva hyperemia, cornea moderately turbid. No foreign matter found in the oral and nasal cavity. The nail bed of both hands were cyanotic. The left forearm was wrapped by bandage. After the bandage was removed, multiple parallel incisions with different depth and direction were seen from the left elbow joint to the front of the left wrist joint. The wound margin was neat, the wound angle was sharp. The shallow one reached deep into the skin epidermis, the deep one reached deep into the subcutaneous fat, and partial wound formed scab. No special findings were found in the remaining postmortem surface examination. The intracardiac blood was extracted by cardiac puncture for the measurement of percentage carboxyhemoglobin, and its content was 17.5%. The corpse was immediately stored in the -30 ℃ ice coffin after the postmortem surface examination (Figure 4).
Figure 4: Incised Wound of Left Upper Limb.
Medico-Legal Autopsy and Microscopic Observation Because we must obtain the informed consent of her family, the autopsy was carried out 7 days after the discovery of the body. During this period, the body was kept in the ice coffin without power failure or removal.
After thawing at room temperature, the body was examined and no obvious corruption was found. The body surface and all organs are slightly bright red. Except for the incisive wound seen on the left upper limb, no forensic International Journal of Forensic Sciences 4 Wang B, et al. Carbon Monoxide Poisoning Death in the Wild Environment in Tropical Areas: a Case Report. Int J Forens Sci 2023, 8(2): 000294.
Copyright© Wang B, et al.
pathological changes of mechanical injury and mechanical asphyxia were found in all parts and organs of the body. Macroscopic and histopathological examination of brain (including cerebrum, cerebellar and brainstem), heart, lung, liver, kidney, pancreas, spleen, adrenal gland and other important organs mainly found vascular dilation, congestion and tissue edema. We had not detected other injuries or diseases of important organs.
Medico-Legal Toxicological Analysis During autopsy, the cardiac blood of the deceased was extracted for examination of common toxic drugs, drugs and carboxyhemoglobin. The methods and results are as follows: • Qualitative and quantitative detection of toxic (drug) components: Refer to the relevant technical specifications for forensic toxicological examination and judicial identification of the People’s Republic of China (standard number: SF/ZJD0107005-2016, SF/ZJD0107008-2010, SF/ZJD0107014-2015, SF/ ZJD0107018-2018) for qualitative detection of common toxic (drug) components, and the results showed that no toxic (drug) components were detected.
• Qualitative and quantitative detection of common drugs: refer to the relevant technical specifications and industrial standards for forensic toxicological examination and judicial identification of the People’s Republic of China (standard number: SF/ZJD0107005-2016, SF/T0116-2021, SF/T0114-2021) to conduct qualitative detection of common drug components, and the results showed that no drug components were detected.
• Carboxyhemoglobin detection: Carboxyhemoglobin was detected in cardiac blood, and the content percentage was 16.98% (reference value 0.0% - 1.5%).
Discussion Based on comprehensive analysis of various information, the death time of the deceased should exceed 12 hours. The most special aspect of this case is that the CO poisoning death site is located in the wild in tropical areas. Due to the high temperature all the year round in tropical areas, no additional heating equipment is usually needed, so there will be no CO poisoning caused by the burning of firewood, coal and other carbonaceous organic substances for heating. However, according to the on-site prompt, the case occurred in a tent built in the field. The deceased may burn charcoal in the tent and cause CO poisoning.
Forensic pathologists generally believe that COHb ≥ 50% can be confirmed as CO poisoning death. In this case, COHb in cardiac blood was detected in two different institutions, and the content was both about 17%, which could at least confirm that the body had CO poisoning (COHb ≥ 10% must be exogenous CO poisoning) [10,11]. The clinical manifestations of CO poisoning are often non-specific and depend on several factors, mainly including (1) the concentration of CO, (2) the duration of continuous exposure, and (3) the health status of individuals (lung ventilation, physical condition, and respiratory speed and efficiency) . The overall volume of the tent is small enough for the dead to climb from the inside to the outside. It was conceivable that the consciousness of the deceased had not completely disappeared after poisoning, and she climbed out of the tent at the last stage of her life, active or unconscious behavior. The head, face, upper limbs and trunk of the deceased were all exposed outside the tent. After she climbed out of the tent, there was still breathing behavior. At this time, there was a level 2-4 wind outside, so the COHb content detected in the blood of the deceased was low. This may be the main reason for the lower content of carboxyhemoglobin in the body of the deceased. In addition, studies have shown that with the prolongation of postmortem time, COHb in the body will decrease, and low temperature is conducive to the stability of COHb, while with the increase of temperature, the content of COHb can decrease due to the release of CO . Therefore, the environmental conditions of the dead body may accelerate the release of CO from COHb in the body, which also constitutes the reason for the low content of COHb detected in the blood of the dead body. The content of COHb in heart blood after 7 days of frozen preservation of the corpse only decreased from 17.5% at the time of discovery to 16.98%, with a small change. On the other hand, it confirmed that low temperature is conducive to the stability of COHb in the corpse. Some studies also suggest that under the same conditions, women with CO poisoning sometimes detect relatively lower COHb in the blood. On the one hand, it may be due to the smaller lung capacity of women, on the other hand, it may be due to poorer tolerance [11,12].
We often describe that COHb is cherry red. In this case, the content of COHb is relatively small, so the cherry red livor mortis of the corpse is also relatively light, and the muscle tissue does not show obvious cherry red visible to the naked eye. In addition, cherry red is not unique to carbon monoxide poisoning, but also needs to be distinguished from cyanide poisoning, drowning, freezing and frozen corpses . In particular, the color of livor mortis observed by naked eyes often depends on the empirical judgment of forensic scholars, and other phenomena cannot be objectively ruled out. Therefore, other possible causes of CO poisoning death should still be ruled out.
However, after a comprehensive autopsy, toxicological examination and histopathological examination, no evidence of its own disease and death caused by common toxic drugs International Journal of Forensic Sciences 5 Wang B, et al. Carbon Monoxide Poisoning Death in the Wild Environment in Tropical Areas: a Case Report. Int J Forens Sci 2023, 8(2): 000294.
Copyright© Wang B, et al.
and drugs was found. Although multiple incisions were found on her left upper limb and the wound was locally scabbed, the wound was not enough to cause death based on the comprehensive analysis of its severity and the information that no blood stain was found in the field survey. According to other autopsy and field investigation, the evidence of death due to mechanical injury and mechanical asphyxia is also insufficient. Consequently, after excluding deaths caused by other causes, all kinds of evidence point to the death of the deceased from CO poisoning. CO poisoning first affects tissues and organs with high oxygen demand, such as brain and heart.
Severe hypoxia of brain tissue can cause severe brain edema and lead to death. Severe myocardial ischemia caused by acute CO poisoning can directly lead to fatal myocardial infarction, and cardiac conduction system can also cause sudden cardiac arrest and death due to ischemia [6,14].
In addition, the multiple incisions of different length and depth and parallel arrangement found from the left elbow to the left wrist of the deceased in this case have the characteristics of “hesitation marks” common in typical suicide cases in terms of their location and characteristics . According to the investigation of the case, the deceased had a suicidal tendency. However, blood scab had formed on the cut wound, and there were bandages on it, no blood was found at the scene, so it was not consistent with the formation of the deceased before her death, above all which further explained the planning process of the deceased for suicide. Combined with the above findings and the field investigation, it seems that the nature of the case is to ignite a charcoal fire in the tent to commit suicide. The reason for finding half of her body climbing out of the tent at the scene of her death may be the unconscious behavior after CO poisoning or the regret behavior. Although the carbon monoxide alarm found in the on-site inspection also seems to indicate that the deceased was worried about the risk of CO poisoning and bought the device, we can’t prove whether one of the batteries fell off accidentally or intentionally. Nevertheless, it is clear that the evidence obtained in this case can exclude the possibility of homicide .
All in all, there are few accidental deaths of CO poisoning caused by carbon fire heating in the outdoor environment of tropical areas. Especially because of the different climatic conditions from other regions, it sometimes leads to difficulties in forensic identification. To sum up, the principle of “comprehensive consideration and specific case analysis” should be followed for death cases suspected of CO poisoning in actual cases. Combined with site survey, system inspection and attention to identify whether there are other factors involved. After analyzing the case comprehensively, fully and scientifically, then make a conclusion, so as to make an objective, fair and realistic appraisal conclusion .
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15. Haliga RE, Morărașu BC, Șorodoc V, Lionte C, Sîrbu O, et al. (2022) Rare Causes of Acute Coronary Syndrome: Carbon Monoxide Poisoning. Life (Basel) 12(8): 1158. 16. Racette S, Kremer C, Desjarlais A, Sauvageau A (2008) Suicidal and homicidal sharp force injury: a 5-year retrospective comparative study of hesitation marks and defense wounds. Forensic Sci Med Pathol 4(4): 221-227. 17. Hampson NB (2008) Stability of carboxyhemoglobin in stored and mailed blood samples. The American Journal of Emergency Medicine 26(2): 191-195. |
190487 | https://users.math.msu.edu/users/zhan/Notes1.pdf | Convergent Sequences Definition 1. A sequence of real numbers (sn) is said to converge to a real number s if ∀ε > 0, ∃N ∈N, such that n > N implies |sn −s| < ε.
(1) When this holds, we say that (sn) is a convergence sequence with s being its limit, and write sn →s or s = limn→∞sn. If (sn) does not converge, then we say that (sn) is a divergent sequence.
We first show that one sequence (sn) can not have two different limits. Suppose sn →s and sn →t. Let ε > 0. Then ε 2 > 0. Since sn →s, by definition there is N1 ∈N such that for n > N1, |sn −s| < ε 2. Since sn →t, by definition there is N2 ∈N such that for n > N2, |sn −t| < ε 2. Here we use N1 and N2 in the two statements because the N coming from the two limits may not be the same. Let N = max{N1, N2}. If n > N, then n > N1 and n > N2 both hold. So |sn −s| < ε 2 and |sn −t| < ε 2, which by triangle inequality imply that |s −t| ≤|sn −s| + |sn −t| < ε 2 + ε 2 = ε.
Now |s−t| < ε holds for every ε > 0. We then conclude that |s−t| = 0 (for otherwise |s−t| > 0, we then get a contradiction by choosing ε = |s −t|). So s = t, and the uniqueness holds.
We will use the following tools to check whether a sequence converges or diverges.
1. the definition 2. basic examples 3. limit theorems 4. boundedness and subsequences.
We have stated the definition. Now we consider some examples.
Example 1. Let s ∈R. If sn = s for all n, i.e., (sn) is a constant sequence, then lim sn = s.
Proof. For any given ε > 0 we simply choose N = 1. If n > N, then |sn −s| = 0 < ε.
Example 2. We have 1 n →0.
Proof. Let ε > 0. By Archimedean property, there is N ∈N such that 1 N < ε. If n > N, then | 1 n −0| = 1 n < 1 N < ε.
Example 3. The following two sequences are divergent 1 (i) (sn) = ((−1)n) = (−1, 1, −1, 1, −1, 1, . . . ); (ii) (sn) = (n) = (1, 2, 3, 4, 5, 6, . . . ).
Proof. (i) We use the notation of subsequence and statement that will be proved later. Suppose n1 < n2 < n3 < · · · is a strictly increasing sequence of indices, then (snk) is a subsequence of (sn). We will prove a theorem, which asserts that, if (sn) converges to s, then any subsequence of (sn) also converges to s.
The sequence (sn) = ((−1)n) contains two constant sequences (1, 1, 1, . . . ) (with nk = 2k) and (−1, −1, −1, . . . ) (with nk = 2k−1), which converge to different limits. So the original (sn) can not converge.
(ii) We use the following theorem. If (sn) is convergent, then it is a bounded sequence. In other words, the set {sn : n ∈N} is bounded. So an unbounded sequence must diverge. Since for sn = n, n ∈N, the set {sn : n ∈N} = N is unbounded, the sequence (n) is divergent.
Remark 1. This example shows that we have two ways to prove that a sequence is divergent: (i) find two subsequences that convergent to different limits; (ii) show that the sequence is unbounded. Note that the (sn) in (i) is bounded and divergent. The (sn) in (ii) is divergent, but lim sn actually exists, which is +∞, and its every subsequence also tends to +∞. We will define that limit later.
Now we state some limit theorems.
Theorem 1 (Theorem 9.1). Every convergent sequence is bounded.
Proof. Let (sn) be a sequence that converges to s ∈R. Applying the definition to ε = 1, we see that there is N ∈N such that for any n > N, |sn −s| < 1, which then implies that |sn| ≤|s|+1.
Let M = max{|s1|, |s2|, . . . , |sN|, |s| + 1}.
The maximum exists since the set is finite. Then for any n ∈N, |sn| ≤M (consider the case n ≤N and n > N separately), i.e., −M ≤sn ≤M. So {sn : n ∈N} is bounded.
Theorem 2 (Theorem 9.3). If (sn) converges to s and (tn) converges to t, then (sn + tn) converges to s + t.
Proof. Let ε > 0. Then ε 2 > 0. Since sn →s, there is N1 ∈N such that for n > N1, |sn−s| < ε 2.
Since tn →t, there is N2 ∈N such that for n > N2, |tn −t| < ε 2. Let N = max{N1, N2}. If n > N, then n > N1 and n > N2 both hold, and so |sn −s| < ε 2 and |tn −t| < ε 2, which together imply (by triangle inequality) that |(sn + tn) −(s + t)| ≤|sn −s| + |tn −t| < ε 2 + ε 2 = ε.
So we have the desired convergence.
Theorem 3 (Theorem 9.4). If (sn) converges to s and (tn) converges to t, then (sn·tn) converges to s · t.
2 Discussion. We need to bound |sntn −st| from above for big n. We write sntn −st = sntn −snt + snt −st = sn(tn −t) + t(sn −s).
By triangle inequality, we get |sntn −st| ≤|sn(tn −t)| + |t(sn −s)| = |sn||tn −t| + |t||sn −s|.
Since tn →t and sn →s, we know that |tn −t| and |sn −s| can be arbitrarily small if we choose n big enough. Thus, if |sn| and |t| are not too big, then we can control the sum on the RHS (righthand side). In fact, the size of |sn| can be controlled because of Theorem 9.1.
Proof. Since (sn) is convergent, by Theorem 9.1, there is M > 0 such that |sn| ≤M for every n. We may choose M big such that M ≥|t|. Let ε > 0. Then ε 2M > 0. Since sn →s, there is N1 ∈N such that for n > N1, |sn −s| < ε 2M . Since tn →t, there is N2 ∈N such that for n > N2, |tn −t| < ε 2M . Let N = max{N1, N2}. If n > N, then n > N1 and n > N2 both hold, and so |sn −s| < ε 2M and |tn −t| < ε 2M , which together with |sn| ≤M and |t| ≤M imply that |sntn −st| ≤|sn(tn −t)| + |t(sn −s)| = |sn||tn −t| + |t||sn −s| ≤M|tn −t| + M|sn −s| < M ε 2M + M ε 2M = ε.
Corollary 1. If (sn) converges to s, k ∈R, and m ∈N, then (ksn) converges to ks and sm n converges to sm.
Proof. For the sequence (ksn), we apply Theorem 9.4 to the sequence (tn) with tn = k for all n. For the sequence (sm n ) we use induction. In the induction step, note that sm+1 n = sn ∗sm n and apply Theorem 9.4 to tn = sm n Corollary 2. If (sn) converges to s and (tn) converges to t, then (sn −tn) converges to s −t.
Proof. We write sn + tn = sn + (−1)tn and apply Theorem 9.3 and the previous corollary.
From this corollary we see that sn →s iffsn −s →0. By the Theorem below, the latter statement is equivalent to that |sn −s| →0.
Theorem 4. (a) Suppose two sequences (sn) and (tn) satisfy that tn →0 and |sn| ≤|tn| for all but finitely many n. Then sn →0.
(b) For any sequence (sn), sn →0 if and only if |sn| →0.
Proof. (a) Let N0 ∈N be such that |sn| ≤|tn| for n > N0. Let ε > 0. Since tn →0, there is N1 ∈N such that for n > N1, |tn −0| < ε. Let N = max{N0, N1}. For n > N, |sn| ≤|tn| and |tn −0| < ε, which imply that |sn −0| = |sn| ≤|tn| = |tn −0| < ε.
(b) From (a) we know that if |sn| = |tn| for all n, then sn →0 ifftn →0. We then apply this result to tn = |sn| and use that ||sn|| = |sn|.
3 Lemma 1 (Lemma 9.5). If (sn) converges to s such that s ̸= 0 and sn ̸= 0 for all n, then (1/sn) converges to 1/s.
Discussion. We need to bound |1/sn −1/s| from above for big n. We write 1 sn −1 s = s −sn sns = |sn −s| |sn||s| .
Since sn →s, |sn −s| can be arbitrarily small if we choose n big enough. Thus, if |sn| and |s| are not too close to 0, then we can control the size of the RHS. This means that we need a positive lower bound of the set {|s1|, |s2|, . . . }.
Proof. Since s ̸= 0, we have |s| 2 > 0. Since sn →s, applying the definition to ε = |s| 2 , we get N ∈N such that for n > N, |sn −s| < |s| 2 , which then implies by triangle inequality that |sn| ≥|s| −|sn −s| > |s| −|s| 2 = |s| 2 . Let m = min{|s1|, |s2|, . . . , |sN|, |s| 2 }. Then m exists and is positive since the set is a finite set of positive numbers.
Let ε > 0. Then m|s|ε > 0. Since sn →s, there is N′ ∈N such that n > N′ implies that |sn −s| < m|s|ε, which together with |sn| ≥m for all n implies that 1 sn −1 s = |sn −s| |sn||s| ≤|sn −s| m|s| < m|s|ε m|s| = ε.
Theorem 5 (Theorem 9.6). Suppose (sn) converges to s and (tn) converges to t. If s ̸= 0 and sn ̸= 0 for all n, then (tn/sn) converges to t/s.
Proof. By Lemma 9.5, (1/sn) converges to 1/s. Applying Theorem 9.4 to the sequences (1/sn) and (tn), we get the conclusion.
Example 4. Derive lim 3n+1 7n−4 and lim 4n3+3n n3−6 Solution. We write 3n + 1 7n −4 = 3 + 1/n 7 + (−4) ∗1/n, 4n3 + 3n n3 −6 = 4 + 3 ∗(1/n)2 1 + (−6) ∗1/n.
We have shown that lim 1/n = 0. So (i) lim(3 + 1/n) = 3 + 0 = 3 and lim(7 + (−4) ∗1/n) = 7 + (−4) ∗0 = 7, which imply that lim 3n+1 7n−4 = lim(3 + 1/n)/ lim(7 + (−4) ∗1/n) = 3/7; (ii) lim(4 + 3 ∗(1/n)2) = 4 + 3 ∗02 = 4 and lim(1 + (−6) ∗1/n) = 1 + (−6) ∗0 = 1, which imply that lim 4n3+3n n3−6 = lim(4 + 3 ∗(1/n)2)/ lim(1 + (−6) ∗1/n) = 4.
We now state some theorems about the relation between limits and orders.
Theorem 6 (Exercise 8.9). (a) If (sn) converges to s, and there is N0 ∈N such that sn ≥0 for all n > N0, then s ≥0.
(b) Suppose (sn) converges to s and (tn) converges to t. If there N0 ∈N such that sn ≤tn for all n > N0, then s ≤t.
4 Proof. (a) We prove by contradiction. Suppose s < 0. Let ε = |s| = −s > 0. Since sn →s, there is N ∈N such that for n > N, |sn −s| < ε, which implies that sn < s + ε = 0. Let n = max{N, N0} + 1. Then n > N0 and n > N. From n > N0 we get sn ≥0; from n > N we get sn < 0. This is the contradiction.
(b) Applying (i) to the sequence (tn −sn) we conclude that its limit t−s is nonnegative.
For x ∈[0, ∞) and n ∈N, the power root x1/n is defined as the unique y ∈[0, ∞) such that yn = x. The uniqueness of such y follows from the fact that if 0 ≤y1 < y2, then yn 1 < yn 2 . The existence follows from the “Intermediate Value Theorem” for continuous function f(x) = xn, which will be stated and proved later.
We now just accept the existence of x1/n for any x ∈[0, ∞). It is clear that 0 ≤x1 < x2 implies that 0 ≤x1/n 1 < x1/n 2 . We restrict our attention to [0, ∞) although in the case that n is an odd number, we can also define x1/n for x < 0.
When n = 2, x1/2 is often written as √x. We have the following theorem.
Theorem 7 (Example 5). Suppose (sn) converges to s and sn ≥0 for all n. Then (√sn) converges to √s.
Discussion We want to bound |√sn−√s| from above for big n. It is useful to note the equality (√sn −√s)(√sn + √s) = (√sn)2 −(√s)2 = sn −s.
Taking absolute value, we get |√sn −√s| · |√sn + √s| = |sn −s|.
If √s > 0, then |√sn −√s| = |sn −s| √sn + √s ≤|sn −s| √s .
Proof. By Theorem 6, s ≥0. First suppose s > 0. Then √s > 0. Let ε > 0. Then √sε > 0.
Since sn →s, there is N ∈N such that for n > N, |sn −s| < √sε, which implies that |√sn −√s| = |sn −s| √sn + √s ≤|sn −s| √s < √sε √s = ε.
We leave the proof in the case s = 0 as an exercise. Note that for x ≥0, √x < ε iffx2 < ε.
5 |
190488 | https://www.arsdcollege.ac.in/wp-content/uploads/2020/03/30-04-April-Electricity_Magnetism_IInd_Sem_L2.pdf | Faraday’s Law of Induction 8.1 Faraday’s Law In the previous chapter, we have shown that steady electric current can give steady magnetic field because of the symmetry between electricity & magnetism.
We can ask: Steady magnetic field can give steady electric current.
× OR Changing magnetic field can give steady electric current.
✓ Define : (1) Magnetic flux through surface S: Φm = ˆ S ⃗ B · d ⃗ A Unit of Φm Weber (Wb) : 1Wb = 1Tm2 (2) Graphical: Φm = Number of magnetic field lines passing through surface S Faraday’s law of induction: Induced emf |E| = N ¯ ¯ ¯ ¯ ¯ dΦm dt ¯ ¯ ¯ ¯ ¯ where N = Number of coils in the circuit.
Name of Faculty: Dr. Amit K Vishwakarma Name of Course: B.Sc Physical Science (Electronics) Subject: Electricity & Magnetism Semester: II 30th March -04th April 8.2. LENZ’ LAW 99 ⃗ B = Constant ⃗ B = Constant ˆ B = Constant ⃗ B = Constant ⃗ A = Constant ˆ A = Constant dB/dt ̸= 0 A = Constant dA/dt ̸= 0 ⃗ A = Constant d ˆ A/dt ̸= 0 E = 0 ∴|E| > 0 ∴|E| > 0 ∴|E| > 0 Note : The induced emf drives a current throughout the circuit, similar to the function of a battery. However, the difference here is that the induced emf is distributed throughout the circuit. The consequence is that we cannot define a potential difference between any two points in the circuit.
Suppose there is an induced current in the loop, can we define ∆VAB?
Recall: ∆VAB = VA −VB = iR > 0 ⇒ VA > VB Going anti-clockwise (same as i), If we start from A, going to B, then we get VA > VB.
If we start from B, going to A, then we get VB > VA.
∴We cannot define ∆VAB !!
This situation is like when we study the interior of a battery.
A battery The loop provides the energy needed to drive the charge carriers around the circuit by chemical reactions.
changing magnetic flux.
sources of emf non-electric means 8.2 Lenz’ Law (1) The flux of the magnetic field due to induced current opposes the change in flux that causes the induced current.
8.3. MOTIONAL EMF 100 (2) The induced current is in such a direction as to oppose the changes that produces it.
(3) Incorporating Lentz’ Law into Faraday’s Law: E = −N dΦm dt If dΦm dt > 0, Φm ↑ ⇒ E appears ⇒ Induced current appears.
⇒⃗ B-field due to induced current ⇒ change in Φm so that = ⇒ Φm ↓ (4) Lenz’ Law is a consequence from the principle of conservation of energy.
8.3 Motional EMF Let’s try to look at a special case when the changing magnetic flux is carried by motion in the circuit wires.
Consider a conductor of length L moving with a velocity v in a magnetic field ⃗ B.
8.3. MOTIONAL EMF 101 Hall Effect for the charge carriers in the rod: ⃗ FE + ⃗ FB = 0 ⇒ q ⃗ E + q⃗ v × ⃗ B = 0 (where ⃗ E is Hall electric field) ⇒ ⃗ E = −⃗ v × ⃗ B Hall Voltage inside rod: ∆V = − ˆ L 0 ⃗ E · d⃗ s ∆V = −EL ∴ Hall Voltage : ∆V = vBL Now, suppose the moving wire slides without friction on a stationary U-shape conductor.
The motional emf can drive an electric cur-rent i in the U-shape conductor.
⇒ Power is dissipated in the circuit.
⇒ Pout = V i (Joule’s heating) (see Lecture Notes Chapter 4) What is the source of this power?
Look at the forces acting on the conducting rod: • Magnetic force: ⃗ Fm = i⃗ L × ⃗ B Fm = iLB (pointing left) • For the rod to continue to move at constant velocity v, we need to apply an external force: ⃗ Fext = −⃗ Fm = iLB (pointing right) ∴ Power required to keep the rod moving: Pin = ⃗ Fext · ⃗ v = iBLv = iBL dx dt = iB d(xL) dt ( xL = A, area enclosed by circuit) = i d(BA) dt ( BA = Φm, magnetic flux) 8.3. MOTIONAL EMF 102 Since energy is not being stored in the system, ∴ Pin + Pout = 0 iV + idΦm dt = 0 We ”prove” Faraday’s Law ⇒ V = −dΦm dt Applications : (1) Eddy current: moving conductors in presence of magnetic field Induced current ⇒ Power lost in Joule’s heating ³E2 R ´ ⇒ Extra power input to keep moving To reduce Eddy currents: (2) Generators and Motors: Assume that the circuit loop is rotating at a constant angular velocity ω, (Source of rotation, e.g. steam produced by burner, water falling from a dam) 8.3. MOTIONAL EMF 103 Magnetic flux through the loop Number of coils ↓ ΦB = N ´ loop ⃗ B · d ⃗ A = NBA cos θ ↓ changes with time! θ = ωt ∴ ΦB = NBA cos ωt Induced emf: E = −dΦB dt = −NBA d dt(cos ωt) = NBAω sin ωt Induced current: i = E R = NBAω R sin ωt Alternating current (AC) voltage generator Power has to be provided by the source of rotation to overcome the torque acting on a current loop in a magnetic field.
⃗ τ = ⃗ µ z }| { Ni ⃗ A × ⃗ B ∴τ = NiAB sin θ 8.4. INDUCED ELECTRIC FIELD 104 The net effect of the torque is to oppose the rotation of the coil.
An electric motor is simply a generator operating in reverse.
⇒ Replace the load resistance R with a battery of emf E.
With the battery, there is a current in the coil, and it experiences a torque in the B-field.
⇒ Rotation of the coil leads to an induced emf, Eind, in the direction opposite of that of the battery.
(Lenz’ Law) ∴ i = E −Eind R ⇒ As motor speeds up, Eind ↑, ∴i ↓ ∴ mechanical power delivered = torque delivered = NiAB sin θ ↓ In conclusion, we can show that Pelectric = i2R + Pmechanical Electric power input Mechanical power delivered 8.4 Induced Electric Field So far we have discussed that a change in mag-netic flux will lead in an induced emf distributed in the loop, resulting from an induced E-field.
However, even in the absence of the loop (so that there is no induced current), the induced E-field will still accompany a change in magnetic flux.
8.4. INDUCED ELECTRIC FIELD 105 ∴ Consider a circular path in a region with changing magnetic field.
The induced E-field only has tangential components. (i.e. radial E-field = 0) Why?
Imagine a point charge q0 travelling around the circular path.
Work done by induced E-field = q0Eind | {z } force · 2πr |{z} distance Recall work done also equals to q0E, where E is induced emf ∴ E = Eind2πr Generally, E = ˛ ⃗ Eind · d⃗ s where ¸ is line integral around a closed loop, ⃗ Eind is induced E-field, ⃗ s is tangential vector of path.
∴ Faraday’s Law becomes ˛ C ⃗ Eind · d⃗ s = −d dt ˆ S ⃗ B · d ⃗ A L.H.S.
= Integral around a closed loop C R.H.S.
= Integral over a surface bounded by C Direction of d ⃗ A determined by direction of line integration C (Right-Hand Rule) 8.4. INDUCED ELECTRIC FIELD 106 ”Regular” E-field Induced E-field created by charges created by changing B-field E-field lines start from +ve and end on −ve charge E-field lines form closed loops can define electric potential so that we can discuss potential difference between two points Electric potential cannot be defined (or, potential has no meaning) ⇓ ⇓ Conservative force field Non-conservative force field The classification of electric and magnetic effects depend on the frame of reference of the observer. e.g. For motional emf, observer in the reference frame of the moving loop, will NOT see an induced E-field, just a ”regular” E-field.
(Read: Halliday Chap.33-6, 34-7) |
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19 3.4 Other Units for Solution Concentrations
Learning Objectives
By the end of this section, you will be able to:
Define the concentration units of mass percentage, volume percentage, mass-volume percentage, parts-per-million (ppm), and parts-per-billion (ppb)
Perform computations relating a solution’s concentration and its components’ volumes and/or masses using these units
In the previous section, we introduced molarity, a very useful measurement unit for evaluating the concentration of solutions. However, molarity is only one measure of concentration. In this section, we will introduce some other units of concentration that are commonly used in various applications, either for convenience or by convention.
Mass Percentage
Earlier in this chapter, we introduced percent composition as a measure of the relative amount of a given element in a compound. Percentages are also commonly used to express the composition of mixtures, including solutions. The mass percentage of a solution component is defined as the ratio of the component’s mass to the solution’s mass, expressed as a percentage:
We are generally most interested in the mass percentages of solutes, but it is also possible to compute the mass percentage of solvent.
Mass percentage is also referred to by similar names such as percent mass, percent weight, weight/weight percent, and other variations on this theme. The most common symbol for mass percentage is simply the percent sign, %, although more detailed symbols are often used including %mass, %weight, and (w/w)%. Use of these more detailed symbols can prevent confusion of mass percentages with other types of percentages, such as volume percentages (to be discussed later in this section).
Mass percentages are popular concentration units for consumer products. The label of a typical liquid bleach bottle (Figure 1) cites the concentration of its active ingredient, sodium hypochlorite (NaOCl), as being 7.4%. A 100.0-g sample of bleach would therefore contain 7.4 g of NaOCl.
Example 1
Calculation of Percent by Mass A 5.0-g sample of spinal fluid contains 3.75 mg (0.00375 g) of glucose. What is the percent by mass of glucose in spinal fluid?
Solution The spinal fluid sample contains roughly 4 mg of glucose in 5000 mg of fluid, so the mass fraction of glucose should be a bit less than one part in 1000, or about 0.1%. Substituting the given masses into the equation defining mass percentage yields:
The computed mass percentage agrees with our rough estimate (it’s a bit less than 0.1%).
Note that while any mass unit may be used to compute a mass percentage (mg, g, kg, oz, and so on), the same unit must be used for both the solute and the solution so that the mass units cancel, yielding a dimensionless ratio. In this case, we converted the units of solute in the numerator from mg to g to match the units in the denominator. We could just as easily have converted the denominator from g to mg instead. As long as identical mass units are used for both solute and solution, the computed mass percentage will be correct.
Check Your Learning A bottle of a tile cleanser contains 135 g of HCl and 775 g of water. What is the percent by mass of HCl in this cleanser?
Answer:
14.8%
Example 2
Calculations using Mass Percentage “Concentrated” hydrochloric acid is an aqueous solution of 37.2% HCl that is commonly used as a laboratory reagent. The density of this solution is 1.19 g/mL. What mass of HCl is contained in 0.500 L of this solution?
Solution The HCl concentration is near 40%, so a 100-g portion of this solution would contain about 40 g of HCl. Since the solution density isn’t greatly different from that of water (1 g/mL), a reasonable estimate of the HCl mass in 500 g (0.5 L) of the solution is about five times greater than that in a 100 g portion, or 5 ×× 40 = 200 g. In order to derive the mass of solute in a solution from its mass percentage, we need to know the corresponding mass of the solution. Using the solution density given, we can convert the solution’s volume to mass, and then use the given mass percentage to calculate the solute mass. This mathematical approach is outlined in this flowchart:
For proper unit cancellation, the 0.500-L volume is converted into 500 mL, and the mass percentage is expressed as a ratio, 37.2 g HCl/g solution:
This mass of HCl is consistent with our rough estimate of approximately 200 g.
Check Your Learning What volume of concentrated HCl solution contains 125 g of HCl?
Answer:
282 mL
Volume Percentage
Liquid volumes over a wide range of magnitudes are conveniently measured using common and relatively inexpensive laboratory equipment. The concentration of a solution formed by dissolving a liquid solute in a liquid solvent is therefore often expressed as a volume percentage, %vol or (v/v)%:
Example 3
Calculations using Volume Percentage Rubbing alcohol (isopropanol) is usually sold as a 70%vol aqueous solution. If the density of isopropyl alcohol is 0.785 g/mL, how many grams of isopropyl alcohol are present in a 355 mL bottle of rubbing alcohol?
Solution Per the definition of volume percentage, the isopropanol volume is 70% of the total solution volume. Multiplying the isopropanol volume by its density yields the requested mass:
Check Your Learning Wine is approximately 12% ethanol (CH3CH2OH) by volume. Ethanol has a molar mass of 46.06 g/mol and a density 0.789 g/mL. How many moles of ethanol are present in a 750-mL bottle of wine?
Answer:
1.5 mol ethanol
Mass-Volume Percentage
“Mixed” percentage units, derived from the mass of solute and the volume of solution, are popular for certain biochemical and medical applications. A mass-volume percent is a ratio of a solute’s mass to the solution’s volume expressed as a percentage. The specific units used for solute mass and solution volume may vary, depending on the solution. For example, physiological saline solution, used to prepare intravenous fluids, has a concentration of 0.9% mass/volume (m/v), indicating that the composition is 0.9 g of solute per 100 mL of solution. The concentration of glucose in blood (commonly referred to as “blood sugar”) is also typically expressed in terms of a mass-volume ratio. Though not expressed explicitly as a percentage, its concentration is usually given in milligrams of glucose per deciliter (100 mL) of blood (Figure 2).
Parts per Million and Parts per Billion
Very low solute concentrations are often expressed using appropriately small units such as parts per million (ppm) or parts per billion (ppb). Like percentage (“part per hundred”) units, ppm and ppb may be defined in terms of masses, volumes, or mixed mass-volume units. There are also ppm and ppb units defined with respect to numbers of atoms and molecules.
The mass-based definitions of ppm and ppb are given here:
Both ppm and ppb are convenient units for reporting the concentrations of pollutants and other trace contaminants in water. Concentrations of these contaminants are typically very low in treated and natural waters, and their levels cannot exceed relatively low concentration thresholds without causing adverse effects on health and wildlife. For example, the EPA has identified the maximum safe level of fluoride ion in tap water to be 4 ppm. Inline water filters are designed to reduce the concentration of fluoride and several other trace-level contaminants in tap water (Figure 3).
Example 4
Calculation of Parts per Million and Parts per Billion Concentrations According to the EPA, when the concentration of lead in tap water reaches 15 ppb, certain remedial actions must be taken. What is this concentration in ppm? At this concentration, what mass of lead (μg) would be contained in a typical glass of water (300 mL)?
Solution The definitions of the ppm and ppb units may be used to convert the given concentration from ppb to ppm. Comparing these two unit definitions shows that ppm is 1000 times greater than ppb (1 ppm = 103 ppb). Thus:
The definition of the ppb unit may be used to calculate the requested mass if the mass of the solution is provided. However, only the volume of solution (300 mL) is given, so we must use the density to derive the corresponding mass. We can assume the density of tap water to be roughly the same as that of pure water (~1.00 g/mL), since the concentrations of any dissolved substances should not be very large. Rearranging the equation defining the ppb unit and substituting the given quantities yields:
Finally, convert this mass to the requested unit of micrograms:
Check Your Learning A 50.0-g sample of industrial wastewater was determined to contain 0.48 mg of mercury. Express the mercury concentration of the wastewater in ppm and ppb units.
Answer:
9.6 ppm, 9600 ppb
Key Concepts and Summary
In addition to molarity, a number of other solution concentration units are used in various applications. Percentage concentrations based on the solution components’ masses, volumes, or both are useful for expressing relatively high concentrations, whereas lower concentrations are conveniently expressed using ppm or ppb units. These units are popular in environmental, medical, and other fields where mole-based units such as molarity are not as commonly used.
Key Equations
Chemistry End of Chapter Exercises
Consider this question: What mass of a concentrated solution of nitric acid (68.0% HNO3 by mass) is needed to prepare 400.0 g of a 10.0% solution of HNO3 by mass?
(a) Outline the steps necessary to answer the question.
(b) Answer the question.
2. What mass of a 4.00% NaOH solution by mass contains 15.0 g of NaOH?
3. What mass of solid NaOH (97.0% NaOH by mass) is required to prepare 1.00 L of a 10.0% solution of NaOH by mass? The density of the 10.0% solution is 1.109 g/mL.
4. What mass of HCl is contained in 45.0 mL of an aqueous HCl solution that has a density of 1.19 g cm–3 and contains 37.21% HCl by mass?
5. The hardness of water (hardness count) is usually expressed in parts per million (by mass) of CaCO3, which is equivalent to milligrams of CaCO3 per liter of water. What is the molar concentration of Ca2+ ions in a water sample with a hardness count of 175 mg CaCO3/L?
6. The level of mercury in a stream was suspected to be above the minimum considered safe (1 part per billion by weight). An analysis indicated that the concentration was 0.68 parts per billion. Assume a density of 1.0 g/mL and calculate the molarity of mercury in the stream.
7. In Canada and the United Kingdom, devices that measure blood glucose levels provide a reading in millimoles per liter. If a measurement of 5.3 mM is observed, what is the concentration of glucose (C6H12O6) in mg/dL?
8. A throat spray is 1.40% by mass phenol, C6H5OH, in water. If the solution has a density of 0.9956 g/mL, calculate the molarity of the solution.
9. Copper(I) iodide (CuI) is often added to table salt as a dietary source of iodine. How many moles of CuI are contained in 1.00 lb (454 g) of table salt containing 0.0100% CuI by mass?
10. A cough syrup contains 5.0% ethyl alcohol, C2H5OH, by mass. If the density of the solution is 0.9928 g/mL, determine the molarity of the alcohol in the cough syrup.
11. D5W is a solution used as an intravenous fluid. It is a 5.0% by mass solution of dextrose (C6H12O6) in water. If the density of D5W is 1.029 g/mL, calculate the molarity of dextrose in the solution.
12. Find the molarity of a 40.0% by mass aqueous solution of sulfuric acid, H2SO4, for which the density is 1.3057 g/mL.
Glossary
mass percentage
: ratio of solute-to-solution mass expressed as a percentage
mass-volume percent
: ratio of solute mass to solution volume, expressed as a percentage
parts per billion (ppb)
: ratio of solute-to-solution mass multiplied by 109
parts per million (ppm)
: ratio of solute-to-solution mass multiplied by 106
volume percentage
: ratio of solute-to-solution volume expressed as a percentage
Solutions
Answers to Chemistry End of Chapter Exercises
(a) The dilution equation can be used, appropriately modified to accommodate mass-based concentration units:
This equation can be rearranged to isolate mass1 and the given quantities substituted into this equation. (b) 58.8 g
114 g
1.75 × 10−3M
95 mg/dL
2.38 × 10−4 mol
0.29 mol
License
3.4 Other Units for Solution Concentrations Copyright © 2016 by Rice University. All Rights Reserved.
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190490 | https://www.worldatlas.com/ancient-world/ancient-trade-routes-that-connected-the-world.html | Ancient Trade Routes That Connected the World
Ancient empires did not survive in isolation. Every prosperous nation relies on trade—trade in food, services, goods, workers, and other resources. While these commodities can travel the globe in hours today, getting them across the world in ancient times was much more time-consuming and hazardous.
Lack of infrastructure, treacherous terrain, fear of ambush, and unfavorable climates were just some of the many hazards that traders of the time faced. To ensure steady and safe supply routes, resourceful and wealthy empires constructed their own paths. The most successful of these ancient arteries changed the development of civilization and continues to echo through history. Below, we look at the impactful trade routes that connected and shaped the world.
The Silk Road
Although the Silk Road has become a common historical legend, this famous path wasn’t a single road, nor did it just trade silk. In reality, it was a series of routes weaving between East Asia and Europe to transport many different goods, as well as ideas, cultural beliefs, and knowledge.
The history of the Silk Roads goes back to 139 BC when General Zhang Qian of China was sent West on a diplomatic mission. While China’s enemies captured Qian on that first mission, he brought back such detailed reports that the Emperor Han sent him on the road again in 119 BC. The Han dynasty began trading along this established pathway, setting in motion a link between Europe and Asia that would continue for more than 1,500 years.
At some point in the 1st century BC, silk was introduced to Rome and quickly became one of the most popular commodities on the route, eventually giving it its name. In addition to silk, traders carried precious stones, spices, tea, textiles, horses, and other valuable commodities. Not every trader made the entire journey; some stopped at the many cities and ports that sprang up along the way, and a few travelled alone, forming merchant caravans for safety.
As the Silk Roads developed, they spread into a vast network of highways and maritime routes from China to the Indian subcontinent, the Iranian Plateau, the Caucasus, Turkey, North Africa, Russia, and Eastern Europe. The routes allowed trade to flow and cultures to mingle, producing vibrant multicultural cities, new technologies, and the spread of world religions. Today, parts of the Silk Roads are preserved as UNESCO World Heritage Sites. These include ancient trading towns such as Samarkand in Uzbekistan and Asia's 5,000 km Tian-shan corridor.
The Incense Route
Incense was a precious and popular commodity in the ancient world. It was used to anoint kings, perform religious rituals, heal, and cleanse. It was one of the most sought-after goods, making it one of the most lucrative. The Arabian Kingdoms that produced, traded, and transported fragrant resins like frankincense and myrrh gained significant wealth and influence thanks to the Incense Route.
Also known as the Ancient Caravan Route, this trading pathway is one of the oldest and longest in the world. It operated from the second millennium BC into the sixth century and covered over 2,000 km from the Arabian Peninsula to the Mediterranean. Around 3,000 tons of incense passed through the Ancient Caravan Route each year, with camels carrying the traders and their precious cargo to their destination. According to Roman scholar Pliny the Elder, the journey took 62 days to complete.
The route facilitated substantial advancements in infrastructure across the inhospitable desert. Not only did a serviceable road need to be built along the way, but also fortified cities to shelter and service travel-weary traders. Many trading hubs are now protected as UNESCO World Heritage Sites, recognized for their historical, architectural, and engineering value.
The Amber Road
Much like the Silk Road, the Amber Road wasn’t a single highway but rather a loose network of routes transporting amber from northern Europe to the Mediterranean. Active from the Bronze and Iron Ages right through to the era of the Roman Empire, these amber roads connected the Celtic and Germanic people in northern Europe with peoples to the south at a time when amber was a highly prized commodity.
Amber is the fossilized resin of coniferous trees, which were plentiful along the shores of the Baltic Sea at the time. Transported south, it was used by prehistoric societies and later the Romans for jewellery, ornaments, decoration, and other high-value items. Its glowing hue and rarity gave amber a special status in ancient life. The gem often appears in Roman mythology and legends as having healing and/or mystical properties.
In Roman times, the major amber highways ran from the Baltic Sea through Austria, Hungary, and Slovenia before reaching the Roman port city of Aquileia, where it was processed and made into jewellery worn by wealthy Romans across the empire. Today, tourists can visit the National Archaeological Museum of Aquileia to see an extensive collection of ancient amber artifacts, including rings, statues, and other pieces.
The Spice Routes
Also known as the Maritime Silk Roads, the Spice Routes turned the world’s oceans and seas into a thriving international trade highway. In operation as early as 2,000 BC, these seafaring routes were a vital bridge between East and West, bringing together vastly different societies to share produce, goods, and cultures.
The Spice Routes covered an area of over 15,000 km, running from Japan, Indonesia, and India through the Middle East and into Europe. Busy ports developed along the way, changing the geopolitical map and increasing the influence and wealth of ancient coastal cities. The trade routes take their name from the most profitable commodity that travelled them, spices. Exotic condiments such as cinnamon from Sri Lanka and cassia from China were in high demand and priced high. Spices were not just used to flavor food; they were valued for their medicinal qualities and used in many healing and religious rituals. Other goods that flowed along the spice routes included ivory, silk, metals, and gemstones.
Sailors took a significant risk travelling the spice routes, transporting precious cargo at the mercy of the weather and marauding pirates. For many traders, it was an acceptable pay-off thanks to the high prices such exotic goods commanded in Western markets.
The Tin Route
Researchers recently uncovered exciting new evidence of one of the oldest trade routes in the world: The Tin Route. The first commodity to be traded across the European continent, tin was vital to Bronze Age peoples who combined it with copper alloy to make bronze tools and weapons. The richest tin deposits were in Cornwall and Devon in southern England, so you may ask how European Bronze Age people acquired their tin.
Archaeologists and historians have hotly debated this question over the years, but now a new study from Durham University’s Department of Archaeology sheds new light on the ancient tin trade. Researchers looked at tin ingots recovered from four Mediterranean shipwrecks—three off the coast of Israel, dating back to 1300 BC, and one off the coast of France from around 600 BC. Looking at its composition and chemistry, the scientists were able to trace the tin from the wrecks back to southeastern England, indicating that the ships had been part of a trade network carrying the metal from Britain to the continent.
This important discovery is credible evidence for a maritime tin route, but researchers continue to fill in the gaps around how the tin was transported overland. Historical records and archeological discoveries show that by 1,300 BC, almost all of Europe and the Mediterranean had access to bronze, hinting at a thriving ancient trade route that swept from Britain’s Jurassic Coast to Continental Europe.
Connecting Civilizations
With new information coming to light about Europe’s Tin Route, it’s clear that there’s still much to learn about how ancient societies traded and transported goods. What is clear is that without these busy commercial connections, our present world would look profoundly different. Whether introducing Eastern markets to Western spices, fostering the development of multi-cultural port cities, or determining the future of empires, trade routes were about much more than profit. As archeologists and historians continue to unearth their secrets, these connections also build a bridge between the past and the future.
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Catherine Morris September 20 2025 in History
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190491 | https://www.quora.com/What-is-a-method-to-determine-the-number-of-digits-in-the-integer-K-from-log-K | Something went wrong. Wait a moment and try again.
Single Digits
Logarithmic Functions
Positive Integers
Functions (mathematics)
Number Theory
Last Digit
Decimal Digits
5
What is a method to determine the number of digits in the integer K from log(K)?
Matt Laine
Senior Software Engineer
·
Author has 3.8K answers and 22.6M answer views
·
5y
Don’t change the bloody question. It originally said LG K rather than log(K).
If you’re working in binary, the answer is LG k rounded up to the nearest whole number.
John K WilliamsSon
Accredited (MS Educ) nerd who loves talking about math
·
Author has 9K answers and 23.2M answer views
·
5y
I have often used the common logarithm, the base 10 logarithm often written log(k) to determine the number of digits in a positive integer. This was especially true in the early days of DOS BASIC programming where we did not yet have string length functions.
calculate log(K)
truncate (get rid of the decimal part of) the log value
add one
the result is the number of digits in that base 1
I have often used the common logarithm, the base 10 logarithm often written log(k) to determine the number of digits in a positive integer. This was especially true in the early days of DOS BASIC programming where we did not yet have string length functions.
calculate log(K)
truncate (get rid of the decimal part of) the log value
add one
the result is the number of digits in that base 10 number.
Here are a few samples:
Image Source: TI-84+ CE graphing calculator
NOTE: If you will be buying a TI-84 because your professor uses a...
Assistant
Bot
·
1y
To determine the number of digits d in an integer K using logarithms, you can use the following formula:
d=⌊log10(K)⌋+1
Explanation:
Logarithm Base 10: The logarithm base 10 of a number K gives you the power to which 10 must be raised to obtain K. For example, if log10(K)=n, then K is approximately 10n.
Floor Function: The floor function ⌊x⌋ gives the greatest integer less than or equal to x. This is used to find the integer part of log10(K).
Adding 1: Since the number of digits is one more than the integer part of the logarithm, you add 1 to the resul
To determine the number of digits d in an integer K using logarithms, you can use the following formula:
d=⌊log10(K)⌋+1
Explanation:
Logarithm Base 10: The logarithm base 10 of a number K gives you the power to which 10 must be raised to obtain K. For example, if log10(K)=n, then K is approximately 10n.
Floor Function: The floor function ⌊x⌋ gives the greatest integer less than or equal to x. This is used to find the integer part of log10(K).
Adding 1: Since the number of digits is one more than the integer part of the logarithm, you add 1 to the result of the floor function.
Example:
If K=1000:
Calculate log10(1000):
log10(1000)=3
Apply the formula:
d=⌊3⌋+1=3+1=4
So, K=1000 has 4 digits.
Note:
For K=0 or negative integers, the concept of digits does not apply in the same way, since digits are defined for positive integers.
Related questions
What is the digit 7 in 1,876?
What is the last two digit method for determining if an integer has been squared before?
What are all digits of
π
?
What is the method for determining if a number is not an integer without rounding it?
What is
π
100,000 digits?
Nick Sugden
Lives in The United Kingdom (1965–present)
·
Author has 157 answers and 101.7K answer views
·
5y
To get the number of digits in any base take the log of the number, divide by the log of the base you want and round up to the next whole number. It doesn’t matter whether you use natural logarithms or base 10 logarithms.
log(k)/log(base)
e.g. 9 in binary (1001) is log(9)/log(2) = 3.169 rounded up is 4 digits.
Note the case where the answer is a whole number - you still round it up to the next one.
e.g. 8 in binary is log(8)/log(2) = 3 rounds up to 4 (8 is 1000 in binary)
10000 in base 10 is log(10000)/log(10) = 4 rounds up to 5.
9999 in base 10 is log(9999)/log(10) = 3.99996 rounds up to 4
Henk Verhelle
Master in Theoretical Computer Science, Ghent University (Graduated 1998)
·
Author has 429 answers and 491.3K answer views
·
4y
Related
How do you find the number of digits in a number using a log?
log10(10) = 1
log10(100) = 2
=> If there are k digits in a number n, then floor(log10(n)) + 1 = k
and log10(x) = ln(x) / ln(10) if we work with natural logarithm (base e)
=> number of digits in n = floor( ln(n)/ln(10) ) + 1
with floor(x) is rounding down to the integer smaller or equal than x
Joe Wezorek
30 years of professional software engineering
·
Author has 1.9K answers and 5.8M answer views
·
7mo
Related
How do you use the logarithm function to count the digits in a number in C++?
Well, conceptually the log base-10 of
n
rounded up basically is the number of digits in
n
, but there are edge case issues with powers of 10 themselves. Since their base-10 logs are exact integers, “rounding up” doesn’t do anything. Therefore, it’s less error prone to add one to the log base-10 rounded down. Conveniently casting to an integer in C-like languages will round down:
int num_digits_using_log(uint64_t n) { return static_cast<int>(std::log10(n)) + 1;}
But, generally speaking, it is a bad idea performance-wise to do floating point operations when you are dealing with integers, unless
Well, conceptually the log base-10 of n rounded up basically is the number of digits in n, but there are edge case issues with powers of 10 themselves. Since their base-10 logs are exact integers, “rounding up” doesn’t do anything. Therefore, it’s less error prone to add one to the log base-10 rounded down. Conveniently casting to an integer in C-like languages will round down:
int num_digits_using_log(uint64_t n) { return static_cast<int>(std::log10(n)) + 1;}
But, generally speaking, it is a bad idea performance-wise to do floating point operations when you are dealing with integers, unless you really can’t avoid it. So the all integer version is likely faster:
int num_digits(uint64_t n) { int count = 0; while (n) { ++count; n /= 10; } return count;}
And indeed the following program confirms this. I’m adding the digits and printing out the sum so the compiler doesn’t optimize away what we are trying to time:
```
include #include #include #include namespace r = std::ranges;namespace rv = std::ranges::views; int num_digits_using_log(uint64_t n) { return static_cast(std::log10(n)) + 1;} int num_digits(uint64_t n) { int count = 0; while (n) { ++count; n /= 10; } return count;} int main() { using namespace std::chrono; // Timing for num_digits_using_log auto start_1 = high_resolution_clock::now(); auto sum_of_digits_1 = r::fold_left( rv::iota(1ll, 1000000000ll) | rv::transform(num_digits_using_log), 0, std::plus<>() ); auto end_1 = high_resolution_clock::now(); auto duration_1 = duration_cast( end_1 - start_1 ).count(); std::println("sum (float) => {} , {} ms", sum_of_digits_1, duration_1); // Timing for num_digits auto start_2 = high_resolution_clock::now(); auto sum_of_digits_2 = r::fold_left( rv::iota(1ll, 1000000000ll) | rv::transform(num_digits), 0, std::plus<>() ); auto end_2 = high_resolution_clock::now(); auto duration_2 = duration_cast( end_2 - start_2 ).count(); std::println("sum (integer) => {} , {} ms", sum_of_digits_2, duration_2);}
```
The above yields the following on my computer:
sum (float) => 298954297 , 6014 ms
sum (integer) => 298954297 , 4692 ms
Pretty consistently the floating point version is taking about 25% to 30% more time.
Related questions
Why does reversing a one-digit integer always result in one less than the original number?
What is the method for determining if the logarithm of a given number is an integer?
What is the method for finding the number of multiples of a given number between two integers?
What is the method for calculating the first two digits of an integer?
For positive integers
n
,
how do I find the integer part of
∑
2
n
k
=
1
√
n
2
+
k
?
Maloth Rajender
6y
Related
In log k=37.2, what is the value of k?
Here the given question is
log k =37.2,than what is the answer for
value of k?
The answer is k= 1.43137928x10^16
Ans:
When this type of questions are asks first of all, We have to know the values of e power X
So,
The given question is log k= 37.2
Then k can be written as
K= e^37.2
K =1.43137928x10^16
Thank you...
Patrick Young
Author has 13K answers and 48.6M answer views
·
6y
Related
What is the value of k if log k =.8?
You didn’t specify your base, which is important. Most people use log for a base of 10 and ln for a base of e. But… some people also use log for a base of e.
—————-
If it’s log base 10, then log k =0.8 can be rewritten as k=10^0.8 by raising both sides of the equation by a power of 10. The value of k would then be 6.31.
If you’re using natural logs, you’d have a different base. You’d then have k=e^0.8 after taking the inverse log of both sides and you’d get k=2.23.
David Smith
BSc (Hons) in Mathematics & Computer Science, University of Bristol (Graduated 1986)
·
Upvoted by
Jeremy Collins
, M.A. Mathematics, Trinity College, Cambridge · Author has 3.6K answers and 4M answer views
·
5y
Related
How do I find the first three digits of N^K, where N is less then 10^9 and K is less then 10^7?
This sounds like an old question designed to use log tables!
Look up log10N, multiply it by K then look up the anti-log (i.e. 10x).
Example: calculate the first three digits of 23.763124.7.
Look up log1023.76 to find ≈1.3758.
Multiply by 4.7 to give ≈6.4663.
Look up the anti-log, 106.4663≈2.926⋅106.
The first three digits of 23.763124.7 are 292.
You have to know how to use log tables because they usually only tabulate logarithms between 0 and 1 (i.e. from log101 to log1010) and leave you to handle powers of 10.
If you’re not using tables
This sounds like an old question designed to use log tables!
Look up log10N, multiply it by K then look up the anti-log (i.e. 10x).
Example: calculate the first three digits of 23.763124.7.
Look up log1023.76 to find ≈1.3758.
Multiply by 4.7 to give ≈6.4663.
Look up the anti-log, 106.4663≈2.926⋅106.
The first three digits of 23.763124.7 are 292.
You have to know how to use log tables because they usually only tabulate logarithms between 0 and 1 (i.e. from log101 to log1010) and leave you to handle powers of 10.
If you’re not using tables then presumably you have a calculator which makes life much simpler.
[If you’re really only interested in the first three digits of the result then after multiplying by K you can discard the digits to the left of the point, keeping only digits to the right.]
John Stephenson
Analyst programmer since the days of mainframes and magnetic tape reels.
·
Author has 2.4K answers and 2.8M answer views
·
5y
Related
How do you find the number of digits of a given integer?
Convert the number to a double floating point. Take the log to the base 10 of it, reduce it to integer only by using the floor() function or by casting it back to an integer. Add 1 to it, and you have the answer.
The following 'C' function will do it:
int numdigits(int n){ return( (int) log10((double) n) + 1);}
Let’s go a step further and make it much smarter. How about printing the length of the number were it to be printed in any base, not just 10.
int numdigits(int n, int b){ return (int) (log((double) n)/log((double) b) + 1);}
numdigits(53452,10) returns 5
but
numdigits(53452,2) returns 16. The
Convert the number to a double floating point. Take the log to the base 10 of it, reduce it to integer only by using the floor() function or by casting it back to an integer. Add 1 to it, and you have the answer.
The following 'C' function will do it:
int numdigits(int n){ return( (int) log10((double) n) + 1);}
Let’s go a step further and make it much smarter. How about printing the length of the number were it to be printed in any base, not just 10.
int numdigits(int n, int b){ return (int) (log((double) n)/log((double) b) + 1);}
numdigits(53452,10) returns 5
but
numdigits(53452,2) returns 16. The number of bits required to hold 53452.
Luqman Khan
Author has 771 answers and 777.9K answer views
·
4y
Related
What is k if logk-log(k-2) =log5?
using the log division rule (log x - log y = log(x/y))
Reverse the log, by raising the power of everything by 10
Multiply everything by (k-2) and expand brackets
using the log division rule (log x - log y = log(x/y))
Reverse the log, by raising the power of everything by 10
Multiply everything by (k-2) and expand brackets
Christopher F Clark
Mentored 12-ish software engineers, taught programming
·
Author has 10.9K answers and 35.4M answer views
·
4y
Related
How do you write a C program to find the number of digits in a number?
There are a variety of options. Are you just doing integers or are you trying to do real numbers (which computing people tend to call floating point and are usually of type “double” in C++)? And do you care only about digits before the decimal point in a real number or do you care about digits after too?
The logarithm method is good for digits before the decimal point, but doesn’t work well for those after the decimal point. In fact, figuring out the number of digits after the decimal point is really difficult, so I am going to assume that if this is a homework problem, you don’t care about tha
There are a variety of options. Are you just doing integers or are you trying to do real numbers (which computing people tend to call floating point and are usually of type “double” in C++)? And do you care only about digits before the decimal point in a real number or do you care about digits after too?
The logarithm method is good for digits before the decimal point, but doesn’t work well for those after the decimal point. In fact, figuring out the number of digits after the decimal point is really difficult, so I am going to assume that if this is a homework problem, you don’t care about that.
In fact, the most common care for this is figuring out how many digits there are in an integer. This is even true in most practical cases too. Now, assuming you don’t want to just use the logarithm, let’s see if we can’t motivate this to be a loop problem.
First, let dispense with a special case, that your number is 0. How many digits are in 0?
Let’s assume next that your number is positive. If it is negative, you can easily make it positive. (Or, if you want a more likely pedantically correct version, make all non-zero numbers negative, and fiddle with the algorithm I am about to give to make it work. Why do I suggest that? What do you know about 2s complement arithmetic?)
Ok, so now we have a positive number greater than 0, that we want to find out how many digits it has.
So, what if the number is between 1 and 9? How many digits does that have? Still easy.
But what if the number is bigger than 9. Well, then it must be at least 10. So, what happens if we divide that number by 10 (throwing away the remainder or equivalently ignoring the part after the decimal point). If the number is between 10 and 99 what do we get?
Ok, so in C, if you have to integers, and you divide them, you get exactly that result, an integer result with the remainder (parts after the decimal point) thrown away.
What happens to numbers from 100 to 999 is you do that?
Ok, do you see the pattern?
If the number is positive and less than ten, you know how many digits it has. So, that’s the base case. If it is 10 or bigger then, if we divide by 10, we get a number with one fewer digits.
Now, we can do our invariant and induction parts. I’m going to write this in pseudo code, but it will be close enough to C that you can easily fix it.
int number = xxx; // get the number from somewhere, xxxif (number < 0) number = (- number); // we only want positive numbersint digits = 0; // we think the number has at least this many digits// double invariant = log(number);while (10 <= number) { // here is our loop invariant "p" // assert(digits + log(number) == invariant); // here want approx = number = number / 10; // decrease t, amount of digits left in number digits++; // reestablish p, account for one digit removed }assert(number < 10);// now we have a number between 0 and 9 in number// and digits is how many divisions we had to do to get there// so, just return the correct value, it is a simple formula of digits
Ranjan Tarafder
M.Sc. in Mathematics in Mathematics & Master of Science in Mathematics, Science (Graduated 2019)
·
5y
Related
What is k, if log(p) /2=log(q) /4=log(r) /8=k,pqr=100?
log(p)/2=log(q)/4=log(r)/8=k
That means,
log(p)/2=k,log(q)/4=k,log(r)/8=k
logp=2k,logq=4k,logr=8k
We have,
pqr=100
Taking log both sides, we get
log(pqr)=log(100)
logp+logq+logr=log100
2k+4k+8k=100
14k=100
k=100/14
k=50/7
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Intermediate/Advanced Algebra
Vectors: Finding line thats perpendicular to 2 lines
Thread starterH.Bisho18
Start dateSep 1, 2019
H
H.Bisho18
New member
Joined Apr 3, 2019 Messages 14
Sep 1, 2019
#1
I'm stuck on 16.
I've written 2 vector equations but i'm just not sure how to get a line to pass through both of them perpendicularly
Here's my working so far but its not much, pretty stumped :/
thanks
Dr.Peterson
Elite Member
Joined Nov 12, 2017 Messages 16,844
Sep 1, 2019
#2
Write an expression for the vector from P to Q, as a function of your two variables [MATH]\lambda[/MATH] and [MATH]\mu[/MATH]. What must be true if this vector is perpendicular to the vectors you called a and b? You might think in terms of either the scalar product or the vector product.
pka
Elite Member
Joined Jan 29, 2005 Messages 11,990
Sep 1, 2019
#3
H.Bisho18 said:
View attachment 13436
I'm stuck on 16. I've written 2 vector equations but i'm just not sure how to get a line to pass through both of them perpendicularly
Click to expand...
This is the most confuted question. These are two skew lines.
ℓ(s)=(1,1,2)+s<2,1,−1>&ℓ(t)=(1,1,4)+y<1,1,−2>\displaystyle \ell(s)=(1,1,2)+s<2,1,-1>~\:\&~\;\ell(t)=(1,1,4)+y<1,1,-2>ℓ(s)=(1,1,2)+s<2,1,−1>&ℓ(t)=(1,1,4)+y<1,1,−2>
ℓ(s):x=1+2 s,y=1+s,z=2−s&ℓ(t):x=1+t y=1+t z=4−2 t\displaystyle \ell(s):\:x=1+2s,\:y=1+s,\:z=2-s~\:\&~\;\ell(t):\:x=1+t\:y=1+t\:z=4-2t ℓ(s):x=1+2 s,y=1+s,z=2−s&ℓ(t):x=1+t y=1+t z=4−2 t
to H.Bisho18, can you show that these two are skew lines?
Two skew lines have a unique common perpendicular.
That is the P Q‾\displaystyle \overline{PQ}P Q that you are asked to find.
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190493 | https://pubs.acs.org/doi/10.1021/acs.jpcc.3c02011 | Supporting Information
Role of La 5p in Bulk and Quantum-Confined Solids Probed by the La 5p 54f 1 3D1 Excitonic Final State of Resonant Inelastic X-ray Scattering Chun-Yu Liu, ∗,†,‡
Ping Feng, ¶,§
Kari Ruotsalainen, †
Karl Bauer, †
Régis Decker, †
Maximilian Kusch, †
Katarzyna E. Siewierska, † Annette Pietzsch, † Markus Haase, ∥Yan Lu, ¶,§, Frank M. F. de Groot, ⊥ and Alexander Föhlisch ∗,†,‡†Institute for Methods and Instrumentation for Synchrotron Radiation Research (PS-ISRR), Helmholtz-Zentrum Berlin für Materialien und Energie GmbH, Albert-Einstein-Str. 15, 12489 Berlin, Germany
‡Institute of Physics and Astronomy, University of Potsdam, Karl-Liebknecht-Strasse 24-25, 14476 Potsdam, Germany
¶Department for Electrochemical Energy Storage, Helmholtz-Zentrum Berlin für Materialien und Energie, Hahn-Meitner-Platz 1, 14109 Berlin, Germany
§Institute of Chemistry, University of Potsdam, Potsdam 14476, Germany
∥Department of Inorganic Chemistry, Functional Nanomaterials, University of Osnabrück, Barbarastr. 7, 49074, Osnabrück, Germany
⊥
Department of Chemistry, Princetonplein 1, 3584 CC Utrecht, Netherlands
S1
Figure S1. X-ray absorption spectrum (XAS) overview and zoom-in pre-edge multiplets 3P1, 3D 1. This study focuses on the excitation at the 3D 1.
3D1
3P 1
3D1S2
Figure S2. High-resolution resonant inelastic x-ray scattering (RIXS) spectra of benchmark samples and fixed-Lorentzian fitted result. (a) La metal, (b) La 2O 3, (c) SrLaAlO 4, (d) LaPO 4 microrod (bulk), (e) LaPO 4:Ce,Tb (quantum dot), (f) LaF 3. The fitted result is based on the least-square method. S3
Figure S3. High-resolution resonant inelastic x-ray scattering (RIXS) spectra of benchmark samples and fixed-Gaussian fitted result. (a) La metal, (b) La 2O 3, (c) SrLaAlO 4, (d) LaPO 4 microrod (bulk), (e) LaPO 4:Ce,Tb (quantum dot), (f) LaF 3, (g) LaAlO 3. The fitted result is based on the least-square method. S4
Table S1. Fixed-Lorentzian two-Voigt fitted parameters of Fig. S2. The gamma ( γ,
Lorentzian part) is set to the values from LaAlO 3 (γD1=0.103 eV, γD2=0.169 eV). The Gaussian parameters σD1 and σD2, representing combined chemical broadening factors, are coupled. The energy separation of the two peaks is fixed to 0.212 eV (with the exception of LaF 3, as free parameter). The two-Voigt peaks’ intensity ratio is fixed to 1:1. The rest parameters are set free.
Sample 3D 1 center (eV) σD1 (eV) 3D 2 center (eV) σD2 (eV)
La Metal 18.615±0.002 0.249±0.003 18.827±0.002 0.249±0.003
LaPO 4 microrod 18.721±0.003 0.221±0.005 18.933±0.003 0.221±0.005
La 2O 3 18.722±0.003 0.187±0.004 18.934±0.003 0.187±0.004
SrLaAlO 4 18.861±0.002 0.076±0.004 19.073±0.002 0.076±0.004
LaAlO 3 18.956±0.001 0.027±0.003 19.168±0.001 0.027±0.003
LaPO 4:Ce,Tb (quantum dot) 19.379±0.003 0.199±0.004 19.591±0.003 0.199±0.004
LaF 3 19.435±0.003 0.078±0.003 19.551±0.005 0.078±0.003
Table S2. Fixed-Gaussian two-Voigt fitted parameters of Fig. S3. The sigma ( σ,
Gaussian part) is fixed to the spectral resolution. The Lorentzian parameters γD1 and γD2 ,representing combined lifetime broadening factors, are allowed to vary but coupled. The energy separation of the two peaks is fixed to 0.212 eV (with the exception of LaF 3,as free parameter). The two-Voigt peaks’ intensity ratio is fixed to 1:1. The rest parameters are set free.
Sample 3D 1 center (eV) σ ( eV) γD1 (eV) 3D 2 center (eV) γD2 (eV)
La Metal 18.594±0.003 0.023 0.377±0.007 18.806±0.003 0.377±0.007
LaPO 4 microrod 18.695±0.003 0.021 0.337±0.008 18.907±0.003 0.337±0.008
La 2O 3 18.695±0.003 0.021 0.285±0.007 18.907±0.003 0.285±0.007
SrLaAlO 4 18.832±0.003 0.019 0.194±0.008 19.044±0.003 0.194±0.008
LaAlO 3 18.936±0.004 0.025 0.129±0.008 19.148±0.004 0.129±0.008
LaPO 4:Ce,Tb (quantum dot) 19.362±0.003 0.025 0.296±0.006 19.574±0.003 0.296±0.006
LaF 3 19.423±0.005 0.023 0.167±0.005 19.635±0.005 0.167±0.005 S5
Figure S4. (a)-(c) Scanning electron microscopy (SEM) image of microrod and (d) X-ray diffraction of the LaPO 4 microrod.
Materials:
Sodium dihydrogen phosphate dihydrate (NaH 2PO 4, ≥98%), lanthanum nitrate
(La(NO 3) 3∙6H 2O, 99.99%), phosphoric acid (H 3PO 4, ≥99.999%), trioctylam ine (98%) and tris(ethylhexyl) phosphate (97%) were purchased from Sigma-Aldrich.
Synthesis:
The synthesis follows the method from Wang. et al. In a typical run, 1.299 g of La(NO 3) 3∙6H 2O and 360.0 mg of NaH 2PO 4 were each dissolved in 15 ml deionized water while stirring. Then the NaH 2PO 4 solution was added dropwise to the former to make a 30 ml mixture. After stirring for 1 h, this solution was poured into a 45 ml teflon-lined stainless-steel reactor and heated in an electrical oven at 180 ℃ for 12 h. After cooling to room temperature, the product was collected by centrifugation for 300 seconds at 9000 RPM. The solids were freeze-dried afterwards. S6
Figure S5. (a)-(c) Transmission electron microscopy (TEM) images of the LaPO 430 nm nanorod and (d) X-ray diffraction of the LaPO 4 30 nm nanorod.
Synthesis:
The synthesis follows the method from Riwotzki. et al. 5 mmol of La(NO 3) 3∙ 6H 2Owas dissolved in 50 mL of tris(ethylhexyl) phosphate. This solution was then added to another 30 mL of tris(ethylhexyl) phosphate which contains 5.0 mmol of crystalline H 3PO 4 and 15 mmol of trioctylamine. The transparent liquid was degassed under vacuum and filled with nitrogen three times. After that, the solution was heated to 200 °C under nitrogen. The temperature was then maintained for 16 h. After the solution was cooled to room temperature, methanol was added, resulting in the precipitation of nanocrystals. The product was collected by centrifugation for 300 seconds at 9000 RPM. The solids were then freeze-dried. 50 mg of the dried powder was redispersed in 10 ml 2-propanol to get the colloidal solution. S7
Figure S6. TEM images of the quantum dot (4 nm LaPO 4 nanoparticles).
The sample is directly acquired from Klein. et al. and the synthesis and characterization can be found in the reference. S8
Figure S7 Diffuse reflectance UV-Vis spectra (DRS).
Fig. S7. shows the DRS raw data (in reflectance mode) of the LaPO 4 of different sizes. Only clear band-absorption behavior (drop in reflectance) is observed in the bulk-like samples (microrod and hydrate). Even though band signals from the nanoparticles are obscured by the background which originates from the baseline correction, calibration and scattering coefficient’s dependence on particle size, no clear drop in reflectance indicates no strong band absorption is present in the energy region we scan. (The reflectance rise close to 6 eV is assigned to artifact, which could potentially come from the absorption of the reference material, absorption and scattering coefficient’s dependence on wavelength and particle size, etc.). Dedicated experimental design and data analysis will be required to get bias-free DRS spectra. DRS can be theoretically described by the Kubelka-Munk Model:
𝐹𝐹 (𝑅𝑅 ) = 𝐾𝐾 𝑆𝑆 = (1 − 𝑅𝑅 ) 2
2𝑅𝑅
Several error-contributing factors are listed below:
LaPO 4 possesses slight luminescence, e.g. when excitation is at 251nm(~4.9eV)
Scattering coefficient depends on particle size: smaller particle size will lead to lower reflectance.
Absorption coefficient is dependent on incoming photon wavelength.
The signal is prone to artifacts due to BaSO 4 reference and baseline correction (especially at high energy regions).
Illuminated volume, especially nanoparticles, is hard to define for normalization.
Sample thickness plays a role, generally large sample amount is required.
General commercial UV-Vis instrument requires dedicated blank reference and source normalization at high energy side (> 5 eV) and doesn’t cover higher energy range where VUV is required (> 6 eV).
F(R) = Kubelka-Munk function
K = absorption coefficient
S = scattering coefficient
R = measured reflectance S9
Generally, DRS is rather a technique suitable for well-established semiconductors such as TiO 2 nanoparticles and is not an ideal tool for wide-gap insulators.
References:
K. Wang, W. Yao, F. Teng, and Y. Zhu, RSC Adv. 5, 56711 (2015).
K. Riwotzki, H. Meyssamy, A. Kornowski, and M. Haase, J. Phys. Chem. B 104, 2824 (2000).
J. Klein, S. M. Beladi-Mousavi, M. Schleutker, D. H. Taffa, M. Haase, and L. Walder, Adv. Opt. Mater. 9, 1 (2021).
S. Landi, I. R. Segundo, E. Freitas, M. Vasilevskiy, J. Carneiro, and C. J. Tavares, Solid State Commun. 341, 114573 (2022). |
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P:02
J
ANTONIE VAN LEEUWENHOEK
Born: 24.10.1632 - Died: 26.8.1723
Delft, Holland
I .J
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P:04
Paniker's Textbook of
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Paniker's Textbook of
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nor the author(s)/editor(s) assume any liability for any injury and/or damage to persons or property arising from or related to use of material in
this book.
This book is sold on the understanding that the publisher is not engaged in providing professional medical services. If such advice or services are
required, the services of a competent medical professional should be sought.
Every effort has been made where necessary to contact holders of copyright to obtain permission to reproduce copyright material. If any have
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Paniker's Textbook of Medical Parasitology
First Edition: 1988
Second Edition: 1989, Reprint: 1991
Third Edition: 1993
Fourth Edition: 1997, Reprint: 1999
Fifth Edition: 2002, Reprint: 2003, 2004
Sixth Edition: 2007, Reprint: 2011
Seventh Edition: 2013
Eighth Edition: 2018
ISBN: 978-93-5270-186-5
Printed at: Ajanta Offset & Packagings Ltd., Faridabad, Haryana.
P:08
FOREWORD
This is a great pleasure to write the foreword to the eighth edition of Paniker's Textbook of Medical
Parasitology dealing with medically important parasites vis-a-vis human diseases caused by them.
The parasitic infections (protozoa! and helminthic) are still major cause of high morbidity as well
as mortality of substantial number of population residing in the developing world of tropical and
subtropical regions. The clinical presentations of parasitic diseases have also significantly evolved
with the passage of time. Malaria caused by Plasmodium vivax has never been life-threatening but
now it is presenting with renal failure as well as acute respiratory distress syndrome (ARDS) thereby
leading to fatal consequences. On the other hand, some of the infections such as dracunculiasis
have been eradicated from India and others are the next targets being in the pipeline.
There are a number of novel diagnostic techniques, which are being designed for rapid
diagnosis of various parasitic diseases and accurate identification of their causative pathogens. The non-invasive imaging
techniques, both MRI and CT scans, are proving to be very useful tools for an early diagnosis thereby delineating the extent
of disease in a particular patient. Therefore, to cope up with the changing epidemiological scenario and newer diagnostic
modalities, medical students and professionals involved in the patient care need updates from time to time. Dr Sougata
Ghosh (Editor), has done a remarkable job of going through the voluminous information and presenting it in a very lucid,
concise and reproducible manner.
This edition will ideally be suited for medical students and resident doctors, who are preparing for various examinations
and entrance tests. I feel the present edition will also be appreciated by students and teaching faculties in all disciplines
of medicine. The chapter on pneumocystosis has been removed, however, on sporozoa dealing with diseases caused by
different species of microsporidia, traditionally retained in this edition, despite the fact that it has also been shifted now to
the kingdom fungi like Pneumocystis jirovecii.
The unique feature of the textbook is that it has many illustrations, photographs of cl inical specimens and
photomicrographs with an easy-to-read and understand format. This will help the students to memorize the information
given in the text easily as well as to use the same in medical practice. Each chapter has key points with a set of multiple choice
questions (MCQs), which will help a student for better understanding and preparation before the examination. Although it
is meant for medical graduates, recent advances mentioned in this book will also be useful for the postgraduates.
The original author, Professor CK Jayaram Paniker, was an experienced and enthusiastic medical teacher, and we
recently lost him. Moreover, he was a legendary microbiologist and the author of numerous valuable textbooks, part icularly
co-author of Ananthanarayan's Textbook of Microbiology. His name has been retained as such in the title of the eighth
edition of this textbook is a great honor and real tribute to him thereby continuing his legacy to attain more heights in the
field of medical parasitology even in his physical absence. I hope that this textbook will continue to benefit the medical
students and faculties for many years as it has done during the last three decades.
Jagdish Chander
Professor and Head
Department of Microbiology
Government Medical College and Hospital
Chandigarh, India
P:10
PREFACE TO THE EIGHTH EDITION
The previous editions of Paniker's Textbook of Medical Parasitology have been widely accepted by the medical students and
teachers across India and abroad for almost three decades.
Medical science is not a static art. Methods of diagnosis and treatment of parasitic infections change constantly. To keep
pace with these developments, all the chapters of present edition have been thoroughly revised and expanded, providing
up-to-date epidemiological data, new diagnostic methods and recent treatment guidelines of parasitic infections.
In the current edition, many new tables, flow charts and photographs of specimens and microscopic view pictures have
been added for better comprehension of the subject.
Recent advances such as vaccinology of malaria and leishmaniasis, malarial drug resistance, new treatment protocols
of different parasitic infections are the salient features of the book.
The aim of the contents of the book remains same in this edition, that is compact yet informative and useful for both
graduate and postgraduate students.
Like the last edition, the present edition is also designed in a colorful format, which can be easily read and comprehended.
Important points and terms have been highlighted by making them bold and italic. At the end of each chapter, the
must-know facts are given as \"Key Points\" in box formats for quick recapitulation.
Important multiple choice questions (MCQs) and review questions from various university examinations' papers have
been added to test and reinforce understanding of the topics by the students.
Sougata Ghosh
P:12
PREFACE TO THE FIRST EDITION
Parasitic infections continue to account for a large part of human illness. Antimicrobial drugs and vaccines that have made
possible the effective control of most bacterial and viral diseases have not been as successful against parasitic infections.
The numbers of persons afflicted by parasites run into many millions. Malaria still affects over 500 millions, pinworm
and whipworm 500 millions each, hookworm 800 millions and roundworm a billion persons. Filariasis, leishmaniasis and
schistosomiasis remain serious public health problems. Infections due to opportunist parasites are becoming increasingly
evident in the affluent countries.
In recent years, there has been a resurgence in the study of parasitic infections. Much new knowledge has been gained
making possible precise diagnosis and more effective control of parasites and the diseases, they cause.
Thi s textbook attempts to present the essential information on parasites and parasitic diseases, with emphasis on
pathogenesis, epidemiology, diagnosis and control. Every effort has been made to incorporate recent advances in the
subject.
It is hoped that medical student s, teachers and physicians will find the book useful. Their comments and suggestions
for improvement of the book will be most welcome.
SHANTHI, East Hill Road
Kozhikode, Kerala-673 006
CK Jayaram Paniker
P:14
ACKNOWLEDGMENTS
I gratefully acknowledge the help of the Principal, Government Medical College, Kolkata; Director, Calcutta School of
Tropical Medicine, Kolkata, West Bengal, India; and all my developmental colleagues for their valuable suggestions.
Lastly, I want to thank my parents, wife and my son Anindya Ghosh, for their emotional support, whenever I needed
during preparations of the manuscript.
I solicit the comments and suggestions for the faculties and students for improvement of the book and many be
e-mailed to s_ghosh2006@rediffmail.com
I owe my special thanks to Shri Jitendar P Vij (Group Chairman), Mr An kit Vij (Group President) and Mr Sabyasachi
Hazra (Commissioning Editor, Kolkata Branch) of M/s Jaypee Brothers Medical Publishers (P) Ltd, New Delhi, India, for their
professional help and guidance to bring out the present edition of the book.
P:16
General Introduction: Parasitology
Parasites
Host 7
Zoonosis 2
Host-parasite Relationships 2
Life Cycle of Parasites 3
Sources of Infection 3
Modes of Infection 4
Pathogenesis 4
Immunity in Parasitic Infection 5
Immune Evasion 5
Vaccination 5
Laboratory Diagnosis 6
Protozoa
General Features 10
Structure 10
Cytoplasm 10
Nucleus 10
Terminologies Used in Protozoology 10
Reproduction 11
Life Cycle 11
Classification of Protozoa 11
CONTENTS
1
10
Amebae 15
Entamoeba histolytica 15
Nonpathogenic Intestinal Ameba 24
Pathogenic Free-living Amebae 26
Intestinal, Oral and Genital Flagellates
Giardia lamblia 32
Trichomonas 36
Chilomastix mesnili 38
Enteromonas hominis 38
Retortamonas intestina/is 38
Dientamoeba fragilis 39
32
P:17
Paniker
's Textbook of Medical Parasitology
Hemoflagellates
Zoo
log
ical Classification of Flagellates 41
Gene
ral Characteristics 41
Try
panoso
mes 42
Leishmania 52
Malaria and Babesia
Malaria 66 Classification 66
Causative Agents of Human Ma
laria 66
Malaria Parasite
66
Babesia Species
86
Classification 86
History and Distribution 86
Habi
tat 86
Morpho
logy 86
Life Cycle 86 Pathogenicity and
Clinical Features
87
L
a
b
o
ratory
D
iagnosis 87
Trea
t ment 88
Prophylaxis 88
Coccidia
Toxoplasma gondii 90
/sospora be/Ii
96
Cryptosporidium parvum
97
Cyclospora cayetanensis 100
8/astocystis hominis 101
Sarcocystis 102
41
66
90
Microspora 104
Hi
st
o
ry
a
nd
D
i
stri
buti
on 104
Mo
r
p
hology 704
Life Cycle 105
Cl
inical Features 105
Laboratory Diagnosis
105
Treatment 106
Prophylax
is 706
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Contents
Balantidium coli 107
History and Distribution 107
Habitat 107
Morphology 107
Life Cycle 108
Pathogenesis 108
Clinical Features 109
Laboratory Diagnosis 709
Treatment 709
Prophylaxis 709
Helminths: General Features
Phylum Platyhelminthes 7 71
Phylum Nemathelminthes (Nematoda) 7 72
Important Features of Helminths 112
Zoological Classification of Helminths 7 7 3
111
Cestodes: Tapeworms 115
Classification of Cestodes 115
Tapeworms: General Characteristics 115
Pseudophyllidean Tapeworms 117
Cyclophyllidean Tapeworms 122
Trematodes: Flukes 141
Classification ofTrematodes 141
Flukes: General Characteristics 747
Life Cycle 142
Blood Flukes 743
Hermaphroditic Flukes: Liver Flukes 150
Intestinal Flukes 156
Lung Flukes 760
Nematodes: General Features
General Characteristics 764
Life Cycle 764
Modes of Infection 765
Classification 765
Larva Migrans 765
164
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Paniker's Textbook of Medical Parasitology
Trichinella spiralis
Common Name 170
History and Distribution 170
Habitat 770
Morphology 110
Life Cycle 7 77
Pathogenicity and Clinical Features 172
Diagnosis 172
Treatment 174
Prophylaxis 174
1 S. Trichuris trichiura
Common Name 175
History and Distribution 175
Habitat 775
Morphology 175
Life Cycle 7 76
Pathogenicity and Clinical Features 177
Laboratory Diagnosis 7 78
Treatment 178
Prophylaxis 178
Strongyloides stercoralis
History and Distribution 180
Habitat 180
Morphology 180
Life Cycle 182
Pathogenicity and Clinical Features 783
Laboratory Diagnosis 184
Treatment 185
Prophylaxis 185
170
175
180
Hookworm 187
History and Distribution 187
Ancylostoma duodenale 187
Necator americanus 189
Pathogenicity and Clinical Features of Hookworm Infection 190
Laboratory Diagnosis 797
Treatment 192
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Prophylax
is 793
Other Hookworms 193 Trichostrongyliasis 793
Enterobius vermicularis
Common Name 795 History and Distribution 795
Habitat 795
Mo
rphology 795
Life Cycle 796 Pathogenicity and
Clinical
Features 196
Laboratory Diagnosis 797
Treatment 798
Prophylaxis 7 99
Ascaris /umbricoides
Common Name 200
Hi
story and Di
stributi
on 200
H
abit
at
200
Morphology 200
Life Cycle 201
Pathogenicity and Clinical Features 203
Laboratory Diagnosis 205
Treatment 205 Prophylaxis 205
Filarial Worms
Lymphatic Filariasis 270
Subcutaneous Fi
l
a
rias
is 2 7 9
Dracunculus medinensis
Common Name 225
Histo
ry and
D
i
stribution 225
Habitat 225
Morph
o
logy 225
Life Cycle 226
Pathoge
nicity and Clinical Features 227
Laboratory Diagnosis 227
Treatment 227
Prophylax
is 229
Contents
195
200
208
225
P:21
Paniker
's Textbook of Medical Parasitology
Miscellaneous Nematodes
Angiostrongylus cantonen
sis 230
Cap
illaria philippinensis 231
Gnathos
toma spinigerum 231
Anisa
kias
is 232
Diagnostic Methods in Parasitology
Examinati
on
of
Stool 234
Examination of Blood 240 Sputum Examination 242 Urine or Body Fluids Examin
ation 243
Tissue
B
iopsy 243
M
uscle
Biopsy 243
Duode
nal Ca
p
s
ule Tec
hnique
{En
terotes
t) 243
Sigmoidosco
py Ma
t
e
rial 244
Uroge
nital Speci
m
en 244
Cul
t ure Me
thods 244
Animal
I
nocul
ation 245
Xeno
diag
nos
is 245
I
m
muno
log
ical
Diag
nos
is 246
Skin
Tests 247
Mo
lecu
l
ar Me
tho
ds 247
230
234
Index 249
J
P:22
r
CHAPTER 1
General Introduction:
• INTRODUCTION
Medical parasitology deals with the parasites, which cause
human infections and the diseases they produce.
• It is broadly divided into two parts:
Protozoology
Helminthology.
• The pioneer Du tch microscopist, Antonie 11an
Leeuwenhoek of Holland in 1681, first introduced single
lens microscope and observed Giardia in his own stools.
• Louis Pastuer in 1870, first published scientific study on
a protozoa( disease leading to its control and prevention
during investigation of an epidemic silk worm disease in
South Europe.
• A seminal discovery was made in 1878 by Patrick Manson
about the role of mosquitoes in filariasis. This was the first
evidence of vector transmission.
• Afterwards, Laveran in Algeria discovered the malarial
parasite (1880), and Ronald Ross in Secunderabad and
Calcuna in India, showed its transmission by mosquitoes
(1897). A large nwnber of vector-borne disease have since
then been identified.
• By mid 20th century, with dramatic advances in antibiotics
and chemotherapy, insecticides and antiparasitic drugs,
and improved lifestyles, all infectious diseases seemed
amenable to control.
• PARASITES
Parasites are living organisms, which depend on a living host
for their nourishment and survival. They mulriply or undergo
development in the host.
• The term \"parasite\" is usually applied to Protozoa
(unicellular organisms) and Helminths (multicellular
organisms) (Flow chart 1).
• Parasites can also be classified as:
Ectoparasite: Ectoparasites inhabit only the body
surface of the host without penetrating the tissue. Lice,
ticks and mites are examples of ectoparasites. lhe
Parasitology
term infestation is often employed for parasitization
with ectoparasites.
Endoparasite: A parasite, which lives within the
body of the host and is said to cause an infection is
called an endoparasite. Most of the protozoan and
helminthic parasites causing human disease are
endoparasites.
Free-living parasite: It refe rs to nonparasitic stages
of active existence, which live independent of the
host, e.g. cystic stage of Naegleriafowleri.
Endoparasites can further be classified as:
Obligate parasite: The parasite, wh ich cannot
exist without a host, e.g. Toxoplasma gondii and
Plasmodium.
Facultative para.site: Organism which may either
live as parasitic form or as fre e-living form, e.g.
Naegleriafowleri.
Accidental parasites: Parasites, which infect an
unusual host are known as accidental parasites.
Echinococcus granulosus infects man accidentally,
giving rise to hydatid cysts.
Aberrant parasites: Parasites, which infect a host
where they cannot develop further are known as
aberrant or wandering parasites, e.g. Toxocara canis
(dog roundworm) infecting lhwnans.
• HOST
Host is defined as an organism, which harbors the parasite
and provides nourishment and shelter to latter and is
relatively larger than the parasite.
• The host may be of the following types:
Definitive host: The host, in which the adult parasite
lives and undergoes sexual reproduction is called the
definitive host, e.g. mosquito acts as definitive host
in malaria.
The definitive host may be a human or any other
living being. However, in majority of human parasitic
infections, man is the definitive host (e.g. fil aria,
roundworm, hookworm).
P:23
Paniker's Textbook of Medical Parasitology
Flow chart 1: Types of parasites
Parasite
Protozoa
(unicellular)
Kingdom-Protista
Helminths
(multicellular)
Kingdom-Animalia
Amebae Flagellates Sporozoa Ciliates
•
En/amoeba
Naeglena
Giardia
Trichomonas
Plasmodium
Babesia
Toxoplasma
Balanlidium
Nematodes
•
Ascaris
Ancylostoma
Intermediate host: The host, in which the larval
stage of the parasite lives or asexual multiplication
takes place is called the intermediate host. In some
parasites, two different intermediate hosts may be
required to compl ete different larval stages. These
are known as first and second intermediate hosts,
respectively (Box 1).
Paratenic host: A host, in which larval stage of the
parasite rema ins viable without furth er development
is referred as a paratenic host. Such host transmits
the infection to another host, e.g. fish for plerocercoid
larva of D. lalum.
Reservoir host: In an e ndemic area, a parasitic
infection is continuously kept up by the presence
of a host, which harbors the parasite and acts as an
important source of infection to other susceptible
hosts, e.g. dog is the reservoir host of hydatid disease.
Accidental host: The host, in which the parasite is
not usually found, e.g. man is an accidental host for
cystic echinococcosis.
• ZOONOSIS
The word zoonosis was introduced by RudolfVirchow in 1880
to include the diseases shared in nature by man and animals.
• Later, in 1959, the World Health Organization (WHO)
defined wonosis as those diseases and infeclions, which are
naturally transmitted between vertebrate animals and man.
Cestodes
•
Taenia
Echinococcus
Trematodes
+
Fasciola
Schistosoma
Box 1: Parasites with man as intermediate or secondary host
• Plasmodium spp.
• Babesia spp.
• Toxoplasma gondii
• Echinococcus granulosus
• Echinococcus multilocu/aris
• Taenia solium
• Spirometra spp.
• It is of following types:
Protozoal zoo noses, e.g. toxoplasmosis, leishmaniasis,
balanlidiasis and cryptosporidiosis.
I lelminthic zoonoses, e.g. hydatid disease, taeniasis.
Anthropozoonoses: Infectio ns tra nsm itted to
man from lower vertebrate an imals, e.g. cystic
echinococcosis.
Zooanthroponoses: Infections transmitted from man
to lower vertebrate animals, e.g. human n1berculosis
to cattle.
• HOST-PARASITE RELATIONSHIPS
Host-parasite re lationships a re o f fo ll owing types
(Flow chart 2):
Symbiosis
• Commensalism
• Parasitism.
P:24
General Introduction: Parasitology
Flow chart 2: Host-parasite relationships
Host-parasite relationships
i 1 i
Symbiosis Commensalism Parasitism
• Both host and parasite are
dependent upon each other
• None of them suffers any
harm from the association
• Only the parasite derives benefit
from the association without
causing any injury to the host
• The parasite derives benefits and
the host is always harmed due to
the association
• A commensal is capable of living
an independent life also
• The parasite cannot hve an
independent life
• LIFE CYCLE OF PARASITES
• Direct life cycle: When a parasite requires only single
host to complete its development, it is called as direct life
cycle, e.g. Entamoeba histolytica requires only a human
host to complete its life cycle (Table 1).
• Indirect life cycle: When a parasite requires two or more
species of host to complete its development, the life
cycle is called as indirect life cycle, e.g. malarial parasite
requires both human host and mosquito to complete its
life cycle (Tables 2 and 3).
• SOURCES OF INFECTION
Contaminated soil and water:
Soil polluted with embryonated eggs (roundworm,
whipworm) may be ingested or infected larvae in
soil, may penetrate exposed skin (hookworm).
lnfeclive forms of parasites present in water may be
ingested (cyst of ameba and Giardia).
Water con taining the intermediate host may be
swallowed (cyclops containing guinea worm larva).
Infected la rvae in water may enter by penetra ting
exposed skin (cercariae of schisotosomes).
Free-living parasites in water may directly enter
through vuln erable sites (Naegleria may enter
through nasopharynx).
Food:
Ingestion of contami nated food or vegetables
conraining infeclive stage of parasite (amebic cysts,
Toxoplasma oocysts, Echinococcus eggs).
Ingestion of raw or undercooked meat harboring
infeclive larvae (measly pork containing cysticercus
cellulosae, the larval stage of Taenia solium).
Vectors: A vector is an agent, usually an arthropod that
transmits an infection from man to man or from other
animals to man, e.g. female Anopheles is the vector of
malarial parasite.
Vectors can be:
Biological vectors: The term biological vector refers
to a vector, which not only assists in the transfer of
Table 1: Parasites having direct life cycle (requiring no intermediate
host)
Protozoa Helminths
• Entamoeba histalytica • Ascaris lumbricaides
• Giardia lambfia • Enterobius vermicularis
• Trichomonas vagina/is • Trichuris trichiura
• Balantidium coli • Ancylostama duodenale
• Cryptosporidium parvum • Necator americanus
• Cyclospora cayeranensis • Hymenolepis nana
• /sospara be/Ii
• Microsporidia
parasites but the parasites undergo developmen t or
multiplicaLion in their body as well. They are also
called as true vectors. Example of true vectors are:
• Mosquito: Malaria, filariasis
• Sandflies: Kala-azar
• Tsetse flies: Sleeping sickness
• Reduviicl bugs: Chagas disease
• Ticks: Babesiosis.
Mechanical vectors: The term mechanical vector
refers to a vector, which assists in the transfer of
parasitic form between hosts but is not essential in
the life cycle of the parasite. Example of mechanical
vectors is:
• Housefly: Amebiasis
In biological vectors, a certain period has to elapse after
the parasite enters the vector, before it becomes infective. This
is necessary because the vector can transmit the infecLion only
after the parasite multiplies to a certain level or undergoes
a developmental process in its body. This inte rval between
the entry of the parasite into the vector and the Lime it takes
to become capable of transmitting the infection is called the
extrinsic incubation period.
Animals:
Domestic:
• Cow, e.g. T. saginata, Sarcocystis
P:25
Paniker's Textbook of Medical Parasitology
Tab le 2: Parasites having indirect life cycle requiring one intermediate Box 2: Parasites causing autoinfection
host and one definitive host
Parasite Definitive host Intermediatehost
Protozoa
Plasmodium spp. Female Anopheles Man
mosquito
Babesia Tick Man
Lei sh mania Man. dog Sandfly
Trypanosoma brucei Man Tsetse fly
Trypanosoma cruzi Man Triatomine bug
Toxoplasma gondii Cat Man
Cestodes
Taenia solium Man Pig
Taenia saginara Man Cattle
Echinococcus granulosus Dog Man
Trematodes
Fascia/a hepatica Man Snail
Fascia/apsis buski Man, pig Snail
Schistosoma spp. Man Snail
Nematodes
Trichinella spiralis Man Pig
Wuchereria bancrofti Man Mosquito
Brugia malayi Man Mosquito
Dracunculus medinensis Man Cyclops
Table 3: Parasites having indirect life cycle requiring two intermediate
host and one definitive host
Parasite lntermediate hosts Definitive host
Fascia/a spp. Snail, plant Man
Clonorchis sinensis Snail, fish Man
Diphyllobothrium latum Cyclops, fish Man
Paragonimus westermani Snail, crustacean Man
• Pig, e.g. T. solium, Trichinella spiralis
• Dog, e.g. Echinococcus granulosus
• Cat, e.g. Toxoplasma, Opisthorchis.
Wild:
• Wild game animals, e.g. trypanosomiasis
• Wild felines, e.g. Paragonimus westermani
• Fish, e.g. fish tapeworm
• Molluscs, e.g. liver flukes
• Copepods, e.g. guinea worm.
Carrier: A person who is infected with parasite without
any clinical or subclinical disease is known as carrier.
He can tra nsmit parasite to others. For example, all
• Hymenolepis nana
• Enterobius vermicularis
• Taenia solium
• Strongyloides stercoralis
• Capillaria philippinensis
• Cryptosporidium parvum
anthroponotic infec tions, vertical tra nsmission of
congenital infections.
• Self(autoinfection) (Box 2):
Finger-to-mouth transmission, e.g. pinworm
Internal reinfection, e.g. Strongyloides.
• MODES OF INFECTION
Oral transmission: The most common method of
transmission is through oral route by contaminated food,
water, soiled fingers, or fomites. Many intestinal parasites
enter the body in this manner; the infective stages being
cysts, embryonated eggs, or larval forms. Infection with E.
histolytica and other intestinal protozoa occurs when the
infective cysts are swallowed.
Skin transmission: Entry through skin is another
important mode of transmission. Hookworm infection
is acquired, when the larvae enter the skin of persons
walking barefooted on contaminated soil. Schistosomiasis
is acquired when the cercarial larvae in water penetrate
the skin.
Vector transmission: Many parasitic diseases are
transmitted by insect bite, e.g. malaria is transmitted by
bite offemale Anopheles mosquito, filariasis is transmitted
by bite of Culex mosquito. A vector could be a biological
vector or a mechanical vector.
Direct tra nsmission: Parasitic infection may be
transmitted by person-to-person contact in some cases,
e.g. by kissing in the case of gingivaJ amebae and by sexual
intercourse in trichomoniasis.
Vertical transmission: Mother to fetus transmission may
take place in malaria and toxoplasmosis.
Iatrogenic transmission: It is seen in case of transfusion
malaria and toxoplasmosis after organ transplantation.
• PATHOGENESIS
Parasitic infections may remain inapparcnt or give rise to
clinical disease. A few organisms, such as E. histolytica may
live as surface cornmensals, without invading the tissue.
• Clinical infection produced by parasite may take many
forms: acute, subacute, chronic, latent, or recurrent.
• Pathogenic mechanisms, which can occur in parasitic
infections are:
Lytic necrosis: Enzymes produced by some parasite
can cause lyric necrosis. E. histolylica lyses intestinal
cells and produces amebic ulcers.
P:26
Tra uma: Attachment of h ookworms on jejunal
mucosa leads to trauma tic damage of villi and
bleeding at the site of attachment.
Allergic manifestations: Cli nical illness may be caused
by host immune response to parasitic infection, e.g.
eosinophilic pneumonia in Ascaris infection and
anaphylactic shock in rupture ofhydatid cyst.
Physical obstruction: Masses of roundworm cause
intestinal obstruction. Plasmodium f alciparum
malaria may produce blockage of brain capillaries in
cerebral malaria.
inflammatory reaction: Clinical illness may be caused
by inflammatory changes and consequent fibrosis,
e.g. lymphadenitis in filariasis and urinary bladder
granuloma in Schistosoma haemalobium infection.
Neoplasia: A few parasitic infection have been shown
to lead to malignancy. The liver flu ke, Clonorchis may
induce bile duct carcinoma, and S. haematobium
may cause urinary bladder cancer.
Space occupying lesions: Some parasites produce
cystic lesion that may compress the surrounding
tissue or organ, e.g. hydatid cyst.
• IMMUNITY IN PARASITIC INFECTION
Like other infectious agents, pa rasi tes also e li cit
immu noresponses in the host, both humoral as well as
cellular (Fig. 1). But immunological protection against
parasitic in fections is much less e fficient, than it is against
bacteria l or viral infections. Several factors may contribute
LO this:
Compared to bacteria a nd viruses, parasites a re
enormously larger or more complex structurally and
antigenically, so that immune system may not be able to
focus attack on the protective anrigens.
• Many protozoan parasites arc intracellular in location,
and this p rotects them from immu nological attack.
Several protozoa and helminths live inside body cavities.
1h.is location limits the efficiency of immunological attack.
• Once the parasitic infection is completely eliminated, the
host becomes again susceptible to reinfection. This type
of immunity to reinfection is dependent on the continued
presence ofresidual parasite population and is known as
\"premunition\".
• Antibodies belonging to different immunoglobulin
classes are produced in response to parasitic infections.
Selective tests for immunoglobulin M (IgM) are helpful in
differentiating current infections from old in fections.
Excessive IgE response occurs in h elminthiasis. A
characteristic cellular response in helminth parasite is
eosinophilia both local and systemic {Fig. 1).
• Parasites have evolved to be closely adapted to the host
and most pa rasitic infections are chronic and show a
degree of host specificity. For example, malarial parasites
General Introduction: Parasitology
Fig. 1: Eosinophils surrounding schistosomulum
(an example of immune attack in bloodstream)
Box 3: Parasites exhibiting antigenic variations
• Trypanosoma brucei gombiense
• Trypanosoma brucei rhodesiense
• Plasmodium spp.
• Giardia lamblia.
of human, bird and rodents are confined to their own
particular species.
• Parasites like trypanosomes exhibit antigenic variation
within the host. This genetic switch protects them from
antibodies. Similar mechanism may be operative in the
recrudescences in human malaria (Box 3).
• Some parasites adopt antigenic disguise. Their surface
antigens are so closely similar to host components that
they are not recognized as foreign by the immune system.
• Some infections may produce immunodeficiency due to
extensive damage to the reticuloendothelial system, as in
case of visceral leishmaniasis.
The fact that immunity normally plays an important role
in the containment of parasitic infections is illustrated by the
florid manifestations caused by opportunistic parasites such
as Pneumocystis jirovecii and T. gondii, when the immune
response is inadequate as in acquired immunodeficiency
syndrome (AIDS) and other immunodeficiencies.
• IMMUNE EVASION
All animal pathogens, including parasitic protozoa and worms
have evolved effective mechanism to avoid elimination by the
host defense system as described in Table 4 .
• VACCINATION
No effective vaccine for humans has so far been developed
against parasites d ue to their complex life cycles, adaptive
responses and antigenic variation, great progress has been
P:27
Paniker's Textbook of Medical Parasitology
Table 4: Parasite escape mechanisms
Parasite escape mechanisms Example
Intracellular habitat Malarial parasite, Leishmania
Encystment Toxop/asma
Trypanosoma cruzi
Resistance to microbial phagocytosis Leishmania
Masking of antigens Schistosomes
Variation of antigen Trypanosomes
Plasmodium spp.
Suppression of immune response Trichinella spirahs
Schistosoma mansoni
Malarial parasite
Interference by polyclonal Trypanosomes
activation
Sharing of antigens between parasite Schistosomes
and host-molecular mimicry
Continuous turnover and release of Schistosomes
surface antigens of parasite
made in identifying protective antigens in malaria and some
other infections, with a view to eventual development of
prophylactic vaccines.
• LABORATORY DIAGNOSIS
Most of the parasitic infection cannot be conclusively
diagnosed. On the basis of clinical features and physical
examination laboratory diagnosis depends upon:
• Microscopy
• Culture
. Serological test
. Skin test
. Molecular method . Animal inocuJation . Xenodiagnosis
• Imaging . Hematology .
Microscopy
An appropriate clinical specimen should be collected for
definitive diagnosis of parasitic infections.
• Following specimens are usually examined to establish a
diagnosis:
Stool
Blood
Urine
Sputum
Cerebrospinal fluid (CSF)
Tissue and aspirates
Genital specimens.
Stool Examination
Examination of stool is very important fo r the detection
of intestinal infections like Giardia, Enlamoeba, Ascaris,
Ancylostoma, etc.
Cysts and lrophozoites of E. histolytica, C. lamblia can be
demonstrated in feces. Eggs of roundworm and tapeworm
are also found in stool. The larvae are found in the feces in S.
slercoralis infection (Table 5).
For furth er details, refer to Chapter 23.
Blood Examination
Examination of blood is of vital importance for demonstrating
parasites which circuJate in blood vessels (Table 6). Malarial
parasite is confirmed by demonstration of its morphological
stages in the blood.
Urine Examination
The characteristic lateral-spined eggs of S. haematobium
and trophozoites of T. vagina/is can be detected in urin e.
Microfilaria of W bancrofti are often demonstrated in the
chylous urine (Box 4).
Sputum Examination
lhe eggs of P. westermani are commonly demonstrated in the
sputum specimen. Occasionally, larvaJ stages of S. s/ercoralis
and A. lumbricoides may also be found in sputum.
Cerebrospinal Fluid Examination
Some protozoa like T. brucei, Naegleria, Acanthamoeba,
Balamulhia and Angiostrongylus can be demonslrated in the
CSF.
Tissue and Aspirates Examination
The larvae of Trichinella and eggs of Schistosoma can
be demonstrated in the muscle biopsy specimens. By
histopathological examination of brain, Naegleria and
Acanthamoeba can be detected. In ka la-azar, LeishmanDonovan (LO) bodies can be demonstrated in spleen and
bone marrow aspirate. Trophozoites of Giardia can be
demonstrated in intestinal aspira tes. Trophozoites of E.
histolytica can be detected in liver pus in cases of amebic liver
abscess.
Genital Specimen Examination
Trophozoites of T. vagina.Lis are found in the vaginal and
urethral discharge. Eggs of E. vermicularis are found in anal
swabs.
P:28
General Introduction: Parasitology
Table 5: Parasites and t heir developmental stages found in stool
Cysts/Trophozoites
• Entamoeba histolytica
• Giardia lamblia
• Balantidium coli
• Sarcocystis spp.
• lsospora be/Ii
• Cyclospora cayetanensis
• Cryptosporidium parvum
Eggs
Cestodes
• Taenia spp.
• Hymenolepis nana
• Hymenolepis diminuta
• Oipy/idium caninum
• Oiphylloborhrium /atum
Trematodes
• Schistosoma spp.
• Fasciolopsis buski
• Fascia/a hepatica
• Fascia/a gigantica
• Clonorchis sinensis
Table 6: Parasites found in peripheral blood film
Protozoa
• Plasmodium spp.
• Babesia spp.
• Trypanosoma spp.
• Leishmania spp.
Box 4: Parasites found in urine
• Schistosoma haematobium
• Wuchereria bancrofti
• Trichomonas vagina/is
Culture
Nematodes
• Wuchereria bancrafti
, Brugia malayi
• Loaloa
• Mansonella spp.
Larvae Adult worms
• Gasrrodiscoides hominis Strongyloides stercoralis • Taenia solium
• Heterophyes heterophyes
• Metagonimus yokogawai
• Opisthorchis spp.
Nematodes
• Trichuris trichiura
• Enterobius vermicularis
• Ascaris lumbricoides
• Ancylostoma duodenale
• Necatoramericanus
• Trichosrrongylus orientalis
• Taenia saginata
• Oiphyllobothrium latum
• Ascaris /umbricoides
• Enrerobius vermicularis
• Tr/chine/la spiro/is
Table 7: Antigen detection in parasitic diseases
• Galactose lect in antigen
• Giardia-specific antigen 65
• WKK and rk39 antigen
• HRP-2 antigen
• Vivax specific pLDH
• 200 kDa Ag and OG4C3 antigen
Entamoeba histo/ytica
Giardia lamblia
Leishmania donovani
Plasmodium falciparum
Plasmodium vivax
Wuchereria bancrofti
Abbreviations, Ag, antigen; HRP-2, histidine-rich protein 2; pLDH, P. folciparum lactate
dehydrogenase; rk39, recombinant kinesin 39; WKK, Witebsky, Klingenstein and Kuhn
by rapid immunochroma tographic test. Filarial antigens
are de tected in curre nt infection by e nzyme-linke d
immunosorbcnt assay (ELISA) (Table 7).
Some p a rasites like Leishmania, Entamoeba a nd
Trypanosoma can be cultured in Lhe laboratory in various
axenic and polyxenic media.
Antibody Detection
The foll owing a ntibody detection procedures are useful
in detecting various parasitic infections like amebiasis,
echinococcosis and leishmaniasis in man:
Serological Tests
Serological tests are helpful for the detection and surveillance
of many protozoa! and helminthic infections. These tests are
basically of two types:
Tests for antigen detection
Tests for antibody detection.
Antigen Detection
Malaria antigen like P. f alciparum lactate dehydrogenase
(pLDI I) and histidine-rich protein 2 (HRP-2) are detected
• Complement fixation test ( CFT)
• Indirect hemagglutination (IHA)
• Indirect immunofluoresccnt antibody (IFA) test
• Rapid immunochromatographic test (ICT)
• Enzyme-linked immunosorbent assay test (ELISA).
Skin Test
Skin tests are perfo rmed by injecting parasitic an tigen
intradermally and observing the reaction. In immediate
hypersensitivity reaction, wheal and flare response is seen
within 30 minutes of infection, whe reas erythema and
P:29
Paniker's Textbook of Medical Parasitology
Box 5: Important skin tests done in parasitology
• Cason i's test done in hydatid disease
• Montenegro test or leishmanin test done in kala-azar
• Frenkel's test done in toxoplasmosis
• Fairley·s test done In schistosomiasis
• Bachman intradermal test done in trichinellosis.
in duration seen after 48 hours of injection is called as delayed
hypersensitivity reaction (Box 5).
Molecular Diagnosis
Molecular methods most frequently used to diagnose human
parasitic infection are deoxyribonucleic acid (ONA) probes,
polymerase chain reaction (PCR) and microarray technique.
1hese tests are very sensitive and specific.
Animal Inoculation
It is useful for the detection of Toxoplasma, Trypanosoma and
Babesia from the blood and other specimens.
Xenodiagnosis
Some parasitic infection like Chagas disease caused by T.
cruzi can be diagnosed by feeding the larvae of reduviid bugs
with patient's blood and then detection of amastigotes of T.
cruzi in their feces.
Imaging
Imaging procedures like X-ray, ulcrason ography (USG),
computed tomography (CT) scan and magnetic resonance
imaging (MRI) are now being extensively used for diagnosing
variou s parasitic infections like n eurocysticercosis and
hydatid cyst disease.
Hematology
Anemia is frequently seen in hookworm infection and malaria.
Eosinophilia is frequentl y present in helminthic infections.
HypergammaglobuJinemia occurs in visceral leishmaniasis.
Leukocytosis is seen in am ebic liver abscess.
KEY POINTS
• Leeuwenhoek in 1681, first observed the parasite Giardia
in stools. Laveran in 1880, discovered malarial parasite and
Ronald Ross in 1897 showed the transmission of malaria by
mosquitoes.
• Protozoa belong to kingdom Protista and helminths belong to
kingdom Animalia.
• Definitive host: The host in which the adult stage lives or the
sexual mode of reproduction takes place.
• Intermediate host: The host in which the larval stage of the
parasite lives or the asexual multiplication takes place.
• Zoonoses: Diseases which can be transmitted to humans
from animals, e.g. malaria, leishmaniasis, trypanosomiasis
and echinococcosis.
• Parasites like trypanosomes exhibit antigenic variation within
the host.
• Parasites like Ascaris and Echinococcus cause allergic
manifestations in the host.
• Innate immunity against parasite may be genetic or by
nonspecific direct cell-mediated or by complement activation.
• Acquired immunity in parasitic infections is by generating
specific antibodies and effector T-cells against parasitic
antigens.
• Diagnosis of parasitic infections are made by direct
identification of parasite in specimens like stool, blood,
urine, bone marrow, CSF, sputum, etc.
• Serological tests are also useful in diagnosis by detection of
parasite-specific antibody and antigen.
• Other diagnostic modalities include imaging, molecular
methods like PCR, skin test and xenodiagnosis.
REVIEW QUESTIONS
Write short notes on:
a. Parasites
b. Host
c. Host-parasite relationship
d. Zoonoses
e. Immune evasion mechanism of the parasites.
Discuss briefly the laboratory diagnosis of parasites.
Describe immunity in parasitic infections.
Differentiate between:
a. Direct and indirect life cycle
b. Definitive host and intermediate hosts
MULTIPLE CHOICE QUESTIONS
Definitive host is one
a. In which sexual multiplication takes place and harbors adult
form
b. In which asexual multiplication takes place and harbors adult
form
c. In which sexual multiplication takes place and harbors larval
form
d. In which asexual multiplication takes place and harbors adult
form
Autoinfection is seen in all except
a. Hymeno/epis nano
b. Enterabius vermicularis
c. Taenia so/ium
d. Ascaris lumbricoides
P:30
Antigenic variation is exhibited by
a. Entamoeba
b. Schistosoma
c. Trypanosoma
d. Leishmania
Which parasite enters, the body by piercing the skin
a. Trichuris trichiura
b. Ascaris
c. Necator americanus
d. Plasmodium
Which parasitic infection leads to malignancy
a. Babesiosis
b. Clonorchis sinensis
c. Trypanosoma cruzi
d. Schistosoma haematobium
Xenodiagnosis is useful in
a. Wuchereria bancrofti
b. Trypanosoma cruzi
c. Trichinella spiralis
d. All of the above
The following are zoonotic disease except
a. Leishmaniasis
b. Balantidiasis
c. Scabies
d. Taeniasis
General Introduction: Parasitology
Two hosts are required in
a. Taenia solium
b. Entamoeba histolytica
c. Trichuris trichiura
d. Giardia
Which of the following parasite passes its life cycle t hrough
three hosts
a. Fascia/a hepatica
b. Fascia/a buski
c. Schistosoma haematobium
d. Clonorchis sinensis
Man is the intermediate host for
a. Strongyloides stercoralis
b. Plasmodium vivax
c. Entamoeba histolytica
d. Enterobius vermicularis
Answer
a 2. d
a 9. d
C
b
C 5. b 6. d 7. C
P:31
CHAPTER 2
• INTRODUCTION
• Single-celled eukaryotic microorganisms belonging to
kingdom Pro tis ta are classified as Protozoa ( Greek protos:
first; zoon: animal).
• Parasitic protozoa are adapted to different host species.
• Out of 10,000 species of parasitic protozoa, man harbours
only about 70 species.
• GENERAL FEATURES
• The single protozoa! cell performs all functions.
• Most of the protozoa are completely nonpathogenic
but few may cause major diseases such as malaria,
leishmaniasis and sleeping sickness.
• Protozoa like Cryptosporidium parvum and Toxoplasma
gondii are being recognized as opportunistic pathogens
in patients affected with human immunodeficiency virus
(lllV) and in those undergoing immunosuppressive
therapy.
• Protozoa exhibit wide range of size (1- 150 µ111), shape and
structure; yet all possess essential common features
• The differences between protozoa and metazoa arc given
in Table l .
• STRUCTURE
the typical protozoan cell is bounded by a trilaminar unit
membrane, supported by a sheet of contractile fibrils enabling
the cell to move and change in shape.
• CYTOPLASM
It has two portions:
Ectoplasm: Outer homogeneous part that serves as the
organ for locomotion and for engulfment of food by
producing pseudopodia is called as the ectoplasm. It also
helps in respiration, discharging waste material, and in
providing a protective covering of cell.
Endoplasm The inner granular portion of cyLOplasm
that contains nucleus is called endoplasm. The
Table 1: Differences between protozoa and metazoa
Protozoa
Morphology Unicellular;
a single \"cell-like unit\"
Physiology A single cell performs
Example
all the functions:
reproduction, digestion,
respiration, excretion. etc.
Ameba
Multicellular;
a number of cells, making
up a complex individual
Each special cell performs
a particular function
Tapeworm
endoplasm shows number of structures: the Golgi bodies,
endoplasmic reticulum, food vacuoles and contractile
vacuoles. Contractile vacuoles serve to regulate the
osmotic pressure.
• NUCLEUS
The nucleus is usually single but may be double or multiple;
some species having as many as 100 nuclei in a single cell.
lhe nucleus contains one or more nucleoli or a central
ka1yosome.
• The chromatin may be distributed along periphery
(peripheral chromatin) or as condensed mass around the
karyosome.
• TERMINOLOGIES USED IN PROTOZOOLOGY
• Chromatoid body: Extranuclear chromatin material
is called chromatoid body (e.g. as found in Entamoeba
histolytica cyst).
Karyosome: It is a deoxyribonucleic acid (DNA) containing
body, situated peripherally or centrally within the nucleus
and found in intestinal ameba, e.g. E. histolytica E.coli.
Kinetoplast: Nonnuclear DNA present in addition to
nucleus is called kinetoplast. It is seen in trypanosomes.
Flagellum originates near the kinetoplast. Point of origin
of flagellum is called as basal body.
1
P:32
• Cilia: These are fi ne, needle-like filaments, covering the
entire surface of the body and are found in ciliates, e.g.
Balantidium coli.
• Trophozoile (trophos: nourishment): Active feeding and
growing stage of the protozoa is called the trophozoites.
It derives nutrition from the environment by diffusion,
pinocytosis and phagocytosis.
• REPRODUCTION
Reproduction can be:
• Asexual reproduction
• Sexual reproduction.
Reproduction usually occurs asexually in protozoans;
however, sexual reproduction occu rs in c ili ates and
sporozoans.
Asexual Reproduction
Binary fission: It is a method of asexual reproduction,
by which a single parasite divides either longitudinally or
transversalJy into two or more equal number of parasites.
Mitotic division of nucleus is followed by division of the
cytoplasm. In amebae, division occurs along any plane,
but in flagellates, division is along longitudinal axis and in
ciliates, in the transverse plane (Fig. 1).
Multiple fission or schizogony: Plasmodium exhibits
schizogony, in wh ich n ucle us undergoes several
successive divisions within the schizont to produce large
number of merozoiles (Fig. I).
Endodyogeny: Some protozoa like Toxoplasma, multiply
by internal budding, resulting in the formation of two
daughter cells.
Longitudinal
binary fission
(Flagellates)
Binary fission
(Ameba)
Protozoa
Sexual Reproduction
• Conjugation: In ciliates, the sexual process is conjugation,
in which two organisms join together and reciprocally
exchange nuclear material (e.g. Balanlidium coli).
• Gametogony or syngamy: In Sporozoa, male and female
gametocytes are produced, which after fertilization form
th e zygote, which gives rise to numerous sporozoites by
sporogony (e.g. Plasmodium).
• LIFE CYCLE
Single host: Pro tozoa like intestinal fl agellates and
ciliates require only one host, within which they multiply
asexually in trophic stage and transfer from one host to
another by the cystic form.
• Second host: In some protozoa like Plasmodium, asexual
method of reproduction occurs in on e host (man )
and sexual method of reproduction in another host
(mosquito).
• CLASS I Fl CATION OF PROTOZOA
Protozoan parasites of medical importance have been
classified into kingdom Protista, subkingdom Protozoa which
is further divided into the following four phyla (Table 2):
l. arcomastigophora
Apicomplexa
Microspora
Ciliophora
The important protozoan pathogens of human a re
summarized in Table 3.
Multiple fission
(schizogony)
(
Transverse
binary fission
(Ciliates)
Plasmod1um Red
blood cell
Daughter
Nuclei
Disrupts cell wall and is released
Fig. 1: Asexual reproduction in protozoans
P:33
Table 2: Classification of protozoa
Phylum Subphylum
Sarcomastigophora Mastigophora (having
one or more flagella)
Superclass class
Zoomastigophorea
Sarcodina Rhizopoda Lobosea
Apicomplexa
Clliophora
Microspora
(pseudopodia present)
Sporozoea
Kinetofragminophorea
M icrosporea
Subcl
Gymn
Coccid
Piropla
Vestib
P:34
lass
namebia
dia
asmia
buliferia
order
Kinetoplastida
Retortamonadida
Diplomonadida
Trichomonadida
Amebida
Schizopyrenida
Eucoccidia
Piroplasmida
Trichostomastida
Microsporidia
Suborder
-Trypanosomatina
Enteromonadina
Diplomonadina
Tubulina
Acanthopodina
Eimeriina
Hemosporina
Trichostomatina
Apansporoblastina
Genus
Trypanosoma
Leishmania
Rerortamonas
Chllomastix
Enteromonas
Giardia
Trichomonas
• Dientamoeba
• Entamoeba
• Endolimax
• /odamoeba
Acanthamoeba
Naegleria
Cryptosporidium
• /sospora
• Sarcocystis
• Toxoplasma
Plasmodium
Babesia
Balantidium
Enterocytozoon
Encephalitozoon
Microsporum
~ ::I
~
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;t
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tr
0
0
';It:\"
0
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iil
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P:35
Table 3: Principal protozoan pathogens of man
species Habitat Disease
Entamoeba Large intestine Amebic dysentery,
histolytica amebic liver abscess
Naegleria fowleri CNS Amebic
meningoencephalltis
Acanthamoeba CNS, eye Encephalitis, keratitis
Giardia lamblia Small intestine Malabsorption, diarrhea
Trichomonos Vagina, urethra Vaginitis, urethritis
vagina/is
Trypanosoma brucei Blood, lymph node, Sleeping sickness
CNS
T rypanosomo cruzi Macrophage of bone Chagas disease
marrow, nerves,
heart, colon, etc.
Leishman,a Reticuloendothelial Kala-azar, Postkala-azar
donovani system dermal leishmaniasis
Leishmania tropica Skin Cutaneous leishmaniasis
(oriental sore)
Leishmania Naso-oral mucosa Mucocutaneous
braziliensis leishmaniasis (espundia,
chiclero's ulcer)
Plasmodium spp. RBC Malaria
Babesia microti RBC Babesiosis
lsospora be/Ii Intestine Diarrhea in AIDS
Cryptosporidium Intestine Diarrhea in AIDS
parvum
Balantidium coli Large intestine Dysentery
Protozoa
from the nagellates by the loss of the flagella. Two groups of
amebae are of medical importance:
Amebae of the alimentary canal: The most important
of these is E. histolylica, whic h causes intestinal and
extraintestinal arnebiasis. Amebae are aJso present in the
mouth.
Potentially pathogenic free-living amebae: Several
species of saprophytic arnebae are found in soil and water.
Two of these, (I) Naegleria and (2) Acanthamoeba are of
clinical interest because they can cause eye infections and
fatal rneningoencephalitis.
Flagellates
These protozoa have whip-like appendages called flagella as
the organs of locomotion. 1he fibriJJar structure of flagelJa is
identical with that of spirochetes and it has been suggested
that they may have been derived from symbiotic spirochetes,
which have become endoparasites. In some species, the
flagellum runs parallel to the body surface, to which it is
connected by a membrane called the undulating membrane.
FlageUates parasitic for man are divided into two groups:
Kinetoplastida: These possess a kinetoplast from which
a single flagellum arises. They are the hemoflagellates
comprising the trypanosomes and Leishmania, which are
transmitted by blood-sucking insects and cause systemic
or locaJ infections.
Flagellates without kinetoplast: Th ese bear multiple
fl agella. Giardia, Trichomonas and other luminal
flagellates belong to this group. Because most of them
live in the intestine, they are generally called intestinal
Abbreviations: AIDS, acquired immunodeficiency syndrome; CNS, central nervous flagellates.
system; RSC, red blood cell
Phylum Sarcomastigophora
Phylum Sarcomastigophora has been subdivided into two
subphyla based on their modes of locomotion:
Sarcodina (sarcos meaningflesh or body): It includes those
parasites, which have no permanent locomotory organs,
but move about with the aid of temporary prolongations
of the body called pseudopodia (e.g. amebae).
Masrigophora (mastix meaning whip or flagellum): It
includes those protozoa which possess whip-like flagella
(e.g. Trypanosoma and Trichomonas).
Amebae
These protean animalcules can assume any shape and
crawl a long surfaces by means of foot-like projections
called pseudopodia (literally meaning false feet). They are
structurally very simple and are believed to have evolved
Phylum Apicomplexa
Phylum Apicomplexa was formerly known as Sporozoa.
Members of this group possess, at some stage in their life
cycle, a structure called the apical complex serving as the
organ of attachment to host cells.
• 1hey are tissue parasites.
• 1hey have a complex life cycle with alternating sexual and
asexual generations.
• To this group, belongs the malarial parasites (Suborder:
Hemosporina, Family: Plasmodiidae), Toxoplasma,
Sarcocystis, lsospora, and Cryptosporidium ( Under rhe
Suborder: Eimeriina), Babesia ( Under the Subclass:
Piroplasma) and the unclassified Pneumocystis jiro11ecii.
Phylum Ciliophora
These protozoa are motile by means of cilia, which cover their
entire body surface. The onJy human parasite in this group is
Balantidium coli, which rarely causes dysentery.
P:36
Paniker's Textbook of Medical Parasitology
Phylum Microspora
Phylum Microspora contains many minute intracellular
protozoan parasites, which frequently cau se di sease in
immunodeficient subjects. They m ay also cause illness in the
immunocompetent, rarely.
The zoological classification of protozoa is complex and
is subject to frequent revisions. The classification described
in the chapter is an abridged version of the classification
proposed in 1980 by the Commillee on Systematics and
Evolution of the Society of Protozoologists, as applied to
protozoa of med ical importance.
IMPORTANT POINTS TO REMEMBER
• Only protozoan parasite found in lumen of human small
intestine: Giardia /amblia.
• Largest protozoa: Balantidium coli.
• Most common protozoan parasite: Toxoplasma gondii.
KEY POINTS OF PROTOZOA
• Protozoa are single-celled, eukaryotic microorganisms
consisting of cell membrane, cytoplasm and nucleus.
• Some protozoa have kinetoplast and flagella or cilia.
• Amebae move about with temporary prolongations of the
body called pseudopodia.
• Hemoflagellates comprising of Trypan \ osoma and
Leishmania possess a single flagellum and kinetoplast.
• Luminal flagellates like Giardia and Trichomonas bea r
multiple flagella without kinetoplast.
• Balantidium coli belongs to the Phylum Ciliophora, which is
motile by cilia that cover its entire body surface.
• Trophozoites are active feeding and growing stage of
protozoa.
• Cysts are resting or resistant stage of protozoa bounded by
tough cell wall.
• Protozoa multiply by both asexual and sexual modes of
reproduction.
• Malaria parasite, Toxoplasma and Cryptosporidium belong
to phylum Apicomplexa or Sporozoa, which possess apical
complex at some stage of their life cycle and have a complex
life cycle with alternating sexual and asexual generations.
• Microspora are intracellular protozoan parasites, which
cause disease in immunodeficient patients.
REVIEW QUESTIONS
Define Protozoa and describe their general characteristics.
Write short notes on:
a. Classification of Protozoa
b. Reproduction in Protozoa
Differentiate between Protozoa and Metazoa.
MULTIPLE CHOICE QUESTIONS
Protozoa belong to kingdom
a. Monera
b. Protista
c. Plantae
d. Animalia
All are intercellular parasites except
a. Leishmania
b. Plasmodium
c. Toxoplasma
d. None of the above
Non-nuclear DNA present in addition to nucleus in protozoan
parasite is
a. Chromatid body
b. Karyosome
c. Kinetoplast
d. Basal body
Entamoeba histolytica trophozoites multiply by
a. Binary fission
b. Schizogony
c. Gametogony
d. All of the above
In humans, malarial parasites multiply by
a. Binary fission
b. Budding
c. Gametogony
d. Schizogony
Which of the following is not a flagellate
a. Naegleria
b. Leishmania
c. Giardia
d. Dientamoeba
Answer
b 2. d 3. C 4. a 5. d 6. a
.l
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• INTRODUCTION
The word ameba is derived from the Greek word \"amibe\"
meaning change.
Amebae are structurally simple protozoans which
have no fixed shape. They are classified under Phylum:
Sarcomastigophora, Subphylum: Sarcodina, Superclass:
Rhizopoda and Order: Amebida.
1he cytoplasm of ameba is bounded by a membrane and
can be differentiated into an outer ectoplasm and inner
endoplasm.
Pseudopodia are formed by the ameba by thrusting out
ectoplasm, followed by endoplasm. TI1ese are employed
for locomotion and engulfment of food by phagocytosis.
• Reproduction occurs by fission and budding. Cyst is
formed in unfavorable conditions and is usually the
infective form for vertebrate host (e.g. Entamoeba
histolytica).
• Amebae are classified as either free-living or inteslinaJ
amebae (Table 1).
• A few of the free-living amebae occasionally act as human
pathogens producing meningoencephalitis and other
infections, e.g. Naegleria and Acanthamoeba
The parasitic amebae inhabit the alimentary canal
Table 1: Classification of amebae
Intestinal amebae
• Entamoeba histolytica
• Entamoeba dispar
• Entamoeba coli
• Entamoeba polecki
• Entamoeba hartmanni
• Entamoeba gingivalis
• Endolimax nana
• /odamoeba butschlii
Note: All intestinal amebae are
nonpathogenic, except Entomoeba
histolytica
Free-living amebae
• Naegleria fowleri
• Acanthamoeba spp.
• Balamuthia mondrillaris
Note: All free-living amebae
are opportunistic pathogens
• ENTAMOEBA HISTOLYTICA
History and Distribution
£. histolytica was discovered by Losch in 1875, wh o
demonstrated me parasite in the dysenteric feces of a patient
in St. Petersburg in Russia.
• In 1890, William Osler reported the case of a young man
with dysentery, who later died of liver abscess.
• Councilman an d Lafleur in 1891 established the
pathogen esis of intestinal and hepa tic amebiasis and
introduced the terms \"amebic dysentery\" and \"amebic
live r abscess''.
E. histolytica is worldwide in prevalence, being m uch
more common in the tropics than elsewhe re. It has been
found wherever sanitation is poor, in all climatic zones
from Alaska (61°N) to Straits of Magellan (52°S).
• lt has been reported d1at about l 0% of world popula tion
and 50% of the inhabitants of developing countries may
be infected with the parasite.
• The infection is not uncommon even in affluent countries,
about l % of Americans being reported to be infected.
While the majority of infected humans (80-99%) a re
asymptomatic, invasive amebiasis causes disabling illness
in an estimated 50 million of people and causes 50,000
deaths annually, mostly in the tropical belt of Asia, Africa
and Latin America.
• It is the third leading parasitic cause of mortality, after
mala ria and schisLosomiasis.
• Epidemiologically, India can be d ivided into three regions,
depending on the prevalence of intestinal amebiasis:
l. High prevalence states (>30%): Chandigarh, Tamil
adu and Maharashtra.
2 . Moderate prevalence sla te s (10-30%): Punjab,
Rajasd1an, Uttar Pradesh, Delhi, Biha r, Assam, West
Bengal, Andhra Pradesh, Karnataka and Kerala.
Low prevalence states (<10%): Haryana, Gujarat,
Himachal Pradesh, Madhya Pradesh, Odisha, Sikkim
and Puducherry.
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Paniker's Textbook of Medical Parasitology
Morphology
E. histolytica occurs in three forms (Figs IA to E):
Trophozoite
Precyst
Cyst.
Trophozoite
Trophozoite is the vegetative or growing stage of the parasite
(Fig. IA). lt is the only form present in tissues.
.
•
•
.
1t is irregular in shape and varies in size from 12- 60 µm;
average being 20 µrn.
It is large and actively motile in freshly-passed dysenteric
stool, while smaller in convalescents and carriers.
The parasite, as it occurs free in the lumen as a commensal
is generally smaller in size, about 15-20 µm and has been
called the minuta form
Cytoplasm: Outer ectoplasm is clear, transparent and
refractile. Inner endoplasm is fin ely granular, having
a ground glass appearance. The endoplasm contains
nucleus, food vacuoles, erythrocytes, occasionally
leukocytes and tissue debris.
Pseudopodia are finge r-like projections formed by
sudden jerky movements of ectoplasm in one direction,
followed by the streaming in of the whole endoplasm.
Typical ameboid motility is a crawling or gliding
movement and not a free swimming one. The direction
of movement may be changed suddenly, with another
pseudopodium being formed at a different site, when
the whole cytoplasm flows in the direction of the new
pseudopodium. The cell has to be attached to some
surface or particle for it to move. In culture tubes, the
trophozoites may be seen crawling up the side of the glass
tube.
• Pseudopod ia formation and motility are inhibited at low
temperatures.
• Nucleus is spherical 4- 6 µm in size and contains central
karyosome, surrounded by clear halo and anchored to
the nuclear membrane by fine radiating fibrils called
the Linin network, giving a cartwheel appearance. The
nucleus is not clearly seen in the living trophozoites, but
can be clearly demonstrated in preparations stained with
iron hematoxylin.
Ectoplasm
Endoplasm
Ingested
erythrocytes
• 111e nuclear membrane is lined by a rim of chromatin
distributed evenly as small granules.
• The trophozoites from acute dysenteric stools often
contain phagocytosed erythrocytes. This feature is
diagnostic as phagocytosed red cells are not fou nd in any
other commensaJ intestinal amebae.
• The trophozoites divide by binary fission in every 8 hours.
• Trophozoiles survive up to 5 hours at 37°C and are killed
by drying, heat and chemical sterilization. Therefore, the
infection is not transmitted by trophozoites. Even if live
trophozoites from freshly-passed stools are ingested,
they are rapidly destroyed in stomach and cannot initiate
infection.
Precystic Stage
Trophozoites undergo encystm ent in the intestinal lumen.
Encystment does not occur in the tissues nor in feces outside
the body.
• Before encystment, the trophozoite extrudes its food
vacuoles and becomes round or oval, about 10- 20 µmin
size. This is the precystic stage of the parasite (Fig. 18).
• It contains a large glycogen vacuole and two chromatid
bars.
• It then secretes a highly retractile cyst wall around it and
becomes cyst.
Cystic Stage
The cyst is spherical in shape about 10-20 µmin size.
.
.
.
The early cyst contains a single nucleus and two
other structures: (1) a mass of glycogen a nd (2) 1- 4
chromatoid bodies or chromidial bars, which are cigarshaped refractile rods with rounded ends (Fig. l C). The
ch romatoid bodies are so called because they stain with
hematoxylin, like chromatin.
As the cyst matures, the glycogen mass and chromidial
bars disappear and the nucleus undergoes two successive
mitotic divisions to form two (Fig. lD) and then four
nuclei. 1he mature cyst is, thus quadrinucleate (Fig. IE).
The cyst wall is a highly refractile membrane, which
makes it highly resistant to gastric juice and unfavorable
environmental conditions.
Chromidial
bar
Glycogen
mass
a Nucleus . m m
Figs 1 A to E: Entamoeba histolytica. (A) Trophozoite; (B) Precystic stage; (C) Uninucleate c . D
(D) Binucleate cyst; and (E) Mature quadrinucleate cyst yst;
l
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• The nuclei and chromidial bodies can be made out in
unstained films, but they appear more prominently in
stained preparations.
With iron hematoxylin stain, nuclear chromatin and
chromaroid bodies appear deep blue or black, while the
glycogen mass appears unstained.
When stained with iodine, the glycogen mass appears
golden brown, the nuclear chromatin and karyosome
bright yellow, and the chromatoid bodies appear as clear
space, being unstained.
Life Cycle
£. histolytica passes its life cycle only in one host man
(Flowchart 1 and Fig. 2).
Infective Form
Mature quadrinucleate cyst passed in feces of convalescents
and carriers. The cysts can remai n viable u nder moist
conditions for about IO days.
Man acquires infection by
ingestion of cysts in
contaminated food and water
Amebae
Mode of Transmission
Man acquires infec tion by swallowing food and water
contaminated with cysts.
• As the cyst wall is resistant to action of gastric juice, the
cysts pass through the stomach undamaged and enter the
small intestine.
• Excystation: When the cyst reaches cecum or lower part
of the ileum, due to the alkaline medium, the cyst wall is
damaged by trypsin, leading to excystation.
Flow chart 1: Life cycle of Entamoeba histolytica (schematic)
r···········• Trophozo1tes in colon
Metacystic trophozoites I
t
Metacyst·in small intestine I
t
Cysts ingested
t----------- •
Precyst
Cysts 1- 4 • nuclei
;
Passed in • feces
~ ------- Cysts in contaminated food or water ~---)
Ingested cyst
from contaminated
food or water
!
Fig. 2: Life cycle of Entamoeba histolytica
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Paniker's Textbook of Medical Parasitology
• The cytoplasm gets detached from the cyst wall a nd
ameboid movements appear causing a tear in the cyst
wall, through which quadrinucleate ameba is liberated.
This stage is called the metacyst (Fig. 2).
• Metacystic trophoz oites: The nuclei in the metacyst
immediately undergo division to form eight nuclei, each
of which gets surrounded by its own cytoplasm to become
eight small amebulae or melacystic trophozoites.
• if excystation takes place in the small intestine, the
me tacysric trophozoites do not colonize there, but are
carried to the cecum.
• The optimal habitat for the me tacystic trophozoite is the
submucosal tissue of cecum and colon, where they lodge
in the glandular crypts and grow by binary fission (Fig. 2).
• Some develop inro precystic forms and cysts, which are
passed in feces to repeat the cycle.
• The entire life cycle is, thus completed in one host.
r n m ost of the cases, E. histolytica remains as a commensal
in the large intestine witho ut causing any ill effects. Such
per ons become carriers or asymptomatic cyst passers and
a re responsible for maintenance and spread of infection in
the community. Sometimes, the infection may be activated
and clinical disease ensues. Such latency and reactivation a re
the characteristics of amebiasis.
Pathogenesis and Clinical Features
• E. hislolytica causes intestinal a nd extraintestinal
amebiasis.
• incubationperiod is highly variable. On an average, it
ranges from 4 days to 4 months.
• Amebiasis can present in different forms and degree of
everity, depending on the organ affected and the extent
of damage caused.
Intestinal Amebiasis
The lumen-dwelling amebae do not cause any illness. They
cause disease o nly when they invade the intestinal tissues.
This happens only in about 10% of cases of infection, the
remaining 90% being asymptomatic.
Not all strain s of£. hislolylica are pathogenic or invasive.
Differentiation between pathogenic and nonpathogenic
strains can be made by susceptibility to complementmediated lysis and phagocytic activity or by the use
of genetic markers or monoclonal antibodies and
zymodeme analysis.
Adherence: Amebic lectins (Gal/ Gal Ac lectin,
260 kDa surface protein of E. histolytica) mediates
adherence to glycogen receptors of colonic mucosa.
Cytolysis: The metacystic rrophozoites peneo·ate the
columnar epithelial cells in the crypts ofliebe rkuhn
in the colon. Penetration of the ameba is facilitated
Box 1: Factors affecting virulence of Entamoeba histolytica
• Amebic cysteine proteinases, which inactivate complement factor C3
and degrade cellular matrix and lg A is an important virulence factor.
• Amebic lectin (Gal/GalNAc lectin) and ionophore protein are other
virulence factors.
• Host factors such as stress, malnutrition, alcoholism, corticosteroid
therapy and immunodeficiency influence the course of infection.
• Glycoproteins in colonic mucus block the attachment of trophozoites
to epithelial cells, therefore alteration in the nature and quality of
colonic mucus may influence virulence.
• Virulence may also be conditioned by the bacterial flora in the colon.
• Based on electrophoretic mobility of six lsoenzymes (acetylglucosaminidase, aldolase, hexokinase, NAD-diaphorase, peptidase and
phosphoglucomutase), E. histolyrica strains can be classified into at
least 22 zymodemes. Of these only nine are invasive and the rest are
noninvasive commensals.
• It has been proposed that pathogenic and nonpathogenic strains
though morphologically identical may represent two distinct species:
(1) the pathogenic strains being E. histalytica and (2) the nonpathogenic
strains reclassified as E. dispar. Trophozoites of£ dispar contain bacteria,
bur no red blood cells (RBCs).
by the motili ty of the trophozoites and the tissue
lytic activity of the am ebic cysteine proteases like
histolysi.n, cathepsin B, metallocollagenase. Cysteine
proteases degrade the extracellular matrix (ECM)
component of host cells and immunoglobulin A (lgA)
(Box 1) and also inactivates complement C3.
Ameba,p ores are ionophore proteins of ameba
capable of inserting ion channels into liposomes
causing lysis of targe t cell membrane of host cells.
Tissue necrosis is also caused by the lysosomal enzymes
of the in0ammatory cells surro unding the trophozoites and
proinflammatory cytokines like interleukin-8 (IL-8) and
tumor necrosis factor-a (T F-a) released from these cells.
Mucosa! penetration by the ameba produces discrete
ulcers with pinhead center and raised edges. Sometimes,
the invasion remains superficial and heals spontaneously.
More often, the ameba penetrates to submucosaJ layer
and multiplies rapidly, causing lytic necrosis and thus
forming an abscess. The abscess breaks down to form an
ulcer.
• Amebic ulcer is the typical lesion seen in intestinal
amebiasis (Fig. 3). The ulcers are multiple and are
confined to the colon, being most numerous in the cecum
and next in the sigmoidorectal region. the intervening
mucous membrane between the ulcers remains healthy.
• Ulcers appear initially on the mucosa as raised nodules
with pouting edges measuring pinhead to l inch. They
later break down discharging brownish necrotic material
containing large numbers of trophozoites.
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Amebae
Figs 3A and B: (A) Intestinal amebiasis: Specimen showing amebic ulcer in colon; (B) Flask-shaped amebic ulcer
• The typical amebic uJcer is flask-shaped in cross section,
with mouth and neck being narrow and base large and
rounded.
Multiple uJcers may coalesce to form large necrotic lesions
with ragged and undermined edges and are covered
with brownish slough. Base is formed by muscular coat
(Figs 3A and B).
• The ulcers generally do not extend deeper than
submucosal layer, but amebae spread laterally in the
submucosa causing extensive undermining and patchy
mucosa! loss. Amebae are seen at the periphery of the
lesions and extending into the surrounding healthy
tissues. Occasionally, the ulcers may involve the muscular
and serous coats of the colon, causing perforation and
peritonitis. Blood vessel erosion may cause hemorrhage.
• The superficial lesions generally heal without scarring, but
the deep ulcers form scars which may lead to strictures,
partial obstruction and thickening of the gut wall.
• Ameboma: Occasionall y, a gra n ulomatous
pseudotumoral growth may develop on the intestinal
wall by rapid invasion from a chronic ulcer. This amebic
granuloma or ameboma may be mistaken for are
maUgnant tumor. Amebomas are most frequent at cecum
and rectosigmoid junction (Box 2).
Systemic manifestations of ameboma are rectal tenesmus,
high fever, abdominal discomfort, anorexia and nausea.
Clinical f eatures of intestinal amebiasis: The clinical picture
covers a wide spectrum from noninvasive carrier state to
fulminant colitis (Box 3).
• The incubation period is highly variable from 1- 4 months.
• l he clinical course is characterized by prolonged latency,
relapses and intermissions.
• The typical manifestation of intestinal amebiasis is amebic
dysentery. This may resemble bacillary dysentery, but
can be differentiated on clinical and laboratory grounds.
Box 2: Lesions in chronic intestinal amebiasis
• Small superficial ulcers involving only the mucosa.
• Round or oval-shaped with ragged and undermined margin and flaskshaped in cross section.
• Marked scarring of intestinal wall with thinning, dilatation and
sacculation.
• Extensive adhesions with the neighboring viscera.
• Format ion of tumor-like masses of granulation tissue (ameboma}.
Box 3: Complications and sequelae of intestinal amebiasis
• Fulminanr amebic colitis:
Toxic megacolon
Perianal ulceration
Perforation and generalized peritonitis
• Ameblc appendicitis
• Ameboma
• Extraintestinal amebiasis:
Amebic hepatitis
Amebic liver abscess
Pulmonary amebiasis
Cerebral amebiasis
Splenic abscess
Cutaneous amebiasis
Genitourinary amebiasis
Pericardia! amebiasis
Compared to bacillary dysentery, it is usually insidious in
onset and the abdominal tenderness is less and localized
(Table 2).
• 1he stools are large, fou l-smelling and brownish black,
often with blood streaked mucus intermingled with
feces. The red blood cells (RBCs) in stools are clumped
and reddish-brown in color. Cellular exudate is scanty.
Charcot-Leyden crystals are often present. E. histolylica
trophozoites can be seen containing ingested eryth rocyces.
• The patient is usually afebrile and nontoxic.
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Paniker's Textbook of Medical Parasitology
Table 2: Differential features of amebic and bacillary dysentery
features amebic Bacillary dysentery
Clinical
Onset Slow Acute
Fever Absent Present
Toxicity Absent Present
Abdominal Localized Generalized
tenderness
Tenesmus Absent Present
Stool
Frequency 6-8 per day Over 10 per day
Odor Offensive Nil
Color Dark red Bright red
Nature Feces mixed with blood Blood and mucus with
and mucus I ittle or no feces
Consistency Not adherent Adherent to container
Reaction Acid Alkaline
Microscopy
Cellular exudates Scanty Abundant
Red blood cells Clumped, yellowish Discrete or in rouleaux,
brown bright red
Macrophages Few Several, some with
ingested red blood cells
Eosinophils Present Absent
Charcot-Leyden Present Absent
crystals
Motile bacteria Present Absent
Ameba Motile trophozoites with Absent
ingested red blood cells
• In fulminant colitis, there is confluent ulceration a nd
necrosis of colon. The patient is febrile and toxic.
• Intestinal amebiasis does not always result in dysentery
Quite often, the re may be on ly diarrh ea or vague
abdominal symptoms popularly called \"uncomfortable
belly\" or \"growling abdomen''.
• Chronic involvement of the cecum causes a condition
simulating appendicitis.
Extraintestinal Amebiasis
The various extraintestinal lesions in amebiasis have been
summarized in Flow chart 2 and depicted in Figure 4.
He11atic amebiasis: Hepatic invo lvement is the most
common excraintestinal complication of amebiasis. Although
trophozoites reach the liver in most ca es of amebic dysentery,
only in a small proportion do they manage to lodge a nd
Flow chart 2: Sites affected in amebiasis
Lungs
+
Subphrenic
abscess
Diaphragm j
I Pericardium
Peritoneum
Skin Liver - ---+-• Stomach
Portal circulation
Intestine
Inferior vena
cava
Spleen
Peritoneum I- Primary infection 1------t• Suprarenal _ in colon Kidney
General
Perianal skin
Genitals
Fig. 4: Specimen showing amebic liver abscess
multiply there. 1n the tropics, about 2- 10% of the individuals
infected with E. histolytica suffer from hepatic complications.
• The history of amebic dysentery is absent in more than
50% of cases.
Several patients wi1.h amebic colitis develop an enlarged
tender liver without detectable impairment of liver
fun ction or fever. lhis acute hepatic involvement (amebic
hepatitis) may be due to repeated invasion by amebae
from an active colonic infection or to toxic substances
from the colon reaching the liver. It is probable that
liver damage may not be caused directly by the amebae,
but by lysosomal enzyme of lysed polymorphonuclear
neutrophils and monocytes a nd cytokines from the
inflammatory cells surrounding the trophozoites.
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Amebic liver abscess:
• In about 5-10% of persons with intestinal amebiasis, liver
abscesses may ensue (Fig. 4). The center of the abscess
contains thick chocolate brown pus ( anchovy sauce pus),
which is liquefied necrotic liverussue. ltis bacteriologically
sterile and free of ameba. At the periphery, there is almost
normal liver tissue, which contains invading ameba
(Flow chart 3A).
• Liver abscess may be multiple or more often solitary,
usually located in the upper right lobe of the liver. Cardinal
signs of amebic liver abscess is painful hepatomegaly.
Fever is present in most cases. Anorexia, nausea, weight
loss and fatigue may also be present. About third-fourth
cases of amebic liver abscess have leukocytosis (>10,000/
µL) and increased serum transaminases. Jaundice
develops only when lesions are multiple or when they
press on the biliary tract.
• Untreated abscesses tend to rupture into the adjacent
tissues through the diaphragm into the lung or pleural
cavity, pericardium, peritoneal cavity, stomach, intestine,
or inferior vena cava or externally through abdominal wall
and skin.
• Amebic liver abscess is 10 times more frequent in adults
than in children and three times more frequent in males
than in females.
Pulmonary amebiasis: Very rarely, primary amebiasis of
the lung may occur by direct hematogenous spread from the
colon bypassing the liver, but it most often follow extension
Amebae
of hepatic abscess through the diaphragm and therefore, the
lower part of the right lung is the usual area affected (Fig. 5).
• J lepatobronchial fistula usually results with expectorauon
of chocolate brown sputum. Amebic empyema develop
less often.
• The patient presents with severe pleuritic chest pain,
dyspnea and nonproducuve cough.
Metastatic amebiasis: Involvement of distant organs is by
hematogenous spread and through lymphatics. Abscesses
in kidney, brain, spleen and adrenals have been noticed.
Spread to brain leads to severe destruction of brain tissue and
is fatal.
Cutaneous amebiasis: It occurs by direct extension around
anus, colostomy site, or discharging sin uses from amebic
abscesses. Extensive gangrenous destruction of the skin
occurs. The lesion may be mistaken for condyloma or
epithelioma.
Genitourinary amebiasis: The prepuce and glans a re
affected in penile amebiasis which is acquired through anal
intercourse. Similar lesions in females may occur on vulva,
vagina, or cervix by spread from perineum. The destructive
ulcerative le ions resemble carcinoma.
Laboratory Diagnosis
Diagnosis of Intestinal Amebiasis
Stool examination: Intestinal amebiasis has to be
differentiated from bacillary dysentery (Table 2). The stool
Flow charts 3A and B: (A) Laboratory diagnosis of amebic liver abscess; (BJ Laboratory diagnosis of Entamoeba histolytica
•
Microscopy
of pus or aspirate
+
Stool examination
• Microscopy
• Macroscopy
• Iodine stained
preparatron
• Trichome stained
preparations to
demonstrate trophozoite
or c st
A. Laboratory diagnosis of amebic liver abscess l I
•
Histopatholoigical
examination
of pus or aspirate
Amebic liver abscess I
l
Serodiagnosis
• IHA
• ELISA
• Latex agglut1na11on
test
•
Radiological
examination
• X-ray
• USG
• CT scan
• MRI
B. Laboratory diagnosis of Entamoeba histolytica
Intestinal amebiasis I
1 ., +
Stool culture
Media used
• Boeck and Drbohlav
• NIH polygenlc
• Craig's • Nelson's
• Robinson's
Mucosal scrapings
Wet mount
Stained preparation
+
Serodiagnosis
• IHA
• ELISA • Latex agglutination
test
•
Stool
examination
Molecular • diagnosis
• DNA probe
Abbreviations: CT. computed tomography; DNA. deoxyribonucleic acid: ELISA, enzyme-linked immunosorbent assay;
IHA, indirect hemagglut,nation: MRI. magnetic resonance imaging; USG, ultrasonography
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Paniker's Textbook of Medical Parasitology
Lung-+- -...,.,..-+-
abscess ~ ------~
Subdiaphragmatic--- -+
abscess
Liver----
abscess
Amebic-----M,
ulcers colon
Periappendiceal-- - --
abscess
Fig. 5: Lesions of amebiasis
Ameboma
colon
should be collected into a wide mouth container and examined
without delay. It should be inspected macroscopically as well
as microscopically (Flow chart 38).
• Macroscopic appearance: The stool is fou l-smelli ng,
copious, semiliquid, brownish -black in color and
intermingled with blood and mucus. It does not adhere
to the container.
• Microscopic appearance:
saline preparation:
• The cellular exudate is scanty and consists of only
the nuclear masses pyknotic bodies) of a few
pus cells, epithelial cells and macrophages.
• The RBCs are in clumps and yellow or brown -red
in color.
• Charcot-Leyden crystals are often present. These
are diamond-shaped, clear and refractilc crystals
(Fig. 6).
• Actively motile trophozoites throwin g pseudopodia can be demonstrated in freshly-passed
stool. Presence of ingested RBCs clinches the
identity of E. hislolytica. ucleus is not visible but
a faint outline may be detected.
• Cyst has a smooth and thin cell wall and contains
round refractile chromatoid bars. Glycogen mass
is not visible.
..
Fig. 6: Charcot-Leyden crystals
Iodine preparation:
• For the d emonstration of cysts or d ead
trop hozoites, stained preparations may be
required for the study of the nuclear character.
Iodine-stained pre pa ration is commonl y
employed for this purpose. The trophozoite of E.
histolytica stains yellow to light brown. ucleus
is clearly visible witl1 a central karyosome. The
cytoplasm of the cystic stage shows smooth
and hyaline appearance. uclear chromatin
and karyosome appear bright yellow. Glycogen
masses stain golden brown and chromatoid
bars are not stained. Trichrome stain is useful
to demonstrate intracellular features of both
trophozoites and cysts.
• Since excretion of cysts in the stool is often
interminent, at least three consecutive specimens
should be examined (Fig. 7).
Mucosal scrapings: Scraping obtained by sigmoidoscopy is
often contributory. Examination method includes a direct
wet mount and iron hematoxylin and immunofluorescent
staining with anti-E. hislolytica antibodies.
Stool culture: Stool culture is a more sensitive method in
diagnosing chronic and asymptomatic intestinal amebiasis.
Culture of stools yields higher positivity for E. histolytica
as compared to direct examination.
Polyxenic culture is done in enriched medium which contains
bacteria, protozoa, serum, starch, etc. for nourishment of the
ameba.
Media used for polyxenic culture include:
• Boeck and Drbohlav's biphasic medium
• ill polygenic medium
• Craig's medium
P:45
, ·~ .. ~ ..
Left- E. histolytica trophozoite, minuta form-smaller,
no ingested erythrocytes. Right-Trophozoite, magna
(tissue invading) form containing ingested erythrocytes.
Left-E. hislo/ytica uninucleate cyst. Middle-Binucleate cyst
Right-quadrinudeate cyst.(Heidenhain's hematoxytin
stain. Magnification 2000X)
Fig. 7: Entamoeba histolytica as it appears in laboratory specimen
• Nelson's medium
• Robinson's medium
• Balamuth's medium.
Axenic culture is done in medium that does not require
presence of other microorganisms. Diamond's axenic
medium is commonly used. Axenic cultures are used for:
• Studies ofpathogenicity
• Antigenic characterization
• Drug sensitivity of ameba.
To obtain growth in these media 50 mg of formed stools
or 0.5 mL of liquid stool containing cyst or trophozoites of
ameba is inoculated and incubated at 37°C.
Serodiagnosis: Serological tests become positive only in
invasive amebiasis.
Antibody detection: Amebic antibodies appear in serum only
in late stages of intestinal amebiasis. Test for antibodies in
scrum help in diagnosis of mainly extra intestinal infections.
Serological tests include indirect hemagglutination
assay (IHA), indirect fluorescent a ntibody (IFA), enzymelinked immunosorbent assay (ELISA), counter-current
immunoelectrophoresis (CIEP) and latex agglutination tests.
Serum with antibody titer of 1:256 or more by IHA and 1:200
by IFA are considered to be significant.
Amebic a ntigen detectton: Amebic antigen in serum a re
detected only in patients with active infections and disappears
after clinical cure. Antigen like Lipophosphoglycan (LPG)
amebic lectin, serine rich E. histolylica protein (SREHP) are
detected using monoclonal antibodies by ELISA.
Amebae
Stool antigen detection: Detection of coproantigen of E.
histolytica in stool by microwell EUSA is more sensitive than
stool examination and culture.
Commercially available ELISA tests like Techlab E.
histolytica ll to detect Entamoeba antigen are more easily
performed and are being used with increasing frequency.
Molecular diagnosis: Recently, deoxyribonucleic acid (DNA)
probes and radioimmunoassay have been used to detect E.
histolytica in stool. It is a rapid and specific method.
Real-lime polymerase chain reaction (RTPCR) is a sensitive
test for detection of E. histolytica from pus of liver abscess.
Diagnosis of Extraintestinal Amebiasis
Microscopy: Microscopic examination of pus aspirated from
liver abscess may demonstrate trophozoite of E. histolytica in
less than 20% cases. Tn case ofliver abscess, when diagnostic
aspiration is done, the pus obtained from the center of the
abscess may not contain ameba as they are confined to the
periphery. The fluid draining after a day or two is more likely
to contain the trophozoite. Aspirates from the margins of the
abscess would also show the trophozoites. Cysts are never
seen in extraintestinal lesions.
Liver biopsy: Trophozoite of E. histolytica may b e
demonstrated in liver biopsy specimen, in case of hepatic
amebiasis or amebic hepatitis.
Serological test: Serological test, are of immense value in the
diagnosis of hepatitis amebiasis.
Craig (1928) was the first to report a complement.fixation
test in amebiasis. Subsequently a number of different
serological tests have been developed including:
• indirect hemagglutination (IHA)
• Latex agglutination (LA)
• Gel diffusion precipitation (GDP)
• Cellulose acetate membrane precipitation (CAP) test
• Counter-current immunoelectrophoresis (CJE)
• Enzyme linked immunosorbent assay (ELJSA)
While IHA and LA are highly sensitive, they often give
false-positive results. They remain positive for several years
even after successful treatment. Gel precipitation tests are
less sensitive, but more specific. ELISAs are both sensitive
and specific and tests like GDP and CIE become negative
within 6 months of successful treatment.
Stool examination: It is not of much value as E. histolytica
cyst can be detected in stool in less than 15% cases of amebic
hepatitis.
Radiological examination:
On X-ray, the right lobe of the liver is generally found to be
situated at a higher level.
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Paniker's Textbook of Medical Parasitology
• Radioisotope scan of the liver may locate the spaceoccupying lesions.
• Ultrasonography (USG), computed tomography (CT)
scan, or magnetic resonance imaging (MRI) of liver may
be found useful in detection of amebic liver abscess (Plow
chart 3A).
The diagnosis of amebic liver abscess is based on the
detection (generally by USG or CT) of one or more spaceoccupying lesions in the liver and a positive serologic test for
antibodies against £. histolytica antigens. When a patient has
a space-occupying lesion of the liver and a positive amebic
serology, it is highly sensitive (>94%) and highly specifi c
(>95%) for the diagnosis of amebic liver abscess {Flow
chart 3A).
Immunity
Infection with invasive strains includes both humoral and
cellular immune responses. Local and systemic antibodies
can be demonstrated within a week of invasive infection.
All classes of immunoglobulins are produced but IgG is
predominant.
Immunoglobulin A plays an important role in Immoral
immunity to E. histolytica to resist Gal/GalNAc lectin.
Infection confers some degree of protection as evidenced
by the very low frequency of recurrence of invasive colitis
and liver abscess in endemic areas. The course and severity
of amebiasis does not seem to be a ffected by human
immun odefi cien cy virus (HIV) infection. Serological
response i hardly ever seen in infection with noninvasive
zymodemes.
Treatment
Three classes of drug are used in Lhe treatment of amebiasis:
Luminal amebicides: Diloxanide furoate, iodoquinol,
paromomycin and tetracycline act in the intestinal lumen
but not in tissues.
Tissue amebicides: Emetine, chloroquine, etc. are
effecLive in systemic infection, but less effective in the
intestine. Dosage of chloroquine in amebic liver abscess
is l g for 2 days followed by 5 g daily for 3 weeks.
Both luminal and tissue amebicides: Metronidazole and
related compounds like rinidazole and om idazole act on
both sites and are the drug of choice fo r treating amebic
colitis and amebic liver abscess.
No te: Although metronidazole a nd tinidazole act on both
the sites but neither of them reach high levels in th e gut
lumen; therefore, patients with amebic colitis or amebic
liver abscess should also receive treatment with a luminal
agent (paromomycin or iodoquinol) to ensure eradication of
infection (Table 3). Paromomycin is the preferred agent.
• Asymptomatic individuals with docum ented E. histolytica
infection should also be treated because of the risks of
Table 3: Recommended dosages of antiamebic drugs
Drug Dosage Duration (In days}
Amebic co/iris or amebic liver abscess
Tinidazole 2 g/day orally 3
Metron idazole 750 mg three times a day, 5-10
orally or intravenous (IV)
Intestinal amebiasis
Paromomycin 30 mg/kg four times a day, 5-10
orally in three divided doses
lodoquinol 650 mg orally, three times 20
a day
developing amebic colitis or amebic liver abscess in the
future and risk of transmitting the infection to others.
Paromomycin or iodoquinol in the doses listed in the
Table 3 should be used in these cases.
• Oral rehydration and electrolyte replacement should be
done wherever necessary.
Aspiration of liver abscess can be done as an adjunct to
medical treatment in case of imminent rupture.
Prophylaxis
General prophylaxis is as for all fecal-oral infections. Food
and water have to be protected from contamination with
human excreta.
Detection and treatment of carriers and their exclusion
from food handling occupations will help in limiting the
spread of infection.
• Health education and inclusion of healthy personal habits
helps in control.
• NONPATHOGENIC INTESTINAL AMEBA
Entamoeba Coli
E. coli was first described by Lewis (1870) and Cunningham
{1871) in Kolkata and its presence in healthy persons was
reported by Grassi (1878).
.
.
.
It is worldwide in distribution and a nonpathogenic
conunensal .intestinal ameba.
It is larger than£. histolytica about 20-50 µm with sluggish
motility and contains ingested bacteria but no red cells.
The nucleus is clearly visible in unstained films and has a
large eccentric karyosome and thick nuclear membrane
lined with coarse granules of chromatin (Figs 8A and B).
Cysts are large, 10- 30 µm in size, with a prominent
glycogen mass in the early stage. The chromatoid bodies
are splinter-Like and irregular. The mature cyst has eight
nuclei (Fig. 8C).
The life cycle is the same as in E. histolytica except that
it remains a luminal commensal without tissue invasion
and is nonpathogenic.
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----~ -- Nucleus with
eccentric
nucleolus
....::.i'---+-- Phagocytosed
bacteria
Nucleus
m
Nucleus
Chromatoid
bodies
Amebae
Cyst membrane
Figs 8A to C: Schematic diagram of the morphological forms of Entamoeba coli (Heidenhain's hematoxylin magnification 2000X).
(A) Vegetative form; (B) Binucleate cyst; and (C) Eight-nucleate cyst
Entamoeba Hartmanni
E. hartmanni occurs wherever E. histolytica is found. le is
now considered ro be a separate species of nonparhogenic
commensal intestinal ameba.
• It is much smaller than E. histolytica, the trophozoirc
measuring 4- 12 µm and cyst 5-10 µmin size (Fig. 9).
• Trophozoites do nor ingest red cells and their motility is
less vigorous.
• 1he cyst resembles that of Endolimax nana.
Differential features of cyst and trophozoites of E. coli, E.
hartmanni and E. histolytica are shown in Table 4 .
Entamoeba Gingivalis
E. gingivalis was the first ameba of humans, discovered by
Gros in 1849.
• It is global in disn·ibution.
• Only the trophozoite is fou nd; the cystic stage being
apparently absent.
The trophozoite is about 10-20 µm, actively motile with
multiple pseudopodia.
• The cytoplasm contains food vacuoles with ingested
bacteria, leukocytes and epithelial cells.
• Nucleus is round with central karyosome lined by coarse
chromatin granules.
• The ameba lives in gingival tissues and is abundant
in unhygienic mouths. [t is a commensal a nd is not
considered to cause any disease.
• It is transmitted by direct oral contact.
• E. gingivalis have been found in bronchial washings and
vaginal and cervical smears, where it can be mistaken for
E. histolytica.
___ .....,.. _ _ Nucleus with
central nucleolus
Fig. 9: Trophozoite of Entamoeba hartmanni
Endolimax Nana
This common commensal ameba is widely distributed.
• It lives in rhe human intestine.
• The trophozoite is small (nana: small), less than 10 µmin
size with a sluggish motility (Fig. lOA).
• The nucleus has conspicuous karyosome connected to
nuclear membrane by one or none coarse strands.
• The cyst is small, oval and quadrinucleate with glycogen
mass and chromidial bars, which are inconspicuous or
absent (Fig. 108).
• It is nonpathogenic.
lodamoeba Butsch/ii
This is widely distributed, though less common than E. coli
and E. nana.
The trophozoice is small, 6- 12 µm, with conspicuous
nucleus (Fig. llA).
• The prominent karyosome is half the size of the nucleus,
having bull's eye appearance.
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Paniker's Textbook of Medical Parasitology
Table 4: Differential features of intestinal Entamoeba
E. hlstolytica
12- 60
Active
E. coll
20-50
Sluggish
E.hartmanni__ ....,.
4--12
Active
Trophozoire
Size (µm)
Motility
Pseudopodia
Cytoplasm
Finger-shaped, rapidly extruded
Clearly defined into ectoplasm and
endoplasm
Short, blunt, slowly extruded
Differentiation, not distinct
Finger-shaped, rapidly extruded
Clearly defined into ectoplasm and
endoplasm
Inclusions Red blood cells (RBCs) present, no
bacteria
Bacteria and other particles, no RBCs Bacteria and other particles, no RBCs
Nucleus
Karyosome
Nuclear membrane
Cyst
Not clearly visible in unstained films
Small, central
Visible in unstained films
Large, eccentric
Not visible in unstained films
Small, eccentric
Delicate, with fine chromatin dots Thick, with coarse chromatin granules Coarse chromatin granules
Size (µm) 10-15
4
10-30
8
5-10
Nucle 4 i in mature cyst
Glycogen mass Seen in uninucleate, but not in
quadrinucleate stage
Seen up to quadrinucleate stage Seen in uninucleate, but not in
quadrinucleate stage
Chromidial 1-4 with crounded ends Splinter-like with angular ends Many with irregular shape
Figs 1 OA and B: Endolimax nana. (A) Vegetative form: and
(B) Quadrinucleate cyst
• The cyst is oval, uninucleate and has a prominent iodine
staining glycogen mass (iodophilic body). Hence, the
name lodamoeba. It is nonpathogen.ic {Fig. llB).
The comparative morphology of amebae infecting
humans is illustrated in Figure 12.
• PATHOGENIC FREE-LIVING AMEBAE
Among the numerous types of free-living amebae fo und in
water and soil, a few are potentially pathogenic and can cause
human infections.
• Primary amebic meningoencephalitis: It is caused by
ameboflagellate Naegleria (the brain-eating Amoeba).
• Granulomatous amebic encephalitis and chronic
amebic keratitis: It is caused by Acanthamoeba.
A few instances of granulomatous amebic encephalitis
(GAE) caused by lyptomyxid ameba like Balamuthia have also
been reported. While primary amebic meningoencephalitis
El
B
~ ~ ~--Food vacuole
containing
bacteria
Nucleus
__::,,,-.-~ ---Glycogen
containing
vacuole
Fig s 11 A and B: lodamoeba butschlii.
(A) Vegetative form: and (B) Cyst
(PAM) and c hron ic amebic keratitis (CAK) occur in
previously healthy individual, GAE has been associated with
immunodeficient patients.
The term amphiwic has been used for organisms, which
can multiply both in the body of a host (endozoic) and in freeliving (exozoic) conditions.
Naegleria Fowleri
It is the only species of genus Naegleria, which infects man.
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::::
:c: . (.) ... II)
~ .Q
Trophozoite Nucleus
@
0
Fig. 12: Comparative morphology of amebae infecting humans.
showing trophozoite and cyst stages, as wel l as enlarged
representation of their nuclear structure
N. Jowleri causes the disease primary amebic
meningoencephalitis (PAM), a brain infection that leads to
destruction of brain tissue.
History and Distribution
N. Jowleri is named after Malcolm Fowler, who along with
Carter described it first from Australia in 1965.
• N. fowleri is a heat-loving (thermophilic) ameba that
thrives in warm water at low oxygen tension and is
commonly found in warm freshwater (e.g. lakes, rivers,
and springs) and soil.
• It is worldwide in distribution.
Amebae
Morphology
N. Jowleri occurs in three forms:
Cyst
Ameboicl trophozoite form
Flagellate trophozoite form.
Trophozoite stage: The rrophozoites occur in two forms, (I)
ameboid and (2)jlagellate.
Ameboid form: The ameboid form is about 10-20 µm,
showing rounded pseudopodia (lobopodia), a spherical
nucleus with big endosome and pulsating vacuoles.
With electron microscopy, vacuoles appear to be densely
granular in contrast to highly vacuolated body of ameba
and are called as ameboslomes. They are used for
engulfing RBCs and white blood cells (WBCs) and vary in
number, depending on the species.
• Ameboid form is the feeding, growing, and replicating
form of the parasite, seen on the surface of vegetation,
mud and water.
• It is the invasive stage of the parasite and the infective
form of the parasite.
Flagellate form: The biflagella te form occurs when
trophozoites are transferred to distilled water .
• This transformation of trophozoites co bijlagellate pearshaped form occurs within a minute.
• tje flagellate can revert to the ameboid form, hence N.
Jowleri is classified as amebojlagellate.
Cyststage:Trophozoites encyst due co unfavorable conditions
like food deprivation, desiccation, cold temperature, etc.
The cyst is spherical 7-10 µm in diameter and has a
smooth double wall.
They are the resting or the dormant form and can resist
unfavorable conditions, such as drying and chlorine up
to SO ppm.
The cyst can withstand moderate heat ( 45°C), but die at
chlorine levels of2 ppm and salinity of0.7%.
Cysts and flagellate forms of N. Jowleri have never been
found in tissues of cerebrospinal fluid {CSF).
Life Cycle
Typ ically, infection occurs when people go swimming
or diving in warm freshwater river or ponds and poorly
mainta ined swimming pools or nasal irrigati on usi ng
contaminated tap water (Fig. 13).
• The life cycle of N. Jowleri is completed in the external
environment.
• The ameboid form of trophozoite multiplies by binary
In the last 10 years from 2002 to 2011, 32 infections were fission.
reported in the United States (US), and in India, a total of •
17 cases have been reported so far.
Under unfavorable conditions, it forms a cyst and which
undergoes excystation in favorable conditions.
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Paniker's Textbook of Medical Parasitology
(~
Flagellate form
Man acquires infection
during swimming
Fig. 13: Life cycle of Naegleria fowleri
• Flagellate form of trophozoite helps in the spread of N.
Jowleri to new water bodies. Since the ameboid form is
the invasive stage, hence, the flagellate forms revert to
ameboid forms to become infective to man.
Pathogenicity and Clinical Features
Patients a re mostly previously healthy young adults or
children.
• Human infection comes from water containi ng the
amebae and usually follows swimming or diving in ponds.
• The amebae invade the nasal mucosa and pass through
the olfactory nerve branches in the cribriform plate into
the meninges, and brain to in itiate an acute purulent
meningitis and encephalitis, called as primary amebic
meningoencephalitis (PAM).
• the incubation period varies from 2 days to 2 weeks.
• In the incubation period, the patient experiences anosmia.
• TI1e disease advances rapidly, causing fever, headache,
vomiting, stiff neck, ataxia, seizure and coma.
• Cranial nerve palsies, especially of the third, fourth and
sixth nerves have also been documented.
• The disease almost always ends fatally within a week
(average 5 days).
Laboratory Diagnosis
The diagnosis of PAM is based on the find ing of motile
Naegleria trophozoites in wet moums of freshly obtained CSE
Cerebrospinal fluid examination: The CSF is cloudy to
purulent, with prominent neutrophilic leukocytosis, elevated
protein and low glucose, resembling pyogenic meningitis.
• Wet film examination of CSF may show trophozoites.
• Cysts are not found in CSF or brain.
• At autopsy, trophozoites can be demonstrated in brain
histologicalJy by immunofluorescent staining.
Culture: N. Jowleri can be grown in several kinds of liquid
axenic media or nonnutrient agar plates coated with
Escherichia coli. Both trophozoites and cysts occur in culture.
Molecular diagnosis: Newer tests based on PCR technology
are being developed.
Treatment
The drug of choice is amphotericin B intravenously. It can
also be instilled directly into the brain.
• Treatment combining miconazole and suJfadiazine has
shown limited success, only when administered early.
• More than 95% cases of PAM are fatal despite of treatment . Acanthamoeba Species
A. culbertsoni {formerly, HartmannelLa culbertsoni) is the
species most often responsible for human infection but other
species like A. polyphagia, A. castellanii and A. astromyx have
also been reported.
Distribution
This is an opportunistic protozoan pathogen found worldwide
in the environment in water and soil.
• Approximately, 400 cases have been reported worldwide.
Morphology
Acanthamoeba exists as active trophozoite form and a
resistant cysUc form.
• The trophozoite is large, 20- 50 µmin size and characterized
by spin e-like pseudopodia (acanthopodia).
• It differs from Naegleria in not having a flagellate stage
and in formin g cysts in tissues (Table 5).
• The polygonal double-walled cysts are highly resistant.
• The cysts are present in all types of environment, all over
the world.
Life Cycle
• Both trophozoitcs and cysts are infecUve.
Human beings acquire by inhalation ofcystortrophozoite,
ingestion of cysts, or through traumatized skin or eyes
(Fig. 14).
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Table 5: Differential features of Naegleria and Acanthamoeba
AmnCNmol6o
Disease Primary amebic Granulomatous amebic
meningoencephalitis encephalitis (GAE) and
(PAM) keratitis
portalof entry Nose Upper respiratory tract,
cornea
Clinical course Acute Subacute or chronic
Pathogenicity Acute suppurative Granulomatous
inflammation inflammation
Morphological Three stages: (1) Two stages: (1)
forms trophozoite, (2) cyst and trophozoite and (2) cyst
(3) nagellate form flagellate form absent
Trophozoite 10-20 µm, with a single 20-S0 µm, with spine-like
pseudopodia pseudopodia
Cyst 7- 10 µm, round with 1S- 25 µm, polygonal
smooth wall double-walled with
wrinkled surface
Nuclear By promitosis, nucleolus Nuclear membrane
division divides, nuclear dissolves
membrane persists
WBCinCSF Predominantly Predominantly
neutrophils lymphocytes
Abbrtviorions: CSF, cerebrospinal fluid; WBC, white blood cell
Man acquires infection
by inhalation and ingestion
of trophozoites and
cysts
Trophozoite showing
spinous acanthopodia
Fig. 14: Life cycle of Acanthamoeba culbertsoni
• After inhalation of aerosol or dust containing trophozoites
and cysts, the trophozoites reach the lungs and from
there, they invade the central nervous system through the
bloodstream, producing granulomatous encephalitis
(GAE).
Pathogenesis and Clinical Features
• Infectio n usually occurs in patients with immunodeficiency, diabetes, malignancies, malnutrition, systemic
lupus erythematosus (SLE), or alcoholism.
Amebae
• The parasite spreads hematogenously into central nervous
system. Subsequent invasion of the connective tissue and
induction of proinflammatory responses lead to neuronal
damage that can be fatal within days.
• A postmortem biopsy reveals severe edema and
hemorrhagic necrosis.
Clinical Disease
It presents chiefly as two chronic conditions: (1) keratitis and
(2) encephalitis.
• Acanthamoeba keratitis: An infection of the eye that
typically occurs in healthy persons and develops from the
entry of the amebic cyst through abrasions on the cornea.
Majority of such cases have been associated with the
use of contact lenses.
The picture resembles that of severe herpetic keratitis
with a slow relapsing course, but the eye is severely
painful in the amebic infection.
Unilateral photophobia, excessive tearing, redness
and foreign body sensation are the earliest signs and
symptoms; disease is bilateral in some contact lens
users.
Keratitis and uveitis can result in permanent visual
impairment or blindness.
• Granulomatous amebic encephalitis: It is a serious
infection of the brain and spinal cord that typically occurs
in persons with a compromised immune system.
Granulomatous amebic encephalitis is believed to
follow inhalation of the dried cysts.
The incubation period is long and the evolution of the
illness is slow.
Clinical picture is that of intracranial spaceoccupying lesions with seizures, pareses and mental
deterioration.
• Disseminated infection: In immunocompromised states
like acquired immunodeficiency syndrome (AIDS), a
widespread infection can affect skin lungs, sinuses, and
other organs independently or in combination.
Laboratory Diagnosis
• Diagnosis of amebic lceratitis is made by demonstration of
the cyst in corneal scrapings by wet mount, histology and
culture. Growth can be obtained from corneal scrapings
inoculated on nutrient agar, overlaid with live or dead
Escherichia coli and incubated at 30°C.
Rapid diagnosis can be made by identification of ameba or
cyst in corneal scraping by fluorescent microscopy using
calcofluor white staining and !FA Lest ([FAT) procedure.
• Diag nosis of GAE is made by demonstra tion of
trophozoites and cysts in brain biopsy, culture and
immunofluorescence microscopy using monoclonal
antibodies.
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Paniker's Textbook of Medical Parasitology
Cerebrospinal fluid shows lymphocytic pleocytosis,
slightly elevated protein levels, and normal or slightly
decrea ed glucose levels.
Computed LOmography scan o f brain provides
inconclusive findings.
Treatment
In acanlhamoeba keralitis, current Lherapy involves topical
administration ofbiguanide or chlorhexidine with or without
diamidine agent. In severe cases, where vision is Lhreatened,
penetrating keratoplasty can be done.
o effective trealmenl is available for \"GAE~ Multidrug
com binations including pentamidine, sulfadiazin e,
rifampicin and lluconazole are being used with limited
success.
Balamuthia Mandrillaris
B. mandrillaris, a leptomixid free-living am eba, is a newly
identified species reported to cause GAE.
Morphology
It exists in ameboid trophozoite stage. The flagellate stage is
absent
• It is relatively large (12-60 µm), irregul ar in shape and
actively motile by broad pseudopodia.
• Cyst of B. mandrillaris are usually spherical (6-20
µm), surrounded by a three-layered cyst wall: ( 1) outer
irregular ectocyst, (2) a middle mesocyst and (3) an inner
endocyst round wall. Under light microscopy, it appears
to have two walls: (1) an outer irregular wall and (2) an
inner smooth wall.
• Infection is transmitled through respiratory tract, skin
lesions, or eyes.
• Life cycle is similar to that of Acanthamoeba spp.
Clinical Disease
It causes granulomatous amebic encephalitis in both healthy
and immunocompromi sed hosts particularly in children and
elderly.
Laboratory Diagnosis
Laboratory diagnosis is done by identifying trophozoites of B.
mandrillaris in the CSF and trophozoites and cysts in brain
tissue.
Polymerase chain reaction also gives reliable diagnosis.
KEY POINTS OF AMEBAE
• E. histolytica is found in human colon and is mainly
asymptomatic.
• Cyst contains glycogen mass and 1-4 chromatid bars.
• Pathogenic strains are identified by genetic markers and
zymodeme analysis.
• Stools: In amebic dysentery, stool is copious, foul-smelling,
brownish black often with blood-streaked mucus.
• Amebic ulcers: Typical ulcers are discrete, flask-shaped,
with ragged undermined margin, found in cecum and
sigmoidorectal region.
• Amebic granuloma or ameboma may develop from chronic
ulcers.
• Extraintestinal complications: Amebic hepatitis and liver
abscess are the most common.
• Abscesses in other organs such as lung, brain, spleen and
genitourinary tract may result from hematogenous spread or
by direct spread from hepatic lesion.
• Diagnosis: By demonstration of trophozoites and cyst in
stool and also by serological tests and imaging techniques in
hepatic amebias1s.
• Treatment: By metronidazole or tinidazole along with
parmomycin, d1loxanide furoate, or chloroquine.
• E. hartmanni, E. coli, E. gingivalis, E. nana, and lodamoeba
are commensals and nonpathogenic amebae.
• Naeg/eria and Acanthamoeba are pathogenic free-living
ameba.
• N. fowleri occurs in three forms: (1) cyst, (2) trophozoite and
(3) nagellate. It causes PAM.
• Acanthamoeba species cause amebic keratltis and also GAE
in immunocompromised subjects.
REVIEW QUESTIONS
Describe briefly the life cycle and laboratory diagnosis of
Entamoeba histolytica.
Write short notes on:
a. Extraintestinal amebiasls
b. Free-living amebae
Differentiate between:
a. Amebic dysentery and bacillary dysentery
b. Enramoeba histolytica and Entamoeba coli
c. Naegleria and Acanthamoeba
MULTIPLE CHOICE QUESTIONS
The main reservoir of Entamoeba histolytica is
a. Man
b. Dirty water
c. Dog
d. Monkey
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The infective form of Entamoeba histolytica is
a. Trophozoite
b. Bi nucleate cyst
c. Quadrinucleate
d. None of the above
The pathogenicity of Entamoeba histolytica is indicaded by
a. Zymodeme pattern
b. Size
c. Nuclear pattern
d. ELISA test
M/C site for extra intestinal amebiasis is
a. Liver
b. Lung
c. Brain
d. Spleen
Amoebic liver abscess can be diagnosed by demonstrating
a. Cyst in the sterile pus
b. Trophozoites in the pus
c. Cyst in the intestine
d. Trophozoites in the feces
Stool of amoebic dysentry has all of the following characteristics
except
a. Charcot-Leyden crystals
b. Pyknotic bodies
c. RBCs
d. Ghost cell
The term ameboma is used to describe
a. Amebic liver abscess
b. Skin lesion due to draining amebic abscess
Amebae
c. Granuloma at ileocecal junction
d. None of the above
True statement regarding Entamoeba histolytica is
a. The trophzoites are infective to man
b. Mature cyst has eccenteric nucleolus
c. It can cause primary amebic encephalitis
d. Cyst are resistant to chlorine concentration used in drinking
water
All are nonpathogenic ameba living in the lumen of large
intestine except
a. Entamoeba coli
b. Entamoeba hartmanni
c. Endolimax nana
d. En tamoeba gingivalis
1 O. Chronic amebic keratitis in seen in
a. Entamoeba histolytica
b. Acanthamoeba
c. Naegleria fowleri
d. Hemoflagellates
11 . Etiologic agent of granulomatous amebic encephalitis is
a. Entamoeba histolytica
b. Acanthamoeba
c. Naegleria
d. Dientamoeba fragillis
Answer
a
d
C
d
a
b
a
b
b 6. d 7. C
P:54
CHAPTER 4
Intestinal, Oral and Genital
Flagellates
• INTRODUCTION
Parasitic protozoa, which possess whip-like flagella as their
organs of locomotion are called as flagella tes and classified
as:
Phylum: Sarcomasrigophora
Subphylum: Mastigophora
Class: Zoomastigophora (mastix: whip)
Depending on their habitat, they can be considered
under:
Lumen-dwellingflagellates: Flagellates fow1d in the
alimentary tract and urogenital tract (Table 1).
Hemojlagellates: Flagellates found in blood a nd
tissues (Table 1).
Most luminal flagellates are nonpathogenic commensals.
Two of them cause clinical diseases: (1) Giardia lamblia,
which can cause diarrhea, and (2) Trichomonas vaginalis,
which can produce vaginitis and urethritis.
Table 1: Flagellates
Group
Lumen-dwelling
flagellates
• Giardia lamblia
• Trichomonas vagina/is
• Trichomonas tenax
• Trichomonas hominis
• Chilomastix mesnili
• Enteromonas hominis
• Retortamonas
intestinalis
• Dientamoeba fragilis
Hemoflagellates • Leishmania spp.
• Trypanosoma brucei
• Trypanosoma cruzi
Habitat
• Duodenum and
jejunum
• Vagina and urethra
• Mouth
• Large intestine (cecum)
• Large intestine (cecum)
• Large intestine (colon)
• Large intestine (colon)
• Large intestine (cecum
and colon)
• Reticuloendothelial
cells
• Connective tissue and
blood
• Reticuloendothelial
cells and blood
• GIARD/A LAMBLIA
History and Distribution
It is one of the earliest protozoan parasites to have been
recorded.
• The flagellate was first observed by Dutch scientist
Antonie van Leeuwenhoek (1681) in his own stools.
• It is named Giardia after Professor Giard of Paris and
lamblia after Professor Lambie of Prague, who gave a
detailed description of the parasite.
• It is the most common protozoan pathogen and is
worldwide in distribution.
Endemicity is very high in areas with low sanitation,
especially tropics and subtropics. Visitors to such places
frequently develop traveler's diarrhea caused by giardiasis
through contaminated water.
Habitat
G. lamblia lives in the duodenum and upper jejunum and is
the only protozoan parasite found in the lumen of the human
small intestine (Box 1).
Morphology
It exists in two forms:
Trophozoite (or vegetative form)
Cyst (or cystic form).
Box 1: Protozoa found in small intestine
• Giardia Iambi/a
• lsospora be/Ji
• Cyclospora cayeranensis
• Cryptosporidium parvum
• Sarcocystis hominis and suihominis
P:55
Trophozoite
The trophozoite is in the shape of a tennis racket (heartshaped or pyriform-shaped) and is rounded anteriorly and
pointed posteriorly (Figs I and 2A and B).
.
.
.
.
It measures 15 pm x 9 mcgwide and 4 mcg thick .
Dorsally, it is convex; and ventrally, it has a concave
sucking disk, which helps in its attachmem to the
intestinal mucosa.
It is bilaterally symmetrical and possesses:
One pair of nuclei
Four pairs of flagella
Blepharoplast, from which the flagella arise (four
pairs)
One pair of axostyles, running along the mid line
Two sausage-shaped parabasal or median bodies,
lying transversely posterior to the sucking disk.
'The trophozoite is motile, with a slow oscillation about
its long ax.is, often resemblingfalling leaf
~
Ventral
aspect
0~
Lateral 5
aspect
0
Fig. 1: Giardia lamblia in duodenal fluid wet preparation.
Magnification 1500X
Sucking-- • ~ =-i\"~
disc
Basal bodies
of flagella
Nucleus
Para basal
body
m
Flagella
(4 pairs)
Sucking
disc
Nucleus
Flagella
(4 pairs)
m
Intestinal, Oral and Genital Flagellates
Cyst
It is the infective form of the parasite (Fig. 2C).
• The cyst is small and oval, measuring 12 mcgx 8 mcgand
is surrounded by a hyaline cyst wall.
• Its internal structure includes two pairs of nuclei grouped
at one end. A young cyst contains one pair of nuclei.
• The axostyle lies diagona lly, forming a dividing line
within cyst wall.
Remnants of the flagella and the sucking disc may be seen
in the young cyst.
Life Cycle
Giardia passes its life cycle in one host.
Infective Form
Mature cyst.
Mode of Transmission
• Man acquires infection by ingestion of cysts in contaminated water and food.
• Ingestion of as far as 10 cysts is sufficient to cause infection
in a man.
Children are commonly affected.
• Direct person-to-person transmission may also occur in
children, male homosexuals and mentally ill persons.
Enhanced susceptibility to giardiasis is associated with
blood group A, achlorhydria, use of Cannabis, chronic
pancreatitis, malnutrition, and immune defects such as
19A deficiency and hypogammaglobulinemia.
• Within half an hour of ingestion, the cyst hatches out
into two trophozoites, which multiply successively by
binary fission and colonize in the duodenum (Fig. 3).
• The trophozoites live in the duodenum and upper part
of jejunum, feeding by pinocytosis.
c
Axostyle
Thick cyst wall
Nuclei
Nucleoli
wv.- 4- Parabasal
bodies
Figs 2A to C: Trophozoite. (A) Ventral view; (B) Lateral view: and (Cl Quadrinucleate cyst
P:56
Paniker's Textbook of Medical Parasitology
Man acquires infection by
ingestion of cyst in
contaminated food
and waler
Trophozo,tes multiply by
binary fission.Remain
adhered to duodenal
mucosa
Excyslalion occurs in
the duodenum and 2
trophozoites hatch out
Fig. 3: Life cycle of Giardia lamblia
• During unfavorable conditions, encystment occurs usualJy
in colon (Fig. 3).
• Cysts are passed in stool and remain viable in soil and
water for several weeks.
• There may be 200,000 cysts passed per gram of feces.
• Injective dose is 10-100 cysts.
Pathogenicity and Clinical Features
G. lamblia is typically seen within the crypts of duodenal
and jejuna! mucosa. It does not invade the tissue, but remains
tightly adhered to intestinal epith elium by means of the
sucking disk.
• They may cause abnormalities of villous architecture
by cell apoptosis and increased lymphatic infiltration
of lamina propria. Loss of brush border epithelium
of intestine leads to deficiency of enzymes including
disaccharides.
• Variant-specific surface proteins (VSSPs) of Giardia play
an important role in virulence and infectivity of the
parasite. Antigenic variation helps the parasite in evasion
of host immune system.
Box 2: Protozoan parasites causing diarrhea
• Giard/a lamblia
• Entamoeba histolytica
• Cyclospora cayetanensis
• Cryptosporidium parvum
• lsospora be/Ii
• Often they are asymptomatic, but in some cases,
Giardia may lead to mucus diarrhea,fat malabsorption
(stearo rrhea), dull e pigastric pa in , belching and
flatulence. The stool contains excess mucus and fat but
no blood (Box 2).
• Children may develop chronic diarrhea, malabsorption
of fat, vitamin A, vitamin B12, folic acid, protein, sugars
like xylose disacch arides, weight loss a nd sprue-like
syndrome. Chronic giardiasis may be due to failure to
develop irnmunoglobulin A (IgA) against specific Giardia
antigen.
• Occasionally, Giardia may colonize the gallbladder,
causing biliary colic and jaundice.
• Incubation period is variable, but is usually about 2 weeks.
P:57
L
t
Laboratory Diagnosis
Stool Examination
Giardiasis can be diagnosed by id entification of cysts of
Giardia lamblia in the formed stools and the trophozoites
and cysts of the parasite in diarrheal stools (Flow chart 1).
• On macroscopic examination, fecal specimens containing
G. lamblia may have an offensive odor, are pale colored
and fatty, and float in water.
• On microscopic examination, cysts and trophozoites
can be fow1d in diarrheal stools by saline and iodine wet
preparations.
• Often multiple specimens need to be examined and
concentration techniques like formal ether or zinc acetate
are used. In asymptomatic carriers, only the cysts are
seen.
Enterotest (String Test)
A useful method for obtaining duodenal specimen is
enterotest. A coiled thread inside a small weighted gelatin
capsule is swallowed by the patient, after attaching the free
end of the thread in the cheek. The capsule passes through
the stomach to the duodenum. After 2 hours, the thread is
withdrawn, placed in saline, and is mechanically shaken.1 he
centrifuged deposit of the saline is examined for Giardia. The
use of enterotest is not recommended because of the very
high cost of the test.
Serodiagnosis
Antigen d etection: Enzyme-linked immunosorbent assay
(ELISA), immunochromatographic strip tests and indirect
immunofluorescence (IIF) tests using monoclonal antibodies
have been developed for detection of Giardia antigens in
feces (Flow chart I).
• The presence of antigen indicates active infection.
• Commercially available ELISA kits (ProSpecT/ Giardia
kit) detects Giardia-specific antigen 65 (GSA 65).
• 1l1e sensitivity of the test is 95% and specificity is I 00%,
when compared Lo conventional microscopy.
Intestinal, Oral and Genital Flagellates
• the test may be used for quantification of cysts and in
epidemiological and control studies, but not for routine
use.
Antibody detection: Indirect immunofluorescence test and
ELISA are used to detect antibodies against Giardia.
• Demonstration of antibodies is useful in the epidemiological and pathophysiological studies.
• lhese tests cannot differentiate between recent and past
infection and lack sensitivity and specificity.
Molecular Method
Deoxyribonucleic acid (D A) probes and polymerase chain
reaction (PCR) have been used to demonstrate parasitic
genome in the stool specimen (Flow chart l ).
Treatment
Metronidazole (250 mg, thrice daily for 5-7 days) and
tinidazole (2 g single dose) are the drugs of choice.
• Cure rates with metronidazole are more than 90%.
• Tinidazole is more effective than metronidazole.
• Furazolidone (100 mg QID x 7 days) and nitazoxanide
(500 mg BO x 3 days) are preferred in children, as they
have fewer adverse effects.
• Paromomycin, an oral aminoglycoside, can be given to
symptomatic pregnant females (500 mgTDS x 5 days).
Note: Only symptomatic cases need treatment.
Prophylaxis
Giardiasis can be prevented by following measures:
• Proper disposal of waste water and feces.
• Practice of personal hygiene like handwashing before
eating and proper disposal of diapers.
• Prevention of food and water contamination. Community
chJorination of water is ineffective for inactivating cysts.
Boiling of water and filtration by membrane filters are
required.
Flow chart 1: Laboratory diagnosis of Giardia tamblia
•
Stool examination
· Macroscopic
examination
• Microscopic
examination of
stained prepration
•
Laboratory diagnosis I
j
•
Enterotest
(string test)
Serological test
• Antigen detection
-ELISA
• IIF test • Antibody detection
-ELISA - IIF test
•
Molecular diagnosis
• DNA probe · PCR
Abbreviations: DNA, deoxyribonucleic acid; ELISA. enzyme-linked immunosorbent assay: IIF, indirect immunofluorescence; PCR, polymerase chain reaction
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Paniker's Textbook of Medical Parasitology
KEY POINTS OF GIARD/A LAMBLIA
• Giardia is the only protozoan parasite found in the lumen of
the human small intestine (duodenum and jejunum).
• Trophozoites a re pear-shaped, bilaterally symmetrical with
two nuclei, fo ur pairs of flagella and a ventral concave
sucking disk. They exhibit motility resembling a \"falling leaf\".
• Ellipsoid cysts contain four nuclei with remnants of flagella.
• Infective form: Ellipsoid cysts.
• Clinical features: Mostly asymptomatic but in some cases
may cause diarrhea, dull epigastric pain and malabsorption.
Stool contains excess mucus but no blood.
• Diagnosis: By microscopic demonstration of trophozoites
or cysts in stool, enterotest a nd s e rodiagnosis by ELISA
(ProSpecT/Giardia antigen assay).
• Treatment Metronidazole and tinidazole are the drugs of
choice.
• TRICHOMONAS
Trichomonas differs from other flagellates, as they exist only
in rrophozoite stage. Cystic stage is not seen.
Genus trichomonas has th ree species, which occur in
humans (Figs 4A to C):
l. T. vaginalis (Fig. 4A)
T. hominis (Fig. 4B)
T. tenax (Fig. 4C)
Trichomonas Vagina/is
History and Distribution
T: vaginalis was fi rst o bserved by Donne (1836) in vaginal
secretion.
Prevalence of trichomoniasis varies from 5% patients at
hospitals to 75% in sexual workers.
Morphology
It is pear-shaped or ovoid and measures 10-30 µm in length
and 5-10 µmin breadth with a short undulating membrane
reaching up to the middle of the body (Fig. 4A).
• It has four anterior flagella and fifth running along the
outer margin of the undulating membrane, which is
supported at its base by a flexible rod, costa.
• A prominent axostyle runs throughout the length of the
body and projects posteriorly like a tail.
• the cytoplasm shows prominent siderophilic granules,
which are most numerous alo ngside the axostyle and
costa.
• It is motile with a rapidj erky or twitching type movement.
Habitat
In females, it lives in vagina and cervix and may also be found
in Bartholin 's glands, urethra and urinary bladder. Ln males,
it occurs mainly in the anterior ureth ra, but may also be
fou nd in the prostate and preputial sac.
Life Cycle
Life cycle of T. vaginalis is com pleted in a single host eithe r
male or female.
Mode of transmission:
• The trophozoite cannot survive outside and so infection
has to be transmitted d irectly from person-to-person.
Sexual transmission is the usual mode of infection
(Box 3).
• Trichomoniasis often coexists wilh other sexually transmitted d iseases like candidiasis, gonorrhea, syph ilis, or
human immunodeficiency virus (HIV).
• Babies may get infected d uring birth.
• Vaginal pH of more than 4.5 facilitates infection.
Figs 4A to C: Trichomonas species. (A) T. vagina/is; (B) T. hominis; and (C) T. tenax
P:59
Box 3: Protozoa transmitted by sexual contact
, Trichomonas vagina/is
, Giardia Iambi/a
, Entamoeba histolytica
Fomites such as towels have been implicated in transmission.
• Trophozoites divide by binary fission.
As cysts are not formed, the lrophozoite itself is the
infective form.
• Incubation period is roughly 10 days.
Pathogenesis
T. vaginalis particularly infects squamous epithelium and not
columnar epithelium. It secretes cysteine proteases, adhesins,
lactic acid and acetic acid, which disrupt the glycogen levels
and lower the pH of Lhe vaginal fluid.
It is an obligate parasite and cannot live without close
association with the vaginal, urethral, or prostatic tissues.
Parasite causes petechial hemorrhage and mucosa!
capillary dilation (strawberry mucosa), metaplastic
changes and desquamation of the vaginal epithelium.
Intracellular edema and so called chicken-like epithelium,
is the characteristic feature oftrichomoniasis.
Clinical Features
Infection is ofte n asymptomatic, particularly in males,
although some may develop urethritis, epididymitis and
prostatitis.
• In females, it may produce severe pruritic vaginitis
with an offensive, yellowish green, often frothy discharge, dysuria and dyspareun ia. Cervical erosion is
common. Endometritis and pyosalpingitis are infrequent
complications.
• Ra rely, neonatal pneumonia and conj unctivitis h ave
been reported in infants born to infected mothers.
The incubation period of trichomoniasis is 4 days to
4 weeks.
Laboratory Diagnosis
Microscopic examination
Wet mount:
Vaginal or urethral discharge is examined microscopically
in saline wet mount preparation for characteristic jerky
and twitching motility and shape. In males, trophozoites
may be found in urine or prostatic secretions. An
abundance of leukocytes is seen.
Permanent stain:
Fixed smears may be stained with acridine orange,
Papanicolaou and Giemsa stains.
Intestinal, Oral and Genital Flagellates
Direct fluorescent antibody:
• Direct fluorescen t antibody (DFA) is another method of
detection of parasite and is more sensitive than the wet
mount.
Culture: Culture is recommended when direct microscopy is
negative and is considered as a \"gold standard\" as well as the
most sensitive (95%) method for the diagnosis of T. vagina/is
infection.
• It grows best at 35-37°C under anaerobic conditions. The
optimal pl I for growth is 5.5-6.0.
• It can be grown in a variety of solid or liquid media,
tissue culture and eggs. Cysteine-peptone-liver-maltose
(CPLM) medium and plastic envelope medium (PEM) are
often used.
Serology: Enzyme-linked immunosorbenl assay is used for
demonstration of T. vagina/is antigen in vaginal smear using
a monoclonal antibody for 65 kDA surface polypeptide of
T. vaginalis.
Rapid immunochromatographic tests (lCTs) are now
available for detection of Antigen like OSOM Trichomonas
rapid test, Xenostrip-Tv.
Molecular method: Deoxyribonucleic acid hybridization and
PCR are also highly sensitive (97%) and specific (98%) tests
for the diagnosis of trichomoniasis.
Sensitive and specific commercially available ucleic
acid amplification test (NAAT) has been developed (Aptima
Trichomonas vagina/is assay).
Treatment
Simultaneous treatment of both partners is recommended as
it is an STD.
Metronidazole 2 g orally as a single dose or 500 mg orally
twice a day for 7 days is the drug of choice.
In patients not responding to treatment with standard
regime, the dose ofmetronidazole may be increased or it
may be administered parenterally.
• In pregnancy, metronidazole is safe in 2nd and 3rd
trimesters.
Prophylaxis
Prevention is same as for other sexually transmitted diseases.
• Avoidance of sexual contact with infected partners and
use of barrier meth od during intercourse prevent the
disease.
Patient's sexual partner should be tested for T. vagina/is
when necessary.
Trichomonas Tenax
T. tenax, also known as T. buccalis, is a harmless commensal
which lives in mouth, in the periodontal pockets, carious
tooth cavities and, less often, in tonsillar crypts.
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Paniker's Textbook of Medical Parasitology
• It is smaller(S-10 µm) than T. vagina/is.
• It is transmitted by kissin g, through salivary droplets and
fomites. There are sporadic reports of its involvement in
respiratory infections a nd th oracic absce ses.
• Beller oral hygiene rapidly eliminates the infection and
no therapy is indicated.
Trichomonas Hominis
T. hominis measures 8- 12 µm, pyriform-shaped, and carries
five anterior flagella and a n undulating m embran e th at
extends the full length of the body.
• It is a very harmless commensal of the cecum.
• Microscopic examination of stool will reveal motile
trophozoite of T. hominis.
• Transmission occurs in rrophic form by fecal-oral route.
KEY POINTS OF TRICHOMONAS
• Trichomonas occurs only in trophozoite form, which is pearshaped, with five flagella and an undulating membrane.
• The motility is rapid jerky or twitching type.
• Habitat: Vagina and cervix in female and urethra in males.
• Clinical features: Often asymptomatic in males. In females,
it leads to pruritic vaginitis with greenish yellow discharge,
strawberry mucosa and dysuria.
• Diagnosis: By wet mount microscopy of vaginal or urethral
discharge, culture (gold standard), PCR and by demonstration
of antigen in vaginal smear by ELISA.
• Treatment: Metronidazole is the drug of choice and simultaneous treatment of both partners is recommended.
• CH/LOMASTIX MESNILI
This occurs as trophozoites and cysts (Fig. 5).
• The trophozoite is pear-shaped measuring 5-20 µm in
length and 5- 10 µmin breadth .
Flagella
Cytostome
Trophozoite Cyst
Clear
hyaline
knob
Cytostome
Nucleus
Fig. 5: Trophozoite and egg of Ch//omastix mesnifi
• At the anterior end, it has a spherical nucleus.
• A distinct spiral groove is seen on one side of the nucleus.
• The cysts are lemom-shaped having a spiral projection
at the anterior end. lt measures 5-10 µmin length and
4-6 µm in breadth and is surrotmded by a thick cyst wall.
• Both rrophozoites and cysts are demonstrated in the
semi-formed srool.
• It is a harmless commensal of cecum where the organism
feeds on bacteria and food debris. Since infection is
acquired through ingestion of cysts, prevention depends
on improved personal hygiene.
• ENTEROMONAS HOMINIS
£. hominis is a nonpathogenic commensal that lives in the
large inte tine, mainly in the cecum.
.
.
It exists in two forms: (1) trophozoile, and (2) cyst (Fig. 6).
the trophozoite is pear-shaped, with three anterior and
one posterior flagella.
It measures 5-10 µmin length and3-6 µmin breadth.
1he cytoplasm contains numerous bacteria and an
anteriorly placed nucleus but no cytostoma.
It shows jerky forward movements.
The cyst is oval in shape, measuring 5-8 µm in length and
4- 6 µm in breadth.
It contains 2-4 nuclei.
The cyst of E. fzominis may mimic a two-nucleated
cyst of E. nana.
lnfection occurs through fecal-oral route by ingestion of
cysts in contaminated food and water.
Diagnosis is made by identification of trophozoites or
cysts in the stool by iron hematoxylin stain.
• RETORTAMONAS INTESTINALIS
Wenyon and O'Connor first observed the parasite in stool in
Egypt.
Nucleus __:,n-r-3 anterior
flagella
-----Posterior
flagella
T rophozoite
Nuclei
Cyst
Fig. 6: Trophozoite and cyst of Enteromonas hominis
1
I
P:61
• R. intestinalis is a small non pathogenic flagellate found in
the large intestine.
• It also exists in two forms: (1) trophozoite, and (2) cyst.
• The trophozoite is elongated, pyriform in shape,
measuring 5- 10 mcm in length and 3-4 µmin breadth.
The cytoplasm is granular and vacuolated.
It has a cleft-like cytosome, spherical nucleus and
central karyosome.
Two minute blepharoplasts are present near nucleus,
from which two flagella origina te.
The trophozoite multiplies by binary fission.
• The cyst is ovoid or pyriform in shape, measuring 6 µm
in length and 3 µmin breadth.
• Water and food contaminated by cysts are the main
source of infection.
Diagnosis is made by identifying the cysts and trophozoites in the direct wet mount and iron hematoxylin-stained
specimen of stool.
• DIENTAMOEBA FRAG/L/S
D. Jragilis was previously considered as an amoeba but
has now been reclassified as an amoebojlagellate, based
on e lectron microscopic study and antigenic similarity to
Trichomonas.
• It is unique as it has only trophowile stage but no cyst
stage.
• The name Dien/amoeba Jragilis is derived from the
binucleate nature of trophozoite (Dien/amoeba) and the
fragmented appearance (fragilis) of its nuclear chromatin.
• It is seen worldwide and is reported to be the most
common intestinal protozoan parasite in Canada.
• It lives in colonic mucosa] crypts, feeding on bacteria. It
does not invade tissues, but may rarely ingest red blood
cells (RBCs).
• The trophozoite is 7- 12 µm in diameter. It is motile
with broad hyaline leaf-like pseudopodia. They have
1- 4 nuclei; the binucleace form being the most common
(Fig. 7). The nuclear chromatin is present as 3-5 granules
in the center, with no peripheral chromatin on the n uclear
membrane.
• In the absence of cyst stage, its mode of transmission
is not clear. Possibly, it is transmitted from person-toperson by th e fecal-oral route or by the eggs of Enterobius
vermicularis and other nematodes, which may serve as a
vector.
Formerly believed to be nonpathogenic, it has now been
associated with a variety of symptoms like intermittent
diarrhea, abdominal pain, flatulence, anorexia, nausea,
malaise a nd fatigue.
• High incidence is seen among children between 2 years
and 10 years of age.
Intestinal, Oral and Genital Flagellates
~ o
l..• >.--~~lr-lngested
bacteria
' ...
Fig. 7 : Trophozoite of Dientamoeba fragilis
Laboratory diagnosis is made by demonstration of
trophozoites in stool. At least three stool specimens
should be collected over a period of7 days.
Metronidazole, iodoquinol, paromomycin and tetracycline have been used for treatment
REVIEW QUESTIONS
Descri be briefly the life cycle and laboratory diagnosis of
Giardia Jamblia.
Write short notes on:
a. Trichomonas vagina/is
b. Dientamoeba fragilis
MULTIPLE CHOICE QUESTIONS
Normal habitat of Giardia is
a. Duodenum and jejunum
b. Stomach
c. Cecum
d. Ileum
All of the following protozoans are found in small intestine
except
a. Giardia lamblia
b. Balantidium coli
c. Cyclospora caytanensis
d. lsospora be/Ii
The following is true of giardiasis except
a. Fever and presence of blood and mucus in stool
b. Acute or chronic diarrhea
c. Duodenum and jejunum are the prime sites of involvement
d. Giardia cysts are resistant to dessication
Giardia lamblia was discovered by
a. Giard
b. Robert hook
c. Leeuwenhoek
d. Losch
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Paniker's Textbook of Medical Parasitology
S. Drug of choice in giardia.sis is
a. Metronidazole
b. Albendazole
c. Thiabendazole
d. Diloxanide furoate
True about Giardia is
a. May cause traveller's diarrhea
b. Giardia inhabits ileum
c. Trophozoites are infective to man
d. Encystment of trophozoites occur in jejunum
Which one following test is used for diagnosis of Giardia lamblia
infections
a. Enterotest
b. Casoni's test
c. Parasight F test
d. Na pier's test
Motility of Trichomonas vagina/is is described as
a. Amoeboid
b. Jerky
c. Falling leaf
d. Lashing
Vaginal discharge in Trichomonas vaginitis is
a. Colorless
b. Yellow
c. Curd- white
d. Blood stained
All of the following protozoan can be transmitted by sexual
contact except
a. Trichomonas vagina/is
b. Entamoeba histolytica
c. Enteromonas hominis
d. Giardia lamblia
Answer
a 2. b
b 9. b
a
C
C 5. a 6. a 7. a
P:63
I
CHAPTER 5
• INTRODUCTION •
The blood and tissue flagellates belong to the family
Trypanosomatidae.
The family consists of six genera, of which two genera
Trypanosoma and Leishmania are pathogenic to humans.
• ZOOLOGICAL CLASSIFICATION OF
FLAGELLATES
Phylum: Sarcomastigophora
Subphylum: Mastigophora
Class: Kinetoplastidea
Order: Trypanosomatida
Family: Trypanosomatidae
Genera: Leishmania and Trypanosoma
• GENERAL CHARACTERISTICS
• They live in the blood and tissues of man and other
vertebrate hosts and in the gut of the insect vectors.
• Members of this family have a single nucleus, a kinetoplast
and a single flagellum (Fig. I).
• Nucleus is round or oval and is situated in the central part
of the body.
• Kineloplast consists of a deeply staining parabasal body
and adjacent dot-Like blepharoplast. The parabasal body
and blepharoplast are connected by one or more thin
fibrils (Fig. l ).
• Flagellum is a thin, hair-like structure, which originates
from the blepharoplast. The portion of the flagellum,
which is inside the body of the parasite and extends from
the blepharoplasl to surface of the body is known as
axoneme. A free flagellum at the anterior end traverses
on the surface of the parasite as a narrow undulating
membrane (Fig. 1).
• Hemoflagellates exist in two or more of fo ur morphological stages. These forms were formerly called the
Blepharoplast
Undulating Flagellum
membrane
Fig. 1 : Basic morphology of hemoflagellates
Note: Parabasal body and blepharoplast together constitute
the klnetoplast.
leishmanial, leptomonad, crithidial and trypanosomal
stages. But as these names are also given to different
genera within the family, they were changed to amastigote, promastigote, epimasligote and trypomastigote. The
names of the stages are formed by the suffix mastigote,
combined with various prefixes, referring to the
arrangement of the Oagella in relation to the position of
the nucleus and its point of emergence from the celJs
(Table 1).
• Staining characteristics of trypanosomes: For smears of
body fluids, Romanowsky's Wrights stain, Giemsa stain
and Leishman's stain are suitable for identifying internal
structures. The cytoplasm appears blue, the nucleus and
flagellum appear pink, and the kinetoplast appears deep
red. For tissue section, hematoxylin-eosin staining is
done for demonstrating structures of the parasite.
• AU members of the family have similar life cycles. They all
require an insect vector as an intermediate host.
• Multiplication in both the vertebrate and invertebrate
host is by binary fission. No sexual cycle is known.
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Table 1: Differences between various morphological stages of hemoflagellates
Morphological
characteristics
Amastlgote Promastigote
Lanceolate in shape.
Kinetoplast is anterior to
Epimastigote Trypomastlgote
This stage is elongated, spindle·
shaped with a central nucleus.
Rounded or ovoid, without
any external flagellum. The
nucleus, kinetoplast and
axial filaments can be seen.
The axoneme extends up to
the anterior end of the cell
the nucleus (antinuclear
kinetoplast) near the
anterior end of the cell, from
which flagellum emerges.
There is no undulating
membrane
Elongated, with the
kinetoplast placed more
posteriorly, though close to
and in front of the nucleus
(juxtanuclear kinetoplast). The
flagellum runs alongside the
body as a short undulating
membrane, before emerging
from the anterior end
The kinetoplast is posterior to the
nucleus (postnuclear kinetoplast)
and situated at t he posterior end
of the body. The flagellum runs
alongside the entire length of the
cell to form a long undulating
membrane before emerging as a
free flagellum from the anterior end
Seen in
Schematic
illustration
Trypanosoma cruzi and
Leishmania as intracellular
form in vertebrate host
It is t he infective stage of
Leishmania, found in the
insect vector as well as in
cultures in vitro
It is the form in which
Trypanosoma bruce/ occur in
salivary gland of the vector
tsetse fly and Trypanosoma
cruzi in the midgut of the
vector reduviid bug.
Note: This stage is lacking in
Leishmania
This is the infective stage of
trypanosomes found in arthropod
vector and in the blood of infected
vertebrate.
Note: This stage is lacking in
Leishmania
Abbreviations: A, axoneme; B, blepharoplast; F, flagellum; K, kinetoplast; N, nucleus; P, parabasal body; U, undulating membrane
Note: Besides the stages described in the table, some transitional stages have been recognized. These include the spheromostigote, a motile round form with free Oagellum,
which is a transitional stage from amastigote to promastigote, seen in the genus Trypanosoma and the paramastigote, a transitional form leading to the infective promastlgore
in Leishmania.
• TRYPANOSOMES
General Characters
All members of the genus Trypanosoma (trypanes: to
bore, soma: body), exist al sometime in their life cycle, as
trypomastigote stage with an elongated spindle-shaped body,
central nucleus, a posterior kinetoplast and long-undulating
membrane. Volutin granules are found in cytoplasm. Some
trypanosomes such as T. cruzi assume amastigote forms in
vertebrate hosts. In addition to the typical forms, cells with
atypical features are frequently found, a condition known as
polymorphism.
• Trypanosomapass their life cycle in twohosts:(l) vertebrate
hosts (definitive hosts) and (2) insect vectors (intermediate
hosts). 1herefore called as digenetic parasites. The vector
becomes infective to the vertebrate host only a fter an
extrinsic incubation period, during which the parasite
undergoes development and multiplication.
• In the vector, the trypanosomes follow one or two modes
of development and are accordingly classified into two
groups: (1) Salivaria and (2) Stercoraria.
l. Salivaria ( anterior station): In salivaria, the trypanosomes migrate to mouth parts of the vectors, so that
infection is transmined by the ir bite (inoculative
transmission). Examples a re T. gambiense a nd
T. rhodesiense causing African trypanosomiasis, which
are transmitted by the bite of tselse flies.
Stercoraria (posterior station): In stercoraria, the
trypanosomes migrate to the hindgut and are passed
in feces (stercorarian tra nsmission), e.g. T. cruzi
causing Chagas disease, which is acquired by rubbing
the feces of the vector bug into the wound caused by
its bite and T. lewisi, the rat trypanosome, which is
transmitted by ingestion of feces of infected rat fleas.
• Distribution: Human trypan osomiasis is strictly
restricted to certain geographical regions; the African
and South American trypanosomiasis being seen only in
ll1e respective continents. This is due to the vector being
confined to these places alone.
African trypanosomiasis (sleeping sickness)
South American trypanosomiasis (Chagas disease).
Classification of Trypanosomes
Trypanosomes Infecting Man
• Trypanosoma brucei complex, causing African trypanosomiasis or sleeping sickness, subspecies are:
Trypanosoma brucei gambiense: It causing West
African sleeping sickness.
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Trypnnosoma brucei rhodesiense: It causing East
African sleeping sickness.
• Trypanosoma cruzi, causing South American trypanosomiasis or Chagas disease.
Trypanosoma rangeli, a nonpathogenic trypanosome
causing human infection in South America.
Trypanosomes of Animals
• Trypanosoma brucei brucei, causing the economically
important disease \"nagana\" in African cattle.
Trypanosoma evansi, causing the disease \"surra\"
in horses, camels and elephants. It is transmitte d
mechanically by biting flies and also by vampire bats. This
infection is found in India.
• Trypanosoma equiperdum, causing \"stallion's disease\"
in horses and mules. It is transmitted by sexual contact,
without the need for an insect vector.
• Trypanosoma lewisi, causing harmless infection of rats
all over the world. The vector is rat flea. A trypanosome
resembling Trypanosoma lewisi was reported from
Madhya Pradesh in India in peripheral blood of two
persons with short-term fever.
Trypanosoma Brucei Gambiense
(West African Trypanosomiasis)
History and Distribution
Trypanosomiasis is believed to have been existing in tropical
Africa from antiquity (Fig. 2).
Fig. 2: Geographical distribution of trypanosomiasis In Africa. Lines
indicate areas endemic for Trypanosoma gambiense and dots
represent Trypanosoma rhodeslense
Hemoflagellates
• Trypanosome was first isolated from the blood of a
steamboat captain on the Gambia river in 1901 (hence,
the nan1e gambiense) by Forde.
• Dutton, in 1902, proposed the name Trypanosoma
gambiense.
• It is endemic in scattered foci in West and Central Africa
between 15° land 18°S latin1des.
Habitat
Trypanosomes live in man and other vertebrate host. They
are essentially a parasite of connective tissue, where they
multiply rapidly and then invade regional lymph nodes,
blood and finally may involve central nervous system.
Morphology
Vertebrate forms: In the blood of vertebrate host, T. brucei
gambiense exists as trypomastigoce form, which is high ly
pleomorphic.
It occurs as a long slender f orm, a stumpy short broad form
with anenuated or absent flagellum and an intermediate
form.
• The trypomastigotes are about 15-40 µm long and 1.5-
3.5 µm broad.
• In fresh blood films, trypornastigotes are seen as colorless,
spindle-shaped bodies that move rapidly, spinning
around the red cells.
• In smears stained with Giemsa or other Romanowsky's
stain, the cytoplasm appears pale blue and the nucleus
appears red. The kinetoplast a ppears as a deep red dot
and volutin granules stain deep blue. The undulating
membrane appears pale blue and the flagellum red.
Insect forms: In insects, it occurs in two forms:
I. Epimastigotes
Metacyclic trypomastigore forms.
Antigenic Variation
Trypanosomes exhibit unique antigenic variation of their
glycoproteins.
• There is a cyclical fluctuation in the trypanosomes in the
blood of infected vertebrates after every 7- 10 days.
• Each successive wave represents a varinnt antigenic
type (VAT) of trypomastigote possessing variant-specific
surface antigens (VSSAs) or variant surface glycoprotein
(VSG) coat antigen.
It is estimated that a single trypanosome may have as
many as 1,000 or more VSG genes that help to evade
immune response. Besides this, trypanosomes have
other mechanisms also that help them to evade host
immune respon es.
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Paniker's Textbook of Medical Parasitology
Life Cycle
Host: T. brucei gambiense passes its life cycle in two hosts:
l . Vertebrate host: Man, game animals and other domestic
animals.
Invertebrate host: Tsetse fly.
Both male and female tsetse fl y of Glossina species
( G. palpalis) are capable of transmitting the disease ro humans.
These flies dwell on the banks of shaded streams, wooded
Savanna and agricultural areas.
/11/ectiveform: Meracyclic trypomastigote forms are infective
ro humans.
Mode of transmission:
• By bite of tsetse fly.
• Congenital transmission has also been recorded.
Reservoirs: Man is the only reservoir host, although pigs and
others domestic animals can act as chronic asymptomatic
carriers of the parasite.
Development in man and other vertebrale hosts:
• Metacyclic slage (infective form) of Lrypomastigotcs arc
inoculated into a man (definitive host) through skin when
an infected tsetse fly takes a blood meal (Fig. 3).
Epimastigote form
Short stumpy form ingested
by Tsetse fly during blood meal
Invades bloodstream
Short stumpy form
Tsetse fly
(Vector)
Man
(Definitive host)
Intermediate form
Fig. 3: Ufe cycle of Trypanosoma brucei
Metacyclic trypomatigote form
infective f\
Transferred to man by bite
of infected Tsetse fly
Metacychc
trypomastigote form
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• l h e pa rasite transforms into slender forms that multiply
asexua lly for l-2 days before e n tering th e peripheral
blood and lymphatic circulation.
• Th ese becom e \"stumpy\" via inte rmediate fo rms and
enter the bloodstream.
• In chronic infection, th e parasite invades the central
nervous system.
• Trypomastigotes (short plumpy fo rm) are ingested by
tsetse fly (male or female) d uring blood meal.
Development in tsetse fly:
• In the midgut of th e fly, short stumpy trypomastigotes
develop into long, slender forms a nd multiply.
• After 2- 3 weeks, they migrate to the salivary glands, where
they develop into epimastigotes, which multiply and fill
the cavity of the gland and eventually transform into the
infective metacyclic trypomastigotes (Fig. 3).
• Development of the infective stage within the tsetse fly
requires 25-50 days (extrinsic incubation period).
• l hereafter, the fly remains infective throughout its life of
about 6 months.
Pathogenicity and Clinical Features
T. brucei gambiense causes African trypan osomiasis (West
African sleeping sickness).
The illness is chronic and can persist for many years.
• There is an initial period of parasitemia, following which
parasite is localized predominantly in the lymph nodes.
• A painless chancre (trypanosomal chancre) appears on
skin at the site of bite by tsetse fly, followed by intermittent
fever, chills, rash, anemia, weight loss and headache.
• Systemic trypanosomiasis w ithout central nervous
system involvement is referred to as stage 1 disease. In
this stage, there is hepatosplenomegaly and lymphadenopathy, particularly in the posterior cervical region
(Wi11terbottom's sign).
• Myocarditis develops frequently in patients with stage I
disease and is especially common in T. brucei rhodesiense
infections.
• Hematological manifestations seen in stage I include
anemia, moderate leukocytosis and thrombocytopenia.
High levels of immunoglobu lins mainly immunoglobulin
M (lgM) are a constant feature.
• Stage Tl disease involves invasion of central n ervous
system. With the invasion of central nervous system, which
occurs after several months, the \"sleeping sickness\" tarts.
This is marked by increasing headache, mental dullness,
ap athy and day time sleepiness. The patient falls into
profound coma followed by death from asthenia (Box 1).
• Histopathology shows chronic meningoencephalitis. The
meninges are heavily infiltrated with lymphocytes, plasma
cells and rnorula cells, which are atypical plasma cells
containing mulberry-shaped masses oflgA. Brain vessels
Hemoflagellates
show perivascular cuffing. This is followed by infiltration
of the brain and spinal cord, neuronal degeneration and
microglial proliferation.
Abnormalities in cerebrospina l fluid (CSF) include raised
intracra nial pressure, pleocytosis and raised total prote in
concentrations.
Trypanosoma Brucei Rhodesiense
(East African Trypanosomiasis)
• It is found in Eastern and Central Africa (Uganda,
Tanzania, Zambia and Mozambique) (Fig. 2).
• Stephens and Fantham discovered T brucei rhodesiense
in 1910 from the blood of a patient in Rhodesia sufferin g
from sleeping sickness.
• The principal vector is G. morsitans, G. palpalis and G.
swynr1ertoni, which live in tl1e open savannah cow1tries.
• Although the disease is usually transmitted by the vector
from ma n-to-man, the disease is actually a zoonosis, with
the reservoir being wild game animals like bu sh buck,
antelope and domestic animals like cattle.
Its morphology, habitat and life cycle is similar lo T. brucei
gambiense (Fig. 3).
• 1 he d ifference between T. brucei gambiense and T. brucei
rhodesiense are detailed in Table 2 .
Box 1: Clinical staging of human African trypanosomiasis (HAT)
• Stage/: Characterized by hematogenous and lymphatic dissemination
of the disease.
• Stage II: Characterized by central nervous system involvement.
Table 2: Differences between West African and East African trypanosomiasis
Characteristics West African East African
Organism T. brucei gambiense T. brucei rhodesiense
Distribution West and Central East and Central Africa
Africa
Vector Tsetse ny (Glossina Tsetse fly(Glossina
pa/pa/is group) morsitans group)
Reservoir Mainly humans Wild and domestic
animals
Virulence Less More
Course of disease Chronic (late central Acute {early central
nervous system nervous system
invasions); months invasion); less than
to years 9 months
Parasitemia Low High and appears early
Lymphadenopathy Early, prominent Less common
Isolation in rodents No Yes
Mortality Low High
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Paniker's Textbook of Medical Parasitology
Pathogenesis and Clinical Features
T. brucei rhodesiense causes East African sleeping sickness
(Table 2).
• East Africa n trypanosomiasis is more acute tha n rhe
Gambian form and appears a fter an incubation period
of 4 weeks.
• It may end fatally within an year of onset, before the
involvement of central nervous sysrem develops.
• Pathological features are similar in both diseases with
some variations:
Ede ma, myocarditis and weakn ess a re more
prominent in East African sickness (Box 2) .
Headache, diffuse muscle and joint pain are present
in majority of the patients.
Lymphadenitis is less prominent.
Febrile paroxysms arc more frequent and severe.
1here is a larger quantity of parasite in the peripheral
blood.
Central nervous system involvement occurs early.
Mania and delusions may occur but the marked
somnolence, which occurs in T. brucei garnbiense
infection is lacking.
Laboratory Diagnosis
The diagnosis of both types of African trypanosomiasis is
similar (Flow chart 1).
Box 2: Parasites causmg myocarditis
• Trypanosoma brucei rhodesiense
• Trypanosoma cruzi
• Toxoplasma gondii
• Echinocaccus granulosus
• Trichinella spiralis
Nonspecific findings:
• Anemia and monocytosis.
• Raised erythrocyte sedimentation rate (E R) due to rise
in gamma globulin levels.
• Reversal of albumin:globulin ratio.
• Increased CSF pressure and raised cell count and proteins
inCSF.
Specific findings: Definitive diagnosis of sleeping sickness
is established by th e demonstra tion of trypanosomes in
peripheral blood, bone marrow, lymph node, CSF an d
chancre Ouid.
Microscopy:
• Wet mount preparation of lymph node aspirates and
chancre fluid are used a a rapid method for demonstration oftrypano omes. These specimens are aJso examined
for parasites after fixing and staining with Giemsa stain.
• Examination of Giemsa-stained thick peripheral blood
smears reveals the presence of the trypomastigotes
(Fig. 4).
Thrombocyte fragments
Erythrocyte
undulating
membrane
Nucleus
Fig. 4: Trypanosoma rhodes/ense, blood smear Giemsa stain,
magnification 1100X
Flow chart 1: Laboratory diagnosis of trypanosomiasis
•
Microscopy
Detection of
Trypanosomes 1n-
• Wet mount preparation
of lymph node aspirate
• Giemsa-stained thick
peripheral blood
smear or concentrated
blood smear
• Wet mount, stained
smear of CSF
•
Culture
In Weinman's
or Tobie's
medium
Laboratory diagnosis I
I
Imaging
CT scan
Shows cerebral
edema
MRI
Shows white matter
enhancement
•
Serodiagnosis
Antibody detection
IHA
IIF
ELISA
CATT
CFT
Antigen detection
ELISA
• •
Molecular diagnosis Others
PCR • animal Inoculation
• Blood examination
reveals anemia,
monocytosis,
raised ESR and
reversed albumin
globulin ratio
Abbreviations: CATT. card agglutination trypanosomlasis test: CT, computed tomography: CFT. complement fixation test: CSF, cerebrospinal fluid: ELISA, enzymelinked immunosorbent assay; ESR, erythrocyte sedimentation rate: IHA, indirect hemagglutination: IIF, indirect immunofluorescence; MRI, magnetic resonance
Imaging: PCR, polymerase chain reaction
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• If parasitemia is low, then examination of concentrated
blood smear is a highly sensitive method. Different concentration techniques employed are buffy coat examination, differential centrifugation, membrane filtration and
ion exchange column chromatography.
• Examination of wet mount and stained smear of the CSF
may also showtrypanosomes (Flowchart l).
Culture: The organisms are difficult to grow, hence culture is
not routinely used for primary isolation of the parasite. However, it can be cultivated in Weinman's or Tobie's medium.
Animal inoculation: Inoculation of specimens from suspected cases to white rat or white mice is a highly sensitive
procedure for detection of T. brucei rhodesiense infection.
Serodiag11osis:
Antibody detection: Almost all patients with African trypanosomiasis have very high levels of total serum IgM antibodies
and later, CSF IgM antibodies. Various serological methods
have been developed to detect these antibodies and are as
follows:
• Indirect hemagglutination (IHA)
• Indirect immunotluorescence (llF)
• Enzyme-linked immunosorbent assay (ELISA)
• Card agglutination trypanosomiasis test (CATT)
• Complement fixation test (CFT)
Specific antibodies are detected by these tests in serum
within 2-3 weeks of infection. Specific antibodies in CSF are
demonstrated by UF and ELISA. These serological tests are
useful for field use and mass screening (Flow chart 1).
Antigen detection: Antigens from serum and CSF can be
detected by ELISA.
Molecular diagnosis: Polymerase chain reaction (PCR)
assays for detecting African trypanosomes in humans have
been developed, but none is commercially available.
Imaging: Computed tomography (CT) scan of the brain
shows cerebral edema and magnetic resonance imaging
(MRI) shows white matter enhancement in patients with late
stage central nervous systems involvement {Flow chart 1).
Blood incubation infectiuity test: For differentiation between
the \"hwnan strains\" and \"animal strains\" of T. brucei, the blood
incubation infectivity test {BUT} had been widely used.
• The strain is incubated with oxalated human blood and
then inoculated into the multimammate rat or other
Hemotlagellates
Table 3: Treatment of human African trypanosomiasis
Causative organism
T. brucei gambiense
(West African)
T. brucei rhodesiense
(East African)
clinical stage
I (normal CSFJ II (abnormal CSF}
Pentamidlne Eflornithine
Su ram in Melarsoprol
Abbreviation: CSF, cerebrospinal fluid
lsoenzyme study: More recently their differentiation is based
on isoenzymes, deoxyribonucleic acid (DNA) and ribonucleic
acid (RNA) characterisLics (Flow chart 1).
Treatment
In the initial stages, when central nervous system is not
involved, i.e. stage I, pentamidine is the d rug of choice
for gambiense h uman African trypanosomiasis (HAT)
and suramin is the drug of choice for rhodesiense HAT.
Dose:
• Pentamidine: Dose 3- 4 mg/ kg of body weight, intramuscularly daily for 7- 10 days.
Suramin: Dose 20 mg/ kg of body weight in a course of
five injections intravenously, at an interval of 5- 7 days.
Suramin does not cross blood-brain barrier but it is
nephrotoxic.
• In patients with central nervous system involvement,
melarsoprol (Mel-B) is the drug of choice, as it can
cross the blood-brain barrier. Dose: 2-3 mg/ kg/ per day
(maximwn 40 mg) for 3-4 days (Table 3).
Prophylaxis
Control is based on early diagnosis and treatment of cases to
reduce the reservoir of infection.
• Control of tsetse fly population (most important preventive measure) by wide spraying of insecticides, traps
and baits impregnated with insecticides.
• No vaccine is available.
Trypanosoma Cruzi
T. cruzi is the causative organism of Chagas disease or South
American trypanosomiasis.
susceptible rodents. History and Distribution • lhe infectivity of \"animal strains\" will be neutralized by
human blood, while \"human strains\" retain infectivity It is a zoonotic disease and is limited to South and Central
after incubation with human blood. America.
• In vitro culture systems are now employed instead of • Carlos Chagas, investigating malaria in Brazil in 1909,
rodents for testing infectivity. accidentally found this trypanosome in the intestine of a
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Paniker's Textbook of Medical Parasitology
triatomine bug and then in the blood of a monkey bitten
by the infected bugs.
• Chagas named the parasite T cruzi after his mentor
Oswaldo Cruz and the disease was named as Chagas
disease in his hon or.
Habitat
• In humans, T cruzi exists in both amastigote a nd
trypomastigote forms:
Amastigotes are the intracellular parasites. They
are found in muscular tissue, nervous tissue and
reticuloendothelial system (Box 3).
Trypomastigotes are found in the peripheral blood.
• In reduviid bugs, epimastigote forms are found in the
midgut and metacyclic trypomastigote forms are present
in hindgut and feces.
Morphology
Amastigote: Amastigotes are oval bodies measuring 2-4 µm
in diameter having a nucleus and kinetoplast (Fig. 5A).
• Flagellum is absenc.
Morphologically, it resembles th e amastigote of
Leishmania spp., hence, it is freque ntly called as
leishmanial form.
• Multiplication of the parasite occurs in this stage.
• This form is fo und in muscles, nerve cells and
reticuloenodothelial systems.
Trypomastigote: Trypomastigotes are nonmultiplying forms
found in the peripheral blood of man and other mammalian
hosts (Fig. 5B).
• In the blood, they appear either as long, thin flagellates
about (20 mcm long) or short stumpy form (15 µm long).
• Posterior end is wedge-shaped.
• In stained blood smears, they are shaped-like alphabet
\"C''; \"U''; or \"S'; having a free flagellum of about one-third
the length of the body.
• These forms do not multiply in humans and are taken up
by the insect vectors.
Epimastigote form: Epimasrigote forms are found in the
insect vector, the reduviid bug and in culture also (Pig. 5C).
• It has a kinetoplast adjacent to the nucleus.
• An undulating membrane runs along the anterior half
o f the parasite.
• Epimastigotes divide by binary fission in hindgut of the
vector.
Life Cycle
Host: 1: cruzi passes its life cycle in two hosts (Pig. 6):
Definitive host: Man.
2 . .Intermediate host (vector}: Reduviid bug or triaLOmine
bugs.
Box 3: Obligate intracellular parasites
• Trypanosoma cruzi
• Leishmania spp.
• Plasmodium spp.
• Babesia spp.
• Toxoplasma gondii
• Microsporidia
Nucleus Parabasal
body m
Figs SA to C: Trypanosoma cruzi. (A) Amastigote;
(B) Trypomastigote; and (C) Epimastigote
Reservoir host: Armadillo, cat, dog and pigs.
Infective form: Metacyclic trypomastigotes forms are the
infective forms found in feces ofreduviid bugs.
• The parasite occurs in three d iffere nt but overlapping
infection cycles, a sylvatic zoonosis in wild animals such
as armadillos and opossums, peridomestic cycle in dogs,
cats, a nd other domestic a nimals and domestic cycle
in humans. Different vector species are active in these
infection cycles.
The vectors important in hwnan infection are the reduviid
bugs adapted to living in human habitations, mainly
Triatoma infestans, Rhodnius prolix.us and Panstrongylus
megistus. These are large (up to 3 cm long) night-biting
bugs, which typically defecate while feeding. The feces of
infected bugs contain the metacyclic trypomastigote.
Mode of transmission:
• Transmission of infection to man and other reservoir
hosts takes place when mucus membranes, conjunctiva,
or wound on the surface of the skin is contaminated by
feces of the bug containing metacyclic trypomastigotes.
• T. cruzi can also be transmitted by the blood transfusion,
organ transplantation and vertical transmission, i.e. from
mother LO fe tus or very rarely by ingestion of contaminated
food or drink.
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Trypomastigote ingested
by reduviid bug
during blood meal
Reduviid bug (Vector)
Hemoflagellates
Metacyclic trypomastigote
(Infective form to man)
\
Shed in
feces
,,_ _ \
Man acquires
Man (Definitive host) infection by rubbing
the bug feces
Trypomastigote formed
and released
in blood bloodstream
(Infective form to reduviid bug)
Amast1gote passes
through promastigote
and epimastigote stages
T rypomastigote
h~
:
(<)
Transforms into
amastigote form
Fig. 6: Life cycle of Trypanosoma cruzi
Development in man:
• The metacyclic trypomastigotes introduced in human
body by bite of reduviid bugs invade the reticuloendothelial system and spread to other tissues.
• After passing through promastigote and epimastigote
forms, they again become trypomastigotes, which are
released into the bloodstream and are the infective stage
for triatomine bug. No multiplication occurs in this stage.
Multiplication takes place only intracellularly in the
amastigote form and to some extent as promastigote or
epimastigotes (Pig. 6).
Development in reduviid bugs:
Bugs acquire infection by feeding on an infected
mammalian host.
• Most triatomine bugs are nocturnal.
• The trypomastigotes are transformed into epimastigotes
in the midgut, from where they migrate to the hindgut
and multiply.
These, in turn, develop into nondividing metacyclic
trypomastigotes (infective form), which are excreted in
feces (stercorarian transmission).
• The development of T cruzi in the vector takes 8- 10 days,
which constitutes the extrinsic incubation period.
Pathogenicity and Clinical Features
The incubation period of T. cruzi in man is 1-2 weeks. The
disease manifests in acute and chronic fo rm.
Acute cha gas disease: Acute phase occurs soon after infection
and may last for 1- 4 months.
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Paniker's Textbook of Medical Parasitology
• lt is seen often in children under 2 years of age.
• First sign appears within a week after invasion of parasite.
• \"Chagoma\" is the typical subcutaneous lesion occurring
at the site of inoculation. Inoculation of the parasite
in conjunctiva causes unilateral, painless edema of
periocular tissues in the eye called as Romana's sign,. This
is a classical finding of the acute Chagas disease.
• In few patients, there may be generalized infection with
fever, lymphadenopathy and hepatosplenomegaly.
• The patient may die of acute myocarditis and
meningoencephalitis.
• Usually within 4-8 weeks, acute signs and symptoms
resolve spontaneously and patients, then enter the
asymptomatic or indeterminate phase of chronic T. cruzi
infection.
Chronic chagas disease: The chronic form is found in adults
and older children and becomes apparent years or even
decades after the initial infection.
• ln chronic phase, T. cruzi produces inflammatory
response, cellular destruction and fibrosis of muscles
and nerves that control tone of hollow organs like heart,
esophagus, colon, etc. Thus, it can lead to cardiac myopathy and megaesophagu.s and megacolon (dilatation of
esophagus and colon).
Congenital infection: Congenital transmission is possible
in both acute and chronic phase of the disease causing
myocardial and neurological damage in the fetus.
Laboratory Diagnosis
Diagnosis is done by demonstration of T. cruzi in blood or
tissues or by serology.
Microscopy:
• The diagnosis of acute Chagas disease requires detection
of parasites.
• Microscopic examination of fresh anticoagulated blood
or the buffy coat is the simplest way to see motile
organisms.
• In wet mount, trypomastigotes are faintly visible but their
snake-like motion against red blood cells (RBCs) makes
their presence apparent.
• Trypomastigotes can a lso be seen in thick and thin
peripheral blood smear, stained with Giemsa stain
(Box 4) (Fig. 7).
• Microhematocrit containing acridine orange as a stain
can also be used.
• When used by experienced personnel, all these methods
yield positive results in a high proportion of cases of acute
Chagas disease.
Note: Serologic testing plays no role in diagnosing acute
Chagas disease.
Culture: ovy, MacNeal and nicolle (NNN) medium or its
modifications are used for growing T. cruzi.
• This medium is inoculated with blood and other
specimens and incubated at 22-24°C.
• The fluid from the culture is examined microscopically
by 4th day and then every week for 6 weeks.
• Epimastigotes and trypomastigotes are found in the
culture.
• Culture is more sensitive than smear microscopy.
Animal inoculation: Guinea pig or mice inoculation may
be done with blood, CSF, lymph node aspirate, or any other
tissue material and the trypomastigote is looked for in its
blood smears in a few days after successful inoculation.
Xenodiagnosis: This is the method of choice in suspected
Chagas disease, if other examinations are negative, especially
during the early phase of the disease onset.
The reduviid bugs are reared in a trypanosome-free
laboratory and starved for 2 weeks. They a re then fed
on patient's blood. If trypomastigotes a re ingested,
they will multiply and develop into epimastigotes and
trypomastigotes, which can be found in the feces of the bug
2 weeks later.
Histopathology: Biopsy examination of lymph nodes and
skeletal muscles and aspirate from chagoma may reveal
amastigotes of T. cruzi.
Box 4: Protozoan parasites detected in peripheral blood film
• Trypanosoma cruzi
• Trypanosoma brucei rhodesiense
• Trypanosoma brucei gambiense
• Leishmania spp.
, Plasmodium spp.
• Babesia spp.
Blepharoplast (large)
Fig. 7: Trypanosoma cruzi, blood smear Giemsa stain,
magnification 1100X
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Serology:
Antigen detection: T. cruzi antigen can be detected in urine
and sera in patients with chronic Chagas disease. ELISA has
been developed for detection of antigens.
Antibody d etection: Antibodies (IgG) against T. cruzi may be
detected by the following tests:
• Indirect hemagglutination
• Complement fixation test (Machado-Guerreiro test)
• Enzyme-linked immunosorbent assay
• Indirect immunofluorescence
• Direct agglutination test (DAT): It is a simple test being
recommended for field use.
• Chagas radioimmune precipitation assay (RIPA) is a
highly sensitive and specific confirmatory method for
detecting antibodies of T. cruzi.
The disadvantage of the antibody based rests is that they
may be false positive with other disease like leishmaniasis
and syphilis.
fntradermal test: The antigen \"cruzin\" is prepared from
T. cruzi culture and used for the intradermal test. A delayed
hypersensitivity reaction is seen.
Molecular diagnosis: Polymerase chain reaction is
available that detects specific primers, which have been
developed against T. cruzi kinetoplastic or nuclear DNA. The
disadvantage of the test is that it is not commercially available.
Other tests:
• Electrocardiography (ECG) and chest X-ray are useful
for diagnosis and prognosis of cardiomyopathy seen in
chronic Chagas disease. the combination of right bundle
branch block (RBBB) and left anterior fascicular block is
a typical feature of Chagas heart disease.
• Endoscopy helps in visualization of megaesophagus in
Chagas disease.
Treatment
o effective specifi c treatment is available for treating
Chagas disease. Nifurtimox and benznidazole have been
used with some success in both acute and chronic Chagas
disease. These drugs kill only the extracellular trypanosomes
but not the intracellular forms.
Dose: Nifurtimox: 8-10 mg/ kg for adults and 15 mg/ kg for
children. The drug should be given orally in four divided
doses each day for 90- 120 days.
Benznidazole: 5- 10 mg/ day orally for 60 days.
Prophylaxis
• Application of insecticide to control the vector bug.
• Personal protection using insect repellant and mosquito
net.
Hemoflagellates
Table 4: Differences between T. cruzi and T. rangeli
Trypanosoma cruzi
• Pathogenic
• 15-20 µm long
• C or U-shaped
• Kinecoplast: Large and terminal
• Primary reservoirs:
Opossums, dog, cats and wild
rodents
Trypanosoma rangeli ___ -.1
Nonpathogenic
30 µm long, more slender and
longer
Not C or U-shaped
Kinteoplast: Small and subterminal
• Primary reservoir:
Wild rodents
• Improvement in rural housing and environment to
eliminate breeding places of bugs.
Trypanosoma Rangeli
T. rangeli was first described by Tejera in 1920 while examining the intestinal content of reduviid bug (R. proli.xus).
.
.
.
.
It is nonpathogenic.
T. rangeli infections are encountered in most areas
where T. cruzi infection also occurs (Mexico, Central
America and northern South America).
Morphologically, it is similar to T. cruzi, except that it
is slender and long (26-36 µm long) and has a smaller
kinetoplast (Table 4).
It is commonly found in dogs, cars and humans .
Infection is transrnined by botb bite of triatomine bug
and fecal contamination from reduviid bug.
T. rangeli multiplies in human blood by binary fission .
Intracellular stage is typically absent.
•
•
.
T. rangeli can circulate in blood of infected animals for
a long period, unlike T. cruzi.
Although T. rangeli appears to be a normal commensal,
they do reduce the life span of reduviid bug.
Diagnostic methods are similar to that of T. cruzi .
KEY POINTS OF TRYPANOSOMES
• Trypanosomes follow one of the two developmental modes
in vectors. In Sa/ivaria: The trypanosomes migrate to mouth
parts of vector tsetse fly, e.g. T. gambiense, T. rhodesiense. In
Stercoraria: The trypanosomes migrate to hindgut of vector
bug, e.g. T. cruzi.
• T. brucei gambiense causes West African sleeping sickness
manifested by fever, hepatosplenomegaly and posterior
cervical lymphadenopathy with chronic central nervous
system invasion.
• T. brucei rhodesiense causes East African sleeping sickness
manifested by fever, early and acute central nervous system
invasion, with loss of weight and myocarditis.
• Diagnosis: By detection of trypanosomes in wet mount
preparations of lymph node aspirates or blood or by serology
and PCR.
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Paniker's Textbook of Medical Parasitology
• Drug of choice: For stage I, HAT by T. brucei gambiense is
pentamidine and by T. brucei rhodesiense is suramin. In stage
II, the drug of choice is melarsoprol in both cases.
• South American trypanosomiasis (Chagas disease) is caused
by T. cruzi.
• It is transmitted by wound or conjunctiva! contamination of
feces of the reduviid bugs.
• Clinical features: \"Chagoma· is the typical subcutaneous
lesion commonly on face (Romana's sign) in Chagas disease.
Damage to nerve cells and muscles leads to mega esophagus,
megacolon and cardiac myopathy.
• Diagnosis: By demonstration of T. cruzi in blood or tissue or
by serology and xenodiagnosis.
• Treatment: Nifurtimox and benznidazole.
• LEISHMAN/A
General Characteristics
The genus Leishmania is named after Sir william Leishman,
who discovered the flagella te protozoa causing kala-azar,
the Indian visceral leishmaniasis (VL).
• All members of the genus Leishmania are obligate
intracellular parasites that pass their life cycle in two hosts:
{l) The mammalian host, and (2) the insect vector, female
sandfly.
• ln humans and other mammalian hoses, they multiply
within macrophages, in which they occur exclusively in
the amastlgote form, having an ovoid body containing a
nucleus and kinetoplast.
• In the sand.fly, they occur in the promasligote form, with
a spindle-shaped body and a single flagellum arising from
anterior end.
• Leishmaniasis has an immense geographical distribution
in the tropics and subtropics of the world, extending
through most of the Central and South America, part of
North America, Central and South-East Asia, India, China,
the Mediterranean region and Africa.
• The disease affects the low socioeconomic group of
people. Overcrowding, poor ventilation and collection of
organic material inside house facilitate its transmission.
• Across the tropics, three different diseases are caused by
various species of genus Leishmania. These are:
I. Visceral leishmaniasis: The species L. donouani
complex infecting internal organs {liver, spleen and
bone marrow) of human is the causative parasite.
Cutaneous leishmaniasis: The species L. tropica
complex, L. aethiopica, L. major and L. mexicana
complex are the causative parasite.
Mucocutaneous leishmaniasis: It is caused by the
L. braziliensis complex.
Classification
The genus Leishmania includes a numbe r of different
varieties and subspecies, which differ in several features such
as antigenic structure, isoenzymes, and other biochemical
characteristics, growth properties, host specificity, etc.
(Table 5).
Leishmania species can also be classified on the basis of
geographical distribution as given in Tables sand 6.
The various manifestations of leishmaniasis a nd
Leishmania species causing them have been summarized
in Flow chart 2.
Old World Leishmaniasis
Leishmania Donovani
L. donouani causes VL or kala-azar. It also causes the
condition, Post-kala-azar dermal leishmaniasis (PKDL).
History a nd distribution: Sir William Leishman in 1900
observed the parasite in spleen smears of a soldier who died
of \"dumdum fever\" or kala-azar contracted at Dum Dum
Calcutta. Leishman reported this finding from London
In the same year, Donovan also reported the same
parasite in spleen smears of patients from Madras. The name
Leishmania donouani was, therefore given to chis parasite.
The amastigote forms of the parasite as seen in smears from
patients are called Leishman-Donovan (LD) bodies.
• Visceral leishmaniasis or kala-azar is a major public
health problem in many parts of world. According to the
World Health Organization (Wl-1O), a total of 500,000
cases of VL occur every year. Of these new cases, 90% are
fow1d in the Indian subcontinent and Sudan and Brazil.
• The disease occurs in endemic, epidemic, or sporadic
forms. Major epidemics of the disease are currendy found
in India, Brazil and Sudan {Fig. 8).
• lhe resurgence of kala-azar in India, beginning in the
mid 1970s, assumed epidemic proportions in 1977 and
involved over 110,000 cases in humans. Initially, the
disease was confined to Bihar (Muzaffarpur, Samastipur,
Vaishali a nd Sitamarhi). Since the n, the cases are
increasing and involving newer areas. The epidemic
extended to West Bengal and first outbreak occurred in
1980 in Malda district.
• At present, the disease has established its endemicity
in 31_ districts in Biha r, 11 districts in West Bengal, five
districts in Jharkhand and three districts in Uttar Pradesh.
Sporadic cases have been reported from Tamil Nadu,
Maharashtra, Karnataka and Andhra Pradesh.
Habitat: The amastigote (LD body) of L. donouani is found in
the reticuloendothelial system. They are found mostly within
P:75
Table 5: Leishmania species involved in human disease
species Disease Geographical
distribution
Leishman/a donovani Visceral leishmaniasis Middle East, Africa and
(kala-azar or dumdum Indian subcontinent
fever)
Leishmania infantum Visceral leishmaniasis, Mediterranean coast,
cutaneous Middle East and China
leishmaniasis
Leishman/a chagasi Visceral leishmaniasis Tropical Sout h
America
Leishmania tropica Cutaneous Middle East and
leishmaniasis (oriental Central Asia
sore, Baghdad boil)
Leishmania major Cutaneous Africa, Indian
leishmaniasis subcontinent and
Central Asia
Leishman/a aethiopica Cutaneous and Ethiopia and Kenya
diffuse cutaneous
leishmaniasis
Leishmania braziliensis Mucocutaneous Tropical South
complex leishmaniasis America
(Espundla)
Leishmania mexicana Mucocutaneous Central America and
complex leishmaniasis Amazon basin
(Chiclero's ulcer)
Table 6: Classification of Leishman/a based on geographical
distribution
Old world leishmanlasis New world leishmanlasis
• Leishman/a donovani • Leishman/a braziliensis complex
• Leishman/a infancum • Leishmania mexicana complex
• Leishmania tropica • Leishmania chagasi
• Leishmania major • Leishmania peruviana
• Leishmania aethiopica
the macrophages in the spleen, liver, bone marrow and less
often in other locations such as skin intestinal mucosa and
mesenteric lymph nodes.
Morphology: The parasite exists in two forms (Figs 9A and B):
Amastigoteform: In humans and other mammals.
Promastigoteform: In the sandOy and in artificial culture.
Amastigote: The amastigote form (LD body) is an ovoid or
rounded cell, about 2- 4 µmin size (Fig. 9A).
• It is typically intracellular, being found inside macrophages, monocytes, neutrophils, or endothelial cells.
They are also known as LD bodies.
Hemoflagellates
Vector Reservoir Transmission
Phlebotomus Humans Anthroponotic,
argentipes, occasionally zoonotic
Phlebotomus orientalis
Phlebotomus Dog, fox, jackal and Zoonotic
pemiciosus, wolf
Phlebotomus ariasi,
Phlebotomus paporasi
Lutzomyia longipalpis Fox and wild canines Zoonotic
Phlebotomus sergenti Humans Anthroponotic
Phlebotomus papatosi, Gerbil Zoonotic
Phlebotomus duboscqi
Phlebotomus longipes, Hydraxes Zoonotic
Phlebotomus pedifer
Lutzomyia umbratilis Forest rodents and Zoonotic
peridomestic animals
Lutzomyia olmeca, Forest rodents and Zoonotic
Lutzomyia marsupials
flaviscutellata
• Smears stained with Leishman, Giemsa, or Wright's
stain show a pale blue cytoplasm enclosed by a limiting
membrane.
• The large oval nucleus is stained red. Lyi ng at the right
angles to nucleus, is the red or purple-stained kinetoplast
• In well-stained preparations, the ki netoplast can be
seen consisting of a pa rabasal body and a dot-like
blepharoplastwith a delicate thread connecting the two.
The axoneme arising from the blepharoplast extends to
the anterior tip of the cell.
• Alongside the kinetoplast a clear unstained vacuole can
be seen.
Flagellum is absent.
Promastigote: It is a flagellar stage and is present in insect
vector, sandfly and in cultures.
• The promastigotes, which are initially short, oval or pearshaped forms, subsequently become long spindle-shaped
cells, 15- 25 µm in length and 1.5-3.5 µm in breadth
(Fig. 9B).
• A single nucleus is situated at the center. lhe kinetoplast
lies transversely near the anterior end.
lhe flagellum is single, delicate and measures 15- 28 µm.
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Paniker's Textbook of Medical Parasitology
Flow chart 2: Distribution and disease caused by Leishmania spp.
t
Leishmania I
I
•
Old world leishmaniasis
t
Visceral leishmaniasis
(Kala-azar)
L. donovanl complex
L. infantum
•
Cutaneous
leishmaniasis
L. Tropica complex
comprising
• L. tropica
• L. aethiopica
• L. major
Fig. 8: Geographical distribution of visceral leishmaniasis.
Endemic areas shaded; dots indicate sporadic cases
• Giemsa or Leishman -sLained films show pale blu e
cytoplasm with a pink nucleus and bright red kinetoplast.
• A vacuole is present near the root of the flagellum.
• There is no undulating membrane.
• Promastigote forms, which develop in artificial cultures,
have the same morphology as in th e sandtly.
Life cycle: l. donovani completes irs life cycle in two hosts
(Fig. 10):
Definitive host: Man, dog and other mammals.
Vector: Female sand.fly (Phlebolomus species) (Table 7).
Infective form: Promastigote form present in midgut of
female sandily.
Mode of transmission:
• Humans acquire by bite of an infected female sand.fly.
It can also be transmitted vertically from mother to fetus,
by blood transfusion and accidental inocu lation in the
laboratory.
Incubation period: Usually 2-8 months, occasionally, it may
be as short as 10 days or as long as 2 years.
t
New world leishmaniasis j
I
•
Visceral leishmaniasis Cutaneous leishmaniasis
L. chagasi I or mucocutaneous leishmaniasis
L. mexicana complex
L braziliensis complex
,_ _ _ __ Vacuole - -----ct't0--- 11--- Blepharoplast---r--v
c:::::.- - + -Parabasal body- --!'-'~~,
Figs 9A and B: Morphology of Leishmania donovani. (A) Amastigote
[Leishman-Donovan (LO) body]; and (B) Promastigote
• The sandlly regurgitates the promastigotes in Lhe wound
caused by its proboscis.
• These are engulfed by the cells of reticuloendothelial
system (macrophages, monocytes and polymorphonuclear leukocytes) and change into amastigote (LD body)
within the cells.
The amastigote multiplies by binary fission, producing
numerous daughter cells that distend the macrophage
and rupture it. 1he liberated daughter cells are in turn,
ph agocytosed by other macrophages and histiocytes.
Small number of LO bodies can be found in peripheral
blood inside neutrophils or monocytes (Fig. 10).
When a vector sandtly feeds on an infected person, the
amastigotes present in peripheral blood and tissue fluids
enter the insect along with its blood meal. In the midgut
(stomach) of the sandfly, the amastigote elongates and
develops into the promastigore form (Fig. 10).
The promastigore multiples by longitudinal binary fission
and reaches enormous numbers. They may be seen as
large roselles with their flagella entangled.
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Hemoflagellates
Stomach
'°'\"\"'---
..,.__~
Amastigotes become
promastigotes accumulate in
which multiply pharynx and block it 7·~'-\"
Sandfly
(Intermediate host Man acquires infection by
bite of female sandfly
Amastigote ingested ey~r
Man
(Definitive host)
eeeee
eoeee
e
0 eP
Amastigotes in
peripheral blood
0
I
Promastigote deposited in
punctured wound
Phagocytosed by
macrophage
Fig. 10: Life cycle of Leishmania donovani
• In the sandfly, they migrate from the midgut to the
pharynx and hypostome, where they accumulate and
block th e passage.
• Such blocked sandflies have difficulty in sucking blood.
Wh en they bite a person and attempt to suck blood, plugs
of adherent parasites may get dislodged from the pharynx
and they are deposited in the punctured wound. It cakes
about 10 days for the promastigotes to reach adequate
numbers after ingestion of the amastigotes, so as to block
the buccal cavity and pharynx of the sandfly. This is,
therefore, the duration of extrinsic incubation period.
This period is also synchronous with the gonadotropic
cycle of the vector, so that amastigotes ingested during a
single blood meal, are ready to be transmitted when the
sandlly takes the next blood meal after its eggs have been laid.
Pathogenicity: L. donovani causes VL or kala-azar.
• Kala-azar is a reticuloendotheliosis resulting from the
invasion of reticuloendothelial system by L. donovani.
• The parasitized macrophages disseminate the infection
to all parts of the body.
• Three major surface membrane proteins of Leishmania,
n amely (1) gp63, (2) lipophosphoglycan (LPG) and
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Paniker's Textbook of Medical Parasitology
Table 7: Vector species responsible for transmission of Leishmania Box 5: Causes of anemia in kala-azar
donovani
Coun ----~-- Phlebotomus species
India • P. argentipes
China, Bangladesh • P. chineses
P. sergenti
Sudan and Africa
Mediterranean countries
Middle East and Russia
Central Asia
South America
• P. pernicious
P. orientalis {Sudan)
P. longicuspis
P. sergenti
• P. pernicious
P.paparasii
P. major
P. tobbi
• P. perfulievi
P. papatasii
• P. papatasii
• P. longipalpis
P. intermudias
P. lutzi
(3) glycosylphosphatidylinositols (CP!s) give protection against hydrolytic enzymes of macrophage
phagolysosome.
In the spleen, liver a nd bone marrow particularly, the
amastigotes multiply e normously in the fixed macrophages to produce a \"blockade\" of the reticuloendothelial
system. thisleads to a marked proliferation and destruction ofreticuloendothelial tissue in these organs.
Spleen:
The spleen is the most a ffected organ. It is grossly
enlarged and the capsule is thicke ned du e to
perisplenitis.
Spleen is soft and friable and cuts easily due to
absence of fibrosis.
The cut section is red or chocolate in color due to the
dilated and engorged vascular spaces.
The trabeculae are thin and atrophic.
Microscopically, the reticulum cells a re greatly
increased in numbers and are loaded with LD bodies.
Lymphocytic infiltration is scanty, but plasma cells
are numerous.
Liver:
the liver is enlarged.
11,e Kupffer cells and vascular endothelial cells are
heavily parasitized, but hepatocytes are not affected.
Liver function is, therefore, not seriously affected,
although prothrombin production is commonly
decreased.
The sinusoidal capiJlaries are dilated and engorged.
Some degree of fatty degeneration is seen. The cut
surface may show a \"nutmeg\" appearance.
• Splenic sequestration of red blood cells (RBCs)
• Decreased erythropoiesis due to replacement of bone marrow with
parasitized macrophages
• Autoimmune hemolysis
• Hemorrhage
• Marrow suppression by cytokines.
• Bone marrow:
The bone marrow is heavily infiltrated with parasirized
macrophages, which may crowd the hematopoielic
tissues.
• Peripheral lymph nodes and lymphoid tissues of the
nasopharynx and intestine are hypertrophic, although
this is not seen in Indian cases.
• Severe anemia with hemoglobin levels of 5-10 g/dL
may occur in kala-azar, as a result of infiltration of the
bone marrow as well as by the increased destruction of
erythrocytes due to hypersplenism. Autoantibodies to red
cells may contribute to hemolysis (Box 5).
• Leukopeniawith marked neutropeniaand thrombocytopenia are frequently seen. Antibodies against white blood
cells (WBCs) and platelets suggest an autoimmune basis
for the pancytopenia observed in kala-azar.
Ecological types: the epidemiology and clinical features ofVL
and the ecology of the parasite are very different in different
geographical areas. The different clinical syndromes have,
therefore been considered to be distinct entities and the
parasite causing them have been given separate species or
subspecies status, as listed here:
• Indian visceral leishmaniasis: Caused by L. donouani
producing the anthroponotic disease kala-azar and its
sequel PKDL. The disease is not zoonotic; human beings
being the only host and reservoir. Vector is the sandfly,
P. argentipes.
• Mediterranean leishmania.sis: Middle Eastern
leishmaniasis caused by L. donovani infantum affecting
mostly young children. It is a zoonolic disease; the
reservoir being dog and wild canines such as foxes, jackals
and wolves. Vectors are P. pernicious and P. papatasii.
• American (New World) visceral leishmaniasis: Caused
by L. chagasi. It is present is most parts of Lalin America
a nd resembles the disease caused by L. infanlum. The
main vector is L. longipalpis.
Clinical features of kala-az ar:
• 11,e onset is typically insidious. The clinical illness begins
with high-grade fever which may be remittent with twice
daily spikes or intermittent or less commonly continuous.
Splenomegaly starts early and is progressive and massive
(Fig. I l). It is usually soft and nontender.
Hepatomegaly is moderate.
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Fig. 11 : Kala-azar spleen showing a greatly enlarged organ
• l ymphadenopathy is common in most endemic areas
except Indian subcontinent.
• Skin becomes dry, rough and darkly pigmented (hence,
the name kala-azar).
• The hair becomes thin and brittle.
• Cachexia with marked anemia, emaciation and loss of
weight is seen.
• Hematological abnormalities:
Anemia is most always present and is usually severe
leukopenia
111rombocytopenia is associated with epistaxis, gum
bleedin g, gastrointestinal (GI) bleeding.
• Asciles and edema may occur due to hypoalbuminemia.
• Renal involvement is also common.
• In late stage of huma n immunodeficiency virus (I IIV)
infection VL can present as opportunistic infection. HIV
coinfection rate is 5% in India and 20% in African countries.
• Secondary infections such as herpes, measles, pneumonia, tuberculosis, bacillary dysentery may occur.
• Most un treated patients die in about 2 years, due to some
intercurrent disease such as dysentery, diarrhea and
tuberculosis.
Post-kala-azar dermal leishmariiasis: About 3- 10% cases
of palients of VL in endemic areas develop PKDL, about an
year or 2 after recovery from the systemic illn ess.
• Post-kala-azar dermal leishmaniasis is seen mainly in
lndia and East Africa and not seen elsewhere. The Indian
and African diseases differ in several aspects; important
features of PKDL. in these two regions are listed in
Table 8.
Post-kala-azar dermal leishmaniasis is a nonulcerative
lesion of skin. lhe lesions are of three types:
Depigmented or hy popigmented macules: These
commonly appear on th e face, the trunk and
extremities and resemble tuberculoid leprosy.
Hemoflagellates
Fig. 12: Erythematous patches (Butterfly distribution)
Table 8: Differences between post-kala-azar dermal leishmaniasis
(PKDL) of India and East Africa
Characreristics India East Africa
Incidence 5% 50%
Time interval between Occurs after visceral Occurs during visceral
visceral leishmaniasis leishmaniasis. May leishmaniasis
and PKDL take 3- 5 years
Age group affected Any age Most ly children
Appearance of rash Rashes appear Rashes may appear
after visceral during visceral
leishmaniasis leishmaniasis
Spontaneous cure Not seen Seen
Duration of treatment 60-120 days 60 days
with sodium
stibogluconate
Erythemalous patches: These are distributed on the
face in a \"butterfly distribution\" (Fig. 12).
Nodula r lesion: Both of the ea rlier me ntio ned
lesions may develop into painless yellowish pink
nonulcerating granulomatous nodules.
• The parasite can be demonsrrated in the lesions.
Diagnosis of post-kala-azar dermal leishmaniasis:
• The nodular lesions are biopsied and amastigote forms
are demonstrated in stained sections.
Th e biopsy material can be cultured or animal inoculation
can be done.
• lmmunodiagn osis has no role in the diagnosis of PKDL.
Treatment ofpost-kala-azar dermal leishmaniasis:
• Liposomal amphotericin-8 (AmBisome) 2.5 mg/ kg/ day
for 20 days or sodium stibogluconate (SSG) 20 mg/ kg/day
for 40- 60 days are given.
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Paniker's Textbook of Medical Parasitology
Immunity:
• The immune response in VL is very complex.
• There is increased production of proinflammatory
cytokines and chemokines. Interleukin-IO (IL-10) and
transforming growth factor-B (TGF-B) are the dominant
cytokines.
• The most important immunological feature in kala-azar
is the marked suppression of cell-med iated immunity
to leishmanial antigens. This makes unrestricted intracellular multiplication of the parasite possible. Cellular
responses to tuberculin and other antigens are also
suppressed and may be regained some 6 weeks after
recovery from the disease.
• 1n contrast, there is an overproduction of immunoglobulins, both specific antileishmanial antibodies as well as
nonspecific polyclonal IgG and lgM. Circulating immune
complexes are demonstrable in serum.
Laboratory diagnosis: Laboratory diagnosis of kala-azar
depends upon direct and indirect evidences {Flow chart 3).
Direct eviden ce:
Microscopy:
• Demonstration of amastigotes in smears of tissue
aspirates is the gold standard for diagnosis ofVL.
• For microscopic demonstration of the parasite, the
materials collected are:
Peripheral blood
Bone marrow
Splenic aspirate
Enlarged lymph node.
• The smears are stained by Leishman, Giemsa, or Wright's
stains and examined under oil immersion objective.
• Amastigote parasite can be seen within the macrophages,
often in large numbers. A few extracellular forms can also
be seen.
• Peripheral blood smear:
Peripheral blood contains the amastigotes present
inside circulating monocytes and less often in
neutrophils, but the numbers are so scanty that a
direct blood smear may not show them.
Chances of detecting them are somewhat improved
by examination of a thick blood film.
It is best to examine huffy coat smear, although even
these are not often found positive.
Buffy coat smears show a diurnal periodicity, more
smears being positive when collected during the day
than at night.
• Bone marrow aspirate:
Bone marrow aspirate is the most common diagnostic specimen collected.
Generally, the sternal marrow is aspirated by
puncturing the sternum at the level of the 2nd or 3rd
intercostal space, using a sternal puncture needle.
Bone marrow samples can also be obtained by
puncturing the Iliac crest.
• Splenic aspirates:
Splenic aspirates are richer in parasites and therefore,
are more valuable for diagnosis.
Flow chart 3: Laboratory diagnosis of kala-azar
Laboratory diagnosis
•
Direct evidence I
• l
Demonstration of Culture
LO bodies In NNN medium
In stained or Schneider's
smears of thick blood liquid medium
film, splenic, bone to demonstrate
marrow, and promastigote
lymph node aspirate form
•
Indirect evidence I
• t l
Animal Serodiagnosis Molecular • Nonspecific
Inoculation
In hamster
or mice
Detection of
antigen
ELISA
--------'
diagnosis serum test
• DNA probe • Aldehyde test
• PCR • Chopra's antimony
test. The tests are
positive in
hypergammaglobulinemia
Detection of antibody
• CFT using WKK antigen
•DAT
• IFAT
• CIEP
• DOT-ELISA
• ICT using rK39 antigen
• •
Skin test Blood picture
Leishmanln • Anemia or • Progressive
Montenegro leukopenia
test • Reverse
albumin:
globulin ratio
Abbreviations: CFT, complement fixation test: CIEP. counter immunoelectrophoresis: DAT, direct agglutination test; DNA, deoxyribonucleic acid; ELISA, enzymelinked ,mmunosorbent assay; ICT, immunochromatographlc test; !FAT, indirect immunofluorescent antibody test: LD, Leishman-Donovan; NNN, Novy, MacNeal and
Nicolle; PCR, polymerase chain reaction; rK39, recombinant kinesin 39
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Hemoflagellates
Volutin
granules
Nucleus Lymphocyte
Flagella
Parasite from
disrupted cell
Nucleus
Characteristic
clusters in ' 0
cultures
Dividing
parasite ) Ingested - ~ -..:c_--''--- Nucleus of
a liver
parenchymal
cell
Commencing parasites
division
Figs 13A and B: Leishmania donovani. (A) Culture form (Giemsa stain, magnification 1100X); and (B) Liver smear
(Giemsa stain. magnification 1100X)
But, the procedure can sometimes cause dangerous
bleeding and therefore, should be done carefully and
only when a marrow examination is inconclusive.
• Lymph node aspirates: Lymph node aspirates are not
useful in the diagnosis of Indian kala-azar, although it is
employed in VL in some other countries.
Comparison of aspiration biopsies: Although splenic
aspiration is the most sensitive method (98% positive),
bone marrow puncture (50- 85%, positive) is a safer
procedure when compared to spleen puncture, as there
is risk of hemorrhage in splenic puncture particularly in
patients with advanced stage of disease with soft enlarged
spleen. Splenic aspiration is contraindicated in patients
with prolonged prothrombin time, or if platelet count
is less than 40,000/mm3
• Liver biopsy is also not a safe
procedure and carries a risk of hemorrhage. Lymph node
aspiration is positive in 65% of cases of African kala-azar,
but not useful in cases of Indian kala-azar.
Culture: Different tissue materials or blood are cultured on
N N medium (described by Novy, MacNeal and Nicolle).
this is a rabbit blood agar slope consisting of two parts of salt
agar and one part of defibrinated rabbit blood. The material
is inoculated into the water of condensation and culture
is incubated at 22- 24°C for 1-4 weeks. At the end of each
week, a drop of culture fluid is examined for promastigotes
under high power objective or phase contrast illumination
(Figs 13A and B). Other biphasic medium, like Schneider's
drosophila tissue culture medium with added 30% fetal calf
serum can also be used.
Animal inoculation: Anima l inoculation is not used for
routine diagnosis.
-
'
••
•• •• • ' .
•
••• . .. ,. ••J••
•••• ···-· •• ,,
Fig. 14: Leishman-Donovan (LD) body in spleen smear of
experimentally infected animal (Giemsa stain)
• When necessary, Chinese golden hamster is the animal
employed.
The material is inoculated intraperitoneally or intradermally into the skin of nose and feet.
the inoculated animals are kepl at 23- 26°C.
• In positive cases, the amastigote can be demonstrated in
smears taken from ulcers or nodules developing at the
sites of inoculation or from the spleen (Fig. 14).
• Animal inoculation is a very sensitive method, but takes
several weeks to become positive.
Indirect evidences:
Serodiagnosis:
• Detection of antigen: The concentration of antigen in
the serum or oth er body fluids is very low. ELISA and
PCR have been developed for detection of leishmanial
antigen.
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Paniker's Textbook of Medical Parasitology
• Two noninvasive antigen detection test in urine for VL
are under evaluation.
Detection of antibodies:
Complement fixation test was the first serological
test used to detect serum antibodies in VL. The
antigen originally used, was prepared from human
tubercle bacillus by Witebsky, Klingenstein a nd
Kuhn (hence, called WKK antigen). CFT using WKK
antigen becomes positive early in the disease, within
weeks of infection. Positive reaction also occurs in
other conditions, including tuberculosis, leprosy and
tropical eosinophilia.
Specific leishmanial antigens prepared from cultures
have been used in a number of tests to demonstrate
specific antibodies. These tests include:
• Indirect immunofluorescent antibody test (!FAT)
• Counter immunoelectrophoresis (CIEP)
• ELISA and DOT-ELISA
• Direct agglutination test (DAT)
rk 39 test: A specific rapid immunchromatographic
test (JCT) method for antibody has been developed
using a recombinant leishmanial antigen rk 39
consisting of 39 amino acids conserved in kinesin
region of L. infantum. The sensitivity of the test is 98%
and specificity is 90%.
Note: The direct aggl utin ation test fo r antileishmanial
antibody has been found to be highly specific and sensitive
for diagnosis of kala-azar. However, rk 39 antibody test is more
useful and easy to perform and recommended by National
Vector Borne Disease Control Programme (NVBDCP) in
India.
Molecular diagnosis: A number of molecular diagnosis
me thods have been develo ped, which he lp in species
identification of Leishmania. The methods include Western
b lot and PCR. The use of PCR is confined to specialized
laboratories and is yet to be used for routine diagnosis ofVL
in endemic areas.
Nonspecific serum tests: These tests are based on the greatly
increased globulin content of serum in the disease.
• 11,e two tests widely used are:
J. Napier's aldehyde or Jormogel test
Chopra's antimony test.
• Napier aldehyde test: l mL of clear serum from the
patient is taken in a small test tube, a drop of formalin
(40% formaldehyde) is added, shaken and kept in a rack
at room temperature.
A control tube with normal serum is also set up.
A positive reaction is jellification and opacification
of the test serum, resembling the coagulated white of
egg appearing within 3-30 minutes.
About 85% of patients with disease of 4 months or
more give positive reaction.
Aldehyde test is always negative in cutaneous
leishmaniasis (CL).
The test merely indicates greatly increased serum
gamma-globulin and thus, is nonspecific.
• Chopra's antimony test: It is done by taking 0.2 mL of
serum diluted 1:10 with distilled water in a Dreyer's tube
and overlaying with few d rops of 4% solution of urea
stibamine. Formation of tlocculcnt precipitate indicates
positive test.
The reaction is said to be more sensitive than the
aldehyde test.
• Both the tests give false-positive reactions in several
other disease such as multiple myeloma, cirrhosis of
liver, tuberculosis, leprosy, schistosomiasis, African
trypanosomiasis, etc. where hypergammaglobulinemia
exists.
Skin test:
• Lelshmanin skin test (Montenegro test):
It is delayed hypersensitivity test.
This was first discovered by Montenegro in South
America and hence, named after him.
0.1 mL of killed promastigote suspension (l 06 washed
promastigotes/ mL) is injected intradermally on the
dorsoventral aspect of forearm.
Positive result is indicated by an induration and
erythcma of 5 mm or more after 48-72 hours.
Positive result indicates prior exposure to leishmanial
parasite.
In active kala-azar, this test is negative and becomes
positive usually 6-8 weeks after cure from the disease.
Blood picture:
• Complete blood count shows normocytic normochromic
anemia and thrombocytopenia.
Leukocyte count reveals leukopenia accompanied by
a relative increase of lymphocytes and monocytes.
Eosinophil granulocytes are absent. During the course
of disease, there is a progressive diminution of leukocyte
count falling to l ,000/mm3 of blood or even below that.
• The ratio of leukocyte to erythrocyte is greatly altered and
may be about 1:200 to 1:100 (normal 1:750).
• Serum shows hypergammaglobulinemia and a reversal
of the albumin: globulin ratio.
• Liver function tests show mild elevations ofliver enzymes.
• Erythrocyte sedimentation rate is elevated.
Treatment: Kala-azar responds to Lreatrnent better than other
forms of VL. The standard treatment consists of pentavalent
antimonial compound, which is the drug of choice in most
of the endemic regions of the world, but there is resistance
to antimony in Bihar in India, where amphotericin-Bdeoxycholate or miltefosine is preferred.
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Pentavalent antimonial compound: Two pentavalent antimonial (Sbv) preparations are available:
Sodium stibogluconate (100 mg of Sbv/mL) (SSG)
Meglumine antimoniate (85 mg of Sb\"/mL).
Dosage: The daily dose is 20 mg/ kg by rapid intravenous (TV)
infusion or intramuscular (IM) injection for 20- 30 days. Cure
rates exceed 90% in most of the old world, except in Bihar
(India) due to resistance (cure rate 36%).
Amphotericin-B:
• Amphotericin-B is currently used as a first-line drug in
Bihar. ln other parts of the world, it is used when initial
antimonia1 treatment fails.
• Dosage: 0.75-1.0 mg/ kg on alternate days for a total of
15 infusions.
Note: Fever with chills is almost seen in all patients, using
amphotericin-8 infusions.
• Liposomal amphotericin-B (AmBisome): It has been
developed and used extensively to treat VL in all parts of
the world. It is the only drug approved by the US Food and
Drug Administration (FDA) for the tream1ent of VL; dose
being 3 mg/ kg daily. By using liposoma1 amphotericin-B,
higher doses can be given, improving the cure, without
toxicity (Box 6).
• Current recommendation in India isl 0 mg/ kg single dose.
Paromomycin: Paromomycin is an intramuscular aminoglycoside antibiotic with anrileishmanial activity.
Dosage: It is given in a dose of 11 mg/ kg daily for 21 days.
Millefosine: Milcefosine is the first oral drug, approved for the
treatment of leishmaniasis.
Dosage: 50 mg daily for 28 days for patients weighing less than
25 kg, and twice daily for patients weighing more than 25 kg.
Prophylaxis:
• Early detection and treatment of all cases.
• Integrated insecticidal spraying to reduce sandfly
population.
• Destruclion of animal reservoir host in cases or zoonotic
kala-azar.
Box 6: Advantages of drugcoadministrations in visceral leishmanias,s
• Increase activity by additive and synergistic effect.
• Reduce length of treatment, toxicity. drug-dose burden.
• Reduce resistant cases and improve patient compliance.
• Improve success in treating human immunodeficiency virus (HIV)•
leishmanlasis coinfected cases.
• Regime of coadministrated drug include:
AmBisome + Paromomycin
AmBisome + Miltefosine
Paromomycin + Miltefosine
Hemoflagellates
• Personal prophylaxis by using anrisandfly measures like,
using thick clothes, bed nets, window mesh, or insect
repellants and keeping the environment clean.
• No vaccine is available at present against kala-azar.
• Candidate vaccine: Many 2nd generation subunit vaccines are under trial in rodent models, e.g. hydrophilic
acctylated su rface protein Bl (HASBl), kinetoplastid
membrane protein JI (KMPII) and Leishlll.
Leishmanla tropica Complex
• lt includes three species:
Leishmania tropica
Leislzmania major
Leishmania aethiopica.
• All these species cause old world cutaneous
leishmanlasis. The disease is also known as oriental
sore, Delhi boil, Bagdad boil, or Aleppo button.
History and distribution: Cunningham (1885) first observed
the parasite in the tissues of a Delhi boil in Calcutta.
• Russian military surgeon, Borovsky (1891) gave an
accurate description of its morphology and Luhc (1906)
gave the name L. tropica.
• L. tropica and L. major arc found in Middle-East, India,
Afghanistan, Eastern Mediterranean countries and North
Africa.
• L. aethiopica occurs in Ethiopia and Kenya.
• In India, CL is restricted to the dry western hair of the
lndo-Gangetic plains including dry areas bordering
Pakistan, extending from Amritsar to Kutch and Gujarat
plains. To the East, the cases have been reported from
Delhi and Varanasi in uttar Pradesh.
Habitat: L. tropica causing CL (old world CL) are essentially
the parasite of skin. The amastigote forms occur in the
rcticulocndothelial cells of the skin, whereas promastigote
forms arc seen in sand fly vector.
Morphology: Morphology of L. tropica complex is indistinguishable from that of L. donovani.
Life cycle: The life cycle of L. tropica is similar to that of
L. donouani except:
Vectors: The vectors of L. tropica complex are Phlebotomus
sandflies. The following species of sand flies acr as vector:
• P. sergenti-L. tropica
• P. papatasi- L. major
• P. longipes- L. aethiopica
Mode of transmission:
• The most common mode of infection is through bite of
sandflies.
• Infection may also sometimes occur by direct contact.
• Infection may be transmitted from man-to-man or
animal-to-man by direct inoculation or amasligotes.
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Paniker's Textbook of Medical Parasitology
• infection may also occur by autoinocu lation.
• The amastigotes are present in th e skin, wi thin large
mononuclear cells, neutrophils, inside capillary endothelial cells, and also free in the tissues.
• They are ingested by sandflies feeding near the skin
lesions.
• In the midgut of the sand.fly, the amastigotes develop into
promastigotes, which replicate profusely.
• These a re in turn transmitted to the skin of persons
bitten by sandflies in the skin, the promastigotes are
phagocytosed by mononuclear cells, in which they
become amastigotes and multiply.
• However, they remain confined to the skin, without being
transported to the internal organs, as is the case in VL.
lncubation period: Incubation period varies from 2-8 months.
Pathology: Amastigote forms are found in histiocytes and
endothelial cells. there is an inflammatory granulomatous
reaction with infiltration of lymphocyte and plasma cells.
Early lesions are papular, followed by ulceration necrosis.
Papule and ulcer are the main pathological lesions. They heal
over months to years, leaving scars.
clinical f eatures: L. tropica causes old world cutaneous
leishmaniasis.
• Features of the disease vary with epidemiological pattern
from region-to-region.
• Three distinct patterns of old world CL have been
recognized.
• The anthroponotic urban type causing painless dry
ulcerating lesions, leading to disfiguring scars, caused by
the species L. tropica.
This is prevalent from the Middle East to NorthWestern India. The most important vector is P. sergenti.
IL is seen mainly in children in endemic areas and is
called as oriental sore or Delhi boil.
It begins as a raised papule, which grows into a
nodule that ulcerates over some weeks.
Lesions may be single or multiple and vary in size
from 0.5 to more than 3 cm. Lymphatic spread and
lymph gland involvement may be palpable and may
precede the appearance of th e skin lesion.
The margins of the ulcer are raised and indurated.
The ulcer is usually painless unl ess secondary
bacterial infection occurs.
There may be satellite lesions, especially in L. major
and L. tropica infections.
The dry ulcers usually heal spontaneously in about
an year.
• The zoonotic rural type causing moist ' which are
inflamed, often multiple, caused by L. major.
The incubation period is usually less than 4 months.
Lesions due to L. major heal more rapidly than L.
tropica
This is seen in the lowland zones of Asia, Middle East
and Africa.
Gerbils, rats and other rodents are the reservoirs.
P. papatasi is the most important vector.
• Diffuse cutaneous leishmaniasis: The nonulcerative and
often diffuse lesions caused by L. aethiopica and seen in
the highlands of Ethiopia and Kenya are known as diffuse
cutaneous leishmaniasis (DCL).
P. /ongipes is the usual vector.
It is a rare form of disease, where nodular lesions
although restricted to skin are disseminated on the
face and extremities from initial localized papule.
It is cha racterized by low humoral as well as cellmediated immunity.
the lesions last for years or even for entire age.
It is difficult to treat.
Leishmaniasis recidiuans is a type of lesion seen in
persons with a high degree of cell-mediated immunity to
the parasite. The lesions are chronic with alternating periods
of activity and healing, characterized by a central scar with
peripheral activity. The lesions resemble those of lupus or
tuberculoid leprosy. Parasites are very scanty in the lesions.
Leishmanin test is strongly positive. Chemotherapy is not
very useful. Better results follow local application of heat.
Laboratory diagnosis:
Microscopy:
• Smear is made from the material obtained from the
indurated edge of nodule or sore and stained by Giemsa
or Leishman stain.
• Amastigotes are found in large numbers inside the
macrophages.
• Definitive d iagnosis is mad e by d emonstration of
amastigote in the smear collected from the lesion.
Culture: Promastigote forms can be isolated by culture of the
aspirate material in NN 1 medium.
Skin test: Leishman.in skin test is helpful. Positive leishmanin
test in children under 10 years of age from endemic areas is
highly suggestive of the disease. The skin test is negative in
diffuse CL.
Serology: these are of limited value as the patient shows no
detectable levels of circulating antibodies.
Treatment: The specific treatment of CL is same as VL.
• Antimony-resistant diffuse CL can be treated with
pentamidine.
• Topical treatment consists of a paste of 10% charcoal in
sulfuric acid or liquid nitrogen.
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Prophylaxis:
Control of sandfly population by insecticides and
sanitation measures.
Personal protection by use of protective clothing and use
of insect repeUants.
Elimination of mammalian reservoir.
New World Leishmaniasis
L. Braziliensis Complex and L. Mexicana Complex
History and distribution: Lindenberg and ParanJ10s (1909)
first described amascigotes in the ulcers of skin in a man in
Brazil. Vian na (1911) named the species as L. braziliensis.
• L. braziliensis complex and L. mexicana complex cause
new world leishmaniasis in Central and South America.
Habitat: These occur as intracellular parasite. The amastigote
form is seen inside the macrophages of skin and mucous
membran e of the nose and buccal cavity. The promastigote
form occurs in vector species Lutzomyia.
Morphology: Morphology of amastigoce and promascigote
forms of both the parasites is same as that of the other two
species of Leishmania.
Life cycle: The life cycle of Leishmania species causing the
new world cutaneous and mucocutaneous leishmaniasis is
similar to that of L. donovani except:
• Amastigotes are found in the reticuloendothelial cells and
lymphoid tissues of skin, but not in the internal organs.
• The infection is transmitted to man from animals by bite
of sand fly vectors of genus Lutzomyia.
• Sylvatic rodents and domestic animals are the common
sources and reservoir of infection.
• Direct transmission and aucoinfection also occurs
man-co-man.
Clinical features: L. mexicana complex leads to cutaneous
leishmaniasis which closely resembles the old world CL.
However a specific lesion of caused by L. mexicana is chiclero
ulcer which is characterized by ulcerations in pinna.
• Chiclero ulcer is also called as self healing sore of Mexico.
• L. braziliensis complex causes both mucocutaneous
leishmaniasis (espundia) and \"CL''.
• L. braziliensis causes the most severe and destructive
form of cutaneous lesion.
• It involves the nose, mouth and larynx.
• The patient experiences a nodule at the site of sand fly bite
with symptoms consistent with oriental sore.
• Subsequent mucocutaneous involvement leads co
nodules inside the nose, perforation of the nasal septum,
and enlargement of the nose and lips (espundia).
Hemoflagellates
• If the larynx is involved, the voice changes as well.
• Ulcerated lesions may lead to scarring an d tissue
destruction that can be disfiguring.
• The disease occurs predominantly in Bolivia, Brazil and
Peru.
• L. mexicana, L. amazonensis also cause DCL similar to
chat of L. aethiopica in individuals with defective cellmediated immunity. Montenegro skin test is negative.
Pian bois: It is also known as \"forest yaws''.
It is caused by L. braziliensis guyanensis and is
characterized by appearance of single or multiple painless
dry persistent ulcers appear all.
Laboratory diagnosis:
Microscopy: Amastigotes are demonstrated in smears taken
from lesions of skin and mucous membrane. L. mexicana
amastigotes are larger than those of L. braziliensis and their
k:inetoplast is more centrally placed.
Biopsy: Amastigotes can also be demonstrated from slit-skin
biopsy.
Culture: Culturing material obtained from ulcers in N N
medium demonstrates promastigotes. L. mexicana grows
well in comparison to L. braziliensis, which grows slowly.
Serology: Antibodies can be detected in serum by IFA test,
which is positive in 89- 95% of cases. ELISA is also a sensitive
method to detect antibody; being positive in 85% of cases.
Skin test: Leishmanin test is positive in cutan eous and
mucocutaneous leishmaniasis.
Treatment: Treatment with a pentavalent antimonial compound is moderately effective for mild mucocutaneous
leishmaniasis.
Amphotericin -B is the best alternative drug currently
available.
In case of respiratory complications, glucocorticoids can
be used.
Prophylaxis:
• Due to sylvatic and rural nature of the disease, control is
often difficult.
• Use of insect repellants, spraying of insecticides and
screening are advisable.
• Forest workers should use protective clothing and other
protective measures.
• A recently developed polyvalent vaccine using five
Leishmania strains has been reported to be successful in
reducing the incidence of CL in Brazil.
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Paniker's Textbook of Medical Parasitology
KEY POINTS OF LEISHMAN/A
• Visceral leishmaniasis (kala-azar) is caused by L. donovani
and L. intantum.
• Vector of kala-azar is sandfly (argentipes).
• Amastigote forms (LD body) are found in macrophages and
monocytes in human.
• Promastigote forms with a single flagellum is found in vector
sandfly and artificial culture.
• Clinical features: Kala-azar: Fever, hepatosplenomegaly,
marked anemia, darkly pigmented skin, weight loss,
cachexia, etc.
• Post-kala-azar dermal leishmaniasis: Seen after 1 - 2 years
of treatment in 3-10% cases and is a nonulcerative lesion
of skin.
• Diagnosis: By demonstrations of LO bodies in peripheral
blood, bone marrow aspirate, splenic aspirate and lymph
node aspirate; culture done in NNN medium; aldehyde test;
detection of specific antigen and antibody by IIF, ELISA, DAT
and rapid rk 39 antibody detection test.
• Blood picture: Anemia, thrombocytopenia, leukopenia with
relative lymphocytosis and hypergammaglobulinemia.
• Treatment: Sodium stibogluconate, amphotericin-B and oral
miltefosine.
• Old world CL (oriental sore) is caused by L. tropica and the
vectors are P. sergenti and P. papatasi.
• New world mucocutaneous (espundia) and CL are caused by
L. brazifiensis and L. mexicana. Vector is sandfly of genus
Lutzomyia.
REVIEW QUESTIONS
Describe briefly the life cycle and laboratory diagnosis of:
a. Trypanosoma brucei gambiense
b. Trypanosoma cruzi
c. Leishmania donovani
Write short notes on:
a. Sleeping sickness
b. Chagas disease
c. Antigenic variations of Trypanosoma brucei gambiense
d. Morphological stages of hemoflagellates
e. Trypanosoma rangeli
f. Kala-azar
g. Post-kala-azar dermal leishmaniasis
h. Cutaneous leishmaniasis
i. Diffuse cutaneous leishmaniasis
Differentiate between:
a. East African trypanosomiasis and West African trypanosomiasis
b. Trypanosoma cruzi and Trypanosoma rangeli
MULTIPLE CHOICE QUESTIONS
Vector for Trypanosoma cruzi is
a. Reduviid bug
b. Tsetse fly
c. Sandfly
d. Hard tick
All of the following are obligate intracellular parasite except
a. Plasmodium
b. Trypanosoma cruzi
c. Toxoplasma gondii
d. Trypanosoma brucei gambiense
Romana's sign occurs in
a. Babesiosis
b. Leishmaniasis
c. Trypanosomiasis
d. Schisotosomiasis
Vector for T. brucei gambiense is
a. Sandfly
b. Reduviid bug
c. Tsetse fly
d. House fly
S. Winterbottom sign in sleeping sicknens refers to
a. Unilateral conjunctivitis
b. Posterior cervical lymphadenitis
c. Narcolepsy
d. Trasient erythema
The drug that can clear trypanosomes from blood and lymph
nodes and is active in late nervous system stages of African
sleeping sickness is
a. Emetine
b. Melarsoprol
c. Nifurtimox
d. Suramin
Which of the following is not true about West African
trypanosomiasis.
a. Primary reservoirs are human
b. Low parasitemia
c. Illness is usually chronic
d. Minimal lymphadenopathy
Chronic infections with which of the following hemoflagellates
may be associated with megaesophagus or megacolon
a. Trypanosoma gambiense
b. Trypanosoma cruzi
c. Leishmania donovani
d. Leishmania tropica
True about visceral leishmaniasis is/are
a. Caused by Leishmania tropica
b. Post leishmaniasis dermatitis develops in 20% of patients
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c. Antimonial compounds are useful
d. Vector is tsetse fly
Which of the following is most severely affected in kala-azar
a. Spleen
b. Liver
c. Lymph nodes
d. Bone marrow
LD bodies are
a. Amastigotes of Leishmania donovani inside RBCs
b. Giant cells seen in leishmaniasis
c. Degenerative lesions seen in leishmaniasis
d. Amastigotes of Leishmania donovani inside macrophages
In a case of kala-azar, aldehyde test becomes positive after
a. 2 weeks
b. 4weeks
c. 8 weeks
d. 12 weeks
Mucocutaneous leishmaniasis is caused by
a. Leishmania braziliensis
b. Leishmania donovani
c. Leishmania tropica
d. None of the above
Chiclero's ulcer is caused by
a. Leishmania mexicana complex
b. Leishmania braziliensis complex
c. Leishmania trapica
d. Leishmania infantum
Answer
a
b
d
C
C
a
C
d
b
d
Hemoflagellates
b
a
d
a
P:88
CHAPTER 6
MALARIA
• INTRODUCTION
Procozoan parasites characterized by the production of sporelike oocysts containing sporozoites were known as sporozoa.
.
•
•
.
They live inlracellularly, at least during part of their life
cycle.
At some stages in their life cycle, they possess a structure
called the apical complex, by means of which they anach
co and penetrate host cells.
These protozoa are therefore grouped under the Phylum
Apicomplexa.
The medically important parasites in this group are the
malaria parasites, Coccidia, and Babesia.
The Phylum Apicomplexa includes two classes viz.
(1) hematozoa and (2) coccidia and three orders-
(1) eimeriida, (2) hemosporida and (3) piroplasmida
(Table l).
Note: Many minute intracellular protozoa formerly grouped
as sporozoa have bee n reclassified beca use of some
strucrural differences. These are now called microspora. they
infect a large spectrum of hosts including vertebrates and
invertebrates. Infection is mostly asymptomatic, but clinical
illness is often seen in the immunodeficient.
Table 1: Phylum Apicomplexa (Sporozoa)
Class
Hematozoa
Coccidia
Order
Hemosporida
Piroplasmida
Eimeriida
Genera
• Plasmodium
• Babesia
• Toxoplasma
• Cyclospora
• Cryptosporidium
• lsospora
• Sarcocystis
• CLASSIFICATION
Malaria parasite belongs to:
Phylum: Apicomplexa
Class: Sporozoa
Order: Hemosporida
Genus: Plasmodium.
• The genus Plasmodium is classified into two subgenera:
(1) P. vivax, (2) P. malariae and P. ovale belong to the
subgenus Plasmodium while P. falciparum belongs to
subgenus Laverania because it differs in a number of
aspects from tl1e other three species.
• P. vivax, P. malariaeand P. ovaleare closely related to other
primate malaria parasites. P.falciparum is more related to
bird malaria parasites and appears to be a recent parasite
of humans, in evolutionary terms. Perhaps for this reason,
falciparum infection causes the most severe form of
malaria and is responsible for nearly all fatal cases.
• P. knowlesi, a parasite of long-tailed Macaque monkeys
may also affect man.
• CAUSATIVE AGENTS OF HUMAN MALARIA
• Plasmodium vivax: Benign tertian malaria
• Plasmodiumfalciparum: Malignant tertian malaria
• Plasmodium malariae: Benign q uartan malaria
Plasmodium ovate: Benign tertian malaria.
• MALARIA PARASITE
History and Distribution
Malaria has been known from ancient times. Seasonal
intermittent fevers with chills and shivering, recorded in
the religious and medical texts of ancient Indian, Chinese
and Assyrian civilizations, are believed to have been malaria
(Fig. I).
P:89
• The name malaria (mal: bad, aria: air) was given in the
18th century in Italy, as it was thought to be caused by foul
emissions from marshy soil.
• The specific agent of malaria was discovered in red blood
cells (RBCs) of a patient in 1880 by Alphonse Laueran, a
French army surgeon in Algeria.
• ln 1886, Golgi in Italy described the asexual development
of the parasite in RBCs (erythrocytic schizogony), which
therefore came to be called as Golgi cycle.
• three different species of malaria parasite infecting man:
(1) P. vivax, (2) P. malariae, and (3) P. falciparum were
described in Italy between 1886 and 1890. The fourth
species, P. ovale was identified only in 1922.
• The mode of transmission of the disease was established
in 1897, when Ronald Ross in Secunderabad, India
identified the developing stages of malaria parasites in
mosquitoes. This led to various measures for th e control
and possible eradication of malaria by mosquito control.
Both Ross (1902} and Laveran (1907) won the Nobel Prize
for their discoveries in malaria.
• Incidence of malaria is more in poor population in rural
areas, also in urban areas having bad sanitary condition.
An epidemic can develop wh en there are changes in
environmental, economic and social conditions such as
migrations and heavy rains following draughts.
• The re lative prevalence of the four species of malaria
parasites varies in different geographical regions (Fig. 1):
l. P. uiuax is the most widely distributed, being most
common in Asia, North Africa, and Central and South
America.
P. Jalciparum, the predominant species in Africa,
Papua New Guinea and Haiti, is rapidly spreading in
Southeast Asia and India.
P. malariae is present in most places but is rare,
except in Africa.
P. ouale is virtually confined to West Africa where it
ranks second after P.falciparum (Fig. 1).
D Areas where Areas where
malaria Is absent malaria Is present
• Areas with
limited risk of
malaria
Fig. 1: Global distribution of malaria
Malaria and Babesia
• Malaria may occur in endemic as well as epidemic
patterns. It is described as endemic, when it occurs
constantly in an area over a period of several successive
years and as epidemic, when periodic or occasional sharp
rises occur in its incidence.
Th e World Health Organization (WHO} h as
rec01mnended the classification of endemicity depending
on the spleen or parasite rate in a statistically significant
sample in the populations of children (2-9 years) and
adults. According to this:
Hypoendemic (transmission is low): Spleen or
parasite rate less than 10%
Mesoendemic (transmission is moderate): Spleen or
parasite rate 11-50%
Hyperendemic (transmission is intense but seasonal):
Spleen or parasite rate 51-75%
Holoendemic (transmission of high intensity): Spleen
or parasite rate more than 75%.
In India, malaria is a major public health threat. In India,
about 27% population lives in high transmission (>l
case/ 1,000 population) and about 58% in low transmission
(0- 1 case)/ 1,000 population) area.
• In spite of decline of total number of malaria cases, the
number of cases of P. Jalciparum malaria has increased.
Vectors
Human malaria is transmitted by over 60 species of female
Anopheles mosquito.
• The male mosquito feeds exclusively on fruits and juices,
but the female needs at least two blood meals, before the
first batch of eggs can be laid.
• Out of 45 species of Anopheles mosquito in India, only
few are regarded as the vectors of malaria. These are An.
culicifacies, An. jluviatilis, An. stephensi, An. minimus, An.
philippinensis, An. sundaicus, etc.
Life Cycle
Malaria parasite passes its life cycle in two hosts:
Definitive host: Female Anopheles mosquito.
Intermediate host: Man.
• The life cycle of ma larial parasite comprises of two
stages-(1) an asexual phase occurring in humans,
which act as the intermediate host and (2) a sexual phase
occurring in mosquito, which serves as a definitive host
for the parasite (Fig. 2).
Asexual Phase
• In this stage, the malaria parasite multiplies by division
or splitting a process designated to as schizogony (from
schizo: to split, and gone: generation).
P:90
Paniker's Textbook of Medical Parasitology
Mosquito injects
sporozoites during
blood meal
Mature
sporozoites
oocyst rup
, wh
tures
ich reach ./
~ \"
the salivary gland of mosquito • l
I
I
I
I
I
/
/
______ r
...
' ' ' \
, Ookinete penetrates
' B the epithelial lining of , mosquito stomach
Ookinete wall
\ Fertilization--Q ' , occurs,
\
\
\
I
Sporozoites
infect liver cell
Sch1zont formed
~ :_. Ruptured schizont
merozoites liberated
Merozoites invade RBC
, zygote formed 1
', ~ - 0 1 '- ~.:.. I
' , Microgamete Macrogamete..- 1 \"'- •• ,-p.1 ~ ~11~- - ----'-Late trophozoite
Early trophozoite formed
.., --. ___ -.. ,, - J\ ~ :r'---- - -l.Mature schizont
Midgut Gametogony of mosquito . ~~ Female Male Mature schizont burst
releasing merozoites
Fig. 2: Life cycle of the Plasmodium vivax
Abbreviation: RBC, red blood cell
Because this asexual phase occurs in man, it is also called
the vertebrate, intrinsic, or endogenous phase.
• In humans, schizogony occurs in two locations- (!) in the
red blood cell ( erythrocytic schizogony) and (2) in the liver
cells ( exoerythrocytic schizogony or the tissue phase).
• Because schizogony in the liver is an essential step before
the parasites can invade erythrocytes, it is called preerythrocytic schizogony.
• The products of schizogony, whether erythrocytic or
exoerythrocytic, are called merozoites (meros: a part,
zoon: animal).
Sexual Phase
• Female Anopheles mosquito represents definitive host, in
which sexual forms takes place. Although the sexual forms
of the parasite (gametocytes) originate in human RBCs.
• Maturation and fertilization take place in the mosquito,
giving rise to a large number of sporozoites (from sporos:
seed). Hence, this phase of sexual multiplication is called
sporogony. It is also called the invertebrate, extrinsic, or
exogenous phase.
Thus, there is an alternation of hosts as the asexual phase
takes place in humans followed by sexual phase in mosquito.
Human Cycle (Schizogony)
Human infection comes through the bite of the infective
fem ale Anopheles mosquito (Fig. 2).
• The sporozoites, which are infective forms of the parasite
are present in the salivary gland of the mosquito.
• they are injected into blood capillaries when the mosquito
feeds on blood after piercing the skin.
• Usually, 10- 15 sporozoites are injected at a time, but
occasionally, man y hundreds may be introduced.
The sporozoites pass into the bloodstream, where many
are destroyed by the phagocytes, but some reach the liver
and enter the parenchymal cells (hepatocytes).
P:91
Pre-erythrocytic (tissue) stage or exoery throcytic stage:
Within an hourofbeing injected into the body by the mosquito,
the sporozoites reach the liver and enter the hepatocytes to
initiate the stage of pre-erythrocytic schizogony or merogony.
• The sporozoites, which are elongated spindle-shaped
bodies, become rounded inside the liver cells.
• They enlarge in size and undergo repeated nuclear
division to form several daughter nuclei; each of which is
surrounded by cytoplasm.
• this stage of the parasite is called the pre-erythrocytic or
exoerythrocylic schizont or meront.
• the hepatocyte is distended by the enlarging schizont and
the liver cell nucleus is pushed co the periphery.
• Mature liver stage schizonts arc spherical ( 45-60 µm),
multinucleate and contain 2,000- 50,000 uninucleate
merozoites.
• Unlike erythrocytic schizogony, there is no pigment in
liver schizonts. These normally ruprure in 6-15 days and
release thousands of merozoites into the bloodstream.
• The merozoites infect the erythrocytes by a process of
invagination.
• Prepatent period: The interval between the entry of the
sporozoites into the body and the first appearance of the
parasites in blood is called the prepatent period.
• The duration of the pre-erythrocytic phase in the liver, the
size of the mature schizont and the number of merozoites
produced vary with the species of the parasite (Table 2).
Latent stage: In P. vivax and P. ovate, two ki nds of
sporozoites are seen, some of which multiply inside
hepatic cells to form schizonts and others persist and
remain dormant (resting phase).
• Relapse: The resting forms are ca lled hypnozoites
(hypnos: sleep). From time to time, some are activated to
become schizonts and release merozoites, which go on
infecting RBCs producing clinical relapse.
• Recrudescence: In P. falciparum and P. malariae, initial
tissue phase disappears completely, and no hypnozoites
are found. However, small numbers of erythrocytic
parasites persist in the bloodstream and in d ue course of
time, they multiply to reach significant numbers resulting
in clinical disease (short-term relapse or recrudescence).
Ery throcytic stage: t he merozoiles released by pre-erythrocytic
schizonts invade the RBCs.
the receptor for merozoites is glycophorin, which is a
major glycoprotcin on the red cells. The differences in the
glycophorins of red cells of different species may account
for the species specificiry of malaria parasites.
Merozoites are pear-shaped bodies, about 1.5 µmin length,
possessing an apical complex (rhoptery). They attach to
the erythrocytes by their apex and then the merozoites
lie within an intraerythrocytic parasitophorous vacuole
formed by red cell membrane by a process of invagination.
Malaria and Babesia
• In the erythrocyte, the merozoite loses its internal
organelles and appears as a rounded body having a
vacuole in the center with the cytoplasm pushed to the
periphery and the nucleus at one pole. These young
parasites are, therefore called the ringforms or young
trophowites.
• The parasite feeds on the hemoglobin of the erythrocyte.
it does not metabolize hemoglobin completely and
therefore, leaves behind a hematin-globin pigment called
the malaria pigment or hemozoin pigment, as residue
(Box 1).
• The malaria pigment released when the parasitized
cells rupture is taken up by reticuloendothelial cells.
Such pigment-laden cells in the internal organs provide
histological evidence of previous malaria infection.
• As the ring form develops, it enlarges in size becoming
irregular in shape and shows ameboid motility. This is
called the ameboidform or late trophozoiteform.
When the ameboid form reaches a certain stage of
development, its nucleus starts dividing by m itosis
followed by a division of cytoplasm to become mature
schizonts or meronts.
• A mature sch izont contains 8-32 merozoites and
hemozoin. The mature schizont bursts releasing the
merozoites into the circulation.
The merozoites invade fresh erythrocytes within which
they go through the same process of development. 1h is
cycle of erythrocytic schizogony or merogony is repeated
sequentially, leading to progressive increase in the
parasitemia, till it is arrested by the development of host
immune response.
Table 2: Features of pre-erythrocytic schizogony in human malaria
parasites
P. vivax P. falciparum P. malariae P. ovale
Pre-erythrocytic 8 6 15 9
stage (days)
Diameter of 45 60 55 60
pre-erythrocytic
schizont (µm)
No. of merozoites 10,000 30,000 15,000 15,000
in pre-erythrocytic
schizont
Box 1: Appearance of malaria pigments in different species
• P. vivax: Numerous fine golden-brown dust-like particles
• P. falciparum: Few 1-3 solid blocks of black pigment
• P. malariae: Numerous coarse dark-brown particles
• P. ovale: Numerous blackish-brown particles.
P:92
Paniker's Textbook of Medical Parasitology
• The rupture of the mature schizont re leases la rge
quantities of pyrogens. This is responsible for the febrile
paroxysms characterizing malaria.
• The interval between the entry of sporozoites into the
host and the earliest manifestation of clinical illness is
the incubation period (Box 4). This is different from
prepatent period, which is the time taken from entry of the
sporozoites to the first appearance of malaria parasite in
peripheral blood.
V,
~
·o
N
0
.t:
0..
e
tV,
'E
0
N
:c
0
(J)
V,
.,
.
0
.9
.,
E
(!) \"'
-
't::
(l)
w
<I)
~ _J
<I)
<ii
E
<I)
u..
P. vivax P falciparum
@
.
•
. ~ •
• In P. falciparum, erythrocyric schizogony always takes
place inside the capillaries and vascular beds of internal
organs. Therefore, in P. falciparum infections, schizonrs
and merozoites are usually not seen in the peripheral
blood.
• The erythrocytic stages of all th e fo ur species of
Plasmodium arc shown in Figure 3.
P malariae P ovate
V
.,.... .. -! . · ... ·: .; . : • • -• •
• I
.
' . ,
Fig. 3: Malaria parasites-Erythrocytic stages of the four species (Giemsa stain. Magnification 2000X)
P:93
Gametogony
After a few erythrocytic cycles, some of the merozoites that
infect RBCs do not proceed to become trophozoites or
schizonts but instead, develop into sexually differentiated
forms, the gametocytes.
• They grow in size till they almost fill the RBC, but the
nucleus remains undivided.
• Development of gametocytes generally takes place within
the internal organs and only the mature forms appear in
circulation.
• The mature gametocytes are round in shape, except in P.
Jalciparum, in which they are crescent-shaped .
• In all species, the female gametocyte is larger
(macrogametocyte) and has cytoplasm staining dark blue
with a compact nucleus staining deep red. In the smaller
male gametocyte (microgametocyte), the cytoplasm
stains pale blue or pink and the nucleus is larger, pale
stained and diffuse. Pigment granules are prominent.
• Female gametocytes are generally more numerous than
the male.
• Gametocyte appears in circulation 4-5 days after the first
appearance of asexual form in case of P. vivax and 10-12
days in P.falciparum.
• A person with gametocytes in blood is a carrier or
reservoir.
• the gametocytes do not cause any clinical illness in the
host, but are essential for transmission of the infection.
• A gametocyte concentration of 12 or more per mm3 of
blood in the human host is necessary for mosquitoes to
become infected.
The Mosquito Cycle (Sporogony)
When a fema le Anopheles mosquito ingests parasitized
erythrocytes along with its blood meal, the asexual forms of
malaria parasite are digested, but the gametocytes are set free
in the midgut (stomach) of mosquito and undergo further
development.
The nuclear material and cytoplasm of the male
gametocytes divides to produce eight microgametes with
long, actively motile, whip-like filaments (ex.flagellating
malegametocytes) (Fig. 4).
• At 25°C, the cxflagellation is complete in 15 minutes for
P. vivax and P. ovale and 15-30 minutes for P.falciparum.
1he female gametocyte does not divide but undergoes a
process of maturation to become the female gamete or
macrogamete. It is fertilized by one of the microgametes
to produce the zygote (Fig. 4).
• Fertilization occurs in 0.5-2 hours after the blood meal.
The zygote, which is initially a motionless round body,
gradually elongates and within 18-24 hours, becomes a
vermicular motile form with an apical complex anteriorly.
This is called the ookinete (travelling vermicule).
Malaria and Babesia
Female gametocyte Male gametocyte
~ l l
0 flagellation I
-
'
Macrogamete Microgamete
Fig. 4: Schematic diagram showing formation of microgamete and
macrogamete
• It penetrates the epithelial lining of the mosquito
stomach wall and comes to lie just beneath the basement
membrane.
• It becomes rou nded into a sp here with an elastic
membrane. 1his stage is called the oocyst, which is yet
another multiplicatory phase, within which numerous
sporozoites are formed.
• tje mature oocyst, which may be about 500 µm in size,
bulges into body cavity of mosquito and when it ruptures,
the sporozoites enter into the hemocele or body cavity,
from where some sporozoites move to the salivary glands.
The mosquito is now infective and when it feeds on
humans, the sporozoites are injected into skin capillaries
to initiate human infection.
• Extrinsic incubation period: The time taken for
completion of sporogony in the mosquito is about 1- 4
weeks (extrinsic incubation period), depending on the
environmental temperature and the species.
Types of Malarial Parasites
Plasmodium Vivax
P. vivax has the widest geographical distribution, extending
through the tropics, subtropics and temperate regions. It
is believed to account for 80% of all malaria infections. lt is
the most common species of malaria parasite in Asia and
America, but is much less common in Africa. It causes benign
tertian malaria with frequent relapses.
• The sporozoites of P. vivax are narrow and slightly curved.
On entering the liver cells, thesporozoites initiate two types
of infection. Some develop promptly into exoerythrocytic
schizonts, while others persist in the dormant state for
varying periods as hypnozoites. There may be two distinct
types of sporozoites: ( 1) the tachysporozoites (tachy: fast),
which develops into the primary exoeryth rocytic schizont
and (2) the bradysporozoite (brady: slow) which becomes
the hypnozoite.
P:94
Paniker's Textbook of Medical Parasitology
• The pre-erythrocytic schizogony lasts for 8 days and
the average number of merozoites per tissue schizont is
10,000.
• Merozoites of P. vivax preferentially infect reticulocytes
and young erythrocytes.
• All stages of erythrocytic schizogony can be seen in
peripheral smears (Fig. 5).
• The degree of parasitization is not generally heavy, each
infected red cell usually having only one trophozoi te
and not more than 2-5% of the red cells being affected.
Reticulocytes are preferentially infected.
• The rrophozoite is actively motile, as indicted by its name
vivax. The ring fo rm is well-defined, with a prominent
central vacuole. One side of the ring is thicker and the
other side thin . Nucleus is situated on the thin side of the
ring (Signet ring appearance). The ring is about 2.5-3 µm
in diameter, about a third of the size of an erythrocyte.
The cytoplasm is blue and the nucleus red in stained
films. 1l1e ring develops rapidJy to the ameboid form and
accumulates malarial pigment (Figs 6 and 7).
Erythrocyte
........... : .. .. ........
. ... . .., . :. -
I .·
Commencing
chromatin
division
...... .. .......
•
Young
ring
stage
Further
chromatin
division
• The infected erythrocytes are enlarged and show red
granules known as Schujfner's dots on the surface.
They become irregular in shape, lose their red color and
present a washed out appearance. A few of the parasitized
erythrocytes retreat into the blood spaces of the internal
organs.
• The schizont appears in about 36-40 hours. It occupies
virtually the whole of the enlarged red cell. The schizont
matures in the next 6-8 hours, with the development of
merozoites, each with its central nucleus and surrounding
cytoplasm. The pigment granules agglomerate into
a few dark brown collections at the center, and with
the merozoites around it, this stage presents a rosette
appearance. There are about 12-24 ( usually 16) merozoites
per schizont.
Erythrocytic schizogony takes approximately 48 hours.
The red cell, which now measures about 10 µm in
diameter is heavily stippled and often distorted. It bursts
to liberate the merozoites and pigment. 1l1e pigment is
phagocytosed by reticuloendothelial cells .
. .
• . ........
. .
Older ring
stage with
Schuffner's
dots
.... ·.:- ..... ~=::..
· ..::~t~ ·;.,
. ~ . \"•
~. . •\"' •.:. .
Schizont
Adult ring in
enlarged cell,
Schuffner's
dots marked
Schizont mature
form prior to
merozoite liberation
.... : .. , .. .. ;; -· .. -~· . .. .. .. ... .. .. .. .. .... .. .. : .. :.,.· .. :.:_ .: .. =·~ .. ,. ............... .,
:. .. :: .. :: .. :.:.. .. ,.
Female
gametocyte
early stage
Female
gametocyte
mature
Male
gametocyte
Fig. 5: Plasmodium vlvax (Giemsa stain, magnification 2000X)
P:95
•
Leishman's, X1000 Oil
Fig. 6: Malarial parasite in blood fi lm-Ring stage of P. vivax
Source: Mohan H. Textbook of Pathology, 6th edition. New Delhi:
Jaypee Brothers Medical Publishers; 2010. p. 189.
• The merozoites measure about 1.5 µm and have no
pigment.
• Gametocytes appear early, usually within 4 days after
the trophozoites first appear. Both male and female
gametocytes are large, nearly filling the enlarged red cell.
The macrogametocyte has dense cytoplasm staining deep
blue and a small compact nucleus. lhe microgametocyte
has pale-staining cytoplasm and a large diffuse nucleus.
Pigment granules are prominent in the gametocytes.
Plasmodium Falciparum
The name Jalciparum comes from the characceristic sickle
shape of the gametocytes of this species (Jalx: sickle, parere: to
bringforth). 1his is the highly pathogenic of all the plasmodia
and hence, the name malignant tertian or pernicious malaria
for its infection.
• The disease has a high rate of complications and unless
treated, is often fatal. The species is responsible for almost
all deaths caused by malaria.
Schizogony: The sporozoites are sickle-shaped. the tissue
phase consists of only a single cycle of pre-erythrocytic
schizogony. No hy pnozoites occur. The ma ture liver
schizont releases about 30,000 merozoites.
• They attack both young and mature erythrocytes and
so the population of cells affected is very large. Infected
erythrocytes present a brassy coloration.
Ringform: The early ring form in the erythrocyte is very
d elicate and tiny, measuring only a one-sixth of the
red cell diameter. Rings are often seen attached along
th e margin of the red cell, the so-called form applique
or accole. Binucleate rings (double chromatin) are
Malaria and Babesia
Lelshman·s, X1000 Oil
Fig. 7: Malarial parasite in blood film- Ameboid form of P. vivax
Source: Mohan H. Textbook of Pathology, 6th edition. New Delhi:
Jaypee Brothers Medical Publishers; 2010. p. 189.
common resembling stereo headphones in appearance.
Several rings may be seen within a single erythrocyte.
In course of time, the rings become larger, about a third
of the size of the red cell and may have I or 2 grains of
pigment in its cytoplasm (Figs 8 and 9).
• The subsequent stages of the asexual cycle- late
trophozoite, early and mature schizoncs- a re not
ordinarily seen in peripheral blood, except in very severe
or pernicious malaria. The presence of P. falciparum
schizonts in peripheral smears indicates a grave prognosis
(Box 2).
• The mature schizont is smaller than in an y other species
and has 8-24 (usually 16) merozoites. The erythrocytic
schizogony takes about 48 hours or less, so that the
periodicity of febrile paroxysms is 36-48 hours.
Very high intensity of parasitization is seen in Jalciparum
malaria. In very severe infections, the rate of parasitized
cells may even be up to 50%.
• The infected erythrocytes are of normal size. They show a
few (6- 12) coarse brick-red dots which are called Maurer's
clefts. Some red cells show basophilic stippling.
Gametogony: It begins after several gen erations of
schizogony. Gametocytes are seen in circulation about
10 days after the ring stage first appears. The early
gametocytes seldom appear in peripheral circulation. The
mature gametocytes, which are seen in peripheral smears
are curved oblong structures, described as crescentic,
sickle, sausage, or banana-shaped. They are usually
referred co as crescents (Fig. 10).
• The male gamecocytes are broad and sausage-shaped or
kidney-shaped, with blunt rounded ends as compared
to the female gametocytes, which are thinner and more
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Paniker's Textbook of Medical Parasitology
Erythrocyte
. . '
. ~
Mature ring
and Maurer's
dots
Advanced merozoite
development with
commencing pigmentation
• '
Marginal
ring form
..
' .. .. Trophozoite amoeboid
stage commencing
chromatin division
Schizont mature with
centralized pigment
Rarely seen in peripheral circulation
t)
Young
ring stage
Nuclear
division
•
u
t)
•
Ring forms with double
chromation dots
Merozoite
development
Rarely seen in peripheral
circulation
Female gametocyte
(crescent)
Male gametocyte
(crescent)
Fig. 8: Plasmodium falciparum (Giemsa stain, magnification 2000X)
Fig. 9: Malarial parasite in blood film-Ring stage of P. falciparum
Source: Mohan H. Textbook of Pathology, 6th edition. New Delhi:
Jaypee Brothers Medical Publishers; 2010. p. 189.
Box 2: Pathogenesis of malignant malaria
• Late stage schizonts of P. falciparum secrete protein on the surface of
RBCs to form knob-like protuberances in erythrocyte's cell membrane.
These knobs produce specific adhesive Plasmodlum falciparum
erythrocyte membrane protein-1 (PfEMP-1 ) so that infected RBCs
become sticky.
• Sometime Inflammatory cytokines particularly IFN-y produced
by the malaria parasite upregulate the expression of endothelial
cytoadherence receptors like thrombospondin, E-selectin, VCAM-1,
ICAM-1 in capillaries in the brain, chondroitin sulfate B in placenta
and (D36 in most other organs. The infected RBCs stick inside and
eventually block capillaries and venules. This phenomenon is called
cytoadherence. At the same stage these P. falciparum infected RBCs
adhere to uninfected RBCs to form rosettes.
• This process of cytoadherence and rosetting causes capillary plugging
and decrease microclrculatory flow in vital organs like brain, kidney,
lungs, spleen, intestine, bone marrow and placenta resulting in serious
complications such as cerebral malaria.
• Other virulence factors of P. falciparum are histidine-rich protein II
(HRP II) and glycosylphosphatidylinositol (GPIJ.
Abbreviations: ICAM-1, intercellular adhesion molecule-1; IFN-y, interferon
gamma; RBCs, red blood cells; VCAM-1 , vascular cell adhesion molecule-1
P:97
Leishman's, X1000 Oil
Fig. 1 O: Malarial parasite in blood film-Gametocytes of P. falciparum
Source: Mohan H. Textbook of Pathology, 6th edition. New Delhi:
Jaypee Brothers Medical Publishers; 2010. p. 189.
typically crescentic, with sharply rounded or pointed ends.
The mature gametocyte is longer than the diameter of the
red cell and so produces gross distortion and sometimes
even apparent disappearance of the infected red cell. The
red cell is often seen as a rim on the concave side of the
gametocyte. The cytoplasm in the female gametocyte is
deep blue, while in the male it is pale blue or pink. The
nucleus is deep red and compact in the female, with the
pigment granules closely aggregated around it, while in
the male, it is pink, large and diffuse, with the pigment
granules scattered in the cytoplasm.
• Falciparum crescents can survive in circulation for up to
60 days, much longer than in other species. Gametocytes
are most numerous in the blood of young children, 9
months to 2 years old. They, therefore serve as the most
effective source of infection to mosquitoes.
Plasmodium Malariae
This was the species of malaria parasite first discovered by
Laveran in 1880 and the name malariae is the one given by
him. It causes quartan malaria, in which febrile paroxysms
occur every 4th day, with 72 hours interval between the bouts.
The disease is generally mild, but is notorious for its
long persistence in circulation in undetectable levels, for
50 years or more. Recrudescence may be provoked by
splenectomy or immunosuppression.
• The development of the parasite, in man and mosquito is
much slower than with other species. Chimpanzees may
be naturally infected with P. malariae and may constitute
a natural reservoir for quartan malaria.
• P. malariae occurs in tropical Africa, Sri Lanka, Burma
an d parts oflndia, but its diso·ibution is patchy.
Malaria and Babesia
• The sporozoites are relatively thick. Pre-erythrocytic
schizogony takes about 15 days, much longer than
in other species. Each schizont releases about 15,000
merozoites. Hypnozoites do not occur. The long latency of
the infection is believed to be due to long time survival of
few erythrocytiv forms in some internal organs.
• P. malariae preferentially infects older erythrocytes and
the degree of parasitization is low.
The ring forms resemble those of P. vivax, although
thicker and more intensely stained. The old rrophozoites
are sometimes seen stretched across the erythrocyte as
a broad band. These band forms are a unique feature of
P. malariae. Numerous large pigment granules are seen
(Fig. 11).
• The schizonts appear in about 50 hours and mature
during the next 18 hours. The mature schizon t has an
average of e ight merozoites, which usually present a
rosette appearance.
• The infected erythrocytes may be of the normal size
or slightly smaller. Fine stippling, called Ziemann's
stippling, may be seen with special stains. The degree of
parasitization is lowest in P. malariae.
• Erythrocytic schizogony takes 72 hours.
• The gametocytes develop in the internal organs and
appear in the peripheral circulation when fully grown.
Gametocytes occupy nearly the entire red cell. The male
has pale blue cytoplasm with a large diffuse nucleus,
while the female has deep blue cytoplasm and a small
compact nucleus.
Plasmodium Ova/e
This parasite produces a tertian fever resembling vivax
malaria, but with milder symptoms, prolonged latency and
fewer relapses.
• It is the rarest of all plasmodia infecting humans and is
seen mostly in tropical Africa, particularly along the West
Coast.
• The pre-erythrocytic stage extends for 9 days. Hepatocytes
containing schizonts usually have enlarged nuclei. The
mature liver schizont releases about 15,000 merozoites.
Hypnozoites are present.
• The trophozoites resemble those in vivax malaria, but are
usually more compact, with less ameboid appearance.
Schuffner's dots appear earlier and are more abundant
and prominent than in vivax infection (Fig. 12).
• The infected erythrocytes are slightly enlarged. In thin
films, many of them present an oval shape with fimbriated
margins. This oval appearance of the infected erythrocyte
is the reason for the name ovate given to this species.
• The schizonts resemble those of P. malariae, except that
the pigment is darker and the erythrocyte is usually oval,
with prominent Schuffner's dots.
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Paniker's Textbook of Medical Parasitology
Erythrocyte
Schizont,
commencing
daisy form
Ring form with
eccentric nucleus
Schizont. mature pigment
centrally clumped
daisy form
,,
-·~ . --
Commencement of band
form dividing chromatin
pigment accumulation
Female gametocyte
compact chromatin
Band form
Note: Chromatin on one
side or band
Male gametocyte
diffuse chromatin
Fig. 11: P/asmodium malariae stages of erythrocytic schizogony (Giemsa stain, magnification 2000X)
Erythrocyte
Commencing
chromatin division
Daisy form
of the parasite
• 0
Young ring
stage
Older ring
stage
Further chromatin
division
Schizont oval form
or erythrocyte
persisting
.. + •• \";,• •••• • • .. ,. .. . >· ••• . :;. . . .. . . . ~
... , ..... :-:! ,., , ' • . _;~-·
-<·· ,.\" •• •• • .,. . ,, .. , . . \"\"\"' ., ..,. .. ,.. • • - I I • I · •
Female
gametocyte
Adult ring in enlarged
oval erythrocyte
Schuffner's erythrocyte
..
<f . ....
..
Merozoite development
Note: Continued oval
form and Schuffner's dots
Male
gametocyte
Fig. 12: Plasmodium ovate stages of erythrocytic schizogony (Giemsa stain, magnification 2000X)
P:99
Malaria and Babesia
Mixed Infections Pathogenesis
ln endemic areas it is not uncommon to find mixed infections
with two or more species of malaria parasites in the same
individual.
Clinical manifestations in malaria are caused by products of
erythrocytic schizogony and the host's reaction to them.
• Toe disease process in malaria occurs due to the local
or systemic response of the host to parasite antigens
and tissue hypoxia caused by reduced oxygen delivery
because of obstruction of blood flow by the parasitized
erythrocytes.
• Mixed infection with P. vivax and P. falciparum is the
most common combination with a tendency for one or
the other to predominate.
• The clinical picture may be atypical with bouts of fever
occurring daily.
• Diagnosis may be made by demonstrating the
characteristic parasitic forms in thin blood smears.
The characteristics of the four species of plasmodia
infecting man are listed in Table 3.
Liver is enla rged and congested. Kupffer cells are
increased and filled with parasites. Hemozoin pigments
are also found in the parenchymal cells (Fig. 13).
Parenchymal cells show fatty degeneration, atrophy and
centrilobular necrosis.
Table 3: Comparison of the characteristics of plasmodia causing human malaria
P. vivax P. lalclparum P.malariae P. ovale
Hypnozoites Yes No No Yes
Erythrocyte preference Reticulocytes Young erythrocytes, but can Old erythrocytes Reticu locytes
infect all stages
Stages found in peripheral blood Rings, trophozoites, Only rings and gametocytes As in vivax As In vivax
schizonts, gametocytes
Ring stage Large, 2.5 µm, usually single, Delicate, small, 1.5 µm, double Similar to vivox, but Similar to vivax, more
prominent chromatin chromatin, and multiple rings thicker compact
common, accole forms found
Late trophozoite Large irregular, actively Compact, seldom seen in Band form Compact, coarse
ameboid, prominent vacuole blood smear characteristic pigment
Schizont Large filling red cell Small, compact, seldom seen Medium size Medium size
in blood smear
Number of merozoites 12- 24 in irregular grape-like 8-24 grape-like cluster 6-12 in daisy-head or 6-12 irregularly
cluster rosette pattern arranged
Microgametocyte Spherical, compact, pale blue Sausage or banana-shaped As in vivax As in vivax
(male gametocyte) cytoplasm, diffuse nucleus pale blue or pink cytoplasm,
large diffuse nucleus
Macrogametocyte Large, spherical, deep blue Crescentic, deep blue As in vivax As in vivax
(female gametocyte) cytoplasm, compact nucleus cytoplasm, compact nucleus
Infected erythrocyte Enlarged, pale, with Normal size, Maurer's clefts, Normal, occasionally Enlarged, oval
Schuffner's dots sometimes basophilic Ziemann's stippling fimbriated, prominent
stippling Schuffner's dots
Duration of schizogony (days) 2 2 3 2
Prepatent period (days) 8 5 13 9
Average incubation period (days) 14 12 30 14
Appearance of gametocyte after 4-5 10 12 11- 14 5-6
parasite patency (days)
Duration of sporogony in 9-10 10-12 25- 28 14-16
mosquito (25°CJ (days)
Average duration of untreated 4 2 40 4
infection (years)
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Paniker's Textbook of Medical Parasitology
,....,_...- - --Brain
Liver - ---- - -1
(Encephalopathy)
Heart
(Congestive
heart failure)
(Hepatomegaly) '\t:t-,--..,.--,--- Spleen
(Splenomegaly)
Kidneys
(Hemoglobinunc
nephrosis)
Fig. 13: Major pathological changes in organs in malaria
Box 3: causes of anemia in malaria
• Destruction of large number of RBCs by complement-mediated and
autoimmune hemolysis.
• Suppression of erythropoiesis in t he bone marrow.
• Increased clearance of both parasitized and nonparasitized RBCs by
the spleen.
• Failure of the host to recycle the iron bound in hemozoin pigment.
• Antimalarial therapy in G6PD deficient patients.
Abbreviations: G6PD, glucose-6-phosphate dehydrogenase; RBCs, red
blood cells
• Spleen is soft, moderately enlarged and congested in
acute infection. In chronic cases, spleen is hard with a
thick capsule and slate gray or dark brown or even black
in color due to dilated sinusoids, pigment accumulation
and fibrosis (Fig. 13).
Kidneys are enlarged and congested. Glomeruli
frequently contain malarial pigments and tubules may
contain hemoglobin casts (Fig. 13).
• The brain in P. Jalciparum infectio n is congested.
Capilla ries of the brain are plugged with parasilized RBCs.
The cul surface of the brain shows slate gray cortex with
multiple punctiform hemorrhage in subcortical white
matter.
• Anemia: Afte r few paroxysms of fever, normocytic and
normochromic anemia develops. Anemia is caused by
destrucrion of large numbe r of red cells by complementmediated autoimmune hemolysis. Spleen also plays an
active role by phagocytic removal of a large number of
both infected and uninfected RBCs. Excess removal of
uninfected RBCs may account for up co 90% of erythrocyte
lo s (Box3).
Box 4: Incubation period
, It is the time interval between the bite of infective mosquito and the
first appearance of clinical symptom s. The duration of incubation
period varies with the species of the parasite .
• The average incubation periods of different species of Plasmodium are
as follows:
P. vivax: 14 (12- 17) days
P. falc,parum: 12 (8- 14) days
P. ovale: 14 (8-31) days
P. malarioe: 28 (18-40) days.
The incubation period is to be distinguished from the prepatent
period, which is the interval between the entry of the parasites into the
host and the time when they first become detectable in blood.
The re is also decreased erythropoiesis in bone marrow
due to rumor necrosis factor (T F) toxiciry and failure of the
host to recycle the iron bound in hemozoin pigments.
• Cytokines like T F, interleukin {IL)-1 and inte rferon
(IFN)-gamma play an important role in the pathogenesis
of end-organ d isease of malaria.
Clinical Features
Benign Malaria
• Incubation period: 12- 17 days {Box4).
the typical clinical feature of malaria consists of periodic
bouts of fever with chill and rigor, followed by anemia,
splenomegaly and hepatomegaly.
• The classic febrile paroxysm comprises of three d istinct
stages- {!) cold stage, (2) hot stage and (3) sweating stage.
I . Cold stage: The patient feels incense cold with chill
and rigor along with lassitude, headache and nausea.
This stage lasts for 15 minutes to I hour.
Hot stage: The patient feels intensely hot. The
temperarure mounts to 4 1 •c or higher. Headache
persists but nausea commonly diminishes. This stage
lasts fo r 2- 6 hours.
Sweating stage: Profuse sweating follows the hot
stage and the temperature comes down to normal.
The skin is cool and moist. The patient usually falls
asleep to wake up refreshed.
The paroxysm usually begins in the early afternoon and
lasts for 8-12 hours. lhe febrile paroxysm synchronizes
with the erythrocytic schizogon y.
The periodicity is approximately 48 hours in tertian
malaria (in P. uiuax, P. falciparum and P. ovale) and
72 hours in quanan malaria {in P. malariae).
Quotidian periodiciry, with fever occurring at 24 hour
intervals may be due to two broods of tertian parasites
maturing on successive days or due to mixed infection.
Regular periodiciry is seldom seen in primary attack, but
is established usually only after a few days of continuous,
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remittent, or intermittent fever. True rigor is typically
present in vivaxmalaria and is less common infalciparum
infection.
There can be both hypoglycemia or hyperglycemia in
malaria.
Sometimes, there may be hyperkalemia due to red cell
lysis and fall in blood pH.
infection with P. vivax usually follows a chronic course
with periodic relapses, whereas P. ovale mala ria is
generally mild. Although P. malariaemalaria is less severe,
but it may lead to renal complications. Relapse mainly
occurs in inadequately treated cases after an interval of
8-40 weeks or more.
Malignant Tertian Malaria
incubation period: 8-14 days.
The most serious and fatal type of malaria is malignant
tertian malaria caused by P Jalciparum. Falciparum malaria
if not treated timely or adequately, severe life-threatening
complications may develop. In severe Jalciparum malaria,
parasitic load is very high and more than 5% red cells are
affected. The term pernicious malaria also have been applied
to these conditions thar include cerebral malaria, blackwater
fever, algid malaria and septicemic malaria (Box 5).
• Cerebral malaria: It is the most common complication
of malignant malaria.
The initial symptoms are nonspecific with fever,
headache, pain in back, anorexia and nausea.
Anemia: The patient may be anemic and mildly
jaundiced.
Hepatosplenomegaly: Liver and spleen are enlarged
and nomender.
Thrombocytopenia is common.
After 4- 5 days of high fever, cerebral malaria is
manifested by features of diffuse symme tric
encephalopathy like headache, confusion, increased
muscle tone, seizures, paralysis, slowly lapsing to
coma.
Box 5: Compl ications of falciparum malaria
• Cerebral malaria
• Algid malaria
• Septicemic malaria
• Blackwater fever
• Pulmonary edema
• Acute renal failure
• Hypoglycemia (<40 mg/dl)
• Severe anemia (Hb<S g/dl, PCV<l 5%)
• Hyperpyrexia
• Metabolic acidosis and shock
• Bleeding disturbances
• Hyperparasitemia.
Abbreviations: Hb, hemoglobin; PCV, packed cell volume
Malaria and Babesia
Retinal hemorrhages may be seen in 15% of adults.
Hypoglycemia is common in patients fo llowing
quinine therapy or with) hyperparasitemia.
In 10% of cases renal dysfunction progressing to
acute renal failure may occur.
Other complications include metabolic acidosis,
pulmonary edema and shock.
Even with treatment, death occurs in 15% of children
and 20% of adults who develop cerebral malaria.
This occurs particularly when nonimmune persons
have remained untreated or inadequately treated for
7-10 days after development of the primary fever.
The basic pathogenesis of cerebral malaria is due to
erythrocyte sequestration in microvasculature of
various organs.
Late stage schizonts of P.falciparum secrete a protein on
the surface ofRBCs to form knob-like deformities. This knob
produces specific adhesive proteins [Plasmodiumfalciparum
erythrocyte membrane protein-I (PfEMP-1)]. which promote
aggregation of infected RBCs to other noninfected RBCs and
receptors of capillary endothelial cells. These sequestrated
RBCs cause capillary plugging of cerebral microvasculature,
which results in anoxia, ischemia and hemorrhage in brain.
• Blackwater fever: A syndrome called b lackwater
fever (malarial hemoglobinuria) is sometimes seen in
Jalciparum malaria, particularly in patients, who have
experienced repeated past infections and inadequate
treatment with quinine. An autoimmune mechanism has
been suggested.
Patients with glucose-6-phosphate dehydrogenase
(G6PD) deficiency may develop this condition after
taking oxidant drugs, even in the absence of malaria.
Clinical manifestations include fever, prostration
and hemoglobinuria (black colored urine), bilious
vomiting and prostration, with passage of dark red or
blackish urine.
The pathogenesis is believed to be massive
intravascular hemolysis caused by antierythrocyte
antibodies, lead ing to massive abso rption of
hemoglobin by the renal tubules (hemoglobinuric
ne phrosis) producing blackwater fever.
Complications of blackwater fever include renal
failure, acute liver failure and circulatory collapse.
Algid malaria: This syndrome is characterized by
peripheral circulatory failure, rapid thready pulse with
low blood pressure and cold clammy skin. There may be
severe abdominal pain, vomiting, diarrhea and profound
shock.
Septicemic malaria: It is characterized by high
continuous fever with dissemination of the parasite to
various organs, leading to multiorgan fa ilure. Death
occurs in 80% of the cases.
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Paniker's Textbook of Medical Parasitology
Merozoite-induced Malaria
Natural malaria is sporozoite-induced, the infection being
transmitted by sporozoites introduced through the bite of
vector mosquitoes. Injection of merozoites can lead to direct
infection of red cells and erythrocytic schizogony with clinical
illness. Such merozoite-induced malaria may occur in the
following situations:
• Tra11sfusio11 malaria: Blood transfusion can accidentally
transmit malaria, if the donor is infected with malaria.
The parasites may remain viable in blood bank for 1-2
weeks. As this condition is induced by direct infection of
red cells by the merozoites, pre-erythrocytic schizogony
and hypnozoites are absent. Relapse does not occur and
incubation period is short.
Table 4 enumerates the differences between mosquitoborne malaria and blood transfusion malaria.
Congenital ma laria: A natural fo rm of merozoiteinduced malaria, where the parasite is transmitted
transplacentally from mother to fetus.
• Renal transplantation may lead to malaria if the donor
had parasitemia.
• Shared syringes among drug addicts may be responsible.
Tropical Splenomegaly Syndrome
Tropical splenomegaly syndrome (TSS) or hyper-reactive
malarial splenomegaly (HMS) is a benign condition seen in
people of malaria endemic areas mainly tropical Africa, new
Guinea and Vietnam.
ft happens from abnormal immunological response to
repeated malaria infection.
• Tropical splenomegaly syndrome is characterized by high
level of immunoglobulin M (IgM) against malaria due to
polyclonal activation of 8-cells, decreased C3 and massive
splenomegaly. Malaria parasite is absent in peripheral
blood.
Table 4: Difference between mosquito-borne malaria and blood
transfusion malaria
Mosquito-borne
mo/aria
Mode of transmission Mosquito bite
Infective stage Sporozolte
Incubation period Long
Pre-erythrocytic Present
schizogony
Hypnozoites May be present
Blood transfusion
malaria
Blood or blood products
transfusion
Trophozoite
Short
Absent
Absent
Severity
Relapse
Comparatively less More complications seen
May occur Does not occur
Radical treatment Required Not required
• A normocytic normochromic anemia is present which
does not respond to hematinics or antihelminthics.
Spleen and liver are enlarged, congested, with dilated
sinusoids and marked lymphocytic infiltration. umerous
pigment-laden Kupffer cells dot the liver. Changes are also
seen in bone marrow, kidneys and adrenals.
Tropical splen omegaly syndrome differs from various
other types of splenomegalies seen in the tropics in its
response to antimalarial treatment.
Immunity
Immunity in malaria could be two types: (1) innate immunity
and (2) acquired immunity.
Innate Immunity
It is the inherent, nonimmu ne mechanism of host
resistance against malarial parasite.
innate immunity could be due ro:
Duffy negative red blood cells: The invasion of red
cells by merozoites requires the presence of specific
glycoprotein receptors on the erythrocyte surface. It
has been found duffy blood group negative persons
are protected from P. vivax infection. Duffy blood
group is absent in West Africa where P. vivax malaria
is not prevalent.
Nature of hemoglobin: Hemoglobin E provides
natural protection against P. vivax. P.falciparum docs
not multiply properly in sickled red cells containing
lfbS. ickle cell anemia trait is very common in Africa,
where falciparum malaria is hyperendemic and
offers a survival advantage. HbF present in neonates
protects them against all Plasmodium species.
Glucose-6-phospha le dehydrogenase deficiency:
Innate immunity to malaria has also been related
to G6PD deficiency found in Mediterranean coast,
Africa, Middle East and India.
Human leukocyte a ntigen-B53: Human leukocyte
antigen-B53 (HL/\-853) is protected from cerebral
malaria associated with protection from malaria.
rutritional status: Patients with iron deficiency and
severe mat nutrition are relatively resistant to malaria.
Pregnancy: Falciparum malaria is more severe in
pregnancy, particularly in primigravida and may be
enhanced by iron supplementation.
Splenectomy: The spleen appears to play an
important role in immunity against malaria.
Splenectomy enhances susceptibility to malaria.
Acquired Immunity
Infection with malaria parasite induces specific immunity
involving both humoral and cellular immunity, which can
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bring about clinical cure but cannot eliminate parasites from
the body.
It can prevent superinfection, but is not powerful enough
to defend against reinfection. This type of resistance
in an infected host, which is associated with continued
asymptomatic parasite infection is called premunition.
This type of immunity disappears once the infection is
eliminated.
Humoral immunity: Circulating antibodies (IgM, lgG and
IgA) against asexual forms give protection by inhibiting red
cell invasion and antibodies against sexual forms reduce
transmission of malaria parasite.
• Acquired antibody-mediated immunity is transferred
from mother to fetus across the placenta and is evident in
endemic areas where infants below the age of 3 months
are protected by passive maternal antibodies.
Young children are highly susceptible to malaria. As they
grow up, they acquire immunity by subclinical or clinical
infections, so that incidence of malaria is low in older
children and adults.
Cellular immunity: Sensitized T cells release cytokines that
regu.late macrophage activation and stimulate B cells to
produce antibodies. The activated macrophages inside liver,
spleen and bone marrow phagocytose both parasitized and
nonparasitized RBCs.
Clinical note: Protective immunity against malaria is species
specific, stage specific and strain specific.
Recrudescence and Relapse
Recrudescence
In P. falciparum and P. malariae infections after the primary
attack, sometimes there is a period of latency, during which
there is no clinical illness. But some parasites persist in some
erythrocytes, although the level of parasitemia is below
the feve r threshold or sometimes below the microscopic
threshold. Erythrocytic schizogony is repeated at a low level
in the body when the number of parasites attain a significant
level, fresh malarial attack develops. This recurrence of clinical
malaria caused by persisting P. Jalciparum and P. malariae is
called recrudescence. Recrudescence may be due to waning
immunity of the host or possibly due to antigenic variation.
In P.Jalciparu.m infections, recrudescences are seen for 1-2
years, while in P. malariae infection, they may last for long
periods, even up to 50 years (Table 5).
Relapse
It is seen in inadequately treated P. vivax and P. ovale
infections. In both these species, two kinds of sporozoites are
seen, some of which multiply inside heparocytes promptly
Malaria and Babesia
Table 5: Differences between recrudescence and relapse
Recrudescence
Seen in P. falciparum and P. malariae
Due to persistence of the parasite at
a subclinical level in circulation
Occurs within a few weeks or
months of a previous attack
Can be prevented by adequate drug
therapy or use of newer antimalarial
drugs in case of drug resistance
Relapse
Seen in P. vivax and P. avale
Due to reactivation of
hypnozoites present in liver cells
Occurs usually 24 weeks to 5
years after the primary attack
Can be prevented by giving
primaquine to eradicate
hypnozoites
to form schizonts and others which remain dormant. These
latter forms a re called hypnozoites (from hypnos: sleep).
Hypnozoites remain inside the hepatocytes as uninucleated
forms, 4- 5 µm in diameter, for long periods. Reactivation of
hypnozoites leads to initiation of fresh erythrocytic cycles
and new anacks of malarial fever. Such new attacks of malaria,
caused by dormant ex:oerythrocytic forms, reactivated usually
from 24 weeks to 5 years after the primary attack are called
relapses (Table 5).
Laboratory Diagnosis
Demonstration of Parasite by Microscopy
Diagnosis of malaria can be made by demonstration of
malarial parasite in the blood (Box 6).
Two types of smears are prepared from the peripheral
blood. One is called thin smear and the other is called thick
smear.
Thin smears: 1hey are prepared from capillary blood of
finger tip and spread over a good quality slide by a second
slide held at an angle of 30-45° from the horizontal such
that a tail is formed.
• A properly made thin film will consist of an unbroken
smear of a single layer ofred cells, ending in a tongue,
which stops a little short of the edge of the slide.
• Thins smears are air dried rapidly, fixed in alcohol
and stained by one of the Romanowsky stains such as
Leishman, Giemsa, Field's, or JSB stain (named after
Jaswant Singh and Bhattacharjee).
• Thins smears are used for detecting the parasites and
determining the species.
thick smears: thhey can be made on the same slide of thin
smear or separately.
• In a thick film, usually three drops of blood are spread
over a small area (about 10 mm).
• The amount of blood in thin smear is about 1- 1.5 µL,
while in a thick smear it is 3-4 µL.
• The thick film is dried and kept in a Koplin jar for 5-10
minutes for dehemoglobinization.
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Paniker's Textbook of Medical Parasitology
Box 6: Morphological feature of malaria parasites in blood smear
. In P. vivax, P. ovate and P. matariae all asexual forms and gametocytes
can be seen in peripheral blood. In P. fatciparum infection, only ri ng
form alone or with gametocytes can be seen.
• Ring forms of all species appear as streaks of blue cytoplasm with
detached nuclear dots. They are large and compact in P. vivax, P. ovate,
and P. matariae and fine delicate with double chromatin (head-phone
appearance). In P. fa/ciparum, multiple rings with •accole\" forms are
seen.
• Gametocytes are banana-shaped (crescents) in P. falciparum and round
in P. vivax, P. ovate and P. matariae.
• Enlarged red blood cells (RBCs) with intracellular coarse brick-red
stippling (Schuffner's dots) are characteristic in P. vivax. In P. falciparum,
RBCs are normal in size with large red dots (Maurer's dots) and
sometimes, with basophilic stippling. Careful search in blood should be
made for mixed infections.
Box 7: Quantification of parasites
Quantification of parasites can be done by thick smear. The counting of
parasites are done to an approximate number in the following method:
• + = 1- 1 O parasite per 100 thick film fields
• ++ = 1 1 - 100 parasite per 100 thick film
• +++ = 1- 1 O parasite per thick film field
• ++++= More than 10 parasite per thick film field.
• It is not fixed in m ethan ol.
• 1hick film is stained similar to thin film.
• 1he stained film is examined under the oil immersion
m icroscope.
• 1he thick film is more sensitive, when examined by
an experienced pe rson, because it concentrates 20- 30
layers of blood cells in a small area.
• Thick film is more suitable for rapid d etectio n of
malarial parasite, pa rticularly wh en they are few (as
low as 20 parasites/ µL) (Box 7).
• 1he dehemoglobinized and stained thick film does
no r show any red cells, but only le ukocytes, a nd, when
present, the parasite s. But the parasites are o fte n
disto rte d in form, a nd as the diagnostic c hanges in
blood cells such as enlargement and stippling canno t
be made out, species identification is difficult.
• Thin film is examin ed first a t the tail e nd a nd if
pa rasites are found, there is n o need for examining
thick film. If parasites a re not detected in thin film,
then thick film should be examined.
• leis recomme nded that 200 oil immersio n fields should
be examined before a thick film is declar e d negative
(Fig. 14).
Quantitative Buffy Coat, Smear
The quantitative buffy coal {QBC) test is a novel method for
diagnosing malaria, wherein a small qua ntity of blood (50- 110
µL) of blood is spun in QBC centrifuge at 12,000 revolutions
per minutes for 5 minutes.
Multiple rings Erythroblast Gametocyte
Fig. 14: Malarial parasite, Ptasmodium fatciparum, in the peripheral
blood showing numerous ring stages and a crescent of gametocyte.
The background shows a normoblast
Source: Mohan H. Textbook of Pathology, 6th edition. New Delhi:
Jaypee Brothers Medica l Publishers; 2010. p. 314.
• Red blood cell conta ining malaria p arasites are less d ense
than normal RBCs and concentrate just be low the buffy
coat of leukocytes at the top of the e rythrocycic column.
• Precoating of the tube with a cridine ora nge induc es a
fluorescen ce on the pa rasites, which can the n be readily
visua lized u nde r the oil immersio n microscope because
the parasite contains de oxyribo nucleic acid (D A), but
the m ature RBCs do not contain DNA and ribonucleic acid
(RNA). The nucleus of the parasite is d etected by acridine
orange stains a nd appears as fluorescing greenish-yellow
against red background.
• The adva ntage of QBC is that it is faster and more se nsitive
than thick blood sm ear.
• The disadvantage of the test is that it is less sensitive than
thick film and is expensive.
• A care ful sm ear e xamina tion still remains as the \"go ld
standard \" in m alaria diagnosis.
Microconcentration Technique
In microconcentra lio n technique, blo od sample is collected
in micro h ematocrit tube a nd c entrifuged a t hig h s p e ed.
The sedime nt is m ixe d with no rma l serum a nd sm ear is
prepared . Though it increases the positivity rate, it cha nges
the m o rphology of the parasite.
Culture of Malaria Parasites
Th e o rigina l m ethod of p etridish c ulture employed a
candle ja r to provide a n atmosphe re of 3% oxygen a nd
P:105
10% carbon dioxide and a relatively simple self-culture
medium (RPMl1640) supplemented with human, rabbit,
or calf serum Lo maintain infected erythrocytes. Fresh
red cells were added periodically for continuation of the
growth and multiplication of plasmodia. The continuous
flow method devised by Trager enables the prolonged
maintenance of stock cultures.
• Computer-co ntrolled cultu re sysLems, introduced
subsequently, provide a steady abundant supply of
parasites. Several culture lines have been established
from blood of infected Aotus monkey or directly from
human patients.
• Schizogony proceeds normally in culture. Gametocytes
are formed infrequently. Pre-erythrocytic stages of some
species have been obtained in tissue cultures. Plasmodia
retain their infectivity in culture.
• Culture of plasmodia provides a source of the parasites for
study of their antigenic structure, in seroepidemiologic
surveys, drug sensitivity tests and studies in
immunoprophylaxis.
Serodiagnosis
Serodiagnosis is not helpful in clinical diagnosis because they
will not differentiate between an active and past infection. It
is used mainly for seroepidemiologicaJ survey and to identify
the infected donors in transfusion malaria. The tests used
are indirect hemagglutination (IHA), indirect fluorescent
antibody (!FA) test and enzyme-linked immunosorbent assay
(ELISA).
Newer Methods of Diagnosis (Box 8)
Fluorescence microscopy:
Kawamoto technique: Fluorescent dyes like acridine
orange or benzothiocarboxy purine are used, which stain tl1e
parasites entering the RBCs but not white blood cells (WBCs).
This is a method of differential staining.
• Acridine orange stains DNJ\ as fluorescent green and
cytoplasmic RNA as red.
Box 8: Laboratory diagnosis of malaria
• Demonstration of malarial parasites in thick and thin blood smear
examination by Leishman, Giemsa, or JSB stain.
• lmmunofluorescence staining and QBC smear.
• Rapid immunochromatographic test (ICT) for detection of malaria
antigen (PfHRP-2 and pLDH).
• Molecular diagnosis: DNA probe and PCR.
• Routine blood examination for Hb, PCV and blood sugar.
Abbreviations: DNA, deoxyribonucleic acid; Hb, hemoglobin; JSB, Jaswant
Singh and Bhattacharjee; PCR, polymerase chain reaction; PCV, packed cell
volume; PfHRP-2; P/asmodium falciparum histidine rich protein-2; pLDH,
parasite lactate dehydrogenase; QBC, quantitative buffy coat
Malaria and Babesia
• The stained slide is examined unde r fl uorescent
microscope.
• The method is mainly used for mass screening in field
laboratory.
Rapid antigen detection tests: Rapid diagnostic test are based
on the detection of antigens using immunochromatographic
methods. These rapid antigen detection tests have been
developed in different test formats like the dipstick, card and
cassette bearing monoclonal antibody, directed against tje
parasite antigens. Several kits are available commercially,
which can detect Plasmodium in 15 minutes (Fig. 15).
Parasite-F test: This test is based on detection of histidine
rich protein-2 (HRP-2) antigen produced by the asexual
stages of P. falciparum expressed on the surface of red cells.
• Monoclonal antibody produced against HRP-2 antigen
(Pf band) is employed in the test strip.
• Advantage: It is widely popular and has high sensitivity
(98%) and specificity.
The test is said to detect low asexual parasitemia of
more than 40 parasites/ µL.
The test can be performed within IO minutes.
• Disadvantage: Plasmodium falciparum HRP-2 (PfHRP2) antigen detection test cannot detect the other three
malaria species.
Tt remains positive up to 2 weeks after cure.
In P.falciparum infection, PfHRP-2 is not secreted in
gametogony stage. Hence in \"carriers'; the Pf band
may be absent.
Dual antigen test: The test detects parasite lactate
dehydrogenase (pLDH) produced by trophozoites a nd
gametocytes of all plasmodimn species and PfHRP-2 antigen
produced by P.falciparum simultaneously.
• Thus, one band (Pv band) is genus specific (Plasmodium
specific) and other is Plasmodium Jalciparum specific (Pf
band).
Fig. 15: Rapid ICT Kit for dual antigen
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Paniker's Textbook of Medical Parasitology
• This test is a rapid two-site sandwich immunoassay used
for specific detection and differentiation of P. falciparum
and P vivax. malaria in areas with high rates of mixed
infection.
• The \"Pv\" ban d can be used for monitoring success of
antimala rial therapy in case of stained alone P. vivax
infection as the test will detect only live parasites and
therefore will be negative, if the parasite has been kiUed
by the treatment.
• The disadvantage of the test is that it is expensive and
cannot differentiate between P. vivax, P ovale and
P. malariae.
Molecular Diagnosis
Deoxyribonucleic acid probe: Deoxyribonucleic acid probe is
a highly sensitive method for the diagnosis of malaria. 1t can
detect less than IO parasites/ µL of blood.
Polymerase chain reaction: Polymerase chain reaction {PCR)
is increasingly used now for species specification and for
detection of drug resistance in malaria.
• Chloroquine resistance in P.Jalciparum is due to mutation
in the Plasmodium Jalciparum chloroquine resistance
transporter (PfCRT), a transporter gene in the parasite.
• Point mutation in another gene Plasmodium falciparum
multidrug resistance protein l (PfMDRI) is responsible
for resistance in vitro.
• Pyrimethamine and sulfadoxine resistances are associated
with point mutations in dihydrofolate reductase (DHFR)
and dihydropteroate synthase (DHPS) genes, respectively.
• Mutation in PfATPase gene is associated with reduced
susceptibility to artemisinin derivatives.
Other Tests
• Measurement of hemoglobin and packed cell volume
(PCV), in case of heavy parasitemia, particularly in
children and pregnant woman.
• Total WBC and p latelet count in severe Jalciparum
malaria.
• Measurement of blood glucose to detect hypoglycemia,
particularly in young children and pregnant women with
severefalciparum malaria and patients receiving quinine.
• Coagulation tests like measurement of antithrombin TI!
level, plasma fib rinogen, fibrin degradation products
(FDPs), partial thromboplastin time (PTT), if abnormal
bleeding is suspected in falciparum malaria.
• Urine for free hemoglobin, ifblackwater fever is suspected.
• Blood urea and serum creatinine to monitor renal failure.
• Glucose-6-phosphate dehydrogenase screening before
treatment with an antioxidant drug like primaquine.
Treatment
Antimalarial drugs are used with various objectives like
clinical cure, prevention ofrelapse, prevention of transmission
and prophylaxis.
Therapeutic
Objective is to eradicate the erythrocytic cycle a nd clinical
cure.
Radical Cure
Objective is to eradicate the exoerythrocytic cycle in liver to
prevent relapse.
Gametocidal
Objective is to destroy gametocytes to prevent mosquito
transmission and thereby reducing human reservoir.
Chemoprophy/axis
Objective is to prevent infections in nonimmune person
visiting endemic areas.
The most commonly used antimalarials are chloroquine,
amodiaquine, quinine, pyrime thamin e, doxycycline,
sulfadoxine, proguanil and primaquine. Newer antimalarial
like artemisinin, lumefantrine, mefloquine, halofantrine are
now commonly used for multidrug-resistant P. Jalciparurn
infections.
Treatment of Uncomplicated Malaria
Positive P. vivax, P. ovate and P malariae cases are treated
with chloroquine 25 mg/kg divided over 3 days.
Vivax malaria relapses due to the presence ofhypnozoites
in the liver. The relapse rate of vivax malaria in India is
about 30%.
• For prevention of relapse, primaquine is given in a dose
of 0.25 mg/ kg daily for 14 days under supervision.
• Primaquine is contraindicated in G6PD deficiency
patients, infants and pregnant women.
• In case of chloroquine resistance: Quinine is given in a
dose of 600 mg 8 hourly for 7 days along with doxycycline
100 mg/ day.
Treatment of Complicated (Falciparum) Malaria
Due to emergence of drug resistance of falciparum malaria is
based on area resistant or sensitive antimalarial drugs.
• Artemisinin-based combination therapy: According to
revised malaria drug policy in India arremisinin-based
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combination therapy (ACT) (artemisinin + sulfadoxine
pyrimethamine) should be given to all microscopically
positive Jalciparum cases for 3 days in all over India
except North-eastern states. This is accompanied by
single dose of primaquine 45 mg (0. 75 mg/kg) on day
2 as gametocidal drug.
In North -eastern states consid ering resistan t to
sulfadoxine - pyrimethamine drugs, Technical Advisory
Committee on Malaria recommended artemether (20
mg+ lumefantrine) as per age specific dose schedule.
Note: According to revised Malaria Drug Policy 2013, there
is no scope for presumptive treatment. Production and sale
of artemisinin as monotherapy has been banned in India as
it can lead to developmenr of parasite resistance to the drug.
Drug resi-Stance of malarial parasite:
• A drug resistant parasite is defined as a parasite that will
survive and multiply in a dosage that normally cures
the infection. Such resistance may be relative (yielding
to increased doses of the drug tolerated by the host)
or complete (withstanding a maximum dose tolerated by
the host).
Malaria and Babesia
Malaria Vaccine
Malaria vaccine is an area of intensive research. Over past
decades, there has been a significant progress in malaria
vaccine development. A completely effective vaccine is
not yet available for malaria, although several vaccines are
under development. SPf66 (a cocktail of four antigens, three
asexual blood stage antigens + circumsporozoite of Pf) was
te ted extensively in endemic areas in the 1990s, but clinical
trials showed it to be insufficiently effective. Other vaccine
candidates targeting the blood stage of parasite's life cycle
using merozoite surface protein 1 (MSP 1), MSP2, MSP 13 and
ring-infected erythrocyte surface antigens (RESAs) have also
been in insufficient on their own. Several potential vaccines
targeting the pre-erythrocytic stage are being developed, witl1
RTS,S/ASOl showing the most promising results. The RTS,S/
ASOl(commercial name, mosquirix) was engineered using
genes from the outer protein of P. falciparam and a portion
of hepatitis B virus, plus a chemical adjuvant (ASOl) to boost
irrunune response.
Vector Control Strategies • Resistance arises from spontaneous point mutations
in the genome or gene duplications. The emergence of •
resistance can be prevented by use of combination of
drugs with different mechanisms of action and different
drug target.
Residual spraying: Sprayin g of residual in secticides, e.g.
dichlorodiphenyltrichloroethane (DDT), malathion and
fenitrothin in the indoor surfaces of the house is highly
effective against adult mosquitos.
Space application: Insecticidal formulation is sprayed
into the atmosphere by ultra-low volume in the form of
mist or fog to kill insects (pyrethrwn extracts).
Individual protection: Man-vector contact can be
reduced by other preventive measures such as the use
of repellants, protective clothing, bed net, preferably
impregnated with long-acting repellant, mosquito coils
and screening of house.
• Three levels of resistance (R) are defined by the WHO:
RI: Following treatment, parasitemia clears but
recrudescence occurs.
RII: Following treatment, there is a reduction but not •
a clearance of parasitemia.
R11/: Following treatment, there is no reduction of
pa rasi ternia.
the earlier method of classifying resistance is based on
counting trophozoites in blood film daily for 7 days after
tJeatment and monitoring the patie nt for any subsequent Antilarval Measures
recrudescence. All patients with afalciparum parasitemia of
more than one rrophozoite per high power field (+++or over} •
in areas of suspected drug resistance, should be checked for a
decrease and clearing of parasites following treatment.
Prophylaxis
Chemoprophylaxis
It is recommended for travelers going to endemic areas as
short-term measure.
Chloroquine (300 mg) or mefloquine (400 mg) weekly
should be given 1 week and 2 weeks before travel to endemic
area respectively.
Alternatively doxycycline (100 mg) daily can be given
from day l before travel.
Old antilarval measures such as oiling the collection of
standing water or dusting them with Paris green have
now become promising with the increase of insecticide
resistance.
Source reduction: Mosquito breedin g sites can be
reduced by proper drainage, filling of land, water level
management, intermittent irrigation, etc.
Integrated Control
In order to reduce too much dep endence on residual
insectici des, increasing empha sis is being put on
integrated vector control methodology, which includes
bioenvironmental and personal protection measures.
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Paniker's Textbook of Medical Parasitology
Malaria Control Programs
In India, the National Malaria Control Programme was
introduced in 1953, with the objective of the ultimate
eradication of the disease and opera ted successfully for
5 years, bringing down the an nual incidence of malaria from
75 million in 1958 to 2 milJion.
• By 1961, the incidence dropped to an all time low of
50,000 cases and no deaths. However, there have been
setbacks from 1970 and by 1976, the incidence rose to
6.4 million cases. With the implementation of modified
plan of operation in 1977, the upsurge of malaria
cases dropped down to 2.1 million cases in 1984. Since
then, the epidemiological situation has not shown any
improvement.
• Malaria control added impe tus as \"roll-back malaria
initiative\" launched jointly by WHO, United Nations
Children's Fund (UNICEF), United Nations Development
Programme (UNDP) and the World Bank in 1998.
Accordingly, National Vector Borne Disease Control
Programme (NVBDCP) is implemented by Directorate
of Health Services jointly with Mission Directorate and
National Rural Health Mission (NRHM). ational goal
established under the program is to reduce the number
of cases and deaths recorded in 2000, by 50% or more in
2010 and by 75% or more by 2015.
BABESIA SPECIES
• INTRODUCTION
Babesia is intraerythrocytic sporozoan parasites th at
morphologically resemble Plasmodium and cause tick-borne
malaria like illness in domestic and wild animals.
It causes opportunistic infection in humans.
• CLASSIFICATION
Order: Piroplasmida
Family: Babesiidae
Species: Medically important Babesia species are:
• B. rnicroti (rodent strain)
• B. clivergen.s (cattle strain)
• 8. ho vis ( cattle strain)
• HISTORY AND DISTRIBUTION
Babesia is so named aJter Babes, who in 1888 described the
intraerythrocytic parasite in the blood of cattle and sheep in
Romania.
• In 1893, the parasite was shown to cause the tick-borne
disease, Texas fever, an acute hemolytic disease of cattle
in southern United States of America (USA).
• This was the first arthropod-borne disease to have been
identified.
• In 2009, more than 700 cases were reported from endemic
state of USA.
• Prevalence of B. rnicroti is underestimated because young
healthy individuals typically experience a mild selflimiting disease and may not seek medical attention.
• HABITAT
The parasite is present in erythrocytes and resembles the ring
stage of P. Jalciparum.
• MORPHOLOGY
Trophozoites are pleomorph ic 2-5 µm in diameter found
inside the red cells. The shape may be pyriform, ameboid, or
spindle-Like, usually in pairs and are often mistaken as ring
form of Plasmodiu:m (Fig. 16).
Merozoites may be spherical or oval or pyriform bodies,
found in pairs.
• LIFE CYCLE
Definitive Host
l xodid ticks.
Intermediate Host
Man or other mammals.
Infective Form
Sporozoites are the infective form for humans.
Mode of Transmission
Infection in vertebrate occurs through bite of the nymphal
stage of Txodid ticks. Transmission occurs during May to
Fig. 16: Trophozoites of Babesia microti in human blood smear
P:109
September. Incubation period is 1-6 weeks. Babesiosis can
also be transmitted via blood transfusion. Transovarian
transmission in ticks also occurs.
• In their life cycle, merogony takes place in vertebrate
hosts and sporogony in the invertebrates.
• Man acquires infection by bite of the infected ticks
(definitive host).
• Sporozoites present in the salivary glands of tick a re
introduced in man or other mammals (intermediate
host).
• Sporozoites change to trophozoites in the circulation,
which then invade the RBCs and multiply asexually
by binary fission or schizogony to form four or more
trophozoites. ewly formed trophozoites are released by
rupturing erythrocytes and invade new erythrocytes.
• Some of the sporozoites grow slowly inside red cells and
become folded like an accordion. These are thought be
gametocytes.
• Female ticks become infected by feeding the host blood.
• In the digestive tract of tick, the gametocytes multiply
sexually and later migrate to the salivary glands where
they divide by multiple fission into smaller forms known
as \"vcrmicules''.
• Vermicules undergo secondary schizogony to produce
sporozoites, which are the infective forms for human.
• PATHOGENICITY AND CLINICAL FEATURES
Hemolysis of the infected erythrocytes is primarily responsible
for many clinical manifestations.
• There is accumulation of parasites in the capillaries ofliver,
spleen and kidneys which leads to cellular degeneration
and necrosis.
• The illness develops 1-6 weeks after the tick bite.
• This may be subclinical or mild self-limiting or acute
illness, resembling malaria.
• In acute disease, there is malaise, fatigue, fever, myalgia,
arthralgia, dry cough and anorexia. Fever exceeds 38°C
and can reach 40.6°C accompanied by chill and sweat.
• Less common syndromes are neck stiffness, sore throat,
abdominal pain, jaundice and anemia.
• Severe babesiosis is associated with parasitemia levels of
more than 4% infected RBCs and requires hospitalization.
Fatality rate is 5% among hospitalized cases but is higher
(20%) among immunocompromised patients.
• Complications of acute babesiosis are renal failure,
dissemin ated intravascular coagulation (DIC), acute
respiratory distress syndrome (ARDS) and congestive
cardiac failure (CCF).
• Risk factors for complication are severe anemia ( <10 g%)
and high levels of parasitemia.
• LABORATORY DIAGNOSIS
Microscopy
Malaria and Babesia
Diagnosis of babesiosis is primarily done by examination of
blood films stained with Leishman or Giemsa stain.
• Babesia appears as intraerythrocytic round or pyriform,
or ring form simulating P.falciparum (Fig. 16).
• The ring forms are the most common and lacks the central
hemozoin deposit, typical of P.falciparum.
Other distinguishing features are the absence of schizonts
and gametocytes and presence of tetrads (maltose
crosses), which are pathognomonic of B. microti or B.
duncani (Table 6).
Polymerase Chain Reaction
If parasite cannot be identified by microscopy, amplificarion
of babesia1 18S rRNA by PCR is recommended.
Serology
It is useful to confirm the diagnosis. An !FA for B. microti is
available.
Immunoglobulin M titer of more than 1:64 and IgG titer
more than 1:1024, signify active or recent infection. Titer
declines over 6-12 months.
Blood Picture
Parasitemia levels typically range from 1 % to 20% in
immunocompetent patients but can reach up to 85% in
asplenic patients.
Table 6: Differential features of malaria and babesiosis
Characteristics Malaria Babesiosis
Distribution Worldwide North America and Europe
Vector Anopheles Tick
mosquito
Reservoir Man Rodent and cattle
No. of parasites per 1-3 1-12
red blood cell (RBC)
Schizont Present Absent
Gametocyte Present Absent
Pigment in Present Absent
trophozoite
Antigenic variation None Profound
Level of parasltemia Correlate with Does not correlate w ith
severity of disease severity of disease
Animal inoculation Negative Positive
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Paniker's Textbook of Medical Parasitology
• Reticulocyte count is elevated.
• Thrombocytopenia is common.
• White blood cell count may be normal or slightly
decreased.
Other Tests
Liver function tests such as serum glutamic pyruvate
transaminase (SGPT) and alkaline phosphatase yield elevated
value.
• Urine analysis m ay detect hemoglobi nuria, excess
urobilinogen and proteimuia.
• In renal complications, increased blood urea nitrogen
(BUN) and serum crealinine are found.
• TREATMENT
B. microti infection appears to be mild and self-limiting. Most
of the patients recover without any specific chemoLherapy,
with only symptomatic treatment.
• In acute cases chemotherapy is required.
• Atovaquone 750 mg twice daily, along with azithromycin
500 mg- 1 g/day for a period of 7- 10 days is effective.
Alternatively, clindamycin (300- 600 m g, 6 hourly)
along with quinine (650 mg 6-8 h ourly) may be given
intravenously.
• lnfulminantcases,exchangetransfusionisrecommended.
• PROPHYLAXIS
No vaccine is available at present. There is no role of
chemotherapy. Individuals who reside or travel in endemic
areas, should wear protective clothing and apply tick
repellents.
Individuals with history of symptomatic babesiosis or
with positive antibody titer should be indefinitely deferred
from donating blood.
KEY POINTS OF PLASMODIUM AND BABES/A
• Malaria parasite belongs to the genus Plasmodium.
• Four species of Plasmodium cause malaria in man- (1) P.
vivax, (2) P. falciparum, (3) P. malariae and (4) P. ovale.
• Definitive host: Anopheles mosquito (sexual phase of life
cycle).
• Intermediate host: Man (asexual phase of life cycle).
• Infective form: Sporozoites present in salivary gland of
mosquito.
• P. vivax and P. ovate cause benign tertian malaria, P.
falciparum causes malignant tertian malaria and P. matariae
causes benign quartan malaria.
• Acute falciparum malaria is the most dangerous and fatal
form and is due to heavy parasitization of RBCs which cause
blockage of capillary and venules by cytoadherence.
• Clinical features: Typical picture of malaria consist of
periodic bouts of fever with rigor followed by anemia and
splenomegaly. Febrile paroxysms comprise of cold stage, hot
stage and the sweating stage.
• Tropical splenomegalysyndrome is a chronic benign condition
resulting from abnormal immunological response to malaria.
• Relapse of malaria occurs in P. vivax and P. ovate infection
due to persistence of dormant stage hypnozoites in liver.
Recrudescence occurs commonly in P. falciparum and P.
matariae due to persistence of parasite in circulation at a
subclinical level.
• Diagnosis: By demonstration of parasite in thick and thin
smear of peripheral blood and also by detection of malaria
antigen by rapid ICT.
• Treatment: Chloroquine, sulfadoxine and pyrimethamine
along with primaquine. In chloroquine resistance, quinine or
artemisinin are used.
• Babesia spec/es comprising 8. microti, 8. divergens and 8.
bovis, are intraerythrocytic sporozoan parasite resembling
plasmodia. They cause opportunistic infections in humans.
• Mode of transmission: Through bite of lxodid ticks.
• Reservoirs: Rodents and cattle.
• Clinical features: Mild and self-limiting. In immunocompromised patients, it causes anemia, jaundice,
hemoglobinuria, respiratory failure. etc.
• Diagnosis: By examination of stained blood films for
intraerythrocytic parasites, reticulocytosis, increased SGPT,
alkaline phosphatase, hemoglobinuria.
• Treatment: Atovaquone + azithromycin. Alternatively,
clindamycin and quinine may be given.
REVIEW QUESTIONS
Describe briefly the life cycle and laboratory diagnosis of:
a. Plasmodium vivax
b. Plasmadium falciparum
Write short notes on:
a. Clinical features of malaria
b. Cerebral malaria
c. Blackwater fever
d. Malignant tertian malaria
e. Prophylaxis of malaria
f. Treatment of malaria
g. Rapid detection test
h. Babesiosis
Differentiate between:
a. Different malarial parasites
b. Recrudescence and relapse
c. Malaria and Babesiosis
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MULTIPLE CHOICE QUESTIONS
Old RBCs are preferentially infected by
a. Plasmodium falciparum
b. P/asmodium malariae
c. Plasmodium vivax
d. Plasmodium ova/e
The infective form of the malaria parasite is
a. Oocyst
b. Sporozoite
c. Bradyzoite
d. Tachyzoite
Prolonged parasitism in malaria is due to
a. Antigenic variation
b, lntracellularity of parasite
c. lmmunosuppression
d. Sequestration
Malaria pigment is formed by
a. Parasite
b. Bilirubin
c. Hemoglobin
d. All of the above
Schuffner's dot in RBCs are sesen in infection with
a. Plasmodium vivax
b. Plasmodium falciparum
c. Plasmodium malariae
d. Plasmodium ovale
Quartan malaria is caused by
a. Plasmodium vivax
b. Plasmodium falciparum
c. Plasmodium malariae
d. Plasmodium ovale
Malaria and Babesia
Malaria is not seen in patients with
a. G6PD deficiency
b. Sickle cell trait
c. Duffy negative blood group
d. All of the above
Which plasmodial infection is more often associated with
nephritic syndrome
a. Plasmodium vivax
b. Plasmodium falciparum
c. Plasmodium malariae
d. Plasmodium ovale
Which is the treatment of choice for benign tertian malaria
a. Sulfamethoxazole - pyrimethamine
b. Quinine
c. Mefloquine
d. Chloroquine
Gametocidal pernicious malaria may occur in
a. Plasmodium vivax
b. Plasmodium falciparum
c. Plasmodium malariae
d. Plasmodium ovale
Babesiosis is transmitted by
a. Ticks
b. Mites
c. Flea
d. Mosquito
Maltose cross is a characteristic feature of
a. Cryptococcus neoformans
b. Babesia microti
c. 8/astomycosis
d. Micrococcus
Schlzonts of Plasmodium fa/ciparum are not found in peripheral Answer
blood because
a. Schizonts are absent in the life cycle
b. Schizonts are killed by antibodies
c. Schizonts develop only in capillaries of internal organs
d. None of the above
Crescent-shaped or banana-shaped gametocytes are seen in
infection with
a. Plasmodium vivax
b. Plasmodium falciparum
c. Plasmodium malariae
d. Plasmodium ovale
b
b
b
d
b
C
C
d
a
12 b
C
a
C
b
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CHAPTER 7
• INTRODUCTION
The coccidia are unicellular protozoa and belong to the
Phylum Apicomplexa.
.
•
They live intraceUularly, at least duringa part of their life
cycle, and at some stage in their life cycle, they possess a
structure called the apical complex, by means of which
they attach to and penetrate host cells; hence included in
Phylum Apicomplexa.
All coccidian have a sexual sporogonic phase and an
asexual schizogonicphase.
Many of them also show an alteration of h osts-a
definitive host and an intermediate host.
Many parasites considered in this chapter have acquired
great prominence due to their frequent association with
human immunodeficiency virus (I-ITV) infection.
• TOXOPLASMA GONDII
History and Distribution
Toxoplasma gondii is an obligate intracellular coccidian
parasite, first described in 1908 by Nicolle and Manceaux in
B
a small orth American rodent called gundi ( Ctenodactylus
gundi).
• Its importance as a human pathogen was recognized
much later, when Janku in 1923 observed the cyst in the
retina of a chi! d with hydrocephalus and microphthalmia.
• The name Toxoplasma is derived from the Greek word
Toxon meaning arc or brow referring to the curved shape
of the trophowite.
• Toxoplasma is now recognized as the most common
protozoan parasite globally, with the widest range of hosts
spread over 200 species of birds, reptiles and mammals,
including humans.
Morphology
T. gondii occurs in three forms (Figs IA to C):
l. Trophozoite
Tissue cyst
Oocyst.
• The trophozoite and tissue cyst represent stages in asexual
multiplication (schizogony), while the oocyst is formed
by sexual reproduction (gametogony or sporogony).
Figs 1 A to C: Toxoplasma gondii. (A) Smear from peritoneal fluid of infected mouse, showing crescentic tachyzoites-extracellular trophozoites
and intracellular form within macrophage; (B) Thick-walled tissue cyst containing rounded forms bradyzoites; and (C) Oocyst containing two
sporocysts with sporozoites inside
P:113
• All three forms occur in domestic cats and other
felines, which are the definiti ve hosts and support both
schizogony and gametogony.
• Only the asexual forms, trophozoites and tissue cysts are
present in other animals, including humans and birds,
which are the intermediate hosts.
• All the three forms are infectious roman.
Trophozoites (Tachyzoites)
The trophozoite is crescent-shaped, with one end pointed
and the other end rounded.
• It measures 3- 7 µm in length. The nucleus is ovoid and is
situated at the blunt end of the parasite.
• Electron microscopy reveals an apical complex at the
pointed end (Fig. 2).
• The trophozoite stains well with Giemsa sta in, the
cytoplasm appearing azure blue and the nucleus red
(Fig. 3).
• The actively multiplying trophozoite is seen intracellularly
in various tissues during early acute phase of infection.
Extracellular trophozoites can also be seen in impression
smears.
• It can invade any nucleated cell and replicate within
cytoplasmic vacuoles by a process called endogony
(internal budding), wherein two daughter trophozoites
are fo rmed, each surrounded by a membrane, while
still within the parent cell. When the host cell becomes
distended with the parasite, it disintegrates, releasing the
trophozoites that infect other cells.
• During acute infection, the proliferating trophozoite
within host cell may appear rounded and enclosed by the
host cell membrane. This is called pseudocyst or colony
and can be differentiated from tissue cysts by staining
reactions.
Fig. 2: Toxopfasma gondii. Trophozoite (tachyzoite), fine structure
seen by electron microscopy
Coccidia
The rapidly proliferating trophozoites in acute infection
are called tachyzoites.
The trophozoites are susceptible to drying, freeze-thawing
and gastric digestion.
Tissue Cyst
Tissue cysts are the resting form of the parasite.
• They are found during chronic stage of the infection and
can be found in the brain (most common site), skeletal
muscles and various other organs.
• The cyst wall is eosinophilic and stains with silver, in
contrast to the pseudocyst.
• With periodic acid-Schiff (PAS) stain, the cyst wall stains
weakly, and the parasites inside are stained deeply. The
slowly multiplying parasites within the cyst are called
bradyzoites.
• The cyst is round or oval, 10- 20 µmin size and contains
numerous bradyzoites. Cysts remain viable in tissue for
several years.
• In immunologically normal hosts, the cysts remain
silent, but in the immunodeficient subjects, they may get
reactivated, leading to clinical disease.
• It is relatively resistant and when the raw or undercooked
meat containing the cysts is eaten, infection occurs.
• The cyst wall is disrupted by peptic or tryptic digestion
and the released parasites initiate infection by invading
intestinal epithelial cells.
• lhey reach various tissues and organs through blood and
lymphatic dissemination.
• Cysts are susceptible to desiccation, freezing, and thawing,
and heat above 60°C.
-:;:>' ~'' I \\\
.
.... ~' ,~ ~
~·· J ~,~
... - II
II_, -,-
Fig. 3: Toxopfasma gondii. Trophozoite grows in tissue culture. Smear
shows trophozoites arranged in different patterns-singly, in cluster,
or as rosette (Giemsa stain)
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Paniker's Textbook of Medical Parasitology
Oocyst
Oocysts develop only in definitive hosts- in the intestine of
cats and other felines but not in humans.
• lt is oval in shape and measures 10-12 µm in diameter.
Each cyst is surrounded by a thick resistant wall.
• The oocysts are formed by sexual reproduction
(gametogony).
• Cats shed millions of oocysts per day in feces for about 2
weeks during the primary infection. The freshly passed
oocyst is not infectious.
• 1hey undergo sporulation in the soil with formation of
two sporocysts, each containing four sporozoites. The
sporulaced oocysc is infective.
• Oocyst is very resistant co environmental conditions and
can remain infective in soil for about a year.
• Whe n the infective oocyst is ingested, it releases
sporozoites in the intestine, which initiates infection.
Life Cycle
Host: T. gondii completes its life cycle in two hosts (Fig. 4).
l. Definitive hosts: Cats and other felines, in which both
sexual and asexual cycles take place.
Cat acquires infection by
ingestion of rodent meat
containing tissue cyst
Tissue cyst formed in
birds, rats, etc
~ _____. Undergo schizogony
l,I.J,.Aol (asexual cycle) In
Bradyzoites mucosal cells
released
Enteric cycle
Contaminated soil containing
sporutated oocyst
ingested by
rats, birds, etc
Intermediate hosts: Man an d other mammals, in which
only the asexual cycle takes place.
T. gondii has 1'•vo types of life cycles:
Enteric cycle
Exoenteric cycle.
Enteric Cycle (Feline Cycle)
Enteric cycle occurs in cat and other definitive hosts (Fig. 4).
• Both sexual reproduction (gametogony) and asexual
reproduction (sch izogony) occur within the mucosa!
epithelial cells of the small intestine of the cat.
• Cat acquires infection by ingestion of tissue cysts in the
meat of rats and other animals or by ingestion of oocysts
passed in its feces.
• The bradyzoites are released in the small intestine and
they undergo asexual multiplication (schizogony) leading
to formation of merozoites.
• Some merozoites enter extrainrestinal tissues resulting in
the formation of tissue cysts in other organs of the body.
• Other merozoites transform into male and female
gametocytes and sexual cycle (gametogony) begins, with
the formation of microgamete and macrogamete.
n Excys1a1;01J
on srna111 0cc
II.Ian acquires Infection Bradyzoite
~ released
by ingestion of from tissue cyst
contaminated food ~ and water containing 'VI
sporulated oocyst or Sporozoite released
by ingestion of undercooked from oocyst
meat containing tissue cyst 1-
Man-<lead end
(cycle end)
~
Exoenteric \~
cycle ~i
!!!.
Fig. 4: Life cycle of Toxoplasma gondii
P:115
• A macrogamete is fertilized by motile microgamete
resulting in the formation of an oocyst, which passes
through maturation stages (sporulation) in the soil after
being excreted from host through feces.
• A mature oocyst containing eight sporozoites is the
infective form which may be ingested by rats or other
mammals to repeat the cycle.
Exoenteric Cycle (Human Cycle)
Exoenteric cycle occurs in humans, mice, rats, sheep, cattle,
pigs and birds, which are the intermediate hosts.
•
.
.
.
•
.
•
.
.
Humans acquire infection after:
Eating uncooked or undercooked infected meat,
particularly lamb and pork containing tissue cysts.
Ingestion of mature oocysts through food, water,
or fingers contaminated with cat feces directly or
indirectly.
Intrauterine infection from mother to fetus
(congenital loxoplasmosis).
Blood transfusion or transplantation from infected
donors.
Sporozoites from the oocysts and bradyzoites from the
tissue cysts enter into the intestinal mucosa and multiply
asexually and tachyzoiles are formed (endodyogeny).
Tachyzoites continue to multiply and spread locally by
lymphatic system and blood.
Some tachyzoites also spread to distant extraintestinal
organs like brain, eye, liver, spleen, lung and skeletal
muscles and form tissue cysts. The slowly multiplying
forms inside the tissue cysts a re known as bradyzoiles,
which remain viable for years.
Th e dormant bradyzoites inside the cyst may be
reactivated in immune suppression causing renewed
infection in the host.
Human infection is a dead end for the parasite (Pig. 4) .
Human roxoplasmosis is a zoonosis.
The full natural cycle is maintained predominantly by cats
and mice.
Mice eat materials contaminated with oocysts shed in
cat's feces. Tissue cysts develop in mice.
When such mice are eaten by cats, they get infected and
again shed oocysts in feces.
Pathogenicity and Clinical Features
The outcome of Toxoplasma infection depends on the
immune status of the infected person.
• Active progression of infection is more likely in
immunocompromised individuals. Toxoplasmosis has
acquired great importance as one of the major fatal
Coccidia
complications in acquired immunodeficiency syndrome
(AIDS).
Most human infections are asymptomatic.
Clinical toxoplasmosis may be congenital or acquired.
Congenital Toxoplasmosis
Congenital toxoplasmosis results when T gondii is transmitted
transplacentally from mother to fetus (Box 1).
.
•
.
.
.
.
This occurs when the mother gets primary toxoplasma
infection, whether clinical or asymptomatic, during the
pregnancy.
The risk of fetal infection rises with progress of gestation;
from 25%, when the mother acquires primary infection
in 1st trimester to 65% in the 3rd trimester. Conversely,
the severity of feral damage is highest, when infection is
transmitted in early pregnancy.
Mothers with chronic or latent Toxoplasma infection,
acquired earlier, do not ordinarily infect their babies. But
in some women with latent or chronic infection, the tissue
cyst may be reactivated during pregnancy and liberate
trophozoites, which may infect the fetus in utero.
Most infected newborns are asymptomatic at birth
and may remain so throughout. Some (0.3-1 %) develop
clinical manifestations of toxoplasmosis within weeks,
months and even years after birth.
The manifestations of congenital toxoplasmosis include
chorioretinitis, cerebra l calcifications, convulsions,
strabismus, deafness, blindness, mental retardation,
microccphaly and hydrocephalus.
A few children are born with manifestations of acute
toxoplasmosis, which may include fever, jaundice,
petechial rashes, microphthalmia, cataract, glaucoma,
lymphadenopathy, hepatosplenomegaly, myocarditis,
cerebral calcifications and chorioretinitis.
Acquired Toxoplasmosis
infection acquired postnatally is mostly asymptomatic.
• The most common manifestation of acute acquired
toxoplasmosis is lymphadenopathy; the cervical lymph
nodes being most frequently affected.
• Fever, headache, myalgia and splenomegaly are often
present. the illness may resemble m ildflu and is selflimited, although the lymphadenopathy may persist.
Box 1: Parasites which can be transmitted from mother to fetus
• Toxoplasma gondii
• Plasmodium spp.
• Trypanosoma cruzi.
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Paniker's Textbook of Medical Parasitology
• In some cases, there may be a typhus-like exanthema
with pneumonitis, myocarditis and meningoencephalitis,
which may be fatal.
Ocular Toxoplasmosis
Another type of toxoplasmosis is ocular.
• It may present as uveitis, choroiditis, or chorioretinitis.
• Some cases may be so severe that they require enucleation.
Toxoplasmosis in lmmunocompromised Patients
Toxoplasmosis is the most serious a nd often fatal in
immunocompromised pati ents, particularly in AIDS,
whether it may be due to reactivation of latent infection or
new acquisition of infections.
• In these patients, involvement of brain is most common.
• Clinical manifestations include encephalitis, altered
mental state, seizures, cerebellar signs, meningismus and
ne uropsychiatric manifestations.
• Besides central nervous system involvement, other organs
involved are lungs, pancreas, gastrointestinal tract, eyes,
heart and liver.
• Toxoplasma pneu monia can be confused with
Pneumocystis pneumonia.
Host Immunity
Host defense against Toxoplasma infection involves both
hum oral (antibody-mediated) and cellular responses. Specific
immunoglobulin G (lgG) antibody can lyse extracellular
trophozoites, but activated T cells and natural killer cells
appear to be more important in containing the infection and
preventing clinical disease.
Laboratory Diagnosis
The diagnosis of acute toxoplasmosis is made mainly by
demonstration of rrophozoites and cysts in tissue and body
fluids and by serology (Flow chart 1).
Microscopy
Tachyzoites and tissue cysts can be detected in various
specimens like blood, sputum, bone marrow aspirate,
cerebrospinal fluid (CSF), amniotic fluid, and biopsy material
from lymph node, spleen and brain.
• Smear made from earlier specimens is stained by Giemsa,
PAS, or Gomori methenamine silver (GMS) stain.
• Tachyzoites appear as crescent-shaped structures with
blue cytoplasm and dark nucleus.
• Tachyzoites or cyst can also be demonstrated effectively
by fluorescent conjugated antibody technique in tissue
biopsy or impression smear.
• Presence of only tissue cysts does not differentiate
between active and chronic infection.
• The presence of cysts in placenta or tissues of newborn
establishes congenital Toxoplasma infection.
Animal Inoculation
Toxoplasma can be isolated by inoculating body fluids, blood,
or tissue specimens by inrraperitoneal inoculation in mice or
Flow chart 1: Laboratory diagnosis of Toxop/asma gondii
Laboratory diagnosis
t
Microscopy
Tachyzoites and
tissue cysts detected
in blood, sputum and
bone marrow aspirates
Stains used:
-Giemsa
PAS
-GMS
t
Serodiagnosis
• Antibody detection:
Test for detecting
lgG antibody:
• ELISA
• IFAT
• Latex agglutination test
• Sabin-Feldman dye test
Test for detecting lgM antibody:
• Double sandwich lgM ELISA
• lgM-ISAGA
Test for detecting lgA antibody:
• Double sandwich lgA ELISA
• Antigen detection:
by ELISA
Molecular
diagnosis
• PCR
t
Imaging
• MRI and CT
scan for central
nervous system
involvement
• USG for
congenital
toxoplasmosis
•
Others
• Animal
inoculation
• Skin test of
Frenkel
Abbreviations: CT. computed tomography: ELISA, enzyme-linked lmmunosorbent assay: GMS. Gomorl methenamme silver: IFAT. indirect fluorescent antibody test;
lgM-ISAGA, immunoglobulin M-immunosorbent agglutination assay; MRI. magnetic resonance imaging; PAS, periodic acid-Schiff; PCR. polymerase chain reaction:
USG. ultrasonography
P:117
in tissue culture. Mice should be examined for Toxoplasma in
d1eir peritoneal exudate after 7-10 days of inoculation.
Serodiagnosis
Serology is the mainstay for diagnosis of toxoplasmosis.
Antibody detection: Diagnosis of acute infection with T.
gondiican be made by detection of thesimultan eous presence
oflgM and lgG antibodies.
• Tests for detecting IgG antibody include:
Enzyme-linked immunosorbent assay (ELISA)
Sabin-Feldman dye test
Indirect fluorescent antibody test (!FAT)
Latex agglutination test.
• Positive lgG titer (>l:10) can be detected as early as 2-3
weeks after infection. Peak level of antibody is observed
in blood 4-8 weeks after infection.
• A positive IgM antibody titer indicates an early primary
infection. The serum lgM titer can be measured by
double-sandwich lgM ELISA or IgM-im munosorbent
agglutination assay (lgM-ISAGA). Both assays are equally
specific and sensitive. egative IgM titer and positive IgG
titer indicate distant infection.
• The double-sandwich lgA EUSA test is used for detecting
congenital infection in newborns.
Antigen detection: Detection of antigen by ELISA indicates
recent Toxoplasma infection.
• In AIDS and other immunocompromised patients,
an tigen detection is very useful.
• Detection of antigen in amniotic fl uid is h elpful to
diagnose congenital toxoplasmosis.
Skin Test of Frenkel
Diluted toxoplasmin is injected intradermally and delayed
positive reaction appears after 48 hours. This test is not very
reliable for diagnosis of ToxopLasma.
Sabin-Feldman Dye Test
This was the first serological test for Toxoplasma antibody to
be described by Sabin and Feldman (1948).
Principal:The test is based on specific inhibition by antibody,
of the staining of trophozoites by alkaline methylene blue dye.
Technique: Equal volumes of diluted patient's serum are
incubated with live trophozoites and normal human serum
(accessory factor) for an hour at 37°C. Later, a drop of alkaline
methylene blue dye is added to each tube and is examined
under microscope. If less than 50% of the tachyzoites first
take up sta in and the cytoplasm remains colorless, the
test is considered to be positive. lhe presence of 90- 100%
tachyzoites, deeply swollen and stained with blue color, shows
Coccidia
me test to be negative. It denotes the absence of Toxoplasma
antibodies. The highest dilution of the serum, which inhibits
staining up to 50%, is the titer.
Limitation: The test is reported to give false-positive reaction
in Sarcocystis, Trichomonas vaginalis and Trypanosoma
lewisi infections. It cannot differentiate between recent and
past infection.
Molecular Methods
Deoxyribonucleic acid (DNA) hybridization techniques and
polymerase chain reaction (PCR) are increasingly used to
detect Toxoplasma from different tissues and body fluids.
• B, gene of T. gondii can be detected by PCR of the amniotic
fluid in case of congenital toxoplasmosis.
Imaging
Magne tic resonance imaging (MRl) and compu ted
tomography (CT) scan are used to diagnose toxoplasmosis
with central nervous system involvement.
• Ultrasonography (USG) of the fetus in utero at 20-24
weeks of pregnancy is useful for diagnosis of congenital
toxoplasmosis.
Treatment
Congenital Toxoplasmosis
eonates with congenital infection are treated the oral
pyrimethamine (1 mg/ kg) daily and sulfadiazine (100 mg/
kg) with folinic acid for 1 year. Systemic corticosteroid may be
added to reduce chorioretinitis.
lmmunocompetent Patients
Immunologically competent adults and older children, who
have only lymphadenopathy, do not require specific therapy
unless th ey have persistent severe symptoms.
• Patients with ocular toxoplasmosis are treated for 1
month with pyrimethamine plus either sulfadiazine or
clindamycin (600 mg QID).
• Folinic acid should be administered concomitantly to
avoid marrow suppressive effect of pyrimethamine.
lmmunocompromised Patients
Acquired immunodeficiency syndrome patients who are
seropositive for T. gondii and have a CD4€ T-lymphocyte count
below less than 100/ µL, should receive primary prophylaxis
against Toxoplasma encephalitis.
• Trimemoprim-sulfamemoxazole is the drug of choice. If
trimeilioprim-sulfamemoxazole cannot be tolerated by
patients, dapsone-pyrimethamine is the recommended
alternative drug of choice.
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Paniker's Textbook of Medical Parasitology
• Prophylaxis against Toxoplasma encephalitis s hould
be d iscontinued in pa tients who have responded
to antiretroviral therapy (ART) and whose cd4€
T-lymphocyte count has been above 200/µL for 3 months.
Prophylaxis
Individua ls at risk, particularly pregnant women, children
and immunocompromised persons should avoid contact
with cat and its feces.
• Proper cooking of meal.
• Proper washing of hands and washing of vegetables and
fru its before eating.
• Blood or blood products from seropositive persons should
not be given and screening for T. gondii antibody should
be done in all blood ban ks.
Control
It is difficult to control roxoplasmosis because of wide range
of animal reservoirs. Currently, there is no effective vaccine
available fo r humans. A genetically engineered vaccine is
under development for use in cats.
KEY POINTS OF TOXOPLASMA GOND/1
• Obligate intracellular parasite.
• Exists in three forms: (1) trophozoite, (2) tissue cyst, and (3)
oocyst.
• Definitive host: Cat family (enteric cycle).
• Intermediate host: Human (exoenteric cycle).
• Human infection occurs by ingestion of food containing
oocyst and tissue cyst.
• Congenital infection can a lso occur.
• Clinical features: Acute encephalopathy, fever, chorioretinitis,
lymphadenopathy, myocarditis, hepatosplenomegaly.
• Disseminated infection in AIDS.
• Diagnosis: By demonstration of parasite in tissue specimen,
ELISA, IFAT, Sabin-Feldman dye test, lgM-ISAGA.
• Treatment: Congenita l infection is treated with pyrimethamine
and sulfadiazine. For primary prophylaxis. trimethoprimsulfamethoxazole is the drug of choice.
• ISOSPORA BELLI
History and Distribution
lsospora belli is a coccidian parasite which can cause diarrhea
in humans.
• It was originally described by Virchow in 1860 but it was
named in 1923.
• The name belli (from bellium meaning war) was given for
its associatio n with war, because several cases of infection
with this parasite were seen among troops stationed in
Middle East d u ring the First World War.
• It is more common in tropical and subtropical coumries.
Morphology
Oocysts of I. belli are elongated-ovoid and measure 25 µm x
15 µm.
.
.
.
Each oocyst is surrounded by a thin smooth two-layered
cyst waU (Figs SA and B).
Immature oocysts seen in the feces of patients con tain
two sporoblasts.
The oocysts mature outside the body.
On maturation, the sporoblasr convert into sporocysts.
Each sporocyst contains four crescent-shaped sporozoites
(Figs 6A and B).
The sporulated oocyst containing eight sporowiles is the
infective stage of the parasite.
Sporoblast
a
Figs SA and B: Oocysts of lsospora be/Ii.
(A) Immature cyst: and (B) Mature cyst
m
Sporocyst
Figs 6A and B: Oocysts of lsospora be/Ii. (A) Oocyst showing two
sporoblasts; and (B) Mature oocyst with two sporocysts containing
sporozoites
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Life Cycle
I. belli completes its life cycle in one host.
.
.
•
•
.
Man gets infection by ingestion of food a nd water
contaminated with sporulated oocyst.
When a sporulated oocyst is swallowed, eight sporozoites
are released from the two sporocysts in the small intestine
and invade the intestinal epithelial cells.
In the epithelium, the sporozoites transform in to
trophozoites, which multiply asexually (schizogony) to
produce a number of (merowites). tje merozoites invade
adjacent epithelial cells to repeat asexual cycle.
Some of the trophozoites undergo sexual cycle
(gametogony) in the cytoplasm of enterocytes and
transform into macrogametocyles and microgametocytes.
After fertilization, a zygote is formed, which secretes a cyst
wall and develops into an immature oocyst.
These immature oocysts are excreted with feces and
mature in the soil.
Incubation period: 1- 4 days.
Clinical Features
Infection is usually asymptomatic.
• Clinical illness includes abdominal discomfort, mild
fever, diarrhea and malabsorption.
• The diarrhea is usually watery and does not contain
blood or pus and is self-limiting. However, protracted
diarrhea, lasting for several years can be seen in
immunocompromised persons, particularly in the HIV
infected.
Laboratory Diagnosis
Stool Examination
Indirect evidence:
• High fecal fat content.
• Presence of fatty acid crystals in stool.
• Presence of Charcot-Leyden crystals in stool.
Direct evidence: It may be d ifficu lt to demonstrate the
transparent oocyst in saline preparation of stool.
• Stool concentration techniques may be required when
direct wet mount of stools are negative.
• the staining techniques used are modified Ziehl-Neelsen
(ZN) stain or Kinyoun acid-fast staining of stool smear. In
these methods, pink-colored acid-fast large oocyst (>25
µm) can be demonstrated. the stool smear can also be
stained by auramine-rhodaminc and Giemsa stains.
Duodenal Aspirates
After repeatedly negative stool examinations, duodenal
aspirate examination or enterotest can be performed to
demonstrate oocyst.
Coccidia
Intestinal Biopsy
Upper gastrointestinal endoscopy may provide biopsy
specimens for demonstration of oocysts.
Others
Eosinophilia, which is generally not seen with other enteric
protozoan infections, is detectable in case of isosporiasis.
Treatment
• o treaunent is indicated in self-limiting infection in
immunocompetenc persons.
Immunodeficient patients with diarrhea and excreting
oocysts in the feces should be treated with cotrimoxazole
( trimethoprim-su I famethoxazole) in a dose of two tablets,
four times a day for 10 days followed by two tablets two
times a day for 3 weeks.
• For patients intolerant to sulfonamides, pyrimethamine
50-75 mg/day is given.
Relapses can occur in persons with AIDS and necessitate
maintenance therapy with cotrimoxazole one tabl et
thrice a week.
• CRYPTOSPORIDIUM PARVUM
History and Distribution
Cryptosporidia were first observed in the gastric mucosa!
crypts of laboratory mice byTyzzer in 1907.
• Its importance as a pathogen causing diarrhea in animals
was recognized in 1971 and the first case of human
infection was reported in 1976.
• Cryptosporidium has assumed great importance as a
frequent cause of intractable diarrhea, in AIDS patients
and immunocompromised subjects.
• It is worldwide in distribution.
• Two species of Cryptosporidium, C. hominis a nd
C. parvum mostly cause human infections.
Habitat
C. parvum inhabits the small intestine. It may also be found in
stomach, appendix, colon, rectum and pulmonary tree.
Morphology
The infective form of Lhe parasite is oocyst.
• The oocyst is sphe rical or oval and measures about 5 µm
in diameter.
• Oocyst does not stain with iodine and is acid-fast.
• TI1e wall of the oocysts is thick, but in 20% cases, wall may
be thin. These thin-walled oocysts are responsible for
autoinfeclion.
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Paniker's Textbook of Medical Parasitology
Figs 7 A and B: Oocysts of Cryptosporidium parvum. (A) Thick-walled
oocyst; and (B) Thin-walled oocyst
• Both thin-walled and thjck-walled oocyst contain fo ur
crescent-shaped sporozoites (Figs 7A and B).
• Oocyst can remain viable in the environment fo r
long periods, as it is very hard and resistant to most
disinfectants and temperature up to 60°C.
• It can survive chlorinated water, but sequential application
of ozone and ch lorine has been fo und effective in
eliminating the cysts.
Life Cycle
The parasite complete its life cycle, sexual and asexual phases
in a single host (monoxenous) (Fig. 8).
Suitable Host
Man.
Reservoirs
Man, cattle, cat and dog.
Mode of Transmission
Man acquires infection by:
• Ingestion of food and water contaminated with feces
containing oocysts.
• Autoinfection.
Infective Form
Sporulated oocysts.
• The oocyst contains four sporozoites, which are released
in the intestine.
• The sporozoites develop into trophozo ites within
parasitophorous vacuoles in the brush border of the
intestine.
• The trophozo ites undergo asexual multip licati on
(schizogony) to produce type I meronts.
• Eight merozoites are released from each type l meront.
These merozoites enter adjacent epithelial cells to repeat
schizogony or form type II meronts, which undergo
gametogony.
• Four merozoires are released from each type II meront.
The merozoites enter host cell to form sexual stagesmicrogamele and macrogamete.
• After fertilization, the zygote formed develops into
the oocyst. Th e oocyst undergoes sporogony to form
sporulated oocyst, wh ich contains four sporozoites.
Sporulated oocysts are released into the feces and transmit
th e infection from one person to another. Some of the
oocysts have a thin wall surroundin g four sporozoites and
are called as thin-walled oocysts. These oocysts infect the
same host and maintajn the cycle of autoinfection.
• The oocysts are fully mature on release and are infective
immediately without further development {Fig. 8).
Pathogenicity and Clinical Features
• Humans get infection either by ingestion of contaminated
food and water with feces or by direct contact with
infected animals. Human-to-human transmission can
also occur lncubation period is 2-14 days.
• Clinica l manifestations of C. parvum infection vary
depending upon the immune status of the host.
Infection in healthy immunocompetent p ersons
may be asym ptomatic or cause a self-limiting febrile
illness, with watery diarrhea in conjunction with
abdominal pain, nausea and weight loss. It can also
cause childhood and traveler's diarrhea, as well as
waterborne outbreaks (Box 2).
In immunocompromised hosts, especially those
with AIDS and cd4€ T-cell counts below 100/mcL,
diarrhea can be chronic, persistent, and remarkably
profuse, causing significant fluid and electrolyte
depletion, weight loss, emaciation and abdominal
pain. Stool volume may range from I L/day to 25 L/
day. Biliary tract involvement can manifest as right
upper quadrant pain, sclerosing cholangitis, o r
cholecystitis.
Laboratory Diagnosis
Stool Examination
Diagnosis is made by demonstration of the oocysts in feces.
• A direct wet mount reveaJs colorless, spherical oocyst of
4- 5 µm, contajning large and small granules.
• The oocysts are difficult to visualize in unstained wet
preparations.
• A n umber of staining techniques have been employed
for demonstration of oocysts of C. parvum in th e stool
specimen. Modified ZN staining is the method of choice
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Coccidia
--
@Ji / -
Autoinfectio -- n ('. (!R ff Sporozoites
~~\" ~ released
Thick-walled Thin-walled
sporulated oocyst oocyst in
/ in feces feces
Th
00
ick-walled
unsporulated oocyst
' Zygote
Thin-walled
unsporulated
oocyst
Mlcrogamete Macrogamete 'z~q;-
~v:
4 merozoites
released
.,,,,.
Sporozoite attaches to
brush border epithelium
of ;mM\"\"' \
Sporozoite develops
into trophozoite
o9e\
~,~al'I ~--., ~cfo ~, ;,,,<:'a \o<:<
J
Undergoes schlzogony
(asexual cycle) in
,._rt> 0 #' mucosal cells ~ I::,~
~ ;::-0 {f- ~q;
-~§ t$' Ci
~
~
Type
I meront
. ..,__.
---D~ ~ relaa~d
Type II meront
Fig. 8: Life cycle of Cryptosporidium parvum
Box 2: Parasites causing traveler's diarrhea
• Cryptosporidlum parvum
• Entamoeba histolyt/ca
Giardia Iambi/a
Cyc/ospora cayetanensis
and by this method oocysts appear as red acid-fast
spheres, against a blue background (Figs 9A and B). Yeast
closely resembles oocysts of C. parvum in shape and size
but can be differentiated by using acid-fast stain, as they
are not acid-fast and appear blue in color. The staining
can also be used for demonstration of oocysts in other
specimens like sputum, bronchial washing, etc.
• If oocysts, load is less and cannot be demonstrated even
after examination of three wet mounts of stool specimen,
concentration techniques like Sheather's sugar floatation
technique and zinc sulfate tloatation technique can be
applied.
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Paniker's Textbook of Medical Parasitology
Figs 9A and B: Oocysts of Cryptosporidium parvum. (A) Acid-fast stain; and (B) Ziehl-Neelsen stain
• Fluorescent staining with auramine-phenol or acridine
orange has also been reported to be a useful technique.
• Definitive identification can be made by indirect
immunofluorescence microscopy using specific
monoclonal antibody.
Histopathological Examination
Cryptosporidium can also be identified by light and electron
microscopy at the apical surface of intestinal e pithelium
from biopsy specimen of the small bowel Uejunum being the
preferred site).
Serodiagnosis
Antibody specific to C. parvum can be demonstrated within 2
months of acute infection.
• Anti-oocyst antibody persists for at least one year and
can be demonstrated by ELISA or immunofluorescence.
• An ELISA for detection of Cryptosporidium antigens
in stools using monoclonal antibody has also been
developed and is highly sensitive and specific.
Molecular Diagnosis
For seroepidemiological srudy, western blot technique is
employed by using a 17 kDa and 27 kDa sporozoite antigen.
• Polymerase chain reaction technique has also been
applied to detect viable cysts.
Treatment
ochemotherapeuticagenteffectiveagainst Crypt.osporidium
has been identified, although nitaz oxanide (500 mg BO x
3 days) or paromomycin may be partially effective in few
patients with Al OS. Improvement in immune status with ART
can lead to amelioration of cryptosporidiosis. Other treatment
methods include supportive therapy with fluid, electrolytes
and nutrient replacement.
KEY POINTS OF CRYPTOSPOR/0/UM PARVUM
• Sexual and asexual cycle in a single host.
• Infective form: Sporulated oocyst in food and water.
• Clinical features: Self-limited diarrhea with abdominal pain
in healthy persons. Chronic persistent watery diarrhea in
immunocompromised hosts.
• Diagnosis: Demonstration of round oocyst in stool by direct
microscopy, fluorescent microscopy and modified acid-fast
stain.
• Treatment: Supportive therapy with electrolytes and fluids
and early ART in AIDS patients.
• CYCLOSPORA CAYETANENSIS
• It is a coccidian parasite.
• It was first reported from Nepal, where it caused seasonal
outbreaks of prolonged diarrhea, with peak prevalence in
the warm rainy months.
Morphology
The morphological form found in the feces is an oocyst.
• The oocyst is a nonrefractile sphere, measuring 8- 10 µm
in diameter.
• It contains two sporocysts.
• Each sporocyst contains two sporozoites. Hence, each
sporulated oocyst contains four sporozoites.
Life Cycle
Oocyst shed in feces sporulares outside the host.
• The sporulated oocysts are infectious to humans.
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• Man acquires infection by ingestion of food and water
contaminated with feces-containing oocysts.
• Excystation of the sporocyst releases crcscenti c
sporozoites measuring 9 µm x 1.2 µm.
• The sporozoites infect enterocytes in the small intestine.
• The sporozoites develop into unsporulated oocysts, which
are excreted in feces.
Pathogenicity and Clinical Features
Infection is through fecal-oral route by ingestion of
contaminated water and vegetables.
• Incubation period is of 1- 7 days.
• Histopathological examination of the enterocytes shows
features of acute and chronic inflammation with blunting
and atrophy of villi and hyperplasia of crypts.
• It causes prolonged diarrhea with abdominal pain, lowgrade fever and fatigue.
• Like other coccidian parasites the infection is more severe
in immunocompromised hosts, especially with Al OS.
Diagnosis
Stool Examination
Diagnosis is by direct wet mount demonstration of oocysts
in feces.
• The oocysts can be stained by ZN stain. Oocysts of
Cyclospora are acid-fast and stain red in color.
• Under ultraviolet illumination, unstained oocysts of
C. cayetanensis are autofluorescent.
Histopathology
Biopsy specimen from jejunum shows villous atrophy and
blunting of villi along with other inflammatory changes.
• The parasite can a lso be seen in small bowel biopsy
material by electron microscopy.
Treatment
Cyclosporiasis is treated with cotrimoxazole (trimethoprim
160 mg/sulfamethoxazole 800 mg) twice daily for 7 days.
HIV-infected patients may require long-term suppressive
maintenance therapy.
• BLASTOCYSTIS HOMINIS
Blastocystis hominis was previously considered a yeast, but
recently it has been reclassified as a protozoan (Fig. 10).
Habitat
It is a strict anaerobic protozoa found in large intestine of
humans.
Coccidia
Fig. 1 O: Blastocystis hominis
Morphology
B. hominis has three morphological forms:
Vacuolated f orm is usually seen in stool specimen. It
measures 8 µm in diameter and is characterized by its
large central vacuole, which pushes the cytoplasm and the
nucleus to the periphery. It multiplies by binary fission.
Ameboidform is a polymorphous cell slightly larger than
the vacuolated form occasionally seen in the feces. Ir
multiplies by sporulation.
Granular f orm measures 10-60 µm in diameter and is
seen exclusively in old cultures.
Pathogenicity and Clinical Features
The parhogenicity of B. hominis is doubtful. However,
recent studies have shown the parasite to be associated with
diarrhea.
• Clinical manifestations include diarrhea, abdominal
pain, nausea, vomiting, fever and chills.
More than half of the patients suffering from infection
with B. hominis has been found to be immunologically
compromised.
Diagnosis
The condition is diagnosed by demonstration of the organism
in stool smear stained by Giemsa or iron hematoxylin or
trichrome stains.
Treatment
If diarrheal symptoms are prominent, either metronidazole
(750 mg thrice a day for 10 days) or iodoquinol (650 mg thrice
a day for 20 days) can be used.
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Paniker's Textbook of Medical Parasitology
• SARCOCYSTIS
Three species of genus Sarcocystis can infect humans:
S. hominis (transmitted through cattle)
S. suihominis ( transmitted th rough pig)
S. Lindemanni.
• Humans are the definitive host of S. hominis and
S. suihominis and the intermediate host for S. lindemanni.
• Sarcocyslis species produce cyst in the muscle of the
intermediate hoses. These cysts, called sarcocysts, contain
numerous merozoites (bradyzoiles) (Fig. 11).
• When sarcocyst is eaten by the defin itive host, the
merozoites are released in the intestine, where they
develop inro male and female gametes.
• After fertilization, the zygote develops into an oocyst
conraining two sporocysts, each having four sporozoites
(Fig. 12).
• These oocysts are shed in feces and are ingested by
intermediate host.
• ln the intermediate hosts, the sporozoites invade rhe
bowel wall and reach the vascular endothelial walls,
where they undergo schizogony producing merozoites
(tachywites).
• These spread to muscle fibers and develop into sarcocysts.
• Cow is the intermed iate host fo r S. hominis. Human
infection is acquired by eating raw or undercooked beef.
Oocysts are shed in human feces, which contaminate
grass and fodder eaten by cows.
Bradyzoites
Fig. 11: Sarcocyst
Fig. 12: Oocyst of Sarcocystis hominis
• In the case of S. suihominis, the pig is the intermediate
hosr and human infection is obtained through eating
contaminated pork. Human infection with S. hominis and
S. suihominis is related to food habits.
• Humans are the intermediate host in S. lindemanni; the
definitive host of which is nor yet known. It is believed that
S. lindemanni may not be a single species but a group of as
yet unidentified species. Humans apparently get infected
by ingestion of oocysts. Sarcocysts develop in the human
skeletal muscles and myocardium.
Clinical Features
• Intestinalsarwcystosis is usually asymptomatic. Patients
may have nausea, abdominal pain and diarrhea.
• Muscular sarcocystosis is also usually asymptomatic but
may cause muscle pain, weakness, or myositis, depending
on the size of the cyst.
Laboratory Diagnosis
Stool Examination
Characteristically sporocysts or occasionally oocysts can be
demonstrated in feces of human beings. Species identification
is not possible with microscopy.
Muscular Sarcocystosis
Diagnosis can be made by demonstration of sarcocysts in
the skeletal muscle and cardiac muscle by biopsy or during
autopsy.
Treatment
1
o specific treatment is available for sarcocystosis.
Prophylaxis
• By avoiding eating raw or w1dercooked beef or pork.
• By avoidance of contamination of food and drink with
feces of cat, dog, or other carnivorous animals.
REVIEW QUESTIONS
Describe the life cycle, clinical features and laboratory diagnosis
of Toxoplasma gondii.
Discuss in brief life cycle of Cryptosporidium parvum.
Write short notes on:
a. Congenital toxoplasmosis
b. Cryptosporidium parvum
c. Sabin-Feldman dye test
d. Sarcocyst
P:125
MULTIPLE CHOICE QUESTIONS
Route of transmission of Toxoplasma
a. Blood
b. Feces
c. Urine
d. None
Toxoplasma gondii lives inside the
a. Lumen of small intestine
b. Lumen of large intestine
c. Reticuloendothelial cell and many ot her nucleated cell
d. RBC
Oocyst of toxoplasma is found in
a. Cat
b. Dog
c. Mosquito
d. Cow
Toxoplasmosis in the fetus can be best confirmed by
a. lgM antibodies in the mother
b. lgM antibodies in the fetus
c. lgG antibodies in the mother
d. lgG antibodies in the fetus
Intermediate hosts of toxoplasmosis are
a. Sheep
b. Cattle
c. Pigs
d. All of the above
The following statements regarding congenital toxoplasmosis
are correct except
a. Most severe form of congenital infection occurs, if it is acquired
in 1st trimester
b. Chorioretinitis and hydrocephalus are common manifestations
in congenital infections
c. Presence of Toxoplasma-specific lgM antibodies in an infant
are suggestive of congenital infection
d. Most severe form of congenital infection occur if it is acquired
in 3rd trimester
Coccidia
Frenkels' skin test is positive in
a. Spinal cord compression
b. Toxoplasmosis
c. Pemphigus
d. Pemphigoid
In humans, cryptosporidiosis presents as
a. Meningitis
b. Diarrhea
c. Pneumonia
d. Asymptomatic infect ion
Which stain demonstrates the oocyst of Cryptosporidium best
a. Hematoxylin-eosin
b. Gram's stain
c. Kinyoun modified acid fast stain
d. Modified trichrome stain
1 O. All of the following cause diarrhea except
a. Entamoeba histolytica
b. Giardia lamblia
c. Naegleria fowleri
d. Cyc/ospora caytanensis
The oval oocyst of /sospora be/Ii found in human feces measures
a. 1-3 µm x 5- 7 µm
b. 3-5 µm x 8-10 µm
c. 5-8 µm x 10- 15 µm
d. 22- 33 µm x 10-15 µm
Stool in lsospora belfi infection may contain all except
a. High fecal content
b. Blood
c. Fatty acid crystals
d. Charcot-Leyden crystals
Answer
a
b
C
C
a
C
b
d
d
b
d 7. b
P:126
CHAPTER 8
• INTRODUCTION
Microsporidia are classified under Phylum Microspora. They
are minute, intracellular, Gram-positive, spore -forming
protozoa.
• Microsporidia are also classified based on their habitat
and the infections caused by them (Table 1).
• HISTORY AND DISTRIBUTION
Microsporidia are of historical interest as they are the first
protozoan parasite to have been successfully studied and
controlled by Louis Pasteur in 1863, during an investigation of
silkworm disease epidemic in France. It was this experience,
which led Pasteur to his epochal work on human and animal
diseases that formed the foundation of microbiology. The
Table 1: Classification of Microsporidia
Species Habitat and infection caused
causative agent of the silkworm disease (pebrine) is Nosema
bombycis, a microsporidian parasite.
• Microsporidia had been known as animal parasite for
long, but their role as human pad1ogens was recognized
only in the mid 1980s with the spreading of acquired
immunodeficiency syndrome (AlDS).
• Some nine genera and 13 species are associated
with h uman disease, particularly in the huma n
immunodeficien cy virus (HIV) infected a nd other
immunocompromised subjects.
• MORPHOLOGY
Microsporidia are unicellular, obligate intracellular parasite.
• They reproduce in host cells by producing spores
(sporogony).
Genus
Enterocytozoon E. bieneusi Small intestine epithelium (leading to diarrhea and wasting). Also found in biliary tract of patients with
cholecystitis. Rarely spreads to respiratory epithelium
Encephalitozoon E. intestinalis
E.hellem
E. cuniculi
Small intestine epithelium (causing diarrhea and wasting). Also causes sinusit is, cholangitis and
bronchiolitis
Conjunctiva! and corneal epithelium (causing keratoconjunctivitis). Also causes sinusitis, respiratory tract
disease and disseminated infection
• Small intestine epithelium (causing diarrhea)
• Corneal and conjunctiva I epithelium (causing keratoconjunctivitis). Rarely, may cause hepatitis and
renal infection
P/eistophora P. ronneafier Skeletal muscle (causing myositis)
Brachia/a , 8. vesicularum • Skeletal muscle (causing myositis)
• 8. conori • Muscles (smooth and cardiac)
Trachipleistophora • T. hominis • Cornea and conjunctiva I epithelium (leading to keratoconjunctivitis). Also causes myositis
• T. anthropophtheria , Brain
Vittaforma V. corneae Corneal stroma (causing stromal keratitis)
Nosema N. ocularum Corneal stroma (causing stromal keratitis)
Microsporidium • M. ceylonensis Corneal stroma (causing stromal keratitis)
• M. africanum
P:127
Polar sac--- - ~ +--=-
Exospore---\"
Endospore
Plasma membrane
Fig. 1: Microsporidian spore
Box 1: Acid-fast parasitic organisms
• Microsporidia (spore)
• Cyclospora cayetanensis (oocyst)
• lsospora be/Ii (oocyst)
• Cryptosporidium parvum (oocyst)
• Spores are 2-4 µmin size and oval to cylindrical in shape,
with a polar filament or tubule {Fig. 1 ).
• The spores are th e infective stage of microsporidia and the
only stage of life cycle capable of existing outside the host
cell.
The polar tubule is an extrusion mechanism for injecting
infective spore contents into the host cell.
• Spores are surround by thick double-layered cyst wall:
Outer laye r (exospore ) is prote inaceous and
electron-dense
Inn e r la yer ( endospore) is ch i ti n o us and
electronlucent.
• Spores are Gram-positive and acid-fast (Box l ).
• LIFE CYCLE
Infection in host is probably by ingestion or inhalation of
spores.
• In the duodenum, th e spore with its nuclear material
is injected through the polar tubule into the host cell
(enterocyte).
• Inside the cell, the microsporidia multiply by repeated
binary fission (merogony) and produce large number of
spores (sporogony).
Microspora
Box 2: Parasites causing opportunistic infections in immunocompromised patients [Human immunodeficiency virus (HIV)-
positive cases]
• Microsporidia
• Cyclospora cayetanensis
• lsospora be/Ii
• Cryptosporidium parvum
• Toxoplasma gondii
• Strongyloides stercoralis
• Entamoeba histolytica
• During sporogony, a thick spore wall is formed tha t
provides environmental protection to the cyst.
The spores are then liberated free from the host cell and
infect other cells.
• CLINICAL FEATURES
They can cause wide range of opportunistic illness in patients
with HIVand other immunocompromised diseases (Box 2).
• In patients with AIDS, Enterocytozoon bieneusi and
Encephalitozoon intestinalis lead to protracted and
debilitating diarrhe a in 10-40% of cases.
E. intestinalis may also cause sinusitis, cholangitis and
bronchiolitis.
Infection with Pleistophora can lead to myositis and
E. he/Lem can cause superficial keratoconjunctivitis,
sinusitis, respiratory disease and disseminated infection.
Stromal ke rati tis associated with trauma has been
reported in infections with Nosema, Vittaforma and
Microsporidium in imrmmocompetent patients.
• LABORATORY DIAGNOSIS
Microscopy
Diagnosis of microsporidiosis is made by demonstration of
the spores in stool, urine, cerebrospinal fluid (CSF), or small
intestine biopsy specimen.
• The spores can be stained with Gram's stain, periodic
acid-Schiff (PAS) stain , or modified trichrome stain.
No te: Spores of microsporidia stain poorly with
hematoxylin and eosin stain.
Although intracellular spores can be visualized by light
microscopy, electron microscopy is the gold standard.
• Iden tificatio n o f species and genera o f microspo ridia is
based on electron microscopy of spore morphology.
• Direct fluorescent method using monoclonal antibody
is also used for detection of microsporidia in clin ical
samples.
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Paniker's Textbook of Medical Parasitology
Cell Culture
Microsporidia spores can be cultured in monkey and rabbit
kidney cells and human fetal lung fibroblast.
Molecular Diagnosis
Microsporidial deoxyribonucleic acid (DNA) can be amplified
an d detected by polymerase chain reaction (PCR).
• TREATMENT
There is no specific and effective drug for microsporidia.
• Intestinal microsporidia may be treated wi th
m eu·onidazole and albendazole.
• For superficial keratoconjunctiviti s, topical therapy with
fumagillin suspension can be used.
• PROPHYLAXIS
Improved personal hygi ene and sanitation, especially in
immunocompromised persons can prevent microsporidia.
KEY POINTS OF MICROSPORIDIA
• Microsporidia are intracel lular spore-forming protozoa, which
belong to Phylum Microspora.
• Spores of microsporidia are oval or cylindrical in shape with
polar filaments or tubules.
• Mode of infection: By ingestion or inhalation of spores.
• Reproduction: Microsporidia multiply by both merogony and
sporogony.
• Clinical features: Protracted and debilitating diarrhea and
disseminated infection in eyes, muscles and lungs.
• Diagnosis: By demonstration of spores in stool, urine and
CSF by Gram·s, PAS, or modified trichrome stains. Serological
diagnosis includes direct fluorescent antibody test. PCR is
also very useful. Electron microscopy is useful in species in
identification of microsporidia.
• Treatment: There is no specific and effective treatment.
Intestinal microsporidia can be treated with metronidazole
and albendazole. Topical therapy with fumagillin suspension
is used for superficial keratoconjunctivitis.
REVIEW QUESTIONS
Describe briefly the laboratory diagnosis of Microsporidia.
Write short note on the morphology of Microsporidia species.
MULTIPLE CHOICE QUESTIONS
All are true about Microsporidia except
a. First protozoan parasite studied by Louis Pasteur
b. Causative agent of silk worm disease
c. Extracellular spore-forming protozoa
d. Cause infection in immunocompromised subjects
Laboratory diagnosis of Microsporidia can be done by all except
a. Modified trichrome stain
b. Hematoxylin and eosin-stain
c. Direct fluorescent antibody
d. Electron microscopy
Enterocytozoon bieneusi preferentially infects
a. Brain
b. Conjunctiva
c. Kidneys
d. Small intestine
Microsporidial keratoconjunctivitis is commonly caused by
a. Enrerocytozoon bieneusi
b. Vittaforma
c. Encephalitozoon hellem
d. Encepha/itozoon intestinalis
Answer
C 2. b 3. d 4. C
P:129
CHAPTER 9
• INTRODUCTION
Balantidium coli belongs to Lhe Phylum Ciliophora and
Family Balantiididae.
• It is the only ciliate protozoan parasite of humans.
• It is the largest protozoan parasite of humans.
• Largest protozoan parasite residing in the large intestine
of man: Balantidium coli.
• HISTORY AND DISTRIBUTION
lt was first described by Malmsten in 1857, in the feces of
dysenteric patients.
• It is present worldwide, but the prevalence of the infection
is very low.
• lhe most endemic area is 1ew Guinea, where there is a
close association between man and pigs.
• HABITAT
coli resides in the large lntestlne of man, pigs and monkeys.
• MORPHOLOGY
B. coli occurs in two stages: (1) trophozoite and (2) cyst (Figs
IA and B).
Trophozoite
The trophozoite lives in the large intestine, feeding on cell
debris, bacteria, starch grains and other particles.
•
.
.
.
The trophozoite is actively motile and is invasive stage of
the parasite found in dysenteric stool.
ft is a large ovoid cell, about 60- 70 µmin length and 40- 50
µm in breadth. Very large cells, measuring up to 200 µm
are sometimes seen.
The cell is enclosed within a delicate pellicle showing
longitudinal striations.
The motility of trophozoite is due to the presence of short
delicate cilia over the entire surface of the body.
,,,,_ ,....,..--Cytostome
1=--',--- Cytopharynx
.i.='--'o--Food
vacuole
Contractile--+~ ~ .I
vacuole
--- - .;.--Micronucleus--lL--- ----;
t--J..--Macronucleus~~-\"'111--\"II
h---Cilia
~~---Cytopyge
Figs 1 A and B: Morphology of Balantidium coli.
(A) Trophozoites: and (B) Cyst
• Its anterior end is narrow and posterior end is broad.
• At the anterior end, there is a groove (peristome) leading
to the moulh (cytostome), and a short funnel-shaped
gullet (cytopharynx).
• Posteriorly, there is a small anal pore (cytopyge).
• The cilia around the mouth are larger (adoral cilia).
• The cell has two nuclei: (1) a large kidney-shaped
macronucleus, and (2) lying in its concavity a small
micronucleus.
• The cytoplasm has one or two contractile vacuoles and
several food vacuoles.
Cyst
The cyst is spherical in shape and measures 40-60 µm in
diameter .
It is surrounded by a thick and transparent double-laye red
wall.
• The cytoplasm is granular. Macronucleus, micronucleus
and vacuoles are also present in the cyst.
• 1he cyst is the infective stage of 8. coli.
It is found in chronic cases and carriers.
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Paniker's Textbook of Medical Parasitology
• LIFE CYCLE
B. coli passes its life cycle in one host only (monoxenous).
Natural Host
Pig.
Accidental Host
Man.
Reservoirs
Pig, monkey and rat.
Infective Form
Cyst.
Mode of Transmission
• Balantidiasis is a zoonosis. Human beings acquire
infection by ingestion of food and water contaminated
with feces containing the cysts of 8. coli.
Reservoir:Pig
• Infection is acquired from pigs and other animal reservoirs
or from human carriers.
Once the cyst is ingested, excystation occurs in the small
intestine (Fig. 2).
• From each cyst, a single trophozoite is produced which
migrates to large intestine.
• Liberated trophozoites multiply in the large intestine by
transverse binary fission. Sexual union by conjugation
also occurs infrequently, during which reciprocal
exchange of nuclear material takes place between two
trophozoites enclosed within a single cyst wall.
• Encystation occurs as the trophozoite passes down the
colon or in the evacuated stool. In this process, the cell
rounds up and secretes a tough cyst wall around it.
The cysts remain viable in feces for a day or 2 and may
contaminate food and water, thus it is transmitted to other
human or animals.
• PATHOGENESIS
In a healthy individual, B. coli lives as lumen commensal and
is asymptomatic.
• Clinical disease occurs only when the resistance of host is
lowered by predisposing factors such as malnourishment,
Fig. 2: Life cycle of Balantidium coli
P:131
alcoh o lism, a chlorhydri a, concurrent infection by
Trichuris trichiura, or any bac terial infection.
• Clinical disease results when the trophozoites burrow
into the intestinal mucosa, set up colonies and in itia te
inflammatory reaction. This leads to mucosal ulcers and
submucosal abscesses, resembling lesions in amebiasis.
• Unlike E. histolytica, B. coli does not invade liver or any
other extra intestinal sites.
• CLINICAL FEATURES
Most infections a re asymptomaric.
• Symptomatic disease or balantidiasis resembles
amebiasis causing d iarrhea or frank dysentery with
abdominal colic, tenesmus, nausea and vomiting.
• Balantidium ulcers may be secondarily in fected by
bacteria.
• Occasionally, intesti nal perforation peritonitis and even
death may occur.
• Rarely, the re may be involvement of genital and urinary
tracts.
• In chronic balantictiasis, patients have diarrhea alternating
with constipation.
• LABORATORY DIAGNOSIS
Stool Examination
Diagnosis of B. coli infection is established by demonstration
of trophozoites and cysts in feces.
• Motile trophozoites occur in d iarrheic feces and cysts are
found in formed stools.
• The trophozoites can be easily recognized by their large
size, macronucleus and rapid-revolving motility.
• The cysts can also be recognized in the formed stools by
their round shape and presence of la rge macro nucleus.
Biopsy
When stool examination is negative, biopsy specimens and
scrapings from intestinal ulcers can be examined for presence
of trophozoites and cysts.
Culture
B. coli can also be cultured in vitro in Locke's egg albumin
medium or IH polyxenic medium s uch as Entamoeba
histolytica, but it is rarely necessa1y (Box 1).
• TREATMENT
Tetracycline is the drug of choice a nd is given 500 mg, four
times daily for 10 d ays. Alternatively, doxycycline can be
Balantidium Coli
Box 1: Parasites which can be cultured in laboratory
• Balantidium coli
• Entamoeba hlstolytica
• Acanthamoeba spp.
• Giardia lamblia
• Trichomonas vagina/is
, Trypanosoma spp.
• Leishman/a spp.
given. Metronidazole and ni troimidazole have also been
reported to be useful in some cases.
• PROPHYLAXIS
• Avoidance of contamination of food a nd water with
human or animal feces.
Prevention of human-pig contact.
• Treatment of infected pigs.
• Treatment of individuals shedd ing 8. coli cysts.
KEY POINTS OF BALANTIDIUM COLI
• It is the only ciliate parasite of humans.
• Largest protozoan parasite residing in large intestine.
• It occurs in two stages: (1) trophozoite and (2) cyst.
• Trophozoite is oval-shaped with a slightly pointed anterior end
with a groove, peristome leading to the mouth, cytostome.
Rounded posterior end has a small anal pore, cytopyge
and has a large kidney-shaped macronucleus and small
micronucleus.
• Cyst: It is the infective stage of the parasite.
• Mode of infection: Infection is acquired from pigs and other
animals by ingestion of cysts in contaminated food and drink.
• Infection leads to mucosa! ulcers and submucosal abscess
in intestine.
• Clinical features: Most infections are asymptomatic. In mild
infections, it causes diarrhea, abdominal colic, tenesmus,
nausea and vomiting.
• Diagnosis: Based on demonstration of trophozoites and cysts
in feces and examination of biopsy specimens and scrapings
from intestinal ulcers.
• Treatment: Tetracycline is the drug of choice.
• Prophylaxis: Avoiding contamination of food and water and
treatment of infected pigs and persons.
REVIEW QUESTIONS
Write short notes on the morphology of Balantidium coli along
with suitable illustration.
Discuss briefly the life cycle and laboratory diagnosis of
Balantidium coli.
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MULTIPLE CHOICE QUESTIONS
Largest protozoa! parasite is
a. Entamoeba histolytica
b. Trichomonas vagina/is
c. Leishmania donovani
d. Balantidium coli
The infective form of Balantidium coli is
a. Tachyzoites
b. Cyst
c. Sporozoite
d. Trophozoite
Which of the following acts as the main reservoir of Balantidium
coli infection
a. Man
b. Monkey
c. Pig
d.Cow
Drug of choice for treating balantidiasis
a. Doxycycline
b. Tetracycline
c. Metronidazole
d. Pentamidine
Answer
d 2. b 3. C 4. b
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CHAPTER 10
• INTRODUCTION
The helminth ic parasites are multicellular (metazoa)
bilaterally symmetrical animals having three germ layers
(triploblaslic melazoa) and belong to the kingdom Metazoa.
• The term helminth (Greek helmins-worm) originally
referred to intestinal worms, but now comprises many
other worms, including tissue parasites as well as many
free-living species.
Helminths, which occur as parasite in humans belong to
two phyla (Table 1):
Phylum Platyhelminthes (flatworms): Tt includes
two classes:
i. Class: Cestoda (tapeworms)
ii. Class: Trematoda (flukes or digeneans)
Phylum Nemathelminthes: lt in clu des class
Nematoda and two subclasses:
i. Subclass: Adenophorea (Aphasmidia)
ii. Subclass: Secernenrea (Phasmidia).
• The differences between cestodes, trematodes and
nematodes have been summarized in Table 2.
Table 2: Differences between cestodes, trematodes and nematodes
• PHYLUM PLATYHELMINTHES
The Platyhelminthes are tape-like, dorsoventrally flattened
worms.
• They either lack alimentary canal (as in cestodes) or their
alimentary canal is incomplete, lacking an anus (as in
trematodes).
Table 1: General features of helminths
Nematohelminthes
(Nematode)
Helminths
Platyhelminthes
(cestode. trematode)
• Body Elongated, cylindrical, Dorsoventrally flated leaf like
unsegmented or tape like segmented or
unsegmented
• Sex Separate (diecious)
• Body cavity Present
• Alimentary Complete
canal
Mostly hermaphrodite except
schistosomes (diecious)
Absent
Incomplete or absent
Cestodes Trematodes Nematodes
Shape
Head end
Alimentary canal
Body cavity
Sex
Life cycle
Tape-like, segmented
Suckers present; some have attached hooks
Absent
Absent, but inside is filled with spongy
undifferentiated mesenchymatous cells, in
the midst of which lie the viscera
Not separate: Hermaphrodite (monoecious)
Requires two host except Hymenolepis (one
host) and Diphyllobothrium (three host)
Leaf-like unsegmented
Suckers are present but no hooks
Present but incomplete, no anus
Same as cestodes
Not separate: Hermaphrodite except
Schistosoma
Requires three host except
schistosomes (two host)
Elongated, cylindrical, unsegmented
Hooks and sucker absent. Well-developed
buccal capsule with teeth or cutting
plates seen in some species
Complete w ith anus
Present and known as pseudocele. Viscera
remains suspended in the pseudocele
Separate (diecious)
Requires one host except filarial worms
(two host) and Dracunculus (two host)
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• Body cavity is absent, viscera is suspended in gelatinous
matrix.
• They are mosdy hermaphrodites (monoecious).
• Phylwn platyhelminthes includes two classes:
Class: Cestoda
Class: Trematoda.
Class Cestoda
Cestodes have tape-like, dorsoventrally flattened,
segmented bodies.
• They do not possess an alimentary system.
• The head carries suckers and some also have hooks.
• They possess scolex, neck and proglottids.
• They are monoecious and body cavity is absent.
• They are oviparous.
Class Trematoda
Trematodes have flat or fleshy, leaf-like unsegmented bodies.
• The alimentary canal is present but is incomplete, i.e.
without an anus.
• They possess suckers but no hooks.
• The sexes are separate in the schistosomes, while the
other flukes are hermaphroditic.
• They are oviparous.
• PHYLUM NEMATHELMINTHES (NEMATODA)
ematodes are elongated, cylindrical worms with an
unsegmented body.
•
•
•
•
•
They possess a relative ly well-developed complete
alimentary canal, with an anus.
Body cavity is present.
The head does not have suckers or hooks, but may have a
buccal capsule with teeth or cutting plates.
The sexes are separate (diecious) .
They are either oviparous or larviparous .
• IMPORTANT FEATURES OF HELMINTHS
Adult Worms
Helminths have an outer protective covering, the cuticle or
Integument, which may be tough and armed with spines or
hooks. The cuticle of live helminths is resistant to intestinal
digestion.
• The mouth may be provided with teeth or cutting plates.
Many helminths possess suckers or hooks for attachment
to host tissues.
• They do not possess organs of locomotion, but in some
species the suckers assist in movement.
• Locomotion is generally by muscular contraction and
relaxation.
• Many helmjnths have a primitive nervous system.
• The excretory system is better developed.
• The greatest development is seen in the reproductive
system. Helminths may be monoecious (with functioning
male and female sex organs in the same individual) or
diecious (the two sexes, male and female, separate). ln
the hermaphroditic helminths, both male and fema le
reproductive systems are present in the same worm and
self-fertiljzation as well as cross-fertilization takes place
(e.g. Taenia solium). In the diecious species, males and
females are separate, the male being smaller than the
female (e.g. Ascaris lumbricoides). Rarely, the female is
parthenogenic, being able to produce fertile eggs or larvae
without mating with males (e.g. Strongyloides).
Eggs
The eggs or larvae are produced in enormous nwnbers-as
many as 200,000 or more per female per day.
Various helminths have distinct morphology of eggs,
whlch can be used to differentiate the helminths (discussed
in the respective chapters).
Larval Forms
There are various larval forms ofhelminths found in man and
other hosts. These forms are as follows:
• Cestodes: Th e various larval forms are cysticercus,
coenurus, coracidiwn, cysticercoid, procercoid, hydatid
cyst and plerocercoid forms.
• Trematodes: The various larval forms are miracidium,
cercaria, redia, metacercaria and sporocyst
• Nematodes: The various larval forms are microfilaria,
filariform larva and rhabditiform larva .
Multiplication
Helminths differ from protozoans in their inability to multiply
in the body of the host. Protozoans multiply in the infected
person, so that disease could result from a single infection.
But helminths, apart from very rare exceptions, do not
multiply in the human body, therefore, a single infection
does not generally leads to disease. Heavy worm load follows
multiple infections. Sometimes, multiplication occurs within
larval forms in Platyhelminthes.
Life Cycle
• Cestodes: They complete their life cycle in two different
hosts, except Hymenolepis nana, which completes its life
cycle in a single host and Diphyllobothrium latum which
completes its life cycle in three hoses.
• Trematodes: They complete their life cycle in one
de finitive host (man) and two intermediate hosts.
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Fresh water snail or mollusc act as first intermediate host
and fish or crab act as second intermediate host except
schistosomes which require two hosts: (I) one definitive
host (man) and (2) other intermediate host (snail).
• Nematodes: Nematodes require only one host to
complete their life cycle except filarial nematodes and
Dracunculus medinensis, which complete their Life cycle
in two hosts.
Pathogenecity: The pathological lesions in helminthic
diseases are due to direct damage caused by helminths
or due to indirect damage by host response, for example
allergic response of the host to th e helminths. Many
helmmths cause malnutrition of the host. Malnutrition
interferes with antibody production.
• ZOOLOGICAL CLASSIFICATION
OF HELMINTHS
Phylum Platyhelminthes
Class Trematoda
• B1ood fl ukes (sexes separate, infection by cercarial
penetration).
Family: Schistosomatidae (schistosomes)
• Hermaphroditic flukes (bisexual, infection by ingestion
of cercariae).
family Fasciolidae (large flukes, cercariae encyst on
aquatic vegetation)
• Genus: Fasciola, Fasciolopsis
Family: Paramphistomatidae (large ventral sucker
posteriorly)
• Genus: Gastrodiscoides
Family: Echinostomatidae (coUar of spines behind
oral sucker, cercariae encyst in mollusc or fish)
• Genus: Echinostoma
Family: Triglotrematidae (testes side-by-side behind
ovary, cercariae encyst in Crustacea)
• Genus: Paragonimus
Family: Opisthorchidae (testes in tandem behind
ovary, cercariae encyst in fish)
• Genus: Clonorchis, Opisthorchis
Family: Dicrocoelida (testes in front of ovary,
cercariae encyst in insects)
• Genus: Dicrocoelium
Family: Heterophyidae (minute flukes, cercarial
encyst in fish)
• Genus: Helerophyes, Metagonimus.
Helminths: General Features
Class Cestoda
Order: Pseudophyllidea (scolex has grooves)
Genus: Diphyllobothrium
Order: Cyclophyllidea (scolex has suckers)
Family: Taeniidae (proglottid longer than broad,
numerous testes, one genital pore, la rva in
vertebrates)
• Genus: Taenia, Multiceps, Echinococcus
Family: Hymenolepididae (transverse proglottids,
one genital pore, larva in insects)
• Genus: Hymenolepis
Family: Dilepidiidae (two genital pores)
• Genus: Dipylidium.
Phylum Nemathelminthes
It includes class Nematoda which is further divided into:
• Subclass: Adenophorea or Aphasmidia (no phasmids, no
caudal papillae in male)
Subclass: Secernentea or Phasmidia (phasmids present,
numerous caudal papilJae).
Detailed classification of class Nematodes is given in
Chapter 13.
KEY POINTS OF HELMINTHS
• Helminths are multicellular and bilateral symmetrical
parasite.
• Helminths are divided into two broad phyla-the cylindrical
worms belonging to phylum Nematohelminthes (class
Nematoda) and flat tape or leaf like helminths belonging to
phylum platyhelminthes (class Cestoda and Trematoda).
• Sexes are separate in Nematodes. Cestodes and trematodes
are hermaphrodites.
• Trematodes are cestodes require two or three hosts.
Nematodes requires one host except filarial worms which
require two host.
REVIEW QUESTIONS
Short notes on:
a. General features of helminths
b. Phylum Nematoda
Differentiate between:
a. Trematodes and nematodes
b. Cestodes and nematodes
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Paniker's Textbook of Medical Parasitology
MULTIPLE CHOICE QUESTIONS
Digestive tract is completely absent in
a. Trematodes
b. Cestodes
c. Nematodes
d. All of the above
Sexes are always separate in
a. Cestodes
b. Trematodes
c. Nematodes
d. None of the above
Nematodes are different iated from other worms by the
following except
a. Absent fragmentation
b. Flat or fleshy leaf-like worm
c. Separate sexes
d. Cylindrical body
Which of the following worm requires two intermediate host
a. Taenia saginata
b. Oiphyllobothrium /arum
c. Hymenolepis nana
d. Echinococcus granulosus
Which of the following statement is true in respect to trematodes
a. Dorsoventrally flattened
b. Intermediate host is snail
c. Hermaphrodite except schistosomes
d. All of the above
Answer
b 2. C 3. b 4. b 5. d
P:137
CHAPTER 11
l~-
• INTRODUCTION
Cestodes (Greek kestos-girdle or ribbon) a re multisegm ented , dorsoventrally fla tte ned tape -like wo rms wh ose
sizes vary from a few millime ters to several meters. The adult
worms a re foun d in the small intestine of humans.
• CLASSIFICATION OF CESTODES
Systemic Classification
Cestod e s b elong to Phylum Platyhelminthes and class
Cestoidea. The class Cestoidea inclu des two orders:
Pseudophyllidea
Cyclophyllidea
For detailed classification see Table l.
Classification of Cestodes Based on the Form of
Parasite Important to Man
the de tailed classification is given in Table 2.
• TAPEWORMS: GENERAL CHARACTERISTICS
Adult Worms
• The adult worm con sists of three parts:
Head (scolex)
e ek
Trunk (strobila) {Figs IA to D).
Head (Seo/ex)
It is the organ of attachment to the intestina l m ucosa o f the
defi nitive h ost, human or animal (Figs lA to D).
• In p arasites of th e order Cyclophyllid e a, the scolex
poss e sses four suckers (or a cetabula). In s ome
Cyclophyllidea like Taenia solium, scolex h a s an apical
Table 1: Classification of medically important Cestodes
order Family_ Genus
Pseudophyllidea Diphyllobothriidae • Diphyllobothrium
• Spirometra
Cyclophyllidea Taeniidae • Taenia
• Echinococcus
Hymenolepididae Hymeno/epis
Dipylidiidae Dipylidium
C I '
A - )
D
-
-
B -
-
-
-
-)
-
-
Figs 1 A to D: Tapeworm. (A) Scolex or head; (B) Neck, leading to the
region of growth below, showing immature segme nts; (C) Mature
segments; and (D) Gravid segments filled with eggs
protrusion called as the rostellum. The rostellum may or
may not be armed with hooks.
• In parasites of the order Pseudophyllide a, the scolex does
not possess suckers b ut possesses a pair of longitudinal
grooves ca lled as bothria, by which it attach es to th e
intestine of the host.
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Table 2: Classification of Cestodes based on the form of parasite important to man
order
Pseudophyllidea
Cyclophyllidea
Heads
Proglottld
Adult wonn-,, in human intestine
Diphyllobothrium latum the fish tapeworm
• Taenia saginata, the beef tapeworm
• Taenia solium, the pork tapeworm
• Hymenolepis nana, the dwarf tapeworm
• Hymenolepis diminuta, the rat tapeworm
(rare)
• Dipylidium caninum, the double-pored dog
tapeworm (rare)
Taenia solium
4 suckers
2 rows or hooks
Longer than broad
7-12 uterine
branches on
each side
Taenia saginata
4 suckers
No hooks
Longer than broad
15-30 uterine
branches
on each side
Larval stage seen In humans
• Spirometra mansoni
• Spirometra theileri
• Spirometra erinacei (larval stage causing sparganosis)
• Taenia solium, the pork tapeworm (larval form can cause cysticercus cellulosae)
• Echinococcus granulosus, the dog tapeworm (larval form causes hydatid disease
in man)
• Echinococcus mulrlloculoris (larval stage causes alveolar or multilocular hydatid
disease)
• Mulriceps mulriceps and other species (larval stage may cause coenurosis in man)
Hymenolepis
nana
4 suckers
single row of
20- 30 hooks
Broader than
long
Hymenolepis
diminuta
4 suckers
No hooks
Broader than
long
Diphyllobothrium
latum
2 Suctorial grooves
or bothria, no
suckers, No hooks
Broader than
long
Uterus coiled
Echinococcus
granulosus
4 suckers
2 rows
of hooks
Longer than
broad
Fig. 2: Differences between heads and proglottids of various Cestodes
Neck
IL is the part, immediately behind the head and is the region of
growth from where the segments of the body (proglottids) are
being generated continuously.
Trunk (Strobila)
The trunk also called as strobila is composed of a chain of
proglottids or segments (Figs IA to D).
• The proglonids near the neck, are the young immature
segments, behind them are the mature segments, and at
the hind end, are the gravid segments.
• Tapeworms are hermaphrodites ( monoecious) and
every mature segment con tains both male and female
sex organs. In the immature segments, the reproductive
organs are not well-developed. They are well-developed in
the mature segments. The gravid segments are completely
occupied by the uterus filled with eggs.
• Tapeworm do not have a body cavity or alime ntary
canal.
• Rudimentary excretory and nervous systems are present.
'The differences between heads and proglottids of various
Cestodes have been illustrated in Figure 2.
Eggs
The eggs of Cyclophyllidea and Pseudophyllidea are different
from each other (Table 3).
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Cestodes: Tapeworms
Table 3: Differences between eggs of Orders Cyclophyllidea and • PSEUDOPHYLLIDEAN TAPEWORMS
Pseudophyllidea
Cyclophyllidean egg
• Covered by two layers: (1) egg
shell and (2) embryophore
• Spherical
• Embryonated from the
beginning
• Eggs are not operculated and
the embryo is not ciliated
Pseudophyllidean egg
• Covered by one layer: egg shell
• Ovoid in shape
• Freshly-passed eggs in feces are
unembryonated
• Eggs are operculated and the
embryo is ciliated
• The embryo inside the egg is called the oncosphere
(meaning hooked ball) because it is spherical and has
hooklets.
• Oncospheres of human tapeworms typically have three
pairs of hooklets and so, are called hexacanth (meaning
six-hooked) embryos.
Life Cycle
Cestodes complete their life cycle in two hosts: (1) defi nitive
host and (2) intermediate host.
• Humans are the definitive host for most tapeworms,
which cause human infection. An important exception
is the dog tapeworm, Echinococcus granulosus, for which
dog is the definitive host and man is the intermediate host.
In Taenia solium, man is ordinarily the definitive host, but
its larval stages can also develop in the human body.
Cestodes complete their life cycle in two different hosts.
Exceptions are:
1/ymenolepis that requires only one host, man and
Diphyllobothrium that requires three hosts, (I)
definitive host: man; (2) first intermediate host:
Cyclops; and (3) second intermediate host: fish.
• Clinical disease can be caused by the adult worm or the
larval form. In general, adult worm causes only minimal
disturbance, while the larvae can produce serious illness,
particularly when they lodge in critical areas like the brain
or the eyes.
• Ps eudophyllidean tapeworms have a central unbranched
convoluted uterus, which opens through a pore, possess
ventrally situated genital pores, and produce operculated
eggs that give rise to ciliated larvae.
In Cyclophyllidean tapeworms, the uterus is branched
and does not have an opening. They have lateral genital
pores and produce nonoperculated eggs that yield larvae,
which are not ciliated. Their larvae are called \"bladder
worms\" and occur in four varieties: (1) cysticercus, (2)
cysticercoid, (3) coenurus and ( 4) Echinococcus.
Diphyllobothrium Latum
Common Name
Fish tapeworm/ Broad tapeworm.
History and Distribution
The head of the worm was found by Bonnet in 1777, and its
life cycle was worked out by Janicki and Rosen in 1917.
Diphyllobothriasis (infection with Diphyllobothrium)
occurs in Central and orthern Europe, particularly in the
Scandinavian coun tries. It is also found in Siberia, Japan,
orth America and Central Africa.
In countries like India, where fish is eaten only after
cooking, the infection does not occur.
Longest cestode infecting man: Diphyllobothrium latum
• Smallest cestode infecting man: Hymenolepis nana.
Habitat
The adult worm is found in the small intestine, usually in the
ileum, where it lies fo lded in several loops with the scolex
embedded in the mucosa.
Morphology
Adull worm: It is ivory-colored and very long, measuring up
to 10 meters or more. It is the largest tapeworm inhabiting the
small intestine of man.
• As in all cestodes, the adult worm has three parts: (1)
scolex, (2) neck and (3) strobila.
Scolex(head) is spatulate or spoon-shaped, about 2-3 mm
long and l mm broad. It carries two slit-like longitudinal
sucking grooves (bothria), one dorsal and the other
ventral. the scolex lacks suckers and hooks ( Fig. 3A).
• eek is thin, unsegmented and is much more longer than
the head.
• Strobila consists of 3,000-4,000 proglottids, consistin g of
immature, mature and gravid segments in that order from
front to backwards.
• The mature proglottid is broader than long, about 2-4 mm
long and l 0-20 mm broad and is practically filled with
male and female reproductive organs (Fig. 3B).
• The testes are represented by numerous minute follicles
situated laterally in the dorsal plane.
• The female reproductive organs are arranged along the
midline, lying ventrally. The ovary is bilobed . The large
rosette-like uterus Lies convoluted in the center.
1hree genital openings are present ventrally along the
midline-the openings of the vas deferens, vagina and
uterus in that order, from front to backwards.
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Paniker's Textbook of Medical Parasitology
........ --........
/ '
,' ' 2 Suctorial
Neck
1 ' grooves or
\ ,' bothria \ I
', ___ .,' ....
Strobila
m
Figs 3A and B: Diphyllobothrium latum. (A) Adult worm showing
spatulate scolex, neck and strobila; and (B) Mature proglottid
......,,,..._----Knob
Fig. 4: Operculated egg of Diphyllobothrium latum
• The fertilized ova develop in the uterus and are discharged
periodically through the uterine pore.
• The terminal segments become dried up after delivering
many eggs and are discharged in strands of varying
lengths in the feces.
Egg: D. la.tum is a prolific egg layer and a single worm may
pass about a million eggs in a day.
• Egg is broadly ovoid, about 65 µm by 45 µm, with a thick,
light brown shell (Fig. 4).
• It has an operculum at one end a nd often a small knob
at the other.
• The freshly-passed egg contains an immature embryo
surrounded by yolk granules. The eggs are resistant to
chemicals but are killed by drying. The embryo with six
hooklets inside the egg is called the oncosphere.
• The egg does not.fl.oat in saturated salt solution and is bile
stained.
• They are not infective to humans.
Larval stages: There are three stages of larval development:
l. First stage larva (coracidium)
Second stage larva (procercoid)
Third stage larva (plerocercoid).
Life Cycle
Definitive hosts: Man, dog and cat. Man is the optimal host.
First intermediate host: Freshwater copepod, mainly of
genera Cyclops or Diaptomus.
Second intermediate host: Freshwater fish (salmon, trout,
etc.).
Tnfective form to human: Third stage plerocercoid larva.
• The adult worm lives in the small intestine. It lays
operculated eggs which are passed along with the feces
in water (Fig. 5).
• The freshly-passed egg contains an immature embryo
surrounded by yolk granules. The embryo with six
hooklets (hexacanth embryo) inside the egg is called the
oncosphere.
• In water, it matures in about l 0- 15 days and ciliated first
stage larva, called coracidium emerges through the
operculum.
• Coracidium (first stage larva) can survive in water for
about 12 hours, by which time it should be ingested by the
fresh water crustacean copepod Cyclops, which is the first
intermediate host (Fig. 5).
• In the midgut of the Cyclops, the coracidium casts off its
ciliated coat and by means of its six hooklets, penetrates
into the hemocele (body cavity). In about 3 weeks, it
becomes transformed into the elongated second stage
larva about 550 µm long, which is called the procercoid
larva.
• Procerco id larva has a rounded caudal appendage
(cercomer) which bears the now useless hooklets.
• If the infected Cyclops is now eaten by a freshwater
fish (second in termediate host), the procercoid larva
penetrates the intestine of the fish and grows.
In the fish, procercoid larva looses its caudal appendage
and develops into the third stage larva called the
plerocercoid larva or sparganum (Fig. 5).
• Plerocercoid larva has a glistening white flattened
unsegmented vermicule, with a wrinkled surface, is about
1-2 cm long, a nd possesses rudimentary scolex. lhis is
the stage infective for humans.
Man gets infection by eating raw or undercooked fish
containing plerocercoid larva.
• The larva develops into adult worm in the small intestine.
• The worm atta ins maturity in about 5-6 weeks and starts
laying eggs, which are passed along with the feces. The
cycle is thus repeated.
The adult worm may Live for about 10 years or more.
P:141
Plerocercoid larva / , ... ,,,,'\"fish
Infected cyclops eaten by fresh -·~ 1' '\"\"-\"\"' host)
Man acquires infection
by ingestion of infected
freshwater fishes
Cestodes: Tapeworms
Fresh water fish
2nd intermediate host
Adult worm lives in
Man small intestine of man
Procercoid larva
,,~Ops f Cyct\"\"'
Definitive host
LIFE CYCLE OF
DIPHYLLOBOTHRIUM LATUM
Cyclops
Feces
1st intermediate host
Oncosphere penetrates
intestine of cyclops
(l
g,
tt.
%.%
~? i.c;-
£, ~ I>) <JI ,. 0
~
Coracidium
ingested by
cyclops
(1st intermediate
host)
Water
-~t::klets
~
~
Coracidium {ciliated 1st
}
. . stage larva) emerging
Ciliated through the operculum epithelium
Coracidium
Fig. 5: Life cycle of Diphyl/obothrium latum
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Paniker's Textbook of Medical Parasitology
Pathogenicity and Clinical Features
The pathogenic effects of diphyllobothriasis depend on the
mass of the worm, absorption of its byproducts by the host and
deprivation of the host's essential metabolic intermediates.
• In some persons, infection may be entirely asymptomatic,
while in others there may be an evidence of mechanical
obstruction.
• Transient abd ominal discomfort, diarrhea, nausea,
weakness, weight loss and anemia are the usual
manifestations. Patients may be frightened by noticing
the strands of proglottids passed in their feces.
• A kind o f pernicious anemia, sometimes caused by the
infection, is called bothriocephalus anemia. This was
forme rly be lieved to be racia lly determined, being
com mon in Finland and rare elsewhere. The anemia
develops because the tapeworm absorbs large quantity of
vitamin 812 and interferes with its ilea! absorption, leading
to vitamin B 12 deficiency.
• In severe cases, patients may exhibit neurologic seq uelae
of vitamin 8 12 deficiency.
Laboratory Diagnosis
Stool microscopy: Eggs are passed in very la rge number in
feces, and the refore, their demonstration in feces offers an
easy me thod of diagnosis. The proglottids passed in feces can
also be identified by their morphology.
Serodiagnosis: A coproantigen detection test is available to
diagnose d iphyllobothriasis.
Treatment
• Praziquantel in a single dose of 10 mg/ kg is effective.
• Eggs are oval, operculated, bile stained and not infective to
man.
• Infective stage: Plerocercoid larva.
• Mode of transmission: Man gets infection by consuming
uncooked or undercooked fish containing third stage
plerocercoid larva.
• Clinical features: Abdominal discomfort, nausea and
megaloblastic anemia.
• Diagnosis: Stool microscopy for egg and coproantigen test.
• Treatment: Praziquantel and if required, vitamin B12•
Spirometra
Genus Spirometra belongs to Diphyll obothriidae family.
Species of this genera which are medically important a re- S.
mansoni, S. theileri and S. erinacei.
• Spirometra along with other Diphyllobothrium tapeworms
that are not normal human parasite, can accidentally
infect man and cause disease called as sparganosis.
• The disease is so named because itis caused by sparganum
(plerocercoid larva) of the parasite.
Distribution
Sparganosis has been reported mostly from Japan and
Southeast Asia; less often from America and Australia. A few
cases have been reported from India also.
Habitat
Adult worms live in the intestinal tract of cats and dogs.
• Parenteral vitamin 812 should be given, ifB12 deficiency is Life Cycle
present.
Prophylaxis
Infection can be prevented by:
• Proper cooking of fish.
• Deep freezing (- 10°C for 24-48 hours) of fish, if it is to be
consumed raw.
Prevention of fecal pollution of narural waters.
• Periodical deworming of pet dogs and cats.
KEY POINTS OF 0/PHYLLOBOTHRIUM LATUM
• Longest tapeworm found in man.
• Adult worm up to 10 meters in length having spoon-shaped
head with two slit-like grooves (bothria).
• Definitive host: Man (optimal host), dogs and cats.
• First intermediate host: Cyclops.
• Second intermediate host: Freshwater fish.
Definitive host: Dog and cat.
First intermediate host: Cyclops.
Second intermediate host: Snakes, frogs and fishes.
• Adult worms live in the intestinal trac t of dogs and cats
and produce large number of eggs which pass out along
with feces in water (Fig. 6).
• Eggs hatch in fresh water to release ciliated first stage
larva called as coracidium.
• The coracidium is ingested by Cyclops (fir L intermediate
host), whe re it develops into second stage larva called as
procercoid larva.
• When the infected Cyclops is ingested by fish, snakes,
amphibians (second intermediate host), the procercoid
larva migrates to various organs of the body and develops
into plerocercoid larva (sparganum Larva). This is the
infective stage of the larva for dogs and cats (definitive
host) (Fig. 6).
P:143
Infected fish, ~'
ingested by dogs, cats Adult worms
I in intestine of cats and dogs
(Definitive host)
Larva migrate to tissues,
and form plerocercoid
(sparganum) larva
' _ _..,,.,
Infected cyclops eaten by fresh
water fish, frog (2nd intermediate host)
I
Eggs, passed ''\"\
®
Eggs hatch
in freshwater to
Cestodes: Tapeworms
Infection occurs due to
Ingestion of infected Cyclops
Ingestion of raw, infected fish
Local application of raw, infective
flesh to skin, conjunctiva, or
vagina (Used as a poultice)
Sparganosis
Sparganum larvae
(plerocercoid larva)
develops in tissues
''\"'/ \"''°osls
Man-dead end
( cycle ends)
Fig. 6: Life cycle of Spirometra
• When a cat or dog eats the second intermediate host,
the plerocercoid larva develops into adult worms in th e
intestine.
• Man acts as an accidental host and gets infection by:
Ingestion of Cyclops containing procercoid larva.
Ingestion of plerocercoid larva present in uncooked
meat of animals or birds, frogs.
Local application of raw flesh of infected animals on
skin or mucosa. 1l1e last method follows the practice
prevalent among the Chinese, of applying split frogs
on skin or eye sores as a poultice.
Sparganosis: The term sparganosi.s is used for ectopic
infection by sparganum (plerocercoid larva) of Spirometra
and some Diphyllobothrium species.
• The sparganum (L3 larva) are liberated from the Cyclops
in the human intestin e. They penetrate the intestinal wall
and migrate to subcutaneous tissue, where they become
encysted and develop into spargana.
• The sparganum is usually found in the subcutaneous
tissues in various pans of the body, but may
also be present in the peritoneum, abdominal viscera, or
brain.
Laboratory Diagnosis
Diagnosis is usually possible only after surgical removal of the
nodules and demonstration of the worm.
Treatment
Definitive treatment is surgical removal of the nodule.
Prophylaxis
Human's sparganosis is prevented by:
• Properly filtering and boiling drinking water.
• Eating properly cooked flesh.
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Paniker's Textbook of Medical Parasitology
• CYCLOPHYLLIDEAN TAPEWORMS
Taenia Saginata and Taenia Solium
Common Name
• Taenia saginata: Beef tapeworm
• Taenia solium: Pork tapeworm.
History and Distribution
T. saginata has been known as an intestjnal parasite of man
from very ancient times. But it was only in 1782 when Goeze
differentiated it from the pork tapeworm, T. solium. ltslife cycle
was elucidated when Leuckart, in 1861, first experimentally
demonstrated that cattle serve as the intermediate host for
the worm.
• The name Taenia is derived from the Greek word meaning
tape or band. It was originally used to refer to most
tapeworms, but is now restricted to the members of the
Genus Taenia.
• 1: saginata is worldwide in distribution, but the infection
is not found in vegetarians and those who do not eat beef.
• T. solium is also worldwide in distribution except in the
countries and commun ities, which proscribe pork as
taboo.
Habitat
The adult worms of both T. saginata and T. solium(Fig. 7) live in
the human small intestine, commonly in the jejunum (Box I).
Morphology
Adult worm of T. saginata: The adult 7: saginata worm
is opalescent white in color, ribbon-like, dorsoventrally
flattened and segmented, measuring 5- 1 0 meters in length.
• The adult worm consists of head (scolex), neck and
strobila (body). The general fea tures of adult worm are
similar to any cyclophyllidean cestodes.
• Scolex: The scolex (head) of T. saginata is about 1-2 mm
in diamete r, quadrate in cross-section, bearing four
hemispherical suckers situated at its four angles. They may
be pigmented. The scolex has no rostellum or booklets
(which are present in T. solium). T. saginata is, therefore
called the unarmed tapeworm. the suckers serve as the
sole organ for attachment (Fig. 8).
• TI1e neck is long and narrow. The strobila (trunk) consists
of 1,000-2,000 proglottids or segmen ts- immature,
mature and gravid.
• The gravid segments are nearly four times long as they are
broad, about 20 mm long and 5 mm broad. The segment
contains male and female reproductive structures.
The testes are numerous, 300- 400 (twice as many as in
T solium). The gravid segment has 15- 30 lateral branches
Box 1: Cestodes living in small intestine
• Diphyl/oborhrium latum
• Toenia solium
• Taenia saginara saginara
• Taenia saginara asiacica
• Hymenolepis nano
Fig. 7: Adult worm of T. so/ium
Hooklets --=:::::--:
(2 rows)
Suckers
Taenia saginata Taenia solium
Fig. 8: Scolex of Taenia saginata and Taenia solium
(as against 7-13 in T. solium). It differs from 1: solium also
in having a prominent vaginal sphincter and in lacking
the accessory ovarian lobe . The common genital pore
opens on the lateral wall of the segments.
• The gravid segments break away and are expelled singly,
actively forcing their way out through the anal sphincter.
As there is no uterine opening, the eggs escape from the
uterus through its ruptured wall.
Adult worm ofT. solium:
• The adult worm is usually 2- 3 meters long.
The scolex of T solium is small and globuJar about l mm
in diameter, with four large cup-like suckers (0.5 mm in
P:145
Table 4: Difference between Taenia saginata and Taenia so/ium
Taenia saginata Taenia solium
Length 5-10 meter 2- 3 meter
Seo/ex Large quadrate Small and globular
Rostellum and hooks Rostellum and hooks are
are absent present
Suckers may be Suckers not pigmented
pigmented
Neck Long Short
Proglottids 1,000-2,000 Below 1,000
Measurement 20mmx 5mm 12mmx6mm
(gravid segment)
Expulsion Expelled singly Expelled passively in
chains of 5 or 6
Uterus Lateral branches 15-30 Lateral branches 5- 10
on each side; thin and on each side; thick and
dichotomous dendritic
Vagina Present Absent
Accessory lobe of Absent Present
ovary
Testes 300-400 follicles 150- 200 follicles
Larva Cysticercus bovis; Cysticercus cellulosae;
present in cow not in present in pig and also
man in man
Egg Not infective to man Infective to man
Definitive host Man Man
Intermediate Cow Pig, occasionally man
host
Disease Causes intestinal Causes intestinal taeniasis
taeniasis and cysticercosis
diameter), and a conspicuous rounded rostellum, armed
with a double row of alternating round and small daggershaped hooks, 20- 50 in number.
• 111e neck is short and half as thick as the head.
• The proglottids number less than a 1,000. They resemble
those of T. saginata in general. 1he gravid segments are
twice as long as broad, 12 mm by 6 mm. The testes are
composed of 150-200 follicles. 1here is an accessory lobe
for the ovary. The vaginal sphincter is absent. The uterus
has only 5-10 (under 13) thick lateral branches. A lateral
thick-lipped genital pore is present, alternating between
the right and left sides of adjacent segments.
• The gravid segments a re not expelled singly, but pass
passively out as short chains. The eggs escape from the
ruptured wall of the uterus.
The other differentiating features of T sagina/a and T.
so/ium are given in Table 4.
Cestodes: Tapeworms
Eggs: Eggs of both species are indistinguishable.
• 1he egg is spherical, measuring 30- 40 mcm in diameter.
• lt has a thin hyaline embryonic membrane around it,
which soon disappears after release.
The inner embryophore is radially striated and is yellowbrown due to bile staining (Figs 9A and B).
• In the center is a fully-developed embryo (oncosphere)
with three pairs of hooklets (hexacanth embryo).
• The eggs do not float in saturated salt solution.
• 1l1e eggs of T. saginata are infective only to cattle and not
to humans, whereas the eggs of T. solium are infective to
pigs and humans too.
Larva: The larval stage of Taenia is called as cysticercus.
• Cysticercus bovis is the larva ofT. saginal.a (Fig. IO).
• Cysticercus cellulosae is the larva of T. solium (Fig. 12).
Cysticercus bovis:
• It is the larval form of T. saginala.
1l1e name cysticercus in derived from the Greek, kystisb/adder and kerkos- tail.
The larva (cysticercus bovis) is infective stage for humans.
• 1l1e cysricercus is an ovoid, milky-white opalescent fluidfilled vesicle measuring about 5 mm x 10 mm in diameter,
and contains a single invaginated scolex (bladder worm).
• the cysticerci are found in the muscles of mastication,
cardiac muscles, diaphragm and tongue of infected cattle
(Fig. IO).
They can be seen on visual inspection as shiny white dots
in the infected beef (measly beef) (Fig. 11).
• Cysricercus bovis is unknown in humans.
Cysticercus cellulosae:
• lt is d1e larval form of T. solium and also the infective form
of the parasite.
• It can develop in various organs of pig as well as in man.
• the cysticercus cellulosae or \"bladder worm\" is ovoid
opalescent milky-white, measuring 8-10 mm in breadth
and 5 mm in length.
The scolex of th e larva, with its suckers, lies invaginated
within the bladder and can be seen as a d1ick white spot.
It remains viable for several months (Fig. 12).
Life Cycle of Taenia Saginata
T. saginata passes its life cycle in two' hosts (Fig. 13):
Definitive host: Humans are the definitive hosts and
harbor the adult worm.
Intermediate host: Cattle (cow or buffalo) are the
intermediate host and harbor the larval stage of the worm.
Infective stage: Cysticercus bovis (larval stage) is the infective
stage to man, while eggs are infective to cattle.
• 1l1e adult worm lives in the small intestine of man. The
gravid segments from the adult worm break away and are
expelled singly. They actively force their way out through
the anal sphincter.
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Paniker's Textbook of Medical Parasitology
Egg or gravid
proglottid
passed in
feces
man'\"'\"'\"\" iafeolioo by \
ingestion of unde\ __ r_coo_k_ed_p_o_rk _________ rl
Pig (Intermediate host) ~
Ingested by pig
O~e,l> ('\'('I F;;
:of-lo\" ~e~ e~ \'i>~ .. _ .-=-.oe{\u~• ~eie
Oncosphere
penetrates
In 01.10 0{\ co,;,9
ruptures,
the wall of
intestine
Fig. 14: Life cycle of Taenia solium
They are fi ltered out principally in the muscles, where
they develop into the larval stage, cysticercus cellulosae in
about 60-70 days.
• ln h umans, it is a dead end and the larvae die without
funher development.
Intestinal infection with 1: solium occurs only in persons
eating undercooked pork and usually in persons of low
socioeconomic condition with poor sanita tion . It is
uncommon in Jews and Mohammedans, who are not
generally pork eaters. But cysticercosis may occur in any
person residing in endemic areas, even in vegetarians
because the mode of infection is contamination of food or
drink with egg deposited in soil.
Eggs of T. solium are infective to pigs as well as to man.
Pathogenicity and Clinical Features
Intestinal taeniasis: It can be caused by both T. saginata and
T solium.
• The adult worm, in spite of its large size, causes surprisingly
little inconvenience to the patient.
• When the infection is symptomatic, vague abdominal
discomfort, indigestion, nausea, diarrhea and weight
loss may be present. Occasional cases or acute intestinal
obstruction, acute appendicitis and pancreatitis have also
been reported.
Cysticercosis: It is caused by larval stage (cysticercus
cellulosae) of T solium.
• Cysticercus cellulosae may be solitary or more often
multiple.
• Any organ or tissue may be involved, the most common
being subcutaneous tissues and muscles. It may also
affect the eyes, brain, and less often the heart, liver, lungs,
abdominal cavity and spinal cord.
• The cysticercus is surrounded by a fibrous capsule except
in the eye and ventricles of rhe brain.
• The larvae evoke a cellular reaction starting with
infiltration of neutrophils, eosinophils, lymphocytes,
plasma cells, and at times, giant cells. This is followed by
fibrosis and death of the larva with eventual calcification.
• The clinical features depend on the site affected:
Subcutaneous nodules are mostly asymptomatic.
Muscular cystlcercosis may cause acute myositis.
Neurocysticercosis (cysticercosis of brain) is
the most common and most serious form of
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cysticercosis. About 70% of adult-onset epilepsy is
due to neurocysticercosis. Other clinical features
of neurocysticercosis a re increased intracranial
tension, hydrocephalus, psychiatric disturbances,
meningoencephalitis, transient paresis, behavioral
disorders, aphasia and visual disturbances. 1t is
considered as the second most common cause of
intracranial space occupying lesion (lCSOL) after
tuberculosis in India.
In ocular cysticercosis, cysts are fou nd in vitreous
humor, subretinal space and conjunctiva. The
condition may present as blurred vision or loss of
vision, iritis, uveitis and palpebral conjunctivitis.
Laboratory Diagnosis
Stool examination:
Eggs:
• Microscopic examination of feces shows characteristic
eggs of Taenia in 20- 80% of patients.
• Formol- e ther sedimentation me t hod of stool
concentration is useful.
• Eggs can also be detected by cellophane swab method
(NIH swab) in 85-95% patients.
• Species identification cannot be made from the eggs,
since the eggs of T. saginata and T. solium are similar
(Flowchart 1).
Cestodes: Tapeworms
Proglonids:
Species identification can be done by examining with a hand
lens, the gravid proglortid pressed between two slides, when
branching can be made out (15- 20 lateral branches in T.
saginata; under 13 in T. solium).
Scolex:
Definitive diagnosis can also be established by demonstration
of unarmed scolex in case of T. saginala after anthelmintic
treatment.
Detection of Taenia antigen in feces: Antigen capture
enzyme-linked immunosorbenl assay (ELISA) using
polyclonal antisera against Taenia are employed to detect
coproantigen in feces since 1990 and is more sensitive
than microscopy (specifici ty 100% and sensitivity 98%). The
drawback of the test is that it cannot differentiate between T.
saginata and T.solium (Flow chart 1).
Serodiagnosis: Specific antibodies to adult stage antigen
in se rum can be d emonstrated by ELISA, indirect
immunofluorescence test and indirect hemagglutination
(IHA) test (Flowchart I).
Molecular diagnosis: Both deoxyribonucleic acid (D A)
probes and polymerase cha in reaction (PCR) technique are
used to detect and differentiate between eggs and proglottids
of T. saginata and T. solium (Flow chart 1). It can also
differentiate between the two subspecies of T. saginata, viz. T.
saginata saginata and T. saginata asiatica.
Flow chart 1: Laboratory diagnosis of Taenia spp.
+
Taeniasis
Stool examination
a) Eggs:
Shows characteristic eggs of
Taenia but species identification
cannot be done
Concentration method:
Formol ether sedimentation
method
b) Proglottids:
Species identification possible by
examining proglottids
c) Taenia antigen (Coproantigen)
More sensitive than microscopy.
Cannot differentiate between
Taenia so/ium and Taenia
saginata
Laboratory diagnosis
Serodiagnosis
can be done by
• ELISA
• IHA
Molecular diagnosis
• Done by DNA probes
and PCR
Biopsy
Definitive method
of diagnosis
• Species and subspecies
identification possible
+
C sticercosis
Serodiagnosis
• Antibody detection by
ELISA
EITB
• Antigen detection by
ELISA using
monoclonal antibodies
Imaging methods
• X-ray
• CT scan
• MRI scan
AbbreviaUons: CT. computed tomography; DNA. deoxyribonucleic acid: EITB, enzyme-linked immunoelectrotransfer blot: ELISA. enzyme-linked immunosorbent
assay: IHA. indirect hemagglutlnal.lon; MRI. magnetic resonance imaging; PCR, polymerase chain reaction
P:148
Paniker's Textbook of Medical Parasitology
Laboratory Diagnosis of Cysticercosis
Diagnosis of cysticercosis is based on the following {Flow
chart 1):
• Biopsy: Definitive diagnosis of cysticercosis is by biopsy
of the lesion and its microscopic examination to show the
invaginated scolex with suckers and hooks.
• Imaging methods:
X-ra.y: Calcified cysticerci can be detected by
radiography of subcutaneous tissue and muscles
particularly in the buttocks and thigh. X-ray of the
skull may demonstrate cerebral calcified cyst.
Computed tomography (CT) scan of brain is the
best method for detecting dead calcified cysts. The
cysticcrcal lesions appear as small hypodensities
(ring or disk-like) with a bright central spot (Figs 15A
and B).
Magnetic resonance imaging (MRI) scan of the
brain is more helpful in detection of nonca lcified
cysts and ventricular cysts. It also demonstra tes
spinal cysticerci.
• Serology:
Antibody detection: Anticysticercus antibodies in
serum or cerebrospinal fluid (CSF) can be detected
by \"ELISA\" and enzyme-linked immunoelectrotrasfer
blot (EITB) tests.
Antigen detection: Antigen can be detected in serum
and CSF by ELISA, using monoclonal antibodies and
indicate recent infection.
• Others:
Ocular cysticercosis can be made out by ophthalmoscopy.
Eosinophilia: Usually occurs in early stage of
cysticercosis, but is not constant.
Figs 15A and B: (A) Computed tomography (CT) scan shows multiple
calcified cysts of cysticercus cellulosae in the brain parenchyma; and
(B) CT scan of brain shows clear cyst wall in a cyst,cercal lesion
Treatment
Intestinal taeniasis: Single dose of praziquantel (10- 20 mg/
kg) is the drug of choice.
• iclosamide (2 g), single dose, is another effective drug.
• Purgation is not considered necessary.
Cysticercosis:
• For cysticercosis, excision is the best method, wherever
possible.
• Asymptomatic neurocysticercosis requires no treatment.
• For symptomatic cerebral cysticercosis, praziquantel in
a dose of 50 mg/ kg in three divided doses for 20- 30 days
and albendazole in a dose of 400 mg twice daily for 30
days may be administered.
• Corticosteroids may be given along with praziquantel or
albendazole to reduce the inflammatory reactions caused
by the dead cysticerci.
• In addition, antiepileptic drugs should be given until the
reaction of Lh e brain has subsided.
• Operative intervention is indicated for hydrocephalus.
Prophylaxis
• Beef and pork to be eaten by man should be s ubjected to
effective inspection for cysticerci in slaughter house.
• Avoidance of eating raw or undercooked beef and pork.
The critical thermal point of cysticercus is 56°C for 5
minutes.
• Maintenance of clean personal habits and general
sanitary measures.
• For control of cysticercosis, preven tion of fecal
contamination of soil, proper disposal of sewage and
avoidance of eating raw vegetables grown in polluted soil
are useful measures.
• Detection and treatment of persons harboring adult worm,
as they can develop cysticercosis due to autoinfection.
KEY POINTS OF TAENIA SAG/NATA
• Most common, large ribbon-like tapeworm.
• Rostellum and hooks absent (unarmed tapeworm).
• 1,000- 2,000 proglottids with 15-30 dichotomously
branched uterus.
• Definitive host: Man.
• Intermediate host: Cow.
• Mode of infection: Undercooked (measly) beef containing
cysticercus bovis
• Eggs are not infective to human.
• Asymptomatic, clinical features occur occasionally- abdominal
discomfort, indigestion.
• Diagnosis: Eggs or proglottids in stool, serodiagnosis,
molecular diagnosis.
• Treatment: Praziquantel is the drug of choice and excision in
case of cysticercosis.
• Prophylaxis: By avoidance of eating undercooked beef.
P:149
KEY POINTS OF TAENIA SOLIUM
• Smaller than T. saginata with rostellum and hooks (armed
tapeworm).
• Less than 1,000 proglottids with 5- 10 th ick dend ritic
bra nched uterus.
• Definitive host: Man.
• Intermediate host: Pig, occasionally man (in case of
cysticercosis).
• Mode of infection: Undercooked (measly) pork containing
cysticercus cellulosae; autoinfection and egg in contaminated
vegetable, food and water.
• Eggs are infective to human.
• Clinical features: Adult worm is asymptomatic. Larva l forms
cause cystic lesion in subcutaneous tissue, muscle, brain
(neurocysticercosis) and eye.
• Diagnosis: Intestinal taeniasis-egg or proglottids in stool;
cysticercosis- biopsy, X-ray, CT scan, MRI and serology.
• Treatment: Praziquantel, albendazole, a ntiepileptics in
neurocysticercosis.
• Prophylaxis: By avoidance of eating undercooked pork and
raw vegetables.
Taenia Saginata Asiatica
T. saginala asiatica is closely relared to T. saginata and is
found mainly in Asia.
• It is morphologically similar to T. saginata except:
Tt is smaller than T. saginata.
Intermediate host is pig (not cow).
Its cysticerci are located primarily in liver of rhe pig
(not muscle).
• Clinical features, diagnosis and treatment are similar to
that of T. saginata.
Multiceps Multiceps (Taenia M ulticeps)
Tapeworms of the Genus Multiceps (M. multiceps, M. serialis,
M. glomeralus, etc.) are widespread natural parasites of dogs
and other canines.
Definitive host: Dog, wolf and fox.
Intermediate host: Sheep, cattle, horses and other ruminants.
• Humans act as accidental intermediare host.
• Humans get infec ted by ingesting food or water
contaminated with dogs feces containing eggs.
Oncospheres ha tch out from the eggs, pene trate the
intestine and migrate to various organs, usually central
nervous system (CNS) where it transforms into th e larval
stage called as coenurus.
Coenurus is a roughly spherical or ovoid bladder worm,
up to 3 cm in size, and bearing multiple invaginated
protoscolices (hence, the name multiceps).
Cestodes: Tapeworms
• In sheep, coenurus is typically seen in the brain and spinal
cord. Affected sheep develop cerebellar ataxia, giving the
disease its name \"staggers''.
• Human coenurosis has been reported from Africa, Europe
and the United States of America (USA). The sites affected
mainly are the orbit, brain and subcutaneous tissue.
• Clinical disease is due to pressure effects, symptoms
being headache, vomiting, paresis and seizures and also
due to allergic reactions.
• Surgical removal, where feasible is the only mode of
treatment.
Echinococcus Granulosus
Common Name
Dog tapeworm.
History and Distribution
Hydatid cysts had been described by Hippocrates and other
ancient physicians.
• Adult £. granulosus was described by Hartmann in the
small intestine of dog in 1695 and the larval form (hydatid
cysts) was recognized in 1782 by Goeze.
• The disease is prevalent in most parts of the world, though
it is most extensive in the sheep and cattle-raising areas of
Australia, Africa and South America. It is also common in
Europe, China and the Middle East.
It is a significant health problem in India. It is seen more
often in temperate than in tropical regions.
Habitat
• lhe adult worm lives in the jejunum and duodenum of
dogs and other canine carnivora ( wolf and fox).
• The larval stage (hydatid cyst) is found in humans and
herbivorous animals (sheep, goat, cattle and horse).
Morphology
Adult worm: It is a small tapeworm, measuring only 3-6 mm
in length.
It consists of a scolex, a short neck and strobUa.
The scolex is pyriform, with four suckers and a prominent
rostellwn bearing two circular rows of hooklets (25-30).
The neck is short than the rest of the worm (3 mm x 6 mm).
The strobila is composed of only three proglottids: (1)
the anterior immature, (2) the middle mature and (3) the
posterior gravid segment {Figs 16A to C).
The terminal proglottid is longer and wider than the rest
of the worm and contains a branched uterus filled with
eggs.
The adult worm lives for 6-30 months.
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Paniker's Textbook of Medical Parasitology
m
Hooklets (2 rows)
Scolex
Neck
....=....----Immature
proglottid
-••a..-- Mature
proglottid Strobila
Gravid
proglottid
m
'
Figs 16A to C: Echinococcus granutosus. (A) Schematic diagram of adult worm; (B) Microscopic appearance of
scolex of Echinococcus; and (C) Microscopic appearance of scolex in tongue
Egg:
• The eggs of Echinococcus are indistinguishable from those
of Taenia species.
• It is ovoid in shape and brown in color.
• It contains an embryo with three pairs of hookJets.
Larval form: The larval form is found within the hydalid cyst
developing inside various organs of the intermediate host.
• lt represents the structure of the scolex of adult worm and
remains invaginated within a vesicular body.
• After entering the definitive host, the scolex with suckers
and rostellar hookJets, becomes exuaginated and develops
into adult worm.
Life Cycle
The worm completes its life cycle in tvvo hosts {Fig. 17):
Definitive hosts: Dog (optimal host), wolf, jackal and fox.
Intermed iate host: Sheep and cattle. Sheep is the ideal
intermediate host.
• Man acts as an accidental intermediate host (dead end).
• The larval stage of the parasite is passed in intermediate
hosts, including man, giving rise to hydatid cyst.
The adult worm lives in the small intestine of dogs and
other canine animals. These animals discharge numerous
eggs in the feces.
• Intermediate hosts (sheep and cattle) ingest them while
grazing.
Human infection follows ingestion of the eggs due to
intimate handling of infected dogs or by eating raw
vegetables or other food items contaminated with dog
feces.
• The ova ingested by man or by sheep and cattle are
liberated from the chitinous wall by gastric juice liberating
the hexacanth embryos which penetrate the intestinal
wall and enter the portal uenules, to be carried to the liver
along the portal circulation.
• These are trapped in hepa tic sinusoids, where they
eventually develop into hydatid cyst. About 75% of
hydatid cyst develops in liver, which acts as the first filter
for embryo.
• However, some embryo which pass through the liver,
enter the right side of heart and are caught in pulmonary
capillaries (forming pulmonary hydatid cysts), so that
the lung acts as the second.filler.
• A few enter the systemic circulation and get lodged in
various other organs a nd tissues such as the spleen,
kidneys, eyes, brain, or bones.
When sheep or cattle harboring hydatid cysts die or are
slaughtered, dogs may feed on the carcass or offal. Inside
the intestine of dogs, the scolices develop into lhe adult
worms that mature in about 6-7 weeks and produce eggs
to repeat the life cycle.
When infection occurs in humans accidentally, the cycle
comes to a dead end because the human hydatid cysts are
unlikely to be eaten by dogs.
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Cestodes: Tapeworms
.... Carcasses of infected sheep
with hydatid cyst ingested
by dog (Definitive host)
'
Adult worm in small
intestine
Egg passed in feces
\ ~ ~ ~\_ ) -✓))f~ n' , - Egg -ingested - by sheep (intermediate host)
J
Hexacanth embryos hatch
in the duodenum
Fig. 17: Life cycle of Echinococcus granulosus
Man (accidental host)
Hydatid cyst forms in
liver, lungs, etc.
Pathogenesis
Evolution of hydatid cyst: At the site of deposition, the
embryo slowly develops into a hollow bladder or cyst filled
with fl uid (Figs 18 to 20). This becomes the hydatid cyst
(Greek hydatis: a drop of water).
Pericyst (outer)
• It enlarges slowly and reaches a diameter of 0.5- 1 cm
in about 6 months. The growing cyst evokes host tissue
reaction leading to the deposition of fi brous capsule
around it.
• The cyst wall secreted by the embryo consists of three
indistinguishable layers (Figs 18 and 19):
l. Pericyst is the outer host inflammatory reaction
consisting of fibroblastic proliferation, mononuclear
cells, eosinophils and giants cells, eventually
Ectocyst
(Intermediate)
Hydatid sand
Fig. 18: Hydatid cyst in the liver
Brood
capsules
Scolex
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Paniker's Textbook of Medical Parasitology
Brood capsule Hooklets
·'
Fig. 19: Microscopy shows three layers in the wall of hydatid cyst. lnbox in the right photomicrograph shows a scolex with a row of hooklets
Source: Mohan H. Textbook of Pathology. 6th edition. New Delhi: Jaypee Brothers Medical Publishers: 2010. p. 617.
Figs 20A to C: Hydatid cyst of the liver- typical look
Source: Bhat S. SRB's Manual of Surgery, 4th edition. New Delhi: Jaypee Brothers Medical Publishers: 2012. p. 639.
developing into dense fibrous capsule which may
even calcify.
Ectocyst is the intermediate layer composed of
characteristic acellular, chitinous, laminated hyaline
material. It has the appearance of the white of a hard
boiled egg.
Endocyst is the inner germinal layer which is cellular
and consists of number of nucle i embedded in a
protoplasmic mass and is extremely thin (22- 25 µm).
The germinal layer is the vital layer of the cyst and is
the site of asexual reproduction giving rise to brood
capsules with scolices. It also secretes hydatid fluid,
which fills the cyst.
Hydatidfluid: The interior of the cyst is filled with a clear
colorless or pale yellow fluid called as hydatidfluid.
pH of the fluid is 6.7 (acidic).
Composition: It contains salts (sodium chloride
0.5%, sodium sulfate, sodium phosphate, and salts of
succinic acid) and proteins.
It is antigenic and highly toxic so that its liberation
into circulation gives rise to pronounced eosinophilia
or may even cause anaphylaxis.
The fluid was used as the antigen for Casoni's
intradermal test.
• A granular deposit or hydatid sand is found at the
bottom of the cyst, consisting of free brood capsules and
protoscolices a nd loose hook.lets.
Brood capsules: From the germinal layer, small knob-like
excrescences or gemmu les protrude into the lumen of the
cyst. 1hese enlarge, become vacuolated, and a rc fi lled with
fluid. These are called as brood capsules.
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• They are initially attached to the germinal layer by a stalk,
but later escape free into the fluid-filled cyst cavity.
• From the inner wall of the brood capsules, protoscolices
(new larvae) develop, which represent the head of the
potential worm, comple te with invaginated scolex,
bearing suckers and hook.lets.
• Seve ra l thou sands of protoscolices d evelop into a
mature hydatid cyst, so that this represents an asexual
reproduction of great magnitude.
• Inside mature hydatid cysts, further generation of cyst,
daughter cysts and granddaughter cysts may develop.
The cyst grows slowly often taking 20 years or more
to become big enough to cause clinical illness and is
therefore, particularly seen in man.
Acephalocysts: Some cysts are sterile and may never produce
brood capsules, while some brood capsule may nut produce
scolices. -These are called acephalocysts.
Fate of hydatid cysts: The cyst may get calcified or
spontaneously evacuated followi ng inflammatory reaction.
Hydatid cyst of liver may rupture into lung or other body
cavity producing disseminated hydatid lesions.
Clinical Features
• Most of the times infection is asymptomatic and
accidentally discovered.
• Clinical disease develops only when the hydatid cyst
has grown big enough to cause obstructive symptoms.
Disease results mainly from pressure effects caused by
the enlarging cysts.
• In about half the cases, the primary hydatid cyst occurs
in liver (63%) {Figs 20A to C), mostly in the right lobe.
Cestodes: Tapeworms
Hepatomegaly, pain and obstructive jaundice are the
usual manifestations.
The next common site is the lung (25%) (most common
being the lower lobe of the right lung). Cough, hemoptysis,
chest pain, pneumothorax and dyspnea constitute the
clinical picture.
• In the kidney (2%), hydatid cyst causes pain and
hematuria.
• Other sites affected include spleen (1 %), brain (I%),
pelvic organs, orbit and bones (3%).
Cerebral hydatid cysts may present as focal epilepsy.
When hydatid cyst is formed inside the bones, the
laminated layer is not well-developed because
of confinement by dense osseous tissues. The
parasite migrates along the bony canals as naked
excrescences that erode the bone tissue. This is called
osseous hydatid cyst. Erosion of bone may lead to
pathological fractures.
• Apart from pressure effec ts, anothe r pathogenic
mechanism in hydatid disease is hypersensitivity to the
echinococcal antigen. The host is sensitized to the antigen
by minute amounts of hydatid fluid seeping through the
capsule. I lypersensitivity may cause urticaria. But if a
hydatid cyst ruptures spontaneously or during surgical
interference, massive release of hydatid fluid may cause
severe, even fatal anaphylaxis.
Laboratory Diagnosis
Imaging: Radiological examinations and other imaging
techniques such as ultrasonography (USG), CT scan and MRI
reveal the diagnosis in most cases of cystic echinococcosis
(Flow chart 2).
Flow chart 2: Laboratory diagnosis of Echinococcus granulosus
Laboratory Diagnosis
•
Imaging techniques
• USG: Diagnostic
procedure of choice
• CT scan: For
extrahepatic disease
· MRI: For cysts in spinal
vertebrae and cardiac
cysts
• X-ray: For cysts of
bones and lungs
• IV pyelogram: For
renal cysts
•
Examination of cyst fluid
• Reveals-Scolices, brood
capsules and hooklets
• Diagnostic puncture of
cyst is not recommended
Casoni's test
• Immediate
hypersensitivity
skin test
• Abandoned due
to nonspecificity
•
Serodiagnosis
1) Antibody detection
Tests detecting antibody against
antigen B (8 and 16 KDA)
• IHA
• Indirect immunofluorescence
• ELISA
Tests detecting antibody against
hydatid fluid fraction 5 antigen
• CFT
• Precipitation test
2) Antigen detection
• Double diffusion
• CIED
Others
• Blood-shows
eosinophilia
• Molecular diagnosis
by DNA probes and
PCR
Abbreviations: CT, computed tomography; CFT. complement fixation test; CIED, cardiac implantable electronic device; DNA, deoxyribonucleic acid; ELISA,
enzyme-linked immunosorbent assay; IHA, indirect hemagglutlnatlon; IV, Intravenous; MRI, magnetic resonance imaging; PCR. polymerase chain reaction; USG,
ultrasonography
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Paniker's Textbook of Medical Parasitology
Fig. 21 : Computed tomography (CT) scan shows a large noncalcified
hydatid cyst in right hepatic lobe
Source: Dr Soma Sarkar
• Ultrasonography is the diagnostic procedure of choice.
Cyst wall typically shows double echogenic lines separated
by a hyp oechoic layer (double contour). Pathogenic
findings include daughter cysts and the \"water-lily\" sign
due to detached endocyst floating within the cavity.
• Computed tomography scan is superior for the detection
of extrahepatic disease (Figs 21 and 22).
• Magnetic resonance ima.ging appears to add diagnostic
benefit for cysts, especially at difficult sites such as spinal
vertebrae and cardiac cysts.
• Plain X-rays permit the detection of hydatid cyst in lung
and bones. In cases where long bones are involved, a
mottled appearan ce is seen in the sk.iagram (Fig. 23).
• Intravenous (IV) pyelogram is often helpful for detection
of renal hydatid cyst.
Examination of cyst fluid: Examination of aspirated cyst
fluid under microscope after trichome staining reveals
scolices, brood capsules and hooklets. Exploratory puncture
of the cyst to obtain cystic fluid should be avoided as it may
cause escape of hydatid fl uid and consequent anaphylaxis.
lherefore, fluid aspirated from surgically removed cyst should
only be examined (Flow chart 2).
Casoni's intradermal test: It is an immediate hypersensitivity
{Type 1) skin test introduced by Casoni in 1911, using
fresh sterile hydatid fluid. The antigen in hydatid fluid is
collected from animal or human cysts and is sterilized by
Seitz or membrane filtration. The fluid is injected (0.2 mL)
intradermally in one arm and an equal volume of saline as
control is injected in th e other arm. In a positive reaction,
a large wheal of about 5 cm in diameter with multiple
pseudopodia like projections appears within half an hour at
Fig. 22: Computed tomography (CT) scan showing a hydatid cyst with
noncalcified wall in right lower lobe of lung
Source: Dr Himanshu Roy
Fig. 23: Chest X-ray shows homogenous radiopaque opacity
involving right lower lung with costophrenic angle
Source: Dr Soma Sarkar
the test side and fades in about an hour. A secondary reaction
consisting of edema and induration appears after 8 hours.
lhe test is almost abandoned now due to nonspecificity and
has been supplemented by serological tests (Flow chart 2).
Serology:
Antibody detection:
• Detection of serum antibodies using specific antigens {8
and 16 kDa) from hydatid fluid are frequently used to
support the clinical diagnosis of cystic echi nococcosis.
The tests include indirect hemagglutination (IHA),
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indirect immunol1uorescence and ELISA. In hepatic
cysts, the sensitivity of test is relatively superior (85-98%)
than pulmonary cyst (50-60%).
• The slid e latex agglutination test a n d immu ne
e lectroph oresis using hydatid flu id fraction 5 an tigen
are also widely used. Precipitin test a nd complement
fixation test (CFT) with hydatid antigen have a lso been
fou nd to be positive. CFT is not ve ry sensitive and
fa lse-positive reaction is seen in those receiving neural
antirabic vaccine. CFT is useful after surgica l remova l of
cysts, when a negative test has a better progn ostic value
(Plow chart 2).
Antigen detection: Specific e chinococcal antigen in sera
and in CSF can be detected by double diffusion and counter
immunoelectrophoresis (CIEP) technique (Flow chart 2).
Blood examination: It may reveal a generalized eosinophilia
of20-25%.
Excretion of the scolices: Excretion of scolices into the
sp utum or urine may be observed in pulmonary or renal cyst,
respectively and can be demonstrated by acid-fast staining or
lactophenol cotton blue (LPCB) stain ing.
Specific molecular diagnostic: Specific molecula r diagnostic
me tho d s have been d eveloped involving DNA p robes
and PCR, but the ir applica tio n is lim ited by the ir technical
complexity.
Treatment
Traditionally surgical removal was conside red as the best
mode of treatment of cysts. Currenlly, ultrasound staging is
recommended and management depends on the stage.
In early stages, the treatment o f choice is puncture,
aspiration, injection and reaspiration (PAIR).
Punc ture, aspiration, injection and reaspiration,
considered as a controversial p rocedure earlier, is now
widely used in early stages of the disease (Box 2).
• Th e basic steps involved in PAlR includ e:
Ultrasound or CT-guided puncture of the cyst.
Aspira tion of cyst fluid .
Infusion of scolicidal agen t (usua lly 95% e thanol;
alternatively, hypertonic saline) (Box 3).
Reaspiration of the fluid after 5 minutes.
• Great care is taken to avoid spillage and cavities are
sterilized with 0.5% silver nitrate or 2.7% sod ium chloride
for prop hylaxis of second ary peritoneal echinococcosis
due to inadvertent spillage of fluid du ring PAIR (Box 4).
Albendazole (1 5 mg/ kg in rwo divided doses) is initiated
4 days before the procedure and continued for 4 weeks
afterwards.
Surgery: It is the treatment of choice for complicated E.
granulosus cysts like those communicating with the biliary
tract and in those cysts where PAIR is not possible.
Cestodes: Tapeworms
Box 2: Indications of puncture, aspiration, injection and reaspiration
(PAIR)
• Cysts with internal echoes on ultrasound (snowflake sign) multiple cysts,
cysts with detached laminar membrane.
. Contraindications of PAIR for superficially located cysts, cysts with
multiple thick internal septal divisions (honeycombing pattern), cysts
communicating with biliary tree.
Box 3: Scolicidal agents and their complications
• Cetrimide: It can cause acidosis
• Alcohol 95%: It can cause cholangitis
• Hypertonic saline: Hypernatremia
• Sodium hypochlorite: Hypernatremia
• Hydrogen peroxide.
Note: In cases with biliary communication only hypertonic saline
(1 5- 20%) is used.
Box 4: Echinococcus species and the diseases caused by them
• Echinococcus granulosus: Hydatid disease
• Echinococcus multilocularis: Alveolar or multilocular hydatid disease
• Echinococcus vogeli and Echinococcus o/igarthrus: Polycystic hydatid
disease
The p referred su rgical approach is pericystectomy. For
pulmonary cyst, treatment consists of wedge resection or
lobectomy.
Recurrence after surgery is common.
Pre and postoperative chemotherapy with albendazole
for 2 years after curative surgery is recommended.
• Positron emission tomography (PET) scan ning can be
used to follow d isease activity.
• Other new treatment modalities includ e laparoscopic
hydatid liver surgery and percutaneous the rmal ablation
(PTA) of the germin al layer of the cyst using rad iofrequency
ablation device.
Chemotherapy: Chem otherapy with ben zimidazole agents
are restricted to resid ua l, postsu rgical and inoperable cysts.
Albendazole (400 mg BO for 3 months) and p raziquantel (20
mg/ kg/ day for 2 weeks) have proved beneficial.
Prophylaxis
E. granulosus infection can be prevented by:
Ensuring pet dogs do nor eat animal carcass or offal.
• Periodical deworming of pet dogs.
• Destruc tion of stray and infected dogs.
• Maintaining persona l hygiene such as washing of hands
afte r touching dogs and avoidance of kissing pet d ogs.
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Paniker's Textbook of Medical Parasitology
KEY POINTS OF ECHINOCOCCUS GRANULOSUS
• Echinococcus causes hydatid cyst in man.
• Smaller than other cestodes
• It measures 3-6 mm and consists of pyriform shaped, scolex,
short neck and strobila consists of 3 proglottids.
• Eggs are similar to taenia
• Larval form is called hydatid cyst which develops inside
various organs of the intermediate host
• Hydatid cyst consists of three layers-pericyst, ectocyst and
endocyst and filled with hydatid fluid
• Hydatid cyst may be a symptomatic or may cause pressure
effect and anaphylactic reactions.
• Laboratory diagnosis by USG, CT scan, MRI and rays.
• Treatment option includes surgery, PAIR and chemotherapy
with albendazole praziquantel.
Echinococcus Multilocularis
This causes the rare but serious condition of alveolar or
multilocular hydatid disease in humans (Box 5).
• It is found in the northern parts of the world, from Siberia
in the East to Canada in the West.
The adult worm is smaller than E. granuLosus a nd lives
in the intestines of foxes, dogs and cats which are the
definitive host.
Rodents are the main intermediate hosts.
Human infection develops from eating fruits or vegetables
contaminated with their feces.
E. multilocularis leads to multilocular hydatid cyst.
The liver is the most commonly affected organ. The
multilocular infiltrating lesion appears like a grossly
invasive growth, without any fluid or free brood capsule
or scolices which can be mistaken for a malignant tumor.
Patients present with upper quadrant and epigastric pain.
Liver enlargement and obstructive jaundice may also be
present. It may also metastasize to the spleen, lungs and
brain in 2% cases.
The prognosis is very grave and if untreated, 70% cases
progress to dealt.
Surgical resection, when possible, is the best method of
treatment. Albendazole therapy is recommended for 2
years after curative surgery. In those cases, where surgery
is not possible, indefinite treatment with albendazole is
recommended.
Hymenolepis Nana
Common Name
Dwarf tapeworm.
Box 5: Malignant hydatid disease
• It is a misnomer, as it is a benign condition.
• It is caused by Echinococcus multilocu/aris (alveolaris). It presents with
multiple small cysts in both lobes of the liver.
• It is difficult to treat and mimics clinically and prognosis wise to
malignancy; hence the name.
• Patients die of liver failure.
History and Distribution
The name I lymenolepis refers to the thin membrane covering
the egg (Greek hymen-membrane, lepis-rind or covering)
and nana to its small size (nan.us-dwarf). It was first
discovered by Bilharz in 1857.
• It is cosmopolitan in distribution bur is more common in
warm than in cold cl imates.
• Infection is most common in school children and
institutional populations.
• I lymenolepis nan.a is the smallest and the most common
tapeworm found in the human intestine.
It is unique that it is the only cestode which completes its
life cycle in one host- humans.
Habitat
The adult worm lives in the proximal ileum of man. H. nan.a
var.jraterna is found in rodents like mice and rats, where they
are found in the posterior part of the ileum.
Morphology
Adult worm: H. nan.a is the smallest intestinal cesrode that
infects man.
• It is 5- 45 mm in length a nd less than l mm thick. The
scoLex has four suckers and a retractile rostellum with a
single row of hook.lets (Fig. 24).
• The long slender neck is followed by the strobila consisting
of 200 or more proglottids, which are much broader than
long.
• Genital pores are situated on the same side along the
margins.
• The uterus has lobulated walls and the testis is round and
three in nwnber.
Eggs are released in the intestine by disintegration of the
distal gravid segments.
Egg: The egg is roughly spherical or ovoid, 30-40 µmin size.
• It has a thin colorless outer membrane and inner
embryopfwre enclosing the hexacanth oncosphere (Figs
25Aand B).
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Cestodes: Tapeworms
Fig. 24: Adult worm of Hymenolepis nana
Figs 25A and B: Egg of Hymenolepis nana. (A) As seen under microscope; and (B) Schematic diagram
• The space between two membranes contains yolk
granules and 4-8 thread like polar .filaments arising from
two knobs on the embryophore.
• The eggs float in saturated solution of salt and are nonbile
stained.
• They are immediately infective and unable to survive for
more than l O days in external environment.
Life Cycle
Host: Man.
• There is no intermediate host.
Mode of transmission: Infection occurs by ingestion of
the food and water contaminated with eggs.
Internal autoinfection may also occur when the eggs
released in the intestine hatch there itself (Fig. 26).
External a.utoinfection occurs when a person ingest
own eggs by fecal oral route.
H. nana is unusual in that it undergoes multiplication in
the body of the definitive host.
When the eggs are swallowed, or in internal autoinfection,
they hatch in the small intestine.
the hexacanth embryo penetrates the intestinal villus and
develops into the cysticercoid larva.
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Paniker's Textbook of Medical Parasitology
1 Man
Ingestion of contaminated
food and water causes
infection
Eggs ingested by rat
Oncosphere is liberated and it
penetrates intestinal wall
Internal autoinfection
(in children) or external
autoinfection
Cysticercoid larva
in rat flea
LIFE CYCLE OF
HYMENOLEPIS NANA
INFECTING RODENTS
'
Rat flea ingest
eggs of Hymenolepis
nana
LIFE CYCLE OF
HYMENOLEPIS NANA
INFECTING MAN
No intermediate
host required
Adult worm in
small intestine
Egg in feces
Fig. 26: Life cycle of Hymeno/epis nana
• This is a solid pyriform structure, with the vesicular
anterior end containing th e invaginated scolex and a
short conical posterior end.
• After about 4 days, the mature larva emerging out of the
villus evaginates its scolex and attaches to the mucosae.
• It starts strobilization, to become the mature worm, which
begins producing eggs in about 25 days.
A different strain of H. nana infects rats and mice. The eggs
passed in rodent feces are ingested by rat fleas (Xenopsylla
cheopis and others), which acts as the intermediate host. The eggs
develop into cysticercoid larvae in the hemocele of these insects.
Rodents get infected when they eat these insects. The murine
strain does not appear to infect man. However, the human strain
may infect rodents, which may, therefore, constitute a subsidiary
reservoir of infection for the human parasite.
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Clinical Features
Hymenolepiasis occurs more commonly in children.
• There are usua!Jy no symptoms but in heavy infections,
there is nausea, anorexia, abdominal pain, diarrhea and
irritability.
• Sometimes pruritus may occur due roan allergic response.
Laboratory Diagnosis
The diagnosis is made by demonstration of characteristic eggs
in feces by direct microscopy. Concentration methods like
salt flotation and formalin ether may be readily used. ELJSA
test has been developed with 80% sensitivity.
Treatment
Praziquantel (single dose of 25 mg/kg) is the drug of choice,
since it acts both against the adult worms and the cysticercoids
in the intestinal villi.
• Ni tazoxanide 500 mg BD fo r 3 days may be used as
alternative.
Prophylaxis
• Maintena nce of good personal hygien e and sanitary
improvements.
• Avoiding of consumption of contaminated food and water.
• Rodent control.
Hymenolepis Diminuta
This is called the rat tapeworm and is a common parasite of
rats and mice.
• The name diminuta is a misnomer, as it is larger than H.
nana being 10-60 cm in length.
• Its life cycle is similar to that of the murine strain of H.
nana.
• Rarely, human infection follows accidental ingestion of
infected rat fleas. Human infection is asymptomatic.
Dipylidium Caninum
This common tapeworm of dogs and cats, it may accidentally
cause human infection, mainly in children.
Morphology
• The adult worm in the intestine is about l 0-70 cm long.
• The scolex has four prominent suckers and a retractile
rostellum with up to seven rows of spines {Figs 27 A to C).
• The mature proglortid has two genital pores, one on
either side, hence th e name Dipylidium (dipylos-two
entrances).
Cestodes: Tapeworms
m
Figs 27A to C: Dipylidium caninum. (A) Scolex showing four suckers
and rostellum with multiple rows of hooklets; (B) Mature proglottid
showing two genital pores, one on either side; and (C) Eggs found in
clusters enclosed in a membrane
Box 6: Parasites requiring as intermediate host
• Hymenolepis diminuta
, Dipylidium caninum
• Hymenolepis nano (murine strain)
• Gravid proglottids are passed out of the anus of the host
singly or in groups.
Life Cycle
Definitive host: Dogs, cats and rarely man.
Intermediate host: Fleas (Box 6).
• Man acquires infection by ingestion of fl ea harboring
cysticercoid larva.
• 1h e eggs or proglottids passed in feces of dogs and cats are
eaten by larval stages of dog and cat fleas, Ctenocephalides
canis and C.felis.
• The embryo develops into a tailed cysticercoid larva.
• When the adult fleas containing the larvae are eaten by
dogs, cats, or rarely humans, infection is transmi tted.
Clinical Features
Human infection is generally asymptomatic, but the actively
motile proglottids passed in srools may raise an alarm.
Diagnosis
the diagnosis is made by detection of proglortids or eggs in
stool.
Treatment
the drug of choice is praziquantel.
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Paniker's Textbook of Medical Parasitology
REVIEW QUESTIONS
Describe briefly:
a. General characters of cestodes
b. Classification of cestodes
Short notes on:
a. Echinococcus granulosus
b. Hymenolepis nana
c. Diphyllobothrium /atum
d. Hydatid cyst
e. Cason i's test
f. Sparganosis
g. Coenurosis
h. Dipylidium caninum
i. Cysticercus cellulosae
j. Neurocysticercosis
Describe morphology, life cycle and laboratory diagnosis of:
a. Taenia solium
b. Taenia saginata
c. Echinococcus granulosus
Differentiate between:
a. Taenia solium and Taenia saginata
b. Taenia saginata saginata and Taenia saginata asiatica
MULTIPLE CHOICE QUESTIONS
Autoinfection is a mode of transmission in
a. Trichinella
b. Cysticercosis
c. Ancylostoma
d. Ascaris
Pigs are reservoir for
a. Taenia solium
b. Diphyllobothrium latum
c. Trichinella spiralis
d. Ancyclostoma
On microscopic examination, eggs are seen, but on saturation
with salt solution eggs are not seen. The eggs are likely to be of
a. Trichuris trichiura
b. Taenia solium
c. Ascaris lumbricoides
d. Ancylostoma duodenale
Which of the following is not a cestodes
a. Diphyllobothrium latum
b. Taenia saginata
c. Schistosoma mansoni
d. Echinococcus granulosus
Consumption of uncooked pork is likely to cause which of the
following helminthic disease
a. Taenia saginata
b. Taenia so/ium
c. Hydatid cyst
d. Trichuris trichiura
All of the following are true about neurocysticerosis, except
a. Not acquired by eating contaminated vegetables
b. Caused by regurgitation of larva
c. Acquired by orofecal route
d. Acquired by eating pork
The longest tapeworm found in man
a. Diphyllobothrium /atum
b. Taenia saginata
c. Taenia solium
d. Echinococus granulosus
Second intermediate host of Diphyl/obothrium latum is
a. Cyclops
b. Man
c. Snail
d. Fresh water fish
Dwarf tapeworm refers to
a. Echinococcus granulosus
b. Loa/oa
c. Hymenolepis nano
d. Schistosoma mansoni
The egg of which of the following parasites consists of polar
filaments arising from either end of the embryophore
a. Taenia saginata
b. Taenia solium
c. Echinococcus granulosus
d. Hymenolepis nana
11 . Coenurus is the larval form of
a. Taenia solium
b. Taenia multiceps
c. Echinococcus granulosus
d. Echinococcus multilocularis
Larval form of Echinococcus granulosus is seen in
a. Dog
b. Man
c. Wolf
d. Fox
The adult worm of Echinococcus granulosus contains
a. 3- 4 segments
b. 50- 100 segments
c. 100- 200 segments
d. 1000-2000 segments
Which skin test is useful for diagnosis of hydatid disease
a. Casoni's test
b. Schick test
c. Dick's test
d. Tuberculin test
Answer
b
d
a
C
b
d
C
b
b
b
a
a
a
a
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CHAPTER 12
• INTRODUCTION
Trematodes are leaf-shaped unsegmented, fl at and broad
helminths (hence the name fluke, from the Anglo-Saxon word
floe meaningflatfish). The name trematode comes from their
having large prominent suckers with a hole in the middle
(Greek trema: hole, eidos: appearance).
• CLASSIFICATION OF TREMATODES
Systemic Classification
Trematodes belong to:
Phy lum: Platyhelminthes
Class: Trematoda
The detailed systemic classification has been given in
Table I .
Table 1: Zoological classification of trematodes
Superfamily
Schistosomatoidea
Paramphistomatoidea
Echinostomatoidea
Opisthorchioidea
Plagiorchioidea
Family
Schistosomatidae
Zygocotylidae
Fasciolidae
• Opisthorchiidae
• Heterophyidae
Paragonimidae
Classification Based on Habitat
Based on habitat, trematodes can be classified as (Table 2):
• Blood flukes
Liver flu kes
• Intestinal llukes
• Lung llukes.
• FLUKES: GENERAL CHARACTERISTICS
They vary in size from 1 mm to several centimeters. Males are
shorter and stouter than females.
• The unique feature of flu kes is the presence of two
muscular cup-shaped suckers (hence called distomata)-
the oral sucker surrounding the mouth at the anterior
end and the ventral sucker or acetabulum in the middle,
ventrally (Fig. 1).
Genus
Schistosomo
• Gastrodiscoides
• Waisonius
• Fasciola
• Fasciolopsis
• Opisthorchis
• Clonorchis
• Heterophyes
• Metagonimus
Paragon/mus
Species
• S. haemarobium
• S. mansoni
• S. japonicum
• S. mekongi
• S. intercalatum
• G.hominis
• W. watsoni
• F. hepatica
• F. buski
• 0. felineus
• 0. viverrini
• C. slnensis
• H. heierophyes
• M. yokogawai
P. westermani
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Paniker's Textbook of Medical Parasitology
All schistosomes live in venous plexuses in the body of
the definitive host, the location varying with the species
(urinary bladder in S. haematobium, sigmoidorectaJ
region in S. mansoni and UeocecaJ region in S. japonicum ).
Schistosoma Haematobium
History and Distribution
This vesical blood fluke, formerly known as bilharzia
haematobium, has been endemic in th e Nile valley in
Egypt for millenia. Its eggs have been found in the renal
pelvis of an Egyptian mummy dating from l ,250-1,000 BC.
Schistosome antigens have been identified by enzyme-linked
immunosorbent assay (ELlSA) in Egyptian mummies of the
Predynastic period, 3,100 BC.
• The adult worm was described in 1851 by Bilharz in Cairo.
Its life cycle, including the larval stage in the snail, was
worked out by Leiper in 1915 in Egypt.
• Although maximally entrenched in the Nile valley, S.
haematobium is also endemic in most parts of Africa and
in West Asia.
• An isolated focus of endemicity in India exists in Ratnagiri
district of Maharashtra.
• About 200 million persons are at a risk of infection and 90
million arc infected by S. haematobium globally.
Habitat
The adult worms live in the vcsicaJ and pelvic plexuses of
veins.
Morphology
Adult worm:
• 1he male is 15 mm long by 0.9 mm thick and covered by a
thick tuberculate tegument.
• It has two muscular suckers: (1) the oral sucker being
small a nd (2) the ventral sucker la rge and prominent.
Beginning immediately behind the ventral sucker and
extending to the caudal end is the gynecophoric canal, in
which the female worm is held (Fig. 3).
• the adult female is long and slender (20 mm by 0.25 mm).
• 1h e gravid worm contains 20-30 eggs in its uterus at one
time and may pass up to 300 eggs a day.
Egg: The eggs are elongated, brownish yellow (about 150 µm
by 50 µm) and nonoperculated. the eggs have characteristic
terminal spine at one pole (Fig. 4).
Mechanism of egg expulsion: The eggs are laid usually in
the small venules of the vesical and pelvic plexuses, though
sometimes they are laid in the mesenteric portal system,
pulmonary arterioles and other ectopic sites.
• 1 he eggs a re laid one behind the other with the spine
pointing posteriorly.
Bifurcated
alimentary canal
Tubercles on back----:v!\"tr
of male parasite ~ ii.'~
Oviduct ===\"Jttf'lfJ~IJ Ovary
Cecum
Schistosoma mansoni
Coupled worms
Giemsa staining, magnification 25X
Fig. 3: Structural details of Schistosoma (coupled)
Fig. 4: Egg of Schistosoma haematobium
• From the vcnules, the eggs make their way through the
vesical wall by the piercing action of the spine, assisted
by the mounting pressure within the venules and a lytic
substance released by the eggs.
• The eggs pass into the lumen of the urinary bladder
together with some extravasated blood.
• TI1cy are discharged in the urine, particularly towards the
end of micturition.
• For some unknown reasons, the eggs are passed in urine
more during midday than at any other time of the day.
• The eggs laid in ectopic sites generally die and evoke local
tissue reactions. They may be found, for instance in rectal
biopsies, but are seldom passed live in feces.
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Life Cycle
S. haematobium passes its life cycle in rwo hosts:
Definitive host: Humans are the only natural definitive
hosts. o animal reservoir is known.
Intermediate host: Freshwater snails (snail of the genus
Bulinus).
Infective form: Cercaria larva.
• The eggs that are passed in urine are embryonated and
hatch in water under suitable conditions to release the
free-living ciliated miracidia.
• Miracidia swim about in water and on encountering a
suitable interm ediate host, penetrate into its tissues and
reach its liver (Fig. 6). The intermediate hosts are snails of
Bulinus species in Africa. In India, the intermediate host is
the limpet, Ferrissia tenuis.
Development in snail: Inside the snail, the miracidia lose their
cilia and in about 4-8 weeks, successively pass through the
stages of the first and second generation sporocysts (Fig. 6).
• Large numbers of cercariae are produced by asexual
reproduction within the second generation sporocyst.
The cercaria has an elongated ovoid body and forked tail
(furcocercous cercaria) (Fig. 5).
• The cercariae escape from the snail into water.
• Swarms of cercariae swim about in water for 1-3 days.
Persons become infected by contact with water containing
cercariae during bathing. Suckers and lytic substances
secreted by cercariae helps them to penetrated intact skin.
Development in man: After pen etrating the skin, the
cercariae loss their tails and become schistosomulae which
travel via peripheral venules to systemic circulation (Fig. 6).
• They then start a long migration, through the vena cava
into the right heart, the pulmonary circulation, the left
heart and the systemic circulation, ultimately reaching
the liver.
• In the intrahepatic portal veins, the schistosomulae grow
and become sexually differentiated adolescents about 20
days after skin penetration.
• They then start migrating against the bloodstream into the
inferior mesenteric veins, ultimately reaching the vesical
Anterior sucker Ventral sucker Forked tail
Fig. 5: Cercaria larva of Schistosoma spp.
Trematodes: Flukes
and pelvic venous plexuses, where they mature, mate and
begin laying eggs.
Eggs start appearing in urine usually 10-12 weeks after
cercarial penetration.
The adult worms may live for 20- 30 years.
Pathogenicity and Clinical Features
Clinical ill ness caused by schistosomes can be classified as
acute and chronic based on the stages in the evolution of the
parasite.
Acute schistosomiasis:
Durin g skin penetration of cercariae, intense irritation and
skin rash may develop at the side of cercarial penetration
(swimmer's itch). It is particularly severe when infection
occurs with cercariae of nonhuman schistosomes.
Anaphylactic or toxic symptoms may develop during
incubation period due to liberation of toxic metabolites
by schistosomules.
Migration of schistosomulae into lungs may cause cough
and mild fever.
Chronic schistosomiasis:
Egg deposition in urinary bladder causes mucosa!
damages lead ing to painless hematuria, dysuria and
proteinuria, particularly in children in endemic areas.
There is innammation of the urinary bladder due
to release of soluble antigens from the eggs causing
pseudoabscesses in the surrounding tissues.
Initially the trigone is involved but ultimately the whole
mucosa is inflamed, ulcerated and thickened. There
is heavy infi ltra tion of macrophages, lymphocytes,
eosinophils and fibroblasts.
Many of the eggs die and become calcified eventually
producing fibrosis ofvesical mucosa and formation of egg
granulomas (sandy patches).
Fibrosis may cause obstructive u ropathies like
hydronephrosis and hydroureter.
Chronic schistosomiasis has been associated wi th
urinary bladder carcinoma (Box 3).
Chronic cystitis may develop due to secondary bacterial
infection.
Chronic infection may result in calculus formation.
Involvement of other organs during schistosomiasis:
Lungs and central n ervous system (spinal cord), skin and
genital organs may be involved.
Box 3: Parasites associated with malignancy
• Schisrosoma haematobium: Bladder carcinoma
• Clonorchis sinensis: Bile duct carcinoma
• Opisthorchis viverrini: Bile duct carcinoma
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Paniker's Textbook of Medical Parasitology
~ ,;s
<J<Y Mature in mtrahepatic
portal veins
Adult worms in
venous plexus
Cercaria sheds its tail
ro
\"\"/= ,,,.
MAN
(Definitive host) S. haematobium S. mansoni S. japonicum
WATER
Penetrate skin of man
(Definitive host)
Free-living ciliated miracidium
hatches In water (16 hours)
Development within snail
(Intermediate host) In 4-8 weeks
Primary sporocysts
Secondary sporocysts
Developing cercariae
within secondary sporocysts
Fig. 6: Life cycle of Schistosoma spp.
• Ectopic lesions in the spinal cord produce a transverse
myelitis-like syndrome.
Schistosomiasis favors urinary carriage of typhoid bacilli.
Laboratory Diagnosis
Urine microscopy: The eggs with characteristic terminal
spines can be demonstrated by microscopic examination
of centrifuged deposits of urine or by filtration of a known
volume of urine through nucleopore filters (Flow chart 1).
• Eggs are more abundant in the blood and pus passed by
patients at the end of micturition.
• Nucleopore filtration method provides quantitative data
on the intensity of infection.
Eggs can also be seen in the seminal fluid in males and
occasionally in feces.
Histopathology: Schistosome infection may also be
diagnosed by demonstrating its eggs in bladder mucosa!
biopsy and rectal biopsy.
Detection of antigen: Another diagnostic method is by
detection of specific schistosome antigens in serum or urine.
Two circulating antigens related to gut of adult schistosomes:
(1) circulating anodic antigen (CAA) and (2) circulating
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Trematodes: Flukes
Flow chart 1: Laboratory diagnosis of Schistosoma haematobium
laboratory diagnosis
' ' l ' •
Demonstration of Detection of antigens Detection of antibody lntradermal skin test Imaging
characteristic egg
• Urine microscopy
(CAA and CCA) by
ELISA
• Complement fixallon test
(CFT)
(Fairley·s test)
The test is group specific
and gives positive result
in all schistosomiasis
• X-ray to demonstrate
bladder and ureteral
• Bladder mucosal calc1ficat1on
biopsy
• Bentonite flocculation test
• Indirect hemagglutinat,on • USG, !VP and cystoscopy
(IHA) for indirect diagnosis
• lmmunofluorescence
• FAST/ELISA
• Enzyme-linked
,mmunoelectrotransfer
blol(EITB)
Abbreviations: CAA, circulating anodic antigen; CCA, circulating cathodic antigen; ELISA. enzyme-linked immunosorbent assay;
FAST, falcon assay screening test; IVP, intravenous pyelogram; USG, ultrasonography
cathodic antigens (CCAs) can be demonstrated by dipstick
assay and ELISA.
The test is very sensitive and specific, but is available only
in specialized laboratories.
Soluble egg antigens (SEAs) can be demonstrated in
serum (Flow chart 1).
Detection of antibody: Several serological tests have been
described for detection of specific antibody, but are not very
useful as they cannot differentiate between present and past
infection. These include complement fixation test (CFT),
bentonite flocculation test, indirect hemagglutination (IHA),
immunofiuorescence and gel diffusion tests.
Two serological tests for detection of antibodies against
Schistosoma haematobium adult worm microsomal antigen
(HAMA)are:(l )thefalronassayscreeningtest(HAMAFAST)/
ELISA and (2) HAMA enzyme-linked immunoelectrotransfer
blot (EITB). Both these tests are highly sensitive and specific
(95% sensitive and 99% specific) (Flow chart 1).
Intradermal skin test (Fairley's test): 11,csc allergic skin tests
are group-specific. The test uses antigen from larvae, adult
forms and eggs of schistosomes from artificially infected
snails and infected laboratory animals.
Imaging:
• X-ray of the abdomen may show bladder and ureteral
calcification.
• Ultrasonography (USG) is also useful in diagnosing
S. haematobium infection. USG may show hydroureter
and hydronephrosis.
• Intravenous pyelogram (TVP) a nd cystoscopy are also
useful in indirect diagnosis of the disease.
Treatment
Prazjquantel ( 40-60 mg per kg in divided doses in a single
day) is the drug of choice.
Metriphonate is th e alternative drug of choice in
schistosomiasis due to S. haematobium (7.5 mg/ kg weekly for
3 weeks).
Prophylaxis
Prophylactic measures include:
• Eradication of the intermediate molluscan hosts by using
molluscicides.
• Prevention of environmental pollution with urine and
feces.
• Effective treatment of infected persons.
• Avoid swimming, bathing and washing in infected water.
Schistosoma Mansoni
History and Distribution
Tn 1902, Manson discovered eggs with lateral spines in the
feces of a West Indian patient that led to the recognition of
this second species of human schistosomes. It was, therefore
named S. mansoni.
• It is widely distributed in Africa, South America and the
Caribbean islands.
Habitat
Adult worm lives in the inferior mesenteric uein.
Morphology
S. mansoni resembles S. haematobium in morphology and
life cycle, except:
• The adult worms are smaller and their integuments
studded with prominent coarse tubercles.
• In the gravid female, the uterus contains very few eggs,
usually 1-3 only.
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Paniker's Textbook of Medical Parasitology
S. mansoni
Ova with a lateral spine
(obtained from stool)
S. haematobium
Ova with a terminal spine
(obtained from urine)
S. japonicum
Ova with a lateral knob
(obtained from stool)
Note: The characteristic surround
of tissue particles
Fig. 7: Schematic diagram to show distinguishing features of eggs of S. mansoni, S. haematobium and S. japonicum
• The prepatent period (the interval between cercarial
penetration and beginning of egg laying) is 4-5 weeks.
• The egg h as a characteristic lateral spine (Fig. 7),
more near to the rounded posterior end. The eggs are
nonoperculated and yellowish brown.
Life Cycle
Definitive host: Humans are the only natural definitive hosts,
though in endemic areas monkeys and baboons have also
been found infected.
Intermediate host: Planorbid freshwater snails of th e genus
Biomphalaria.
Infective form: Fork-tailed cercaria.
In humans, the schistosomulae mature in the liver and
the adult worms move against th e bloodstream into the
venules of the inferior mesenteric group in the sigmoidorectal
area. Eggs penetrate the gut wall, reach the colonic lumen and
are shed in feces.
Pathogenicity and Clinical Features
• Cercarial dermatitis:
Following skin penetration by cercariae: A pruritic
rash called as cercarial dermatitis or swimmers itch
may develop locally. It is a self-limiting disease.
• Katayama/ever:
A.fter4-8 weeks orcercarial invasion a serum sickness
like illness may happened during production of eggs.
lt results from high worm load and egg antigen stimuli
which leads to formation of immune complexes. Sign
and symptoms include high fever, rash, arthralgia,
hepatosplenomegaly, lymphade nopathy and
eosinophilia.
• Intestinal bilharziasis:
During the stage of egg deposition in small intestine,
patients may develop pain in abdomen and bloody
dysentery, which may go on intermittently for many
years.
The eggs deposited in the intestinal wall may cause
microabscesses, granulomas, hyperplasia and
eventual fibrosis. Egg granulomas are found in the
distal part of the colon and rectum. Ectopic lesions
include hepatosplenomegaly and periportal fibrosis,
portal hypertension, as some of the eggs are carried
through portal circulation into liver.
Portal hypertension may cause gastrointestinal
hemorrhage.
Laboratory Diagnosis
Stool microscopy: Eggs with lateral spines may be
demonstrated microscopically in stools. Kato-Katz thick
smear or otl1er concentration methods may be required when
infection is light. Kato-Katz thick smear provides quantitative
data on tl1e intensity of infection, which is of value in assessing
the degree of tissue damage and monitoring the effect of
chemotherapy.
Rectal biopsy: Proctoscopic biopsy of rectal mucosa may
reveal eggs when examined as fresh squash preparation
between two slides.
Serological diagnosis: Serological diagnosis by detecting
schistosomal antigen and antibody is similar to that of
S. haematobium.
Imaging: Ultrasonography is useful to detect hepatosplenomegaly and periportal fibrosis.
Blood examination: Blood examination may reveal
eosinophilia and increased levels of alkaline phosphatase.
Treatment
Praziquantel (single oral dose 40 mg/kg) is the drug of choice.
Oxamniquine (single oral dose 15 mg/ kg) is also effective.
It damages the tegument of male worm and thereby, makes
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the worm more susceptible to le thal action of the immune
system.
Prophylaxis
Same as S. haematobium.
Schistosoma Japonicum
Common Name
Oriental blood Duke.
Distribution
S. japonicum is found in the Far East, Japan, China, Taiwan,
Philippines and Sulawesi.
Habitat
The adult worms are seen typically in the venules of the
superior mesenteric vein draining the ileocecal region.
They are also seen in the intrahepatic portal venules and
hemorrhoidal plexus of veins.
Morphology
Morphologically, they are similar to the schistosomes
described earlier except:
The adult male is comparatively slender (0.5 mm thick)
and does not have cuticular tuberculations.
Trematodes: Flukes
• In the gravid female, the uterus contains as many as JOO
eggs at one time and up to 3,500 eggs may be passed daily
by a single worm.
• The prepatent period is 4- 5 weeks.
• The eggs are smaller and more spherical than those of S.
haematobium and S. mansoni. The egg has no spine, but
shows a lateral small rudimentary knob (Fig. 7).
Differentiating features between the three species of
Schislosoma are illustrated in Table 3.
Life Cycle
Life cycle of S. japonicum is similar to S. haematobium with
the following exceptions:
Definitive host: Man is the definitive host but in endemic
areas, natural infection occurs widely in several domestic
animals and rodents, which act as reservoirs of infection.
lritermed iate host: Amphibian snails of the genus
Oncomelania.
lnfectiveformfor humans: Fork-tailed cercaria.
• Eggs d eposited in the superior mesenteric venules
penetrate the gut wall and are passed in feces.
• They hatch in water and th e miracidia. infect the
intermediate hosts, amphibian snails of the genus
Oncomelania.
• The fork-tailed cercaria, which escapes from the snails is
the infective form for men and other definitive hosts.
Table 3: Differentiating features of S. haematobium, S. mansoni and S. japonicum
Habitat
Morphology
Size: Male
Female
Integument
Number of testes
Ovary
Uterus
Egg
Cephalic glands in cercariae
Distribution
Definitive host
Intermediate host
Schlstosoma haematobium
Veins of the vesical and pelvic plexuses,
less commonly in portal vein and its
mesenteric branches
• 1.Scmx 1 mm
• 2 cm x 0.22 mm
• Finely tuberculated
• 4 5 in groups
• In the posterior one-third of the body
• Contains 20-30 eggs
Elongated with terminal spine
Two pairs oxyphilic and three pairs
basophilic
Africa, Near East, Middle East and India
Man
Snail of genus Bulinus
Schistosoma mansonl
Inferior mesenteric vein and its
branches
• 1 cmx 1 mm
• 1.4 cm x0.25 mm
• Grossly tuberculated
• 8-9 in a zigzag row
• In the anterior half of the body
• 1-3 eggs
Elongated with lateral spine
Two pairs oxyphilic and four pairs
basophilic
Africa and South America
Man
Snail of genus Biomphaloria
Schistosoma japonlcum
Superior mesenterlc vein and its
branches
• 1.2- 2cm x0.5 mm
• 2.6cm x 0.3mm
• Nontubercular
• 6-7 in a single file
• In the middle of the body
• SO or more eggs
Round with small lateral knob
Five pairs oxyphilic, no basophilic
China, Japan and Far East (oriental)
Man (mainly) domestic animals and
rodents (which act as reservoir of
infection)
Amphibian snail of genus Oncomelania
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Paniker's Textbook of Medical Parasitology
Pathogenicity and Clinical Features
Disease caused by S. japonicum is also known as oriental
schistosomiasis or Katayama disease.
• Pathogenesis is almost sim ilar to that of S. mansoni. But
the disease is more severe due to higher egg production.
• During the acute phase of the disease, Katayama/ever is
similar to that seen in S. mansoni.
• Chronic illness is characterized by intestinal m ucosa!
h y p erp lasia, h epatosplenomegaly an d porta l
hypertension. Liver is hard and shows periportal fibrosis
(clay pipestem fibrosis). Portal hypertension leads
to esophageal va rices and gastrointestinal bleeding.
Intestinal disease manifests as colicky abdominal pain,
bloody diarrhea and anemia (Box 4).
• Central nervous system and lung involvement (cor
pulmonale) may occur in 2-4% of cases. Pa rietal lobe
of the brain and spine are commonly affec ted. Severe
epileptic seizures may be observed in these patients.
Laboratory Diagnosis
Similar to that of S. mansoni.
Treatment
S. japonicum infection is more resistant to treatment than
othe r schistosomiasis. A prolonged course of intravenous
tartar emetic gives good results. Praziquantel is the drug of
choice.
Prophylaxis
Same as S. haematobium.
Schistosoma lntercalatum
S. intercalatum was first noted in 1934 in West-Central Africa.
• The eggs are fully embryonated without any opercu.lum
having terminal spines, but are passed exclusively in
stools. The eggs are acid-fast.
• It produces few symptoms involving the mesenteric portal
system.
Box 4: Parasites leading to bloody diarrhea
• Intestinal Schistosomo species:
S. japonicum
S. mansoni
S. intercalarum
S. mekongi.
• Trichuris trichiura
• Entamoeba histolytica
• Balantidium coli.
• Diagnosis is established by detection of the egg in feces
and rectal biopsy.
• Praziquantel is the drug of choice.
KEY POINTS OF SCHISTOSOMES
• Schistosomes are dioecious, sexes a re separate.
• Habitat: In the mesenteric venous plexus (S. mansoni and
S. japonicum) and vesical, and prostatic venous plexus (S.
haematobium).
• Leaf-like unsegmented body with two cup-like suckers with
delicate spines.
• Intestine is bifurcated (inverted Y-shaped}.
• Male is broader than female.
• They produce elongated nonoperc ulated eggs containing
ciliated embryo, miracidium.
• Definitive host: Man.
• Intermediate host: Freshwater snails.
• Infective form: Fork-tailed cercariae.
• Clinical features: Swimmer's itch, Katayama fever, hematuria
and portal hypertension.
• Diagnosis: Detection of eggs in urine or stool, biopsy, imaging,
and detection of antigen and antibody.
• Treatment: Praziquantel is the drug of choice.
• Prophylaxis: Avoidance of bathing in infected water and
e radication of snail.
Schistosoma Mekongi
this species first recognized in 1978 is found in Thailand and
Cambodia, along the Mekong river.
• lt is closely related to S. japonicum but is slightly smaller
and round.
• Man and dog a re the definitive host.
• Man acquires infection in the same way as in S. japonicum.
• HepatosplenomegaJy and asci.tes are the common clinical
finding.
• HERMAPHRODITIC FLUKES: LIVER FLUKES
The adult forms of all hermaphroditic flukes infecting man
reside in the lumen of the biliary, intestinal, or respiratory
tracts. This location gives the flukes suitable protection from
host defense mechanisms and also facilitates dispersal of
eggs to the environment.
• Flukes inhabiting the h uman biliary tract are Clonorchis
sinensis, Fasciola hepatica, less often Opisthorchis species,
and rarely, Dicrocoelium dendriticum.
Fascio/a Hepatica
Common Name
Sheep liver fluke.
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History and Distribution
F. hepatica was the first trematode that was discovered more
than 600 years ago in 1379 by Jehan de Brie.
• It was named by Linnaeus in 1758.
• It is the largest and most common liver fluke found in
man, however its primary host is the sheep and to a less
extent, cattle.
• It causes the economically important disease, \"liver rot';
in sheep.
• It is worldwide in distribution, being fo und mainly in
sheep-rearing areas.
• In India, few cases reported from North India and North
Eastern part of India including Uttar Pradesh (UP), Bihar
and Assam.
• F. gigantica is more prevalent in India than F hepatica.
Habitat
The parasite resides in the liver and biliary passages of the
definitive host.
Morphology
Adult worm:
• It is large in size, flat leaf-shaped fluke measuring 30 mm
long and 15 mm broad, gray or brown in color.
• lt has a conical projection anteriorly containing an oral
sucker and is rounded posteriorly (Figs BA and B).
• The adult worm lives in the biliary tract of the definitive
host for many years-about 5 years in sheep and 10 years
in humans.
• Like all other trematodes, it is hermaphrodite.
Egg: The eggs are large, ovoid, operculated, bile-stained and
about 140 µm by 80 µmin size (Box 5 and Fig. 9).
Trematodes: Flukes
Eggs contain an immature larva, the miracidium.
Eggs do not float in saturated solution of common salt.
Eggs of F. hepatica and Fasciolopsis buski cannot be
differentiated.
• Eggs are unembryonated when freshly passed.
Box 5: Parasites with operculate eggs
• Fascia/a hepatica
• Fascia/a gigantica
• Fascia/apsis buski
• C/anarchis sinensis
• Paraganimus westermani
• Gastradiscaides haminis
• Opistharchis felineus
• Opistharchis viverrini
• Heteraphyes heteraphyes
• Diphy/Jabathrium /atum.
Fig. 9: Egg of Fasciola hepatica
Bf\"f<ie---- Oral sucker
/~ ·•,.,,'4'\"-<----- Intestinal
cecum Ventral sucker - -\"'~r=--?-~
Uterus - -F.;'.:~·..,.- _, j ---
Vitellaria
Ovary
Figs 8A and B: (A) Fasciola hepatica; and (B) Specimen showing Fasciola hepatica
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Paniker's Textbook of Medical Parasitology
Life Cycle
Migrates t @~ · /.,.,oo~,ru, o,cy,ts 11)1.•-.=
in duodenum
Adult worm in
bile ducts
Man and other herbivores get
infection by eating aquatic plants
eocysi wlth metace,~,;,.
Man (Definitive host)
Metacercarfa
Water plants
(2nd intermediate
host)
r
ll encysts on aquatic
vegetations lo become
metacercaria
Water
Snail
(1st intermediate
host)
Development within snail
(First intermediate host)
Sporocyst
First generation redia
Second generation redia
Cercariae
Fig. 10: Life cycle of Fasciola hepatica
Egg embryonates
in water and miracidium
escapes out
I
Miracidium ingested / bysoan
F. hepatica passes its life cycle in one definitive host and two
intermediate hosts.
Mode of infection: 'TI1e definitive host, sheep and man, get
infection by ingestion of metacercariae encysted on aquatic
vegetation.
Adult worm lives in the biliary passage of sheep or man. Definitive host Eggs are laid in the biliary passages and are shed in feces. : Sheep, goat, cattle and man.
Intermediate host: Sna ils of the genus Lymnaea and
Succinea. Encystment occurs on aquatic plants, which act as
second intermediate host.
• lhe embryo matures in water in about 10 days and the
miracidium escapes. It penetrates th e tissues of first
intermediate host, snails of the genus Lymnaea (Fig. 10).
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Box 6: Parasites with aquatic vegetations as the source of infection
• Fascia/a hepatica
• Fasciolopsis buski
• Gastrodiscoides hominis
• Watsonius watsoni.
• In snail, the miracidium progresses through the sporocyst
and the first and second generation redia stages to
become the cercariae in about 1-2 months.
• the cercariae escape into the water and encyst on aquatic
vegetation or blades of grass to become metacercariae,
which can survive for long periods (Box 6).
• Sheep, cattle, or humans eating watercress or other
water vegetation containing the melacercaria become
infected.
the metacercariae excyst in the duodenum of the definitive
host and pierce the gut wall to enter the peritoneal cavity.
TI1ey penetrate the Glisson's capsule, traverse the liver
parenchyma, and reach the biliary passages, where they
matureinto the adult worms in about 3-4 months (Fig. 10).
Pathogenicity
Fascioliasis differs from clonorchiasis in that F. hepatica
is larger and so causes more mechanical damage. In
traversing the liver tissue, it causes parenchymal injury.
As humans are not its primary host, it causes more severe
inflammatory response. Some larvae penetrate right
through the liver and diaphragm ending up in the lung.
• In acute phase during the migration of the larva, patients
present with fever, right upper quadrant pain, eosinophilia
and tender hepatomegaly.
In chronic phase, p a tients may develop biliary
obstruction, biliary cirrhosis, obstructive jaundice,
cholelithiasis and anemia. No association to hepatic
malignancy has been ascribed to Jascioliasis.
Occasionally, ingestion of raw liver of infected sheep
results in a condition called halzoun (meaning
suffocation). The adult worms in the liver attach to the
pharyngeal mucosa, causing edematous congestion of
the pharynx and surrounding areas, leading to dyspnea,
acute dysphagia, deafness and rarely, asphyxiation.
However, this condition is more oflen due to pentastome
larvae. Halzoun is particularly common in Lebanon and
other parts of the Middle East and North Africa.
Diagnosis
Stool microscopy: Demonstration of eggs in feces or aspirated
bile from duodenum is the best method of diagnosis. Eggs of
E hepatica and F. buski are indistinguishable.
Blood picture: It reveals eosinophilia.
Trematodes: Flukes
Serodiagnosis:Serological tests such as immunofl uorescence,
ELISA, immunoelectrophoresis and complement fixation are
helpful in lightly infected individuals for detection of specific
antibody. ELISA becomes positive within 2 weeks of infection
and is negative after treatm ent. In chronic fascio liasis,
Fasciola coproantigen may be detected in stool.
Imaging: Ultrasonography, computed tomography (CT)
scan, endoscopic retrograde cholangiopancreatography
(ERCP) and percutaneous cholangiography may be helpful
in diagnosis.
Treatment
Oral triclabendazole (10 mg/ kg once) is the treatment of
choice.
Alternative drug is bithionol (30-50 mg for 10- 15 days).
Prednisolone at a dose of 10- 20 mg/ kg is used to control
toxemia.
Prophylaxis
Fascioliasis can be prevented by:
Health education.
Control of snails.
Proper disposal of human, sheep and cattle feces.
Proper disinfection of watercresses and other water
vegetations before consumption.
KEY POINTS OF FASCIOLA HEPATICA
• Largest and most common liver flu ke.
• Large leaf-shaped with a dorsoventrally flattened body.
• Hermaphroditic parasite.
• Eggs are ovoid, operculated and bile-stained.
• Definitive host Primary definitive host is sheep, but it is also
found in biliary tract of man.
• first intermediate host Fresh water snails (Lymnaea).
• Second intermediate host Aquatic vegetations.
• Infective form: Metacercariae encysted on raw aquatic
vegetations.
• Clinical features: Acute phase-fever, right upper quadrant
pain and hepatomegaly. Chronic phase-biliary obstruction,
obstructive jaundice, cholelithiasis and anemia.
• Diagnosis: Detection of eggs in stool and aspirated bile, USG,
ERCP and ELISA.
• Treatment: Oral triclabendazole or bithional.
• Prophylaxis: Preventing pollution of water with feces and
proper disinfection.
Dicrocoelium Dendriticum
Also known as the \"lancet Duke\" because of its shape, D.
dendriticum is a very common biliary parasite of sheep and
other herbivores in Europe, North Africa, Northern Asia and
parts of the Far East.
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Paniker's Textbook of Medical Parasitology
Definitive Host
Sheep and other herbivores.
First Intermediate Host
Snails.
Second Intermediate Host
Ants of genus Formica.
• Eggs passed in feces of sheep are ingested by land snails.
• Cercariae appear in slime balls secreted by the snails
and are eaten by ants of the genus Formica, in which
metacercariae develop.
• Herbivores get infected wh en they accidentally eat the
ants while grazing.
• Reports of human infection have come from Europe,
Middle East and China.
• However, spurious infection is more common. ln the
latter, the eggs can be passed in feces for several days by
persons eating infected sheep liver.
• Eurytrema pancreaticum, a related fl uke is commonly
present in the pancreatic duct of cattle, sheep and
monkeys. Occasional human infection has been noticed
in China and Japan.
Clonorchis Sinensis
Common Name
The Chinese liver fluke and oriental liver fluke.
History and Distribution
C. sinensis was first described in 1875 by McConnell in the
biliary tract of a Chinese carpenter in Calcutta Medical
College Hospital.
• Complete life cycle of Clonorchis was worked out by Faust
and Khaw in 1927.
• Human clonorchiasis occurs in Japan, Korea, Taiwan,
China and Vietnam, affecting about 10 million persons.
Habitat
Adult worm lives in the biliary tract and sometimes in the
pancreatic duct.
Morphology
Adult worm: It has a flat, transparent, spatulate body; pointed
anteriorly and rounded posteriorly {Fig. 11).
• It is 10- 25 mm long and 3-5 mm broad.
• The adult worm can survive in the biliary tract for 15 years
or more.
• The hermaphroditic worm discharges eggs into the bile
duct.
/ t-\"\"l't--t-::::,--- Intestinal
ceca
,-,;;.a~r-,.;...:..:,.:.,i,_ Testes
(2)
Fig. 11: Adult worm and egg of Clonorchis sinensis
Eggs: Eggs are flask-shaped, 35 µm by20 µmwith a yellov.rish -
brown (bile-stained) shell.
• It is operculated at one pole and possesses a tiny knob
at the other pole and a small hook-like spine at the other
(Fig. 11).
• Eggs do not float in saturated solution of common salt.
• The eggs passed in feces contain the ciliated miracidia.
Life Cycle
Definitive host: Humans are the principal definitive host, but
dogs and other fish -eating canines act as reservoir hosts.
Intermediate hosts: Two intermediate hosts are required to
complete its life cycle, the first being snail and the second
being.fish.
Infective form: Metacercaria larva.
Mode of infection: Man acquires infection by eating
undercooked freshwater fish carrying metacercariae larvae.
Clonorchis eggs although embryonated do not hatch
in water, but only when ingested by suitable species
of operculate snails (first intermediate host), such as
Parafossarulus, Bulimus, or Alocinma species.
The miracidium develops through the sporocyst and
redia stages to become the lophocercus cercaria with a
large fluted tail in about 3 weeks {Fig. 12).
The cercariae escape from the snail and swim about in
water, waiting to get attached to the second intermediate
host, suitable freshwater fish of the Carp family.
The cercariae shed their tails and encyst under the scales
or in the flesh of the fish to become metacercariae, in
about 3 weeks, which are the infective stage for humans.
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Trematodes: Flukes
Metacercaria excysts in duodenum
Adult worm in
bile ducts
Man (definitive host)
Infected fish ingested by man
t Ingested by snail Fish (2nd
intermediate host) Snail ~
(1st intermediate host) •
Cercaria penetrates under scales of fresh-water
fish and develops into metacercaria Miracidium hatches out in the midgut of snail
Development within snail
(First intermediate host)
1 . Sporocyst
First generation redia
Second generation redia
Cercariae
~
Fig. 12: Life cycle of Clonorchis sinensis
Infection occurs when such fish are eaten raw or
inadequately processed by h uman or other defini tive
hosts. Frozen, dried, or pickled fish may act as source of
infection (Fig. 12).
Infection may also occur through fingers or cooking
uten sils contaminated with the metacercariae during
preparation of the fish for cooking.
• The metacercariae excyst in the duodenum of the
definitive host.
• The adolescaria that come out, enter the common bile
duct through the ampulla of Vater a nd proceed to the
distal bile capillaries, where th ey marure in about a month
and assume the adult form (Fig. 11).
• Adult worms produce an average of 10, 000 eggs per day,
which exit the bile ducts and are excreted in the feces.
The cycle is then repeated.
Pathogenicity
Th e m igration of the lar va up the bile duct induces
desquamation, fo llowed by hyperplasia, a nd sometimes,
adenomatous changes. The smaller bile ducts undergo cystic
dilatation.
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Paniker's Textbook of Medical Parasitology
• The adult worms may obstruct and block the common
bile duct leading to cholangitis.
• Patients in the early stage have fever, epigastric pain,
diarrhea and tender hepatomegaly. This is followed by
biliary colic, jaundice and progressive liver enlargement.
Many infections are asymptomatic.
• Chronic infection may result in calculus formation.
• A few cases go on to biliary cirrhosis and portal
hypertension.
• Some patients with chronic clonorchiasis tend to become
biliary carriers of typhoid bacilli.
• Chronic infection has also been linked with
cholangiocarcinoma.
Diagnosis
1he eggs may be demonstrated in feces (stool microscopy) or
aspirated bile. They do not float in concentrated saline.
• Several serological tests have been described including
complement fixation and gel precipitation but extensive
cross-reactions limit their utility. !HA with a saline extract
of etherized worms has been reported to be sensitive and
specific.
• Intradermal allergic tests have also been described.
Treatment
Drug of choice is praziquantel 25 mg/ kg, three doses in l day.
Surgical intervention may become necessary in cases
with obstructive jaundice.
Prophylaxis
Clonorchiasis can be prevented by:
• Proper cooking of fish.
• Proper disposal of feces.
• Control of snails.
Opisthorchis Species
Some species of Opisth.orchis, which resemble C. sinesis can
cause human infection.
• O.Jelineus, the cat liver nuke, which is common in Europe
and the erstwhile Soviet Union, may infect humans.
• Infection is usually asymptomatic but may sometimes
cause liver disease resembling clonorchiasis.
• 0. viverrini is common in Thailand, where the civet cat
is the reservoir host. Chandler found that 60% of cats
in Calcutta, were infected with the parasite and human
cases have also been reported from India.
• Most of the infected patients have a low worm burden, so
they are asymptomatic.
• Cholangiocarcinoma is epidemiologically related 10 C.
sinensis infection in China and to 0. viverrini infection in
ortheast Thailand.
• the life cycle and other features of Opisthorchis are same
as those of Clonorchis.
• INTESTINAL FLUKES
A number of flukes parasitize the human small intestine.
These include Fasciolopsis buski, 1-/eterophyes, Metagonimus
yokogawai, Watsonius watsoni and Echinostoma. Only one
fluke Gastrodiscoides hominis, parasitizes the hwnan large
intestine.
Fasciolopsis Buski
Common Name
Giant intestinal fluke
History and Distribution
It was first described by Busk in 1843 in the duodenum of an
East Indian sailor, who died in London.
• Ir is the largest and most common intestinal fluke of man
and pigs.
• Mainly found in China and in Southeast Asian countries.
• In India it occurs in Assam, Bengal, Bihar and Odisha.
• Prevalence rate is as high as 22.4% in India.
• Children are more prone to infection than adults as they
enjoy playing in water.
Habitat
The adult worm lives in the duodenum or jejunum of pigs
and man.
Morphology
Adult worm: The adult is a large fleshy worm, 20-75 mm long
and 8- 20 mm broad (Fig. 13) and 0.5-3 mm in thickness.
• Largest trematode infecting humans: Fasciolopsis buski
• Smallest trematode infecting humans: 1-/eterophyes
• It is elongated ovoid in shape, with a small oral sucker
and a large acetabulum. lt has no cephalic cone as in F.
hepatica (Fig. 14).
• The adult worm has a lifespan of about 6 months.
• The two intestinal caeca do not bear any branches
(Fig. 14).
Eggs:
• The operculated eggs are similar 10 those of F. hepatica
(Fig. 15).
• Eggs are laid in the lumen of the intestine in large
numbers, about 25,000 per day.
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Fig. 13: Specimen showing Fascio/opsis buski
Life Cycle
F. buski passes its life cycle in one definitive host and two
intermediate host.
Definitive host: Man and pigs. Pigs serve as a reservoir of
infection for man.
First intermediate host: Snails of the genus Segmentina.
Second intermediate host: Encystment occurs on aquatic
plants, roots of the lotus, bulb of the water chestnut which act
as second intermediate host.
Infective form: Encysted metacercariae on aquatic vegetation.
• The eggs passed in feces of definitive host hatch in waler in
about 6 weeks, releasing the miracidia which swim about.
• On coming in contact wilh a suitable molluscan
intermediate host, snails of the genus Segmentina,
miracidia penetrates its tissues to undergo development
in the next few weeks as sporocyst, first and second
generation rediae and cercariae (Fig. 16).
• The cercariae, which escape from the snail, encyst on
the roots of the lotus, bulb of the water chestnut, water
hyacinth and on other aquatic vegetations.
• When they are eaten by man, the metacercariae excysts
in the duodenum, become allached to the mucosa and
develop into adults in about 3 months (Fig. 16).
Pathogenesis
The pathogenesis of fasciolopsiasis is due to traumatic,
mechanical and toxic effects.
• Larvae that attach to the duodenal and jejuna! mucosa
cause inflammation and local ulceration. Intoxication
and sensitization also account for clinical illness.
Trematodes: Flukes
mlf---\"<----Oral sucker
Pharynx
.::=~7--'~+-Uterus
cr-CTT~ -+--~ Ovary
Vilellaria
-et-- Intestinal
cecum
Fig. 14: Fasciolopsis buski
Fig. 15: Egg of Fasciolopsis buski
• In h eavy infections, the adult worms cause partial
obstruction of the bowel, malabsorption, protein-losing
enteropathy and impaired vitamin 812 absorption.
• ·n,e initial symptoms are diarrhea and abdominal pain.
• Toxic and allergic symptoms appear usually as edema,
asciLes, anemia, prostration and persistent diarrhea.
• Paralytic ileus is a rare complication.
Laboratory Diagnosis
History of residence in endemic areas suggests the diagnosis,
which is con.fumed by demonstration of the egg in feces or of
the worms after administration of a purgative or anthelmintic
drug.
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Paniker's Textbook of Medical Parasitology
Metacercaria excysts in
duodenum and
attaches to intestinal wall
in small intestine I Man (Definitive host) Operculated
Man and other herbivores
eats aquatic plants with
encysted metacercariae
'
Water plants
(2nd intermediate
host)
Metacercaria
I encysts on aquatics
vegetations
beco metacercaria
Free swimming
cercaria escapes
from snail into
water
Water
Snail
(1st intermediate
host)
egg in feces
Egg embryonates
In water and miracidium
\"\"T'\"'
Miracidium ingested
by snail
Development within snail
(First intermediate host)
Sporocyst
First generation redia
Second generation redia
Cercariae
Fig. 16: Life cycle of Fasciolopsis buski
Treatment
Drug of choice is praziquantel.
• Hexylresorcinol and tetrachloroe rh ylene have also been
fow1d useful.
Prophy laxis
• Treatment of infected persons.
• Proper disinfection of water vegetables, by hot water.
• Prevention of polution of water resources from human
and pig feces.
• Community-based praziquantel treatment can be used to
control infection.
• Control of snails.
Heterophyes heterophyes
This is the smallest trematode parasite of man.
• 1he infection is prevalent in the Nile delta, Turkey and in
the Far East.
• The worm has been reported in a dog in India.
• The adult worm lives in the small intestine and has a
lifespan of about 2 months.
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Definitive Hosts
Humans, cats, dogs, foxes and other fish-eating mammals.
First Intermediate Host
Snails of the genera Pirone/la and Cerithidea.
Second Intermediate Host
Fishes, such as the mullet and tilapia; encystment occurs in
fishes.
• Man acquires infection by eating raw or undercooked
fishes containing metacercaria.
• In the small intestine, it can induce mucous diarrhea and
colicky pains.
• Ectopic lesions may occur as granulomas in myocardium,
brain and spinal cord.
• Diagnosis is based on the finding of a minute operculated
egg in the stool.
Drug of Choice
Praziquantel.
Metagonimus Yokogawai
It is found in the Far East, Northern Siberia, Balkan states and
Spain.
Definitive Hosts
Humans, pigs, dogs, cats and pelicans.
First Intermediate Host
Freshwater snail.
Second Intermediate Host
Fish.
• Definitive hosts are infected by eating raw fish containing
the metacercariae.
• Pathogenic effects consist of mucous diarrhea and ectopic
lesions in myocardium and central nervous system as in
heterophyasis.
Drug of Choice
Praziquantel.
Watsonius Watsoni
• This trematode infects various primates in Asia and Africa.
onnal host is the monkey.
• Eggs are operculated.
Trematodes: Flukes
• Infection occurs by ingestion of water plants containing
metacercariae.
• Diagnosis, clinical features, treatment and prophylaxis is
same as that of Heterophyes.
Echinostoma
Echinostomes are medium-sized fl u kes causing small
intestinal infection of rats and dogs.
• Seen in Japan, Philippines and all along the Far East.
• The characteristic feature is a crown of spines on a
disc surrounding the oral sucker, justifying its name
Echinostoma which means \"spiny mouth''.
• Its eggs resemble those of Pasciolopsis. Mild infections are
asymptomatic, but diarrhea and abdominal pain follow
heavy infection.
• E. ilocanum is the species usually seen in h uman
infections.
Gastrodiscoides Hominis
C. hominis is the only fluke inhabiting the human large
intestine (Fig. 17).
lt was discovered by Lewis and McConnell in 1876 in the
cecum of an Indian patient.
lt is a common human parasite in Assam. Cases have also
been reported from Bengal, Bihar and Odisha.
lt also occurs in Viemam, Philippines and some parts of
erstwhile Union of Soviet Socialist Republics (USSR).
The adult worm is pyriform, with a conical anterior end
and a discoidal posterior part. It is about 5- 14 mm long
and 4-6 mm broad.
• The eggs are operculated and measure 150 µm by 70 µm.
Fig. 17: Specimen showing Gastrodiscoides hominis
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Paniker's Textbook of Medical Parasitology
Definitive Host
Man, pigs and monkey. Pigs are the reservoir hosts.
First Intermediate Host
Snails.
Second Intermediate Host
Aquatic plants.
• The miracidia invade th e tissues of the intermediate
molluscan host.
• The cercariae encyst on water plants. Infected persons
develop mucoid diarrhea.
• Man and animals become infected by feeding upon
vegetations harboring the metacercaria.
Drug of Choice
Praziquamel. Tetrachloroethylene is also useful in treatment.
• LUNG FLUKES
Paragonimus Westermani
Common Name
Oriental lung fluke.
History and Distribution
P. westermani was discovered in 1878 by Kerbert in the I ungs
of a Bengal tiger captured in India that died in the zoological
gardens at Amsterdam.
• The parasite is endemic in the Far East-Japan, Korea,
Taiwan, China and South East Asia- Sri Lanka and India.
• Th ere are about 40 species of Paragonimus that infect
mammals.
• ln India, cases have been reported from Assam, Bengal,
TamiI Nadu, Kerala, Manipur, Sikkim, Arunachal Pradesh
and Nagaland.
• P. westermani is the most common species infecting
human.
• Endemic foci of P. westermani and P heterotremus are
present in Manipur.
• lt is an important hwnan pathogen in Central and South
America.
Morphology
Adult worm: The adult worm is egg-shaped about 10 mm
long, 5 mm broad and 4 mm thick and reddish-brown in color
(Fig. 18).
• The integument is covered with scale-like spines.
Fig. 18: Paragonimus westermani morphology
lntestinal---<c::,:..,;;.s,;;..;_, Spine
cecum
~ ::r--H'T\"r:~ 5-.:-- Ventral
sucker
Uterus ---'\"\"\"\"r..:,.-++-~ ~
,...,...__ ...,_ ~~~- Ovary
Testes --.;::-::,,\"61;=+-i---:-,r-
(two)
Fig. 19: Paragonimus westermani
• It has an oral sucker placed anteriorly and a ventral sucker
located towaJds the middle of the body (Fig.19).
• It has two unbranched intestinal caeca which end blindly
in the caudal area.
• They have a lifespan of up to 20 years in humans.
Egg: The eggs are opercu!ated, golden-brown in color and
about 100 µm by 50 µmin size (Fig. 20).
• They are unemb1yonated when freshly laid.
Habitat
Adults worms live in the lungs, usually in pairs in cystic
spaces that communicate with bronchi (Table 4).
Life Cycle
Definitive host: Man. Besides humans, other definitive
hosts include cats, tigers, leopards, foxes, dogs, pigs, beavers,
mongoose, and many other crab-eating mammals and
domestic animals.
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Fig. 20: Egg of Paragonimus westermani
Table 4: Helminths present in lung
Trematode
Paragonimus
westermani
Cestode
Echinococcus granulosus
Dirolilario immitis
Nematode
Capillaria aerophila
First intermediate host: Freshwater snail, belonging lo the
genera Semisulcospira and Brotia.
Second intermediate host: Freshwater crab or crayfish.
Infective form: Metacercariae encysted in crab or crayfish.
Mode of infection: Man acquires infection by eating
undercooked crab or crayfish containing metacercariae.
• The adull worms live in the respiratory tract of the
definitive host.
• Unembryonated eggs escape into the bronchi and are
coughed up and voided in sputum or swallowed a nd
passed in feces {Fig. 21).
• The eggs mature in about 2 weeks and hatch to release
free-swimming miracidia.
• These infect the.first intermediate molluscan host, snails
belonging to the genera Semisulcospira an d Brotia.
• Cercariae that are released from the snails after several
weeks are microcercus, having a short stumpy tail.
• The cercariae that swim about in streams are drawn into
the gill chambers of the second intermediate crustacean
host, crabs or crayfish (Fig. 21).
• lhey encyst in the gills or muscles as metacercariae.
• Definitive hosts are infected when they eat such crabs or
crayfish raw or inadequately cooked.
• The metacercariae excyst in the duodenum and the
adolesca riae penetrate the gut wall, reaching the
abdominal cavity in a few hours.
Trematodes: Flukes
They then migrate up through the diaphragm into the
pleural cavity and lungs finally reaching in the vicinity of
the bronchi, where they develop into adult worms in 2-3
months {Fig. 21).
The worm is hermaphroditic but usually it takes 2 for
fertilization.
Sometimes, the migrating larvae lose their way and reach
ectopic sites such as the mesentery, groin and brain.
Pathogenicity and Clinical Features
Pulmonary features: In the lungs, the worms lie in cystic
spaces surrounded by a fibrous capsule formed by the host
tissues.
The cysts, about a centimeter in diameter are usually in
communication with a bronchus.
Inflammatory reaction to the worms and their eggs lead to
peribronchial granulomatous lesions, cystic dilatation
of the bronchi, abscesses, pneumonitis and eosinophilia.
• Patients present with cough, chest pain and hemoptysis.
The viscous sputum is speckled with the golden-brown
eggs. Occasionally, the hemoptysis may be profuse.
• Chronic cases may resemble pulmonary tuberculosis.
Extrapulmonary f eatures: The clinical features depend on
the site of involvement.
Extrapulmonary infections are more common in P.
mexicanus, P. heterolremus and rare in P. westermani.
• Abdomi1tal paragonimiasis: Occasionally the fluke
migrates to liver and intestinal wall resu lting in enlarge
liver, abdominal tenderness and bloody diarrhea.
• Cerebral paragonimiasis: Encapsulated cyst of
Paragonimus is found in brain and spinal cord.
Symptoms include headache, fever, paralysis, visual
disturbances and convulses seizures.
Laboratory Diagnosis
Microscopy: Demonstration of the eggs in sputum or feces
provides definitive evidence. Sputum examination should be
repeated for 7 consecutive days.
Serology: Complement fixation test is positive only during
and shortly after active infection, while the intradermal test
remains positive for much longer periods.
Parasite-specifi c immunoglob ulin E (lgE) and
antiparagonimus antibodies can be detected in serum.
• Indirect hemagglutination and ELISA tests are highly
sensitive. they become negative within 3-4 months after
successful treatment.
• Serology is of particular importance in egg-negative cases
and in cerebral paragonimiasis.
Imaging: Chest X-ray reveals abnormal shadows (nodular,
cystic, ring infiltrative) in the middle and lower lung field.
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Paniker's Textbook of Medical Parasitology
Metacercaria
excysts in duodenum
Man gets infected by ingestion
of ray
' 00y-<00ke<J c,ab
Man
Crab
(2nd intermediat
Metacercaria host) develops
inside the viscera, muscles.
and gills or crab
Water
Cercaria
penetrates
crab
Free-swimming
cercaria escape
from snail into
water
Snail
(1st intermediate
host)
Development within snail
(First intermediate host)
Sporocyst
2 First generation redia
Second generation redia
Cercariae
Fig. 21: Life cycle or Paragonimus westermani
Prophylaxis
Egg embryonates in water
and free-swimming
miracidium released
• Computed tomography scan of chest also helps in diagnosis
of pulmonary lesions and cerebral lesions. \"Soap-bubble''
like appearance may be seen in cerebral cysts. Adequate cooking of crabs and crayfish and washing lhe
hands after preparing them for food.
Treatment
• Praziquantel (25 mg/ kg TDS for 1-2 days) is the drug of
choice.
• Bithionol and niclofolan are also effective in treatment.
• Treatment of infected persons.
• Disinfection of sputum and feces.
• Eradication of molluscan hosts.
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KEY POINTS OF PARAGONIMUS WESTERMAN(
• Adult worm is egg-shaped, reddish, brown and covered with
scale-like spine.
• Habitat: Cystic spaces in the lung.
• Eggs are oval, operculated and golden brown.
• Definitive hosts: Man and domestic animals.
• First intermediate host: Snails of genera Semisu/cospira
(Melania species).
• Second intermediate host Crab or crayfish.
• Infective form: Encysted metacercaria in crab or crayfish.
• Clinical features: Peribronchial granuloma and cystic dilation
of bronchi. Dyspnea, hemoptysis, pneumonitis, bronchiectasis,
abscess and pneumothorax. Extrapulmonary lesions in brain
and intestine.
• Diagnosis: Ova in sputum, X-ray and CT scan of chest, CFT,
IHA and ELISA.
• Treatment: Praziquantel is the drug of choice.
• Prophylaxis: Adequate cooking of crabs and crayfish,
eradication of molluscan hosts and t reatment of infected
persons.
REVIEW QUESTIONS
Describe briefly:
a. General characters of trematodes
b. Classification of trematodes
c. General characters of schistosomes
Short notes on:
a. Clonorchis sinensis
b. Fasciolopsis buski
c. Paragonimus
d. Opisthorchis species
Describe morphology, life cycle and laboratory diagnosis of
a. Fascia/a hepatica
b. Schistosoma haematobium
Differentiate between Schistosoma haematobium, S. mansoni
and S.japonium.
MULTIPLE CHOICE QUESTIONS
Which of the following flukes is carcinogenic
a. Fascia/a
b. Clonorchis
c. Paragonimus
d. Gastrodiscoides
Trematodes: Flukes
Organism causing biliary tract obstruction
a. Ancylostoma duodenale
b. Clonorchis sinensis
c. Strongyloides stercoralis
d. Enterobius vermicularis
All float in a saturated salt solution except
a. Clonorchis sinensis
b. Fertilized eggs of Ascaris
c. Larva of Strongyloides
d. Trichuris trichiura
Terminal spined eggs are seen in
a. Schistosoma haematobium
b. Schistosoma mansoni
c. Schistosoma japonicum
d. Clonorchis sinensis
Largest trematode infecting humans
a. Fascia/a hepatica
b. Fasciolopsis buski
c. Schistosoma haematobium
d. Paragonimus westermani
The second intermediate host of Fasciola hepatica is
a. Snail
b. Fresh water fish
c. Crab
d. Aquatic plants
Schistosoma japonicum resides in
a. Superior mesenteric vein
b. Inferior mesenteric vein
c. Small intestine
d. Gallbladder
All of the following lead to bloody diarrhea except
a. Schistosomajaponicum
b. Entamoeba histolytica
c. Schistosoma mansoni
d. Schistosoma haematobium
Answer
b
b
b
d
a
a
a
d
P:182
CHAPTER 13
Nematodes: General
Features
• INTRODUCTION
Nematodes are said to be the most worm-like of all helminths. This is because they generally resemble the common
earthworm in appearance, which is considered to be the
prototype of \"worms''. However, taxonomically earthworms
are not nematodes as they are segmented worms of the
Phylum Annelida.
• ematodes are elongated, cylindrical, unsegmented
worms with tapering ends. The name \"nematode\" means
\"thread-like'; from \"nema\" meaning \"thread''.
Unlike u·ematodes and cestodes, all of which are parasitic,
most nematodes are free-living forms found in soil and
water.
• Several species are parasites of plants and are of great
economic importance. Many nematodes parasitize
invertebrate and vertebrate animals.
• The largest nwnber of helminthic parasites of humans
belong to the class of nematodes. There are an estimated
500,000 species of nematodes.
• GENERAL CHARACTERISTICS
They a re cylindrical, or filariform in shape, bilaterally
symmetrical with a secondary triradiate symmetry al the
anterior end.
The adults vary greatly in size, from about a millimeter
(Strongyloides stercoralis) to a meter (Dracuncu.lus
medinensis) in length. Male is genera lly smaller th an
female and its posterior end is curved or coiled ventrally.
Their body is covered wirh a tough outer cuticle, which
may he smooth, striated, bossed, or spiny. 1he middle
layer is hypodermis and the inner layer is the somatic
muscular layer. They move by sinuous jlexion of the body.
• The body cavity is a pseudocele, in which all the viscera
are suspended.
Th e digestive system is comple te, consisting of an
anteriorly placed mouth leading to the esophagus,
which characteristically varies in shape and structure in
different groups. The intestine is lined with a single layer of
Box 1: Types of female nematodes
• Oviparous (laying eggs):
Unsegmented eggs: Ascaris, Trichuris
Segmented eggs: Ancy/ostoma, Necator
Eggs containing larvae: Enterobius
• Viviparous (producing larvae): Trichinella, Wuchereria, Brugia,
Dracunculus.
• Ovoviviparous (laying eggs containing fully formed larvae, which hatch
out immediately): Strongylaides.
columnar cells and leads to the rectum, opening through
the anus. In the male, the rectum and the ejaculatory
duct open into the cloaca.
• Nematodes have simple excret01y and nervous systems.
• The nematodes are diecious, i.e. the sexes are separate.
• The male reproductive system consists of a single delicate
tubule differentiated into testis, vas deferens, seminal
vesicle and ejaculatory duct, which opens into the cloaca.
It also includes copulatory structures such as spicules or
bursa or both.
• The female reproductive system consists of the ovary,
oviduct, seminal receptacle, uterus and vagina.
• Female nematodes may produce eggs (oviparous) or
larvae (viviparous). Some lay eggs containing larvae,
which immedia tely hatch out (ovoviviparous) (Box l).
• LIFE CYCLE
The life cycle of nematodes consists typically of four larval
stages and the aduJt form. The cuticle is shed while passing
from one stage to the other.
• Man is Lhe optimum host for all. the nematodes. They
pass their life cycle in one host, except the superfamilies
Filarioidea and Dracunculoidea, where two hosts are
required. Insect vectors and Cyclops constitute the
second hosts in these superfamilies, respectively.
cmatodes localize in rhe intestinal tract and their eggs
pass our with the feces of the host. They undergo few
developmental changes before they enter new host.
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• MODES OF INFECTION
• By ingestion of:
eggs Ascaris, Enterobius, Trichuris
Larvae within intermediate host: Dracunculus
Encysted larvae in muscle: Trichinella
• By penetration of skin: Ancylostoma, Necator, Strongyloides
• By blood-sucking insects: Filariae
• By inhalation of dust containing eggs: Ascaris, Enterohius.
• CLASSIFICATION
Nematodes can be classified on the basis of the habitat of
the adult worm (Table 1) and zoologically (Table 2).
Zoological Classification
• Phylum: Nemathelminthes (Nematoda)
• Class: Nematoda which is divided into two subclasses
based on the absence or presence of\"phasmids'; which are
caudal chemoreceptors. The two subclasses were earlier
called Aphasmidia and Phasmidia, but now have been
renamed as Adenophorea and Secernentea, respectively
(Table 3).
Detailed zoological classification of nematodes is given
in Table 2.
Table 1: Classification of nematodes on the basis of the habitat of
adult worms
Intestinal human nematodes
Small intestine
• Ascaris /umbricoides (common
roundworm)
• Ancylostoma duodenale (Old
World hookworm)
• Necatoramericanus (American or
New World hookworm)
• Strongyloides stercoralis
• Trichinella spiralis
• Capil/aria philippinensis
Large Intestine
• Trichuris trichiura (whipworm)
• Enterobius vermicularis (thread or pinworm)
Somatic human nematodes
Lymphatics
• Wuchereria bancrofti
• Brugia malayi
• Brugia timori
Skin/subcutaneous tissue
• Loa Joa
• Onchocerca volvulus
• Dracunculus medinensis
(guinea worm)
Mysentery
• Mansonella ozzardi
• Mansonella perstans
Conjunctiva
• Loa loa
Nematodes: General Features
• LARVA MIGRANS
The life cycles of most nematodes parasitizing humans
include larval migration through various tissues and organs
of the body. Sometimes the larvae appear to lose their way
and wander around aimlessly. Th is condition is known as
larva migrans.
• This is generally seen when human infection occurs with
nonhuman species of nematodes. In such infections,
the worm is unable to w1dergo normal development and
complete its life cycle.
• Abnormal or arrested larval migration may also sometimes occur when human parasitic nematodes infect
immune persons. The immunity is sufficient to prevent
the normal progression of infection.
• Larva migrans can be classified into cutaneous or visceral types, depending on whether the larval migration
takes place in the skin or in deeper tissues (Table 4).
Cutaneous Larva Migrans
This condition also known as creeping eruption (also called
ground itch) is caused by nematode larvae that infect by skin
penetration.
Etiology
The most common cause is nonhuman species of hookworm
(Ancylostoma braziliense and A. caninum) (Table 5).
Pathogenesis
Parasite eggs are passed in the feces of infected animals into
the soil, where the larvae hatch out.
• Infection with these hookworms of dogs and cats is
acquired from soil contaminated with excreta of these
animals.
• On coming in contact with human skin, the larvae
penetrate the skin to cause infection.
• Between a few days and a few months after the initial
infection, the larvae migrate beneath the skin.
• In normal animal host, the larvae are able to penetrate the
deeper layers of the skin by reaching there via circulation.
• On ce they enter intestine, they mature sexually and lay
more eggs that are then excreted to repeat the cycle.
• However, in a human host, which is an accidental host
for the parasite, the larvae are unable to penetrate the
basement membrane to invade the dermis, so that the
disease remains confined to the outer layers of the skin.
Clinical Features
• The lar vae produce itching papules, wh ich develop
into serpiginous tunnels in the epidermis. With the
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Paniker's Textbook of Medical Parasitology
Table 2: Zoological classificat ion of nematodes
Subclass Order Su erfamily Family Genus Species
Adenophorea/ Enoplida Trichinelloidea (anterior part of Trichinellidae • Trichinefla • T. spiralis
Aphasmidia (no body narrower than posterior) Trichuridae • Trichuris T. trichiura
phasmids, no caudal • Capillaria C. philippinensis
papillae in male, eggs C. aerophila
usually unsegmented C. hepatica
w ith polar plugs or
hatching in uterus)
Secernentea/ Rhabditida Rhabditoidea (alternation Strongyloididae Strongyloides S. stercoralis
Phasmidia (phasmids of free-living and parasit ic
present, numerous generations, parasitic females
caudal papillae) parthenogenetic)
Strongyl ida • Ancylostomatoidea (prominent • Ancylostomatida e • Ancylostoma • A. duodenale
buccal capsule with teeth or • Metastrongylidae • Necator • N. americanus
cutting plates) • Angiostrongylus • A. cantonensis
• Metastrongyloidea (tissue
parasites, inconspicuous buccal
capsule, have intermediate
hosts)
Ascaridida Ascaridoidea (large worms of gut • Ascarididae • Ascaris • A. lumbricoides
lumen, mouth has three lips)
Oxyurida Oxyuroidea (male has no
caudal bursa, short stout body,
esophagus has prominent bulb,
eggs planoconvex, embryonate
in uterus)
Spirurida • Filarioidea (tissue parasites,
viviparous, insect vector)
• Dracunculoidea (very long
female and small male,
viviparous, larvae escape from
ruptured uterus)
• Gnat hostomatoidea (spiny
body w ith bulbous head)
Table 3: Differences in subclass adenophorea and secernentea
Adenophorea
Phasmid (sensory structure)
Excretory system
Caudal papillae
Infective stage of larva
Absent
Without lateral canals
Absent or few
First larval stage
movements of the larva in the skin, the lesion also shifts,
hence the name \"creeping eruption''. Scratching may
lead to secondary bacterial infection.
• Transient creeping eruptions may be produced sometimes by the human hookworm, Necator americanus.
Gnathostomiasis and sparganosis may produce larva
• Anisakidae
Oxyuridae
• Onchocercidae
• Dracunculid ae
• Gnathostomatidae
• Anisakis
Enterobius
• Wuchereria
• Brugia
• Dirofilaria
• Loa
• Mansonefla
• Onchocerca
• Dracunculus
• Gnathostoma
Secernentea
Present
With lateral canals
Numerous
Third larval stage
• A. simplex
E. vermicularis
• W. bancrofti
• 8.malayi
• D. conjunctivae
• D.immitis
• L. loa
• M. perstans
• M. ozzardi
• M. streptacerca
• 0. volvulus
• D. medinensis
• G. spinigerum
migrans, where the lesions are deeper, subcutaneous or
in the muscles. Loeffler's syndrome may occur in onefourth to one-half of the cases.
• A rapidly moving lesion is produced by Strongyloides
stercoralis particularly in immune persons. This is known
as larva currens.
P:185
Table 4: Animal nematodes infecting man
Visceral larva migrans
It is a syndrome caused by nematodes that are normally parasitic for
nonhuman host species
In human, these nematode larvae do not develop into adult worms,
but, instead, migrate through host tissues and elicit eosinophilic
inflammation
Common causes:
• Toxocara canis (dog roundworm)- most common
• Taxocara cati (cat roundworm)
• Ascaris suum (pig ascaris)
• Angiosrrongylus cantonensis
• Gnathostoma spinigerum
• Anisakis simplex
• Baylisascaris procyonis
Table 5: Etiological agents (cutaneous larva m igrans)
Zoophilic nematode
• Ancylosroma braziliense
• Ancylostoma caninum
• Gnathostoma spinigerum
• Dirofrlaria
• Spirometra
• Uncinaria stenocephala
• Bunostomum phlebotomum
Human nematode
• Strongy/oides srercoralis
• Necator americanus
• Loa loo
Human trematode
• Ectopic infection with Fasciola and
Paragonimus
Nonhelmenthic agents
• Flies of genus Hypoderma and
Gastrophilus
Nematodes: General Features
Cutaneous larva migrans
• It is a serpiginous skin eruption caused by burrowing larvae of animal
hookworms (usually the cat and the cat hookworm)
• The larvae hatch from eggs passed in dog and cat feces and mature in
the soil. Humans become infected after skin contact with contaminated
soil. After larvae penetrate the skin, erythematous lesions form along the
tortuous tracks of their migration. It is also known as creeping eruption
Common causes:
• Ancylosroma braziliense (hookworm of wild and domestic dogs and cats)
• Ancylosroma caninum (dog hookworm found in Australia)
• Uncinaria srenocephala (dog hookworm found in Europe)
• Bunostomum phlebotomum (cattle hookworm)
Table 6: Etiological agents (visceral larva migrans)
Zoophilic nematode Nonhuman nematode
• Taxocara canis • Filariaspp.
• Toxocara cat/ • Dirofrlaria immitis
• Angiostrongylus cantonensis • Brug/a pahangi
• Brugia patei
• Angiostrongylus costaricensis
• Anisakis
• Gnathostoma spinigerum
Human nematode
• Ascaris lumbricoides
• Strongyloides stercoralis
Visceral Larva Migrans
This condi tion is caused by the migration of larvae of
nonhuman species of nematodes that infect by the oral route.
• Creeping myiasis is caused by fli es of the genus
Hypoderma an d Eastrophilus. Etiology
The most common cause is the dog ascarid, Toxocara
canis and less often the cat ascarid, T. cati. Visceral larva
migrans may also be caused by Anisakis, which are large
ascarid parasites of marine animals and also by Gnathostoma
spinigerum, Angiostrongylus cantonensis. Human nematodes
like A. lumbricoides and S. stercoralis may produce visceral
larva migrans, when they get lost in ectopic sites (Table 6).
• Ectopic infections with Fasciola and Paragonimus may
produce creeping lesions on abdominal wall.
Diagnosis
Eosinophilia is rare and occurs only when Loefler's syn -
drome develop.
• Serological tests are not developed.
• On biopsy, larvae are rarely found in the skin lesion.
11iagnosis is based mainly on clinical features.
Treatment
Th iabendazole is useful in treatment. When the lesions are
few, freezing the advancing part of the eruption with ethyl
chloride is effective.
Pathogenesis
When the infective eggs present in the soil contaminated by
dog and cat feces are ingested, the larvae hatch in the small
intestine, penetrate the gut wall, and migrate to the liver.
• They may remain there or migrate to other organs such as
lungs, brain, o r eyes.
• In humans they do not develop into ad ults, but induce
granulomatous lesions, which cause local damage.
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Paniker's Textbook of Medical Parasitology
Table 7: Difference between cutaneous and visceral larva migrans
Cutaneous larva migrans
Skin
Visceral larva migrans
Tissue involved
Infecting organism
Portal of entry
Eosinophilia
Serodiagnosis
Treatment
Mostly by nonhuman nematodes
Penetration of skin
Various organs of body like liver, lungs and eyes
Mainly by dog and cat (Toxocara spp.)
Ingestion of infected eggs
Clinical Features
Mild
Not developed
Thiabendazole
Clinical manifestations depend on the sites affected and the
degree and duration of infection.
• As children are more likely to swallow dirt, this condition
is much more frequent in them.
• Fever, hepatomegaly, pneumonitis, hyperglobulinemia
and pica are th e common findings.
• Patients may develop neurological disturbances (neural
larva migrans) and endophthalmitis (ophthalmic larva
migrans).
• Marked leukocytosis occurs with persistently high
eosinophilia.
Diagnosis
Serological tests, such as passive hemagglutination, bentonite
nocculation, microprecipitation, and more specifi cally,
enzyme-linked immunosorben t assay (ELISA) have been
developed for Lhe diagnosis of toxocariasis ( vi sceral larva
migrans).
Treatment
Diethylcarbamazine (DEC), 100 mg TDS for 3 weeks in an
adult, kills the larva and arrest the disease. Thiabendazole may
be useful in treatment/Prednisolone should be administered
concurrently either topicall y or systemically.
Prophylaxis
Deworming of household pets helps in prevention by limiting
the contam ination of soil.
Differences between cu taneous and v isceral lar va
migrans are given in Table 7.
Persistent high
Well developed
Diethylcarbamazine and prednisolone
KEY POINTS OF CUTANEOUS AND
VISCERAL LARVA MIGRANS
• Sometimes larvae lose their way and wander around
aimlessly in human body, this condition is known as larva
migrans (cutaneous or visceral).
• Mainly caused by nonhuman species of nematodes (zoophilic
helminths), but occasionally by nonhelminthic agents like
mite and larvae of fly (myiasis).
• Man acquires the infection as an accidental host.
• Abnormal migrations also occur sometimes in human
nematodes.
• The helminths are unable to complete their development and
life cycle in man and are arrested at some level in skin or
other organs like lung, liver, etc.
• Pathogenesis: Due to mechanical damage and host's
inflammatory response against parasitic antigen.
• Clinical manifestations: Depend on route of entrance, sites
affected, and degree and duration of infection.
• Diagnosis: Based mainly on clinical features, skin biopsy and
serology.
• Treatment: Symptomatic and specific therapy with antihelminthics.
REVIEW QUESTIONS
Describe briefly:
a. General characters of Phylum Nematoda
b. Systematic classification of nematodes
Short notes on:
a. Classification of nematodes based on habitat
b. Cutaneous larva migrans
c. Visceral larva migrans
d. Viviparous nematodes
e. Larva currens
Differentiate between class Adenophorea and Secernentea.
4 . Enumerate the etiological agents of cutaneous and visceral
larva migrans.
P:187
MULTIPLE CHOICE QUESTIONS
All of the following nematodes are oviparous except
a. Ascaris
b. Ancylostoma
c. Trichinella
d. Enterobius
Nematoda residing in large intestine
a. Necatar
b. Trichinella
c. Strongyloides
d. Trichuris
All of the following are somatic nematodes except
a. Loaloa
b. Capillaria phi/ippinensis
c. Onchacerca vo/vulus
d. Brugia malayi
Most common cause of visceral larva migrans
a. Ancylostoma braziliensis
b. Anisakis simplex
c. Strongyloides stercora/is
d. Toxocara canis
Nematodes: General Features
Cutaneous larva migrans is due to
a. Ancyclostoma braziliensis
b. Wuchereria bancrofti
c. Brugia malayi
d. Dracuncu/us medinensis
A teenager who plays with dogs developed skin rash,
eosinophilia, and an enlarged liver and spleen for 1 year. The
most likely cause of this infection is
a. Trichinosis
b. Schistosomiasis
c. Toxoplasmosis
d. Visceral larva migrans
Answer
c 2. d 3. b 4. d 5. a 6. d
P:188
CHAPTER 14
• INTRODUCTION
• Trichinella spiralis, tissue n ematode, is the causative
agent of trichinosis.
• The name Trichinella is derived from the minute size of
the adult (Greek trichos-hair, ella suffix for diminutive,
spiralis refers to th e spirally coiled appearance of larvae
in muscles).
• COMMON NAME
Trichina worm.
• HISTORY AND DISTRIBUTION
• It was first observed in 1821 in the muscles of a patient at
autopsy by James Paget, who was then a first year medical
student at St Bartholomew's Hospital, London.
• Owen, in 1835, described the encysted larval form in
muscles and named it Trichinella spiralis.
• Virchow discovered its life cycle in 1859.
• The major source of human infection was shown to be the
consumption of inadequately cooked pork.
• Trichinosis is recognized as an important public health
problem in Europe and America, but is mu ch less
common in the tropics and oriental countries.
• Human trichinosis had not been recorded in India till
1996, when the first case was reported from Punjab.
• HABITAT
Adult worms live deeply buried in the mucosa of small
intestine (duodenum or jejunum) of pig, bear, rat, or man. The
encysted larvae are present in the striated muscles of these
hosts. There are no free-living stages.
• MORPHOLOGY
Adult Worm
The adult T. spiralis, a small white worm just visible to the naked
eye, is one of the smallest nematodes infecting humans.
The male measures about 1.5 mm by 0.04 mm and the
female about 3 mm by0.06 mm (twice the length of male).
The anterior half of the body is thin and pointed, welladapted for burrowing into the mucosa! epithelium
(Fig. 1).
1he posterior end of the male has a pair of pear-shaped
clasping papillae (termed as claspers), one on each side
of the cloaca/ orifice that it uses to hold the female worm
during mating (Fig. 1).
Fig. 1: Adult worms of Trichinel/a spiralis (male and female)
P:189
• The female worm is viviparous and discharges larva
instead of eggs.
• The lifespan of the adult worm is very short. The male
worm dies soon after fertilizing the female and the female
dies after 4 weeks to 4 months (16 weeks), the time
required for discharging the larvae.
Larvae
Th e larva becomes encysted in the striated muscle fiber
(Fig. 2) and at the time of encystment measures 1 mm in
length by 36 µm in diameter.
The larva in the cyst is coiled and hence, the name spiralis.
Trichinella Cyst
• Cysts are ovoid 400 mcm by 250 mcm in size.
• The cyst is fo rmed by the tissue reaction around th e
encapsulated larvae.
• Cysts develop preferentially in muscles relatively poor
in glycogen and in hypoxic environment. Therefore, the
diaphragm, biceps, muscles of jaw, ex:traocular muscles,
neck, and lower back, which are constantly active, are the
ones mostly affected.
Cysts are more abundant near the sites of attachment of
muscles to tendons and bones than in other parts. They lie
longitudinally along the muscle fibers.
The deltoid being easily accessible, is chosen for taking
diagnostic muscle biopsies.
• The larva remains infective inside the cyst fo r years and
eventually, most become calcified and die.
• LIFE CYCLE
TrichinelLa is a parasite that has a direct life cycle, which
means it completes all stages of development in one host.
But only a single cycle occurs in one host and for
continuation of the cycle and maintenance of the species,
it is necessary for the infection to be transmitted to another
host of the same species or of different species (Fig. 3).
• Optimum host: Pig.
• Alternate host: Man.
• Infection can pass from-pig-to-pig (facilitated by the
custom of feeding pigs with untreated household garbage,
which may contain bits of pork with infective cysts), ratto-rat and pig-to-rat (Table 1).
• Man is the dead-end of the parasite, as the cysts in human
muscles are unlikely to be eaten by another host.
• Infective form: Encysted larva found in the muscles of
pigs and other animals (Fig. 2).
Mode of infection: Man acqu ires infection main ly
by eating raw or undercooked pork or inadequately
Trichinella Spira/is
Fig. 2: Encysted larva in muscles; infective stage
processed sausages or other meat products containing
the viable larvae.
• When such meat is eaten without adequate cooking, the
cysts are digested by the gastric juice and viable larvae
are released (excystation) in the stomach, duodenum and
jejunum.
• The larvae immediately penetrate the mucosal epithelium
• They m oult four times and rapidly develop into adults,
either male or female, by the 2nd day of infection. withnin
5 days, they become sexually mature.
• The male dies after fertilizing the female. The fertilized
females start releasing motile larvae by the 6th day of
infection.
Larvae continue to be discharged during the remaining
part of the lifespan of the female worm, which ranges
from 4 weeks to 4 months.
• Each female gives birth to approximately 1,000 larvae.
• \"These larvae enter the intestinal lymphatics or mesenteric
venules and are transported in circulatio n to different
parts of the body.
• They get deposited in the muscles, central nervous system
and other sites. The larva dies in most other situations,
except the skeletal muscles, where it grows.
Deposition in the muscles occurs mostly during the
2nd week of infection. Larval development in muscles
takes place during the next 3 or 4 weeks.
• Within 20 days after entering the muscle celJs, the larvae
become encysted. A muscle cell carrying larva of T. spiralis
is called as a nurse cell.
Encysted larvae lie parallel to the muscles of host.
• Encysted larva can survive for months to years. In man,
die life cycle ends here (Fig. 3).
• Smoking, salting or drying the meat does not destroy the
infective larvae. Prolonged freezing (20 days in a normal
freezer or at - 20°C for 3 days) decontami nates the meat.
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Paniker's Textbook of Medical Parasitology
~
Larvae
?
released
\"
from
8 . ~ due to the action of ;g
digestive enzymes ;?
~ \"
_ _,__ Striated
muscle
Man Pig
Adult worms
in small intestine
Larvae die in
other tissues,
except striated
Encysted larva in
striated muscle
muscles
\
Man dead-end
(cycle ends)
encystmen
Larva undergoes
t in muscle
~
and nurse cell-larva
complex formed
t
Fig. 3: Life cycle of Trichinella spiratis
Table 1: Parasites with source of infection
Pork Fish
• Taenia solium • Diphyllobathrium latum
• Trichine/Ja spiralis • C/onorchis sinensis
• Sarcocystis suihominis • Metagonimus yokogawai
• Heterophyes spp.
• Gnathostoma spp.
Beef
• Taenia saginata
• Sarcocystis hominis
• Toxoplasma gondii
• PATHOGENICITY AND CLINICAL FEATURES
The disease caused by T. spiralis is called trichinosis.
• The manifestations vary from asymptomatic infection,
which is very common, to an acute fatal illness, which is
extremely rare.
• The pathology and clinical features vary according to the
stage in the life cycle of the worm (Table 2).
• DIAGNOSIS
Diagnosis of trich inosis can be made by direct and indirect
methods.
Direct Methods
• Detection of spiral larvae in muscle tissue by performing
muscle biopsy. Deltoid, biceps, gastrocnemius, or
pectoralis muscles are usually selected for biopsy (Box 1).
• Detection of adult worms and larvae in the stool during
the diarrheic stage.
• Xenodiagnosis: For xenodiagnosis, biopsy bits are fed to
laboratory rats, which are killed in a month or so, later.
The larvae can be demonstrated more easily in the
muscles of such infected rats (Flow chart 1 ).
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Trichinella Spira/is
Table 2: Stages in the life cycle of Trichinella spiralis (in man)
Stage of intestinal invasion: Stage of muscle invasion: Stage of encystation:
First stage Second stage Fina/stage
Pathology The stage begins with the ingestion of raw
pork containing Infective larvae and ends
with the larvae invading the intestinal
epithelium and developing into adult
The stage begins when new infective larvae are released
from the adult female and ends with the deposition
of the larvae in the muscles. Myositis and basophilic
granular degeneration of muscles occurs in this stage
This stage occurs only in
striated muscle. The infective
larvae become encysted in
t his stage
Clinical Malaise, nausea, vomiting, diarrhea, Fever, myalgia, periorbital edema, weakness of affected
muscle, hemorrhage in subconjunctiva and new beds
(splinter hemorrhages), myocarditis (if heart muscles are
involved) and encephalitis (if central nervous t issue is
involved). Eosinophilia is a constant feat ure of this stage.
The stage is seen 1-4 weeks after infection
All symptoms subside
features abdominal cramps. Onset w ithin 2- 30
hours of ingestion of infective food
Box 1: Muscle biopsy
• Muscle biopsy specimen is collected for demonstration of spiral larvae.
• Specimen: Deltoid, biceps, gastrocnemius, or pectoralis.
• At least 1 gram of muscle should be taken for biopsy, preferably near tendon insertion.
• Examination technique: Muscle fibers are digested with trypsin and mounted on a glass slide and examined under microscope. Young larvae may be
digested and missed during such examination.
A teased preparation of muscle tissue is prepared in a drop of saline solution and it is squeezed between two glass slides.
Muscle tissue is stained with safranin.
+
Direct methods
Flow chart 1: Laboratory diagnosis of Trichinella spiralis
Laboratory diagnosis
I
+
Indirect methods I ... Muscle biopsy
Alternative method
for definitive
diagnosis.
Demonstrates larva
in muscle tissue
Xenodiagnosis History
History of
consumption
of raw or
inadequately
cooked pork-
•
Serology
• Radiological
examination
Calcified cysts can
be detected on
X-ray
Stool examination
May demonstrate
adult worms and
larvae
Indirect Methods
Detection of
antibody by:
• ELISA
(Confirmatory test)
2 weeks earlier
• Bentonite
flocculation test
• Latex fixation test
Blood examination
Differential blood
count shows
eosinophilia (20-95%)
• Raised levels of
muscle enzymes,
including creatine
phosphokinase
• Serology:
Bachman
intradermal test
The test remains
positive for years
after infection
PCR
Uistoty of consumption of raw or inadequately cooked or
processed pork, about 2 weeks earlier along with a recent
episode of gastroenteritis.
There is massive hypergammaglobulinemia with
elevated serum immunoglobulin E (IgE).
Blood examination: It shows eosinophilia (20-95%).
T. spiralis antibody can be detected by enzymelinked immunosorbenc assay (ELISA) test using
TSL-1 secreting antigens obtained from the infective
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Paniker's Textbook of Medical Parasitology
stage larvae. Bentonite flocculation test and latex
fixation test for demonstration of antibodies have
also been widely used. A positive test indicates recent
infection.
• Bachman intra.dermal test: It uses a 1:5,000 or 1:10,000
dilution of the larval antigen. An erythematous wheal
appears in positive cases within 15- 20 minutes. The rest
remains positive for years after infection.
• Radiological examination: Calcified cysts may be
demonstrated on X-ray examination.
• Molecular methods like multiplex polymerase chai n reaction (PCR) are now being used for species identification
of Trichinetla (Flow chart 1).
• TREATMENT
• Mild cases: Supportive treatment consisting of bedrest,
analgesics and antipyretics.
• Moderate cases: Albendazole 400 mg BID for 8 days or
mebendazole 200-400 mg TID for 3 days, then 400 mg
TlD for 8 days.
• Severe cases: Add glucocorticoids like prednisolone to
albendazole or mebendazole.
Note: Mebendazole and albendazole are active against enteric
stage of the parasite, but their efficacy against encysted larva
has not yet been completely demonstrated.
• PROPHYLAXIS
• Proper cooking of pork and other meat likely to be infected.
• The most effective method is to stop the practice offeeding
pigs with raw garbage.
Extermination of rats from pig farms-the spread of
infection.
KEY POINTS OF TR/CH/NELLA SP/RAUS
• One of the smallest nematodes infecting humans (1.5-3 mm).
• Entire life cycle is passed in one host.
• The fema le worm is viviparous.
• Optimum host: Pig.
• Alternate host: Man. Man is t he dead-end for parasite.
• Infective form: Encysted larvae in the striated muscles of
pigs and other animals.
• Larvae remain encysted tightly coiled in striated muscles in
human body.
• Muse/es commonly involved: Diaphragm, pectoralis, deltoid,
biceps and gastrocnemius.
• Pathogenesis: Myositis and basophilic degeneration of the
muscles.
• Clinical features: Malaise, diarrhea, periorbital edema,
muscle weakness, myocarditis, encephalitis.
• Diagnosis: Muscle biopsy for larvae, stool examinat ion for
adult worm or larvae, xenodiagnosis, Bachman intradermal
test, ELISA, X-ray for calcified cyst, PCR.
• Treatment: Albendazole and mebendazole along with
corticosteroids (in case of severe infection).
REVIEW QUESTIONS
Name the various intestinal nematodes and describe briefly
the life cycle of Trichinella.
Write short notes on:
a. Trichinella cysts
b. Laboratory diagnosis of Trichinel/a spiralis
MULTIPLE CHOICE QUESTIONS
Larva found in muscle is
a. Trichinella spiralis
b. Ancylostoma duodena/e
c. Trichuris trichiura
d. Enterobius vermiculoris
Which of the following is not a neuroparasite
a. Taenia solium
b. Acanthamoeba
c. Naegleria
d. Trichinel/a spiralis
Which of the following is viviparous
a. Strongyloides stercoralis
b. Trichinella spiralis
c. Enterobius
d. Ascaris
Best site for taking biopsy for diagnosis of trichinellosis is
a. Deltoid muscle
b. Diaphragm
c. Pectoralis major
d. Liver
Bach man's test is done to diagnose infections with
a. Schistosoma japonicum
b. Trichinella spiralis
c. Trichuris trichiura
d. Ancylostoma duodenale
The larval form of Trichinella can be destroyed by
a. Smoking of meat
b. Deep freezing of meat
c. Drying of meat
d. Salting of meat
Answer
a 2. d 3. b 4. a 5. b 6. b
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CHAPTER 15
• INTRODUCTION
• The name Trichuris means a \"hair-like tail\" (Greek
trichos- hair, oura-tail). lhis name is not quite correct
because it is th e anterior end of the worm that is hair-like
an d not th e tail. the name whipworm is more apt as the
thick posterior part resembles the stock and thin anterior
end resembles the lash of a whip.
• The helminth causes trichiuris in humans, an intestinal
infection caused by invasion of colonic mucosa.
• COMMON NAME
Whipworm.
• HISTORY AND DISTRIBUTION
• Trichuris trichiura, the human whipworm, was first
described by Linnaeus in 1771.
• The antiquity of the whipworm as a human parasite is
indicated by the demonstration of its eggs in colonic
contents of a young man, who died on the Alps some
5,300 years ago and whose well-preserved body was
discovered in 1990.
• It is worldwide in distribution, but is much more common
in the tropics. The infection is widespread in tropical
Afri ca, South America and South-cast Asia. Trichuris
infection is found throughout India.
• Some 800 million people are estimated to be infected
with this worm.
• Whil e wh ipwo rm in fection is extremely frequen t,
whipworm disease is relatively rare.
• HABITAT
T. trichiura lives in the large intestine (Box 1 ). The adult
worms are found attached to the wall of the cecum and less
commonly to the vermiform appendix, colon and anal canal.
• MORPHOLOGY
Adult Worm
The male worm is 30- 45 mm long, while th e female is slightly
larger, about 40-50 mm.
.
.
.
.
The worm is flesh -colored. In shape, it resembles a whip,
with the anterior three-fifth (3/ 5) thin and thread-like and
the posterior two-fifth (2/ 5) th ick and fleshy, appearing
like the handle of a whip (Figs l A and B).
The a nenuated a nterior po rtion, which contains the
capilla ry esophagus, is embedded in the mucosa. The
posterior part contains the intestines and reproductive
organs.
The posterior end of the male is coiled ventrally, while
the hind end of the female is straight, blunt and rounded
(Figs IA and B).
1l1e worm has a lifespan of 5-10 years .
Egg
the egg has a characteristic appearance.
It is brown in color being bile-stained.
• It has a triple shell, the outermost layer of which is
stained brown.
• It is barrel-shaped and about 50 mcmlong and 25 mcmwide
in the middle, with a projecting mucus plug at each pole
containing an unsegmented ovum (Figs 2A and B). The
plugs are colorless.
• The egg floats in saturated salt solution (Boxes 2 and 3).
Box 1: Nematodes present in large intestine
• Enterobius vermicularis
• Trichuris trichiura
• Oesophagosromum spp.
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5
~E!i!i:•==S!!!!!!!!~i!!t~•~,i~11~~~~~ Comma or \;::> ~ ~~~t~hri~eednd
Very thin
IP.I anterior
Iii portion
Figs lA and B: (A) Adult Trichuris trichiura worms (male and female); and (B) Specimens of male and female whipworm
Mucous plug
•
Figs 2A and B: Egg of Trichuris trichiura. (A) As seen under microscope; and (B) Schematic diagram
Box 2: Helminths whose eggs float in saturated salt solution
• Enterobius vermicularis
• Ancylostoma duodenale
• Necatoramericanus
• Ascaris lumbricoides
• Trichuris trichiura
• When freshly passed, the egg contains an unsegmented
ovum. At this stage, it is not infective for humans.
• The fertilized female lays about 5,000 eggs per day.
• LIFE CYCLE
Natural host: Man. No intermediate host is required.
Box 3: Helminths whose eggs do not float in the saturated solution
• Eggs of Taenia solium and Taenia saginata
• Eggs of all intestinal flukes
• Unfertilized eggs of Ascaris lumbricoides
Infectiveform: Embryonated eggs containing rhabditiform
larva.
• Adult female worm lives in large intestine, worm lays eggs
which are discharged in feces.
• The egg undergoes development in soil, optimally under
warm, moist, shady conditions, when the infective
rhabditiform larva develops within the egg in 3-4 weeks.
At lower temperatures, this may be delayed for 3 months
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Trichuris Trichiura
~ Develops into adult worms
Passed down
to cecum
I
liberated larva
in small intestine
Larva liberated through
one of the poles of
egg in small intestine
Man
Egg passed in
feces
,,,_---------------------- Ingested -----
embryonated egg
Man acquires infection by with infective
consuming food and water rhabditiform larva
contaminated with
embryonated egg
Fig. 3: Life cycle of Trichuris trichiura
or more {Fig. 3). These embryonated eggs a re infective
to man.
Mode of transmission: Infection occurs in humans when
the mature embryonaled eggs containing the infective
larvae are swallowed in contaminated food or waler.
The eggs hatch in the small intestine and Lhe larva, which
emerges through the pole of Lhe egg, passes down into
the cecum.
In about 2-3 months, they become man1re adults and lie
embedded in the cecal wall, with the thread-like anterior
portion piercing the mucosa and the thick posterior end
projecting out.
the gravid adult female lays eggs, which are discharged in
feces and the cycle is repeated (Fig. 3).
The entire life cycle can be passed in one host, from the
ingested infective egg to the development of the adults
and the release of their eggs in feces. But for transmission
of infection to other hosts and perpetuation of the species,
the egg has to undergo development in the soil and then
infect another person.
• Huma ns are the only natural host for T. trichiura, but
morphologically similar worms are found to infect pigs
and some monkeys.
• Eggs start appearing in feces usually about 3 months after
infection.
• PATHOGENICITY AND CLINICAL FEATURES
Infection with T. trichiura (trichuriasis, whipworm infection, or
trichocephnliasis) is asymptomatic, except when the worm
load is heavy. Disease may result either due Lo mechanical
effects or allergic reaction.
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CHAPTER 16
• INTRODUCTION
Normand (1876) observed minute cylindrical worms in the
diarrheic feces and intestinal walls of some French soldiers
in Cochinchina. 111ese were named Strongyloides stercoralis
(strongylus- round, eidos-resembling, stercoralis-fecal).
• HISTORY AND DISTRIBUTION
• It is found mainly in the warm moist tropics, but may also
occur in the temperate regions. It is common in Brazil,
Columbia, and in the Far East-Myanmar, Thailand,
Vietnam, Malaysia and Philippines.
• Another species S. fullerborni is widely prevalent in
African monkeys. It infects pygmies in the forests of Zaire
and Zambia. It also ca uses human infection in Papua
New Guinea. Trichostrongylus, a parasite of sheep and
goats, seen in Africa a nd Asia (including India), may
cause human infection, which is usually asymptomatic
(Table 1).
• HABITAT
The adult worm is found in the small intestine (duodenum
and jejunum) of man (Box l).
• Largest nematode known to cause human infection:
Ascaris lumbricoides.
• Smallest nematode known to cause human infection:
Strongyloides stercoralis.
• MORPHOLOGY
Adult Worms
Female Worm
111e female worm is thin, transparent, about 2.5 mm long and
0.05 mm wide (Fig. 1).
• It has a cylindrical esophagus occupying the anterior
one-third of the body and the intestines in the posterior
Table 1: Difference between fi lariform larva of hookworm and
Strongyloides
Hookworm Strongy/oides
• Esophagus extended up to 25% • Esophagus extended up to 40%
of the total body length of the total body
• Sheathed • Nonsheathed
• Tail: Pointed • Tail: Forked
Box 1: Nematodes present in small intestine
• Strangyloides stercoralis
• Ascaris lumbricoides
• Ancylostoma duodenale
• Necator americanus
• Trichinella spiralis
• Trichostrongylus spp.
• Capillaria philippinensis.
two-thirds, opening through the anus situated ventrally, a
little in front of the pointed tail tip.
• The reproductive system contains paired uteri, vagina
and vulva. The paired uteri lead to the vulva situated at
the junction of the middle and posterior thirds of the
body. In the gravid female, the uteri contain thin-walled
transparent ovoid eggs, 50 µm by 30 µmin size.
• the worm is ovoviviparous.
• the individual worm has a lifespan of 3 or 4 months, but
b eca use it can cause a utoinfection, the infection m ay
persist for years.
Male Worm
the male worm is shorter and broader than female measuring
0.6- 1 mm in length and 40-50 mcmin width.
• The copulatory spicules, which penetrate th e female
during copulation, are located on each side of the
gubernaculum (Fig. 1).
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• They are not seen in human infection because they do
not have penetrating power, therefore do not invade the
intestinal wall.
Eggs
Eggs are conspicuous within the uterus of gravid female.
• Each uterus contains 8- 10 eggs arranged anteroposteriorly
in a single row (Fig. 1).
• They are oval and measure 50-60 mcmin length and
30-35 µrn in breadth (Fig. 2).
• As soon as the eggs are laid, they hatch out to rhabditiform
larva (first stage larva). Thus, it is the larva and not the
egg, which is excreted in feces and detected on stool
examination and not egg.
Larva
Rhabditiform Larva (L 1 Stage) (Fig. 3A)
t his is d1e first stage of larva. Eggs hatch ' 10 form Ll larva
in the smaJI intestine.
• It is the most common form of the parasite found in the
feces.
• It measures 0.25 mm in length, with a relatively short
muscular double bulb esophagus (fig. 3B).
• The Ll larva migrates into the lumen of the intestine and
passes down the gut to be released in feces.
Filariform Larva (L3 Stage)
This is the third stage of larva.
• Ll larva moults twice to become the L3 larva.
• It is long and slender and measures 0.55 mm in lengd1
with a long esophagus of uniform width and notched tail
(Fig. 3C).
• It is the infective stage of the parasite to man.
Strongyloides Stercoralis
Anal opening f
o'
Vulval
Esophagus
Fig. 1: Adult worm (male and female)
Rhabditiform
larva
Fig. 2: Egg of Strongyloides stercoralis
\"Double bulb\"
esophagus
Long, slender
esophagus
ll
Notched tail
Fig s 3A to C: Larvae of Strongyloides stercoralis. (A and B) Rhabditiform Larva (Courtney Dr Anita Nandi); (C) Filariform larva
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Paniker's Textbook of Medical Parasitology
Filariform
larva
penetrates
skin of man
(Definitive host
• LIFE CYCLE
In the small intestine,
~ larvae mature into
/ adult worms
Larva enters circulation and
via heart, lungs, respiratory
tree, and esophagus,
reach small intestine
I
~ .... .... .... ........
\"'lt,t, o,;,
Penetrates the ~Ct,·
perianal skin 1
01)
Man
Soil
Direct cycle
Rhabditiform larva directly
metamorphose
to infective filariform larva in soil
\"Double bulb\"
esophagus
Indirect cycle
Egg in soil
Fig. 4: Life cycle of Strongy/oides stercoralis
Infective Form
Filariform larva.
• Mode of infecti.on:
,,
Adult female embedded in
the mucosa of small intestine
.,, ,,, ,,,.
/ ,,,
~
Egg containing
the larva
J
0 ) habditiform
larva immediately
released and
passes out
in feces
The life cycle of S. stercoralis is complex because of the
multiplicity of pathways through which it can develop. It is
unique among human nematodes as it has a parasitic cycle
and a Cree-living soil cycle, in which it can persist for long
periods in soil by feeding on soil bacteria, p assing through
several generations (Fig. 4 and Flow chart 1).
Penetration of skin by th e third stage filariform larva,
when a person walks barefoot
Natural Host
Man, although dogs and cats are foun d in fected with
morphologically indistinguishable strains.
Autoinfection (Box 2).
• The adult female worm is found in the human intestine
embedded in the mucosa of the duodenum and upper
jejunum.
• Since only the female worms are seen in the intestine, it
was earlier believed that they are parthenogenetic and
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Flow chart 1: Life cycle of Strongyloides stercoralis
Female worm in intestine lays eggs
l
Rhabditiform larvae hatch out
Develop into filariform
larvae in ut
Pass through feces
into soil
Passed in
feces Direct cycle
Develop into
filariform
larvae in soil
Indirect cycle
Develop into
free-living males
and females
Penetrate
gut wall
\"internal
reinfection\"
Autoinfection
by piercing
perianal and
perinea! skin
Parasitic phase I
Box 2: Autoinfection
Penetrate
skin of
feet to infect
another host
Females lay eggs!
l
Rhabditiform
larvae hatch out
Become Develop into
filariform free-living
larvae which males and
infect humans females --...--....
Free-living phase I
• External autoinfection: S. srercora/1s has a cycle of autoinfection.
Here the rhabditiform larvae mature into the infective third stage
larvae during their passage down the gut. These filariform larvae
cause reinfection by piercing the perianal and perinea! skin during
defecation. The larvae wander in the dermis of the perianal region
for sometime, causing a radiating perianal creeping eruption, a form
of cutaneous larva migrans. They ultimately enter the lymphatics or
venules and are carried to the right heart and the lungs to complete
the life cycle as earlier.
• Internal autoinfection: In this type of autoinfection, seen typically
in immunodeficient hosts, the rhabditiform larvae mature into the
infective filariform larvae in the bowel itself. The filarlform larvae
penetrate the deeper layers of the intestine, to reach the mesenteric
venules and are carried in circulation to complete the life cycle. This
mode of autoinfection is called internal reinfection. It may lead to very
heavy infection causing serious and sometimes even fatal illness.
can produce offsprings without being fertilized by the
male. But, it has now been established that parasitic males
do exist. They can be demonstrated in experimentally
infected dogs. They are not seen in human infections
because they do not invade the intestinal wall and so are
eliminated from the bowel soon after the females begin to
oviposit. However, the majority of females are probably
parthenogenetic.
• The eggs laid in the mucosa hatch immediately, releasing
rhabditiform larva.
Strongy/oides Stercoralis
The rhabditiform larva migrates into the lumen of the
intestine and passes down the gut to be released in feces.
• The rhabditifo rm larva may even metamorphose into
.filariform larva during passage through the bowel.
These filariform larvae may penetrate colonic mucosa or
peria nal skin without leaving the host and going to the
soil, thus providing a source of autoinfection. 1l1is ability
to cause a utoinfection expla ins the persistence of the
infection in patients fo r long periods, even 30-40 years,
after leaving the endemic areas.
The rhabditiform larva voided with the feces may undergo
two types of development in the soil (Flow chart 1):
Direct development
Indirect development.
Direct development: l h e rhabditiform larva on reaching
the soil moults twice to become the infective filariform
larva.
Each rhabditiform larva gives rise to one filariform
larva. When a person walks barefoot on soil containing
the infective filariform larvae, they penen·ate the skin
and enter the circulation.
The larvae are carried along the venous circulation to
the right side of the heart and to the lungs.
Here, they escape from the pulmonary capillaries
into the alveoli, migrate up the respiratory tract to
the pharynx, and are swallowed, reaching their final
destination, small intestine.
In the intestine, they mature into adult parasitic
females and males in 15-20 days. Female worms then
burrow into the mucosa of the intestine and lays eggs.
1he rhabditiform larvae hatch out immediately and
enter into lumen of the bowel. They are excreted in
the feces and thus, the life cycle is repeated.
• Fre e-living phase/ indirect d e v elopment: Th e
rhabditiform larva passed in stools develop in moist soil
into free-living males and females.
They mate in soil.
The fertilized female lays eggs, which hatch to release
th e next generation of rhabditiform larvae.
These may repeat the free-living cycle or may develop
into the fila riform larvae, which infect humans and
initiate the parasitic phase.
• PATHOGENICITY AND CLINICAL FEATURES
Strongyloidiasis (infection caused by S. stercoralis) is generally
benign and asymptomatic. Blood eosinophilia and larvae in
stool being the only indications of infection.
Sometimes it may cause clinical manifestations, which
may be severe and even fatal, particularly in those with
defective immune response.
• The clinical disease may have cutaneous, pulmonary an d
intestinal manifestations.
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Paniker's Textbook of Medical Parasitology
Cutaneous Manifestations
There may be dermatitis, with erythema and itching at the
site of penetration of the filariform larva, particularly when
large numbers of larvae enter the skin.
• In those sensitized by prior infection, there may be an
allergic response.
• Pruritus an d urticaria, particularly around th e
perianal skin and buttocks, are sym ptoms of chroni c
strongyloidiasis.
• the term larva currens (meaning racing larvae) has been
applied to the rapidly progressing linear or serpiginous
urticaria! tracks caused by migrating filariform larvae.
These often follow autoinfection and start perianally.
Pulmonary Manifestations
When the larva escape from the pulmonary capillaries into
the alveoli, small hemorrhages may occur in the alveoli and
bronchioles.
Bronchopneumonia may be present, which may progress
to chronic bronchitis and asthmatic symptoms in some
patients.
• Larva of Strongyloides may be found in the sputum of
these patients.
Intestinal Manifestations
The symptoms may resemble those of peptic ulcer or of
malabsorption syndrome.
• Mucus diarrhea is often present. In heavy infection, the
mucosa may be honeycombed with the worm and there
may be extensive sloughing, causing dysenteric stools.
• Other manifestations are protein-losing enteropathy and
paralytic Beus.
Hyperinfection
In debilitated individuals a nd particula rly in those with
cellular immune defects, extensive internal reinfection takes
place, leading to an enormous number of adult worms in the
intestines and lungs and larvae in various tissues and organs.
This is known as hyperinfection.
• Severe malnutrition, lepromatous leprosy, lymphoreticular
malignancies, acquired immunodefi ciency syndrome
(AIDS), immunosuppressive drugs and other situations,
in which cell-mediated immunity is defective, predispose
co this condition.
Hyperinfection is an importan t hazard of steroid therapy
and other instances of prolonged immunosuppression as
in transplant patients.
During hype ri nfection, the filariform larvae may enter
in to arterial circulation and lodge in various organs, e.g.
heart, lungs, brain, kidney, pancreas, liver a nd lymph
nodes. Manifestations depend on the sites affected.
Brain abscess, meningitis and peritonitis are major fatal
complications.
It has been reported that circulating Strongyloides larvae
may carry intestinal bacteria, causing septicemia.
• LABORATORY DIAGNOSIS
Microscopy
Direct wet mount of stool: Demonstration of the
rhabditiform larvae in freshly passed stools is the most
important method of specific diagnosis. Larvae found in
stale stools have to be differentiated from larvae hatched
from hookworm eggs (Flow chart 2).
Concentration methods of stool examinatum: Stool may
be concentrated by formol-ether concentration method
or Baermann's funnel gauze method and examined for
larvae more efficiently. Baermann's test requires a special
appa ratus and relies on th e principal that larva will
actively migrate out of the feces on a wire mesh covered
with several layers of gauge.
Larvae may sometimes be present in sputum or duodena]
asp.iratcs and jejuna] biopsies.
Flow chart 2: Laboratory diagnosis of Strongyloides stercoralis
Microscopy
Direcl wet mount of stool:
Demonstrates rhabditiform
larva ( definitive diagnosis}
Stool concentrations methods:
• Formol ether concentration
• Baermann's funnel gauze
Demonstration of larva in
sputum or duodenal aspirates
or jejunal biopsies
Laboratory diagnosis
•
Stool culture
Done when larvae are
scanty in stools
Methods used:
• Agar plate culture
• Charcoal culture
method
I
Serology
Done using
S/rongy/oides or
filarial antigens
Methods used:
• Complement fixation
• Indirect hemagglutination
• ELISA
Radiological
imaging
Abbreviations: ELISA, enzyme-linked immunosorbent assay; lgE, immunoglobulin E
•
Blood
examination
• Peripheral
eosinophilia
• Raised serum
lgE levels
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Stool Culture
When larvae are scanty in stools, diagnosis may be facilitated
by stool culture.
Culture techniques used:
• Agar plate culture
• Cha rcoal culture method.
• The larvae develop into free-living forms and m ultiply in
charcoal cultures set up with stools. Large number of freeliving larvae and adult worms can be seen after 7-1 O days.
• Serial examinations and the use of agar plate detection
method improves the sensitivity of stool diagnosis.
Serology
Serological tests have been described, using Strongyloides or
filarial antigens.
• Complement fixaLion, indirect hemagglutination and
enzyme-linked immunosorbent assay (ELISA) have been
reported.
• Enzyme-linked immunosorbent assay has a sensitivity of
95% and should be used when microscopic examinations
a re negative.
• Limitations of serological tests:
Larval antigens are not freely available.
There is extensive cross-reactions with other
helminthic infections.
Imaging
Radiological appearances in intestinal and pulmonary
infection are said to be characteristic and helpful in diagnosis.
Others
• Pe ripheral eosinophilia (>500/ cu mL of blood) is a
constant finding. However, in severe hyperinfection,
eosinophilia may sometimes be absent.
Total serum immunoglobulin (lg) E antibody level is
e levated in more than half of the patients (Flow chart 2).
• TREATMENT
All cases of strongyloidiasis, whether symptomatic or not,
should be treated to prevent severe invasive disease.
• lvermcctin (200 mg/ kg daily for 2 days) is more effective
than albendazole (400 mg daily for 3 days).
• For disseminated srrongyloidiasis, treatment with
ivermectin should be extended for at least 5-7 days.
• PROPHYLAXIS
Strongyloidiasis can be prevented by:
• Prevention of contamination of soil with feces.
Strongyloides Stercoralis
Avoiding contact with infective soil and contaminated
surface waters.
• Treatment of all cases.
KEY POINTS OF STRONGYLOIDES STERCORAL/5
• It is the smallest nematode infecting man.
• Adult worm lives in duodenum and jejunum of man.
• Females are ovoviviparous.
• Egg is ovoid, thin-walled and transparent.
• Natural host: Man (optimal host).
• Infective form: Third stage filariform larva.
• Mode of transmission: Penetration through the skin by the
filariform larva in soil. Autoinfection can occur.
• Clinical features: Generally benign and asymptomatic,
but may cause cutaneous, pulmonary and intestinal
manifestations.
• Diagnosis: By demonstrating larva or adult females in stool
or by demonstrating larval antigen by serological methods
like ELISA.
• Technique for stool concentration: Baermann's technique
and formal-ether concentration.
• Techniques for stool culture: Agar plate culture, charcoal
culture.
• Treatment: Drug of choice is ivermectin or a lbendazole.
REVIEW QUESTIONS
Classify intestinal nematodes and describe briefly the life cycle
of Strongyloides.
Short notes on:
a. Strongyloides
b. Hyperlnfection
c. Larva currens
Differentiate between filariform larvae of hookworm and
Strongyloides.
MULTIPLE CHOICE QUESTIONS
Parasites penetrating through skin for entry into the body are
a. Trichinella
b. Strongyloides
c. Roundworm
d. Trichuris trichiura
Larval form of the following parasites is found in stool except
a. Strongyloides stercoralis
b. Ancylostoma duodenale
c. Ascaris lumbricoides
d. Necator americanus
Autolnfection is seen with
a. Cryptosporidium
b. Strongyloides
c. Giardia
d. Gnathostoma
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Paniker's Textbook of Medical Parasitology
The term larva currens is used for migrating larva of
a. Stronglyloides stercoralis
b. Necator americanus
c. Ancylostoma duodonale
d. Hymeno/epis nano
Smallest nematode known to cause infection in man is
a. Trichinella spiralis
b. Strongyloides stercoralis
c. Ancy/ostoma duodenale
d. Trichuris trichiura
Infective form of Strongyloides is
a. Eggs
b. Rhabditiform larva
c. Filariform larva
d. Cercaria larva
Baerman n's funnel gauze method is used for detection of larva
of
a. Necator
b. Strongyloides
c. Ancy/ostoma
d. Ascaris
Strongyloides can be cultured in /by
a. NNN medium
b. Harada Mori method of stool culture
c. Agar plate culture
d. Hockmeyer's medium
Answer
b
b
C
C
b
b
a
C
P:203
CHAPTER 17
• HISTORY AND DISTRIBUTION
Hookworms have been known since very ancient times. 11,ey
have been referred to in the Ebers Papyrus (Circa 1600 BC).
• Two species of hookworms are human parasites:
(l)Ancylostomaduodenaleand(2)Necatoramericanus.
• Ancylostoma duodenale ( Greek ankylos-hooked, stomamouth) was originally described by Dubini in 1843 in
Italy. ·n,e life cycle of the worm was worked out by Looss
in 1898 in Egypt.
• lhe second species Necator americanus was identified by
Stiles in 1902 in specimens obtained from Texas, United
States of America (USA). lhe name literally means the
\"American murderer\" (Latin neca.tor-murderer). It is
called the American or the \" ew World\" hookworm
and A. duodenale the \"Old World\" hookworm. But, it is
believed that N. americanus actually originated in Africa
and was transported to America with the slave trade.
• Hookworm disease is prevalent throughout the tropics
and subtropics. Even though it has been controlled in the
advanced countries, it is estimated that it still affects some
900 million people, causing the loss of about 9 million
liters of blood overall each day (Box 1).
• A. duodenale was prevalent along the Mediterranean
coast of Europe and Africa, in northern India, China and
Japan, while N. americanus was prevalent in Central and
South America, Central and Southern Africa, Southern
India, the Far East and the Southern Pacific region.
Box 1: Conditions favoring hookworm infection
• Presence of infected persons.
• Dispersal of eggs in soi l due to indiscriminate defecation and
inadequate processing of excreta.
• Appropriate environmental factors facilitating development of eggs
in soil, and opportunity for the larva to infect people through their
exposed skin surfaces.
Note: These conditions prevail throughout the year in most parts of the
tropics, but in subtropical areas, these conditions exist only seasonally,
being limited to the warmer months.
• However, in more recent times, movement of infected
persons has blurred the geograp hic differences in
distribution of the two species. For example, A. duodenale
is now commonly seen along with N. americanus in South
India and South East Asia.
• ANCYLOSTOMA DUODENAL£
Habitat
The adult worms live in the small intestines of infected
persons, mostly in the jejunum less often in tl1e duodenum,
and infrequently in rhe ileum.
Morphology
Adult Worm
They are relatively stout cylindroidal worms.
• They are pale pink or greyish white, but may appear
reddish-brown due to ingested blood.
• the body is curved with the dorsal aspect concave and
the ventral aspect convex. The anterior end is somewhat
conslricted and bent dorsally in the same direction of
general body curvature. this cervical curvature gave it the
name hookworm (Fig. 1 ).
• The mouth is not at the Lip but directed dorsally. The
prominent buccal capsule, reinforced with a hard chitinlike substance carries six teeth, four hook-like teeth
ventrally and two knob-like with a median cleft dorsally.
Male worm: The male worm is smaller than female worm8- 11 mm in lengtl1 and 0.4 mm thick.
The poste rior end of 1hc male is expanded into a
copulat0ry bursa which consists of lhree lobes, one dorsal
and two lateral. there are 13 fleshy chitinous rays, five
each in lateral lobes and three in dorsal lobe. TI1e dorsal
ray is partially divided al the tip and each division is
tripartite. The pattern of the rays helps in distinguishing
between different species.
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Paniker's Textbook of Medical Parasitology
Buccal----
capsule
Esophagus
Vulva----1.:z.
opening
Anal pore
&>1r----Buccal
capsule
Esophagus
\\\"'llt--c;;,-Copulatory
spicules
Copulatory
bursa
Fig. 1: Adult worm of Ancylostoma duodenale (male and female)
Table 1: Distinguishing features of male and female worms of
Egg
'The egg of hookworm is:
Oval or e lliptical, measuring 60 µm by 40 mcm
• Colorless, not bile stained.
• Surrounded by a thin transparent hyaJineshell membrane.
• Floats in saturated salt solution.
• When released by the worm in the intestine, the egg
contains an unsegmen ted ovum.
During its passage down the intestine, the ovum develops.
When passed in feces, the egg contains a segmented
ovum, usually with four or eight blastomeres.
• There is a clear space between the segmented ovum and
the egg shell (Figs 2A and B).
• A single female worm lays about 25,000-30,000 eggs in a
day and some 18- 54 million during its life time.
Life Cycle
Life cycle of Ancylostoma is completed in a single host (Fig. 3).
Ancylostoma duodenale--~-- ____ Definitive Host
Male Female
Size Smaller, about 8-11 mm Larger, 10-13 mm in
in length length
Copulatory Present Absent
bursa
Genital opening Opens in cloaca along Opens at the junction of
with anus the middle and posterior
third of body
Posterior end Expanded in like Tapering
umbrella
• The cloaca into which the rectum and genital cana l open
is situated within the copula tory bursa.
• There are two long retractile bristle-like copulatory
spicules, the tips of which project from the bursa.
Female worm: The female worm is larger, 10- 13 mm long and
0.6 mm thick.
• Its hind end is conoid, with a subte rminal anus situated
ventrally.
• The vulva opens ventra lly at the junction of the middle
and posterior thirds of the body.
• The vagina leads to two intricately coiled ovarian tubes
which occupy the hind and middle pan s of the worm.
• During copulation the male attaches its copulatory bursa
to the vulva. The copulali ng pair therefore presents a
Y-shaped appearance.
• Sexes are easily differentiated by d1e ir size, the shape of
the posterior end and the position of the genital opening
(Table 1).
Humans are the only natura l host. No intermediate host is
required like other helminths (Box 2).
Infective Form
Third-stage filariform larva.
• Adult worm inhabiting the small intestine of man attach
themselves to the mucous membrane by means of their
mouth parts. The female worm lays eggs.
• The eggs containing segmented ova with four blastomeres,
are passed out in the feces of infected person (Fig. 3).
Eggs freshly passed in feces are not infective for h umans.
• When deposited in the soil, the embryo develops inside
the eggs. Its development takes place optimally in sandy
loamy soil with decaying vegetation under a moist, warm,
shady environm ent.
• In about 2 days, a rhabditiform larva, measuring 250 mcm
in length hatches out of the egg. It feeds on bacteria and
other organic matter in the soil and grows in size (Fig. 3).
• It moults twice, on the 3rd and 5th days after hatching to
become the third-stage infective filariform larva (Fig. 3).
• Filariform larva is about 500-600 µm long, with a sharp
pointed tail. The filariform larva is nonfeeding. They can
live in the soil for 5-6 weeks, with their heads waving in
th e air, waiting for the ir hosts. They can a lso ascend on
blades of grass or other vegetation, being carried in
capillary water films on their surface. Direct sun light,
d1ying, or salt water can kill the larva.
• Mode of infection:
When a person walks barefooted on soil conta ining
the filariform larva, they penetrate the skin and enter
P:205
Hookworm
•
Figs 2A and B: Egg of Ancy/ostoma duodenale. (A) As seen under microscope; and (B) Schematic diagram
Box 2: Helminths requiring no intermediate host
• Ancylostoma duodenale
• Necatoramericanus
• Ascaris lumbricoides
• Trichuris trichiura
• Enterobius vermicularis
• Hymenolepis nano
the subcutaneous tissue. The common sites of entry
are the skin between t.he toes, the dorsum of the foot
and the medial aspect of the sole. In farm workers
and miners, the larvae may penetrate the skin of the
hands.
Rarely, infection may take place by the oral route,
the filariform larva being carried on contaminated
vegetables or fruits. The larvae may penetrate the
buccal mucosa co reach the venous circulation and
complete their migration via the lungs.
Transmammary and transplacental transmission
has been also reported for Ancylostoma, but not for
Necator.
Inside the human body, the larvae are carried along the
venous circulation to the right side of the heart and to the
lungs. Here, they escape from the pulmonary capillaries
into the alveoli, migrate up the respiratory tract to
the pharynx, and are swallowed, reaching their final
destination, small intestine.
During migration or on reaching the esophagus, they
undergo third moulting.
They feed, grow in size, and undergo a fourth and.final
moulting in the small intestine and develop the buccal
capsule, by which they attach themselves to the small
intestine and grow into adults.
• There is no multiplication in the host and a single infective
larva develops into a single adult, male or female.
It takes usually about 6 weeks from the time of infection
for the adult worms to become sexually mature and
start laying eggs. Bur sometimes, there may be an arrest
in development and the process may take much longer,
6 months or more.
Alternatively, the larvae may be swallowed and may
develop directly into adults in the small intestine without
a tissue phase.
• NEGATOR AMER/GANUS
Morphology
The adult worms are slightly smaller than A. duodenale, the
male being 7- 9 mm by 0.3 mm and the female 9-11 mm by
0.4mm.
• the anterior end is bent in a direction opposite to the
general curvature of the body, while in A. duodenale the
bend is in the same direction.
• They have a smaller buccal capsule with two pairs
of semilunar cutting plates instead of teeth as in A.
duodena le.
The copulatory bursa of the male is long and wide. The
copulatory spicules are fused at the ends to form a barbed
tip.
In female, the vulva is placed in the middle of the body or
anterior to it (Figs 4A to C).
The eggs of N. americanus are identical with those of
A. duodena le. 1beir life cycle is similar to that of A. duodenale.
The lifespan of Necator is much longer being abour 4- 20 years
than in Ancylostoma, where it is of2-7 years.
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Paniker's Textbook of Medical Parasitology
Settle in small
intestine and
larvae reach
pharynx and
are ultimately
develop into .....__
adult worms ......_
I swallowed
Man
(Definitive host)
7
Penetrates
skin of man
(Definitive host)
Soil
J ·S 'g @ ~
;/' ~
... ?s ~ I lt
re q;f'-·-$\" ~ 0
Q
/
Rhabditifo,:'--1 ---®
larva hatches out Egg containing
rhabditiform larva
Fig. 3: life cycle of Ancylostoma duodena/e
The differe ntiating features of A. duodenale and N.
americanus have been discussed in Table 2 and differentiating
features between fil ariform larva of both species has been
discussed in Table 3 .
• PATHOGENICITY AND CLINICAL FEATURES
OF HOOKWORM INFECTION
Effects Due to Migrating Larva
Ground itch: Larvae may give rise to severe itching at the
site of penetration. It is more common in N. americanus
than in A. duodena/e.
• Creeping eruption: ft is formed due to subcutaneous
migration of filariform larvae. There is reddish itchy
papule along the path traversed by them.
Respiratory system: Mild transient pneumonitis, or
bronchitis occurs when larvae break out of pulmonary
capillaries into alveoli.
Effect Due to Adult Worm
• Early hookworm infection: Adult wo rms produce
epigastric pain, dyspepsia, nausea, vomiting and diarrhea.
Chronic hookworm infection: It leads to iron deficiency
anemia and protein energy malnutrition resulting from
P:207
A.duodenale N.americanus
~~R ~ DR
~ s ~ cs
Figs 4A to C: Major distinguishing features between Ancylostoma
duodenale and N. americanus. (A) Adult female in Ancy/ostomaanterior curvature uniform with body curve; in Necator anterior
curvature in opposite direction to body curve. Vulva opens at junction
of middle and posterior thirds in Ancy/ostoma; in (Necator) it opens a
little in front of the middle; (B) Buccal capsule, (Ancy/ostoma) has two
pairs of hook-like teeth ventrally and a dental plate with median cleft
dorsally; (Necator) has two pairs of semilunar cutting plates instead of
teeth; and (C) Copulatory bursa. In (Ancy/ostoma), the dorsal ray (DR)
is single with a split end, making a total of 13 rays; (Necator) has a
paired dorsal ray, making a total of 14 rays. Copulatory spicules (CS)
separate in (Ancylostoma); they are fused at the tip in (Necator)
blood loss. Adult worms attach themselves to intestinal
wall by buccal capsule and teeth and suck blood.
A duodenale ingests 0.15- 0.25 mL of blood and N.
americanus 0.03 ml of blood per day. 1hey also secrete
anticoagulants at the attachment site so that bleeding
from these sites continue. There is also interference of
absorption of iron, vitamin B12 and folic acid.
The pathogenesis and clin ical features has been described
in Flow chart 1.
• LABORATORY DIAGNOSIS
Direct Methods
• Demonstration of characteristic oval segmented
hookworm eggs in feces by direct wet microscopy or by
Hookworm
Table 2: Differentiating features of two species of hookworm
Ancylostoma duodenale Necator amerlcanus
Adult worms
Size Large and thicker Small and slender
Shape Head bent in same Head bent in opposite
direction as body direction
Buccal capsule Four ventral teeth and Two ventral and
two dorsal knob-like two dorsal chitinous
teeth cutting plates
Copulatory bursa 13 rays, two separate 14 rays, two spicules
spicules, dorsal ray single fused at the tip, dorsal
ray split
Caudal spine in Present Absent
female
Vulval opening Situated behind the Situated in anterior to
middle of the body middle part of body
Pathogenicity More Comparatively less
Eggs Similar Similar
First and second Similar Similar
stage larva
Egg/day 15,000-20,000 6,000-1 1,000
Rate of Faster Slower
development
Pulmonary More common Less common
reaction
Blood loss/worm 0.2 mUday 0.03 ml/day
Iron loss (mg/day) 0.76 mg 0.45mg
Male:female ratio 1 :1 1.5:1
Life span 2-7 years 4-20 years
Table 3: Differential features of filariform larva (third-stage larva)
Ancylostoma duodenale Necator americanus
Size 720µm
Head Slightly conical
Buccal cavity Short, lumen larger
Sheath Faint culticular striations
Intestine No gap between
esophagus and intestine
Posterior end A small retractile body is
of intestine present
Esophageal Not prominent
spears
Tail Long and blunt
660µm
Rounded
Larger, lumen shorter
Prominent striation
A gap is present between
esophagus and intestine
No retractile body
Prominent
Short and pointed
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Paniker's Textbook of Medical Parasitology
Flow chart 1: Clinical disease in hookworm
Clinical disease j
I +
Larva
+
Ground itch
• When the filariform larvae
enter the skin, they cause
severe local itching
• An erythematous papular
rash develops which later
becomes vesicular. It occurs
when large number of larvae
penetrate the skin
• More common in infection with
Necator than with Ancylostoma
• Self-limiting condition, lasting for
2-4 weeks
i
Creeping eruption
(cutaneous larva
migrans)
----- • It is a condition in which the
filariform larvae wander about
the skin and produce a reddish
itchy papule along the path
traversed by them
• More common in infections
with animal hookworms than
with human hookworms
concentration m ethods is the best m ethod of diagnosis. In
stool samples examined 24 hours or more after collection,
the eggs may have hatched and rhabditiform larvae
m ay be present. These h ave to be differentiated from
Strongyloides larvae.
• Egg counts give a m easure of the intensity of infection.
Modified Kato-Katz smear technique is a useful m ethod
for quanti tative estimation of eggs in stool. A count of
less than five eggs per mg of feces seldom causes chmcal
disease, while coun ts of 20 eggs or more are associated
with significant anemi a (Box 3). Egg cow1ts of SO or m ore
represent m assive infection.
• Adult hookworms m ay sometimes be seen in feces. Eggs of
A. duodenale and N. americanus cannot be differentiated
by m orphology. Thus specific diagnosis can only be made
by studyin g morphology of adult worms.
• Duodenal contents m ay reveal eggs or adult worms.
Stool culture: Ha rada-Mori m ethod of stool culture is
carried out 10 dem on sn·ate th ird-stage filariform larvae
which helps in distinguishing A. duodenale and N.
americanus (Flow chart 2).
Indirect Methods
• Blood examination revea ls microcyti c, hypochromic
anemia and eosinophilia.
l
Respiratory I
manifestations
• Occurs when larvae break
out of the pulmonary
capillaries and enter
the alveoli
• Manifests as bronchitis
and bronchopneumonia
• Rarely, Loeffler syndrome
can be seen
•
Adult worm !
I
• It is responsible for hook
worm disease
Adult worm sucks blood
leading to microcytic
hypochromic anemia
• Patient develops epigastric
pain, dyspepsia, vomiting and
diarrhea. The stool becomes
reddish or black in color
• Symptoms and signs of
anemia are present, viz.
exertional dyspnea,
palpitation, dizziness,
generalized puffy edema,
dry brittle hair and
koilonychia
• Severe hookworm anemia
commonly leads to cardiac
failure
Box 3: Causes of anemia in hookworm infection
• Blood sucking by the parasite for their food.
• Chronic hemorrhages from the punctured sites from jejuna! mucosa.
• Deficient absorption of vitamin Bl 2 and folic acid.
• Depression of hematopoietic system by deficient intake of proteins.
• Average blood loss by the host per worm per day is 0.03 ml with
N. americanus and 0.2 ml with A. duodenale.
• With iron deficiency, hypochromic microcytic anemia is caused and
with deficiency of both iron and v1tam1n Bl 2 or fohc acid, d1morph1c
anemia is caused.
• Secretion of anticoagulants at the site of attachment.
• Stool examination may show occult blood and CharcotLeyden crystals (Plow chart 2).
• Ch est X-ray may sh ow pulmonary infiltrates in the
migratory phase.
• TREATMENT
• For specific anlihelminthic treatment, the most practical
and effective drug is alben dazole ( 400 mg single dose)
or mebendazole (500 m g once). Pyrantel pamoate (11
mg/ kg x 3 days) i s al so effective and can be used in
pregnancy. Thiabendazole is less effective. The old drug
tetrachloroethylene is active, but toxic. Bephen ium
P:209
Hookworm
Flow chart 2: Laboratory diagnosis of hookworm
Direct methods
Laboratory diagnosis
I
+
Blood examination
Microcytic hypochromic
anemia and eosinophilia
+
Indirect methods
l
Stool examination
To demonstrate presence
of occult blood and
Charcot-Leyden crystals
Chest
X-ray
Demonstration of eggs
In feces by direct
Demonstration of adult
Worm in feces
Stool culture
By Harada-Mori method
wet microscopy or by
concentration method or in
duodenal aspirate
or duodenal aspirate
(specific diagnosis)
hydroxynaphthoate is active against Ancylostoma but not
against Necator.
• Treatment of hookworm disease also includes relief of
anemia. In hookworm disease, the intestinal absorption
of iron is apparently normal so that oral administration
of iron can correct the anemia, but in severe cases, a
preliminary packed cell transfusion may be needed.
When the hemoglobin level is very low, antihe lminthic
drugs should not be used before correcting the anemia.
• PROPHYLAXIS
• Prevention of soil pollution with feces and proper disposal
of night soil and use of sanitary lan-ines.
• Use of footwear to prevent entry of larva through the skin
of the foot. Gloves give similar protection to the hands of
farm workers.
Treatment of patients and carriers, preferably a ll at the
same time, limit to the source of infection.
• OTHER HOOKWORMS
Ancylostoma ceylanicum naturally parasitizes cats and wild
felines in South-East Asia, but can occasionally infect man.
A.braziliense, a parasite of cats and dogs and some o ther
species of animal ancylostomes have been reported to infect
man, but rhey tend to cause creeping eruption (Jarva m igrans)
rather than intestinal infection.
• TRICHOSTRONGYLIASIS
• Trichostrongylus species, normally parasitic in sheep and
goats, can also cause human infections.
This is particularly like ly, where the use of night soil as
manure is prevalent.
The infection is present in some parts of India.
1he life cycle is similar to that of hookworms.
• Human infection is usually acquired by ingestion of leafy
vegetables carrying the third-stage larva.
• Adults attach themselves to small intestinal mucosa,
suck blood and live for long periods. Infection is mostly
asymptomatic but epigastric discomfort and anemia with
marked eosinophilia occur in massive infections.
• The eggs passed in feces resemble hookworm eggs, but
are larger, with more pointed ends and show greater
segmentation with 16-32 blastomeres.
Metronidazole is effective in treatment.
KEY POINTS OF HOOKWORM
• A. duodenale is the Old World hookworm and N. americanus
is the New World hookworm.
• Adult worm live in small intestine Uejunum and duodenum).
• In A. duodena le, the anterior end is bent dorsally in the same
direction of body curvature, hence the name hookworm. The
mouth contains six teeth, four hook-like teeth ventrally and
two knob-like dorsally. Posterior end of male has a copulatory
bursa.
• Female is longer than male with tapering end.
• Eggs are oval, colorless. not bile-sta ined, and float in
saturated salt solution a nd contain segmented ovum with
four blastomeres.
• Natural host: Humans. life cycle is completed in a single host.
• Infective form: Third-stage filariform larva.
• Portal of entry: Penetration of skin.
Contd ...
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Contd ...
• Clinical features: Ground itch, creeping eruption (cutaneous
larva migrans), bronchitis and bronchopneumonia in lung,
hypochromic microcytic or dimorphic anemia and intestinal
symptoms like epigastric pain, dyspepsia, nausea and pica.
• Diagnosis: Done by demonstration of characteristic egg in
the feces by direct microscopy or by concentration methods
or by demonstration of adult worms in stool or duodenal
aspirate.
• Treatment: Albendazole, mebendazole and pyrantel
pamoate. Oral iron in anemia.
REVIEW QUESTIONS
Name the helminths that do not require any intermediate host
and describe briefly the life cycle of Ancylostoma duodena le.
Short notes on:
a. Causes of anemia in hookworm infection
b. Clinical disease in hookworm infection
c. Trichostrongyliasis
d. Prevention of hookworm infection
Differentiate between:
a. Male and female of Ancylosromaduodenale
b. Ancylostoma duodenale and Necator americanus
c. Filariform larvae of Ancylosroma and Necaror
MULTIPLE CHOICE QUESTIONS
Highest incidence of anemia in the tropics is due to
a. Hookworm
b. Thread worm
c. Ascaris
d. Guinea worm
The average blood loss per worm in ancylostomiasis is
a. 0.2 ml/day
b. 2 ml/day
c. 0.33 ml/day
d. 1 ml/day
Which of the following does not cause biliary tract obstruction
a. Ascaris lumbricoides
b. Ancy/ostoma duodenale
c. Clonorchis sinensis
d. Fascia/a hepatica
Which of the following stages of Ancylostoma duodenale is
infective to human beings
a. Rhabditiform larva
b. Filariform larva
c. Eggs
d. Adult worm
A 6-year-old girl is emaciated with a hemoglobin level of 6 g/dl.
Her face appears puffy with swollen eyelids and edema over feet
and ankles. There are no laboratory facilities available. The most
likely cause of the child's condition is
a. Schistosomiasis
b. Cercarial dermatitis
c. Ascariasis
d. Hookworm disease
All of the following are characteristics of Ancylostoma except
a. Its copulatory bursa has 13 rays
b. Caudal spine is present in females
c. Head is bent in a direction opposite to body
d. Vulval opening is situated in the middle of the body.
Answer
a 2. a 3. b 4. b 5. d 6. C
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CHAPTER 18
• INTRODUCTION
The name Enterobius 11ermicularis means a tiny worm living
in the intestine (Greek enteron-intestine, bias-life and
vermiculus-small worm). The term Oxyuris means \"sharp
tail'; a feature of the female worm, from which the name
\"pinworm\" is also derived.
• COMMON NAME
Pinworm, seatworm, threadworm.
• HISTORY AND DISTRIBUTION
Enterobius vermicularis, formerly called Oxyuris vermicularis
has been known from ancient times.
• Leuckart (1865) first described the complete life cycle of
the parasite.
• lt is worldwide in distribution. Unlike the usual situation,
where helminthic infections are more prevalent in the
poor people of the tropics, E. vermicularis is one worm
infestation which is far more common in the affluent
nations in the cold and temperate regions (cosmopolitan).
• Enterobius uermicularis is considered to be world's most
common parasite, which specially affects the children.
• HABITAT
Adult worms are found in the cecum, appendix and adjacent
portion of ascending colon.
• MORPHOLOGY
Adult Worm
1l1e adults are short, white, fusiform worms with pointed
ends, .looking like bits of white thread.
• The mouth is surrounded by three wing-like curicular
expansions (cervical alae), which are transversely striated.
A ~r----Cervical
alae
Double bulb
esophagus
00:::.-r::tH - Eggs in
uterus
,__ __ Posterior
and straight
Cervical
alae
Posterior
one-third
Fig. 1: Adult worm of Enterobius vermicularis (male and female)
• The esophagus has a double bulb structure, a feature
unique to this worm (Fig. 1).
Female Worm
The female is 8-13 mm long and 0.3-0.5 mm thick.
• Its posterior third is drawn into a thin pointed pin-like
tail (Fig. 1).
• The vulva is located just in front of the middle third of the
body and opens into the single vagina, which leads to the
paired uteri, oviducts and ovaries. In the gravid femaJe,
virtually the whole body is filled by the distended uteri
carrying thousands of eggs.
• The worm is oviparous.
• Females survive for 5- 12 weeks.
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Male Worm
The mal e worm is 2-5 mm long and 0.1-0.2 mm thick.
• Its posterior end is tightly curved ventrally, sharply
truncated and carries a prominent copulatory spicule
(Fig. 1).
• Males live for about 7- 8 weeks.
Egg
The egg is colorless and not bile-stained.
• It floats in saturated salt solution.
• It has a characteristic shape, being elongated ovoid,
flattened on one side and convex on the other (planoconvex), measuring 50- 60 µm by 20-30 µm (Fig. 2).
• The eggshell is double-layered and relatively thick, though
transparent. The outer albuminous layer makes the eggs
stick to each other and to clothing and other objects.
• The egg contains a tadpole-shaped coiled embryo, which
is fully formed, but becomes infectious only 6 hours after
being deposited on the skin. Under cool moist conditions,
the egg remains viable for about 2 weeks (Fig. 2).
• A single female worm lays 5,000-17,000 eggs.
• LIFE CYCLE
Enterobius verm icuLaris is monoxenous, passing its entire life
cycle in the human host. It has no intermediate host and does
not undergo any systemic migration (Box 1).
Natural Host
Man.
Fig. 2: Planoconvex egg of Enterobius vermicularis containing
tadpole-shaped embryo
Box 1: Nematodes not showing systemic migration in man
• Enterobius vermiculoris
• Trichuris trichiuro.
Infective Form
Embryonated Eggs
• Mode of infection: Man acquires infection by ingesting
embryonated eggs containing larva by means of:
Contaminated fingers
Autoinfection.
• Eggs laid on perianal skin containing infective larvae are
swallowed and hatch out in the intestine.
• They moult in the ileum and enter the cecum, where they
mature into adults.
• It takes from 2 weeks to 2 months from the time the eggs
are ingested, to the development of the gravid female,
ready to lay eggs.
• The gravid female migrates down the colon to the rectum.
At night, when the host is in bed, the worm comes out
through the anus and crawls about on the perianal and
perineal skin to lay its sticky eggs. The worm may retreat
into the anal canal and come out again to lay more eggs.
• The female worm may wander into the vulva, vagina
and even into the uterus and fallopian tubes, sometimes
reaching the peritoneum.
• the male is seldom seen as it does not migrate. It usually
dies after mating and is passed in the feces.
• When all the eggs are laid, the female worm dies or gets
crushed by the host during scratching. The worm may
often be seen on the feces, having been passively carried
from the rectum. The eggs, however, are only infrequently
found in feces, as the female worm lays eggs in the
perianal area and not the rectum.
• Crawling of the gravid female worm leads to pruritus and
the patient scratches the affected perianal area. These
patients have eggs of E. vermicularis on fingers and under
nails leading to autoinfection (Fig. 3).
• Autoinfection: Ingestion of eggs due to scratching of
perianal area with fingers leading to deposition of eggs
under the nails. This type of infection is mostly common
in children. This mode of infection occurs from anus to
mouth.
• Retro infection: In this process, the eggs laid on the perianal
skin immediately hatch into the infective stage larva and
migrate through the anus to develop into worms in the
colon. This mode of infection occurs from anus to colon.
• PATHOGENICITY AND CLINICAL FEATURES
Enterobiasis occurs mostly in children. It is more common
in females than in males. About one-third of infections are
asymptomatic.
• The worm produces intense irritation and pruritus of the
perianal and perinea! area (pruritus ani), when it crawls
out of the anus to lay eggs. This leads to scratching and
excoriation of the skin around the anus.
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Enterobius Vermicularis
V \"\"\"\"'''' lato ''\"\" woon
Liberated larva
migrate towards cecum Man
Adult worms in \ (
The egg shell is
dissolved by the
digestive juices
large intestine
Soil
Eggs laid at the
perianal skin by
the gravid female
and larva liberated Freshly laid
In small intestine
.
~- i '\"'\"\"' ~ eo,~\"\"\"''\" \"'\"\"· ll\"iJ
Egg (with infective larva) food, ~~~~-~~?lhin1/ \'Y
swallowed by man~ Embryonated egg
(definitive host) with infective larva in soil
Fig. 3: Life cycle of Enterobius vermicularis
• As the worm migrates out at night, it disturbs sleep.
Nocturnal enuresis is sometimes seen.
• The worm crawling into the vulva and vagina causes
irritation and a mucoid discharge. lt may migrate up to
the uterus, fallopian tubes and into the peritoneum. This
may cause symptoms of chronic salpingitis, cervicitis,
peritonitis and recurrent urinary tract infections.
• The worm is sometimes found in surgically removed
appendix and has been claimed to be responsible for
appendicitis.
• LABORATORY DIAGNOSIS
Pinworm infestation can be suspected from the history of
perianal pruritus. Diagnosis depends on the demonstration
of the eggs or adult worms (Flowchart 1).
Demonstration of Eggs
• Eggs are present in the feces only in a small proportion
of patients and so feces examination is not useful in
diagnosis.
• They are deposited in large numbers on the perianal and
perineal skin at night and can be demonstrated in swabs
collected from the sites early morning, before going to the
toilet or bathing. Swabs from perianal folds are most often
positive.
• The eggs may sometimes be demonstrated in the dirt
collected from beneath the finger nails in infected
children.
NIH Swab Method
The NIH swab [named after National Institutes of Health,
United States of America (USA)) has been widely used for
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Flow chart 1: Laboratory diagnosis of Enteroblus vermicu/aris
Laboratory diagnosis
Under finger
nails
Detection of egg
I
NIH swab method
collection of specimens. This consists of a glass rod at one
end of which a piece of transparent cellophane is attached
with a rubber band. The glass rod is fixed on a rubber stopper
and kept in a wide test tube. The cellophane part is used
for swabbing by rolling over the perianal area (Fig. 4). It is
returned to the test tube and sent to the laboratory, where the
cellophane piece is detached, spread over a glass side and
examined microscopically.
Scotch Tape Method
Another method for collection of specimens is with scotch
tape (adhesive transparent cellophane tape) held sticky
side out, on a wooden tongue depressor. The mounted tape
is firmly pressed against the anal margin, covering all sides
(Fig. 5). The tape is transferred to a glass slide, sticky side
down, with a drop of toluene for clearing and examined
under the microscope.
Demonstration of Adult Worm
The adult worms may sometimes be noticed on the surface
of stools.
• Tuey may occasionally be found crawling out of the anus
while the children are asleep.
• They may be detected in stools collected after an enema
and may be in the appendix during appendectomy
(Box2).
Note: Unlike the other intestinal nematodes, Enterobius
infection is not associated with eosinophilia or with elevated
immunoglobulin E (lgE).
• TREATMENT
Pyrantel pamoate ( l l mg/kg once, maximum lg), albendazole
( 400 mg once) or mebendazole (100 mg once) can be used for
single dose therapy, while piperazine has to be given daily for
1 week.
l
Scotch tape method I
Rubber stopper
Test tube
Rubber band
Detection of adult
worm
Stool sample
Fig. 4 : National Institutes of Health (NIH) swab. A piece of transparent
cellophane is attached with rubber band to one end of a glass rod,
which is fixed on a rubber stopper and kept in a wide test tube
Fig. S: Scotch tape method (press the sticky side of the tape against
the skin across the anal opening)
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Box 2: Infectious parasites which may be present in a fecal sample
• Enterobius vermicularis
• Strongyloides stercoralis
• Taenia solium
• Hymenolepis nano
• Entamoeba histolytica
• Giardia lamblia
• Cryptosporidium parvum
• It is necessary to repeat the treatment after 2 weeks to take
care of autochthonous infections and ensure elimination
of all worms.
• As pinworm infection usually affects a group, it is
advisable to treat the whole family or group of children,
as the case may be.
• PROPHYLAXIS
• Maintenance of personal and community hygiene such as
frequent hand washing, finger nail cleaning and regular
bathing.
• Frequent washing of night clothes and bed linen.
KEY POINTS OF ENTEROBIUS VERMICULARIS
• Adult worm lives in cecum and appendix.
• Mouth is surrounded by three wing-like cervical alae.
Esophagus has a double bulb structure.
• Worm is oviparous.
• Eggs are colorless, not bile-stained; plano-<:onvex in shape.
• Natural host: Humans. E. vermicularis passes its entire life
cycle in human host. No intermediate host is required.
• Infective form: Embryonated egg containing infective larva.
• Mode of infection: By ingestion of eggs or autoinfection. Seen
mostly in children and among family members.
• Clinical features: Pruritus ani, nocturnal enuresis. Sometimes,
salpingitis, peritonitis, appendicitis, etc. may be seen.
• Diagnosis: Detection of eggs by NIH swab and cellophane
scotch tape method. Detection of adult worm in finger nails or
from stool after enema.
• Treatment: Mebendazole, albendazole, or pyrantel pamoate.
Enterobius Vermicularis
REVIEW QUESTIONS
List the parasites causing autoinfection and describe briefly the
life cycle of Enterobius vermicularis.
Short notes on:
a. Egg of Enterobius vermicularis
b. Laboratory diagnosis of Enterobius vermicularis
c. NIH swab
MULTIPLE CHOICE QUESTIONS
Most common presenting symptom of thread worm infection
amongst the following is
a. Abdominal pain
b. Rectal prolapse
c. Urticaria
d. Vaginitis
Which one of the following does not pass through the lungs
a. Hookworm
b. Ascaris
c. Strongyloides
d. Enterobius vermicularis
Infection with which of the following parasites may cause
enuresis
a. Ascaris lumbricoides
b. Enterobius vermicularis
c. Trichinella spiralis
d. Wuchereria bancrofti
History of mild intestinal distress, sleeplessness, itching, and
anxiety is seen in preschool child attending play school.
Possible parasite agent causing these manifestations is
a. Trichomonas vagina/is
b. Enterobius vermicu/aris
c. Ascaris lumbricoides
d. Necator americanus
The common name for Enterobius vermicularis is
a. Threadworm
b. Pinworm
c. Seatworm
d. Whipworm
Which of the following nematodes lays eggs contaning larvae
a. Trichinella spiralis
b. Enterobius vermicularis
c. Brugia malayi
d. Ascaris lumbricoides
Answer
a 2. d 3. b 4. b 5. C 6. b
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CHAPTER 19
• COMMON NAME
Roundworm.
• HISTORY AND DISTRIBUTION
Ascaris lumbricoides has been observed and described from
very ancient times, when it was sometimes confused with the
earthworm.
• Its specific name lumbricoides is derived from its
resemblance with earthworm (Lumbricus, meaning
earthworm in Latin).
• It is the most common of human helminths and is
distributed worldwide. A billion people are estimated
to be infected with roundworms. The individual worm
burden could be very high, even up to over a thousand.
An editorial in the Lancet in 1989 observed that if all the
roundworms in all the people worldwide were placed
end-to-end they would encircle the world 50 times.
• The incidence may be as high as 80- 100% in rural areas
with poor sanitation.
• HABITAT
Adult worms live in the small intestine (85% in jejunum and
15% in ileum).
The roundworm, Ascaris lumbricoides is the largest
nematode parasite in the human intestine.
• MORPHOLOGY
Adult Worm
They are large cylindrical worms, with tapering ends, the
anterior end being more pointed than the posterior (Fig. 1).
• They are pale pink or flesh colored when freshly passed in
stools, but become white outside the body.
• The mouth at the anterior end has three finely toothed
lips, one dorsal and two ventrolateral (Figs 2A to E).
Fig. 1: Specimen of Ascaris /umbricoides
Male Worm
• The adult male worm is little smaller than female. It
measures 15-30 cm in length and 2-4 mm in thickness
(Figs 2A to E).
• Its posterior end is curved ventrally to form a hook and
carries two copulatory spicules (Figs 2A to E).
Female Worm
The female is larger than male, measuring 20-40 cm in length
and 3-6 mm in thickness.
• Its posterior extremity is straight and conical.
• The vulva is situated mid-ventrally, near the junction
of the anterior and middle thirds of the body. A distinct
groove is often seen surrounding the worm at the level
of the vulvar opening. This is called the vulvar waist or
genital girdle and is believed to facilitate mating (Figs 2A
to E). The vulva leads to a single vagina, which branches
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Ascaris Lumbricoides
~ Dorsalllp
(one)
. ...,...,---Paplllla
'-.:\"¥:;:,-<--r==--ventral lips
Vulvar
waist cl
(two)
.-ll!E\"\"--tt- - Anal
El
Copulatory
spicules
opening
II
Figs 2A to E: Ascaris lumbrlco/des. (A) Adult female and male worms; (B) Anterior end of worm. Head-on view, showing one dorsal and two
ventral lips with papillae; (C) Posterior end of female, showing anal opening, a little above the conical tip; (D) Posterior end of male, showing
two protruding copulatory spicules(s); and (E) Specimen showing Ascaris lumbricoides, male and female
into a pair of genital tubules that lie convoluted through
much of the posterior two-thirds of the body. The genital
tubules of the gravid worm contain an enormous number
of eggs as many as 27 million at a time (Box 1).
• A single worm lays up to 200,000 eggs per day. The eggs
are passed in feces.
Egg
Two types of eggs are passed by the worms: (1) fertilized and
(2) unfertilized.
The fertilized eggs, laid by females, inseminated by
mating with a male, are embryonated and develop into
the infective eggs (Figs 3A to C).
The unfertilized eggs, are laid by uninseminated female.
These are nonembryonated and cannot become infective
(Fig. 3D).
Note: Stool samples may show both fertilized and unfertilized
eggs, or either type alone (Table 1).
• LIFE CYCLE
Life cycle of Ascaris involves only one host.
Box 1: Parasites with bile-stained eggs
• Ascaris lumbricoldes
• Clonorchis sinensis
• Trichuris trichiura
• Fasc/ola hepatlca
• Taenia solium
• Fasclo/opsls busk/
• Taenia saginata.
Natural Host
Man. There is no intermediate host.
Infective Form
Embryonated eggs.
• Mode of transmtsston:
Infection occurs when the egg containing the
infective rhabditiform larva Is swallowed. A frequent
mode of transmission is through fresh vegetables
grown in fields manured with human feces (night
soil). Infection may also be transmitted through
contaminated drinking water.
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c
Figs 3A to D: Types of Ascaris eggs found in stools. (A) Fertilized egg surface focus. showing outer mamillary coat; (8) Fertilized egg, median
focus, showing unsegmented ovum surrounded by three layers of coats; (C) Decorticated fertilized egg, the mamillary coat is absent; and (D)
Unfertilized egg, elongated, with atrophic ovum
Table 1: Feat ures of roundworm egg
Type of egg
Unfertilized egg
(Fig. 4A)
Fertilized eggs
(Fig.48)
Main feature
• Elliptical in shape
• Narrower and longer
• 80 µm x 55 µm in size
• Has a thinner shell with an irregular coating of albumin
• Contains a small atrophied ovum with a mass of disorganized highly refractile granules of various size
• Does not float in salt solution
• Round or oval in shape
• Size 60-75 µm x 40-45 µm
• Always bile-stained
• Golden brown in color
• Surrounded by thick smooth translucent shell with an outer coarsely mammillated albuminous coat. a thick transparent middle
layer and the inner lipoidal vitelline membrane
• Some eggs are found in feces without the outer mammillated coat They are called the decorticated eggs (Fig. 3C)
In the middle of the egg is a large unsegmented ovum, containing a mass of coarse lecithin granules. It nearly fills the egg, except
for a clear crescentic area at either poles
Floats in saturated solution of common salt
m
Figs 4A and B: (A) Unfertilized egg of Ascaris; and (B) Fertilized egg of Ascaris
P:219
Children playing about in mud can transmit eggs to
their mouth through dirty fingers (geophage).
Where so il contamination is heavy due to
indiscriminate defecation, the eggs sometimes get
airborne along with windswept dust and are inhaled.
The inhaled eggs get swallowed.
Development in Soil
The fertilized egg passed in feces is not immediately infective.
It has to undergo a period of incubation in soil before
acquiring infectivity.
• The eggs are resistant to adverse conditions and can
survive for several years.
• The development of the egg in soil depends on the nature
of the soil and various environmental factors. A heavy
clayey soil and moist shady location, with temperature
between 20°c and 30°C are optimal for rapid development
of the embryo.
• The development usually takes from J 0-40 days, during
which time the embryo moults twice and becomes the
infective rhabditiforrn larva, coiled up within the egg.
Development in Man
When the swallowed eggs reach the duodenum, the larvae
hatch out.
•
.
•
•
•
•
The rhabditlform larva, about 250 µm in length and 14
µm in diameter, are actively motile.
They penetrate the intestinal mucosa, enter the portal
vessels and are carried to the liver.
They then pass via the hepatic vein, inferior vena cava,
and the right side of the heart and in about 4 days reach
the lungs, where they grow and moult twice.
After development in the lungs, in about 10-15 days, the
larvae pierce the lung capillaries and reach the alveoli.
They crawl up or are carried up the respiratory passage to
the throat and are swallowed.
The larvae moult.finally and develop into adults in the
upper part of the small intestine. They become sexually
mature in about 6-12 weeks and the gravid females start
laying eggs to repeat the cycle (Fig. 5).
The adult worm has a lifespan of 12-20 months.
• PATHOGENICITY AND CLINICAL FEATURES
Disease caused by A. lumbricoides is called as ascariasis.
• Clinical manifestations in ascariasis can be caused either
by the migrating larvae or by the adult worms.
Symptoms Due to the Migrating Larvae
The pathogenic effects of larval migration are due to allergic
reaction and not the presence of larvae as such. Therefore,
Ascaris Lumbricoides
the initial exposure to larvae is usually asymptomatic, except
when the larval load is very heavy.
• When reinfection occurs subsequently, there may be
intense cellular reaction to the migrating larvae in the
lungs, with infiltration of eosinophils, macrophages and
epithelioid cells.
• This Ascaris pneumonia is characterized by lowgrade fever, dry cough, asthmatic wheezing, urticaria,
eosinophilia and mottled lung infiltration in the chest
radiograph.
• The sputum is often blood-tinged and may contain
Charcot-Leyden crystals. The larvae may occasionally be
found in the sputum, but are seen more often in gastric
washings. This condition is called Loejfler's syndrome.
• The clinical features generally clear in 1 or 2 weeks,
though it may sometimes be severe and rarely, even fatal.
Loeffler's syndrome can also be caused by hypersensitivity
to other agents, both living and nonliving (Box 2).
Symptoms Due to the Adult Worm
Clinical manifestations due to adult worm vary from
asymptomatic infection to severe and even fatal consequences.
• Asymptomatic infection: Generally seen in mildly
infected cases; however, it is not unusual to find children
apparently unaffected in spite of heavy infestation with
the worms.
• The pathological effects, when present, are caused by
spoliative action, toxic action, mechanical effects and
wandering effects .
The spoliative or nutritional effects are usually seen
when the worm burden is heavy. The worms may be
present in enormous numbers, sometimes exceeding
500, in small children, occupying a large part of the
intestinal tract. This interferes with proper digestion
and absorption of food. Ascariasis may contribute
to protein-energy malnutrition and vitamin A
deficiency. Patients have loss of appetite and are often
listless. Abnormalities of the jejuna! mucosa are often
present, including broadening and shortening of villi,
elongation of crypts and round cell infiltration of
lamina propria. These changes are reversed when the
worms are eliminated .
the toxic effects are due to hypersensitivity to the
worm antigens and may be manifested as fever,
urticaria, angioneurotic edema, wheezing and
conjunctivitis. These are more often seen in persons
who come into contact with the worm occupationally,
as in laboratory technicians and abattoir workers
{who become sensitive to the pig ascarid, A. suum),
than in children having intestinal infestation.
The mechanical effects are the most important
manifestations of ascariasis. Mechanical effects can
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Paniker's Textbook of Medical Parasitology
Larva burrows through the
mucous membrane of the
small intestine
Rhabditiform larva
liberated in the
duodenum
Man acquires infection
by Ingestion of food and water
contaminated with embryonated
eggs
Reach the lungs, trachea
and pharynx. From here
they are swallowed and
reach small intestine.
Man
Soil
Contamination of
vegetables
Adult worms
in small
intestine of
man
Fertilized egg containing
Unfertilized unsegmented ovum
egg passed in feces
Rhabditiform larva develops
in soil within the egg
Fig. 5: Life cycle of Ascaris lumbricoides
Box 2: Parasites causing pneumonitis or Loeffler's syndrome
• Migrating larvae of:
Ascaris lumbricoides
Strongyloides stercoralis
Ancylostoma duodena/e
Necator americanus
• Echinococcus granulosus
• Eggs of Paragonimus westermani
• Cryptosporidium parvum
• Trichomonas tenax
• Entamoeba histolytica.
be due to masses of worms causing luminal occlusion
or even a single worm infiltrating into a vital area.
The adult worms Jive in the upper part of the small
intestine, where they maintain their position due to
their body muscle tone, spanning the lumen.
They may stimulate reflex peristalsis, causing
r ecurr en t and often severe colicky pain in the
abdomen. The worms may be clumped together
into a mass, filling the lumen, leading to volvulus,
intussusception, or intestinal obstruction and
intestinal perforation.
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Ectopic ascariasis (Wanderlust): The worms
are restless wanderers, apparently showing great
inquisitiveness, in that they tend to probe and
insinuate themselves into any aperture they find on
the way. The wandering is enhanced when the host is
ill, particularly when febrile, with temperature above
39°C. The male worm is more responsive to illness
of the host, than the female. The worm may wander
up or down along the gut. Going up, it may enter the
opening of the biliary or pancreatic duct causing
a cute biliary obstruction or pancreatitis. It may
enter the liver parenchyma, where it may lead to liver
abscesses. The worm may go up the esophagus and
come out through the mouth or nose. It may crawl
into the trachea and the lung causing respiratory
obstruction or lung abscesses. Migrating downwards,
the worm may cause obstructive appendicitis.
It may lead to peritonitis when it perforates the
intestine, generally at weak spots such as typhoid
or tuberculous ulcers or through suture lines. This
tendency makes preoperative deworming necessary
before gastrointestinal surgery in endemic areas. The
wandering worm may also reach kidneys.
• LABORATORY DIAGNOSIS
Detection of Parasite
Adult Worm
The adult worm can occasionally be detected in stool or
sputum of patient by naked eye.
Barium meal may reveal the presence of adult worm in
the smalJ intestine.
A plain abdominal film may reveal masses of worms
in gas-filled loops of bowel in patients with intestinal
obstruction.
PancreaticobiJiary worms can be detected by ultrasound
(more than 50% sensitive) and endoscopic retrograde
cholangiopancreatography (ERCP; 90% sensitive).
Larvae
In the early stages of infection, when migrating larvae
cause Loeffler's syndrome, the diagnosis may be made by
demonstrating the larvae in sputum, or more often in gastric
washings.
• Presence of Charcot-Leyden crystals in sputum and an
attendant eosinophilia supports the diagnosis. At this
stage, no eggs are seen in feces.
• Chest X-ray may show patchy pulmonary infiltrates.
Eggs
Definitive diagnosis of ascariasis is made by demonstration
of eggs infeces.
Ascaris Lumbricoides
• Ascarids are prolific egg layers. A single female may
account for about three eggs per mg of feces. At this
concentration, the eggs can be readily seen by microscopic
examination of a saline emulsion of feces. Both fertilized
and unfertilized eggs are usually present. Occasionally,
only one type is seen. The fertilized eggs may sometimes
appear decorticated. The unfertilized eggs are not
detectable by salt floatation.
• Rarely when the infestation is light, eggs are demonstrable
only by concentration methods.
• Eggs may not be seen if only male worms are present, as
may occasionally be the case. Fecal films often contain
many artifacts resembling Ascaris eggs and care must be
taken to differentiate them.
• Eggs may be demonstrative in the bile obtained by
duodenal aspirates (Flow chart 1).
Serological Tests
Ascaris antibody can be detected by:
• Indirect hemagglutination (IHA)
• Indirect fluorescent antibody (IFA)
• Enzyme-linked irnmunosorbent assay (ELISA).
• Serodiagnosis is helpful in extraintestinal ascariasis like
Loeffler's syndrome (Flow chart 1).
Blood Examination
Complete blood count may show eosinophilia in early stage of
invasion (Flowchart 1).
• TREATMENT
Several safe and effective drugs are now available for
treatment of ascariasis. These include pyrantel pamoate (11
mg/ kg once; maximum 1 g), albendazole (400 mg once),
mebendazole (100 g twice daily for 3 days or 500 mg once),
or ivermectin (150-200 mg/kg once). These medications are
contraindicated in pregnancy; however, pyrantel pamoate is
safe in pregnancy.
• Partial intestinal obstruction should be managed with
nasogastric suction, intravenous fluid administration and
instillation of piperazine through the nasogastric tube.
• Complete obstruction requires immediate surgical
intervention.
• PROPHYLAXIS
• Ascariasis can be eliminated by preventing fecal
contamination of soil. The Ascaris egg is highly resistant.
Therefore, the use of night soil as manure will lead to
spread of the infection, unless destruction of the eggs is
ensured by proper composting. Treatment of vegetables
and other garden crops with water containing iodine 200
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Paniker's Textbook of Medical Parasitology
i
Eggs
• Definitive diagnosis of
ascariasis is made by
demonstration of eggs
in feces
• Rarely, when the
infestation is light,
eggs are demonstrable
only by concentration
methods
Flow chart 1: Laboratory diagnosis of Ascaris lumbricoides
Laboratory Diagnosis !
Detection of Parasite
l
Larva
+
In lhe early stage of
infection,when migrating
larva cause Loeffler's
syndrome, the diagnosis
may be made by
demonstrating the larvae
in sputum, or in gastric
washings
Adult worm
• Can occasionally be
seen by naked eye in
stool or sputum of
patient
• X-ray, Barium meals
and ultrasound
imaging may help in
diagnosis
l
Serodiagnosis
Ascans antibodies can be
detected by
•ELISA
• IHA
• IFA
Serodiagnosis is helpful in
extra intestinal
ascariasis like Loeffler's
syndrome
l
Blood examination
Eosinophilia may be
seen in early stages
of infection
Abbreviations: ELISA, enzyme-linked immunosorbent assay; IFA, indirect fluorescent antibody; IHA, indirect hemagglutination
ppm for 15 minutes kills the eggs and larvae of Ascaris and
other helminths.
• Avoid eating raw vegetables.
• Improvement of personal hygiene.
• Treatment of infected persons especially the children.
KEY POINTS OF ASCARIS LUMBRICOIDES
• A. lumbricoides is the largest nematode infecting human.
• Adult worm is cylindrical resembling an earthworm.
• Male is little smaller than female. Posterior end of male is
curved ventrally to form a hook with two copulatory spicules.
Posterior end of female is conical and straight.
• Fertilized eggs are bile-stained, round or oval, surrounded by a
thin translucent wall with outer mammillated coat containing
a large unsegmented ovum. Unfertilized eggs are elliptical,
longer with an outer thinner irregular mammillated coat,
containing a small atrophied ovum and retractile granules.
• Natural host: Man.
• Infective form: Embryonated egg containing rhabditiform
larva.
• Clinical features: Spoliative action-protein and vitamin A
deficiency. Toxic action-utricaria and angioneurotic edema.
Mechanical action- intestinal obstruction, intussusception,
volvulus, intestinal perforation. In lungs- Ascaris can cause
pneumonia (Loeffler's syndrome).
• Diagnosis: Demonstration of eggs in stool, finding of larvae in
sputum, finding of adult worm in stool or sputum.
• Treatment: Albendazole, mebendazole, ivermectin, or pyrantel
pamoate.
Fig. 6: Adult worms of Toxocara canis
• OTHER ROUNDWORMS
Toxocara
Toxocara canis and T. cati, natural parasites of dogs and cats
(Fig. 6), respectively can cause aberrant infection in human s
leading to visceral larva migrans.
• Infection is acquired in puppies by transmission of larvae
transplacentally or lactogenically (through breast milk),
but in kittens, only Jactogenic transmission is recorded.
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Ascaris Lumbricoides
Box 3: Geohelminths
• Soil-transmitted intestinal nematodes are called Geohelminths. In all of them, eggs passed in feces undergo maturation in soil. They are classified
into three categories based on their life cycle:
Direct: Ingested infective eggs directly develop into adults in the intestine, e.g. whipworms.
Modified direct: Larvae from ingested eggs penetrate intestinal mucosa enter bloodstream and through the liver, heart, lungs, bronchus and
esophagus, reach the gut to develop into adults, e.g. roundworms.
Skin penetrating: Infective larvae in soil penetrate host skin, reach the lung, and proceed to the gut as in the modified direct method, e.g.
hookworms.
• Geohelminths pose a serious health problem in poor countries, particularly among children. Their control requires general measures such as personal
hygiene, sanitation and health education, besides provision of diagnostic and treatment facilities.
• Older animals are infected by ingestion of mature eggs in
soil or of larvae by eating infected rodents, birds, or other
paratenic hosts.
• Eggs are shed in feces and become infective in 2-3 weeks.
• Human infection is by ingestion of eggs.
• Larvae hatch out in the small intestine, penetrate the
mucosa, and reach the liver, lungs, or other viscera. They
do not develop any further.
• Mostinfectionsareasymptomatic, but in some, particularly
in young children, visceral larva migrans develops,
characterized by fever, hepatomegaly, cough, pulmonary
infiltrates, high eosinophilia and hyperglobulinemia. In
some, the eye is affected (ophthalmic larva migrans).
Baylisascaris
Baylisascaris procyonis, an ascarid parasite of raccoons in
North America, is known to cause serious zoonotic infections
leading to visceral larva migrans, ophthalmic larva migrans
and neural larva migrans. Complications include blindness
and central nervous system lesions ranging from minor
neuropsychiatric conditions to seizures, coma and death
(Box3).
REVIEW QUESTIONS
Name the parasites causing pneumonitis and describe briefly
the life cycle and laboratory diagnosis of Ascaris lumbricoides.
Short notes on:
a. Clinical manifestations of ascariasis
b. Loeffler's syndrome
c. Surgical complications of ascariasis
d. Toxoca riasis
e. Geohelminths
Differentiate between fertilized and unfertilized egg of Ascaris
lumbricoides.
MULTIPLE CHOICE QUESTIONS
Which of the following parasites does not penetrat e human skin
a. Ascaris Jumbricoides
b. Ancylostoma duodenale
c. Strongyloides stercora/is
d. Schistosoma haematobium
The common nam e for Ascaris lumbricoides is
a. Roundworm
b. Hookworm
c. Threadworm
d. None of the above
3 . The largest intestinal nematode infecting humans is
a. Necator americanus
b. Ascaris lumbricoides
c. Enterobius vermicularis
d. None of the above
All of the following are correct regarding fertilized egg of Ascaris
except
a. It is always b ile-stained
b. Covered by an outer mamilliated coat
c. Floats in saturated solution of salt
d. Does not float in saturated solution of salt
All of the following parasites have bile-stained eggs except
a. Ascaris
b. Clonorchis
c. Taenia so/ium
d. Enterobius
Loeffier's syndrome may b e seen in infection with
a. Ancy/ostoma duodena/e
b. Ascaris lumbricoides
c. Trichinella spiralis
d. Trichuris trichiura
Answer
a 2. a 3. b 4. d 5. d 6. b
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CHAPTER 20
• INTRODUCTION
Nematodes belonging to the superfamily Filarioidea are
slender thread-like worms (Latin,filum and thread), which are
transmitted by the bite of blood-sucking insects. .
•
•
•
.
•
The filarial worms reside in the subcutaneous tissues,
lymphatic system, or body cavities of humans (Table 1).
The adult worm generally measures 80-100 mm in length
and 0.25- 0.30 mm in breadth; the female worm being
longer than the males.
The tail of the male worm has perianal papillae and
unequal spicules but no caudal bursa.
The female worms are viviparous and give birth to larvae
known as microfilariae.
The microfilariae released by the female worm, can be
detected in the peripheral blood or cutaneous tissues,
depending on the species.
In some species, the microfilariae retain their egg
membranes which envelop them as sheath. They are
known as sheathed microfilariae.
In some other species of filarial nematodes, the egg
membrane is ruptured and is known as unsheathed
microfilariae.
Once the microfilariae are classified on the basis of sheath
as \"sheathed\" or \"unsheathed'; their further differentiation
can be done on the characteristic arrangement of nuclei
(Flow chart 1 and Table 2).
• Periodicity: Depending on when the largest number of
microfilariae occur in blood, filarial worms can exhibit
nocturnal, dJurnal perlodJcity or no periodicity at all
(Box 1).
The basis of periodicity is unknown but it may be an
adaptation to the biting habits of the vector.
• The life cycle of filarial nematodes is passed in two hosts:
(1) definitive host is man and (2) intermediate host are
the blood-sucking arthropods.
• The microfilariae complete their development in the
arthropod host to produce the infective larval stages.
Table 1: Classification of filarial worm based on location in body
Lymphatk filarlasls Subcutaneous serous cavity filarlasis
• Wuchereria
bancrofti
• Brugia malayi
• Brugia timari
filarlasls
• Loaloa
• Onchocerca
volvulus
• Mansonella
streptocerca
• Mansonella perstans
• Mansonella ozzardi (They are
virtually nonpathogenic)
These are transmitted to humans by arthropod, which
are their vectors also during the next feed. Adult worms
live for many years whereas microfilariae survive for 3-36
months.
• Eight species of filarial worms infect humans, of them six
are pathogenic-(1) Wuchereria bancrofti, (2) Brugia
malayi and (3) B. timori cause lymphatic filariasis; ( 4)
Loa loa causes malabar swellings and allergic lesions; (5)
Onchocerca volvulus causes eye lesions and dermatitis;
(6) Mansonella streptocerca leads to skin diseases; and
two of them, (7) M. ozzardi and (8) M. perstans are
virtually nonpathogenic (Table 3).
• Infection with any of the filarial worms may be called
.filariasis, but traditionally, the term filariasis refers to
lymphatic filariasis caused by Wuchereria or Brugia
species.
• Adult filarial worm contains an endosymbiotic Rickettsialike a-proteobacterium of the genus Wolbachia spp. This
has got definite role in the pathogenesis of filariasis and
has become a target for antifilarial chemotherapy.
• Wolbachia spp. along with filarial antigen activates the
release of proinflammatory and chemotactic cytokines.
These include cellular infiltration and amplification of
inflammatory processes. Toll-like receptors (TLRs) play
an important role in the process.
P:225
Filarial Worms
Flow chart 1: Differentiating feat ures of various microfilariae on the basis of presence of nuclei in tail end
+
Sheathed microfilariae
I
Nuclei do not
extend up to the
tail ti
Tail end
Nuclei extend
up to the tail tip
M icrofilariae
I +
Unsheathed microfilariae
Nuclei extend
up to the tail tip
Tail end
I
Nuclei do not
extend up to
the tail tip
Nuclei present in
a row up to
the tail tip
Two nuclei at
the tip of tail
Mansonella
perstans
Mansonella
streptocerca
Wuchereria
bancrofti
Loaloa Brugia
malayi
Table 2: Head and tail ends of microfilariae found in humans
Species Wuchereria Brugia malayi Loaloa
bancrofti
Shape
Posterior end
(5;' ~ ~
Tail nuclei
~ Nuclei do not
~ 2 nuclei at the tip Nucl
~ ei form
extend to the tip of the tail continuous row
of tail in the tip of the
tail
Anterior end
~ ~ ~
Size 300 x 8 µm 220 x6 µm 270 x 8 µm
Sheathed/unsheathed Sheathed Sheathed Sheathed
Habitat Blood Blood Blood
Mansonella
ozzardi
Mansonella Mansonella
perstans ozzardi
Onchocerca
volvu/us
Onchocerca
volvulus
~ ~ _,.r
I
Nuclei extend to Nuclei do not Nuclei do not
the tip of the tail extend to the tip extend to the tip of
of the tail the tail
180 X 4 µm 220 X 4 µm 200 x 360 µm
Unsheathed Unsheathed Unsheathed
Blood Blood Skin, eye
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Paniker's Textbook of Medical Parasitology
Box 1: Different types of periodicity exhibited by m,crofilariae
• Nocturnal periodicity:When the largest number of microfilariae occur in blood at night, e.g. Wuchereria bancrofti
• Diurnal periodicity:When the largest number of microfilariae occur in blood during day, e.g. Loa loa
• Nonperiodic: When the microfilariae circulate at constant levels during the day and night, e.g. Onchocerca volvulus
• Subperiodic or nocturnally subperiodic: When the microfilariae can be detected in the blood throughout the day but are detected in higher numbers
during the late afternoon or at night.
Note:The microfilariae are found in capillaries and blood vessels of lungs during the period when they are not present in the peripheral blood.
Table 3: Filarial nematodes infecting humans
Parasite
I. Lymphatic filariasis
Wuchereria bancrofti
Brugia malayi
Brugia timori
II. Subcutaneous filariasis
Loaloa
Onchocerca volvulus
Mansonella streptocerca
Ill. Serous cavity filariasis
Mansonella ozzardi
Mansonella perstans
Location In body
adult
Lymphatics
Lymphatics
Lymphatics
Connective tissue,
conjunctiva
Mlcrofilaria
Blood
Blood
Blood
Blood
Subcutaneous nodules Skin, eyes
Subcutaneous Skin
Peritoneum and pleura Blood
Peritoneum and pleura Blood
• LYMPHATIC FILARIASIS
Wuchereria Bancrofti
History and Distribution
Filariasis has been known from antiquity. Elephantiasis had
been described in India by Sushruta and in Persia by Rhazes
and Avicenna.
• Elephantiasis-painful, disfiguring swelling of the legs
and genital organs-is a classic sign of late-stage disease.
• The term Malabar leg was applied to the condition by
Clarke in 1709 in Cochin.
• Microfilaria was first observed by Demarquay (1863) in
the hydrocele fluid of a patient from Havana, Cuba. The
genus is named after Wucherer, a Brazilian physician
who reported microfilariae in chylous urine in 1868.
Charocterlstlcs of mlcrofilarla Periodicity of
micrafilaria
Sheathed, pointed tail tip free Nocturnal
of nuclei
Sheathed, blunt tail tip with Nocturnal
two terminal nuclei
Sheathed, longer than Nocturnal
Mf.malayi
Sheathed, nuclei extending Diurnal
up to pointed tail tip
Unsheathed, blunt tail tip free Nonperiodic
of nuclei
Unsheathed blunt tail tip Nonperiodic
with nuclei
Unsheathed, pointed tail tip Nonperiodic
without nuclei
Unsheathed, pointed tail tip Nonperiodic
with nuclei
Principal vector
Cu/ex qumquefasciatus
Manson/a spp.
Anopheles barbirostris
Chrysops spp.
Simulium spp.
Culicoides
Culicoides
Culicoides
Microfilaria was first demonstrated in human blood in
Calcutta by Lewis (1872).
• In 1876, Bancroft first reported and described adult
female worm and in 1888, adult male worm was described
by Bourne.
• Manson (1878) in China identified the Culex mosquito
as the vector. This was the first discovery of insect
transmission of a human disease. Manson (1879) also
demonstrated the nocturnal periodicity of microfilariae
in peripheral blood.
• W. bancrofti is distributed widely in the tropics and
subtropics of sub-Saharan Africa, South-East Asia, India
and the Pacific islands. The largest number of cases of
filariasis occurs in India (Fig. 1).
• In India, the endemic areas are mainly along the sea coast
and along the banks of the large rivers, though infection
occurs virtually in all states, except in the north-west.
P:227
Fig. 1: Geographical distribution of Wucherer/a bancrofti
Habitat
The adult worms reside in the lymphatic system of man. The
microfilariae are found in blood.
Morphology
Adult worm: The adults are whitish, translucent, thread-like
worms with smooth cuticle and tapering ends.
•
•
•
•
•
The female is larger (70-100 x 0.25 mm) than the male
(25-40 X 0.1 mm).
The posterior end of the female worm is straight, while
that of the male is curved vertically and contains two
spicules of unequal length.
Males and females remain coiled together usually in the
abdominal and inguinal lymphatics and in the testicular
tissues (Fig. 2).
The female worm is viviparous and directly liberates
sheathed microfilariae into lymph.
The adult worms live for many years, probably 10-15 years
or more.
Mtcro.filariae: The microfilaria has a colorless, translucent
body with a blunt head, and pointed tail {Fig. 3).
• It measures 250-300 µm in length and 6-10 µm in
thickness. It can move forwards and backwards within the
sheath which is much longer than the embryo.
• It is covered by a hyaline sheath, within which it can
actively move forwards and backwards as sheath is much
longer than the embryo.
• When stained with Leishman or other Romanowsky stains,
structural details can be made out. Along the central axis
of the microfilaria, a column of granuJes can be seen,
which are called somatic cells or nuclei. The granules are
absent at certain specific locations-a feature which helps
in the identification of the species. The specific locations
are as following (Fig. 3):
Filarial Worms
Fig. 2: Adult worm of Wucherer/a bancrofti
---...--Sheath _,...,.....---.,..---,,,,, Stylet
Anterior V-spot
Fig. 3: Morphology of Microfilaria bancrofti
At the head end is a clear space devoid of granules,
called the cephalic space. In Micro.ft/aria bancrofti,
the cephalic space is as Jong as it is broad, while in
Micro.ft/aria malayi, it is longer than its breadth. With
vital stains, a stylet can be demonstrated projecting
from the cephalic space (see Fig. 9).
In the anterior half of the microfilaria, is an oblique
area devoid of granules called the nerve ring.
Approximately midway along the length of the
microfilaria is the anterior V-spot, which represents
the rudimentary excretory system.
The posterior V-spot (tail spot) represents the cloaca
or anal pore.
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Paniker's Textbook of Medical Parasitology
The genital cells (G-cells) are situated anterior to the
anal pore.
The internal (central) body of Manson extending from
the anterior V-spot to G-cell one, representing the
rudimentary alimentary system.
The tail tip, devoid of nuclei in Mf bancrofti
(distinguishing feature), bears two distinct nuclei in
Mf malayi (see Fig. 9).
• Microfilariae do not multiply or undergo any further
development in the human body. If they are not taken up
by a female vector mosquito, they die.
• Their lifespan is believed to be about 2- 3 months.
• Tt is estimated that a microfilarial density of at least 15 per
drop of blood is necessary for infecting mosquitoes.
Periodicity
• The microfilariae circulate in the bloodstream.
• In India, China and many other Asian countries, they
show a nocturnal periodicity in peripheral circulation;
being seen in large numbers in peripheral blood only at
night (between 10 pm and 4 am).
• This correlates with the night biting habit of the vector
mosquito.
Infective larva
• Periodicity may also be related to the sleeping habits of
the hosts. lt has been reported that if the sleeping habits
of the hosts are reversed over a period, the microfilariae
change their periodicity from nocturnal to diurnal.
• Nocturnal periodic rnicrofilariae are believed to spend the
day time mainly in the capillaries of the lung and kidneys
or in the heart and great vessels.
• In the Pacific islands and some parts of the Malaysian
archipelago, the microfilariae are nonperiodic or
diurnal subperiodic, such that they occur in peripheral
circulation at all times, with a slight peak during the late
afternoon or evening. This is related to the day-biting
habits of the local vector mosquitoes (some authors
separate the subperiodic Pacific type of W. bancrofti as
a distinct species designated W. pacifica, but this is not
widely accepted).
Life Cycle
Wuchereria bancrofti passes its life cycle in two hosts (Fig. 4):
Definitive host: Man. No animal host or reservoir is
known for W. bancrofti.
Intermediate host: Female mosquito, of different species
acts as vectors in different geographic areas. The major
deposited on the skinl_,,, ___ ._
of man when
mosquito bites
(
Infective 3rd-stage
of mosquito
Mosquito
lymphatic system
and lymph nodes
X
larva lying in the
proboscis sheath
(intermediate host)
Man
(definitive host)
2nd-stage
larva
Short 1st-stage
larva Ingested by female
mosquito during
blood meal
Fig. 4: life cycle of Wuchereria bancrofti
P:229
Box 2: Parasites with mosquito as intermediate host
• Wuchereria bancrofti
• Brugia spp.
• Mansonella spp.
• Dirofilaria spp.
vector in India and most other parts of Asia is Cu/ex
quinquefasciatus (C.fatigans) (Box 2).
Infective form: Actively motile third-stage filariform larva is
infective to man.
Mode of transmission: Humans get infection by bite of
mosquito carrying filariform larva.
Development in mosquito: When a vector mosquito feeds on
a carrier, the microfilariae are taken in with the blood meal
and reach the stomach of the mosquito.
• Within 2-6 hours, they cast off their sheaths ( exsheathing),
penetrate the stomach wall and within 4- 17 hours migrate
to the thoracic muscles where they undergo further
development.
• During the next 2 days, they metamorphose into thefirststage larva, which is a sausage-shaped with a spiky tail,
measuring 125-250 x 10-15 µm (Fig. 4).
• Within a week, it moults once or twice, increases in
size and becomes the second-stage lar va, measuring
225-325 x 15-30 µm {Fig. 4).
• In another week, it develops its internal structures and
becomes the elongated third-stage filariform larva,
measuring 1,500-2,000 x 15-25 µm. It is actively motile
and is the infective form {Fig. 4).
• It enters the proboscis sheath of the mosquito, awaiting
opportunity for infecting humans on whom the mosquito
feeds.
• There is no multiplication of the microfilaria in the
mosquito and one rnicrofilaria develops into one infective
larva only.
• The time taken from the entry of the microfilaria into
the mosquito till the development of the infective thirdstage larva located in its proboscis sheath, constitutes
the extrinsic incubation period. Its duration varies with
environmental factors such as temperature and humidity,
as well as with the vector species. Under optimal
conditions, its duration is 10-20 days.
• When a mosquito with infective larvae in its proboscis
feeds on a person, the larvae get deposited, usually in
pairs, on the skin near the puncture site.
Development in man: The larvae enter through the puncture
wound or penetrate the skin by themselves.
• The infective dose for man is not known, but many larvae
fail to penetrate the skin by themselves and many more
are destroyed in the tissues by immunological and other
Filarial Worms
Table 4: Differences between classical and occult filariasis
Cause
Basic lesion
Organs
involved
Classkal filariasis
Due to adult and
developing worms
Lymphangitis,
lymphadenitis
Lymphatic vessels and
lymph node
Occult filariasis
Hypersensitivity to mlcrofilarial
antigen
Eosinophilic granuloma
formation
Lymphatic system, lung, liver,
spleen, joints
Microfilaria Present in blood Present in tissues but not in
blood
Serological Complement fixation Complement fixation test
test test not so sensitive highly sensitive
Therapeutic No response Prompt response to
response diethylcarbamazine (DEC)
defense mechanisms. A very large number of infected
mosquito bites are required to ensure transmission to
man, perhaps as many as 15,000 infective bites per person.
• After penetrating the skin, the third-stage larvae enter the
lymphatic vessels and are carried usually to abdominal
or inguinal lymph nodes, where they develop into adult
forms {Fig. 4).
• There is no multiplication at this stage and only one adult
develops from one larva, male or female.
• They become sexually mature in about 6 months and
mate.
• The gravid female worm releases large numbers of
microfilariae, as many as 50,000 per day. They pass
through the thoracic duct and pulmonary capillaries to
enter the peripheral circulation.
• The microfilariae are ingested with the blood meal by
mosquito and the cycle is repeated.
Prepatent period: The period from the entry of the infective
third-stage larvae into the human host till the first appearance
of microfilariae in circulation is called the biological
incubation period or the prepatent period. This is usually
about 8-12 months.
Clinical incubation period: The period from the entry of the
infective larvae, till the development of the earliest clinical
manifestation is called the clinical incubation period. This is
very variable, but is usually 8-16 months, though it may often
be much longer.
Pathogenesis
Infection caused by W. bancrofti is termed as wuchereriasis or
bancroftian filariasis.
The disease can present as (Table 4):
• Classical filariasis
• Occult filariasis.
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Classical filariasis:
Pathogenesis:
• It occurs due to blockage of lymph vessels and lymph
nodes by the adult worms. The blockage could be due to
mechanical factors or allergic inflammatory reaction
to worm antigens and secretions. The affected lymph
nodes and vessels are infiltrated with macrophages,
eosinophils, lymphocytes and plasma cells. The vessel
walls get thickened and the lumen narrowed or occluded,
leading to lymph stasis and dilatation of lymph vessels.
The worms inside lymph nodes and vessels may cause
granuloma formation, with subsequent scarring and
even calcification. Inflammatory changes damage the
valves in lymph vessels, further aggravating lymph
stasis. Increased permeability of lymph vessel walls lead
to leakage of protein-rich lymph into the tissues. This
produces the typical hard pitting or brawny edema of
filariasis. Fibroblasts invade the edematous tissues, laying
down fibrous tissue, producing the nonpitting gross
edema of elephantiasis. Recurrent secondary bacterial
infections cause further damage.
• Animal models have been developed, such as experimental filarial infection in cats with Brugia pahangi
or Br. malayi. These have helped in understanding
the pathogenesis of the disease, but in cats and other
animals, filarial infection does not cause elephantiasis.
Elephantiasis is a feature unique to human filariasis,
apparently caused by human erect posture and
consequent hydrodynamic factors affecting lymph flow.
Clinical manifestations: The most common presentations
of lymphatic filariasis are asymptomatic (subclinical)
microfilaremia, acute adenolymphangitis (AOL) and chronic
lymphatic disease.
• Most of the patienLs appear clinically asymptomatic but
virtually all of them have subclinical disease including
microscopic hematuria or proteinuria, dilated lymphatics
(visualized by imaging) and in men with W. bancrofli
infection, scrotal Iymphangiectasia (detected by
ultrasound).
• Acute adenolymphangitis is cha racterized by high
fever, lymphatic inflammation (lymphangitis and
lymphadenitis) and transient local edema.
Fever is of high grade, sudden in onset, associated
with rigors and last for 2 or 3 days.
Lymphangitis is inflamed lymph vessels seen
as red streaks underneath the skin. Lymphatics
of the testes and spermatic cord are frequently
involved, with epididymo-orchitis and funiculitis.
Acute lymphangitis is usually caused by allergic
or inflammatory reaction to filarial infection, but
may often be associated with streptococcal infection
also.
Lymphadenitis: Inflammation oflymph nodes. Most
common affected lymph nodes being inguinal nodes
followed by axillary nodes. The lymph nodes become
enlarged, painful and tender.
Lymphedema: This follows successive attacks of
lymphangitis and usually starts as swelling around
the ankle, spreading to the back of Lhe foot and leg.
It may also affect the arms, breast, scrotum, vulva, or
any other part of body. Initially, the edema is pitting
in nature, but in course of time, becomes hard and
non pitting.
Lymphangiovarix: Dilatation of lymph vessels
commonly occurs in the inguinal, scrotal, testicular
and abdominal sites.
The lymphangitis and lymphadenitis can involve both
the upper and lower extremities in both bancroftian
and brugian filariasis but involvement of genital
lymphatics occurs exclusively with W. bancrofti
infection. The genital involvement can be in the form
of funiculitis, epididymitis and hydrocele formation.
• Hydrocele: This is a very common manifestation of
filariasis. Accumulation of fluid occurs due to obstruction
of lymph vessels of the spermatic cord and also by
exudation from the inflamed testes and epididymis. The
fluid is usually clear and straw colored but may sometimes
be cloudy, milky, or hemorrhagic. The hydrocele may be
unilateral or bilateral and is generally small in size in
the early stage, but may occasionally assume enormous
proportions in association with elephantiasis of the
scrotum. The largest reported hydrocele weighed over 100
kilograms.
• Lymphorrhagia: Rupture of lymph varices lead ing
to release of lymph or chyle and resulting in chyluria
(Fig. 5), chylous diarrhea, chylousascitesand chylothorax,
depending on the involved site.
• Elephantiasis: This is a delayed sequel to repeated
lymphangitis, obstruction and lymphedema. Repeated
leakage of lymph into tissues first results in lymphedema,
then to elephantiasis, in which there is nonpitting
brawny edema with growth of new adventitious tissue
and thickened skin, cracks, and fissures with secondary
bacterial and fungal infections, commonly seen in leg but
may also involve other parts of body {Fig. 6).
Clinical features of filarlasis
• Asymptomatic microfilaremia, acute adenolymphangitis, lymphadenitis
, Lymphedema, lymphangiovarix, chronic funiculitis, epldidymiltis
hydrocele, elephantiasis, chylothorax, chyluria
Occultfllariasis:
• It occurs as a result of hypersensitivity reaction to
microfilarial antigens, not directly due to lymphatic
involvement.
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Fig. 5: Chylous urine
• Microfilariae are not found in blood, as they are destroyed
by the allergic inflammation in the tissues.
• Clinical manifestations:
Massive eosinophilia (30-80%)
Hepatosplenomegaly
Pulmona ry symptoms like dry nocturnal cough,
dyspnea and asthmatic wheezing.
Occult filariasis has also been reported to cause
arthritis, glomerulonephritis, thrombophlebitis,
tenosynovitis, etc.
Classical features of lymphatic filariasis are absent.
• Meyers Kouwenaar syndrome is a synonym for occult
filariasis.
• Tropical pulmonary eosinophilia:
This is a manifestation of occult filariasis which
presents with low-grade fever, loss of weight, and
pulmonary symptoms such as dry nocturnal cough,
dyspnea and asthmatic wheezing.
Children and young adults are more commonly
affected in areas of endemic filariasis including the
Indian subcontinent.
There is a marked increase in eosinophil count
(>3000 µm which may go up to 50,000 or more).
Chest X-ray shows mottled shadows similar to
miliary tuberculosis.
It is associated with a high level of serum
immunoglobulin E (IgE) and filarial antibodies.
Serological tests with filarial antigen are u su a lly
strongly positive.
Th e co ndition resp onds to treatment with
diethylcarbamazine (DEC), which acts on
microfilariae.
Filarial Worms
Fig. 6: Elephantiasis of the legs
Laboratory Diagnosis
The diagnosis of filariasis depends on the clinical features,
history of exposure in endemic areas and on laboratory
findings.
The laboratory tests that can be used for diagnosis has
been described in Flow chart 2.
Demonstration of microfilaria:
• Microfilaria can be demonstrated in blood, chylous urine
(Fig. 6) exudate of lymph varix and hydrocele fluid.
Peripheral blood is the specimen of choice.
• The method has the advantage that the species of the
infecting filaria can be identified from the morphology of
the microfilaria seen. It is also the method used for carrier
surveys.
• In India and other areas, where the prevalent filarial
species is nocturnally periodic, it is best to collect \"night
blood\" samples between 10 pm and 4 am.
• Microfilaria can be demonstrated in unstained as well as
stained preparations and in thick as well as thin smears
(Fig. 7).
Unstained film:
• Examination under the low power microscope shows
the actively motile microfilariae lashing the blood cells
around.
• The timing of blood collection is critical and should be
based on the periodicity of the microfilariae.
• The examination may be conveniently made the next
morning as microfilariae retain their viability and motility
for a day or 2 at room temperature.
Stained.film: A \"thick and thin\" blood smear is prepared on a
clean glass slide and dried.
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Flow chart 2: Laboratory diagnosis of Wuchereria bancrofti
laboratory diagnosis I
l I I l
Direct evidence
Detection of microfllariae
By examination of a thick
and thin blood smear,
stained with Giemsa
stain
By examination of
unstained mount of blood
under microscope
By acridlne orange -
microhematocrit tube
technique
Indirect evidence
• Eosinophilia in blood
• Elevated serum lgE
levels
Detection of adult worm
• Lymph node biopsy
• On X-ray (if worms are
calcified)
• High frequency
ultrasound and Doppler
within the scrotum
Note: Adult worms have a
distinctive pattern of
movement (termed the
ti/aria dance sign) within
the lymphatic vessel
lmmunodiagnosis
Antigen detection
ELISA
• ICT
• Both tests have
sensitivity of 93-100%
and specificity of
100% and sample
can be collected
during day time
Antibody detection
• CFT
• IHA
• IFA
These test have low
sensitivity and specificity
Molecular diagnosis
• Done by PCR
• The test is positive
only when microfilaria
are present in
peripheral blood.
Negative in chronic
filariasis
Abbreviations: CFT, complement fixation test; ELISA, enzyme-linked immunosorbent assay; ICT, immunochromatographic test; IFA, indirect
fluorescent antibody; lgE, immunoglobulin E; IHA, indirect hemagglutination; PCR, polymerase chain reaction
Fig. 7: Microfilaria in blood film
Source: Mohan H. Textbook of Pathology, 6th edition. New Delhi:
Jaypee Brothers Medical Publishers; 2010. p. 190.
• The thick part of the smear is dehemoglobinized by
applying distilled water. The smear is fixed in methanol
and stained with Giemsa, Leishman, or polychrome
methylene blue stains. Microfilariae may be seen under
the low power microscope in the thick film.
• The morphology of microfilariae can be studied in thin
film. The microfilaria of W. bancrofti are sheathed and
appear as smooth curves in stained smear and are 298
µm long and 7.5-10 µmin diameter (Fig. 7).
• By using a micropipette for taking a known quantity of
blood (20-60 rnm3
) for preparing the smear and counting
the number of microfilariae in the entire stained smear,
microfilaria counts can be obtained.
Concentration techniques: When the microfilaria density is
low, concentration techniques are used:
• Knott's concentration technique: Anticoagulated blood
(1 mL) is placed in 9 mL of 2% formalin and centrifuged
500 x g for 1 minute. The sediment is spread on a slide to
dry thoroughly. The slide is stained with Wright or Giemsa
stain and examined microscopically for microfilariae.
• Nucleopore filtration: In the filtration methods used
at present, larger volumes of blood, up to 5 mL, can be
filtered through millipore or nucleopore membranes
(3 µm diameter). The membranes may be examined
as such or after staining, for microfilariae. The filter
membrane technique is much more sensitive, so that
blood can be collected even during day time for screening.
The disadvantages of the technique are the cost and the
need for venipuncture.
• Dtethylcarbamazine provocation test: A small dose of
DEC (2 mg per kg body weight) induces microfilariae to
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Box 3: Parasites found in urine
• Wuchereria bancrofti
• Schistosoma hematabium
• Trichamanas vagina/is.
appear in peripheral blood even during day time. For
surveys, blood samples can be collected 20-50 minutes
after the administration of one 100 mg tablet of DEC to
adults.
• Other specimens: Microfilaria may be demonstrated in
centrifuged deposits of lymph, hydrocele fluid, chylous
urine or other appropriate specimens. Usually 10-20 mL
of the first early morning urine is collected for examination
and demonstration (Box 3).
Biopsy: Adult filarial worms can be seen in sections of biopsied
lymph nodes, but this is not employed in routine diagnosis.
Skin test: Intradermal injection offilarial antigens (extracts of
microfilariae, adult worms and third-stage larvae of 8. malayi
or of the dog filaria Dirofilaria irnrnitis) induce an immediate
hypersensitivity reaction. But, the diagnostic value of the skin
test is very limited due to the high rate of false-positive and
negative reactions.
Imaging techniques:
mtrasonography: High frequency ultrasonography (USG)
of scrotum and female breast coupled with Doppler imaging
may result in identification of motile adult worm (filaria
dance sign) within the dilated lymphatics.
• Adult worm may be visualized in the lymphatics of the
spermatic cord in up to 80% of the infected men with
microfilaria associated with W. bancrofti.
Radiology:
• Dead and calcified worms can be detected occasionally
by X-ray.
• In tropical pulmonary eosinophilia (TPE), chest
X-ray shows mottled appearance resembling miJiary
tuberculosis.
• Intravenous u rography, retrograde pyelography,
lymphangiography and lymphoscintigraphy may be used
to demonstrate abnormal lymphatic urinary fistula.
Serodiagnosis:
Demonstra tion of antibody: Several serological tests,
including complement fixation, indirect hemagglutination
(IHA), indirect fluorescent antibody (IPA), immunodiffusion
and immunoenzyme tests have been described.
• Indirect immunofluorescence and enzyme-linked
irnmunosorbent assay (ELISA) detect antibodies in over
95% of active cases and 70% of established elephantiasis.
Disadvantages: Antibody detection test cannot differentiate
between current and past infections.
Filarial Worms
Demonstration of circulating antigen: Highly sensitive
and specific test for detection of specific circulating filarial
antigen (CFA) have been developed for detection of recent
bancroftian filariasis.
• The Trop-bio test is a semiquantitative sandwich ELISA
for detection of CPA in serum or plasma specimen.
• Imrnunochromatographic test (JCT) is a new and rapid
filarial antigen test that detects soluble W. bancrofti
antigens using monoclonal antibody (AD/2) in the
serum of infected humans.
• Both assay have sensitivities of 93-100% and specificities
approaching 100%.
• Specific IgG4 antibody against W. bancrofti antigen
WbSXP-1 have been used to develop ELISA for detecting
circulating filarial antigen in sera of patients with filariasis.
• There is however, extensive cross-reactivity between
filarial antigens and antigens of other helminths,
including intestinal roundworm, thus interpretation of
serological findings can be difficult.
Advantages: Antigen detection tests are more sensitive than
microscopy and can differentiate between current and past
infections.
Molecular diagnostic technique: Polymerase chain reaction
(PCR) can detect filarial deoxyribonucleic acid (DNA) from
patient's blood, only when circulating microfilaria are present
in peripheral blood but not in chronic carrier state.
• Usually the test provides sensitivities that are up to tenfold
greater than parasitic detection by direct examination and
is 100% specific.
Indirect evidences: Eosinophilia (5-15%) is a common finding
in filariasis. Elevated serum IgE levels can also be seen.
Treatment
Diethylcarbamazine is the drug of choice. It is given orally
in a dose of 6 mg/ kg body weight daily for a period of 12 days
amounting to a total of 72 mg of DEC per kg of body weight.
It has both macro and microfilaricidal properties. Following
treatment with DEC severe allergic reaction (Mazzotti
reaction) may occur due to death of microfilariae. It kills both
microfilaria and adult worm.
Antihistamines or corticosteroids may require to control
the allergic phenomenon.
The administration of DEC can be carried oul in three
ways:
l. Mass therapy: In this approach, DEC is given to almost
everyone in community irrespective of whether they
have rnicrofilarernia disease manifestation or no signs
of infection except those under 2 years of age, pregnant
women and seriously-ill patients. The dose recommended
is 6 mg/ kg body weight. In some countries it is used
alone and in some, with albendazole or ivermectin. Mass
therapy is indicated in highly endemic areas.
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Paniker's Textbook of Medical Parasitology
Selective treatment: Diethylcarbamazine is given only to
those who are microfilaria-positive. In India, the current
strategy is based on detection and treatmelll of human
carriers and filarial cases. The recommended dose in the
Indian program is DEC 6 mg/ kg of body weight daily for
12 doses, to be completed in 2 weeks. In endemic areas,
treatment must be repeated every 2 years.
Diethylcarbamazine medicated salts: Common salt
medicated with 1-4 gram of DEC per kg has been used for
filariasis conLrol in Lakshadweep island, after an initial
reduction in prevalence had been achieved by mass or
selective treatment of microfilaria carriers.
Ivermectin: In doses of 200 µg/kg can kill the microfilariae
but has no effect on adults. It is not used in India. It is used in
regions of Africa.
Tetracyclines or doxycycline for 4-8 weeks also have an
effect in the treatment offilariasis by inhibiting endosymbiotic
bacteria (Wolbachia species) that are essential for the fertility
of the worm.
Supportive treatment:
• Chronic condition may not be curable by antifilarial drugs
and require other measures like elevation of the affected
limb, use of e lastic bandage and local foot care reduce
some of the symptoms of elephantiasis.
• Surgery is required for hydrocele.
• Medical management of chyluria includes bed rest, high
protein diet with exclusion of fat, drug therapy with DEC
and use of abdominal binders.
• Surgical management of refractory case includes
endoscopic sclerotherapy using silver nitrate.
Prophylaxis
The two major measures in prevention and control offilariasis
are:
l. Eradication of the vector mosquito.
Detection and treatment of carriers.
Eradication of vector mosquito:
• Antilarval measures: The ideal method of vector
control would be elimination of breeding places by
providing adequate sanitation and underground waste
water disposal system. However, this involves a lot of
expenditure, hence current approach in India is to restrict
the antilarval measures to urban areas by:
Chemical control: Using antilarval chemicals like:
• Mosquito larvicidal oil
• Pyrosene oil-E
• Organophosphorous larvicides like temephos,
fenthion, etc.
Removal of Pistia plant: Mainly restricted to control
of Mansonia mosquitoes leading to brugian filariasis.
• Anliadult measures: Adult mosquitoes can be restricted by
use of dichlorodiphenyltrichloroethane (DDT), dieldrin
and pyrethrum. However, vector mosquitoes of filariasis
have become resistant to DDT and dieldrin. Pyrethrum,
as a space spray, is still being used.
• Personal prophylaxis: Using mosquito nets and mosquito
repcllants is the best method.
KEY POINTS OF WUCHERER/A BANCROFT/
• Adult worm is white, thread-like with smooth cuticle and
tapering end.
• The female worm is viviparous. The embryo (microfilaria) is
colorless, sheathed, with tail-tip free of nuclei and actively
motile.
• Microfilaria in blood shows nocturnal periodicity (10 pm to 4
am).
• Definitive host: Man.
• Intermediate host: Cu/ex quinquefasciatus (C. fatigans).
• Microfilaria do not multiply in man. When taken up by vector
mosquito, it undergoes stages of development and become
third-stage filariform larva which is the infective form.
• Pathogenesis: Adult worm causes mechanical blockage of
lymphatic system and allergic manifestations.
• Clinical features: Early stage-fever, malaise, urticaria,
fugitive swelling, lymphangitis. Chronic stage-lymphadenitis,
lymphangiovarix, chyluria, hydrocele and elephantiasis.
Tropical pulmonary eosinophilia occurs due to hypersensitivity
reaction to filarial antigen.
• Diagnosis: Demonstration of microfilaria in peripheral blood or
chylous urine. Demonstration of adult worm in biopsy, Doppler
USG and X-ray. Demonstration of filarial antigen and antibody.
• Treatment: Drug of choice is DEC and ivermectin. Supportive
and surgical management in some cases.
Detection and treatment of carriers: The recommended
treatment is DEC 6 mg per kg body weight daily for 12
days, the drug being given for 2 weeks, 6 days in a week.
Brugia Malayi
History and Distribution
• the genus Brugia was named after Brug, who in 1927
described a new type of microfilaria in the blood of natives
in Sumatra.
• The adult worm of 8. malayi was described by Rao and
Maplestone in India (1940).
• Besides 8. malayi, the genus includes B. timori, which
parasitizes humans in Timor, Indonesia and a number of
animal species, such as B. pahangi and 8. patei infecting
dogs and cats.
• The geographical distribution of B. malayi is much more
restricted than that of W. bancrofti. It occurs in India and
Far-East, Indonesia, Philippines, Malaysia, Thailand,
Vietnam, China, South Korea and Japan.
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Fig. 8: Geographical distribution of Brugia malayi
In India, Kerala is the largest endemic area, particularly
the districts of Quilon, Alleppey, Konayam, Emakulam
and Trichur. Endemic pockets occur in Assam, Orissa,
Madhya Pradesh and West Bengal. B. malayi and
W. bancrofti may be present together in the same
endemic area, as in Kerala. In such places, B. malayi
tends to be predominantly rural and W. bancrofti urban
in distribution (Fig. 8).
Morphology
Adult worms:
• The adult worms of B. malayi are generally similar to
those of W. bancrofti, though smaller in size.
Microfilariae: The microfilariae of B. malayi, although
sheathed are different in a number of respects from
Micro.ft/aria bancrofti.
• Mf malayi is smaller in size, shows kinks and secondary
curves, its cephalic space is longer, carries double stylets
at the anterior end, the nuclear column appears blurred in
Giemsa-stained films and the tail tip carries two d istinct
nuclei, one terminal and the other subterminal (Fig. 9
and Table 5).
Life Cycle
the life cycle of 8. malayi is similar to that or W. bancrofli;
however, the intermediate host of Brugia are vectors of genera
Mansonia, Anopheles and Aedes. In India, main vectors are
Mansonia annulifera and M. uniformis.
• Pathogenicity, clinical features, laboratory diagnosis and
treatment are similar to W. bancrofti.
Filarial Worms
Table S: Distinguishing features of Mf. bancrofti and Mt. malayi
Features Mf. bancrofti Mf.malayi
Length 250-300 µm 175- 230µm
Appearance Graceful, sweeping Kinky, with secondary
curves curves
Cephalic space Length and breadth Almost twice as long as
equal broad
Stylet at anterior Single Double
end
Excretory pore Not prominent' Prominent
Nuclear column Discrete nuclei Blurred
Tail tip Pointed, free of nuclei Two distinct nuclei, are at
tip, the other subterminal
Sheath Faintly-stained Well stained
• Prevention: The breeding of Mansonia mosquito is
associated with certain plants such as Pistia. In absence
of these plants, mosquito cannot breed. Thus in countries
like Sri Lanka and India where M. annulifera is the chief
vector of B. malayi, the transmission of the parasite can be
effectively reduced by removal of these plants in addition
to the antilarval, antiadult and self prophylaxis methods
described in W. bancrofti.
Brugia Timori
Brugia timori is limited to Timor and some other islands of
Eastern Indonesia.
• The vector of B. timori is Anopheles barbirostris, which
breeds in rice fields and is a night feeder.
• Definitive host: Man. No animal reservoir is known.
• The microfilaria is larger than Mf malayi. The sheath
of Mf timori fails to take Giemsa stain with 5-8 nuclei
present in the tail.
• 11,e lesions produced by 8. timori are milder than those of
bancroftian or malayan filariasis. A characteristic lesion is
the development of draining abscesses caused by worms
in lymph nodes and vessels along the saphenous vein,
leading to scarring.
• SUBCUTANEOUS FILARIASIS
Loa Loa
Common Name
African eyeworm.
History and Distribution
Loa Loa, causing loiasis, \"fugitive swellings'' or \"Calabar
swellings'; was first detected in the eye of a patient in West
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Paniker's Textbook of Medical Parasitology
Large regular~ and smooth
body waves
Single---~H
style!
Length and
breadth of
cephalic
space equal
Sheath faintly ___ _
stained
Body nuclei----
discrete
Tail up,-----+tP
pointed
Microfilaria bancrofti
Kinky, small
and irregular
body waves
Length of
cephalic space
more than breadth
1----Well-stained
sheath
I- Body nuclei
blurred and
squeezed
11-+.~+--Tail tip rounded
with two nuclei
at tail-tip
Mlcrofl/aria malayi
Fig. 9: Schematic diagram showing distinguishing features of Microfilaria bancrofti and Microfi/aria ma/ayi
Indies in 1770. But at present, it is limited to its primary
endemic areas in the forests of West and Central Africa, where
about 10 million people are affected.
Life Cycle
Life cycle is completed in two hosts:
1.
Morphology 2
·
Adult worm: The adult worm is thin and transparent,
measuring about 30- 70 mm in length and 0.3-0.5 mm in
thickness. •
• In infected persons, they live in the subcutaneous tissues,
through which they wander. They may also occur in the •
subconjunctival tissue.
• Adults live for 4-l 7 years.
Micro.filaria: The microfilariae are sheathed with column of •
nuclei extending completely to the tip of the tail.
• They appear in peripheral circulation only during the day •
from 12 noon to 2 pm diurnal periodicity).
Definitive host: Man
Intermediate host or vectors: Day-biting flies (mango
flies) of the genus Chrysops, ( C. dimidiata, C. silacea and
other species) in which the microfilariae develop into the
infective third-stage larvae.
Infection is transmitted to man through the bite of infected
Chrysops during their blood meal.
The infective third-stage larvae enter the subcutaneous
tissue, moult, and develop into mature adult worm over
6-12 months and migrate in subcutaneous tissues.
Female worms produce sheathed microfilaria which have
diurnal periodicity.
The microfilaria is ingested by Chrysops during its blood
meal.
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• They cast off their sheaths, penetrate the stomach wall and
reach thoracic muscles where they develop into infective
larvae.
• Development in Chrysops is completed in about l O days.
Pathogenicity and Clinical Features
The pathogenesis of loiasis depends on the migratory habit
of the adult worm.
• The ir wanderings through subcutaneous tissues set
up temporary foci of inflammation, which appear as
swellings, of up to 3 cm in size, usually seen on the
extremities. These are the Calabar swellings or fugitive
swellings, because they disappear in a few days, only to
reappear elsewhere.
• Ocular manifestations occur when the worm reaches the
subconjunctival tissues during its wanderings. l h e ocular
lesions include granulomata in the bulbar conjunctiva,
painless edema of the eyelids and proptosis.
• Complications like nephropathy, encephalopathy and
cardiomyopathy can occur but are rare.
Laboratory Diagnosis
Diagnosis rests on the appearance of fugitive swelling in
persons exposed to infection in endemic area.
• Definitive diagnosis requires the detection of microfilaria
in peripheral blood or the isolation of the adult worm
from the eye.
• Microfilariae may be shown in peripheral blood collected
during the day.
• The adult worm can be demonstrated by removal from
the skin or conjunctiva or from a subcutaneous biopsy
specimen from a site of swelling.
• High eosinophil count is common.
Treatment
Diethylcarbamazine (8-10 mg/ kg per day for 21 days) is
effective against both the adult and the microfilarial forms
of l. Loa, but requires multiple courses. It has to be used with
caution as severe adverse reactions may develop following
the sudden death oflarge numbers of microfilariae.
• Simultaneous administration of corticosteroids
minimizes such reaction.
• lvermectin or albendazole although not approved by
Food and Drug Administration (FDA) for this purpose,
is effective in reducing microfilarial loads. lvermectin is
contraindicated in patients with heavy microfilaremia
(>5,000 microfilaria/m L).
• Treatment by surgical removal of the adult worms is
rarely done.
Filarial Worms
KEY POINTS OF LOA LOA
• Loa loa is also known as African eyeworm and causes loiasis.
• Vectors: Day-biting flies (Chrysops).
• Microfilaria is sheathed and nuclei extend up to tail tip.
• Microfilaria appears during the day (diurnal periodic).
• Clinical features: Subcutaneous swellings (Calabar swellings),
ocular granuloma, edema of eyelid and proptosis.
• Diagnosis: Demonstration of adult worm from s kin and
conjunctiva. Demonstration of microfilaria in peripheral blood
during day. High eosinophil count
• Treatment: Diethylcarbamazine with simultaneous
administration of corticosteroid of other drugs which may be
used. lvermectin or albendazole.
Onchocerca Volvulus
History and Distribution
Onchocerca volvulus, the \"convoluted filaria'; or the \"blinding
filaria\" producing onchocerciasis or \"river blindness\" was
first described by Leuckart in 1893.
• It affects about 40 million people, mainly in tropical Africa,
but also in Central and South America. A small focus of
infection exists in Yemen and South Arabia.
• Onchocerciasis is the second major cause of blindness in
the world.
Habitat
The adult worms are seen in nodules in subcutaneous
connective tissue of infected persons.
Morphology
Adult worm: The adult worms are whitish, opalescent, with
transverse striations on the cuticle (Fig. 10).
Fig. 1 0: Onchocerca volvulus
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• the posterior end is curved, hence the name Onchocerca,
which means \"curved tail''.
• the male worm measures about 30 mm in length and
0.15 mm in thickness and the female measures 50 cm by
0.4mm.
Microfilaria: The microfilariae are unsheathed and
non periodic.
• They measure about 300 by 0.8 µm.
• The microfilaria is found typically in the skin and
subcutaneous lymphatics in the vicinity of parent worms.
• They may also be found in the conjunctiva and rarely in
peripheral blood.
Life Cycle
Life cycle is completed in two hosts:
l. Definitive host: Humans are the only definitive host.
Intermediate hosts: Day-biting female black flies of th e
genus Simulium (black flies).
The vector Simulium species breed in \"fast-flowing
rivers\"; and therefore, the disease is most common along the
course of rivers. Hence, the name \"river blindness''.
• The female black flies are \"pool feeders\" and suck in
blood and tissue fluids. Microfilariae from the skin and
lymphatics are ingested and develop within the vector,
becoming the infective third-stage larvae, which migrate
to its mouth parts.
• The extrinsic incubation period is about 6 days. Infection
is transmitted when an infected Simulium bites a person.
• The prepatent period in man is 3- 15 months.
• The adult worm lives in the human host for about 15 years
and the microfilariae for about 1 year.
Pathogenicity and Clinical Features
Pathogenesis depends on the host's allergic and inflammato1y
reactions to the adult worm and microfilariae.
• The infective larvae deposited in the skin by the bite of the
vector develop at the site to adult worms. Adult worms are
seen singly, in pairs, or in tangled masses in subcutaneous
tissues. They may occur in the subcutaneous nodules or
free in the tissues.
• Th e subcutaneous nodule or onchocercoma is a
circumscribed, firm, nontender tumor, formed as a result
of fibroblastic reaction aroun d the worms. 1 odules vary
in size from a few mm to about 10 cm. 111ey tend to occur
over anatomical sites where the bones are superficial,
such as the scalp, scapulae, ribs, elbows, iliac crest,
sacrum and knees. the nodules are painless and cause no
trouble except for their unsightly appearance
• Microfilariae cause lesions in tl1e skin and eyes.
The skin lesion is a dermatitis with pruritus,
pigmentation, atrophy and fibrosis. In an immunologically hyperactive form of onchodennatitis called
as Sowdah, the affected skin darkens as a result of
intense inflammation, which occurs as result of
clearing of microfilariae from blood.
Ocular manifestati ons range from photophobia
to gradual blurring of vision, progressing to total
blindness. Lesions may develop in all parts of the eye.
The most common early finding is conjunctivitis with
photophobia. Other ocular lesions include punctale
or sclerosing keratitis, iridocyclitis, secondary
glaucoma, choroidoretinitis and optic atrophy.
Laboratory Diagnosis
Microscopy: The microfilariae may be demonstrated by
examination of skin snip from the area of maximal microfilariaJ
density such as iliac crest or trapezius region, which is placed
on a slide in water or saline. the specimen is best collected
around midday. this method is specific and most accurate.
• Microfilariae may also be shown in aspirated material
from subcutaneous nodules.
• ln patients with ocular manifestations, microfilariae may
be found in conjunctival biopsies.
• Adult worms can be detected in the biopsy material of the
subcutaneous nodule.
Serology: Serological tests are useful for the diagnosis of cases
in which microfiJariae are not demonstrated in the skin.
• Enzyme-linked immunosorbent assay is more sensitive
than skin snip tests. The test detects antibodies against
specific onchocercal antigen.
• A rapid card test using antigen 0Vl 6 to detect IgG4 in
serum has been evaluated.
Molecular diagnosis: Polymerase chain reaction from skin
snips is done in specialized laboratories and is highly sensitive
and specific.
Prophylaxis
In 1974, World Health Organization (WHO) launched a
control program in West Africa using aerial larvicide for vector
control and treatment of patients with ivermectin. This is
believed to have prevented blindness in millions of children.
Treatment
• Chemotherapy with ivermectin is the main stay of
treatment. Ivermectin is given orally in a single dose
of 150 µg/ kg either yearly or semiannually. ln areas of
Africa coendemic for 0. volvutus and Loa Loa, however,
ivermectin is contraindicated because of severe posttrea011enl encephalopathy seen in patients.
• Diethylcarbamazine and suramin have also been used.
DEC destroys microfilariae, but usually causes an intense
reaction (Mazzotti reaction) consisting of pruritus, rash,
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lymphadenopathy, fever, hypotension and occasionally,
eye damage.
• A 6 week course of doxycycline is macrofilariastatic,
rendering the female worm sterile as it targets the
Wolbachia endosymbiont offilarial parasites.
• Surgical excision is recommended when nodules arc
located on the head due to the proximity of the worm to
the eyes.
KEY POINTS OF ONCHOCERCA VOLVULUS
• Onchocerca volvulus, produces onchocerciasis or \"river
blindness\".
• The adult worm is white with transverse striation on the
cuticle. The posterior end is curved.
• Microfilaria is unsheathed, tail-tip free of nuclei and
nonperiodic.
• Definitive host: Humans.
• Intermediate host: Female black flies (Simulium).
• Clinical features: Subcutaneous nodule formation
(onchocercoma). Ocular manifestations-sclerosing keratitis,
secondary glaucoma. optic atrophy, chorioretinitis. It is the
second major cause of blindness in world.
• Diagnosis: Demonstration of microfilaria from skin snips
and aspirated material form subcutaneous nodules.
Demonstration of lgG4 antibody and PCR.
• Treatment: lvermectin is the drug of choice except in areas
coendemic for 0. volvulus and L. loa.
Mansonella Streptocerca
Also known as Acanthocheilonema, Dipetalonema, or
Tetrapetalonema streptocerca, this worm is seen only in West
Africa.
• 1he adult worms live in the dermis, just under the skin
surface.
• The unsheathed microfiliariae are found in the skin.
• Culicoides species are the vectors.
Chimpanzees may act as reservoir hosts.
Infection may cause dermatitis with pruritus and
hypopigmented macuJes.
• Diagnosis is made by demonstration of the microfilariae
in skin clippings.
• Ivermectin (single dose of 150 µg/ kg) is effective in
treating streptocerciasis.
• SEROUS CAVITY FILARIASIS
Mansonella Ozzardi
Mansonella ozza rdi is a New World filaria seen only in Central
and South America and th e West Indies.
• the adult worms are found in the peritoneal and pleural
cavities of humans.
Filarial Worms
• The nonperiodic unsheathed microfilariae are found in
the blood.
• Culicoides species are the vectors.
• Infection does not cause any illness.
• Diagnosis is made by demonstrating microfilariae in
blood.
• lvermectin (single dose 6 mg) is effective in treatment.
Mansonel/a Perstans
Also known as Acanthocheilonema, Dipetalonema, or
Tetrapetalonema perslans, this worm is extensively distributed
in tropical Africa and coastal South America.
• The adult worms live in the body cavities of humans,
mainly in peritoneum, less often in pleura, and rarely in
pericardium.
• The microfilariae arc unsheathed and subperiodic.
• Vectors are Culicoides species.
African primates have been reported to act as reservoir
hosts.
Infection is generally asymptomatic, though ii has been
claimed that it causes transient abdominal pain, rashes,
angioedema and malaise.
Diagnosis is by demonstration of the microfilariac in
peripheral blood or serosal effusion.
• Doxycycline (200 mg twice a day for 6 weeks) targeting
the Wolbachia endosymbiont in M. perstans is the first
effective treatment.
Zoonotic Filariasis
Filariae naturally parasitic in domestic and wild animals may
rarely cause accidental infection in man through the bite of
their vectors.
• In such zoonotic filariasis, the infective larvae develop
into adults, but do not mature to produce microfilariae.
The worm dies and the inflammatory reaction around the
dead worm usually causes clinical manifestations.
Brugia Pahangi
A parasite of dogs and cats in Malay ia may infect man and
cause lymphangitis and lymphadenitis.
Dirofilaria lmmitis
The dog \"heartworm\" is a common parasite of dogs, widely
distributed in the tropics and subtropics. When humans get
infected, the worm lodges in the right heart or branches of
the pulmonary artery. l he dead worm becomes an em bolus
blocking a small branch of the pulmonary artery, producing
a pulmonary infarct. The healed infarct may appear as a
\"coin lesion\" on chest radiography and can be mistaken for
malignancy.
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Dirofilaria Repens
A natural parasite of dogs, it may sometimes infect humans,
causing subcutaneous and subconjunctival nodules. Many
Dirojilaria species may form nodules in human conjunctiva
and are collectively calied Dirofilarla conjunctivae.
REVIEW QUESTIONS
Name the species of filarial worms that infect humans and
describe briefly the life cycle and laboratory diagnosis of
Wuchereria bancrofti.
Short notes on:
a. Microfilariae
b. Periodicity of microfilariae
c. Pathogenesis of lymphatic filariasis
d. Tropical pulmonary eosinophilia
e. Filariasis
f. Preventive measures in filariasis
g. Brugia malayi
h. Loaloa
i. Onchocerca volvulus
Differentiate between:
a. Occult and classical filariasis
b. Micron/aria bancrofti and Micron/aria malayi
MULTIPLE CHOICE QUESTIONS
All are true regarding filariasis except
a. Man is an intermediate host
b. Caused by Wuchereria bancrofti
c. Involves lymphatic system
d. DEC is used in treatment
All of the following are true about Brugia malayi except
a. The intermediate host in India is Mansonia mosquito
b. The tail tip is free from nuclei
c. Nuclei are blurred, so counting is difficult
d. Adult worm is found in the lymphatic system
Hydrocele and edema in foot occurs in
a. Wuchereria bancrofti
b. Brugia malayi
c. Brugia timori
d. Onchocerca volvulus
In w hich stage of filariasis are microfilaria seen in peripheral
blood
a. Tropical eosinophilia
b. Early adenolymphangitis stage
c. Late adenolymphangitis stage
d. Elephantiasis
Diurnal periodicity is seen in larvae of
a. Brugia malayi
b. Wuchereria bancrofti
c. Loa loa
d. Mansonella perstans
Which of the following microfilariae is unsheathed
a. Mf. loa
b. Mf. bancrofti
c. Mf. malayi
d. Mf. perstans
All of t he following parasites can be detected in urine sample
except
a. Wuchereria bancrofti
b. Schistosoma haematobium
c. Trichomonas vaginalis
d. Giardia lamblia
Fugitive or calabar swelling is seen in infection with
a. Onchocerca volvulus
b. Loa Joa
c. Wuchereria bancrofti
d. Brugia timori
River blindness is the name given to disease caused by
a. Loaloa
b. Onchocerca volvulus
c. Toxoplasma gondii
d. Acanthamoeba culbertsoni
The filarial worm which can be seen in conjunctiva is
a. Brugia malayi
b. Loaloa
c. Onchocerca volvulus
d. None of the above
Answer
a 2. b
b 9. b
a
b
b 5. C 6. d 7. d
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CHAPTER 21
• COMMON NAME
Guinea worm.
• HISTORY AND DISTRIBUTION
The guinea worm has been known from antiquity. It is
believed to have been the \"fiery serpent\" in the Bible, which
tormented th e Israelites on the banks of the Red Sea.
• The technique of extracting the worm by twisting it on a
stick, still practiced by patients in endemic areas is said to
have been devised by Moses. The picture of the \"serpent
worm\" on a stick may have given rise to the physician's
symbol of caduceus.
• Galen named the disease dracontiasis, (Greek dracodragon or serpent). Avicenna called it the Medina worm
as it was prevalent there. l lence, the name Dracunculus
medinensis (Dracunculus being the diminutive of Draco).
• The worm was present in tropical Africa, the Middle East
in Arabia, Iraq, Iran, and in Pakistan and India. ln India,
it was seen in the dry areas in Rajas than, Gujarat, Madhya
Pradesh, Andhra Pradesh, Maharashtra, Tamil Nadu
and Karnataka (fig. 1). About 50 million people were
estimated to be infected with the worm.
Fig. 1: Geographical distribution of Dracunculus medinensis
infection (before its eradication)
• The infection has been eradicated from India and all of
Southeast Asia region by 2000.
• The disease still remains endemic in 13 African countries
including Sudan (highest incidence), Niger, etc.
• HABITAT
The adult females of D. medinensis are usually found in the
subcutaneous tissue of the legs, arms and back in man.
• MORPHOLOGY
Adult Worm
The adult female is a long, cylindrical worm with smooth
milky-while cuticle resembling a long piece of white twine. It
has a blunt anterior end and a tapering recurved tail (Fig. 2).
It measures about a meter (60-120 cm) in length and 1-2
mm in thickness.
• The body of the gravid female is virtually filled with the
branches of an e normous uterus, containing some 3
million embryos.
• The female worm is viviparous (Box 1).
• The male worm, which is rarely seen, is much smaller
than female being 10-40 mm long and 0.4 mm thick.
Female worm survives for about a year, whereas life span
of male worm is not more than 6 months.
Larva
The larva measures 500- 750 µmin length and 15- 25 µmin
breadth.
• It has a broad anterior end and a slender filiform tail
which extends for a third of the entire body length (Fig. 3).
• The cuticle shows prominent striations.
• The larva swims about with a coiling and uncoiling
motion.
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Fig. 2: Adult worm of Dracuncu/us medinensis
Box 1: Viviparous nematodes
• Dracunculus medinensis
• Trichinella spiralis
• Wuchereria bancrofti
• Brugia malayi
• Brugia timori
Ovoviviparous nematodes
• Strongyloides stercoralis.
• LIFE CYCLE
D. medinensis passes its life cycle in two hosts:
l. Definitive host: Man
Intermediate host: Cyclops, in which embryos undergo
developmental changes. There is no animal reservoir
(Table l ).
Infective Form
1hird-stage larva present in the hemocele of infected Cyclops.
• Mode of transmission: Humans get infected by drinking
unfiltered water containing infected Cyclops.
• Incubation period: About 1 year.
• The adult worm, which is viviparous discharges larvae,
which are ingested by the freshwater crustacean. Cyclops,
the intermediate host.
Development of Adult Worm in Man
When water containing infected Cyclops is swallowed by
man, the Cyclops is killed by the gastric acidity and the guinea
worm larvae present in its hemocele are released.
• The larvae penetrate the wall of the duodenum and reach
the retroperitoneal and subcutaneous connective tissues.
Fig. 3: Larva of Dracunculus medinensis
Table 1: Parasites requiring one intermediate host to complete their
life cycle
Intermediate host
Man
Pig
Cow
Snail
Cyclops
Sandfly
Tsetse fly
Chrysops
Mosquito
Tick
Triatomine bug
Flea
Parasite
• Plasmodium species
• Echinococcus granu/osus
• Echinococcus multilocularis
• Taenia multiceps
• Taenia solium
• Taenia saginata asiatica
• Sarcocystis suihominis
• Trichinella spiralis
• Taenia saginata
• Sarcocystis hominis
Schistosoma species
Dracunculus medinensis
Leishmania species
T rypanosoma species
Loa/oa
• Wuchereria bancrofti
• Brugia spp.
• Mansonella spp.
Babesia species
Trypanosoma cruzi
• Hymenolepis nano
• Hymenolepis diminuta
• Dipylidium caninum
• Here, the larvae develop into male and female adults in
about 3- 4 months and mate.
• After mating, the male worms die in the tissues and
sometimes become calcified.
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• In a nother 6 months time, the fertilized female worm
grows in size, matures, and migrates within the connective
tissues throughout the body, to fin ally reach a site where it
is likely to come into contact with water.
• The most common site involved is the leg, but other sites
such as arms, shoulder, breast, buttocks, or genitalia may
also be affected.
• At this site, it secretes a toxin that causes a blister
formation, which eventually ruptures, discharging a
milky-white fluid containing numerous LI stage larvae.
• this process continues for 2-3 weeks, till aU the larvae are
released.
Development of Larvae in Cyclops
The larvae swim about in water, where they survive for about
a week.
• They are swallowed by the freshwater copepod Cyclops,
which is the intermediate host (Fig. 4).
• The larvae penetrate the gut wall of the Cyclops and enter
its body cavity, where they molt twice.
• In about 2-4 weeks, they develop into the infective thirdstage larvae (L3).
• The entire life cycle takes about a year, so that all the
infected p ersons develop the blisters and present with
clinical man ifestations at about the same time of the year
(Fig. 4).
• PATHOGENICITY AND CLINICAL FEATURES
D. medinensis causes dracunculiasis or dracunculosis.
• Infection induces no illness till the gravid female worm
comes to lie under the skin, ready to discharge its embryos.
• The body fluid of the adult worm is toxic and leads to
blister formation.
• A few hours before the developme nt of the blister,
there may be constitutional symptoms such as nausea,
vomiting, intense pruritus and urticaria! rash.
Dracunculus Medinensls
• The blister develops initially as a reddish papule with a
vesicular center and surrounding induration.
The most common sites for blister formation are the feet
between the metatarsal bones or on the ankles.
• The fluid in the blister is a sterile yellowish liquid with
polymorphs, eosinophils and mononuclear cells.
• The local discomfort diminishes with the rupture of the
blister and release of the embryos.
• Seconda.ry bacterial infection is frequent. Sometimes, it
may lead to tetanus.
• Sometimes, the worm travels to unusual sites such as the
pericardium, the spina l canal, or the eyes, with serious
effects.
• Dracunculiasis lasts usually for 1-3 months.
• LABORATORY DIAGNOSIS
• Detection of adult worm: Diagnosis is evident whe n
the tip of the worm projects from the base of the ulcer.
Calcified worms can be seen by radiography.
• Detection of larva: By bathin g the ulcer with water, the
worm can be induced to release the embryos (LI larvae),
which can be examined under the microscope.
• Skin test: An intradermal test with guinea worm antigen
elicits positive response.
Serological test: Enzyme-linked immunosorbent
assay (ELISA) and immunofluorescence assay (lFA) are
frequemJy used to detected antibodies to D. medinensis
(Flow chart 1).
• TREATMENT
• Antihistaminics and steroids are of help in the initial stage
of allergic reaction.
• Metronidazole, niridazole and thiabendazole are useful
in treatment.
Flow chart 1: Laboratory diagnosis of Dracunculus medinensis
Laboratory diagnosis
~ + + + + t Detection of adult Detection of X-ray Skin test Serological test Blood test worm larva Calcified worms Guinea worm to detect antibodies: reveals
From the base Under the can be seen in antigen injected · ELISA eosinophilia
of ulcer microscope radiography intradermally elicits • IFA positive response
Abbreviations: ELISA, enzyme-linked immunosorbent assay; IFA, immunofluorescence assay
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Cyclops are digested
in stomach and L3
larvae released
Man (Definitive host)
Cyclops containing
L3 stage larva
Larvae reach the
retroperitoneal
and subcutaneous
connective tissues,
and mature
into adult worms
Man
(Definitive host)
Adult worm in the
subcutaneous tissue
Gravid female in
subcutaneous tunnel
ready to discharge
larvae on contact
with water
Adult female
discharging
larvae in
Water water (L 1 stage)
Cyclops
(Intermediate host)
Larva penetrate the gut
wall of Cyclops (intermediate host)
and enter the body cavity
Fig. 4: Life cycle of Dracuncu/us medinensis
Motile L 1 stage
larva in water
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Fig. 5: Ancient technique of removing adult worm from blister
• For removal of the worm, the best method is the ancient
technique of patiently twisting it around a stick. It may
take 15-20 days to extract the whole worm but if care
is taken not to snap the worm , this method is safe and
effective (Fig. 5).
• Surgical removal of the worm under anesthesia is another
m ethod of treatment.
• PROPHYLAXIS
• Provision of protected piped water supply is the best
method of prevention or else boiling or fil tering water
through a cloth and then consuming water.
• Destroying Cyclops in water by chemical treatment wilh
Abate (temephos).
• Not allowing infected persons to bathe or wade in sources
of drinking water.
Note: Because of its simple life cycle, localized distribution,
and the absence of animal reservoirs, guinea worm infection
was eradicable. Measures to eliminate the infection have been
successful. Global eradication of the infection is imminent.
1
KEY POINTS OF DRACUNCULUS MEDINENSIS
• Guinea worm infection has been eradicated from India.
• Adult females are found in subcutaneous tissue of man.
• Female worm is viviparous releasing thousands of motile firststage larvae into the water.
• Definitive host: Humans.
Dracunculus Medinensis -
• Intermediate host: Cyclops, in which larvae undergo
development changes to become third-stage larvae.
• Infective form to humans: Cyclops containing L3 larvae.
• Clinical features: Pruritus, urticaria! rash, blister formation in
skin and cellulitis.
• Diagnosis: Detection of adult worm and larval form in ulcer.
Demonstration of dead worm by X-ray. Serology-ELISA and
IFA.
• Treatment: Antihistaminics and steroids in initial stage.
Metronidazole and niridazole are useful. Surgical removal of
the worm.
REVIEW QUESTIONS
List viviparous nematodes and describe briefly the life cycle and
laboratory diagnosis of Dracunculus medinensis.
Short notes on:
a. Pathogenicity and clinical features of dracunculosis
b. Tissue nematodes
c. Prophylaxis of guinea worm infection
MULTIPLE CHOICE QUESTIONS
Which of the following parasite does not enter into the body by
skin penetration
a. Dracunculus
b. Necator americanus
c. Ancylostoma duodenale
d. Strongyloides
Definitive host for Guinea worm is
a. Man
b. Cyclops
c. Snail
d. Cyclops and man
Guinea worm is
a. Enterobius
b. Trichuris
c. Dracunculus
d. Taenia solium
Cyclops is the source of infection in
a. Dracunculus
b. Spirometra
c. Both
d. None
Answer
a 2. a 3. C 4. C
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CHAPTER 22
• ANG/OSTRONGYLUS CANTONENSIS
Common Name
Rat lungvvorm.
History and Distribution
Angiostrongylus cantonensis causes eosinophilic meningoencephalitis (cerebral angiostrongyliasis) in humans.
• This condition was first reported from Taiwan in 1945.
• Since then, hundreds of cases have occurred in Taiwan,
Thailand, Indonesia and the Pacific islands.
• Human infection has also been recorded in lndia, Egypt,
Cuba and the United States of America (USA).
Habitat
The adult worm is present in the branches of pulmonary
artery in rats.
Morphology
• It is about 20 mm long and 0.3 mm thick.
• Eggs of Angiostrongylus resemble those of hookworms.
Life Cycle
Natural host: Rats.
Intermediate hosts: Molluscs, slugs and snails.
Jnfectiveform: Third-stage larvae. .
.
.
The eggs hatch in the lungs and the larvae which migrate
up the trachea are swallowed and expelled in the feces.
The larvae infect molluscs, slugs and snails, which are the
intermediate hosts. Crabs, freshwater prawns and frogs
have also been fow1d to be naturally infected (Box 1).
The larva undergoes two molts .
In about 2 weeks, the infective third-stage larvae develop,
which can survive in the body of the intermediate host for
about a year.
Rats become infected when they eat the molluscs.
Box 1: Nematodes with crabs and crayfishes as source of infection
• Angiosrrongy/us canronensis
• Paragonimus westermani
• In the rat, the larvae penetrate the gut wall to enter the
venules and are carried in circulation to the brain, where
they develop into young adults in about a month.
• These penetrate the cerebral venules and reach the
pulmonary artery, where they lodge, mature, and start
laying eggs.
• Human infection is acquired by eating infected molluscs
and other intermediate hosts containing the third-stage
larvae. Infection may also occur through raw vegetables
or water contaminated with the larvae.
• The larvae penetrate the gut and are carried to the brain,
but they are unable to develop further.
• They die and induce an inflammatory reaction in the
brain and meninges to produce meningoencephalitis.
• The incubation period is about 2-3 weeks.
Clinical Features
Patients present with intense headache, fever, neck stiffness,
convulsions and various degrees of pareses.
• The worm may also cause ocular complications.
• Infection docs not seem to confer immunity, as second
attacks have been recorded.
• Fatality is rare.
Diagnosis
Peripheral eosinophilia and high cerebrospinal fluid (CSF)
eosinophilia (up to 90%) are constant features.
• Larvae and adult worms may be seen in CSF (Table 1).
Treatment
Most cases recover sponta neously, only some develop
residual pareses.
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Table 1: Parasites found in cerebrospinal fluid
Protozoa Helminths
• Trypanosoma brucei spp. Angiasrrongylus cantonensis
• Naegleria fowleri
• Acanthamoeba spp.
• Anthelmintic treatment is not recommended, as the
disease is due to dead larvae.
• The drugs may even enhance the illness due to destruction
of more larvae.
Nole: Angiostrongylus costaricensis, inhabiting the mesenteric
arteries of wild rodents in Costa Rica in Central America,
may cause human infections. The disease presents as
inflammation of the lower bowels and is known as abdominal
angiostrongyliasis.
• CAPILLARIA PHILIPPINENSIS
C. phillippinensisis a small nematode, about 3- 4 mm long. It
belongs to the superfamilyTrichuroidea.
History and Distribution
It has been responsible for several fatal cases of diarrheal
illness in the Philippines from 1963.
• It has also been reported from Thailand, Japan, Iran and
Egypt.
Habitat
The adult worn inhabits the small intestine particularly the
jejunum.
Life Cycle
Definitive host: Birds (fish-eating birds)
Intermediate host: Fish.
• Its life cycle has not been worked out.
• Human infection is believed to occur by eating infected
fish, which are the intermediate hosts harboring the
infective larvae.
• Autoinfection is stated to be responsible for the high
degree of infection in man.
Clinical Features
lhe clinical disease consists of malabsorption syndrome with
severe diarrhea, borborygmi and abdominal pain.
Seriou cases may be fatal in 2 weeks to 2 months.
Miscellaneous Nematodes
Diagnosis
Diagnosis is made by detection of the eggs, larvae and adults
in stools. The eggs resemble those of Trichuris trichiura, but
are smaller.
Treatment
Mebendazole is useful in treatment.
ote: C. hepatica is a common parasite of rats, which may
occasionally infect man causing hepatitis that may be fatal.
• GNATHOSTOMA SPINIGERUM
History and Distribution
Gnathostoma spi11igerum, originally described from gastric
tumors of a tiger, parasitizes dogs, tigers, lions, cats and their
wild relative .
Gnathostomiasis is a zoonotic infection of man.
• lluman infections have been reported from 7hailand and
other countries in the Far East.
Cases of human infection with G. spinigerum and a related
species G. hispidum have also been reported from India.
Morphology
It is a small spirurid nematode. The female (25- 55 mm) is
longer than the male ( I 0-25 mm).
• Th e eggs a re oval, b rown, unsegmented bearing a
transparent knob-like thickening at one end {Fig. 1).
Life Cycle
Definitive host: Dog, cat and other carnivorous animals
First intermediate lwst: Cyclops
Second interm ediate host: Freshwater fish and frog
Fig. 1: Adult worm and egg of Gnathostoma splnigerum
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Table 2: Helminths causing central nervous system (CNS) infection
Cestodes Trematodes Nematodes
• Taenia solium • Schistosoma japonicum • Trichinella spiralis
• Taenia mult,ceps • Paragonimus westermani • Angiostrongylus cantonensis
• Spirometra spp.
• Echinococcus granulosus
• Echinococcus multi/ocularis
Paratenic host: Birds and humans.
Adult worm resides in the tumors or granulomatous
lesions of the sromach wall of cat and dog. Eggs are laid
in the tumors.
• They pass into gastric lumen by means of an aperture and
are discharged in feces into water, where they hatch into
first-stage larva.
• L1 larvae are ingested by Cyclops (first intermediate host)
in which the second-stage larvae develop.
• Cyclops is eaten by fishes, frogs and snakes, in which the
third-stage larvae develop (L3).
• When the third-stage larvae are eaten by cats, dogs, or
other suitable hosts, the larvae develop into adults inside
their body.
• When other hosts that are not suited to be a definitive host
(reptiles, buds or mammals) get infected, the larva does
not undergo any furth er development and such a host is
paratenic.
• Humans get infected by eating unde rcooked fish
containing third-stage larvae, but further development of
the worm cannot proceed normally in paratenic host.
The larvae migrate in the tissues of in fected persons,
causing indurated nodules or abscesses and creeping
eruplion (larva migrans) (Table 2).
Clinical Features
The migration of larvae in the tissues of the infected persons
leads to indurated nodules or abscesses and creeping
eruption.
• When the nodules are superficial, they can be incised and
the larvae can be removed.
thhe wandering larvae may reach the brain or eyes causing
severe damage.
Diagnosis
An intradermal test using the larval or adult antigens has
been described.
• Th e lesion can be biopsied and the presence of typical
larva confirms the diagnosis.
• Toxocara canis
• Toxocara coli
• Gnathostoma spinigerum
• Strongyloides stercoralis
Table 3: Parasites with fishes as the source of infection
Freshwater fish
• Gnathostoma spinigerum
• Capillaria philippinensis
• Clonorchis s/nensis
• Heterophyes heterophyes
• Metagonimus yokogawai
• D1phyllobothrium /arum
Treatment
Marine fish
Anisakis simplex
• Incision of the lesion and removal of larva.
Albendazole, mebendazole in high closes has also been
recommended.
• ANISAKIASIS
Anisakis species are nematode parasites of marine mammals
like dolphins, seals and whales.
Anisakiasis is common in Japan and other places like
etherland and USA where fresh or undcrtreatcd fish is a
popular food (Table 3).
Life Cycle
Dejinitive host: Dolphin, seals and whales
Intermediate host: Sea fishes
• 1he eggs are passed in seawater, hatch a nd infect marine
crustacea (krill).
Marine fish eats the infccrcd krill and the infective larvae
remain in the fish's viscera and flesh.
• Wh en human s consume un cooked or improperly
preserved fish containing the infective larvae, they
penetrate the gut wall at the level of the throat, stomach,
or intestine, leading to local inflammation and granuloma
formation.
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Clinical Features
infection with the larva of anisakis is known as anjsakJasls or
herring worm disease.
• Local inflammation and granuloma formation is present
at the level of throat, stomach, or intestine, depending on
the level of penetration of gut wall.
• The illness varies according to the site involved, such as
throat irritation or acute gastric or bowel symptoms.
• No case has been reported from India.
Treatment
Endoscopic surgical treatme nt of gastric and intestina l
anisaldasis is the method of choice.
Prophylaxis
Proper cooking of sea fish.
REVIEW QUESTIONS
Short notes on:
a. Anisakiasis
b. Gnathostoma spinigerum
c. Angiostrongylus cantonensis
d. Paratenic host
Miscellaneous Nematodes -
MULTIPLE CHOICE QUESTIONS
Rat lung worm is the common name of
a. Paragonimus westermani
b. Toxocara can is
c. Angiostrongylus cantonensis
d. Mansonella streptocerca
2 . Paratenic host for Angiostrongylus cantonensis is
a. Rat
b. Man
c. Frog
d. Camel
All of the following parasites are found in CSF except
a. Naegleria
b. Acanthamoeba
c. Angiostrongylus
d. Trypanosoma
Definitive host for Capillaria philippinensis is
a. Man
b. Rat
c. Birds
d. Fish
Answer
c 2. b 3. d 4. C
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CHAPTER 23
Diagnostic Methods in
Parasitology
• INTRODUCTION
Laboratory procedures play an important role in the
diagnosis of parasitic infections, both for confirmation of
clinical suspicion and for identifying unsuspected infections.
The principles of laboratory diagnosis are the same as in
bacterial and viral infections, but the relative importance of
the different methods varies greatly.
• While isolation of the infecting agent and detection of
specific antibodies are the major methods in bacteriology
and virology, they are of much less importance in
parasitology than morphological identification of the
parasite by microscopy.
• Compared to bacteria a nd viruses, parasites are very
large and possess distinctive shape and structure, which
enables their sp ecific diagnosis on morphological
grounds.
• Due to tl1eir complex antigenic structure and extensive
cross-reactions, serological diagnosis is of limited value
in parasitic infections.
• Although many pathogenic parasites can be grown in
laboratory cultures, this method is not suitable for routine
diagnosis because of its relative insensitivity and the delay
involved.
• Morphological diagnosis of parasites consists of two steps:
(1) detection of the parasite or its parts in clinical samples
and (2) its identification.
l. Detection depends on collection of the appropriate
samples and the ir examinatio n by suitable
techniques.
Identification requires adequate skill and expertise
in recognizing the parasite in its various stages and its
differentiation from morphologically similar artifacts.
A description of the common diagnostic techniques in
parasitology is given here.
• EXAMINATION OF STOOL
Collection of Fresh Stool Specimen
• All stool specimens should be collected in a suitable,
clean, wide mouthed container like a plastic container
with a light-fitting lid, waxed cardboard box, or match box.
• All fresh specimens should be handled carefully because
each specimen represents a potential source of infectious
material.
• The specimen should not be contaminated with water,
urine, or disinfectants.
Liquid stools should be examined or preserved within
30 minutes of passage. Soft stools should be examined or
preserved within 1 hour of passage and formed stool should
be examined or preserved within 24 hours of passage.
• Normally passed stools are preferable, although samples
obtain ed after purgative (sodium sulfate) or high saline
enema may also be used.
• Examination of fresh specimens is necessary for observing
motility of protozoan parasites.
• Stool should be examined for its consistency, color, odor
and presence of blood or mucus.
• Tn some instances, parasites may be seen on gross
inspection, as in the case of roundworm, pinwonn , or
tapeworm proglottids.
Microscopic Examination
• 111e microscope should be equipped with a micrometer
eyepiece, as it is often essential to measure the size of
p arasites. For example, the differentiation between
cysts of the pathogenic Entamoeba histolytica and the
nonpathogenic E. hartmanni is based entirely on their
sizes.
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• Microscopy should also include contributory findings
such as the presence of Charcot-Leyden crystals and
cellular exudates such as pus cells, red blood cells (RBCs)
and macrophages.
• For detection of parasites, ii is best to employ a
combination of methods, as different methods serve
different purposes.
• The methods include examination of: (i) wet mounts, (ii)
thick smears, and (iii) permanent-stained preparations.
• Various concentration methods can be used to increase
the sensitivity of microscopic examination.
• If there is a delay in examination, use of preservatives
such as formalin, sodium acetate and polyvinyl alcohol is
recommended.
Wet Mounts
• Unstainedwet.film:The unstained wet film is the standard
preparation and is made by emulsifying a small quantity
of stool in a drop of (0.85%) saline placed on a slide and
applying a coverslip (22 mm x 22 mm) on top, avoiding
air bubbles. A proper preparation should be just dense
enough for newspaper print to be read through it. If the
feces con tains mucus, it is advisable to prepare films using
the mucus part. 1he entire field under coverslip should be
systematically examined with low-power objective (l 0X)
under lowlight intensity. Any suspicious object may then
be examined with the high-power objective.
• Wet saline mounts: Wet saline mounts are particularly
useful for detecting live motile trophozoites of E.
histolytica, Balantidium coli and Giardia lamblia. Eggs
of helminths a re also readily seen. Rhabditiform larvae
of Strongyloides stercoralis are detected in freshly passed
stool.
• Eosin staining: Eosin 1 % aqueous solution, can be used
for staining wet films. Eosin stains everything except living
protoplasm. Trophozoites and cysts of protozoa, as well as
helminth larvae and thin-walled eggs stand out as pearlywhite objects against a pink backgrou nd and can be easily
detected. Chromatoid bodies and nuclei of amebic cysts
can be seen prominently. Eosin also indicates the viability
of cysts; live cysts are unstained and dead ones are stained
pink.
• Iodine staining: Iodine staining or wet mounts is another
standard method of examination. Either Lugol's iodine
diluted (5 g iodine, 10 g potassium iodide and 100 mL of
distilled water) or Dobell and O'Connor iodine solution
(1 g iodine, 2 g potassium iodide and 50 ml of distilled
water) are used. Iodine helps to confirm the identiry of
cysts, as it prominently stains the glycogen vacuoles and
nuclei. Protozoan cyst stained with iodine show yellowgold cytoplasm, brown glycogen material and pale
refractile nuclei.
Diagnostic Methods in Parasitology -
Thick Smears
These are not useful for routine examination, b ut are valuable
in surveys for intestinal helminth eggs.
The method described by Kato and Miura in 1954 is
known as the Kato thick smear technique.
• About 50 mg stool is taken on a slide and covered with a
special wettable cellophane coverslip soaked in glycerin
containing aqueous malachite green.
• The preparation is left for about a n hour at room
temperature, during which the glycerin clears the stool,
enabling the helminth eggs to be seen d istinctly under
low-power magnification.
• This method is, however not useful for diagnosis of
protozoa or helminth larvae.
Permanent Stained Smears
Permanent stained smears a re examined normally under oil
immersion {lO0X} objective.
• Confirmation of the intestin al p rotozoan, both
trophozoites and cysts, is the primary purpose of this
technique.
• Helminthic eggs and larvae take up too much stain and
usually cannot be identified.
• Permanent smear can be prepared with both fresh and
polyvinyl alcohol preserved stool specimen.
• The two methods commonly used are: (1) the ironhematoxylin stain and (2) Wheatley's trichrome stain. The
iron-hematm xylin .. is the older method, but is more difficult.
.lron-hematoxylin stain
Procedure:
• Fecal smear on a slide is fixed in Schaudinn's
solution for 15 minutes and is immersed
successively for 2-5 minutes in 70% alcohol, 70%
alcohol containing a trace of iodine, and then
50% alcohol for 2-5 minutes.
• It is washed in water for 5- l O minutes and
imme rsed in 2% aqu eous ferric ammonium
sulfate solution for 5- 15 minutes.
• It is again washed in water for 3-5 minutes and
stained with 0.5% aqueous hematoxylin for 5- 15
minutes.
• It is washed for 2-5 minutes and differentiated
in saturated aqueous solution of picric acid for
10- 15 minutes.
• It is then washed fo r 10- 15 minutes and
dehydrated by passing through increasing
strengths of alcohol, cleared in toluene or xylol
and mounted.
Trlchrome stain (Wheatley's method)
• The trichrome technique of Wheatley for stool
specimens is a modification of Gomori's original
staining procedure for tissue.
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Paniker's Textbook of Medical Parasitology
Box 1: Reagents of trichrome stain
• Chromotrope 2R: 0.6 g
• Light green SF: 9.3 g
• Phosphotungstic acid: 0.7 g
• Acetic acid (glacial): 1.0 ml
• Distilled water: 100 ml.
• lt is a quicker and simpler method, whi ch
produces uniformly well-stained smears of the
intestinal protozoa, human cells, yeast cells and
artifact material in about 45 minutes or less.
Procedure:
• 1he smear is fixed in Schaudinn's solution and
taken successively through alcohol, as earlier.
• Trichrome stain (chromotrope 2R, light green SF,
phosphotungstic acid in glacial acetic acid and
distilled waler) is then applied for 5-10 minutes,
differentiated in acid-alcohol dehydrated, cleared
and mounted (Box l).
Modified trichrome stain for microsporidia:
This staining method is based on the fact that stain
penetration of the microsporidial spore is very
difficult, thus more dye is used in d1e chromotrope
2R than that routinely used to prepare Wheatley's
modification of Lrichrome method and the staining
time is much longer (90 m inutes).
Other sta ining techniques are used for specia l
purpose. For exampl e, modified acid-fast or
Giemsa stain is employed for detection of oocysts of
Cryptosporidium and fsospora.
Modified Ziehl-Neelsen ( acid-fast) stain (hot method):
Oocysts of Cryptosporidium and Isospora in fecal
specimens may be difficult to detect, without special
staining. Modified acid-fast stains are recommended
to demonstrate these organisms.
Application of heat to the carbolfuchsin assists in
the staining and the use of a milder decolorizer (5%
sulfuric acid) allows the organisms lo retain more of
their pink-red color.
Kinyoun's acid-fast stain (cold method):
Cryptosporidium and Isospora h ave been
recognized as causes of severe dia rrhea in
immunocompromised hosts but can also cause
diarrhea in immunocompetent hosts.
Kinyoun's acid-fast stains are recommended to
demonstrate these organisms.
Unlike th e Ziehl-Neelsen modified acid-fast stain,
Kinyoun's stain does not require th e heating of
reagents for staining (Box 2).
Procedure:
Smear 1- 2 drops of specimen on the slide and allow
it to air dry.
Box 2: Reagents of Kinyoun's acid-fast stain
• 50% ethanol (add 50 ml of absolute ethanol and 50 ml of distilled
water).
• Kinyoun's carbolfuchsin:
Solution A: Dissolve 4 g of basic fuchsin in 20 ml of 95% ethanol.
Solution 8: Dissolve 8 g of phenol crystals in 100 ml of distilled
water.
Mix solution A and B, and store at room temperature.
1 % sulfuric acid.
• Alkaline methylene blue.
• Dissolve 0.3 g of methylene blue in 30 ml of 95% ethanol, and add
100 ml of dilute (O.O1 %) potassium hydroxide.
Fix with absolute methanol for l minute .
Flood the slide with Kinyoun's carbolfuchsin and
stain it for 5 minutes.
Rinse the slide briefly (3-5 seconds) with 50% elhanol.
Rinse the slide thoroughly with water.
Oecolorize by using 1 % sulfuric acid for 2 minutes or
until no more color runs from the slide.
Rinse the slide with water (it may take less than 2
minutes; do not destain too much) and drain.
Counterstain with methylene blue for l minute.
Rinse the slide with water and air dry.
Examine with the low or high dry objective. To see
internal morphology, use the oil objective (1 00X).
• Auramine O stainfor coccidia:
Coccidia are acid-fast organisms and also stain well
with phenolized auramine 0 .
The size and typical appearance of Cryptosporidium,
Cyclospora and /sospora oocysts enable auramine
0 -stained slides to be examined at low-power under
the l0X objective.
The entire sample area can usually be examined in
less than 30 seconds.
The low cost of the reagenrs, the simple staining
protocol and the rapid microscopic exam ination also
make this staining method suitable for screening
unconcentrated stool specimens. Concentrated
sediment from fresh or nonpolyvinyl alcoholpreserved stool may also be used.
Concentration Methods
When the parasites are scanty in stools, routine microscopic
examination may fail to detect them. It is then necessary to
selectively concentrate the protozoan cysts and helminth
eggs and larvae. Concentration may be done using fresh or
preserved feces. Several concenn·ation techniques have been
described.
They can be classified as Lhe floatation or sedimentation
methods.
I
j
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In floatation method, the feces are suspended in a solution
of high specific gravity, so that parasitic eggs and cysts
float up and get concentrated at the surface.
• In sedimentation method, the feces a re suspended in a
solution with low specifi c gravity, so that the eggs and
cysts get sedimented at the bottom, either spontaneously
or by centrifugation.
Floatation Methods
• Saturated salt solution technique
Procedure:
A simple and popular method is salt floatation using
a saturated solution of sodium chloride, having a
specific gravity of 1.2. About 2 ml of the salt solution
is taken in a flat bottomed tube (or penicillin bottle)
and 1 g of feces is emulsified in it.
1he container is then fi lled completely to the brim
with the salt solution.
A slide is placed on the container, so that it is in
contact with the surface of the solution without any
intervening a ir bubbles. After standing for 20-30
minutes, the slide is removed, without jerking,
reve rsed to bring the wet surface on lop, and
examined under the microscope.
A coverslip need not to be applied, if examination is
done immediately. Any delay in exam ination may
cause salt crystals to develop, interfering with clarity
of vision.
This simple method is quite useful for detecting th e
eggs of the common nematodes such as roundworm,
hookworms and whipworm, but is not applicable for eggs
of tapeworms, unfertilized egg of Ascaris lumbricoides,
eggs of trematodes and protozoan cysts.
• Zinc sulfate centrifugal floatation
Procedure:
Make a fine suspension of about 1 g of feces in 10 m L
of water and strain through gauze Lo remove coarse
particles.
Collect the liquid in a small test tube and centrifuge
for 1 minute at 2,500 revolutions per minute. Pour
off the supernatant, add water, resuspend, and
centrifuge in the same manner, repeating the process,
till the supernatant is clear.
Pour off the clear supernatant, add a small quantity
of zinc sulfate solution (specific gravity 1.18- 1.2) and
resuspend the sediment well.
Add zinc sulfate solutio n to a little below the brim
and centrifuge at 2,500 revolution per minute for 1
minute (Fig. lA).
Take samples care fully from the surface, using a
wire loop, transfer to slide and examine under the
microscope (Fig. 18). A drop of dilute iodine helps to
bring out the protozoan cysts in a better way.
Diagnostic Methods in Parasitology
This technique is useful for protozoan cysts and eggs of
nematodes and small tapeworms, but it does not detect
unfertilized roundworm eggs, nematode larvae, and eggs
of most trematodes and large tapeworms.
Sugar floatation technique:
Sheather's sugar floatation technique is recommended
for the detection of cryptosporidia infection.
Sedimentation Methods
m
B
Formal-ether sedimentation technique
Formol-ether concentration method has been the most
widely used sedimentation method (Fig. IC).
Procedure:
Emulsify 1-2 g feces in 10 mL of water and let large
particles sediment. Take the supernatant and spin at
2,500 revolutions per minute for 2-3 minutes.
Discard the upernatant. Add 10% formol-saline, mix
well and let it stand for l O minutes.
Add 3 mL ether and shake well. Spin at 2,500
revolutions per minute for 2-3 minutes. Four layers
Zinc sulfate
Sediment
/ 0 7
Ethyl
acetate
Debris/fat
Formalin
Sediment
Figs 1 A to C: (A) Zinc sulfate floatation concentration technique; (B)
Method used to remove surface film in the zinc sulfate floatation
concentration procedure: and (C) Formol-ether sedimentation
technique
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Paniker's Textbook of Medical Parasitology
will form-(1) a top layer of ether, (2) a plug of debris
at the interface, (3) the formalin-saline layer and (4)
the sediment at the bottom (Fig. IC).
Carefully detach the debris from the sides of the tube
and discard the top three layers.
Suspend the sediment in a few drops of flujd and
examine wet mount and iodine preparation.
As eth er is inflammable an d explosive, its use can be
hazardous. Ethyl acetate can be conveniently used in
its place, with equally good results.
The method is useful for all helminth eggs and
protozoan cysts.
• Baermann concentration method
Procedure:
Another me thod of examination of stool specimen
suspected of having small numbers of Strongyloides
larvae is the use of a modified Baermann apparatus
(Fig. 2).
The Baermann technique, which involves using a
funnel apparatus, relies on the principle that active
larvae migrate from a fresh fecal specimen that has
been placed on a wire mesh with several layers of
gauze, which are in contact with tap water.
Larvae migrate through the gauze into the water and
settle to the bottom of the funnel, where tl1ey can be
collected and examjned.
Besides being used for patient's stool specimens, this
technjque can be used to examine soil specimens for
the presence oflarvae.
Egg Counting Methods
A semiquantitative assessment of the worm burden can be
made by estimating the number of eggs passed in stools. This
is done by egg counts and by relating the counts to the number
of worms present by assuming the number of eggs passed per
worm per day.
However, these are at best approximations and only a
rough indication of worm burden can be obtained. Egg counts
help to classify helminth infections as heavy, moderate, or
light. Egg counts can be done by different methods.
• The standard wet mount gives rough indication of the
number of eggs. Ordinarily, 1-2 mg of feces is used for
preparing a wet film, and if all the eggs in the film are
counted. The number of eggs per gram of feces can be
assessed.
• The modified Kato thick smear technique using 50 mg of
stool cleared by glycerin-soaked cellophane coverslip can
be used for egg counting.
• McMaster's egg counting chamber can also be used.
In this method, eggs in 20 mg of stool are concentrated
by salt floatation on the squared grid on the roof of the
chamber, which can be coun ted.
Soil or
fecal material
Gauze
Wire screen
Water---\i,--,,i..---41k'I
Rubber tubing _ __ _
Clamp - --e::::::::11•
Container _ ,. __
Fig. 2: Baermann concentration method
Box 3: Hatching test for schistosoma eggs
This test is used to demonstrate the viability of the miracidia within
the schistosome eggs recovered from the urine or stool. Fecal or urine
specimens must be processed without any preservative. The specimens
are placed in 10 volumes of dechlorinated or spring water. Living
miracidia may be released by hatching within few hours. The specimens
are examined microscopically for presence of miracidia, which indicates
active infection.
ln Stoll's dilution technique, 4 g of feces is mixed
thoroughly with 56 ml of N/10 sodium hydroxide using
beads in a rubber stoppered glass tube. Using a pipetle,
exactly 0.075 mL of the sample is transferred to a slide,
cover glass is applied, and all the eggs present are counted.
The number multiplied by 200 gives the number of eggs
per gram of feces. this figure requires to be corrected
for the consistency of feces, by multiplying by 1 for hard
formed feces, by 2 for mushy formed feces, by 3 for loose
stools and by 4 for liquid stools. Watery stools are unfit for
counting.
Special techniques have been described fo r particular
purposes as for example, Bell's dilution-filtration count
for schistosome eggs (Box 3).
Scotch tape method:This is a simple and effective metl1od
fo r detection of eggs and female worms of Enterobius
vermicularis a nd occasiona lly eggs of Taenia solium,
T. saginata and Schistosoma mansoni. In this method,
a piece of transparent adhesive tape is pressed firmly
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Et Use a piece or clear cellophane
B
tape approximately 4 inches long
Press the sticky side or the tape against
the skin across the anal opening
Diagnostic Methods in Parasitology
m Hold the tape between thumbs and
forefingers with sticky side facing upward
m Place the sticky side of the tape down
against the surface or a clean glass slide
Figs 3A to D: Method for collection of a cellophane (scotch) tape preparation for pinworm diagnosis. This method dispenses with the tongue
depressor, requiring only tape and a glass microscope slide. The tape must be pressed deep into the anal crack
against perianal skin, and the adhesive surface of the tape
is spread on a glass slide (Figs 3A to D). The slide is then
placed under microscope and observed for parasitic eggs.
A drop of toluene or xylol may be placed between the
tape and the slide to clear the preparation. The specimen
should be collected for 3 consecutive days at night or early
in the morning.
Fecal Culture
Fecal culture is not used for routine diagnosis, but for
species identification, for example in differentiation between
Ancylostoma and Necator.
Harada-Mori Filter Paper Strip Culture
1he test detects light infection with hookworm, S. stercoralis,
Trichostrongylus spp, as well as to facilitate species
identification of helminths.
The Harada -Mori culture method uses strips of filter
paper on which feces is smeared in the middle third. The
paper strips are kept in conical centrifuge tubes with water
at the bottom, in which the strips dip (Fig. 4). The tubes are
kept at room temperature in the dark for 7-10 days, during
which time the larvae develop and fall into the water at the
bottom, from which they can be collected. Also, caution
must be exercised in handling the paper strip itself, since
infective Strongyloides larvae may migrate upwards, as well as
downwards on the paper strip.
Fig. 4: Harada-Mori tube method and petri dish culture method
Agar Plate Culture for Strongyloides
Agar plate cultures are used to recover larvae of S. stercoralis
and appear to be more sensitive. Approximately, 2 g fecal
specimens are inoculated onto agar plates. Then the plates
are sealed with tape to prevent accidental infection and
placed in room temperature for 2 days. In positive cases,
larvae will crawl over the agar, making visible tracks over it.
For further confirmation of larvae, the plates are examined
microscopically.
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Paniker's Textbook of Medical Parasitology
Charcoal Culture
Charcoal culrures are simple and efficient. oftened feces is
mixed with 5- 10 parts of moistened charcoal granules and
packed into a suitable container and kept covered. In 7-10
days, the larvae hatch out and come to the surface. They can
be collected by applying a pad of soft cotton cloth on the
surface for half an hour. The cloth is removed and kept upside
down on a sedimentation flask fi lled to the brim with warm
water. The larvae fall to the bottom of the flask, while the
charcoal particles remain on the cloth.
• EXAMINATION OF BLOOD
Next to feces, the largest number of parasites are found in
blood. Blood examination i the routine diagnostic method
in malaria, filariasis, African trypanosomiasis and babesiosis.
It is sometimes positive in Chagas di ease and rarely, in kalaazar and toxoplasmosis. Blood examination is done in the
following ways.
Examination for Malarial Parasites
1he standard diagnostic method in malaria is the examination
of stained blood fiJms- boLh thin and thick smears.
Collection of Blood
For demonstration of malarial parasites, blood should be
collected not during the peak of fever, but optimally several
hours after it. Bouts of fever follow the synchronous rupture
of large number of parasitized erythrocytes, releasing their
membrane shreds and contents. the emerging merozoites
parasitize other erythrocytes and initiate a fresh cycle of
erythrocylic schiwgony. The timing is particularly important in
P. Jalciparum infections, as here the late stages of schizogony
are not seen in peripheral circuJarion.
• In practice, the rule is to take a blood smear when a
uspected malaria patient is first seen and then again
subsequently afrer a bout of fever.
Smears should invariably be collected before starting
antimalarial treatment.
Thin smear:
• A thin smear is prepared from finger prick or in infants
from heel prick blood or ethylene diaminetetra-acetic
acid (EDTA) anticoagulated venous blood can also be
used, provided blood fi lms are made within 30 minutes.
• A small drop (10-15 µL ) is spread on a clean grease-free
slide with a spreader to give a uniform smear, ideally a
single celJ thick smear. The margins of the smear should
be well short of the sides of the slide, and the rail should
end a little beyond the center of its length.
• The thin smear displays blood cells and parasites clearly.
Its only disadvantage is that only a small volume of blood
can be surveyed, so that a light infection could be missed.
• If the smear are prepared from a nticoaguJated blood,
which is more than an hour old, the morphology of both
parasites and infected RBCs may not be typical.
• After drying, the smear is stained with Giemsa or
Leishman stain.
• For Giemsa stain, the smear is fixed in methanol for
3-5 minutes. After drying, Giemsa stain, diluted 1 drop
in l mL of buffered water, pH 7-7.2, is applied for 30-45
minutes. The slide is then flushed gently with tap water,
dried and examined under the oil immersion objective.
The cytoplasm of malarial parasites is stained blue and the
chromatin dot is stained red.
• For Leishman's sta in, prior fixation is not necessary as
the stain is an alcoholic solution, which fixes as it stains.
Leishman stain is applied fo r 1-2 minutes and diluted with
rwice its volume of buffered water, pH 7-7.2 and is kept for
10-15 minutes. the smear is then dried and examined.
lleporting of thin blood.films:
In malignant tertian malaria, only the ring stage and
gamctocyte are seen in peripheral smear, while in benign
tertian malaria, all stages of schizogony and gametocytes
can be seen.
Thin smear examination enables the appreciation of
changes in the e rythrocytes, such as enla rgement,
alteration of shape, fimbriation, red cells stippling
(Schuffner's dots) as seen with P. vivax, and irregular
stippling (Maurer's clefts), as seen in mature P. Jalciparum
trophozoites.
Any marked increase in white cell numbers and if
indicated perform a differential white cell count.
Parasitized erythrocytes are seen most often in the upper
and lower margins of the tail of the smear.
Count the percentage of parasitized red cells, when there
is high falciparum malaria parasitemia ( +++ or more
para ites seen in the thick film) to monitor a patient's
response to treatment.
• A min imum of 100 fi elds should be examined before a
negative report is given.
Thick smear:
• Thick smears have the advan tage that a larger quantity of
blood can be tested. Increased volume of blood present
on thick film may allow the malaria parasite to be detected
even with low para itemia. Compared with a thin film, a
thick film is about 30 times more sensitive and can detect
about 20 parasites/ µL of blood.
• The disadvantages are that the red cells arc lysed and the
morphology of the para ices is di toned, so that species
identification becomes difficult.
P:257
• A big drop or blood (20-30 µL) from finger or heel prick
is collected on a clean grease-free slide and spread with
th e corner of a nother clean slide to form a uniforml y
thick smear of about 1 cm2
• The thickness of the smear
should be such that the hands of a wristwatch can be seen
through it, but not the figures on the dial.
• The smear is dried in a horizontal position, kept covered
from dust.
• Thick smears have to be dehemoglobinized before
staining.
• They can be stained with Giemsa or l eishman's stains
as described earlier. Wright's stain and )SB stain (so
called because it was devised by Jaswant Singh and
Bhattacharjee, in 1944) are very useful fo r staining large
numbers of thick films as in malaria surveys.
Wright's stain consists of two solutions:
l. Solution A contains methylene blue and azure B in
phosphate buffer.
solution B contains eosin in phosphate buffer. The
film is immersed in solution A for 5 seconds, washed
in tap water, immersed in solution B for 5 seconds,
washed, dried and examined. Staining times may
need adjustment. If the smear is too blue, stain longer
in solution B; if too pink, in solution A.
Jaswant Singh and Bhattacharj ee stain also consists of
two solutions:
l. Th e fi rst contains me thylene blue, potassi um
dichromate, sulfu ric acid, potassium hydroxide and
water.
The second solution is aqueous eosin.
For staining, the smear is immersed in solution 1 for 10
seconds, washed for 2 seconds in acidulated water pH 6.2-
6.6, stained in solution 2 for 1 second, washed in acidulated
water, immersed again in solution l and washed.
Reporting of thick blood films:
• Select an area that is well-stained and not too thick.
• Examine for malaria parasites and malaria pigment under
oil immersion objective (l00X).
• Examine at least I 00 high-power microscope fields for
parasites.
• Report the approximate number of parasites ( trophozoites,
schizonls a nd gametocytes) and also whether malaria
pigment is present in white cells or not.
• The plus sign scheme that can be used to repon parasite
numbers are described in Box 4.
Box 4: Plus sign scheme for reporting parasite numbers
• 1- 1 0 per 100 high-power fields: +
• 11-10 per 100 high-power fields: ++
• 1-1 0 in every high-power field:+++
• More than 10 in every high-power field: ++++.
Diagnostic Methods in Parasitology
Combined thick and thin blood.films:
• Combined thick and thin smears can be taken on the
same slide.
Draw a thick line with a glass-marking pencil on a slide,
dividing it into two'un equal parts. The thick smear is
made on the smaller part and the th in smear drawn on
the larger.
• Thick smear is first dehemoglobinized and the two are
then stained together. An easy method is to add undiluted
Leishman stain over the thin smear, and then the diluted
stain flooded over to the thick smear also.
• Do not allow the methanol to contact the thick film when
fixing the thin film.
• The stained thin smear is examined first. If the thin
smear is negative, the thick smear should be searched for
parasites.
When a slide is positive for malarial parasites, the report
should indicate the species, the developm ental stages
found and the density of parasites in the smear.
Examination for Microfilaria
Microfilariae may be detected in peripheral blood, both in
unstained mounts and in stained smear (Table I and Box 5).
Wet Mount
• Two or three drops of blood are collected on a clean glass
slide and mixed with two drops of water to lyse the red
cells.
The preparation is covered with a coverslip and sealed.
The preparation is examined under the low-power
microscope for the motile microfilariae, which can be
seen wriggling about, swirling the blood cells in their
neighborhood.
Table 1: Parasites found in peripheral blood film
Protozoa Nematodes
• Plasmodium spp. • Wuchereria bancrofti
• Babes,a spp. • Brugia spp.
• Leishmania spp. • Loaloa
• Trypanosoma spp. • Mansonella ozzardi
Box 5: Time of collection
In case of nocturnal periodic microfilariae, blood should be collected
between 10 PM and 2 AM. In subperiodic nocturnal infection, the time
of collection of blood should be between 8 PM and 10 PM and for
subperiodic diurnal infection the time of collection should be ideally
between 2 PM and 6 PM.
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Paniker's Textbook of Medical Parasitology
• The examination may conveniently be deferred till next
morning, as microfilariae retain their viability and motility
for 1 or 2 days at room temperature.
• By using a simple counting chamber, microfilariae in the
wet mount can be counted.
Stained Smears
• A thick smear is prepared as for malaria, dehemoglobinized, and stained with Leishman's, Giemsa, or
Delafield's hematoxylin stains.
• Stained smears have the advantage that the morphology
of microfilariae can be studied and species identification
can be made. Thus, for differentiation b etween Mf
bancrofti and Mf malayi stained smears are necessary.
• Sometimes, microfilariae may be seen in thin smears also.
• By using a measured quantity of blood for preparing
smears, as for example with a 20 cubic mm pipette and
counting the total number of microfilariae in the smear,
microfilaria counts can be obtained. Multiplying the
number of microfilariae in a 20 cubic mm smear by 50
gives the count per mL of blood.
Concentration Methods
These methods have been developed to recover low numbers
of microfilariae from blood and employ venous blood.
• Sedimentation method:
In sedimentation method, the sample of blood
is first lysed with acetic acid, saponin, or other
lytic substance, or by freeze-thawing, and then
centrifuged.
The sediment is stained and the microfilariae are
counted.
• Membrane filtration concentration:
In membrane filtration method, a measured quantity
(1- 5 mL) of blood is collected into an anticoagulant
solution and passed through membrane filters fixed
on syringes with Swinney filter holder. Blood cells
and proteins sticking on to the filter are washed away
by repeatedly passing saline through it.
The filter is removed, placed on a slide, stained with
dilute Giemsa stain and examined under low-power
microscope for microfilariae.
Millipore and nucleopore membrane filters (5 ~un
porosity) are available for this purpose; the latter
being more sensitive, as it can screen larger volumes
of blood.
Membrane filtration recovers most species of
microfilariae; however, because of their small size,
Mansonella perstans and M. ozzardi may not be
recovered. Membranes with smaller pores (3 ~un)
have been suggested to recover these two species.
The membrane filter method is much more sensitive
than the finger prick method as the blood samples
are taken during day, it also give reliable results even
with nocturnal periodic microfilariae.
However, the method has the disadvantages that
venipunclure is necessary, membranes are costly,
and microfilariae may not be in a satisfactory
condition for detailed morphological study.
The number of microfilariae counted divided by 10
gives the number of microfilariae per mL of blood.
This is the most sensitive method of detecting small
numbers of microfilariae, but it is expensive for
routine use.
• Microhematocrit tube method:
Capillary blood is collected in two heparinized
capillary tubes or about 100 µL is first collected into
EDTA an ticoagulant, and then transferred to plain
capillary tubes.
The blood is centrifuged in a microhematocrit
centrifuge.
The huffy coat is exami ned microscopically for
motile microfilariae.
In areas where the species is known and Mansonella
microfilariae are not found, tl1is is a rapid technique
for detecting microfilariae.
• Buffy coat blood film:
The buffy coat con taining white blood cells (WBCs)
and platelets obtained after centrifugation of whole
anticoagulated blood and the layer of RBCs just
below the buffy coat layer, can be used to prepared
thick and thin blood films in suspected infections
with filaria, Leishmania, Trypanosoma and malaria.
TI1e sensitivity of this method is much higher than
that of routine thick film.
Diethylcarbamazine Provocation Test
Oral administration of diethylcarbamazine (DEC; 100 mg
or 2 mg/ kg of body weight) brings about mobilization of
microfilariae into peripheral blood. Blood collected 20-50
minutes after the drug is given, will show microfilariae so
that blood collection can be done during day time. This is a
great advantage for surveys. But the drug may cause febrile
reactions, particularly in brugiasis. It cannot be used in
areas endemic for onchocerciasis because of the danger of
provoking severe reactions.
• SPUTUM EXAMINATION
Sputum is examined commonly for the demonstration of eggs
of Paragonimus westermani, and sometimes for detection
oftrophozoites of E. histolytica in amebic pulmonary abscess.
Rarely, the larval stages of hookworm, A. lumbricoides, or
P:259
Box 6: Parasites found in sputum
• Paragonimus wesrermani
• Enramoeba his10/ytica (trophozoites in case of pulmonary abscess)
• Pneumocysris jirovecii
• Rarely migrating larvae of Ascaris lumbricoides
• Rarely migrating larvae of Suongyloides stercorahs
• Rarely migrating larvae of Ancylostoma duodenale
• Rarely migrating larvae of Necator americanus.
S. stercoraLis or the cestode hooklets may be seen in sputum
samples (Box 6).
• Concentrated stained prepa rations of induced sputum
are commonly used to de tect P. jirovecii and differentiate
trophozoite a nd cyst fo rms from o th er possible
causes of pne umonia, pa rti cula rly in an acqu ired
immunodeficiency syndrome (AJDS) pa tient.
• ormally, direct saline mount preparation is done fo r
microscopy.
• If the s putum is thick, equa l volume of 3% N-aceryl
cysteine or 3% sodium hydroxide is added to the sputum
to liquefy the specimen and afte r centrifugation, the
sedimenr is examined for mic roscopic examination under
low (!OX) and high (40X) powe r magnifications.
• In a Paragonim us s pp. infection, the s putum may be
viscous and tinged with brownish necks, which a re
cluste rs of eggs (iron.fili11gs) and may be streaked with
blood.
• URINE OR BODY FLUIDS EXAMINATION
• Large volume of urine samples should be allowed to settle
for 1- 2 hours.
• About 50 mL of the bottom sediment of Lhe sample is
taken for centrifugation.
• The highly concentrated sediment after centrifugation is
examined for direct wet mount microscopy.
May show eggs of Schistosoma and Trichomonas vagina Lis.
Mic rofi laria may be de tected from chylo us urine in
lymphatic filariasis.
• TISSUE BIOPSY
Tissue biopsies and fine-needle aspirations a rc taken from
cutaneous ulcers of trypan osomiasis or leishmaniasis and
from skin nodules of o nchocerciasis and post-kala-azar
de rma l leishmaniasis (PKDL).
A skin snip can be o btained to d iagnose subcutaneous
filariasis or leishmaniasis by grasping with a fo rceps or
elevating a portion of skin with the tip of needle. Tip of the
small cone of the skin is, then sliced with a sha rp blade o r
razor.
Diagnostic Methods in Parasitology
• Wet mount prepara rio n of lymph node as pirate and
chancre fluid are used as rapid me thods for demonstration
of rrypanosomes.
• Biopsies from liver, spleen, bone marrow and lymph nodes
a re taken in visceral leishmaniasis for demon tration of
Leishman-Donovan (LD) bodies.
• All biopsy tissues must be submitted to the laboratory
without the addition of formalin fixative. If there is delay
in transport or processing, the specimen should be placed
in polyvinyl alcohol fixative. In soft specimens, a small
partshould be scraped and examined as direct saline wet
mount.
• Impression smears can be made from freshly cut tissue
specimens on a glass sli de and examined after fixatio n
with Schaudinn' solution. Trichrome or othe r stains can
be used.
• The residual pa rt of the biopsy specimen may be processed
for histopathological examination.
• Adult fila rial worms can sometimes be found in section of
biopsied lymph node.
Corneal scrapings arc usef ul in diagnosis ofacanlhamoeba
keratitis.
• MUSCLE BIOPSY
Spira l la rval form of Trichinella spiralis, larva l form
of T. solium (cyslicercus cellulosae) a nd am astigote o f
Trypanosoma cruzi can be demonstrated in skele tal muscle
biopsy. In trichinosis, muscle biopsy (gascrocnemius, deltoid
and biceps) specimen must be examined by compressing the
tissue be tween two slides and checking the prepa ration under
low-power (lOX) objective. this me thod does no t become
positive until 2-3 weeks after the illness.
• DUODENAL CAPSULE TECHNIQUE
(ENTEROTEST)
Enlerotest is a simple method of sampling duodenal contents.
• the device is composed of a length of nylo n ya rn-co iled
inside a gelatin capsule.
t he end of the yarn is affixed to the patient's face.
• The capsule is then swallowed and the gelatin d issolves in
the stomach.
• The weighted string is carried into the duodenum by
peristal is.
• Bile-stained mucus is then retrieved after 3-4 hours and
duo denal contents adhe rent to the yarn is scrapped off
and examined under microscope as we t mount or a
stained smear after preservation in formalin or polyvi nyl
alcohol.
• Usually 4- 5 drops of ma terial is obtained.
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Paniker's Textbook of Medical Parasitology
• Enterotest is used for detecting trophozoites of Ciardia,
larvae of Strongyloides, eggs of Liver flukes and oocysts of
lsospora.
• SIGMOIDOSCOPY MATERIAL
Material obtained from sigmoidoscopy is useful in the
diagnosis of E. histolytica that cannot be diagnosed by
routine examination for at least 3 days.
• Material from the intestinal mucosa should be aspirated
or scraped and not to be collected by cotton swabs.
• the material should be processed immediately.
• 1n heavy infection of Trichuris, sigmoidoscopy may show
white bodies of the worms hanging from the inflamed
mucosa of large intestine.
• UROGENITAL SPECIMEN
The de tection of T. uaginalis is usually based on wet
preparation of vaginal and urethral discharges and prostatic
specimens. Specimens should be collected in small volume
of0.85% saline and should be sent immediately for detection
of actively motile organisms, as the jerky movements of
Trichomonas begin to dimjnish with time.
• CULTURE METHODS
Many parasites can now be grown in culture, but this has not
become a routine diagnostic method in parasitic infections
(Box 7). It is sometimes employed for accurate identification
of the parasite species. It is more often employed for
obtajning large yields of the parasite as a source of antigen,
animal inocula tion, drug-sensitivity testing, experimental
or physiological studies and teaching purposes. Some of the
culture methods used for different parasites are indicated
here.
Ameba
E. histolylica and other intestinal amebae can be grown
in diphasic or monophasic media, media containing other
microorganisms, o r axenic cultures.
• Boeck and Drbohlau diphasic medium, the classical
culture medium for ameba has been modified by vario us
workers (Box 8).
The medium as used now, is basically an egg slant,
with an overlay of sterile serum or liver extract in
buffered saline.
A loopful of sterile rice powder is added to the
medium just before inoculation with fresh feces or its
saline centrifugal sediment.
Cultures can be obtained from feces-containing cysts
or trophozoites.
Box 7: Parasites which can be cultured in the laboratory
• Entamoeba histolytica
• Giardia /amblia
• Trichomonas vagina/is
• Leishmania spp.
• Trypanosoma spp.
• Acanthamoeba spp.
• Naegleria fowleri
• Balanridium coli
• Plasmodium spp.
Box 8: Composition of Boeck and Drbohlav medium (Locke's solution)
• Sodium chloride: 9 g
• Potassium chloride: 0.4 g
• Calcium chloride: 0.2 g
• Sodium bicarbonate: 0.2 g
• Glucose: 2.5 g
• Distilled water: 1000 ml
• Egg: Four (clean and washed)
Box 9: Composition of Balamuth's medium
• Liver concentrate powder: 1 part
• Egg yolk medium: 9 part
• Phosphate buffer
• Tribasic potassium phosphate: 212 g
• Monobasic potassium: 136 g
• Distilled water
The cultures are incubated at 37°C and subcultured at
48-hour intervals.
Arnebae can be demonstrated in the Liquid phase in
unstained mounts or stained smears.
• Balamuth's monophasic liquid medium is also used
commonly for cultivation of amcbae and other intestinal
protozoa. This is an egg yolk-Liver extract infusion medium
(Box9).
Both protozoa and bacteria present in stools grow in
the earlier media.
Bacteria l growth can be reduced by addition of
penicillin o r other antibioti cs that do not inhibit
protozoa.
Axenic cultures (pure cultures without bacter.ia
or other microorganisms) were first developed by
Diamond in 1961. Axenic cu ltivation has enabled
precise antigenic and b iochemical studies on
amebae.
coli grows well in Balamuth's medium. G. lamblia
had been established in association with Candida
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and Saccharomyces, but axenic cultu res were
developed in 1970.
T. uaginalis grows very well in several commercially
available media such as trypticase serum media.
Naegleria and Acantham oeba from cerebrospinal
fl uid (CSF) can be grown on agar plates heavily
seeded with Escherichia coli.
Leishmania and Trypanosomes
• Nouy-MacNeal-Nicolle medium: The classica l NouyMacNeal-Nicolle (NNN} medium first described in 1904
fo r cultivation of Leishmania, is equally satisfactory for
trypanosomes a lso. This is a defibrinated rabbit blood
agar medium (Box 10). Several modifications of this
medium have been introduced.
Two bottles of culture are aseptically inoculated with
0.1 mL of specimen in each and incubated at 24°C for
4 weeks.
The primary culture is examined every 4 days for
promasrigotes in leishmaniasis and for epimastigote
stages in trypanosomiasis for up to 30 days.
• Schneider's insect tissue culture medium: It is
recommended in vitro culture of Leishmania. this medium
is said to the more sensitive than NN medium (Box 11).
Malaria Parasites
• Cultivation of malaria parasites was first obtained by Bass
and Jones in 1912. A simple method of cultivation is as
follows:
About 10- 12 mL of defibrinated or heparinized blood
rich in ring forms of malaria parasite, mixed with
0.2 mL of 50% dexLrose solution are incubated at 37°C
in a sterile test tube in an upright position.
The blood separates into the erythrocytes below,
plasma above and the huffy coat in between.
Malaria parasites grow in the erythrocyte layer
immediately below the huffy coat.
Smears are collected from this layer at intervals,
without tilting the tube.
Box 10: Composition of Novy-MacNeal-Nicolle (NNN) medium
• Bactoagar (Difeo): 1.4 g
• Sodium chloride: 0.6 g
• Double distilled water: 90 ml
• Defibrinated rabbit blood (10%):10 ml.
Box 11 : Composition of Schneider's insect tissue culture medium
• Schneider's Drosophila tissue culture medium: 80 ml
• Fetal calf serum: 20 ml
• Antibiotic-antimycotic solution: 1.2 ml.
Diagnostic Methods in Parasitology
Segmented schizonts a re usually observed after
incubation for 24-36 hours.
• The breakthrough in cultivation of malarial parasites came
in 1976 when Trager and Jensen successfully maintained
P. Jalciparum in con ti n uous cul tu res in h uman
erythrocytes using Roswell Pa rk Me moria l Institute
(RPMI) 1640 medium.
The cultures are incubated at 38°C with 10% human
serum at pH 6.8-7.2 under an atmosphere with 7%
carbon dioxide and 1-5% oxygen.
A continuous flow system is used in which the
mediwn flows slowly and continuously over the layer
of erythrocytes. The method has been applied to
various species of Plasmodia.
It has been employed for preparation of antigens,
drug-sensitivity studies, vaccine tests and man y other
purposes.
• ANIMAL INOCULATION
Anima l inoculation is not a routine diagnostic procedure
in parasitic infections, but can be used in some instances
because of its sensitivity.
Toxoplasmosis: Animal inoculation can be used for
isolating Toxoplasma gondii from infected pe rsons.
Lymph node or other biopsy materials are inoculated
intrapcritoneally into im munosuppressed mice.
Peritoneal fluid obtained 7-10 days later, may show the
parasite in Giemsa-stain ed smears. However, serial
passages may be necessary for its isolation. Brain smears
may be examined for cysts after sacrificing the mice 3-4
weeks after inoculation. Seroconversion of the animal
inoculation also inclicates a positive result.
• Visceral leishmaniasis: Bone marrow, liver, spleen, or
lymph node aspirates from kala-azar patients, injected
intraperitoneally into hamsters is a very sensitive method
fo r diagnosing visceral leishman iasis. Eve n a single
amastigote can establish the infection in the anima l.
Spleen smears taken 4-6 weeks later show Leishmania
donovani (LO) bodies.
• Try panosomiasis: Blood from patients with trypanosomiasis can be injected intraperitoneally or into the
tail vein of mice, rats and guinea pigs, etc. These animals
are susceptible to infection by T. brucei rhodesiense.
Parasitenlia can be demonstrated in 2 weeks.
• XENODIAGNOSIS
This method involves the diagnostic infection of a vector, in
wh ich the parasite multiplies and can be demonstrated. In
T. cruzi, diagnosis may be established by letting the vector
reduviid bug feed on suspected patients. In 4-5 weeks, live
flagellate forms can be seen in the feces of the bugs .
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Paniker's Textbook of Medical Parasitology
• IMMUNOLOGICAL DIAGNOSIS
Serology
Several serological tests have been developed for detection of
antibodies to parasites using antigens from cultured parasites
or from natural or experimental infections in animals or
humans. In some cases, antigens are obtained from related
parasites or even sometimes from bacteria. Advances in
cultivation of parasites have made parasitic antigens more
readily available. Cloning of parasitic antigens promises to be
a new source.
In some instances, diagnosis is attempted by serological
demonstration of parasitic antigens in blood, tissues, or
secretions of suspected patients.
Virtually, all types of serological reactions have been
used. However, serodiagnosis in parasitic infections has only
limited value due to various factors:
• Parasites are complex antigenically and exhibit wide
range of cross-reactions, so that serological tests are not
sufficiently specific.
• Another difficulty is in distinguishing between past and
current infections. This has been solved partly by looking
for immunoglobulin M (IgM) antibody, as in amebiasis
and toxoplasmosis.
• In general, indirect hemagglutination (IHA), enzymelinked immunosorbent assay (ELISA) and counterimmunoelectrophoresis (CIEP) are most sensitive;
indirect immunofluorescence (IF), direct agglutination
test (DAT) and complement fixation test (CFT) a re
moderately sensitive; and simple precipitation in gel
and coated particle agglutination tests are least sensitive.
Serology has not been ve ry useful in the diagnosis of
individual cases, but has been valuable as a screening
method in epidemiological surveys. However, in some
infections where parasites are seldom demonstrable in
patients, for example in toxoplasmosis and hydatidosis,
serology is of great help. Listed here are some of th e
applications of serology.
Amebiasis
Serology is of no value in the diagnosis of acute amebic
dysentery or luminal amebiasis. But in invasive amebiasis,
particularly in liver abscess, serology is very useful.
• Indirect hemagglu.tination is mos! widely employed. Titers
of 1:256 or more are sign ificant in cases of amebic liver
abscess and have prognostic value.
• Tech Lab E. histolytica test was able to detect galactose
lectin (GalNAc) antigen in almost all patients of amebic
liver abscess.
Giardiasis
Enzyme-Linked immunosorbent assay and indirect
immu.nofluorescence (IIF) test have been developed for
detection of Giardia.
• Commercially available ELISA (ProSpec T / Giardia)
kit detects Giardia-speciflc antigen 65 (GSA 65). The
sensitivity of the test is 95% and specificity is 100%, when
compared to conventional microscopy.
Trypanosomiasis
Serological rests used to detect trypanosomiasis are IHA,
indirect fluorescent antibody (/FA} and ELISA.
• Specific antibodies are detected by these tests in the
serum within 2- 3 weeks infection.
• Specific antibodies can be demonstrated by !FA and
ELISA in CSF.
Leishmaniasis
Indirect hemagglu.tinalion, CIEP and DOT-ELISA are usually
positive in kala-azar.
• Complement tesl using Witebsky, Klingenstein and Kuhn
(WKK} antigen from the acid-fast Kedrowsky bacillus are
relatively less sensitive.
• Indirect fluorescent antibody rest is positive very early in
the disease, even before the appearance of symptoms and
becomes negative within 6 months of cure.
• rK39 micro ELISA test is a qualitative immunochromatographic assay for detection of antibodies to
Leishmania.
Malaria
Indirect immunofluorescence, ELISA and IHA are sensitive
and specific, but are not useful for diagnosis of acute malaria
because antibodies persist for some years after cure.
• A negative test may, however help to exclude malaria.
• Serological tests are useful in epidemiological surveys for
malaria.
• Molecular assays such as antigen capture for detection of
hislidine-rich protein II (HRP-2) and Plasmodium lactate
dehydrogenase (pLDH} have been applied for developing
rapid dipstick tests (e.g. ParaSight-F in malignant tertian
malaria).
Toxoplasmosis
Serological tests offer the most useful diagnostic method in
roxoplasmosis.
• The original Sabin-Feldman dye test, though very specific
and sensitive, is no longer in use. IIF IHA and CFT were
P:263
other useful tests. The dye test remains positive for life,
while CFT becomes negative soon after active infection.
• At present, ELISA is routinely used in Toxoplasma
serology. It is very informative, as it provides titers of IgM
and IgG antibodies separately for better interpretation of
the results.
Cryptosporidiosis
Indirect fluorescent antibody and ELISA using purified oocysts
as antigens have been used to detect circulating antibodies
specific to Cryptosporidium parvum.
Intestinal Helminths
Antibod ies can be demo nstrated in most in testi nal
helminthiases, but extensive cross-reactions limit their use
in diagnosis.
Trichinosis
Serology is very useful in diagnosis of trichinosis. Bentonite
flocculation slide tests and CFT become positive 3-4 weeks
after infection.
• Indirect immunofluorescence becomes positive even
earlier.
• Enzyme-linked immunosorbent assay is also available.
Demonstration of seroconversion is diagnostic.
Toxocariasis
lligh titers in serological tests are obtained in visceral larva
migrans, but specificity is low d ue to cross-reactions wilh
intestinal nematode antigens.
Filariasis
Indirect lzemagglutination and bentonite flocculation tests
with antigen from Dirofllaria immitis gives positive reaction
in patients, and high titers in tropical pulmonaryeosinophilia.
But cross-reactions arc frequent.
lmmunochromatographic card test (JCT) is a new and
rapid filaria l antigen test that detects soluble Wuchereria
bancrofti antigens in the serum of infected humans.
Echinococcosis
Several serological tests have been developed using hydatid
fl uid or scolex antigens from hydatid cysts in sheep. IHA, JJF,
CI EPand ELISA are very sensitive. Cross-reactions occur with
cysticercosis.
Diagnostic Methods in Parasitology
• SKIN TESTS
lntradermal tests have been used in many parasitic infections.
They are sensitive and persist for many years, sometimes even
for life. But specificity is relatively low.
• Casoni's test: This test had been used widely in the
diagnosis of hydatid disease since its original description
in 1911. The antigen is sterile hydatid flu id drawn from
hydatid cysts from cattle, sheep, pig, or humans, filtered
and tested for sterility. Intradermal injection of 0.2 mL
of the antigen induces a wheal and flare reaction within
20 minutes in positive cases. A saline control is used.
False-positive tests are seen in schistosomiasis and some
other conditions. Casoni's test is now largely replaced by
serological tests.
• l eishma nin (Montenegro) test: lhis test is used to measure
delayed hypersensitivity. Leishmania test is sensitive and
relatively specific. The antigen is obtained from cultured
Leishmania and co nsists of killed promastigotes in
phenol saline. lntradermal injection of 0.1 mL induces
a papule of 5 mm or more in diameter in 48-72 hours.
This delayed hypersensitivity test is positive in cutaneous
leishmaniasis and negative in diffuse cutaneous and
visceral leishmaniasis.
• Fairley's lest: This skin test is group-specific and gives
positive results in all schistosomiasis. The intradermal
allergic test uses antigen infected snails, cercariae, eggs
and adult schistosomes from experimentally in fected
laboratory animals.
Skin test in Bancroftian filariasis: Intradermal injection
of filarial antigens (extracts of microfilariae, adult worms
and third-stage larvae of Brugia malayi, or the dog filaria,
Dirofilaria immitis) induce an immediate hypersensitivity
reaction, but the diagnostic value of the skin test is very
limited due to the high rate of false-positive and negative
reactions.
• MOLECULAR METHODS
ucleic acid-based diagnostic tests are mainly available in
specialized or reference centers. Nucleic acid probes and
amplification techniques such as polymerase chain reaction
(PCR) and multiplex PCR, western blot and deoxyribonucleic
acid (D A) hybridization techniques are increasingly used
to detect parasites in specimens of blood, stool, or tissue from
patients.
• These test are useful for detecting subspecies or stain level
identification which is important for epidemiological
studies and are also used to detect parasitic drug
resistance. For example, specific 17 kDa and 27 kDa
P:264
Paniker's Textbook of Medical Parasitology
sporozoite antigens are employed for seroepidemiological
studies in cryptosporidiosis using western blot technique.
• Deoxyribonucleic acid probe is a highly sensitive method
for the diagnosis of malaria. It can detect even less than
10 parasite/ µL of blood.
• B, geneofT. gondiican be detected byPCRoftheamniotic
fluid in case of congenital toxoplasmosis. PCR have been
developed for detecLion of filarial DNA from patients
blood. If parasite cannot be identified by microscopy,
amplification of babesial 18S ribonucleic acid (RNA) by
PCR is recommended.
• Drug resistances in malaria are detected now by PCR
techniques. PCR is increasingly used now fo r species
specification and for detection of drug resistance in
malaria. Chloroquine resistance in P.falciparum has been
attributed to mutation in the Plasmodium Jalciparum
chloroquine resistance transporter (PfCHT), a transporter
gene in the parasite. Poirit mutation in another gene
Plasmodium falciparum multidrug resistance protein
1 (PfMDHl) has also been implicated in determining
resistance in vitro. Pyrimethamine and sulfadoxine
resistances are associated with point mutations in
dihydrofolate reductase (DHFR) and dihydropteroate
synth ase (DHPS) genes respectively. Mutation in
PfATPase gene is associated with reduced susceptibility
to artemisinin derivatives.
REVIEW QUESTIONS
Enumerate the various methods employed for examination of
stools and describe in detail the concentration methods of stool
examination.
Describe various skin test s used for diagnosis in many parasitic
infections.
Write short notes on:
a. Scotch tape method
b. Blood examination for malarial parasite
c. Blood examination for microfilaria
d. Enterotest
e. Casoni's test
f. Floatation method of stool examination
MULTIPLE CHOICE QUESTIONS
time of collection of blood is important in
a. Microfilaria
b. Trypanosoma spp.
c. Leishmania spp.
d. Babesia spp.
Modified acid-fast stain is used for the diagnosis of
a. Entamoeba histolytica
b. Toxoplasma gondii
c. Cryptosporidium parvum
d. Leishmania donovani
Sputum examination is commonly done for detecting the eggs
of
a. Strongyloides stercoralis
b. Entamoeba histolytica
c. Paragonimus westermani
d. Ascaris lumbricoides
larval forms of which parasite can be found in muscle biopsy
a. Ascaris lumbricoides
b. Taenia so/ium
c. Trichuris trichiura
d. Ancylostoma duodenale
Answer
a 2. C 3. C 4. b
P:265
INDEX
Page numbers followed by b refer to box,frefer to figure.Jc refer to flow chart and I refer to table
A
Abscess, splenic 19
Acanthamoeba 12, l3, IS, 26, 28,291,231,
233, 244
culbertsoni, life cycle of 29f
keralicis 29, 30
Acanthocheilonema 223
Acanthopodia 28
Accidental host 2, 121
Acephalocysts 133
Acetabulum 14 l
Acid-fast
parasitic organisms 105b
stain I OOJ, 236
Acidosis, metabolic 79
Acquired immunodeficiency syndrome 5,
13, 29, 93, 104, 184 243
Adenolymphangitis, acute 214
Adenophorea 166
Adoral cilia 107
Adult Trichuris rrichiura worms l 76f
Adult worm 112, 144, 151, 154, 156, 160,
170, 175, 180, 181, 198, 203
African trypanosomiasis 42, 46
Agar plate culture 239
Albendazole 128, 135
Alimentary canal, amebae of 13
Alphonse laveran 67
Amastigote 42, 48, 48f, 53, 54f
Ameba 244
classification of 151
drug sensitivity of 23
Amebapores 18
Amebiasis 20fc, 246
cutaneous 19, 21
genitourinary 19, 21
hepatic 20
lesions of 22f
metastatic 2 I
pulmonary 19, 21
Amebic
antigen detection 23
appendicitis 19
colitis 24
cysteine proteases 18
dysentery 13, LS, 19, 20
encephalitis, granulomacous 26, 29
granuloma 30
hepatitis 19, 20
keratitis 26, 29
lectin 18
liver abscess 13, 15, 19, 20f, 21, 2lfc, 24
meningoencephalitis 13, 26-29
ulcer 18, 19f
flask-shaped 19/
Amebida 12
Ameboflagellate 27
Ameboma 19, 30
Amebostomes 27
American rrypanosomiasis 47
American visceral leishmaniasis 56
Amoeboflagellate 39
Amphotericin B 28, 61
Ampulla of Vater 155
Ancylostoma 6, 140, 165, 189
brazilie11se 165, 167
caninum 167
ceyla11iwm 193
duodenale 3, 7, 165, 176, 180, 187-189,
192,194,207,229, 243
adult worm of 188/
egg of 189]
life cycle of 190/
Anemia 46, 57
causes of 56b, 78b, 192b
dimorphic 192
severe 56, 87
Angiostrongyliasis, abdominal 231
Angiosrrongylus cantonensis 167, 230-233
Animal inoculation 8, 47, 50, 59, 94, 245
Anisakiasis 232, 233
Anisakis simplex 167, 232
Anodic antigen, circulating 147
Anopheles barbirostris 2 LO, 219
Anthroponotic urban type 62
Anthropozoonoses 2
Antiamebic drugs 24 I
Antibody
demonstration of 217
detection 7, 23, 35, 51, 60, 95, 128, 147
Antigen 7
detectio11 7, 35, 47, 51, 59, 95, 128, 135,
146
tests, rapid 83
Anti-oocyst antibody 100
Apansporoblastina 12
Aphasmidia 165
Apicomplexa, phylum 66, 661
Appendicitis 178
Artcmisinin-based combination therapy 84
Ascariasis, ectopic 205
Ascaris 6, 8, 140, 207
eggs, types of 202f
fertilized egg of 202/
lumbricoirles 3, 7,112,165, 167, 176,180,
189, 194, 199-201, 20 If, 204, 206f c,
207, 243,248
life cycle of 204/
pneumonia 203
suum 167
unfertilized egg of 202/
Ascites 57
Aspirates, splenic 58
Aspiration 135, 135b
biopsies 59
Atovaquone 88
Autoimmune hemolysis 56
Axoneme 41, 42, 53
Azithromycin 88
B
Babesin 4, 12, 66
bovis 86
microti 13, 86, 86/
Babesiosis 87,871
Bachman intradermal test 174
Bacillary dysentery 20, 201
Bacterial infection, secondary 227
Baermann concenrration method 238, 238/
Balamuth's medium 23
composition of 244b
Balamuth's monophasic liquid medium 244
Balamuthia 26
mnndrillaris 15, 30
Balantidiasis I 09
Balantidium 12, 109
coli 3, 7, 11, 13, 14, 39, 107, 107f, 109,
150,244
life cycle of 108/
Bancroftian filariasis 213, 247
Basal body 10
Basophilic stippling 73
Baylisascaris 207
procyonis 167
Bell's dilution-filtration count 238
Bentonite flocculation tests 247
Benznidazole 51
Bile
duct carcinoma 145
staining 123
Bilharziasis 143
Biliary
cirrhosis 156
obstruction, acute 205
passage 152
tract 142, 154
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Paniker's Textbook of Medical Parasitology
Binary fission 11, J 6, 41
Binucleate cyst 16f, 25f
Biopsy 217
BiLhionol 153
Blackwater fever 79
Bladder
carcinoma 145
containing seeds 142
worm ll7, 123
Blastocyslis hominis 10 I, 10 If
Blastomeres 188
Blepharoplast 42, 53
Blinding filaria 221
Blister format ion 227
Blood 13, 142
collection of 240
examination 6, 135
fluke 141-143
incubation infectivity test 47
loss 178
picture 87
smear 82b
transfusion malaria 801
urea nitrogen 88
Boeck and Drbohlav diphasic medium 244,
244b,
Bone marrow 56
aspirate 58
macrophage of 13
Bothriocephalus anemia 120
Bradyzoites 9 l, 93, 102
Brain 21, 104,232
pa renchyma l28f
Bronchi 161
Brugia malayi 4, 7,165,208,210,218, 219f,
224,226
Brugia pahangi 167, 223
Brugia patei 167
Brugia Limori 165,208,210, 2 19,226
Buffy coat blood film 242
Bunostomum phlebotomum 167
C
Cachexia 57
Calabar swellings 2 19, 22 l
Calcofluor white staining 29
Candidate vaccine 61
Capil/aria a eropltila 16 1
Capillaria philippinensis 4, 165, 180, 231,
232
Card agglutination trypanosomiasis test
46, 47
Cardiac implantable electronic device 133
Cartwheel appearance 16
Casoni's inrradermal 1es1 134
Casoni's test 247
Cat liver fluke 156
Cathodic antigen, circulating 147
Caudal papillae l 66
Cecum 18
Cellular exuda1es 20,235
Cellulose acetate membrane precipitation
test 23
Central nervous system 13, 46, 129, 150, 17 1
infection 232t
Centrilobular necrosis 77
Cercarial dermatitis 148
Cerebral
amebiasis 19
angiostrongyliasis 230
malaria 79
paragonimiasis 161
Cerebrospinal fluid 6, 27-29, 45-47, 128,
230,231,245
Cestodes 4 , 112, l l 5
classification of 115, l 16t
living 122b
Chagas disease 13, 42, 47
acute 49, 50
chronic 50
Chagas radioimmune precipitation assay 5 1
Chagoma 50
Chancre
painless 45
trypanosomal 45
Charcoal culture 240
Charcot-Leyden crystals 19, 20, 22, 22f, 235
Chemoprophylaxis 84, 85
Chiclero's ulcer 53, 63
Chilomastix 12
mesnili 32, 38
egg of 38f
Chinese live r fluke 154
Chocolate brown sputum 21
Cholangiocarcinoma 156
Cholangitis 156
Chopra's antimony test 60
Chromatoid bodies 10, 16
Chrysops 220, 221
Chylous urine 215/
Cilia 11
Ciliophora 11, 12
Cloaca 164
Clonorchis 113, 141,207
sinensis 4, 7, 143, 145, 151, 154, 172, 194,
20 1,232
egg of 154f
life cycle of 155/
Coccidia 12, 66, 90
Coenurns 117, 129
Colon 13, 18
Complement fixation test 7, 46, 47, 58, 133,
135,216,246
Complete blood count 205
Congestive cardiac failure 87
Conjunctiva 165
Conjunctiva! biopsies 222
Conjunctiva] epithelium 104
Coproantigen, detection of 23
Copulatory spicules 200
Coracidium 11 8, 120
Cornea 29, 104
Corneal stroma 104
Counter-currem immunoelectrophoresis
23
Craig's medium 22
Creeping myiasis 167
Crescentic tachyzoites 90f
Crustacea 232
Cruzin 51
Cryptosporidiosis 247
Cryptosporidium 12, 14, 66, 100, 236
paruum 3, 4, 7, 13, 10, 97, 99, 100, 105,
199,204,248
life cycle of 99f
oocysrs of 98f, 100/
Clenodactylus gundi 90
Cu/ex quinquefasciatus 2 10, 213
Culicoides 2!0, 223
Cutaneous leishmaniasis, diffuse 53, 62
cyclophyllidean 117, 1171
tapeworms 122
cyclops 118, 232
Cyclospora 66, 101,236
cayetanensis 3, 7,39,99, 100,105
Cylindrical esophagus 180
Cyst 14, 16, 17, 26, 27, 29, 30, 39, 107, 107f
fluid 134
mature 96/
uninucleate 16f
cysteine-pep1one-liver-maltose 37
cysticerci in muscles 124}
cysticercosis 126, 128, 140
Cysticercus 117, 123
bouis 123, 124f
cellulosae 123, 124f
cysts of I 28f
Cytoadherence 74
Cytolysis 18
Cytopharynx 107
Cytoplasm 10, 16
Cytopyge 107
Cytos10me 1 07
D
Deoxyribonucleic acid 8, 21, 35, 47, 58, 83,
127, 133,217,247
Dermatitis 184
Diamond's axenic medium 23
Diarrhea 13, 34b, 97
bloody 15011
Oichlorodiphenyltrichloroethane 85, 218
Dicrocoelium dendriticum 150, 153
Dienlamoeba 12, 39
fragilis 32, 39
trophozoite of 39{
P:267
Died1ylcarbamazine 168, 215, 217, 222
medicated salts 218
provocation test 216, 242
Dihydrofolate reductase 84, 248
Dihydropteroate synthase 84,248
Dipetalonema 223
Diphyllobothrium 113, 115
latum 4, 7, 112,116,117, 118f. L22, 151,
172,232
life cycle of l 19f
Dipylidium 113, 115
caninum 7, 116, 139, 139f. 226
Direct agglutination test 51, 58, 60,246
Direct fluorescent meLhod 105
Dirofilaria 167
conjunctivae 224
immitis 161,167,223
repens 224
Disseminated intravascular coagulaLion 87
Distomata 141
Doxycycline 218
Dracunculiasis 227
Dracunculus medinensis 4, 164, 165, 225,
226, 227fc, 229
adult worm of 226/
infection 225.f
larva of 226.f
life cycle of 228/
Dumdum fever 52, 53
Duodenal aspirates 97, 184, 205
Duodenal capsule technique 243
Duodenwn 156
Dysentery 13
E
East Africa n trypanosomiasis 45,451
Echinococcosis 247
£chinococcus 8, I 15, 117
granulosus 2, 4, 46, 116, 117, 129, 130.f.
133/ c, 140, 16 1,204,226,232
life cycle of 131/
multilocularis 2, 116, 136, 226, 232
Echinostoma 113, 156, 159
Echinostomatoidea 141
Ectocyst 132
Ect0pa rasite 1
Ectopic infection 146, 167
Ectoplasm Io
Edema 46, 57
painless 221
Elephantiasis 210,214, 215f
Embryophore, inner 123
Encephalitis 13
granulomatous 29
£ncephalitowon 12, 104
inleslinalis 105
Encephalopamy, diffuse symmeLric 79
Encysted larvae 165
Endemic foci 160
Endocyst 132
Endodyogeny 11
Endogony 91
Endolimax nana 15, 25, 26.f
Endoparasite I
Endoplasm 10
Endoscopy 5 1
Endospore 105
Entamoeba 6, 12
coli LS, 24, 25/
gingivalis 15, 25
hartmanni 15, 25
trophozoite of 25f
hisloly tica 3, 6, 7, 10, 13, 15, 16.f. 18/J,
2 1/c, 23.f. 99, 105, 109, 150, 199,
204,234,243,244, 248
life cycle of 17!, 17/c
polecki 15
Enteric cycle 92
Enterobius vermicularis 3, 4, 6, 7, 39, 165,
175, 176, 189, 195, 196, 196); 198/ c,
199,207
adult worm of 195/
life cycle of 197 /
Enterocyte 105
J;'nterocylozoon bieneusi 105
Enteromonadina 12
Enteromonas 12
hominis 32, 38, 40
cyst of 38/
Enterotest 35
Enzyme-linked immunosorbent assay 7,
21,23,35, 46,47,51,58,83,94, 95,
127, 133, 147, 168, 173, 185,205,
206,2 16,217,227,246,247
Eosinophil count 215
Eosinophilia 128, 178
peripheral 185
Eosinophils 5(
Epilepsy, focal 133
Epimastigotes 42, 43, 45, 48, 48)
Erythematous patches 57/
Erythrocyte
mature 73
sedimentation rate 46
sequestration 79
surface antigens, ring-infected 85
Erythrocytic schizogony 68, 69, 76); 240
Escherichia coli 29
Esophagus, double bulb 181
Espundia 63
Ethylene diamineterra-acetic acid 240
Eucoccidia 12
Eurytrema pancreaticum 154
ExcystaLion 17
Exilagellating male gametocytes 71
Exoenteric cycle 93
Exoerythrocytic
schizogony 68
schizont 69
stage 69
Extrinsic incubation period 45, 55
Eyes 232
F
Fairley's lCSt 147, 247
Index
Falciparum malaria, complications of 79b
Falcon assay screening test 147
Fasciola 113, 141, 167
gigantica 7, 151
hepatica 4, 7, 143, 150, 15 1, 151.f. 153,
194,201
egg of 151/
life cycle of 152f
Fascioliasis 153
Fasciolidae 14 1
Fasciolopsis 113, 141
buski 4, 7, 143, 151, 153, 156, 157!, 201
egg of 157(
life cycle of 158/
Fast-flowing rivers 222
Fat malabsorption 34
Ferrissia tenuis 145
Fever 20
high-grade 56
Fibrin degradation products 84
Filarial antigen, circulating 2 17
Filarial worm 208
classification of 208t
Filariasis 208, 247
lymphatic 2 10
subcutaneous 210,219
Filariform 183
larva 181, 181/. 184, 188, 19lt, 213
third-stage 188
Flagella 13
Flagellates 32, 321
zoological classification of 41
Flagellwn 4 1, 42
Floatation method 237
Flukes 141
Fluorescent antibody
direct 37
indirect 83, 205, 206
Fluorescent staining 100
Formogcl test 60
Formol-ether sedimentation technique 237,
237f
Fragilis 39
Free-living soil cycle 182
Frenkel, skin test of 95
Fulminant amebic colitis 19
Furcocercous cercaria 145
Fusiform worms 195
P:268
Paniker's Textbook of Medical Parasitology
G
Gametocytes 68, 71
Gametogony 11, 71, 73, 90, 97
Gastric washings 205
Gastrodiscoides 11 3, 141
hominis 7,151, 153, 156, 159, l59f
Gastrointestinal tract J 42
Gastrophilus 167
Gelatin capsule 243
Gelminths 112
Genital flagellates 32
Geohelminths 207b
Giant intestinal fluke 156
Giardia 6, 12, 13
lamblia 3, 5-7, 13, 14, 32, 33/, 35.f. 99,
109, 199,244
life cycle of 34f
Giardiasis 246
Giardia-specific antigen 35, 65
Giemsa stain 46, 59/, 9 lf, 240, 241
Glisson's capsule 153
Glucose-6-phosphate dehydrogenase 78, 79
deficiency 80
Glycogen
mass of 16
vacuole, large 16
Glycophorin 69
Glycoproteins 18
Glycosylphosphatidylinositol 56, 74
Gnathostoma spinigerum 167, 231, 232
egg of 23lf
Gnalhostomiasis 166
Golgi 67
cycle 67
Gomori methenamine silver 94
Gram's stain I 05
Granules, column of 211
Granuloma formation 214
Ground glass appearance 16
Guinea worm 165
Gymnamebia 12
Gynecophoric canal 142
H
Harada-Mori filter paper strip culture 239
Harada-Mori tube method 192, 239f
I Iartmannella culbertsoni 28
llearr 13
I leidenhain's hematoxylin magnification
25f
llelminths L, 111, U lt, 113
zoological classification of 113
Hemagglutination, indirect 7, 83, 127, 133,
205,206
l lemonagellates 13, 14, 32, 41
stages of 4 21
Hemoglobin 79, 83
nature of 80
Hemoglobinuric nephrosis 79
Hemoptysis 161
Hemorrhage 56
Hemosporina 12
I lemozoin pigment 69, 77
I lcpatic lobe, right 134f
Hermaphrodites 112, 116
Hermaphroditic flukes 143, 150
Hermaphroditic trematode, morphology
of l42f
Herring worm disease 233
Heterophyes 113, 141, 156
heterophyes 7, 15 L, 158, 232
l leterophyidae J 4 l
1 lexacanth 117
embryo 118, 123, 130
oncosphere 136
Histidine rich protein 7, 74, 83
J lookworm 187
diagnosis of l93f c
filariform larva of .l 80t
infection 187b, 190, 192/J
Host-pa rasite relationships 2, 3Jc
Human African trypanosomiasis 4511, 47
treatment of 47/
Human
acquire infection 93
hookworm 166
immunodeficiency virus 10, 24, 36, 57,
105b
infection 230
large intestine 159
leukocyte antigen 80
malaria 66
parasites 69t
nematode 167
trematode 167
Humoral immunity 8 1
Hydatid
cyst 130, 131, 131/, 132}; 134}, 136
fate of 133
disease, ma lignant 136b
fluid 132
sand 132
Hyd rocelc 214
Hymenolepiasis 139
Ilymenolepis 113, 115
diminuta 7, 116,139,226
nana 3, 4, 7,112,116, 122, 136, 139, 189,
199,226
adult worm of l37f
egg of 137f
life cycle of 138J
llypergammaglobulinemia 60
Hyperinfection 184
Hypnozoites 69, 71, 81
reactivation of 81
Ilypochromic microcytic anemia 192
Hypoglycemia 79
Iatrogenic transmission 4
Iliac crest 58
Immature cyst 96f
Immunity 5, 24, 58, 80
lmmunochromatographic card test 58,216,
2 17,247
lmmunofluorescence
assay 227
indirect 35
I mmunoglobulin
E 198, 215
M 5, 80,246
Indian visceral leishmaniasis 56
Indirect fluorescent antibody 23, 216, 217,
246
test 94, 95
Indirect hemagglutination 21, 23, 46, 47, 51,
216, 217,246,247
assay 23
Indirect immunofluorescence 47, 51,246,
247
Infective rhabditiform 201
larva 176
lnflammatory reaction 5
Innate immunity 80
lntercellular adhesion molecule 74
Inte rferon gamma 74
Intestinal
amebiasis 18, 19, 19/, 21, 24
chronic 19b
sequelae of 19b
bilharziasis 148
biopsy 97
entamoeba 261
flagellates 13, 32
flukes 141,142, 156,1 76
helminths 247
human nematodes 165
invasion, stage of 173
sarcocystosis l 02
taeniasis 126, 128
Intestine 13
large 13,107, 142,165, 175, 175b
small 13, 32b, 122b, 142, 165, 180, 180h,
200
lntradermal
allergic tests .156
skin test 147
test 51
Intravenous pyelogram 134, 147
Iodamoeba 12, 26
butschlii 15, 25, 26
Iodine staining 235
Iodophilic body 26
Iodoquinol 24
Iron-hematoxylin stain 235
lsoenzym e study 47
P:269
Isospora 12, 66, 236
belli 3, 7, 13, 96, 105
oocysts of 96/
Itching pa pules 165
lvermectin 218,222
J
Jaundice, obstructive 136
JejunaJ biopsies 184
Jejunum 129, 156, 187
K
Kala-azar 13, 52, 53, 55, 56, 56b, 57/. 58/ c
Karyosome 10
Katayama
disease 150
fever 148, 150
Kato d1ick smear technique 235
modified 238
Kato-Katz smear tech nique 192
Kawamoto technique 83
Keratitis 13, 29
stromal 104
Keratoconjunctiviris 104
Kidney 21
Kinetoplast I 0, 41 , 42, 53
Kinetoplastida 12, 13
Kinyoun's acid-fast stain 97,236
reagents of 236b
Kinyoun's carbol fuchsin 236
Knott's concentration technique 216
Kupffer cells 56
L
Lactophenol cotton blue 135
Lancet fluke I 53
Larva 165,171, 225
currens 166, 184
detection of 227
development of 227
infective stage of 166
migrans 165, 232
cutaneous 167, 167t, 168, 1681
third-stage I 9 lt
Latex agglutination test 23, 95
Laverania 66
Leishman's stain 240, 24 1
Leishman-Donovani body 54/. 59/
Leishmania 4, 12, 13, 41 , 52, 245
aethiopica 53, 61
braziliensis 1 3
complex 53
classification of 531
donovani 7, 13, 52, 53, 59/. 248
life cycle of 55/
morphology of 54/
transmission of 561
infantum 53
major 53, 61
mexican a complex 53
peruviana 53
tropica 13, 53, 61
complex 61
Leishmaniasis 246
cutaneous 13, 52, 53
mucocutaneous 13, 52, 53, 63
Leishmanin skin test 60,247
Lepromatous leprosy 184
Leukopenia 56, 57
Lieberkuhn, crypts of 18
Lipophosphoglycan 23, 55
Liposomal amphotericin-B 61
Liver 56, 131!, 151
abscesses 21
biopsy 23
fluke 141,142, 150
rot 151
Loaloa 7, 165,167,208, 2 10,219,221, 222,
224, 226,241
Lobopodia 27
Lobosea 12
Locke's solution 244b
Locomotion 112
Loeffler's syndrome 166, 203, 204b
Lugol's iodine 235
Lumbricoides 200
Lumen-dwelling flagellates 32
Luminal amebicides 24
Lung 150, 160
flukes 141, 142, 160
right lower lobe of 13'\l
Lutzomyia 63
jlaviscutellata 53
longipalpis 53
olmeca 53
umbratilis 53
Lymph node 13
aspirates 59
peripheral 56
Lymphadenitis 214
Lymphadenopathy 45, 57, 93
Lymphangiovarix 214
Lymphangitis 2 14
Lympheclema 214
l.ymphoreticular malignancies 184
Lymphorrhagia 214
M
Machado-Guerreiro test 5 1
Macrogamete 92, 98
Macrogametocyte 7 l, 73
Macules, hypopigmented 223
Mala ria 13, 66, 78/J, 83/J, 871
congenital 80
Control Programs 86
global distribution of 67/
initiative, roll-back 86
merozoite-incluced 80
organs in 78/
Index
parasite 14, 66, 70!, 73/. 74!, 82b, 82!, 240,
245
culture of 82
pigment 69, 69b
sep ticemic 79
tertian malignant 79
vaccine 85
Malarial parasite
drug resistance of 85
types of 71
Malnutrition, severe 184
Mansonella
ozzardi 165, 208, 210,223, 241
perstans 165, 208, 210, 223, 224, 242
strepLocerca 208, 210, 223, 233
Mass d1erapy 217
Mastigophora 12, 13
Mastigote 41
Maurer's clefts 73
Mazzotti reaction 222
McMaster's egg counting 238
Melarsoprol 47
Membrane filtration concentration 242
Meningoencephalitis 230
Merogony 69, 105
Merozoites 68
Mesoendemic 67
Metacyclic trypomastigotes 43, 49
Metacyst 18
Metacystic trophozoites 18
Metagonimus 141
yokogawai 7, 143, 156, 159, 172, 232
Metazoa 10
Metriphonate 147
Metronidazole 24
Meyers Kouwenaar syndrome 215
Microabscesses 148
Microconcentration technique 82
Microfilaria 208,2 11 ,219,241, 248
bancrofti 211, 220J
morphology of 21 V
demonstration of 215
malayi 220f
Microgamete 92, 98
Microhematocrit rube method 242
Microspora 11, 12, 14, 66, 104
Microsporidia 3, 48, I 05, I 06
classification of 1041
infective stage of 105
Microsporidium 104, 105
Microsporum 12
Migrating larva 190, 203
Mild flu 93
Monocytosis 46
Montenegro test 60, 247
P:270
Paniker's Textbook of Medical Parasitology
Mosquito-borne malaria 80t
Motile bacteria 20
Motile nophozoites 20
Mucus plug 175
Multiceps multiceps .129
Multilocular hydatid 136
Multiple fission 11
Murine strain 139
Muscle 104, 171
biopsy 172, 173/J, 243
invasion, stage of 173
Muscular cysticercosis 126
Myocarditis 46, 46b
Myositis 104
N
Naegleria 12, 15,291,233
fowleri I , 13, 15, 26, 231, 244
life cycle of 28/
Napier's aldehyde test 60
ational Rural I lealth Mission 86
National Vector borne Disease Control
Programme 86
Necator 165
americanus 3, 7, 165-167, 176,180,187,
189,192,204,207,229,243
Nelson's medium 23
Nematodes 111 - 113, 164
classification or 1651
zoological classification of 166t
Nematohelminthes 11 l
Neoplasia 5
Nerves 13
Neural larva migrans 168,207
Neurocysticercosis l26
Neutropenia 56
Nifurtimox 51
Nitazoxanide LOO
Nocturnal enuresis 197
Noncalcified hydatid cyst 134/
Nonspecific serum tests 60
Normocytic normochromic anemia 60
Nosema bombycis 104
ovy-Macneal-NicolJe medium 245, 245b
ucleic acid amplificaLion test 37
Nuclepore filtration 216
Nucleus LO, 16, 41, 42
0
Ocular cysticercosis 127, 128
Ocular toxoplasmosis 94
Onchocerca volvulus 165, 208, 210, 221,
22 11; 223
Onchocercoma 222
Onchodermatilis 222
Oncosphere 117, 11 8
Oocysr 7 1, 90, 92, I 05
mature 96/
spheri cal 98
thin-walled 98
Ookine te 7 1
Operculate snails 154
Operculum 118
Ophthalmic larva migrans 168,207
Opisthorchioidea 141
Opislhorchis I 13, 14 1
felineus 143, 151
viverrini 143, 145, 151
Opportunistic infections 105b
Oral flagellates 32
Ovarian lobe, accessory 122
Oxyuris vermicularis 195
p
Packed cell volume 79, 83, 84
Pancreatic duct 154
PancreaLitis 205
Panstrongylus megistus 48
Parabasal body 42, 53
Paragonimiasis, abdominal 161
Paragonimus 113, 14 1
westermani 4, 143, 151, 160, 160f. I 61,
163,204,230,232,233,242,243,
248
egg of 161/
life cycle of 162/
Paramphistomatidae 141
Parasite I, 2/J, 3t, 7b, 115, 201/J, 204b
aberrant l
accidental l
detection of 205
escape mechanisms 6t
exhibiting antigenic variations 5b
P test 83
facultative l
free-living l
infectious 199b
lactate dehydrogenase 83
life cycle of 3
quantification of 82/J
types of 2/c
Parasitic diseases 7t
Parasitology l
Paratenic host 2
Paromomycin 24, 61, 100
Pelvic
plexuses 144
venous plexuses 145
Pentamidine 47
Peribronchial granulomatous lesions 161
Pericardia! amebiasis 19
Peri cyst I 31
Peridomestic cycle 48
Periodic acid-Schiff stain 91, 105
Peripheral blood 71, 82/
Peristome 107
Petri dish culture meLhod 239/
Phasmid 166
Phlebotomus
argenlipes 53
ariasi 53
longipes 53
orientalis 53
papatasi 53
pedifer 53
perniciosus 53
sergenti 53
Pinworm 165
Piroplasmia 12
Pistia plant, removal of 218
Plagiorchioidea 141
Planoconvex egg 196/
Plasmodium l, 11, 12, 66
falciparum 5, 7, 66, 67, 69, 73, 74/, 77, 78,
82/, 83, 88, 89
chloroquine resistance transporter
84,248
erythrocyte membrane protein-I 74,
79
histidine-rich protein 83
lactate dehydrogenase 7
multidrug resistan ce protein 84, 248
lactate clehyd rogenase 246
malariae 66, 67, 69, 75,, 77, 78, 89
stages 76/
ovale 66,67,69, 75, 77, 78,89
uivax 7, 66, 67, 69, 7 l, 72f. 73f. 77, 78, 88,
89
life cycle of 68/
Plastic envelope medium 37
Platyhelminthes 111, 115
Pleistophora 104
Plerocercoid larva 118
Pneumocystis
jirovecii 5, 243
pneumonia 94
PneumoniLis 204b
Polar tubule 105
Polymerase chain reaction 8, 35, 83, 84, 87,
127,133,216,217,247
Portal hypertension 148, 156
Post-kala-azar dermal leishmaniasis 13, 52,
57, 57t, 243
treatment of 57
Praziquantel 128, 147, 148, 158
Precysr 16
Pre-erythrocytic schizogony 68, 691
Primaquine 84, 85
Procercoid larva 118, 120
Proglonids 116, 127
Promastigote 42, 53, 54/
Protein 34
merozoite surface 85
Protozoa I, 410,, 32b
1
P:271
classification of 11, I 2t
transmitted 37 b
Protozoan parasites 34h
Protozoology 1
Pruritus ani 196
Pseudocele 164
Pseudocyst 91
Pseudophyllidean 117, 1171
tapeworms l J 7
Pseudopodia I 0, 13, 15, J 6
Pseudotumoral growth granulomatous 19
Pulmonary capillaries 130
Pyknotic bodies 22
Pyriform 129, 159
Pyrimethamine 95
Q
Quadrinucleate ameba 18
Quadrinucleate cyst 26J; 33(
mature 16
Quantitative buffy coat 82, 83
Quartan malaria 75
Quinine 84
R
Rat fleas 138
Rat tapeworm 139
Rectal biopsy 146, 148
Red blood cell 13, 19, 20, 39, 67, 68, 74, 78,
00,82,235
splenic sequestration of 56
Renal transplantation80
Respiratory distress syndrome, acute 87
Reticulocyces 72
Reliculoendothelial system 13
Retortamonadida 12
Retortamonas 12
intestinalis 32, 38
Retroinfection 196
Rhabditiform larva 181, 18 1{. 183, 188, 203
Rhizopoda 12
Rhodnius prolixus48
Ribonucleic acid 82, 248
River blindness 222
RK39 tesr 60
Robinson's medium 23
Rodent feces 138
Romana's sign 50
Roundworm egg 202t
s
Sabin.Feldman dye test 95, 246
Salivary gland 68
Salpingitis, chronic 197
Sandy patches 145
Sn rcocyst is 12, 66, I 02
homl11is 32, J 72, 226
oocyst of 102(
suihominis 172, 226
Sarcocystosis, muscular 102
Sarcodina 12, 13
Sarcomastigophora 11, 12, 15
Saturated sail solution 176h
technique 237
Schaudinn's solution 236
Schistosoma 141, 144/
eggs 238b
hematobium 5, 7, 143·145, 147/c, 149,
207,217
egg of 144/
intercalatum 150
japonicum 143, 149, 232
mansoni 143, 1<17, 149
mekongi 150
Schistosomatidae 141
Schistosomes 143, 143/J, I so
morphology of 143/
Schistosomiasis 145
acuce 145
ch ronic 145
Schizogony 11, 68, 73, 90, 97, 98
Schneider's drosophila tissue culture
medium 59
Schneider's insect tissue culture medium
245, 245h
Schuffner's docs 72
Scotch tape method 198, l 98/ ; 238
Segmemina 157
Serological tests 7, 205
Serous cavity filariasis 210, 223
Serpent worm 225
erurn glutamic pyruvate transaminase 88
Sheep liver fluke 150
Sigmoidorectal region 18, 144
Sigmoido copy 178
Silkworm disease J 04
Skeletal muscle 104, 171
Skin 13
snip 243
test 7, 8h, 60, 62, 63,217,227,247
transmission 4
Sleeping sickness 13, 42
Slender thread•like worms 208
Smooth curves 216
Somatic cells 211
Somatic human nematodes 165
outh American trypanosorniasis 42
Sparganosis 120, 121, 166
Sparganum ll8
larva 120
Spherical nucleus 38
Spirometra 115, 120
erinacei 116
life cycle of 121/
theileri 116
Spirurid nematode 231
Index -
Spleen 21, 56, 78
Spoliative effects 203
Sporoblasts 96
porocyst 96, 142, 153, 154
Sporogony 68, 71, 90, 104, 105
Sporozoa 12, 661
Sporozoites 68, 71
StalJion's disease 43
Scercoraria 42
StoU's dilution technique 238
Strawberry mucosa 37
String test 35
Strobila 116, 11 7
Strongyloides 165,229,239
stercora/is 4, 105, 164,165, 167, 180,
184/ c, 185, 199,204,207,226,232,
243, 248
egg or 181/
larvae of 181/
life cycle or 182); 183/ c
Strongyloidiasis 185
Sugar floatation technique 237
Suihominis 32
Sulfadiazine 95
Suppurative inflammation, acute 29
Swimmer's itch 145
Sylvatic zoonosis 48
Syngamy 11
Syscemic lupus erythematosus 29
T
Tachyzoites 91, 93
Taenia 115
antigen, detection or 127
egg of 124/
multiceps 129, 226, 232
saginata 4, 7, 11 6, 122, 123t, 140, 172,
176,201,226
asiatica 122, 129, 226
life cycle of 125/
solium 2·4, 7, 112, 116, ll 7, l 22, l 22/ ,
1231, 140, 199, 201, 207, 226, 229,
232
adult worm or 122/
eggof 176
life cycle of 125, 126/
Tapeworm 115, llSJ; 122, 129,
Tetracyclines 218
Tetrapetalonema
perstans 223
streplocerca 223
Thromhocytopenia 56, 57, 60
Tick•borne disease 86
Tinidazole 24
Tiny knob 154
Tissue 6
amebicides 24
biopsy 243
~ ... ~- - - -!~ii.i=l.lo,oiii•
P:272
B!aniker's Textbook of Medical Parasitology
cyst 90, 91, 93
hypoxia 77
necrosis 18
Toxic megacolon 19
Toxocara canis 167, 206, 232, 233
adult worms of 206/
Toxocara cati 167,206,232
Toxocariasis 247
Toxoplasma 11, 12, 14, 66, 90, 94
e ncephalitis 95, 96
gondii 1, 2, 4, 5, 10, 14, 46, 48, 90, 90{.
91/, 93, 94/c, 96, I 05, l 72, 248
lire cycle of 92f
inrection 93, 94
pneumonia 94
Toxoplasmosis 94, 245, 246
acquired 93
acute 93
congenital 93, 95
Trachipleistophora I 04
Transfusion malaria 80
Transovarian transmission 87
Transverse binary fission I 08
Trau ma 5
Traveler' diarrhea 99/,
Trematodes 4, 112, 141, 143/,
classification of 141
zoological classification of 14 l/
Triatoma inrestans 48
Trichina worm 170
Trichinel/a 140, 164
cyst 171
Spira/is 4, 7, 46, 165, 170, 172, 173/< , 174,
180, 199,207, 226,232,243
adult worms or 170/
life cycle of 172/, l 73t
Trichinosis 247
Trichomonadida 12
Trichomonas 12, 13, 36, 39
hominis 32, 36/, 38
tenax 32, 36/. 37, 204
vagina/is 3, 6, 7, 13, 36, 36); 95, 109, 217,
244
Trichostomatina 12
Trichostrongyliasis 193
Trichostrongylus orienlalis 7
Trichrome stain 235
modified LOS, 236
reagents of 236b
Trichuris 175,229
lrichiura 3, 7, l09, J 50, 165, 175, J 76,
178, 178f c,189, 196,20l,207
egg of 176/
life cycle or L 77]
Triclabenda1ole 153
Tripartite 187
Trop-bio test 2 I 7
Trophozoite 11, 14, 16, 16/; 27/ , 29, 33,33{,
38/, 39, 86/; 90, 91, 91{. 107, 107/
extracellula r 90{
Tropical pulmonary eosinophilia 215, 2 17
Trypa11osoma 12, 13, 41, 233, 241, 244, 248
brucei 4, 13, 32, 42,231
brucei 43
lire cycle or 44/
gambiense 5, 42, 50
rhodesiense 5, 43, 45, 46, 50
cruzi 4, 13, 32, 43, 46-48, 48{. 50, SOJ, 51,
93,226
life cycle or 49f
equiperdum 43
evansi 43
gambiense 43, 43/
lewisi 43
infections 95
rangeli 43, 51
rhodestense43f. 46[
Trypanosomatidae 41
Trypanosomatina 12
Trypanosomes 42, 5 1
classification or 42
Trypanosomiasis 43{. 46/< , 245, 246
Trypomastigote 42, 48, 48]
Tsetse fly 45
Tubercles 147
Tubulina 12
Typhus-like examhema 94
u
Uncinaria stenoceplzala 167
United Nations Children's Fund 86
United ations Development Programme
86
Upper respiratory tract 29
Urethra 13
Urethritis 13
Urinary bladde r 144
carcinoma 145
Urine 6, 7/J
V
Vaccination 5
Vacuole 53
Vagina 13
Vaginal sphincter, prominent 122
Vaginitis 13
Vascular cell adhesion molecule- I 74
Vector mosquito, eradication of 2 18
Vector transmission 4
Vermicules 87
Vertebrate host 44
Visceral larva migrans 167, 1671, 168, 1681,
206,207
Visceral leishmaniasis 52, 53, 54/, 61 h, 245
Viviparous nematodes 226/J
w
Water plants, ingestion of 159
Watsonius watsoni 153, 156, 159
West African trypanosomiasis 43, 451
Western blot 100
Wet saline mounts 235
Wheatley's lrichrome stain 235
Whip-like flage lla 32
Whipworm 165, 175, 1761
White blood cell 29, 83, 242
Winterbonom's sign 45
Wolbachia 208,223
Wright's tain 241 ·
wuchereria 164
y
bancrofti 4, 7, 165, 199,208, 2 10, 211{.
212,213, 2 16/,, 2 17,218,224,226,
24 1
adult worm or 21J/
life cycle of 212/
Young erythrocytes 72
Young trophowites 69
z
Ziehl-Neelsen stain 100(
modified 97, 98, 236
Ziemann's stippling 75
Zinc sulfate noatation concentration
technique 237/
Zooa11throponoses 2
Zoomastigophorea 12
Zoonoses 2, 8
Zoonotic filariasis 223
Zoophilic nematode 167
Zygocotylidae 141
Zygote 7 1
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How to expand product of n n factors.
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Asked 6 years, 10 months ago
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I have a product say
F(a,n,x)=∏j=0 n(1−a n−2 j x)F(a,n,x)=∏j=0 n(1−a n−2 j x)
I want to expand and hope to have general terms of the coefficients. I did for n=2,3,4,5,6,7,8...n=2,3,4,5,6,7,8... I see it will be different for n n even or n n odd. We have
F(a,2,x)=1−x 3+(a 2+1+a−2)x 2+(−a 2−1−a−2)x F(a,2,x)=1−x 3+(a 2+1+a−2)x 2+(−a 2−1−a−2)x
F(a,3,x)=1+x 4+(−a 3−a−a−3−a−1)x 3+(a 4+2+a−2+a 2+a−4)x 2+(−a 3−a−a−3−a−1)x F(a,3,x)=1+x 4+(−a 3−a−a−3−a−1)x 3+(a 4+2+a−2+a 2+a−4)x 2+(−a 3−a−a−3−a−1)x
The coefficients of a a gets more interesting as n n grows. I am interested in the coefficient of a.a. Does anyone know how to expand product of n n factors.
binomial-coefficients
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edited Nov 12, 2018 at 16:34
LearnerLearner
asked Nov 11, 2018 at 23:31
LearnerLearner
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By "coefficients of a a" do you mean "coefficients of x k x k in terms of a a"?YiFan Tey –YiFan Tey 2018-11-12 01:06:45 +00:00 Commented Nov 12, 2018 at 1:06
@ YiFan yes you can say that.Learner –Learner 2018-11-12 01:23:09 +00:00 Commented Nov 12, 2018 at 1:23
There should be a development of this product involving the a 2 a 2-binomial coefficients. See here.René Gy –René Gy 2018-11-12 18:38:50 +00:00 Commented Nov 12, 2018 at 18:38
@Thanks René Gy Learner –Learner 2018-11-13 02:52:05 +00:00 Commented Nov 13, 2018 at 2:52
Apply the q q-binomial formula∏k=0 n−1(1+q k t)=∑k=0 n q k(k−1)/2(n k)q t k∏k=0 n−1(1+q k t)=∑k=0 n q k(k−1)/2(n k)q t k to n+1 n+1, a n x a n x and a−2 a−2 instead of n n, t t and q q.darij grinberg –darij grinberg 2018-11-14 00:44:23 +00:00 Commented Nov 14, 2018 at 0:44
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When expanding the product F(a,n,x)F(a,n,x) in terms of x x the coefficients of x k x k are polynomials P n,k(a)P n,k(a) in a a. Here we expand F(a,n,x)F(a,n,x) in order to see the coefficients of a a in P n,k(a)P n,k(a) explicitly.
We obtain
F(a,n,x)=∏j=0 n(1−a n−2 j x)=∑S⊆{0,1,…,n}(−x)|S|a n|S|∏j∈S a−2 j=1+∑k=1 n+1∑S⊆{0,1,…,n}|S|=k(−x)|S|a n|S|∏j∈S a−2 j=1+∑k=1 n+1(−x)k a n k∑0≤j 1<⋯<j k≤n a−2(j 1+⋯+j k)=1+∑k=1 n+1(−x)k a n k∑l=k(k−1)/2 k(2 n−k+1)/2∑0≤j 1<⋯<j k≤n j 1+⋯+j k=l a−2(j 1+⋯+j k)=1+∑k=1 n+1(−1)k∑l=k(k−1)/2 k(2 n−k+1)/2⎛⎝⎜⎜∑0≤j 1<⋯<j k≤n j 1+⋯+j k=l 1⎞⎠⎟⎟a n k−2 l x k=1+∑k=1 n+1(−1)k∑l=0 k(n−k+1)⎛⎝⎜⎜∑0≤j 1<⋯<j k≤n j 1+⋯+j k=l+k(k−1)/2 1⎞⎠⎟⎟a k(n−k+1)−2 l x k=1+∑k=1 n+1(−1)k∑l=0 k(n−k+1)⎛⎝⎜⎜∑0≤q 1≤⋯≤q k≤n q 1+⋯+q k=l 1⎞⎠⎟⎟a k(n−k+1)−2 l x k(1)(2)(3)(4)(5)(6)(7)(8)(1)F(a,n,x)=∏j=0 n(1−a n−2 j x)(2)=∑S⊆{0,1,…,n}(−x)|S|a n|S|∏j∈S a−2 j(3)=1+∑k=1 n+1∑S⊆{0,1,…,n}|S|=k(−x)|S|a n|S|∏j∈S a−2 j(4)=1+∑k=1 n+1(−x)k a n k∑0≤j 1<⋯<j k≤n a−2(j 1+⋯+j k)(5)=1+∑k=1 n+1(−x)k a n k∑l=k(k−1)/2 k(2 n−k+1)/2∑0≤j 1<⋯<j k≤n j 1+⋯+j k=l a−2(j 1+⋯+j k)(6)=1+∑k=1 n+1(−1)k∑l=k(k−1)/2 k(2 n−k+1)/2(∑0≤j 1<⋯<j k≤n j 1+⋯+j k=l 1)a n k−2 l x k(7)=1+∑k=1 n+1(−1)k∑l=0 k(n−k+1)(∑0≤j 1<⋯<j k≤n j 1+⋯+j k=l+k(k−1)/2 1)a k(n−k+1)−2 l x k(8)=1+∑k=1 n+1(−1)k∑l=0 k(n−k+1)(∑0≤q 1≤⋯≤q k≤n q 1+⋯+q k=l 1)a k(n−k+1)−2 l x k
The coefficients of a a in P n,k(a)P n,k(a) are given in (8) as the blue marked sum times (−1)k(−1)k.
We conclude the absolute value of the coefficient gives the number of integer partitions of l l into k k parts with largest part at most n n.
Comment:
In (2) we note the product (1) consists of n+1 n+1 factors and from each factor we choose either 1 1 or −a n−2 j x−a n−2 j x. We represent each choice as subset S⊆{0,1,…,n}S⊆{0,1,…,n}.
In (3) We reorder the summands according to the size k k of S S. We also extract the term 1 1 which represents the case S=∅S=∅. In this case we have chosen always 1 1 from each of the n+1 n+1 factors.
In (4) we can factor out −x−x and a n a n and thanks to k k we can explicitly write the elements of S={j 1,j 2,…,j k}S={j 1,j 2,…,j k} for each specific choice.
In (5) we do again a reordering by organizing the summands according to the sum j 1+j 2+⋯+j k j 1+j 2+⋯+j k of the k k-tupels.
We observe the smallest sum comes from the k k-tupel (0,1,2,…,k−1)(0,1,2,…,k−1) which gives ∑j=0 k j=k(k−1)/2∑j=0 k j=k(k−1)/2
while the k k-tupel with the largest sum is (n−k+1,n−k+2,…,n)(n−k+1,n−k+2,…,n) which gives ∑j=n−k+1 n j=∑j=1 n j−∑j=1 n−k j=n(n+1)2−(n−k)(n−k+1)2=k(2 n−k+1)2.∑j=n−k+1 n j=∑j=1 n j−∑j=1 n−k j=n(n+1)2−(n−k)(n−k+1)2=k(2 n−k+1)2.
In (6) we factor out a−2(j 1+⋯+j k)=a−2 l a−2(j 1+⋯+j k)=a−2 l.
In (7) we shift the index of l l to start from 0 0.
In (8) we finally change the index variables j t=q t+t−1,1≤t≤k j t=q t+t−1,1≤t≤k. We have j 1+j 2+⋯+j k q 1+(q 2+1)+⋯+(q k+k−1)q 1+q 2+⋯+q k=l+k(k−1)2=l+k(k−1)2=l j 1+j 2+⋯+j k=l+k(k−1)2 q 1+(q 2+1)+⋯+(q k+k−1)=l+k(k−1)2 q 1+q 2+⋯+q k=l
This way we get an inequality chain containing ≤≤ symbols only which admits a nice interpretation via partitions.
Example F(a,3,x)F(a,3,x):
We evaluate the expression (8) for the case n=3 n=3. We obtain
F(a,3,x)=1+∑k=1 4(−1)k∑l=0 k(4−k)⎛⎝⎜⎜∑0≤q 1≤⋯≤q k≤3 q 1+⋯+q k=l 1⎞⎠⎟⎟a k(4−k)−2 l x k=1−∑l=0 3⎛⎝⎜⎜∑0≤q 1≤3 q 1=l 1⎞⎠⎟⎟a 3−2 l x+∑l=0 4⎛⎝⎜⎜∑0≤q 1≤q 2≤3 q 1+q 2=l 1⎞⎠⎟⎟a 4−2 l x 2−∑l=0 3⎛⎝⎜⎜∑0≤q 1≤q 2≤q 3≤3 q 1+q 2+q 3=l 1⎞⎠⎟⎟a 3−2 l x 3−∑l=0 0⎛⎝⎜⎜∑0≤q 1≤q 2≤q 3≤q 4≤3 q 1+q 2+q 3+q 4=l 1⎞⎠⎟⎟a−2 l x 4=1−(a 3+a+a−1+a−3)x+(a 4+a 2+2+a−2+a−4)x 2−(a 3+a+a−1+a−3)x 3+x 4 F(a,3,x)=1+∑k=1 4(−1)k∑l=0 k(4−k)(∑0≤q 1≤⋯≤q k≤3 q 1+⋯+q k=l 1)a k(4−k)−2 l x k=1−∑l=0 3(∑0≤q 1≤3 q 1=l 1)a 3−2 l x+∑l=0 4(∑0≤q 1≤q 2≤3 q 1+q 2=l 1)a 4−2 l x 2−∑l=0 3(∑0≤q 1≤q 2≤q 3≤3 q 1+q 2+q 3=l 1)a 3−2 l x 3−∑l=0 0(∑0≤q 1≤q 2≤q 3≤q 4≤3 q 1+q 2+q 3+q 4=l 1)a−2 l x 4=1−(a 3+a+a−1+a−3)x+(a 4+a 2+2+a−2+a−4)x 2−(a 3+a+a−1+a−3)x 3+x 4
in accordance with OPs calculation.
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edited Jun 12, 2020 at 10:38
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answered Nov 18, 2018 at 18:49
Markus ScheuerMarkus Scheuer
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@ Markus. Thanks for time and effort I really appreciate it.Learner –Learner 2018-11-19 20:50:08 +00:00 Commented Nov 19, 2018 at 20:50
@Learner: You're welcome.Markus Scheuer –Markus Scheuer 2018-11-19 20:54:33 +00:00 Commented Nov 19, 2018 at 20:54
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190498 | https://education.ti.com/en/customer-support/knowledge-base/financial-calculators/product-usage/11235 | Solution 11235: Computing Interest Rate Using the BA II PLUS™ Family Financial Calculator.
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Solution 11235: Computing Interest Rate Using the BA II PLUS™ Family Financial Calculator.
How do I compute for the interest rate using a BA II PLUS family calculator?
To compute the interest rate of a loan using a BA II PLUS family calculator (includes the BA II PLUS AND BA II PLUS PROFESSIONAL), use the Time Value of Money feature and follow the example below:
For example:
What is the interest rate on a mortgage for $75,000 with payments of $576.69 each month for 360 months?
N = 360
I/Y = ?
PV = $75,000
PMT = -$576.69
FV = 0
1) Press the [2nd] key and the [I/Y] key. (This enters the P/Y worksheet.)
2) Set P/Y to 12 for monthly payments by entering 12 and pressing the [ENTER] key. This also sets the C/Y (compounding periods) to monthly.
3) Press the [2nd] key and the [CPT] key. (This exits the P/Y worksheet.)
4) Press the [2nd] key and the [FV] key. (This clears the TVM worksheet.)
5) Input 75000 and press the [PV] key. (This stores 75000 to the Present Value register.)
6) Input 576.69, press the [+/-] key, and press the [PMT] key. (This stores -576.69 to the Payment register.)
7) Input 360 and press the [N] key. (This stores 360 to the N register.)
8) Press the [CPT] key and the [I/Y] key.
The interest rate is 8.50%.
Please see the BA II PLUS or BA II PLUS PROFESSIONAL guidebooks for additional information.
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190499 | https://alpha.chem.umb.edu/chemistry/ch115/Mridula/CHEM%20116/documents/chapter_11au.pdf | Intermolecular Forces Chapter 11 Intermolecular Forces, Liquids, and Solids Chemistry, The Central Science, 10th edition Theodore L. Brown; H. Eugene LeMay, Jr.; and Bruce E. Bursten Intermolecular Forces States of Matter The fundamental difference between states of matter is the distance between particles.
Intermolecular Forces States of Matter Because in the solid and liquid states particles are closer together, we refer to them as condensed phases.
Intermolecular Forces The States of Matter • The state a substance is in at a particular temperature and pressure depends on two antagonistic entities: The kinetic energy of the particles The strength of the attractions between the particles Intermolecular Forces Intermolecular Forces Intermolecular Forces The attractions between molecules are not nearly as strong as the intramolecular attractions that hold compounds together.
Intermolecular Forces Intermolecular Forces They are, however, strong enough to control physical properties such as boiling and melting points, vapor pressures, and viscosities.
Intermolecular Forces Intermolecular Forces These intermolecular forces as a group are referred to as van der Waals forces.
Intermolecular Forces van der Waals Forces • Dipole-dipole interactions • Hydrogen bonding • London dispersion forces Intermolecular Forces Ion-Dipole Interactions • A fourth type of force, ion-dipole interactions are an important force in solutions of ions.
• The strength of these forces are what make it possible for ionic substances to dissolve in polar solvents.
Intermolecular Forces Dipole-Dipole Interactions • Molecules that have permanent dipoles are attracted to each other.
The positive end of one is attracted to the negative end of the other and vice-versa.
These forces are only important when the molecules are close to each other.
Intermolecular Forces Dipole-Dipole Interactions The more polar the molecule, the higher is its boiling point.
Intermolecular Forces London Dispersion Forces While the electrons in the 1s orbital of helium would repel each other (and, therefore, tend to stay far away from each other), it does happen that they occasionally wind up on the same side of the atom.
Intermolecular Forces London Dispersion Forces At that instant, then, the helium atom is polar, with an excess of electrons on the left side and a shortage on the right side.
Intermolecular Forces London Dispersion Forces Another helium nearby, then, would have a dipole induced in it, as the electrons on the left side of helium atom 2 repel the electrons in the cloud on helium atom 1.
Intermolecular Forces London Dispersion Forces London dispersion forces, or dispersion forces, are attractions between an instantaneous dipole and an induced dipole.
Intermolecular Forces London Dispersion Forces • These forces are present in all molecules, whether they are polar or nonpolar.
• The tendency of an electron cloud to distort in this way is called polarizability (to give physical polarity to).
Intermolecular Forces Factors Affecting London Forces • The shape of the molecule affects the strength of dispersion forces: long, skinny molecules (like n-pentane tend to have stronger dispersion forces than short, fat ones (like neopentane).
• This is due to the increased surface area in n-pentane that allows the molecules to make contact over the entire surface area.
Intermolecular Forces Factors Affecting London Forces • The strength of dispersion forces tends to increase with increased molecular weight.
• Larger atoms have larger electron clouds, which are easier to polarize.
Intermolecular Forces Comparing London attractions n-pentane, C5H12 m.p. -130ºC b.p.
36ºC n-nonane, C9H20 m.p.
-54ºC b.p.
151ºC Intermolecular Forces Explaining macroscopic behavior by reasoning about intermolecular forces name formula melting point lauric acid C11H23COOH 44° C myristic acid C13H27COOH 58° C palmitic acid C15H31COOH 63° C stearic acid C17H35COOH 70° C Intermolecular Forces • As the number of carbons increases in a series of fatty acids… • the melting point increases.
• This is because… • as the number of carbons increases, the chains get longer. When the chains are longer, the molecules are bigger.
• The larger the molecule, the greater the dispersion forces.
• When the attractive forces holding particles together is greater, you have to get to a higher temperature to break those forces, so the melting point is higher.
Intermolecular Forces Which Have a Greater Effect: Dipole-Dipole Interactions or Dispersion Forces?
• If two molecules are of comparable size and shape, dipole-dipole interactions will likely be the dominating force.
• If one molecule is much larger than another, dispersion forces will likely determine its physical properties.
Intermolecular Forces How Do We Explain This?
Intermolecular Forces • Water has a high boiling point, high specific heat and high heat of vaporization indicating that intermolecular forces between the water molecules are quite strong. • HF and NH3 also behave the same way Intermolecular Forces Hydrogen Bonding • The dipole-dipole interactions experienced when H is bonded to N, O, or F are unusually strong.
• We call these interactions hydrogen bonds.
Intermolecular Forces • Hydrogen bonding is a special type of molecular attraction between the hydrogen atom in a polar bond and nonbonding electron pair on a nearby small electronegative ion or atom (usually F, O or N).
Intermolecular Forces Hydrogen Bonding Hydrogen bonding arises in part from the high electronegativity of nitrogen, oxygen, and fluorine.
Also, when hydrogen is bonded to one of those very electronegative elements, the hydrogen nucleus is exposed.
Intermolecular Forces In which of the following substances is hydrogen bonding likely to play an important role in determining physical properties: methane (CH4), hydrazine (H2NNH2), methyl fluoride (CH3F), or hydrogen sulfide (H2S)?
In which of the following substances is significant hydrogen bonding possible: methylene chloride (CH2Cl2) phosphine (PH3) hydrogen peroxide (HOOH), or acetone (CH3COCH3)?
Page 444 Intermolecular Forces Summarizing Intermolecular Forces Intermolecular Forces Summarizing From weakest to strongest Increasing strength of attractions +1 -1 London forces Dipole-dipole forces Hydrogen bonding Ion-ion forces Intermolecular Forces Complications • Any molecules that experience one type of attraction, also experience all the weaker types of attractions • HCl molecules experience: Hydrogen bonding (which is the strongest form of dipole-dipole interactions), and London dispersion forces Intermolecular Forces • List the substances BaCl2, H2, CO, HF, and Ne in order of increasing boiling points.
• The attractive forces are stronger for ionic substances than for molecular ones • The intermolecular forces of the remaining substances depend on molecular weight, polarity, and hydrogen bonding. The molecular weights are H2 (2), CO (28), HF (20), and Ne (20). • The boiling point of H2 should be the lowest because it is nonpolar and has the lowest molecular weight. • The molecular weights of CO, HF, and Ne are roughly the same. Because HF can hydrogen bond, however, it should have the highest boiling point of the three. • Next is CO, which is slightly polar and has the highest molecular weight. Finally, Ne, which is nonpolar, should have the lowest boiling point of these three. • The predicted order of boiling points is therefore: Intermolecular Forces (A)Identify the intermolecular forces present in the following substances, and (B) select the substance with the highest boiling point: CH3CH3, CH3OH, and CH3CH2OH Answers: (a) CH3CH3 has only dispersion forces, whereas the other two substances have both dispersion forces and hydrogen bonds; (b) CH3CH2OH Intermolecular Forces Intermolecular Forces Affect Many Physical Properties The strength of the attractions between particles can greatly affect the properties of a substance or solution.
Intermolecular Forces Intermolecular Forces Viscosity • Resistance of a liquid to flow is called viscosity.
• It is related to the ease with which molecules can move past each other.
• Viscosity increases with stronger intermolecular forces caused by increase in the molecular weight, and decreases with higher temperature.
Intermolecular Forces Explaining macroscopic behavior: viscosity • Viscosity is the resistance to flow in liquids • Viscosity is dependent on more than just molecule size – it also depends on the kinds of attractive forces between molecules • Viscosities of various liquids at 20ºC (in centipoise) Liquid Viscosity (cp) Type of IM attraction water 1.002 H-bonding & London tetrachloromethane (CCl4) 0.969 London olive oil 84 London oleic acid 25 London glycerine 1490 H-bonding & London Intermolecular Forces Surface Tension Surface tension results from the net inward force experienced by the molecules on the surface of a liquid.
Intermolecular Forces Intermolecular Forces Phase Changes Intermolecular Forces Energy Changes Associated with Changes of State • Heat of Fusion: Energy required to change a solid at its melting point to a liquid.
Intermolecular Forces Energy Changes Associated with Changes of State • Heat of Vaporization: Energy required to change a liquid at its boiling point to a gas.
Intermolecular Forces • Heat of Sublimation: Is the enthalpy change required to transform a solid directly into gaseous state. Intermolecular Forces • A refrigerator contains an enclosed gas that can be liquefied under pressure. The liquid absorbs heat as it subsequently evaporates and the refrigerator cools in the process. The vapor is then recycled through a compressor.
• The heat absorbed by the liquid during vaporization is released during condensation. This heat is dissipated through cooling coils at the back of the unit. Intermolecular Forces Energy Changes Associated with Changes of State • The heat added to the system at the melting and boiling points goes into pulling the molecules farther apart from each other.
• The temperature of the substance does not rise during the phase change.
Intermolecular Forces Intermolecular Forces • Calculate the enthalpy change upon converting 1.00 mol of ice at –25° ° ° °C to water vapor (steam) at 125° ° ° °C under a constant pressure of 1 atm. The specific heats of ice, water, and steam are 2.09 J/g-K, 4.18 J/g-K and 1.84 J/g-K, respectively. For H2O, ∆ ∆ ∆ ∆Hfus = 6.01 kJ/mol and ∆ ∆ ∆ ∆Hvap = 40.67 kJ/mol.
∆ ∆ ∆ ∆H = mc∆ ∆ ∆ ∆T At the phase change ∆ ∆ ∆ ∆H = m x enthalpy of fusion or vaporization Intermolecular Forces • What is the enthalpy change during the process in which 100.0 g of water at 50.0° ° ° °C is cooled to ice at –30.0° ° ° °C? The specific heats of ice, water, and steam are 2.09 J/g-K, 4.18 J/g-K and 1.84 J/g-K, respectively. For H2O, ∆ ∆ ∆ ∆Hfus = 6.01 kJ/mol and ∆ ∆ ∆ ∆Hvap = 40.67 kJ/mol.
Intermolecular Forces Please print another copy of the lab schedule.
Intermolecular Forces Critical Temperature and Pressure • The highest temperature at which a distinct liquid phase can form is called the critical temperature.
• The critical pressure is the pressure required to bring about liquefaction at this critical temperature • Above the critical temperature the motion energies of the molecules are greater than the attractive forces that lead to the liquid state. • At critical temperature the properties of the gas and liquid phases become the same resulting in only one phase: the supercritical fluid.
Intermolecular Forces It is useless to try to liquefy the gas if it is above its critical temperature, it needs to be cooled to this temperature before it can be liquefied.
The nonpolar low molecular weight substances have lower critical temperatures.
Substance Critical temp K Critical pressure atm Water 647 217 Ammonia 405 111 H2S 374 89 Propane 370 42 Phosphine 324 64 CO2 304 73 O2 154 50 Argon 150 48 N2 126 33 Intermolecular Forces Vapor Pressure • At any temperature, some molecules in a liquid have enough energy to escape.
• As the temperature rises, the fraction of molecules that have enough energy to escape increases.
Intermolecular Forces Comparing a liquid at two temperatures From Chemistry & Chemical Reactivity 5th edition by Kotz / Treichel. C 2003. Reprinted with permission of Brooks/Cole, a division of Thomson Learning: www.thomsonrights.com. Fax 800-730-2215.
Particle level The effect of temperature on the distribution of kinetic energies in a liquid Intermolecular Forces Vapor Pressure As more molecules escape the liquid, the pressure they exert increases.
Intermolecular Forces Vapor Pressure The liquid and vapor reach a state of dynamic equilibrium: liquid molecules evaporate and vapor molecules condense at the same rate.
Intermolecular Forces Vapor Pressure • The boiling point of a liquid is the temperature at which its vapor pressure equals atmospheric pressure.
• The normal boiling point of a liquid is the temperature at which its vapor pressure is 760 torr.
Intermolecular Forces Focus on Liquid to Gas: For a 1.00-mol sample of liquid water 0 20 40 60 80 100 120 0 2000 4000 6000 8000 Energy added (Joules) Temperature (Celsius) C D Intermolecular Forces What happens if you add more heat energy to liquid water at 100ºC?
• It boils • What is boiling?
There is a statistical range of kinetic energies (velocities) of particles in the liquid Some particles will always have enough energy to break away from attractive forces that keep them in liquid →evaporation (vapor pressure) As temperature rises, eventually it is high enough that so many particles can break away that their gas pressure (vapor pressure) equals the pressure of the surroundings →boiling • Boiling continues with no change in temperature until all liquid particles have converted to gas phase • Normal boiling point is temperature at which vapor pressure reaches atmospheric pressure when patm = 1atm Intermolecular Forces Phase Diagrams Phase diagrams display the state of a substance at various pressures and temperatures and the places where equilibria exist between phases.
Intermolecular Forces Heating curve vs. phase diagram vs. vapor pressure curve • Heating curve Temperature vs. heat energy added Characteristic up-across-up-across shape Cooling curve is how temperature changes as you remove energy • Phase diagram Pressure vs. temperature All 3 phases shown with boundaries between them • Vapor pressure curve The liquid-gas portion of the full phase diagram Vapor pressure line is the boundary between L and G All liquids (and solids too) have vapor pressure Liquid to gas: As you add energy, the temperature changes. As the temperature changes, the vapor pressure changes.
Intermolecular Forces Phase Diagrams • The AB line is the liquid-vapor interface.
• It starts at the triple point (A), the point at which all three states are in equilibrium.
Intermolecular Forces Phase Diagrams It ends at the critical point (B); which is at the critical temperature and critical pressure, above this critical temperature and critical pressure the liquid and vapor are indistinguishable from each other.
Intermolecular Forces Phase Diagrams Each point along this line is the boiling point of the substance at that pressure.
Intermolecular Forces Phase Diagrams • The AD line is the interface between liquid and solid.
• The melting point at each pressure can be found along this line.
Intermolecular Forces Phase Diagrams • Below A the substance cannot exist in the liquid state.
• Along the AC line the solid and gas phases are in equilibrium; the sublimation point at each pressure is along this line.
Intermolecular Forces Compare phase diagrams of H2O and CO2 Intermolecular Forces Phase Diagram of Water • Note the high critical temperature and critical pressure: These are due to the strong van der Waals forces between water molecules.
Intermolecular Forces Phase Diagram of Water • The slope of the solid– liquid line is negative.
This means that as the pressure is increased at a temperature just below the melting point, water goes from a solid to a liquid.
Intermolecular Forces H2O: an unusual phase diagram • Unusual behavior • At same T, as you increase p, substance changes from solid to liquid • Liquid more dense than solid • Exhibits triple point where all three phases coexist From Chemistry & Chemical Reactivity 5th edition by Kotz / Treichel. C 2003. Reprinted with permission of Brooks/Cole, a division of Thomson Learning: www.thomsonrights.com. Fax 800-730-2215.
Intermolecular Forces • If you have this equilibrium and increase the pressure on it, according to Le Chatelier's Principle the equilibrium will move to reduce the pressure again. That means that it will move to the side with the smaller volume. Liquid water is produced Intermolecular Forces • since water fills a smaller volume when it's liquid, rather than solid, it will go to a lower melting point -- allowing more solid to become liquid.
Intermolecular Forces Intermolecular Forces Phase Diagram of Water □Water is a substance whose liquid form is more compact an its solid form Intermolecular Forces Phase Diagram of Carbon Dioxide Carbon dioxide cannot exist in the liquid state at pressures below 5.11 atm; CO2 sublimes at normal pressures.
Intermolecular Forces Referring to Figure describe any changes in the phases present when is (a) kept at 0°C while the pressure is increased from that at point 1 to that at point 5 (vertical line), (b) kept at 1.00 atm while the temperature is increased from that at point 6 to that at point 9 (horizontal line).
Intermolecular Forces Types of materials • Molecular Non-polar molecules • Octane, C8H18 • Fats (e.g., olive oil) Polar molecules • Water, H2O • Ammonia, NH3 • Acetic acid (vinegar is an aqueous solution of it), CH3COOH • Ionic • Sodium chloride (table salt), NaCl • Sodium bicarbonate (baking soda), NaHCO3 • Copper (II) sulfate pentahydrate, CuSO4⋅5H2O • Metallic • Copper metal, Cu • Aluminum foil, Al • Others that defy categorization Network • Quartz, SiO2 • Sand, SiO2 • Diamond, C Polymeric • Any plastic, such as high density polyethylene (HDPE) Amorphous • Glasses Mixtures • Butter, see Notice that the names of these categories are based on the type of bonding Intermolecular Forces Properties that molecular materials exhibit • Most are liquids or gases at room temperature • Smallest molecules are gases at room temperature • Only very large molecules are solids at room temperature • All have relatively low melting points (near or below room temperature) • Most feel soft • Chemical composition is usually carbon, hydrogen, oxygen, nitrogen and a few others (“organic”) • In liquid state, usually do not conduct electricity • Some can dissolve in water and others cannot Intermolecular Forces Properties that ionic materials exhibit • All are solids at room temperature • Very high melting points • Do not conduct electricity in solid state • Conduct electricity in liquid state • Crystalline • Brittle, break along flat/planar surfaces • When they contain transition metals, usually are colored; when they do not contain transition metals, usually are white • Generally called “salts” because they can be made from mixing together an acid and a base • Some can dissolve in water and others cannot Intermolecular Forces Properties that metallic materials exhibit • Lustrous (shiny) • Malleable (can be pounded into a pancake) • Ductile (can be bent) • Conduct electricity • Sometimes rust (oxidize) • Never dissolve in water Intermolecular Forces Solids • We can think of solids as falling into two groups: Crystalline—particles are in highly ordered arrangement.
Intermolecular Forces Solids Solids can be: Amorphous—no particular order in the arrangement of particles or Crystalline Intermolecular Forces Intermolecular Forces Intermolecular Forces Unit Cells • We can think of a crystalline solid as being built of bricks.
• Each brick is a unit cell. • Each crystalline solid is represented by a crystal lattice. The crystal lattice is like a scaffolding for the solid.
Intermolecular Forces A Unit Cell • The unit cell is the fundamental concept in solid state chemistry • It is the smallest representation of structure which carries all the information necessary to construct unambiguously an infinite lattice.
Intermolecular Forces Crystalline Solids .
Intermolecular Forces • There are three kinds of unit cells • Primitive cubic • Body - centered cubic • Face - centered cubic Intermolecular Forces • Primitive cubic : When lattice points are at the corners only • Body centered cubic: When the lattice point also occurs at the center of the unit cell • Face centered cubic: When the cell has lattice points at the center of each face as well as at each corner.
Intermolecular Forces Crystalline Solids There are several types of basic arrangements in crystals, such as the ones shown above.
Intermolecular Forces Crystal Structure of Sodium Chloride • NaCl is a face centered cubic as we can center either the Na+ ion or the Cl- ion on the lattice point of a face centered cubic unit cell Intermolecular Forces Some examples of ionic solids (particle level illustrations) Sodium chloride (NaCl) Ammonium chloride (NH4Cl) From Chemistry & Chemical Reactivity 5th edition by Kotz / Treichel. C 2003. Reprinted with permission of Brooks/Cole, a division of Thomson Learning: www.thomsonrights.com. Fax 800-730-2215.
Intermolecular Forces Attractions in Ionic Crystals In ionic crystals, ions pack themselves so as to maximize the attractions and minimize repulsions between the ions.
Intermolecular Forces • Equal sized spheres can be arranged like this Intermolecular Forces • Hexagonal close packing Intermolecular Forces Cubic close packing Intermolecular Forces • In each of these structures there are 12 equidistant neighbors.
• Thus the particles are thought to have a coordination number of 12.
• The coordination number is the number of particles immediately surrounding a particle in the crystal. Intermolecular Forces • In both types of close packing74% of the total volume of the structure is occupied by the spheres and 26% is empty space. Intermolecular Forces • In comparison each sphere in the body centered cubic structure has a coordination number of 8 and only 68% of the space is occupied. In the primary cubic structure the coordination number is 6 and only 52% of the space is occupied. Intermolecular Forces • The higher the coordination number the greater the packing efficiency Intermolecular Forces Crystalline Solids We can determine the empirical formula of an ionic solid by determining how many ions of each element fall within the unit cell.
Intermolecular Forces Ionic Solids What are the empirical formulas for these compounds?
(a) Green: chlorine; Gray: cesium (b) Yellow: sulfur; Gray: zinc (c) Green: calcium; Gray: fluorine CsCl ZnS CaF2 (a) (b) c) Intermolecular Forces • Start here Intermolecular Forces Types of Bonding in Crystalline Solids Intermolecular Forces Covalent-Network and Molecular Solids • Diamonds are an example of a covalent-network solid in which atoms are covalently bonded to each other.
They tend to be hard and have high melting points.
Intermolecular Forces Covalent-Network and Molecular Solids • Graphite is an example of a molecular solid in which atoms are held together with van der Waals forces.
They tend to be softer and have lower melting points.
Intermolecular Forces Metallic Solids • Metals are not covalently bonded, but the attractions between atoms are too strong to be van der Waals forces.
• In metals, valence electrons are delocalized throughout the solid. |
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