Vibration isolation reduces the transmission of unwanted mechanical oscillations between an apparatus and its surroundings. Compliant supports and damping can reduce transmitted motion, but their effectiveness depends on the disturbance frequency and the support's resonance.

Mechanical impedance is the ratio of a driving force complex amplitude to the resulting velocity complex amplitude at a specified point and in a specified direction. Its units are $\mathrm{N}\mathrm{s}{\mathrm{m}}^{-1}$. The direction of the force, the time convention and whether one is describing a travelling component or a total field must be stated. A positive real travelling-wave mechanical impedance transfers mean power $⟨P⟩=Z|v{|}^2/2$ when $|v|$ is the peak velocity wave amplitude.

Motion after launch under a specified gravitational field, with negligible drag force and no thrust. In a constant downward gravitational acceleration $g$, horizontal velocity is constant and vertical position obeys $y(t)=y_0+v_{y0}t-g{t}^2/2$. Eliminating time gives a parabola when horizontal velocity is nonzero. Over larger distances the changing direction and magnitude of gravity require an orbital model instead.

Section of Classical Electromagnetism by Richard Fitzpatrick by whole-work author Richard Fitzpatrick; no separate chapter byline is established. The University of Texas at Austin hosts the author’s teaching text.

Online text: farside.ph.utexas.edu/teaching/jk1/Electromagnetism/node60.html

Verified title and author byline: farside.ph.utexas.edu/teaching/jk1/Electromagnetism/index.html

Online revision: 2014-06-27; this is not an original publication date.

Cited passages: Complete Permanent Ferromagnets section, particularly equations (709) and (712)–(716); Outward surface-normal discussion following equation (710); Whole-work index byline; existing reusable source scope also verified in physics-bibliography.bigb.

Section of Classical Electromagnetism: An intermediate level course by Richard Fitzpatrick by whole-work author Richard Fitzpatrick; no separate chapter byline is established. The University of Texas at Austin hosts the author’s teaching text.

Online text: farside.ph.utexas.edu/teaching/em/lectures/node73.html

Verified title and author byline: farside.ph.utexas.edu/teaching/em/lectures/lectures.html

Online revision: 2006-02-02; this is not an original publication date.

Cited passages: Complete section; equations (856)–(857) and surrounding averaging definition; Equations (860)–(864), current decomposition and H definition; Whole-work title and author byline on lectures.html.

Blood is a fluid containing biological cells that transports substances through a circulatory system. Its viscosity depends on its cellular content and deformation rate. Treating it as a Newtonian fluid in the Hagen-Poiseuille equation is an approximation that depends on the vessel size and flow conditions.

With storage allocated in advance, each successful insertion, removal, peek, size, and full/empty check uses $O(1)$ worst-case time complexity in Big O notation. This assumes constant-cost indexed array access, bounded machine word index arithmetic and fixed-cost scalar item reads and writes. Each operation touches a bounded number of slots and fields regardless of $n$ and $C$; no shifting, scanning or resizing occurs. Rejected operations also take constant work. Producing a copy of all $n$ live items takes $O(n)$ work under the same scalar-cost assumptions.

The preallocated array occupies $O(C)$ scalar slots plus constant bookkeeping. Allocation and any initial slot construction are separate setup costs. Copying a large item, running an expensive destructor or callback, allocating an item, or growing a dynamic array changes the operation cost and must be charged separately. The bound is worst-case structural work, with no need for amortized analysis, and says nothing about a measured wall-clock deadline.

These formulas describe sequential calls. Wrapping array indices implies neither atomic operations nor safe access from multiple software threads. Publication of an item's contents and reuse of its slot require a protocol justified under the relevant memory model, including any necessary acquire-release synchronization. The kernel documentation discusses barriers under explicit producer/consumer restrictions; Boost's documentation separately requires mutual exclusion for shared access involving writes. Neither property follows merely from choosing a circular buffer.^[ref][ref]

Continue from step 9 with a proposed insertion of $H$. A rejecting operation try_enqueue(H) returns full, writes nothing and leaves $(C,D,E,F,G)$ with $h=2,n=5,w=2$. The next dequeue() returns $C$, leaving $(D,E,F,G)$ with $h=3,n=4,w=2$. Rejection requires the caller to decide whether to retry or abandon $H$.

An overwriting operation append_latest(H) instead writes $B_h\leftarrow H$, advances $h\leftarrow (h+1)\mathrm{mod}C$, and keeps $n=C$. It can report the discarded label $C$ as part of its specified result. At step 10 this gives $(D,E,F,G,H)$ with $h=3,n=5,w=3$. The next dequeue() returns $D$, leaving $(E,F,G,H)$ with $h=4,n=4,w=3$. Below full capacity, append_latest follows the ordinary insertion rule without discarding anything.

The lower bars in the trace diagram expose the consequence: $C$ entered at step 4 and remains a member until step 11 under rejection, but only until step 10 under overwrite. $H$ never becomes a member in the rejecting branch; it does in the overwriting branch. The same physical slot can therefore hold different logical items over time. These interfaces are observably different, so representation independence cannot justify exchanging them while retaining the same promised results.

Boost's documented push_back behavior chooses overwrite when full, implemented by assignment to the old element. Its discussion of iterator invalidation illustrates why a reused address need not still identify the original item. Assignment, object destruction, and release of an owned resource are separate language-level concerns; the membership bars make no claim about those lifetimes.^[ref]

A rejecting FIFO queue suits work that must remain pending until explicitly removed. An overwriting circular buffer suits a bounded recent history where losing the oldest retained item is the stated policy. Neither storage wrap nor bounded capacity chooses between those requirements.

Start with $C=5$, $h=0$, $n=0$ and immutable labels. Perform enqueue(A), enqueue(B), dequeue(), enqueue(C), enqueue(D), dequeue(), enqueue(E), enqueue(F), enqueue(G). The removals return $A$ and $B$. The interleaved insertions and removals make the front advance before the array fills; slot reuse does not require waiting for the FIFO queue to become empty.

At step 7, $E$ occupies physical slot $4$ and the next-write array index wraps to $0$. At step 8, $F$ reuses slot $0$, giving the physical layout in the preceding diagram. At step 9, $G$ replaces the stale contents of slot $1$: the physical array is $(F,G,C,D,E)$ but the logical sequence is $(C,D,E,F,G)$. Now $h=w=2$, yet $n=5$ makes the state full rather than empty.

The displayed $h,n,w$ values are related by $w=(h+n)\mathrm{mod}5$ after every call. They are shown together to explain the invariant; an implementation need store only two. For the rejecting FIFO queue, every successful removal returns the earliest accepted label not already removed. The representation independence example supplies the general proof through its abstraction function; this trace makes the wrapped physical storage visible.

With front-plus-count storage, empty means $n=0$ and full means $n=C$; every allocated slot is usable. With only the two wrapped array indices $h,w$, the equality $h=w$ describes both possibilities. Indeed $w-h\equiv n(\mathrm{mod}C)$, and the two allowed counts $0,C$ have the same remainder. Recovering the count as $(w-h)\mathrm{mod}C$ therefore loses the full state. A count, a separate full flag, or a different index representation must resolve that ambiguity.

Another convention reserves one slot: for a physical array of length $C\ge 2$, empty means $w=h$ and full means $(w+1)\mathrm{mod}C=h$. Its usable capacity is $C-1$, and $(w-h)\mathrm{mod}C$ now recovers its count because $0\le n<C$. A five-slot array then stores at most four items.

This is the convention of the kernel's documented CIRC_* helpers, which additionally require power-of-two array lengths. It is not a universal property of circular buffers; our capacity-five count-based example uses all five slots and does not use those helpers.^[ref]

The smallest count-based case, $C=1$, makes the issue vivid: both $h$ and $w$ are always zero, but $n$ still distinguishes one live item from none. Capacity zero is excluded from our formulas because remainder modulo zero is undefined.

Let the capacity be a positive integer $C$, and allocate exactly $C$ array slots $B_0,\dots ,B_{C-1}$. Store the front array index $h$ and count $n$. The representation invariant isThe logical sequence and next-write array index areHere $\mathrm{mod}$ denotes the nonnegative remainder from modular arithmetic. The count determines which slots are live: leftover bits in other slots are not members of the FIFO queue. If $n=0$, $h$ has no current front item; it identifies the position where the next successful insertion begins. If $n=C$, $w=h$ points at a live item, so writing there requires an explicit overwrite policy.

For $n<C$, enqueue(x) writes $B_w\leftarrow x$, then increments $n$, leaving $h$ fixed. For $n>0$, dequeue() saves and returns $B_h$, then sets $h\leftarrow (h+1)\mathrm{mod}C$ and $n\leftarrow n-1$. These are sequential updates, completed before the next call. A rejecting interface tests $n=C$ or $n=0$ first and returns full or empty, respectively, without changing state. peek returns $B_h$ only when $n>0$; size returns $n$.

One can instead store $w$ and $n$, recovering $h=(w-n)\mathrm{mod}C$. On insertion, advance $w$ and increment $n$; on removal, keep $w$ fixed and decrement $n$. Thus the next-write array index is not necessarily the front array index.

Names such as head and tail require checking the actual API: the kernel documentation calls the insertion position head and the removal position tail.^[ref]

The diagram is the state after step 8 of circular buffer operation trace. Reading the physical array from slot zero would give $F,B,C,D,E$, which includes a removed label and puts $F$ first. Reading from $h$ for exactly $n$ items gives $(C,D,E,F)$. The next insertion uses slot $1$, not slot $0$. This distinction also explains why the live sequence may need two contiguous pieces of the array: here slots $2$ through $4$, followed by slot $0$. No items need to move to make this wrap work.

The chromatographic mobile phase supplied to a chromatography apparatus.

The Hill reaction couples light-driven electron transfer from water to an added electron acceptor with molecular oxygen release in an isolated photosynthetic preparation. The chloroplast dye-reduction experiment uses DCPIP.^[ref]

In 1937, Robin Hill's Oxygen Evolved by Isolated Chloroplasts described a spectroscopic molecular oxygen measurement in illuminated isolated chloroplasts: oxygenation of muscle hemoglobin provided the indicator. This was an oxygen-binding readout. The publicly accessible first paragraph establishes that method; it does not specify enough preparation or control details to reconstruct the entire historical experiment here.^[ref]

The experimental idea is powerful because isolation asks how much of photosynthesis remains functional outside an intact biological cell. Breaking a chloroplast envelope does not logically imply destroying every internal thylakoid membrane. Conversely, finding an active preparation cannot establish that its chloroplast envelope remains intact. Activity and structural integrity require different evidence.

The double boundary formed by the outer chloroplast envelope membrane and inner chloroplast envelope membrane.^[ref]

Unsigned experiment in the department-compiled manual.

Cited passages: printed pp. 15–18, equations (3)–(7), Procedure and Table 1.

Online text: openstax.org/books/chemistry-2e/pages/12-6-reaction-mechanisms

Section of Chemistry 2e by Paul Flowers and others. Paul Flowers, Klaus Theopold, Richard Langley and William R. Robinson are the whole-book authors; no separate section author is credited. OpenStax publishes and hosts the book. Its publication date is 14 February 2019; no separate section revision date is established.

Online text: openstax.org/books/chemistry-2e/pages/5-3-enthalpy

Book metadata: openstax.org/books/chemistry-2e/pages/preface

Cited passages: §5.3, enthalpy conventions and Hess’s Law.