# zei'i'au VUhU3 experimental cmavo

unary mekso operator: (analytically continued) Riemann zeta function zeta(z), for complex-valued input z.

## On grammatical class:

bai'i (exp!)
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 (exp!)
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 (exp!)
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 (exp!)
unary mekso operator: nth Bernoulli number Bn of the second kind (B1 = +1/2 = >0).
boi'ai (exp!)
unary mekso operator: immediately convert number into a single digit.
ca'ei'a (exp!)
Unary mekso operator: unit vector/normalization of the argument (vector) X.
ca'o'e (exp!)
mekso 4-nary operator: spherical harmonics on colatitudinal/polar angle a and azimuthal/longitudinal angle b of unassociated order c and associated order d.
ca'oi (exp!)
binary operator: complex number from argument and phase, (r, \phi) \mapsto r ei \phi
ci'ai'u (exp!)
unary mex operator: n-set; maps a nonnegative integer 'a' to the set \1, \dots ,a\ (the intersection of the set of all natural numbers with the closed ordered interval [1,a] such that a geq 1).
ci'au'i (exp!)
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 (exp!)
mekso operator (binary): projection function; the Bth term/entry ("element") of tuple A
cu'au'ei (exp!)
mekso binary/unary operator: multinomial coefficient/binomial coefficient/choose
da'a'au (exp!)
mekso operatory: prime mark append
dei'au'o (exp!)
mekso binary operator: Lambert product-log W function; W(a, b)
di'ei'o'au (exp!)
mathematical ternary operator: Dirichlet convolution (a×b)(c)
du'a'e (exp!)
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 (exp!)
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 (exp!)
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 (exp!)
mathematical ordered n-ary operator: (pointwise) functional left composition; X1 \circ X.
fa'ai'ai (exp!)
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'au (exp!)
mathematical unary operator: map notation
fau'i (exp!)
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'ei (exp!)
binary mekso operator: divided by (fraction): a/(b...)
ga'ai (exp!)
unary mekso operator: Lorentz-Einstein gamma factor +1/((1-(|X|2)) for input X.
gau'a (exp!)
mekso (no-more-than-4-ary) operator: Gaussian function f(x, a, b, c) = c e-((x-a).
gu'au'i (exp!)
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 (exp!)
mekso operator: associated Legendre polynomial in a with unassociated order b and associated order c
ji'e'ai (exp!)
mekso unary operator: determinant, det(A)
ji'i'u (exp!)
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 (exp!)
mekso, at-most-5-ary; rounding function. (See notes).
joi'i (exp!)
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.
ku'au'a (exp!)
mekso (n+1)-ary operator: q-analog converter - the ath analog of b (quoted operator) applied to operands c, d, ...
lau'au (exp!)
mekso unary operator: for input X, this outputs X/(1+X).
ma'au (exp!)
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 (exp!)
unary mekso operator: signum function
ma'o'e (exp!)
mathematical operator: vague/elliptical/general/generic operator
me'ei'o (exp!)
mekso n-ary operator: interleave sequences
mu'ai'au (exp!)
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 (exp!)
unary mekso operator: measure of the complement; 1 - x1.
nei'au (exp!)
unary mekso operator: (-1)x
no'au'au (exp!)
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 (exp!)
mekso operator: part of number/projection (one sense); the X2 part of X1
pau'ei (exp!)
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 (exp!)
at-most-3-ary mekso operator: "integer exponent" for X1 divided by X2 in algebraic structure X3
pi'au'e (exp!)
mekso ternary operator: extract digit from number; X2nd macrodigit/term of number/tuple X1 when X1 is expressed in base/basis X3.
pi'ei (exp!)
mathematical/mekso binary operator: vector or function inner product over a field; the inner product of A and B over field C
pi'ei'au (exp!)
mathematical ternary operator: not-greater-prime-counting function
pi'u'e (exp!)
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'oi (exp!)
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)
ra'i'e (exp!)
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 (exp!)
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)
sau'i (exp!)
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.
se'au (exp!)
mathematical quinary operator; big operator: left sequence notation/converter - operator a, sequence b defined as a function on index/argument/variable/parameter c, in set d, under ordering e
se'i'a'o (exp!)
unary mathematical operator: identity function id(a) = a
si'oi'e (exp!)
n-ary mekso operator: Logistical growth/cumulative function, sigmoid function; (X3 / (1 + e^(-X.
su'i'e (exp!)
su'i'o (exp!)
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'i'e (exp!)
mekso unary operator: Kleene star - X1*
te'au'u (exp!)
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 (exp!)
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 (exp!)
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 (exp!)
unary mekso operator: natural exponentiation operator exp, where exp(a) = ea \forall a.
te'oi'i (exp!)
mekso ordered/non-commutative n-ary operator: tensor product/exterior product (of tensors); letting "@" denote the tensor product, A1 @ A2 @...@ An .
to'ei'au (exp!)
binary mathematical operator: Jordan totient function Ja(b)
vau'i (exp!)
mekso n-ary operator: reciprocal of the successive ordered-difference of the reciprocal of each of x1, x; id est output is ((\dots ((x1^-1 - x
vi'oi'au (exp!)
mekso unary operator: the set of all fixed points of function a
vo'au'u (exp!)
quaternary mathematical operator: (left) convolution (a★b)(c) in structure d
xa'ai (exp!)
mekso operator: the bth branch of the (possibly multivalued) function a
xo'ei (exp!)
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 (exp!)
At-most-unary mekso operator: like {xo'ei} but for selma'o XOhEhOhE, rather than just PA
za'ei (exp!)
mekso binary operator: right-handed vectorial cross product (ordered input), a×b
zi'a'o (exp!)
mathematical operator: the empty/null [one sense]/trivial [one sense]/blank operator
zu'oi (exp!)
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
cu'a
unary mathematical operator: absolute value/norm |a|.
de'o
binary mathematical operator: logarithm; [log/ln a to base b]; default base 10 or e.
fa'i
unary mathematical operator: reciprocal; multiplicative inverse; [1/a].
fe'a
binary mathematical operator: nth root of; inverse power [a to the 1/b power].
fe'i
n-ary mathematical operator: divided by; division operator; [(((a / b) / c) / ...)].
fu'u
n-ary mathematical operator: elliptical/unspecified mathematical expression (mex) operator.
ge'a
mathematical operator: null mathematical expression (mex) operator (used in >2-ary ops).
gei
trinary mathematical operator: order of magnitude/value/base; [b * (c to the a power)].
ju'u
binary mathematical operator: number base; [a interpreted in the base b].
ne'o
unary mathematical operator: factorial; a!.
pa'i
binary mathematical operator: ratio; [the ratio of a to b].
pi'a
n-ary mathematical operator: operands are vectors to be treated as matrix rows.
pi'i
n-ary mathematical operator: times; multiplication operator; [(((a * b) * c) * ...)].
re'a
unary mathematical operator: matrix transpose/dual; AT.
ri'o
trinary mathematical operator: [integral of a with respect to b over range c].
sa'i
n-ary mathematical operator: operands are vectors to be treated as matrix columns.
sa'o
trinary mathematical operator: [derivative of a with respect to b of degree c].
si'i
trinary mathematical operator: [sigma summation of a using variable b over range c].
su'i
n-ary mathematical operator: plus; addition operator; [(((a + b) + c) + ...)].
te'a
binary mathematical operator: to the power; exponential; [a to the b power].
va'a
unary mathematical operator: additive inverse; [- a].
vu'u
n-ary mathematical operator: minus; subtraction operator; [(((a - b) - c) - ...)].
su'ifa'uvu'u (comp!)
mekso operator: plus or minus with order important, (((a±b)±c)±...±z)
su'ijavu'u (comp!)
mekso operator: plus or minus, (((a±b)±c)±...±z)
bai'ei (exp!)
unary mathematical operator: successor/augment/increment (by one), succ(a) = a++ = a+1
cu'ai (exp!)
binary mathematical operator: vector norm/magnitude of vector a in structure (normed vector space) b.
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.
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.
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.
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.
gu'ai (exp!)
binary operator: left group action g.x
jau'au (exp!)
unary mathematical operator: length/number of components/terms of/in object/array/formal string/sequence/word/text in some alphabet/base/basis which includes each digit; number of digits/components/entries
ka'au (exp!)
mekso unary operator: cardinality (#, | |)
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'oi (exp!)
unary operator: primorial a#
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.
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'oi (exp!)
unary mathematical operator: predecessor/diminish/decrement (by one), \operatornamepred(a) = a-- = a-1
pi'ei'oi (exp!)
mathematical ternary operator: prime-generating function.
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).
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.
ru'ei (exp!)
n-ary operator: n-ary magma/group/ring operator a*b = ab`
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.
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 (exp!)
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 (exp!)
binary mekso operator: mod(ulus)/remainder; X1 \% X2, \,\,\, X1 (mod(X2)).
zei'i'au (exp!)
unary mekso operator: (analytically continued) Riemann zeta function zeta(z), for complex-valued input z.

## In definition:

cmavrvu'u
x1 is the selma'o "VUhU".