Representation learning learns a transformation of observations into features that make a subsequent task easier. Writing $z=f_{\theta }(x)$ and a prediction as $g_{\varphi }(z)$ separates the learned representation from the predictor; training may optimize both together. The objective can come from supervised learning or unsupervised learning.

For example, a neural network trained to classify images can learn edge-like and shape-sensitive features instead of receiving a fixed list of hand-measured quantities. The representation can sometimes be reused on a second task with fewer labeled examples. Such transfer is a hypothesis to evaluate: information useful for the original task can omit a distinction essential to the next. A distributed representation describes how information is encoded; representation learning describes how the encoding is obtained.^[ref]

A distributed representation encodes an object through a pattern across several components, with each component reusable across many objects. In contrast to allocating a separate indicator to every object, shared components let related objects share information. For example, three binary attributes can encode eight combinations even though there are only three coordinates.

In machine learning, the coordinates are often learned real-valued features rather than named attributes. A word embedding can encode several aspects of word usage in one vector; no single coordinate needs to have a simple human interpretation. Combinatorial capacity alone does not guarantee useful learning: the training objective must make the representation's geometry relevant to the task.^[ref]

A feature is an input attribute or computed representation used by a machine learning system. An example can be represented by a vector whose coordinates are individual feature values; a design matrix commonly places examples in rows and these coordinates in columns. A feature can be measured directly or derived from other measurements.^[ref]

For example, classifying manufactured parts might use measured length and width as two feature inputs. Their ratio is another possible feature, exposing a shape relationship to a simple model. Choosing a representation changes which rules are easy to learn. A variable available only after the desired prediction time can introduce data leakage, while a distributed representation can learn useful coordinates without assigning each one a human-readable meaning.

PyTorch is a software framework for numerical tensor operations and machine learning, including automatic differentiation and tools for training neural networks. Its array operations provide the calculations; its differentiation machinery records how a result depends on selected inputs. A computed gradient is distinct from the later optimization step that uses it.^[ref]

For example, this small Python calculation differentiates $y=x^2+3x$ at $x=2$:

import torch
x = torch.tensor(2.0, requires_grad=True)
y = x * x + 3 * x
y.backward()
print(x.grad)  # tensor(7.)

The result agrees with $dy/dx=2x+3$. The call to backward() computes and accumulates derivatives; it does not itself change x to minimize y. In a training loop, parameter updates and clearing old gradients are separate operations.^[ref]

A useful consequence of its dynamically constructed computation is that executed control flow determines the recorded graph. Recording a new graph on the next execution makes it possible to differentiate the computation that actually ran. Disabling gradient tracking or detaching a value changes that connection and must be deliberate.^[ref]

A generative model describes how observations can be produced. It can specify a probability distribution $p_{\theta }(x)$ explicitly, or provide a procedure that generates samples whose distribution is determined by parameters $\theta$. Producing observations and assigning a tractable numerical probability to every observation are separate capabilities.

An original two-bit example defines a joint probability distribution through conditional probabilities. LetGenerate $X_1$ first, then generate $X_2$ using the applicable conditional probability. The four possible outputs have probabilitieswhich sum to one. The procedure generates both complete observations and a dependence between their components. Always outputting the most probable observation, 11, would produce a different distribution: it would discard all the other outcomes rather than sample the stipulated model.

A generative adversarial network gives a different implementation. A generator transforms an input random variable into a candidate observation, while a discriminator supplies a training signal by distinguishing generated samples from training examples. The original paper develops this as an adversarial learning procedure.^[ref]

The model's samples must be assessed against the intended distribution or task. A visually plausible observation alone does not establish that the model represents the full range of observations well; distributional coverage and the probability of different outcomes matter alongside individual sample appearance.

A turbine extracts mechanical work from a flowing fluid through interaction with a rotating set of blades. Steam, hot gas or water changes its momentum while passing through the rotor, producing torque on the shaft. A steam turbine can drive an electrical generator in a thermal power station.

For steady axial through-flow in an ideal control-volume model, shaft torque is $\tau =\dot{m}(r_1{v}_{\theta 1}-r_2{v}_{\theta 2})$, the rate at which fluid angular momentum is removed. Here $v_{\theta }$ is the tangential velocity component and $r$ the radius. Shaft power is $P=\tau \Omega$ at angular speed $\Omega$. The relation connects blade-induced changes in flow direction with useful output; losses and operating conditions determine actual conversion efficiency.

An electric motor converts electrical energy into mechanical work, usually rotation, through electromagnetic forces. For a coil with $N$ turns, area $A$ and electric current $I$ in a uniform magnetic field $B$, the torque magnitude iswhere $\theta$ is the angle between the coil's normal and the magnetic field. Repeated rotation requires the current or effective field arrangement to keep supplying torque in the intended direction. Electrical input power must cover useful output plus losses.

A coil with $N=100$, $I=0.5\mathrm{A}$, $A=0.01{\mathrm{m}}^2$ and $B=0.2\mathrm{T}$ has maximum torque $0.1\mathrm{N}\mathrm{m}$. At angular speed $\omega$, instantaneous mechanical power is $P=\tau \omega$; the maximum-torque value alone does not specify a motor's sustained output.

Michael Faraday's 1821 rotation experiment made the conversion visible: a current-carrying wire moved around a fixed magnet while a pool of Mercury maintained electrical contact. The Royal Institution preserves his 1822 apparatus, consisting of a wire, glass vessel and fixed bar magnet connected through the conducting liquid. Continuous rotation, rather than a single deflection, established a route from steady electric current to repeated mechanical motion.^[ref]

Fumarate hydratase, also called fumarase, is the enzyme that reversibly interconverts fumarate plus water with L-malate. The activity is EC4.2.1.2:In the hydration direction, water adds across the carbon-carbon double bond of fumarate. In the reverse direction, water is eliminated. Its lyase classification describes this transformation; the mere presence of water does not make the reaction hydrolysis.^[ref]

For an invented inventory, begin with ten micromoles of fumarate and let six micromoles undergo hydration. Four micromoles of fumarate remain and six micromoles of L-malate form. Each retains a four-carbon skeleton, so the counted carbon inventory remains $4(4+6)=40$ micromoles of carbon atoms. The chosen extent does not determine chemical equilibrium or enzyme kinetics.

A 1995 X-ray crystallography study of Escherichia coli fumarase C found a four-subunit assembly at $2.0\text{A˚}$ resolution. A competitive tungstate derivative helped locate the active site, which included atoms from three of the four protein subunits.^[ref] The example shows why counting subunits and locating a catalytic site are different tasks: several chains can cooperate to build one local reaction environment.

Lung compliance measures how much lung volume changes for a change in distending pressure. For a static pressure-volume relation, the local value isThe relevant pressure is transpulmonary pressure, not alveolar pressure or airway-opening pressure alone. The pleural pressure reference separates the lung's elastic behaviour from that of the surrounding chest wall.

OpenStax connects lung expansion to pressure changes, elastic recoil and pulmonary surfactant, which modifies surface-tension effects.^[ref] A compliant lung needs a smaller pressure increment for the same small volume increment, but that statement alone does not describe recoil, airway patency or overall respiratory function.

A finite pressure-volume ratio is a secant slope, while $dV/d{P}_L$ is a local slope. In an original nonlinear example, set $x=V-V_0$ and $P_L-P_{L,0}=5x+5x^3$ when volume is measured in litres and pressure in centimetres of water. Local compliance is $1/(5+15x^2)$. It falls from $0.20$ at $x=0$ to $0.05\mathrm{L}(\mathrm{cm}{\mathrm{H}}_2\mathrm{O}{)}^{-1}$ at $x=1$ L. Across that entire one-litre interval, however, the secant compliance is $1/10=0.10$ in the same units. These are constructed values, not clinical reference ranges.

Static versus dynamic lung compliance distinguishes elastic behaviour from measurements during airflow. A simplified lumped dynamic model adds an airway resistance term proportional to volume-flow rate: pressure depends on both $V$ and $R\dot{V}$. A pressure ratio obtained while air is moving can therefore differ from the static elastic slope. Inflation history and hysteresis also require the direction and volume interval to be specified.

In a 1959 primary study, researchers used an intra-esophageal balloon connected to a differential pressure transducer, with a mouthpiece pressure connection, to obtain pressure differences relevant to lung mechanics.^[ref] Such a surrogate pressure measurement has its own positioning and calibration assumptions. Spirometry supplies volume information; volume measurement alone does not establish lung compliance.

Report the pressure references, volume range, flow conditions and measurement history alongside a compliance value. A single number is a property of the specified experiment and state, not an invariant description of every breath.

A tubular lumen is the continuous internal passage of a tube-shaped anatomical structure. In a renal tubule, its lining epithelium separates the tubular fluid from surrounding interstitial fluid. The apical membrane faces the tubular lumen, so movement along the tube and movement across its wall follow different paths.^[ref]

A straight cylinder gives a useful scale estimate. If the internal radius is $10\mu \mathrm{m}$ and the length is $1\mathrm{mm}$, the luminal volume isThis is an ideal geometric example; real renal tubules change shape and diameter along their length.

Experiments on isolated perfused collecting ducts exploit the same compartment distinction. Fluid is supplied through the tubular lumen, while a separate surrounding solution contacts the outside of the epithelium. Changing the conditions permits measurement of water crossing the wall. In a 1995 study, the increased membrane permeability during vasopressin exposure accompanied movement of aquaporin-2 toward the apical membrane.^[ref]