Singleton-Optimal LRCs and Perfect LRCs via Cyclic and Constacyclic Codes

Locally repairable codes (LRCs) have emerged as an important coding scheme in distributed storage systems (DSSs) with relatively low repair cost by accessing fewer non-failure nodes. Theoretical bounds and optimal constructions of LRCs have been widely investigated. Optimal LRCs via cyclic and constacyclic codes provide significant benefit of elegant algebraic structure and efficient encoding procedure. In this paper, we continue to consider the constructions of optimal LRCs via cyclic and constacyclic codes with long code length. Specifically, we first obtain two classes of $q$-ary cyclic Singleton-optimal $(n, k, d=6;r=2)$-LRCs with length $n=3(q+1)$ when $3 \mid (q-1)$ and $q$ is even, and length $n=\frac{3}{2}(q+1)$ when $3 \mid (q-1)$ and $q \equiv 1(\bmod~4)$, respectively. To the best of our knowledge, this is the first construction of $q$-ary cyclic Singleton-optimal LRCs with length $n>q+1$ and minimum distance $d \geq 5$. On the other hand, an LRC acheiving the Hamming-type bound is called a perfect LRC. By using cyclic and constacyclic codes, we construct two new families of $q$-ary perfect LRCs with length $n=\frac{q^m-1}{q-1}$, minimum distance $d=5$ and locality $r=2$

Significant Conditions on the Two-electron Reduced Density Matrix from the Constructive Solution of N-representability

We recently presented a constructive solution to the N-representability problem of the two-electron reduced density matrix (2-RDM)---a systematic approach to constructing complete conditions to ensure that the 2-RDM represents a realistic N-electron quantum system [D. A. Mazziotti, Phys. Rev. Lett. 108, 263002 (2012)]. In this paper we provide additional details and derive further N-representability conditions on the 2-RDM that follow from the constructive solution. The resulting conditions can be classified into a hierarchy of constraints, known as the (2,q)-positivity conditions where the q indicates their derivation from the nonnegativity of q-body operators. In addition to the known T1 and T2 conditions, we derive a new class of (2,3)-positivity conditions. We also derive 3 classes of (2,4)-positivity conditions, 6 classes of (2,5)-positivity conditions, and 24 classes of (2,6)-positivity conditions. The constraints obtained can be divided into two general types: (i) lifting conditions, that is conditions which arise from lifting lower (2,q)-positivity conditions to higher (2,q+1)-positivity conditions and (ii) pure conditions, that is conditions which cannot be derived from a simple lifting of the lower conditions. All of the lifting conditions and the pure (2,q)-positivity conditions for q>3 require tensor decompositions of the coefficients in the model Hamiltonians. Subsets of the new N-representability conditions can be employed with the previously known conditions to achieve polynomially scaling calculations of ground-state energies and 2-RDMs of many-electron quantum systems even in the presence of strong electron correlation

Extensive nonadditive entropy in quantum spin chains

We present details on a physical realization, in a many-body Hamiltonian system, of the abstract probabilistic structure recently exhibited by Gell-Mann, Sato and one of us (C.T.), that the nonadditive entropy $S_q=k [1- Tr \hat{\rho}^q]/[q-1]$ ($\hat{\rho}\equiv$ density matrix; $S_1=-k Tr \hat{\rho} \ln \hat{\rho}$) can conform, for an anomalous value of q (i.e., q not equal to 1), to the classical thermodynamical requirement for the entropy to be extensive. Moreover, we find that the entropic index q provides a tool to characterize both universal and nonuniversal aspects in quantum phase transitions (e.g., for a L-sized block of the Ising ferromagnetic chain at its T=0 critical transverse field, we obtain $\lim_{L\to\infty}S_{\sqrt{37}-6}(L)/L=3.56 \pm 0.03$). The present results suggest a new and powerful approach to measure entanglement in quantum many-body systems. At the light of these results, and similar ones for a d=2 Bosonic system discussed by us elsewhere, we conjecture that, for blocks of linear size L of a large class of Fermionic and Bosonic d-dimensional many-body Hamiltonians with short-range interaction at T=0, we have that the additive entropy $S_1(L) \propto [L^{d-1}-1]/(d-1)$ (i.e., $\ln L$ for $d=1$, and $L^{d-1}$ for d>1), hence it is not extensive, whereas, for anomalous values of the index q, we have that the nonadditive entropy $S_q(L)\propto L^d$ ($\forall d$), i.e., it is extensive. The present discussion neatly illustrates that entropic additivity and entropic extensivity are quite different properties, even if they essentially coincide in the presence of short-range correlations.Comment: 9 pages, 4 figures, Invited Paper presented at the international conference CTNEXT07, satellite of STATPHYS23, 1-5 July 2007, Catania, Ital

Why is the CMB fluctuation level 10^{-5}?

We explore the qualitative changes that would occur if the amplitude Q ~ 10^{-5} of cosmological density fluctuations were different. If is less than about 10^{-6}, the cosmological objects that form would have so low virial temperatures that they may be unable to cool and form stars, and would be so loosely bound that even if they could produce a supernova explosion, they might be unable to retain the heavy elements necessary for planetary life. If Q is greater than about 10^{-4}, dense supermassive galaxies would form, and biological evolution could be marred by short disruption timescales for planetary orbits. If Q were still larger, most bound systems would collapse directly to supermassive black holes. These constraints on Q can be expressed in terms of fundamental constants alone, and depend only on the electromagnetic and gravitational coupling constants, the electron-proton mass ratio and the matter-to-photon ratio. We discuss the implications for inflation and defect models, and note that the recent anthropic upper bounds on the cosmological constant Lambda would be invalid if both Q and Lambda could vary and there were no anthropic constraints on Q. The same applies to anthropic bounds on the curvature parameter Omega.Comment: Revised to match accepted version. 8 pages, with 1 figure included. Color figure and related links at http://www.sns.ias.edu/~max/Q.html (faster from the US), from http://www.mpa-garching.mpg.de/~max/Q.html (faster from Europe) or from [email protected]. ApJ, in pres

P-q theory power components calculations

The “Generalized theory of the instantaneous reactive power in three-phase circuits", proposed by Akagi et al., and also known as the p-q theory, is an interesting tool to apply to the control of active power filters, or even to analyze three-phase power systems in order to detect problems related to harmonics, reactive power and unbalance. In this paper it will be shown that in three phase electrical systems the instantaneous power waveform presents symme-tries of 1/6, 1/3, 1/2 or 1 cycle of the power system fundamen-tal frequency, depending on the system being balanced or not, and having or not even harmonics (interharmonics and sub-harmonics are not considered in this analysis). These symme-tries can be exploited to accelerate the calculations for active filters controllers based on the p-q theory. In the case of the conventional reactive power or zero-sequence compensation, it is shown that the theoretical control system dynamic response delay is zero.Fundação para a Ciência e a Tecnologia (FCT) - POCTI/ESE/41170/2001

Chiral heavy fermions in a two Higgs doublet model: 750 GeV resonance or not

We revisit models where a heavy chiral 4th generation doublet of fermions is embedded in a class of two Higgs doublets models (2HDM) with a discrete $Z_2$ symmetry, which couples the "heavy" scalar doublet only to the 4th generation fermions and the "light" one to the Standard Model (SM) fermions - the so-called 4G2HDM introduced by us several years ago. We study the constraints imposed on the 4G2HDM from direct searches of heavy fermions, from precision electroweak data (PEWD) and from the measured production and decay signals of the 125 GeV scalar, which in the 4G2HDM corresponds to the lightest CP-even scalar h. We then show that the recently reported excess in the $\gamma\gamma$ spectrum around 750 GeV can be accommodated by the heavy CP-even scalar of the 4G2HDM, H, resulting in a unique choice of parameter space: negligible mixing (sin\alpha ~ O(0.001)) between the two CP-even scalars h,H and heavy 4th generation quark and lepton masses m_t',m_b' < 400 GeV and $m_{\nu'},m_{\tau'}$ > 900 GeV, respectively. Whether or not the 750 GeV \gamma \gamma resonance is confirmed, interesting phenomenology emerges in q' - Higgs systems (q'=t',b'), that can be searched for at the LHC. For example, the heavy scalar states of the model, S=H,A,H^+, may have BR(S -> q'q') ~ O(1), giving rise to observable q'q' signals on resonance, followed by the flavor changing q' decays t'->uh (u=u,c) and/or b'->dh (d=d,s,b). This leads to distinct high jet-multiplicity signatures, with or without charged leptons, of the form q'q' -> (nj + mb + lW)_S (j and b being light and b-quark jets, respectively), with n+m+l =6-8 and unique kinematic features. It is also shown that the 4G2HDM can easily accommodate the interesting recent indications of a percent-level branching ratio in the lepton-flavor-violating (LFV) decay $h \to \tau \mu$ of the 125 GeV Higgs, if confirmed.Comment: 13 pages, late

SSD의 긴 꼬리 지연시간 문제 완화를 위한 강화학습의 적용

학위논문(박사)--서울대학교 대학원 :공과대학 컴퓨터공학부,2020. 2. 유승주.NAND flash memory is widely used in a variety of systems, from realtime embedded systems to high-performance enterprise server systems. Flash memory has (1) erase-before-write (write-once) and (2) endurance problems. To handle the erase-before-write feature, apply a flash-translation layer (FTL). Currently, the page-level mapping method is mainly used to reduce the latency increase caused by the write-once and block erase characteristics of flash memory. Garbage collection (GC) is one of the leading causes of long-tail latency, which increases more than 100 times the average latency at 99th percentile. Therefore, real-time systems or quality-critical systems cannot satisfy given requirements such as QoS restrictions. As flash memory capacity increases, GC latency also tends to increase. This is because the block size (the number of pages included in one block) of the flash memory increases as the capacity of the flash memory increases. GC latency is determined by valid page copy and block erase time. Therefore, as block size increases, GC latency also increases. Especially, the block size gets increased from 2D to 3D NAND flash memory, e.g., 256 pages/block in 2D planner NAND flash memory and 768 pages/block in 3D NAND flash memory. Even in 3D NAND flash memory, the block size is expected to continue to increase. Thus, the long write latency problem incurred by GC can become more serious in 3D NAND flash memory-based storage. In this dissertation, we propose three versions of the novel GC scheduling method based on reinforcement learning. The purpose of this method is to reduce the long tail latency caused by GC by utilizing the idle time of the storage system. Also, we perform a quantitative analysis for the RL-assisted GC solution. RL-assisted GC scheduling technique was proposed which learns the storage access behavior online and determines the number of GC operations to exploit the idle time. We also presented aggressive methods, which helps in further reducing the long tail latency by aggressively performing fine-grained GC operations. We also proposed a technique that dynamically manages key states in RL-assisted GC to reduce the long-tail latency. This technique uses many fine-grained pieces of information as state candidates and manages key states that suitably represent the characteristics of the workload using a relatively small amount of memory resource. Thus, the proposed method can reduce the long-tail latency even further. In addition, we presented a Q-value prediction network that predicts the initial Q-value of a newly inserted state in the Q-table cache. The integrated solution of the Q-table cache and Q-value prediction network can exploit the short-term history of the system with a low-cost Q-table cache. It is also equipped with a small network called Q-value prediction network to make use of the long-term history and provide good Q-value initialization for the Q-table cache. The experiments show that our proposed method reduces by 25%-37% the long tail latency compared to the state-of-the-art method.낸드 플래시 메모리는 실시간 임베디드 시스템으로부터 고성능의 엔터프라이즈 서버 시스템까지 다양한 시스템에서 널리 사용 되고 있다. 플래시 메모리는 (1) erase-before-write (write-once)와 (2) endurance 문제를 갖고 있다. Erase-before-write 특성을 다루기 위해 flash-translation layer (FTL)을 적용 한다. 현재 플래시 메모리의 write-once 특성과 block erase특성으로 인한 latency 증가를 감소 시키기 위하여 page-level mapping방식이 주로 사용 된다. Garbage collection (GC)은 99th percentile에서 평균 지연시간의 100배 이상 증가하는 long tail latency를 유발시키는 주요 원인 중 하나이다. 따라서 실시간 시스템이나 quality-critical system에서는 Quality of Service (QoS) 제한과 같은 주어진 요구 조건을 만족 시킬 수 없다. 플래시 메모리의 용량이 증가함에 따라 GC latency도 증가하는 경향을 보인다. 이것은 플래시 메모리의 용량이 증가 함에 따라 플래시 메모리의 블록 크기 (하나의 블록이 포함하고 있는 페이지의 수)가 증가 하기 때문이다. GC latency는 valid page copy와 block erase 시간에 의해 결정 된다. 따라서, 블록 크기가 증가하면, GC latency도 증가 한다. 특히, 최근 2D planner 플래시 메모리에서 3D vertical 플래시 메모리 구조로 전환됨에 따라 블록 크기는 증가 하였다. 심지어 3D vertical 플래시 메모리에서도 블록 크기가 지속적으로 증가 하고 있다. 따라서 3D vertical 플래시 메모리에서 long tail latency 문제는 더욱 심각해 진다. 본 논문에서 우리는 강화학습(Reinforcement learning, RL)을 이용한 세 가지 버전의 새로운 GC scheduling 기법을 제안하였다. 제안된 기술의 목적은 스토리지 시스템의 idle 시간을 활용하여 GC에 의해 발생된 long tail latency를 감소 시키는 것이다. 또한, 우리는 RL-assisted GC 솔루션을 위한 정량 분석 하였다. 우리는 스토리지의 access behavior를 온라인으로 학습하고, idle 시간을 활용할 수 있는 GC operation의 수를 결정하는 RL-assisted GC scheduling 기술을 제안 하였다. 추가적으로 우리는 공격적인 방법을 제시 하였다. 이 방법은 작은 단위의 GC operation들을 공격적으로 수행 함으로써, long tail latency를 더욱 감소 시킬 수 있도록 도움을 준다. 또한 우리는 long tail latency를 더욱 감소시키기 위하여 RL-assisted GC의 key state들을 동적으로 관리할 수 있는 Q-table cache 기술을 제안 하였다. 이 기술은 state 후보로 매우 많은 수의 세밀한 정보들을 사용 하고, 상대적으로 작은 메모리 공간을 이용하여 workload의 특성을 적절하게 표현 할 수 있는 key state들을 관리 한다. 따라서, 제안된 방법은 long tail latency를 더욱 감소 시킬 수 있다. 추가적으로, 우리는 Q-table cache에 새롭게 추가되는 state의 초기값을 예측하는 Q-value prediction network (QP Net)를 제안 하였다. Q-table cache와 QP Net의 통합 솔루션은 저 비용의 Q-table cache를 이용하여 단기간의 과거 정보를 활용 할 수 있다. 또한 이것은 QP Net이라고 부르는 작은 신경망을 이용하여 학습한 장기간의 과거 정보를 사용하여 Q-table cache에 새롭게 삽입되는 state에 대해 좋은 Q-value 초기값을 제공한다. 실험결과는 제안한 방법이 state-of-the-art 방법에 비교하여 25%-37%의 long tail latency를 감소 시켰음을 보여준다.Chapter 1 Introduction 1 Chapter 2 Background 6 2.1 System Level Tail Latency 6 2.2 Solid State Drive 10 2.2.1 Flash Storage Architecture and Garbage Collection 10 2.3 Reinforcement Learning 13 Chapter 3 Related Work 17 Chapter 4 Small Q-table based Solution to Reduce Long Tail Latency 23 4.1 Problem and Motivation 23 4.1.1 Long Tail Problem in Flash Storage Access Latency 23 4.1.2 Idle Time in Flash Storage 24 4.2 Design and Implementation 26 4.2.1 Solution Overview 26 4.2.2 RL-assisted Garbage Collection Scheduling 27 4.2.3 Aggressive RL-assisted Garbage Collection Scheduling 33 4.3 Evaluation 35 4.3.1 Evaluation Setup 35 4.3.2 Results and Discussion 39 Chapter 5 Q-table Cache to Exploit a Large Number of States at Small Cost 52 5.1 Motivation 52 5.2 Design and Implementation 56 5.2.1 Solution Overview 56 5.2.2 Dynamic Key States Management 61 5.3 Evaluation 67 5.3.1 Evaluation Setup 67 5.3.2 Results and Discussion 67 Chapter 6 Combining Q-table cache and Neural Network to Exploit both Long and Short-term History 73 6.1 Motivation and Problem 73 6.1.1 More State Information can Further Reduce Long Tail Latency 73 6.1.2 Locality Behavior of Workload 74 6.1.3 Zero Initialization Problem 75 6.2 Design and Implementation 77 6.2.1 Solution Overview 77 6.2.2 Q-table Cache for Action Selection 80 6.2.3 Q-value Prediction 83 6.3 Evaluation 87 6.3.1 Evaluation Setup 87 6.3.2 Storage-Intensive Workloads 89 6.3.3 Latency Comparison: Overall 92 6.3.4 Q-value Prediction Network Effects on Latency 97 6.3.5 Q-table Cache Analysis 110 6.3.6 Immature State Analysis 113 6.3.7 Miscellaneous Analysis 116 6.3.8 Multi Channel Analysis 121 Chapter 7 Conculsion and Future Work 138 7.1 Conclusion 138 7.2 Future Work 140 Bibliography 143 국문초록 154Docto

Finite-size scaling and boundary effects in two-dimensional valence-bond-solids

Various lattice geometries and boundaries are used to investigate valence-bond-solid (VBS) ordering in the ground state of an S=1/2 square-lattice quantum spin model---the J-Q model, in which 4- or 6-spin interactions Q are added to the Heisenberg exchange J. Ground state results for finite systems (with up to thousands of spins) are obtained using a projector QMC method. Great care has to be taken when extrapolating the order parameter to infinite size, in particular in cylinder geometry. Even though strong 2D VBS order exists and is established clearly with increasing system size on L*L lattices (or Lx* Ly lattices with a fixed Lx/Ly), only short-range VBS correlations are observed on long cylinders (when Lx -> infinity at fixed Ly). The correlation length increases with Ly, until long-range order sets in at a "critical" Ly. This width is large even when the 2D order is strong, e.g, for a system where the order parameter is 70% of the largest possible value, Ly=8 is required for ordering. Correlation functions for small L*L lattices can also be misleading. For a 20%-ordered system results for L up to 20 appear to extrapolate to a vanishing order parameter, while for larger L the behavior crosses over and extrapolates to a non-zero value (with exponentially small finite size corrections). The VBS order also exhibits interesting edge effects related to emergent U(1) symmetry, which, if not considered properly, can lead to wrong conclusions for the thermodynamic limit. The finite-size behavior for small L*L lattices and long cylinders is similar to that predicted for a Z2 spin liquid. The results raise concerns about recent works claiming Z2 spin liquid ground states in frustrated 2D systems, in particular, the Heisenberg model with nearest and next-nearest-neighbor couplings. VBS state in this system cannot be ruled out.Comment: 26 pages, 28 figures. v2: final, published versio