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2. Evolutionary dynamics of rock-paper-scissors game in the patchy network with mutations
- Author
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Tina Verma and Arvind Kumar Gupta
- Subjects
Hopf bifurcation ,education.field_of_study ,General Mathematics ,Applied Mathematics ,Population ,Evolutionary game theory ,Biodiversity ,General Physics and Astronomy ,Statistical and Nonlinear Physics ,Metapopulation ,symbols.namesake ,Transcritical bifurcation ,Evolutionary biology ,Mutation (genetic algorithm) ,symbols ,education ,Evolutionary dynamics ,Mathematics - Abstract
Connectivity is the safety network for biodiversity conservation because connected habitats are more effective for saving the species and ecological functions. The nature of coupling for connectivity also plays an important role in the co-existence of species in cyclic-dominance. The rock-paper-scissors game is one of the paradigmatic mathematical model in evolutionary game theory to understand the mechanism of biodiversity in cyclic-dominance. In this paper, the metapopulation model for rock-paper-scissors with mutations is presented in which the total population is divided into patches and the patches form a network of complete graph. The migration among patches is allowed through simple random walk. The replicator-mutator equations are used with the migration term. When migration is allowed then the population of the patches will synchronized and attain stable state through Hopf bifurcation. Apart form this, two phases are observed when the strategies of one of the species mutate to other two species: co-existence of all the species phase and existence of one kind of species phase. The transition from one phase to another phase is taking place due to transcritical bifurcation. The dynamics of the population of species of rock, paper, scissors is studied in the environment of homogeneous and heterogeneous mutation. Numerical simulations have been performed when mutation is allowed in all the patches (homogeneous mutation) and some of the patches (heterogeneous mutation). It has been observed that when the number of patches is increased in the case of heterogeneous mutation then the population of any of the species will not extinct and all the species will co-exist.
- Published
- 2021
3. Evolutionary dynamics in the rock-paper-scissors system by changing community paradigm with population flow
- Author
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Junpyo Park
- Subjects
Hopf bifurcation ,education.field_of_study ,General Mathematics ,Applied Mathematics ,Population ,General Physics and Astronomy ,Robustness (evolution) ,Statistical and Nonlinear Physics ,Fixed point ,symbols.namesake ,symbols ,Outflow ,Statistical physics ,Balanced flow ,Evolutionary dynamics ,education ,Multistability ,Mathematics - Abstract
Classic frameworks of rock-paper-scissors game have been assumed in a closed community that a density of each group is only affected by internal factors such as competition interplay among groups and reproduction itself. In real systems in ecological and social sciences, however, the survival and a change of a density of a group can be also affected by various external factors. One of common features in real population systems in ecological and social sciences is population flow that is characterized by population inflow and outflow in a group or a society, which has been usually overlooked in previous works on models of rock-paper-scissors game. In this paper, we suggest the rock-paper-scissors system by implementing population flow and investigate its effect on biodiversity. For two scenarios of either balanced or imbalanced population flow, we found that the population flow can strongly affect group diversity by exhibiting rich phenomena. In particular, while the balanced flow can only lead the persistent coexistence of all groups which accompanies a phase transition through supercritical Hopf bifurcation on different carrying simplices, the imbalanced flow strongly facilitates rich dynamics such as alternative stable survival states by exhibiting various group survival states and multistability of sole group survivals by showing not fully covered but spirally entangled basins of initial densities due to local stabilities of associated fixed points. In addition, we found that, the system can exhibit oscillatory dynamics for coexistence by relativistic interplay of population flows which can capture the robustness of the coexistence state. Applying population flow in the rock-paper-scissors system can ultimately change a community paradigm from closed to open one, and our foundation can eventually reveal that population flow can be also a significant factor on a group density which is independent to fundamental interactions among groups.
- Published
- 2021
4. Comments on the paper 'Asymptotic behavior for a fourth-order parabolic equation involving the Hessian. Z. Angew. Math. Phys., (2018) 69: 147'
- Author
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Jun Zhou and Hang Ding
- Subjects
Hessian matrix ,symbols.namesake ,Fourth order ,Applied Mathematics ,General Mathematics ,symbols ,General Physics and Astronomy ,Applied mathematics ,Finite time ,Mathematics ,Energy functional ,Blowing up - Abstract
In this note, we make two revisions of the paper [2]. The first one is the asymptotic behavior of the energy functional as $$t\rightarrow T$$ (see [2, Theorem 1.6]), where T is the blow-up time. The second one is the equivalent conditions for the solutions blowing up in finite time or existing globally (see [2, Theorem 1.8]).
- Published
- 2019
5. Corrigendum to the papers on Exceptional orthogonal polynomials: J. Approx. Theory 182 (2014) 29–58, 184 (2014) 176–208 and 214 (2017) 9–48
- Author
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Antonio J. Durán
- Subjects
Numerical Analysis ,Pure mathematics ,Applied Mathematics ,General Mathematics ,Hilbert space ,Approx ,symbols.namesake ,TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES ,TheoryofComputation_ANALYSISOFALGORITHMSANDPROBLEMCOMPLEXITY ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,Orthogonal polynomials ,symbols ,Analysis ,Mathematics - Abstract
We complete a gap in the proof that exceptional polynomials are complete orthogonal systems in the associated Hilbert spaces.
- Published
- 2020
6. Erratum to the paper 'L∞(L∞)-boundedness and convergence of DG(p)-solutions for nonlinear conservation laws with boundary conditions'
- Author
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Christian Henke and Lutz Angermann
- Subjects
Conservation law ,Pure mathematics ,Lemma (mathematics) ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Lebesgue integration ,Computational Mathematics ,Nonlinear system ,symbols.namesake ,Convergence (routing) ,symbols ,Boundary value problem ,Affine transformation ,Constant (mathematics) ,Mathematics - Abstract
In the paper (HA14), unfortunately, a computational error occurred in one estimate. Although the wrong estimate does not affect the main results, we want to present the necessary corrections. Essentially, Lemma 5.2 has to be corrected and, since it is used in the proof of Theorem 5.1, the proof of this theorem also requires an adaptation. (i) The corrected formulation of Lemma 5.2 is as follows. Lemma 5.2 For Lagrange finite elements with a shape-regular family of affine meshes { T n h } h>0 there is a constant C > 0 independent of q and h such that for all w ∈ Wh and q = 2m, m ∈N: CΛq−2 p (∇w,∇Ip h (wq−1))T ∫ T ‖∇w‖l2‖w‖ q−2 0,∞,T dx, ∀T ∈ T n h , (5.1) where Λp = ‖ ∑ndof i=1 |φi|‖0,∞,T is the Lebesgue constant.
- Published
- 2015
7. A Note on Recent Papers by Grafakos and Teschl, and Estrada
- Author
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Adam Nowak and Krzysztof Stempak
- Subjects
Hankel transform ,Partial differential equation ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Function (mathematics) ,Transplantation ,symbols.namesake ,Radial function ,Fourier transform ,Fourier analysis ,symbols ,Analysis ,Mathematics - Abstract
We indicate how recent results of Grafakos and Teschl (J Fourier Anal Appl 19:167–179, 2013), and Estrada (J Fourier Anal Appl 20:301–320, 2014), relating the Fourier transform of a radial function in $$\mathbb R^n$$ and the Fourier transform of the same function in $$\mathbb R^{n+2}$$ and $$\mathbb R^{n+1}$$ , respectively, are located within known results on transplantation for Hankel transforms.
- Published
- 2014
8. Von Neumann Geometry and E(∞) Quantum Spacetimefn1fn1This paper is dedicated to Itmor Procaccia, a personal friend and a fellow fighter for all those things which make life worth living like beauty, peace and justice, on the occasion of his birthday
- Author
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M.S. El Naschie
- Subjects
Physics::General Physics ,Mathematics::Operator Algebras ,General Mathematics ,Applied Mathematics ,General Physics and Astronomy ,Statistical and Nonlinear Physics ,Absolute geometry ,Geometry ,Quantum spacetime ,Quantum differential calculus ,General Relativity and Quantum Cosmology ,symbols.namesake ,Von Neumann algebra ,symbols ,Foundations of geometry ,Noncommutative quantum field theory ,Continuous geometry ,Synthetic geometry ,Mathematics ,Mathematical physics - Abstract
In this paper, it is shown that von Neumann continuous geometry may be regarded as the first attempt towards formulating a general quantum spacetime geometry akin to that of Cantorian spacetime E(∞) and noncommutative geometry.
- Published
- 1998
9. A weighted uniform $L^{p}$--estimate of Bessel functions: A note on a paper of Guo
- Author
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Krzysztof Stempak
- Subjects
symbols.namesake ,Cylindrical harmonics ,Bessel process ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Struve function ,Bessel polynomials ,symbols ,Calculus ,Bessel function ,Lommel function ,Mathematics - Published
- 2000
10. Remarks on DiPerna’s paper 'Convergence of the viscosity method for isentropic gas dynamics'
- Author
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Gui-Qiang Chen
- Subjects
Discrete mathematics ,Isentropic process ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Vacuum state ,Finite difference method ,Euler equations ,Binary entropy function ,symbols.namesake ,Riemann hypothesis ,Compact space ,Mathematics Subject Classification ,symbols ,Mathematics - Abstract
Concerns have been voiced about the correctness of certain technical points in DiPerna’s paper (Comm. Math. Phys. 91 (1983), 1–30) related to the vacuum state. In this note, we provide clarifications. Our conclusion is that these concerns mainly arise from the statement of a lemma for constructing the viscous approximate solutions and some typos; however, the gap can be either fixed by correcting the statement of the lemma and the typos or bypassed by employing the finite difference methods. In [Di], DiPerna found a global entropy solution of the isentropic Euler equations for the following exponents in the equation of state for the pressure: γ = 1 + 2/(2m+ 1), m ≥ 2 integer. (1) He divided his arguments into the following two steps. 1. Compactness framework Assume that a sequence of approximate solutions (ρ (x, t),m (x, t)), 0 ≤ t ≤ T , satisfies: (i). There exists a constant C(T ) > 0, independent of > 0, such that 0 ≤ ρ (x, t) ≤ C, |m (x, t)/ρ (x, t)| ≤ C; (ii). For all weak entropy pairs (η, q) of the isentropic Euler equations, the measure sequence η(ρ ,m )t + q(ρ ,m )x is contained in a compact subset of H −1 loc (R× [0, T ]). If γ satisfies (1), then the sequence (ρ (x, t),m (x, t)) is compact in Lloc(R× [0, T ]). The reason for the restriction on the number γ is that, in such a case, any weak entropy function is a polynomial function of the Riemann invariants (w, z). This is the key step in DiPerna’s arguments and is also his main contribution to the compensated compactness method in this aspect. Received by the editors May 16, 1996. 1991 Mathematics Subject Classification. Primary 35K55, 35L65; Secondary 76N15, 35L60, 65M06.
- Published
- 1997
11. On the Paper 'A Note on Spaces of Absolutely Convergent Fourier Transforms' by Björn G.Walther (this Issue). Letter to the Editor
- Author
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S. V. Kislyakov
- Subjects
Algebra ,symbols.namesake ,Letter to the editor ,Partial differential equation ,Fourier transform ,Fourier analysis ,Applied Mathematics ,General Mathematics ,Mathematics education ,symbols ,Absolute convergence ,Analysis ,Mathematics - Published
- 2014
12. Response to Bucy’s comment on a paper by Udwadia and Kalaba
- Author
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Firdaus E. Udwadia and Robert E. Kalaba
- Subjects
Rank (linear algebra) ,Independent equation ,General Mathematics ,Mathematical analysis ,General Engineering ,General Physics and Astronomy ,Equations of motion ,symbols.namesake ,Theory of equations ,Simultaneous equations ,Lagrange multiplier ,Costate equations ,symbols ,Applied mathematics ,Mathematics ,Numerical partial differential equations - Abstract
We disagree with Bucy’s comments (Bucy 1994) on our (Udwadia &; Kalaba 1992) paper. The reasons are as follows. 1. Just as the Gibbs-Appell equations (which use quasi-coordinates) are not the same as the Lagrange equations (which use Lagrange multipliers), the equations of motion obtained by us are not the same as the Gibbs-Appell equations. Some of the steps required to obtain the Gibbs-Appell equations are (Pars 1979; Whittaker 1937): (1) choice of quasi-coordinates; (2) elimination of certain quasicoordinates in terms of other preferred quasi-coordinates; (3) setting up of the Gibbs function; and (4) differentiation of the Gibbs function with respect to the preferred quasi-coordinates. None of these steps is used in obtaining our equations of motion. 2. Yet, the equations of motion obtained by us are equivalent to the Lagrange equations with multipliers (Kalaba et al. 1995) and to the Gibbs-Appell equations (Udwadia & Kalaba 1996). By equivalent we mean each set of equations implies the other. In fact all these different sets of equations are simply different, yet equivalent, ways of stating D’Alembert’s principle for bilinear constraints. Thus what applies to one set of equations applies to the others. Any deficiency in the equations derived in our paper (say regarding rank changes of the matrix A) is therefore present in the Gibbs-Appell equations and Lagrange’s equations as well, because of their equivalence.
- Published
- 1996
13. Correction to the paper: NevanlinnaCartan theory over function fields and a Diophantine equation
- Author
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Junjiro Noguchi
- Subjects
Algebra ,symbols.namesake ,Pure mathematics ,Diophantine set ,Diophantine geometry ,Applied Mathematics ,General Mathematics ,Diophantine equation ,symbols ,Function (mathematics) ,Legendre's equation ,Thue equation ,Mathematics - Published
- 1998
14. An operator valued function space integral: A sequel to Cameron and Storvick’s paper
- Author
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D. L. Skoug and G. W. Johnson
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,Multiple integral ,Integral representation theorem for classical Wiener space ,Mathematical analysis ,Riemann integral ,Riemann–Stieltjes integral ,Singular integral ,Fourier integral operator ,Volume integral ,symbols.namesake ,symbols ,Daniell integral ,Mathematics - Abstract
Recently Cameron and Storvick introduced and studied an operator valued function space integral related to the Feynman integral. The main theorems of their study establish the existence of the function space integral as a weak operator limit of operators defined at the first stage by finite-dimensional integrals. This paper provides a substantial strengthening of their existence theorem giving the function space integrals as strong operator limits rather than as weak operator limits.
- Published
- 1971
15. Book Review: The lost notebook and other unpublished papers
- Author
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Richard Askey
- Subjects
symbols.namesake ,Applied Mathematics ,General Mathematics ,symbols ,Ramanujan's sum ,Mathematics - Published
- 1988
16. An addendum to the paper ‘Dynamic response of an infinite plate subjected to a steadily moving transverse force’
- Author
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David H. Y. Yen and Clifford C. Chou
- Subjects
Applied Mathematics ,General Mathematics ,Mathematical analysis ,General Physics and Astronomy ,Addendum ,Transverse force ,Integral transform ,Physics::Fluid Dynamics ,symbols.namesake ,Classical mechanics ,Mach number ,Plate theory ,symbols ,Initial value problem ,Supersonic speed ,Mathematics - Abstract
The dynamic response of an infinite, elastic plate to a steadily moving transverse force is studied under the so-called improved plate theory. Analytic solutions for the subsonic and the intersonic cases are obtained using the method of integral transforms. The usual difficulty of the nonuniqueness of the steady-state solutions is circumvented by formulating the problems as initial value problems and then seeking the large time limits of the solutions of such initial value problems. The subsonic and intersonic solutions obtained here are similar to that obtained previously for the supersonic case. However, in contrast with the supersonic solution, which contains two families of Mach lines, the subsonic solution contains no Mach lines and the intersonic solution contains only one family of Mach lines.
- Published
- 1974
17. On positive eigenvalues of one-body schrödinger operators: Remarks on papers by agmon and simon
- Author
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K.-H. Jansen and H. Kalf
- Subjects
symbols.namesake ,Applied Mathematics ,General Mathematics ,symbols ,Eigenvalues and eigenvectors ,Schrödinger's cat ,Mathematical physics ,Mathematics - Published
- 1975
18. The Application of Basic Numbers to Bessel's and Legendre's Functions (Second Paper)
- Author
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F. H. Jackson
- Subjects
Bessel process ,Legendre wavelet ,General Mathematics ,Legendre's equation ,Legendre function ,symbols.namesake ,Bessel polynomials ,Struve function ,symbols ,Calculus ,Applied mathematics ,Legendre's constant ,Legendre polynomials ,Mathematics - Abstract
n/a
- Published
- 1905
19. ON THE DISTRIBUTION OF THE CORRELATION COEFFICIENT IN SMALL SAMPLES. APPENDIX II TO THE PAPERS OF 'STUDENT' AND R. A. FISHER. A COOPERATIVE STUDY
- Author
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H. E. Soper, A. W. Young, A. Lee, Karl Pearson, and B. M. Cave
- Subjects
Statistics and Probability ,Distribution (number theory) ,Correlation coefficient ,Intraclass correlation ,Applied Mathematics ,General Mathematics ,Fisher transformation ,Correlation ratio ,Agricultural and Biological Sciences (miscellaneous) ,Spearman's rank correlation coefficient ,Pearson product-moment correlation coefficient ,symbols.namesake ,Statistics ,symbols ,Statistics, Probability and Uncertainty ,General Agricultural and Biological Sciences ,Mathematics - Published
- 1917
20. Some remarks on A. C. Schaeffer's paper on Dirichlet series
- Author
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Hwa S. Hahn
- Subjects
symbols.namesake ,General Mathematics ,Dirichlet's principle ,Calculus ,symbols ,Applied mathematics ,30.24 ,Dirichlet's energy ,Dirichlet series ,Mathematics - Published
- 1960
21. Two papers on similarity of certain Volterra integral operators
- Author
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Stanley Osher
- Subjects
Algebra ,symbols.namesake ,Similarity (network science) ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,symbols ,Microlocal analysis ,Volterra integral equation ,Fourier integral operator ,Mathematics - Published
- 1967
22. Development of Explicit Schemes for Diffusive SEAIR COVID-19 Epidemic Spreading Model: An Application to Computational Biology
- Author
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Muhammad Shoaib Arif, Yasir Nawaz, and Muhammad Usman Ashraf
- Subjects
Partial differential equation ,Stability criterion ,Computer science ,General Mathematics ,Finite difference method ,Stability (learning theory) ,General Physics and Astronomy ,COVID-19 ,General Chemistry ,Explicit scheme ,symbols.namesake ,Jacobian matrix and determinant ,Taylor series ,symbols ,General Earth and Planetary Sciences ,Applied mathematics ,Consistency ,General Agricultural and Biological Sciences ,Convergence ,Stability ,Linear stability ,Characteristic polynomial ,Research Paper - Abstract
In this contribution, a first-order time scheme is proposed for finding solutions to partial differential equations (PDEs). A mathematical model of the COVID-19 epidemic is modified where the recovery rate of exposed individuals is also considered. The linear stability of the equilibrium states for the modified COVID-19 model is given by finding its Jacobian and applying Routh-Hurwitz criteria on characteristic polynomial. The proposed scheme provides the first-order accuracy in time and second-order accuracy in space. The stability of the proposed scheme is given using the von Neumann stability criterion for standard parabolic PDEs. The consistency for the proposed scheme is also given by expanding the involved terms in it using the Taylor series. The scheme can be used to obtain the condition of getting a positive solution. The stability region of the scheme can be enlarged by choosing suitable values of the contained parameter. Finally, a comparison of the proposed scheme is made with the existing non-standard finite difference method. The results indicate that the non-standard classical technique is incapable of preserving the unique characteristics of the model's epidemiologically significant solutions, whereas the proposed approaches are capable of doing so. A computational code for the proposed discrete model scheme may be made available to readers upon request for convenience.
- Published
- 2021
23. Observations on a paper by Rosenblum
- Author
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S. Cater
- Subjects
Complex conjugate ,Applied Mathematics ,General Mathematics ,Hilbert space ,Uniform limit theorem ,Combinatorics ,symbols.namesake ,Operator (computer programming) ,Skew-Hermitian matrix ,Bounded function ,symbols ,Normal operator ,Complex number ,Mathematics - Abstract
M. Rosenblum in [2] presented a most ingenious proof of the Fuglede and Putnam Theorems by means of entire vector valued functions [1, p. 59]. We will demonstrate that some curious properties of bounded Hilbert space operators can be derived from Rosenblum's argument and similar arguments. Throughout this text we mean by an "operator" a bounded linear transformation of a Hilbert space into itself. Given an operator A we mean by "exp A " the uniform limit of the series I+A +A 2/2 1 +A3/3! +A4/4! + * * * . We let A * denote the adjoint of the operator A, and let z* denote the complex conjugate of the complex number z. A "normal" operator is an operator which commutes with its adjoint. A critical fact in the Rosenblum proof is that given a normal operator A and any complex number z, exp (izA) exp (iz*A *) exp (izA +iz*A *) = exp (iz*A *) exp (izA), and this operator is unitary because i(zA +z*A *) is skew hermitian. Our first result states, among other things, that the converse is true; if the above equations hold for a fixed operator A and all complex numbers z, then A is normal.
- Published
- 1961
24. Global optimization in Hilbert space
- Author
-
Benoît Chachuat, Boris Houska, Engineering & Physical Science Research Council (EPSRC), and Commission of the European Communities
- Subjects
Technology ,Optimization problem ,Mathematics, Applied ,0211 other engineering and technologies ,CONVEX COMPUTATION ,010103 numerical & computational mathematics ,02 engineering and technology ,ELLIPSOIDS ,01 natural sciences ,90C26 ,93B40 ,Convergence analysis ,0102 Applied Mathematics ,Branch-and-lift ,CUT ,Mathematics ,65K10 ,021103 operations research ,Full Length Paper ,Operations Research & Management Science ,0103 Numerical and Computational Mathematics ,Bounded function ,Physical Sciences ,symbols ,49M30 ,Calculus of variations ,INTEGRATION ,SET ,Complexity analysis ,Complete search ,Operations Research ,General Mathematics ,APPROXIMATIONS ,Set (abstract data type) ,symbols.namesake ,Applied mathematics ,ALGORITHM ,0101 mathematics ,INTERSECTION ,Global optimization ,0802 Computation Theory and Mathematics ,Science & Technology ,Infinite-dimensional optimization ,Hilbert space ,Computer Science, Software Engineering ,Constraint (information theory) ,Computer Science ,Software - Abstract
We propose a complete-search algorithm for solving a class of non-convex, possibly infinite-dimensional, optimization problems to global optimality. We assume that the optimization variables are in a bounded subset of a Hilbert space, and we determine worst-case run-time bounds for the algorithm under certain regularity conditions of the cost functional and the constraint set. Because these run-time bounds are independent of the number of optimization variables and, in particular, are valid for optimization problems with infinitely many optimization variables, we prove that the algorithm converges to an \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document}ε-suboptimal global solution within finite run-time for any given termination tolerance \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon > 0$$\end{document}ε>0. Finally, we illustrate these results for a problem of calculus of variations.
- Published
- 2017
25. On the Zeros of Dirichlet L-Functions.II (With Corrections to Ön the Zeros of Dirichlet L-Functions.I' and the Subsequent Papers)
- Author
-
Akio Fujii
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,Dirichlet L-function ,Dirichlet's energy ,Dirichlet eta function ,Class number formula ,symbols.namesake ,Dirichlet kernel ,Dirichlet's principle ,symbols ,General Dirichlet series ,Dirichlet series ,Mathematics - Published
- 1981
26. Correction to the Paper 'A Problem Concerning Orthogonal Polynomials'
- Author
-
G. Szegö
- Subjects
Gegenbauer polynomials ,Applied Mathematics ,General Mathematics ,Discrete orthogonal polynomials ,Mathematical analysis ,Classical orthogonal polynomials ,Algebra ,symbols.namesake ,Wilson polynomials ,Orthogonal polynomials ,Hahn polynomials ,symbols ,Jacobi polynomials ,Koornwinder polynomials ,Mathematics - Published
- 1936
27. On the Distribution of the Correlation Coefficient in Small Samples. Appendix II to the Papers of 'Student' and R. A. Fisher
- Author
-
Karl Pearson, A. W. Young, A. Lee, H. E. Soper, and B. M. Cave
- Subjects
Statistics and Probability ,Correlation coefficient ,Distribution (number theory) ,Intraclass correlation ,Applied Mathematics ,General Mathematics ,Fisher transformation ,Correlation ratio ,Agricultural and Biological Sciences (miscellaneous) ,Pearson product-moment correlation coefficient ,symbols.namesake ,Statistics ,symbols ,Statistics, Probability and Uncertainty ,General Agricultural and Biological Sciences ,Mathematics - Published
- 1917
28. Application of Bernoulli Polynomials of Negative Order to Differencing (Second Paper)
- Author
-
B. F. Kimball
- Subjects
Classical orthogonal polynomials ,symbols.namesake ,Difference polynomials ,General Mathematics ,Discrete orthogonal polynomials ,Mathematical analysis ,symbols ,Applied mathematics ,Order (ring theory) ,Bernoulli process ,Mathematics ,Bernoulli polynomials - Published
- 1934
29. Application of the Theory of Relative Cyclic Fields to both Cases of Fermat's Last Theorem (Second Paper)
- Author
-
H. S. Vandiver
- Subjects
Pure mathematics ,Fermat's little theorem ,Proofs of Fermat's little theorem ,Applied Mathematics ,General Mathematics ,Regular prime ,Fermat's theorem on sums of two squares ,Wieferich prime ,Fermat's factorization method ,symbols.namesake ,Fermat's theorem ,symbols ,Mathematics ,Fermat number - Published
- 1927
30. A Correction to the Paper 'On Effective Sets of Points in Relation to Integral Functions'
- Author
-
V. Ganapathy Iyer
- Subjects
Applied Mathematics ,General Mathematics ,Mathematical analysis ,Line integral ,Riemann–Stieltjes integral ,Riemann integral ,Fourier integral operator ,Volume integral ,symbols.namesake ,Improper integral ,symbols ,Coarea formula ,Daniell integral ,Mathematics - Published
- 1938
31. A remark on Neuwirth and Newman’s paper: 'Positive 𝐻^{1/2} functions are constants'
- Author
-
Shinji Yamashita
- Subjects
Combinatorics ,Lemma (mathematics) ,symbols.namesake ,Applied Mathematics ,General Mathematics ,Blaschke product ,symbols ,Function (mathematics) ,Absolute value (algebra) ,Boundary values ,Decomposition theorem ,Mathematical physics ,Mathematics - Abstract
PROOF. By a theorem of Rudin a function gEH' in U whose boundary values are real a.e. on I can be analytically continued to D [3, p. 59]. The lemma follows on applying Rudin's result to gi= (1/2) (fl+f2) and g2=(i/2) (fi-f2). PROOF OF THEOREM 1. By a well-known decomposition theorem [2, p. 87], f(z)=B(z)F2(Z), where B(z) is a Blaschke product and F(z) EH1. Since the boundary values of B (z) have absolute value one a.e. on K, we have a.e. on I, f(ei0)= |f(eio)I, or B(ei0)F2(ei0) = F2(ei0) |, and hence
- Published
- 1969
32. Volterra's Integral Equation of the Second Kind, with Discontinuous Kernel, Second Paper
- Author
-
Griffith C. Evans
- Subjects
Applied Mathematics ,General Mathematics ,Mathematical analysis ,Summation equation ,Electric-field integral equation ,Integral equation ,Volterra integral equation ,symbols.namesake ,Integro-differential equation ,Kernel (statistics) ,Improper integral ,symbols ,Daniell integral ,Mathematics - Published
- 1911
33. Integers represented as the sum of one prime, two squares of primes and powers of 2
- Author
-
Haiwei Sun and Guangshi Lü
- Subjects
Discrete mathematics ,Applied Mathematics ,General Mathematics ,Short paper ,MathematicsofComputing_GENERAL ,Prime number ,Prime (order theory) ,Algebra ,symbols.namesake ,Integer ,symbols ,Idoneal number ,Prime power ,Sphenic number ,Mathematics - Abstract
In this short paper we prove that every sufficiently large odd integer can be written as a sum of one prime, two squares of primes and 83 83 powers of 2 2 .
- Published
- 2008
34. Higher order Turán inequalities for the Riemann $\xi$-function
- Author
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Dimitar K. Dimitrov, Fábio Rodrigues Lucas, Universidade Estadual Paulista (Unesp), and Universidade Estadual de Campinas (UNICAMP)
- Subjects
Degree (graph theory) ,Applied Mathematics ,General Mathematics ,Entire function ,Mathematical analysis ,Short paper ,Function (mathematics) ,Maclaurin coefficients ,Riemann ξ function ,Combinatorics ,Riemann hypothesis ,symbols.namesake ,Jensen polynomials ,symbols ,Order (group theory) ,Shape function ,Laguerre-Pólya class ,Turán inequalities ,Mathematics - Abstract
Submitted by Vitor Silverio Rodrigues (vitorsrodrigues@reitoria.unesp.br) on 2014-05-27T11:25:28Z No. of bitstreams: 0Bitstream added on 2014-05-27T14:41:41Z : No. of bitstreams: 1 2-s2.0-79951846250.pdf: 494002 bytes, checksum: 56b6ee8beddda3e7dae971355d44a19f (MD5) Made available in DSpace on 2014-05-27T11:25:28Z (GMT). No. of bitstreams: 0 Previous issue date: 2011-03-01 Item merged in doublecheck by Felipe Arakaki (arakaki@reitoria.unesp.br) on 2015-12-11T17:28:11Z Item was identical to item(s): 71803, 21370 at handle(s): http://hdl.handle.net/11449/72321, http://hdl.handle.net/11449/21804 Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) The simplest necessary conditions for an entire function ψ(x) =∞ ∑ k=0 γk xk/k! to be in the Laguerre-Pólya class are the Turán inequalities γ2 k- γk+1γk-1 ≥ 0. These are in fact necessary and sufficient conditions for the second degree generalized Jensen polynomials associated with ψ to be hyperbolic. The higher order Turán inequalities 4(γ2 n - γn-1γn+1)(γ2n +1 - γnγn+2) - (γnγn+1 - γn-1γn+2) 2 ≥ 0 are also necessary conditions for a function of the above form to belong to the Laguerre-Pólya class. In fact, these two sets of inequalities guarantee that the third degree generalized Jensen polynomials are hyperbolic. Pólya conjectured in 1927 and Csordas, Norfolk and Varga proved in 1986 that the Turán inequalities hold for the coefficients of the Riemann ψ-function. In this short paper, we prove that the higher order Turán inequalities also hold for the ψ-function, establishing the hyperbolicity of the associated generalized Jensen polynomials of degree three. © 2010 American Mathematical Society. Departamento de Ciências de Computação e Estatística IBILCE, Universidade Estadual Paulista, 15054-000 São José do Rio Preto, SP Departamento de matemática Aplicada IMECC UNICAMP, 13083-859 Campinas, SP Departamento de Ciências de Computação e Estatística IBILCE, Universidade Estadual Paulista, 15054-000 São José do Rio Preto, SP FAPESP: 03/01874-2 FAPESP: 06/60420-0 CNPq: 305622/2009-9 CAPES: DGU-160
- Published
- 2011
35. Spectral cluster estimates for Schrödinger operators of relativistic type
- Author
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Yannick Sire, Cheng Zhang, and Xiaoqi Huang
- Subjects
Applied Mathematics ,General Mathematics ,Eigenfunction ,Type (model theory) ,Wave equation ,Sobolev space ,Kernel (algebra) ,symbols.namesake ,Operator (computer programming) ,symbols ,Cluster (physics) ,Schrödinger's cat ,Mathematics ,Mathematical physics - Abstract
This paper is dedicated to L p bounds on eigenfunctions of a Schrodinger-type operator ( − Δ g ) α / 2 + V on closed Riemannian manifolds for critically singular potentials V. The operator ( − Δ g ) α / 2 is defined spectrally in terms of the eigenfunctions of − Δ g . We obtain also quasimodes and spectral clusters estimates. As an application, we derive Strichartz estimates for the fractional wave equation ( ∂ t 2 + ( − Δ g ) α / 2 + V ) u = 0 . The wave kernel techniques recently developed by Bourgain-Shao-Sogge-Yao [4] and Shao-Yao [27] play a key role in this paper. We construct a new reproducing operator with several local operators and some good error terms. Moreover, we shall prove that these local operators satisfy certain variable coefficient versions of the “uniform Sobolev estimates” by Kenig-Ruiz-Sogge [18] . These enable us to handle the critically singular potentials V and prove the quasimode estimates.
- Published
- 2021
36. Splines of the Fourth Order Approximation and the Volterra Integral Equations
- Author
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D.E. Zhilin, A.G. Doronina, and I. G. Burova
- Subjects
Polynomial ,Series (mathematics) ,General Mathematics ,Type (model theory) ,Integral equation ,Volterra integral equation ,symbols.namesake ,Continuation ,Computer Science::Graphics ,symbols ,Applied mathematics ,Focus (optics) ,Mathematics ,Interpolation - Abstract
This paper is a continuation of a series of papers devoted to the numerical solution of integral equations using local interpolation splines. The main focus is given to the use of splines of the fourth order of approximation. The features of the application of the polynomial and non-polynomial splines of the fourth order of approximation to the solution of Volterra integral equation of the second kind are discussed. In addition to local splines of the Lagrangian type, integro-differential splines are also used to construct computational schemes. The comparison of the solutions obtained by different methods is carried out. The results of the numerical experiments are presented.
- Published
- 2021
37. On Lacunas in the Spectrum of the Laplacian with the Dirichlet Boundary Condition in a Band with Oscillating Boundary
- Author
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Denis Borisov
- Subjects
Statistics and Probability ,Applied Mathematics ,General Mathematics ,Operator (physics) ,Mathematical analysis ,Spectrum (functional analysis) ,Boundary (topology) ,Function (mathematics) ,symbols.namesake ,Amplitude ,Dirichlet boundary condition ,symbols ,Flat band ,Laplace operator ,Mathematics - Abstract
In this paper, we consider the Laplace operator in a flat band whose lower boundary periodically oscillates under the Dirichlet boundary condition. The period and the amplitude of oscillations are two independent small parameters. The main result obtained in the paper is the absence of internal lacunas in the lower part of the spectrum of the operator for sufficiently small period and amplitude. We obtain explicit upper estimates of the period and amplitude in the form of constraints with specific numerical constants. The length of the lower part of the spectrum, in which the absence of lacunas is guaranteed, is also expressed explicitly in terms of the period function and the amplitude.
- Published
- 2021
38. Logarithmic Potential and Generalized Analytic Functions
- Author
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O.V. Nesmelova, Vladimir Gutlyanskiĭ, Vladimir Ryazanov, and A.S. Yefimushkin
- Subjects
Statistics and Probability ,Dirichlet problem ,Pure mathematics ,Applied Mathematics ,General Mathematics ,Harmonic (mathematics) ,Unit disk ,Sobolev space ,Riemann hypothesis ,symbols.namesake ,Harmonic function ,symbols ,Neumann boundary condition ,Analytic function ,Mathematics - Abstract
The study of the Dirichlet problem in the unit disk 𝔻 with arbitrary measurable data for harmonic functions is due to the famous dissertation of Luzin [31]. Later on, the known monograph of Vekua [48] has been devoted to boundary-value problems (only with Holder continuous data) for the generalized analytic functions, i.e., continuous complex valued functions h(z) of the complex variable z = x + iy with generalized first partial derivatives by Sobolev satisfying equations of the form 𝜕zh + ah + b $$ \overline{h} $$ = c ; where it was assumed that the complex valued functions a; b and c belong to the class Lp with some p > 2 in smooth enough domains D in ℂ. The present paper is a natural continuation of our previous articles on the Riemann, Hilbert, Dirichlet, Poincar´e and, in particular, Neumann boundary-value problems for quasiconformal, analytic, harmonic, and the so-called A−harmonic functions with boundary data that are measurable with respect to logarithmic capacity. Here, we extend the corresponding results to the generalized analytic functions h : D → ℂ with the sources g : 𝜕zh = g ∈ Lp, p > 2 , and to generalized harmonic functions U with sources G : △U = G ∈ Lp, p > 2. This paper contains various theorems on the existence of nonclassical solutions of the Riemann and Hilbert boundary-value problems with arbitrary measurable (with respect to logarithmic capacity) data for generalized analytic functions with sources. Our approach is based on the geometric (theoretic-functional) interpretation of boundary-values in comparison with the classical operator approach in PDE. On this basis, it is established the corresponding existence theorems for the Poincar´e problem on directional derivatives and, in particular, for the Neumann problem to the Poisson equations △U = G with arbitrary boundary data that are measurable with respect to logarithmic capacity. These results can be also applied to semilinear equations of mathematical physics in anisotropic and inhomogeneous media.
- Published
- 2021
39. Stability and collapse of the Lyapunov spectrum for Perron–Frobenius operator cocycles
- Author
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Anthony Quas and Cecilia González-Tokman
- Subjects
Pure mathematics ,Mathematics::Dynamical Systems ,Dense set ,Applied Mathematics ,General Mathematics ,Blaschke product ,Banach space ,Lyapunov exponent ,Fixed point ,symbols.namesake ,Unit circle ,symbols ,Invariant measure ,Mathematics ,Analytic function - Abstract
In this paper, we study random Blaschke products, acting on the unit circle, and consider the cocycle of Perron-Frobenius operators acting on Banach spaces of analytic functions on an annulus. We completely describe the Lyapunov spectrum of these cocycles. As a corollary, we obtain a simple random Blaschke product system where the Perron-Frobenius cocycle has infinitely many distinct Lyapunov exponents, but where arbitrarily small natural perturbations cause a complete collapse of the Lyapunov spectrum, except for the exponent 0 associated with the absolutely continuous invariant measure. That is, under perturbations, the Lyapunov exponents become 0 with multiplicity 1, and $-\infty$ with infinite multiplicity. This is superficially similar to the finite-dimensional phenomenon, discovered by Bochi \cite{Bochi-thesis}, that away from the uniformly hyperbolic setting, small perturbations can lead to a collapse of the Lyapunov spectrum to zero. In this paper, however, the cocycle and its perturbation are explicitly described; and further, the mechanism for collapse is quite different. We study stability of the Perron-Frobenius cocycles arising from general random Blaschke products. We give a necessary and sufficient criterion for stability of the Lyapunov spectrum in terms of the derivative of the random Blaschke product at its random fixed point, and use this to show that an open dense set of Blaschke product cocycles have hyperbolic Perron-Frobenius cocycles. In the final part, we prove a relationship between the Lyapunov spectrum of a single cocycle acting on two different Banach spaces, allowing us to draw conclusions for the same cocycles acting on $C^r$ functions spaces.
- Published
- 2021
40. On Some Properties of the New Generalized Fractional Derivative with Non-Singular Kernel
- Author
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Khalid Hattaf
- Subjects
Lyapunov function ,Article Subject ,Non singular ,General Mathematics ,Science and engineering ,General Engineering ,Engineering (General). Civil engineering (General) ,01 natural sciences ,010305 fluids & plasmas ,Fractional calculus ,010101 applied mathematics ,symbols.namesake ,Exponential stability ,Kernel (statistics) ,0103 physical sciences ,QA1-939 ,symbols ,Applied mathematics ,TA1-2040 ,0101 mathematics ,Mathematics - Abstract
This paper presents some new formulas and properties of the generalized fractional derivative with non-singular kernel that covers various types of fractional derivatives such as the Caputo–Fabrizio fractional derivative, the Atangana–Baleanu fractional derivative, and the weighted Atangana–Baleanu fractional derivative. These new properties extend many recent results existing in the literature. Furthermore, the paper proposes some interesting inequalities that estimate the generalized fractional derivatives of some specific functions. These inequalities can be used to construct Lyapunov functions with the aim to study the global asymptotic stability of several fractional-order systems arising from diverse fields of science and engineering.
- Published
- 2021
41. Existence and Uniqueness of the Global L1 Solution of the Euler Equations for Chaplygin Gas
- Author
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Zhen Wang, Tingting Chen, and Aifang Qu
- Subjects
Continuous function ,General Mathematics ,Weak solution ,010102 general mathematics ,General Physics and Astronomy ,Euler system ,Absolute continuity ,Lebesgue integration ,01 natural sciences ,010101 applied mathematics ,symbols.namesake ,symbols ,Local boundedness ,Applied mathematics ,Initial value problem ,Uniqueness ,0101 mathematics ,Mathematics - Abstract
In this paper, we establish the global existence and uniqueness of the solution of the Cauchy problem of a one-dimensional compressible isentropic Euler system for a Chaplygin gas with large initial data in the space L loc 1 . The hypotheses on the initial data may be the least requirement to ensure the existence of a weak solution in the Lebesgue measurable sense. The novelty and also the essence of the difficulty of the problem lie in the fact that we have neither the requirement on the local boundedness of the density nor that which is bounded away from vacuum. We develop the previous results on this degenerate system. The method used is Lagrangian representation, the essence of which is characteristic analysis. The key point is to prove the existence of the Lagrangian representation and the absolute continuity of the potentials constructed with respect to the space and the time variables. We achieve this by finding a property of the fundamental theorem of calculus for Lebesgue integration, which is a sufficient and necessary condition for judging whether a monotone continuous function is absolutely continuous. The assumptions on the initial data in this paper are believed to also be necessary for ruling out the formation of Dirac singularity of density. The ideas and techniques developed here may be useful for other nonlinear problems involving similar difficulties.
- Published
- 2021
42. On the Finite Time Blowup of the De Gregorio Model for the 3D Euler Equations
- Author
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Thomas Y. Hou, De Huang, and Jiajie Chen
- Subjects
symbols.namesake ,Mathematics - Analysis of PDEs ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,FOS: Mathematics ,symbols ,Finite time ,Analysis of PDEs (math.AP) ,Mathematics ,Euler equations - Abstract
We present a novel method of analysis and prove finite time asymptotically self-similar blowup of the De Gregorio model \cite{DG90,DG96} for some smooth initial data on the real line with compact support. We also prove self-similar blowup results for the generalized De Gregorio model \cite{OSW08} for the entire range of parameter on $\mathbb{R}$ or $S^1$ for H\"older continuous initial data with compact support. Our strategy is to reformulate the problem of proving finite time asymptotically self-similar singularity into the problem of establishing the nonlinear stability of an approximate self-similar profile with a small residual error using the dynamic rescaling equation. We use the energy method with appropriate singular weight functions to extract the damping effect from the linearized operator around the approximate self-similar profile and take into account cancellation among various nonlocal terms to establish stability analysis. We remark that our analysis does not rule out the possibility that the original De Gregorio model is well posed for smooth initial data on a circle. The method of analysis presented in this paper provides a promising new framework to analyze finite time singularity of nonlinear nonlocal systems of partial differential equations., Comment: Added discussion in Section 2.3 and made some minor edits. Main paper 57 pages, Supplementary material 29 pages. In previous arXiv versions, the hyperlinks of the equation number in the main paper are linked to the supplementary material, which is fixed in this version
- Published
- 2021
43. Approximating a common solution of extended split equality equilibrium and fixed point problems
- Author
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J. M. Ngnotchouye, F. U. Ogbuisi, and F. O. Isiogugu
- Subjects
TheoryofComputation_MISCELLANEOUS ,Iterative method ,Applied Mathematics ,General Mathematics ,Numerical analysis ,Hilbert space ,TheoryofComputation_GENERAL ,Extension (predicate logic) ,Fixed point ,symbols.namesake ,Monotone polygon ,Convergence (routing) ,symbols ,Applied mathematics ,Equilibrium problem ,Mathematics - Abstract
In this paper, we study an extension of the split equality equilibrium problem called the extended split equality equilibrium problem. We give an iterative algorithm for approximating a solution of extended split equality equilibrium and fixed point problems and obtained a strong convergence result in a real Hilbert space. We further applied our result to solve extended split equality monotone variational inclusion and equilibrium problems. The result of this paper complements and extends results on split equality equilibrium problems in the literature.
- Published
- 2021
44. Mapped Regularization Methods for the Cauchy Problem of the Helmholtz and Laplace Equations
- Author
-
Hojjatollah Shokri Kaveh and Hojjatollah Adibi
- Subjects
Cauchy problem ,Laplace transform ,General Mathematics ,MathematicsofComputing_NUMERICALANALYSIS ,General Physics and Astronomy ,General Chemistry ,Spectral galerkin ,Regularization (mathematics) ,Tikhonov regularization ,symbols.namesake ,Helmholtz free energy ,Singular value decomposition ,symbols ,General Earth and Planetary Sciences ,Applied mathematics ,Initial value problem ,General Agricultural and Biological Sciences ,Mathematics - Abstract
In this paper, Spectral Galerkin Method is applied for Cauchy problem of Helmholtz and Laplace equations in the regular domains. It is well known that these problems have severely ill-posed solutions. Accordingly, regularization methods are required to overcome the ill-posedness issue. In this paper, we utilize the regularization method based upon mapped methods. These methods include Tikhonov and truncated singular value decomposition methods and additionally several new filters of regularization which are introduced. Finally, some test examples are given to demonstrate the capability and efficiency of the proposed method.
- Published
- 2021
45. EXISTENCE OF SOLUTIONS TO FRACTIONAL DIFFERENTIAL EQUATIONS WITH FRACTIONAL-ORDER DERIVATIVE TERMS
- Author
-
Ai Sun, Tongxiang Li, Qingchun Yuan, and You-Hui Su
- Subjects
Computer simulation ,Iterative method ,General Mathematics ,010102 general mathematics ,Fixed-point theorem ,Derivative ,Function (mathematics) ,01 natural sciences ,010101 applied mathematics ,Nonlinear system ,symbols.namesake ,Green's function ,symbols ,Applied mathematics ,Point (geometry) ,0101 mathematics ,Mathematics - Abstract
The study in this paper is made on the nonlinear fractional differential equation whose nonlinearity involves the explicit fractional order D0+β u(t). The corresponding Green's function is derived first, and then the completely continuous operator is proved. Besides, based on the Schauder's fixed point theorem and the Krasnosel'skii's fixed point theorem, the sufficient conditions for at least one or two existence of positive solutions are established. Furthermore, several other sufficient conditions for at least three, n or 2n-1 positive solutions are also obtained by applying the generalized AveryHenderson fixed point theorem and the Avery-Peterson fixed point theorem. Finally, several simulation examples are provided to illustrate the main results of the paper. In particularly, a novel efficient iterative method is employed for simulating the examples mentioned above, that is, the interesting point of this paper is that the approximation graphics for the solutions are given by using the iterative method.
- Published
- 2021
46. Rarefaction Wave Interaction and Shock-Rarefaction Composite Wave Interaction for a Two-Dimensional Nonlinear Wave System
- Author
-
Sisi Xie and Geng Lai
- Subjects
Conservation law ,Equation of state ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Rarefaction ,01 natural sciences ,Shock (mechanics) ,010104 statistics & probability ,Nonlinear system ,Riemann hypothesis ,symbols.namesake ,Method of characteristics ,symbols ,Order (group theory) ,0101 mathematics ,Mathematics - Abstract
In order to construct global solutions to two-dimensional (2D for short) Riemann problems for nonlinear hyperbolic systems of conservation laws, it is important to study various types of wave interactions. This paper deals with two types of wave interactions for a 2D nonlinear wave system with a nonconvex equation of state: Rarefaction wave interaction and shock-rarefaction composite wave interaction. In order to construct solutions to these wave interactions, the authors consider two types of Goursat problems, including standard Goursat problem and discontinuous Goursat problem, for a 2D self-similar nonlinear wave system. Global classical solutions to these Goursat problems are obtained by the method of characteristics. The solutions constructed in the paper may be used as building blocks of solutions of 2D Riemann problems.
- Published
- 2021
47. Infinitely many solutions for a class of fractional Robin problems with variable exponents
- Author
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Ramzi Alsaedi
- Subjects
Class (set theory) ,Work (thermodynamics) ,General Mathematics ,variational methods ,robin ,Mathematics::Spectral Theory ,Type (model theory) ,variable exponents ,Euler equations ,symbols.namesake ,Continuation ,fracional sobolev spaces ,Operator (computer programming) ,QA1-939 ,symbols ,Applied mathematics ,Boundary value problem ,Mathematics ,Variable (mathematics) - Abstract
In this paper, we are concerned with a class of fractional Robin problems with variable exponents. Their main feature is that the associated Euler equation is driven by the fractional $ p(\cdot)- $Laplacian operator with variable coefficient while the boundary condition is of Robin type. This paper is a continuation of the recent work established by A. Bahrouni, V. Radulescu and P. Winkert [ 5 ].
- Published
- 2021
48. On Admissible Locations of Transonic Shock Fronts for Steady Euler Flows in an Almost Flat Finite Nozzle with Prescribed Receiver Pressure
- Author
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Zhouping Xin and Beixiang Fang
- Subjects
35A01, 35A02, 35B20, 35B35, 35B65, 35J56, 35L65, 35L67, 35M30, 35M32, 35Q31, 35R35, 76L05, 76N10 ,Shock (fluid dynamics) ,Astrophysics::High Energy Astrophysical Phenomena ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Nozzle ,Mathematical analysis ,Boundary (topology) ,Euler system ,01 natural sciences ,Physics::Fluid Dynamics ,010104 statistics & probability ,symbols.namesake ,Mathematics - Analysis of PDEs ,FOS: Mathematics ,Free boundary problem ,Euler's formula ,symbols ,Boundary value problem ,0101 mathematics ,Transonic ,Astrophysics::Galaxy Astrophysics ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
This paper concerns the existence of transonic shock solutions to the 2-D steady compressible Euler system in an almost flat finite nozzle ( in the sense that it is a generic small perturbation of a flat one ), under physical boundary conditions proposed by Courant-Friedrichs in \cite{CourantFriedrichs1948}, in which the receiver pressure is prescribed at the exit of the nozzle. In the resulting free boundary problem, the location of the shock-front is one of the most desirable information one would like to determine. However, the location of the normal shock-front in a flat nozzle can be anywhere in the nozzle so that it provides little information on the possible location of the shock-front when the nozzle's boundary is perturbed. So one of the key difficulties in looking for transonic shock solutions is to determine the shock-front. To this end, a free boundary problem for the linearized Euler system will be proposed, whose solution will be taken as an initial approximation for the transonic shock solution. In this paper, a sufficient condition in terms of the geometry of the nozzle and the given exit pressure is derived which yields the existence of the solutions to the proposed free boundary problem. Once an initial approximation is obtained, a further nonlinear iteration could be constructed and proved to lead to a transonic shock solution., 53 pages
- Published
- 2020
49. Area‐Minimizing Currents mod 2 Q : Linear Regularity Theory
- Author
-
Jonas Hirsch, Camillo De Lellis, Salvatore Stuvard, and Andrea Marchese
- Subjects
Pure mathematics ,multiple valued functions, Dirichlet integral, regularity theory, area minimizing currents mod(p), minimal surfaces, linearization ,Generalization ,General Mathematics ,Dimension (graph theory) ,area minimizing currents mod(p) ,linearization ,minimal surfaces ,Dirichlet integral ,01 natural sciences ,010104 statistics & probability ,symbols.namesake ,Mathematics - Analysis of PDEs ,Mod ,FOS: Mathematics ,49Q15, 49Q05, 49N60, 35B65, 35J47 ,0101 mathematics ,Mathematics ,Applied Mathematics ,010102 general mathematics ,Codimension ,regularity theory ,symbols ,multiple valued functions ,Analysis of PDEs (math.AP) - Abstract
We establish a theory of $Q$-valued functions minimizing a suitable generalization of the Dirichlet integral. In a second paper the theory will be used to approximate efficiently area minimizing currents $\mathrm{mod}(p)$ when $p=2Q$, and to establish a first general partial regularity theorem for every $p$ in any dimension and codimension., 37 pages. First part of a two-papers work aimed at establishing a first general partial regularity theory for area minimizing currents modulo p, for any p and in any dimension and codimension. v3 is the final version, to appear on Comm. Pure Appl. Math
- Published
- 2020
50. On Class of Fractional-Order Chaotic or Hyperchaotic Systems in the Context of the Caputo Fractional-Order Derivative
- Author
-
Ameth Ndiaye and Ndolane Sene
- Subjects
Equilibrium point ,Class (set theory) ,Article Subject ,Phase portrait ,General Mathematics ,Chaotic ,Context (language use) ,Lyapunov exponent ,01 natural sciences ,Stability (probability) ,010305 fluids & plasmas ,Nonlinear Sciences::Chaotic Dynamics ,symbols.namesake ,0103 physical sciences ,QA1-939 ,symbols ,Order (group theory) ,Applied mathematics ,010301 acoustics ,Mathematics - Abstract
In this paper, we consider a class of fractional-order systems described by the Caputo derivative. The behaviors of the dynamics of this particular class of fractional-order systems will be proposed and experienced by a numerical scheme to obtain the phase portraits. Before that, we will provide the conditions under which the considered fractional-order system’s solution exists and is unique. The fractional-order impact will be analyzed, and the advantages of the fractional-order derivatives in modeling chaotic systems will be discussed. How the parameters of the model influence the considered fractional-order system will be studied using the Lyapunov exponents. The topological changes of the systems and the detection of the chaotic and hyperchaotic behaviors at the assumed initial conditions and the considered fractional-order systems will also be investigated using the Lyapunov exponents. The investigations related to the Lyapunov exponents in the context of the fractional-order derivative will be the main novelty of this paper. The stability analysis of the model’s equilibrium points has been focused in terms of the Matignon criterion.
- Published
- 2020
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