# mekso -mek-me'o- gismu

x1 [quantifier/expression] is a mathematical expression interpreted under rules/convention x2.

## On gloss:

li'ai in sense "as a name"
unevaluated mekso as name.

## In definition:

tu'o
null operand (used in unary mekso operations).
su'ifa'uvu'u
mekso operator: plus or minus with order important, (((a±b)±c)±...±z)
su'ijavu'u
mekso operator: plus or minus, (((a±b)±c)±...±z)
bai'i
mekso string operator (ternary): find-and-replace; in string/text/word/sequence X1 formally replace X2 (ordered tuple of terms to be replaced) with X3 (ordered tuple of terms to be respectively substituted)
bai'i'i
mekso operator: in ordered tuple/list/vector/sequence X1, replace the X2th entry with term X3 of appropriate type, and leave all other entries untouched (optional: where the index for the very first/leading/header entry is X4).
be'ei'oi
ternary mekso operator: x1th Bergelson multiplicative interval with exponents bounded from above by function x2 and with sequence of shifts x3, where exponents belong to set x4
bei'u'i
unary mekso operator: nth Bernoulli number Bn of the second kind (B1 = +1/2 = >0).
boi'ai
unary mekso operator: immediately convert number into a single digit.
boi'e'u
mekso operator/function terminator (in Polish notation): inserts exactly enough "{boi}"'s consecutively so as to terminate the most recently uttered operator/function in a mekso expression
boi'oi
preserve formal interpretation of mekso subexpression with no substitutions made
ca'ei'a
Unary mekso operator: unit vector/normalization of the argument (vector) X.
ca'o'e
mekso 4-nary operator: spherical harmonics on colatitudinal/polar angle a and azimuthal/longitudinal angle b of unassociated order c and associated order d.
ci'ai'u
Mekso unary or binary operator: n-set or integer interval; in unary form, it maps a nonnegative integer X1 = n to the set \1, \dots , n\ (fully, officially, and precisely: the intersection of (a) the set of exactly all positive integers with (b) the closed ordered interval [1, n] such that n \geq 1; see notes for other n); in binary form, it maps ordered inputs (X1, X= (m, n) to the intersection of (a) the set of exactly all integers with (b) the closed ordered interval [m, n].
ci'au'i
mekso at-most-4-ary operator: integer lattice ball; the set of all points belonging to the intersection of Zn with the closure of the ball that is centered on X1 and has radius X2 in metric X3, where Z is the set of all integers and where, for any set A and non-negative integer n, An is the set of all n-tuples such that each coordinate/entry/term belongs to A, and where the dimensionality n = X4..
ci'o'au
mekso operator (binary): projection function; the Bth term/entry ("element") of tuple A
cu'au'ei
mekso binary/unary operator: multinomial coefficient/binomial coefficient/choose
da'a'au
mekso operatory: prime mark append
de'au'u (exp!)
mekso ternary operator: positive super-logarithm; the super-logarithm (inverse operator of hyper-operator with respect to "height" of power tower) of a with base b and of order c-2.
dei'au'o
mekso binary operator: Lambert product-log W function; W(a, b)
du'a'e
mekso n-ary ordered operator: structure creator/ordered tuple, 'endow'; the structure formed by underlying set X1 (as) endowed with element, order, quoted operator, etc. X2, X3, ...
du'a'o
mekso binary operator: extract substructure/underlying set/endowing operator; the substructure (general sense; includes just operator, order, set, etc.) of X1 (structure; explicitly given by {du'a'e}) which is formed by collecting the ith entries of that {du'a'e}-tuple in order together into their own {du'a'e}-tuple (or by extracting them naked into the ambient environment if X2 is a singleton) for all i in set X2
du'ei
mekso binary operator: left-handed vectorial cross product (ordered input), -a \times b = b \times a (if using right-hand convention - notice the negative sign/operator or order).
fa'ai'ai
mekso k-ary operator, for natural k and 1 < k < 5: ordered input (f, g, S, m) where f and g are functions, S is a set of positive integers or "ro" (="all"), and m is 0 or 1 (as a toggle); output is a function equivalent to the function f as applied to an input ordered tuple with g applied to the entries/terms with indices in S (or to all entries/terms if S="ro") if m=0, or g left-composed with the same if m=1.
fa'ei
Unary mekso operator: reverse finite ordered sequence, tuple, list, string, etc.
fau'i
mekso ternary operator: inverse function of input function X1 with respect to its input X2, taken on branch or restricted domain X3 ("domain" being of X1).
fe'au'u (exp!)
mekso ternary operator: positive super-root; the bth super-root (inverse operator of hyper-operator with respect to base) of a of order c-2.
fe'ei
binary mekso operator: divided by (fraction): a/(b...)
fei'i (exp!)
mekso variable-arity (at most ternary) operator: number of prime divisors of number X1, counting with or without multiplicity according to the value X2 (1 xor 0 respectively; see note for equality to -1 and for default value), in structure X3.
fi'au
mekso operator: continued fraction, Kettenbruch notation; for ordered input (X1, X, where: X1 is an ordered pair of functions and X2 is a free or dummy variable/input/index which ranges through set X3 in order(ing) X4, the result is K(X for Kettenbruch notation K.
ga'ai
unary mekso operator: Lorentz-Einstein gamma factor +1/((1-(|X|2)) for input X.
gau'a
mekso (no-more-than-4-ary) operator: Gaussian function f(x, a, b, c) = c e-((x-a).
ga'u'au
mekso n-ary operator: append contravariant (upper) indices to tensor
gei'au (exp!)
mekso 7-ary operator: for input (X1 = z, X2 = (ai)= (bj)= p, X5 = q, X6 = h1, X= h2), this word/function outputs/yields \sumn=0^\infty (((\prodi = 1p (ne'o'o(ai,n,1,h= 1q (ne'o'o(bj,n,1,h; by default, X6 = 1 = X7 unless explicitly specified otherwise.
gei'i'e
mekso unary operator: converts a string of digits which includes {pi} to the same string of digits without {pi}; if {pi} is not present in the original/input string, the output is identical
gu'au'i
mekso operator, variable arity - algebraic structure order of X1; OR: order of/(size of) period of element X1 in algebraic structure X2 under operator/of type X3
je'e'e
mekso operator: associated Legendre polynomial in a with unassociated order b and associated order c
ji'e'ai
mekso unary operator: determinant, det(A)
ji'i'u
mekso, at-most-5-ary operator: a rounding function; ordered input list is (x,n,t,m,b) and the output is sgn(x) bt roundn (b(-t) abs(x)), with rounding preference n and where the fractional part of b(-t) abs(x) being equal to 1/2 causes the roundn ( ) function to map b(-t) abs(x) to the nearest integer of form 2Z+m, for base b (determined by context if not explicitly input) and some integer Z (determined by context).
ji'i'u'u
mekso, at-most-5-ary; rounding function. (See notes).
jo'ei'i (exp!)
nonlogical connective (and mekso operator) - symmetric difference of sets
joi'i
mekso string operator (n-ary): formal right-concatenation; X1 + X, where Xi is a string/word/text/character/letteral/lerfu/quoted utterance (quote appropriately iff necessary; preserve and be careful about the use-vs.-mention distinction) for all i.
ka'au (exp!)
mekso unary operator: cardinality (#, | |)
kei'ai
mekso style converter: elementwise application of operator
kei'au
mekso operator: finite result set derived from/on set A with/due to operator/function B under ordering of application C
kei'i
non-logical connective/mekso operator - of arity only 1 xor 2: set (absolute) complement, or set exclusion (relative complement). Unary: X2 ^C; binary: X1\setminusX.
ku'au'a
mekso (n+1)-ary operator: q-analog converter - the ath analog of b (quoted operator) applied to operands c, d, ...
ku'oi'u
closing bracket/terminator for mekso expression interpretation modifiers
lau'au
mekso unary operator: for input X, this outputs X/(1+X).
li'ei'au
ternary mekso operator: retrieves/gets/outputs the X2th entry/term from ordered list X1 under indexing rules X3.
ma'au
Binary mekso operator: uniform probability A(X2)u(X for input (X1,X where X1 is a number and X2 is a set or space. (See notes for details).
mai'u
unary mekso operator: signum function
mau'au
mekso: conversion of operator/function to operand
me'ei'o
mekso n-ary operator: interleave sequences
mu'ai'au
mathematical/logical/mekso ternary operator: μ (mu) operator: outputs the most extreme extended-natural number that satisfies relationship/predicate A, where extremeness is bounded by B and of a version determined by C; error output is -1
mu'au
unary mekso operator: measure of the complement; 1 - x1.
na'au (exp!)
converts an unevaluated mekso expression into a sumti referencing its evaluated result (if sensible/defined)
nei'au
unary mekso operator: (-1)x
ne'o'a (exp!)
mekso ternary operator: the generalized incomplete (factorial-extending) Pi function; for input (X1, X this word outputs the definite integral of t^X1 e^-t with respect to t from X2 to X3 (see notes for default values).
ne'o'au (exp!)
mekso quaternary operator: polygamma function; for input X1, X, outputs the (-X2)th derivative of Log(ne'o'a(X1, X)) with respect to X1.
ne'o'o (exp!)
mekso quaternary operator – Pochhammer symbol: with/for input (X1, X, this word/function outputs \prodk = 0^X2 - 1 (X; by default, X4 = 1 unless explicitly defined otherwise.
ni'a'au
mekso n-ary operator: append covariant (lower) indices to tensor
no'au'au
mathematical/mekso binary operator: the zero/identity-element/(primitive (-))constant operator; outputs the identity-element of structure A (contextless default: the additive group of integers) regardless of the input value of B (except blank or ill-defined values)
pau'a'u
mekso operator: part of number/projection (one sense); the X2 part of X1
pau'au (exp!)
ternary mekso operator: p-adic valuation; outputs (positive) infinity if x1 = 0 and, else, outputs sup(Set(k: k is a nonnegative integer, and ((1 - x3)x divides x1)), where pn is the nth prime (such that p1 = 2).
pau'ei
mekso operator: power set - produces the set of all subsets of set X1 that are of (any) size (that is) X2 [a nonnegative integer or transfinite/infinite number; default: su'o no].
pei'e'a
at-most-3-ary mekso operator: "integer exponent" for X1 divided by X2 in algebraic structure X3
pi'au'e
mekso ternary operator: extract digit from number; X2nd macrodigit/term of number/tuple X1 when X1 is expressed in base/basis X3.
pi'ei
mathematical/mekso binary operator: vector or function inner product over a field; the inner product of A and B over field C
pi'u'e
mekso n-ary operator: generate ordered tuple/list from inputs; pi'u'e(x1, x= (x1, x, pi'u'e(x1, x= (x1, x, etc.
po'i'ei (exp!)
n-ary mekso operator: for an input of ordered list of ordered pairs ((X1, Y, it outputs formal generalized rational function (x - X1)^Y in the adjoined indeterminate (here: x).
po'i'oi
mekso at-most-3-ary operator: convert to polynomial; X1 (ordered list of algebraic structure (probably field) elements) forms the (ordered list of) coefficients of a polynomial/Laurent-like series with respect to indeterminate X2 under ordering rule X3 (default for finite list: the first entry is the coefficient of the highest-degree term and each subsequent entry is the next lesser-degree coefficient via counting by ones and wherein the last entry is the constant term)
pu'e'ei (exp!)
mekso binary operator: generate span; outputs span(X1, X= spanX; set of all (finite) sums of terms of form c v, where v is an element of algebraic structure X1 (wherein scalar multiplication and summation is defined), and c is a scalar belonging to ring X2.
ra'i'e
ternary mekso/mathematical operator: radical; for input (x,y,z), it outputs the largest y-th-power-free product of prime divisors of x in structure (ring) z.
rai'i
mekso (2 or 3)-ary operator: maximum/minimum/extreme element; ordered list of extreme elements of the set underlying ordered set/structure X1 in direction X2 of list length X3 (default: 1)
sai'ei (exp!)
turns PA into CAI; intensity attitude modifier expressed by a mekso.
sau'au (exp!)
mekso binary operator: del, nabla, quabla, partial-derivative vector/tensor operator; outputs the functional-valued formal-covector (or analog thereof) of partial derivatives with respect to X1 (tensor in the same format and order), each degree X2 (default: 1).
sau'i
mekso n-ary operator: reciprocal of the sum of the reciprocal of each of X1, X2, ..., Xn (for any natural number n); 1/((1/X1) + (1/X.
si'oi'e
n-ary mekso operator: Logistical growth/cumulative function, sigmoid function; (X3 / (1 + e^(-X.
su'i'e
su'i'o
mekso unary or binary operator: ordered inputs (n, b) where n and b are nonnegative integers and b > 1; output is the ultimate digital root of n in base-b.
tai'e'i
mekso unary operator: basic Schlafli symbol composer (defined only on ordered lists)
tai'i'au (exp!)
8-ary mekso operator: the X1th nonnegative sum of X2 mutually-distinct perfect X3th-powers (i.e.: of integers) in X4 mutually truly-distinct ways, requiring exactly X5 terms to be negative in each sum (counting with(out^X6) multiplicity), requiring exactly X7 terms to be repeated between sums (counting with(out^X8) multiplicity), according to the usual ordering of the integers.
tai'i'e
mekso unary operator: Kleene star - X1*
te'au'u
mekso ternary operator: Knuth up-arrow notation: a \uparrow \dots \uparrow b of order/with c-2 arrows ("\uparrow") initially, evaluated from right to left; the cth hyperoperator on a by b.
te'i'ai
6-ary mekso/mathematical operator: Heaviside function/step/Theta function of a, of order b, in structure c, using distribution d, within approximated limit e, with value f_b at 0
tei'au
4-ary mekso operator: Taylor expansion/polynomial term; for ordered input (X1, X, output is the X3th Taylor polynomial term of at-least-X3-smooth function X2 which was expanded around point X4 and which is evaluated at point X1, namely (1/(X3!)) × (D^X.
te'o'a
unary mekso operator: natural exponentiation operator exp, where exp(a) = ea \forall a.
te'oi'i
mekso ordered/non-commutative n-ary operator: tensor product/exterior product (of tensors); letting "@" denote the tensor product, A1 @ A2 @...@ An .
te'oi'oi
terminator, mekso: terminates the listing of an ordered sequence of indices for a tensor
va'au'au (exp!)
Binary mekso operator: group-theoretic conjugation (group action): maps inputs (X1, X to X2^(-X= \phi(X. Default: X3 = 1.
vei'e
mekso binary operator – quotient from integer-division: sgn(X1) sgn(X2) ((abs(X1) - (abs(X1) \% X.
vei'o (exp!)
binary mekso operator: form quotient space X1/X.
vei'u
binary mekso operator: mod(ulus)/remainder; X1 \% X2, \,\,\, X1 (mod(X2)).
vi'oi'au
mekso unary operator: the set of all fixed points of function a
xa'ai
mekso operator: the bth branch of the (possibly multivalued) function a
xa'ei'o
binary mekso operator: Let the inputs X1 and X2 be sets in the same universal set O; then the result of this operator applied to them is X1^c \cup X, where for any A \subseteq O, Ac = O \setminus A.
xa'ei'u
binary mekso operator: Let the inputs X1 and X2 be sets in the same universal set O; then the result of this operator applied to them is X1^c \cap X, where for any A \subseteq O, Ac = O \setminus A.
xau'e'o
mekso convention default specification/definition (explicit)
xau'o'o
mekso convention cancellation
xe'au
mekso clausal referent bracket initializer
xi'ei
binary mekso operator: for ordered list X1, this word outputs the same ordered list except the indices/subscripts have been relabelled/redefined/reindexed according to rule X2 (see notes).
xo'ei
unary mekso operator: produces a string of n consecutive "xo'e"'s, treated as digits (concatenated into a single string of digits)
xo'e'o'ei
At-most-unary mekso operator: like {xo'ei} but for selma'o XOhEhOhE, rather than just PA
xoi'u (exp!)
non-logical connective (mekso set operator): regardless
za'ei
mekso binary operator: right-handed vectorial cross product (ordered input), a×b
ze'ai'au
unary mekso operator: reverse ordered list/tuple X1.
zei'i'au (exp!)
unary mekso operator: (analytically continued) Riemann zeta function zeta(z), for complex-valued input z.
zoi'ai
non-mekso quote/name substitution for ordered collection of prescriptions, descriptions, definitions, etc.
zu'oi
mekso; binary operator: z-score for the X1 quantile; X2 (default: 1) acts as the descriptor toggle (see notes).
cia'o'e
mekso 4-nary operator: spherical harmonics on colatitudinal/polar angle a and azimuthal/longitudinal angle b of unassociated order c and associated order d.
xua'ai
mekso operator: the bth branch of the (possibly multivalued) function a

## In notes:

cmaci
x1 is a mathematics of type/describing x2.
dilcu
x1 is a quotient of 'x2/x3' [dividend x2 divided by divisor x3], leaving remainder x4.
fancu
x1 is a function/single-valued mapping from domain x2 to range x3 defined by expression/rule x4.
frinu
x1 is a fraction, with numerator x2, denominator x3 (x2/x3).
jalge
x1 (action/event/state) is a result/outcome/conclusion of antecedent x2 (event/state/process).
namcu
x1 (li) is a number/quantifier/digit/value/figure (noun); refers to the value and not the symbol.
parbi
x1 (me'o, fraction) is a ratio/rate of x2 (quantity) with respect to x3 (quantity), [x2:x3].
pilji
x1 is the product/total/result of factors/multiplicands (x2 and x3) x2 multiplied by x3.
brima'o
x1=c1 (zo) is a cmavo with a predicate role or semantic structure which expresses predicate x2=b1 (quote/du'u) between arguments x3=b3 (ordered list), the said cmavo being of form or in category/class x4=c2 in/and belonging to language/dialect x5=c3.
sapna'u
x1 is a scalar in structure/set x2 with properties (potentially including magnitude, etc. in a given metric and coordinate system) x3; x1 is a simple number; x1 is a number that lacks
kurti
x1 (proposition) is data/information being an example of deep formalization of subject x2 gathered by method x3 (proposition)
bu'o'e (exp!)
elliptical/unspecified/vague letteral/symbol
cu'oi'e (exp!)
convert number to statistical odds selbri; event x1 (nu) has statistical odds (n) of occurring (versus not occuring) under conditions x2.
ju'au
semi-mathematical binary operator: named number base operator/interpreter
la'e'au
the specific referent of [following sumti] defined/specified by the grammar
ni'ai
x1 is a number/value such that the abstraction is true, under mathematical system x2; x1 binds to ke'a within the abstraction
noi'a'u
PA nonrestrictive/incidental relative clause; attaches to a PA number/numeral/digit with the ke'a referring to that PA number/numeral/digit.
poi'a'u
PA restrictive relative clause; attaches to a PA number/numeral/digit with the ke'a referring to that PA number/numeral/digit.
ra'ei (exp!)
semi-discursive: and so forth, and so on, et cetera, continuing similarly
xau'e
symbol string to number/variable
xau'o (exp!)
text to number/variable
xo'au
pro-numeral: the most-recently mentioned full/complete numerical or mathematical string/expression.
zau'e'u
text affirmation/negation mode toggle
zoi'o'e
elliptical/unspecified/vague string/text/word
cpolinomi'a
x1 is a formal polynomial with coefficients x2 (ordered list) of degree x3 (li; nonnegative integer) over structure/ring x4 (to which coefficients x2 all belong) and in indeterminant x5.
enfa
x1 abstractly pertains to an exponential/root/logarithmic relationship between the elements of x2 (ordered pair) which are related via concrete relationship x3.
fancuvitno
x1 is a fixed point of function x2
kleivmu
x1 (set) is the relative set complement of x2 (set) in/from/with respect to/relative to set x3 (set; default: the relevant universal set).