Fourier Transforms on the Line and in Space

This paper presents a new method for the design of two-channel conjugate quadrature (CQ) filter banks in which halfband filter and spectrum factorization are not required. Instead, a CQ filter is directly optimized subject to the perfect reconstruction and possibly other constraints (such as number of vanishing moments (VM)). We develop a design strategy in that the solution is approached sequentially with each update confined to within a small vicinity of the current iterate where the problem at hand behaves like a convex one, thus the update can be obtained as a solution of a convex problem. Four design scenarios are considered, namely the least squares designs with or without VM requirement, and equiripple designs with or without VM requirement. The simulation studies demonstrate that the proposed method is reliable to design high-order CQ filters with improved performance.

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.

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.

In this paper, the most common method of probability density function integration Gaussian Integration is analyzed in real and complex methods. This paper begins by clarifying the meaning and significance of integrating the probability density function. After this, it elucidates and derive the mathematical methods for the analysis of integrals, namely the multiple integral method in real analysis and the residue theorem method in complex analysis. At the same time, the paper also makes it clear to analyze the most typical Gaussian integral in the probability density function. Since then, this integration method has been used respectively for real analysis and complex analysis. Different means of analysis including Pinch criterion or Residue theorem have been applied comprehensively. In this session, different forms and formats of Gaussian integrals are explored to a certain extent. Finally, the benefits of complex analysis and real analysis for the synthesis of probability density function including Gaussian integral are summarized, and the possible applications and prospects of this new analysis method are prospected. However, the limitations of the current approach have also been mentioned.

The argument principle is one of the basic theorems from the classical complex analysis and it's crucial role in complex analysis is very nicely underlined in the book of A. F. Beardon Complex Analysis – the Argument Principle in Analysis and Topology, John Wiley and Sons, 1979. We shall generalize a result of T. Radó and P.V. Reichelderfer from Continuous Transformations in Analysis, Springer-Verlag, Berlin, Göttingen, Heidelberg, 1955, p. 399, and a recent paper of P. Duren, w. Hengartner and R. S. LaugesenAmer. Math. Monthly 103(5) (1996), 411-415, showing that the argument principle is an entirely topological theorem (because it holds for continuous, open, discrete mappings in the plane) and hence this theorem belongs more to topology than to complex analysis. We also present as an application a generalization of Rouché's theorem.

Reflection and transmission coefficients for an electromagnetic wave are usually computed using the frequency representation: a monochromatic plane wave decomposition [1-2]. Even though this method is a powerful tool, the physical meaning can be hidden by some mathematical complexity coming from Fourier integrals, complex analysis and matrix manipulation. Furthermore, the Maxwell equations, when the fields are only dependent on x and t, are very simple and can be solved using elementary calculation [3]. It will be shown that the physical propagation mechanism, which includes initial conditions, transmission and reflection, can be completely built up within the space-time, rather than in the frequency domain. This would shed a new light on traditional well known results.

Theoretical physicists have predicted that the scaling limits of many two-dimensional lattice models in statistical physics are in some sense conformally invariant. This belief has allowed physicists to predict many quantities for these critical systems. The nature of these scaling limits has recently been described precisely by using one well-known tool, Brownian motion, and a new construction, the Schramm-Loewner evolution (SLE). This book is an introduction to the conformally invariant processes that appear as scaling limits. The following topics are covered: stochastic integration; complex Brownian motion and measures derived from Brownian motion; conformal mappings and univalent functions; the Loewner differential equation and Loewner chains; the Schramm-Loewner evolution (SLE), which is a Loewner chain with a Brownian motion input; and applications to intersection exponents for Brownian motion. The prerequisites are first-year graduate courses in real analysis, complex analysis, and probability. The book is suitable for graduate students and research mathematicians interested in random processes and their applications in theoretical physics.

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

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.

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 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.

This study develops an optimal formulation for fortified gluten-free extruded snacks with added sea buckthorn powder, with the objective of expanding the range of products available for both patients with coeliac disease and the healthy population. This required comprehensive physicochemical and organoleptic analyses of raw materials, including rice, buckwheat flour, and sea buckthorn powder, as well as the use of mathematical modelling based on compositional planning to optimise the proportions of the components. The following optimal ratios were identified: 30.10% rice flour, 74.95% buckwheat, and 21.95% sea buckthorn powder, which ensured increased nutritional value and improved the organoleptic properties of the final product. These results demonstrate the possibility of using the developed formulation in industry, reducing dependence on imported analogues, and expanding the range of gluten-free products. The novelty of this study lies in its scientifically based approach to the creation of highly nutritious snacks with antioxidant and vitamin functions through the use of mathematical modelling and complex analysis of raw materials, which contributes to the development of technologies and the strengthening of domestic production. Further value of this study lies in its development of a scientifically based approach to the creation of gluten-free products with high nutritional value. The results obtained contribute to the advancement of gluten-free production and the expansion of the range of functional foods. The practical significance of the results obtained includes the possibility of using the developed recipe in production, which will solve the problem of providing coeliac patients with gluten-free products and overcome consumer dependence on the purchase of expensive imported analogues, while simultaneously developing the processing industry of Kazakhstan.

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.