Quadratic Space Research Articles

Height pairings on orthogonal Shimura varieties

Let $M$ be the Shimura variety associated to the group of spinor similitudes of a quadratic space over $\mathbb{Q}$ of signature $(n,2)$. We prove a conjecture of Bruinier and Yang, relating the arithmetic intersection multiplicities of special divisors and complex multiplication points on $M$ to the central derivatives of certain $L$-functions. Each such $L$-function is the Rankin–Selberg convolution associated with a cusp form of half-integral weight $n/2+1$, and the weight $n/2$ theta series of a positive definite quadratic space of rank $n$. When $n=1$ the Shimura variety $M$ is a classical quaternionic Shimura curve, and our result is a variant of the Gross–Zagier theorem on heights of Heegner points.

Sparsity Preserving Algorithms for Octagons

Known algorithms for manipulating octagons do not preserve their sparsity, leading typically to quadratic or cubic time and space complexities even if no relation among variables is known when they are all bounded. In this paper, we present new algorithms, which use and return octagons represented as weakly closed difference bound matrices, preserve the sparsity of their input and have better performance in the case their inputs are sparse. We prove that these algorithms are as precise as the known ones.

NC-link: A New Linkage Method for Efficient Hierarchical Clustering of Large-Scale Data

In various disciplines, hierarchical clustering (HC) has been an effective tool for data analysis due to its ability to summarize hierarchical structures of data in an intuitive and interpretable manner. A run of HC requires multiple iterations, each of which needs to compute and update the pairwise distances between all intermediate clusters. This makes the exact algorithm for HC inevitably suffer from quadratic time and space complexities. To address large-scale data, various approximate/parallel algorithms have been proposed to reduce the computational cost of HC. However, such algorithms still rely on conventional linkage methods (such as single, centroid, average, complete, or Ward's) for defining pairwise distances, mostly focusing on the approximation/parallelization of linkage computations. Given that the choice of linkage profoundly affects not only the quality but also the efficiency of HC, we propose a new linkage method named NC-link and design an exact algorithm for NC-link-based HC. To guarantee the exactness, the proposed algorithm maintains the quadratic nature in time complexity but exhibits only linear space complexity, thereby allowing us to address million-object data on a personal computer. To underpin the extensibility of our approach, we showthat the algorithmic nature of NC-link enables single instruction multiple data (SIMD) parallelization and subquadratic-time approximation of HC. To verify our proposal, we thoroughly tested it with a number of large-scale real and synthetic data sets. In terms of efficiency, NC-link allowed us to perform HC substantially more space efficiently or faster than conventional methods: compared with average and complete linkages, using NC-link incurred only 0.7%-1.75% of the memory usage, and the NC-link-based implementation delivered speedups of approximately 3.5 times over the centroid and Ward's linkages. With regard to clustering quality, the proposed method was able to retrieve hierarchical structures from input data as faithfully as in the popular average and centroid linkage methods. We anticipate that the existing approximation/parallel algorithms will be able to benefit from adopting NC-link as their linkage method for obtaining better clustering results and reduced time and space demands.

A critical case of Rallis inner product formula

Let $\pi$ be a genuine cuspidal representation of the metaplectic group of rank $n$. We consider the theta lifts to the orthogonal group associated to a quadratic space of dimension $2n+1$. We show a case of regularised Rallis inner product formula that relates the pairing of theta lifts to the central value of the Langlands $L$-function of $\pi$ twisted by a character. The bulk of this paper focuses on proving a case of regularised Siegel-Weil formula, on which the Rallis inner product formula is based and whose proof is missing in the literature.

Phenomenological applications of rational approximants

We illustrate the powerfulness of Padé approximants (PAs) as a summation method and explore one of their extensions, the so-called quadratic approximant (QAs), to access both space- and (low-energy) time-like (TL) regions. As an introductory and pedagogical exercise, the function [Formula: see text] is approximated by both kind of approximants. Then, PAs are applied to predict pseudoscalar meson Dalitz decays and to extract [Formula: see text] from the semileptonic [Formula: see text] decays. Finally, the [Formula: see text] vector form factor in the TL region is explored using QAs.

Functional error estimators for the adaptive discretization of inverse problems

So-called functional error estimators provide a valuable tool for reliably estimating the discretization error for a sum of two convex functions. We apply this concept to Tikhonov regularization for the solution of inverse problems for partial differential equations, not only for quadratic Hilbert space regularization terms but also for nonsmooth Banach space penalties. Examples include the measure-space norm (i.e., sparsity regularization) or the indicator function of an ball (i.e., Ivanov regularization). The error estimators can be written in terms of residuals in the optimality system that can then be estimated by conventional techniques, thus leading to explicit estimators. This is illustrated by means of an elliptic inverse source problem with the above-mentioned penalties, and numerical results are provided for the case of sparsity regularization.

Solution sets for equations over free groups are EDT0L languages

We show that, given an equation over a finitely generated free group, the set of all solutions in reduced words forms an effectively constructible EDT0L language. In particular, the set of all solutions in reduced words is an indexed language in the sense of Aho. The language characterization we give, as well as further questions about the existence or finiteness of solutions, follow from our explicit construction of a finite directed graph which encodes all the solutions. Our result incorporates the recently invented recompression technique of Jeż, and a new way to integrate solutions of linear Diophantine equations into the process. As a byproduct of our techniques, we improve the complexity from quadratic nondeterministic space in previous works to [Formula: see text] here.

Regularized theta lifts and (1,1)-currents on GSpin Shimura varieties

We introduce a regularized theta lift for reductive dual pairs of the form $(Sp_4,O(V))$ with $V$ a quadratic vector space over a totally real number field $F$. The lift takes values in the space of $(1,1)$-currents on the Shimura variety attached to $GSpin(V)$, and we prove that its values are cohomologous to currents given by integration on special divisors against automorphic Green functions. In the second part to this paper, we will show how to evaluate the regularized theta lift on differential forms obtained as usual (non-regularized) theta lifts.

A generalization of Watson transformation and representations of ternary quadratic forms

Let L be a positive definite (non-classic) ternary Z-lattice and let p be a prime such that a 12Zp-modular component of Lp is nonzero isotropic and 4⋅dL is not divisible by p. For a nonnegative integer m, let GL,p(m) be the genus with discriminant pm⋅dL on the quadratic space Lpm⊗Q such that for each lattice T∈GL,p(m), a 12Zp-modular component of Tp is nonzero isotropic, and Tq is isometric to (Lpm)q for any prime q different from p. Let r(n,M) be the number of representations of an integer n by a Z-lattice M. In this article, we show that if m⩽2 and n is divisible by p only when m=2, then for any T∈GL,p(m), r(n,T) can be written as a linear summation of r(pn,Si) and r(p3n,Si) for Si∈GL,p(m+1) with an extra term in some special case. We provide a simple criterion on when the extra term is necessary, and we compute the extra term explicitly. We also give a recursive relation to compute r(n,T), for any T∈GL,p(m), by using the number of representations of some integers by lattices in GL,p(m+1) for an arbitrary integer m.

A Posteriori Error Analysis of Two-Step Backward Differentiation Formula Finite Element Approximation for Parabolic Interface Problems

This paper studies a residual-based a posteriori error estimates for linear parabolic interface problems in a bounded convex polygonal domain in $$\mathbb {R}^2$$R2. We use the standard linear finite element spaces in space which are allowed to change in time and the two-step backward differentiation formula (BDF-2) approximation at equidistant time step is used for the time discretizations. The essential ingredients in the error analysis are the continuous piecewise quadratic space---time BDF-2 reconstruction and Scott---Zhang interpolation estimates. Optimal order in time and an almost optimal order in space error estimates are derived in the $$L^{\infty }(L^{2})$$Lź(L2)-norm using only energy method. The interfaces are assumed to be of arbitrary shape but are smooth for our purpose. Numerical experiments are performed to validate the asymptotic behaviour of the derived error estimators.

On some branches of the Bruhat–Tits tree

We explicitly compute the largest subtree, in the local Bruhat–Tits tree for [Formula: see text], whose vertices correspond to maximal orders containing a fixed order generated by a pair of orthogonal pure quaternions. In other words, we compute the set of maximal integral valued lattices in a ternary quadratic space, whose discriminant is a unit, containing a pair of orthogonal vectors, extending thus previous computations by Schulze-Pilot. The maximal order setting makes these computations simpler. The method presented here can be applied to arbitrary sub-orders or sublattices. The shape of this subtree is described, when it is finite, by a set of two invariants. In a previous work, the first author showed that determining the shape of these local subtrees allows the computation of representation fields, a class field determining the set of isomorphism classes, in a genus of Eichler orders, containing an isomorphic copy of a given order. Some further applications are described here.

Stochastic Optimization for Kernel PCA

Kernel Principal Component Analysis (PCA) is a popular extension of PCA which is able to find nonlinear patterns from data. However, the application of kernel PCA to large-scale problems remains a big challenge, due to its quadratic space complexity and cubic time complexity in the number of examples. To address this limitation, we utilize techniques from stochastic optimization to solve kernel PCA with linear space and time complexities per iteration. Specifically, we formulate it as a stochastic composite optimization problem, where a nuclear norm regularizer is introduced to promote low-rankness, and then develop a simple algorithm based on stochastic proximal gradient descent. During the optimization process, the proposed algorithm always maintains a low-rank factorization of iterates that can be conveniently held in memory. Compared to previous iterative approaches, a remarkable property of our algorithm is that it is equipped with an explicit rate of convergence. Theoretical analysis shows that the solution of our algorithm converges to the optimal one at an O(1/T) rate, where T is the number of iterations.

On the one-mode quadratic Weyl operators

It has been known, in [7], that one-mode quadratic Weyl operators are well-defined as unitary operators acting on the quadratic Fock space. These operators were defined by their action on the finite particles space. However, their action on the domain of the quadratic exponential vectors is still unknown. In this paper, we provide a new reformulation (w.r.t. [1]) of the quadratic exponential vectors and we compute the action of the one-mode quadratic Weyl operators on the set of these exponential vectors. Then, we prove the independence of the one-mode quadratic Weyl operators parameterized by the principal domain associated with the quadratic Heisenberg group obtained in [7]. This significant contribution to the program of developing the quadratic white noise calculus constitutes a step toward the C⁎-representation of the renormalized square of white noise algebra.

Optimal partitioning of data chunks in deduplication systems

Deduplication is a special case of data compression in which repeated chunks of data are stored only once. For very large chunks, this process may be applied even if the chunks are similar and not necessarily identical, and then the encoding of duplicate data consists of a sequence of pointers to matching parts. However, not all the pointers are worth being kept, as they incur some storage overhead. A linear, sub-optimal solution of this partition problem is presented, followed by an optimal solution with cubic time complexity and requiring quadratic space.

Dynamic Monitoring and Control of Irreversible Chronic Diseases with Application to Glaucoma

To effectively manage chronic disease patients, clinicians must know (1) how to monitor each patient (i.e., when to schedule the next visit and which tests to take), and (2) how to control the disease (i.e., what levels of controllable risk factors will sufficiently slow progression). Our research addresses these questions simultaneously and provides the optimal solution to a novel linear quadratic Gaussian state space model. For the new objective of minimizing the relative change in state over time (i.e., disease progression), which is necessary for management of irreversible chronic diseases, we show that the classical two-way separation of estimation and control holds, thereby making a previously intractable problem solvable by decomposition into two separate, tractable problems while maintaining optimality. The resulting optimization is applied to the management of glaucoma. Based on data from two large randomized clinical trials, we validate our model and demonstrate how our decision support tool can provide actionable insights to the clinician caring for a patient with glaucoma. This methodology can be applied to a broad range of irreversible chronic diseases to optimally devise patient-specific monitoring and treatment plans.

Total Space in Resolution

We show quadratic lower bounds on the total space used in resolution refutations of random $k$-CNFs over $n$ variables and of the graph pigeonhole principle and the bit pigeonhole principle for $n$ holes. This answers the open problem of whether there are families of $k$-CNF formulas of polynomial size that require quadratic total space in resolution. The results follow from a more general theorem showing that, for formulas satisfying certain conditions, in every resolution refutation there is a memory configuration containing many clauses of large width.

Efficiency of lambda-encodings in total type theory

AbstractThis paper proposes a new typed lambda-encoding for inductive types which, for Peano numerals, has the expected time complexities for basic operations like addition and multiplication, has a constant-time predecessor function, and requires only quadratic space to encode a numeral. This improves on the exponential space required by the Parigot encoding. Like the Parigot encoding, the new encoding is typable in System F-omega plus positive-recursive type definitions, a total type theory. The new encoding is compared with previous ones through a significant case study: mergesort using Braun trees. The practical runtime efficiency of the new encoding, and the Church and Parigot encodings, are compared by two translations, one to Racket and one to Haskell, on a small suite of benchmarks.

An energy‐aware performance analysis of SWIMM: Smith–Waterman implementation on Intel's Multicore and Manycore architectures

SummaryAlignment is essential in many areas such as biological, chemical and criminal forensics. The well‐known Smith–Waterman (SW) algorithm is able to retrieve the optimal local alignment with quadratic time and space complexity. There are several implementations that take advantage of computing parallelization, such as manycores, FPGAs or GPUs, in order to reduce the alignment effort. In this research, we adapt, develop and tune the SW algorithm named SWIMM on a heterogeneous platform based on Intel's Xeon and Xeon Phi coprocessor. SWIMM is a free tool available in a public git repository https://github.com/enzorucci/SWIMM. We efficiently exploit data and thread‐level parallelism, reaching up to 380 GCUPS on heterogeneous architecture, 350 GCUPS for the isolated Xeon and 50 GCUPS on Xeon Phi. Despite the heterogeneous implementation obtaining the best performance, it is also the most energy‐demanding. In fact, we also present a trade‐off analysis between performance and power consumption. The greenest configuration is based on an isolated multicore system that exploits AVX2 instruction set architecture reaching 1.5 GCUPS/Watts. Copyright © 2015 John Wiley & Sons, Ltd.

Formula omitted] cubic splines on Powell–Sabin triangulations

A bivariate C1 cubic spline space on a triangulation with Powell–Sabin refinement which extends the well-known C1 quadratic spline space and has a nested structure is introduced. A construction of a locally supported basis that forms a partition of unity is presented based on choosing particular triangles and line segments in the domain. Further, it is shown how these objects can be determined in order to obtain nonnegative basis functions under a natural restriction on the Powell–Sabin refinement. Geometrically intuitive B-spline representation is proposed which makes these splines a useful tool for CAGD applications.

Dynamic Personalized Monitoring and Treatment Control of Glaucoma

To e↵ectively manage chronic disease patients, clinicians must know (1) how to monitor each patient (i.e., when to schedule the next visit and which tests to take), and (2) how to control the disease (i.e., what levels of controllable risk factors will su ciently slow progression). Our research addresses these questions simultaneously and optimally by employing linear quadratic Gaussian state space systems modeling and optimal control of measurement adaptive systems. For the new quadratic objective of minimizing disease progression over time, we show that the classical two-way separation of estimation and control holds, thereby making a previously intractable problem solvable by decomposition into two separate, tractable problems while maintaining optimality. The resulting optimization is applied to the management of glaucoma. Based on data from two large randomized clinical trials, we found that fast-progressing patients, who are most at risk for disease progression, show an average of 38% to 58% less loss of peripheral vision per year, leading to 21% to 32% better quality of vision after 10 years when compared to existing treatment controls attained in the clinical trials. This methodology can be applied to a broad range of chronic diseases to optimally devise patient-specific monitoring and treatment plans.