For a passive scalar in a planar shear flow $u(y,t)$, let $m_j(y,t)={\int }_{\mathbb{R}}{x}^j\chi (x,y,t)dx$. Integrating the advection-diffusion equation by parts in $x$ gives the displayed hierarchy for $j\ge 0$, with negative-index terms omitted. Sufficient decay and finite moments justify the integrations. Insulating walls give Neumann boundary conditions for every moment. Solving the first two equations then determines the long-time growth of the second moment without solving the full scalar field.

For a map $f:\mathbb{N}\to \mathbb{N}$ from the cofinite topology to the tail topology on the natural numbers, continuity is equivalent to every set $\{n:f(n)\ge k\}$ being empty or cofinite. If the image is unbounded, every such set is nonempty and cofinite, so $f(n)\to \infty$. If it is bounded, its maximum $M$ exists, and the inverse image of the tail at $M$ forces $f(n)=M$ eventually, with $f(n)\le M$ always. Conversely either condition makes every tail preimage empty or cofinite.

If a Hermitian interaction-picture Hamiltonian satisfies $\hat{T}V(t){\hat{T}}^{-1}=V(-t)$ for an antiunitary operator, its Dyson series conjugates $(-i{)}^n$ to $i^n$. The substitution $t_j\mapsto -t_j$ converts ordered integration to anti-ordered integration, giving $\hat{T}S{\hat{T}}^{-1}=S^{†}$. Antiunitarity then gives the displayed reciprocity: the time-reversed process has the reversed outgoing state as its incoming state and vice versa. It does not merely reverse the momenta while leaving their incoming/outgoing roles fixed.

For a charged spin-zero complex scalar field, time reversal preserves charge and can include an intrinsic phase. In its free-field mode expansion of a free field, antiunitarity conjugates the plane-wave coefficients. Relabel spatial momentum to its negative and compare the annihilation and creation terms: $\hat{T}a(\mathbf{p}){\hat{T}}^{-1}={\eta }_{T}a(-\mathbf{p})$ and $\hat{T}{b}^{†}(\mathbf{p}){\hat{T}}^{-1}={\eta }_T{b}^{†}(-\mathbf{p})$. Particle annihilation remains annihilation; unlike charge conjugation, this operation does not exchange particle and antiparticle operators.

For $n$ distinct points in the real plane, $L_{\ge k}$ counts lines containing at least $k\ge 2$ points. The Szemerédi–Trotter theorem applied to these lines gives the bound by absorbing its additive line-count term; bounded $k$ follows by counting point pairs. Consequently, summing occupancies of lines with at least $k$ points by dyadic ranges gives $O(n^2/k^2+n\log n)$.

A linear system is homogeneous when its forcing is zero. Its solution set is the kernel of a linear map, a vector subspace. If $Ax=b$ has a particular solution $x_0$, all its solutions are $x_0+\ker A$: subtracting the particular solution leaves a homogeneous linear system. The rank-nullity theorem gives the dimension of this freedom.

The augmented matrix of a linear system $Ax=b$ appends the right-hand side as one extra column. Elementary row operations preserve its solution set. The linear system is consistent exactly when the coefficient matrix rank equals the augmented matrix rank, because this is equivalent to $b$ lying in the span of the columns of $A$. A larger augmented matrix rank produces a zero coefficient row with a nonzero right-hand side during Gaussian elimination.

For matrices representing linear maps on finite-dimensional vector spaces, $AX=B$ has a solution precisely when the image of a linear map represented by $B$ is contained in that represented by $A$. Necessity follows by applying the equation to vectors. For sufficiency choose an $A$-preimage of each column of $B$ and assemble those vectors as the columns of $X$. All solutions are $X_0+N$ with $AN=0$, equivalently every column of $N$ lies in the kernel of a linear map represented by $A$. When $B$ has at least one column, a solution is unique precisely when $A$ has trivial kernel.

An operation that takes effect indivisibly relative to the specified competing accesses. An atomic increment protects the whole read-modify-write; separate atomic loads and stores do not make their combined sequence an atomic operation.

// Target: computer-science.bigb

Rules specifying which shared-data observations and operation orders are allowed. Sequential consistency is the model used in the two diagrams. A language memory model can permit weaker ordering and impose additional conditions on conflicting accesses. Correct use of an indivisible flag must therefore be checked against that memory model, including happens-before relationships for the data guarded by the flag.

A group of participants cannot proceed because each waits for a resource whose release depends on another participant in the group. Consider two initially free mutex objects A and B. T1 acquires A then tries B; T2 acquires B then tries A. If each obtains its first mutex, both wait permanently for the other, with release reachable only after the second acquisition. Each mutex still enforces mutual exclusion.^[ref]

Each software thread performs four indivisible operations: acquire its first mutex, acquire its second, release the second, release the first. An unavailable acquisition blocks; neither participant releases early. A state records each participant's completed-operation count and both owners. Exhaustive exploration finds 19 reachable states with opposite orders, including exactly one nonterminal state with no enabled operation: both counts are $1$, T1 owns A, and T2 owns B. Giving both participants the order A then B leaves 16 reachable states and no such blocked state. Unshown grid positions are unreachable. These are tiny sequential consistency models, without timeouts, cancellation, recursive acquisition or other resources.

A requirement of eventual progress, with no finite prefix ruling out its eventual satisfaction. For starvation freedom, a persistent requester must eventually enter its critical section. Establishing such a liveness property requires explicit scheduling and protocol assumptions in addition to mutual exclusion. Exhaustive finite thread interleaving checks alone do not establish eventual service for repeatedly competing software threads.^[ref]

A requirement whose violation has a finite execution prefix that no continuation can repair. Two software threads simultaneously inside a protected critical section already violate mutual exclusion; later leaving does not erase that violation. This contrasts with a liveness property, for which a long finite wait alone does not establish failure.^[ref]

Mutual exclusion allows at most one participating software thread at a time to execute a protected critical section. A mutex is an object used to enforce that rule; the critical section is the region of the program that needs the rule. Several different critical sections can use the same mutex to protect the same shared data. The benefit is reasoning about a complete update without another cooperating software thread observing or changing its intermediate state.^[ref]

For a small lost update, shared x initially equals $0$. Each of two software threads, T1 and T2, uses its own local r and performs these three steps:Assume sequential consistency: each step is indivisible, each software thread keeps the displayed order, and a read sees the latest preceding write to x. In the thread interleaving T1 read, T2 read, T1 add, T2 add, T1 write, T2 write, both local values become $1$ and both writes store $1$. Final $x=1$ instead of $2$: a lost update despite two correctly executed local additions. Separate indivisible reads and writes are insufficient; the complete read, modify and write must be protected.

The binomial coefficient $\left(\genfrac{}{}{0ex}{}{6}{3}\right)=20$ counts the thread interleavings preserving the two three-step orders: choose T1's three positions, with the remaining positions belonging to T2. Exhaustive enumeration gives final $x=1$ in 18 orders and $x=2$ in the two serial orders. These counts are not probabilities: no random scheduler has been specified. With the same mutex acquired before reading and released after writing, only the two complete increment orders remain, and both give $2$. Alternatively, an atomic read-modify-write increment updates x in one indivisible step. Protecting only the write still allows both software threads to read $0$ beforehand.

Mutual exclusion is a safety property: two participants must never be inside the protected critical section together. Eventual entry for every persistent requester is a liveness property, expressed here as starvation freedom. A protocol that never admits anyone satisfies the first requirement while failing the second. Nor does a mutex alone promise which waiting software thread enters next, or that a producer runs before a consumer. Such application ordering needs an additional synchronization protocol.^[ref]

The sequential consistency assumptions delimit this calculation. In C++, unsynchronized conflicting accesses to an ordinary shared object can be a data race, which makes behavior undefined; the 20 schedules are not a prediction for that program. An atomic operation with relaxed ordering also supplies no acquire-release synchronization for other data. A real mutex must supply both exclusive entry and the required happens-before ordering between its users.^[ref]

An inhibitory postsynaptic potential is a synaptic potential associated with decreased likelihood of a postsynaptic action potential. It can involve hyperpolarization or depolarization; an inhibitory conductance can even produce no measurable voltage change at rest. See synaptic potential and shunting inhibition for worked examples and sources.

An excitatory postsynaptic potential is a synaptic potential associated with increased likelihood of a postsynaptic action potential. Its size and effect depend on the receiving neuron's state; the definition concerns firing, not a fixed neurotransmitter identity. See synaptic potential for the mechanism and source.

Synaptic conductance describes how strongly activated synaptic ion channels conduct electric current for a given driving force. In the passive approximation, $I_{\mathrm{syn}}=g_{\mathrm{syn}}(E_{\mathrm{syn}}-V)$, with inward-positive electric current. It is the electrical conductance contribution of a synapse; its postsynaptic voltage response also depends on the driving force, the other membrane electrical conductances, and the capacitance. The synaptic potential model and shunting inhibition example show why a nonzero synaptic conductance can have zero electric current at rest.

Constraints or interactions make the strengthening of some inputs limit others. In developmental models, competition together with input correlations can segregate ON and OFF subregions or inputs from the two eyes; correlations alone do not specify the full biological developmental mechanism.

Geometric quantization associates quantum states and observables with a symplectic manifold. A first step uses a complex line bundle with a connection on a vector bundle whose curvature is proportional to the symplectic form divided by $\hslash$; its existence imposes an integrality condition. A real polarization is an integrable half-dimensional distribution with Lagrangian submanifolds as leaves. Requiring sections to be covariantly constant along the leaves reduces the variables to a configuration-space description. Restricting the curvature to a Lagrangian leaf gives zero, permitting local flat sections; global sections can require trivial holonomy. Exact Lagrangian graphs also describe local semiclassical phases through their generating functions.

A symplectic capacity is monotone under symplectic embeddings, scales by $c(X,a\omega )=ac(X,\omega )$ for $a>0$, and has positive unit-ball capacity and finite unit-cylinder capacity. It is invariant under symplectomorphisms. A normalized symplectic capacity additionally assigns $\pi$ to both unit domains.