On the lack of compactness in the 2D critical Sobolev embedding

This paper is devoted to the description of the lack of compactness of $H^1_{rad}(\R^2)$ in the Orlicz space. Our result is expressed in terms of the concentration-type examples derived by P. -L. Lions. The approach that we adopt to establish this characterization is completely different from the methods used in the study of the lack of compactness of Sobolev embedding in Lebesgue spaces and take into account the variational aspect of Orlicz spaces. We also investigate the feature of the solutions of non linear wave equation with exponential growth, where the Orlicz norm plays a decisive role.Comment: 38 page

Lack of compactness in the 2D critical Sobolev embedding, the general case

This paper is devoted to the description of the lack of compactness of the Sobolev embedding of $H^1(\R^2)$ in the critical Orlicz space {\cL}(\R^2). It turns out that up to cores our result is expressed in terms of the concentration-type examples derived by J. Moser in \cite{M} as in the radial setting investigated in \cite{BMM}. However, the analysis we used in this work is strikingly different from the one conducted in the radial case which is based on an $L^ \infty$ estimate far away from the origin and which is no longer valid in the general framework. Within the general framework of $H^1(\R^2)$, the strategy we adopted to build the profile decomposition in terms of examples by Moser concentrated around cores is based on capacity arguments and relies on an extraction process of mass concentrations. The essential ingredient to extract cores consists in proving by contradiction that if the mass responsible for the lack of compactness of the Sobolev embedding in the Orlicz space is scattered, then the energy used would exceed that of the starting sequence.Comment: Submitte

Energy scattering for 2D critical wave equation

We investigate existence and asymptotic completeness of the wave operators for nonlinear Klein-Gordon and Schr\"odinger equations with a defocusing exponential nonlinearity in two space dimensions. A certain threshold is defined based on the value of the conserved Hamiltonian, below which the exponential potential energy is dominated by the kinetic energy via a Trudinger-Moser type inequality. We prove that if the energy is below or equal to the critical value, then the solution approaches a free Klein-Gordon solution at the time infinity. The interesting feature in the critical case is that the Strichartz estimate together with Sobolev-type inequalities can not control the nonlinear term uniformly on each time interval, but with constants depending on how much the solution is concentrated. Thus we have to trace concentration of the energy along time, in order to set up favorable nonlinear estimates, and then to implement Bourgain's induction argument. We show the same result for the "subcritical" nonlinear Schr\"odinger equation.Comment: 33 pages, submitte

Absence of the Dissenting Arbitrator’s Signature and its Effect on the Validity of the Arbitration Award

من المشاكسات الإجرائية المعتادة من بعض المُحكمين غير المُحايدين عدم قيامهم بالتوقيع على حكم التحكيم، وذلك عندما يتعدد أعضاء الهيئة، ويتجه الأغلبية نحو الحكم ضد الطرف الذي يميل إليه المحكم. فالمحكم المشاكس يستهدف بذلك التشكيك في صحة إجراءات صدور الحكم، وبالتالي التيسير على الطرف الذي يميل إليه في أن يُبطل هذا الحكم. على أن عدم توقيع المحكم قد يكون لسببٍ آخر مشروع: غير إرادي، كما في حالة وفاته؛ أو إرادي، كما في حالة عدم إشراكه في المداولة. تختلف الدول، وكذلك لوائح تحكيم مراكز التحكيم، حول مدى تطلّب توقيع كل أعضاء هيئة التحكيم، وعند التخفّف، مدى ذلك التخفّف (عددًا، وسببًا). وقد كان المُشرع القطري - قبل صدور قانون التحكيم الجديد - يتخفف من عدد التوقيعات، فيكتفي بالأغلبية، ولم يكن يتطلب تسبيب عدم توقيع الأقلية، مكتفيًا بذكر أنهم لم يوقعوا 2). على أن قانون التحكيم القطري، الصادر بالقانون رقم 2 لسنة 2017 ، قد غير من موقفه، قليلاً؛ حيث / (المادة 202 1)، وذلك نقلاً عن القانون النموذجي. / اكتفى بتوقيع الأغلبية، لكنه تطلب ذكر سبب عدم توقيع الأقلية (المادة 31 وبذلك يَحِقّ التساؤل: هل هذا النص آمر، وبالتالي يجوز الاتفاق على اللجوء إلى خلافه (كما في حال الاتفاق على اللجوء إلى تحكيم غرفة التجارة الدولية)؟ ثم، مَن الذي يثبت السبب؟ وما الموقف لو أن السبب لم يُصرح به؟ ثم، هل كل أحوال التصريح عن السبب كافية ليصدر الحكم صحيحًا بالأغلبية؟ يحاول الباحث الإجابة عن ذلك في ضوء الآراء المختلفة للفقه والقضاء المقارن، وبما يهدي القاضي القطري، بمناسبة مراقبته سبب عدم توقيع أقلية التحكيم على حكم التحكيم.Refusal to sign the arbitration award is considered as one of the procedural controversies engendered by dissenting minority arbitrators. This happens when the arbitral tribunal is made up of more than one arbitrator. The dissenting arbitrator aims, thereby, at questioning the validity of the decision and therefore facilitating the annulment of the decision by the losing party. However, the absence of signature may be due to another legitimate reason: involuntary; namely in case of the arbitrator being taken ill, or voluntary; namely in case of the arbitrator’s absence from the deliberations or proceedings. National laws on arbitration as well as the arbitration rules set by arbitration centers or institutions, in varying degrees, require the signature of all the tribunal members. The Qatari Legislator- under the old regime only required the majority’s signature. Nevertheless, an adjustment of this position has been made in the new Qatari Arbitration Law, Act 2 of 2017. According to this Law, the signature of the majority of the tribunal is deemed to be sufficient, but no reason for the absence of the minority’s signature is required according to Article 31/1. This is based on the UNCITRAL Model Law. Consequently, we may ask this question: is the award binding? Who has to provide the reasons for the absence of the minority signature? What is the consequence for the absence of reasons? Through exploring the motive behind not signing the arbitration award by the arbitration minority, this paper aims at answering the above-mentioned questions in the light of the views evoked in the jurisprudence and comparative law literature

Economic Reform, Banks’ Efficiency and Market Structure in Egypt

Egypt during the period of 2000 until 2010 witnessed major activities of economic cycles, financial reform, bank reform, development and restructuring that changed the face, structure and size of banks and the banking industry. New regulations, new laws, and Nazif’s government that was appointed in 2003, fostered the change especially during worldwide recession, economic boom, tighter regulation on the financial sector and financial crises. This thesis assessed the changes that happened in the Egyptian-banking sector during the period of 2004 until 2010, in term of efficiency change and industry structure. The research investigated the hypotheses that the lower the number of banks and the larger capital available for them, the higher the efficiency. The research used Data Envelopment analysis (DEA) methodologies to test the hypotheses on 27 banks out of 39 banks in Egypt that account for 90% of banking activities in Egypt. The input variables were the available funds and the owners’ equity. While the output variables were net loans, securities, operating profits and net profit. Output oriented tests were conducted using the BCC (Banker – Charnes – Cooper), CCR (Charnes – Cooper – Rhodes) and Window analysis. In addition to the full sample, segmentation between private vs. public and international vs. local & regional were conducted. Moreover, the research also includes a descriptive analysis of the banking activities and changes that occurred during the period of study, and conducted a concentration index for the banking sector. The research indicates that there was slight improvement in efficiency during the period of study, and that having more funds, resources may lead to improvements in efficiency. Although the banking industry increased in size (deposits, loans, profits and branches), the concentration ratio increased slightly, which indicates the dominance of few banks on the sector. The thesis recommends that banks and the central bank could work on different dimensions to improve efficiency by having better reach, products and better assets utilization, and continue in the reform to reduce the concentration ratio, especially that the public sector still has ownership in 17 banks out of the sample of the 27 banks

A scalable mining of frequent quadratic concepts in d-folksonomies

Folksonomy mining is grasping the interest of web 2.0 community since it represents the core data of social resource sharing systems. However, a scrutiny of the related works interested in mining folksonomies unveils that the time stamp dimension has not been considered. For example, the wealthy number of works dedicated to mining tri-concepts from folksonomies did not take into account time dimension. In this paper, we will consider a folksonomy commonly composed of triples and we shall consider the time as a new dimension. We motivate our approach by highlighting the battery of potential applications. Then, we present the foundations for mining quadri-concepts, provide a formal definition of the problem and introduce a new efficient algorithm, called QUADRICONS for its solution to allow for mining folksonomies in time, i.e., d-folksonomies. We also introduce a new closure operator that splits the induced search space into equivalence classes whose smallest elements are the quadri-minimal generators. Carried out experiments on large-scale real-world datasets highlight good performances of our algorithm

On-Chip Nanoscale Plasmonic Optical Modulators

In this thesis work, techniques for downsizing Optical modulators to nanoscale for the purpose of utilization in on chip communication and sensing applications are explored. Nanoscale optical interconnects can solve the electronics speed limiting transmission lines, in addition to decrease the electronic chips heat dissipation. A major obstacle in the path of achieving this goal is to build optical modulators, which transforms data from the electrical form to the optical form, in a size comparable to the size of the electronics components, while also having low insertion loss, high extinction ratio and bandwidth. Also, lap-on-chip applications used for fast diagnostics, and which is based on photonic sensors and photonic circuitry, is in need for similar modulator specifications, while it loosens the spec on the modulator’s size. Silicon photonics is the most convenient photonics technology available for optical interconnects application, owing to its compatibility with the mature and cheap CMOS manufacturing process. Hence, building modulators which is exclusively compatible with this technology is a must, although, Plasmonics could be the right technology for downsizing the optical components, owing to its capability in squeezing light in subwavelength dimensions. Hence, our major goal is to build plasmonic modulators, that can be coupled directly to silicon waveguides. A Plasmonic Mach-Zehnder modulator was built, based on the orthogonal junction coupling technique. The footprint of the modulator is decreased to 0.6 4.7, extinction ratio of 15.8 dB and insertion loss of 3.38 dB at 10 volts was achieved in the 3D simulations. The voltage length product for the modulator is 47 V. The orthogonal junction coupler technique minimized the modulator’s footprint. On the other hand, photonic sensors favorably work in the mid-infrared region, owing to the presence of a lot of molecules absorption peaks in this region. Hence, III-V semiconductor media is used for this type of applications, owing to the availability of laser sources built of III-V media, and to the lower losses that these materials have in mid-infrared region. Hybrid plasmonic waveguide, formed of doped InAs, AlAs and GaAs is studied extensively. Based on this waveguide an electro-absorption modulator is built. The device showed an extinction ratio of 27 dB at 40 length, and 1.2 dB of insertion loss. The small device footprint predicts a much lower energy consumption