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Abstract: This paper identifies a natural Hamiltonian on a ten dimensional complex Lie algebra that unravels the mysteries encountered in Kowalewski’s famous paper on the motions of a rigid body around its fixed point under the influence of gravity. This system reveals that the enigmatic conditions of Kowalewski, namely, two principal moments of inertia equal to each other and twice the value of the remaining moment of inertia, and the centre of gravity in the plane spanned by the directions corresponding to the equal moments of inertia, are both necessary and sufficient for the existence of an isospectral representation \(\frac{dL(\lambda )}{dt}=[M(\lambda ),L(\lambda )]\) with a spectral parameter \(\lambda \) . This representation then yields a crucial spectral invariant that naturally accounts for all the integrals of motion, known as Kowalewski type integrals in the literature of the top. This result is fundamentally dependent on a preliminary discovery that the equality of two principal moments of inertia and the placement of the centre of mass in the plane spanned by the corresponding directions is intimately tied to the existence of another integral of motion on whose zero level surface the above spectral representation resides. The link between mechanical tops and Hamiltonian systems on Lie algebras is provided by an earlier result in which it is shown that the equations of mechanical tops with a linear potential, (heavy tops, in particular) can be represented on certain coadjoint orbits in the semi-direct product \({\mathfrak {g}}={\mathfrak {p}}\rtimes {\mathfrak {k}}\) induced by a closed subgroup K of the underlying group G. The passage to complex Lie algebras is motivated by Kowalewski’s mysterious use of complex variables. It is shown that the complex variables in her paper are naturally identified with complex quaternions and the representation of \(\mathfrak {so}(4,{{\mathbb {C}}})\) as the product \(\mathfrak {sl}(2,{{\mathbb {C}}})\times \mathfrak {sl}(2,{{\mathbb {C}}})\) . The paper also shows that all the equations of Kowalewski type can be solved by a uniform integration procedure over the Jacobian of a hyperelliptic curve, as in the original paper of Kowalewski. PubDate: 2021-10-15
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Abstract: We construct the edge-localized stationary states of the nonlinear Schrödinger equation on a general quantum graph in the limit of large mass. Compared to the previous works, we include arbitrary multi-pulse positive states which approach asymptotically a composition of N solitons, each sitting on a bounded (pendant, looping, or internal) edge. We give sufficient conditions on the edge lengths of the graph under which such states exist in the limit of large mass. In addition, we compute the precise Morse index (the number of negative eigenvalues in the corresponding linearized operator) for these multi-pulse states. If N solitons of the edge-localized state reside on the pendant and looping edges, we prove that the Morse index is exactly N. The technical novelty of this work is achieved by avoiding elliptic functions (and related exponentially small scalings) and closing the existence arguments in terms of the Dirichlet-to-Neumann maps for relevant parts of the given graph. We illustrate the general results with three examples of the flower, dumbbell, and single-interval graphs. PubDate: 2021-10-07
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Abstract: We consider some boundary behavior effect for bianalytic functions related to the Dirichlet problem solvability. It is proved that there exist such Jordan domains (even with infinitely smooth but not analytic boundaries) where non-constant bianalytic functions may can tend to zero near the boundary only sufficiently slow. More precisely, we prove that for any \(\alpha \) and \(\beta \) such that \(0<\alpha<\beta <1\) , there exists a Jordan domain \(D=D(\alpha ,\beta )\) possessing the following two properties: (i) there exists a non-constant function of the class \({\mathrm {Lip}}_\alpha ({{\overline{D}}})\) which is bianalytic in D and vanishes identically on the boundary \(\partial D\) of D; (ii) every arc containing in \(\partial D\) is a uniqueness set for functions bianalytic in D and belonging to the class \({\mathrm {Lip}}_\beta ({{\overline{D}}})\) . PubDate: 2021-10-05
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Abstract: In this paper we continue the stability analysis of the model for coinfection with density dependent susceptible population introduced in Andersson et al. (Effect of density dependence on coinfection dynamics. arXiv:2008.09987, 2020). We consider the remaining parameter values left out from Andersson et al. (Effect of density dependence on coinfection dynamics. arXiv:2008.09987, 2020). We look for coexistence equilibrium points, their stability and dependence on the carrying capacity K. Two sets of parameter value are determined, each giving rise to different scenarios for the equilibrium branch parametrized by K. In both scenarios the branch includes coexistence points implying that both coinfection and single infection of both diseases can exist together in a stable state. There are no simple explicit expression for these equilibrium points and we will require a more delicate analysis of these points with a new bifurcation technique adapted to such epidemic related problems. The first scenario is described by the branch of stable equilibrium points which includes a continuum of coexistence points starting at a bifurcation equilibrium point with zero single infection strain #1 and finishing at another bifurcation point with zero single infection strain #2. In the second scenario the branch also includes a section of coexistence equilibrium points with the same type of starting point but the branch stays inside the positive cone after this. The coexistence equilibrium points are stable at the start of the section. It stays stable as long as the product of K and the rate \({\bar{\gamma }}\) of coinfection resulting from two single infections is small but, after this it can reach a Hopf bifurcation and periodic orbits will appear. PubDate: 2021-09-30
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Abstract: The classical theorems of Mittag-Leffler and Weierstrass show that when \((\lambda _n)_{n \geqslant 1}\) is a sequence of distinct points in the open unit disk \(\mathbb {D}\) , with no accumulation points in \(\mathbb {D}\) , and \((w_n)_{n \geqslant 1}\) is any sequence of complex numbers, there is an analytic function \(\varphi \) on \(\mathbb {D}\) for which \(\varphi (\lambda _n) = w_n\) . A celebrated theorem of Carleson [2] characterizes when, for a bounded sequence \((w_n)_{n \geqslant 1}\) , this interpolating problem can be solved with a bounded analytic function. A theorem of Earl [5] goes further and shows that when Carleson’s condition is satisfied, the interpolating function \(\varphi \) can be a constant multiple of a Blaschke product. Results from [4] determine when the interpolating function \(\varphi \) can be taken to be zero free. In this paper we explore when \(\varphi \) can be an outer function. PubDate: 2021-09-24
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Abstract: In this paper, conformable fractional resonant Schrödinger equation is studied. A new auxiliary equation approach is implemented to derive the solutions to the governing equation. Fractional complex traveling wave transform and homogeneous balance technique are the key procedures to implement the method. The predicted solution are set in finite series form of some functions satisfying an ODE of first order second degree. Many kinds of solutions covering exponential, hyperbolic and trigonometric functions are derived based on the suggested method. PubDate: 2021-09-22
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Abstract: In this paper we develop a compartmental model of SIR type (the abbreviation refers to the number of Susceptible, Infected and Recovered people) that models the population dynamics of two diseases that can coinfect. We discuss how the underlying dynamics depends on the carrying capacity K: from a simple dynamics to a more complex. This can also help in understanding the appearance of more complicated dynamics, for example, chaos and periodic oscillations, for large values of K. It is also presented that pathogens can invade in population and their invasion depends on the carrying capacity K which shows that the progression of disease in population depends on carrying capacity. More specifically, we establish all possible scenarios (the so-called transition diagrams) describing an evolution of an (always unique) locally stable equilibrium state (with only non-negative compartments) for fixed fundamental parameters (density independent transmission and vital rates) as a function of the carrying capacity K. An important implication of our results is the following important observation. Note that one can regard the value of K as the natural ‘size’ (the capacity) of a habitat. From this point of view, an isolation of individuals (the strategy which showed its efficiency for COVID-19 in various countries) into smaller resp. larger groups can be modelled by smaller resp. bigger values of K. Then we conclude that the infection dynamics becomes more complex for larger groups, as it fairly maybe expected for values of the reproduction number \(R_0\approx 1\) . We show even more, that for the values \(R_0>1\) there are several (in fact four different) distinguished scenarios where the infection complexity (the number of nonzero infected classes) arises with growing K. Our approach is based on a bifurcation analysis which allows to generalize considerably the previous Lotka-Volterra model considered previously in Ghersheen et al. (Math Meth Appl Sci 42(8), 2019). PubDate: 2021-09-21
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Abstract: The aim of this paper is to study the non-cooperative elliptic Schrödinger systems arising in Bose–Einstein condensation phenomena and some nonlinear optical materials. The more delicate case of systems of negative potentials is considered. We prove the existence and multiplicity of nontrivial solutions for the above system in space dimensions \(N\ge 3\) . Our proofs are based on symmetric mountain pass method, the monotone iterative method, as well as suitable Schrödinger test-function arguments. PubDate: 2021-09-20
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Abstract: We consider three classes of functions defined using the class \({\mathcal {P}}\) of all analytic functions \(p(z)=1+cz+\cdots \) on the open unit disk having positive real part and study several radius problems for these classes. The first class consists of all normalized analytic functions f with \(f/g\in {\mathcal {P}}\) and \(g/(zp)\in {\mathcal {P}}\) for some normalized analytic function g and \(p\in {\mathcal {P}}\) . The second class is defined by replacing the condition \(f/g\in {\mathcal {P}}\) by \( (f/g)-1 <1\) while the other class consists of normalized analytic functions f with \(f/(zp)\in {\mathcal {P}}\) for some \(p\in {\mathcal {P}}\) . We have determined radii so that the functions in these classes to belong to various subclasses of starlike functions. These subclasses includes the classes of starlike functions of order \(\alpha \) , parabolic starlike functions, as well as the classes of starlike functions associated with lemniscate of Bernoulli, reverse lemniscate, sine function, a rational function, cardioid, lune, nephroid and modified sigmoid function. PubDate: 2021-09-18
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Abstract: In the first part of this work, we establish the existence and uniqueness of a local mild solution to deterministic convective Brinkman–Forchheimer (CBF) equations defined on the whole space, by using properties of the heat semigroup and fixed point arguments based on an iterative technique. Moreover, we prove that the solution exists globally. The second part is devoted for establishing the existence and uniqueness of a pathwise mild solution upto a random time to the stochastic CBF equations perturbed by Lévy noise by exploiting the contraction mapping principle. Then by using stopping time arguments, we show that the pathwise mild solution exists globally. We also discuss the local and global solvability of the stochastic CBF equations forced by fractional Brownian noise. PubDate: 2021-09-18
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Abstract: In a self contained presentation we discuss the Gamma Function Spaces. As an application we investigate the boundedness, compactness and closed range of the weighted composition operator on Gamma Spaces. PubDate: 2021-09-14
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Abstract: In this paper, fractional nonlinear-coupled evolution equations and fractional the dimensionless wave equation are proposed and discussed. An efficient algorithm, the q-homotopy analysis transform method, was used to solve such problems. The algorithm used gives an approximate solution in the form of a convergent series, which is somewhat similar to the exact solution while reducing the difficulty of many other approaches. The uniqueness theorem of the expected problem is discussed. The normal frequency of the fractional solution to this problem varies according to the difference of the fractional derivative. PubDate: 2021-09-14
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Abstract: Let A be a closed operator on a separable Hilbert space \(\mathcal {H}\) . In this paper we obtain sufficient conditions for the existence of a solution to the Lyapunov operator equation \( A^*X+X^*A=I\) , under the assumption that it is singular (without a unique solution). Specially, if A is a self-adjoint operator, we derive sufficient conditions for the solution X to be symmetric. We also show that these results hold in the bounded-operator setting and in \(C^*-\) algebras. By doing so, we generalize some known results regarding solvability conditions for algebraic equations in \(C^*-\) algebras. We apply our results to study some functional problems in abstract analysis. PubDate: 2021-09-07 DOI: 10.1007/s13324-021-00596-z
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Abstract: Two different ways of constructing the discrete equations and the corresponding physical laws of continuous Birkhoffian systems are respectively proposed in this paper. The corresponding mathematical methods and geometric structures are formulated and compared. The determining equations of the Noether symmetries are obtained via the Lie point transformations acting on the difference equations. Two types of the discrete conserved quantities of the systems are presented using the structure equation satisfied by the gauge functions. The algorithms can be developed based on these two approaches applied to the nonholonomic systems with symmetries. As a result, the geometric structure and the Noether invariants are numerically preserved. The numerical simulations based on the two approaches demonstrate the high precision and the long-time stability of the algorithms compared with the standard Runge–Kutta method. PubDate: 2021-09-06 DOI: 10.1007/s13324-021-00594-1
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Abstract: In this paper, we prove some inequalities that relate the sup-norm of a univariate complex coefficient polynomial and its derivative, when there is a restriction on its zeros. We further obtain the polar derivative generalizations of the obtained results. The obtained results produce various inequalities that are sharper than the previous ones while taking into account the placement (absolute value) of the zeros and the extremal coefficients of the polynomial. Moreover, some concrete numerical examples are presented, showing that in some situations, the bounds obtained by our results can be considerably sharper than the ones previously known. PubDate: 2021-08-25 DOI: 10.1007/s13324-021-00591-4
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Abstract: The initial-boundary value problem for the nonlinear Schrödinger equation on the half-line with initial data in Sobolev spaces \(H^s(0, \infty )\) , \(1/2< s\leqslant 5/2\) , \(s\ne 3/2\) , and Robin boundary data of appropriate regularity is shown to be locally well-posed in the sense of Hadamard. The proof is through a contraction mapping argument and hence relies crucially on certain estimates for the forced linear counterpart of the nonlinear problem. In particular, the essence of the analysis lies in the pure linear initial-boundary value problem, which corresponds to the case of zero forcing, zero initial data, and nonzero boundary data. This problem, which is studied by taking advantage of the solution formula derived via the unified transform of Fokas, holds an instrumental role in the overall analysis as it reveals the correct function space for the Robin boundary data. PubDate: 2021-08-23 DOI: 10.1007/s13324-021-00589-y
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Abstract: We introduce and study the variable generalized Hölder spaces of holomorphic functions over the unit disc in the complex plane. These spaces are defined either directly in terms of modulus of continuity or in terms of estimates of derivatives near the boundary. We provide conditions of Zygmund type for imbedding of the former into the latter and vice versa. We study mapping properties of variable order fractional integrals in the frameworks of such spaces. PubDate: 2021-08-21 DOI: 10.1007/s13324-021-00587-0
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Abstract: This paper aims at proving the boundedness property of multidimensional singular integral operators associated with \((\varphi ,\psi )\) -harmonic functions, which are connected by the use of two orthogonal basis of the Euclidean space \({{\mathbb {R}}}^m\) . Besides, necessary and sufficient conditions for the solvability of the \({\overline{\partial }}\) -problem for such \((\varphi ,\psi )\) -harmonic functions are described. The basic devices that make it possible to state and prove both results are borrowed from Clifford analysis. PubDate: 2021-08-20 DOI: 10.1007/s13324-021-00590-5
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Abstract: In this paper, we aim to discuss the Noether property of the Riemann boundary value problems in a Banach algebra of continuous functions over simple closed curves and its direct approximate solution through approximation of the principal coefficient, establishing a bound for the error of approximate solution of the problem to the exact solution. PubDate: 2021-08-18 DOI: 10.1007/s13324-021-00582-5
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