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Elliptic polylogarithms can be defined as iterated integrals on a genus-one Riemann surface of a set of integration kernels whose generating series was already considered by Kronecker in the 19th century. In this article, we employ the Schottky parametrization of a Riemann surface to construct higher-genus analogues of Kronecker's generating series, which we refer to as Schottky–Kronecker forms. Our explicit construction generalizes ideas from Bernard's higher-genus construction of the Knizhnik-Zamolodchikov connection. Integration kernels generated from the Schottky–Kronecker forms are defined as Poincaré series. Under technical assumptions, related to the convergence of these Poincaré series on the underlying Riemann surface, we argue that these integration kernels coincide with a set of differentials defined by Enriquez, whose iterated integrals constitute higher-genus analogues of polylogarithms. Enriquez' original definition is not well-suited for numerical evaluation of higher-genus polylogarithms. In contrast, the Poincaré series defining our integration kernels can be evaluated numerically for real hyperelliptic curves, for which the above-mentioned convergence assumptions can be verified. We numerically evaluate several examples of genus-two polylogarithms, thereby paving the way for numerical evaluation of hyperelliptic analogues of polylogarithms. [ABSTRACT FROM AUTHOR]

We study Hilbert Poincaré series associated to general seed functions and construct Cohen’s kernels and double Eisenstein series as series of Hilbert Poincaré series. Then we calculate the Rankin–Cohen brackets of Hilbert Poincaré series and Hilbert modular forms and extend Zagier’s kernel formula to totally real number fields. Finally, we show that the Rankin–Cohen brackets of two different types of Eisenstein series are special values of double Eisenstein series up to a constant. [ABSTRACT FROM AUTHOR]

AbstractThe present paper deals with the investigation of the structure of general fiber product rings R×TS , where R, S and T are local rings with common residue field. We show that the Poincaré series of any R-module over the fiber product ring R×TS is bounded by a rational function. In addition, we give a description of depth(R×TS) , which is an open problem in this theory. As a biproduct, using the characterization of the Betti numbers over R×TS obtained, we provide certain cases of the Cohen-Macaulayness of R×TS and, in particular, we show that R×TS is always non-regular. Some positive answers for the Buchsbaum-Eisenbud-Horrocks and Total rank conjectures over R×TS are also established. [ABSTRACT FROM AUTHOR]

We introduce and study 'non-abelian' Poincaré series for the group G=\operatorname {SU}(2,1), that is, Poincaré series attached to a Stone-Von Neumann representation of the unipotent subgroup N of G. Such Poincaré series have in general exponential growth. In this study we use results on abelian and non-abelian Fourier term modules obtained by Bruggeman and Miatello [ Representations of \mathrm {SU}(2,1) in Fourier term modules , Springer, Cham, 2023] to compute the inner product of truncations of these series with square integrable automorphic forms, in connection with their Fourier expansions. As a consequence, we obtain general completeness results for \operatorname {SU}(2,1) that, in particular, generalize those valid for the classical holomorphic (and antiholomorphic) Poincaré series for \operatorname {SL}(2,\mathbb {R}). [ABSTRACT FROM AUTHOR]

For a connected reductive group G over a nonarchimedean local field F of positive characteristic, GenestierLafforgue and Fargues-Scholze have attached a semisimple parameter Lss to each irreducible representation π. Our first result shows that the Genestier-Lafforgue parameter of a tempered π can be uniquely refined to a tempered L-parameter L(π), thus giving the unique local Langlands correspondence which is compatible with the GenestierLafforgue construction. Our second result establishes ramification properties of Lss (π) for unramified G and supercuspidal π constructed by induction from an open compact (modulo center) subgroup. If Lss is pure in an appropriate sense, we show that Lss (π) is ramified (unless G is a torus). If the inducing subgroup is sufficiently small in a precise sense, we show Lss is wildly ramified. The proofs are via global arguments, involving the construction of Poincaré series with strict control on ramification when the base curve is P1 and a simple application of Deligne’s Weil II. [ABSTRACT FROM AUTHOR]

In the article, we study the Oberdieck derivative defined on the space of weak Jacobi forms. We prove that the Oberdieck derivative maps a Jacobi form to a Jacobi form. Moreover, we study the adjoint of the Oberdieck derivative of a Jacobi cusp form with respect to the Petersson scalar product defined on the space of Jacobi forms. As a consequence, we also obtain the adjoint of the Jacobi–Serre derivative (defined in an unpublished work of Oberdieck). As an application, we obtain certain relations among the Fourier coefficients of Jacobi forms. [ABSTRACT FROM AUTHOR]

In this article we study the delta invariant of reduced curve germs via topological techniques. We describe an explicit connection between the delta invariant of a curve embedded in a rational singularity and the topological Poincaré series of the ambient surface. This connection is established by using another formula expressing the delta invariant as 'periodic constants' of the Poincaré series associated with the abstract curve and a 'twisted' duality developed for the Poincaré series of the ambient space. [ABSTRACT FROM AUTHOR]

The equation |$x^2 + 1 = 0\mod p$| has solutions whenever p  = 2 or |$4n + 1$|⁠. A famous theorem of Fermat says that these primes are exactly the ones that can be described as a sum of two squares. The roots of the former equation are equidistributed is a beautiful theorem of Duke, Friedlander and Iwaniec. The angles associated to the representation of such prime as a sum of squares are equidistributed is a famous theorem of Hecke. We give a natural way to associate between roots and angles and prove that the joint equidistribution of the sequence of pairs of roots and angles is equidistributed as well. Our approach involves an automorphic interpretation, which reduces the problem to the study of certain Poincare series on an arithmetic quotient of |$SL_2(\mathbb{R})$|⁠. Since our Poincare series have a nontrivial dependence on their Iwasawa θ -coordinate, they do not factor into functions on the upper half plane, as in the case studied by Duke et al. Spectral analysis on these higher dimensional varieties involves the nonspherical spectrum, making this paper the first complete study of a nonspherical equidistribution problem, with an arithmetic application. A couple of notable challenges we had to overcome were that of obtaining pointwise bounds for nonspherical Eisenstein series and utilizing a non-spherical analogue of the Selberg inversion formula, which we believe may have further implications beyond this work. [ABSTRACT FROM AUTHOR]

The loop group |$G((z^{-1}))$| of a simple complex Lie group |$G$| has a natural Poisson structure. We introduce a natural family of Poisson commutative subalgebras |$\overline{{\textbf{B}}}(C) \subset{{\mathcal{O}}}(G((z^{-1}))$| depending on the parameter |$C\in G$| called classical universal Bethe subalgebras. To every antidominant cocharacter |$\mu $| of the maximal torus |$T \subset G$|⁠ , one can associate the closed Poisson subspace |${{\mathcal{W}}}_{\mu }$| of |$G((z^{-1}))$| (the Poisson algebra |${{\mathcal{O}}}({{\mathcal{W}}}_{\mu })$| is the classical limit of so-called shifted Yangian |$Y_{\mu }(\mathfrak{g})$| defined in [ 1 , Appendix B]). We consider the images of |$\overline{{\textbf{B}}}(C)$| in |${{\mathcal{O}}}({{\mathcal{W}}}_{\mu })$|⁠ , denoted by |$\overline{B}_{\mu }(C)$|⁠ , that should be considered as classical versions of (not yet defined in general) Bethe subalgebras in shifted Yangians. For regular |$C$| centralizing |$\mu $|⁠ , we compute the Poincaré series of these subalgebras. For |$\mathfrak{g}=\mathfrak{g}\mathfrak{l}_{n}$|⁠ , we define the natural quantization |${\textbf{Y}}^{\textrm{rtt}}(\mathfrak{g}\mathfrak{l}_{n})$| of |${{\mathcal{O}}}(\operatorname{Mat}_{n}((z^{-1})))$| and universal Bethe subalgebras |${\textbf{B}}(C) \subset{\textbf{Y}}^{\textrm{rtt}}(\mathfrak{g}\mathfrak{l}_{n})$|⁠. Using the RTT realization of |$Y_{\mu }(\mathfrak{g}\mathfrak{l}_{n})$| (invented by Frassek, Pestun, and Tsymbaliuk), we obtain the natural surjections |${\textbf{Y}}^{\textrm{rtt}}(\mathfrak{g}\mathfrak{l}_{n}) \twoheadrightarrow Y_{\mu }(\mathfrak{g}\mathfrak{l}_{n})$|⁠ , which quantize the embedding |${{\mathcal{W}}}_{\mu } \subset \operatorname{Mat}_{n}((z^{-1}))$|⁠. Taking the images of |${\textbf{B}}(C)$| in |$Y_{\mu }(\mathfrak{g}\mathfrak{l}_{n})$|⁠ , we recover Bethe subalgebras |$B_{\mu }(C) \subset Y_{\mu }(\mathfrak{g}\mathfrak{l}_{n})$| proposed by Frassek, Pestun, and Tsymbaliuk. [ABSTRACT FROM AUTHOR]

Laurent solutions of systems of linear ordinary differential equations with truncated power series as coefficients are considered. The Laurent series in the solutions are also truncated. As a means for constructing such solutions, induced recurrent systems are used; earlier, an algorithm for the case when the induced recurrent system has a nonsingular leading matrix was proposed. For the series in solutions, this algorithm finds the maximum possible number of terms that are invariant with respect to any prolongation of the truncated coefficients of the original system. Results on extending the applicability of the earlier proposed algorithm to the case when the leading matrix is singular using the EG-elimination algorithm as an auxiliary tool. An implementation of the proposed algorithm in the form of a Maple procedure is given and examples of its use are presented. [ABSTRACT FROM AUTHOR]

The main theorem of this article is a Runge type theorem proved for k-differentials (k ≥ 2) . The integrability in the L 1 - norm is defined for k-differentials in Section 2. We consider k-differentials which are integrable in the defined L 1 - norm on the Riemann surface, and are holomorphic on an open subset of that surface. We will show those k-differentials can be approximated by meromorphic k-differentials. The proof applies a generalized form of the Poincaré series map. This generalized form is proved in Section 3. Section 2 contains the definition of the Poincaré series and its convergence, with particular focus on the convergence of the Poincaré series for rational functions, which is applied in the main theorem. Sections 3 and 4 contain the new results proved in this paper. The statement and proof of the main theorem are in Section 4. [ABSTRACT FROM AUTHOR]

Traces of singular moduli can be approximated by exponential sums of quadratic irrationals. Recently Andersen and Duke used theory of Maass forms to estimate generalized twisted traces with power-saving error bounds. We establish an asymptotic formula with effective error bounds for such traces. Our methods depend on an explicit bound for sums of Kloosterman sums on Γ 0 (4) . [ABSTRACT FROM AUTHOR]

Numerical methods for the computation of the parabolic cylinder function U (a , z) for real a and complex z are presented. The main tools are recent asymptotic expansions involving exponential and Airy functions, with slowly varying analytic coefficient functions involving simple coefficients, and stable integral representations; these two main methods can be complemented with Maclaurin series and a Poincaré asymptotic expansion. We provide numerical evidence showing that the combination of these methods is enough for computing the function with 5 × 10 − 13 relative accuracy in double precision floating point arithmetic. [ABSTRACT FROM AUTHOR]

We show that the values of elliptic Dedekind sums, after normalization, are equidistributed mod 1. The key ingredient is a non-trivial bound on generalized Selberg–Kloosterman sums for discrete subgroups of PSL 2 (C) using Poincaré series. [ABSTRACT FROM AUTHOR]

Let N ⊴ G be a pair of finite subgroups of SL2(ℂ) and V a finite-dimensional fundamental G-module. We study Kostant’s generating functions for the decomposition of the SL2(ℂ)-module Sk(V ) restricted to N ◃ G in connection with the McKay–Slodowy correspondence. In particular, the classical Kostant formula was generalized to a uniform version of the Poincaré series for the symmetric invariants in which the multiplicities of any individual module in the symmetric algebra are completely determined. [ABSTRACT FROM AUTHOR]

There exist several equivalent equations for the Poincaré series of a collection of valuations on the ring of germs of functions on a complex analytic variety. We give definitions of the Poincaré series of a collection of valuations in the real setting (i.e., on the ring of germs of functions on a real analytic variety), compute them for the case of one curve valuation on the plane and discuss some of their properties. [ABSTRACT FROM AUTHOR]

In [20], Rohrlich proved a modular analog of Jensen's formula. Under certain conditions, the Rohrlich–Jensen formula expresses an integral of the log-norm $\log \Vert f \Vert $ of a ${\mathrm {PSL}}(2,{\mathbb {Z}})$ modular form f in terms of the Dedekind Delta function evaluated at the divisor of f. In [2], the authors re-interpreted the Rohrlich–Jensen formula as evaluating a regularized inner product of $\log \Vert f \Vert $ and extended the result to compute a regularized inner product of $\log \Vert f \Vert $ with what amounts to powers of the Hauptmodul of $\mathrm {PSL}(2,{\mathbb {Z}})$. In the present article, we revisit the Rohrlich–Jensen formula and prove that in the case of any Fuchsian group of the first kind with one cusp it can be viewed as a regularized inner product of special values of two Poincaré series, one of which is the Niebur–Poincaré series and the other is the resolvent kernel of the Laplacian. The regularized inner product can be seen as a type of Maass–Selberg relation. In this form, we develop a Rohrlich–Jensen formula associated with any Fuchsian group $\Gamma $ of the first kind with one cusp by employing a type of Kronecker limit formula associated with the resolvent kernel. We present two examples of our main result: First, when $\Gamma $ is the full modular group ${\mathrm {PSL}}(2,{\mathbb {Z}})$ , thus reproving the theorems from [2]; and second when $\Gamma $ is an Atkin–Lehner group $\Gamma _{0}(N)^+$ , where explicit computations of inner products are given for certain levels N when the quotient space $\overline {\Gamma _{0}(N)^+}\backslash \mathbb {H}$ has genus zero, one, and two. [ABSTRACT FROM AUTHOR]

Let Γ be a congruence subgroup of Sp 2 n (ℤ). Using Poincaré series of K -finite matrix coefficients of integrable discrete series representations of Sp 2 n (ℝ) , we construct a spanning set for the space S m (Γ) of Siegel cusp forms of weight m ∈ ℤ > 2 n . We prove the non-vanishing of certain elements of this spanning set using Muić's integral non-vanishing criterion for Poincaré series on locally compact Hausdorff groups. Moreover, using the representation theory of Sp 2 n (ℝ) , we study the Petersson inner products of corresponding cuspidal automorphic forms, thereby recovering a representation-theoretic proof of some well-known results on the reproducing kernel function of S m (Γ). Our results are obtained by generalizing representation-theoretic methods developed by Muić in his work on holomorphic cusp forms on the upper half-plane to the setting of Siegel cusp forms of a higher degree. [ABSTRACT FROM AUTHOR]

Systems of linear ordinary differential equations with the coefficients in the form of infinite formal power series are considered. The series are represented in a truncated form, with the truncation degree being different for different coefficients. Induced recurrent systems and literal designations for unspecified coefficients of the series are used as a tool for studying such systems. An algorithm for constructing Laurent solutions of the system is proposed for the case where the determinant of the leading matrix of the induced system is not zero and does not contain literals. The series included in the solutions are still truncated. The algorithm finds the maximum possible number of terms of the series that are invariant with respect to any prolongations of the truncated coefficients of the original system. The implementation of the algorithm as a Maple procedure and examples of its usage are presented. [ABSTRACT FROM AUTHOR]

Let 한 be a field and let I be a monomial ideal in the polynomial ring Q = 한[x1,…,xn]. In her thesis, Taylor introduced a complex which provides a finite free resolution for Q/I as a Q-module. Later, Gemeda constructed a differential graded structure on the Taylor resolution. More recently, Avramov showed that this differential graded algebra admits divided powers. We generalize each of these results to monomial ideals in a skew polynomial ring R. Under the hypothesis that the skew commuting parameters defining R are roots of unity, we prove as an application that as I varies among all ideals generated by a fixed number of monomials of degree at least two in R, there is only a finite number of possibilities for the Poincaré series of 한 over R/I and for the isomorphism classes of the homotopy Lie algebra of R/I in cohomological degree larger or equal to two. [ABSTRACT FROM AUTHOR]

We use Poincaré series for massive Maass-Jacobi forms to define a "massive theta lift", and apply it to the examples of the constant function and the modular invariant j-function, with the Siegel-Narain theta function as integration kernel. These theta integrals are deformations of known one-loop string threshold corrections. Our massive theta lifts fall off exponentially, so some Rankin-Selberg integrals are finite without Zagier renormalization. [ABSTRACT FROM AUTHOR]

We study the Poincaré series of the mixed and pure trace rings of generic matrices. These series are known to be rational functions. We obtain an explicit formula in lowest terms in the case of 2 × 2 matrices; a denominator, which we presume but have not been able to prove to be in lowest terms, in the case of 3 × 3 matrices; and a conjectured denominator in the general case. [ABSTRACT FROM AUTHOR]

For a hyperplane arrangement in a real vector space, the coefficients of its Poincaré polynomial have many interpretations. An interesting one is provided by the Varchenko-Gel'fand ring, which is the ring of functions from the chambers of the arrangement to the integers with pointwise addition and multiplication. Varchenko and Gel'fand gave a simple presentation for this ring, along with a filtration and associated graded ring whose Hilbert series is the Poincaré polynomial. We generalize these results to cones defined by intersections of halfspaces of some of the hyperplanes and prove a novel result for the Varchenko-Gel'fand ring of an arrangement: when the arrangement is supersolvable the associated graded ring of the arrangement is Koszul. [ABSTRACT FROM AUTHOR]

We introduce the class of vector-spread monomial ideals. This notion generalizes that of t -spread ideals introduced by Ene, Herzog and Qureshi. In particular, we focus on vector-spread strongly stable ideals, we compute their Koszul cycles and describe their minimal free resolution. As a consequence the graded Betti numbers and the Poincaré series are determined. Finally, we consider a generalization of algebraic shifting theory for such a class of ideals. [ABSTRACT FROM AUTHOR]

In this work we find the groups of Hochschild cohomologies for the Chinese monoid algebra and derive its Hilbert and Poincaré series. In order to obtain this result we construct the Anick resolution via the algebraic discrete Morse theory and Gröbner–Shirshov basis for the Chinese monoid. [ABSTRACT FROM AUTHOR]

We rewrite the worldsheet torus partition function of the Thermal AdS CFT by isolating the boundary parameters. Using this, we show that the spectrum of the Euclidean BTZ black hole and Lorentzian A d S 3 can be extracted – the latter as a zero temperature limit. A similar procedure recovers the Lorentzian BTZ spectrum proposed in an earlier work. We then use our expression to construct a boundary modular invariant expression as a Poincaré series. • Comparing boundaries. We point out that the comparison of boundary geometries of thermal A d S 3 (TAdS) and Euclidean BTZ (EBTZ) is rather subtle. The boundary tori are characterised not only by their modular parameter T but also a Kähler modulus ρ constituted by the volume and B-field. • Euclidean spectra. Accounting for the subtlety, we show that the partition function calculated by Maldacena, Ooguri and Son [5] can be expanded to reveal either the TAdS spectrum (58) or the EBTZ spectrum (66). An important way in which the latter interpretation differs from the former is that it diagonalises the hyperbolic current algebra generator. • B-field. In both interpretations, we observe that correct momentum quantisation fixes an additive constant in the B-field, thereby disallowing large gauge transformations. We find that the EBTZ B-field differs from that expected [13] for Lorentzian BTZ. This, together with observations from our previous work [8] , indicates that black hole microstates (as opposed to its equilibrium density matrix) are better studied directly in the Lorentzian signature. • Lorentzian spectra. The Lorentzian A d S 3 spectrum was originally obtained [5] by interpreting the TAdS partition function as the free energy of a gas of strings. We show that the same is obtained simply as a low temperature limit followed by a Wick rotation. • Boundary modular invariance. We rewrite the Euclidean partition function in a manner that separates out its dependence on boundary parameters T and ρ (see (74) and (75)). This allows us to formulate a boundary modular invariant (100), akin to a Farey tail expansion [17]. [ABSTRACT FROM AUTHOR]

We prove the rationality of the Poincaré series of multiplier ideals in any dimension thus extending the main results for surfaces of Galindo and Monserrat and Alberich-Carramiñana et al. Our results also hold for Poincaré series of test ideals. In order to do so, we introduce a theory of Hilbert functions indexed over R which gives a unified treatment of both cases. [ABSTRACT FROM AUTHOR]

Using elementary hyperbolic geometry, we give an explicit formula for the contraction constant of the skinning map over moduli spaces of relatively acylindrical hyperbolic manifolds. [ABSTRACT FROM AUTHOR]

We consider a local (or standard graded) ring R with ideals \mathcal {I}', \mathcal {I}, \mathcal {J}', and \mathcal {J} satisfying certain Tor-vanishing constraints. We construct free resolutions for quotient rings R/\langle \mathcal {I}', \mathcal {I}\mathcal {J}, \mathcal {J}'\rangle, give conditions for the quotient to be realized as a fiber product, and give criteria for the construction to be minimal. We then specialize this result to fiber products over a field k and recover explicit formulas for Betti numbers, graded Betti numbers, and Poincaré series. [ABSTRACT FROM AUTHOR]

A modular grid is a pair of sequences (fm)m$ (f_m) _m$ and (gn)n$ (g_n) _n$ of weakly holomorphic modular forms such that for almost all m and n, the coefficient of qn$q^n$ in fm$f_m$ is the negative of the coefficient of qm$q^m$ in gn$g_n$. Zagier proved this coefficient duality in weights 1/2 and 3/2 in the Kohnen plus space, and such grids have appeared for Poincaré series, for modular forms of integral weight, and in many other situations. We give a general proof of coefficient duality for canonical row‐reduced bases of spaces of weakly holomorphic modular forms of integral or half‐integral weight for every group Γ⊆SL2(R)$\Gamma \subseteq {\text{\rm SL}}_2(\mathbb {R})$ commensurable with SL2(Z)${\text{\rm SL}}_2(\mathbb {Z})$. We construct bivariate generating functions that encode these modular forms, and study linear operations on the resulting modular grids. [ABSTRACT FROM AUTHOR]

Let k be a field. Let m ∈ N>0 be a positive integer. Let f ∈ k[x1, . . ., xm] be a polynomial with degree d ≥ 1 and associated hypersurface H := H(f) := Spec(k[x1, . . ., xm](f). In this article, we firstly provide a structure property of the weighted-homogeneity of f in terms of the jet schemes LH of H. As a by-product, we deduce from this property a new and very effective method for the computation of the motivic Poincaré power series PH(T) := ∑n≥0 [Ln(H)]Tn ∈ K0(Vark)[[T]] associated with a homogeneous hypersurface H with a single isolated singularity at the origin o (and more generally with a specific class of isolated quasi-homogeneous hypersurface singularities). With this point of view we obtain various consequences. For the considered class of varieties, our method provides a characteristic-free proof of the rationality of PH(T) which does not use motivic integration nor the existence of resolutions of singularities; we obtain a precise description of the numerator and the possible poles in the rational expression of PH(T); when the field is assumed to be of characteristic zero, this allows us to prove the validity of the motivic monodromy conjecture. [ABSTRACT FROM AUTHOR]

For approximate summation of Poincaré theta series in the Schottky model of real hyperelliptic curves, modifications of the Bogatyrev and Schmies algorithms are proposed that reduce the number of summed terms without losing accuracy. [ABSTRACT FROM AUTHOR]

New algorithms for approximate summation of Poincaré theta series in the Schottky model of real hyperelliptic curves are proposed. As a result, for the same output accuracy estimate, the amount of computations is reduced by several times in the case of slow convergence and by tens of percent in the usual situations. For the sum of the Poincaré series over the subtree on descendants of a given node, a new estimate in terms of the series member at this node is obtained. [ABSTRACT FROM AUTHOR]

Earlier it was described to which extent the Alexander polynomial in several variables of an algebraic link in the Poincaré sphere determines the topology of the link. It was shown that, except some explicitly described cases, the Alexander polynomial of an algebraic link determines the combinatorial type of the minimal resolution of the curve and therefore the topology of the corresponding link. The Alexander polynomial of an algebraic link in the Poincaré sphere coincides with the Poincaré series of the corresponding set of curve valuations. The latter one can be defined as an integral over the space of divisors on the E8-singularity. Here, we consider a similar integral for rational double point surface singularities over the space of Weil divisors called the Weil-Poincaré series. We show that, except a few explicitly described cases, the Weil-Poincaré series of a collection of curve valuations on a rational double point surface singularity determines the topology of the corresponding link. We give analogous statements for collections of divisorial valuations. [ABSTRACT FROM AUTHOR]

The field of mathematical morphology offers well-studied techniques for image processing and is applicable for studies ranging from materials science to ecological pattern formation. In this work, we view morphological operations through the lens of persistent homology, a tool at the heart of the field of topological data analysis. We demonstrate that morphological operations naturally form a multiparameter filtration and that persistent homology can then be used to extract information about both topology and geometry in the images as well as to automate methods for optimizing the study and rendering of structure in images. For illustration, we develop an automated approach that utilizes this framework to denoise binary, grayscale, and color images with salt and pepper and larger spatial scale noise. We measure our example unsupervised denoising approach to state-of-the-art supervised, deep learning methods to show that our results are comparable. [ABSTRACT FROM AUTHOR]

We construct a filtration of chiral Hodge cohomolgy of a K3 surface |$X$|⁠ , such that its associated graded object is a unitary representation of the |$\mathcal{N}=4$| superconformal vertex algebra with central charge |$c=6$| and its subspace of primitive vectors has the property; its equivariant character for a symplectic automorphism |$g$| of finite order acting on |$X$| agrees with the McKay–Thompson series for |$g$| in Mathieu moonshine. [ABSTRACT FROM AUTHOR]

A twin cam mixer (TCM) system equipped with two triangular cams is built in this study. The effect of operating conditions on chaotic advection in TCM is investigated. Particle Image Velocimetry (PIV) is employed to investigate the flow characteristics under co‐ and counter‐rotating modes. Based on the good agreement between simulation and experimental results, chaotic advection under various operating conditions is investigated numerically. To evaluate the chaotic advection, time series Poincaré section, recording point density, and combined mixing index are proposed. The results show that counter‐rotating mode is superior to co‐rotating mode regarding chaotic advection. A more global chaotic advection can be obtained with the cams in TCM rotating at a higher average angular velocity, higher disturbance quantity, and lower disturbance frequency. The chaotic advection in TCM under different waveform protocols has no significant difference, but it is all better than that under constant protocol. [ABSTRACT FROM AUTHOR]

We prove power-saving bounds for general Kloosterman sums on Sp (4) associated to all Weyl elements via a stratification argument coupled with p-adic stationary phase methods. We relate these Kloosterman sums to the Fourier coefficients of Sp (4) Poincare series. [ABSTRACT FROM AUTHOR]

We estimate the bottom of the L 2 spectrum of the Laplacian on locally symmetric spaces in terms of the critical exponents of appropriate Poincaré series. Our main result is the higher rank analog of a characterization due to Elstrodt, Patterson, Sullivan and Corlette in rank one. It improves upon previous results obtained by Leuzinger and Weber in higher rank. [ABSTRACT FROM AUTHOR]

The present note is part of a series of articles targeting the theory of Abel maps associated with complex normal surface singularities with rational homology sphere links (Nagy and Némethi in Math Annal 375(3):1427–1487, 2019; Nagy and Némethi in Adv Math 371:20, 2020; Nagy and Némethi in Pure Appl Math Q 16(4):1123–1146, 2020). Besides the general theory, by the study of specific families we wish to show the power of this new method. Indeed, using the general theory of Abel maps applied for elliptic singularities in this note we are able to prove several key properties for elliptic singularities (e.g. the statements of the next paragraph), which by 'old' techniques were not reachable. If (X ~ , E) → (X , o) is the resolution of a complex normal surface singularity and c 1 : Pic (X ~) → H 2 (X ~ , Z) is the Chern class map, then Pic l ′ (X ~) : = c 1 - 1 (l ′) has a (Brill–Noether type) stratification W l ′ , k : = { L ∈ Pic l ′ (X ~) : h 1 (L) = k } . In this note we determine it for elliptic singularities together with the stratification according to the cycle of fixed components. E.g., we show that the closure of any W (l ′ , k) is an affine subspace. For elliptic singularities we also characterize the End Curve Condition and Weak End Curve Condition in terms of the Abel map, we provide several characterization of them, and finally we show that they are equivalent. [ABSTRACT FROM AUTHOR]

For each odd prime p, we produce a 2‐generated pro‐p group G whose normal Hausdorff spectra hspec⊴S(G)=hdimGS(H)∣H⊴cG\begin{equation*}\hskip7pc \operatorname{hspec}_{\trianglelefteq }^{\mathcal {S}}(G) = {\left\lbrace \operatorname{hdim}_{G}^{\mathcal {S}}(H)\mid H\trianglelefteq _\mathrm{c} G \right\rbrace}\hskip-7pc \end{equation*}with respect to five standard filtration series S$\mathcal {S}$, namely the lower p‐series, the dimension subgroup series, the p‐power series, the iterated p‐power series and the Frattini series, are all equal to the full unit interval [0,1]. Here hdimGS:{X∣X⊆G}→[0,1]$\operatorname{hdim}_G^{\mathcal {S}} : \lbrace X\mid X \subseteq G \rbrace \rightarrow [0,1]$ denotes the Hausdorff dimension function associated to the natural translation‐invariant metric induced by the filtration series S$\mathcal {S}$. [ABSTRACT FROM AUTHOR]

Rotationally reciprocating mixing, in which a plate impeller slowly rotates back and forth for only half a rotation, shows excellent fluid mixing performance at Re > 40. On the other hand, since vertical flow disappears at Re < 10, two-dimensional fluid mixing proceeds in the horizontal cross section of a cylindrical vessel, producing segmented mixing regions with a pair of central poor mixing zones. In this study, we investigated the usefulness of an anchor-type impeller in the expectation of enhancing two-dimensional fluid mixing under a laminar regime by utilizing the fluid flow through the central clearances. First, we examined the effect of the clearance size on the spreading behavior of tracer particles by numerical simulation. Time series of Poincaré sections are used to classify the tracer particles depending on their spreading behavior, which is then quantified by newly defined mixing ability. It is then elucidated that even a narrow central clearance could avoid the segmentation of the mixing region, while a vast central clearance produces additional isolated mixing regions in the clearance due to stable vortex flows. The tracer particles could uniformly spread in the entire circular cross section by increasing the impeller width and adjusting the flow rate in the central and wall clearances without segmentation. Furthermore, the experimental observation of streaklines could also demonstrate that the uniformity of the fluid mixing performance is improved considerably. [ABSTRACT FROM AUTHOR]

In this paper, we consider ℤ r {\mathbb{Z}^{r}} -graded modules on the Cl ⁡ (X) {\operatorname{Cl}(X)} -graded Cox ring ℂ ⁢ [ x 1 , ... , x r ] {\mathbb{C}[x_{1},\ldots,x_{r}]} of a smooth complete toric variety X. Using the theory of Klyachko filtrations in the reflexive case, we construct a collection of lattice polytopes codifying the multigraded Hilbert function of the module. We apply this approach to reflexive ℤ s + r + 2 {\mathbb{Z}^{s+r+2}} -graded modules over any non-standard bigraded polynomial ring ℂ ⁢ [ x 0 , ... , x s , y 0 , ... , y r ] \mathbb{C}[x_{0},\ldots,x_{s},\allowbreak y_{0},\ldots,y_{r}]. In this case, we give sharp bounds for the multigraded regularity index of their multigraded Hilbert function, and a method to compute their Hilbert polynomial. [ABSTRACT FROM AUTHOR]

In every dimension d ≥ 2, we give an explicit formula that expresses the values of any Schwartz function on Rd only in terms of its restrictions, and the restrictions of its Fourier transform, to all origin-centered spheres whose radius is the square root of an integer. We thus generalize an interpolation theorem by Radchenko and Viazovska [Publ. Math. Inst. Hautes Études Sci. 129 (2019), pp. 51–81] to higher dimensions. We develop a general tool to translate Fourier uniqueness and interpolation results for radial functions in higher dimensions, to corresponding results for non-radial functions in a fixed dimension. In dimensions greater or equal to 5, we solve the radial problem using a construction closely related to classical Poincaré series. In the remaining small dimensions, we combine this technique with a direct generalization of the Radchenko–Viazovska formula to higher-dimensional radial functions, which we deduce from general results by Bondarenko, Radchenko and Seip [ Fourier interpolation with zeros of zeta and L-functions , arXiv:2005.02996, 2020] [ABSTRACT FROM AUTHOR]