Convex Form Research Articles

Strengthened convexity of positive operator monotone decreasing functions

We prove a strengthened form of convexity for operator monotone decreasing positive functions defined on the positive real numbers. This extends Ando and Hiai's work to allow arbitrary positive maps instead of states (or the identity map), and functional calculus by operator monotone functions defined on the positive real numbers instead of the logarithmic function.

Robust Probabilistic-Constrained Optimization for IRS-Aided MISO Communication Systems

Taking into account imperfect channel state information, this letter formulates and solves a joint active/passive beamforming optimization problem in multiple-input single-output systems with the support of an intelligent reflecting surface. In particular, we introduce an optimization problem to minimize the total transmit power subject to maintaining the users' signal-to-interference-plus-noise-ratio coverage probability above a predefined target. Due to the presence of probabilistic constraints, the proposed optimization problem is non-convex. To circumvent this issue, we first recast the proposed problem in a convex form by adopting the Bernstein-type inequality, and we then introduce a converging alternating optimization approach to iteratively find the active/passive beamforming vectors. In particular, the transformed robust optimization problem can be effectively solved by using standard interior-point methods. Numerical results demonstrate the effectiveness of jointly optimizing the active/passive beamforming vectors.

Ntbox: An r package with graphical user interface for modelling and evaluating multidimensional ecological niches

Abstract Biodiversity studies rely heavily on estimates of species' distributions often obtained through ecological niche modelling. Numerous software packages exist that allow users to model ecological niches using machine learning and statistical methods. However, no existing package with a graphical user interface allows users to perform model calibration and selection based on convex forms such as ellipsoids, which may match fundamental ecological niche shapes better, incorporating tools for exploring, modelling, and evaluating niches and distributions that are intuitive for both novice and proficient users. Here we describe an r package, NicheToolBox (ntbox), that allows users to conduct all processing steps involved in ecological niche modelling: downloading and curating occurrence data, obtaining and transforming environmental data layers, selecting environmental variables, exploring relationships between geographic and environmental spaces, calibrating and selecting ellipsoid models, evaluating models using binomial and partial ROC tests, assessing extrapolation risk, and performing geographic information system operations via a graphical user interface. A summary of the entire workflow is produced for use as a stand‐alone algorithm or as part of research reports. The method is explained in detail and tested via modelling the threatened feline species Leopardus wiedii. Georeferenced occurrence data for this species are queried to display both point occurrences and the IUCN extent of occurrence polygon (IUCN, 2007). This information is used to illustrate tools available for accessing, processing and exploring biodiversity data (e.g. number of occurrences and chronology of collecting) and transforming environmental data (e.g. a summary PCA for 19 bioclimatic layers). Visualizations of three‐dimensional ecological niches modelled as minimum volume ellipsoids are developed with ancillary statistics. This niche model is then projected to geographic space, to represent a corresponding potential suitability map. Using ntbox allows a fast and straightforward means by which to retrieve and manipulate occurrence and environmental data, which can then be implemented in model calibration, projection and evaluation for assessing distributions of species in geographic space and their corresponding environmental combinations.

Effectiveness of a soil mapping geomatic approach to predict the spatial distribution of soil types and their properties

A soil map (1:50,000 scale) was recently produced in Sardinia (Italy) using a cost-effective GIS approach. In this study we aimed to verify, in two pilot areas and by means of statistical analysis, the effectiveness of the adopted methodology in representing and predicting the spatial distribution of soil types and properties. We focused on evaluation of 1) the influence of landforms and parent materials on soil types (WRB Reference Soil Groups) and selected soil properties and 2) the suitability of the adopted methodology for calibrating a model to predict land unit composition in terms of different soil types.Leptosols, Regosols and Cambisols were prevalent on slopes, with Leptosols being more frequent on convex slopes and Regosols and Cambisols on concave slopes. In flat areas, soil types mainly depended on the type and age of parent material, with Regosols and Cambisols prevailing on Holocene deposits and highly developed soils (mainly Luvisols) largely prevailing on Pleistocene deposits. On hard rock, Leptosols were very frequent on terrigenous metamorphic rock and frequent on granite. Besides Leptosols, Regosols occurred more frequently than Cambisols on both parent materials. Landforms strongly influenced soil depth and available water capacity. Soils on plains were deeper than those on slopes, where convex forms had shallower soils than concave forms. A similar trend applied to the available water capacity. The parent material had a significant effect on topsoil properties (thickness, texture, pH and organic carbon content) of soils belonging to the same WRB Reference Soil Group (analysis done on the most relevant WRB Reference Soil Groups, i.e. Leptosols, Regosols and Cambisols).We calibrated and tested stepwise multiple linear regressions (MLR) and general linear models (GLM) to predict the composition of map units in terms of different WRB Reference Soil Groups. The two models gave very similar results, with distinct distribution patterns that were coherent with the relationships observed between soil groups and specific combination of terrain attributes and parent materials. Results showed that both models were more reliable in predicting the absence rather than presence of a given soil type.

Fast Globally Optimal Transmit Antenna Selection and Resource Allocation Scheme in mmWave D2D Networks

Transmit antenna selection (TAS) at base station has been widely studied and employed in many communication networks (such as those using massive multiple-input multiple-output (MIMO) systems). Thanks to the small size of microstrip antenna elements applicable to millimeter-wave (mmWave) frequencies, the implementation of TAS in small user equipments (UEs) employing switchable directional antennas is recently becoming popular. In this paper, we consider device-to-device (D2D) communications underlaying a cellular network wherein each D2D UE is equipped with switchable transmit antennas. By employing Generalized Bender’s Decomposition (GBD) algorithm, we obtained the solution to the problem of globally optimal transmit antenna selection and channel allocation to D2D UEs together with transmit powers of cellular and D2D UEs. Although we reformulated the non-convex primal subproblem of the proposed GBD-based method into convex form, we further managed to obtain the corresponding closed-form solution through analytical manipulations; this extensively increased (at least 60 times) the execution speed of the proposed method as shown in the simulation results.

Functions with Linear Price Elasticity for Forecasting Demand and Supply

Abstract In theoretical demand and supply analyses, functions with constant price elasticity are still frequently used although price elasticity is known to change in response to price. We relax the assumption of constant price elasticity to linear price elasticity which allows us to model demand and supply that decreases or increases with price. Quantity functions with linear price elasticity have been used in economics before but only to a limited extent since they have not been sufficiently theoretically studied. This paper overcomes this gap by identifying and studying all possible functional forms with linear price elasticity as well as their inverses, actually plotted as demand and supply curves. We find that quantity (demanded or supplied) as a function of price with linear price elasticity is a product of an exponential and a power function of price, while the price as a function of quantity involves the Lambert W function. Hence, the class of functions with linear price elasticity is heterogeneous: it contains reversible and irreversible functional forms as well as convex and non-convex functional forms. The class’ heterogeneity provides several modelling and research opportunities.

Viscous conservation laws in 1D with measure initial data

The one-dimensional viscous conservation law is considered on the whole line u t + f ( u ) x = ε u x x , ( x , t ) ∈ R × R + ¯ , ε > 0 , \begin{equation*} u_t + f(u)_x=\varepsilon u_{xx},\quad (x,t)\in \mathbb {R}\times \overline {\mathbb {R}_{+}},\quad \varepsilon >0, \end{equation*} subject to positive measure initial data. The flux f ∈ C 1 ( R ) f\in C^1(\mathbb {R}) is assumed to satisfy a p − p- condition, a weak form of convexity. In particular, any flux of the form f ( u ) = ∑ i = 1 J a i u m i f(u)=\sum _{i=1}^Ja_iu^{m_i} is admissible if a i > 0 , m i > 1 , i = 1 , 2 , … , J . a_i>0,\,m_i>1,\,\,i=1,2,\ldots ,J. The only case treated hitherto in the literature is f ( u ) = u m f(u)=u^m [Arch. Rat. Mech. Anal. 124 (1993), pp. 43–65] and the initial data is a “single source”, namely, a multiple of the delta function. The corresponding solutions have been labeled as “source-type” and the treatment made substantial use of the special form of both the flux and the initial data. In this paper existence and uniqueness of solutions is established. The method of proof relies on sharp decay estimates for the viscous Hamilton-Jacobi equation. Some estimates are independent of the viscosity coefficient, thus leading to new estimates for the (inviscid) hyperbolic conservation law.

Between a beam and catenary: Influence of geometry on gravitational stresses and stability of natural rock arches

The self-weight of freestanding natural arches imparts stresses on the lintel and abutments, setting the conditions for rock fracture and erosion. Rock is weak in tension, therefore an ideal arch form is one that minimizes tensile stresses and supports the weight of the lintel in compression, e.g. an inverted catenary. However, rock mass structure, including bedding, cross-bedding and other discontinuities, often imparts strong control on the geometry of arches, leading to forms that are less favorable from a stress perspective. Here we analyze a suite of nineteen arch models created from ground- and drone-based photogrammetry to assess the three-dimensional static stress field under gravitational loading. The models represent a range of arch lengths from 4 to 88 m, as well as a variety of forms, and use material properties previously calibrated from dynamic analysis of ambient vibrations. Our results demonstrate that arches shaped like beams have relatively high tensile stresses and a nearly symmetrical statistical distribution of principal stresses, while those with convex forms have comparably lower tensile stresses and statistical distributions favoring compressive stresses. In-situ observations of tensile cracks frequently correspond to the location of predicted tensile stresses in our models. We calculated the ratio of mean principal stresses for each arch, which is theoretically 1 for a flat prismatic beam and approaches infinity for an ideal inverted catenary. We found several arches with mean principal stress ratio around 1 and that values increased with lintel convexity. These results indicate that while self-sculpture might attempt to create ideal stress-based forms, discontinuities can control lintel geometry and natural arches evolve with forms that may be less favorable for long-term stability. The mean principal stress ratio is a simple metric to classify and compare arches, useful for assessment of arch stability supporting conservation and hazard analyses.

Intelligent Reflecting Surface: Joint Optimal Training Sequence and Refection Pattern

We investigate channel estimation for an intelligent reflecting surface (IRS) aided wireless communication system with a multi-antenna access point (AP), an multi-element IRS, and a set of multiple users, each with a single antenna. Our goal is to minimize mean square error (MSE) of estimating the channels of direct and cascaded links from the AP to each user via the IRS elements by jointly optimizing the training sequence of the AP and reflection pattern of the IRS. To solve this challenging problem, we first find the joint optimal structure and reflection pattern for the training sequence, and then, we convert the problem into a convex form to design the optimal training sequence efficiently. To reduce the computational complexity, we also propose a suboptimal training sequence in closed form. Numerical results show that the proposed scheme outperforms baseline schemes.

Optimal Training Design for MIMO Amplify-and-Forward Two-Way Relay Systems in the Presence of Interference

In this paper, we study the optimal training design for multiple-input multiple-output (MIMO) amplify-and-forward (AF) two-way relay systems in the presence of interference. In order to address the interference issue while accurately estimating the channels, we minimize the total transmit power consumed by two source nodes and a relay subject to the constraint on mean square error (MSE) of the channel estimation, which is a nonconvex problem. To solve this challenging problem, we first derive the optimal structure of the training signals, and then, we convert the original problem into a tractable convex form, from which the optimal training scheme is designed efficiently. We further consider a practical special case, for which the optimal training scheme is designed with a lower complexity and the useful insights into the optimal training design are obtained. Also, to reduce the computational complexity of the optimal training design, we propose an asymptotically optimal training scheme in the high signal-to-interference-plus-noise ratio (SINR) regime with a low complexity. Simulation results demonstrate that the proposed schemes significantly outperform the existing schemes.

Joint Content Push and Transmission in NOMA With SWIPT Caching Helper

This letter investigates the joint content push and transmission design relying on the cache-enabled helper. The helper is also equipped with the function of simultaneous wireless information and power transfer. With the help of non-orthogonal multiple access (NOMA) technique, we propose a two-stage transmission scheme. Under the minimum quality of service requirements, the sum rate optimization problem is formulated. Owing to the non-convexity nature of this problem, a series of approximation techniques, such as semi-definite relaxation, successive convex approximation and arithmetic-geometric mean inequality, are employed, after which the original optimization problem is converted into the convex form. Simulation results manifest the fast convergence property of the proposed algorithm, and demonstrate the superiority of the NOMA-based two-stage transmission strategy compared with a benchmark scheme.

Discrete Element Simulation of Durability for Bionic Wheels of Mars Rover

Four kinds of bionic surfaces (convex, scale-like, ribbed and concave shape) for Mars rover wheels were designed to improve wheel durability. A discrete element wheel-soil simulation system was developed, and three-dimensional discrete element analysis and soil bin tests were conducted. The results show that the masses of the convex and scale-like forms increased by 5.4% and 17%, respectively, and that the tangential cumulative forces decreased by about 40% and 70%, respectively. The wear mass decreased by 53% and 47%, respectively, compared with a prototypical wheel. The simulation shows that the durability increased significantly. The average tangential cumulative forces of the concave and ribbed shapes increased by more than 50%, and the wear mass decreased by 35% and 32%, respectively, compared with a prototypical wheel. The simulation shows that the results were not definitive. The analysis shows that the durability of the wheels can be improved via bionic optimization method. These results therefore provide a basis and reference for the design and optimization of the Mars rover’s wheel surface.

Resource Allocation for Non-Orthogonal Multiple Access-Enabled Fog Radio Access Networks

Non-orthogonal multiple access (NOMA) has been considered as a promising communication technology to enhance the spectral efficiency and support massive connections in fog radio access networks (F-RANs). In this paper, with the aim of maximizing the weighted sum rate while taking co-channel interference into consideration, a joint resource block (RB) and power allocation problem is formulated. To solve this problem, we first propose the optimal resource allocation scheme. Specifically, the monotonic optimization is applied and an outer polyblock approximation algorithm is proposed to get the global optimal solution. In order to reduce the computational complexity, we then propose the suboptimal resource allocation scheme. In particular, the original problem is decomposed into separated RB and power allocation problems. The RB allocation problem is modeled as a many-to-one matching game and a modified swap-enabled matching algorithm is proposed. The power allocation problem is converted into a convex form through some approximations and solved by a successive convex approximation algorithm. Simulation results demonstrate that the suboptimal scheme can achieve almost the same performance as the optimal scheme, while requiring much less computational complexity. In addition, the superiority of NOMA-enabled F-RANs over the conventional OMA-enabled F-RANs is verified.

Optimal secure transmit design for wireless information and power transfer in V2X vehicular communication systems

Simultaneous wireless information and power transfer (SWIPT) in vehicle to everything networks is a perspective technology which helps prolong the lifetime of sensors installed on roadside devices and vehicles. In this study, we consider a secure vehicular communication system with SWIPT under a non-linear EH model where the energy vehicular receivers are able to collect energy from the radio frequency signals as they are idle. We formulate the secure transmit signal design as an optimization problem which aims to maximize the minimum fairness energy power at the energy vehicular users. In addition, the considered optimization problem also considers different quality of service requirements: a maximum tolerable capacity at energy vehicular receivers and a minimum needed signal-to-interference-ratio at information vehicular receivers. To obtain a manageable solution, we reexpress the referred problem by variable substitutions and replacing a non-convex constraint with a convex form. Then, we adopt the semidefinite relaxation (SDR) technique to obtain the optimal solution of the original problem. To decrease calculational complexity, we also propose a suboptimal secure beamforming scheme for facilitating wireless energy transfer and providing physical layer security. Simulation results indicate that the developed secure transmit signal algorithm can obtain better performance gains than other algorithms.

Novel robust observer‐controller synthesis method for Takagi‐Sugeno systems

AbstractFor the output feedback control problem of a T‐S fuzzy system, an observer‐controller synthesis method is proposed. First, with parallel distributed compensation fuzzy observer and controller, an augmented error system is obtained. Based on Lyapunov function approaches, system stability and robust conditions are achieved. Then, by dividing the non‐convex conditions into observer‐dominant and controller dominant parts, the conditions are converted into convex form to facilitate the numerical solving process. Also, input saturation and system measurement disturbance are considered in the synthesis procedure. Finally, a numerical and a vessel control example are adopted to make comparisons with existing methods, with the results clearly showing the conservatism reduction effects and good control performance of the proposed method.

A new double crystal calibration system for absolute x-ray emission measurements down to ∼1 keV energies.

Over the past few years, work has been conducted at AWE to accurately characterize x-ray diffraction crystals to allow for absolute measurements of x-ray emission for our Orion opacity campaigns. Diffraction crystals are used in spectrometers on Orion to record the dispersed spectral features emitted by the laser produced plasma to obtain a measurement of the plasma conditions. Previously, based on a Manson x-ray source, our calibration system struggled to attain a high signal at the low energies required in calibration for the use of aluminum as a tracer for higher atomic number experiments. Here, we present data from the newly commissioned CTX400 x-ray source, a twin anode water cooled system, showing it to be a bright source even for ∼1 keV energies. Rocking curve measurements for three of the most commonly used crystals, namely, pentaerythritol, cesium acid phthalate, and germanium, are presented for both convex and flat forms.

Efficient resource allocation algorithms for high energy efficiency with fairness among users in OFDMA networks

In this article, we devise two efficient radio resource allocation algorithms in order to increase fair energy efficiency among users of OFDMA networks. The objective function is considered to be the well-known max-min function in order for allocating radio resources fairly in terms of energy efficiency, by taking the maximum transmit power, minimum rate requirements, and subchannel assignment constraints into account. The resulting problem is a nonconvex mixed-integer nonlinear problem for which two algorithms are proposed. In the first proposed algorithm, by reallocating the transmit power on subchannels, we obtain a suitable subchannel assignment with high fair energy efficiency. In the second proposed algorithm, by applying generalized fractional programming, mathematically modifying the objective function and constraints to a continuous and convex form and employing sequential convex programming, we allocate power and subchannel jointly. The proposed algorithms efficiently utilize radio resources and therefore, yield a far better performance than available solutions.

Nanoindentation Properties of 18CrNiMo7-6 Steel after Carburizing and Quenching Determined by Continuous Stiffness Measurement Method

In this work, the nanomechanical properties involving the indentation size effect (ISE) and yield strength of a surface-modified layer of 18CrNiMo7-6 steel after case hardening were investigated via nanoindentation experiments. The experimental results showed that the hardness increased with an increase in strain rate; the contact stiffness versus indentation depth curves take the form of upper convexity due to residual compressive stress relaxation. On the basis of the Ruiz-Moreno model, a modified model considering the cutoff parameter as a function of indentation depth was proposed. This model was able to better describe the ISE of the surface-modified layer. With the Hough transform error angle of 0.1° as the critical value (h0.1° is the corresponding depth), when h > h0.1°, the yield strength calculated by the Ma model started to disperse at the depth of h0.1°. These results provide useful insight into the local mechanical properties of 18CrNiMo7-6 steel after carburizing and quenching treatment.

Robust Synthesis Algorithm for Directional Modulation Signal with Array Manifold Vectors Uncertainty

Directional modulation (DM) has become a new research hotspot of physical layer security (PLS) communication at the transmitter side. In this paper, we propose a robust synthesis algorithm for DM signal under the condition of the array manifold vectors perturbation. This algorithm optimizes the constraints of sidelobe level and Euclidean distance of constellation points by considering the worst case performance of array manifold vectors. Furthermore, we also design an active constellation extension (ACE) method to relax the equality constraint of desired modulation symbols into a robust inequality constraint at the desired direction. These constraints can be reformulated in a convex form with l2 and l∞ regularization, which are computationally tractable. Simulation results show better performance of the proposed robust algorithm compared with the benchmark synthesis algorithms in the presence of array manifold vectors uncertainty.

On Sum of Squares Representation of Convex Forms and Generalized Cauchy--Schwarz Inequalities

A convex form of degree larger than one is always nonnegative since it vanishes together with its gradient at the origin. In 2007, Parrilo asked if convex forms are always sums of squares. A few ye...