On the Limits of Depth Reduction at Depth 3 Over Small Finite Fields

Recently, Gupta et.al. [GKKS2013] proved that over Q any $n^{O(1)}$-variate and $n$-degree polynomial in VP can also be computed by a depth three $\Sigma\Pi\Sigma$ circuit of size $2^{O(\sqrt{n}\log^{3/2}n)}$. Over fixed-size finite fields, Grigoriev and Karpinski proved that any $\Sigma\Pi\Sigma$ circuit that computes $Det_n$ (or $Perm_n$) must be of size $2^{\Omega(n)}$ [GK1998]. In this paper, we prove that over fixed-size finite fields, any $\Sigma\Pi\Sigma$ circuit for computing the iterated matrix multiplication polynomial of $n$ generic matrices of size $n\times n$, must be of size $2^{\Omega(n\log n)}$. The importance of this result is that over fixed-size fields there is no depth reduction technique that can be used to compute all the $n^{O(1)}$-variate and $n$-degree polynomials in VP by depth 3 circuits of size $2^{o(n\log n)}$. The result [GK1998] can only rule out such a possibility for depth 3 circuits of size $2^{o(n)}$. We also give an example of an explicit polynomial ($NW_{n,\epsilon}(X)$) in VNP (not known to be in VP), for which any $\Sigma\Pi\Sigma$ circuit computing it (over fixed-size fields) must be of size $2^{\Omega(n\log n)}$. The polynomial we consider is constructed from the combinatorial design. An interesting feature of this result is that we get the first examples of two polynomials (one in VP and one in VNP) such that they have provably stronger circuit size lower bounds than Permanent in a reasonably strong model of computation. Next, we prove that any depth 4 $\Sigma\Pi^{[O(\sqrt{n})]}\Sigma\Pi^{[\sqrt{n}]}$ circuit computing $NW_{n,\epsilon}(X)$ (over any field) must be of size $2^{\Omega(\sqrt{n}\log n)}$. To the best of our knowledge, the polynomial $NW_{n,\epsilon}(X)$ is the first example of an explicit polynomial in VNP such that it requires $2^{\Omega(\sqrt{n}\log n)}$ size depth four circuits, but no known matching upper bound

A performance model of speculative prefetching in distributed information systems

Previous studies in speculative prefetching focus on building and evaluating access models for the purpose of access prediction. This paper investigates a complementary area which has been largely ignored, that of performance modelling. We use improvement in access time as the performance metric, for which we derive a formula in terms of resource parameters (time available and time required for prefetching) and speculative parameters (probabilities for next access). The performance maximization problem is expressed as a stretch knapsack problem. We develop an algorithm to maximize the improvement in access time by solving the stretch knapsack problem, using theoretically proven apparatus to reduce the search space. Integration between speculative prefetching and caching is also investigated, albeit under the assumption of equal item sizes

Landau diamagnetism revisited

The problem of diamagnetism, solved by Landau, continues to pose fascinating issues which have relevance even today. These issues relate to inherent quantum nature of the problem, the role of boundary and dissipation, the meaning of thermodynamic limits, and above all, the quantum-classical crossover occasioned by environment-induced decoherence. The Landau Diamagnetism provides a unique paradigm for discussing these issues, the significance of which are far-reaching. Our central result is a remarkable one as it connects the mean orbital magnetic moment, a thermodynamic property, with the electrical resistivity, which characterizes transport properties of materials.Comment: 4 pages, 1 figur

Can re-entrance be observed in force induced transitions?

A large conformational change in the reaction co-ordinate and the role of the solvent in the formation of base-pairing are combined to settle a long standing issue {\it i.e.} prediction of re-entrance in the force induced transition of DNA. A direct way to observe the re-entrance, i.e a strand goes to the closed state from the open state and again to the open state with temperature, appears difficult to be achieved in the laboratory. An experimental protocol (in direct way) in the constant force ensemble is being proposed for the first time that will enable the observation of the re-entrance behavior in the force-temperature plane. Our exact results for small oligonucleotide that forms a hairpin structure provide the evidence that re-entrance can be observed.Comment: 12 pages and 5 figures (RevTex4). Accepted in Europhys Lett. (2009

The modern technology of iron and steel production and possible ways of their development

В изменяющейся мировой обстановке на рынке сырых материалов для черной металлургии разрабатывается ряд новых технологий по производству чугуна и стали, альтернативных существующим технологиям, которые способны обеспечить экономически устойчивую работу металлургических компаний. В дополнении к этому фокусируется внимание на экономии энергии и снижении выбросов парниковых газов в целях решения важнейших вопросов охраны окружающей среды. Изменение состояния окружающей среды ставит новые проблемы перед металлургической промышленностью, потребляющей значительные энергетические и топливные ресурсы. Отрасль вынуждена сосредоточить свое внимание на сокращении всех видов энергии, что приведет и к снижению выброса парниковых газов. Разработка альтернативных технологических процессов производства чугуна и стали способна обеспечить металлургическим компаниям экономически выгодную и устойчивую работу в производстве стали. Для оценки воздействий деятельности металлургических компаний на окружающую среду Инженерно-консалтинговой компанией ХАТЧ (НАТСH, Сanada) были разработаны новые методики моделирования, позволяющие квалифицированно и качественно оценивать риски в потреблении энергии и выбросах СО2 в металлургической промышленности. Методика для анализа выбросов углеродсодержащих парниковых газов названа G-CAP ™ (Зеленый Дом — Борьба с загрязнением воздуха углекислым газом), а для анализа энергоэффективности — En-MAPTM (Планирование действий при управлении энергией). Оценка существующего положения в большинстве интегрированных заводов показала, что они располагают возможностями по экономии энергии и борьбы с загрязнением атмосферы парниковыми газами, лучшие из этих заводов исчерпали эти возможности даже при высоких ценах на квоты выбросов СО2. В этом контексте важно оценить те важные особенности альтернативных технологий получения чугуна и стали, которые разработаны к настоящему времени. Эта статья содержит сравнительную оценку энергоэффективности и выбросов ПГ для некоторых выбранных альтернативных технологий производства чугуна и стали, которые рассматриваются для их реализации. Для этого применены методики G-CAP ™ и G-CAP ™ , элементы которых были разработаны в компании HATCH с основной целью количественной и квалификационной оценки потенциала экономии энергии и сокращения выбросов СО2 в металлургической промышленностиIn the changing global market scenario for raw materials for the steel industry, a number of novel iron and steelmaking process technologies are being developed to provide the steel companies with economically-sustainable alternatives for iron and steel-making. In addition, the steel industry is also focusing on reduction of energy consumption as well as green-house gas (GHG) emissions to address the crucial subject of climate change. Climate change is presenting new risks to the highly energy and carbon-intensive, iron and steel industry. The industry needs to focus on reduction of energy consumption as GHG emissions to address climate change. Development of alternate iron and steelmaking process technologies can provide steel companies with economically-sustainable alternatives for steel production. For managing climate change risks, novel modelling tools have been developed by Hatch to quantify and qualify potential energy savings and CO2 abatement within the iron and steel industry. The tool developed for abatement of greenhouse gas carbon is called G-CAPTM (Green-House Gas Carbon Abatement Process) while that developed for improving energy efficiency is called En-MAPTM (Energy Management Action Planning). Evaluation of existing operations have shown that most integrated plants have GHG and energy abatement opportunities; on the other hand, the best-in-class plants may not have a lot of low-risk abatement opportunities left, even at high CO2 price. In this context, it is important to assess these critical issues for the alternate iron and steelmaking technologies that have been developed. This paper presents a comparative evaluation of energy-efficiency and GHG emissions for some selected iron- and steelmaking technologies that are being considered for implementation. In this work, Hatch’s G-CAP™ and En-MAP™ tools that were developed with the main objective of quantifying and qualifying the potential energy savings and CO2 abatement within the iron and steel industry, were employed in the evaluation conducted

Thermal conductivity of comets

A value is described for the thermal conductivity of the frost layer and for the water-ice solid debris mixture. The value of the porous structure is discussed as a function of depth only. Graphs show thermal conductivity as a function of depth and temperature at constant porosity and density

Magnetic structure of EuFe2As2 determined by single crystal neutron diffraction

Among various parent compounds of iron pnictide superconductors, EuFe2As2 stands out due to the presence of both spin density wave of Fe and antiferromagnetic ordering (AFM) of the localized Eu2+ moment. Single crystal neutron diffraction studies have been carried out to determine the magnetic structure of this compound and to investigate the coupling of two magnetic sublattices. Long range AFM ordering of Fe and Eu spins was observed below 190 K and 19 K, respectively. The ordering of Fe2+ moments is associated with the wave vector k = (1,0,1) and it takes place at the same temperature as the tetragonal to orthorhombic structural phase transition, which indicates the strong coupling between structural and magnetic components. The ordering of Eu moment is associated with the wave vector k = (0,0,1). While both Fe and Eu spins are aligned along the long a axis as experimentally determined, our studies suggest a weak coupling between the Fe and Eu magnetism.Comment: 7 pages, 7 figure

Diffusion at constant speed in a model phase space

We reconsider the problem of diffusion of particles at constant speed and present a generalization of the Telegrapher process to higher dimensional stochastic media ($d>1$), where the particle can move along $2^d$ directions. We derive the equations for the probability density function using the ``formulae of differentiation'' of Shapiro and Loginov. The model is an advancement over similiar models of photon migration in multiply scattering media in that it results in a true diffusion at constant speed in the limit of large dimensions.Comment: Final corrected version RevTeX, 6 pages, 1 figur