Calculus Rules Research Articles

Constraint qualifications and optimality conditions for nonsmooth multiobjective mathematical programming problems with vanishing constraints on Hadamard manifolds via convexificators

In this paper, we introduce the notion of convexificators in the framework of Hadamard manifolds. We derive some calculus rules for convexificators on Hadamard manifolds and employ them to investigate nonsmooth multiobjective programming problems with vanishing constraints on Hadamard manifolds (abbreviated as, (NMPVC)). The Abadie constraint qualification (abbreviated as, (ACQ)), as well as the generalized (ACQ) (abbreviated as, (GACQ)), are introduced for (NMPVC) in terms of convexificators. Further, (GACQ) is employed to establish the well-known Karush-Kuhn-Tucker (abbreviated as, KKT) type necessary Pareto efficiency criteria for (NMPVC). Moreover, we introduce several other constraint qualifications for (NMPVC) in the Hadamard manifold setting using convexificators and establish interrelations among them. We have furnished non-trivial examples in the framework of some well-known Hadamard manifolds to illustrate the significance of our results. As far as the best of our knowledge is concerned, constraint qualifications and optimality conditions for (NMPVC) have not yet been studied in the framework of Hadamard manifolds via convexificators.

Roles of Earth’s Albedo Variations and Top-of-the-Atmosphere Energy Imbalance in Recent Warming: New Insights from Satellite and Surface Observations

Past studies have reported a decreasing planetary albedo and an increasing absorption of solar radiation by Earth since the early 1980s, and especially since 2000. This should have contributed to the observed surface warming. However, the magnitude of such solar contribution is presently unknown, and the question of whether or not an enhanced uptake of shortwave energy by the planet represents positive feedback to an initial warming induced by rising greenhouse-gas concentrations has not conclusively been answered. The IPCC 6th Assessment Report also did not properly assess this issue. Here, we quantify the effect of the observed albedo decrease on Earth’s Global Surface Air Temperature (GSAT) since 2000 using measurements by the Clouds and the Earth’s Radiant Energy System (CERES) project and a novel climate-sensitivity model derived from independent NASA planetary data by employing objective rules of calculus. Our analysis revealed that the observed decrease of planetary albedo along with reported variations of the Total Solar Irradiance (TSI) explain 100% of the global warming trend and 83% of the GSAT interannual variability as documented by six satellite- and ground-based monitoring systems over the past 24 years. Changes in Earth’s cloud albedo emerged as the dominant driver of GSAT, while TSI only played a marginal role. The new climate sensitivity model also helped us analyze the physical nature of the Earth’s Energy Imbalance (EEI) calculated as a difference between absorbed shortwave and outgoing longwave radiation at the top of the atmosphere. Observations and model calculations revealed that EEI results from a quasi-adiabatic attenuation of surface energy fluxes traveling through a field of decreasing air pressure with altitude. In other words, the adiabatic dissipation of thermal kinetic energy in ascending air parcels gives rise to an apparent EEI, which does not represent “heat trapping” by increasing atmospheric greenhouse gases as currently assumed. We provide numerical evidence that the observed EEI has been misinterpreted as a source of energy gain by the Earth system on multidecadal time scales.

Stability criteria and calculus rules via conic contingent coderivatives in Banach spaces

This paper addresses the study of novel constructions of variational analysis and generalized differentiation that are appropriate for characterizing robust stability properties of constrained set-valued mappings/multifunctions between Banach spaces important in optimization theory and its applications. Our tools of generalized differentiation revolve around the newly introduced concept of ε-regular normal cone to sets and associated coderivative notions for set-valued mappings. Based on these constructions, we establish several characterizations of the central stability notion known as the relative Lipschitz-like property of set-valued mappings in infinite dimensions. Applying a new version of the constrained extremal principle of variational analysis, we develop comprehensive sum and chain rules for our major constructions of conic contingent coderivatives for multifunctions between appropriate classes of Banach spaces.

Some calculus rules, generalized convexity via convexifactors and their applications

Some calculus rules, generalized convexity via convexifactors and their applications

Higher-order KKT optimality conditions through contingent derivatives for constrained nonsmooth vector equilibrium problems

In this paper, we deal with some higher-order optimality conditions for local strict efficient solutions to a nonsmooth vector equilibrium problem with set, cone and equality constraints. For this aim, the concept of m−stable and m−steady functions (m≥2 and integer) for single-valued functions and some constraint qualifications of higher order in terms of contingent derivatives are proposed accordingly. We analyze the sum calculus rule of mth-order adjacent set, mth-order interior set, asymptotic mth-order tangent cone and asymptotic mth-order adjacent cone. Subsequently, we employ the obtained calculus rules to treat KKT necessary and sufficient optimality conditions of higher order in terms of contingent derivatives for the mth-order (local) strict efficient solutions to such problem. Simultaneously, we employ these rules to study KKT higher-order optimality conditions for such efficient solutions for a nonsmooth vector optimization problem with constraints. Some illustrative examples are provided to demonstrate the main results of the literature.

RÓWNANIA POLA ELEKTROMAGNETYCZNEGO W ŚRODOWISKU NIELINIOWYM

The paper proposes electromagnetic field equations from the point of view of their adaptation to numerical methods. Maxwell's equations with partial derivatives are used, written concerning field vectors, which most fully reproduce the picture of physical processes in electrical engineering devices. The values of these vectors provide comprehensive information about the field at any spatio-temporal point. The concept of creating mathematical models of electrical devices adequate to physical processes has been developed. Mathematical transformations are carried out according to the rules of differential calculus. Dynamic processes in the elements of electrotechnical devices were analyzed using the apparatus of mathematical modeling. An algorithm for implementing differential equations with partial derivative numerical methods using computer simulation was implemented. The obtained results made it possible to understand the nature of electromagnetic phenomena in nonlinear media. The paper provides calculations of the field parameters in a flat ferromagnetic plate and the groove of the rotor of an electric machine.

The physical mechanism of stochastic calculus in random walks

Stochastic differential equations (SDEs) play an important role in fields ranging from physics and biology to economics. The interpretation of stochastic calculus in the presence of multiplicative noise continues to be an open question. Commonly, the choice of stochastic calculus rules is largely based on empirical knowledge and lacks quantitative substantiation. In this study, we introduce a functional method that quantitatively links stochastic calculus rules to the underlying physical mechanisms in random walks. For a given diffusion coefficient, we construct three models to exemplify the physical features of conventional stochastic calculus. Our work provides a new perspective for quantitatively addressing state-dependent noise and aims to contribute to the understanding of the physical factors underlying uncertainty in SDEs.

Projectional Coderivatives and Calculus Rules

Projectional Coderivatives and Calculus Rules

Calculus and Fine Properties of Functions of Bounded Variation on RCD Spaces

We generalize the classical calculus rules satisfied by functions of bounded variation to the framework of {textrm{RCD}} spaces. In the infinite dimensional setting, we are able to define an analogue of the distributional differential and, on finite dimensional spaces, we prove fine properties and suitable calculus rules, such as the Vol’pert chain rule for vector valued functions.

Density of subalgebras of Lipschitz functions in metric Sobolev spaces and applications to Wasserstein Sobolev spaces

We prove a general criterion for the density in energy of suitable subalgebras of Lipschitz functions in the metric Sobolev space H1,p(X,d,m) associated with a positive and finite Borel measure m in a separable and complete metric space (X,d).We then provide a relevant application to the case of the algebra of cylinder functions in the Wasserstein Sobolev space H1,2(P2(M),W2,m) arising from a positive and finite Borel measure m on the Kantorovich-Rubinstein-Wasserstein space (P2(M),W2) of probability measures in a finite dimensional Euclidean space, a complete Riemannian manifold, or a separable Hilbert space M. We will show that such a Sobolev space is always Hilbertian, independently of the choice of the reference measure m so that the resulting Cheeger energy is a Dirichlet form.We will eventually provide an explicit characterization for the corresponding notion of m-Wasserstein gradient, showing useful calculus rules and its consistency with the tangent bundle and the Γ-calculus inherited from the Dirichlet form.

FP²: Fully in-Place Functional Programming

As functional programmers we always face a dilemma: should we write purely functional code, or sacrifice purity for efficiency and resort to in-place updates? This paper identifies precisely when we can have the best of both worlds: a wide class of purely functional programs can be executed safely using in-place updates without requiring allocation, provided their arguments are not shared elsewhere. We describe a linear _fully in-place_ (FIP) calculus where we prove that we can always execute such functions in a way that requires no (de)allocation and uses constant stack space. Of course, such a calculus is only relevant if we can express interesting algorithms; we provide numerous examples of in-place functions on datastructures such as splay trees or finger trees, together with in-place versions of merge sort and quick sort. We also show how we can generically derive a map function over _any_ polynomial data type that is fully in-place. Finally, we have implemented the rules of the FIP calculus in the Koka language. Using the Perceus reference counting garbage collection, this implementation dynamically executes FIP functions in-place whenever possible.

Modified Backpropagation Algorithm with Multiplicative Calculus in Neural Networks

Backpropagation is one of the most widely used algorithms for training feedforward deep neural networks. The algorithm requires a differentiable activation function and it performs computations of the gradient proceeding backwards through the feedforward deep neural network from the last layer through to the first layer. In order to calculate the gradient at a specific layer, the gradients of all layers are combined via the chain rule of calculus. One of the biggest disadvantages of the backpropagation is that it requires a large amount of training time. To overcome this issue, this paper proposes a modified backpropagation algorithm with multiplicative calculus. Multiplicative calculus provides an alternative to the classical calculus and it defines new kinds of derivative and integral forms in multiplicative form rather than addition and subtraction forms. The performance analyzes are discussed in various case studies and the results are given comparatively with classical backpropagation algorithm. It is found that the proposed modified backpropagation algorithm converges in less time to the solution and thus provides fast training in the given case studies. It is also shown that the proposed algorithm avoids the local minima problem.

Semantic Information and the Complexity of Deduction

AbstractIn the chapter “Information and Content” of their Impossible Worlds, Berto and Jago provide us with a semantic account of information in deductive reasoning such that we have an explanation for why some, but not all, logical deductions are informative. The framework Berto and Jago choose to make sense of the above-mentioned idea is a semantic interpretation of Sequent Calculus rules of inference for classical logic. I shall argue that although Berto and Jago’s idea and framework are hopeful, their definitions do not capture what is intended. This is so because the definitions are solely based on the logical complexity of an argument and they fail to capture the richness of the non-logical content of that argument. Then I will suggest some amendments to address this problem. Finally, I will extend the application of the definitions to first-order logic. It shall be observed that in some classical deductions, applying contraction may lead to hiding some epistemic impossibilities. This happens when formulae which contribute different propositions to the proof get contracted at the quantified level.

Fluctuation-dissipation relation, Maxwell-Boltzmann statistics, equipartition theorem, and stochastic calculus

We derive the fluctuation-dissipation relation and explore its connection with the equipartition theorem and Maxwell-Boltzmann statistics through the use of different stochastic analytical techniques. Our first approach is the theory of backward stochastic differential equations, which arises naturally in this context, and facilitates the understanding of the interplay between these classical results of statistical mechanics. Moreover, it allows to generalize the classical form of the fluctuation-dissipation relation. The second approach consists in deriving forward stochastic differential equations for the energy of an electric system according to both Itô and Stratonovich stochastic calculus rules. While the Itô equation possesses a unique solution, which is the physically relevant one, the Stratonovich equation admits this solution along with infinitely many more, none of which has a physical nature. Despite of this fact, some, but not all of them, obey the fluctuation-dissipation relation and the equipartition of energy.

Modern perspectives in Proof Theory.

Modern perspectives in Proof Theory.

Multi-composition rule of convex subdifferential calculus

In this paper, a general formula concerning the multi-composition rule of convex subdifferential calculus is provided in the setting of Banach spaces under an appropriate regularity condition. As an application, this calculus rule is applied to obtain necessary and sufficient Karush–Kuhn–Tucker type optimality conditions for constrained convex minmax location problems with perturbed minimal time functions and set-up costs.

Fourth-order tensor calculus operations and application to continuum mechanics

This paper provides a presentation of differential calculus involving fourth-order tensors in consideration of the cyclic class-based tensor operations introduced in an earlier paper on fourth-order tensor algebraic operations. Three classes of cyclic second-order tensor operators, denoted by dot, cross, and star, with separate definitions for double contraction, quadruple contraction, and tensor products were recognized. The present paper examines the differentiation of second-order isotropic tensor functions with respect to a second-order tensor, taking into account the three classes of second-order tensor operators. Product rules, chain rules, and identities necessary to maintain consistency are defined. In combining the proposed calculus rules with the second-order tensor operators, a comprehensive set of proofs for the power derivatives is first presented. A closed-form solution for the second-order tensor derivative of an isotropic tensor function is also detailed. Some of the new solutions are an outcome of capabilities offered by new operators and identities provided in the literature. The paper is arranged so as to be a useful reference work with regard to the mathematics of fourth-order tensors in absolute notation, particularly for application in the field of continuum mechanics in engineering.

A Generalized Newton Method for Subgradient Systems

This paper proposes and develops a new Newton-type algorithm to solve subdifferential inclusions defined by subgradients of extended real-valued prox-regular functions. The proposed algorithm is formulated in terms of the second order subdifferential of such functions that enjoy extensive calculus rules and can be efficiently computed for broad classes of extended real-valued functions. Based on this and on the metric regularity and subregularity properties of subgradient mappings, we establish verifiable conditions ensuring the well-posedness of the proposed algorithm and its local superlinear convergence. The obtained results are also new for the class of equations defined by continuously differentiable functions with Lipschitzian gradients ([Formula: see text] functions), which is the underlying case of our consideration. The developed algorithms for prox-regular functions and their extension to a structured class of composite functions are formulated in terms of proximal mappings and forward–backward envelopes. Besides numerous illustrative examples and comparison with known algorithms for [Formula: see text] functions and generalized equations, the paper presents applications of the proposed algorithms to regularized least square problems arising in statistics, machine learning, and related disciplines. Funding: Research of P. D. Khanh is funded by Ho Chi Minh City University of Education Foundation for Science and Technology [Grant CS.2022.19.20TD]. Research of B. Mordukhovich and V. T. Phat was partly supported by the U.S. National Science Foundation [Grants DMS-1808978 and DMS-2204519]. The research of B. Mordukhovich was also supported by the Air Force Office of Scientific Research [Grant 15RT0462] and the Australian Research Council under Discovery Project DP-190100555. This work was supported by the Air Force Office of Scientific Research [Grant 15RT0462].

Regularly abstract convex functions with respect to the set of Lipschitz continuous concave functions

Given a set of functions defined on a set X, á function is called abstract -convex if it is the upper envelope of its -minorants, i.e. such its minorants which belong to the set ; and f is called regularly abstract -convex if it is the upper envelope of its maximal (with respect to the pointwise ordering) -minorants. In the paper we first present the basic notions of (regular) -convexity for the case when is an abstract set of functions. For this abstract case a general sufficient condition based on Zorn's lemma for a -convex function to be regularly -convex is formulated. The goal of the paper is to study the particular class of regularly -convex functions, when is the set of real-valued Lipschitz continuous classically concave functions defined on a real normed space X. For an extended-real-valued function to be -convex it is necessary and sufficient that f be lower semicontinuous and bounded from below by a Lipschitz continuous function; moreover, each -convex function is regularly -convex as well. We focus on -subdifferentiability of functions at a given point. We prove that the set of points at which an -convex function is -subdifferentiable is dense in its effective domain. This result extends the well-known classical Brøndsted-Rockafellar theorem on the existence of the subdifferential for convex lower semicontinuous functions to the more wide class of lower semicontinuous functions. Using the subset of the set consisting of such Lipschitz continuous concave functions that vanish at the origin we introduce the notions of -subgradient and -subdifferential of a function at a point which generalize the corresponding notions of the classical convex analysis. Some properties and simple calculus rules for -subdifferentials as well as -subdifferential conditions for global extremum points are established. Symmetric notions of abstract -concavity and -superdifferentiability of functions where is the set of Lipschitz continuous convex functions are also considered.

Automatic Algorithm Programming Model Based on the Improved Morgan's Refinement Calculus

The automatic algorithm programming model can increase the dependability and efficiency of algorithm program development, including specification generation, program refinement, and formal verification. However, the existing model has two flaws: incompleteness of program refinement and inadequate automation of formal verification. This paper proposes an automatic algorithm programming model based on the improved Morgan's refinement calculus. It extends the Morgan's refinement calculus rules and designs the C++ generation system for realizing the complete process of refinement. Meanwhile, the automation tools VCG (Verification Condition Generator) and Isabelle are used to improve the automation of formal verification. An example of a stock's maximum income demonstrates the effectiveness of the proposed model. Furthermore, the proposed model has some relevance for automatic software generation.