Type Formula Research Articles

Poincaré-Lelong Type Formulas and Segre Numbers

Let E and F be Hermitian vector bundles over a complex manifold X and let g:E→F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$g:E\\rightarrow F$$\\end{document} be a holomorphic morphism. We prove a Poincaré-Lelong type formula with a residue term Mg\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M^g$$\\end{document}. The currents Mg\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M^g$$\\end{document} so obtained have an expected functorial property. We discuss various applications: If F has a trivial holomorphic subbundle of rank r outside the analytic set Z, then we get currents with support on Z that represent the Bott-Chern classes c^k(E)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\hat{c}}_k(E)$$\\end{document} for k>rankE-r\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k >\\mathrm{rank \\,}E-r$$\\end{document}. We also consider Segre and Chern forms associated with certain singular metrics on E. The multiplicities (Lelong numbers) of the various components of Mg\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M^g$$\\end{document} only depend on the cokernel of the adjoint sheaf morphism g∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$g^*$$\\end{document}. This leads to a notion of distinguished varieties and Segre numbers of an arbitrary coherent sheaf, generalizing these notions, in particular the Hilbert-Samuel multiplicity, in case of an ideal sheaf.

Multidimensional continuous Bessel wavelet transform: properties and applications

In the present paper we consider the new concept of the multidimensional continuous Bessel wavelet transform, and we develop its fundamental properties. Also, we establish for the novel proposed continuous wavelet transform a Calderon's reproducing type formula and a reconstruction formula. In the end, we formulate the Heisenberg's uncertainty principles associated to this new wavelet transform.

A Simons Type Formula for Spacelike Submanifolds in Semi-Riemannian Warped Product and its Applications

A Simons Type Formula for Spacelike Submanifolds in Semi-Riemannian Warped Product and its Applications

Pricing of timer volatility-barrier options under Heston’s stochastic volatility model

Timer options are financial instruments that enable investors to exercise their rights on a random maturity date the realized variance reaches the level of variance budget. These options provide a stable investment opportunity for investors under the unpredictable and complex financial markets, such as global financial crisis or COVID-19 pandemic, which can induce the drastic changes of the volatility for the underlying asset. Meanwhile, in the financial markets, investors who invest in standard timer options may face the problems caused by the postponement of the exercising time for too low volatility, compared to standard vanilla options. In this regard, to overcome such disadvantages, we propose timer volatility-barrier options, which are activated and expired when the volatility arrives at a relatively low barrier level, with the original properties of the standard timer options. In this paper, by making use of the method of images, we derive an analytical formulas for the timer volatility-barrier options so that the volatility process can be driven by Heston stochastic volatility model, and verify the pricing accuracy of the timer options by comparing our solutions with those obtained from Monte Carlo simulations. Finally, we conduct numerical studies on the timer volatility-barrier options to examine the pricing sensitivities with respect to the various model parameters, and implement the discussion for pricing formula of double volatility barrier type of the timer options.

Failure of the Lie-Trotter type formula for q-exponential operators

We give various examples which show that the Lie-Trotter type formula for q-exponential operators of Rajagopal-Tsallis is, in general, false.

A Comprehensive Assessment of the Environmental Impact of Different Infant Feeding Types: The Observational Study GREEN MOTHER.

To observe and compare the environmental impacts of different types of infant feeding, considering the use of formula, infant feeding accessories, potentially increased maternal dietary intake during breastfeeding (BF) and food consumption habits. An observational cross-sectional multicentre study conducted in the Barcelona Metropolitan Area of the Catalan Institute of Health. Data were collected from 419 postpartum women on infant feeding type (formula milk and accessories), maternal dietary intake (24-h register) and food consumption habits from November 2022 to April 2023. The environmental impacts (climate change (CC), water consumption and water scarcity) of the infant feeding types and maternal diet were calculated using the IPCC, ReCiPE and AWARE indicators, respectively. The differences in impacts were calculated by Kruskal-Wallis test. Significant differences for the three environmental impacts were observed. The CC impact of formula milk and feeding accessories was 0.01 kg CO2eq for exclusive BF, 1.55 kg CO2eq for mixed feeding and 4.98 kg CO2eq for formula feeding. While BF mothers consumed an extra 238 kcal, no significant differences were found related to maternal diet across feeding types. Exclusive BF was the most sustainable type of infant feeding, considering formula and infant feeding accessories. In our study, the difference between the impacts of BF and non-BF mothers' diet was insignificant. Offer informative and educational support for midwives and other healthcare professionals on BF and a healthy, sustainable diet to transfer this knowledge to the general public. Raise the general public's awareness about BF and a healthy, sustainable diet. To reduce environmental impacts through behavioural changes. STROBE. Patients of the Catalan Health Service reviewed the content of the data collection tools. (for the whole GREEN MOTHER project): NCT05729581 (https://clinicaltrials.gov).

Composition of some positive linear integral operators

Abstract This article is devoted to constructing sequences of integral operators with the same Voronovskaja formula as the generalized Baskakov operators, but having different behavior in other respects. For them, we investigate the eigenstructure, the inverses, and the corresponding Voronovskaja type formulas. A general result of Voronovskaja type for composition of operators is given and applied to the new operators. The asymptotic behavior of differences between the operators is investigated, and as an application, we obtain a formula involving Euler’s gamma function.

Optimization of the annular fin arrangement in phase change heat storage units based on response surface methodology

Latent heat thermal energy storage (LHTES) technology is of increasing industrial interest due to its advantages such as high heat storage density and near-constant temperature during the phase change process. However, the latent heat storage technology is impacted by the low thermal conductivity of the phase change material (PCM), leading to delays and heat loss in the heat transfer process within LHTES. This limitation reduces the efficiency of heat transfer and diminishes the performance of the heat storage unit. To address this issue, this paper employs numerical simulation and response surface methodology to investigate different types of annular fin phase change heat storage units, aiming to derive the optimal configuration for enhancing unit performance. Therefore, the effect of fin type and number on the melting and solidification process is analyzed in detail. The research results indicate that among the five different types of fins studied, the total time required for the unit to complete melting and solidification is the least with type “e” fins, only 51.78 h, demonstrating that type “e” fins have the best effect on improving unit performance. For type “e” fins, when the number of fins n is 15, it can most effectively shorten the melting and solidification time of the phase change heat storage unit, reducing it to only 47.81 h. In addition, the response surface methodology was utilized to derive the optimal fitting formulas for different types of annular fins and to validate these formulas, and the error between the validation analysis and the numerical simulations was within 5%, which confirms the reliability of the fitting formulas developed in this study. This approach facilitates the selection of various annular fin models for subsequent experimental and simulation studies, thereby reducing the time and cost associated with model selection.

On the asymptotic behavior of solutions to bilinear Caputo stochastic fractional differential equations

In this paper, we focus on investigating the asymptotic behavior of solutions in a mean square sense to bilinear Caputo stochastic fractional differential equations (CSFDEs). The main tools in the proof include a variation of the constant formula for CSFDEs, the Jordan normal form of a matrix, the summation formula of Djrbashian type, and constructing a weighted norm in the associated Banach space.

A Girsanov transformed Clark-Ocone-Haussmann type formula for $$L^1$$-pure jump additive processes and its application to portfolio optimization

AbstractWe derive a Clark-Ocone-Haussmann (COH) type formula under a change of measure for $$ L^1 $$ L 1 -canonical additive processes, providing a tool for representing financial derivatives under a risk-neutral probability measure. COH formulas are fundamental in stochastic analysis, providing explicit martingale representations of random variables in terms of their Malliavin derivatives. In mathematical finance, the COH formula under a change of measure is crucial for representing financial derivatives under a risk-neutral probability measure. To prove our main results, we use the Malliavin-Skorohod calculus in $$ L^0 $$ L 0 and $$ L^1 $$ L 1 for additive processes, as developed by Di Nunno and Vives (2017). An application of our results is solving the local risk minimization (LRM) problem in financial markets driven by pure jump additive processes. LRM, a prominent hedging approach in incomplete markets, seeks strategies that minimize the conditional variance of the hedging error. By applying our COH formula, we obtain explicit expressions for locally risk-minimizing hedging strategies in terms of Malliavin derivatives under the market model underlying the additive process. These formulas provide practical tools for managing risks in financial market price fluctuations with $$L^1$$ L 1 -additive processes.

Lie product type formulas for continuous multilinear operators

In this paper we prove various Lie product type formulas for continuous multilinear operators. A sample result: Let k≥2 be a natural number, X1,..., Xk, Y be Banach algebras with unit and T:X1×⋅⋅⋅×Xk→Y a continuous k-linear operator such that T(1,...,1)=1. Let also (an)n∈N be a sequence of natural numbers such that limn→∞⁡an=∞. Then for all x1∈X1,..., xk∈Xk we havelimn→∞⁡[T(ex1an,...,exkan)]an=eT(x1,1,...,1)+T(1,x2,1,...,1)+⋅⋅⋅+T(1,...,1,xk).

Quantum (or q$$ q $$‐) operator equations and associated partial differential equations for bivariate Laguerre polynomials with applications to the q$$ q $$‐Hille‐Hardy type formulas

Based on the extensive application of the ‐series and ‐polynomials including (for example) the ‐Laguerre polynomials in several fields of the mathematical and physical sciences, we attach great importance to the equations and related application issues involving the ‐Laguerre polynomials. The mission of this paper is to find the general ‐operational equation together with the expansion issue of the bivariate ‐Laguerre polynomials from the perspective of ‐partial differential equations. We also give some applications including some ‐Hille‐Hardy type formulas. In addition, we present the Rogers‐type formulas and the ‐type generating functions for the bivariate ‐Laguerre polynomials by the technique based upon ‐operational equations. Moreover, we derive a new generalized Andrews‐Askey integral and a new transformation identity involving the bivariate ‐Laguerre polynomials by applying ‐operational equations.

Probabilistic Type 2 poly-Bernoulli Polynomials

The main purpose of this article is to introduce the probabilistic type 2 poly-Bernoulli polynomials under the condition that Y is a random variable. This means that we will consider the probabilistic extension of the type 2 poly-Bernoulli polynomials and study to obtain some new results. Furthermore, we also define the probabilistic unipoly-Bernoulli polynomials and numbers attached to p, and investigate their interesting basic properties. Based on these new definition, we derive some meaningful formulae of probabilistic type 2 poly-Bernoulli polynomials and probabilistic unipoly-Bernoulli polynomials and numbers attached to p.

On Spectral Dai-Yuan Structured Technique for Solving Nonlinear Least-Squares Optimisation

Handling nonlinear least squares (NLS) problems as unconstrained optimisation problems has led to the development and adaptation of numerous numerical approaches. Calculating the Hessian matrix expensively for each iteration is one of the problems with the current numerical approaches for solving NLS problems. Some methods designed to rely solely on first-derivative information while ignoring second-order details tend to underperform when dealing with problems involving non-zero residuals. This paper suggests a modification to the Dai-Yuan type (DY) formula by deriving a spectral parameter for the proposed search direction. The sufficient descent results of the new formula are established using the standard assumptions under the strong Wolfe line search. Furthermore, experiments are conducted on some benchmark functions to demonstrate the computational efficiency of the proposed technique. The results highlight the effectiveness of the algorithm compared to existing methods.Keywords: Conjugate gradient, convergence theory, Hessian matrix, optimisation, Quasi-Newton, spectral, structured secant method

Asymptotically compatible energy of variable-step fractional BDF2 scheme for the time-fractional Cahn–Hilliard model

Abstract A novel discrete gradient structure of the variable-step fractional BDF2 formula approximating the Caputo fractional derivative of order $\alpha \in (0,1)$ is constructed by a local-nonlocal splitting technique, that is, the fractional BDF2 formula is split into a local part analogue to the two-step backward differentiation formula (BDF2) of the first derivative and a nonlocal part analogue to the L1-type formula of the Caputo derivative. Then a local discrete energy dissipation law of the variable-step fractional BDF2 implicit scheme is established for the time-fractional Cahn–Hilliard model under a weak step-ratio constraint $0.3960\le \tau _{k}/\tau _{k-1}<r^{*}(\alpha )$, where $\tau _{k}$ is the $k$th time-step size and $r^{*}(\alpha )\ge 4.660$ for $\alpha \in (0,1)$. The present result provides a practical answer to the open problem in [SINUM, 57: 218-237, Remark 6] and significantly relaxes the severe step-ratio restriction [Math. Comp., 90: 19–40, Theorem 3.2]. More interestingly, the discrete energy and the corresponding energy dissipation law are asymptotically compatible with the associated discrete energy and the energy dissipation law of the variable-step BDF2 method for the classical Cahn–Hilliard equation, respectively. To the best of our knowledge, such type energy dissipation law is established at the first time for the variable-step L2 type formula of Caputo’s derivative. Numerical examples with an adaptive stepping procedure are provided to demonstrate the accuracy and the effectiveness of our proposed method.

Asymptotic expansions for variants of the gamma and Post–Widder operators preserving 1 and xj

Recently, the authors constructed operators acting on a space of functions defined on and preserving 1 and for a given . To this end, they considered suitable modifications of the Post–Widder and gamma operators. They proved convergence results and established the associated Voronovskaja type formulas. In this paper, we give asymptotic expansions for the sequences of these operators.

Topological recursion for Kadomtsev–Petviashvili tau functions of hypergeometric type

AbstractWe study the ‐point differentials corresponding to Kadomtsev–Petviashvili (KP) tau functions of hypergeometric type (also known as Orlov–Scherbin partition functions), with an emphasis on their ‐deformations and expansions. Under the naturally required analytic assumptions, we prove certain higher loop equations that, in particular, contain the standard linear and quadratic loop equations, and thus imply the blobbed topological recursion. We also distinguish two large families of the Orlov–Scherbin partition functions that do satisfy the natural analytic assumptions, and for these families, we prove in addition the so‐called projection property and thus the full statement of the Chekhov–Eynard–Orantin topological recursion. A particular feature of our argument is that it clarifies completely the role of ‐deformations of the Orlov–Scherbin parameters for the partition functions, whose necessity was known from a variety of earlier obtained results in this direction but never properly understood in the context of topological recursion. As special cases of the results of this paper, one recovers new and uniform proofs of the topological recursion to all previously studied cases of enumerative problems related to weighted double Hurwitz numbers. By virtue of topological recursion and the Grothendieck–Riemann–Roch formula, this, in turn, gives new and uniform proofs of almost all Ekedahl–Lando–Shapiro–Vainshtein (ELSV)‐type formulas discussed in the literature.

Polymeromorphic Itô–Hermite functions associated with a singular potential vector on the punctured complex plane

We provide a theoretical study of a new family of orthogonal functions on the punctured complex plane solving the eigenvalue problems for some magnetic Laplacian perturbed by a singular vector potential with zero magnetic field modeling the Aharonov–Bohm effect. The functions are defined by their β-modified Rodrigues type formula and extend the polyanalytic Itô–Hermite polynomials to the polymeromorphic setting. Mainly, we derive their different operational representations and give their explicit expressions in terms of special functions. Different generating functions and integral representations are obtained.

Bernstein–Kantorovich operators, approximation and shape preserving properties

Inspired by the construction of Bernstein and Kantorovich operators, we introduce a family of positive linear operators Kn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{K}_n$$\\end{document} preserving the affine functions. Their approximation properties are investigated and compared with similar properties of other operators. We determine the central moments of all orders of Kn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathcal {K}}}_n$$\\end{document} and use them in order to establish Voronovskaja type formulas. A special attention is paid to the shape preserving properties. The operators Kn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathcal {K}}}_n$$\\end{document} preserve monotonicity, convexity, strong convexity and approximate concavity. They have also the property of monotonic convergence under convexity. All the established inequalities involving convex functions can be naturally interpreted in the framework of convex stochastic ordering.

Hurewicz and Dranishnikov-Smith theorems for asymptotic dimension of countable approximate groups

We establish two main results for the asymptotic dimension of countable approximate groups. The first one is a Hurewicz type formula for a global morphism of countable approximate groups f:(Ξ,Ξ∞)→(Λ,Λ∞), stating that asdimΞ≤asdimΛ+asdim([ker⁡f]c). This is analogous to the Dranishnikov-Smith result for groups, and is relying on another Hurewicz type formula we prove, using a 6-local morphism instead of a global one. The second result is similar to the Dranishnikov-Smith theorem stating that, for a countable group G, asdim G is equal to the supremum of asymptotic dimensions of finitely generated subgroups of G. Our version states that, if (Λ,Λ∞) is a countable approximate group, then asdim Λ is equal to the supremum of asymptotic dimensions of approximate subgroups of finitely generated subgroups of Λ∞, with these approximate subgroups contained in Λ2.