In the present paper, we study the Lagrange dualities of the DC infinite optimization problems. By using the properties of the epigraph of the conjugate functions, we provide characterizations for the DC infinite optimization problem to have the stable weak Lagrange dualities and the stable Lagrange dualities. Applications to DC semi-infinite programming, DC conic programming and convex infinite programming are given.

Predicting single-cell perturbation responses requires mapping between two unpaired single-cell data distributions. Optimal transport (OT) theory provides a principled framework for constructing such mappings by minimizing transport cost. Recently, Wasserstein-2 (W2) neural optimal transport solvers (e.g. CellOT) have been used for this prediction task. However, W2 OT relies on the general Kantorovich dual formulation, which involves optimizing over two conjugate functions, leading to a complex min-max optimization problem that converges slowly. To address these challenges, we propose a novel solver based on the Wasserstein-1 (W1) dual formulation. Unlike W2, the W1 dual simplifies the optimization to a maximization problem over a single 1-Lipschitz function, thus eliminating the need for time-consuming min-max optimization. While solving the W1 dual only reveals the transport direction and does not directly provide a unique optimal transport map, we incorporate an additional step using adversarial training to determine an appropriate transport step size, effectively recovering the transport map. Our experiments demonstrate that the proposed W1 neural optimal transport solver can mimic the W2 OT solvers in finding a unique and "monotonic" map on 2D datasets. Moreover, the W1 OT solver achieves performance on par with or surpasses W2 OT solvers on real single-cell perturbation datasets. Furthermore, we show that W1 OT solver achieves 25∼45× speedup, scales better on high dimensional transportation task, and can be directly applied on single-cell RNA-seq dataset with highly variable genes. Our implementation and experiments are open-sourced at https://github.com/poseidonchan/w1ot.

Abstract We characterize the set of semiclassical measures corresponding to sequences of eigenfunctions of the attractive Coulomb operator $$\widehat{H}_{\hbar }{:}{=}-\frac{\hbar ^2}{2}\Delta _{\mathbb {R}^3}-\frac{1}{|x|}$$ H ^ ħ : = - ħ 2 2 Δ R 3 - 1 | x | . In particular, any Radon probability measure on the fixed negative energy hypersurface $$\Sigma _E$$ Σ E of the Kepler Hamiltonian H in classical phase space that is invariant under the regularized Kepler flow is the semiclassical measure of a sequence of eigenfunctions of $$\widehat{H}_{\hbar }$$ H ^ ħ with eigenvalue E as $$\hbar \rightarrow 0$$ ħ → 0 . The main tool that we use is the celebrated Fock unitary conjugation map between eigenspaces of $$\widehat{H}_{\hbar }$$ H ^ ħ and $$-\Delta _{\mathbb {S}^3}$$ - Δ S 3 . We first prove that for any Kepler orbit $$\gamma $$ γ on $$\Sigma _E$$ Σ E , there is a sequence of eigenfunctions that converge in the sense of semiclassical measures to the delta measure supported on $$\gamma $$ γ as $$\hbar \rightarrow 0$$ ħ → 0 , and we finish using a density argument in the weak-* topology.

Based on two Bateman conjugate functions, a class of finite-energy spatiotemporally localized null electromagnetic fields in vacuum have been constructed. These wavepackets share with the basic Hopfion, as well as higher-order Hopf-Ranada solutions, topological properties dealing with linked and knotted electromagnetic field lines. Contrary to the Hopf-Ranada solutions, however, they are unidirectional and devoid of energy backflow, the latter being a counterintuitive phenomenon whereby, even for forward-propagating plane wave components, the energy locally propagates backward. Conservation laws for electromagnetic chirality and helicity for these novel wavepackets have been studied.

The present paper deals with the problem of constructing discrete counterparts of conjugate harmonic functions in the scope of hypercomplex analysis. Specifically, let us denote by Δh the discrete Laplacian, by Dh the discrete Dirac operator such that (Dh)2=-Δh, and by ∂s a time-derivative. Our approach starts with the correspondence between the null solutions w=u-e0v of the Dirac type operator e0∂s+Dh and the pair of solutions (u,v) of the following semidiscrete Cauchy-Riemann type system: ∂su(x,s)=-Dhv(x,s)∂sv(x,s)=-Dhu(x,s),(x,s)∈hZn×(0,∞).Hereby, e0 stands for a Clifford basis element, satisfying e02=-1 and e0Dh+Dhe0=0. In a nutshell, we prove that the solutions of such system can be characterized in terms of the null solutions of the semidiscrete Laplacian ∂s2+Δh. Afterwards, we show that a similar formulation on discrete space-time lattices arises from the aforementioned semidiscrete formulation on hZn×(0,∞) by means of sampling using Bessel functions of the first kind. These results provide us the building blocks to characterize the discrete analogue of the Riesz-Hilbert transform in terms of the ’boundary behavior’ of discrete conjugate harmonic functions, generated from operator semigroups of Poisson type. We end this paper by discussing some open problems of future research.

Given an analytic function f=u+iv in the unit disk D, Zygmund’s theorem gives the minimal growth restriction on u which ensures that v is in the Hardy space h1. This need not be true if f is a complex-valued harmonic function. However, we prove that Zygmund’s theorem holds if f is a harmonic K-quasiregular mapping in D. Our work makes further progress on the recent Riesz-type theorem of Liu and Zhu (Adv. Math., 2023), and the Kolmogorov-type theorem of Kalaj (J. Math. Anal. Appl., 2025), for harmonic quasiregular mappings. We also obtain a partial converse, thus showing that the proposed growth condition is the best possible. Furthermore, as an application of the classical conjugate function theorems, we establish a harmonic analogue of a well-known result of Hardy and Littlewood.

Artificial intelligence (AI) is a fundamental breakthrough in contemporary science. At its innovative core is the quest to simulate human intelligence processes using machines and computer systems. A topical moral discussion lies in the development of sexual robots that can perform conjugal functions. However, the Bible presents the concept of sexuality as a privilege exercised within the precincts of the same species in heterosexual relationships. Hence, according to the Scriptures, human beings can only have a sexual relationship with fellow human beings of the opposite biological gender. However, the rise in robotic technology that includes sexuality raises fundamental questions. What are the ethical implications of AI sexual innovation for marriage covenants? Towards what cause should contemporary theology and ethics relate to this innovation within AI? What is the place of sexuality in humans? Can sexuality with a human-like robot be understood as a biblical and legitimate alternative in the face of rising sexually transmitted diseases? This theoretical paper seeks to interrogate this line of AI innovation from an ethical and theological assessment.

Interactions between β-lactoglobulin and polyphenols: Mechanisms, properties, characterization, and applications.

This paper presents a detailed method for the numerical solution of direct and inverse problems arising in the framework of the gas lift process of oil production, which are described by a hyperbolic system of differential equations. The direct problem is solved using high-precision second-order difference schemes that ensure stability and accuracy of calculations in the space-time domain. The inverse problem, in turn, is formulated as an optimal control problem, where minimizing the target functional is implemented using the gradient method. To find the gradient of the conjugate function, a conjugate problem is used, the construction of which is based on the principles of Lagrangian identities and duality, which provides strict mathematical validity. Numerical experiments have confirmed the high efficiency of the proposed method in solving inverse problems and optimizing key parameters of the gas lift process. The results show that the use of the conjugate equation method contributes to a significant increase in oil recovery while minimizing the cost of injected gas, which makes the method economically and technologically advantageous. Additionally, it is demonstrated that the method makes it possible to correctly approximate the initial conditions even in difficult conditions associated with changes in the physical parameters of the process.

We show that the complex conjugate function z↦z¯ cannot be pointwise approximated by holomorphic polynomials on the Alice Roth’s Swiss cheese QR⊂C. Moreover, under some extra hypotheses, we also show that the complex conjugate cannot be pointwise approximated either by functions holomorphic on QR.

First, we introduce the generalized q-de la Vallée Poussin means. Then, using these new means, we extend a result of Leindler (Acta Sci Math (Szeged) 29:147–162, 1968) and one of Duman (Constr Math Anal 4(2):135–144, 2021) on uniform summability of Fourier series. In addition, we use the same means to determine the degree of approximation of a 2π-periodic function and its corresponding conjugate function in the norm of Hölder.

This paper introduces a lot of shadowing properties that have to do with the uniform convergence of a sequence {Hm : m ∈ N} of continuous self-maps over the metric space (M,ϱ). The paper gives a proof for the sequence {Hm : m ∈ N} that has (average-shadowing property, almost average-shadowing, asymptotic average-shadowing) that converges uniformly to H : M→M on (M, ϱ), then H also has (average-shadowing property, almost average-shadowing). Thus, we study the average chain-transitive by uniform convergence. However, we show that if H and F are conjugate maps, then H is the average chain-mixing if and only if F is the average chain-mixing.

In this paper we use the trigonometric mean R1(?), for 1 < ? < 2, to estimate a sharper degree of approximation of conjugate functions in H?p?space. This paper also generalizes some results of Nigam [12] and [16]. In addition, a particular result is derived from our results as corollary.

In this paper we study how Lagrange duality is connected to optimization problems whose objective function is the difference of two convex functions, briefly called DC problems. We present two Lagrange dual problems, each of them obtained via a different approach. While one of the duals corresponds to the standard formulation of the Lagrange dual problem, the other is written in terms of conjugate functions. When one of the involved functions in the objective is evenly convex, both problems are equivalent, but this relation is no longer true in the general setting. For this reason, we study conditions ensuring not only weak, but also zero duality gap and strong duality between the primal and one of the dual problems written using conjugate functions. For the other dual, and due to the fact that weak duality holds by construction, we just develop conditions for zero duality gap and strong duality between the primal DC problem and its (standard) Lagrange dual problem. Finally, we characterize weak and strong duality together with zero duality gap between the primal problem and its Fenchel-Lagrange dual following techniques used throughout the manuscript.

We present a concise point of view on the first and the second Korn’s inequality for general exponent p p and for a class of domains that includes Lipschitz domains. Our argument is conceptually very simple and, for p = 2 p = 2 , uses only the classical Riesz representation theorem in Hilbert spaces. Moreover, the argument for the general exponent 1 > p > ∞ 1>p>\infty remains the same, the only change being invoking now the q q -Riesz representation theorem (with q q the harmonic conjugate of p p ). We also complement the analysis with elementary derivations of Poincaré-Korn inequalities in bounded and unbounded domains, which are essential tools in showing the coercivity of variational problems of elasticity but also propedeutic to the proof of the first Korn inequality.

Rational approximations of the conjugate function on the segment $[-1,~1]$ by Abel--Poisson sums of conjugate rational integral Fourier-Chebyshev operators with restrictions on the number of geometrically different poles are investigated. An integral representation of the corresponding approximations is established.Rational approximations on the segment $[-1,~1]$ of the conjugate function with density $(1-x)^\gamma,$ $\gamma\in (1/2,~1),$ by Abel-Poisson sums are studied. An integral representation of approximations and estimates of approximations taking into account the position of a point on the segment $[-1,~1]$ are obtained. An asymptotic expression as $r\to 1$ for the majorant of approximations, depending on the parameters of the approximating function is established. In the final part, the optimal values of parameters which provide the highest rate of decrease of this majorant are found. As a corollary we give some asymptotic estimates of approximations on the segment $[-1,~1]$ of the conjugate function by Abel-Poisson sums of conjugate polynomial Fourier-Chebyshev series.

On the Rational Approximations of the Conjugate Function on a Segment by Abel–Poisson Sums of Fourier–Chebyshev Integral Operators

Continuous imbedding theorems in Musielak spaces

The influence of metal-free thiophene spacer chain on optoelectronic analysis by TD-DFT method for efficient dye-sensitized solar cells with enhanced non-linear optical activity

We study problems arising in real-time auction markets, common in e-commerce and computational advertising, where bidders face the problem of calculating optimal bids. We focus upon a contract management problem where a demand aggregator is subject to multiple contractual obligations requiring them to acquire items of heterogeneous types at a specified rate and where they will seek to fulfill these obligations at minimum cost. Our main results show that, through a transformation of variables, this problem can be formulated as a convex optimization problem, for both first and second price auctions. The resulting duality theory admits rich structure and interpretations. Additionally, we show that the transformation of variables can be used to guarantee the convexity of optimal bidding problems studied by other authors, who did not leverage convexity in their analysis. The main point of our work is to emphasize that the natural convexity properties arising in first and second price auctions are not being fully exploited. Finally, we show direct analogies to problems in financial markets: the expected cost of bidding in second price auctions is formally identical to certain transaction costs when submitting market orders, and that a related conjugate function arises in dark pool financial markets.