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We consider natural variants of Lehmer's unresolved conjecture that Ramanujan's tau-function never vanishes. Namely, for $n>1$ we prove that $$\tau(n)\not \in \{\pm 1, \pm 3, \pm 5, \pm 7, \pm 691\}.$$ This result is an example of general theorems for newforms with trivial mod 2 residual Galois representation, which will appear in forthcoming work of the authors with Wei-Lun Tsai. Ramanujan's well-known congruences for $\tau(n)$ allow for the simplified proof in these special cases. We make use of the theory of Lucas sequences, the Chabauty-Coleman method for hyperelliptic curves, and facts about certain Thue equations., Comment: To appear in JNT Prime. For more general results, see arXiv:2005.10354

Wigner distribution is a tool for signal processing to obtain instantaneous spectrum of a signal. By using Wigner distribution analysis, another representation of the Euler product can be obtained for Dirichlet series of the Ramanujan tau function. From which, it can be proved that the Ramanujan tau function never become zero for all numbers.

We construct a broad class of solutions of the KP-I equation by using a reduced version of the Grammian form of the $\tau$-function. The basic solution is a linear periodic chain of lumps propagating with distinct group and wave velocities. More generally, our solutions are evolving linear arrangements of lump chains, and can be viewed as the KP-I analogues of the family of line-soliton solutions of KP-II. However, the linear arrangements that we construct for KP-I are more general, and allow degenerate configurations such as parallel or superimposed lump chains. We also construct solutions describing interactions between lump chains and individual lumps, and discuss the relationship between the solutions obtained using the reduced and regular Grammian forms.

A characterization of the Kadomtsev-Petviashvili hierarchy of type C (CKP) in terms of the KP tau-function is given. Namely, we prove that the CKP hierarchy can be identified with the restriction of odd times flows of the KP hierarchy on the locus of turning points of the second flow. The notion of CKP tau-function is clarified and connected with the KP tau function. Algebraic-geometrical solutions and in particular elliptic solutions are discussed in detail. The new identity for theta-functions of curves with holomorphic involution having fixed points is obtained., 44 pages, no figures

Let $\tau(n)$ be Ramanujan's tau function, defined by the discriminant modular form \[ \Delta(z) = q\prod_{j=1}^{\infty}(1-q^{j})^{24}\ =\ \sum_{n=1}^{\infty}\tau(n) q^n \,,q=e^{2\pi i z} \] (this is the unique holomorphic normalized cuspidal newform of weight 12 and level 1). Lehmer's conjecture asserts that $\tau(n)\neq 0$ for all $n\geq 1$; since $\tau(n)$ is multiplicative, it suffices to study primes $p$ for which $\tau(p)$ might possibly be zero. Assuming standard conjectures for the twisted symmetric power $L$-functions associated to $\tau$ (including GRH), we prove that if $x\geq 10^{50}$, then \[ \#\{x < p\leq 2x: \tau(p) = 0\} \leq 1.22 \times 10^{-5} \frac{x^{3/4}}{\sqrt{\log x}},\] a substantial improvement on the implied constant in previous work. To achieve this, under the same hypotheses, we prove an explicit version of the Sato-Tate conjecture for primes in arithmetic progressions., Comment: 16 pages, fixed typographical errors and minor computational details. To be published in the International Journal of Number Theory

Using the so-called Schubert decomposition, we present a finite-dimensional twisted description of the tau functions of the universal character (UC ) hierarchy using Grassmannians. Moreover, from the standpoint of the relation between the UC and Kadomtsev–Petviashvili hierarchies, we study the expansion of this tau function in terms of the action of Abelian groups on finite-dimensional Grassmannians. We use the Gekhtman–Kasman determinant formula involving exponentials of finite-dimensional matrices, which naturally leads to the structure of two finite Grassmannians. Using the Gekhtman–Kasman-type formula, we consider some concrete examples: rational, soliton, and mixed solutions.

The note corrects the aforementioned paper (Bertola in Commun Math Phys 294(2):539–579, 2010). The consequences of the correction are traced and the examples updated.

In this paper we report some explicit evolutionary PDEs of the Drinfeld-Sokolov hierarchy of type $$E_6^{(1)}$$ , and show how the unknown functions in these PDEs are related to the tau function. Moreover, for this hierarchy we compute its topological solution of formal series up to a certain degree, whose coefficients of monomials give the Fan-Jarvis-Ruan-Witten invariants for the E6 simple singularity. Based on such results we also derive several explicit evolutionary PDEs and some low-degree terms of the topological solution for the Drinfeld-Sokolov hierarchy of type $$F_4^{(1)}$$ .

Ramanujan's last letter to Hardy explored the asymptotic properties of modular forms, as well as those of certain interesting $q$-series which he called \emph{mock theta functions}. For his mock theta function $f(q)$, he claimed that as $q$ approaches an even order $2k$ root of unity $\zeta$, \[\lim_{q\to \zeta} \big(f(q) - (-1)^k (1-q)(1-q^3)(1-q^5)\cdots (1-2q + 2q^4 - \cdots)\big) = O(1),\] and hinted at the existence of similar statements for his other mock theta functions. Recent work of Folsom-Ono-Rhoades provides a closed formula for the implied constant in this radial limit of $f(q)$. Here, by different methods, we prove similar results for all of Ramanujan's 5th order mock theta functions. Namely, we show that each 5th order mock theta function may be related to a modular bilateral series, and exploit this connection to obtain our results. We further explore other mock theta functions to which this method can be applied., Comment: 15 Pages

Sayılar Teorisi denince akla gelen bilim insanlarından birisi olan Srinivasa Ramanujan sadece yaşadığı yüzyıla değil modern matematiğin tarihi zirve yaptığı 21. yüzyıla da süren etkisiyle çok önemli başarılara imza atmıştır. Altı bölümden oluşan bu çalışmada Ramanujan'ın sonuçlarına temel oluşturan Ramanujan tau fonksiyonu çalışılmıştır. Modüler formların tanıtılması ile başlayan çalışmanın ikinci bölümünde Ramanujan tau fonksiyonu tanımlanıp temel özellikleri incelenmiştir. Üçüncü bölümde parçalanış fonksiyonu verilmiş ve Ramanujan'ın bu fonksiyon üzerine önemli sonuçlar verdiği kongrüanslar incelenmiştir. Dördüncü bölüme Ramanujan konjektürleri incelenmiştir. Beşinci bölümde ise Ramanujan tau fonksiyonun Fourier katsayılarının asallığı üzerine yapılan çalışmalar üzerinde durulmuştur. Çalışmanın altıncı ve son bölümünde ise Ramanujan tau fonksiyonuna dair güncel bir çalışma incelenmiştir. Çalışma derleme niteliğindedir. Srinivasa Ramanujan, one of the scientists that comes to mind when it comes to NumberTheory, has achieved significant success not only in the century he lived in, but also in the 21st century, when modern mathematics reached its historical peak. In this study, which consists of six chapters, the Ramanujan tau function, which forms the basis for Ramanujan's results, is studied. In the second part of the study, which started with the introduction of modular forms, the Ramanujan tau function was defined and its basic properties were examined. In the third chapter, the partition function is given and the congruences on which Ramanujan gives important results on this function are examined. In the fourth chapter, Ramanujan conjectures are examined. In the fifth chapter, the studies on the primality of the Fourier coefficients of the Ramanujan tau function are emphasized. In the sixth and last part of the study, a recent study on the Ramanujan tau function was examined. The study is a compilation.

In this paper, we study monotone set functions defined as the composition of an additive measure with a strictly increasing function. This function is a unary operator in continuous-valued logic, called the tau function, and it is a generator function-based parametric mapping. We provide a necessary and sufficient condition for the equality of two tau functions that are induced by different generator functions. Using the tau function and its properties, we introduce a new monotone measure that we call the tau-additive measure. This measure is computationally simple and it can be viewed as an upper or lower probability depending on the parameter settings of the tau function. We present the parameter-dependent submodularity and supermodularity of the tau-additive measure and show how this measure can be constructed from a set function on a finite set. This procedure is analogous to how the well-known lambda-additive measure can be constructed, but our method is computationally simpler. We demonstrate that the tau-additive measure can be used to approximate the lambda-additive measure. Lastly, exploiting these theoretical results, we present an application in the area of human resource management.

In Alexandrov's work [4,5] it has been shown that the extended partition function exp⁡(Fo,ext+Fc) introduced by Buryak in [14,15] is a tau-function of the KP hierarchy. In this work, we compute the affine coordinates of this tau-function on the Sato Grassmannian, and rewrite the Virasoro constraints as recursions for the affine coordinates in the fermionic picture. As applications we derive some formulas for the extended partition function and the connected n-point functions using methods developed by Zhou in [48] based on the boson-fermion correspondence.

For an arbitrary solution to the Burgers--KdV hierarchy, we define the tau-tuple $(\tau_1,\tau_2)$ of the solution. We show that the product $\tau_1\tau_2$ admits Buryak's residue formula. Therefore, according to Alexandrov's theorem, $\tau_1\tau_2$ is a tau-function of the KP hierarchy. We then derive a formula for the affine coordinates for the point of the Sato Grassmannian corresponding to the tau-function $\tau_1\tau_2$ explicitly in terms of those for $\tau_1$. Applications to the analogous open extension of the generalized BGW tau-function and to the open partition function are given., Comment: Minor changes: improved presentations; corrected typos; added references. 21pages

The paper has two relatively distinct but connected goals; the first is to define the notion of Padé approximation of Weyl–Stiltjes transforms on an arbitrary compact Riemann surface of higher genus. The data consists of a contour in the Riemann surface and a measure on it, together with the additional datum of a local coordinate near a point and a divisor of degree g. The denominators of the resulting Padé-like approximation also satisfy an orthogonality relation and are sections of appropriate line bundles. A Riemann–Hilbert problem for a square matrix of rank two is shown to characterize these orthogonal sections, in a similar fashion to the ordinary orthogonal polynomial case. The second part extends this idea to explore its connection to integrable systems. The same data can be used to define a pairing between two sequences of line bundles. The locus in the deformation space where the pairing becomes degenerate for fixed degree coincides with the zeros of a “tau” function. We show how this tau function satisfies the Kadomtsev–Petviashvili hierarchy with respect to either deformation parameters, and a certain modification of the 2-Toda hierarchy when considering the whole sequence of tau functions. We also show how this construction is related to the Krichever construction of algebro-geometric solutions.

Using the embedding of the moduli space of generalized $GL(n)$ Hitchin’s spectral covers to the moduli space of meromorphic Abelian differentials we study the variational formulæ of the period matrix, the canonical bidifferential, the prime form and the Bergman tau function. This leads to residue formulæ which generalize the Donagi–Markman formula for variations of the period matrix. The computation of second derivatives of the period matrix reproduces the formula derived in [2] using the framework of topological recursion.

In this paper, we state some q-analogs of the famous Ramanujan Master Theorem. As an application, we compute some values of Jackson’s q-integrals that involve q-special functions.

We study the rational Picard group of the projectivized moduli space of holomorphic n-differentials on complex genus g stable curves. We define (n - 1) natural classes in this Picard group that we call Prym-Tyurin classes. We express these classes as linear combinations of boundary divisors and the divisor of n-differentials with a double zero. We give two different proofs of this result, using two alternative approaches: an analytic approach that involves the Bergman tau function and its vanishing divisor and an algebro-geometric approach that involves cohomological computations on the universal curve., 26 pages, 1 figure, comments are welcome

Using the bilinear formalism, we consider multicomponent and matrix Kadomtsev-Petviashvili hierarchies. The main tool is the bilinear identity for the tau function realized as the vacuum expectation value of a Clifford group element composed of multicomponent fermionic operators. We also construct the Baker-Akhiezer functions and obtain auxiliary linear equations that they satisfy.

In this work, we present a solution of the Boussinesq equation, which models shallow water waves. A powerful inverse scattering approach of Deift–Tomei–Trubowitz was published in 1982 for solving this equation in the Schwartz class. We rewrite the corresponding Lax pair in a matrix form and show that a recently developed theory of evolutionary nodes is applicable. As a result, we can locally solve this equation with analytic initial conditions on the line, providing a formula for the tau function, which defines where the solutions exist. More generally, arbitrary analytic solution is shown to arise from a vessel.

We exhibit a generating function of spin Hurwitz numbers analogous to (disconnected) double Hurwitz numbers that is a tau function of the two-component BKP (2-BKP) hierarchy and is a square root of a tau function of the two-component KP (2-KP) hierarchy defined by related Hurwitz numbers., Some minor changes, following the referee's report. To appear in Manuscripta Math

We show that if $E/\mathbb{Q}$ is an elliptic curve with a rational $p$-torsion for $p=2$ or $3$, then there is a congruence relation between Ramanujan's tau function and $E$ modulo $p$. We make use of such congruences to compute the Iwasawa invariants of $2$-adic and $3$-adic Mazur--Tate elements attached to Ramanujan's tau function. We also investigate numerically the Iwasawa invariants of the Mazur--Tate elements attached to an elliptic curve with additive reduction at a fixed prime number., To appear in the Ramanujan Journal

In this article, we consider the generalised two-parameter Cauchy two-matrix model and the corresponding integrable lattice equation. It is shown that with parameters chosen as $$1/k_i$$ , $$k_i\in {\mathbb {Z}}_{>0}$$ ( $$i=1,\,2$$ ), the average characteristic polynomials admit $$(k_1+k_2+2)$$ -term recurrence relations, which can be interpreted as spectral problems for integrable lattices. The tau function is then given by the partition function of the generalised Cauchy two-matrix model as well as Gram determinant. The simplest solvable example is given.

We prove a number of results regarding odd values of the Ramanujan $$\tau $$ -function. For example, we prove the existence of an effectively computable positive constant $$\kappa $$ such that if $$\tau (n)$$ is odd and $$n \ge 25$$ then either $$\begin{aligned} P(\tau (n)) \; > \; \kappa \cdot \frac{\log \log \log {n}}{\log \log \log \log {n}} \end{aligned}$$ or there exists a prime $$p \mid n$$ with $$\tau (p)=0$$ . Here P(m) denotes the largest prime factor of m. We also solve the equation $$\tau (n)=\pm 3^{b_1} 5^{b_2} 7^{b_3} 11^{b_4}$$ and the equations $$\tau (n)=\pm q^b$$ where $$3\le q < 100$$ is prime and the exponents are arbitrary nonnegative integers. We make use of a variety of methods, including the Primitive Divisor Theorem of Bilu, Hanrot and Voutier, bounds for solutions to Thue–Mahler equations due to Bugeaud and Győry, and the modular approach via Galois representations of Frey–Hellegouarch elliptic curves.

Post-translational modifications (PTMs) on tau have long been recognized as affecting protein function and contributing to neurodegeneration. The explosion of information on potential and observed PTMs on tau provides an opportunity to better understand these modifications in the context of tau homeostasis, which becomes perturbed with aging and disease. Prevailing views regard tau as a protein that undergoes abnormal phosphorylation prior to its accumulation into the toxic aggregates implicated in Alzheimer's disease (AD) and other tauopathies. However, the phosphorylation of tau may, in fact, represent part of the normal but interrupted function and catabolism of the protein. In addition to phosphorylation, tau undergoes another forms of post-translational modification including (but not limited to), acetylation, ubiquitination, glycation, glycosylation, SUMOylation, methylation, oxidation, and nitration. A holistic appreciation of how these PTMs regulate tau during health and are potentially hijacked in disease remains elusive. Recent studies have reinforced the idea that PTMs play a critical role in tau localization, protein-protein interactions, maintenance of levels, and modifying aggregate structure. These studies also provide tantalizing clues into the possibility that neurons actively choose how tau is post-translationally modified, in potentially competitive and combinatorial ways, to achieve broad, cellular programs commensurate with the distinctive environmental conditions found during development, aging, stress, and disease. Here, we review tau PTMs and describe what is currently known about their functional impacts. In addition, we classify these PTMs from the perspectives of protein localization, electrostatics, and stability, which all contribute to normal tau function and homeostasis. Finally, we assess the potential impact of tau PTMs on tau solubility and aggregation. Tau occupies an undoubtedly important position in the biology of neurodegenerative diseases. This review aims to provide an integrated perspective of how post-translational modifications actively, purposefully, and dynamically remodel tau function, clearance, and aggregation. In doing so, we hope to enable a more comprehensive understanding of tau PTMs that will positively impact future studies.

In the work of Mukhin and Varchenko from 2002 there was introduced a Wronskian map from the variety of full flags in a finite dimensional vector space into a product of projective spaces. We establish a precise relationship between this map and the Plücker map. This allows us to recover the result of Varchenko and Wright saying that the polynomials appearing in the image of the Wronsky map are the initial values of the tau-functions for the Kadomtsev-Petviashvili hierarchy.

This is a short review of the Kadomtsev-Petviashvili hierarchies of types B and C. The main objects are the $L$-operator, the wave operator, the auxiliary linear problems for the wave function, the bilinear identity for the wave function and the tau-function. All of them are discussed in the paper. The connections with the usual Kadomtsev-Petviashvili hierarchy (of the type A) are clarified. Examples of soliton solutions and the dispersionless limit of the hierarchies are also considered., Comment: 26 pages, no figures. arXiv admin note: text overlap with arXiv:2012.04482

Let (−A, B, C) be a linear system in continuous time t > 0 with input and output space \(\mathbb {C}\) and state space H. The function ϕ(x)(t) = Ce−(t+2x)AB determines a Hankel integral operator \(\Gamma _{\phi _{(x)}}\) on \(L^2((0, \infty ); \mathbb {C})\); if \(\Gamma _{\phi _{(x)}}\) is trace class, then the Fredholm determinant \(\tau (x)=\det (I+ \Gamma _{\phi _{(x)}})\) defines the tau function of (−A, B, C). Such tau functions arise in Tracy and Widom’s theory of matrix models, where they describe the fundamental probability distributions of random matrix theory. Dyson considered such tau functions in the inverse spectral problem for Schrodinger’s equation − f″ + uf = λf, and derived the formula for the potential \(u(x)=-2{{d^2}\over {dx^2}}\log \tau (x)\) in the self-adjoint scattering case (Commun Math Phys 47:171–183, 1976). This paper introduces a operator function Rx that satisfies Lyapunov’s equation \({{dR_x}\over {dx}}=-AR_x-R_xA\) and \(\tau (x)=\det (I+R_x)\), without assumptions of self-adjointness. When − A is sectorial, and B, C are Hilbert–Schmidt, there exists a non-commutative differential ring \({\mathcal A}\) of operators in H and a differential ring homomorphism \(\lfloor \,\,\rfloor :{\mathcal A}\to \mathbb {C}[u,u', \dots ]\) such that u = −4⌊A⌋, which extends the multiplication rules for Hankel operators considered by Poppe and McKean (Cent Eur J Math 9:205–243, 2011).

We study thermal correlation functions of Jackiw-Teitelboim (JT) supergravity. We focus on the case of JT supergravity on orientable surfaces without time-reversal symmetry. As shown by Stanford and Witten recently, the path integral amounts to the computation of the volume of the moduli space of super Riemann surfaces, which is characterized by the Brezin-Gross-Witten (BGW) tau-function of the KdV hierarchy. We find that the matrix model of JT supergravity is a special case of the BGW model with infinite number of couplings turned on in a specific way, by analogy with the relation between bosonic JT gravity and the Kontsevich-Witten (KW) model. We compute the genus expansion of the one-point function of JT supergravity and study its low-temperature behavior. In particular, we propose a non-perturbative completion of the one-point function in the Bessel case where all couplings in the BGW model are set to zero. We also investigate the free energy and correlators when the Ramond-Ramond flux is large. We find that by defining a suitable basis higher genus free energies are written exactly in the same form as those of the KW model, up to the constant terms coming from the volume of the unitary group. This implies that the constitutive relation of the KW model is universal to the tau-function of the KdV hierarchy., 36 pages, v2: typos corrected, footnotes and references added, v3: several explanations added, typos corrected, published version

Fordy and Xenitidis [ J. Phys. A: Math. Theor. 50 (2017) 165205. ( doi:10.1088/1751-8121/aa639a )] recently proposed a large class of discrete integrable systems which include a number of novel integrable difference equations, from the perspective of Z N graded Lax pairs, without providing solutions. In this paper, we establish the link between the Fordy–Xenitidis (FX) discrete systems in coprime case and linear integral equations in certain form, which reveals solution structure of these equations. The bilinear form of the FX integrable difference equations is also presented together with the associated general tau function. Furthermore, the solution structure explains the connections between the FX novel models and the discrete Gel’fand–Dikii hierarchy.

We study special circle bundles over two elementary moduli spaces of meromorphic quadratic differentials with real periods denoted by $\mathcal Q_0^{\mathbb R}(-7)$ and $\mathcal Q^{\mathbb R}_0([-3]^2)$. The space $\mathcal Q_0^{\mathbb R}(-7)$ is the moduli space of meromorphic quadratic differentials on the Riemann sphere with one pole of order 7 with real periods; it appears naturally in the study of a neighbourhood of the Witten's cycle $W_1$ in the combinatorial model based on Jenkins-Strebel quadratic differentials of $\mathcal M_{g,n}$. The space $\mathcal Q^{\mathbb R}_0([-3]^2)$ is the moduli space of meromorphic quadratic differentials on the Riemann sphere with two poles of order at most 3 with real periods; it appears in description of a neighbourhood of Kontsevich's boundary $W_{-1,-1}$ of the combinatorial model. The application of the formalism of the Bergman tau-function to the combinatorial model (with the goal of computing analytically Poincare dual cycles to certain combinations of tautological classes) requires the study of special sections of circle bundles over $\mathcal Q_0^{\mathbb R}(-7)$ and $\mathcal Q^{\mathbb R}_0([-3]^2)$; in the case of the space $\mathcal Q_0^{\mathbb R}(-7)$ a section of this circle bundle is given by the argument of the modular discriminant. We study the spaces $\mathcal Q_0^{\mathbb R}(-7)$ and $\mathcal Q^{\mathbb R}_0([-3]^2)$, also called the spaces of Boutroux curves, in detail, together with corresponding circle bundles., Comment: 34 pages, 17 figures; minor linguistic/aesthetic improvements

We use a q-series identity by Ramanujan to give a combinatorial interpretation of Ramanujan's tau function which involves t-cores and a new class of partitions which we call (m,k)-capsids. The same method can be applied in conjunction with other related identities yielding alternative combinatorial interpretations of the tau function., 14 pages, 3 figures

Sato introduced the tau-function to describe solutions to a wide class of completely integrable differential equations. Later Segal-Wilson represented it in terms of the relevant integral operators on Hardy space of the unit disc. This paper gives another representation of the tau-functions by the Weyl functions for 1d Schr\"odinger operators with real valued potentials, which will make it possible to extend the class of initial data for the KdV equation to more general one., Comment: 36 pages, 1 figure erased

Background & Objective: Neurodegenrative diseases are among the most widespread lifethreatening disorders around the world in elderly ages. The common feature of a group of neurodegenerative disorders, called tauopathies, is an accumulation of microtubule associated protein tau inside the neurons. The exact mechanism underlying tauopathies is not well-understood but several factors such as traumatic brain injuries and genetics are considered as potential risk factors. Although tau protein is well-known for its key role in stabilizing and organization of axonal microtubule network, it bears a broad range of functions including DNA protection and participation in signaling pathways. Moreover, the flexible unfolded structure of tau facilitates modification of tau by a wide range of intracellular enzymes which in turn broadens tau function and interaction spectrum. The distinctive properties of tau protein concomitant with the crucial role of tau interaction partners in the progression of neurodegeneration suggest tau and its binding partners as potential drug targets for the treatment of neurodegenerative diseases. Conclusion: This review aims to give a detailed description of structure, functions and interactions of tau protein in order to provide insight into potential therapeutic targets for treatment of tauopathies.

We show that the dual partition function of the pure $\mathcal N=2$ $SU(2)$ gauge theory in the self-dual $\Omega$-background (a) is given by Fredholm determinant of a generalized Bessel kernel and (b) coincides with the tau function associated to the general solution of the Painleve III equation of type $D_8$ (radial sine-Gordon equation). In particular, the principal minor expansion of the Fredholm determinant yields Nekrasov combinatorial sums over pairs of Young diagrams.

Hafner and Stopple proved a conjecture of Zagier on the asymptotic expansion of a Lambert series involving Ramanujan’s tau function with the main term involving the nontrivial zeros of the Riemann zeta function. Recently, Chakraborty et. al. have extended this result to any cusp form over the full modular group and also over any congruence subgroup. The aim here is to study the asymptotic behaviour of a similar Lambert series involving the coefficients of Maass cusp forms over the full modular group.

In this paper, the author proves some basic hypergeometric series which utilizes the same ideas that Margaret Jackson used to give a proof of Ramanujan’s [Formula: see text] summation formula.

We describe the magnetic phase of SU(N) $${\mathcal {N}}=2$$ super-Yang–Mills theories in the self-dual $$\Omega $$ -background in terms of a new class of multi-cut matrix models. These arise from a non-perturbative completion of topological strings in the dual four-dimensional limit which engineers the gauge theory in the strongly coupled magnetic frame. The corresponding spectral determinants provide natural candidates for the $$\tau $$ -functions of isomonodromy problems for flat spectral connections associated with the Seiberg–Witten geometry.

The modular function h ( τ ) = q ∏ n = 1 ∞ ( 1 − q 16 n ) 2 ( 1 − q 2 n ) ( 1 − q n ) 2 ( 1 − q 8 n ) is called a level 16 analogue of Ramanujan's series for 1 / π . We prove that h ( τ ) generates the field of modular functions on Γ 0 ( 16 ) and find its modular equation of level n for any positive integer n. Furthermore, we construct the ray class field K ( h ( τ ) ) modulo 4 over an imaginary quadratic field K for τ ∈ K ∩ H such that Z [ 4 τ ] is the integral closure of Z in K, where H is the complex upper half plane. For any τ ∈ K ∩ H , it turns out that the value 1 / h ( τ ) is integral, and we can also explicitly evaluate the values of h ( τ ) if the discriminant of K is divisible by 4.

We study multimatrix models, which may be viewed as integrals of products of tau functions depending on the eigenvalues of products of random matrices. We consider tau functions of the two-component Kadomtsev–Petviashvili (KP) hierarchy (semi-infinite relativistic Toda lattice) and of the B-type KP (BKP) hierarchy introduced by Kac and van de Leur. Such integrals are sometimes tau functions themselves. We consider models that generate Hurwitz numbers HE,F, where E is the Euler characteristic of the base surface and F is the number of branch points. We show that in the case where the integrands contain the product of n > 2 matrices, the integral generates Hurwitz numbers with E ≤ 2 and F ≤ n+2. Both the numbers E and F depend both on n and on the order of the factors in the matrix product. The Euler characteristic E can be either an even or an odd number, i.e., it can match both orientable and nonorientable (Klein) base surfaces depending on the presence of the tau function of the BKP hierarchy in the integrand. We study two cases, the products of complex and the products of unitary matrices.

In this paper, we construct a new integrable equation called Mobius-Toda equation which is a generalization of q -Toda equation. Meanwhile its soliton solutions are constructed to show its integrable property. Further the Lax pairs of the Mobius-Toda equation and a whole integrable Mobius-Toda hierarchy are also constructed. To show the integrability, the bi-Hamiltonian structure and tau symmetry of the Mobius-Toda hierarchy are given and this leads to the existence of the tau function.

We use Ramanujan’s mixed modular equations to evaluate some new values of the parameter $$J_n$$ involving Ramanujan’s theta-function $$f(-q)$$ . The values $$J_n$$ are then used to evaluate new explicit values of the Ramanujan–Selberg continued fraction.

Tau is an intrinsically disordered protein with a central role in the pathology of a number of neurodegenerative diseases. Tau normally functions to stabilize neuronal microtubules, although the mechanism underlying this function is not well understood. Of note is that the interaction between tau and soluble tubulin, which has implications both in understanding tau function as well as its role in disease, is underexplored. Here we investigate the relationship between heterogeneity in tau-tubulin complexes and tau function. Specifically, we created a series of truncated and scrambled tau constructs and characterized the size and heterogeneity of the tau-tubulin complexes formed under nonpolymerizing conditions. Function of the constructs was verified by tubulin polymerization assays. We find that, surprisingly, the pseudo-repeat region of tau, which flanks the core microtubule-binding domain of tau, contributes largely to the formation of large, heterogeneous tau tubulin complexes; additional independent tubulin binding sites exist in repeats two and three of the microtubule binding domain. Of particular interest is that we find positive correlation between the size and heterogeneity of the complexes and rate of tau-promoted microtubule polymerization. We propose that tau-tubulin can be described as a “fuzzy” complex, and our results demonstrate the importance of heterogeneous complex formation in tau function. This work provides fundamental insights into the functional mechanism of tau, and more broadly underscores the relevance of heterogeneous and dynamic complexes in the functions of intrinsically disordered proteins.

For any non-negative integer n and non-zero integer r, let $$p_r(n)$$ denote Ramanujan’s general partition function. In this paper, we prove many infinite families of congruences modulo 7 for the general partition function $$p_r(n)$$ for negative values of r by using q-identities.