PREDICTIONS.pptx [Sola lettura] - uniroma2.it€¦ · • When+there+are+mul9ple+differentkind+of+...
Transcript of PREDICTIONS.pptx [Sola lettura] - uniroma2.it€¦ · • When+there+are+mul9ple+differentkind+of+...
PREDICTIONS
• Will + base form à predic9ons based on some (scien9fic) evidence
• The ac9on poten9al will cause the signal to travel across the axon.
• Will + base form à predic9ons based on hypothesis (if clause 1)
• If the companion star in the binary is close enough or massive enough, it will begin to tear the outer layers of the red giant away.
PREDICTIONS
• Be going to + base form à predic9ons based on stronger (scien9fic) evidence
• In the forma9on of NaCl, Na is going to give electrons and Cl is going to get them.
• When there are mul9ple different kind of alleles, for example in blood types, we are going to have a combina9on between co-‐dominant and recessive genes.
PREDICTIONS
• Be bound to + base form à predic9on of certainty
• Posi9vely and nega9vely charged par9cles are bound to a5ract each other.
• Possessing two different alleles, he is bound to be a heterozygous genotype
• Get is also possible: • The molecule is bound to get spli5ed during the process
PREDICTIONS
• Be likely to + base form à predic9ons of probability
• Likelihood à probability • Which of these elements is most likely to be a good conductor of electricity?
• These are the orbitals where the electron is most unlikely to be found.
• Spin relaxa9on is likely to occur in these processes.
PREDICTIONS • May, might, could à regulate the degree of possibility in predic9on of events
• May à more formal, high possibility • HMMs may be employed for decoding the iden9ty of visual s9muli from recorded neuronal spiking ac9vi9es.
• The electron may interact with the electron cloud, resul9ng in a small angular devia9on
• Bacteria may become an9bio9c resistant through muta9on of their gene9c material.
PREDICTIONS
• The next stage (on a temporale line corresponding to ti+1 ) may involve…
• Here the predic9on is stronger, we have material enough to state our conclusions
• What if the system is caho9c (i.e. logis9c map, atmosphere, metastasis diffusion, etc.)?
• Predic9ons rela9ve to state ti+2 ? • What if there is also more than one possibility?
PREDICTIONS
• Might à lower possibility. • We are referring to a ti+2 state of a system • The next stage might involve invasion, breach of the extracellular membrane, and intravasa9on.
• As for incomplete dominance, where no trait is dominant, you might have mixing or blending of the traits.
PREDICTIONS
• The hypothesis/predic9on on the state is less strong:
• If we consider this variable, then it might (well) turn out that ….
• We might (well) consider the fact that… • You can add well to make it stronger
PREDICTIONS
• Might à used in the past to show the event has not occurred.
• The process might have started with invasion (àbut it did not turn out like that).
• Might à hypothesis when the situa9on is not ‘real’:
• If A were B, it might do that (A is not B) à here may is not possible
• If A had been B, it might have done …
PREDICTIONS
• May/might as well à the only thing le\ to do • We are arranging the system so that electrons may pass through the foils of copper as well.
• Try as + subject + may/might à there is no way out
• Try as they may the par9cles will never follow this path.
• Try as they might… (same thing in the past).
PREDICTIONS
• Could à remote possibility, predic9on of uncertainty
• Assuming that current could flow out of the posi9ve terminal, through the circuit and into the nega9ve terminal of the source is a wrong conven9on.
• Secondary colonies could re-‐metastasize to form ter9ary and quaternary colonies.
PREDICTIONS
• Could à predic9ons with compara9ves in order to express possibility or impossibility
• Here poten9al levels could be higher (not may or might)
• The point could not be worse for measuring the output of ….
• The atom could gain more negaEve par9cles than it did previously.
PREDICTIONS
• Could à we are moving further in 9me, our predic9ons are less and less strong
• At this level the effect of …. could be …. • It is currently unclear if secondary colonies could re-‐metastasize to form ter9ary and quaternary colonies
PREDICTIONS
• Future perfect à we look at a state back from a future state
• In three steps the system will have completed its cycle
• By the end of the tenth step it will be producing energy for a big while.
• May, might and could are also possible to regulate the degree of probability
PREDICTIONS
• Is/are to be à formal disposi9ons • Par9cles are to assemble in that region • On the point of /about to à immediatly a\er • The atom is on the point of yielding one electron
• At this level the signal is about to be sent
PREDICTIONS
• We can use adverbs to reinforce the predic9on:
• Could possibly à stronger predic9on • Will definitely à certainty • Will probably à less sure • Highly likely à strongly probable • Adjec9ves are also possible: • May/might well à stronger predic9on
EXERCISES
• Choose any topics of your area and make predic9ons using will, going to and the future perfect (both simple and con9nuous).
• Make predic9ons using may/might /could for sta9ng events’ possibili9es .
• Use bound to, likely to and other future expressions to build appropriate texts. Use adverbs or adjec9ves to reinforce your points.
TEXT BUILDING: CANCER (1)
• Let’s consider any cells of any 9ssues of the body; as that 9ssue is growing the cells will experience mitosis and replicate themselves making copies of each other, which may experience mitosis as well; as soon as they realize the neighborhood is becoming too crowded they will undergo contact inhibi9on. Some of them will experience a licle defect and kill themselves –actually they are kind of make way for other healthy cells.
CANCER
• The cell might even kill itself if it realizes that there is something wrong. There is actually a cellular mechanism for this, namely apoptosis. There is some type of outside influence on the cell: it self-‐recognizes that there might be some outer damage and just destroys itself. It is a regular circumstance, even when there is a muta9on.
CANCER
• Even when muta9on is rela9vely infrequent in certain types of 9ssues, there will be on the overall a hundred billion of cells experiencing apoptosis (and so new cells) per day. So we will experience a lot of muta9ons. One of the muta9ons may keep the cell from destroying itself, another muta9on will make it replicate faster than its neighbor, so that, through mitosis, the cell is bound to make a bunch of copies of itself.
CANCER
• All this body of cells is going to be essen9ally defec9ve. It is mostly a body, a big lump of defec9ve cells called neoplasm, or tumor. If this lump is not replica9ng out of control, a lot faster than its neighbor cells, if it is not harming the environment, we will call it a benign tumor, which essen9ally means a harmless one. But one of these may experience another muta9on that makes it go like crazy, so that it may well become invasive, breaking the DNA replica9on scheme.
CANCER
• As a consequence, muta9ons become more frequent un9l one of these muta9ons will allow itself to break off to travel to other parts of the body. Once infiltrated, it will start to take over all other cells. In this case we say the cell has metastasized. They are known as cancer cells. Cancer is a whole class of muta9ons where the cells start exhibi9ng this fast invasive growth and these metastasis because of DNA replica9on.
CANCER
• We can think of it as the by-‐product of broken mitosis or, even more specifically, broken DNA replica9on. The cell starts replica9ng taking over resources and spreading through the body. If every malignancy remained at its ini9al loca9on, most solid cancers could be treated effec9vely with surgery.
CANCER (2)
• Metastasis, or the spread of cancer to mul9ple new loca9ons, however, makes it increasingly hard to track and remove new colonies of afflicted cells. Anyway, an unexpected diversity of assumpEons across experts has lead to a striking lack of agreement over the basic stages and sequence of metastasis.
CANCER
• The en9re process could possibly follow some stages. At the primary stage, an individual renegade cell may become malignant and begin uncontrollable prolifera9on: it will learn to avoid the immune system and an9-‐growth signals and form the ini9al (primary) tumor with access to its own blood supply. This stage might be followed by a stage of detachment (migra9on).
CANCER
• Here tumor cells will get disconnected from the primary colony to begin their journey through the body. The next stage might involve invasion, breach of the extracellular membrane (ECM), and intravasa9on. At this stage the cancer cell will have to overcome several obstacles to ac9vely reach and penetrate the wall of a blood or lymph vessel (intravasa9on).
CANCER
• In so doing it gains access to the body's transport system that will carry it to unaffected 9ssues. Next, blood or lymph will carry the cancer cell to new loca9ons—the stage of migra/on (transport). The extravasa/on stage may be considered as intravasa9on in reverse: the cancer cell will escape the blood or lymph vessel that has carried it by penetra9ng its wall and will invade the new organ or 9ssue.
CANCER
• This stage will be followed by coloniza/on of the 9ssue/organ and cell prolifera/on. Early stages of this infesta9on may be described as micrometastasis, or forma9on of a small secondary tumor, usually without its own blood supply (no angiogenesis occurs at early stages) and with a balance of prolifera9on and apoptosis.
CANCER
• When the secondary tumor manages to ac9vate growth of its own blood vessels (angiogenesis) that provide small colonies with sufficient nutrients, it will enter the stage of macrometastasis, culmina9ng in the growth of a large secondary tumor. It is currently unclear if secondary colonies could re-‐metastasize to form ter9ary and quaternary colonies (doced line indica9ng a cyclic process).
MODELING SEQUENCES
• The concept of predictability is intrinsic to language
• In English language predic9ons can be, as seen, independent of or dependent on the 9me of speaking, at any point on a temporal axis
• The same idea is present in the past (past simple/present perfect/past perfect)
• Can we model temporal sequences to quan9fy real world’s varia9ons?
MODELING SEQUENCES
• Is a state of a system bound to depend, for instance, on the immediately prior stage? Do transi9on probabili9es from one state to another vary across the sequence?
• Math modeling, numerical methods • Dealing with caho9c systems. • For example atmosphere and weather forecas9ng, chemical reac9ons, lasers, par9cle accelerators.
MODELING SEQUENCES
• Some9mes we are not able to solve par9al differen9al equa9ons analy9cally. So we ini9alize them from data in order for rates of change to be determined.
• Rates of change predict the state of the system a short 9me into the future by an increment in 9me, or step.
MODELING SEQUENCES
• In order to find new rates of change we apply the equa9ons to this new state.
• The new rates of change allow us to make predic9on at a further 9me step into the future.
• The length of this procedure depends on the distance between the points on a (computa9onal) grid.
PREVIOUS OUTCOMES AND PAST TENSES
• ti-‐1 : present perfect • The past event is relevant to the present state (ti) at the 9me of speaking (à if ti-‐1 affects ti when quan9ta9vely measuring it or qualita9vely describing it).
• The immediately previous state has brought about a change in the actual posi9on, due to….
PREVIOUS OUTCOMES AND PAST TENSES
• Past perfect: the past event has already happened before the past event we are actually considering. We want to emphasise, for instance, the way ti-‐2 had effected ti-‐1 before the lacer has effected ti .
• Before genng into the previous posi9on, the system had already shown a similar behaviour.
PREVIOUS OUTCOMES AND PAST TENSES
• Dura9on: present perfect conEnuous • We want to emphasize the fact that a variable has been effec9ng the system for an amount of 9me.
• The size of the x deriva9ve has been modeling the trend since the very beginning of the simula9on
• It has been causing this effect since three steps. • For à dura9on • Since à beginning of the dura9on
TENSE PERCOLATE
ti-‐n ti-‐2 ti-‐1 ti ti+1 Ti+2 Ti+n
Dura9on forms à ti
Past perfect
Present perfect
Time of speaking
Will May
Might Could
Future perfect ti ß
EXERCISE
• Considering, every 9me, the present state of a system, write a 250-‐line text in your scien9fic area making predic9ons and, at the same 9me, referring to the previous events. Compound your statements forwards and backwards so as to produce a coherent text. Include data analysis (equa9ons and graph) and write an appropriate introduc9on and conclusion.
READING
• When trying to quan9ta9vely embracing the massive varia9on in the natural, or real, world one is bound to tease out their underlying pacerns. The idea of expecta9on is then going to be vital to this modeling –that is, es9ma9on of (random) probability of some event. The expected value will thus converge on something, or the ra9o of something to its counterpart, as the number of trials increases.
READING
• If observa9on of all events be con9nued for the en9re infinity, it will be no9ced that everything in the world is governed by precise ra9os and a constant law of change. Since also the probabili9es away from averages has been seen to follow a (binomial) distribu9on, the concept of independence may well be made for a basic condi9on when dealing with large numbers.
READING
• This essen9ally means that outcome of previous events is not going to change the probability of future events. Anyway, most things in the physical world are dependent on prior outcomes, or they are condi9oned -‐that is, they are depending variables. So the law of large numbers might apply the depending variables as well. As long as any state is reachable, when running in a sequence we will get equilibrium.
READING
• No macer where we start, once we begin the sequence, the number of 9mes we visit each state is going to converge to some specific ra9os, or probability. This therefore disproves that only independent events could converge on predictable distribu9ons. We can thus model sequences of random events using states and transi9ons between states. This has become known as Markov chain.
READING COMPREHENSION
• Answer the following qes9ons • Are pacerns and expecta9ons going to be connected? In what way?
• In what way does independence affect probability?
• Why and in what way can we model sequences? Refer to the text and support your arguments with further examples.
MODELING(4)
• Since metastasis show a sequen9al nature, the sequence of possible metasta9c events may be modeled with a Mathema9cal code, namely the simple yet powerful Markov chain. Math modeling is a way of making these events predictable. Let’s consider an ordered sequence of random variables X0, X1, …, XK, which correspond to the sequence of events.
MODELING
• Each of these variables will take values (X0 = x0, X1 = x1, …, Xk= xk) from a set of states Σ = {1, 2, …, N} that may correspond to the expert proposed stages of metastasis (e.g., primary tumor, detachment, invasion, etc).
• This modeling involves condi9on probabili9es.
MODELING
• Markov chains are o\en described by a sequence of directed graphs where the edges of graph n are labeled by the probabili9es of going from one state at 9me n to the other states at 9me n+1,
• The same informa9on is represented by the transi9on matrix from 9me n to 9me n+1.
• (read P(A|B) as “the probability of A given B”)
MODELING
• P(A|B) is the probability of A given B • P(A|B) is the probability that the event A may occur given the fact that the event B has occurred.
• Or: assumed to • AssumpEons ó predicEons ó expectaEons
MODELING
• We could noEce that a Markov chain is a sequence of random variables X1, X2, X3, ... with the Markov property, namely that, given the present state, the future and past states are independent. Formally,
MODELING
• Thus, the sequence of random variables are likely to follow a homogeneous Markov process under these condi9on probabili9es:
where ρ is the probability that malignancy will go from one state to another and λ may be assumed as a the vector component.
MODELING
• This implies that each metasta9c stage is bound to depend only on the immediately prior stage, and that the transi9on probabili9es from one stage to another will not vary across the sequence. In this way, the Markov process is likely to be determined by a vector and a matrix
MODELING
• and
• where ρij is the condi9onal probability of transi9on from stage i to stage j so that
MODELING
• Since the probability of transi9oning from one state to another one must be 1, this matrix is a right stochas9c matrix. Anyway, diagonal elements in a stochas9c matrix defining a Markov process might be posi9ve. Because this analysis focuses on the sequence of metasta9c stages rather than the 9ming involved in the transi9on between stages, we could not allow a metasta9c stage to transi9on to itself.
MODELING
• We get
• We may define N states of the process, and assume that it is likely to start at ar9ficially defined stage S (state 0), and to end up at another ar9ficially defined stage E (state N, where N = 28).
MODELING
• In other words, we may start in state S with probability 1 such that
and we will end the chain once state E is reached such that the steady state or sta9onary distribu9on vector of metasta9c stages (π) will be defined as
MODELING
• For data with mi observed transi9ons from state i to any other state, the set of observed counts of transi9ons across all expert stories, {cij}, j = 1,…,N (where i≠j) are bound to follow a mul9nomial distribu9on with expected values {ρij}, j = 1,…,N
• The distribu9on will be in the form
MODELING
where • (read: the probability of A given B equals the binomial coefficient 9mes the productoria for N of the probability cancer cells diffuse).
MODELING
• The prior distribu9on of transi9on probabili9es will be defined by a Dirichlet distribu9on
MODELING(5)
• The posterior expecta9on es9mate of ρij will be given by
• We might also obtain a maximum a posteriori probability es9mate of ρij:
TEXT BUILDING: NEURONS(6)
• A neuron essen9ally transmits signals across its length, depending on the signals it receives. The body of the neuron is cons9tuted by the soma, around the nerve cell nucleus. Neurons shows peculiar structures s9cking out from the soma that keep branching off. These branches off of the soma of the neuron are called the dendrites, which keep splinng off. They tend to be the place where neurons receive their signals.
NEURONS
• Neurons show a kind of tail called axon: this is where the distance of the signal gets traveled. It ends at the axon terminal, where can connect to other dendrites or maybe to other types of 9ssue like muscles. The point at which the soma connects to the axon is called the axon hillock. There are insula9ng cells around the axon, known as Schwann cells, which make up the myelin sheath in the peripheral nervous system.
NEURONS
• The Schwann cells’ intersec9ons are the so-‐called nodes of Ranvier. The general idea is that the neuron gets s9mulated at the dendrites so that the combined effect of the signals gets summed up and travel to the hillock. If they are large enough and meet some threshold level, they will trigger an ac9on poten9al on the axon.
NEURONS
• It will cause the signal to travel down the bonds of the axon and reach other dendrites terminal where it may be connected via synapses, from where other processes might be triggered on, by s9mula9ng other neurons. An interes9ng aspect in neuroscience predic9ons is indeed the voltage poten9al across the membrane of a neuron.
MODELING(7)
• Neurons emit ac9on poten9als, which may be seen as brief and stereotyped electrical events that might be recorded with extracellular electrodes in behaving animals. Hidden Markov Models (HMMs) are actually used in this way. HMMs may be thought of as being useful tools for model-‐based analyses of complex behavioral and neurophysiological data.
MODELING
• They take into account the probabilis9c nature of behavior and brain ac9vity. Actually, the trend in neuroscience is to observe and manipulate brain ac9vity in freely moving animals during natural behaviors and to record from several dozens of neurons at the same 9me.
MODELING
• For example, an experimenter could be interested in recovering from noisy measurements of brain ac9vity the underlying electrical ac9vity of single brain cells using the constraint that ac9vity has to agree with cellular biophysics (such as in the spike sor9ng problem). Or, the experimenter may want to describe the variance in some behavior and relate it to causes encoded in the neural recordings.
MODELING
• The observed variable (the output of the HMM) might be a test subject's decision, or an animal's motor output. What many studies have in common is the quest to iden9fy underlying brain states that correlate with the measured signals. Spike data recorded from single or mul9ple nerve cells (neurons) is amenable to modeling with HMMs.
MODELING
• A popular applica9on of HMMs is decoding informa9on from recorded spike data. An HMM may be used to model neuronal responses in the visual cortex or cor9cal ac9vity as a 9me-‐dependent Poisson process. The Poisson means are hidden parameters. The recorded spike trains might be divided into small 9me bins in which the spike counts are assumed to obey a Poisson distribu9on with 9me-‐dependent mean rate.
MODELING
• These models are likely to be considered as yielding to a temporal segmenta9on of spike data into a sequence of 'cogni9ve' states, each with its dis9nguished vector of Poisson means. In this way HMMs may be employed for decoding the iden9ty of visual s9muli from recorded neuronal spiking ac9vi9es.
MODELING
• They could possibly be obtained from several neurons in the visual cortex of during the presenta9on of different visual s9muli. For each s9mulus, an HMM on a subset of the respec9ve trials might be trained. These trained HMMs, then, may be used to decode s9mulus iden9ty from neural responses by selec9ng the s9mulus for which the HMM gives the largest likelihood for genera9ng the neural responses.
MODELING
• Imagine two spike trains S1 and S2 both represented by vectors of inter-‐spike intervals. One modeling could possibly transform one spike train into the other by (i) adding spike intervals, (ii) removing spike intervals and (iii) changing the dura9on of a spike interval by Δt . There will be a unit cost for (i) and (ii), and a cost of q * Δt for (iii). The 'distance' or dissimilarity between two spike trains may be defined as the minimal cost to transform S1 into S2 using (i-‐iii).
MODELING
• This measure may be calculated efficiently using dynamic programming.
• Instead of considering spike intervals, we might well deal with exact spike 9mes, where the parameter q will determine the cost of shi\ing a spike in 9me. Again, spikes could also be added or removed.
• The two measures may also be realized using a pair HMM
MODELING • In such a pair HMM there will be two states corresponding to unmatched inter-‐spike intervals in the two sequences, which may be dealt with by adding or removing spike inter-‐spike intervals, respec9vely. There might be also one match state in which the two inter-‐spike intervals are going to be matched with associated emission probability in the form ( ) ( ) exp M bΔt = −qΔt , where q is a free parameter and Δt the interval difference. The total costs of adding, removing, and matching intervals will be encoded in the emission and transi9on probabili9es of the pair HMM.
MODELING
• The iden9fica9on and classifica9on of spikes from the raw data is called spike sor9ng. Most spike sor9ng methods consist of two steps. In a first step, spike events will be extracted from the raw data. In a second step these events will be classified. Difficul9es arise when spikes overlap on the recorded trace (arising when neurons are densely packed and fire at high rate).
MODELING
• HMMs thus provide a framework for addressing the spike sor9ng problem. Spikes may be described by independent random variables (the hidden variables), whereas the recorded voltage will be the probabilis9c outcome condi9onal on the state of the hidden variables.
MODELING
• These 'overlaps' are notoriously difficult to sort. The shape of such an overlap can be complex, because the number of different spikes contribu9ng to the shape is unknown, as is the exact 9me delay between them. These models are likely to be considered as yielding to a temporal segmenta9on of spike data into a sequence of 'cogni9ve' states, each with its dis9nguished vector of Poisson means.
EXERCISE
• Choose any systems in your area of interest and describe it emphasising predic9ons where necessary. The system being consistent with numerical predic9ons, refer to its next step behaviour as a func9on of the previous one using appropriate tenses. The text must be coherent within a layout including an introduc9on, a descrip9on of the involved events and a conclusion.
PREDICTION METHODS (9)
• Single-‐step methods (like Eulero’s or Runge-‐Kuca’s) use only the informa9on from one previous point to compute the next one, that is, only the ini9al point (t0 , y0) is used to compute (t1 , y1) and in general yk is used to compute yk+1.
• Using a combina9on of a predictor and a corrector requires only two func9on evalua9ons of f(t,y) per step.
EXERCISE • The Math model for epidemics is as follows: Assume there is a community of L members containing P infected
individuals. Let y(t) denote the number of infected at 9me t. for a mild illness such as cold everyone con9nues to be ac9ve, and the epidemics spreads from those who are infected to those uninfected. Since there are PQ possible contacts between these two groups, the rate of change of y(t) is propor9onal to PQ. So the problem can be stated as
y’ = ky (L – y) with y(0) = 0 • Use L= 25,000, K = .00003, h = .2 with the ini9al condi9on y(0)=250
and compute any approximate solu9ons (for instance Euler’s) over [0,12] wri9ng a coherent text using predic9ons and compounding statements forwards and backwards.
PREDICTION METHODS(10)
• The classical astrophysical N-‐body problem consists of each member of an aggregate of N(i=1,…,N) point masses, having masses mI, experiencing an accelera9on from the gravita9onal acrac9ons of all the other bodies in the system.
PREDICTION METHODS
• The descrip9on of the problem will be completed by specifying the ini9al posi9ons (xi at t = 0) and veloci9es (vi at t=0) for the N par9cles. Solu9ons of this problem may range from the orbit of the moon to the structure of the Kirkwood gaps in the asteroid belt and countless other phenomena. This richness arises from strong nonlinearity in the equa9on as a slight change in ini9al condi9ons may lead to very different outcomes – chaos.
PREDICTION METHODS
• The N-‐body problem involves calcula9ng • 1. the force on each par9cle at a given 9me • 2. determining the new posi9on of the par9cle at a future 9me
PREDICTION METHODS • Methods for advancing par9cle posi9ons may be considered:
• Euler – the simplest possible approach • Runge-‐Kuca – standard for ordinary differen9al equa9on
• Bulirsch-‐Stoer – accurate but limited to few bodies – with Richardson extrapola9on
• Symplec9c map – accurate for very long integra9ons (no close encounters)
• Predictor-‐corrector – large numbers of par9cles with reasonable accuracy (Leapfrog)
PREDICTION METHODS
• In astronomy we o\en want to study large systems (e.g. a galaxy with >100G objects).
• It will take the Sun 250 Myr to orbit the Galaxy so it has only orbited ~18 9mes.
• myr à 1 million years. • Es9mated age of the universeà13,798 myr.
PREDICTOR-‐CORRECTOR
• Addi9onally, there are mul9ple 9me scales, e.g. star clusters need small 9me steps.
• One solu9on to these problems may be to use a so called ‘leapfrog’ method, which is a simple predictor-‐corrector method, with 9me step halving.
• The largest 9me step can be es9mated from
PREDICTOR-‐CORRECTOR
• where ai is the accelera9on of the i-‐th par9cle and η is a small factor
• The method is a simple second-‐order integra9on that will allow us to advance posi9ons and veloci9es defined in intervals separated by Δti /2.
PREDICTOR-‐CORRECTOR
• Here posi9on is going to be advanced at half 9me-‐steps and velocity at full 9me.
• Addi9onally, more sophis9cated versions could be made.
• These are known more generally as as predictor corrector methods.
SCATTERING PREDICTIONS(11)
• If we consider an isolated single atom, elas9c scacering may occur in one of two ways:
• The electron may interact with the electron cloud, resul9ng in a small angular devia9on.
• Alterna9vely, if an electron penetrates the electron cloud and approaches the nucleus, it will be strongly a5racted and may be sca5ered through a larger angle.
SCATTERING PREDICTIONS
• Many electron-‐electron interac9ons are inelas9c. For instance, the nuclear interac9on may result in the genera9on of a bremsstrahlung X-‐ray, or may even result in the displacement of the atom from its site in the crystal, both of which involve some energy loss for the electron.
• May even + base form à predic9ons of fairly unexpected results
EXERCISE Describe the system emphasized by the figure (11/B) using
predic9ons. Here the method is Monte Carlo.
CELLULAR AUTOMATA
• Cellular automata may ideally reproduce a physical system, typically nonlinear dynamic systems
• Space and time are discretized onto an N-dimension grid. Physical quantities get a finite set of values according to transition rules which define the local physics.
CELLULAR AUTOMATA
• There are simple rules of local interactions (neighbour cells) which can reproduce general behaviours of the system (auto-organization at a threshold: always the same slope).
• There exists a transition function that allows the system to evolve while describing the ‘local’ physics.
CELLULAR AUTOMATA
• Sandpile are self-organizing cellular automata: singoli elements interacting using simple local rules, show a scaling global trend. We define a lattice where we randomly insert grains of sand. At a critical state (threshold value) the system gets auto-organized: it has always the same slope. Namely, we observe a power law distribution.
CELLULAR AUTOMATA
• One target of these simulations (sandpile models) can be the study of a 3D-lattice evolution, to which we approximate the solar atmosphere, undergoing a perturbation where relaxation is brought about by a local instantaneous redistribution of energy (avalanche à solar flares).
CELLULAR AUTOMATA
- The lattice, for instance, can be a cube with initial conditions. Perturbation time (the falling grain of sand) is longer than relaxation’s (neighbour cells’ energy re-distribution), which is considered as being instantaneous. At each step a small perturbation (with respect to threshold) is being added. Initially all cells are at the same energy.
CELLULAR AUTOMATA
• A\er a threshold the flare is caused to happen
(12)
where Fi is the energy of the i-‐th element, (1/6) ∑nn Fnn is the average energy of the neighbour cells and δc is the threshold value.
CELLULAR AUTOMATA
• When flares happen, the average energy is being distributed as follows:
• Fi loses a certain amount of energy, while the neighbours earn some.
EXERCISE
• Write a coherent text using the previous informa9on on sandpiles and solar flares. Describe the actual system’s percolate appropriately using the ‘tense percolate’ table as shown above.
IMPLICIT DIFFERENTIATION(13)
• When dealing with curves like a circle with equa9on x2 + y2 = 1, for instance, we cannot explicit x or y, for we have both. In order to find the slope of the tangent line at a point on the circle, say P(√2/2, √2/2), we will have to calculate the deriva9ve in terms of x and y than define y as a func9on of x, for that will give us two solu9ons, namely y = √ (1-‐ x2) and y = -‐ √ (1-‐ x2).
IMPLICIT DIFFERENTIATION
• Consequently there is going to be another way to find the deriva9ve in terms of both. This is called implicit differen9a9on, and it is an applica9on of the chain rule. More formally, given a differen9able rela9on F(x,y)=0, which defines the differen9able func9on y=f(x), it is possible to find the deriva9ve f’ even in the case when we could not symbolically find f.
IMPLICIT DIFFERENTIATION • That means we will apply the operator d/dx (deriva9ve with respect to x) on both sides of the equa9on and solve in terms of the differen9al dy/dx:
• d/dx(x2 + y2) = d/dx(1) • d/dx(2x) + d/dy(y2)*dy/dx = 0 • 2x + 2y*dy/dx = 0 • dy/dx = -‐x/y • (read: the deriva9ve of x squared plus y squared with respect to x is equal to …., so that the deriva9ve of y with respect to x, or the differen9al, is equal to nega9ve x over y).
IMPLICIT DIFFERENTIATION
• Consequently the slope at P equals -‐1, which means we must have found the value of the deriva9ve in terms of both x and y.
• Implicit differen9a9on may well recall the chain rule, something we are likely to use any 9me we have a composi9on of func9ons and we search for the deriva9ve, which tells us the slope at any point along a curve f.
IMPLICIT DIFFERENTIATION
• Given the curve (x-‐y)2 = x+y+1 if we differen9ate it implicitly we will get d/dx [(x-‐y)2 ]= d/dx(x+y+1) which means • 2(x – y) (1 – dy/dx) = 1 * dy/dx
IMPLICIT DIFFERENTIATION
• where 2(x – y) = d/d(x-‐y) [(x – y)2)], that is, the deriva9ve of something squared with respect to that same something; while (1 – dy/dx) = d/dx (x – y), that is, the deriva9ve of something inside the brackets with respect of x and y, which is what we are trying to solve for. And this is just the chain rule! –namely, the deriva9ve of the sub-‐func9on and the deriva9ve of the en9re func9on.
IMPLICIT DIFFERENTIATION
• Finally we get 2x – 2y(1 – dy/dx) = = 1 + dy/dx à dy/dx = 2y-‐2x+1/2y-‐2x-‐1
IMPLICIT DIFFERENTIATION
• EXAMPLE • Given f(x) = sin2 (x), then df/dx = 2sinx * d/dx(sinx) = 2sinx * cosx
• (read: given f of x equal to the sine squared of x, then the deriva9ve of f with respect to x is equal to twice the sine of x 9mes the deriva9ve of the sine of x with respect to x, which is equal to twice the sine of x (or two sine of x) 9mes the cosine of x).
IMPLICIT DIFFERENTIATION
• Why 2sinx? Because we apply the chain rule and we get d/d(sinx) [sin2 (x)] = 2sinx
• Why cosx? Because d/d(sinx) [sinx] = cosx • EXERCISE: differen9ate implicitly x√ y=1; exp(xy2) = x – y; sin(2x-‐7y)=16y; y = xp/q ; x2 + y2 = 100, building up a coherent text.
EXERCISE
• Itera9ve techniques, nonlinear func9ons. Consider f1(x,y) = x2 – 2x – y + 0.5 ; f2(x,y) = x2 + 4y2 – 4. (14) Seek for a method of solu9on for the system of nonlinear equa9ons f1(x,y) = 0, f2(x,y) = 0. No9ce the equa9ons implicitly define curves in the x-‐y plane. Hence a solu9on of the system is a point (p,q) where the two curves cross (i.e. both f1(p,q)=0 and f2(p,q)=0). You may use a fixed-‐point itera9on genera9ng a sequence {(pk, qk)} that converges to the solu9on (p,q). Write a text both coherently describing the procedure and appropriately inser9ng numerals and tables.
REFERENCES • 1. Listening exercise at https://www.khanacademy.org/science/biology/cellular-molecular-biology/stem-cells-and-
cancer/v/cancer. All Khan Academy content is available for free at www.khanacademy.org. Listen to the texts carefully, take notes and build a coherent text.
• 2. Adapted from Divoli et al. “Conflicting Biomedical Assumptions for Mathematical Modeling: The Case of Cancer Metastasis” , PLoS Comput Biol. 2011 Oct; 7(10): e1002132.
• 3. In Davoli et al, quoted. • 4. Adapted from Davoli et al., quoted, and from http://en.wikipedia.org/wiki/Markov_chain • 5. From Davoli et al., quoted. • 6. Listening exercise at https://www.khanacademy.org/science/biology/human-biology/neuron-nervous-system/v/
anatomy-of-a-neuron • 7. Adapted from B. Florian et al., “Hidden Markov Models in the Neurosciences”, Institute of Neuroinformatics,
University of Zurich, online http://cdn.intechopen.com/pdfs-wm/15362.pdf • 8. In Florian et al., quoted. • 9. See J.H. Mathews, “Numerical Methods”, Prentice Hall 1987, p.434-435. • 10. D. Hobbs, “Computational Astrophysics” online http://www.astro.lu.se/~david/teaching/SPH/notes/
ComputationalAstrophysicsL5 • 11- 11/A – 11/B. from http://web.pdx.edu/~jiaoj/phy452/lecture_2_TEM_12.pdf • 12. The values have been proposed by O. Podladchikova, B. Lefebvre in “Lattice Models for Solar Flares and Coronal
Heating”, 2006 International Astronomical Union DOI: 07.2006/reviews.287.5461.2267. • 13. Listening exercise at https://www.khanacademy.org/math/differential-calculus/taking-derivatives/
implicit_differentiation/v/implicit-differentiation-1 and following. Listen to the texts carefully, take notes and build a coherent text.
• 14. In Mathews, quoted, p. 369-70.