This paper presents two results in the realm of conformal Kaehler submanifolds. These are conformal immersions of Kaehler manifolds into the standard flat Euclidean space. The proofs are obtained by making a rather strong use of several facts and techniques developed in Chion and Dajczer (Proc Edinb Math Soc 66:810–833, 2023) for the study of isometric immersions of Kaehler manifolds into the standard hyperbolic space.
We prove that a generic function from (bigcap nolimits _{p<1}H^p) has unbounded Taylor coefficients, this applies also to the Taylor coefficients of its derivatives. Results of similar nature are valid when the space (bigcap nolimits _{p<1}H^p) is replaced by (H^p)((0<p<1)) and by localized versions of such spaces. Moreover, we prove that a generic function from (A(mathbb {D})) has Taylor coefficients outside of (ell ^1), this applies also to the Taylor coefficients of its derivatives. Lastly, we prove that a generic function from (bigcap nolimits _{p<1}h^p) has a harmonic conjugate that does not belong to any (h^q(q>0)).
In this paper we present the notion of arithmetic variety for numerical semigroups. We study various aspects related to these varieties such as the smallest arithmetic that contains a set of numerical semigroups and we exhibit the rooted tree associated with an arithmetic variety. This tree is not locally finite; however, if the Frobenius number is fixed, the tree has finitely many nodes and algorithms can be developed. All algorithms provided in this article include their (non-debugged) implementation in GAP.
We raise the question of the realizability of permutation modules in the context of Kahn’s realizability problem for abstract groups and the G-Moore space problem. Specifically, given a finite group G, we consider a collection ({M_i}_{i=1}^n) of finitely generated (mathbb {Z}G)-modules that admit a submodule decomposition on which G acts by permuting the summands. Then we prove the existence of connected finite spaces X that realize each (M_i) as its i-th homology, G as its group of self-homotopy equivalences (mathcal {E}(X)), and the action of G on each (M_i) as the action of (mathcal {E}(X)) on (H_i(X; mathbb {Z})).
A category equivalent to the category of 3-dimensional cobordisms is defined in terms of planar diagrams. The operation of composition in this category is completely described via these diagrams.
In this paper, we carry out in-depth research centering around the Harnack inequality for positive solutions to nonlinear heat equation on Finsler metric measure manifolds with weighted Ricci curvature (textrm{Ric}_{infty }) bounded below. Aim on this topic, we first give a volume comparison theorem of Bishop-Gromov type. Then we prove a weighted Poincaré inequality by using Whitney-type coverings technique and give a local uniform Sobolev inequality. Further, we obtain two mean value inequalities for positive subsolutions and supersolutions of a class of parabolic differential equations. From the mean value inequality, we also derive a local gradient estimate for positive solutions to heat equation. Finally, as the application of the mean value inequalities and weighted Poincaré inequality, we get the desired Harnack inequality for positive solutions to heat equation.
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