“…However, diagnostic value estimation showed that even algebraic methods allow us to obtain satisfactory results.…”
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“…Mechanical power was computed both with the geometric method, as the area between the inspiratory limb of the airway pressure and the volume, and with two algebraic methods, a comprehensive and a surrogate formula. …”
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“…The first model (Model 1) is a simpler ordinary differential equation (ODE) compartmental system composed of 3 equations with 10 parameters; the second model (Model 2) is a differential algebraic equation (DAE) system with 4 differential equations, 1 algebraic equation and 13 parameters incorporating the formation of the ZA+enzyme+Ag complex. …”
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“…We perform a general algebraic analysis on the possibility of realising slow-roll inflation in the moduli sector of string models. …”
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“…The one-loop amplitudes are computed using process-independent algebraic manipulations of Feynman diagrams and numerical tensor reduction. …”
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“…For a large class of theories, we explicitly compute the Higgs branch HS which reveals various algebraic/geometric aspects of the moduli space such as the dimension of the space, generators of the moduli space and relations connecting them. …”
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“…IntroductionDiophantine EquationsModular ArithmeticPrimes and the Distribution of PrimesCryptographyDivisibilityDivisibilityEuclid's Theorem Euclid's Original Proof The Sieve of Eratosthenes The Division Algorithm The Greatest Common Divisor The Euclidean Algorithm Other BasesLinear Diophantine EquationsThe Postage Stamp Problem Fermat and Mersenne Numbers Chapter Highlights Problems Unique FactorizationPreliminary Results The Fundamental Theorem of Arithmetic Euclid and the Fundamental Theorem of ArithmeticChapter Highlights Problems Applications of Unique Factorization A Puzzle Irrationality Proofs The Rational Root Theorem Pythagorean Triples Differences of Squares Prime Factorization of Factorials The Riemann Zeta Function Chapter Highlights Problems CongruencesDefinitions and Examples Modular Exponentiation Divisibility TestsLinear Congruences The Chinese Remainder TheoremFractions mod m Fermat's Theorem Euler's Theorem Wilson's Theorem Queens on a Chessboard Chapter Highlights Problems Cryptographic ApplicationsIntroduction Shift and Affine Ciphers Secret Sharing RSA Chapter Highlights Problems Polynomial Congruences Polynomials Mod Primes Solutions Modulo Prime PowersComposite Moduli Chapter Highlights Problems Order and Primitive RootsOrders of Elements Primitive Roots DecimalsCard Shuffling The Discrete Log Problem Existence of Primitive Roots Chapter Highlights Problems More Cryptographic Applications Diffie-Hellman Key Exchange Coin Flipping over the Telephone Mental Poker The ElGamal Public Key Cryptosystem Digital Signatures Chapter Highlights Problems Quadratic Reciprocity Squares and Square Roots Mod Primes Computing Square Roots Mod p Quadratic Equations The Jacobi Symbol Proof of Quadratic Reciprocity Chapter Highlights Problems Primality and Factorization Trial Division and Fermat Factorization Primality Testing Factorization Coin Flipping over the Telephone Chapter Highlights Problems Geometry of NumbersVolumes and Minkowski's Theorem Sums of Two Squares Sums of Four Squares Pell's Equation Chapter Highlights Problems Arithmetic FunctionsPerfect Numbers Multiplicative Functions Chapter Highlights Problems Continued Fractions Rational Approximations; Pell's Equation Basic TheoryRational Numbers Periodic Continued Fractions Square Roots of Integers Some Irrational Numbers Chapter Highlights Problems Gaussian Integers Complex Arithmetic Gaussian Irreducibles The Division Algorithm Unique Factorization Applications Chapter Highlights Problems Algebraic IntegersQuadratic Fields and Algebraic IntegersUnits Z[√-2] Z[√3] Non-unique Factorization Chapter Highlights Problems Analytic MethodsΣ1/p Diverges Bertrand's Postulate Chebyshev's Approximate Prime Number Theorem Chapter Highlights Problems Epilogue: Fermat's Last Theorem Introduction Elliptic Curves Modularity Supplementary Topics Geometric SeriesMathematical InductionPascal's Triangle and the Binomial TheoremFibonacci NumbersProblems Answers and Hints for Odd-Numbered ExercisesIndex…”
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“…The first is useful in atmospheric physics, while the dynamics emerging in the second is governed by the Robinson–Trautman equation, describing a Calabi flow on the surface, and known to appear when solving Einstein’s equations for algebraically special Ricci-flat or Einstein spacetimes.…”
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“…Essentially, while the modular sector of the elliptic genus takes care of anomaly cancellation involving the universal B-field, the quasi-Jacobi one accounts for additional B-fields that can be present.Thus once again, diverse mathematical ingredients, namely here the algebraic geometry of fourfolds, relative Gromow-Witten theory pertaining to flux backgrounds, and the modular properties of (quasi-)Jacobi forms, conspire in an intriguing manner precisely as required by stringy consistency.…”
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“…In particular, we derive a result for the DGLAP kernel in terms of the series of Bernoulli numbers found previously by Vogt directly from algebraic all-order expressions. We identify the off-diagonal DGLAP splitting functions and soft-quark Sudakov logarithms as inherent two-scale quantities in the large-x limit. …”
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“…This requires an extension of the description for ordinary <math display="inline"><mo stretchy="false">[</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo stretchy="false">]</mo></math>-7-branes to consistently include <math display="inline"><mrow><mi mathvariant="normal">O</mi><msup><mrow><mn>7</mn></mrow><mrow><mo>+</mo></mrow></msup></mrow></math> planes, which then naturally encodes the dynamics of <math display="inline"><msub><mi mathvariant="fraktur">sp</mi><mi>n</mi></msub></math> gauge algebras, including their <math display="inline"><mi>p</mi></math>-form center symmetries. …”
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“…Amongst these, we show that one—and only one—type of densities is of the quasi-topological kind, namely, such that the equation for f(r) is algebraic. Our arguments do not work for D = 4, in which case there seems to be a single unique GQT density at each order which is not of the quasi-topological kind. …”
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“…We find that, in these densities, the terms that affect the monotonicity of the holographic <math display="inline"><mi>c</mi></math>-function are algebraic in the curvature, and do not involve covariant derivatives of the Riemann tensor. …”
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“…The usefulness of the formalism is illustrated by applying it to purely algebraic computations of the tension of (p,q) strings, and the mass and entropy of extremal blackholes constructed from D-1-branes and D-5-branes. …”
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“…Within this method we are able to derive algebraic expressions for the generalised GSO projections for all sectors that appear in the models. …”
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“…The title of his thesis was The Algebraic Manipulation of Constraints. He has built satellite data acquisition systems, a multiprocessor version of UNIX®, several compilers, mail systems, and window managers. …”
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“…New mathematical concepts and ideas are introduced by prominent mathematicalphysicists and mathematicians, covering among others the fields of Dynamical Systems, Operator Algebras, Partial Differential Equations, Probability Theory, Random Matrices, Condensed Matter Physics, Statistical Mechanics, General Relativity, Quantum Mechanics, Quantum Field Theory, Quantum Information and String Theory. …”
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“…The J-matrix method is an algebraic method of quantum scattering with substantial success in atomic and nuclear physics. …”
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“…The underlying integral equation connecting the sought for pion spectrum and the measured gamma spectrum has been discretized and subsequently reduced to a system of linear algebraic equations. The latter system, however, is known to be ill-posed and must be regularized to obtain a stable solution. …”
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“…Written by internationally renowned experts, this comprehensive introduction to boundary conformal field theory reaches from theoretical foundations to recent developments, with an emphasis on the algebraic treatment of string backgrounds. Topics covered include basic concepts in conformal field theory with and without boundaries, the mathematical description of strings and D-branes, and the geometry of strongly curved spacetime. …”
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