esa-sceva/satcom-chunk-collection huggingface dataset:
## 1 Introduction
coli gidA_ impaired thebiosynthesis of mmm\({}^{5}\)s\({}^{2}\)U. Interestingly, their study also showed identical levels of the same undermodified form of U34 are present in tRNA hydrolysates from _gidA_ and _mmmE_ mutants suggesting GidA and MnmE form a functional complex in which both proteins are interdependent [16]. Further studies done in _E. coli_ have provided additional evidence suggesting the _in vitro_ binding ability of GidA and MnmE and that together these two enzymes are responsible for the proper biosynthesis of mmm\({}^{5}\)s\({}^{2}\)U in bacterial tRNA [23, 24, 25]. Then, another enzyme, MnmC (formerly known as YfcK or TrmC), catalyzes a FAD-dependent deacetylation to 5-aminomethyl (nm\({}^{5}\)), and this group is subsequently methylated to produce methylaminomethyl (mmm\({}^{5}\)) [26, 27, 28, 29]. Additionally, [PERSON] _et al._ showed that GidA and MnmE bind together to modify _Salmonella_ tRNA [30]. One of the most interesting aspects of the GidA/MnmE tRNA modification pathway is its potential role as a pathogenic regulatory mechanism. GidA and MnmE are highly conserved proteins with homologues of GidA and MnmE in humans (MTO1 and GTPBP3, respectively) and yeast (MTO1 and MSS1, respectively) involved in modification of mitochondrial tRNAs [31, 32, 33, 34]. Lack of modification has shown mitochondrial dysfunction in human cells, neurological and developmental dysfunctions in _Caenorhabditis elegans_, and attenuated numerous bacterial pathogens [10, 14, 18, 30, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55]. Additionally, lack of modification has been associated with acute liver failure, lactic acidosis, hypertrophic cardiomyopathy, and mitochondrial encephalomyelitis in humans [56, 57, 58]. Within this review, we summarize the current knowledge of the GidA/MnmE tRNA modification pathway in the pathogenesis of bacterial pathogens.
## 3 Proposed Quality of Service Routing Protocol (QoSRP) in UWSNs
### Underwater Acoustic Channel Detection (UWCD) Algorithm
#### 3.2.1 Decision Rules for the Fused Information in Distributed Cooperative Sensing (DCoS)
In the proposed scheme, the probability of \(\mathcal{FA}_{\delta}\), \(\mathcal{MU}\mathcal{D}\) and \(\mathcal{D}\mathcal{FT}\) using K-out-of-M rule can be written as:  
\[\rho_{\mathcal{D}\mathcal{FT}}=\sum_{\ell=\ell=1}^{|m|}\sum_{\mathcal{J}=1}^{|m| }\rho_{\mathcal{D}\mathcal{FT}}\big{(}\mathcal{SU}_{\mathcal{J}}\big{)}^{\ell} \left(1-\rho_{\mathcal{D}\mathcal{FT}}(\mathcal{SU}_{\mathcal{J}_{\ell}})\right) ^{|m-\ell|} \tag{9}\]  
\[\rho_{\mathcal{M}\mathcal{D}}=1-\sum_{\mathcal{J}=1}^{|m|}\Big{[}\rho_{ \mathcal{D}\mathcal{FT}}(1-\rho_{e})+\left(1-\rho_{e(\mathcal{SU}_{\mathcal{J} _{\ell}})}\right)+\rho_{e(\mathcal{SU}_{\mathcal{J}_{\ell}})}\Big{]} \tag{10}\]  
\[\rho_{\mathcal{FT}\mathcal{A}_{\delta}}=1-\sum_{\mathcal{J}=1}^{|m|}\Big{[}(1 -\rho_{\mathcal{FT}\mathcal{A}_{\delta}})(1-\rho_{e})\mathcal{SU}_{\mathcal{ J}}+\rho_{\mathcal{FT}\mathcal{A}_{\delta}}(\mathcal{SU}_{\mathcal{J}_{\ell}}), \rho_{e(\mathcal{SU}_{\mathcal{J}})}\Big{]} \tag{11}\]  
Equations (9), (10) and (11) illustrate the probability of \(\mathcal{D}\mathcal{FT}\) when all \(m\) secondary users sensed the existence of a \(\mathcal{PU}_{\ell}\) such that \(\mathcal{SU}_{\mathcal{J}}>\mathcal{SU}_{i}\) determine \(\mathcal{H}_{1}\) when \(\mathcal{H}_{1}\) exits, the \(\mathcal{MU}\mathcal{D}\) probability when the secondary users \(\mathcal{SU}_{\mathcal{J}}>\mathcal{SU}_{i}\) decides \(\mathcal{H}_{0}\) but \(\mathcal{H}_{1}\) exists, and the \(\mathcal{FA}_{\delta}\) probability when the secondary users \(\mathcal{SU}_{\mathcal{J}}>\mathcal{SU}_{i}\) decide \(\mathcal{H}_{0}\) but \(\mathcal{H}_{1}\) exists, respectively.
# Lightweight and Privacy-Preserving Multi-Keyword Search over Outsourced Data
## 3 Basic PRMS Construction
The cloud server computes \[E=\mathsf{Score}(\hat{I},\hat{Q})=\hat{I}\odot\hat{Q}\mod p\] (3) where \"\(\odot\)\" denotes the modular vector inner product operation. By properly choosing the elements under the given security parameters \(\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4}\), both the following conditions can be satisfied \[\hat{I}\odot\hat{Q}<p\] (4) and \[\rho =\sum_{i=1,I_{i}\
eq 0,Q_{i}\
eq 0}^{n+2}(\gamma t_{i}I_{i}+m_{i} \delta Q_{i}+m_{i}t_{i})+\] \[\sum_{i=1,I_{i}=0,Q_{i}\
eq 0}^{n+2}(m_{i}\delta Q_{i}+m_{i}t_{i })+\sum_{i=1,I_{i}\
eq 0,Q_{i}=0}^{n+2}(\gamma t_{i}I_{i}+m_{i}t_{i})+\sum_{i=1,I_{ i}=0,Q_{i}=0}^{n+2}m_{i}t_{i}\] \[<\gamma\delta.\] (5) Then, the cloud server computes \[\mathsf{Score}(\vec{I},\vec{Q})=\sum_{i=1}^{n}I_{i}\cdot Q_{i}=\frac{E-(E\mod \delta\cdot\gamma)}{\delta\cdot\gamma}\] (6) If the following search condition is satisfied \[\mathsf{Score}(\vec{I},\vec{Q})\geq\tau\] (7) then the corresponding document \(\overline{F}\) is returned. According to the similarity score \(\mathsf{Score}(\vec{I},\vec{Q})\), the searched documents can be sorted. In this way, the searched documents would be returned if their similarity scores are greater than the given search threshold. A flow chart of the proposed basic PRMS scheme is depicted in Figure 3.  
Figure 3: The flow chart of basic PRMS scheme. **Theorem 1**: _The proposed basic PRMS scheme is correct._  
To compute the similarity score \(\mathsf{Score}(\vec{I},\vec{Q})\), it is required that both \(I_{i}\
eq 0\) and \(Q_{i}\
eq 0\) are satisfied for \(1\leq i\leq n\). Let  
\[E=\mathsf{Score}(\hat{I},\hat{Q})=\hat{I}\odot\hat{Q}\mod p\]0  
Note that  
\[E=\mathsf{Score}(\hat{I},\hat{Q})=\hat{I}\odot\hat{Q}\mod p\]1 \[E=\mathsf{Score}(\hat{I},\hat{Q})=\hat{I}\odot\hat{Q}\mod p\]2 \[E=\mathsf{Score}(\hat{I},\hat{Q})=\hat{I}\odot\hat{Q}\mod p\]3 \[E=\mathsf{Score}(\hat{I},\hat{Q})=\hat{I}\odot\hat{Q}\mod p\]4  
If \(E<p\) and \(\rho<\gamma\delta\) hold, then we have  
\[E=\mathsf{Score}(\hat{I},\hat{Q})=\hat{I}\odot\hat{Q}\mod p\]5 \[E=\mathsf{Score}(\hat{I},\hat{Q})=\hat{I}\odot\hat{Q}\mod p\]6 \[E=\mathsf{Score}(\hat{I},\hat{Q})=\hat{I}\odot\hat{Q}\mod p\]7 \[E=\mathsf{Score}(\hat{I},\hat{Q})=\hat{I}\odot\hat{Q}\mod p\]8 \[E=\mathsf{Score}(\hat{I},\hat{Q})=\hat{I}\odot\hat{Q}\mod p\]9  
Thus, the proposed basic PRMS scheme is correct. As shown in Equation (2), each element \(Q_{i}\) in the query vector \(\vec{Q}\) is randomized with a randomly chosen value \(t_{i}\). Therefore, even when the \(i\)-th keyword does not exist in the dictionary \(D\), the privacy of the query keyword set at \(i\)-th position can still be guaranteed, which means \(Q_{i}=0\) would not be leaked from the search trapdoor element \(\hat{Q_{i}}\).
## 2 Materials and Methods
### Description of the Precipitation Products
#### 2.1.2 Radar Networks
The \"European Operational Program for Exchange of weather RAdar information\" (OPERA) was established in 1999 and constituted of 30 member states operating over 200 radars in the C-, X- and S-band [28]. The goal of this program is to organise a European cooperation of radar-products exchange with the help of the European national weather services and to build radar composites for the European Meteorological Network EUMETNET [14]. Hence, it is worth noting that the French ARAMIS radar network serves as a foundation of the radar coverage for OPERA over France, with the notable exceptions of the X-band radars that are not included; the locations are given in Figure 1. The processing and corrections made on the radar composites can be synthesised in 4 steps:  
* An anomaly-removal module identifying non-weather-related pixels;
* A hit-accumulation clutter filter which detects pixels with an occurrence frequency higher than a reference threshold of 0.6;
* A beam-blockage correction;
* A satellite-based filter of residual non-rain echoes from the probability of the Precipitation Clouds product of the EUMETSAT Nowcasting Satellite Application Facilities. The \(Z\)-\(R\) conversion is based on the same Marshall-Palmer relationship (\(Z=200\cdot R^{1.6}\)) as in the PANTHERE product. [PERSON] et al. [32] mentioned several points of improvement, such as follows:  
* The differences between the radars in the network;
* The contamination by non-meteorological targets (partially corrected as explained earlier and in [28] with the implementation of a hit-accumulation clutter);
* The absence of correction of the vertical profile of reflectivity of the raw scans;
* The use of the single Marshall-Palmer relationship to convert \(Z\) to \(R\). [PERSON] et al. [14] indicated that 3 different OPERA products at 4 km\({}^{2}\) spatial resolution are available: the rain-rate composites with a 15 min timestep, the reflectivity composite with a 15 min timestep, and the hourly rainfall-accumulation composite. For this study, we use the rain-rate composite with a timestep of 15 min.
## 3 Results
### Block Diagram of the Simulator
#### 3.1.1 The Gough-Stewart Platform and Antenna Structure  
Kinematics and dynamics, under the given constraints, were programmed in SimScape Multibody, the controller, motion planner, wind aerodynamic disturbance as well as frictions were programmed in Simulink. In all cases, the initial conditions of the azimuth installation angle were aligned to the North, i.e., \(\pi_{0}=0\).  
The FESTO actuator ESBF-BS-80-1000-40P-S1-R3 was selected due to its robust life cycle performance, with a stroke length of 1000 mm that provides a maximum speed of 1.34 m/s and 5.4 kN maximum force. The mass of the antenna is 50 kg. The polar moments of inertia of each element are computed by SimScape, given material density and geometry. The base and platform radio, where passive Cardan and universal joints are placed, are \(r_{b}=623\) mm and \(r_{a}=238.6\) mm, where the platform shape is an hexagon of 0.3586 mt each side. Friction is modeled in accordance to selected component's data sheet, where the combined friction effects of linear actuation mechanism are viscous and Coulomb friction with the same coefficient of 5 for both frictions, with the corresponding units in IS.
# SPEKS: Forward Private SGX-Based Public Key Encryption with Keyword Search
## 2 Background
### Public Key Encryption with Keyword Search (PEKS)  
[PERSON] et al. [1] first introduced the notion of public key encryption with keyword search (PEKS). Compared to symmetric searchable encryption, PEKS has better performance in data sharing. [PERSON] et al. [15] gives a generic framework of PEKS and shows how to obtain public key based searchable encryption from anonymous identity-based encryption. In the PEKS scheme, a data receiver provides a gateway with a trapdoor function for keywords. Then, a data sender uses the data receiver's public key for encrypting a keyword and sends the ciphertext to the server or the gateway. The latter applies search or test function to the search token and the ciphertext. When the keywords within the search token and the ciphertext match, the search or the test function returns 1; otherwise, 0. The scheme is proven to be secure in the standard model, but under the condition where the number of malicious clients is smaller than a specified value.
###### Abstract  
A three-dimensional profile reconstruction of live cells in dynamic cell cultures is a challenging problem due to the highly scattering nature of cell mediums. Furthermore, it is an interesting problem since these cultures present the optimal in vitro conditions that most closely resemble the cells' natural conditions. In this paper, we report a holographic method used for imaging during the process of treatment of dynamic cell cultures with transforming growth factor beta (TGF-\(\beta\)) and the subsequent epithelial-mesenchymal transition (EMT). The imaging of dynamic cell cultures presents many challenges for holographic techniques due to the highly scattering and high speed nature of the environment. Here we report the algorithmic workflow we used for decreasing the imaging noise due to the presence of cell medium and achieving high speed reconstruction rates in real time. We also report the prominent morphological features we extracted from the obtained depth maps throughout the experiment. We conducted experiments on four different cell lines: ONCO-DG1, HCT-116, MDA-MB-231, and HUVEC. We observed the EMT process throughout a 48 h period after treatment with TGF-\(\beta\) with 6 h intervals for each sample. We show some examples of the reconstructed depth maps of tumor cells during the EMT phase. From these depth maps we extract some morphological parameters and report how they change after the EMT process is completed. The obtained results indicate that the proposed method presents certain advantages from an optical perspective particularly for applications where a dynamical medium is present. These advantages are lower signal-to-noise ratio (SNR) values and a simple setup compared to the setups used in similar studies. Future studies on this application could lead to the development of a model for the EMT process and its effects on cell to cell interactions.  
Keywords:digital holographic microscopy; epithelial-mesenchymal transition; tumor cells +
Footnote †: journal: Applied Physics Letters
## 3 Our Solution: CSS
### Architecture  
This section describes the structure and core components of CSS. CSS is implemented as an independent daemon in the Linux environment and operates directly on top of the operating system of the host machine; CSS has been designed based on the MAPE-K model [13]. CSS has a system call monitor, a task analyzer, and container controller modules. Figure 5 shows the structure of CSS and system components related to resource management operations in detail.  
In the MAPE-K cycle of CSS, three modules operate sequentially. The Bpftrace [14] delivers the system call log collected from the Linux kernel to the system call monitor, and the system call monitor transforms the system call log into a system call pattern and transmits it to the task analyzer. Next, the task analyzer creates a resource limit policy by analyzing the type of resource used by the container and the system resource information together. The container controller creates a _cgroups_ policy for the container based on the resource limit policy and applies the _cgroups_ policy to tasks inside a Singularity container. A detailed description of the CSS core components is as follows.  
1. System call monitor. It is a component with the role of collecting futex system calls to derive the resource usage type of the container. It collects the PIDs of running containers through the namespace and manages them in a tree structure. It also monitors system calls that occur in subprocesses created inside the container. The number of collected system calls is stored in the PID tree for the container. The number of occurrences of system calls is accumulated for each resource management interval, and it is initialized for each cleaning interval. The system administrator sets the resource management interval according to the physical system specification and the cleaning interval according to the size of one job.
2. Task analyzer. It analyzes monitoring data, creates resource management policies, and performs functions corresponding to the Analysis and Plan stages in the MAPE-K model. It analyzes data stored in the PID tree created by the system call monitor at each resource management interval; based on the analysis, the task analyzer creates a _cgroups_ policy that determines CPU quota. Specific techniques for resource management are described in detail in the next section.  
Figure 5: Architecture and resource management operation of CSS.  
Container controller. It has the role of applying the resource management policy created by the task analyzer to the container. The _cgroups_ policy, created by the task analyzer, starts the Singularity container runtime process as root and applies to all processes under it. The _cgroups_ policy applied to the container is changed when the container type is changed, or a new container is added.
# Nanoarchitectures in Management of Fungal Diseases: An Overview
## 4 Nanocarriers in Management of Fungal Infections
### Vesicle Systems
#### 4.1.3 Ethosomes  
Ethosomes are soft, noninvasive, and malleable drug carriers that allow drugs to spread in the systemic circulation and deep skin layers. Ethosomes, mainly composed of 20-45% ethanol, 2-5% phospholipids, and 100% _w/w_ water, enhance the percutaneous absorption of drugs. A high ethanol concentration makes the ethosomes unique, which confers a negative charge to liposomes and reduces vesicle sizes and the improved bioavailability of therapeutic drugs [38].  
[PERSON] et al. formulated FLZ-encapsulated liposomes and ethosomes. Their study examined that the drugs diffused (%) from ethosomes were almost two-fold greater in contrast to liposomes and three-fold greater than a hydroethanolic solution across rat skin. The results demonstrated that the developed ethosomal DDS yielded improved antifungal activity [39].  
[PERSON] et al. developed an ethosome-based subcutaneous delivery system for VCZ with the ultimate goal to enhance their antifungal properties and drug penetration. The outcome reflected that ethosomes attained a similar inhibition rate against an _Aspergillus flavus_ infection at a two-times lesser drug concentration, contrary to the VCZ solution in dimethylsulfoxide (DMSO). For ethosomes, the obtained drug permeability rate across the abdominal skin was about six times higher than VCZ in hydroalcoholic solution. Likewise, the drug deposited on the skin was found to be greater and reliant on the ethanol concentration [40].  
In a study by [PERSON] and [PERSON], clotrimazole (CLO)-loaded Cavamax W7 composite ethosomes were developed for enhanced drug delivery through the epidermis. The Cavamax W7 composite ethosomal optimized gel displayed a higher (88.53% \(\pm\) 2.10%) drug permeation rate in 8 h. The steady-state flux (Jss) was found to be 3.39 \(\pm\) 1.45 \(\mathrm{\SIUnitSymbolMicro m}^{2}/\mathrm{min}\) in contrast to that of ethosomal gel (1.57 \(\pm\) 0.23 \(\mathrm{\SIUnitSymbolMicro g}/\mathrm{cm}^{2}/\mathrm{min}\)) [41].
## 3 Results and Discussion
### Development of MW-FLR Method  
#### 3.2.1 Spectral Characteristics  
The fluorescence spectra (excitation and emission) of DUV were recorded in methanol (Figure 1). The spectra showed that DUV has two excitation wavelengths (\(\lambda_{\mathrm{ex}}\)) at 275 nm and 340 nm and showed maximum emission (\(\lambda_{\mathrm{em}}\)) at 405 nm. The initially achieved intensity of the native fluorescence of DUV was promising for the development of a fluorimetric method for its determination in both capsules and plasma samples. In the subsequent experiments, fluorescence intensities were measured at 405 nm after excitation at 275 nm.
## 2 Past to Present  
Virtual Reality (VR), in simple terms, is a computer-simulated environment which enables the user to interact with and alter their perception as a result of a mixture of sensory information sent to the human brain [11]. VR, as it exists today, is a result of many years of technological and microprocessors development capable of high-level computation at an affordable price. It started in 1960 with [PERSON]'s invention of a multi-sensory simulation device called Sensorama [12]. Sensorama is a stationary version of what is known today as a Head Mounted Display (HMD), where it plays a pre-recorded movie on a stereoscopic 3D display, with stereo sound, fans, scents generator, and a vibrating chair to stimulate the user's senses and \"immerse\" the user into the movie [13]. Fast forward to 1984, the first monochrome stereoscopic HMD called as Virtual Interactive Environment Workstation (VIEW) lab was created by [PERSON] at NASA's Aerospace Human Factors Research Division [11; 12]. VIEW resembled all the elements used today in a HMD including audio, body tracking, tactile and forced feedback, and connections to telerobotic systems, which made it interactive, in contrast to previous devices [14]. In the field of professional education and training, a dramatic surge of interest in VR took place in the 80s [15], which was followed by VR's appearance in higher education in the 90s in projects such as ScienceSpace Worlds, Safety World, Atom World, etc. [16].  
Until the appearance of commercialized high-end VR systems such as Samsung's Oculus Rift and HTC's Vive in early to mid-2010s, VR systems were accessible only to a niche of researchers and for thousands of dollars, which prohibited its commercialization [4; 5]. Today, VR is as cheap as $15, through a constructible HMD made of cardboard, sold by Google under the name of Google Cardboard [17]. According to [PERSON] [5], these commercialized high-end VR systems are tethered to a high-end PC at all times that makes it capable of offering high frame rates, attractive environments, and support different user interaction options. However, they are limited to a small area of the PC's proximity and require a significant amount of computational power, which comes at a great cost [5]. A more affordable VR solution is VR-ready mobile devices (smart phones) that enable the user to have lower VR experience due to the lower processing power, at a much lower cost [2; 5; 18; 19]. Additionally, the user benefits from cordless (no cable) connection, and a higher degree of movement VR experience [5]. The main difference is the lower degree of immersion from the low-end in comparison to the high-end mobile VR HMDs [20; 21].
## 3 Reduction of the Inverse Problem to a Non-Classical Forward Problem
### Variational Formulation  
Let us introduce the functions  
\[v(x,t)=\frac{\partial U}{\partial x}(x,t),\ \ w(x,t)=\frac{\partial b}{ \partial x}(x,t). \tag{20}\]  
In order to construct a weak formulation for \((v,\varphi):=\int_{0}^{1}v\varphi dx\), where \(\varphi\) is a function of \(H^{1}(\Omega)\), we form the inner product of (12) with \(\varphi_{x}\) and integrate by parts with respect to \(x\), using the boundary conditions (14). We perform the same procedure for the pair \((b,\varphi)\). As a result, we obtain  
\[\begin{split}&\left(\frac{\partial v}{\partial t},\varphi\right)+ \
u\left(\frac{\partial v}{\partial x},\varphi_{x}\right)=\beta(t)(w,\varphi _{x})+\beta(t)(\varphi(1)-\varphi(0))g_{B}(t)\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \qquad\qquad\qquad\qquad\qquad\qquad\qquad-f(t)(\varphi(1)-\varphi(0))- \left(\frac{\partial\widehat{u}}{\partial t},\varphi_{x}\right),\\ &\left(\frac{\partial w}{\partial t},\varphi\right)+\
u_{m} \left(\frac{\partial w}{\partial x},\varphi_{x}\right)=\beta(t)(v,\varphi_{x })+\beta(t)(\varphi(1)-\varphi(0))g_{u}(t)\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \qquad\qquad\qquad-\left(\frac{\partial\widehat{B}}{\partial t},\varphi_{x} \right),\\ &(v(x,0),\varphi)=\left(\frac{dU_{0}}{dx},\varphi\right),\ \ (w(x,0),\varphi)=\left(\frac{db_{0}}{dx},\varphi\right).\end{split} \tag{21}\]  
According to (18)-(20), we obtain  
\[\beta(t)=\frac{F^{\prime}(t)-\
u_{m}(w(1,t)-w(0,t))}{g_{u}(t)}, \tag{22}\] \[f(t)=\
u\big{(}v(1,t)-v(0,t)\big{)}+\beta(t)g_{B}(t)-E^{\prime}( t). \tag{23}\]  
**Theorem 2**: _Let the assumptions of Theorem 1 hold. Then, if \(\{v(x,t),w(x,t)\}\) is a variational solution of (21)-(23), then the pair \(\{U(x,t),b(x,t)\}\) is a solution to the problem (12)-(19) and vice versa._  
From the assumptions of the theorem, it follows that \(v=\frac{\partial U}{\partial x}\in H^{1}(\Omega)\), \(w=\frac{\partial b}{\partial x}\in H^{1}(\Omega)\); which is the solution to the variational problem (21) and (22).  
Conversely, the sufficient smoothness of the variational solution of (21) and (22) assures the solution of the forward problem (12)-(19).  
Therefore, if one can obtain an approximate solution of the variational problem (21)-(23), one also has an approximate solution to the original problem (12)-(19). Thus, in the next section, we concentrate on the Galerkin and finite element numerical solution of the variational problem.