Showing papers on "Product (mathematics) published in 2019"

Optimal Forecast Reconciliation for Hierarchical and Grouped Time Series Through Trace Minimization

Abstract: Large collections of time series often have aggregation constraints due to product or geographical groupings. The forecasts for the most disaggregated series are usually required to add-up exactly ...

Sharp phase transition for the random-cluster and Potts models via decision trees

Abstract: We prove an inequality on decision trees on monotonic measures which generalizes the OSSS inequality on product spaces. As an application, we use this inequality to prove a number of new results on lattice spin models and their random-cluster representations. More precisely, we prove that 1. For the Potts model on transitive graphs, correlations decay exponentially fast for $\beta<\beta_c$. 2. For the random-cluster model with cluster weight $q\geq1$ on transitive graphs, correlations decay exponentially fast in the subcritical regime and the cluster-density satisfies the mean-field lower bound in the supercritical regime. 3. For the random-cluster models with cluster weight $q\geq1$ on planar quasi-transitive graphs $\mathbb{G}$, $$\frac{p_c(\mathbb{G})p_c(\mathbb{G}^*)}{(1-p_c(\mathbb{G}))(1-p_c(\mathbb{G}^*))}~=~q.$$ As a special case, we obtain the value of the critical point for the square, triangular and hexagonal lattices (this provides a short proof of the result of Beffara and Duminil-Copin [Probability Theory and Related Fields, 153(3-4):511--542, 2012]). These results have many applications for the understanding of the subcritical (respectively disordered) phase of all these models. The techniques developed in this paper have potential to be extended to a wide class of models including the Ashkin-Teller model, continuum percolation models such as Voronoi percolation and Boolean percolation, super-level sets of massive Gaussian Free Field, and random-cluster and Potts model with infinite range interactions.

Disassembly line balancing problem: a review of the state of the art and future directions

Abstract: The disassembly line balancing (DLB) problem assigns the set of tasks to each workstation for each product to be disassembled and aims at attaining several objectives, such as minimising the number...

Strong Quantum Nonlocality without Entanglement.

Abstract: Quantum nonlocality is usually associated with entangled states by their violations of Bell-type inequalities. However, even unentangled systems, whose parts may have been prepared separately, can show nonlocal properties. In particular, a set of product states is said to exhibit ``quantum nonlocality without entanglement'' if the states are locally indistinguishable; i.e., it is not possible to optimally distinguish the states by any sequence of local operations and classical communication. Here, we present a stronger manifestation of this kind of nonlocality in multiparty systems through the notion of local irreducibility. A set of multiparty orthogonal quantum states is defined to be locally irreducible if it is not possible to locally eliminate one or more states from the set while preserving orthogonality of the postmeasurement states. Such a set, by definition, is locally indistinguishable, but we show that the converse does not always hold. We provide the first examples of orthogonal product bases on ${\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}$ for $d=3$, 4 that are locally irreducible in all bipartitions, where the construction for $d=3$ achieves the minimum dimension necessary for such product states to exist. The existence of such product bases implies that local implementation of a multiparty separable measurement may require entangled resources across all bipartitions.

Version 4 of the SMAP Level-4 Soil Moisture algorithm and data product

Abstract: The NASA Soil Moisture Active Passive (SMAP) mission Level‐4 Soil Moisture (L.4_SM) product provides global, 3‐hourly, 9‐km resolution estimates of surface (0–5 cm) and root zone (0–100 cm) soil moisture with a mean latency of ~2.5 days. The underlying L4_SM algorithm assimilates SMAP radiometer brightness temperature (Tb) observations into the NASA Catchment land surface model using a spatially distributed ensemble Kalman filter. In Version 4 of the L4_SM modeling system the upward recharge of surface soil moisture from below under nonequilibrium conditions was reduced, resulting in less bias and improved dynamic range of L4_SM surface soil moisture compared to earlier versions. This change and additional technical modifications to the system reduce the mean and standard deviation of the observation‐minus‐forecast Tb residuals and overall soil moisture analysis increments while maintaining the skill of the L4_SM soil moisture estimates versus independent in situ measurements; the average, bias‐ adjusted root‐mean‐square error in Version 4 is 0.039 m/m for surface and 0.026 m/m for root zone soil moisture. Moreover, the coverage of assimilated SMAP observations in Version 4 is near global owing to the use of additional satellite Tb records for algorithm calibration. L4_SM soil moisture uncertainty estimates are biased low (by 0.01–0.02 m/m) against actual errors (computed versus in situ measurements). L4_SM runoff estimates, an additional product of the L4_SM algorithm, are biased low (by 35 mm/year) against streamflow measurements. Compared to Version 3, bias in Version 4 is reduced by 46% for surface soil moisture uncertainty estimates and by 33% for runoff estimates. Plain Language Summary Soil moisture is important because of its impact on the land surface water, energy, and nutrient cycles. The low‐frequency microwave observations collected by the NASA Soil Moisture Active Passive (SMAP) satellite are suitable for estimating soil moisture globally. Their sensitivity, however, is limited to the top few centimeters of the soil, and observations are only available every other day depending on location. The SMAP Level‐4 Soil Moisture (L4_SM) data product addresses these limitations by merging the satellite observations into a numerical model of the land surface water and energy balance while considering the uncertainty of the observations and model estimates. The resulting L4_SM data product is publicly disseminated within ~2.5 days from the time of observation and provides global estimates of surface and deeper‐layer (or “root zone”) soil moisture at 3‐hourly temporal and 9‐km spatial resolution. This study presents an overview of recent updates in the L4_SM algorithm and an assessment of the quality of the resulting Version‐4 soil moisture estimates. The algorithm updates reduce bias in L4_SM surface soil moisture and runoff estimates compared to previous versions while otherwise maintaining the product skill, which meets the accuracy requirement specified prior to SMAP's launch. ©2019. The Authors. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited. RESEARCH ARTICLE 10.1029/2019MS001729 Key Points: • Version 4 of the SMAP Level‐4 Soil Moisture product features several model and analysis improvements compared to previous versions • SMAP Level‐4 surface soil moisture, soil moisture uncertainty, and runoff estimates have lower bias in Version 4 than in previous versions • SMAP Level‐4 analysis adjustments are smaller in Version 4 than in previous versions while achieving similar overall product skill Supporting Information: • Supporting Information S1 Correspondence to: R. H. Reichle, rolf.reichle@nasa.gov Citation: Reichle, R. H., Liu, Q., Koster, R. D., Crow, W. T., De Lannoy, G. J. M., Kimball, J. S., et al. (2019). Version 4 of the SMAP Level‐4 Soil Moisture algorithm and data product. Journal of Advances in Modeling Earth Systems, 11. https://doi.org/10.1029/2019MS001729 Received 30 APR 2019 Accepted 29 AUG 2019 Accepted article online 2 SEP 2019

MDS Codes With Hulls of Arbitrary Dimensions and Their Quantum Error Correction

Abstract: The hull of linear codes has promising utilization in coding theory and quantum coding theory. In this paper, we study the hull of generalized Reed–Solomon codes and extended generalized Reed–Solomon codes over finite fields with respect to the Euclidean inner product. Several infinite families of MDS codes with hulls of arbitrary dimensions are presented. As an application, using these MDS codes with hulls of arbitrary dimensions, we construct several new infinite families of entanglement-assisted quantum error-correcting codes with flexible parameters.

Faster quantum simulation by randomization

Abstract: Product formulas can be used to simulate Hamiltonian dynamics on a quantum computer by approximating the exponential of a sum of operators by a product of exponentials of the individual summands. This approach is both straightforward and surprisingly efficient. We show that by simply randomizing how the summands are ordered, one can prove stronger bounds on the quality of approximation for product formulas of any given order, and thereby give more efficient simulations. Indeed, we show that these bounds can be asymptotically better than previous bounds that exploit commutation between the summands, despite using much less information about the structure of the Hamiltonian. Numerical evidence suggests that the randomized approach has better empirical performance as well.

List-decodable Linear Regression

Abstract: We give the first polynomial-time algorithm for robust regression in the list-decodable setting where an adversary can corrupt a greater than 1/2 fraction of examples. For any \alpha < 1, our algorithm takes as input a sample {(x_i,y_i)}_{i \leq n} of n linear equations where \alpha n of the equations satisfy y_i = \langle x_i,\ell^*\rangle +\zeta for some small noise \zeta and (1-\alpha) n of the equations are {\em arbitrarily} chosen. It outputs a list L of size O(1/\alpha) - a fixed constant - that contains an \ell that is close to \ell^*. Our algorithm succeeds whenever the inliers are chosen from a certifiably anti-concentrated distribution D. In particular, this gives a (d/\alpha)^{O(1/\alpha^8)} time algorithm to find a O(1/\alpha) size list when the inlier distribution is a standard Gaussian. For discrete product distributions that are anti-concentrated only in regular directions, we give an algorithm that achieves similar guarantee under the promise that \ell^* has all coordinates of the same magnitude. To complement our result, we prove that the anti-concentration assumption on the inliers is information-theoretically necessary. To solve the problem we introduce a new framework for list-decodable learning that strengthens the ``identifiability to algorithms'' paradigm based on the sum-of-squares method.

Global well-posedness for the two-dimensional Muskat problem with slope less than 1

Abstract: We prove the existence of global, smooth solutions to the two-dimensional Muskat problem in the stable regime whenever the product of the maximal and minimal slope is less than 1. The curvature of these solutions decays to 0 as t goes to infinity, and they are unique when the initial data is C 1 , ϵ . We do this by getting a priori estimates using a nonlinear maximum principle first introduced in a paper by Kiselev, Nazarov, and Volberg (2007), where the authors proved global well-posedness for the surface quasigeostraphic equation.

Separation of variables and scalar products at any rank

Abstract: Separation of variables (SoV) is a special property of integrable models which ensures that the wavefunction has a very simple factorised form in a specially designed basis. Even though the factorisation of the wavefunction was recently established for higher rank models by two of the authors and G. Sizov, the measure for the scalar product was not known beyond the case of rank one symmetry. In this paper we show how this measure can be found, bypassing an explicit SoV construction. A key new observation is that the measure for spin chains in a highest-weight infinite dimensional representation of 𝔰𝔩(N) couples Q-functions at different nesting levels in a non-symmetric fashion. We also managed to express a large number of form factors as ratios of determinants in our new approach. We expect our method to be applicable in a much wider setup including the problem of computing correlators in integrable CFTs such as the fishnet theory, $$ \mathcal{N} $$ = 4 SYM and the ABJM model.

Iterative algorithms for solving some tensor equations

Abstract: We propose a class of iterative algorithms to solve some tensor equations via Einstein product. These algorithms use tensor computations with no matricizations involved. For any (special) initial t...

Topological nonlinear σ -model, higher gauge theory, and a systematic construction of 3 + 1 D topological orders for boson systems

Abstract: A discrete nonlinear $\ensuremath{\sigma}$-model is obtained by triangulate both the space-time ${M}^{d+1}$ and the target space $K$. If the path integral is given by the sum of all the simplicial homomorphisms $\ensuremath{\phi}:{M}^{d+1}\ensuremath{\rightarrow}K$ (i.e., maps without any topological defects), with an partition function that is independent of space-time triangulation, then the corresponding nonlinear $\ensuremath{\sigma}$-model will be called topological nonlinear $\ensuremath{\sigma}$-model which is exactly soluble. These exactly soluble models suggest that phase transitions induced by fluctuations with no topological defects usually produce a topologically ordered state and are topological phase transitions. In contrast, phase transitions induced by fluctuations with all topological defects give rise to trivial product states and are not topological phase transitions. Under the classification conjecture of Lan-Kong-Wen [Phys. Rev. X 8, 021074 (2018)], it is shown that, if $K$ is a space with only nontrivial first homotopy group $G$, which is finite, then these topological nonlinear $\ensuremath{\sigma}$-models can already realize all $3+1\text{D}$ bosonic topological orders without emergent fermions, which are described by Dijkgraaf-Witten theory with gauge group ${\ensuremath{\pi}}_{1}(K)=G$. Under the similar conjecture, we show that the $3+1\text{D}$ bosonic topological orders with emergent fermions can be realized by topological nonlinear $\ensuremath{\sigma}$-models with ${\ensuremath{\pi}}_{1}(K)=$ finite groups, ${\ensuremath{\pi}}_{2}(K)={Z}_{2}$, and ${\ensuremath{\pi}}_{ng2}(K)=0$. A subset of these topological nonlinear $\ensuremath{\sigma}$-models corresponds to 2-gauge theories, which realize and may classify bosonic topological orders with emergent fermions that have no emergent Majorana zero modes at triple string intersections.

Product decomposition of periodic functions in quantum signal processing

Abstract: In some embodiments, one or more unitary-valued functions are generated by a classical computer generating using projectors with a predetermined number of significant bits. A quantum computing device is then configured to implement the one or more unitary-valued functions. In further embodiments, a quantum circuit description for implementing quantum signal processing that decomposes complex-valued periodic functions is generated by a classical computer, wherein the generating further includes representing approximate polynomials in a Fourier series with rational coefficients. A quantum computing device is then configured to implement a quantum circuit defined by the quantum circuit description.

Embedding Tracking Codes in Additive Manufactured Parts for Product Authentication

Abstract: Additive manufacturing (AM) process chain relies heavily on cloud resources and software programs. Cybersecurity has become a major concern for such resources. AM produces physical components, which can be compromised for quality by many other means and can be reverse engineered for unauthorized reproduction. This work is focused on taking advantage of layer-by-layer manufacturing process of AM to embed codes inside the components and reading them using image acquisition methods. The example of a widely used QR code format is used, but the same scheme can be used for other formats or alphanumeric strings. The code is segmented in a large number of parts for obfuscation. The results show that segmentation and embedding the code in numerous layers help in eliminating the effect of embedded features on the mechanical properties of the part. Such embedded codes can be used for parts produced by fused filament fabrication, inkjet printing, and selective laser sintering technologies for product authentication and identification of counterfeits. Post processing methods such as heat treatments and hot isostatic pressing may remove or distort these codes; therefore, analysis of AM method and threat level is required to determine if the proposed strategy can be useful for a particular product.

Private Hypothesis Selection

Abstract: We provide a differentially private algorithm for hypothesis selection. Given samples from an unknown probability distribution $P$ and a set of $m$ probability distributions $\mathcal{H}$, the goal is to output, in a $\varepsilon$-differentially private manner, a distribution from $\mathcal{H}$ whose total variation distance to $P$ is comparable to that of the best such distribution (which we denote by $\alpha$). The sample complexity of our basic algorithm is $O\left(\frac{\log m}{\alpha^2} + \frac{\log m}{\alpha \varepsilon}\right)$, representing a minimal cost for privacy when compared to the non-private algorithm. We also can handle infinite hypothesis classes $\mathcal{H}$ by relaxing to $(\varepsilon,\delta)$-differential privacy. We apply our hypothesis selection algorithm to give learning algorithms for a number of natural distribution classes, including Gaussians, product distributions, sums of independent random variables, piecewise polynomials, and mixture classes. Our hypothesis selection procedure allows us to generically convert a cover for a class to a learning algorithm, complementing known learning lower bounds which are in terms of the size of the packing number of the class. As the covering and packing numbers are often closely related, for constant $\alpha$, our algorithms achieve the optimal sample complexity for many classes of interest. Finally, we describe an application to private distribution-free PAC learning.

Strong quantum nonlocality in multipartite quantum systems

Abstract: Recently, based on some quantum states which are locally irreducible in all bipartitions, strong quantum nonlocality has been presented by Halder et al. [Phys. Rev. Lett. 122, 040403 (2019)]. However, the remaining questions are whether a set of quantum states can reveal strong quantum nonlocality when these states are locally irreducible in all tripartitions or more multipartitions and how to construct strong nonlocality of orthogonal product states in more than tripartite quantum systems. Here we present a general definition of strong quantum nonlocality based on the local irreducibility. Then, using a $3\ensuremath{\bigotimes}3\ensuremath{\bigotimes}3$ quantum system as an example, we show three sets of locally indistinguishable orthogonal product states to explain the general definition of strong quantum nonlocality. Furthermore, in $3\ensuremath{\bigotimes}3\ensuremath{\bigotimes}3\ensuremath{\bigotimes}3$, we construct a set of strong nonlocality of orthogonal product states; these states do not form a complete basis. Our results demonstrate the phenomenon of strong nonlocality without entanglement for the multipartite quantum systems.

Perturbation theory for Moore–Penrose inverse of tensor via Einstein product

Abstract: The spectral norm of an even-order tensor is defined and investigated. An equivalence between the spectral norm of tensors and matrices is given. Using derived representations of some tensor expressions involving the Moore–Penrose inverse, we investigate the perturbation theory for the Moore–Penrose inverse of tensor via Einstein product. The classical results derived by Stewart (SIAM Rev 19:634–662, 1977) and Wedin (BIT 13:217–232, 1973) for the matrix case are extended to even-order tensors. An implementation in the Matlab programming language is developed and used in deriving appropriate numerical examples.

Prime II$_1$ factors arising from irreducible lattices in products of rank one simple Lie groups

Abstract: We prove that if $\Gamma$ is an icc irreducible lattice in a product of connected non-compact rank one simple Lie groups with finite center, then the II$_1$ factor $L(\Gamma)$ is prime. In particular, we deduce that the II$_1$ factors associated to the arithmetic groups $\text{PSL}_2(\mathbb Z[\sqrt{d}])$ and $\text{PSL}_2(\mathbb Z[S^{-1}])$ are prime, for any square-free integer $d\geq 2$ with $d ot\equiv 1 mod{4}$ and any finite non-empty set of primes $S$. This provides the first examples of prime II$_1$ factors arising from lattices in higher rank semisimple Lie groups. More generally, we describe all tensor product decompositions of $L(\Gamma)$ for icc countable groups $\Gamma$ that are measure equivalent to a product of non-elementary hyperbolic groups. In particular, we show that $L(\Gamma)$ is prime, unless $\Gamma$ is a product of infinite groups, in which case we prove a unique prime factorization result for $L(\Gamma)$.

Family of bound entangled states on the boundary of the Peres set

Abstract: Bound entangled (BE) states are strange in nature: a nonzero amount of free entanglement is required to create them but no free entanglement can be distilled from them under local operations and classical communication (LOCC). Even though the usefulness of such states has been shown in several information processing tasks, there exists no simple method to characterize them for an arbitrary composite quantum system. Here we present a $(d\ensuremath{-}3)/2$-parameter family of BE states each with positive partial transpose (PPT). This family of PPT-BE states is introduced by constructing an unextendible product basis (UPB) in ${\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}$ with $d$ odd and $d\ensuremath{\ge}5$. The range of each such PPT-BE state is contained in a $2(d\ensuremath{-}1)$-dimensional entangled subspace, whereas the associated UPB subspace is of dimension ${(d\ensuremath{-}1)}^{2}+1$. We further show that each of these PPT-BE states can be written as a convex combination of $(d\ensuremath{-}1)/2$ rank-4 PPT-BE states. Moreover, we prove that these rank-4 PPT-BE states are extreme points of the convex compact set $\mathcal{P}$ of all PPT states in ${\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}$, namely, the Peres set. An interesting geometric implication of our result is that the convex hull of these rank-4 PPT-BE extreme points---the $(d\ensuremath{-}3)/2$ simplex---is sitting on the boundary between the set $\mathcal{P}$ and the set of non-PPT states. We also discuss consequences of our construction in the context of quantum state discrimination by LOCC.

SCFT/VOA correspondence via Ω-deformation

Abstract: We investigate an alternative approach to the correspondence of four­ dimensional $$ \mathcal{N} $$ = 2 superconformal theories and two-dimensional vertex operator algebras, in the framework of the Ω-deformation of supersymmetric gauge theories. The two­dimensional Ω-deformation of the holomorphic-topological theory on the product four­ manifold is constructed at the level of supersymmetry variations and the action. The supersymmetric localization is performed to achieve a two-dimensional chiral CFT. The desired vertex operator algebra is recovered as the algebra of local operators of the resulting CFT. We also discuss the identification of the Schur index of the $$ \mathcal{N} $$ = 2 superconformal theory and the vacuum character of the vertex operator algebra at the level of their path integral representations, using our Ω-deformation point of view on the correspondence.

The balanced tensor product of module categories

Abstract: The balanced tensor product M⊗AN of two modules over an algebra A is the vector space corepresenting A-balanced bilinear maps out of the product M×N. The balanced tensor product M⊠CN of two module categories over a monoidal linear category C is the linear category corepresenting C-balanced right-exact bilinear functors out of the product category M×N. We show that the balanced tensor product can be realized as a category of bimodule objects in C, provided the monoidal linear category is finite and rigid.

Eco-modular product architecture identification and assessment for product recovery

Abstract: In order to improve the efficiency of disassembly and product recovery of an abandoned product at the end-of-life stage, it is essential to develop modular product architecture by considering manufacturing and recovering processes in early product design stage. In this paper, a novel concept of a design methodology is introduced to develop eco-modular product architecture and assess the modularity of the architecture from the viewpoint of product recovery. Eco-modular product architecture contributes to enhancing product recovery processes by recycling and reusing modules without full disassembly at component or material levels. It leads to less consumption of natural resources and less landfill damage to the environment. Three sustainable modular drivers, namely, interface complexity, material similarity, and lifespan similarity, are introduced to reconstruct the modular architecture of commercial products into the eco-modular architecture. Alternatives of modular architectures are identified by Markov Cluster Algorithm based on these sustainable modular drivers and physical interconnections of the components of product architecture. To select the eco-modular architecture from these alternatives, we propose modularity assessment metrics to identify independent interactions between modules and the degrees of similarity within each module. To demonstrate the effectiveness of the proposed methodology, a case study is performed with a coffee maker.

A decomposition theorem for projective manifolds with nef anticanonical bundle

Abstract: Let X be a simply connected projective manifold with nef anticanonical bundle. We prove that X is a product of a rationally connected manifold and a manifold with trivial canonical bundle. As an application we describe the MRC fibration of any projective manifold with nef anticanonical bundle.

Out-of-time ordered correlators and entanglement growth in the random-field XX spin chain

Abstract: We study out-of-time ordered correlations $C(x,t)$ and entanglement growth in the random-field XX model with open boundary conditions using the exact Jordan-Wigner transformation to a fermionic Hamiltonian. For any nonzero strength of the random field, this model describes an Anderson insulator. Two scenarios are considered: a global quench with the initial state corresponding to a product state of the N\'eel form, and the behavior in a typical thermal state at $\ensuremath{\beta}=1$. As a result of the presence of disorder, the information spreading as described by the out-of-time correlations stops beyond a typical length scale ${\ensuremath{\xi}}_{\text{OTOC}}$. For $|x|l{\ensuremath{\xi}}_{\text{OTOC}}$, information spreading occurs at the maximal velocity ${v}_{\mathrm{max}}=J$ and we confirm predictions for the early-time behavior of $C(x,t)\ensuremath{\sim}{t}^{2|x|}$. For the case of the quench starting from the N\'eel product state, we also study the growth of the bipartite entanglement, focusing on the late- and infinite-time behavior. The approach to a bounded entanglement is observed to be slow for the disorder strengths we study.

Nonconforming Virtual Element Method for $2m$th Order Partial Differential Equations in $\mathbb {R}^n$

Abstract: A unified construction of the $H^m$-nonconforming virtual elements of any order $k$ is developed on any shape of polytope in $\mathbb R^n$ with constraints $m\leq n$ and $k\geq m$. As a vital tool in the construction, a generalized Green's identity for $H^m$ inner product is derived. The $H^m$-nonconforming virtual element methods are then used to approximate solutions of the $m$-harmonic equation. After establishing a bound on the jump related to the weak continuity, the optimal error estimate of the canonical interpolation, and the norm equivalence of the stabilization term, the optimal error estimates are derived for the $H^m$-nonconforming virtual element methods.

A volume preserving flow and the isoperimetric problem in warped product spaces

Abstract: In this article, we continue the work in \cite{GL} and study a normalized hypersurface flow in the more general ambient setting of warped product spaces. This flow preserves the volume of the bounded domain enclosed by a graphical hypersurface, and monotonically decreases the hypersurface area. As an application, the isoperimetric problem in warped product spaces is solved for such domains.