Survey on the 𝑝-adic Hayman conjecture

The field of meromorphic functions on a sigma divisor of a hyperelliptic curve of genus $3$ is described in terms of the gradient of it's sigma function. Solutions of corresponding families of polynomial dynamical systems in $\mathbb{C}^4$ with two polynomial integrals are constructed as an application. These systems were introduced in the work of V. M. Buchstaber and A. V. Mikhailov on the basis of commuting vector fields on the symmetric square of algebraic curves.

Let v be a compact Riemann surface and v' be the complement in v of a nonvoid finite subset. Let M(v') be the field of meromorphic functions in v'. In this paper we study the ramification divisors of the functions in M(v') which have exponential singularities of finite degree at the points of v-v', and one proves, for instance, that if a function in M(v') belongs to the subfield generated by the functions of this type, and has a finite ramification divisor, it also has a finite divisor. It is also proved that for a given finite divisor d in v', the ramification divisors (with a fixed degree) of the functions of the said type whose divisor id d, define a proper analytic subset of a certain symmetric power of v'.

Symbolic computation of Drazin inverses by specializations

In this chapter we start some basic facts on analytic and complex geometry (divisors, blowing-up, big line bundles), we prove the theorem of Siegel-Remmert-Thimm, that the field of meromorphic functions on a connected compact complex manifold is an algebraic field of transcendence degree less than the dimension of the manifold. Then we study in more detail Moishezon manifolds and their relation to projective manifolds. In particular we prove that a Moishezon manifold is projective if and only if it carries a Kähler metric. We end the section 2.2 by giving the solution of the Grauert-Riemenschneider conjecture as application of the holomorphic Morse inequalities from Theorem 1.7.1.KeywordsLine BundleComplex ManifoldAhler ManifoldHolomorphic Line BundleCompact Complex ManifoldThese 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.

Let L ¯ i →X i be a holomorphic line bundle over a compact complex manifold for i=1,2. Let S i denote the associated principal circle-bundle with respect to some hermitian inner product on L ¯ i . We construct complex structures on S=S 1 ×S 2 which we refer to as scalar, diagonal, and linear types. While scalar type structures always exist, the more general diagonal but non-scalar type structures are constructed assuming that L ¯ i are equivariant (ℂ * ) n i -bundles satisfying some additional conditions. The linear type complex structures are constructed assuming X i are (generalized) flag varieties and L ¯ i negative ample line bundles over X i . When H 1 (X 1 ;ℝ)=0 and c 1 (L ¯ 1 )∈H 2 (X 1 ;ℂ) is non-zero, the compact manifold S does not admit any symplectic structure and hence it is non-Kähler with respect to any complex structure.We obtain a vanishing theorem for H q (S;𝒪 S ) when X i are projective manifolds, L ¯ i ∨ are very ample and the cone over X i with respect to the projective imbedding defined by L ¯ i ∨ are Cohen-Macaulay. We obtain applications to the Picard group of S. When X i =G i /P i where P i are maximal parabolic subgroups and S is endowed with linear type complex structure with “vanishing unipotent part” we show that the field of meromorphic functions on S is purely transcendental over ℂ.

Let $X$ be a Berkovich space over a valued field. We prove that every finite group is a Galois group over $\Ms(B)(T)$, where $\Ms(B)$ is the field of meromorphic functions over a part $B$ of $X$ satisfying some conditions. This gives a new geometric proof that every finite group is a Galois group over $K(T)$, where $K$ is a complete valued field with non-trivial valuation. Then we switch to Berkovich spaces over ${\bf Z}$ and use a similar strategy to give a new proof of the following theorem by D. Harbater: every finite group is a Galois group over a field of convergent arithmetic power series. We believe our proof to be more geometric and elementary that the original one. We have included the necessary background on Berkovich spaces over ${\bf Z}$.

In the Langlands program over number fields, automorphic representations and Galois representations are placed into correspondence, using the cohomology of Shimura varieties as an intermediary. Over a function field, the appropriate intermediary is a moduli space of shtukas. We review the shtukas and introduce their local analogues, which play a similar role in the Langlands program over a p p -adic field. Along the way we construct the Fargues-Fontaine curve and discuss perfectoid spaces and diamonds. This survey may be seen as preparatory for the contribution of Fargues-Scholze.

Let k be a field and let X, Y, Z be indeterminates over k. We will denote by Ord X the usual valuation on the meromorphic function field k((X)).

The third chapter contains a geometric description of finite algebraic extensions of the field of meromorphic functions on a Riemann surface. For such surfaces, the geometry of ramified coverings and Galois theory are not only analogous but in fact very closely related to each other. This relationship is useful in both directions. On the one hand, Galois theory and Riemann’s existence theorem allow one to describe the field of functions on a ramified covering over a Riemann surface as a finite algebraic extension of the field of meromorphic functions on the Riemann surface. On the other hand, the geometry of ramified coverings together with Riemann’s existence theorem allows one to give a transparent description of algebraic extensions of the field of meromorphic functions over a Riemann surface.

This paper presents the analysis and determination of microwave filter order using numerical contour integration method. Based on argument principle, the numerical contour integration is evaluated using central finite-differencing for the $S$ parameter derivative and applied to various microwave bandpass filters involving transmission line structures. It is useful to ascertain the filter order that is often not obvious especially for transmission line structures and enables one to know the targeted number of poles to solve for within certain passband. Using several examples, we show that the order or number of poles for bandpass filters may not be determined simply from the filter structures, their number of sections or reflection zeros. The correct order or number of poles can be determined using our numerical contour integration method.

In this paper, we carry out a systematic study to discover the properties of a subclass of meromorphic starlike functions defined using the Mittag–Leffler three-parameter function. Differential operators involving special functions have been very useful in extracting information about the various properties of functions belonging to geometrically defined function classes. Here, we choose the Prabhakar function (or a three parameter Mittag–Leffler function) for our study, since it has several applications in science and engineering problems. To provide our study with more versatility, we define our class by employing a certain pseudo-starlike type analytic characterization quasi-subordinate to a more general function. We provide the conditions to obtain sufficient conditions for meromorphic starlikeness involving quasi-subordination. Our other main results include the solution to the Fekete–Szegő problem and inclusion relationships for functions belonging to the defined function classes. Several consequences of our main results are pointed out.

In this paper we consider the meromorphic function G(z) with a pole of order 1 at -a and analytic function F(z) with a zero -a of order 2 in <TEX>$\mathbb{D}=\{z :{\mid}z{\mid}</TEX><TEX><</TEX><TEX>1\}$</TEX>, where -1 < a < 1. From these functions we obtain the regular simply-connected minimal surface <TEX>$S=\{(u(z),\;{\nu}(z),\;H(z)):z{\in}\mathbb{D}\}$</TEX> in <TEX>$E^3$</TEX> and the harmonic function <TEX>$f=u+i{\nu}$</TEX> defined on <TEX>$\mathbb{D}$</TEX>, and then we investigate properties of the minimal surface S and the harmonic function f.

To address such challenges as an uncertain number of demagnetization poles of the permanent magnet synchronous motor (PMSM) and cases in which the fault cannot be located, this paper proposes a fault identification and location methodology based on the analysis of the motor stator current. First, the influence of the irreversible demagnetization of permanent magnets on the analytical model of the back electromotive force (Back-EMF) of the rotor in a single motor stator slot is analyzed. Moreover, considering the topology of the motor, the influence of the demagnetization fault on the stator phase current and branch current is analyzed. Since the stator phase currents cannot diagnose the partial demagnetization faults of PMSM with some topological structures, the stator branch current is selected as the signal for the identification and localization of the demagnetization fault. Secondly, the demagnetization fault diagnosis and mode recognition of the motor are carried out through the amplitude of the real-time branch current and the harmonic components of the PMSM. A sample database of demagnetization faults is established through calculation and normalization of the residual value of the stator branch current and the branch current of the healthy motor after demagnetization in one pole order. The fault threshold is obtained by analyzing the residual of the branch current of uniform demagnetization and the Pearson correlation coefficient of the fault sample database. Then, the correlation coefficient between the real-time branch current residual value of PMSM and the fault sample database is analyzed, and the number of demagnetization poles and the fault location are determined by the number and location of the calculated correlation coefficient exceeding the threshold. Finally, the feasibility and effectiveness of the proposed method are verified by the finite element analysis (FEA) results.

Let f be a nonconstant meromorphic function in the plane and h be a nonconstant elliptic function. We show that if all zeros of f are multiple except finitely many and T(r, h) = o{T(r, f)} as r→∞, then f′ = h has infinitely many solutions (including poles).

analogs are constructed for n-dimensional compact complex spaces with n algebraically independent meromorphic functions; they are called by the author minischemes. The present part of the work contains a number of theorems on morphisms and monoidal transformations of schemes, as well as the definition of minischeme and of a morphism of minischemes and some consequences of these definitions, including the construction of the product of minischemes.