arithmetics and nomenclature

The nomenclature of mathematics withstands improvement to clarify its utility in high-technology--

• At-sign, @ , for a number "at" a place, position, so-many-tens, powers of ten, characteristic, scale, range, radix-ten exponent; scientific notation [rather than computer-traditional E, double-precision D];
  Eg. 5.12@3 = 5120 ;
  Eg. 5.12@-3 = 0.00512 ;
• Place the exponent of a term, upon, upper-fore of its root term [rather than as formerly over], as exponentiation is conceptually the iteration-extension above multiplication,- even as multiplication is the iteration-extension of addition;
  Eg. ²3 = 9 ;
  Eg. ^-13 = 1/3 ;
  Eg. ^-1sin = 1/sin ;
  Eg. ²f x = ²(f x) = (f x)×(f x) = (f×f) x = (²f) x ;
  Eg. ²log x = ²(log x) ;
• Exponent placement-compounding symbolically increases involvement by deeper application, afore, upon, above, over, behind: where the deeper are more involved functionally internally or argumentive-to, rather than externally usage-of, the function;
• Placing an exponent over, upper-aft its root operation, is the functional composition repetition-iteration notation (consistent with the traditional arc-function, inverse function, operator algebras);
  Eg. sin^-1 = arcsin (unequivocally);
  Eg. f² x = (f f) x = f f x = f(f x) ;
  Eg. log² x = log log x ;
  Eg. f^-1(f x) = f^-1 f x = (f^-1 f) x = (1_f) x = x (in the domain of f; where f is 1-1);
  NB. x² = (x x) = x x = x(x) = x×x = ²x : Strictly nonfunction numeric variables and constants are the same either way;
• Functions may have multiple, parameter-arguments, if so defined,--
  Eg. f x y = f(x,y) , is a two-argument function; generally ≠ (f x) y , unless--
  NB. (f x) y = f_x y , is a progressive-parameterization-application function, taking y as a subsequent separable, argument;
• x/y+z "x over y plus z" is more global than division, x÷y+z ; [formerly x/(y+z)]; (NB. y+z\x may be "y plus z , dividing x");
• Logarithm, £ x = ln x = log_e x "natural log; ponent";
  Cf The entropy of p , 0 < p < 1, is -p £ p (in natural base; binary base is more usual);
• Complex angle, & = ¢ + i$ "ensine equals cosine plus i (times) sine, of ...";
  • outer product, · , (angle summation);
  • inner product, ·¯, (angle difference) [macron over middot];
  • inner division, ÷ , (angle difference);
  • outer division, : , (angle summation);
• Vector correlation, * , the sum of corresponding products; [formerly "dot product"]; the complex conjugate self-product (a)(a');
• Gut arithmetic [formerly matrix arithmetic, but not-much new comes from vector sums of products: whence a gut process];
• Teams of Operators [formerly groups of elements, but compounded elements are non-elemental; cf nuclear fusion is not addition creating new elements; cf sets have non operational elementals];
• Left-Multiplication is a property of addition, not an operation (no more so than parentheses or exponentiation). Addition may be any additive process: Multiplication is the iterative extension property thereof;- which is to say, a process is multiplicative only if it includes natural-number multiplication, eg. matrix A*B , is multiplication only if, n*B already is and = n_i=j*B = [₀ⁿ···_n⁰]*B ;

  Multiplication always includes specifically intrinsic Left Multiplication, which has special necessary properties and is specifically denoted; Other-multiplicative process is distinguished as a, "product", if it does not include the intrinsic: To say, * , is the multiplicative operator on a ring, means, * , includes n*x : {n} = N₊ in the ring set;- And whence, y*x , imputes, y , may be any natural number: and therefor the ring set includes the set of all natural numbers (The ring set need not entirely separate them: cf modulo).... Thus, another-multiplicative process, X, existing on a ring, If , X , does not include intrinsic multiplication, makes for two actual operations, and three operation symbols, +,*,X , on that ring.

• Identity Operator, 1_i=j = 1 : i=j ; else = 0 (assumed for i≠j : multiplicative null, empty, undefined); [cf Kronecker Delta δ_ij] ;
  cf Logic Operator, (i=j) = true : i=j ; else = false (for i≠j) ;
  Eg. Identity Matrix, [1_i=j] = [₀¹···₁⁰] ;
• Infinitesmal, ∂ "par", sometimes ð "eth" a standalone numeric value-holder;
• Derivative, δ "deriv; der of; del or deal of"; δ_xy = (∂/∂x) y "... on x, or with respect to x, of y";
  NB. ∂_i = ∂ξ_i , generalizes dimensionality, eg. ∇ = i(δ_i) + j(δ_j) + ... [rather than prior assumed scalar:vector, x:i, y:j, ...]
• Integration interval, ∫_[x,y] closed, ∫_(x,y) open; or, ∫_[x+,y-] open, ∫_x<ξ<y∂ξ open, etc;
• The notation for specifying a set, eg. {i}, is simpler, or, more inclusive:--
  Eg. {i≤5} or {i:≤5} is the "set of i [that are] less than or equal 5";
  Eg. {i: i; i(+,-); i≤5} is the "set of i where i exists, i is closed on addition and subtraction, i is less than or equal 5";
  Note that i(+,-) is like the computer-logic denotation-as-needed of numeric value or logical "is-defined" value;
• Linguistics of the much-used space-time derivatives:--
  • The First Derivative of, distance [scalar], with respect to time is, speed; eg. ft/sec.
  • The First Derivative of, position [vector], with respect to time is, velocity.
  • The First Derivative of, speed or velocity, with respect to time is, acceleration; eg. ft/sec².
  • The First Derivative of, acceleration [scalar], with respect to time is, pump (not jerk *); eg. ft/sec³.
  • The First Derivative of, acceleration [vector], with respect to time is, swing (not jerk *).
  • The First Derivative of, pump or swing, with respect to time is, jerk (*); eg. ft/sec⁴.
  • The First Derivative of, jerk, with respect to time is, jolt [tbd]; eg. ft/sec⁵.
  • NB. Throw, may be any of the five, scalar or vector; Jounce, is a jousting jolt-hard bounce; Heave, implies massive;
  * (There is no jerk in the continuously changing acceleration vector of circular motion,- without changing speed, focus, distance; the driver pumping brakes, a trapeze swinging, change acceleration smoothly continuously; Jerk is a quickened pump-derivative; Swing, in space-time physics is deemed indistinguishable from artificial gravity acceleration, until measured in another way ... A swing with more directional control-freedom changes its vector with more surprise, aka, jerk,-- cf a child's swing is pumped, not jerked, to increase the lesser derivatives, height, speed, acceleration,- while swing itself is minorly unnoticed: short of gimbaled adolescent amusement park swings; Pumping is equalized by pumping, or leaning in gravity, not by jerking: Jerk is unequalizable, nonlinear difference, metastable, abbreviated, accelerated leaning; cf the Olympic Jerk is instantaneous: the pump-derivative);
• Natural distribution, e^{-π x²/s²}/s , semidiameter s , var. s²/2π , identified by its 'knee/ankle' angular joints and inflections;
• We use now-standard-html special symbol-letters, √, ≈, →, ⇒, π, μ, Σ, ∞, ≤, ≠, ≡, ⇔, ∀, ∃, ... ⊂, ⊆, ∈, etc. (as these exist);
• We do arithmetic-checking-by-9's-and-11's ... There is no radix-2-arithmetic-checking-by-1's: we check-by-3's (binary-11's);
• (We're converting from "half-life" 1/2-remains to "natural-life" 1/e-remains, as equals its average for inverse-exponential).

• Aether, cosmic aether, we discuss as a real (coeternal) infinite-time-integrated infinite-space-defining infinitely-dense manifestation (tbd) of mathematical complexor development, not frame-relativistic "measure-space-time"; "open cosmos";
• Atomic power, commonly misnomered, refers to energy stored in electron orbitals below electrochemical and electroplasma "degenerate electrochemistry" trans-ficial orbitals; eg. a H-1s electron boosted 6.3 eV to 2s , where maintained by rf-laser coupling the groundstate electron-spin potentials, it holds 69K°K till released; We discuss transition to p-e-p nuclear fusion;
• Mass-hole (better-phrase), is a gravitational singularity, mass compressed beneath its nucleon (and quark) roche-threshold, to wave-entropical radius (tbd); a "black hole" not necessarily black; We discuss aether-noodle-fragmentation renormalization;
• Nuclear, refers to atomic nuclear; energy, states, structure, bonds, chemistry, fission, fusion, inner bound electrons, protons, neutrons, deuterons, helons (alphas), ...; We discuss doubly-charged protons p⁺⁺ and negatively-charged neutrons n^-;
• Subatomic, refers to β-exchange between a nucleus and a low-state or groundstate electron orbital, as in the Ho-Dy-163 co-version experiment; electrons, protons, neutrons, and related nuclear-size particles, nuclear fragments; We discuss antihelon stability "sitting" outside the closed 1s-orbital shell; see also Supranuclear;
• Subnuclear, refers to nucleons, energy, interchanges; We discuss aether wavicles, "neutrino chop";
• Supranuclear, refers to electrons "sitting" just outside the nucleus, adsorption below groundstate without absorption-fusion;
• Supraneutronium, ferroneutronium; refer to sub-absolute-zero-cold maximum atomic-density mass; not necessarily degenerate electron nor neutron star densities, and stable nuclear tendencies; not known to occur naturally but maybe made [see article];

• TEMPEST, (military) acronym for various spurious-electronic-signal-emission suppression technologies, techniques, contractual requirements and standards, for operation under occasional (thermo)nuclear-detonation duress, developed in the 1950's-70's;
  Some common interpretations include:--
  • Thermonuclear Electro-Magnetic-Pulse Exception Suppression Technologies, meaning it didn't quit under nuclear assault ... cf reinforced concrete radio horns, fluted ferrite beads (nonsaturating) and spark gaps hung on every electric path, wire, sewer line ... above and beyond good electrical grounding and shielding techniques every engineer must use ...
  • Thermo-Electronic-Magnetic-Pulse Emanation-Suppression Technologies, meaning EMP is not an electromagnetic radiowave but a ripping-hot electron-cascade avalanche pulse (zone-ionization) borne in an expanding magnetic field that couples and suppresses photonic emanation energy-loss,- thus keeping the pulse hot ... (And affects chemistry)
  • Crypto-TEMPEST, was Transient Electro-Magnetic-Pulse Emission-Suppression Techniques, "Single-Point-Failure-Safe" electronic-ware design, preventing misdirection of messages in and after an EMP event not destroying a Crypto device ...
  • TEMPEST Crypto, we do not discuss here, refers to Crypto [secret] messages during or pertaining-to nuclear warfare ...
• Tuple, (mathematics) terminology: a doublet, a 2-vector, eg. a complex number; a single-element consisting of a pair of terms ... Note that we use the term consistently: N-tuples here are N-pair elements [formerly N-part elements we here call, "N-uples"];
• -uple, (mathematics) suffix to a [natural whole] number, meaning the part-count consistency of-or-defining an objectified unit;