Linear Space-time Research Articles

Quantum Black creative geographies: embodiment, coherence and transcendence in a time of climate crisis†

This paper brings together three parallel strands of work—Black Geographies, geographies of Caribbean creative practice, and quantum geographies. The paper begins by considering static linear spacetimes as colonial spacetimes, and draws on Michelle Wright's critique of Middle Passage epistemologies, from Black Studies, to elaborate on this. It then moves through a number of ways in which, over the last couple of decades, I have drawn together insights from Wilson Harris and Karen Barad to explore how quantum mechanics can facilitate a conversation about uncertainty, connectedness, entanglement and the liveliness of always already climate‐changed landscapes in relation to Black embodiment. In pushing briefly into string theory, the paper ends with the possibility of connecting spirituality with materialities, to push towards more politically attuned forms of emancipation.

Finiteness of quantum gravity with matter on a PL spacetime

We study the convergence of the path integral (PI) for general relativity with matter on a picewise linear (PL) spacetime that corresponds to a triangulation of a smooth manifold by using a PI measure that renders the pure gravity PI finite. This measure depends on a parameter p, and in the case when the matter content is just scalar fields, we show that the PI is absolutely convergent for 0,5 and not more than two scalar fields. In the case of Yang–Mills (YM) fields, we show that the PI is absolutely convergent for the U(1) group and 0,5. In the case of Dirac fermions, we show that the PI is absolutely convergent for any number of fermions and a sufficiently large p. When the matter content is given by scalars, YM fields and fermions, as in the case of the standard model (SM), we show that the PI is absolutely convergent for 52,5. Hence one can construct a finite quantum gravity theory on a PL spacetime such that the classical limit is general relativity coupled to the SM.

Nonlinear parallel-in-time simulations of multiphase flow in porous media

As the number of processors on supercomputers has increased dramatically, there is a growing interest in developing scalable algorithms with a high degree of parallelism for extreme-scale reservoir simulation. However, traditional simulators and algorithms for such nonlinear problems are usually based on the family of time-marching methods, where parallelization is restricted to the spatial dimension only. In this paper, we propose a family of parallel-in-time (PinT) reservoir algorithms for solving multiphase flow problems in porous media to fully exploit the parallelism of supercomputers, in which the design of efficient nonlinear and linear space-time algorithms plays an essential role. More precisely, we first introduce a space-time mixed finite element method for the fully-implicit discretization. The nonlinear system arising at each parallel-in-time step is solved by a variant of bound-preserving Newton methods with a nonlinear elimination preconditioner, where the corresponding linear system is solved by the space-time restricted additive Schwarz (stRAS) method. Since the straightforward extension of one-level stRAS to multilevel does not work because of the pollution effects from the temporal direction, we accordingly present a nonstandard V-cycle multilevel stRAS method with a pollution-removing strategy to accelerate the convergence and improve the robustness. Numerical experiments are presented to demonstrate that the aforementioned parallel-in-time algorithm can not only achieve a high degree of parallelism in accurately resolving reservoir transport features in a heterogeneous medium, but also successfully circumvent the convergence and scalability issues associated with the constraint of large time steps and the boundedness requirement of the solution.

Gapped Indexing for Consecutive Occurrences

The classic string indexing problem is to preprocess a string S into a compact data structure that supports efficient pattern matching queries. Typical queries include existential queries (decide if the pattern occurs in S), reporting queries (return all positions where the pattern occurs), and counting queries (return the number of occurrences of the pattern). In this paper we consider a variant of string indexing, where the goal is to compactly represent the string such that given two patterns \(P_1\) and \(P_2\) and a gap range \({[}\alpha , \beta ]\) we can quickly find the consecutive occurrences of \(P_1\) and \(P_2\) with distance in \({[}\alpha , \beta ]\), i.e., pairs of subsequent occurrences with distance within the range. We present data structures that use linear space and query time \({\widetilde{O}}(|P_1|+|P_2|+n^{2/3})\) for existence and counting and \({\widetilde{O}}(|P_1|+|P_2|+n^{2/3}\hbox {occ}^{1/3})\) for reporting. We complement this with a conditional lower bound based on the set intersection problem showing that any solution using \({\widetilde{O}}(n)\) space must use \({\widetilde{\Omega }}(|P_1| + |P_2| + \sqrt{n})\) query time. To obtain our results we develop new techniques and ideas of independent interest including a new suffix tree decomposition and hardness of a variant of the set intersection problem.

String indexing for top-k close consecutive occurrences

The classic string indexing problem is to preprocess a string S into a compact data structure that supports efficient subsequent pattern matching queries, that is, given a pattern string P, report all occurrences of P within S. In this paper, we study a basic and natural extension of string indexing called the string indexing for top-k close consecutive occurrences problem (Sitcco). Here, a consecutive occurrence is a pair (i,j), i<j, such that P occurs at positions i and j in S and there is no occurrence of P between i and j, and their distance is defined as j−i. Given a pattern P and a parameter k, the goal is to report the top-k consecutive occurrences of P in S of minimal distance. The challenge is to compactly represent S while supporting queries in time close to the length of P and k. We give three time-space trade-offs for the problem. Let n be the length of S, m the length of P, and ϵ∈(0,1]. Our first result achieves O(nlog⁡n) space and optimal query time of O(m+k). Our second and third results achieve linear space and query times either O(m+k1+ϵ) or O(m+klog1+ϵ⁡n). Along the way, we develop several techniques of independent interest, including a new translation of the problem into a line segment intersection problem and a new recursive clustering technique for trees.

Time-Domain Analysis of Thin-Wire Structures Based on the Cagniard-DeHoop Method of Moments

Thin-wire structures in the presence or absence of a ground plane are analyzed numerically in the time domain (TD) with the aid of the Cagniard–DeHoop method of moments (CdH-MoM). It is demonstrated that the TD solution of such problems can be cast into the form of discrete-time-convolution equations. Under the assumption of piecewise linear space–time axial current distribution, the elements of the TD impedance array are derived analytically in terms of elementary functions. Their approximations applying to multiconductor transmission lines are discussed. Illustrative numerical examples validating the TD solution are presented.

On the Resolution of a Remarkable Bond Pricing Model from Financial Mathematics: Application of the Deductive Group Theoretical Technique

The classical Cox–Ingersoll–Ross (CIR) bond-pricing model is based on the evolution space-time dependent partial differential equation (PDE) which represents the standard European interest rate derivatives. In general, such class of evolution partial differential equations (PDEs) has generally been resolved by classical methods of PDEs and by ansatz-based techniques which have been previously applied in a similar context. The author here shows the application of an invariant approach, a systematic method based on deductive group-theoretical analysis. The invariant technique reduces the scalar linear space-time dependent parabolic PDE to one of the four classical Lie canonical forms. This method leads us to exactly solve the scalar linear space-time dependent parabolic PDE representing the CIR model. It was found that CIR PDE is transformed into the first canonical form, which is the heat equation. Under the proper choice of emerging parameters of the model, the CIR equation is also reduced to the second Lie canonical form. The equivalence transformations which map the CIR PDE into the different canonical forms are deduced. With the use of these equivalence transformations, the invariant solutions of the underlying model are found by using some well-known results of the heat equation and the second Lie canonical form. Furthermore, the Cauchy initial-value model of the CIR problem along with the terminal condition is discussed and closed-form solutions are deduced. Finally, the conservation laws associated with the CIR equation are derived by using the general conservation theorem.

Proposal of a New Scalar Gauge in Relativity Field Equation

In this paper we propose a new gauge term in addition to the conventional gauge to acquire complete solution for the linear approximated gravitational equation. The calculation to make general form for the linear gravitational equation uses the well-known Nöether’s theorem saying that gauge symmetry is equal to conservation law. The unsolved coefficients in the equation require another condition which is leading to new gauge term. This proposed new gauge is a tensor product by a scalar quantity with a metric tensor having the trace value of 2. The scalar component in the 5th row and column of Kaluza-Klein’s metric tensor can be found as 2 diagonal components in our proposed 4×4 metric tensor. We also show that only a constant scalar gauge can be allowed in the curved space-time although arbitrary gauge can exist in the linear space-time.

Toverline{T} , black holes and negative strings

String theory on AdS3 has a solvable single-trace irrelevant deformation that is closely related to Toverline{T} . For one sign of the coupling, it leads to an asymptotically linear dilaton spacetime, and a corresponding Hagedorn spectrum. For the other, the resulting spacetime has a curvature singularity at a finite radial location, and an upper bound on the energies of states. Beyond the singularity, the signature of spacetime is flipped and there is an asymptotically linear dilaton boundary at infinity. We study the properties of black holes and fundamental strings in this spacetime, and find a sensible picture. The singularity does not give rise to a hard ultraviolet wall for excitations -one must include the region beyond it to understand the theory. The size of black holes diverges as their energy approaches the upper bound, as does the location of the singularity. Fundamental strings pass smoothly through the singularity, but if their energy is above the upper bound, their trajectories are singular. From the point of view of the boundary at infinity, this background can be thought of as a vacuum of Little String Theory which contains a large number of negative strings.

Ranked document selection

Let D be a collection of string documents of n characters in total. The top-k document retrieval problem is to preprocess D into a data structure that, given a query (P,k), can return the k documents of D most relevant to pattern P. The relevance of a document d for a pattern P is given by a predefined ranking function w(P,d). Linear space and optimal query time solutions already exist for this problem. In this paper we consider a novel problem, document selection, in which a query (P,k) aims to report the kth document most relevant to P (instead of reporting all top-k documents). We present a data structure using O(nlogϵ⁡n) space, for any constant ϵ>0, answering selection queries in time O(log⁡k/log⁡log⁡n), and a linear-space data structure answering queries in time O(log⁡k), given the locus node of P in a (generalized) suffix tree of D. We also prove that it is unlikely that a succinct-space solution for this problem exists with poly-logarithmic query time, and that O(log⁡k/log⁡log⁡n) is indeed optimal within O(npolylogn) space for most text families. Finally, we present some additional space-time trade-offs exploring the extremes of those lower bounds.

LAM: Locality affine-invariant feature matching

False match removal is a crucial and fundamental task in photogrammetry and computer vision. This paper proposes a robust and efficient mismatch-removal algorithm based on the concepts of local barycentric coordinate (LBC) and matching coordinate matrices (MCMs), called locality affine-invariant matching (LAM). LAM is suitable for both rigid and nonrigid image matching problems. We define a novel LBC system based on area ratios, which is invariant to local affine transformations. We also present the MCMs based on the coordinates of matches, whose degeneracy is able to indicate the correctness of correspondences. Our LAM method first builds a mathematical model based on the LBCs to extract good matches that preserve local neighborhood structures. Then, LAM constructs local MCMs using the extracted reliable correspondences and identifies the correctness for the remaining matches via minimizing the rank of the MCMs. LAM has linear space and linearithmic time complexities. Extensive experiments on both rigid and nonrigid real datasets demonstrate the power of the proposed method; i.e., LAM is more robust to complex transformations compared to other methods and is two orders of magnitude faster than RANSAC under low inlier rates. The source code of the proposed LAM method will be publicly available in http://www.escience.cn/people/lijiayuan/index.html.

Generalized space time autoregressive (gstar)-artificial neural network (ann) model with multilayer feedforward networks architecture

The GSTAR model is one linear space time model used to model time series data with inter-location linkages. One of the weaknesses of the GSTAR model is that the model has not been able to capture any nonlinear patterns that may arise. The ANN is one model of nonlinear artificial intelligence that has a flexible functional form and is a supervised engine learning that provides a good framework to represent a relationship in time series data. Due to the advantages of such ANN, it can be combined with GSTAR model. In this research conducted study of GSTAR-ANN model and its network architecture. The model is constructed in 3 layers namely input, hidden and output. The GSTAR model is used as input in the training process. Each input will receive an input signal and forward the signal to the hidden layer, then to the output layer. The GSTAR-ANN model has network architecture with one neuron unit at the output layer, p input neuron, q neuron in hidden layer (multilayer feedforward networks).

Computing Nonoverlapping Inversion Distance Between Two Strings in Linear Average Time.

Biological events like inversions are not automatically detected by the usual alignment algorithms. Alignment with inversions does not have a known polynomial time algorithm and Schöniger and Waterman introduced a simplification of the alignment problem with nonoverlapping inversions, where all regions will not be allowed to overlap. They presented an \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \cal O} ( {n^6} )$$ \end{document} algorithm to compute nonoverlapping inversion distance between two strings of length n. The time and space complexities were improved to \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \cal O} ( {n^3} )$$ \end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \cal O} ( {n^2} )$$ \end{document} later by Cho, Vellozo, and Ta. In this article, a linear space and linear average time algorithm to compute the inversion distance between two strings of the same length is presented. The recursive formula for this purpose is new to the best of our knowledge. The space costs of the algorithms to solve the same problem are quadratic in the literature, and thus our original algorithm is the first linear space and linear average time algorithm to solve the inversion distance problem.

Distributed Linear Convolutional Space-Time Coding for Two-Hop Full-Duplex Relay 2x2x2 Cooperative Communication Networks

In this paper, distributed linear convolutional space-time coding (DLC-STC) for two relay full-duplex (FD) $2\times 2\times 2$ cooperative communication networks is presented. A network topology model which consists of two sources, two destinations and one layer of two relays, is considered. In this communication model, two sources may transmit signals to two different destinations with the help of two relays simultaneously, which will cause inter-user interference. As a result, one has to deal with the inter-user interference in addition to the self-loop interference at the relays. Most part of the self-loop interference can be cancelled largely with the self-loop interference techniques at the relays, but the residual part is still a problem that requires proper processing. We consider that the two relays work in FD mode with an amplify-and-forward (AF) protocol, and the relay self-loop interference may not be cancelled completely from the received signals. Distributed space-time coding for this model is studied, for which the residual part of relay self-loop signal is applied to serve as a self-coding signal. With the self-coding property of the AF FD system, a distributed linear convolutional space time code is formed. Simulation experiments are conducted to evaluate the proposed DLC-STC for the considered FD relay communication network model, and the results illustrate the effectiveness of the presented approach.

A Simple Linear Space Algorithm for Computing Nonoverlapping Inversion and Transposition Distance in Quadratic Average Time.

In the sequence alignment problem, it is important to compare DNA sequences to retrieve relevant information and align these sequences. An inversion and a translocation are important operations in comparing DNA sequences in biosequence analysis. The alignment problem with nonoverlapping inversions and translocations is to find an alignment with nonoverlapping inversions and translocations for the given two strings X and Y. This problem has interesting application for finding a common sequence from two mutated sequences. A linear space and quadratic average time algorithm to compute the mutation distance between two strings of the same length under nonoverlapping inversions and transpositions is presented in this article. The recursive formula for this purpose is novel to the best of our knowledge. The space costs of the algorithms to solve the same problem are typically quadratic, and thus, our original algorithm is the first linear space algorithm to solve the mutation distance problem.

Randomized Embeddings with Slack and High-Dimensional Approximate Nearest Neighbor

Approximate nearest neighbor search (ϵ-ANN) in high dimensions has been mainly addressed by Locality Sensitive Hashing (LSH), which has complexity with polynomial dependence in dimension, sublinear query time, but subquadratic space requirement. We introduce a new “low-quality” embedding for metric spaces requiring that, for some query, there exists an approximate nearest neighbor among the pre-images of its k &gt; 1 approximate nearest neighbors in the target space. In Euclidean spaces, we employ random projections to a dimension inversely proportional to k . Our approach extends to the decision problem with witness of checking whether there exists an approximate near neighbor; this also implies a solution for ϵ-ANN. After dimension reduction, we store points in a uniform grid of side length ϵ /√ d ′ , where d ′ is the reduced dimension. Given a query, we explore cells intersecting the unit ball around the query. This data structure requires linear space and query time in O ( d n ρ ), ρ ≈ 1-ϵ 2 i&gt;/log(1ϵ), where n denotes input cardinality and d space dimension. Bounds are improved for doubling subsets via r -nets. We present our implementation for ϵ-ANN in C++ and experiments for d ≤ 960, n ≤ 10 6 , using synthetic and real datasets, which confirm the theoretical analysis and, typically, yield better practical performance. We compare to FALCONN, the state-of-the-art implementation of multi-probe LSH: our prototype software is essentially comparable in terms of preprocessing, query time, and storage usage.

Linear Space-Time Interference Alignment for $K$ -User MIMO Interference Channels

Interference is believed to be the most significant bottleneck for the next-generation wireless networks to achieve high throughout. Interference alignment (IA), as a novel interference management scheme to break through the traditional interference cancelation, not only makes the complete mitigation of interference possible but also achieves a theoretical breakthrough in promoting the wireless network capacity region. In this paper, by combining the space and time, we proposed a linear space-time (LST) IA algorithm based on the extension of the channel in time dimension for $K$ -user multi-input multi-output interference channel. The proposed LST-IA scheme effectively reduces the number of antennas required for eliminating interference completely in systems, and the closed-form solution of precoding matrices and detector matrices is obtained as well. Compared with the classical IA algorithms, the simulation results demonstrate that the proposed scheme shows distinguished advantages in terms of sum-rate and bit error rate in the strong interference communication scenarios.

Robust blind channel estimation algorithm for linear STBC systems using fourth order cumulant matrices

A novel blind channel estimation algorithm, based on fourth-order cumulant matrices, is proposed and applied to linear Space–Time Block Coded (STBC) for Multiple Input Multiple Output systems. Contrary to subspace and Second-Order Statistics (SOS) methods, the presented approach estimates the channel matrix without any modification of the transmitter. It takes advantage of the statistical independence of the signals in front of the space–time encoding. In this paper, the presented algorithm estimates the channel matrix by minimizing a cost function based on the higher cumulant matrices after Zero-Forcing equalization to mitigate the computational complexity and improve the performance. We employ the proposed method to the STBC systems including Spatial Multiplexing, Orthogonal, quasi-Orthogonal and Non-Orthogonal STBC systems. Symbol error rate and Normalized Mean Square Error simulations of the proposed algorithm are shown for a different number of users, signal to noise ratios and different number of symbols per user in comparison with subspace and Second-Order Statistics (SOS) methods. The results show that the presented method performs well and outperforms other methods in estimating the channel matrix from the received data. Moreover, the proposed method presents high convergence speed in estimating the channel matrix.

Bounded, minimal, and short representations of unit interval and unit circular-arc graphs. Chapter II: algorithms

This is the second and last chapter of a work in which we consider the unrestricted, minimal, and bounded representation problems for unit interval (UIG) and unit circular-arc (UCA) graphs. In the unrestricted version (${\rm R{\small EP}}$), a proper circular-arc (PCA) model $\mathcal M$ is given and the goal is to obtain an equivalent UCA model $\mathcal U$. In the bounded version (${\rm B{\small OUND}R{\small EP}}$), $\mathcal M$ is given together with some lower and upper bounds that the beginning points of $\mathcal U$ must satisfy. In the minimal version (${\rm M{\small IN}UCA}$), the circumference of the circle and the length of the arcs in $\mathcal U$ must be simultaneously as small as possible, while the separation of the extremes is greater than a given threshold. In this chapter we take advantage of the theoretical framework developed in Chapter I to design efficient algorithms for these problems. We show a linear-time algorithm with negative certification for ${\rm R{\small EP}}$, that can also be implemented to run in logspace. We develop algorithms for different versions of ${\rm B{\small OUND}R{\small EP}}$ that run in linear space and quadratic time. Regarding ${\rm M{\small IN}UCA}$, we first show that the previous linear-time algorithm for ${\rm M{\small IN}UIG}$ (i.e., ${\rm M{\small IN}UCA}$ on UIG models) fails to provide a minimal model for some input graphs. We fix this algorithm but, unfortunately, it runs in linear space and quadratic time. Then, we apply the algorithms for ${\rm M{\small IN}UIG}$ and ${\rm M{\small IN}UCA}$ (Chapter I) to find the minimum powers of paths and cycles that contain given UIG and UCA models, respectively.

Time-Optimal Top-$k$ Document Retrieval

Let $\mathcal D$ be a collection of $D$ documents, which are strings over an alphabet of size $\sigma$, of total length $n$. We describe a data structure that uses linear space and reports $k$ most relevant documents that contain a query pattern $P$, which is a string of length $p$ packed in $p/\log_\sigma n$ words, in time $O(p/\log_\sigma n+k)$. This is optimal in the RAM model in the general case where $\log D = \Theta(\log n)$, and involves a novel RAM-optimal suffix tree search. Our construction supports an ample set of important relevance measures, such as the number of times $P$ appears in a document (called term frequency), a fixed document importance, and the minimal distance between two occurrences of $P$ in a document. When $\log D = o(\log n)$, we show how to reduce the space of the data structure from $O(n\log n)$ to $O(n(\log\sigma+\log D+\log\log n))$ bits, and to $O(n(\log\sigma+\log D))$ bits in the case of the popular term frequency measure of relevance, at the price of an additive term $O(\log^\varepsilon_\sigma n)$ in the query time, for any constant $\varepsilon>0$. We also consider the dynamic scenario, where documents can be inserted and deleted from the collection. We obtain linear space and query time $O(p(\log\log n)^2/\log_\sigma n+\log n + k\log\log k)$, whereas insertions and deletions require $O(\log^{1+\varepsilon} n)$ time per symbol, for any constant $\varepsilon>0$. Finally, we consider an extended static scenario where an extra parameter $\mathtt{par}(P,d)$ is defined, and the query must retrieve only documents $d$ such that $\mathtt{par}(P,d)\in [\tau_1,\tau_2]$, where this range is specified at query time. We solve these queries using linear space and $O(p/\log_\sigma n + \log^{1+\varepsilon} n + k\log^\varepsilon n)$ time, for any constant $\varepsilon>0$. Our technique is to translate these top-$k$ problems into multidimensional geometric search problems. As a bonus, we describe some improvements to those problems.