2019/06/05 by Vassilis G. Papanicolaou, Papanicolaou, Vassilis G., Eva Kallitsi +3 · 1 citation
Mathematics · #32A15 #32W30 #Caloric theory #Combinatorics #Complex Variables (math.CV) #Conjecture #Differential equation #Entire function #FOS: Mathematics #Function (biology) #Heat equation #Holomorphic and Operator Theory #Mathematical Dynamics and Fractals #Mathematical analysis #Mathematical physics #Mathematics #Meromorphic and Entire Functions #Order (exchange) #Partial differential equation #Physics #Pure mathematics #Thermodynamics #Type (biology) #math.CV #msc:32A15 #msc:32W30
paper · pdf · doi:10.48550/arxiv.1906.02233
38 pages
openalex publication_date 2019/06/05 · arxiv created 2019/06/08 · arxiv updated 2019/06/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
Motivated by the recent proof of Newman's conjecture \citeR-T we study certain properties of entire caloric functions, namely solutions of the heat equation ∂t F = ∂z2 F which are entire in z and t. As a prerequisite, we establish some general properties of the order and type of an entire function. Then, we start our inquiry on entire caloric functions by determining the necessary and sufficient condition for a function f(z) to be the initial condition of an entire solutions of the heat equation and, subsequently, we examine the relation of the z-order and z-type of an entire caloric function F(t, z), viewed as function of z, to its t-order and t-type respectively, if it is viewed as function of t. After that, we shift our attention to the zeros zk(t) of an entire caloric function F(t, z), viewed as function of z. We show that the points (t, z) at which F(t, z) = ∂z F(t, z) = 0 form a discrete set in ℂ2 and we derive the t-evolution equations of the zeros of F(t, z). These are differential equations which hold for all but countably many t ∈ ℂ.