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Verified Quadratic Virtual Substitution for Real Arithmetic [article]

Matias Scharager, Katherine Cordwell, Stefan Mitsch, André Platzer
2021 arXiv   pre-print
This paper presents a formally verified quantifier elimination (QE) algorithm for first-order real arithmetic by linear and quadratic virtual substitution (VS) in Isabelle/HOL.  ...  To our knowledge, this is the first work to formalize VS for quadratic real arithmetic including inequalities.  ...  Thank you also to the anonymous FM reviewers for their useful feedback. This material is based upon work supported by the National Science Foundation under Grant No.  ... 
arXiv:2105.14183v1 fatcat:e2oldaaqozdl7luolm62svg7he

Virtual Substitution for SMT-Solving [chapter]

Florian Corzilius, Erika Ábrahám
2011 Lecture Notes in Computer Science  
In this paper we present an adaptation of the virtual substitution method, providing these abilities.  ...  Currently available decision procedures for real algebra -the first-order theory of the reals with addition and multiplication -do not exhibit these features.  ...  The virtual substitution is used for linear arithmetic in [Bjo10] .  ... 
doi:10.1007/978-3-642-22953-4_31 fatcat:66peapiggrfwbesvb53uekibl4

New Opportunities for the Formal Proof of Computational Real Geometry? [article]

Erika Ábrahám, James Davenport, Matthew England, Gereon Kremer, Zak Tonks
2020 arXiv   pre-print
The purpose of this paper is to explore the question "to what extent could we produce formal, machine-verifiable, proofs in real algebraic geometry?"  ...  Non-Linear Real Arithmetic Constraints with a Conflict Driver Search Using Cylindrical Algebraic Coverings, 2020] might provide trace and outputs that allow the results to be more susceptible to machine  ...  VTS: Virtual Term Substitution would consider a variety of possible values v i for y.  ... 
arXiv:2004.04034v1 fatcat:gmyqt67c2ncypffethxo4ahvga

New Opportunities for the Formal Proof of Computational Real Geometry? (Extended Abstract)

Erika Ábrahám, James Davenport, Matthew England, Gereon Kremer, Zak Tonks
2020 CEUR Workshop Proceedings  
of Non-Linear Real Arithmetic Constraints with a Conflict Driver Search Using Cylindrical Algebraic Coverings, 2020] might provide a trace and outputs that allow the results to be more susceptible to machine  ...  We present the thesis that a new algorithm for ascertaining satisfiability of formulae over the reals via Cylindrical Algebraic Coverings [Ábrahám, Davenport, England, Kremer, Deciding the Consistency  ...  The verification proceeds largely as for CAD, except that step (e) is less trivial in general. VTS: Virtual Term Substitution would consider a variety of possible values 𝑣 𝑖 for 𝑦.  ... 
doi:10.18154/rwth-2021-09565 fatcat:pn4r7ur7zzehjek6wfi5zes52y

New Opportunities for the Formal Proof of Computational Real Geometry? (Extended Abstract)

Erika Ábrahám, James H. Davenport, Matthew England, Gereon Kremer, Zak Tonks
2020 Conference on Automated Deduction  
of Non-Linear Real Arithmetic Constraints with a Conflict Driver Search Using Cylindrical Algebraic Coverings, 2020] might provide a trace and outputs that allow the results to be more susceptible to machine  ...  We present the thesis that a new algorithm for ascertaining satisfiability of formulae over the reals via Cylindrical Algebraic Coverings [Ábrahám, Davenport, England, Kremer, Deciding the Consistency  ...  The verification proceeds largely as for CAD, except that step (e) is less trivial in general. VTS: Virtual Term Substitution would consider a variety of possible values 𝑣 𝑖 for 𝑦.  ... 
dblp:conf/cade/AbrahamDEKT20 fatcat:rii5ajbog5cnph7eelahlx44ou

Hermite Interpolation by Pythagorean Hodograph Quintics

R. T. Farouki, C. A. Neff
1995 Mathematics of Computation  
This algorithm establishes the PH quintics as a viable medium for the design or approximation of free-form curves, and allows a one-for-one substitution of PH quintics in lieu of the widely-used "ordinary  ...  We show that formulating PH quintics as complex-valued functions of a real parameter leads to a compact Hermite interpolation algorithm and facilitates an identification of the "good" interpolant (in terms  ...  In the complex formulation, the curvature can be expressed as k(í) = |r'(0r3Im(r'(0r"(0), and by substituting from (29) it may be verified that the numerator of this expression is proportional to the quadratic  ... 
doi:10.2307/2153373 fatcat:p45qeoad2rawfhktxr7nej3q5m

Hermite interpolation by Pythagorean hodograph quintics

R. T. Farouki, C. A. Neff
1995 Mathematics of Computation  
This algorithm establishes the PH quintics as a viable medium for the design or approximation of free-form curves, and allows a one-for-one substitution of PH quintics in lieu of the widely-used "ordinary  ...  We show that formulating PH quintics as complex-valued functions of a real parameter leads to a compact Hermite interpolation algorithm and facilitates an identification of the "good" interpolant (in terms  ...  In the complex formulation, the curvature can be expressed as k(í) = |r'(0r3Im(r'(0r"(0), and by substituting from (29) it may be verified that the numerator of this expression is proportional to the quadratic  ... 
doi:10.1090/s0025-5718-1995-1308452-6 fatcat:ew7spddaojaonhnawe7bbsahqa

On Superintegral Kleinian Sphere Packings, Bugs, and Arithmetic Groups [article]

Michael Kapovich, Alex Kontorovich
2021 arXiv   pre-print
The same holds for more general objects we call Kleinian Bugs, in which the spheres need not be disjoint but can meet with dihedral angles pi/m for finitely many m.  ...  We extend the Arithmeticity Theorem to Kleinian packings, that is, the superintegral ones come from Q-arithmetic lattices of simplest type.  ...  For any k-arithmetic hyperbolic lattice of simplest type O F poq, there is a totally real, quadratic extension k 1 Ą k, with ring of Figure 2 .  ... 
arXiv:2104.13838v1 fatcat:ih57yictive5foes3gqr3onp3i

Compositionally analyzing a proportional-integral controller family

Ashish Tiwari
2011 IEEE Conference on Decision and Control and European Control Conference  
Such a result is useful for compositionally verifying a system consisting of a plant and a controller.  ...  We use real quantifier elimination methods to solve the absolute always eventually region stability problem. The class of plants found in our solution includes nonlinear and switched plant models.  ...  For example, for verifying stability, the user can provide a template for the Lyapunov function (say a quadratic Lyapunov function) and the solver can find a concrete quadratic function that proves stability  ... 
doi:10.1109/cdc.2011.6160779 dblp:conf/cdc/Tiwari11 fatcat:f5epqb55vfgnzdfliljn6pldka

Arithmetic hyperbolic reflection groups [article]

Mikhail Belolipetsky
2016 arXiv   pre-print
This survey article is dedicated to the study of arithmetic hyperbolic reflection groups with an emphasis on the results that were obtained in the last ten years and on the open problems.  ...  I would like to thank Alice Mark for sending me the preliminary version of her thesis and Igor Dolgachev for sending me his lecture notes. Comments from Richard Borcherds,  ...  I am grateful to Ernest Borisovich Vinberg and Daniel Allcock for their comments and corrections, which helped to significantly improve the quality of this paper.  ... 
arXiv:1506.03111v4 fatcat:muoxxht66zg4vghdxsuem3ypou

Arithmetic hyperbolic reflection groups

Mikhail Belolipetsky
2016 Bulletin of the American Mathematical Society  
This survey article is dedicated to the study of arithmetic hyperbolic reflection groups with an emphasis on the results that were obtained in the last ten years and on the open problems.  ...  The author would like to thank Alice Mark for sending me the preliminary version of her thesis and Igor Dolgachev for sending me his lecture notes.  ...  Acknowledgments The author is grateful to Ernest Borisovich Vinberg and Daniel Allcock for their comments and corrections, which helped to significantly improve the quality of this paper.  ... 
doi:10.1090/bull/1530 fatcat:exxfoxomxzgadkcxdi56ibziby

Quadratic Residues and Non-Residues: Selected Topics [article]

Steve Wright
2016 arXiv   pre-print
A major part of the Disquisitiones deals with quadratic residues and nonresidues.  ...  Beginning with these fundamental contributions of Gauss, the study of quadratic residues and nonresidues has subsequently led directly to many of the key ideas and techniques that are used everywhere in  ...  This makes it virtually impossible for V to be deceived in this manner. 98 We now ask the following question: what has quadratic reciprocity got to do with all of this?  ... 
arXiv:1408.0235v7 fatcat:44ypl6ye6fhktoqzx3mm7xuzvm

A hyperbolic counterpart to Rokhlin's cobordism theorem [article]

Michelle Chu, Alexander Kolpakov
2020 arXiv   pre-print
An analogous result holds for non-compact orientable hyperbolic arithmetic n-manifolds of finite volume that are geometric boundaries, for n ≥ 2.  ...  +1)-manifolds, for any n ≥ 2, thereby establishing that these classes of manifolds have the same growth rate with respect to volume as all compact orientable hyperbolic arithmetic n-manifolds.  ...  Acknowledgements The first author thanks Darren Long for many helpful discussions.  ... 
arXiv:1905.04774v5 fatcat:xoz3iljqvnbyxjnk22x2wlxuoe

Derivation of a Linear Decision Rule for Production and Employment

Charles C. Holt, Franco Modigliani, John F. Muth
1956 Management science  
That these roots satisfy the auxiliary equation (3.4) may be verified by direct substitution.  ...  This is one reason for interest in quadratic criteria.  ... 
doi:10.1287/mnsc.2.2.159 fatcat:l5dcvkvwqbhkdejyk42lbeij5m

Mechanisms of Proton-Proton Inelastic Cross-Section Growth in Multi-Peripheral Model within the Framework of Perturbation Theory. Part 1

Igor Sharf, Andrii Tykhonov, Grygorii Sokhrannyi, Maksym Deliyergiyev, Natalia Podolyan, Vitaliy Rusov
2011 Journal of Modern Physics  
In the vicinity of this point the integrand is substituted for an expression of Gaussian type, which made possible to compute this integral numerically. The paper has two parts.  ...  It is shown that virtuality of all internal lines of the "comb" diagram reduced at the constrained maximum point with energy growth.  ...  Due to negativity of virtual particle squared fourmomentums the magnitude Equation (4) is real and positive.  ... 
doi:10.4236/jmp.2011.212182 fatcat:i7aqe465hvdy5ow5hokotqfjri
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