A generalization of the argument principle

A new method for mode determination of multilayer planar waveguides with complex-valued dielectric permeability is proposed. The method is based on the combination of an argument principle in complex analysis and a quasi-Newton method. The argument principle allows reliable determination of the number of modes within any finite region of the complex plane representing the propagation constants. This makes it possible to locate the propagation constants to the precision sufficient for the usage of the quasi-Newton method. The method involves 'tracking of the phase' of a complex function instead of conventional integration along a contour.

In complex analysis, the winding number measures the number of times a path (counter-clockwise) winds around a point, while the Cauchy index can approximate how the path winds. We formalise this approximation in the Isabelle theorem prover, and provide a tactic to evaluate winding numbers through Cauchy indices. By further combining this approximation with the argument principle, we are able to make use of remainder sequences to effectively count the number of complex roots of a polynomial within some domains, such as a rectangular box and a half-plane.

A number of results concerning infinite series are presented in this chapter. The material spans convergence tests for series of constants, uniform convergence for series of functions, and power series. It is not the intention here to give a comprehensive account of this theory, but rather to review material that is relevant to later chapters. This chapter is intended to make the rest of the book more self contained and remind the reader of some basic results. It is assumed that the reader has a background in real analysis of a single variable and elementary complex analysis; consequently, no attempt is made to prove these results or provide examples. Proofs and examples can be found in textbooks on real analysis such as Apostol (Mathematical Analysis, Addison-Wesley, 2nd ed., 1974), Fulks (Advanced Calculus, 3rd ed., Wiley, 1978), and Little et al. (Real Analysis via Sequences and Series, Springer-Verlag, 2015) and complex analysis (Ahlfors, Complex Analysis, McGraw-Hill, 2nd ed., 1966; Brown and Churchill, Complex Variables and Applications, McGraw-Hill, 7th ed., 2004; Priestley, Introduction to Complex Analysis, Rev. ed., Oxford University Press, 1990). The exception is in the final section on double sequences and series. This material is often not part of courses on real or complex analysis. Here we provide proofs, examples and exercises.

In this paper, the topological nature of the generalization of the classical argument principle well-known in multidimensional complex analysis is discussed. The topological approach offered here ensures some topological results on the structure of pole and zero sets of holomorphic maps of bounded domains in complex manifolds. Some connections with integral representations of holomorphic functions are studied and a geometric interpretation of the Martinelli-Bochner complex-valued, differential-form realization is given.

The group ring ℂ2 of pairs (ξ,η) of classical complex numbers is used as an algebraic basis for a function theory for one four-dimensional variable of the form Z = 1 x + i y + j z + k ct. All of the axioms and properties of the complex analysis are carried forward into four dimensions, including functions, derivatives, integrals, and even the notation. The new analysis properly subsumes and extends the classical complex analysis. The single unexpected divergence from traditional properties is that the sets (ξ, 0) and (0, η) of non invertible elements are interpreted as the zero element (zero set). The zero set can be constrained to lie in the relativistic limit of special relativity; consequently, the new analysis scheme may be applied in any real-world, physical problem. The four-dimensional Cauchy-Riemann equations provide a way to apply continuity conditions to the field equations of physics, while at the same time converting PDEs to easier-to-solve ODEs. Significant simplifications are achieved for many physics field equations. The technique is illustrated on the Emden's equation. Symbolic manipulations in four dimensions are easily handled by any computer algebra program that can perform classical complex algebraic operations.

From the stereographic projection the ancient Greeks used to chart the heavens to modern circle-packing techniques that can chart the inner reaches of the human brain, conformal maps have played a central role in mathematical analysis and applications for over two millennia. It seems a shame, then, that undergraduate mathematics students rarely learn about them, the more so since a basic knowledge of calculus and some facility with trigonometry are enough to make a significant start. In this paper, we use single-variable calculus to establish the conformality of the stereographic projection and the Mercator world map. Along the way, the complex exponential function arises in a natural way that establishes it as a conformal mapping—without using any complex analysis. Furthermore, the exponential function provides a link between the symmetric structures of the two conformal maps. We conclude by indicating how one can construct conformal maps of surfaces of revolution other than the sphere.

Chapter 2 Aspects of Mathematics and Mechanics

From classical complex analysis it is well-known that the index of the Riemann-Hilbert problem is connected with the winding number. How to generalize the winding number to a rotational number or rotation is explained in [nf] and also in [krza]. How to get a connection between the index and the rotation can be found in [dei]. For matrices of singular integral operators a rotation can be defined which is connected with the index of this system of singular integral operators. Because Clifford analysis is a higher dimensional analog to complex analysis there should be some analogous behavior of singular integrals. In fact, we will show that the index of singular integral operators with paravector-valued symbols in Clifford algebras can be connected with the so-called wrapping number which is the higher dimensional analog to the winding number. The proof is rather elementary.

The weighted Kohn Laplacian ◻ φ \Box _\varphi is a natural second order elliptic operator associated to a weight φ : C n → R \varphi :\mathbb {C}^n\rightarrow \mathbb {R} and acting on ( 0 , 1 ) (0,1) -forms, which plays a key role in several questions of complex analysis. We consider here the case of model monomial weights in C 2 \mathbb {C}^2 , i.e., \[ φ ( z , w ) := ∑ ( α , β ) ∈ Γ | z α w β | 2 , \varphi (z,w):=\sum _{(\alpha ,\beta )\in \Gamma }|z^\alpha w^\beta |^2, \] where Γ ⊆ N 2 \Gamma \subseteq \mathbb {N}^2 is finite. Our goal is to prove coercivity estimates of the form \[ ( ∗ ) ◻ φ ≥ μ 2 , (*)\hspace {10pc} \Box _\varphi \geq \mu ^2,\hspace {10pc} \] where μ : C n → R \mu :\mathbb {C}^n\rightarrow \mathbb {R} acts by pointwise multiplication on ( 0 , 1 ) (0,1) -forms, and the inequality is in the sense of self-adjoint operators. We proved in 2015 how to derive from ( ∗ ) (*) new pointwise bounds for the weighted Bergman kernel associated to φ \varphi . Here we introduce a technique to establish ( ∗ ) (*) with \[ μ ( z , w ) = c ( 1 + | z | a + | w | b ) ( a , b ≥ 0 ) , \mu (z,w)=c(1+|z|^a+|w|^b) \qquad (a,b\geq 0), \] where a , b ≥ 0 a,b\geq 0 depend on (and are easily computable from) Γ \Gamma . As a corollary we also prove that, for a wide class of model monomial weights, the spectrum of ◻ φ \Box _\varphi is discrete if and only if the weight is not decoupled, i.e., Γ \Gamma contains at least a point ( α , β ) (\alpha ,\beta ) with α ≠ 0 ≠ β \alpha \neq 0\neq \beta . Our methods comprise a new holomorphic uncertainty principle and linear optimization arguments.

This chapter introduces a few features of the Fourier transform, mainly for its use in the design of Daubechies’ continuous, compactly supported, orthogonal wavelets. The principal use of the Fourier transform in the design of wavelets will consist in transforming “convolutions” of functions into ordinary multiplications of functions, which allow for an analysis easier than with convolutions. The material presented here involves mostly calculus, with only a few results with brief statements—such as Fubini’s theorem on the permutation of the order of multiple integrals [12, pp. 65-66], [21, pp. 384-388], [28, pp. 226-239]—borrowed from the literature in mathematical real analysis and complex analysis.

Domain-specific deployment of large language models (LLMs) remains constrained by prompt brittleness, inference cost, and uneven generalization across task subtypes. This paper presents APOF (Adaptive Prompt Optimization Framework), a closed-loop framework that jointly opti- mizes prompt structure, retrieval context, and inference-time control signals for domain-specific LLM applications. APOF combines three elements: (i) a policy-guided prompt composer that dynamically allocates instruction budget across task facets, (ii) a critic model that estimates prompt-task fitness before expensive decoding, and (iii) an online adaptation module that up- dates prompt policies using delayed feedback from production outcomes. We instantiate APOF in three high-stakes domains—clinical note summarization, legal clause risk classification, and materials-science question answering—using a shared 13B parameter base model and domain adapters.Our experiments include 162,000 annotated instances across public and institutionally cu- rated corpora, 12 baseline methods, and controlled ablation studies. APOF improves macro-F1 by up to 6.8 points over the strongest static prompt baseline, while reducing median latency by 18.4% through pre-decoding prompt pruning and adaptive generation parameters. The frame- work also improves calibration (ECE reduction of 0.041) and demonstrates higher robustness under distribution shift (average relative performance drop 12.3% vs. 21.7% for static meth- ods). We provide mathematical formulations, complexity analysis, and practical deployment recommendations. Results suggest that adaptive prompting, when treated as a structured op- timization problem rather than manual engineering, is a viable path to reliable domain-specific LLM systems.

This paper introduces a new variant of the Shortest Path Problem (SPP) called the Cost-Balanced Path Problem (CBPP). Various real problems can either be modeled as BCPP or include BCPP as a sub-problem. We prove several properties related to the complexity of the CBPP problem. In particular, we demonstrate that the problem is NP-hard in its general version, but it becomes solvable in polynomial time in a specific family of instances. Moreover, a mathematical formulation of the CBPP, as a mixed-integer programming model, is proposed, and some additional constraints for modeling real requirements are given. This paper validates the proposed model and its extensions with experimental tests based on random instances. The analysis of the results of the computational experiments shows that the proposed model and its extension can be used to model many real applications. Obviously, due to the problem complexity, the main limitation of the proposed approach is related to the size of the instances. A heuristic solution approach should be required for larger-sized and more complex instances.

Complex analysis is not an elementary subject, and the author of a book like this has to make some reasonable assumptions about what his readers know already. Ideally one would like to assume that the student has some basic knowledge of complex numbers and has experienced a fairly substantial first course in real analysis. But while the first of these requirements is realistic the second is not, for in many courses with an “applied” emphasis a course in complex analysis sits on top of a course on advanced (multi-variable) calculus, and many students approach the subject with little experience of ∈-δ arguments, and with no clear idea of the concept of uniform convergence. This chapter sets out in summary the equipment necessary to make a start on this book, with references to suitable texts. It is written as a reminder: if there is anything you don’t know at all, then at some point you will need to consult another book, either the suggested reference or another similar volume.

Till date, the sequence of vortices present in the solid corners of steady internal viscous incompressible flows, widely known as Moffatt vortices was thought to be infinite. However, the already existing and most recent geometric theories on incompressible viscous flows that express vortical structures in terms of critical points in bounded domains, indicate a strong opposition to this notion of infiniteness. In this study, we endeavor to bridge the gap between the two opposing stream of thoughts by addressing what might have gone wrong and pinpoint the shortcomings on the assumptions of the existing theorems on Moffatt vortices. We provide our own set of proofs for establishing the finiteness of the sequence of Moffatt vortices by making use of the continuum hypothesis and Kolmogorov scale, which guarantee a non-zero scale for the smallest vortex structure possible in incompressible viscous flows. We point out that the notion of infiniteness resulting from discrete self-similarity of the vortex structures is not physically feasible. The centers of these vortices have been quantified by us as fixed points through Brouwer fixed-point theorem and boundary of a vortex as circle cell. With the aid of these new developments and making use of some existing theorems in topology along with some elementary concept of mathematical analysis, we provide several approaches to delve into this issue. All these approaches converge to the same conclusion that the sequence of Moffatt vortices cannot be infinite; in fact, it is at most finite.

Modeling as a tool solves extremely difficult tasks in life sciences. Recently, schemes of culturing of microalgae have received special attention because of its unique features and possible uses in many industrial applications for renewable energy production and high value products isolation. The goal of this review is to present the use of system analysis theory applied to microalgae culturing modeling and process development. The review mainly focuses on the modeling of the key steps of autotrophic growth under the integral biorefinery concept of the microalgae biomass. The system approach follows systematically a procedure showing the difficulties by modeling of sub-systems. The development of microalgae kinetics and computational fluid dynamics (CFD) studies were analyzed in details as sub-systems in advanced design of photobioreactor (PBR). This review logically follows the trends of the modeling procedure and clarifies how this approach may save time and money during the research efforts. The result of this work is a successful development of a complex PBR mathematical analysis in the frame of the integral biorefinery concept.