scholarly journals A Forgotten Differential Equation Studied by Jacopo Riccati Revisited in Terms of Lie Symmetries

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1312
Author(s):  
Daniele Ritelli
Keyword(s):  
Differential Equation ◽  
Closed Form ◽  
Hypergeometric Functions ◽  
Solution Method ◽  
Complete Solution ◽  
Lie Symmetries ◽  
Parameter Family ◽  
Original Solution ◽  
The One ◽  
Two Parameter

In this paper we present a two parameter family of differential equations treated by Jacopo Riccati, which does not appear in any modern repertoires and we extend the original solution method to a four parameter family of equations, translating the Riccati approach in terms of Lie symmetries. To get the complete solution, hypergeometric functions come into play, which, of course, were unknown in Riccati’s time. Re-discovering the method introduced by Riccati, called by himself dimidiata separazione (splitted separation), we arrive at the closed form integration of a differential equation, more general to the one treated in Riccati’s contribution, and which also does not appear in the known repertoires.

scholarly journals Pentagon integrals to arbitrary order in the dimensional regulator

2021 ◽  
Vol 2021 (6) ◽  
Author(s):  
Nikolaos Syrrakos
Keyword(s):  
Differential Equation ◽  
Closed Form ◽  
Hypergeometric Function ◽  
Arbitrary Order ◽  
Explicit Solutions ◽  
Single Variable ◽  
Boundary Terms ◽  
Special Limit ◽  
Transcendental Functions ◽  
The One

Abstract We analytically calculate one-loop five-point Master Integrals, pentagon integrals, with up to one off-shell leg to arbitrary order in the dimensional regulator in d = 4−2𝜖 space-time dimensions. A pure basis of Master Integrals is constructed for the pentagon family with one off-shell leg, satisfying a single-variable canonical differential equation in the Simplified Differential Equations approach. The relevant boundary terms are given in closed form, including a hypergeometric function which can be expanded to arbitrary order in the dimensional regulator using the Mathematica package HypExp. Thus one can obtain solutions of the canonical differential equation in terms of Goncharov Polylogartihms of arbitrary transcendental weight. As a special limit of the one-mass pentagon family, we obtain a fully analytic result for the massless pentagon family in terms of pure and universally transcendental functions. For both families we provide explicit solutions in terms of Goncharov Polylogartihms up to weight four.

Homogeneous Einstein Manifolds with Vanishing Curvature

2019 ◽  
Vol 62 (3) ◽  
pp. 525-537
Author(s):  
Libing Huang ◽  
Zhongmin Shen
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Homogeneous Spaces ◽  
Parameter Family ◽  
Finsler Metrics ◽  
Einstein Manifolds ◽  
Constant Condition ◽  
Regular Solutions ◽  
Two Parameter

AbstractInfinitely many new Einstein Finsler metrics are constructed on several homogeneous spaces. By imposing certain conditions on the homogeneous spaces, it is shown that the Ricci constant condition becomes an ordinary differential equation. The regular solutions of this equation lead to a two parameter family of Einstein Finsler metrics with vanishing $S$ curvature.

Subjective Time Preference and Willingness to Pay for an Energy-Saving Durable Good

2001 ◽  
Vol 32 (3) ◽  
pp. 133-141 ◽  
Author(s):  
Gerrit Antonides ◽  
Sophia R. Wunderink
Keyword(s):  
Willingness To Pay ◽  
Energy Saving ◽  
Subjective Time ◽  
Durable Good ◽  
Discount Function ◽  
Hyperbolic Discount ◽  
The One ◽  
Two Parameter ◽  
Difference Effect ◽  
Better Than

Summary: Different shapes of individual subjective discount functions were compared using real measures of willingness to accept future monetary outcomes in an experiment. The two-parameter hyperbolic discount function described the data better than three alternative one-parameter discount functions. However, the hyperbolic discount functions did not explain the common difference effect better than the classical discount function. Discount functions were also estimated from survey data of Dutch households who reported their willingness to postpone positive and negative amounts. Future positive amounts were discounted more than future negative amounts and smaller amounts were discounted more than larger amounts. Furthermore, younger people discounted more than older people. Finally, discount functions were used in explaining consumers' willingness to pay for an energy-saving durable good. In this case, the two-parameter discount model could not be estimated and the one-parameter models did not differ significantly in explaining the data.

Płyty kołowe Hencky’ego–Bolle’a spoczywające na podłożu sprężystym Własowa

2020 ◽  
Vol 29 (4) ◽  
pp. 444-453
Author(s):  
Mykola Nagirniak
Keyword(s):  
Differential Equation ◽  
Circular Plate ◽  
Significant Influence ◽  
Single Layer ◽  
Medium Thickness ◽  
Cross Sectional ◽  
Equation Solution ◽  
Integral Characteristics ◽  
Two Parameter ◽  
Plate Deflection

The work presents the equations of the theory of symmetrical plates, resting on one-way, single-layer, two-parameter Vlasov’s subsoil. Two cases of differential equation solution of the plate deflection of thin and medium thickness on the ground substrate were analyzed depending on the size of the integral characteristics UÖD and 6ÖD. The example of loading the circular plate with a Pk load evenly distributed over the edge was considered and shows dimensionless graphs of deflection, bending torques and transverse forces in the plate and in the ground subsoil. The effect of the Poisson’s coefficient of the plate on deflection values and cross-sectional forces was investigated. The Poisson’s number has been shown to have a significant influence on deflection values and bending torque, however shown negligible effect on transverse forces values.

Theoretical accuracy of the last squares method in the isotope dilution analysis

1984 ◽  
Vol 49 (4) ◽  
pp. 805-820
Author(s):  
Ján Klas
Keyword(s):  
Least Squares ◽  
Isotope Dilution ◽  
Least Squares Method ◽  
Isotope Dilution Analysis ◽  
Straight Line ◽  
The One ◽  
Two Parameter ◽  
Theoretical Accuracy

The accuracy of the least squares method in the isotope dilution analysis is studied using two models, viz a model of a two-parameter straight line and a model of a one-parameter straight line.The equations for the direct and the inverse isotope dilution methods are transformed into linear coordinates, and the intercept and slope of the two-parameter straight line and the slope of the one-parameter straight line are evaluated and treated.

THE FORWARD PDE FOR EUROPEAN OPTIONS ON STOCKS WITH FIXED FRACTIONAL JUMPS

2005 ◽  
Vol 08 (02) ◽  
pp. 239-253 ◽  
Author(s):  
PETER CARR ◽  
ALIREZA JAVAHERI
Keyword(s):  
Differential Equation ◽  
Closed Form ◽  
Stock Price ◽  
Arrival Rate ◽  
European Options ◽  
Standard Assumption ◽  
Integro Differential Equation ◽  
Local Variance ◽  
Calendar Dates

We derive a partial integro differential equation (PIDE) which relates the price of a calendar spread to the prices of butterfly spreads and the functions describing the evolution of the process. These evolution functions are the forward local variance rate and a new concept called the forward local default arrival rate. We then specialize to the case where the only jump which can occur reduces the underlying stock price by a fixed fraction of its pre-jump value. This is a standard assumption when valuing an option written on a stock which can default. We discuss novel strategies for calibrating to a term and strike structure of European options prices. In particular using a few calendar dates, we derive closed form expressions for both the local variance and the local default arrival rate.

scholarly journals AN INDUSTRY QUESTION: THE ULTIMATE AND ONE-YEAR RESERVING UNCERTAINTY FOR DIFFERENT NON-LIFE RESERVING METHODOLOGIES

2014 ◽  
Vol 44 (3) ◽  
pp. 495-499 ◽  
Author(s):  
Eric Dal Moro ◽  
Joseph Lo
Keyword(s):  
Closed Form ◽  
Internal Models ◽  
Cape Cod ◽  
Best Estimate ◽  
To Come ◽  
Chain Ladder ◽  
One Year ◽  
The One ◽  
Best Estimates

AbstractIn the industry, generally, reserving actuaries use a mix of reserving methods to derive their best estimates. On the basis of the best estimate, Solvency 2 requires the use of a one-year volatility of the reserves. When internal models are used, such one-year volatility has to be provided by the reserving actuaries. Due to the lack of closed-form formulas for the one-year volatility of Bornhuetter-Ferguson, Cape-Cod and Benktander-Hovinen, reserving actuaries have limited possibilities to estimate such volatility apart from scaling from tractable models, which are based on other reserving methods. However, such scaling is technically difficult to justify cleanly and awkward to interact with. The challenge described in this editorial is therefore to come up with similar models like those of Mack or Merz-Wüthrich for the chain ladder, but applicable to Bornhuetter-Ferguson, mix Chain-Ladder and Bornhuetter-Ferguson, potentially Cape-Cod and Benktander-Hovinen — and their mixtures.

scholarly journals New Approaches to the General Linearization Problem of Jacobi Polynomials Based on Moments and Connection Formulas

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1573
Author(s):  
Waleed Mohamed Abd-Elhameed ◽  
Badah Mohamed Badah
Keyword(s):  
Differential Equation ◽  
Hypergeometric Functions ◽  
Jacobi Polynomials ◽  
Tau Method ◽  
Riccati Differential Equation ◽  
Shifted Jacobi Polynomials ◽  
Linearization Problem ◽  
Linearization Coefficients ◽  
New Formula ◽  
Connection Formulas

This article deals with the general linearization problem of Jacobi polynomials. We provide two approaches for finding closed analytical forms of the linearization coefficients of these polynomials. The first approach is built on establishing a new formula in which the moments of the shifted Jacobi polynomials are expressed in terms of other shifted Jacobi polynomials. The derived moments formula involves a hypergeometric function of the type 4F3(1), which cannot be summed in general, but for special choices of the involved parameters, it can be summed. The reduced moments formulas lead to establishing new linearization formulas of certain parameters of Jacobi polynomials. Another approach for obtaining other linearization formulas of some Jacobi polynomials depends on making use of the connection formulas between two different Jacobi polynomials. In the two suggested approaches, we utilize some standard reduction formulas for certain hypergeometric functions of the unit argument such as Watson’s and Chu-Vandermonde identities. Furthermore, some symbolic algebraic computations such as the algorithms of Zeilberger, Petkovsek and van Hoeij may be utilized for the same purpose. As an application of some of the derived linearization formulas, we propose a numerical algorithm to solve the non-linear Riccati differential equation based on the application of the spectral tau method.

Wavelets in the Solution of Nongray Radiative Heat Transfer Equation

1998 ◽  
Vol 120 (1) ◽  
pp. 133-139 ◽  
Author(s):  
Y. Bayazitoglu ◽  
B. Y. Wang
Keyword(s):  
Transfer Equation ◽  
Solution Method ◽  
Radiative Heat ◽  
Radiative Transfer Equation ◽  
Basis Functions ◽  
Wavelet Basis ◽  
System Of Equations ◽  
Heat Transfer Equation ◽  
One Dimensional ◽  
The One

The wavelet basis functions are introduced into the radiative transfer equation in the frequency domain. The intensity of radiation is expanded in terms of Daubechies’ wrapped-around wavelet functions. It is shown that the wavelet basis approach to modeling nongrayness can be incorporated into any solution method for the equation of transfer. In this paper the resulting system of equations is solved for the one-dimensional radiative equilibrium problem using the P-N approximation.

scholarly journals The holographic conformal manifold of 3d $$ \mathcal{N} $$ = 2 S-fold SCFTs

2021 ◽  
Vol 2021 (7) ◽  
Author(s):  
Nikolay Bobev ◽  
Friðrik Freyr Gautason ◽  
Jesse van Muiden
Keyword(s):  
String Theory ◽  
Three Dimensional ◽  
Parameter Family ◽  
Global Symmetry ◽  
Maximal Supergravity ◽  
All Solutions ◽  
Operator Spectrum ◽  
Type Iib String Theory ◽  
Two Parameter

Abstract We employ a non-compact gauging of four-dimensional maximal supergravity to construct a two-parameter family of AdS4 J-fold solutions preserving $$ \mathcal{N} $$ N = 2 supersymmetry. All solutions preserve $$ \mathfrak{u} $$ u (1) × $$ \mathfrak{u} $$ u (1) global symmetry and in special limits we recover the previously known $$ \mathfrak{su} $$ su (2) × $$ \mathfrak{u} $$ u (1) invariant $$ \mathcal{N} $$ N = 2 and $$ \mathfrak{su} $$ su (2) × $$ \mathfrak{su} $$ su (2) invariant $$ \mathcal{N} $$ N = 4 J-fold solutions. This family of AdS4 backgrounds can be uplifted to type IIB string theory and is holographically dual to the conformal manifold of a class of three-dimensional S-fold SCFTs obtained from the $$ \mathcal{N} $$ N = 4 T [U(N)] theory of Gaiotto-Witten. We find the spectrum of supergravity excitations of the AdS4 solutions and use it to study how the operator spectrum of the three-dimensional SCFT depends on the exactly marginal couplings.

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