pinecone/core-2020-05-10-deduplication huggingface dataset:
An Iterated Eigenvalue Algorithm for Approximating Roots of Univariate Polynomials

AbstractWe discuss an iterative algorithm that approximates all roots of a univariate polynomial. The iteration is based on floating-point computation of the eigenvalues of a generalized companion matrix. With some assumptions, we show that the algorithm approximates the roots within about logρ/ϵχ(P) iterations, where ϵ is the relative error of floating-point arithmetic, ρ is the relative separation of the roots, and χ(P) is the condition number of the polynomial. Each iteration requires an n×n floating-point eigenvalue computation, n the polynomial degree, and evaluation of the polynomial to floating-point accuracy at up to n points.We describe a careful implementation of the algorithm, including many techniques that contribute to the practical efficiency of the algorithm. On some hard examples of ill-conditioned polynomials, e.g. high-degree Wilkinson polynomials, the implementation is an order of magnitude faster than the Bini–Fiorentino implementation mpsolve
Relations Between Roots and Coefficients, Interpolation and Application to System Solving

AbstractWe propose an algebraic framework to represent zero-dimensional algebraic systems. In this framework, we give new interpolation formulæ. We use this good representation of the algebraic systems to develop a generalization of Weierstrass’s method to the multivariate systems. This method allows us to approximate simultaneously all the roots of an algebraic system. We obtain an effective iteration function with a quadratic convergence in a neighbourhood of the solutions. We use this Weierstrass iteration function in a continuation method to obtain a global method. Experiments are exposed to underline the efficiency of the approach
Univariate Polynomials: Nearly Optimal Algorithms for Numerical Factorization and Root-finding

AbstractTo approximate all roots (zeros) of a univariate polynomial, we develop two effective algorithms and combine them in a single recursive process. One algorithm computes a basic well isolated zero-free annulus on the complex plane, whereas another algorithm numerically splits the input polynomial of the n th degree into two factors balanced in the degrees and with the zero sets separated by the basic annulus. Recursive combination of the two algorithms leads to computation of the complete numerical factorization of a polynomial into the product of linear factors and further to the approximation of the roots. The new root-finder incorporates the earlier techniques of Schönhage, Neff/Reif, and Kirrinnis and our old and new techniques and yields nearly optimal (up to polylogarithmic factors) arithmetic and Boolean cost estimates for the computational complexity of both complete factorization and root-finding. The improvement over our previous record Boolean complexity estimates is by roughly the factor of n for complete factorization and also for the approximation of well-conditioned (well isolated) roots, whereas the same algorithm is also optimal (under both arithmetic and Boolean models of computing) for the worst case input polynomial, whose roots can be ill-conditioned, forming clusters. (The worst case complexity bounds for root-finding are supported by our previous algorithms as well.) All algorithms allow processor efficient acceleration to achieve solution in polylogarithmic parallel time
A Probabilistic Algorithm to Test Local Algebraic Observability in Polynomial Time

AbstractThe following questions are often encountered in system and control theory. Given an algebraic model of a physical process, which variables can be, in theory, deduced from the input–output behaviour of an experiment? How many of the remaining variables should we assume to be known in order to determine all the others? These questions are parts of thelocal algebraic observability problem which is concerned with the existence of a non-trivial Lie subalgebra of model’s symmetries letting the inputs and the outputs be invariant.We present a probabilistic seminumerical algorithm that proposes a solution to this problem in polynomial time. A bound for the necessary number of arithmetic operations on the rational field is presented. This bound is polynomial in the complexity of evaluation of the model and in the number of variables. Furthermore, we show that the size of the integers involved in the computations is polynomial in the number of variables and in the degree of the system. Last, we estimate the probability of success of our algorithm
Cancellative Abelian Monoids and Related Structures in Refutational Theorem Proving (Part I)

AbstractWe present superposition calculi in which the axioms of cancellative Abelian monoids and, optionally, the torsion-freeness axiom are integrated. Cancellative Abelian monoids comprise Abelian groups, but also such ubiquitous structures as the natural numbers or multisets. Our calculi require neither extended clauses nor explicit inferences with the theory axioms. Compared with AC-superposition calculi, the number of variable overlaps is significantly reduced by strong ordering restrictions
Cancellative Abelian Monoids and Related Structures in Refutational Theorem Proving (Part II)

AbstractCancellative superposition is a refutationally complete calculus for first-order equational theorem proving in the presence of the axioms of cancellative Abelian monoids, and, optionally, the torsion-freeness axioms. Thanks to strengthened ordering restrictions, cancellative superposition avoids some of the inefficiencies of classical AC-superposition calculi. We show how the efficiency of cancellative superposition can be further improved by using variable elimination techniques, leading to a significant reduction of the number of variable overlaps. In particular, we demonstrate that in divisible torsion-free Abelian groups, variable overlaps, AC-unification and AC-orderings can be avoided completely
Normal Parametrizations of Algebraic Plane Curves

AbstractIn this paper we present a theoretical and algorithmic analysis on the normality of rational parametrizations of algebraic plane curves over arbitrary fields of characteristic zero. If the field is algebraically closed we give an algorithm to decide whether a parametrization is proper and, if not, a normal parametrization is computed. If the field is not algebraically closed the problem is more complicated, and a degenerated situation may appear. We classify the degenerations in strong and weak degenerations, and an algorithm to decide this phenomenon is derived. Furthermore, we prove that if the parametrization is strongly degenerated then the curve cannot be normally parametrized, but weak degenerations can be resolved, and an algorithm to reparametrize the input weakly degenerated parametrization into a non-degenerated one is given. In addition, we show how these results can be applied and improved to the case of real rational curves. In this case, we present an algorithm that decides whether a given real parametrization can be normally parametrized over the reals, and that computes such a parametrization if it exists
Calculating the Galois Group of Y′===AY+++B,Y′===AY Completely Reducible

AbstractWe consider a special case of the problem of computing the Galois group of a system of linear ordinary differential equations Y′=MY, M∈C (x)n×n. We assume that C is a computable, characteristic-zero, algebraically closed constant field with a factorization algorithm. There exists a decision procedure, due to Compoint and Singer, to compute the group in case the system is completely reducible. Berman and Singer (1999, J. Pure Appl. Algebr., 139, 3–23) address the case in which M= [yjsco5390x.gif M 1 * 0 M 2 ], Y′=MiY completely reducible for i= 1, 2. Their article shows how to reduce that case to the case of an inhomogeneous system Y′=AY+B, A∈C (x)n×n, B∈C (x)n, Y′=AY completely reducible. Their article further presents a decision procedure to reduce this inhomogeneous case to the case of the associated homogeneous system Y′=AY. The latter reduction involves using a cyclic-vector algorithm to find an equivalent inhomogeneous scalar equation L(y) =b,L∈C(x)[ D ], b∈C (x), then computing a certain set of factorizations of L in C(x)[D ]; this set is very large and difficult to compute in general. In this article, we give a new and more efficient algorithm to reduce the case of a system Y′=AY+B,Y′=AY completely reducible, to that of the associated homogeneous systemY′=AY. The new method’s improved efficiency comes from replacing the large set of factorizations required by the Berman–Singer method with a single block-diagonal decomposition of the coefficient matrix satisfying certain properties
Modular Termination Proofs for Rewriting Using Dependency Pairs

AbstractRecently, Arts and Giesl developed the dependency pair approach which allows automated termination and innermost termination proofs for many term rewriting systems (TRSs) for which such proofs were not possible before. The motivation for this approach was that virtually all previous techniques for automated termination proofs of TRSs were based on simplification orderings. In practice, however, many rewrite systems are not simply terminating, i.e. their termination cannot be verified by any simplification ordering. In this paper we introduce a refinement of the dependency pair framework which further extends the class of TRSs for which termination or innermost termination can be shown automatically. By means of this refinement, one can now prove termination in a modular way. Thus, this refinement is inevitable in order to verify the termination of large rewrite systems occurring in practice. To be more precise, one may use several different orderings in one termination proof. Subsequently, we present several new modularity results based on dependency pairs. First, we show that the well-known modularity of simple termination for disjoint unions can be extended to DP quasi-simple termination, i.e. to the class of rewrite systems where termination can be shown automatically by the dependency pair technique in combination with quasi-simplification orderings. Under certain additional conditions, this new result also holds for constructor-sharing and composable systems. Second, the above-mentioned refinement of the dependency pair method yields new modularity criteria for innermost termination which extend previous results in this area considerably. In particular, existing results for modularity of innermost termination can easily be shown to be direct consequences of our new criteria
Barnett’s Theorems About the Greatest Common Divisor of Several Univariate Polynomials Through Bezout-like Matrices

AbstractThis article provides a new presentation of Barnett’s theorems giving the degree (resp. coefficients) of the greatest common divisor of several univariate polynomials with coefficients in an integral domain by means of the rank (resp. linear dependencies of the columns) of several Bezout-like matrices. This new presentation uses Bezout or hybrid Bezout matrices instead of polynomials evaluated in a companion matrix as in the original Barnett’s presentation. Moreover, this presentation also allows us to compute the coefficients of the considered greatest common divisor in an easier way than in the original Barnett’s theorems
Quartic Fields and Radical Extensions

AbstractLet K be a field and K(α) be an extension field ofK . If [ K(α) : K ] = 3, charK≠= 3, and the minimal polynomial of α over K isT3−uT−v∈K [T ], it is proved in Kang (2000, Am. Math. Monthly, 107, 254–256) thatK (α) is a radical extension of K if and only if, for somew∈K, 81 v2− 12u3=w2if char K≠= 2, or u3/v2=w2+w if char K= 2. In this paper, we prove a similar result when [ K(α) : K ] = 4, charK≠= 2, and the minimal polynomial of α over K isT4−uT2−vT−w∈K [ T ] with v≠= 0 :K (α) is a radical extension of K if and only if the following system of polynomial equations is solvable in K, 64 X3− 32uX2+ (4 u2+ 16w )X−v2= 0 and 64wX2− (32 uw− 3v2 )X+ (4 u2w+ 16w2−uv2) −Y2= 0. The situation when v= 0 will also be solved
Computing Rational Forms of Integer Matrices

AbstractA new algorithm is presented for finding the Frobenius rational form F∈Zn×nof any A∈Zn×nwhich requires an expected number of O(n4(logn+log|| A ||) +n3(logn+log|| A ||)2) word operations using standard integer and matrix arithmetic (where || A || =maxij| Aij|). This substantially improves on the fastest previously known algorithms. The algorithm is probabilistic of the Las Vegas type: it assumes a source of random bits but always produces the correct answer. Las Vegas algorithms are also presented for computing a transformation matrix to the Frobenius form, and for computing the rational Jordan form of an integer matrix
Algorithmic Computation of Flattenings and of Modular Deformations

AbstractThis paper presents an algorithm for the computation of any jet of the flattening stratum of a module over a local algebra based on an obstruction theory for lifting flatness. It is applied to modular deformations of singular germs. Infinitesimal modular deformations of isolated complete intersection singularities are characterized as flattening strata of its first tangent cohomology. Some examples are discussed that indicate relations with moduli spaces of certain classes of singularities. Implementations are done in Singular
Computing Modular Invariants of p-groups

AbstractLet V be a finite dimensional representation of a p -group, G, over a field,k , of characteristic p. We show that there exists a choice of basis and monomial order for which the ring of invariants, k [ V ]G, has a finite SAGBI basis. We describe two algorithms for constructing a generating set for k [ V ] G. We use these methods to analyse k [2V3 ]U3where U3is the p -Sylow subgroup ofGL3 (Fp) and 2 V3is the sum of two copies of the canonical representation. We give a generating set for k [2 V3]U3forp= 3 and prove that the invariants fail to be Cohen–Macaulay forp> 2. We also give a minimal generating set for k [mV2 ]Z/pwere V2is the two-dimensional indecomposable representation of the cyclic group Z/p
Valuations and Filtrations

AbstractThe classical theory of Gröbner bases, as developed by Bruno Buchberger, can be expanded to utilize objects more general than term orders. Each term order on the polynomial ring k [x] produces a filtration of k [x] and a valuation ring of the rational function fieldk (x). The algorithms developed by Buchberger can be performed by using directly the induced valuation or filtration in place of the term order. There are many valuations and filtrations that are suitable for this general computational framework that are not derived from term orders, even after a change of variables. Here we study how to translate between properties of filtrations and properties in valuation theory, and give a characterization of which valuations and filtrations are derived from a term order after a change of variables. This characterization illuminates the properties of valuations and filtrations that are desirable for use in a generalized Gröbner basis theory
Thomas’ Family of Thue Equations Over Imaginary Quadratic Fields

AbstractWe consider the family of relative Thue equations x3− (t− 1)x2y− (t+ 2)xy2−y3= μ,where the parameter t, the root of unityμ and the solutions x and y are integers in the same imaginary quadratic number field.We prove that there are only trivial solutions (with x, y≤ 1), if t is large enough or if the discriminant of the quadratic number field is large enough or ifRet=−1/2 (there are a few more solutions in this case which are explicitly listed). In the caseRet=−1/2, an algebraic method is used, in the general case, Baker’s method yields the result
Use of donor serum to prevent passive transfer of hyperacute rejection

Organ transplantation in presensitized recipients continues to be contraindicated for heart and kidney recipients due to the risk of hyperacute rejection, which has no known treatment at this time. We tested whether donor serum, which contains soluble MHC class I antigen, is able to neutralize the effect of anti-donor antibody in the recipient and prevent hyperacute or accelerated rejection. A rat model of passive immunization was used to test the role of anti-donor antibody in hyperacute rejection. Seven of 10 recipients of hyperimmune serum (HyS), derived from Lewis rats (RT1l) following 3 ACI (RT1a) skin grafts, developed hyperacute or accelerated rejection. Intravenous injection of ACI serum prior to the HyS administration prevented hyperacute rejection in all recipients tested. When third-party (Wistar-Furth, RT1u) serum was given to Lewis rats injected with HyS, hyperacute rejection was not abrogated. When examining the mechanism of this effect, a simple antibody blocking phenomenon was found to be unlikely since flow cytometry analysis showed that ACI serum needed to be present at > or = 256-fold excess compared to HyS to block anti-ACI antibody binding to RT1.Aa+cells by 50%. We tested whether the RT1.Aa class I antigen in ACI serum had other biologic properties that resulted in the prolonged graft survival. However, removal of RT1.Aa antigen from ACI serum prior to use in the passive transfer model did not abrogate the graft prolongation observed previously. These data suggest that components of donor serum other than MHC class I antigen may be useful for preventing the antibody-mediated component of hyperacute rejection
A high incidence of native portal vein thrombosis in veterans undergoing liver transplantation

The incidence of native portal vein thrombosis (PVT) in liver transplant recipients has been reported to range from 2.1 to 13.8%. We have identified an inordinately high incidence of PVT in a consecutive series of U.S. veterans receiving liver transplants. Between October 1989 and February 1994, 88 consecutive U.S. veterans received 99 orthotopic liver transplants under primary Tacrolimus (Prograf, formerly FK506) based immunosuppression. A number of clinical features were examined in an effort to identify risk factors for PVT and outcome was compared to patients without PVT. Native PVT was present in 23/88 (26%) patients. All of these patients were male U.S. veterans with a mean age of 47 years. When compared to the 65 patients without PVT, we found no significant difference with respect to underlying liver disease, age, Childs-Pugh score (mean = 12), UNOS status as defined prior to April 1995 (95% UNOS 3 or 4), previous abdominal surgery, or liver volume. Median blood loss for patients with PVT (21 units of packed red blood cells) was greater than for those without PVT (14 units, P = 0.04). Portal thrombectomy was performed in 11 patients, 11 patients required mesoportal jump grafts, and 1 patient had an interposition graft. Standard veno-venous bypass was used in 10 patients with single bypass utilized for the remainder. Actuarial patient survival for all patients at 1, 2, and 4 years was 88, 85, and 79%, respectively. There was no significant difference in patients with or without PVT. Patients with PVT had poorer graft survival than patients without PVT (86% vs 65%, 1 year; 81% vs 65%, 2 years; 81% vs 61%, 4 years; P = 0.03); however, this was not related to technical problems with the portal venous inflow. PVT occurred in 26% of U.S. veterans undergoing liver transplantation. These patients bad significantly higher operative blood loss and poorer graft survival. The high incidence of postnecrotic cirrhosis in a predominantly male group of patients with advanced disease, as is evident by the high mean Childs-Pugh score and UNOS status, perhaps accounts for our observations
Effect of ischemia on the canine large bowel: A comparison with the small intestine

Mucosal injury caused by ischemia and reperfusion has been well documented with the small intestine, but little is known about the colon. In the present study, the effect of warm and cold ischemia on the canine colon was studied and compared to that on the small intestine. After in situ flushing, the small intestine and the colon from six beagle dogs were removed and stored for 0.5, 1.5, and 3 hr at 37°C (warm ischemia) or for 1, 6, 12, 24, 36, and 48 hr at 4°C (cold ischemia). Electrophysiology, permeability, biochemistry, and histopathology of the specimens at each ischemic period and after reperfusion in the Ussing chamber were determined. Warm and cold ischemia induced duration-dependent suppression of electrophysiology in both organs, but the colonic mucosa retained higher activity of absorptive enterocytes and cryptic cells than the small intestine. Only the colon showed increased permeability of FITC-conjugated Dextran from the mucosal surface to the submucosal layer after prolonged ischemia. Changes in adenine nucleotides and purine catabolites were not markedly different between the organs. Histopathologic abnormalities during ischemia and after reperfusion were more serious with the small intestine than with the colon. Compared to warm ischemia, hypothermia lessened or delayed these morphofunctional derangements in both organs, which became universally worsened after reperfusion. Colonic mucosa receives morphofunctional derangements from ischemia and reperfusion, but the severity of the damage was much less severe in the colon than in the small intestine
Intestinal neuromuscular function after preservation and transplantation

While it is well known that prolonged preservation of the intestinal graft causes severe mucosal damage after transplantation, little is known about the effect on neuromuscular function. The entire small intestine of adult hound dogs was flushed and preserved with cold lactated Ringer's solution and autotransplanted either immediately (n = 6) or after 24 hr (n = 6). Animals undergoing sham operation (n = 4) were used as a control. Fasting motility and the response of the intestinal smooth muscle and enteric nerves to bethanechol (100 μg/kg/0.5 hr, iv) and cisapride (0.5 mg/kg, iv) were determined by a multiple strain gauge method on Postoperative Days 2, 4, 7, 14, 21, and 28. Compared to the control, immediately transplanted grafts and those preserved for 24 hr developed delayed reappearance of migrating myoelectric complexes (MMC), hypercontractile activity, and reduced response to bethanechol and cisapride administration. Animals in the preservation group developed more abnormal fasting motility after transplantation, but responses to bethanechol and cisapride stimulation were not markedly different from those of the immediate group. The reappearance of MMC occurred 3 weeks postoperatively in the preservation group compared to 2 days in the immediate group. The results of our study indicate that intestinal dysmotility is augmented in prolonged-preservation grafts compared to those with brief preservation. The dysmotility was transient and normalized 3 to 4 weeks after surgery. Preservation and reperfusion injury to the neuromuscular system of intestinal grafts are reversible and are attenuated by simple hypothermia
On the Existence of Hydrotalcite-Like Phases in the Absence of Trivalent Cations

α-Hydroxides of nickel(II) and cobalt(II) are hydrotalcite-like phases, possessing a layered double hydroxide (LDH) structure even though there are no trivalent cations in the lattice. While the LDHs acquire a positive charge on the hydroxide layers by the incorporation of trivalent cations, we suggest that the α-hydroxides acquire a positive charge by partial protonation of the hydroxyl ions according to the equationM(OH)2+xH+→[M(OH)2-x(H 2O)x]x+.As in the LDHs, charge balance is restored by the incorporation of anions in the interlayer region. © 1997 Academic Press