Conjugate Functions Research Articles

Korn and Poincaré-Korn inequalities: A different perspective

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.

Continuous imbedding theorems in Musielak spaces

We provide continuous Sobolev-type imbedding results for Musielak spaces. In the first result the target space is a Musielak space bigger than the considered one, while the second is that of bounded and continuous functions. Continuous imbeddings into a particular Musielak space built upon Sobolev conjugate functions generate serious difficulties and does not holds in general for arbitrary Musielak functions, as it is well known in variable exponent Sobolev spaces. We overcome this problem by imposing only a regularity assumption on the Musielak functions without resorting to other restrictions such as the Δ2/∇2 conditions. Nevertheless, the continuous embedding in the space of bounded continuous functions is obtained by using the convex envelope, which then allows us to use a result already obtained in Orlicz spaces. The results we obtain extend and unify those obtained in classical Sobolev spaces, variable exponent Sobolev spaces as well as those obtained by Donaldson-Trudinger in the setting of Orlicz 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

The pyrene group in conjugation with the thiophene moiety was used to design new organic molecules for the dye-sensitized solar cells (DSSCs) along with non-linear optical (NLO) properties. Five new thiophene chain conjugation molecules (PTRA-SPn, n = 1–5) were formed based on the (E)-2-(5-oxo-4-((5-(pyren-1-yl)thiophen-2-yl)methylene)-2-thioxothiazolidin-3-yl)acetic acid (PTRA). To comprehend the function of conjugation in varying the optoelectronic characteristics of these compounds, the π-spacer chain has been screened. The techniques of density functional theory (DFT) and time-dependent DFT (TD-DFT) have been used to study the electronic structures, Ultraviolet–Visible (UV–Vis) absorptions, and NLO activities of donor-spacer-acceptor (D-π-A) structures. According to the computed results, PTRA-SP5 exhibits a smaller energy gap (Eg) of 2.11 eV. The maximum UV–Vis spectrum has been found in PTRA-SP5 with a wavelength (λmax) of 532 nm. The reorganizational energy (RE) and dye regeneration (ΔGreg) of PTRA-SP5 are the smallest (λ- = 0.2340 eV, λ+ = 0.2122 eV, ΔGreg = 0.21 eV) compared to other sensitizers. The lowest exciton binding energies (Eb) of PTRA-SP2 (0.06 eV) and PTRA-SP3 (0.06 eV) are the lowest ones. The open circuit photovoltage (eVoc) for PTRA-SP5 (1.10 eV) is the highest. The NLO activity of the dipole moment (μnormal) and first-order hyperpolarizability (β) values of PTRA-SP5 achieve higher values (9.00 Debye and 34.82 × 10−30 esu, respectively). In summary, this work refined the impact of chain π-linkers on the photophysical characteristics of D-π-A spacer derivatives and offered some helpful suggestions to create new organic materials for optoelectronic uses.

Convexity in Real-time Bidding and Related Problems

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.

Elliptic Inversions in Taxicab Geometry

The goal of this research is to introduce inversion with respect to an ellipse which is a generalization of the classical circular inversion in taxicab plane and to investigate general properties and basic concepts of this transformation in taxicab geometry. The cross ratio is preserved under the elliptic inversion in taxicab plane though this transformation is not an isometry. Thus some properties such as cross ratio and harmonic conjugates of the elliptic inversions in R_T^2 are also studied.

Relaxed Stationary Distribution Correction Estimation for Improved Offline Policy Optimization

One of the major challenges of offline reinforcement learning (RL) is dealing with distribution shifts that stem from the mismatch between the trained policy and the data collection policy. Stationary distribution correction estimation algorithms (DICE) have addressed this issue by regularizing the policy optimization with f-divergence between the state-action visitation distributions of the data collection policy and the optimized policy. While such regularization naturally integrates to derive an objective to get optimal state-action visitation, such an implicit policy optimization framework has shown limited performance in practice. We observe that the reduced performance is attributed to the biased estimate and the properties of conjugate functions of f-divergence regularization. In this paper, we improve the regularized implicit policy optimization framework by relieving the bias and reshaping the conjugate function by relaxing the constraints. We show that the relaxation adjusts the degree of involvement of the sub-optimal samples in optimization, and we derive a new offline RL algorithm that benefits from the relaxed framework, improving from a previous implicit policy optimization algorithm by a large margin.

Active Kriging-based conjugate first-order reliability method for highly efficient structural reliability analysis using resample strategy

Efficient structural reliability analysis method is crucial to solving reliability analysis of complex structural problems. High-computational cost and low-failure probability problems greatly limit the efficiency in structural reliability analysis problems, causing the safety and reliability of the structure to be questioned. In this work, a highly efficient structural reliability analysis method coupling active Kriging algorithm with conjugate first order reliability method (AK-CFORM) is proposed. Specifically, the resample strategy is considered to reduce the number of samples evaluated in each active learning process; the uniform sampling is used to better balance global and local optimal problems; the conjugate map is used to improve the robustness of analytical first order reliability method; and the approximate numerical differential formula is proposed to solve the problems of non-convergence when solving the gradient of the Kriging surrogate model. Finally, three numerical cases and four engineering cases are used to illustrate the effectiveness and robustness of the proposed method. The results show that the proposed AK-CFORM has greater advantages in the number of calling system response and surrogate model with robust and accurate performance.

Quantum interference features and thermoelectric properties of macrocyclic-single molecules: theoretical and modelling investigation.

The quantum interference effect on the thermoelectric properties of cycloparaphenylacetylene-based molecular junctions was investigated theoretically using a combination of density functional theory (DFT) methods, a tight binding (Hückel) model (TBHM) and quantum transport theory (QTT). Manipulating the unique conjugation function of these molecules not only creates a quantum interference (QI) but it is also a robust strategy for improving the thermoelectric properties of these molecules. QI controls the transport behaviour and decreases the electrical conductance (G) from 0.14 × 10-7 to 0.67 × 10-11 S, as well as enhancing the Seebeck coefficient (S) from 14.4 to 294 μV K-1, and promoting the electronic figure of merit (Z el T) from 0.008 to 1.8, making these molecules promising candidates for thermoelectric applications.

The conjugate operator on variable harmonic Bergman spaces

We prove that the harmonic conjugation operator on variable exponent harmonic Bergman spaces in the unit disc is bounded when the exponent has positive minimum and finite maximum and satisfies the log-Hölder condition. Under these conditions on the exponent, we also give a characterization of variable exponent Bergman spaces in terms of the derivatives of functions in them. Several key lemmas involve the boundedness of various maximal operators in these variable spaces.

In-Depth Structure and Function Characterization of Heterogeneous Interchain Cysteine-Conjugated Antibody-Drug Conjugates.

Antibody-drug conjugates (ADCs), integrating high specificity of antigen-targeting antibodies and high potency of cell-killing chemical drugs, have become one of the most rapidly expanding therapeutic biologics in oncology. Although ADCs were widely studied from multiple aspects, overall structural elucidation with comprehensive understanding of variants is scarcely reported. Here, for the first time, we present a holistic and in-depth characterization of an interchain cysteine-conjugated ADC, focusing on conjugation and charge heterogeneity, and in vitro biological activities. Conjugation mapping utilized a bottom-up approach, unraveled positional isomer composition, provided insights into the conjugation process, and elucidated how conjugation affects the physicochemical and biological properties of an ADC. Charge profiling combined bottom-up and top-down approaches to interrogate the origin of charge heterogeneity, its impact on function, and best practice for characterization. Specifically, we pioneered the utilization of capillary isoelectric focusing-mass spectrometry to decode not only critical post-translational modifications but also drug load and positional isomer distribution. The study design provides general guidance for in-depth characterization of ADCs, and the analytical findings in turn benefit the discovery and development of future ADCs.

AtIAR1 is a Zn transporter that regulates auxin metabolism in Arabidopsis thaliana.

Root growth in Arabidopsis is inhibited by exogenous auxin-amino acid conjugates, and mutants resistant to one such conjugate [indole-3-acetic acid (IAA)-Ala] map to a gene (AtIAR1) that is a member of a metal transporter family. Here, we test the hypothesis that AtIAR1 controls the hydrolysis of stored conjugated auxin to free auxin through zinc transport. AtIAR1 complements a yeast mutant sensitive to zinc, but not manganese- or iron-sensitive mutants, and the transporter is predicted to be localized to the endoplasmic reticulum/Golgi in plants. A previously identified Atiar1 mutant and a non-expressed T-DNA mutant both exhibit altered auxin metabolism, including decreased IAA-glucose conjugate levels in zinc-deficient conditions and insensitivity to the growth effect of exogenous IAA-Ala conjugates. At a high concentration of zinc, wild-type plants show a novel enhanced response to root growth inhibition by exogenous IAA-Ala which is disrupted in both Atiar1 mutants. Furthermore, both Atiar1 mutants show changes in auxin-related phenotypes, including lateral root density and hypocotyl length. The findings therefore suggest a role for AtIAR1 in controlling zinc release from the secretory system, where zinc homeostasis plays a key role in regulation of auxin metabolism and plant growth regulation.

Effectiveness of the Even-Type Delayed Mean in Approximation of Conjugate Functions

Effectiveness of the Even-Type Delayed Mean in Approximation of Conjugate Functions

New embedding results for double phase problems with variable exponents and a priori bounds for corresponding generalized double phase problems

In this paper we present new embedding results for Musielak–Orlicz Sobolev spaces of double phase type. Based on the continuous embedding of W^{1,mathcal {H}}(Omega ) into L^{mathcal {H}_*}(Omega ), where mathcal {H}_* is the Sobolev conjugate function of mathcal {H}, we present much stronger embeddings as known in the literature. Based on these results, we consider generalized double phase problems involving such new type of growth with Dirichlet and nonlinear boundary condition and prove appropriate boundedness results of corresponding weak solutions based on the De Giorgi iteration along with localization arguments.

Basic Properties of Solutions of Boundary Value Problem for the Dirac-Wave Operator

We study local and global properties of solutions of the boundary value problem for the Dirac wave operator. This includes explicit formulas, domain of dependence, range of influence, finite propagation speed and energy estimates. We also introduce the space of Dirac wave homogeneous polynomials and the corresponding Dirac-wave conjugate functions.

Mathematical modeling and optimal motion control of horizontal spindle apparatuses of cotton pickers

The paper presents differential equations of the movement of the horizontal spindle drum compiled using the 2nd kind of Lagrange equations. The task was set on how to control a horizontal spindle drum. The task of the speed of the Pontriyagin maximum principle was set, and the requirement for optimal guidance based on quality criteria was studied. Through the formation of the Hamilton-Pontryagin function, conjugate functions were composed. These conjugate functions made it possible to obtain a solution to the control algorithm. Pontryagin boundary value questions were articulate on the basis of the formed mathematical models. Using Runge-Kutta methods in solving boundary problems, the movement figures in the subject transient operation were identified based on the predetermined criterion, and as a result, moments of inertia of inertia of rotating masses of the horizontal-spindle drum, coefficients of viscosity and elasticity of the drum shaft were determined through the given resistance forces.

Emulsification and encapsulation properties of conjugates formed between whey protein isolate and carboxymethyl cellulose under acidic conditions

In this study, carboxymethyl cellulose (CMC) was used to interact with whey protein isolate (WPI) to prepare conjugates as emulsifiers and embedding agents, which can be used under acidic conditions. Firstly, the effects of ratios and pH values on the formation of WPI-CMC conjugates were investigated. The turbidity and particle size of WPI were reduced in the presence of CMC at pH 4.6 (near the isoelectric point). Then the characterization of physicochemical properties indicated that electrostatic interactions played a major role in the formation of WPI-CMC conjugates, thereby changing the structure and function of conjugates. CMC and WPI reached the optimal aggregation state at pH 4.6 and a ratio of 4:1. The conjugates exhibited excellent emulsifying activity and stability for the oil-in-water emulsions. WPI-CMC conjugates also could provide protection to allicin by preventing degradation under environmental stresses, while maintaining its antioxidant activity.

Regularity properties of k-Brjuno and Wilton functions

AbstractWe study functions related to the classical Brjuno function, namely k-Brjuno functions and the Wilton function. Both appear in the study of boundary regularity properties of (quasi) modular forms and their integrals. We consider various possible versions of them, based on the $$\alpha $$ α -continued fraction developments. We study their BMO regularity properties and their behaviour near rational numbers of their finite truncations. We then complexify the functional equations which they fulfill and we construct analytic extensions of the k-Brjuno and Wilton functions to the upper half-plane. We study their boundary behaviour using an extension of the continued fraction algorithm to the complex plane. We also prove that the harmonic conjugate of the real k-Brjuno function is continuous at all irrational numbers and has a decreasing jump of $$\pi /q^k$$ π / q k at rational points p/q.

Convergence of a conjugate function in Zygmund space by almost Nörlund transform

Abstract The proposed work aims to study the degree of convergence of a function h ~ {\widetilde{h}} , conjugate to a Lebesgue integrable 2 ⁢ π {2\pi} -periodic function h, in the Zygmund class Z r ( ν ) {Z^{(\nu)}_{r}} , r ≥ 1 {r\geq 1} , by almost Nörlund transform. Our results give sharper estimates than the previous existing results.

Acquisition of Visible Satellites Based on Antenna Arrays via Joint $\ell _{1,1}$-Norms Minimization

This work tackles the acquisition of visible satellites based on an antenna array in the presence of impulsive noise. First, the joint angle and number estimation of visible satellites is formulated as a joint <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\ell _{1,1}$</tex-math></inline-formula> -norms minimization problem by exploiting the spatial sparsity of visible satellites. Then, in order to obtain a closed-form solution, a generalized conjugate function for complex-valued matrix variable is introduced and the joint <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\ell _{1,1}$</tex-math></inline-formula> -norms minimization is transformed into a Lagrangian dual function maximization. Furthermore, the dual ascent framework and subgradient technique are employed to obtain a suitable iterative solution, where the convergence rate and steady-state value can be flexibly adjusted. Numerical results are presented to demonstrate the effectiveness of the proposed approach.

A Novel Fast Sparse Bayesian Learning STAP Algorithm for Conformal Array Radar

Space-time adaptive processing (STAP) is an important method of clutter suppression that requires adequate training samples. For an airborne conformal array radar, conventional STAP methods do not have enough training samples to acquire good performance due to the range dependent clutter caused by geometry and the problem of polarization. Sparse-recovery-based STAP (SR-STAP) methods have garnered significant attention in the past few decades because they only require a small number of training samples. Sparse Bayesian Learning (SBL) methods have seen increasing amounts of development due to its robust, self-regularizing nature and because it is not sensitive to user parameters, but it converges slowly. In this paper, a novel fast SBL (NFSBL) method is put forward to increase the rate of convergence. To minimize the SBL penalty function, the proposed method introduces the conjugate function to construct a surrogate function. Additional solution sparsity will be achieved through iteratively minimizing the surrogate function. Then, from the proposed method, we could obtain a more accurate clutter plus noise covariance matrix. Numerical simulation results express that this method could acquire better performance of STAP and improvement in convergence and computational complexity for a conformal array.