Given a real closed polytope P , we first describe the Fourier transform of its indicator function by using iterations of Stokes' theorem. We then use the ensuing Fourier transform formulations, together with the Poisson summation formula, to give a new algorithm to count fractionally-weighted lattice points inside the one-parameter family of all real dilates of P . The combinatorics of the face poset of P plays a central role in the description of the Fourier transform of P . We also obtain a closed form for the codimension-1 coefficient that appears in an expansion of this sum in powers of the real dilation parameter t. This closed form generalizes some known results about the Macdonald solid-angle polynomial, which is the analogous expression traditionally obtained by requiring that t assumes only integer values. Although most of the present methodology applies to all real polytopes, a particularly nice application is to the study of all real dilates of integer (and rational) polytopes.
Following the work of Cano and Díaz, we consider a continuous analog of lattice path enumeration. This allows us to define a continuous version of any discrete object that counts certain types of lattice paths. As an example of this process, we define continuous versions of binomial and multinomial coefficients, and describe some identities and partial differential equations they satisfy. Finally, we illustrate a general method to recover discrete combinatorial quantities from their continuous analogs.
We give a presentation for the group of Artin braids that become trivial when considered over the [Formula: see text]-sphere and explore the relation to Brunnian braids and homotopy groups of spheres.
It is well known that the Collatz Conjecture can be reinterpreted as the Collatz Graph with root vertex 1, asking whether all positive integers are within the tree generated. It is further known that any cycle in the Collatz Graph can be represented as a tuple, given that inputting them into a function outputs an odd positive integer; yet, it is an open question as to whether there exist any tuples not of the form (2, 2, ..., 2), thus disproving the Collatz Conjecture. In this paper, we explore a variant of the Collatz Graph, which allows the 3x+1 operation to be applied to both even and odd integers. We prove an analogous function for this variant, called the Loosened Collatz Function (LCF), and observe various properties of the LCF in relation to tuples and outputs. We then analyse data on the numbers that are in cycles and the length of tuples that represent circuits. We prove a certain underlying unique factorisation monoid structure for tuples to the LCF and provide a geometric interpretation of satisfying tuples in higher dimensions. Research into this variant of the Collatz Graph may provide reason as to why there exist no cycles in the Collatz Graph.
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