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In this paper, we construct generalized L-functions associated to meromorphic modular forms of weight 1 2 for the theta group with a single simple pole in the fundamental domain. We then consider their behavior toward i∞ and relate this to the Riemann zeta function. [ABSTRACT FROM AUTHOR]

Endoscopes are crucial for assisting in surgery and disease diagnosis, including the early detection of cancer. The effective use of endoscopes relies on their optical performance, which can be characterized with a series of metrics such as resolution, vital for revealing anatomical details. The modulation transfer function (MTF) is a key metric for evaluating endoscope resolution. However, the 2020 version of the ISO 8600-5 standard, while introducing an endoscope MTF measurement method, lacks empirical validation and excludes opto-electronic video endoscopes, the largest family of endoscopes. Measuring the MTF of video endoscopes requires tailored standards that address their unique characteristics. This paper aims to expand the scope of ISO 8600-5:2020 to include video endoscopes, by optimizing the MTF test method and addressing parameters affecting measurement accuracy. We studied the effects of intensity and uniformity of image luminance, chart modulation compensation, linearity of image digital values, auto gain control, image enhancement, image compression and the region of interest dimensions on images of slanted-edge test charts, and thus the MTF based on these images. By analyzing these effects, we provided recommendations for setting and controlling these factors to obtain accurate MTF curves. Our goal is to enhance the standard's relevance and effectiveness for measuring the MTF of a broader range of endoscopic devices, with potential applications in the MTF measurement of other digital imaging devices. [ABSTRACT FROM AUTHOR]

This study looks closely into the analysis of the variation of constants formula given by Φ (t) = S (t) Φ (0) + ∫ 0 t S (t − σ) F (σ , Φ (σ)) d σ , for t ∈ [ 0 , T ] , T > 0 , within the context of modular function spaces L ρ . Additionally, this research explores practical applications of the variation of constants formula in variable exponent Lebesgue spaces L p (·) . Specifically, the study examines these spaces under certain conditions applied to the exponent function p (·) and the functions F as well as the semigroup S (t) , utilizing the symmetry properties of the algebraic semigroup. This investigation sheds light on the intricate interplay between parameters and functions within these mathematical frameworks, offering valuable insights into their behavior and properties in L p (·) . [ABSTRACT FROM AUTHOR]

By using modular functions on the upper complex half-plane, we study a class of strain energies for crystalline materials whose global invariance originates from the full symmetry group of the underlying lattice. This follows Ericksen's suggestion which aimed at extending the Landau-type theories to encompass the behavior of crystals undergoing structural phase transformation, with twinning, microstructure formation, and possibly associated plasticity effects. Here we investigate such Ericksen-Landau strain energies for the modelling of reconstructive transformations, focusing on the prototypical case of the square-hexagonal phase change in 2D crystals. We study the bifurcation and valley-floor network of these potentials, and use one in the simulation of a quasi-static shearing test. We observe typical effects associated with the micro-mechanics of phase transformation in crystals, in particular, the bursty progress of the structural phase change, characterized by intermittent stress-relaxation through microstructure formation, mediated, in this reconstructive case, by defect nucleation and movement in the lattice. [ABSTRACT FROM AUTHOR]

The applicability of the distance aggregation problem has attracted the interest of many authors. Motivated by this fact, in this paper, we face the modular quasi-(pseudo-)metric aggregation problem, which consists of analyzing the properties that a function must have to fuse a collection of modular quasi-(pseudo-)metrics into a single one. In this paper, we characterize such functions as monotone, subadditive and vanishing at zero. Moreover, a description of such functions in terms of triangle triplets is given, and, in addition, the relationship between modular quasi-(pseudo-)metric aggregation functions and modular (pseudo-)metric aggregation functions is discussed. Specifically, we show that the class of modular (quasi-)(pseudo-)metric aggregation functions coincides with that of modular (pseudo-)metric aggregation functions. The characterizations are illustrated with appropriate examples. A few methods to construct modular quasi-(pseudo-)metrics are provided using the exposed theory. By exploring the existence of absorbent and neutral elements of modular quasi-(pseudo-)metric aggregation functions, we find that every modular quasi-pseudo-metric aggregation function with 0 as the neutral element is an Aumann function, is majored by the sum and satisfies the 1-Lipschitz condition. Moreover, a characterization of those modular quasi-(pseudo-)metric aggregation functions that preserve modular quasi-(pseudo-)metrics is also provided. Furthermore, the relationship between modular quasi-(pseudo-)metric aggregation functions and quasi-(pseudo-)metric aggregation functions is studied. Particularly, we have proven that they are the same only when the former functions are finite. Finally, the usefulness of modular quasi-(pseudo-)metric aggregation functions in multi-agent systems is analyzed. [ABSTRACT FROM AUTHOR]

This paper presents a dynamic-phasor-based, average-value modeling method for power systems with extensive converter-tied subsystems. In the proposed approach, the overall system model is constructed using modular functions, interfacing both conventional and converter-tied resources. Model validation is performed against detailed Electro-Magnetic Transient (EMT) simulations. The analytical capabilities offered by the proposed modeling method are demonstrated on a modified IEEE 9-bus system. A Graphics Processing Unit (GPU)-based parallel computing approach for the solution of the resulting model is presented and exemplified on a modified IEEE 118-bus system, showing significant improvements in computing efficiency over EMT solvers. A co-simulation approach using a Central Processing Unit (CPU) and a GPU is also presented and exemplified using a modified version of the IEEE 118-bus system, demonstrating the model's parallelization. [ABSTRACT FROM AUTHOR]

In this paper, we present a thorough study of the regularized submodular maximization problem, in which the objective $ f:=g-\ell $ f := g − ℓ can be expressed as the difference between a submodular function and a modular function. This problem has drawn much attention in recent years. While existing works focuses on the case of g being monotone, we investigate the problem with a nonmonotone g. The main technique we use is to introduce a distorted objective function, which varies weights of the submodular component g and the modular component ℓ during the iterations of the algorithm. By combining the weighting technique and measured continuous greedy algorithm, we present an algorithm for the matroid-constrained problem, which has a provable approximation guarantee. In the cardinality-constrained case, we utilize random greedy algorithm and sampling technique together with the weighting technique to design two efficient algorithms. Moreover, we consider the unconstrained problem and propose a much simpler and faster algorithm compared with the algorithms for solving the problem with a cardinality constraint. [ABSTRACT FROM AUTHOR]

We study the Wilson line defect half-indices of 3d N = 2 supersymmetric SU(N) Chern-Simons theories of level k ≤ – N with Neumann boundary conditions for the gauge fields, together with 2d Fermi multiplets and fundamental 3d chiral multiplets to cancel the gauge anomaly. We derive some exact results and also make some conjectures based on expansions of the q-series. We find several interesting connections with special functions known in the literature, including Rogers-Ramanujan functions for which we conjecture integral representations, and the appearance of Appell-Lerch sums for certain Wilson line half-index grand canonical ensembles which reveal an unexpected appearance of mock modular functions. We also find intriguing q-difference equations relating half-indices to Wilson line half-indices. Some of these results also have a description in terms of a dual theory with Dirichlet boundary conditions for the vector multiplet in the dual theory. [ABSTRACT FROM AUTHOR]

In this paper, we present a five-step iterative scheme, at the beginning, it seems complicated and difficult to implement, but in fact it’s not. This scheme is constructed for (λ,ρ) - firmly nonexpansive mappings in modular function spaces. Two different ρ-convergences result has been proved for double schemes under consideration. In our study there is a comparison between these cases through answering the question "which one is faster?” Finally, numerical examples are given by using MATLAB software program. [ABSTRACT FROM AUTHOR]

Our results investigate mock theta functions and quantum modular forms via quantum q-series identities. After Lovejoy, quantum q-series identities are such that they do not hold as an equality between power series inside the unit disc in the classical sense, but do hold at dense sets of roots of unity on the boundary. We establish several general (multivariable) quantum q-series identities and apply them to various settings involving (universal) mock theta functions. As a consequence, we surprisingly show that limiting, finite, universal mock theta functions at roots of unity for which their infinite counterparts do not converge are quantum modular. Moreover, we show that these finite limiting universal mock theta functions play key roles in (generalized) Ramanujan radial limits. A further corollary of our work reveals that the finite Kontsevich–Zagier series is a kind of "universal quantum mock theta function," in that it may be used to evaluate odd-order Ramanujan mock theta functions at roots of unity. (We also offer a similar result for even-order mock theta functions.) Finally, to complement the notion of a quantum q-series identity and the results of this paper, we also define what we call an "antiquantum q-series identity' and offer motivating general results with applications to third-order mock theta functions. [ABSTRACT FROM AUTHOR]

In this work, we present a new weighted method for proving the generalized Hyers-Ulam stability for nonlinear Volterra integral equations in modular spaces. Using the same technique, we also prove the generalized Hyers-Ulam stability for nonlinear functional equations under Δ2 conditions. Fixed-point theorems in modular spaces form the foundation of our main conclusions. [ABSTRACT FROM AUTHOR]

We show that for every finite set of prime numbers $S$ , there are at most finitely many singular moduli that are $S$ -units. The key new ingredient is that for every prime number $p$ , singular moduli are $p$ -adically disperse. We prove analogous results for the Weber modular functions, the $\lambda$ -invariants and the McKay–Thompson series associated with the elements of the monster group. Finally, we also obtain that a modular function that specializes to infinitely many algebraic units at quadratic imaginary numbers must be a weak modular unit. [ABSTRACT FROM AUTHOR]

The Petersson inner product is a natural inner product on the space of modular invariant functions. We derive a formula, written as a convergent sum over elementary functions, for the inner product Es(G, B) of the real analytic Eisenstein series and a general point in Narain moduli space. We also discuss the utility of the Petersson inner product as a distance measure on the space of 2d CFTs, and apply our procedure to evaluate this distance in various examples. [ABSTRACT FROM AUTHOR]

We describe solutions of asymptotically AdS3 Einstein gravity that are sourced by the insertion of operators in the boundary CFT2, whose dimension scales with the central charge of the theory. Previously, we found that the geometry corresponding to a black hole two-point function is simply related to an infinite covering of the Euclidean BTZ black hole [1]. However, here we find that the geometry sourced by the presence of a third black hole operator turns out to be a Euclidean wormhole with two asymptotic boundaries. We construct this new geometry as a quotient of empty AdS3 realized by domes and doors. The doors give access to the infinite covers that are needed to describe the insertion of the operators, while the domes describe the fundamental domains of the quotient on each cover. In particular, despite the standard fact that the Fefferman-Graham expansion is single-sided, the extended bulk geometry contains a wormhole that connects two asymptotic boundaries. We observe that the two-sided wormhole can be made single-sided by cutting off the wormhole and gluing on a "Lorentzian cap". In this way, the geometry gives the holographic description of a three-point function, up to phases. By rewriting the metric in terms of a Liouville field, we compute the on-shell action and find that the result matches with the Heavy-Heavy-Heavy three-point function predicted by the modular bootstrap. Finally, we describe the geometric transition between doors and defects, that is, when one or more dual operators describe a conical defect insertion, rather than a black hole insertion. [ABSTRACT FROM AUTHOR]

We consider a two-stage submodular maximization problem subject to a cardinality constraint and k matroid constraints, where the objective function is the expected difference of a nonnegative monotone submodular function and a nonnegative monotone modular function. We give two bi-factor approximation algorithms for this problem. The first is a deterministic $\left({{1 \over {k + 1}}\left({1 - {1 \over {{e^{k + 1}}}}} \right),1} \right)$ -approximation algorithm, and the second is a randomized $\left({{1 \over {k + 1}}\left({1 - {1 \over {{e^{k + 1}}}}} \right) - \varepsilon ,1} \right)$ -approximation algorithm with improved time efficiency. [ABSTRACT FROM AUTHOR]

We study Apollonian circle packings using the properties of a certain rank 4 indefinite Kac-Moody root system \Phi. We introduce the generating function Z(\mathbf {s}) of a packing, an exponential series in four variables with an Apollonian symmetry group, which is a symmetric function for \Phi. By exploiting the presence of affine and Lorentzian hyperbolic root subsystems of \Phi, with automorphic Weyl denominators, we express Z(\mathbf {s}) in terms of Jacobi theta functions and the Siegel modular form \Delta _5. We also show that the domain of convergence of Z(\mathbf {s}) is the Tits cone of \Phi, and discover that this domain inherits the intricate geometric structure of Apollonian packings. [ABSTRACT FROM AUTHOR]

In 2021 da Silva, Hirschhorn, and Sellers studied a wide variety of congruences for the k -elongated plane partition function d k (n) by various primes. They also conjectured the existence of an infinite congruence family modulo arbitrarily high powers of 2 for the function d 7 (n). We prove that such a congruence family exists — indeed, for powers of 8. The proof utilizes only classical methods, i.e. integer polynomial manipulations in a single function, in contrast to all other known infinite congruence families for d k (n) which require more modern methods to prove. [ABSTRACT FROM AUTHOR]

We prove Rogers–Ramanujan-type identities for the Nahm sums associated with the tadpole Cartan matrix of rank 3. These identities reveal the modularity of these sums, and thereby we confirm a conjecture of Calinescu, Penn and the first author in this case. We show that these Nahm sums together with some shifted sums can be combined into a vector-valued modular function on the full modular group. We also present some conjectures for a general rank. [ABSTRACT FROM AUTHOR]

Objective Scatter Index (OSI) and the Modular Transfer Function (MTF) by HD Analyzer®. Two QoV questionnaires were applied to patients in which both eyes were included: the McAlinden and the Catquest-9 SF. Results: We included 50 eyes (28 patients). The mean follow-up time was 5.4± 1.0 years. Comparing to month 1 after surgery, at the last follow-up visit, there was a decrease in the uncorrected visual acuity (0.14± 0.13 vs 0.08± 0.09 LogMAR, p=0.024), a negative increase in the spherical equivalent (− 0.31± 0.60 vs − 0.02± 0.20, p=0.006) and no changes in the best-corrected visual acuity (p> 0.999). An uncorrected near visual acuity of at least J2 was achieved in 89% of eyes one month after surgery and in 91% of eyes at the last follow-up visit (p=0.829). At the last follow-up, the mean OSI was 5.1± 1.8 and the mean MTF was 17.5± 10.6. Some degree of near vision difficulty was reported by 91% of patients, and 74% of patients reported photic phenomena (halos, glare, starbursts). However, most patients reported that these symptoms caused none to little bothersome. At the last follow-up, 87% of patients were at least fairly satisfied with the surgery. Conclusion: Even after a mean follow-up time of 5 years, patients maintained good uncorrected visual acuity. Even though most patients experienced some degree of near vision difficulty and visual symptoms, globally, our patients were satisfied with their current vision, and the experienced symptoms did not have a significant impact on their daily lives. [ABSTRACT FROM AUTHOR]

Featured Application: TraumaMeter v.873 was designed to increase the accuracy of specific measurements at the spine level. The system meets the characteristics of open-source and modular functions and has been extended to perform angular measurements on lower extremity X-ray studies with high reliability. To establish classifications and to obtain pre- and post-operative information on patient-specific alignments, it is necessary to measure different angular values accurately and precisely, mainly on weight-bearing, full-length anteroposterior X-rays of the lower limbs (LLRs). This study evaluated angular measurements' validity and absolute reliability on LLRs with a self-developed, computer-aided measurement system (TraumaMeter v.873). Eight independent observers measured the preoperative mechanical hip-knee-ankle (mHKA) angle of 52 lower extremities (26 cases) in a blinded fashion on three occasions separated by two weeks. We obtained an intra-observer mean bias error (MBE) of 0.40°, a standard deviation (SD) of 0.11°, and a 95% confidence interval (CI) of 0.37°–0.43°. We also obtained an inter-observer MBE of 0.49°, an SD of 0.15°, and a 95% C of 0.45°–0.53°. The intra-observer MBE for the measurement pair between the second and the first measurement round (T2T1) was 0.43°, the SD was 0.13°, and the 95% CI was 0.39°–0.47°; the MBE between the third and the second round (T3T2) was 0.37°, with an SD of 0.10° and a 95% CI of 0.34°–0.40°; and the MBE between the third and the first round (T3T1) was 0.40°, with an SD of 0.10° and a 95% CI of 0.37°–0.43°. The interobserver MBE for the first round of measurements was 0.52°, with an SD of 0.16° and a 95% CI of 0.48°–0.56°; the MBE for the second round was 0.50°, with an SD of 0.15° and a 95% CI of 0.46°–0.54°; and the MBE for the third round was 0.46°, with an SD of 0.14° and a 95% CI of 0.42°–0.50°. There were no statistically significant differences in the inter-observer errors for the three tests. In the case of the intra-observer analysis, there were differences between T2T1 and between T3T2, but these differences were minimal, with no overlaps in the lower or upper values, respectively, of the confidence intervals. These results led us to conclude that the TraumaMeter v.873 software extension for measuring lower-limb angles in LLRs is an accurate tool with low intra- and inter-observer variability. [ABSTRACT FROM AUTHOR]

Viazovska's solution of the sphere packing problem in eight dimensions is based on a remarkable construction of certain special functions using modular forms. Great mathematics has consequences far beyond the problems that originally inspired it, and Viazovska's work is no exception. In this article, we'll examine how it has led to new interpolation theorems in Fourier analysis, specifically a theorem of Radchenko and Viazovska. [ABSTRACT FROM AUTHOR]

Simple Summary: The health of Amphioctopus fangsiao, a species used in aquaculture or fish farming, can be greatly affected by infections. To understand how their immune system responds, we looked at the changes in their immune cells when exposed to substances that mimic these infections. By studying these responses, we were able to identify key parts of their immune system that change when faced with infection-causing threats. We also discovered important genes, including PKMYT1 (protein kinase, membrane associated tyrosine/threonine 1) and NAMPT (nicotinamide phosphoribosyltransferase), that play a crucial role in this response. Our study gives us a deeper understanding of the immune system of the Amphioctopus fangsiao. The primary influencer of aquaculture quality in Amphioctopus fangsiao is pathogen infection. Both lipopolysaccharides (LPS) and polyinosinic:polycytidylic acid (Poly I:C) are recognized by the pattern recognition receptor (PRR) within immune cells, a system that frequently serves to emulate pathogen invasion. Hemolymph, which functions as a transport mechanism for immune cells, offers vital transcriptome information when A. fangsiao is exposed to pathogens, thereby contributing to our comprehension of the species' immune biological mechanisms. In this study, we conducted analyses of transcript profiles under the influence of LPS and Poly I:C within a 24 h period. Concurrently, we developed a Weighted Gene Co-expression Network Analysis (WGCNA) to identify key modules and genes. Further, we carried out Gene Ontology (GO) and Kyoto Encyclopedia of Genes and Genomes (KEGG) enrichment analyses to investigate the primary modular functions. Co-expression network analyses unveiled a series of immune response processes following pathogen stress, identifying several key modules and hub genes, including PKMYT1 and NAMPT. The invaluable genetic resources provided by our results aid our understanding of the immune response in A. fangsiao hemolymph and will further our exploration of the molecular mechanisms of pathogen infection in mollusks. [ABSTRACT FROM AUTHOR]

In this work we start from near vector spaces, which we endow with some additional properties that allow convex analysis. The seminormed structure used here will also be improved by adding properties such as the null condition and null equality, thus resulting in a new type of space, which is still weaker than the conventional Banach structures: pre-convex regular near-Banach space. On the newly defined structure, we introduce the concept of uniform convexity and analyze several resulting properties. The major outcomes prove a remarkable resemblance to the classical properties resulting from uniform convexity on hyperbolic metric spaces or modular function spaces, including the famous Browder-Gohde fixed point theorem. [ABSTRACT FROM AUTHOR]

Prime factorization is a procedure of determining the prime factors of a given number N that requires super-polynomial time for conventional digital computers. Peter Shor developed a polynomial-time algorithm for quantum computers. Period finding is the key part of the algorithm, which is accomplished with the help of quantum superposition of states and quantum entanglement. The period finding can be also accomplished using classical wave superposition. In this study, we present experimental data obtained on a multi-port spin wave interferometer made of Y3Fe2(FeO4)3. Number 817 was factorized by a sequence of phase measurements. We also present the results of numerical modeling on the prime factorization of larger numbers 334 597 , 1 172 693 , 3 377 663 , and 9 363 239. The results of numerical modeling reveal significant shortcomings of the period-based approach. The major problems are associated with an inability to predict the period of the modular function, significant overhead over classical digital computers in some cases, and phase accuracy requirements. We argue that the same problems are inherent in classical analog and quantum computers. [ABSTRACT FROM AUTHOR]

This article discusses the scientific publication rate in research on disorders of consciousness (DoC). The authors note that there has been an increase in publications on the topic, particularly in the use of neuroimaging techniques such as fMRI, PET scans, and EEG studies. However, they also observe a decrease in publication rates since 2013-2015. The authors suggest that this decline may be due to various factors, including limitations in data interpretation and the cost-benefit ratio of advanced technologies. They emphasize the need for further research and the application of advanced technologies in understanding consciousness. [Extracted from the article]

Motivation 3D chromatin structure plays an important role in regulating gene expression and alterations to this structure can result in developmental abnormalities and disease. While genomic approaches like Hi-C and Micro-C can provide valuable insights in 3D chromatin architecture, the resulting datasets are extremely large and difficult to manipulate. Results Here, we present mariner , a rapid and memory efficient tool to extract, aggregate, and plot data from Hi-C matrices within the R/Bioconductor environment. Mariner simplifies the process of querying and extracting contacts from multiple Hi-C files using a parallel and block-processing approach. Modular functions allow complete workflow customization for advanced users, yet all-in-one functions are available for running the most common types of analyses. Finally, tight integration with existing Bioconductor infrastructure enables complete analysis and visualization of Hi-C data in R. Availability and implementation Available on GitHub at https://github.com/EricSDavis/mariner and on Bioconductor at https://www.bioconductor.org/packages/release/bioc/html/mariner.html. [ABSTRACT FROM AUTHOR]

Let p be a prime number. Continuing and extending our previous paper with the same title, we prove explicit rates of overconvergence for modular functions of the form E k ∗ V (E k ∗) where E k ∗ is a classical, normalized Eisenstein series on Γ 0 (p) and V the p-adic Frobenius operator. In particular, we extend our previous paper to the primes 2 and 3. For these primes our main theorem improves somewhat upon earlier results by Emerton, Buzzard and Kilford, and Roe. We include a detailed discussion of those earlier results as seen from our perspective. We also give some improvements to our earlier paper for primes p ≥ 5 . Apart from establishing these improvements, our main purpose here is also to show that all of these results can be obtained by a uniform method, i.e., a method where the main points in the argumentation is the same for all primes. We illustrate the results by some numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of triangular ideal relative convergence for double sequences of functions defined on a modular space. Based upon this new convergence method, we prove Korovkin theorems. Then, we construct an example such that our new approximation results work. Finally, we discuss the reduced results which are obtained by special choices. [ABSTRACT FROM AUTHOR]

We investigate the existence of solutions for the Fredholm integral equation Φ (ϑ) = G (ϑ , Φ (ϑ)) + ∫ 0 1 F (ϑ , ζ , Φ (ζ)) d ζ , for ϑ ∈ [ 0 , 1 ] , in the setting of the modular function spaces L ρ . We also derive an application of this research within the framework of variable exponent Lebesgue spaces L p (·) subject to specific conditions imposed on the exponent function p (·) and the functions F and G. [ABSTRACT FROM AUTHOR]

We prove the modular convexity of the mixed norm L p (ℓ 2) on the Sobolev space W 1 , p (Ω) in a domain Ω ⊂ R n under the sole assumption that the exponent p (x) is bounded away from 1, i.e., we include the case sup x ∈ Ω p (x) = ∞ . In particular, the mixed Sobolev norm is uniformly convex if 1 < inf x ∈ Ω p (x) ≤ sup x ∈ Ω p (x) < ∞ and W 0 1 , p (Ω) is uniformly convex. [ABSTRACT FROM AUTHOR]

Generalised Eisenstein series are non-holomorphic modular invariant functions of a complex variable, τ, subject to a particular inhomogeneous Laplace eigenvalue equation on the hyperbolic upper-half τ-plane. Two infinite classes of such functions arise quite naturally within different string theory contexts. A first class can be found by studying the coefficients of the effective action for the low-energy expansion of type IIB superstring theory, and relatedly in the analysis of certain integrated four-point functions of stress tensor multiplet operators in N = 4 supersymmetric Yang-Mills theory. A second class of such objects is known to contain all two-loop modular graph functions, which are fundamental building blocks in the low-energy expansion of closed-string scattering amplitudes at genus one. In this work, we present a Poincaré series approach that unifies both classes of generalised Eisenstein series and manifests certain algebraic and differential relations amongst them. We then combine this technique with spectral methods for automorphic forms to find general and non-perturbative expansions at the cusp τ → i∞. Finally, we find intriguing connections between the asymptotic expansion of these modular functions as τ → 0 and the non-trivial zeros of the Riemann zeta function. [ABSTRACT FROM AUTHOR]

In monstrous moonshine, the replication formula and the Hecke operator played a central role. We generalize the replication formula and the Hecke operator to higher genus modular curves, with an eye toward extending moonshine to these cases. Specifically, we extend the definitions of replicates and a Hecke operator to harmonic Maass functions on modular curves of higher genera and obtain uniform proofs for numerous arithmetic properties of Fourier coefficients of modular functions of arbitrary level, which have been proved only for special cases of curves of genus zero or small prime levels. [ABSTRACT FROM AUTHOR]

Celebrating 100 years of the Banach contraction principle, we prove some fixed point theorems having all ingredients of the principle, but dealing with common fixed points of a contractive semigroup of nonlinear mappings acting in a modulated topological vector space. This research follows the ideas of the author's recent papers ['On modulated topological vector spaces and applications', Bull. Aust. Math. Soc. 101 (2020), 325–332, and 'Normal structure in modulated topological vector spaces', Comment. Math. 60 (2020), 1–11]. Modulated topological vector spaces generalise, among others, Banach spaces and modular function spaces. The interest in modulars reflects the fact that the notions of 'norm like' but 'noneuclidean' (and not even necessarily convex) constructs to measure a level of proximity between complex objects are frequently used in science and technology. To prove our fixed point results in this setting, we introduce a new concept of Opial sets using analogies with the norm-weak and modular versions of the Opial property. As an example, the results of this work can be applied to spaces like $L^p$ for $p> 0 $ , variable Lebesgue spaces $L^{p(\cdot)}$ where $1 \leq p(t) , Orlicz and Musielak–Orlicz spaces. [ABSTRACT FROM AUTHOR]

We study the modularity of the function u (τ) = C (τ) C (2 τ) , where C (τ) is Ramanujan's cubic continued fraction. It is an analogue of Ramanujan's function k (τ) = r (τ) r (2 τ) 2 , where r (τ) is the Rogers–Ramanujan continued fraction. We first prove the modularity of u (τ) and express C (τ) and C (2 τ) in terms of u (τ). Subsequently, we find modular equations of u (τ) of level n for every positive integer n by using affine models of modular curves. Finally, we demonstrate that the value of u (τ) generates the ray class field over an imaginary quadratic field modulo 2 for some τ in an imaginary quadratic field. [ABSTRACT FROM AUTHOR]

A new canonical Hopf algebra called the quantum pseudo-Kähler plane is introduced. This quantum group can be viewed as a deformation quantization of the complex two-dimensional plane C 2 with a pseudo-Kähler metric, or as a complexified version of the well-known quantum plane Hopf algebra. A natural class of nicely-behaved representations of the quantum pseudo-Kähler plane algebra is defined and studied, in the spirit of the previous joint work of the author and Frenkel. The tensor square of a unique irreducible representation decomposes into the direct integral of the irreducibles, and the unitary decomposition map is expressed by a special function called the modular double compact quantum dilogarithm, used in the recent joint work of the author and Scarinci on the quantization of 3d gravity for positive cosmological constant case. Then, from the associativity of the tensor cube, and from the maps between the left and the right duals, we construct unitary operators forming a new representation of Kashaev's group of transformations of dotted ideal triangulations of punctured surfaces, as an analog of Kashaev's quantum Teichmüller theory. The present work thus inspires one to look for a Kashaev-type quantization of 3d gravity for positive cosmological constant. [ABSTRACT FROM AUTHOR]

Hadamard spaces have traditionally played important roles in geometry and geometric group theory. More recently, they have additionally turned out to be a suitable framework for convex analysis, optimization and non-linear probability theory. The attractiveness of these emerging subject fields stems, inter alia, from the fact that some of the new results have already found their applications both in mathematics and outside. Most remarkably, a gradient flow theorem in Hadamard spaces was used to attack a conjecture of Donaldson in Kähler geometry. Other areas of applications include metric geometry and minimization of submodular functions on modular lattices. There have been also applications into computational phylogenetics and image processing. We survey recent developments in Hadamard space analysis and optimization with the intention to advertise various open problems in the area. We also point out several fallacies in the existing proofs. [ABSTRACT FROM AUTHOR]

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In 2018, Liuquan Wang and Yifan Yang proved the existence of an infinite family of congruences for the smallest parts function corresponding to the third-order mock theta function ω (q) . Their proof took the form of an induction requiring 20 initial relations, and utilized a space of modular functions isomorphic to a free rank 2 Z [ X ] -module. This proof strategy was originally developed by Paule and Radu to study families of congruences associated with modular curves of genus 1. We show that Wang and Yang's family of congruences, which is associated with a genus 0 modular curve, can be proved using a single-variable approach, via a ring of modular functions isomorphic to a localization of Z [ X ] . To our knowledge, this is the first time that such an algebraic structure has been applied to the theory of partition congruences. Our induction is more complicated, and relies on sequences of functions which exhibit a somewhat irregular 5-adic convergence. However, the proof ultimately rests upon the direct verification of only 10 initial relations, and is similar to the classical methods of Ramanujan and Watson. [ABSTRACT FROM AUTHOR]

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We recently proved that, when integrating out the spacetime dependence with a certain integration measure, four-point correlators O 2 O 2 O p i O p i in 풩 = 4 supersymmetric Yang-Mills theory with SU(N) gauge group are governed by a universal Laplace-difference equation. Here O p i is a superconformal primary with charge p and degeneracy i. These physical observables, called integrated correlators, are modular-invariant functions of Yang-Mills coupling τ. The Laplace-difference equation is a recursion relation that relates integrated correlators of operators with different charges. In this paper, we introduce the generating functions for these integrated correlators that sum over the charge. By utilising the Laplace-difference equation, we determine the generating functions for all the integrated correlators, in terms of the initial data of the recursion relation. We show that the transseries of the integrated correlators in the large-p (i.e. large-charge) expansion for a fixed N consists of three parts: 1) is independent of τ, which behaves as a power series in 1/p, plus an additional log(p) term when i = j; 2) is a power series in 1/p, with coefficients given by a sum of the non-holomorphic Eisenstein series; 3) is a sum of exponentially decayed modular functions in the large-p limit, which can be viewed as a generalisation of the non-holomorphic Eisenstein series. When i = j, there is an additional modular function of τ that is independent of p and is fully determined in terms of the integrated correlator with p = 2. The Laplace-difference equation was obtained with a reorganisation of the operators that means the large-charge limit is taken in a particular way here. From these SL(2, ℤ)-invariant results, we also determine the generalised 't Hooft genus expansion and the associated large-p non-perturbative corrections of the integrated correlators by introducing λ = p g YM 2 . The generating functions have subtle differences between even and odd N, which have important consequences in the large-charge expansion and resurgence analysis. We also consider the generating functions of the integrated correlators for some fixed p by summing over N, and we study their large-N behaviour, as well as comment on the similarities and differences between the large-p expansion and the large-N expansion. [ABSTRACT FROM AUTHOR]

As a new ingredient for analyzing the fine structure of entanglement, we study the symmetry resolution of the modular flow of U(1)-invariant operators in theories endowed with a global U(1) symmetry. We provide a consistent definition of symmetry-resolved modular flow that is defined for a local algebra of operators associated to a sector with fixed charge. We also discuss the symmetry-resolved modular correlation functions and show that they satisfy the KMS condition in each symmetry sector. Our analysis relies on the factorization of the Hilbert space associated to spatial subsystems. We provide a toolkit for computing the symmetry-resolved modular correlation function of the charge density operator in free fermionic theories. As an application, we compute this correlation function for a 1 + 1-dimensional free massless Dirac field theory and find that it is independent of the charge sector at leading order in the ultraviolet cutoff expansion. This feature can be regarded as a charge equipartition of the modular correlation function. Although obtained for free fermions, these results may be of potential interest for bulk reconstruction in AdS/CFT. [ABSTRACT FROM AUTHOR]

String compactifications on an orbi-folded torus with complex structure give rise to chiral fermions, spontaneously broken CP, modular invariance. We show that this allows simple effective theories of flavour and CP where: i) the QCD angle vanishes; ii) the CKM phase is large; iii) quark and lepton masses and mixings can be reproduced up to order one coefficients. We implement such general paradigm in supersymmetry or supergravity, with modular forms or functions, with or without heavy colored states. [ABSTRACT FROM AUTHOR]

Cycle integrals of modular functions are expected to play a role in real quadratic analogue of singular moduli. In this paper, we extend the definition of cycle integrals of modular functions from real quadratic numbers to badly approximable numbers. We also give explicit representations of values of extended-cycle integrals for some cases. [ABSTRACT FROM AUTHOR]

The idea of enriched mappings in normed spaces is relatively a newer idea. In this paper, we initiate the study of enriched mappings in modular function spaces. We first introduce the concepts of enriched ρ -contractions and enriched ρ -Kannan mappings in modular function spaces. We then establish some Banach Contraction Principle type theorems for the existence of fixed points of such mappings in this setting. Our results for enriched ρ -contractions are generalizations of the corresponding results from Banach spaces to modular function spaces and those from contractions to enriched ρ -contractions. We make a first ever attempt to prove existence results for enriched ρ -Kannan mappings and deduce the result for ρ -Kannan mappings. Note that even ρ -Kannan mappings in modular function spaces have not been considered yet. We validate our main results by examples. [ABSTRACT FROM AUTHOR]

We consider the integrated correlators associated with four-point correlation functions O 2 O 2 O p i O p j in four-dimensional N = 4 supersymmetric Yang-Mills theory (SYM) with SU(N) gauge group, where O p i is a superconformal primary with charge (or dimension) p and the superscript i represents possible degeneracy. These integrated correlators are defined by integrating out spacetime dependence with a certain integration measure, and they can be computed via supersymmetric localisation. They are modular functions of complexified Yang-Mills coupling τ. We show that the localisation computation is systematised by appropriately reorganising the operators. After this reorganisation of the operators, we prove that all the integrated correlators for any N, with some crucial normalisation factor, satisfy a universal Laplace-difference equation (with the laplacian defined on the τ-plane) that relates integrated correlators of operators with different charges. This Laplace-difference equation is a recursion relation that completely determines all the integrated correlators, once the initial conditions are given. [ABSTRACT FROM AUTHOR]

On page 330 of his second notebook, Srinivasa Ramanujan recorded a P -Q modular equation of degree 45, proof of which has been given by Bruce C. Berndt via theory of modular forms. We in this paper, give a simple proof of the same using the identities of Ramanujan and also establish few new P -Q modular equations of degree 45. Further using these, we establish certain new modular equations of signature 3. [ABSTRACT FROM AUTHOR]

Pressing environmental research questions demand the integration of increasingly diverse and large‐scale ecological datasets as well as complex analytical methods, which require specialized tools and resources.Computational training for ecological and evolutionary sciences has become more abundant and accessible over the past decade, but tool development has outpaced the availability of specialized training. Most training for scripted analyses focuses on individual analysis steps in one script rather than creating a scripted pipeline, where modular functions comprise an ecosystem of interdependent steps. Although current computational training creates an excellent starting place, linear styles of scripting can risk becoming labor‐ and time‐intensive and less reproducible by often requiring manual execution. Pipelines, however, can be easily automated or tracked by software to increase efficiency and reduce potential errors. Ecology and evolution would benefit from techniques that reduce these risks by managing analytical pipelines in a modular, readily parallelizable format with clear documentation of dependencies.Workflow management software (WMS) can aid in the reproducibility, intelligibility and computational efficiency of complex pipelines. To date, WMS adoption in ecology and evolutionary research has been slow. We discuss the benefits and challenges of implementing WMS and illustrate its use through a case study with the targets r package to further highlight WMS benefits through workflow automation, dependency tracking and improved clarity for reviewers.Although WMS requires familiarity with function‐oriented programming and careful planning for more advanced applications and pipeline sharing, investment in training will enable access to the benefits of WMS and impart transferable computing skills that can facilitate ecological and evolutionary data science at large scales. [ABSTRACT FROM AUTHOR]

In our previous work (Y. Hashimoto, Selberg's zeta function for the modular group in the critical strip, Math. Nachr. 294 (2021) 1899–1904, https://doi.org/10.1002/mana.202000268), we proposed an upper bound of the logarithmic derivative of Selberg's zeta function for the modular group in the critical strip. This paper studies the growth of its square integral for the modular group, co-compact arithmetic groups derived from indefinite quaternion algebras and their subgroups. [ABSTRACT FROM AUTHOR]