This article describes a new method of producing space filling fractal curves based on a hinged t... more This article describes a new method of producing space filling fractal curves based on a hinged tiling procedure. The fractals produced can be generated by a simple L-system. The construction as a hinged tiling has the advantage of automatically implying that the fractiles produced tessellate, and that the Heighway fractal dragon curve, and all the other curves constructed, do not cross themselves. This also gives a new limiting procedure to apply to certain Trouchet tilings. I include the computation of the fractal dimension of one of the curves, and describe an algorithm for computing the sim value of the fractal boundary of these curves. The curves considered have previously been described by [Tab14], but the hinged tiling approach is new, as is the algorithm for computing the sim value. Figure 1. Some Trouchet tiles and examples of their tilings
Let f be a modular form of weight k for a congruence subgroup Γ ⊂ SL 2(Z), and t a weight 0 modul... more Let f be a modular form of weight k for a congruence subgroup Γ ⊂ SL 2(Z), and t a weight 0 modular function for Γ. Assume that near t = 0, we can write f = ∑n≥0bn tn, bn ∈ Z. Let ℓ(z) be a weight k + 2 modular form with q-expansion ∑γnqn, such that the Mellin transform of ℓ can be expressed as an Euler product. Then we show that if [Formula: see text] for some integers ai, di, then the congruence relation bmpr -γpbmpr-1 + εppk+1bmpr-2 ≡ 0 ( mod pr) holds. We give a number of examples of this phenomena.
... These properties are used, for instance, to verify the Bershadsky-Cecotti-Ooguri-Vafa (BCOV) ... more ... These properties are used, for instance, to verify the Bershadsky-Cecotti-Ooguri-Vafa (BCOV) predic-tion for the genus one Gromov ... 1. Marco Aldi (Northwestern University, USA) (Ph.D. student) 2. Gert Almkvist (Lunds University, Sweden) 3. Maiia Bakhova (Louisiana State ...
Modular symbols of weight 2 for a congruence subgroup Γ satisfy the identity {α,γ,(α)}={β,γ(β)} f... more Modular symbols of weight 2 for a congruence subgroup Γ satisfy the identity {α,γ,(α)}={β,γ(β)} for all α,β in the extended upper half plane and γΓ. The analogue of this identity is false for modular symbols of weight greater than 2. This paper provides a definition of transportable modular symbols, which are symbols for which an analogue of the above identity holds, and proves that every cuspidal symbol can be written as a transportable symbol. As a corollary, an algorithm is obtained for computing periods of cuspforms.
This article describes a new method of producing space filling fractal curves based on a hinged t... more This article describes a new method of producing space filling fractal curves based on a hinged tiling procedure. The fractals produced can be generated by a simple L-system. The construction as a hinged tiling has the advantage of automatically implying that the fractiles produced tessellate, and that the Heighway fractal dragon curve, and all the other curves constructed, do not cross themselves. This also gives a new limiting procedure to apply to certain Trouchet tilings. I include the computation of the fractal dimension of one of the curves, and describe an algorithm for computing the sim value of the fractal boundary of these curves. The curves considered have previously been described by [Tab14], but the hinged tiling approach is new, as is the algorithm for computing the sim value. Figure 1. Some Trouchet tiles and examples of their tilings
Let f be a modular form of weight k for a congruence subgroup Γ ⊂ SL 2(Z), and t a weight 0 modul... more Let f be a modular form of weight k for a congruence subgroup Γ ⊂ SL 2(Z), and t a weight 0 modular function for Γ. Assume that near t = 0, we can write f = ∑n≥0bn tn, bn ∈ Z. Let ℓ(z) be a weight k + 2 modular form with q-expansion ∑γnqn, such that the Mellin transform of ℓ can be expressed as an Euler product. Then we show that if [Formula: see text] for some integers ai, di, then the congruence relation bmpr -γpbmpr-1 + εppk+1bmpr-2 ≡ 0 ( mod pr) holds. We give a number of examples of this phenomena.
... These properties are used, for instance, to verify the Bershadsky-Cecotti-Ooguri-Vafa (BCOV) ... more ... These properties are used, for instance, to verify the Bershadsky-Cecotti-Ooguri-Vafa (BCOV) predic-tion for the genus one Gromov ... 1. Marco Aldi (Northwestern University, USA) (Ph.D. student) 2. Gert Almkvist (Lunds University, Sweden) 3. Maiia Bakhova (Louisiana State ...
Modular symbols of weight 2 for a congruence subgroup Γ satisfy the identity {α,γ,(α)}={β,γ(β)} f... more Modular symbols of weight 2 for a congruence subgroup Γ satisfy the identity {α,γ,(α)}={β,γ(β)} for all α,β in the extended upper half plane and γΓ. The analogue of this identity is false for modular symbols of weight greater than 2. This paper provides a definition of transportable modular symbols, which are symbols for which an analogue of the above identity holds, and proves that every cuspidal symbol can be written as a transportable symbol. As a corollary, an algorithm is obtained for computing periods of cuspforms.
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Papers by H Verrill