Brandon Williams
We study the theta decomposition of Jacobi forms of nonintegral lattice index for a representation that arises in the theory of Weil representations associated to even lattices, and suggest possible applications.
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Jacobi forms of theta type
设$L$是正定偶格.引入θ型Jacobi形式，构造了具有特殊易拉回结构的三个塔型Jacobi形式。我们使用θ型Jacobi形式来解释存在的尖点形式的重量24相对于不可约根格$D_{4}$，这是基于额外的对称性的Coxeter图。
On the integral of the fourth Jacobi theta function
我们将Raabe公式推广到$q$ -loggamma函数。由此得到第四Jacobi θ函数在其最小虚零点之间的对数的积分与分拆函数和黎曼ζ函数有关。
A Jacobi theta series and its transformation laws
我们考虑了Jacobi θ级数的推广，并证明了每一个这样的函数是一个拟Jacobi形式。在一定条件下，我们建立了这些函数关于Jacobi群的变换律，并证明了这些函数是Jacobi型。在建立这些结果，我们构造其他功能，也是雅可比形式。这些结果是由顶点算子代数理论中的应用所激发的。
Harmonic Maass-Jacobi forms with singularities and a theta-like decomposition
实解析Jacobi型在数学和物理的各个领域都起着重要的作用，但一直缺乏一个令人满意的理论。在本文中，我们填补了这一空白，通过引入一个空间的调和马斯-雅可比形式的奇异性，其中包括真正的分析雅可比形式从Zwegers的博士论文。我们提供了几个结构的结果，这样的Jacobi形式的空间，我们采用Zwegers的$\widehatμ$ -功能，建立一个θ-样分解。
We evaluate some finite and infinite sums involving $q$-trigonometric and $q$-digamma functions. Upon letting $q$ approach $1$, one obtains corresponding sums for the classical trigonometric and the digamma functions. Our key argument is a theta product formula of Jacobi and Gosper's $q$-trigonometric identities.
Short proofs for $q$-Raabe formula and integrals for Jacobi theta functions
我们将回答一个问题MezIgnon的$q$ -模拟Raabe的积分公式为$0<q<1$，我们将评估一个积分涉及第一个θ函数。此外，我们将重现一些身份的MezIgn短的证明。
In this paper, we consider $L$-functions of modular forms of weight 3, which are products of the Jacobi theta series, and express their special values at $s=3$, $4$ in terms of special values of Kampé de Fériet hypergeometric functions. Moreover, via $L$-values, we give some relations between special values of Kampé de Fériet hypergeometric functions and generalized hypergeometric functions.
H-Harmonic Maass-Jacobi Forms of Degree 1: The Analytic Theory of Some Indefinite Theta Series
在以前的工作中，Zwegers（和Zagier）定义的一元$\widehatμ$ -函数和他的签名格$(r\!+\!1,1)$上的不定θ级数都是海森堡调和的Maaesthet-Jacobi形式。本文将Heisenberg调和性的概念推广到任意多个椭圆变量的Maaubrill-Jacobi形式，得到了符号为$(r\!+\!s,s)$的非退化格的“乘积型”的不定θ级数.因此，我们得到一个干净的推广$\widehatμ$这些负定格。通过对Heisenberg调和Maaesthet-Jacobi型扭点的限制，得到了在Zagier和Zwegers意义下的高阶调和弱Maaesthet-Jacobi型。特别是，我们解释了一些模块化完成，所谓的退化不定θ系列的背景下，更高的深度混合模拟模形式。本文发展的Heisenberg调和Maaesthet-Jacobi型的结构理论也解释了Zwegers二元$\widehatμ$ -函数奇异分裂为亚纯Jacobi型和一元Maaesthet-Jacobi型之和的现象。
Periods of modular forms on $Γ_0(N)$ and products of Jacobi theta functions
推广了\cite{Z1991}关于第一层模形式的一个结果，给出了$Γ_0(N)$上所有Hecke本征形乘以其二元奇周期多项式之和的闭公式，该公式是任意无平方层$N$的Jacobi θ级数的一个乘积.我们还表明，对于$N=2$，~3和~5这个公式完全确定的所有Hecke特征形的所有权$Γ_0(N)$的傅立叶展开。
Optimal Mock Jacobi Theta Functions
我们对权为1的有理系数的最优拟Jacobi形式进行了分类。它们所跨越的空间是34维的，并且允许一个由双曲平面的等距亏格零群参数化的特殊基。我们表明，他们的傅里叶系数可以明确地表示在奇异模量，并获得积极的条件，区分最佳的模拟雅可比形式出现在阴影月光。我们发现所有的Ramanujan模拟θ函数都可以简单地表示为有理系数的最优模拟Jacobi形式。
We study Taylor expansions of Jacobi forms of lattice index. As the main result, we give an embedding from certain space of such forms, whether scalar-valued or vector-valued, integral-weight or half-integral-weight, of any level, with any character, into a product of finitely many spaces of modular forms. As an application, we investigate linear relations among Jacobi theta series of lattice index. Many linear relations among the second powers of such theta series associated with the $D_4$ lattice and $A_3$ lattice are obtained, along with relations among the third powers of series associated…
We give a new proof for a product formula of Jacobi which turns out to be equivalent to a $q$-trigonometric product which was stated without proof by Gosper. We apply this formula to derive a $q$-analogue for the Gauss multiplication formula for the gamma function. Furthermore, we give explicit formulas for short products of $q$-gamma functions.
Number Field Analogue of Jacobi Theta Relation And Zeros of Dedekind zeta function on Re$(s)=1/2$
本文建立Jacobi theta关系的数域类比，并据此证明任意数域的Dedekind zeta函数在临界线Re$(s)=1/2$上均有无穷多个非平凡零点。
We prove that a non--zero Jacobi form of arbitrary level $N$ and square--free index $m_1m_2$ with $m_1|N$ and $(N,m_2)=1$ has a non--zero theta component $h_μ$ with either $(μ,2m_1m_2)=1$ or $(μ,2m_1m_2)\nmid 2m_2$. As an application, we prove that a non--zero Siegel cusp form $F$ of degree $2$ and an odd level $N$ in the Atkin--Lehner type newspace is determined by fundamental Fourier coefficients up to a divisor of $N$.
We study the Taylor expansion around the point $x=1$ of a classical modular form, the Jacobi theta constant $θ_3$. This leads naturally to a new sequence $(d(n))_{n=0}^\infty=1,1,-1,51,849,-26199,\ldots$ of integers, which arise as the Taylor coefficients in the expansion of a related "centered" version of $θ_3$. We prove several results about the numbers $d(n)$ and conjecture that they satisfy the congruence $d(n)\equiv (-1)^{n-1}\ (\textrm{mod }5)$ and other similar congruence relations.
Recently, Romik determined in [9] the Taylor expansion of the Jacobi theta constant θ_3, around the point x = 1. He discovered a new integer sequence, (d(n))_0^\infty=1, 1, -1, 51, 849, -26199, \dots, from which the Taylor coefficients are built, and conjectured that the numbers d(n) satisfy certain congruences modulo various primes. In this paper, we prove some of these conjectures, for example that d(n)\equiv (-1)^{n+1}(mod 5) for all n\geq 1,and that for any prime p\equiv 3 (mod 4), d(n) vanishes modulo p for all large enough n.
The Petersson norm of the Jacobi theta function
我们计算的Jacobi θ函数的Petersson范数，通过一个简单的应用程序的Rankin-Selberg方法和一个小技巧。
We define theta blocks as products of Jacobi theta functions divided by powers of the Dedekind eta-function and show that they give a powerful new method to construct Jacobi forms and Siegel modular forms, with applications also in lattice theory and algebraic geometry. One of the central questions is when a theta block defines a Jacobi form. It turns out that this seemingly simple question is connected to various deep problems in different fields ranging from Fourier analysis over infinite-dimensional Lie algebras to the theory of moduli spaces in algebraic geometry. We give several answers t…
Recently, Funke and Hofmann constructed a singular theta lift of Borcherds type for the dual reductive pair $U(1,1)\times U(p,q)$, $p,q\geq 1$, the input functions of which are harmonic weak Maass forms of weight $k= 2-p-q$. In the present paper, we give an explicit evaluation of the Fourier-Jacobi expansion of the lift. For this purpose, we adapt a method introduced by Kudla in his paper 'Another product for a Borcherds form'. As an application, in the case $U(p,1)$ we recover a new infinite product expansion associated to a Borcherds form, analogous to the case $O(p,2)$ treated by Kudla.
A theory of theta functions to the quintic base
四个五次θ函数的性质与经典的Jacobi零θ函数的性质并行发展。证明了五次θ函数满足Jacobi的四次θ函数恒等式的类似物和Jacobi的符号公式的重复、减少和改变原理的对应物。由此产生的五次变换公式库是用来描述系列多节的模形式在简单的矩阵运算。这些努力最终在一个正式的技术推导出的各种组合生成函数，包括分拆函数的同余模5的权力。进一步分析的五次θ函数进行探索其模块化的性质和他们的连接到爱森斯坦系列。由此产生的关系导致五次θ函数的微分方程的耦合系统。
Functional Equation for Theta Series
本文给出了Siegel-Jacobi空间上的θ级数在Jacobi模群作用下的变换公式。这个公式推广了Mumford在他的书[第189页，Tata Lectures on Theta I（1983），Birkhauser]中得到的公式（5.1）。
Quasi-Jacobi forms, Appell-Lerch functions, and false theta functions as q-brackets of functions on partitions
研究分拆上类theta函数代数，其生成函数涉及theta函数、拟Jacobi形式、Appell-Lerch和及假theta函数。
Modular forms of orthogonal type and Jacobi theta-series
本文考虑任意正定格L的半整指标的Jacobi形式（Eichler和Zagier书中的经典Jacobi形式对应于格A_1=<2>）。我们用Jacobi θ-级数给出了根系奇异权和临界权的Jacobi形式的许多例子。我们描述了Jacobi形式的半整指数的Jacobi提升。在某些情况下，给出了新的反射模形式的加法提升构造。
Romik's Conjecture for the Jacobi Theta Function
该论文针对Romik关于Jacobi theta函数Taylor系数模素数周期性的猜想，在p≡1 mod 4情形下给出简洁代数关系，并推导出有效周期。
Orbits of Jacobi forms and Theta relations
本文提出Jacobi形式轨道的新方法，用于生成Jacobi theta函数恒等式，简化已知theta关系证明并推广Riemann theta关系。
On the Coble quartic and Fourier-Jacobi expansion of theta relations
Q. Ren，S. Sam，G. Schrader，和B. Sturmfels（arXiv：1208.1229），在3属情况下的通用Kummer簇的简化方程。虽然，这些方程中的大多数是从亏格4中θ常数之间的关系的傅里叶-雅可比展开获得的，但更突出的是，Coble的四次曲线是以不同的方式获得的。本文的目的是表明，Coble的四次曲线可以作为亏格4中θ-常数之间关系的Fourier-Jacobi展开式得到。我们还得到一个关系，可以在理想中描述的“普遍库默尔三重”。
In this paper, we study a family of rank two false theta series associated to the root lattice of type $A_2$. We show that these functions appear as Fourier coefficients of a meromorphic Jacobi form of negative definite matrix index. Hypergeometric $q$-series identities are also obtained.
Algebraic independence results for values of Jacobi theta-constants
令$θ_3(τ)=1+2\sum_{ν=1}^{\infty} q^{ν^2}$与$q=e^{iπτ}$和$\Im (τ)>0$表示Jacobi θ函数\[θ(z|τ) \,=\,\sum_{ν=-\infty}^{\infty} e^{πiν^2τ+ 2πiνz} \,.\]的θ nullwert。此外，令$θ_2(τ)=2\sum_{ν=0}^{\infty} q^{{(ν+1/2)}^2}$和$θ_4(τ)=1+2\sum_{ν=1}^{\infty} {(-1)}^νq^{ν^2}$。对于任何偶数$n\geq 6$，这是不是一个权力的2，我们建设性地证明了存在一个非平凡的整数多项式$Q_n(X,Y)$，使\[Q_n\Big( \,\frac{θ_3^4(nτ)}{θ_3^4(τ)},\frac{θ_2^4(τ)}{θ_3^4(τ)}\, \Big) \,=\, 0 \]成立的所有复数$τ$从上半平面的$\mathbb{C}$。这些多项式用于证明$θ_3(nτ)$和$θ_3(τ)$对所有代数数$q=e^{iπτ}$和$0<|q|<1$的代数独立性。结合作者以前的结果，证明了对于这样的代数$q$，对每个整数$n\geq 2$，数$θ_3(nτ)$和$θ_3(τ)$在$\mathbb{Q}$上代数独立.还给出了整数$\ell,m,n\geq 1$的三个数$θ_3(\ellτ)$、…
Theta Series Associated with the Weil-Schroedinger Representation
Andre Weil发现的Weil表示在研究θ级数变换性质中起着重要作用。本文定义了Jacobi群的Weil-Schroedinger表示，并证明了与Weil-Schroedinger表示相关的θ级数是关于Jacobi模群的适当算术子群的Jacobi形式.
We prove the transformation laws of the four Jacobi theta functions using Gordon's proof for the transformation law of the Dedekind eta function.
In this paper, we give some extensions for Ramanujan's circular summation formula with the mixed products of two Jacobi's theta functions. As some applications, we also obtain many interesting identities of Jacobi's theta functions.
On Algebraic Conditions for the Non-Vanishing of Linear Forms in Jacobi Theta-Constants
本文研究雅西θ常数线性形式的非零性，在多种参数条件下证明相关线性无关性结论，填补了已有θ常数代数独立/相关性研究的空白，还建立了对应函数域上的线性无关定理。
Asymptotic behavior of partial and false theta functions arising from Jacobi forms and regularized characters
我们证明了几个渐近结果的部分和假θ函数所产生的Jacobi形式，作为模变量$τ$趋于$0$沿虚轴，和椭圆变量$z$是不受限制的复平面。我们观察到，这些功能表现出斯托克斯现象-这些功能的渐近行为急剧不同的椭圆变量$z$位于复平面内。我们应用这些结果研究了共形域论中的$(1,p)$ -单重顶点算子代数的正则特征和量子维数的渐近展开.特别是，这恢复和扩展了几个已知的结果有关的正则化量子尺寸，这是一个主要的动机来源。
Truncated Theta Series Related to the Jacobi Triple Product Identity
本文证明了Merca关于雅可比三重积截断恒等式猜想的前三种情况及相关恒等式，提出新组合解释，并用组合方法证明了可简化为已知猜想的恒等式。
The theta-block conjecture proposed by Gritsenko--Poor--Yuen in 2013 characterizes Siegel paramodular forms which are simultaneously Borcherds products and additive Jacobi lifts. In this paper, we prove this conjecture for two new infinite series of theta blocks of weights 2 and 3. The proof is based on Scheithauer's classification of reflective modular forms of singular weight.
For a prime $p$ larger than $7$, the Eisenstein series of weight $p-1$ has some remarkable congruence properties modulo $p$. Those imply, for example, that the $j$-invariants of its zeros (which are known to be real algebraic numbers in the interval $[0,1728]$), are at most quadratic over the field with $p$ elements and are congruent modulo $p$ to the zeros of a certain truncated hypergeometric series. In this paper we introduce "theta modular forms" of weight $k \geq 4$ for the full modular group as the modular forms for which the first dim$(M_k)$ Fourier coefficients are identical to certain…
Theta sums of higher index
本文得到了m次正定对称实矩阵Jacobi群的Schroedinger-Weil表示的高指标θ和的一些性质.
The Gross-Prasad conjecture and local theta correspondence
利用局部theta对应将Fourier-Jacobi情形与Beuzart-Plessis建立的Bessel情形联系起来，建立了酉群的局部Gross-Prasad猜想的Fourier-Jacobi情形.为了实现这一点，我们证明了D. Prasad关于局部θ对应的精确描述（几乎）等秩酉对偶的局部朗兰兹对应。
In this paper we construct an infinite family of paramodular forms of weight $2$ which are simultaneously Borcherds products and additive Jacobi lifts. This proves an important part of the theta-block conjecture of Gritsenko--Poor--Yuen (2013) related to the only known infinite series of theta-blocks of weight $2$ and $q$-order $1$. We also consider some applications of this result.
The congruence properties of Romik's sequence of Taylor coefficients of Jacobi's theta function $θ_3$
本文证明了 Jacobi theta 函数 Taylor 系数序列 $d(n)$ 在不同素数幂和 2 的幂模下的最终消失、最终周期及纯周期同余性质，并揭示了这些性质在更精细的多项式层面上成立。
Positivity and tails of Jacobi theta series
该研究利用基本q级数运算，建立雅可比θ级数尾部正性性质，定义相关级数等式，证明对特定整数范围，系数J_{k,n}(m)恒为正，为该领域研究提供了新结论。
We show that Hida's families of $p$-adic elliptic modular forms generalize to $p$-adic families of Jacobi forms. We also construct $p$-adic versions of theta lifts from elliptic modular forms to Jacobi forms. Our results extend to Jacobi forms previous works by Hida and Stevens on the related case of half-integral weight modular forms.
A Serre derivative for even weight Jacobi Forms
利用变形或扭曲的Eisenstein级数，构造了偶权Jacobi形式上的Jacobi-Serre导数，推广了经典的模形式上的Serre导数.作为应用，我们得到了指数$1$ Eisenstein级数$E_{4,1}, E_{6,1}$和一个新定义的$E_{2,1}$的Ramanujan方程.最后，我们将变形的爱森斯坦级数直接与经典的第一雅可比θ函数联系起来。
An algebraic property of Hecke operators and two indefinite theta series
证明了定义模形式上Hecke算子的元素的一个代数性质，这意味着对应于模形式的Petersson标积的周期多项式上的配对是Hecke等变的.由于这一证明，我们得出两个不确定的θ系列身份，可以看作是类似物的Jacobi公式的θ系列与四个平方和。
Hermitian Jacobi Forms Having Modules as their Index and Vector-Valued Jacobi Forms
本文发展了格指数的埃尔米特雅可比形式理论，证明了向量值雅可比形式的theta分解定理，并据此给出了矩阵指数的正交和埃尔米特雅可比形式的良好定义。
Jacobi's triple product, mock theta functions, unimodal sequences and the $q$-bracket
在拉马努金给哈代的最后一封信中，他列举了一种奇怪的新无穷级数的例子，他称之为“模拟θ函数“。事实证明，所有这些例子本质上都是戈登-麦金托什的所谓通用模拟θ函数$g_3(z,q)$的特殊化。在这里，我们表明，$g_3$自然产生的经典Jacobi三重积的倒数-并密切联系到单峰序列，这是连接到模拟模块化和量子模形式的秩生成函数-在一个操作相关的统计物理和分区理论，$q$ -括号布洛赫Okounkov的行动。其次，我们发现$g_3(z,q)$在$q$中推广到整个复平面减去单位圆，并给出了这个泛拟θ函数在单位根处的一个有限公式，与文献中的其他公式相比，它是简单的;我们还指出了其他$q$ -超几何级数的类似公式。最后，我们看看有趣的“量子”行为的模拟θ函数内，外，和单位圆。
Mock Jacobi forms in basic hypergeometric series
我们证明了一些$q$ -级数，如泛仿θ函数，是θ项与权为1/2的仿Jacobi形式的线性和，当它们被限制在扭点上并乘以$q$的适当幂时，它们成为实解析模形式的全纯部分.证明了某些$q$ -级数的线性和是权为1/2的弱全纯模形式，这是由于模Jacobi形式的湮灭或模Jacobi形式的完备化.作为应用，我们得到了划分的秩与曲柄之间的关系。
Some elementary remarks on the powers of a partial theta function and corresponding q-analogs of the binomial coefficients
本文研究了偏theta函数的幂及其对应的二项式系数的q-模拟，推导出了该函数正幂和负幂系数的计算公式。
We define Jacobi forms with complex multiplication. Analogous to modular forms with complex multiplication, they are constructed from Hecke characters of the associated imaginary quadratic field. From this construction we obtain a Jacobi form which specializes to $η(τ)^{26}$ which we present to highlight an open question of Dyson and Serre. We give other examples and applications of Jacobi forms with complex multiplication including constructing theta blocks associated to elliptic curves with complex multiplication and new families of congruences and cranks for certain partition functions.
Negative index Jacobi forms and quantum modular forms
本文考虑一类特殊的负指数亚纯Jaocbi型的傅里叶系数。最近的工作已经做了这样的系数的情况下，雅可比形式的正指数，但几乎没有什么是已知的雅可比形式的负指数。在本文中，我们表明，从两个不同的角度来看，他们的傅里叶系数有一个简单的分解部分θ函数。第一个视角使用李超代数的语言，第二个视角应用椭圆函数的理论。特别是，我们发现了一个新的无限家族的秩曲柄型偏微分方程推广著名的例子Atkin和Garvan。然后，我们描述了这些系数的模块化特性，表明它们是“混合部分θ函数“，沿途确定一类新的量子模块化部分θ函数，这是独立的利益。
In this paper, we explicitly construct mock modular forms whose shadows are Eisenstein series of arbitrary integral and half-integral weight, level and character at the cusps $\infty$ and $0$. As an application, we give explicit construction of harmonic weak Maass forms which are Hecke eigenforms and are the preimages of $Θ^k, k \in \{ 3, 5, 7\}$ under the shadow operator, where $Θ$ is the classical Jacobi theta function.
Two generalizations of Jacobi's derivative formula
本文将著名的Jacobi三重积恒等式推广到更高亏格/维数的情形，其中Jacobi三重积恒等式被认为是θ函数及其导数的特征恒等式.通过应用我们在前一篇论文math.AG/0310085中所发展的结果和方法，得到了Siegel模形式的几个推广。这些推广是特征为零的θ函数及其导数所满足的恒等式。将这些关系的傅里叶展开的系数等于零，得到非平凡的组合恒等式。
A theta operator on Picard modular forms modulo an inert prime
研究了模惰性素数、模p和p-adic模形式的Picard模曲面的约化问题。我们在这种模形式上构造了一个theta算子，并研究了它的极点及其对Fourier-Jacobi展开式的影响。
We revisit and elucidate the $\widehat{A}$-genus, Hirzebruch's $L$-genus and Witten's $W$-genus, cobordism invariants of special classes of manifolds. After slight modification, involving Hecke's trick, we find that the $\widehat{A}$-genus and $L$-genus arise directly from Jacobi's theta function. For every $k\geq 0,$ we obtain exact formulas for the quasimodular expressions of $\widehat{A}_k$ and $L_k$ as ``traces'' of partition Eisenstein series \[ \widehat{\mathcal{A}}_k(τ)= \operatorname{Tr}_k(ϕ_{\widehat{A}};τ)\ \ \ \ \ \ {\text {and}}\ \ \ \ \ \ \mathcal{L}_k(τ)= \operatorname{Tr}_k(ϕ_L;…
We define a theta operator on p-adic vector-valued modular forms on unitary groups of arbitrary signature, over a quadratic imaginary field in which p is inert. We study its effect on Fourier-Jacobi expansions and prove that it extends holomorphically beyond the μ-ordinary locus, when applied to scalar-valued forms.
The Kronecker theta function is a quotient of the Jacobi theta functions, which is also a special case of Ramanujan's $_1ψ_1$ summation. Using the Kronecker theta function as building blocks, we prove a decomposition theorem for theta functions. This decomposition theorem is the common source of a large number of theta function identities. Many striking theta function identities, both classical and new, are derived from this decomposition theorem with ease. A new addition formula for theta functions is established. Several known results in the theory of elliptic theta functions due to Ramanuja…
Quasimodular forms arising from Jacobi's theta function and special symmetric polynomials
本文利用Ramanujan的拟模形式序列构造出Γ_0(4)上的权n拟模形式，其对称函数化身给出对称多项式T_n，从而解决关于T_n的两个猜想，并揭示其与旋量流形Borel-Hirzebruch Â-亏格的联系。
We apply the new framework for modularity of false theta functions developed by the second author and Nazaroglu to study the asymptotic behavior of Taylor coefficients of false Jacobi forms. The examples we study generate moments of the rank for unimodal sequences. For two types of unimodal sequences, we prove asymptotic series for the rank moments.
The transformation laws of algebraic theta functions
本文从代数几何的角度，即从交换格式的模的角度，对θ函数的变换规律作了全面的讨论。这是通过引入θ下降结构，hypoplectic堆栈和半形式束的几何概念来实现的，它们的分析化身是经典变换定律的不同方面的基础。作为一个应用程序，我们奠定了基础的几何理论的模形式的半整数的重量，更一般地说，模形式的值在韦尔表示。我们讨论进一步的应用代数理论的Jacobi形式和理论的行列式丛的阿贝尔计划。
We study generating functions of certain shapes of planar polygons arising from homological mirror symmetry of elliptic curves. We express these generating functions in terms of rational functions of the Jacobi theta function and Zwegers' mock theta function and determine their (mock) Jacobi properties. We also analyze their special values and singularities, which are of geometric interest as well.
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