aa r X i v : . [ m a t h . G M ] S e p Philosophy of Natural Numbers
Yuri Kondratiev
Department of Mathematics, University of Bielefeld,D-33615 Bielefeld, Germany,Dragomanov University, Kyiv, UkraineEmail: [email protected]
Abstract
We discuss an extension of classical combinatorics theory to the case of spatiallydistributed objects.
Keywords:
Combinatorics, Newton polynomials, Stirling operators, correlation functions
AMS Subject Classification 2010:
The set of natural numbers N = { , , , . . . } is a fundamental object in the mathematics.In certain sense N is the root of all modern mathematics. Other mathematical structuresmay be created as a logical development of this object. The latter motivated L. Kroneckerwho summarized ”God made the integers, all else is the work of man”.There is famouscitation from I.Kant: ”Two things fill the mind: the starry heavens above me and themoral law within me”. A mathematician may continue: ”and natural numbers given tomy mind”.From the time of Pithagoras philosophers was trying to see hidden meaning of naturalnumbers and their mystical properties. Considering N as a set of real things in math-ematics we will ask ourself about possible ideas behind these numbers. The myth ofPlato’s Cave served as one of the motivations for developing his concept of a world ofideas and a world of things. In the dialogue ”State” he gives several examples illustratingthis concept. As we know, Plato considered mathematics as one of the most importantbuilding blocks used to construct his philosophical system. Mathematical theories canserve as simple and illustrative tools for the existence of a world of ideas and a world ofthings. In a number of model situations, we are dealing with objects that appear from ourobservations in physics, biology, ecology, etc., yet full understanding of the mathematicalstructures of these models requires consideration of more general mathematical theories,which under some canonical mapping leads to the model situations in question.The first and essentially obvious observation here is the following. A number n ∈ N we interpret as a number of objects (a population) located in a location space X . Forsimplicity we take X = R d . The collection of all n -point subsets (or configurations with n elements) form a locally compact space Γ ( n ) ( R d ). It is the space (quite huge) of ideasfor the number n . Then to N corresponds the setΓ ( R d ) = ∪ ∞ n =0 Γ ( n ) ( R d )1f all finite configurations. We can consider additionally the set Γ( R d ) consisting all locallyfinite configurations. This set may be considered as the space of ideas which correspondsto natural numbers and additionally to the actual infinity which is absent in the classicalframework on natural numbers.In such extension of N we arrive in the main question. Namely, most importantmathematical theories related to natural numbers we need to develop to this new level.It concerns, first of all, the combinatorics that play central role in many mathematicalstructures and applications from probability theory to genetics. In this note we will tryto show such possibility trying to be as much as possible on technically simple ground.To be friendly to more wide audience, we restrict out explanations to descriptions of mainconstructions and formulation of some particular results. For detailed discussions andextended references we refer to the recent paper [2]. The combinatoric is dealing with the set of natural numbers N and relations betweenthem. As an important object we introduce binomial coefficients: (cid:18) nk (cid:19) = n ( n − . . . ( n − k + 1) k !defined for n ∈ N and 0 ≤ k ≤ n . Introducing the falling factorial ( n ) k we can write (cid:18) nk (cid:19) = ( n ) k k ! . These coefficients may be extended using embedding N ⊂ R to polynomials N k ( t ) := (cid:18) tk (cid:19) = t ( t − . . . ( t − k + 1) k ! = ( t ) k k ! , t ∈ R which are called Newton polynomials. For Newton polynomials hold Chu-Vandermondrelations: ( t + s ) n = n X k =0 (cid:18) nk (cid:19) ( t ) k ( s ) n − k . An alternative definition is given by the generation function e λ ( t ) := e t log(1+ λ ) = ∞ X n =0 λ n n ! ( t ) n = ∞ X n =) λ n N n ( t ) . Such transition to continuous variables makes possible to apply in discrete mathematicsmethods of analysis. Note that using many particular generation functions we may createdifferent polynomial systems.Transition to continuous variables makes possible to apply in discrete mathematicsmethods of analysis. In particular, let us define for functions f : R → R differenceoperators ( D + f )( t ) = f ( t + 1) − f ( t ) , ( D − f )( t ) = f ( t − − f ( t ) .
2y a direct computation we obtain D + ( t ) n = n ( t ) n − ,D − ( t ) n = − n ( t − n − . Additionally, D + e λ ( t ) = λe λ ( t ) . In this way we arrive in the framework of difference calculus closely related with thecombinatorics [3]. There are specific questions inside of difference calculus as, e.g., ananalysis of Newton series ∞ X n =0 a n N n ( t )and many others.For functions a : N → R we define b : N → R as b = Ka, b ( n ) = n X k =0 (cid:18) nk (cid:19) a ( k ) . This operator K (aka combinatorial transform) is very useful in combinatorics and itsinverse gives so-called inclusion-exclusion formula: a ( n ) = n X k =0 (cid:18) nk (cid:19) ( − n − k b ( k ) . Note that for a : N → R , a ( j ) = 0 , j = k, a ( k ) = 1( Ka )( n ) = (cid:18) nk (cid:19) = k ! N k ( n ) . Any n ∈ N we interpret as the size of a population. It is convenient in the study ofpopulation models. There is a natural generalization leading to spatial ecological models.Now we would like to consider objects located in a given locally compact space X . Forsimplicity we will work with the Euclidean space R d . For the substitution of N in this situ-ation we can use two possible sets. Denote Γ( R d ) the set of all locally finite configurations(subsets) from R d .Γ( R d ) = { γ ⊂ R d | | γ ∩ K | < ∞ , any compact K ⊂ R d } . It is the first version of the space in the spatial (continuous) combinatoric we will use.Another possibility, is to introduce the set of all finite configurations Γ ( R d ). ThenΓ ( R d ) = ∪ ∞ n =0 Γ ( n ) ( R d ) , where Γ ( n ) ( R d ) denoted the set of all configurations with n elements. We will see that inthe continuous combinatoric the spaces Γ( R d ) and Γ ( R d ) will play very different roles. Itis a specific moment related with transition to the continuum. In this sense N is splitting3n these two spaces of configurations that makes corresponding combinatorics essentiallymore reach and sophisticated.Configuration spaces present beautiful combinations of discrete and continuous prop-erties. In particular, in these spaces we have interesting differential geometry, differentialoperators and diffusion processes etc., see e.g. [1]. From the other hand side, discretenessof an individual configuration makes possible to introduce proper analog of the differencecalculus.Note from the beginning, that the analog of the extension N ⊂ R now naturally playthe pair Γ( R d ) ⊂ M ( R d ) where we have in mind an imbedding of configurations in thespace of discrete Radon measures on R d and, as a result, in the space of all Radon mesureson R d : Γ( X ) ∋ γ γ ( dx ) = X y ∈ γ δ y ( dx ) ∈ M ( R d ) . Therefore, instead of pair N ⊂ R we have Γ( R d ) ⊂ M ( R d ) . As a result, the transition to ”continuous” variables in the considered situation leads tofunctions on M ( R d ). In spatial combinatorics many objects will be measure-valued.Now we will introduce an analog of the generation function from classical combina-torics. For a test function from the Schwarz space of test functions D ( R d ) 0 ≤ ξ ∈ D ( R d )consider a function E ξ ( ω ) = e < ln(1+ ξ ) ,ω> ω ∈ D ′ ( R d )that is a function on the space of Schwarz distributions. The power decomposition w.r.t. ξ gives E ξ ( ω ) = ∞ X n =0 n ! < ξ ⊗ n , ( ω ) n > . Generalized kernels ( ω ) n ∈ D ′ ( R nd ) are called infinite dimensional falling factorials on D ′ ( R d ). Define binomial coefficients (Newton polynomials) on D ′ ( R d ) as (cid:18) ωn (cid:19) = ( ω ) n n ! . Note that these objects are defined now on the very big space of distributions. In particularcases we shall restrict them on the space of configuration or Radon measures.In particular, infinite dimensional Chu-Vandermond relations on configurations is( γ ∪ γ ) n = n X k =0 (cid:18) nk (cid:19) ( γ ) k ⊗ ( γ ) n − k . Theorem 1.
For ω ∈ M ( R d ) ( ω ) = 1( ω ) = ω ( ω ) n ( x , . . . , x n ) = ω ( x )( ω ( x ) − δ x ( x )) . . . ( ω ( x n ) − δ x ( x n ) − · · · − δ x n − ( x n )) .
4n the particular case ω = γ = { x i | i ∈ N } ( γ ) n = n ! (cid:18) γn (cid:19) = X { i ....,i n }⊂ N δ x ⊙ · · · ⊙ δ x n , where δ x ⊙ · · · ⊙ δ x n denotes symmetric tensor product.We have Γ( R d ) ∋ γ γ ( dx ) ∈ M ( R d ) . Due to our construction ( γ ) n ∈ M ( R nd )is a symmetric Radon masure. Therefore, we arrive in measure valued Newton polynomi-als. The latter is the main consequence of continuous combinatoric transition. For any x ∈ γ define an elementary Markov death operator (death gradient) D − x F ( γ ) = F ( γ \ x ) − F ( γ )and the tangent space T − γ (Γ) = L ( R d , γ ). Then for ψ ∈ C ( R d ) D − ψ F ( γ ) = X x ∈ γ ψ ( x )( F ( γ \ x ) − F ( γ ))is the directional (difference) derivative.Similarly, we define for x ∈ R d D + x F ( γ ) = F ( γ ∪ x ) − F ( γ )and the tangent space T − γ (Γ) = L ( R d , dx ). Then for ϕ ∈ C ( R d ) D + ϕ F ( γ ) = Z R d ϕ ( x )( F ( γ ∪ x ) − F ( γ )) dx is another directional (difference) derivative.For ϕ ∈ C ( R d ) define a function E ϕ ( γ ) = exp( < γ, log(1 + ϕ ) > ) , γ ∈ Γ . It is the generation function for the system on falling factorials (Newton polynomials) onΓ: E ϕ ( γ ) = ∞ X n =0 n ! < ϕ ⊗ n , ( γ ) n > . Then D + ψ E ϕ ( γ ) = < ϕψ > E ϕ ( γ ) . An explicit formula for the falling factorials (as measures on ( R d ) n ) is( γ ) n = X { x ,...,x n }⊂ γ δ x ⊙ δ x ⊙ · · · ⊙ δ x n , δ x ⊙ δ x ⊙ · · · ⊙ δ x n denotes the symmetric tensor product of measures.The action of difference derivatives on Newton monomials is given by D + ψ < ϕ ( n ) , ( γ ) n > = n Z R d ψ ( x ) < ϕ ( n ) ( x, · ) , ( γ ) n − ( · ) > dx,D − ψ < ϕ ( n ) , ( γ ) n > = − n X x ∈ γ ψ ( x ) < ϕ ( n ) ( x, · ) , ( γ \ x ) n − ( · ) > . We have polynomial equality ( γ ) n = n X k =1 s nk γ ⊗ k , where s nk : D ′ ( R kd ) → D ′ ( R nd )is a linear mapping.On other side γ ⊗ n = n X k =0 S nk ( γ ) k , where S nk : D ′ ( R kd ) → D ′ ( R nd )is a linear mapping.Kernels s nk and S nk we will call Stirling kernels of first and second kind respectively. Inthe classical combinatorics Stirling coefficients play a very important role.For f ( n ) ∈ D ( R nd ) < ( γ ) n , f ( n ) > = n X k =0 n ! k ! < γ ⊗ k ( x , . . . , x k ) , X i + ...i k = n ( − n + k i . . . i k f ( n ) ( x , . . . , x | {z } i times , . . . , x k , . . . , x k | {z } i k times ) > . For the second kind kernels < γ ⊗ n , f ( n ) > = n X k =0 k ! < ( γ ) k ( x , . . . , x k ) , X i + ...i k = n (cid:18) ni . . . i k (cid:19) f ( n ) ( x , . . . , x | {z } i times , . . . , x k , . . . , x k | {z } i k times ) > . Γ( R d ) Functions G : Γ ( R d ) → R we call quasi-observables. Note that G restricted on Γ ( n ) ( R d )is given by a symmetric kernel G ( n ) ( x , . . . , x n ) and then G = ( G ( n ) ) ∞ n =0 . F : Γ( R d ) → R we call observables. For a quasi-observable G define anoperator ( KG )( γ ) = X η ⊂ γ, | η | < ∞ G ( η ) , γ ∈ Γ( R d )that is an observable. To be well defined we need certain assumptions about G [4].For G , G : Γ ( R d ) → R define( G ⋆ G )( η ) = X η ∪ η ∪ η = η G ( η ∪ η ) G ( η ∪ η ) . Then K ( G ⋆ G ) = KG KG . Let µ ∈ M (Γ( R d )). K : F un (Γ ) → F un (Γ), K ∗ : M (Γ) → M (Γ ) .K ∗ µ = ρ, ρ = ( ρ ( n ) ) ∞ n =0 . The measure ρ is called correlation measure for µ (Fourier transform of µ ).Assume absolute continuity dρ ( n ) ( x , . . . , x n ) = 1 n ! k ( n ) ( x , . . . , x n ) dx . . . x n . We call k ( n ) ( x , . . . , x n ) , n ∈ N correlation functions of the measure µ .Transition from measures to CFs is one of the main technical aspects of the analysison CS in applications to dynamical problems.Alternatively define the Bogoliubov functional B µ ( φ ) = Z Γ( R d ) e <γ, log(1+ φ> ) dµ ( γ ) . Assuming B µ is holomorphic in φ ∈ L ( R d ) we obtain B µ ( φ ) = ∞ X n =0 n ! Z k ( n ) ( x , . . . , x n ) φ ( x ) . . . φ ( x n ) dx . . . dx n . Having developed combinatorial structures in the continuum, we may consider the inversedirection. Namely, how looks like our infinite-dimensional objects in the one dimensionalreduction. Surprisingly, it may give some new structures even in this classical case.Let a, b : N → R . Define a convolution( a ⋆ b )( n ) = X j + k + l = n a ( j + k ) b ( k + l ) .
7s before ( Ka )( n ) = n X k +0 (cid:18) nk (cid:19) a ( k ) . Then K ( a ⋆ b ) = Ka · Kb.
Introduce coherent states e λ ( · ) : N → C , e λ ( n ) = λ n , λ ∈ C . ( Ke λ )( n ) = (1 + λ ) n . The configuration space Γ( R d ) is the space of microscopic states in the classical sta-tistical physics of continuous systems. A measure µ ∈ M (Γ( R d )) is a macroscopic stateof a continuous system in the statistical physics. Coming back we can interpret (a bitnaively) a measure µ ∈ M ( N ) as a state of 0 D system.For example, the Poisson measure for σ > π σ ( n ) = e − σ σ n n ! . Several characteristics we can incorporate in such a case from the analysis on Γ( R d ).Introduce the Bogoliubov functional: B ( λ ) = Z R + (1 + λ ) x dµ ( x ) . (1 + λ ) x = ∞ X n =0 λ n n ! ( x ) n . Theorem 2.
Let µ ∈ M ( R + ) . Then µ ( N ) = 1 iff B ( λ ) has a holomorphic extension. Similarly we can define correlation measures Z N ( Ka )( x ) dµ ( x ) = Z N a ( x ) dρ µ ( x ) .ρ µ ( n ) = 1 n ! Z N ( x ) n dµ ( x ) = ∞ X m = n (cid:18) mn (cid:19) µ ( m ) . The financial support by the Ministry for Science and Education of Ukraine throughProject 0119U002583 is gratefully acknowledged.8 eferences [1] Albeverio, S., Kondratiev, Y.G., R¨ockner, M.: Analysis and geometry on configura-tion spaces. J. Funct. Anal. , 444–500 (1998)[2] Finkelshtein, D., Kondratiev, Y., Lytvynov, E., Oliveira, M.J, Spatial combinatorics,ArXiv 2007.01175v1, 2020[3] Philippe Flajolet, Robert Sedgewick,
Analytic Combinatorics , Cambridge UniversityPress,, 2009[4] Kondratiev, Y.G., Kuna, T.: Harmonic analysis on configuration space. I. Generaltheory. Infin. Dimens. Anal. Quantum Probab. Relat. Top.5