Type Formula Research Articles

Optimal coil configuration analysis for the high-uniformity and large-caliber magnetic field immunity testing system

Power electronic equipment regulated by the International Thermonuclear Experimental Reactor (ITER) organization must pass the relevant steady-state magnetic field immunity test. The magnetic field immunity testing equipment studied in this paper is an interdisciplinary problem involving electrical, electronic information, structure, fluid, materials, computers, etc., and involves the integration of industrial information systems. It needs to be studied by using the analysis method of multi physical coupling. On the basis of fluid analysis, acquisition and control research and power supply design, this paper focuses on the in-depth study of the coil body. Through mathematical derivation, the magnetic field calculation formulas of several types of coil structures, two-coil and three-coil configurations are obtained in this paper. On the premise of economy, the optimal general formula of coil structure parameters is obtained. These formulas establish the relationship between the coil parameters and the required uniform magnetic field. The magnetic field distribution under various coil structures is compared and analyzed, and the distribution law is revealed. The model analysis is proven to be correct and effective by finite element software and experimental results. The research of this paper provides a theoretical basis for the design of an induction coil with a high magnetic field and high-quality uniformity.

A Duistermaat–Heckman type formula for the based loop group

A Duistermaat–Heckman type formula for the based loop group

The operator $$\sqrt{-1}\widehat{c}(V)(d+\delta )$$ and the Kastler–Kalau–Walze type theorems

In this paper, we obtain two Lichnerowicz type formulas for the operators \(\sqrt{-1}\widehat{c}(V)(d+\delta )\) and \(-\sqrt{-1}(d+\delta )\widehat{c}(V)\). And we give the proof of Kastler–Kalau–Walze type theorems for the operators \(\sqrt{-1}\widehat{c}(V)(d+\delta )\) and \(-\sqrt{-1}(d+\delta )\widehat{c}(V)\) on 3,4-dimensional oriented compact manifolds with (resp.without) boundary.

The Green polynomials via vertex operators

An iterative formula for the Green polynomial is given using the vertex operator realization of the Hall-Littlewood function. Based on this, (1) a general combinatorial formula of the Green polynomial is given; (2) several compact formulas are given for Green's polynomials associated with upper partitions of length $\leq 3$ and the diagonal lengths $\leq 3$; (3) a Murnaghan-Nakayama type formula for the Green polynomial is obtained; and (4) an iterative formula is derived for the bitrace of the finite general linear group $G$ and the Iwahori-Hecke algebra of type $A$ on the permutation module of $G$ by its Borel subgroup.

Supervised Nonparametric Classification in the Context of Replicated Point Patterns

A spatial point pattern is a collection of points in space, representing, e.g. observed locations of trees, bird nests, centers of cells in a histological sample, etc. When several independent realizations of the underlying stochastic process are observed, these realizations are referred to as replicated point patterns. The main objective of this paper is to classify a newly observed pattern into one of the existing classes using a supervised nonparametric classification method, namely the Bayes classifier in combination with the k-nearest neighbors algorithms and the kernel regression method. The dissimilarity between a pair of patterns is defined using the functional summaries extracted from the point patterns via the Cramér-von Mises or Kolmogorov-Smirnov type formula. A set of simulation experiments is presented to investigate the performance of the proposed classifier with a dissimilarity measure based on functional summaries, such as the pair correlation function. The application of such a classifier to a real point pattern dataset is also illustrated.

Picard modular forms and the cohomology of local systems on a Picard modular surface

We formulate a detailed conjectural Eichler–Shimura type formula for the cohomology of local systems on a Picard modular surface associated to the group of unitary similitudes \operatorname{GU}(2,1, \mathbb{Q}(\sqrt{-3})) . The formula is based on counting points over finite fields on curves of genus three which are cyclic triple covers of the projective line. Assuming the conjecture we are able to calculate traces of Hecke operators on spaces of Picard modular forms. We provide ample evidence for the conjectural formula. Along the way we prove new results on characteristic polynomials of Frobenius acting on the first cohomology group of cyclic triple covers of any genus, dimension formulas for spaces of Picard modular forms and formulas for the numerical Euler characteristics of the local systems.

Hirzebruch-Riemann-Roch and Lefschetz type formulas for finite dimensional algebras

We introduce the dimension matrix of a bimodule complex and the trace matrix of a chain endomorphism of a bimodule complex, and give the Hirzebruch-Riemann-Roch (HRR) and Lefschetz type formulas for finite dimensional elementary algebras of finite global dimension. Moreover, we compare our HRR and Lefschetz type formulas with Shklyarov's HRR type formulas and Petit's Lefschetz type formulas for dg algebras.

The use of plant raw materials in the production of meat pate

Meat and meat-containing products represent one of the most difficult foundations for creating functional food, although according to the rules of healthy nutrition, meat is one of the most important food products along with vegetables, fruit, dairy products and fish. Implementation of the suggested technology of meat pate production using plant raw materials (broccoli cabbage) at food industry enterprises will expand the range of domestic products made from local natural raw materials. Essential amino acids such as tryptophan, lysine and methionine are also present in this unique product. It is also beneficial for the health of the patients who have undergone various surgical operations, because liver contains glycogen which is an excellent building material for cells and tissue. Broccoli cabbage has the following health benefits: it contains a unique combination of vitamins and minerals, and it also can compete with beef and chicken eggs in protein composition, tryptophan and lysine content. Food fiber and fiber fetlock contained in cabbage regulate the work of the gastrointestinal tract. The suggested formula and production technology of a new type of meat products (pate with the use of broccoli cabbage) allows obtaining a functional product with a reduced calorie rate, which is essential for healthy nutrition.

Curve counting via stable objects in derived categories of Calabi-Yau 4-folds

In our previous paper with Maulik, we proposed a conjectural Gopakumar-Vafa (GV) type formula for the generating series of stable pair invariants on Calabi-Yau (CY) 4-folds. The purpose of this paper is to give an interpretation of the above GV type formula in terms of wall-crossing phenomena in the derived category. We introduce invariants counting LePotier's stable pairs on CY 4-folds, and show that they count certain stable objects in D0-D2-D8 bound states in the derived category. We propose a conjectural wall-crossing formula for the generating series of our invariants, which recovers the conjectural GV type formula. Examples are computed for both compact and toric cases to support our conjecture.

To What Extent do Freshmen University Chemistry Students Master Chemistry Calculations?

The research mapped chemistry-oriented university freshman students' ability to solve chemical calculations. Their success was monitored based on several factors such as their faculty, field of studies, the type of calculation and the assignment type (word problem vs. formula). The results indicate a significant need to change the approach to teaching chemical calculations - the students were rather unsuccessful in the tasks. The obstacles they face include the ability to identify a problem, understand the concepts of calculation and appropriately adjust the correct use of the mathematical apparatus. These findings represent an impulse for teaching in upper-secondary school as well as to introductory university courses.

The Pieri formulas for hook type Schur multiple zeta functions

We study the Pieri type formulas for the Schur multiple zeta functions along with those for the Schur functions. To express these formulas, we introduce a new insertion rule for adding boxes in the Young tableaux and obtain the results for the hook type Schur multiple zeta functions. For the proof, we show certain extended Jacobi-Trudi formulas for the Schur multiple zeta functions.

On Hyperbolic Complex Numbers

For dimensions two, three and four, we derive hyperbolic complex algebraic structures on the basis of suitably defined vector products and powers which allow in a standard way a series definitions of the hyperbolic vector exponential function. In doing so, we both modify arrow multiplication, which, according to Feynman, is fundamental for quantum electrodynamics, and we give a geometric explanation of why in a certain situation it is natural to define random vector products. Through the interplay of vector algebra, geometry and complex analysis, we extend a systematic approach previously developed for various other complex algebraic structures to the field of hyperbolic complex numbers. We discuss a quadratic vector equation and the property of hyperbolically holomorphic functions of satisfying hyperbolically modified Cauchy–Riemann differential equations and also give a proof of an Euler type formula based on series expansion.

On Polynomials Orthogonal with Respect to an Inner Product Involving Higher-Order Differences: The Meixner Case

In this contribution we consider sequences of monic polynomials orthogonal with respect to the Sobolev-type inner product f,g=⟨uM,fg⟩+λTjf(α)Tjg(α), where uM is the Meixner linear operator, λ∈R+, j∈N, α≤0, and T is the forward difference operator Δ or the backward difference operator ∇. Moreover, we derive an explicit representation for these polynomials. The ladder operators associated with these polynomials are obtained, and the linear difference equation of the second order is also given. In addition, for these polynomials, we derive a (2j+3)-term recurrence relation. Finally, we find the Mehler–Heine type formula for the particular case α=0.

Universality for Random Matrices with Equi-spaced External Source: A Case Study of a Biorthogonal Ensemble

We prove the edge and bulk universality of random Hermitian matrices with equi-spaced external source. One feature of our method is that we use neither a Christoffel–Darboux type formula, nor a double-contour formula, which are standard methods to prove universality results for exactly solvable models. This matrix model is an example of a biorthogonal ensemble, which is a special kind of determinantal point process whose kernel generally does not have a Christoffel–Darboux type formula or double-contour representation. Our methods may showcase how to handle universality problems for biorthogonal ensembles in general.

On properties of the Riemann zeta distribution

We examine various properties of positive integers selected according to the Riemann zeta distribution. That is, if ζ(s)=∑n≥11∕ns, s>1, then we consider the random variable Xs with P(Xs=n)=1∕(ζ(s)ns), n≥1. We derive various results such as the analog of the Erdös–Kac central limit theorem (CLT) for the number of distinct prime factors, ω(Xs), of Xs, as s↘1, large and moderate deviations for ω(Xs), and a Berry–Esseen result. In addition, we prove analogs of Erdös–Delange type formulas for expectations of additive and multiplicative functions evaluated at Xs. We also examine various applications using Dirichlet series. Finally, we show how to derive asymptotic distributional results for an integer selected uniformly at random from [N]={1,2,…,N} or according to the harmonic distribution from their analogous asymptotic results for Xs as s↘1.

Topology of clique complexes of line graphs

The clique complex of a graph G is a simplicial complex whose simplices are all the cliques of G, and the line graph L(G) of G is a graph whose vertices are the edges of G and the edges of L(G) are incident edges of G. In this article, we determine the homotopy type of the clique complexes of line graphs for several classes of graphs including triangle-free graphs, chordal graphs, complete multipartite graphs, wheel-free graphs, and 4-regular circulant graphs. We also give a closed form formula for the homotopy type of these complexes in several cases.

A Voronovskaja type formula for Soardi’s operators

In 1991 Soardi introduced a sequence of positive linear operators beta _{n} associating to each function fin Cleft[ 0,1right] a polynomial function which is closely related to the Bernstein polynomials on left[ -1,+1right] . One of the authors already studied the operators beta _{n} in several papers. This paper is devoted to other properties of Soardi’s operators. We introduce a version {tilde{beta }}_{n} which can be expressed in terms of the classical Bernstein operators and present the relations between beta _{n} and {tilde{beta }}_{n}. We derive Voronovskaja-type results for both beta _{n} and {tilde{beta }}_{n}. Furthermore, rates of convergence for {tilde{beta }}_{n}, respectively beta _{n}, are estimated. Finally, we study the first and second moments of beta _{n}.

The Kastler–Kalau–Walze Type Theorems for the Perturbation of the de Rham Hodge Operator

As applications of the noncommutative residue, we prove the Kastler–Kalau–Walze type theorems for the perturbation of the de Rham Hodge operator on 4-dimensional and 6-dimensional compact manifolds. To get these theorems, we give the Lichnerowicz type formulas for the perturbation of the de Rham Hodge operator on compact manifolds without boundary and the calculation formulas of the boundary term on compact mani-folds with boundary. Additionally, some concrete examples of the perturbation of the de Rham Hodge operator are provided for our main theorems. To the best of the authors' knowledge, the Kastler–Kalau–Walze type theorems for the perturbation of the de Rham Hodge operator have not been considered in the literature. Thus, the current study hopes to serve such a need, in this paper, we investigate the Kastler–Kalau–Walze type theorems for the perturbation of the de Rham Hodge operator on 4-dimensional and 6-dimensional compact manifolds with or without boundary. We compute the Lichnerowicz formulas for the perturbation of the de Rham Hodge operator and the Kastler–Kalau–Walze type theorems for the perturbation of the de Rham Hodge operator on n-dimensional compact manifolds without boundary. And then we prove the Kastler–Kalau–Walze type theorems on 4-dimensional and 6-dimensional compact manifolds with boundary for the perturbation of the de Rham Hodge operator. Results: The motivation of this paper is to prove the Kastler–Kalau–Walze type theorems for the perturbation of the de Rham Hodge operator. Specically, we calculate widetilde{mathrm{Wres}}[pi ^+D_{A}^{-1}circ pi ^+ {{D_A^*}^{-1}}], widetilde{mathrm{Wres}}[pi ^+D_{A}^{-1}circ pi ^+ {{D_A}^{-1}}], widetilde{mathrm{Wres}}[pi ^+D_{A}^{-1}circ pi ^+ {{D_A^*}D_{A}{D_A^*}^{-1}}]. The main work of this paper is to prove the Kastler–Kalau–Walze type theorems for the perturbation of the de Rham Hodge operator on 4-dimensional and 6-dimensional compact manifolds with (resp.without) boundary. Theorem 2.2 shows that the Kastler–Kalau–Walze type theorems for the perturbation of the de Rham Hodge operator on n-dimensional compact manifolds without boundary. On 4-dimensional compact manifolds with boundary, the main results of this paper are Theorem 3.8 and Theorem 3.18, which are the Kastler–Kalau–Walze type theorems for the perturbation of the de Rham Hodge operator. On the other hand, on 6-dimensional compact manifolds with 1 boundary, the main results of this paper are Theorem 4.3 and Theorem 4.15, which are the Kastler–Kalau–Walze type theorems for the perturbation of the de Rham Hodge operator. Comparing with the previous conclusion, we use new operator to study. Although the calculation process is more complex, the better conclusions we have. The results are useful for analyzing and interpreting certain physical theories.

A generalized change of variable formula for the Young integral

This article shows an Itô-Wentzell type formula adapted to Young integral. We apply this formula to study differential equations driven by α-Hölder paths with α∈121.We obtain a formula for the composition and the inverse of solution maps associated with a Young differential equation. Also, we prove a Leibniz rule for interchanging derivative and the Young integral, and a substitution formula for Young integrals.We also study the Cauchy problem for first-order semilinear differential equations driven by a Hölder path via the Method of Characteristics. We also prove a version of the structure theorem of characteristics in the Young context. We conclude by showing the explicit solution of a transport equation perturbed by the H-fractional Brownian motion with Hurst parameter 1/2 <H ≤ 1.

An End-to-End Deep Learning Method for Voltage Sag Classification

Power quality disturbances (PQD) have a negative impact on power quality-sensitive equipment, often resulting in great financial losses. To prevent these losses, besides detecting a PQD on time, it is important to classify it, so that appropriate recovery procedures are employed. The majority of research employs machine learning model PQD classifiers on manually extracted features from simulated or real-world signals. This paper presents an end-to-end approach that circumvents the manual feature extraction and uses signals generated from mathematical voltage sag type formulas. We developed a configurable voltage sag generator that was used to form training and validation datasets. Based on the synthetic three-phase voltage signals, we trained several end-to-end LSTM classifiers that classify voltage sags according to ABC classification. The best-performing model achieved an accuracy of over 90% in the real-world dataset.