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Gukov–Pei–Putrov–Vafa conjectured the existence of q-series whose radial limits are Witten–Reshetikhin–Turaev invariants and called them homological blocks. For weakly negative definite plumbed 3-manifolds, Gukov–Pei–Putrov–Vafa and Gukov–Manolescu constructed homological blocks. In this paper, we construct indefinite false theta functions which are candidates of homological blocks for some plumbed 3-manifolds which are not weakly negative definite. Moreover we prove that, for the Poincaré homology sphere, our indefinite false theta function coincides with the original homological block. [ABSTRACT FROM AUTHOR]

We extend a certain type of identities on sums of I-Bessel functions on lattices, previously given by G. Chinta, J. Jorgenson, A. Karlsson and M. Neuhauser. Moreover we prove that, with continuum limit, the transformation formulas of theta functions such as the Dedekind eta function can be given by I-Bessel lattice sum identities with characters. We consider analogues of theta functions of lattices coming from linear codes and show that sums of I-Bessel functions defined by linear codes can be expressed by complete weight enumerators. We also prove that I-Bessel lattice sums appear as solutions of heat equations on general lattices. As a further application, we obtain an explicit solution of the heat equation on Z n whose initial condition is given by a linear code. [ABSTRACT FROM AUTHOR]

Recently, establishing the Hecke-type double sums forms for Eulerian series has received a great deal of attention to seek special cases of q-hypergeometric series. In this paper, with the aid of some basic hypergeometric series identities given by Chen and Wang, and Liu, we discover many new Eulerian series and express them in terms of Hecke-type double sums and Appell–Lerch sums. Based on the modern sense for the mock theta functions, we find that some of these series are mock theta functions and the others are mixed mock theta functions. Furthermore, from the viewpoint of Hecke-type double sums, we find three string function identities with levels 2, 4, and 6. [ABSTRACT FROM AUTHOR]

Mock theta functions were first introduced by Ramanujan in his last paper to Hardy. Moreover, some other mock theta functions were presented in his lost notebook. It is well known that all the classical mock theta functions can be expressed by the universal mock theta functions g 2 (x , q) and g 3 (x , q). In this paper, we establish some generalized mock theta functions and express them in terms of Appell–Lerch sums. Meanwhile, we show that some of Ramanujan's two-parameter mock theta functions are the special cases of these functions. [ABSTRACT FROM AUTHOR]

The Z-function is the real function given by Z (t) = e i θ (t) ζ 1 2 + i t , where ζ (s) is the Riemann zeta function, and θ (t) is the Riemann–Siegel theta function. The function, central to the study of the Riemann hypothesis (RH), has traditionally posed significant computational challenges. This research addresses these challenges by exploring new methods for approximating Z (t) and its zeros. The sections of Z (t) are given by Z N (t) : = ∑ k = 1 N cos (θ (t) − ln (k) t) k for any N ∈ N . Classically, these sections approximate the Z-function via the Hardy–Littlewood approximate functional equation (AFE) Z (t) ≈ 2 Z N ˜ (t) (t) for N ˜ (t) = t 2 π . While historically important, the Hardy–Littlewood AFE does not sufficiently discern the RH and requires further evaluation of the Riemann–Siegel formula. An alternative, less common, is Z (t) ≈ Z N (t) (t) for N (t) = t 2 , which is Spira's approximation using higher-order sections. Spira conjectured, based on experimental observations, that this approximation satisfies the RH in the sense that all of its zeros are real. We present a proof of Spira's conjecture using a new approximate equation with exponentially decaying error, recently developed by us via new techniques of acceleration of series. This establishes that higher-order approximations do not need further Riemann–Siegel type corrections, as in the classical case, enabling new theoretical methods for studying the zeros of zeta beyond numerics. [ABSTRACT FROM AUTHOR]

Recently, we established and used the generalized Littlewood theorem concerning contour integrals of the logarithm of analytical function to obtain new criteria equivalent to the Riemann hypothesis. The same theorem was subsequently applied to calculate certain infinite sums and study the properties of zeroes of a few analytical functions. In this article, we discuss what are, in a sense, inverse applications of this theorem. We first prove a Lemma that if two meromorphic on the whole complex plane functions f(z) and g(z) have the same zeroes and poles, taking into account their orders, and have appropriate asymptotic for large |z|, then for some integer n, d n ln (f (z)) d z n = d n ln (g (z)) d z n . The use of this Lemma enables proofs of many identities between elliptic functions, their transformation and n-tuple product rules. In particular, we show how exactly for any complex number a, ℘(z)-a, where ℘(z) is the Weierstrass ℘ function, can be presented as a product and ratio of three elliptic θ 1 functions of certain arguments. We also establish n-tuple rules for some elliptic theta functions. [ABSTRACT FROM AUTHOR]

We know that the degeneracy of solutions to PDEs, given in terms of theta functions on Riemann surfaces, provides important results about particular solutions, as in the case of the NLS equation. Here, we degenerate the so called finite gap solutions of the Toda lattice equation from the general formulation in terms of abelian functions when the gaps tend to points. This degeneracy allows us to recover the Sato formulas without using inverse scattering theory or geometric or representation theoretic methods. [ABSTRACT FROM AUTHOR]

We know that the degeneracy of solutions to PDEs, given in terms of theta functions on Riemann surfaces, provides important results about particular solutions, as in the case of the NLS equation. Here, we degenerate the so called finite gap solutions of the Toda lattice equation from the general formulation in terms of abelian functions when the gaps tend to points. This degeneracy allows us to recover the Sato formulas without using inverse scattering theory or geometric or representation theoretic methods.

In the well known planted clique problem, a clique (or alternatively, an independent set) of size k is planted at random in an Erdos-Renyi random G(n, p) graph, and the goal is to design an algorithm that finds the maximum clique (or independent set) in the resulting graph. We introduce a variation on this problem, where instead of planting the clique at random, the clique is planted by an adversary who attempts to make it difficult to find the maximum clique in the resulting graph. We show that for the standard setting of the parameters of the problem, namely, a clique of size k = n planted in a random G (n , 1 2) graph, the known polynomial time algorithms can be extended (in a non-trivial way) to work also in the adversarial setting. In contrast, we show that for other natural settings of the parameters, such as planting an independent set of size k = n 2 in a G(n, p) graph with p = n - 1 2 , there is no polynomial time algorithm that finds an independent set of size k, unless NP has randomized polynomial time algorithms. [ABSTRACT FROM AUTHOR]

We study pair correlation functions for planar Coulomb systems in the pushed phase, near a ring-shaped impenetrable wall. We assume coupling constant Γ = 2 and that the number n of particles is large. We find that the correlation functions decay slowly along the edges of the wall, in a narrow interface stretching a distance of order 1/n from the hard edge. At distances much larger than 1 / n , the effect of the hard wall is negligible and pair correlation functions decay very quickly, and in between sits an interpolating interface that we call the “semi-hard edge”. More precisely, we provide asymptotics for the correlation kernel K n (z , w) as n → ∞ in two microscopic regimes (with either | z - w | = O (1 / n) or | z - w | = O (1 / n) ), as well as in three macroscopic regimes (with | z - w | ≍ 1 ). For some of these regimes, the asymptotics involve oscillatory theta functions and weighted Szegő kernels. [ABSTRACT FROM AUTHOR]

We investigate long time asymptotics of the modified Camassa–Holm equation in three transition zones under a nonzero background. The first transition zone lies between the soliton region and the first oscillatory region, the second one lies between the second oscillatory region and the fast decay region, and possibly, the third one, namely, the collisionless shock region, that bridges the first transition region and the first oscillatory region. Under a low regularity condition on the initial data, we obtain Painlevé‐type asymptotic formulae in the first two transition regions, while the transient asymptotics in the third region involves the Jacobi theta function. We establish our results by performing a ∂¯$\bar{\partial }$ nonlinear steepest descent analysis to the associated Riemann–Hilbert problem. [ABSTRACT FROM AUTHOR]

We prove general fomulas for the deviations of two overpartition ranks from the average. These formulas are in terms of Appell–Lerch series and sums of quotients of theta functions and can be used, among other things, to recover any of the numerous overpartition rank difference identities in the literature. We give two illustrations. [ABSTRACT FROM AUTHOR]

We construct a bipartite generalization of Alon and Szegedy's nearly orthogonal vectors, thereby obtaining strong bounds for several extremal problems involving the Lovász theta function, vector chromatic number, minimum semidefinite rank, nonnegative rank, and extension complexity of polytopes. In particular, we answer a question from our previous work together with Letzter and Sudakov, while also addressing a question of Hrubeš and of Kwan, Sauermann, and Zhao. Along the way, we derive a couple of general lower bounds for the vector chromatic number which may be of independent interest. [ABSTRACT FROM AUTHOR]

This study seeks to establish connections between Ramanujan-type Eisenstein series and Borweins’ cubic theta functions, employing the (p, k)-parametrization method introduced by Alaca. Furthermore, the derived identities are applied to provide a novel representation for the evaluation of the discrete convolution sum P 2l+3m=n δ(i)δ(j). [ABSTRACT FROM AUTHOR]

S. Ramanujan in his last letter to G.H. Hardy has introduced seventeen mock theta functions of differenct orders (3,5 and 7) without explicitly mentioning the reason for his levelling of order, later Watson added to this set three more third order mock theta function. Watson also defined bilateral form of some mock theta function of order five and expressed them in terms of lerch transendetal functions. Patial mock theta functions of sixth order were studied by Y.S. Choi and partial mock theta functions of third order were studied by G.E. Andrews. In this paper we have given generalization of partial sixth and third order mock theta functions and it is shown that these generalized partial mock theta functions are Fq functions. q-integral representation of these generalized partial mock theta functions are also given. we have also expressed some bilateral mock theta functions in terms of lerch transcendental functions f(x, ξ; q, p). [ABSTRACT FROM AUTHOR]

We study meromorphic modular forms associated with positive definite binary quadratic forms and their cycle integrals along closed geodesics in the modular curve. We show that suitable linear combinations of these meromorphic modular forms have rational cycle integrals. Along the way we evaluate the cycle integrals of the Siegel theta function associated with an even lattice of signature (1, 2) in terms of Hecke's indefinite theta functions. [ABSTRACT FROM AUTHOR]

Let L be an even lattice of odd rank with discriminant group L ′ / L {L^{\prime}/L} , and let α , β ∈ L ′ / L {\alpha,\beta\in L^{\prime}/L} . We prove the Weil bound for the Kloosterman sums S α , β ⁢ ( m , n , c ) {S_{\alpha,\beta}(m,n,c)} of half-integral weight for the Weil Representation attached to L. We obtain this bound by proving an identity that relates a divisor sum of Kloosterman sums to a sparse exponential sum. This identity generalizes Kohnen’s identity for plus space Kloosterman sums with the theta multiplier system. [ABSTRACT FROM AUTHOR]

An elliptic curve E can be immersed in P N - 1 as a curve of degree N by means of the linear system of |NO|, where O is the origin of E. Well-known classical results going back to Bianchi and Klein say that if N is odd, this immersion is uniquely determined by specifying a full-level N structure. In this paper we show that if N is even, uniqueness of immersion is ensured by specifying a level structure associated with a certain congruence subgroup between Γ (N) and Γ (2 N) . Moreover, we construct, over the complex number field, an immersion by means of suitably chosen theta functions, and write down the quadratic equations satisfied by them. [ABSTRACT FROM AUTHOR]

We denote by $ \Delta _\nu $ Δ ν the Fubini-Study Laplacian perturbed by a uniform magnetic field whose strength is proportional to ν. When acting on bounded functions on the complex projective n-space, this operator has a discrete spectrum consisting on eigenvalues $ \beta _m, \ m\in \mathbb {Z}_+ $ β m , m ∈ Z + . For the corresponding eigenspaces, we give a new proof for their reproducing kernels by using Zaremba's expansion directly. These kernels are then used to obtain an integral representation for the heat kernel of $ \Delta _\nu $ Δ ν . Using a suitable polynomial decomposition of the multiplicity of each $ \beta _m $ β m , we write down a trace formula for the heat operator associated with $ \Delta _\nu $ Δ ν in terms of Jacobi's theta functions and their higher order derivatives. Doing so enables us to establish the asymptotics of this trace as $ t\searrow 0^+ $ t ↘ 0 + by giving the corresponding heat coefficients in terms of Bernoulli numbers and polynomials. The obtained results can be exploited in the analysis of the spectral zeta function associated with $ \Delta _\nu $ Δ ν . [ABSTRACT FROM AUTHOR]

In this paper, we focus on the truncations of three classical theta series of Euler and Gauss, and analyze their combinatorial properties which play a key role in proving these truncated identities. Several interesting partition identities are established bijectively. [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]

We show that sequences of the form |$\alpha n^{\theta } \pmod {1}$| with |$\alpha> 0$| and |$0 < \theta < \tfrac {43}{117} = \tfrac {1}{3} + 0.0341 \ldots $| have Poissonian pair correlation. This improves upon the previous result by Lutsko, Sourmelidis, and Technau, where this was established for |$\alpha> 0$| and |$0 < \theta < \tfrac {14}{41} = \tfrac {1}{3} + 0.0081 \ldots $|⁠. We reduce the problem of establishing Poissonian pair correlation to a counting problem using a form of amplification and the Bombieri–Iwaniec double large sieve. The counting problem is then resolved non-optimally by appealing to the bounds of Robert–Sargos and (Fouvry–Iwaniec–)Cao–Zhai. The exponent |$\theta = \tfrac {2}{5}$| is the limit of our approach. [ABSTRACT FROM AUTHOR]

We prove an equality, predicted in the physical literature, between the Jeffrey–Kirwan residues of certain explicit meromorphic forms attached to a quiver without loops or oriented cycles and its Donaldson–Thomas type invariants. In the special case of complete bipartite quivers we also show independently, using scattering diagrams and theta functions, that the same Jeffrey–Kirwan residues are determined by the the Gross–Hacking–Keel mirror family to a log Calabi–Yau surface. [ABSTRACT FROM AUTHOR]

On page 188 of his lost notebook, Ramanujan introduced distinct classes of exquisite infinite series, representing them in terms of Eisenstein series. The objective of this paper is to develop various differential identities related to Borwein's cubic theta functions and h-functions. Also, we present certain identities containing Eisenstein series of level 6 and h-functions that are used to establish relations involving class one infinite series and h-functions. Additionally, we present a straightforward approach to evaluate a discrete convolution sum by employing relationships involving Eisenstein series and h-functions. [ABSTRACT FROM AUTHOR]

The (p, k)-parametrization method, as introduced by Alaca, offers an innovative approach to derive a variety of new Eisenstein series identities that incorporate Borweins' theta functions. This paper presents new Eisenstein series identities that involve Borweins' cubic theta functions, achieved through the (p, k)-parameterization technique. [ABSTRACT FROM AUTHOR]

In this work we study families of ℤ2 orbifolds of toroidal conformal field theories based on both factorizable and non-factorizable target space tori. For these classes of theories, we analyze their moduli spaces, and compute their partition functions. Building on previous work, we express the calculated partition functions in terms of suitable Siegel-Narain theta functions that allow us to determine their ensemble averages. We express the derived averaged partition functions of the studied families of conformal field theories in a manifest modular invariant finite sum of products of real analytic Eisenstein series. We speculate on a tentative holographic three-dimensional dual bulk interpretations for the considered ℤ2 orbifold classes of ensembles of conformal field theories. [ABSTRACT FROM AUTHOR]

In this short note we prove a lower bound for the MaxCut of a graph in terms of the Lovász theta function of its complement. We combine this with known bounds on the Lovász theta function of complements of H-free graphs to recover many known results on the MaxCut of H-free graphs. In particular, we give a new, very short proof of a conjecture of Alon, Krivelevich and Sudakov about the MaxCut of graphs with no cycles of length r. [ABSTRACT FROM AUTHOR]

The monodromy of the Casimir connection is considered. It is shown that the trace of the monodromy operator over an appropriate space of flat sections gives the Jacobi theta constant and incomplete theta functions. A definition of new objects, namely, incomplete Appell–Lerch sums, is given, and their connection with the trace of the monodromy operator is revealed. [ABSTRACT FROM AUTHOR]

Ramanujan in his notebook recorded two modular equations involving multipliers with moduli of degrees (1,7) and (1,23). In this paper, we find some new Ramanujan-type modular equations involving multipliers with moduli of degrees (3,5) and (1,15), and give concise proofs by employing Ramanujan's multiplier functional equation. [ABSTRACT FROM AUTHOR]

Ramanujan (1916) and Shen (1999) discovered differential equations for classical Eisenstein series. Motivated by them, we derive new differential equations for Eisenstein series of level 2 from the second kind of Jacobi theta function. This gives a new characterization of a system of differential equations by Ablowitz–Chakravarty–Hahn (2006), Hahn (2008), Kaneko–Koike (2003), Maier (2011), Nidelan (2022) and Toh (2011). As application, we show some arithmetic results on Ramanujan's tau function. [ABSTRACT FROM AUTHOR]

The weight of a partition is the sum of its nonzero coordinates. Let R and S be partitions of the same weight and $ {\mathcal {A}}(R,S) $ A (R , S) be the class of all $ (0,1) $ (0 , 1) -matrices with row sums R and column sums S. For a positive integer t, the t-term rank of a matrix $ A\in {\mathcal {A}}(R,S) $ A ∈ A (R , S) , denoted $ \rho _t(A) $ ρ t (A) , is defined as the largest number of 1's in A with at most one 1 in each column and at most t 1's in each row. The term rank partition of A is the partition $$\begin{align*} term\ \rho(A) & :=(\rho_1(A)-\rho_0(A),\rho_2(A) \\ & \quad - \rho_1(A),\ldots ,\rho_t(A)-\rho_{t-1}(A),\ldots), \end{align*} $$ term ρ (A) := (ρ 1 (A) − ρ 0 (A) , ρ 2 (A) − ρ 1 (A) , ... , ρ t (A) − ρ t − 1 (A) , ...) , where $ \rho _0(A)=0 $ ρ 0 (A) = 0. Let θ be a partition of weight equal to the number of nonzero coordinates of S. In this paper, we study conditions for the existence of a matrix $ C\in {\mathcal {A}}(R,S) $ C ∈ A (R , S) with $ term\ \rho (C)=\theta. $ term ρ (C) = θ. [ABSTRACT FROM AUTHOR]

It is known that many (upper) cluster algebras possess different kinds of good bases which contain the cluster monomials and are parametrized by the tropical points of cluster Poisson varieties. For a large class of upper cluster algebras (injective-reachable ones with full rank coefficients), we describe all of their bases with these properties. Moreover, we show the existence of the generic basis for them. In addition, we prove that Bridgeland's representation-theoretic formula is effective for their theta functions (weak genteelness). Our results apply to (almost) all known cluster algebras arising from representation theory or higher Teichmüller theory, including quantum affine algebras, unipotent cells, double Bruhat cells, skein algebras over surfaces, where we change the coefficients if necessary so that the full rank assumption holds. [ABSTRACT FROM AUTHOR]

In this paper certain continued fractions associated with Ramanujan's theta functions have been discussed. [ABSTRACT FROM AUTHOR]

Using a higher-dimensional analog of an identity known to Kronecker, we discover a new Andrews–Crandall-type identity and use it to count the number of integer solutions to x 2 + 2 y 2 + 2 z 2 = n . [ABSTRACT FROM AUTHOR]

Schröter-type theta function identities were very instrumental in proving modular equations. In this paper, by employing a generalization of this identity, we prove for the first time a modular equation of degree 61. Furthermore, new modular equations of degrees 13 and 29 are obtained. [ABSTRACT FROM AUTHOR]

In this paper, we investigate sums of four squares of integers whose prime factorizations are restricted, making progress towards a conjecture of Sun that states that two of the integers may be restricted to the forms 2 a 3 b and 2 c 5 d. We obtain an ineffective generalization of results of Gauss and Legendre on sums of three squares and an effective generalization of Lagrange's four-square theorem. [ABSTRACT FROM AUTHOR]

We are studying stationary random processes with conditional polynomial moments that allow a continuous path modification. Processes with continuous path modification are important because they are relatively easy to simulate. One does not have to care about the distribution of their jumps which is always difficult to find. Among those processes with the continuous path are the Ornstein–Uhlenbeck process, the Gamma process, the process with Arcsin or Wigner margins and the Theta functions as the transition densities and others. We give a simple criterion for the stationary process to have a continuous path modification expressed in terms of skewness and excess kurtosis of the marginal distribution. [ABSTRACT FROM AUTHOR]

The (p, fc)-parametrization technique introduced by Alaca provides an alternative approach for formulating several new Eisenstein series identities involving Borwein's theta functions. As a practical example, we have computed the convolution sum of divisor functions, which exhibit a connection with Eisenstein series. [ABSTRACT FROM AUTHOR]

In recent years, congruence properties for the coefficients of mock theta functions have been studied by mathematicians. Recently, Silva and Sellers proved some congruences modulo 3 and 9 for the coefficients of the third order mock theta function ξ(q) given by Gordon and McIntosh. Motivated by their work, we prove new congruences modulo 9 for the coefficients of ξ(q) based on the formula for the number of representations of an integer as sums of seven squares in this paper. Those congruences involve in_nitely many primes. [ABSTRACT FROM AUTHOR]

Ramanujan's lost notebook contains many mock theta functions and mock theta function identities not mentioned in his last letter to Hardy. For example, we find the four tenth‐order mock theta functions and their six identities. The six identities themselves are of a spectacular nature and were first proved by Choi. We also find eight sixth‐order mock theta functions in the lost notebook, but among their many identities there is only a single relationship like those of the tenth‐orders. Using Appell function properties of Hickerson and Mortenson, we discover and prove three new identities for the sixth‐order mock theta functions that are in the spirit of the six tenth‐order identities. We also include an additional nineteen tenth‐order like identities for various combinations of second‐, sixth‐, and eighth‐order mock theta functions. [ABSTRACT FROM AUTHOR]

We give an exact theta expression of a tuple of Hermitian modular forms as an inverse period mapping of lattice polarized K3 surfaces. Our result gives a non-trivial relation among moduli of K3 surfaces, theta functions and the finite complex reflection group of rank 5. [ABSTRACT FROM AUTHOR]

Two classes of finite trigonometric sums, each involving only sines, are evaluated in closed form. The previous and original proofs arise from Ramanujan's theta functions and modular equations. [ABSTRACT FROM AUTHOR]

We demonstrate how formulas that express Hecke-type double-sums in terms of theta functions and Appell–Lerch functions—the building blocks of Ramanujan's mock theta functions—can be used to give general string function formulas for the affine Lie algebra A 1 (1) for levels N = 1 , 2 , 3 , 4 . [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]

For a simply connected connected simple algebraic group |$G$|⁠ , it is known that a variety |$B_{w_0}^-:=B^-\cap U\overline{w_0}U$| has a geometric crystal structure with a positive structure |$\theta ^-_{\textbf{i}}:(\mathbb{C}^{\times })^{l(w_0)}\rightarrow B_{w_0}^-$| for each reduced word |$\textbf{i}$| of the longest element |$w_0$| of Weyl group. A rational function |$\Phi ^h_{BK}=\sum _{i\in I}\Delta _{w_0\Lambda _i,s_i\Lambda _i}$| on |$B_{w_0}^-$| is called a half-potential, where |$\Delta _{w_0\Lambda _i,s_i\Lambda _i}$| is a generalized minor. Computing |$\Phi ^h_{BK}\circ \theta ^-_{\textbf{i}}$| explicitly, we get an explicit form of string cone or polyhedral realization of |$B(\infty)$| for the finite dimensional simple Lie algebra |$\mathfrak{g}=\textrm{Lie}(G)$|⁠. In this paper, for an arbitrary reduced word |$\textbf{i}$|⁠ , we give an algorithm to compute the summand |$\Delta _{w_0\Lambda _i,s_i\Lambda _i}\circ \theta ^-_{\textbf{i}}$| of |$\Phi ^h_{BK}\circ \theta ^-_{\textbf{i}}$| in the case |$i\in I$| satisfies that for any weight |$\mu $| of |$V(-w_0\Lambda _i)$| and |$t\in I$|⁠ , it holds |$\langle h_t,\mu \rangle \in \{2,1,0,-1,-2\}$|⁠. In particular, if |$\mathfrak{g}$| is of type |$\textrm{A}_n$|⁠ , |$\textrm{B}_n$|⁠ , |$\textrm{C}_n$| or |$\textrm{D}_n$| then all |$i\in I$| satisfy this condition so that one can completely calculate |$\Phi ^h_{BK}\circ \theta ^-_{\textbf{i}}$|⁠. We will also prove that our algorithm works in the case |$\mathfrak{g}$| is of type |$\textrm{G}_2$|⁠. [ABSTRACT FROM AUTHOR]

On pp. 243 and 244 of his second notebook, Ramanujan recorded seven modular equations of degree 11 followed by two more which he had crossed out. The first modular equation in the list was proved by Berndt using theta function identities. The same was also proved by Venkatachaliengar in a different way. To the best of our knowledge, the only available proofs to the other six modular equations in the list are by Berndt. These proofs employ the theory of modular forms. Interestingly these are the first set of modular equations, the proofs to which by Berndt take a shift from the elementary algebraic methods to the theory of modular forms. In this paper, we reprove all these modular equations of degree 11 (except the first one in the list) by elementary algebraic techniques. First, we use two series identities due to Ye and Liu to prove one of the modular equations and its reciprocal. The remaining modular equations are proved by the method of parametrization. [ABSTRACT FROM AUTHOR]

According to the polynomial recursion formalism, the modified Jaulent–Miodek hierarchy is derived in a standard way. The first two nontrivial members in the modified Jaulent–Miodek hierarchy are listed correspondingly. Based on the squared eigenfunctions, an algebraic curve κ n and a Riemann surface S with arithmetic genus n are introduced, then the Dubrovin-type equations are obtained naturally. With the help of the conservation laws, the Baker–Akhiezer functions are defined. Finally, the asymptotic properties of the Baker–Akhiezer functions are analyzed, from which the algebro-geometric solutions of the modified Jaulent–Miodek hierarchy are constructed in term of the Riemann theta function. [ABSTRACT FROM AUTHOR]

We consider ancient noncollapsed mean curvature flows in R4$\mathbb {R}^4$ whose tangent flow at −∞$-\infty$ is a bubble‐sheet. We carry out a fine spectral analysis for the bubble‐sheet function u that measures the deviation of the renormalized flow from the round cylinder R2×S1(2)$\mathbb {R}^2 \times S^1(\sqrt {2})$ and prove that for τ→−∞$\tau \rightarrow -\infty$ we have the fine asymptotics u(y,θ,τ)=(y⊤Qy−2tr(Q))/|τ|+o(|τ|−1)$u(y,\theta ,\tau)= (y^\top Qy -2\textrm {tr}(Q))/|\tau | + o(|\tau |^{-1})$, where Q=Q(τ)$Q=Q(\tau)$ is a symmetric 2 × 2‐matrix whose eigenvalues are quantized to be either 0 or −1/8$-1/\sqrt {8}$. This naturally breaks up the classification problem for general ancient noncollapsed flows in R4$\mathbb {R}^4$ into three cases depending on the rank of Q. In the case rk(Q)=0$\mathrm{rk}(Q)=0$, generalizing a prior result of Choi, Hershkovits and the second author, we prove that the flow is either a round shrinking cylinder or R×$\mathbb {R}\times$2d‐bowl. In the case rk(Q)=1$\mathrm{rk}(Q)=1$, under the additional assumption that the flow either splits off a line or is self‐similarly translating, as a consequence of recent work by Angenent, Brendle, Choi, Daskalopoulos, Hershkovits, Sesum and the second author we show that the flow must be R×$\mathbb {R}\times$2d‐oval or belongs to the one‐parameter family of 3d oval‐bowls constructed by Hoffman‐Ilmanen‐Martin‐White, respectively. Finally, in the case rk(Q)=2$\mathrm{rk}(Q)=2$ we show that the flow is compact and SO(2)‐symmetric and for τ→−∞$\tau \rightarrow -\infty$ has the same sharp asymptotics as the O(2) × O(2)‐symmetric ancient ovals constructed by Hershkovits and the second author. The full classification problem will be addressed in subsequent papers based on the results of the present paper. [ABSTRACT FROM AUTHOR]

Srinivasa Ramanujan developed seventeen fast convergent series for 1/π. Motivated by Ramanujan's series for 1/π many mathematicians have developed many theories for deriving new series for 1/π. In this article we obtain new series for 1/π using Eisenstein series of level three. [ABSTRACT FROM AUTHOR]