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Rectilinearization of functions definable by a Weierstrass system and its applications

2010
*
Annales Polonici Mathematici
*

This paper presents

doi:10.4064/ap99-2-2
fatcat:iaenbancnverxklcv2ma3n3gei
*several**theorems*on*the*rectilinearization*of**functions*definable by*a*convergent Weierstrass system, as well as their applications to decomposition into special cubes and quantifier ... N N201 372336 from*the*Polish Ministry*of*Science and Higher Education. ...*Several**theorems**concerning*rectilinearization*of*quasi-subanalytic*functions*are provided in Section 2. ...##
###
Synchronization of Finite Automata: Contributions to an Old Problem
[chapter]

2002
*
Lecture Notes in Computer Science
*

*The*present paper investigates this and related problems within

*the*general framework

*of*

*a*

*composition*theory for

*functions*

*over*

*a*

*finite*domain N with n elements. ... Our results show that

*the*study

*of*

*functions*

*of*

*several*

*variables*may shed light also to

*the*case where all

*functions*considered are unary. ... With

*the*following sections in mind,

*the*next definition is stated for

*functions*with

*several*

*variables*.

*The*

*variables*still

*range*

*over*

*the*

*set*N , and also

*the*

*function*values are in N . ...

##
###
Page 451 of Mathematical Reviews Vol. 26, Issue 3
[page]

1963
*
Mathematical Reviews
*

Salomaa, Arto 2356

*A**theorem**concerning**the**composition**of**functions**of**several**variables**ranging**over**a**finite**set*. J. Symbolic Logic 25 (1960), 203-208 (1962). ...*A**function*g is said to be generated by*a**set*F*of**functions*if g can be expressed as*a**finite**composition**of**functions*in F. ...##
###
Verification of the Completeness of Unification Algorithms à la Robinson
[chapter]

2010
*
Lecture Notes in Computer Science
*

*The*methodology is based on

*a*previously developed formalization

*of*

*the*

*theorem*

*of*existence

*of*most general unifiers for unifiable terms

*over*first-order signatures. ... This work presents

*a*general methodology for verification

*of*

*the*completeness

*of*firstorder unification algorithmsà la Robinson developed in

*the*higher-order proof assistant PVS. ...

*range*terms whose

*variables*also

*range*in this

*set*and that conform

*a*

*set*

*of*

*variables*disjoint from

*the*domain. ...

##
###
Composition sequences for functions over a finite domain

2003
*
Theoretical Computer Science
*

Diverse problems

doi:10.1016/s0304-3975(01)00227-4
fatcat:ysa7sqmxhjepnkphjjh4j6uy3q
*ranging*from many-valued logics to ÿnite automata can be expressed as questions*concerning**compositions**of**functions**over**a*ÿnite domain. ... Many*of**the*very basic problems turn out to be NP-hard. Also*several*open problems are pointed out. ... D C (f) = max(D(F; f)); but now F*ranges**over*complete*sets**of**functions*. ...##
###
The Arity Gap of Polynomial Functions over Bounded Distributive Lattices

2010
*
2010 40th IEEE International Symposium on Multiple-Valued Logic
*

Let

doi:10.1109/ismvl.2010.29
dblp:conf/ismvl/CouceiroL10
fatcat:5ccf3jori5anxhjgadswadrksq
*A*and B be arbitrary*sets*with at least two elements.*The*arity gap*of**a**function*f:*A*^n \to B is*the*minimum decrease in its essential arity when essential arguments*of*f are identified. ... In this paper we study*the*arity gap*of*polynomial*functions**over*bounded distributive lattices and present*a*complete classification*of*such*functions*in terms*of*their arity gap. ... It is easily observed that gap f = min i =j (ess f − ess f i←j ), where i and j*range**over**the**set**of*indices*of*essential*variables**of*f . ...##
###
Page 3680 of Mathematical Reviews Vol. , Issue 89G
[page]

1989
*
Mathematical Reviews
*

*The*

*theorem*

*of*Tsen-Lang says that if F is an algebraically closed field then K is

*a*C,,-field, i.e., any homogeneous polynomial

*of*degree d

*over*K with more than d"

*variables*has

*a*nontriv- ial solution ... Emphasis is on

*the*use

*of*

*the*CRAY-1 and CRAY-2 supercomput- ers as

*a*mighty tool to provide insight based on results obtained “

*several*orders

*of*magnitude” beyond

*the*

*range*

*of*smaller ma- chines. ...

##
###
Learning Read-Constant Polynomials of Constant Degree Modulo Composites

2013
*
Theory of Computing Systems
*

Boolean

doi:10.1007/s00224-013-9488-6
fatcat:fal6fnzso5dcfligtvyswc5f4i
*functions*that have constant degree polynomial representation*over**a*fixed*finite*ring form*a*natural and strict subclass*of**the*complexity class ACC 0 . ...*the*additional constraint that each*variable*in*the*target polynomial appears in*a*constant number*of*monomials. ... Chattopadhyay was at*the*University*of*Toronto when this work was written up, partially supported by*a*Natural Sciences and Engineering Research Council (NSERC) postdoctoral fellowship, an Ontario Ministry ...##
###
Learning Read-Constant Polynomials of Constant Degree Modulo Composites
[chapter]

2011
*
Lecture Notes in Computer Science
*

Boolean

doi:10.1007/978-3-642-20712-9_3
fatcat:zzpp2pg73bbz5hupfozuy6vdgu
*functions*that have constant degree polynomial representation*over**a*fixed*finite*ring form*a*natural and strict subclass*of**the*complexity class ACC 0 . ...*the*additional constraint that each*variable*in*the*target polynomial appears in*a*constant number*of*monomials. ... Chattopadhyay was at*the*University*of*Toronto when this work was written up, partially supported by*a*Natural Sciences and Engineering Research Council (NSERC) postdoctoral fellowship, an Ontario Ministry ...##
###
Page 1029 of Mathematical Reviews Vol. 54, Issue 4
[page]

1977
*
Mathematical Reviews
*

i¢ xja,=B,, j<m

*of*m linear equations in 7*variables*, where*the**a*, and*the*b, are positive ordinals and*the**variables*x,*range**over*ordinals. ... It follows from*the*previous work*of*Benda, Ketonen and Kanamori that under*the*assumption*of**the**theorem*, these*functions*must dominate (on*a*stationary*set*) every*function*from w, into w,. ...##
###
Page 3002 of Mathematical Reviews Vol. , Issue 84h
[page]

1984
*
Mathematical Reviews
*

*The*author shows that

*the*least intermediate logic containing both L( K) and L(M), where K and M

*range*

*over*

*finite*

*sets*

*of*

*finite*trees, is

*of*

*the*form L(N) for some

*finite*

*set*N

*of*

*finite*trees. ...

*finite*

*sets*

*of*

*finite*trees forms

*a*sublattice

*of*

*the*lattice

*of*all intermediate logics. ...

##
###
Alfred Tarski and undecidable theories

1986
*
Journal of Symbolic Logic (JSL)
*

*Over*

*the*ensuing fifty-five years, Tarski put forth

*a*steady stream

*of*

*theorems*

*concerning*decidability, many with far-reaching consequences. ... Just as

*the*work

*of*

*the*1926 seminar reflected Tarski's profound early interest in decidability, so does his last work,

*A*formalization

*of*

*set*theory without

*variables*,

*a*monograph written in collaboration ... in place

*of*x < y and by restricting

*the*

*variables*to

*range*

*over*

*the*

*set*defined by * (x). ...

##
###
The April meeting of the American Mathematical Society at Chicago

1917
*
Bulletin of the American Mathematical Society
*

Let {F[0, 1, 2, • • -, n; i^i*™- •]} (fc = 1, 2, • • •) be

doi:10.1090/s0002-9904-1917-02974-6
fatcat:trnl2jh7hvf5roh6ymz3qkss6y
*a**set**of*iterative*compositions**of**a**function**of*n + 1*variables*(n = 1, 2, 3, • • •),*the**variables*in*the*entire*set*being notationally distinct ... to*a*class §,*of**the*three*variables*p', p" and x, which*range**over**the*classes ty' and ^3" (general classes) and 36 (*a*linear interval) re- 2. ... MCEWEN: "Determination*of**the**functional*relation between one*variable*and each*of**a*number*of*correlated*variables*by successive approximation." (4) Professor W.*A*. ...##
###
Arithmetical Composition and Inversion of Functions Over Classes

1931
*
Transactions of the American Mathematical Society
*

It is noteworthy that

doi:10.2307/1989515
fatcat:7e2v5t3hrfhvbmlvui6asprf3m
*a*comprehensive theory, among whose instances are*several*extensive theories*of*numerical*functions*, many*of*which have themselves*a*high degree*of*generality, should be comprised ... If one or more*of**the*postulates*of**a*field be modified ( §13), contradicted ( §12), or suppressed ( §5), and if*the*altered*set**of*postulates be selfconsistent,*the**set*defines what we shall call*a*variety ...*Composition*as required for*the*purpose outlined in §7 is*concerned*with*functions**of*matric*variables*. ...##
###
Arithmetical composition and inversion of functions over classes

1931
*
Transactions of the American Mathematical Society
*

It is noteworthy that

doi:10.1090/s0002-9947-1931-1501623-x
fatcat:xxdltxrl4bdafbpdwm2j4hosmu
*a*comprehensive theory, among whose instances are*several*extensive theories*of*numerical*functions*, many*of*which have themselves*a*high degree*of*generality, should be comprised ... If one or more*of**the*postulates*of**a*field be modified ( §13), contradicted ( §12), or suppressed ( §5), and if*the*altered*set**of*postulates be selfconsistent,*the**set*defines what we shall call*a*variety ...*Composition*as required for*the*purpose outlined in §7 is*concerned*with*functions**of*matric*variables*. ...
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