Theta function identities in the study of wavelets satisfying advanced differential equations

In this paper we establish two theta function identities with four parameters by the theory of theta functions. Using these identities we introduce common generalizations of Hirschhorn-Garvan-Borwein cubic theta functions, and also re-derive the quintuple product identity, one of Ramanujan’s identities, Winquist’s identity and many other interesting identities.

Ramanujan’s lost notebook contains many results on mock theta functions. In particular, the lost notebook contains eight identities for tenth order mock theta functions. Previously the author proved the first six of Ramanujan’s tenth order mock theta function identities. It is the purpose of this paper to prove the seventh and eighth identities of Ramanujan’s tenth order mock theta function identities which are expressed by mock theta functions and a definite integral. L. J. Mordell’s transformation formula for the definite integral plays a key role in the proofs of these identities. Also, the properties of modular forms are used for the proofs of theta function identities.

Many questions in the theory of differential equations can be answered in a particularly elegant manner using general concepts of functional analysis. In this and later chapters, functional-analytic methods will be used to derive theorems on existence, uniqueness, and dependence of solutions on parameters. We begin by introducing the concept of a Banach space.KeywordsBanach SpaceGlobal SolutionFixed Point TheoremUniqueness TheoremLipschitz ConditionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Since the pioneering work of Landau and Nakanishi on general Feynman integrals, it has been known that the singularities of the S-matrix, causal Green's functions and related functions are described by the so called Landau equations. These Landau singularities were physically interpreted as the macroscopic causality by the theoretical physicists working in 5-matrix theory, and the notion of essential support was obtained (Chandler-Stapp [3], lagolnitzer-Stapp [5]). In the branch of mathematics, on the other side, the theory of microfunction has evolved and has been applied powerfully to the general theory of partial differential equations (Sato-Kawai-Kashiwara [11]). It contained the essential support theory as the singularity spectrum of a function (i.e. its support viewed as a microfunction), and was far-reaching because of its close connection with the theory of differential equations. Namely the method of microlocal analysis, based on the theory of holonomic systems, has provided a systematic way of handling functions with natural background and found most effective applications to various problems of mathematics, such as the theory of ^-functions and Fourier transformations (Sato [10], Kashiwara [6]). It was then recognized that the Landau equations give holonomic varieties, which led one to the holonomicity postulate of ^-matrix and related quantities (Sato [10]). In the present paper we shall study the holonomy structure of Landau singularities and Feynman integrals from this standpoint. First we review the notion of Landau varieties. In contrast with

3 - LINEAR SYSTEMS

Evaluations of the cubic continued fraction by some theta function identities: Revisited

The Schröter formula is an important theta function identity. In this paper, we will point out that some well-known addition formulas for theta functions are special cases of the Schröter formula. We further show that the Hirschhorn septuple product identity can also be derived from this formula. In addition, this formula allows us to derive four remarkable theta functions identities, two of them are extensions of two well-known Ramanujan's identities related to the modular equations of degree 5. A trigonometric identity is also proved.

In his first and second letters to Hardy, Ramanujan made several assertions about the Rogers-Ramanujan continued fraction F ( q ) F(q) . In order to prove some of these claims, G. N. Watson established two important theorems about F ( q ) F(q) that he found in Ramanujan’s notebooks. In his lost notebook, after stating a version of the quintuple product identity, Ramanujan offers three theta function identities, two of which contain as special cases the celebrated two theorems of Ramanujan proved by Watson. Using addition formulas, the quintuple product identity, and a new general product formula for theta functions, we prove these three identities of Ramanujan from his lost notebooks.

The inability of students to master the material concept of the advanced differential equation gives rise to internal and external factors that are formed in them so that it needs to require encouragement as effective use of learning media so that learning in the classroom becomes increasingly attractive to students. We used online learning media as zoom, offline or online learning videos, E-learning, and others. This research aimed to identify and describe students learning difficulties in advanced differential equation courses in terms of the use of learning media during the Covid-19 pandemic. The type of research used is qualitative research, with a descriptive approach. Results showed that students have great difficulty learning numerical methods to solve first-order differential equations from all three sub-chapters of matter. In this material, students are still less careful in answering questions (58,1%). Because students could not identify the type of methods used to solve problems, students need online learning media that are effective for math learning, such as offline learning videos (71%), as a solution to save internet quotas. So, this research is useful for educators/lecturers in observing the condition of student learning difficulties given the use of learning media used to motivate students to be active in learning advanced differential equation courses during the Covid-19 pandemic.

Fractional-like Hukuhara derivatives in the theory of set-valued differential equations

Adequate predimension inequalities in differential fields

In this paper, we develop a new technique to deduce oscillation of a second-order noncanonical advanced differential equation by using established criteria for second-order canonical advanced differential equations. We illustrate our results by presenting two examples.

SECOND ORDER DIFFERENTIAL EQUATIONS IN BANACH SPACE

We take a new approach to the generation of Jacobi theta function identities. It is complementary to the procedure which makes use of the evaluation of Parseval-like identities for elementary cylindrically-symmetric functions on computer holograms. Our method is more simple and explicit than this one, which was an outcome of the construction of neurocomputer architectures through the Heisenberg model.

How Floquet Theory Applies to Index 1 Differential Algebraic Equations