There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Fractional Operations Confluences Series Expansion and Contiguity Relation Fuchsian Differential Equation and Generalized Riemann Scheme Reduction of Fuchsian Differential Equations Deligne - Simpson Problem A Kac - Moody Root System Expression of Local Solutions Monodromy Reducibility Shift Operators Connection Problem Examples Further Problems

Fractional calculus of Weyl algebra and Fuchsian differential equations

We explain some results of [G. Cotti, B. A. Dubrovin and D. Guzzetti, Isomonodromy deformations at an irregular singularity with coalescing eigenvalues, preprint (2017); arXiv:1706.04808.], discuss...

Results on the extension of isomonodromy deformations to the case of a resonant irregular singularity

ln this paper an introduction to the physics of superconductors is presented. A brief hystorical comment and the general characteristics of these materials are developed mainly based on the zero resistance and the perfect diamagnetic behaviour properties. The quantum effects are qualitatively introduced to contribute to the understanding of the collective phenomena associated with the occurrence of superconductivity. The BCS theory is presented as a complete explanation of this phenomenum in metallic materials. Finally, the new ceramic compounds of high critical temperatures are introduced, specially concerning recent different existent theories up to now.

/pdf/an-introduction-to-the-physics-of-superconductors-2rw8x41ie7.pdf

An introduction to the physics of superconductors

We consider an $n\times n$ linear system of ODEs with an irregular singularity of Poincar\'e rank 1 at $z=\infty$, holomorphically depending on parameter $t$ within a polydisc in $\mathbb{C}^n$ centred at $t=0$. The eigenvalues of the leading matrix at $z=\infty$ coalesce along a locus $\Delta$ contained in the polydisc, passing through $t=0$. Namely, $z=\infty$ is a resonant irregular singularity for $t\in \Delta$. We analyse the case when the leading matrix remains diagonalisable at $\Delta$. We discuss the existence of fundamental matrix solutions, their asymptotics, Stokes phenomenon and monodromy data as $t$ varies in the polydisc, and their limits for $t$ tending to points of $\Delta$. When the deformation is isomonodromic away from $\Delta$, it is well known that a fundamental matrix solution has singularities at $\Delta$. When the system also has a Fuchsian singularity at $z=0$, we show under minimal vanishing conditions on the residue matrix at $z=0$ that isomonodromic deformations can be extended to the whole polydisc, including $\Delta$, in such a way that the fundamental matrix solutions and the constant monodromy data are well defined in the whole polydisc. These data can be computed just by considering the system at fixed $t=0$. Conversely, if the $t$-dependent system is isomonodromic in a small domain contained in the polydisc not intersecting $\Delta$, if the entries of the Stokes matrices with indices corresponding to coalescing eigenvalues vanish, then we show that $\Delta$ is not a branching locus for the fundamental matrix solutions. The importance of these results for the analytic theory of Frobenius Manifolds is explained. An application to Painlev\'e equations is discussed.

Isomonodromy Deformations at an Irregular Singularity with Coalescing Eigenvalues

The tunneling effect predicted by B.Josephson (Nobel Prize, 1973) concerns the Josephson junction: two superconductors separated by a narrow dielectric. It states existence of a supercurrent through it and equations governing it. The overdamped Josephson junction is modeled by a family of differential equations on 2-torus depending on 3 parameters: $B$ (abscissa), $A$ (ordinate), $\omega$ (frequency). We study its rotation number $\rho(B,A;\omega)$ as a function of $(B,A)$ with fixed $\omega$. The phase-lock areas are the level sets $L_r:=\{\rho=r\}$ with non-empty interiors; they exist for $r\in\mathbb Z$ (Buchstaber, Karpov, Tertychnyi). Each $L_r$ is an infinite chain of domains going vertically to infinity and separated by points called constrictions (expect for those with $A=0$). We show that: 1) all the constrictions in $L_r$ lie in its axis $\{ B=\omega r\}$ (confirming a conjecture of Tertychnyi, Kleptsyn, Filimonov, Schurov); 2) each constriction is positive: some its punctured neighborhood in the vertical line lies in $\operatorname{Int}(L_r)$ (confirming another conjecture). We first prove deformability of each constriction to another one, with arbitrarily small $\omega$, of the same $\rho$, $\ell:=\frac B\omega$ and type (positive or not), using equivalent description of model by linear systems of differential equations on $\overline{\mathbb C}$ (Buchstaber, Karpov, Tertychnyi) and studying their isomonodromic deformations described by Painleve 3 equations. Then non-existence of ghost constrictions (i.e., constrictions either with $\rho\neq\ell$, or of non-positive type) with a given $\ell$ for small $\omega$ is proved by slow-fast methods. In Section 6 we present applications of results and elaborated methods and open problems.

pdf/on-families-of-constrictions-in-model-of-overdamped-4blglxnn2t.pdf

On families of constrictions in model of overdamped Josephson junction and Painlev\'e 3 equation

The paper is devoted to non-Schlesinger isomonodromic deformations for reso- nant Fuchsian systems. There are very few explicit examples of such deformations in the literature. In this paper we construct a new example of the non-Schlesinger isomonodromic deformation for a resonant Fuchsian system of order 5 by using middle convolution for a resonant Fuchsian system of order 2. Moreover, it is known that middle convolution is an operation that preserves Schlesinger's deformation equations for non-resonant Fuchsian systems. In this paper we show that Bolibruch's non-Schlesinger deformations of resonant Fuchsian systems are, in general, not preserved by middle convolution.

/pdf/non-schlesinger-isomonodromic-deformations-of-fuchsian-3hhzsfh7zk.pdf

Non-Schlesinger Isomonodromic Deformations of Fuchsian Systems and Middle Convolution

In this paper, we present a general scheme to generate constructive solutions to the Riemann---Hilbert problem via middle convolution and illustrate this approach for a Fuchsian system with four singular points.

Constructive Solutions to the Riemann---Hilbert Problem and Middle Convolution

We consider some properties of double confluent Heun equation related to the Josephson Effect. In particular, we prove that adjacency points of phased-locked areas on a parameter plane can be described via poles of Bessel solution of Painleve 3 equation.

Josephson Effect and Isomonodromic Deformations

https://projecteuclid.org/download/pdf_1/euclid.pja/1430397895

Middle convolution and non-Schlesinger deformations

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