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Linearity and the shared $l=1$ angular eigenvalue giveUsing the relative phases in part (c),Thus each real orbital remains an energy eigenstate with $E=-1/8$ hartree and has definite orbital angular momentum magnitude $\sqrt{2}\hslash$, but neither is an eigenfunction of ${\hat{l}}_z$. A measurement of $l_z$ gives $+\hslash$ or $-\hslash$, each with probability $1/2$. Its mean is zero and its variance is ${\hslash }^2$; a zero mean does not mean that a measurement gives zero.

Wavefunction normalization requires $\alpha >0$. For the normalized probability weight proportional to $e^{-\alpha x^2}$, the Gaussian integral and its derivatives giveSince ${\tilde{\Psi }}^{\prime \prime }=({\alpha }^2{x}^2-\alpha )\tilde{\Psi }$, the kinetic energy and potential energy expectations areThusThe second derivative is positive, and the energy diverges at both ends of $\alpha >0$, so this is the global minimum within the Gaussian function family. The best Gaussian function upper bound isThis is an upper bound, not the exact quartic ground energy.

Quantum tunnelling is penetration of, or transmission through, a finite potential barrier into a classically forbidden region, where the total energy is less than the potential energy. The wavefunction generally remains nonzero there, so the Born rule assigns a nonzero probability of detection. This does not require an increase in total energy; it is the classical relation between a definite position and a definite kinetic energy that cannot be imposed on the quantum state. For a bound quantum harmonic oscillator, the relevant quantity is the probability of finding the particle beyond its classical turning points; it is not an escaping transmitted beam.

Include an intercept $w_0$ and let $\mathbf{z}$ be the chosen numerical feature vector. Logistic regression usesThe logistic function maps every real score into $(0,1)$, allowing a Bernoulli distribution model for the label. Equivalently, the log odds are linear in the features:This gives a smooth differentiable probability model with a linear decision boundary at any fixed probability threshold. The sigmoid output is a fitted conditional probability, not a categorical prediction until a threshold is chosen.

For conditionally independent binary labels, the likelihood isMaximizing it is equivalent to minimizing its negative log-likelihood, the mean binary cross-entropy:It is the loss implied by a Bernoulli distribution for each target, penalizes confident errors strongly and is convex in the linear-logistic parameters. For numerical stability, each contribution can be evaluated as $\log (1+e^{u_i})-y_i{u}_i$ using a stable softplus implementation.

The 10:90 class imbalance does not invalidate this likelihood. Class weighting or resampling may be useful if errors have unequal costs, but changes the training objective and potentially the interpretation of fitted probabilities. Optional ridge regression adds $\lambda \Vert \mathbf{w}{\Vert }^2/2$, usually excluding the intercept, to control overly large weights or complete separation.

Write $q=P_{BI}$ and $p=P_{AI}=1-q$. A path with $n$ visits to $I$ makes $n-1$ unsuccessful excursions back to $B$, followed by one successful step to $A$. Every $B\to I$ step has probability one, soThe visit count has a geometric distribution, with success meaning the final $I\to A$ step.

Most high-dimensional phase space has negligible equilibrium probability. Uniform sampling spends nearly all its work in these irrelevant regions and seldom visits the small subset that dominates the partition function or the numerator of an equilibrium average. The resulting estimator can have enormous variance.

Importance sampling places samples where the product of statistical weight and observable matters. Sampling the Boltzmann distribution, for example, converts a weighted phase-space integral into an equilibrium expectation. If a Markov chain Monte Carlo algorithm is used, adequate exploration of separate free-energy basins and allowance for autocorrelation remain necessary.

Here $\mathbf{j}$ is electric current density, $n$ the number density of independently mobile ions, $Q$ their charge, $T$ absolute temperature and $k_B$ the Boltzmann constant. The sum runs over all nearest-neighbour destinations $j$ of site $i$, and ${\mathbf{r}}_{ij}$ is the displacement from $i$ to $j$. The applied field is $\mathcal{E}$, ${\omega }_0$ is an attempt frequency, and $\Delta E_{ij}$ is the zero-field reaction energy barrier for the indicated jump. With the normalization of the supplied equation,It has units s${}^{-1}$; it is not an equilibrium occupation probability. The current formula is the linear-response result of a small field-induced asymmetry in forward and reverse rates, requiring $|Q{\mathbf{r}}_{ij}\cdot \mathcal{E}|\ll k_{B}T$. Use its explicit factor $1/2$ consistently rather than absorbing it into a redefined jump rate.

The classical transmission probability is $P_{\mathrm{cl}}(E)=\Theta (E-V_0)$; the convention exactly at the threshold does not affect a thermal integral. Below the barrier it is zero, whereas the quantum result permits tunnelling. Above the barrier it is one, whereas quantum reflection is generally nonzero except at transmission resonances. Both descriptions tend to unit transmission at arbitrarily high energy, and both have zero transmitted probability in the zero-energy limit for this positive barrier.

The printed PDF has a plus sign before the second logarithm. Differentiating that expression literally, with $h_i=\sigma (a_i)$ and $\partial h_i/\partial w_j=h_i(1-h_i)z_{ij}$, givesFor example, at $y_i=0$ it is $-h_i{z}_{ij}$, so minimizing this printed objective pushes the predicted probability of class one upwards rather than downwards; the objective is unbounded below as $h_i\to 1$. It is therefore not the usual binary cross-entropy.

If the second sign was intended to be negative, the standard negative log-likelihood is $-y_i\log h_i-(1-y_i)\log (1-h_i)$ and the derivative simplifies instead to $(h_i-y_i)z_{ij}$. Stating both cases preserves the actual question's algebra while identifying the apparent sign error. The logarithms here are natural; another base multiplies the gradients by its constant conversion factor.

For $K$ mutually exclusive classes, the output layer should provide $K$ nonnegative probabilities summing to one, with labels aligned consistently to output coordinates. A softmax activation on logits $a_k$ providesSubtracting ${\max }_{\ell }{a}_{\ell }$ from all logits before exponentiation preserves these probabilities and prevents numerical overflow. Categorical cross-entropy $-{\sum }_k{y}_k\log p_k$ is appropriate for one-hot or probabilistic class targets. Independent sigmoid outputs fit a different setting in which multiple labels can be present simultaneously and their probabilities need not sum to one.

People often underweight base rates when a vivid case description competes with them, but the extent of neglect depends on representation, relevance and task. The normative benchmark is conditional on a known sampling process. If a case is randomly drawn from a population, a prior class proportion $P(H)$ must combine with the case evidence's likelihood ratio. In Bayes' theorem,Changing prior odds while holding diagnostic evidence fixed should ordinarily change posterior odds. A judgment driven only by resemblance to a class violates that benchmark when the stated premises apply.

In Daniel Kahneman and Amos Tversky's prediction experiments, descriptions of people were paired with different proportions of engineers and lawyers in the source group. Judgments based on an occupationally suggestive description were surprisingly insensitive to those proportions despite their relevance to a random draw. Without the individualized description, participants could use the stated proportions. This contrast is especially informative: the issue was not simply inability to read a base rate, but the way case-specific evidence displaced it. Similarity and probability can agree in ordinary situations, which helps explain why the substitution is tempting.^[ref]

The first account is the representativeness heuristic, potentially supplemented by dual-process theory. An occupational description quickly evokes similarity to a stereotypical engineer or lawyer. That judgment does not inherently encode class prevalence or evidential reliability, so it can dominate the probability estimate. More controlled reasoning can integrate the base rate, but may fail to engage, misunderstand the statistical structure or fail to override the compelling response. This account predicts errors especially when representativeness and prior probability conflict, with possible improvement when people deliberately compare alternative hypotheses and their frequencies.

It is too strong to infer from a biased answer that no statistical information was processed. Wim De Neys and colleagues contrasted problems where descriptions and base rates conflicted with control problems where they agreed. Although verbal reports often emphasized the description, measures including decision time, later memory for the base rates and reconsideration behavior indicated sensitivity to the conflict. Statistical information can therefore influence processing without determining the final answer. This challenges a simple account in which analytic monitoring is always absent; it is compatible with incomplete integration or unsuccessful correction.^[ref]

The second account emphasizes ecological rationality and information format. Gerd Gigerenzer and Ulrich Hoffrage argued that algorithms adapted to natural frequencies can perform Bayesian updating more easily than algorithms given separately normalized probabilities. Natural sampling represents the relevant nested sets directly: how many cases belong to each class, and how many within each class produce the evidence. The denominator can then be the whole set of evidence-positive cases, without reconstructing joint counts by multiplying conditional probabilities.

Their 1995 experiments compared probability and frequency versions of Bayesian problems while also varying the information menu. Frequency formats substantially increased the use of Bayesian solutions across the problem set, assessed using solution processes as well as final answers. This supports a representational contribution to apparently poor reasoning; it does not show that everyone is spontaneously Bayesian or that every frequency phrasing helps. The preserved nesting and common reference population matter. Normalizing both classes to independent “out of one hundred” samples can remove precisely the base-rate information the natural-frequency representation was meant to preserve.^[ref]

An original numerical illustration makes the distinction explicit. Suppose a randomly sampled population contains $200$ category-$H$ cases and $800$ non-$H$ cases. A cue occurs in $75\%$ of the first class and $25\%$ of the second. There are then $150$ cue-positive $H$ cases and $200$ cue-positive non-$H$ cases, so $P(H\mid E)=150/(150+200)=3/7$, approximately $0.429$. Reporting $0.75$ would confuse the cue's conditional rate in class $H$ with the probability of class $H$ given the cue. These are constructed teaching counts, not results from a cited experiment.

The two accounts agree that the task representation matters and that naïve answers need not obey the normative model. They disagree in emphasis: the heuristic account highlights substitution and control, while the ecological account highlights a mismatch between the supplied representation and an available inference algorithm. Frequency success alone cannot decide between them, because a clearer nested-set display can also make deliberate correction easier. Conversely, conflict sensitivity constrains an absolute “base rates were never noticed” story without proving successful Bayesian computation.

A discriminating design would cross numerical format with description–base-rate conflict, preserve the sampling assumptions and information content, and measure answers together with process indicators. It should distinguish a failure to encode prevalence, a failure to combine encoded evidence and a rational belief that the stated reference group is irrelevant. Verbal descriptions seldom specify a likelihood ratio, so prior-insensitivity is easier to diagnose than a unique numerical Bayesian answer for every description. If descriptions were selectively rather than randomly sampled, or a base rate were not applicable to the individual, the simple benchmark would change. Accordingly, base-rate neglect is a reproducible conditional pattern, not evidence that people invariably ignore statistics or possess no Bayesian competence; both heuristic competition and representational structure help explain when it occurs.

Human disorders show that motivation depends on several distinguishable operations: initiating activity, translating incentive value into effort, evaluating outcomes and sustaining goal-directed behavior. A person may be able to execute an instructed movement yet fail to initiate useful action, or may respond differently to reward and effort without a global loss of intelligence. These dissociations argue against equating motivation with motor strength or with a single quantity of pleasure.

A striking starting point is auto-activation deficit. Dominique Laplane and colleagues described three people with marked reduction in spontaneous mental and behavioral activity after toxic encephalopathy and bilateral basal ganglia brain lesions, especially involving the globus pallidus on the available scans. Intellectual capacities were relatively preserved and motor impairment could be mild. External stimulation could reverse the apparent inactivity. This contrast suggested a problem in self-activation rather than inability to perform every relevant cognitive or motor operation.^[ref]

The cases were mechanistically informative but not anatomically pure experiments. Toxic injury can affect more than the structures visible on a scan, and reduced initiation can interact with executive, emotional and motor processes. The observation therefore implicates basal ganglia–frontal systems in internally generated activity without showing that one nucleus contains all motivation. It also motivates assessing spontaneous behavior separately from performance after repeated prompting.

Liane Schmidt and colleagues tested a more specific incentive-to-action hypothesis. They contrasted an instructed handgrip task, requiring specified force levels, with an incentive task where effort could earn money. Patients with basal ganglia damage and auto-activation difficulty could use external instructions to produce force but showed abnormal translation of incentives into effort. The apparatus makes the dissociation concrete: an available motor response does not ensure that anticipated reward will appropriately energize it.^[ref]

This helps separate motor capacity from motivational use of that capacity. Nevertheless, monetary incentives also require understanding task contingencies, representing value and maintaining the goal. A reduced force response does not uniquely identify absence of pleasure. The strength of the design is the contrasting instruction condition; the remaining question is which valuation, activation or control operation prevents self-driven use of the preserved response.

Parkinson's disease offers converging evidence while introducing its own confounds. Campbell Le Heron and colleagues tested 39 patients, with and without clinical apathy, ON and OFF their usual dopaminergic medication, alongside 32 controls. Choices varied both monetary reward and required physical effort. Apathy and medication state were associated with different patterns across the reward–effort space, rather than one uniform decrease in willingness to act. This supports distinguishing clinical motivational difficulty from the complete set of effects produced by changing dopamine availability.^[ref]

The design is stronger than comparing raw movement speed alone, but an ON/OFF contrast is not a selective perturbation of one psychological computation. Medication changes motor function and potentially learning, alertness and response vigor. Apathy groups are not randomly assigned, and comprehension, depression, fatigue and motor capacity can differ. Interpreting choice patterns therefore requires suitable calibration and comparison measures. Dopamine sensitivity does not justify the statement that dopamine is simply a pleasure signal.

A small statistical model illustrates why effort-based decision-making experiments must vary both benefits and costs. Suppose acceptance depends on $U=\alpha R-\kappa E$, where $R$ is reward, $E$ effort, $\alpha$ reward gain and $\kappa$ effort cost. With a fixed logistic function as the choice rule, a lower reward gain and a higher effort cost can produce exactly the same acceptance probability at one offer. This single-offer reward-effort ambiguity motivates varying both dimensions to distinguish their predictions within the model. These are alternative explanations to test, not parameter estimates obtained merely by diagnosing apathy.

Patients with prefrontal cortex damage reveal another component: evaluation of actions across outcomes and time. Antoine Bechara and colleagues introduced the procedure now called the Iowa gambling task, in which choices offered different short-term gains and longer-term consequences. Patients with ventromedial prefrontal cortex damage continued making disadvantageous choices despite relatively preserved conventional intellectual abilities. This linked real-world decision difficulty to experimentally observable valuation and learning demands.^[ref]

The result supports a role in using consequences to guide behavior, but the task combines reward learning, loss sensitivity, reversal, uncertainty and human memory. It does not uniquely establish one bodily-feedback mechanism or prove that the patients have no emotional responses. Similarly, anhedonia, reduced initiation and willingness to expend effort should be measured separately rather than treated as synonyms. Several mechanisms can produce superficially similar inactivity, and a change in one response measure does not imply a change in every form of motivation.

Across these disorders, the most useful inference is computational and relational. Basal ganglia injury distinguishes instructed action from self-activation and incentive translation; Parkinsonian comparisons distinguish aspects of reward–effort choice from medication-related motor and motivational effects; prefrontal damage reveals difficulty applying future consequences. These operations interact through connected systems and need not map to one region each. Patient studies have transformed motivation from a vague explanation of behavior into separable hypotheses about initiation, valuation, effort and control, while demanding that the preserved capacities and competing explanations be tested explicitly.

Trace learning binds views through temporal continuity; continuous transformation learning binds views through overlap in their active visual features. Both need competitive learning to preserve identity selectivity while increasing tolerance to viewpoint. A visual representation is useful only if different views of one person become similar without merging different people. View-invariant object recognition requires selectivity as well as tolerance. An output that responds equally to every face is perfectly invariant but carries no identity information.

Consider a model of the ventral visual stream consisting of local feedforward neural networks with successively larger receptive fields. Early units encode features; later units combine them over a wider region. Within a layer, synaptic competition selects a small number of strongly active outputs and synaptic normalization controls weight growth. Let $x_j(t)$ be current input activity and $y_i(t)$ the competitive output. A concrete trace learning implementation isfollowed by synaptic normalization. Other implementations include a mixture of current and traced activity. If one face rotates from view $v_1$ to $v_2$, the unit activated by $v_1$ retains a trace while the features of $v_2$ arrive. Those features strengthen onto the same unit. Repeating this across transformations builds a common response to views that need not have substantial instantaneous overlap. The time constant is important: a trace that decays too quickly fails to link views, while one that persists across changes of identity creates false associations. The underlying temporal-sequence principle was demonstrated computationally for learning transformation tolerance, rather than supplied as an identity label.^[ref]

Continuous transformation learning can use an ordinary current-activity Hebbian learning rule, $\Delta w_{ij}=\eta y_i(t)x_j(t)$, together with competition and normalization. Suppose adjacent views activate overlapping input features. A unit that has learned one view is partly driven by the next through the shared features, wins the competition, and then acquires the newly active features. Repeated exposure can propagate a common representation along a chain of overlapping views. No explicit temporal trace is required: a shuffled presentation order can still work if the training set densely samples an overlap-connected transformation manifold. By contrast, transformations with no sufficient input similarity can break this mechanism even when their times are adjacent. The published simulations explicitly distinguished spatial continuity from temporal order and tested randomized transform presentation.^[ref]

A small constructed example makes the distinction concrete. Let three views have active feature sets $\{1,2\}$, $\{2,3\}$ and $\{3,4\}$. Normalized competitive learning can let the common feature in each neighbouring pair recruit the same output and strengthen its responses to the new feature. The first and third views need not overlap directly. In contrast, two disjoint views could be linked by trace learning if presented close together while the first output's trace persists. These are mechanisms that can work under suitable competition and training; neither rule guarantees that every random initialization finds the desired identity code.

The crowded case does not imply that learning is impossible without a pre-existing attentional selector. Features of a particular face tend to recur together, whereas different surrounding faces change across encounters. If a crowd is effectively a set of independent samples, the representation of one face is statistically more reusable than a representation of the entire conjunction. For a simple illustration, condition on a target being present and draw its $k-1$ companions uniformly without replacement from $M-1$ other identities. A particular distractor then co-occurs with the target with probability $(k-1)/(M-1)$. The target's own visible features recur much more consistently. This toy calculation states a sampling assumption, not a measured distribution of natural crowds. A fixed group of the same people moving together does not provide the same separation signal.

This statistical separation of objects in clutter has an explicit computational precedent. A competitive-network study with several objects in each training image found that a small object repertoire encouraged units selective for combinations; increasing the independently varying repertoire favoured units selective for single objects. Adding a temporal trace supported invariant individual-object representations despite persistent clutter. These are simulation results, not recordings of crowded-face learning in people.^[ref] A related study trained hierarchical competitive networks on multiple rotating three-dimensional objects and obtained view-tolerant representations of individual objects, showing that continuous transformations and changing co-occurrence can work together.^[ref]

In a real crowd, both mechanisms can nevertheless fail. Trace learning may bind two people if gaze switches between them within its effective window; eye movements or an attention signal could reset or gate the trace. Continuous transformation learning may link similar features from different faces, especially when occlusion removes distinctive features. Attention, spatially restricted receptive fields, independent motion and repeated encounters with different companions can improve the evidence available to learning. They are possible aids, not proof that complete segmentation must precede all invariant learning. Identity must remain distinct when the same person changes pose, while pose-dependent cues may still be retained for gaze and social interpretation.

Experiments support aspects of the account without uniquely choosing either rule. Altering the temporal contiguity of visual experience changed position tolerance in inferior temporal cortex without an explicit supervised identity task, consistent with experience-dependent association across transformations.^[ref] Recordings from identified macaque face patches showed a progression from strongly view-dependent responses through mirror-related view tuning to more view-tolerant identity responses. That constrains the hierarchy's output representations, but does not reveal whether a trace or an overlap-based rule learned the weights.^[ref]

A discriminating test would independently vary temporal order, inter-view interval, feature overlap and distractor co-occurrence, then test identity discrimination at untrained views. Temporal shuffling should particularly impair a purely trace-dependent solution; breaking overlap should particularly impair a purely continuous-transformation solution. Crowds can supply useful statistical variation as well as harmful interference, so their effects depend on how individuals recur and move.

Let $p$ be the row player's probability of $T$. Its expected payoffs against $L$ and $R$ are $6-5p$ and $2+p$. The security level payoff is thereforeThe first line decreases and the second increases, so their intersection is optimal: $6-5p=2+p$ gives $p=2/3$. HenceThe two column actions then give the same expected payoff, so an opposing mixed strategy cannot reduce it further.

The row player's unique best response to $L$ is $B$ and to $R$ is $T$. The column player's unique best response to $T$ is $R$ and to $B$ is $L$. Thus the pure Nash equilibria are $(T,R)$ and $(B,L)$.

For a fully mixed equilibrium, let $p=\mathrm{Pr}(T)$ and $q=\mathrm{Pr}(L)$. Row-player indifference requiresColumn-player indifference requiresBoth probabilities lie strictly between zero and one, so these indifference conditions make both supported actions best responses. Hence the remaining equilibrium isIts expected payoffs are $(8/3,18/7)$. There are no other partly mixed equilibria: if either player is pure, the other's best response is unique, forcing both to be pure. If both mix, the displayed linear indifference equations uniquely determine their probabilities. The row player's equilibrium mixture differs from its maxmin mixture in part (a); these are different optimization criteria in this non-zero-sum game.

As literally stated, the claim is false. Give player one two pure actions, with payoffs one and zero against every opposing action. A mixed strategy assigning probability one to the first action has exactly that action's payoffs, so very weakly dominates it under the printed non-strict inequality. The other pure action nevertheless fails to dominate it.

The intended statement becomes true if the mixture gives positive probability to some distinct action, equivalently $p_i<1$. Fix any opposing pure strategy profile at which $s_i$ pays one. Since all payoffs are in $\{0,1\}$,Equality forces every positive-probability supported action to pay one there. At a profile where $s_i$ pays zero, every other action already pays at least zero. Therefore every action in the support of a mixed strategy is no worse than $s_i$ against every opposing profile. Choosing a distinct supported action proves the corrected conclusion: nontrivial mixed very weak dominance implies very weak domination by a distinct pure action. This proof does not insert the missing support condition silently.

Compute $C={\max }_j{c}_j$ and $D={\max }_j{d}_j$ in one pass. Scan the columns again: return the top cell of column $j$ if $c_j=C$ and $a_j\ge b_j$, or its bottom cell if $d_j=D$ and $b_j\ge a_j$. Each test is exactly mutual best response, so any returned cell is a pure Nash equilibrium.

If none is found, part (e) supplies columnsLocate one of each and mix with probability $1/2$ on each of those two columns, while the row player mixes $1/2$ on each row. The two row actions then both have expected payoff $1/2$.

To check all opposing deviations, observe that no column can have $(c_j,d_j)=(1,1)$. Such a column would maximize the column player's payoff in either row; choosing a row-player best response within it would give a pure equilibrium, contrary to the failed scan. Therefore every column has $c_j+d_j\le 1$, and against the equal row mixture its expected payoff is at most $1/2$. The two supported columns attain that bound.

Hence the returned half-half profile is a Nash equilibrium against every column, not only the two supported ones. A constant number of scans takes $O(n)$ running time, with constant auxiliary storage if the output mixture is represented by its two supported column indices and probabilities. This is the construction in equilibrium in a two-row binary game.

A basic measurement is specified by an orthonormal basis $\{|b_j⟩{\}}_{j=0}^{d-1}$, with outcome $j$ associated with the rank-one orthogonal projection $P_j=|b_j⟩⟨b_j|$. For a normalized pure state $|\psi ⟩$, the Born rule givesThe probabilities sum to one because ${\sum }_j{P}_j=I$. Conditional on an outcome with $p(j)>0$, the post-measurement state iswhich represents $|b_j⟩$ up to a global phase. An outcome of zero probability has no conditional state to normalize. For a density matrix $\rho$, the corresponding expressions are $p(j)=\mathrm{Tr}(P_j\rho )$ and $P_j\rho P_j/p(j)$. If the outcome is ignored, the nonselective projective measurement instead produces ${\sum }_j{P}_j\rho P_j$, generally a mixed state.^[ref]

Let $p=|\alpha {|}^2$ and $1-p=|\beta {|}^2$. The first basic measurement has outcome probabilities $p$ and $1-p$, obtained by conjugating the amplitudes in the bras. Conditional on its first outcome, the next computational-basis measurement returns 0 with probability $p$; conditional on its second outcome, that probability is $1-p$. The law of total probability therefore givesConsequently the minimum is $1/2$, attained exactly when $|\alpha |=|\beta |=1/\sqrt{2}$. The relative complex phases do not affect this result. The alternatives after the first measurement form a classical mixture; their amplitudes must not be added coherently as if no measurement occurred.