71 results
Search Results
2. Option pricing formulas under a change of numèraire
- Author
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Antonio Attalienti and Michele Bufalo
- Subjects
Numéraire ,General Mathematics ,lcsh:T57-57.97 ,numèraire ,Process (computing) ,Black–Scholes model ,binomial model ,Binomial distribution ,martingale measures ,Valuation of options ,black-scholes formula ,lcsh:Applied mathematics. Quantitative methods ,Call option ,Node (circuits) ,Binomial options pricing model ,Mathematical economics ,Mathematics - Abstract
We present some formulations of the Cox-Ross-Rubinstein and Black-Scholes formulas for European options obtained through a suitable change of measure, which corresponds to a change of numèraire for the underlying price process. Among other consequences, a closed formula for the price of an European call option at each node of the multi-period binomial tree is achieved, too. Some of the results contained herein, though comparable with analogous ones appearing elsewhere in the financial literature, provide however a supplementary widening and deepening in view of useful applications in the more challenging framework of incomplete markets. This last issue, having the present paper as a preparatory material, will be treated extensively in a forthcoming paper.
- Published
- 2020
3. Control system defined by some integral operator
- Author
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Stanislaw Walczak and Marek Majewski
- Subjects
Discrete mathematics ,implicit function theorem ,General Mathematics ,Existential quantification ,lcsh:T57-57.97 ,010102 general mathematics ,Volterra equation ,Volterra equations ,Nonlinear control ,sensitivity ,01 natural sciences ,Implicit function theorem ,010101 applied mathematics ,Norm (mathematics) ,Control system ,lcsh:Applied mathematics. Quantitative methods ,0101 mathematics ,Mathematics - Abstract
In the paper we consider a nonlinear control system governed by the Volterra integral operator. Using a version of the global implicit function theorem we prove that the control system under consideration is well-posed and robust, i.e. for any admissible control \(u\) there exists a uniquely defined trajectory \(x_{u}\) which continuously depends on control \(u\) and the operator \(u\mapsto x_{u}\) is continuously differentiable. The novelty of this paper is, among others, the application of the Bielecki norm in the space of solutions which allows us to weaken standard assumptions.
- Published
- 2017
4. Semicircular elements induced by p-adic number fields
- Author
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Ilwoo Cho and Palle E. T. Jorgensen
- Subjects
\(C^{*}\)-algebras ,Mathematics::Operator Algebras ,General Mathematics ,lcsh:T57-57.97 ,010102 general mathematics ,Hilbert-space representations ,Algebraic number field ,Free probability ,01 natural sciences ,Prime (order theory) ,free probability ,wighted-semicircular elements ,Combinatorics ,primes ,010104 statistics & probability ,semicircular elements ,lcsh:Applied mathematics. Quantitative methods ,0101 mathematics ,\(p\)-adic number fields \(\mathbb{Q}_{p}\) ,p-adic number ,Mathematics - Abstract
In this paper, we study semicircular-like elements, and semicircular elements induced by \(p\)-adic analysis, for each prime \(p\). Starting from a \(p\)-adic number field \(\mathbb{Q}_{p}\), we construct a Banach \(*\)-algebra \(\mathfrak{LS}_{p}\), for a fixed prime \(p\), and show the generating elements \(Q_{p,j}\) of \(\mathfrak{LS}_{p}\) form weighted-semicircular elements, and the corresponding scalar-multiples \(\Theta_{p,j}\) of \(Q_{p,j}\) become semicircular elements, for all \(j\in\mathbb{Z}\). The main result of this paper is the very construction of suitable linear functionals \(\tau_{p,j}^{0}\) on \(\mathfrak{LS}_{p}\), making \(Q_{p,j}\) be weighted-semicircular, for all \(j\in\mathbb{Z}\).
- Published
- 2017
5. The paired-domination and the upper paired-domination numbers of graphs
- Author
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Włodzimierz Ulatowski
- Subjects
Discrete mathematics ,General Mathematics ,lcsh:T57-57.97 ,Induced subgraph ,paired-domination ,upper paired-domination number ,Graph ,Vertex (geometry) ,Combinatorics ,Dominating set ,lcsh:Applied mathematics. Quantitative methods ,paired-domination number ,Connectivity ,Mathematics - Abstract
In this paper we continue the study of paired-domination in graphs. A paired-dominating set, abbreviated PDS, of a graph \(G\) with no isolated vertex is a dominating set of vertices whose induced subgraph has a perfect matching. The paired-domination number of \(G\), denoted by \(\gamma_{p}(G)\), is the minimum cardinality of a PDS of \(G\). The upper paired-domination number of \(G\), denoted by \(\Gamma_{p}(G)\), is the maximum cardinality of a minimal PDS of \(G\). Let \(G\) be a connected graph of order \(n\geq 3\). Haynes and Slater in [Paired-domination in graphs, Networks 32 (1998), 199-206], showed that \(\gamma_{p}(G)\leq n-1\) and they determine the extremal graphs \(G\) achieving this bound. In this paper we obtain analogous results for \(\Gamma_{p}(G)\). Dorbec, Henning and McCoy in [Upper total domination versus upper paired-domination, Questiones Mathematicae 30 (2007), 1-12] determine \(\Gamma_{p}(P_n)\), instead in this paper we determine \(\Gamma_{p}(C_n)\). Moreover, we describe some families of graphs \(G\) for which the equality \(\gamma_{p}(G)=\Gamma_{p}(G)\) holds.
- Published
- 2015
6. Maillet type theorem for singular first order nonlinear partial differential equations of totally characteristic type. Part II
- Author
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Akira Shirai
- Subjects
formal solution ,Partial differential equation ,Formal power series ,Maillet type theorem ,General Mathematics ,lcsh:T57-57.97 ,Mathematical analysis ,First-order partial differential equation ,Mathematics::Analysis of PDEs ,Order (ring theory) ,Type (model theory) ,Nonlinear system ,Convergence (routing) ,lcsh:Applied mathematics. Quantitative methods ,singular partial differential equations ,Divergence (statistics) ,totally characteristic type ,Mathematics ,nilpotent vector field ,Gevrey order - Abstract
In this paper, we study the following nonlinear first order partial differential equation: \[f(t,x,u,\partial_t u,\partial_x u)=0\quad\text{with}\quad u(0,x)\equiv 0.\] The purpose of this paper is to determine the estimate of Gevrey order under the condition that the equation is singular of a totally characteristic type. The Gevrey order is indicated by the rate of divergence of a formal power series. This paper is a continuation of the previous papers [Convergence of formal solutions of singular first order nonlinear partial differential equations of totally characteristic type, Funkcial. Ekvac. 45 (2002), 187-208] and [Maillet type theorem for singular first order nonlinear partial differential equations of totally characteristic type, Surikaiseki Kenkyujo Kokyuroku, Kyoto University 1431 (2005), 94-106]. Especially the last-mentioned paper is regarded as part I of this paper.
- Published
- 2015
7. Oscillation criteria for third order nonlinear delay differential equations with damping
- Author
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Said R. Grace
- Subjects
Third order nonlinear ,Differential equation ,Oscillation ,General Mathematics ,lcsh:T57-57.97 ,Second order equation ,Mathematical analysis ,Zero (complex analysis) ,Delay differential equation ,oscillation ,third order ,Prime (order theory) ,Combinatorics ,lcsh:Applied mathematics. Quantitative methods ,delay differential equation ,Mathematics - Abstract
This note is concerned with the oscillation of third order nonlinear delay differential equations of the form \[\label{*} \left( r_{2}(t)\left( r_{1}(t)y^{\prime}(t)\right)^{\prime}\right)^{\prime}+p(t)y^{\prime}(t)+q(t)f(y(g(t)))=0.\tag{\(\ast\)}\] In the papers [A. Tiryaki, M. F. Aktas, Oscillation criteria of a certain class of third order nonlinear delay differential equations with damping, J. Math. Anal. Appl. 325 (2007), 54-68] and [M. F. Aktas, A. Tiryaki, A. Zafer, Oscillation criteria for third order nonlinear functional differential equations, Applied Math. Letters 23 (2010), 756-762], the authors established some sufficient conditions which insure that any solution of equation (\(\ast\)) oscillates or converges to zero, provided that the second order equation \[\left( r_{2}(t)z^{\prime }(t)\right)^{\prime}+\left(p(t)/r_{1}(t)\right) z(t)=0\tag{\(\ast\ast\)}\] is nonoscillatory. Here, we shall improve and unify the results given in the above mentioned papers and present some new sufficient conditions which insure that any solution of equation (\(\ast\)) oscillates if equation (\(\ast\ast\)) is nonoscillatory. We also establish results for the oscillation of equation (\(\ast\)) when equation (\(\ast\ast\)) is oscillatory.
- Published
- 2015
8. Spontaneous decay of level from spectral theory point of view
- Author
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Eduard Ianovich
- Subjects
absolutely continuous spectrum ,T57-57.97 ,Spectral theory ,Applied mathematics. Quantitative methods ,Field (physics) ,General Mathematics ,Spectrum (functional analysis) ,Time evolution ,spontaneous decay ,spectral theory ,symbols.namesake ,Quantum mechanics ,Excited state ,self-adjoint operators ,symbols ,Quantum field theory ,Hamiltonian (quantum mechanics) ,quantum field theory ,Eigenvalues and eigenvectors ,Mathematics - Abstract
In quantum field theory it is believed that the spontaneous decay of excited atomic or molecular level is due to the interaction with continuum of field modes. Besides, the atom makes a transition from upper level to lower one so that the probability to find the atom in the excited state tends to zero. In this paper it will be shown that the mathematical model in single-photon approximation may predict another behavior of this probability generally. Namely, the probability to find the atom in the excited state may tend to a nonzero constant so that the atom is not in the pure state finally. This effect is due to that the spectrum of the complete Hamiltonian is not purely absolutely continuous and has a discrete level outside the continuous part. Namely, we state that in the corresponding invariant subspace, determining the time evolution, the spectrum of the complete Hamiltonian when the field is considered in three dimensions may be not purely absolutely continuous and may have an eigenvalue. The appearance of eigenvalue has a threshold character. If the field is considered in two dimensions the spectrum always has an eigenvalue and the decay is absent.
- Published
- 2021
9. Spectral properties of certain operators on the free Hilbert space \mathfrak{F}[H_{1},...,H_{N}] and the semicircular law
- Author
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Ilwoo Cho
- Subjects
Pure mathematics ,T57-57.97 ,jump operators ,shift operators ,Applied mathematics. Quantitative methods ,General Mathematics ,Spectral properties ,Hilbert space ,free hilbert spaces ,symbols.namesake ,jump-shift operators ,separable hilbert spaces ,semicircular elements ,symbols ,Mathematics - Abstract
In this paper, we fix \(N\)-many \(l^2\)-Hilbert spaces \(H_k\) whose dimensions are \(n_{k} \in \mathbb{N}^{\infty}=\mathbb{N} \cup \{\infty\}\), for \(k=1,\ldots,N\), for \(N \in \mathbb{N}\setminus\{1\}\). And then, construct a Hilbert space \(\mathfrak{F}=\mathfrak{F}[H_{1},\ldots,H_{N}]\) induced by \(H_{1},\ldots,H_{N}\), and study certain types of operators on \(\mathfrak{F}\). In particular, we are interested in so-called jump-shift operators. The main results (i) characterize the spectral properties of these operators, and (ii) show how such operators affect the semicircular law induced by \(\bigcup^N_{k=1} \mathcal{B}_{k}\), where \(\mathcal{B}_{k}\) are the orthonormal bases of \(H_{k}\), for \(k=1,\ldots,N\).
- Published
- 2021
10. Certain properties of continuous fractional wavelet transform on Hardy space and Morrey space
- Author
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Bivek Gupta and Amit K. Verma
- Subjects
Pure mathematics ,Mathematics::Functional Analysis ,T57-57.97 ,Applied mathematics. Quantitative methods ,Mathematics::Complex Variables ,General Mathematics ,Mathematics::Classical Analysis and ODEs ,fractional fourier transform ,Fractional wavelet transform ,Hardy space ,Space (mathematics) ,continuous fractional wavelet transform ,Complement (complexity) ,Functional Analysis (math.FA) ,Mathematics - Functional Analysis ,symbols.namesake ,42B10, 42C40, 46E30 ,symbols ,FOS: Mathematics ,hardy space ,morrey space ,Mathematics - Abstract
In this paper we define a new class of continuous fractional wavelet transform (CFrWT) and study its properties in Hardy space and Morrey space. The theory developed generalize and complement some of already existing results.
- Published
- 2021
11. A note on possible density and diameter of counterexamples to the Seymour's second neighborhood conjecture
- Author
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Illia Nalivayko, Oleksiy Zelenskiy, and Valentyna Darmosiuk
- Subjects
Combinatorics ,density of graph ,T57-57.97 ,Conjecture ,Applied mathematics. Quantitative methods ,Computer Science::Discrete Mathematics ,seymour's second neighborhood conjecture ,General Mathematics ,graph theory ,diameter of graph ,Mathematics ,Counterexample - Abstract
Seymour's second neighborhood conjecture states that every simple digraph without loops or 2-cycles contains a vertex whose second neighborhood is at least as large as its first. In this paper we show, that from falsity of Seymour's second neighborhood conjecture it follows that there exist strongly-connected counterexamples with both low and high density (dense and sparse graph). Moreover, we show that if there is a counterexample to conjecture, then it is possible to construct counterexample with any diameter \(k\geq 3\).
- Published
- 2021
12. The achromatic number of K_{6} □ K_{7} is 18
- Author
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Mirko Horňák
- Subjects
Combinatorics ,Achromatic lens ,law ,General Mathematics ,lcsh:T57-57.97 ,complete vertex colouring ,lcsh:Applied mathematics. Quantitative methods ,cartesian product ,achromatic number ,law.invention ,Mathematics - Abstract
A vertex colouring \(f:V(G)\to C\) of a graph \(G\) is complete if for any two distinct colours \(c_1, c_2 \in C\) there is an edge \(\{v_1,v_2\}\in E(G)\) such that \(f(v_i)=c_i\), \(i=1,2\). The achromatic number of \(G\) is the maximum number \(\text{achr}(G)\) of colours in a proper complete vertex colouring of \(G\). In the paper it is proved that \(\text{achr}(K_6 \square K_7)=18\). This result finalises the determination of \(\text{achr}(K_6 \square K_q)\).
- Published
- 2021
13. On the crossing numbers of join products of W_{4}+P_{n} and W_{4}+C_{n}
- Author
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Juraj Valiska and Michal Staš
- Subjects
Mathematics::Combinatorics ,cycle ,General Mathematics ,lcsh:T57-57.97 ,join product ,path ,Computer Science::Computational Geometry ,graph ,Combinatorics ,cyclic permutation ,Computer Science::Discrete Mathematics ,lcsh:Applied mathematics. Quantitative methods ,Join (sigma algebra) ,crossing number ,Mathematics - Abstract
The crossing number \(\mathrm{cr}(G)\) of a graph \(G\) is the minimum number of edge crossings over all drawings of \(G\) in the plane. The main aim of the paper is to give the crossing number of the join product \(W_4+P_n\) and \(W_4+C_n\) for the wheel \(W_4\) on five vertices, where \(P_n\) and \(C_n\) are the path and the cycle on \(n\) vertices, respectively. Yue et al. conjectured that the crossing number of \(W_m+C_n\) is equal to \(Z(m+1)Z(n)+(Z(m)-1) \big \lfloor \frac{n}{2} \big \rfloor + n+ \big\lceil\frac{m}{2}\big\rceil +2\), for all \(m,n \geq 3\), and where the Zarankiewicz's number \(Z(n)=\big \lfloor \frac{n}{2} \big \rfloor \big \lfloor \frac{n-1}{2} \big \rfloor\) is defined for \(n\geq 1\). Recently, this conjecture was proved for \(W_3+C_n\) by Klešč. We establish the validity of this conjecture for \(W_4+C_n\) and we also offer a new conjecture for the crossing number of the join product \(W_m+P_n\) for \(m\geq 3\) and \(n\geq 2\).
- Published
- 2021
14. Some existence results for a nonlocal non-isotropic problem
- Author
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Rachid Bentifour and Sofiane El-Hadi Miri
- Subjects
anisotropic operator ,integro-differential problem ,General Mathematics ,lcsh:T57-57.97 ,Isotropy ,lcsh:Applied mathematics. Quantitative methods ,variational methods ,Mathematical physics ,Mathematics - Abstract
In this paper we deal with the following problem \[\begin{cases}-\sum\limits_{i=1}^{N}\left[ \left( a+b\int\limits_{\, \Omega }\left\vert \partial _{i}u\right\vert ^{p_{i}}dx\right) \partial _{i}\left( \left\vert \partial _{i}u\right\vert ^{p_{i}-2}\partial _{i}u\right) \right]=\frac{f(x)}{u^{\gamma }}\pm g(x)u^{q-1} & in\ \Omega, \\ u\geq 0 & in\ \Omega, \\ u=0 & on\ \partial \Omega, \end{cases}\] where \(\Omega\) is a bounded regular domain in \(\mathbb{R}^{N}\). We will assume without loss of generality that \(1\leq p_{1}\leq p_{2}\leq \ldots\leq p_{N}\) and that \(f\) and \(g\) are non-negative functions belonging to a suitable Lebesgue space \(L^{m}(\Omega)\), \(1\lt q\lt \overline{p}^{\ast}\), \(a\gt 0\), \(b\gt 0\) and \(0\lt \gamma \lt 1.\)
- Published
- 2021
15. Quasilinearization method for finite systems of nonlinear RL fractional differential equations
- Author
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Zachary Denton and Juan Diego Ramírez
- Subjects
Nonlinear system ,lower and upper solutions ,quasilinearization method ,General Mathematics ,lcsh:T57-57.97 ,lcsh:Applied mathematics. Quantitative methods ,Finite system ,fractional differential systems ,Applied mathematics ,Fractional differential ,Mathematics - Abstract
In this paper the quasilinearization method is extended to finite systems of Riemann-Liouville fractional differential equations of order \(0\lt q\lt 1\). Existence and comparison results of the linear Riemann-Liouville fractional differential systems are recalled and modified where necessary. Using upper and lower solutions, sequences are constructed that are monotonic such that the weighted sequences converge uniformly and quadratically to the unique solution of the system. A numerical example illustrating the main result is given.
- Published
- 2020
16. On 2-rainbow domination number of functigraph and its complement
- Author
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Athena Shaminezhad and Ebrahim Vatandoost
- Subjects
Combinatorics ,Domination analysis ,General Mathematics ,cubic graph ,lcsh:T57-57.97 ,lcsh:Applied mathematics. Quantitative methods ,Rainbow ,complement ,functigraph ,Mathematics ,Complement (complexity) ,2-rainbow domination number - Abstract
Let \(G\) be a graph and \(f:V (G)\rightarrow P(\{1,2\})\) be a function where for every vertex \(v\in V(G)\), with \(f(v)=\emptyset\) we have \(\bigcup_{u\in N_{G}(v)} f(u)=\{1,2\}\). Then \(f\) is a \(2\)-rainbow dominating function or a \(2RDF\) of \(G\). The weight of \(f\) is \(\omega(f)=\sum_{v\in V(G)} |f(v)|\). The minimum weight of all \(2\)-rainbow dominating functions is \(2\)-rainbow domination number of \(G\), denoted by \(\gamma_{r2}(G)\). Let \(G_1\) and \(G_2\) be two copies of a graph G with disjoint vertex sets \(V(G_1)\) and \(V(G_2)\), and let \(\sigma\) be a function from \(V(G_1)\) to \(V(G_2)\). We define the functigraph \(C(G,\sigma)\) to be the graph that has the vertex set \(V(C(G,\sigma)) = V(G_1)\cup V(G_2)\), and the edge set \(E(C(G,\sigma)) = E(G_1)\cup E(G_2 \cup \{uv ; u\in V(G_1), v\in V(G_2), v =\sigma(u)\}\). In this paper, \(2\)-rainbow domination number of the functigraph of \(C(G,\sigma)\) and its complement are investigated. We obtain a general bound for \(\gamma_{r2}(C(G,\sigma))\) and we show that this bound is sharp.
- Published
- 2020
17. An inverse backward problem for degenerate two-dimensional parabolic equation
- Author
-
Khalid Atifi, Bouchra Khouiti, and El-Hassan Essoufi
- Subjects
heat equation ,General Mathematics ,lcsh:T57-57.97 ,Degenerate energy levels ,Mathematics::Analysis of PDEs ,Inverse ,Inverse problem ,Regularization (mathematics) ,degenerate equations ,regularization ,Data assimilation ,lcsh:Applied mathematics. Quantitative methods ,Applied mathematics ,inverse problem ,Heat equation ,data assimilation ,adjoint method ,optimization ,Mathematics - Abstract
This paper deals with the determination of an initial condition in the degenerate two-dimensional parabolic equation \[\partial_{t}u-\mathrm{div}\left(a(x,y)I_2\nabla u\right)=f,\quad (x,y)\in\Omega,\; t\in(0,T),\] where \(\Omega\) is an open, bounded subset of \(\mathbb{R}^2\), \(a \in C^1(\bar{\Omega})\) with \(a\geqslant 0\) everywhere, and \(f\in L^{2}(\Omega \times (0,T))\), with initial and boundary conditions \[u(x,y,0)=u_0(x,y), \quad u\mid_{\partial\Omega}=0,\] from final observations. This inverse problem is formulated as a minimization problem using the output least squares approach with the Tikhonov regularization. To show the convergence of the descent method, we prove the Lipschitz continuity of the gradient of the Tikhonov functional. Also we present some numerical experiments to show the performance and stability of the proposed approach.
- Published
- 2020
18. Decompositions of complete 3-uniform hypergraphs into cycles of constant prime length
- Author
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T. Poovaragavan and R Lakshmi
- Subjects
Combinatorics ,Mathematics::Combinatorics ,General Mathematics ,uniform hypergraph ,lcsh:T57-57.97 ,lcsh:Applied mathematics. Quantitative methods ,Cycle decomposition ,cycle decomposition ,Constant (mathematics) ,Nuclear Experiment ,Prime (order theory) ,Mathematics - Abstract
A complete \(3\)-uniform hypergraph of order \(n\) has vertex set \(V\) with \(|V|=n\) and the set of all \(3\)-subsets of \(V\) as its edge set. A \(t\)-cycle in this hypergraph is \(v_1, e_1, v_2, e_2,\dots, v_t, e_t, v_1\) where \(v_1, v_2,\dots, v_t\) are distinct vertices and \(e_1, e_2,\dots, e_t\) are distinct edges such that \(v_i, v_{i+1}\in e_i\) for \(i \in \{1, 2,\dots, t-1\}\) and \(v_t, v_1 \in e_t\). A decomposition of a hypergraph is a partition of its edge set into edge-disjoint subsets. In this paper, we give necessary and sufficient conditions for a decomposition of the complete \(3\)-uniform hypergraph of order \(n\) into \(p\)-cycles, whenever \(p\) is prime.
- Published
- 2020
19. A unique weak solution for a kind of coupled system of fractional Schrödinger equations
- Author
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Fatemeh Abdolrazaghi and Abdolrahman Razani
- Subjects
General Mathematics ,Weak solution ,lcsh:T57-57.97 ,fractional laplacian ,uniqueness ,weak solution ,Schrödinger equation ,Nonlinear system ,symbols.namesake ,lcsh:Applied mathematics. Quantitative methods ,symbols ,Uniqueness ,Fractional Laplacian ,nonlinear systems ,Mathematics ,Mathematical physics - Abstract
In this paper, we prove the existence of a unique weak solution for a class of fractional systems of Schrödinger equations by using the Minty-Browder theorem in the Cartesian space. To this aim, we need to impose some growth conditions to control the source functions with respect to dependent variables.
- Published
- 2020
20. Concentration-compactness results for systems in the Heisenberg group
- Author
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Patrizia Pucci and Letizia Temperini
- Subjects
Heisenberg group, concentration compactness, critical exponents, Hardy terms ,General Mathematics ,lcsh:T57-57.97 ,heisenberg group ,hardy terms ,Compact space ,lcsh:Applied mathematics. Quantitative methods ,Heisenberg group ,concentration-compactness ,critical exponents ,concentration compactness ,Critical exponent ,Mathematical physics ,Mathematics - Abstract
In this paper we complete the study started in [P. Pucci, L. Temperini, Existence for (p,q) critical systems in the Heisenberg group, Adv. Nonlinear Anal. 9 (2020), 895–922] on some variants of the concentration-compactness principle in bounded PS domains \(\Omega\) of the Heisenberg group \(\mathbb{H}^n\). The concentration-compactness principle is a basic tool for treating nonlinear problems with lack of compactness. The results proved here can be exploited when dealing with some kind of elliptic systems involving critical nonlinearities and Hardy terms.
- Published
- 2020
21. Fractional p&q-Laplacian problems with potentials vanishing at infinity
- Author
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Teresa Isernia
- Subjects
Pure mathematics ,ground state solution ,General Mathematics ,media_common.quotation_subject ,lcsh:T57-57.97 ,vanishing potentials ,lcsh:Applied mathematics. Quantitative methods ,fractional \(p\&q\)-laplacian ,Infinity ,Laplace operator ,Mathematics ,media_common - Abstract
In this paper we prove the existence of a positive and a negative ground state weak solution for the following class of fractional \(p&q\)-Laplacian problems \[\begin{aligned} (-\Delta)_{p}^{s} u + (-\Delta)_{q}^{s} u + V(x) (|u|^{p-2}u + |u|^{q-2}u)= K(x) f(u) \quad \text{ in } \mathbb{R}^{N},\end{aligned}\] where \(s\in (0, 1)\), \(1\lt p\lt q \lt\frac{N}{s}\), \(V: \mathbb{R}^{N}\to \mathbb{R}\) and \(K: \mathbb{R}^{N}\to \mathbb{R}\) are continuous, positive functions, allowed for vanishing behavior at infinity, \(f\) is a continuous function with quasicritical growth and the leading operator \((-\Delta)^{s}_{t}\), with \(t\in \{p,q\}\), is the fractional \(t\)-Laplacian operator.
- Published
- 2020
22. Global existence and blow up of solution for semi-linear hyperbolic equation with the product of logarithmic and power-type nonlinearity
- Author
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Wei Lian, Runzhang Xu, and Salik Ahmed
- Subjects
global existence ,Logarithm ,General Mathematics ,lcsh:T57-57.97 ,Mathematical analysis ,logarithmic and polynomial nonlinearity ,Mathematics::Analysis of PDEs ,Type (model theory) ,potential well ,Power (physics) ,Nonlinear system ,Product (mathematics) ,lcsh:Applied mathematics. Quantitative methods ,Hyperbolic partial differential equation ,blow-up ,Mathematics - Abstract
In this paper we consider the semilinear wave equation with the multiplication of logarithmic and polynomial nonlinearities. We establish the global existence and finite time blow up of solutions at three different energy levels (\(E(0)\lt d\), \(E(0)=d\) and \(E(0)\gt 0\)) using potential well method. The results in this article shed some light on using potential wells to classify the solutions of the semilinear wave equation with the product of polynomial and logarithmic nonlinearity.
- Published
- 2020
23. Oscillation criteria for even order neutral difference equations
- Author
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Srinivasan Selvarangam, Sandra Pinelas, Ethiraju Thandapani, and S. A. Rupadevi
- Subjects
even order ,neutral difference equation ,Oscillation ,General Mathematics ,lcsh:T57-57.97 ,Mixed type ,Order (ring theory) ,010103 numerical & computational mathematics ,oscillation ,01 natural sciences ,010101 applied mathematics ,Combinatorics ,Integer ,lcsh:Applied mathematics. Quantitative methods ,asymptotic behavior ,mixed type ,0101 mathematics ,Mathematics - Abstract
In this paper, we present some new sufficient conditions for oscillation of even order nonlinear neutral difference equation of the form \[\Delta^m(x_n+ax_{n-\tau_1}+bx_{n+\tau_2})+p_nx_{n-\sigma_1}^{\alpha}+q_nx_{n+\sigma_2}^{\beta}=0,\quad n\geq n_0\gt0,\] where \(m\geq 2\) is an even integer, using arithmetic-geometric mean inequality. Examples are provided to illustrate the main results.
- Published
- 2019
24. Deformation of semicircular and circular laws via p-adic number fields and sampling of primes
- Author
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Ilwoo Cho and Palle E. T. Jorgensen
- Subjects
circular elements ,General Mathematics ,lcsh:T57-57.97 ,Mathematical analysis ,Sampling (statistics) ,banach \(*\)-probability spaces ,Deformation (meteorology) ,free probability ,primes ,\(p\)-adic number fields ,semicircular elements ,truncated linear functionals ,lcsh:Applied mathematics. Quantitative methods ,Mathematics ,p-adic number - Abstract
In this paper, we study semicircular elements and circular elements in a certain Banach \(*\)-probability space \((\mathfrak{LS},\tau ^{0})\) induced by analysis on the \(p\)-adic number fields \(\mathbb{Q}_{p}\) over primes \(p\). In particular, by truncating the set \(\mathcal{P}\) of all primes for given suitable real numbers \(t\lt s\) in \(\mathbb{R}\), two different types of truncated linear functionals \(\tau_{t_{1}\lt t_{2}}\), and \(\tau_{t_{1}\lt t_{2}}^{+}\) are constructed on the Banach \(*\)-algebra \(\mathfrak{LS}\). We show how original free distributional data (with respect to \(\tau ^{0}\)) are distorted by the truncations on \(\mathcal{P}\) (with respect to \(\tau_{t\lt s}\), and \(\tau_{t\lt s}^{+}\)). As application, distorted free distributions of the semicircular law, and those of the circular law are characterized up to truncation.
- Published
- 2019
25. The intersection graph of annihilator submodules of a module
- Author
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Sh. Payrovi, S. B. Pejman, and S. Babaei
- Subjects
Noetherian ,Simple graph ,Mathematics::Commutative Algebra ,General Mathematics ,Prime ideal ,Computer Science::Information Retrieval ,lcsh:T57-57.97 ,010102 general mathematics ,Computer Science::Computation and Language (Computational Linguistics and Natural Language and Speech Processing) ,0102 computer and information sciences ,Commutative ring ,Intersection graph ,01 natural sciences ,Graph ,Combinatorics ,Annihilator ,prime submodule ,010201 computation theory & mathematics ,intersection annihilator graph ,lcsh:Applied mathematics. Quantitative methods ,Finitely-generated abelian group ,0101 mathematics ,annihilator submodule ,Mathematics - Abstract
Let \(R\) be a commutative ring and \(M\) be a Noetherian \(R\)-module. The intersection graph of annihilator submodules of \(M\), denoted by \(GA(M)\) is an undirected simple graph whose vertices are the classes of elements of \(Z_R(M)\setminus \text{Ann}_R(M)\), for \(a,b \in R\) two distinct classes \([a]\) and \([b]\) are adjacent if and only if \(\text{Ann}_M(a)\cap \text{Ann}_M(b)\neq 0\). In this paper, we study diameter and girth of \(GA(M)\) and characterize all modules that the intersection graph of annihilator submodules are connected. We prove that \(GA(M)\) is complete if and only if \(Z_R(M)\) is an ideal of \(R\). Also, we show that if \(M\) is a finitely generated \(R\)-module with \(r(\text{Ann}_R(M))\neq \text{Ann}_R(M)\) and \(|m-\text{Ass}_R(M)|=1\) and \(GA(M)\) is a star graph, then \(r(\text{Ann}_R(M))\) is not a prime ideal of \(R\) and \(|V(GA(M))|=|\text{Min}\,\text{Ass}_R(M)|+1\).
- Published
- 2019
26. Direct and inverse spectral problems for Dirac systems with nonlocal potentials
- Author
-
Kamila Dębowska and L. P. Nizhnik
- Subjects
nonlocal boundary conditions ,General Mathematics ,lcsh:T57-57.97 ,Dirac (software) ,dirac system ,lcsh:Applied mathematics. Quantitative methods ,nonlocal potential ,Inverse ,inverse spectral problem ,Mathematics ,Mathematical physics - Abstract
The main purposes of this paper are to study the direct and inverse spectral problems of the one-dimensional Dirac operators with nonlocal potentials. Based on informations about the spectrum of the operator, we find the potential and recover the form of the Dirac system. The methods used allow us to reduce the situation to the one-dimensional case. In accordance with the given assumptions and conditions we consider problems in a specific way. We describe the spectrum, the resolvent, the characteristic function etc. Illustrative examples are also given.
- Published
- 2019
27. The existence of consensus of a leader-following problem with Caputo fractional derivative
- Author
-
Ewa Schmeidel
- Subjects
leader-following problem ,General Mathematics ,02 engineering and technology ,caputo fractional differential equation ,nonlinear system ,01 natural sciences ,Leader following ,Schauder fixed point theorem ,Stability theory ,0202 electrical engineering, electronic engineering, information engineering ,Applied mathematics ,0101 mathematics ,schauder fixed point theorem ,Trajectory (fluid mechanics) ,Resolvent ,Mathematics ,lcsh:T57-57.97 ,010102 general mathematics ,Fractional calculus ,ComputingMilieux_GENERAL ,Computer Science::Multiagent Systems ,Nonlinear system ,Kernel (image processing) ,consensus ,lcsh:Applied mathematics. Quantitative methods ,020201 artificial intelligence & image processing - Abstract
In this paper, consensus of a leader-following problem is investigated. The leader-following problem describes a dynamics of the leader and a number of agents. The trajectory of the leader is given. The dynamics of each agent depends on the leader trajectory and others agents' inputs. Here, the leader-following problem is modeled by a system of nonlinear equations with Caputo fractional derivative, which can be rewritten as a system of Volterra equations. The main tools in the investigation are the properties of the resolvent kernel corresponding to the Volterra equations, and Schauder fixed point theorem. By study of the existence of an asymptotically stable solution of a suitable Volterra system, the sufficient conditions for consensus of the leader-following problem are obtained.
- Published
- 2019
28. Global well-posedness of a class of fourth-order strongly damped nonlinear wave equations
- Author
-
Qin Lanlan, Salik Ahmed, Yang Yanbing, and Xu Runzhang
- Subjects
global existence ,Class (set theory) ,General Mathematics ,lcsh:T57-57.97 ,010102 general mathematics ,Mathematical analysis ,Mathematics::Analysis of PDEs ,01 natural sciences ,010101 applied mathematics ,Fourth order ,blow up ,Nonlinear wave equation ,lcsh:Applied mathematics. Quantitative methods ,0101 mathematics ,Finite time ,strong damping ,fourth-order nonlinear wave equation ,Well posedness ,Energy (signal processing) ,Mathematics - Abstract
Global well-posedness and finite time blow up issues for some strongly damped nonlinear wave equation are investigated in the present paper. For subcritical initial energy by employing the concavity method we show a finite time blow up result of the solution. And for critical initial energy we present the global existence, asymptotic behavior and finite time blow up of the solution in the framework of the potential well. Further for supercritical initial energy we give a sufficient condition on the initial data such that the solution blows up in finite time.
- Published
- 2019
29. On edge product cordial graphs
- Author
-
Jaroslav Ivančo
- Subjects
Discrete mathematics ,General Mathematics ,Product (mathematics) ,lcsh:T57-57.97 ,lcsh:Applied mathematics. Quantitative methods ,edge product cordial labelings ,regular graphs ,Edge (geometry) ,dense graphs ,Mathematics ,MathematicsofComputing_DISCRETEMATHEMATICS - Abstract
An edge product cordial labeling is a variant of the well-known cordial labeling. In this paper we characterize graphs admitting an edge product cordial labeling. Using this characterization we investigate the edge product cordiality of broad classes of graphs, namely, dense graphs, dense bipartite graphs, connected regular graphs, unions of some graphs, direct products of some bipartite graphs, joins of some graphs, maximal \(k\)-degenerate and related graphs, product cordial graphs.
- Published
- 2019
30. Graphs with equal domination and certified domination numbers
- Author
-
Mateusz Miotk, Paweł Żyliński, Magdalena Lemańska, Radosław Ziemann, Magda Dettlaff, and Jerzy Topp
- Subjects
Physics::General Physics ,Domination analysis ,General Mathematics ,lcsh:T57-57.97 ,Graph ,Vertex (geometry) ,Combinatorics ,05C69 ,Dominating set ,lcsh:Applied mathematics. Quantitative methods ,FOS: Mathematics ,Mathematics - Combinatorics ,Combinatorics (math.CO) ,certified domination ,Mathematics ,domination - Abstract
A set $D$ of vertices of a graph $G$ is a dominating set of $G$ if every vertex in $V_G-D$ is adjacent to at least one vertex in $D$. The domination number (upper domination number, respectively) of a graph $G$, denoted by $\gamma(G)$ ($\Gamma(G)$, respectively), is the cardinality of a smallest (largest minimal, respectively) dominating set of $G$. A subset $D\subseteq V_G$ is called a certified dominating set of $G$ if $D$ is a dominating set of $G$ and every vertex in $D$ has either zero or at least two neighbors in $V_G-D$. The cardinality of a~smallest (largest minimal, respectively) certified dominating set of $G$ is called the certified upper certified, respectively domination number of $G$ and is denoted by $\gamma_{\rm cer}(G)$ ($\Gamma_{\rm cer}(G)$, respectively). In this paper relations between domination, upper domination, certified domination and upper certified domination numbers of a graph are studied., Comment: 4 figures, 15 pages
- Published
- 2019
31. Oscillatory results for second-order noncanonical delay differential equations
- Author
-
Jozef Džurina, Irena Jadlovská, and Ioannis P. Stavroulakis
- Subjects
Oscillation ,delay ,General Mathematics ,lcsh:T57-57.97 ,010102 general mathematics ,second-order ,Order (ring theory) ,Delay differential equation ,oscillation ,01 natural sciences ,noncanonical ,010101 applied mathematics ,Linear differential equation ,lcsh:Applied mathematics. Quantitative methods ,linear differential equation ,0101 mathematics ,Mathematics ,Mathematical physics - Abstract
The main purpose of this paper is to improve recent oscillation results for the second-order half-linear delay differential equation \[\left(r(t)\left(y'(t)\right)^\gamma\right)'+q(t)y^\gamma(\tau(t))= 0, \quad t\geq t_0,\] under the condition \[\int_{t_0}^{\infty}\frac{\text{d} t}{r^{1/\gamma}(t)} \lt \infty.\] Our approach is essentially based on establishing sharper estimates for positive solutions of the studied equation than those used in known works. Two examples illustrating the results are given.
- Published
- 2019
32. Oscillatory criteria for second order differential equations with several sublinear neutral terms
- Author
-
Blanka Baculíková
- Subjects
Second order differential equations ,Sublinear function ,Oscillation ,sub-linear neutral term ,General Mathematics ,lcsh:T57-57.97 ,Mathematical analysis ,lcsh:Applied mathematics. Quantitative methods ,second order neutral differential equation ,oscillation ,Mathematics - Abstract
In this paper, sufficient conditions for oscillation of the second order differential equations with several sublinear neutral terms are established. The results obtained generalize and extend those reported in the literature. Several examples are included to illustrate the importance and novelty of the presented results.
- Published
- 2019
33. Large and moderate deviation principles for nonparametric recursive kernel distribution estimators defined by stochastic approximation method
- Author
-
Yousri Slaoui
- Subjects
stochastic approximation algorithm ,General Mathematics ,lcsh:T57-57.97 ,Nonparametric statistics ,Estimator ,large and moderate deviations principles ,Stochastic approximation ,Distribution (mathematics) ,Kernel (statistics) ,distribution estimation ,lcsh:Applied mathematics. Quantitative methods ,Applied mathematics ,Moderate deviations ,Mathematics - Abstract
In this paper we prove large and moderate deviations principles for the recursive kernel estimators of a distribution function defined by the stochastic approximation algorithm. We show that the estimator constructed using the stepsize which minimize the Mean Integrated Squared Error (MISE) of the class of the recursive estimators defined by Mokkadem et al. gives the same pointwise large deviations principle (LDP) and moderate deviations principle (MDP) as the Nadaraya kernel distribution estimator.
- Published
- 2019
34. Improved iterative oscillation tests for first-order deviating differential equations
- Author
-
George E. Chatzarakis and Irena Jadlovská
- Subjects
non-monotone argument ,Differential equation ,Oscillation ,General Mathematics ,lcsh:T57-57.97 ,010102 general mathematics ,MathematicsofComputing_NUMERICALANALYSIS ,First order ,01 natural sciences ,nonoscillatory solution ,differential equation ,oscillatory solution ,Argument ,0103 physical sciences ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,lcsh:Applied mathematics. Quantitative methods ,Applied mathematics ,010307 mathematical physics ,0101 mathematics ,MATLAB ,computer ,Mathematics ,computer.programming_language - Abstract
In this paper, improved oscillation conditions are established for the oscillation of all solutions of differential equations with non-monotone deviating arguments and nonnegative coefficients. They lead to a procedure that checks for oscillations by iteratively computing \(\lim \sup\) and \(\lim \inf\) on terms recursively defined on the equation's coefficients and deviating argument. This procedure significantly improves all known oscillation criteria. The results and the improvement achieved over the other known conditions are illustrated by two examples, numerically solved in MATLAB.
- Published
- 2018
35. Backward stochastic variational inequalities driven by multidimensional fractional Brownian motion
- Author
-
Dariusz Borkowski and Katarzyna Jańczak-Borkowska
- Subjects
Fractional Brownian motion ,General Mathematics ,lcsh:T57-57.97 ,Mathematical analysis ,fractional Brownian motion ,backward stochastic differential equation ,Type (model theory) ,backward stochastic variational inequalities ,Stochastic integral ,Variational inequality ,lcsh:Applied mathematics. Quantitative methods ,subdifferential operator ,Uniqueness ,Divergence (statistics) ,Mathematics - Abstract
We study the existence and uniqueness of the backward stochastic variational inequalities driven by \(m\)-dimensional fractional Brownian motion with Hurst parameters \(H_k\) (\(k=1,\ldots m\)) greater than \(1/2\). The stochastic integral used throughout the paper is the divergence type integral.
- Published
- 2018
36. On the stability of some systems of exponential difference equations
- Author
-
Garyfalos Papaschinopoulos, Nikolaos Psarros, and C. J. Schinas
- Subjects
General Mathematics ,lcsh:T57-57.97 ,010102 general mathematics ,Mathematical analysis ,asymptotic behaviour ,Absolute value ,difference equations ,01 natural sciences ,biological dynamics ,Exponential type ,global stability ,Exponential function ,010101 applied mathematics ,centre manifold ,Exponential stability ,Stability theory ,lcsh:Applied mathematics. Quantitative methods ,Uniqueness ,0101 mathematics ,Eigenvalues and eigenvectors ,Mathematics ,Linear stability - Abstract
In this paper we prove the stability of the zero equilibria of two systems of difference equations of exponential type, which are some extensions of an one-dimensional biological model. The stability of these systems is investigated in the special case when one of the eigenvalues is equal to -1 and the other eigenvalue has absolute value less than 1, using centre manifold theory. In addition, we study the existence and uniqueness of positive equilibria, the attractivity and the global asymptotic stability of these equilibria of some related systems of difference equations.
- Published
- 2018
37. Minimal unavoidable sets of cycles in plane graphs
- Author
-
Martina Tamásová and Tomáš Madaras
- Subjects
plane graph ,General Mathematics ,lcsh:T57-57.97 ,0102 computer and information sciences ,01 natural sciences ,Graph ,Planar graph ,Combinatorics ,In plane ,symbols.namesake ,010201 computation theory & mathematics ,lcsh:Applied mathematics. Quantitative methods ,symbols ,set of cycles ,polyhedral graph ,Mathematics ,Polyhedral graph - Abstract
A set \(S\) of cycles is minimal unavoidable in a graph family \(\cal{G}\) if each graph \(G \in \cal{G}\) contains a cycle from \(S\) and, for each proper subset \(S^{\prime}\subset S\), there exists an infinite subfamily \(\cal{G}^{\prime}\subseteq\cal{G}\) such that no graph from \(\cal{G}^{\prime}\) contains a cycle from \(S^{\prime}\). In this paper, we study minimal unavoidable sets of cycles in plane graphs of minimum degree at least 3 and present several graph constructions which forbid many cycle sets to be unavoidable. We also show the minimality of several small sets consisting of short cycles.
- Published
- 2018
38. Adelic analysis and functional analysis on the finite Adele ring
- Author
-
Ilwoo Cho
- Subjects
\(C^{*}\)-algebras ,Measurable function ,Functional analysis ,General Mathematics ,lcsh:T57-57.97 ,010102 general mathematics ,representations ,010103 numerical & computational mathematics ,Operator theory ,Free probability ,01 natural sciences ,Noncommutative geometry ,the Adele ring ,the finite Adele ring ,Combinatorics ,\(p\)-adic number fields ,Adele ring ,lcsh:Applied mathematics. Quantitative methods ,0101 mathematics ,Commutative property ,Mathematics - Abstract
In this paper, we study operator theory on the \(*\)-algebra \(\mathcal{M}_{\mathcal{P}}\), consisting of all measurable functions on the finite Adele ring \(A_{\mathbb{Q}}\), in extended free-probabilistic sense. Even though our \(*\)-algebra \(\mathcal{M}_{\mathcal{P}}\) is commutative, our Adelic-analytic data and properties on \(\mathcal{M}_{\mathcal{P}}\) are understood as certain free-probabilistic results under enlarged sense of (noncommutative) free probability theory (well-covering commutative cases). From our free-probabilistic model on \(A_{\mathbb{Q}}\), we construct the suitable Hilbert-space representation, and study a \(C^{*}\)-algebra \(M_{\mathcal{P}}\) generated by \(\mathcal{M}_{\mathcal{P}}\) under representation. In particular, we focus on operator-theoretic properties of certain generating operators on \(M_{\mathcal{P}}\).
- Published
- 2018
39. Flat structure and potential vector fields related with algebraic solutions to Painlevé VI equation
- Author
-
Mitsuo Kato, Jiro Sekiguchi, and Toshiyuki Mano
- Subjects
Pure mathematics ,Generalization ,Algebraic solution ,General Mathematics ,lcsh:T57-57.97 ,010102 general mathematics ,flat structure ,potential vector field ,algebraic solution ,01 natural sciences ,Flat organization ,Nonlinear Sciences::Exactly Solvable and Integrable Systems ,0103 physical sciences ,lcsh:Applied mathematics. Quantitative methods ,Vector field ,Algebraic function ,010307 mathematical physics ,0101 mathematics ,Algebraic number ,Painlevé VI equation ,Mathematics - Abstract
A potential vector field is a solution of an extended WDVV equation which is a generalization of a WDVV equation. It is expected that potential vector fields corresponding to algebraic solutions of Painleve VI equation can be written by using polynomials or algebraic functions explicitly. The purpose of this paper is to construct potential vector fields corresponding to more than thirty non-equivalent algebraic solutions.
- Published
- 2018
40. Upper bounds for the extended energy of graphs and some extended equienergetic graphs
- Author
-
B. R. Rakshith and Chandrashekar Adiga
- Subjects
energy of a graph ,010304 chemical physics ,Degree (graph theory) ,Spectral radius ,General Mathematics ,lcsh:T57-57.97 ,Inverse ,0102 computer and information sciences ,extended energy of a graph ,01 natural sciences ,Upper and lower bounds ,Graph ,Combinatorics ,Indifference graph ,010201 computation theory & mathematics ,Chordal graph ,extended equienergetic graphs ,0103 physical sciences ,lcsh:Applied mathematics. Quantitative methods ,Adjacency matrix ,Mathematics - Abstract
In this paper, we give two upper bounds for the extended energy of a graph one in terms of ordinary energy, maximum degree and minimum degree of a graph, and another bound in terms of forgotten index, inverse degree sum, order of a graph and minimum degree of a graph which improves an upper bound of Das et al. from [On spectral radius and energy of extended adjacency matrix of graphs, Appl. Math. Comput. 296 (2017), 116-123]. We present a pair of extended equienergetic graphs on \(n\) vertices for \(n\equiv 0(\text{mod } 8)\) starting with a pair of extended equienergetic non regular graphs on \(8\) vertices and also we construct a pair of extended equienergetic graphs on \(n\) vertices for all \(n\geq 9\) starting with a pair of equienergetic regular graphs on \(9\) vertices.
- Published
- 2018
41. Hubtic number in graphs
- Author
-
Veena Mathad, Sultan Senan Mahde, and Shadi Ibrahim Khalaf
- Subjects
Vertex (graph theory) ,050210 logistics & transportation ,021103 operations research ,hubtic number ,General Mathematics ,lcsh:T57-57.97 ,05 social sciences ,0211 other engineering and technologies ,02 engineering and technology ,partition ,Combinatorics ,0502 economics and business ,hub number ,lcsh:Applied mathematics. Quantitative methods ,Partition (number theory) ,Mathematics - Abstract
The maximum order of partition of the vertex set \(V(G)\) into hub sets is called hubtic number of \(G\) and denoted by \(\xi(G)\). In this paper we determine the hubtic number of some standard graphs. Also we obtain bounds for \(\xi(G)\). And we characterize the class of all \((p,q)\) graphs for which \(\xi(G)=p\).
- Published
- 2018
42. On domination multisubdivision number of unicyclic graphs
- Author
-
Joanna Raczek
- Subjects
Domination analysis ,business.industry ,General Mathematics ,domination number ,lcsh:T57-57.97 ,Unicyclic graphs ,0102 computer and information sciences ,02 engineering and technology ,Basis (universal algebra) ,trees ,Characterization (mathematics) ,01 natural sciences ,Constructive ,Combinatorics ,unicyclic graphs ,domination multisubdivision number ,010201 computation theory & mathematics ,domination subdivision number ,lcsh:Applied mathematics. Quantitative methods ,0202 electrical engineering, electronic engineering, information engineering ,020201 artificial intelligence & image processing ,business ,Mathematics ,Subdivision - Abstract
The paper continues the interesting study of the domination subdivision number and the domination multisubdivision number. On the basis of the constructive characterization of the trees with the domination subdivision number equal to 3 given in [H. Aram, S.M. Sheikholeslami, O. Favaron, Domination subdivision number of trees, Discrete Math. 309 (2009), 622-628], we constructively characterize all connected unicyclic graphs with the domination multisubdivision number equal to 3. We end with further questions and open problems.
- Published
- 2018
43. Estimation of the distortion risk premium for heavy-tailed losses under serial dependence
- Author
-
Hakim Ouadjed
- Subjects
Estimation ,Hazard (logic) ,Reinsurance ,050208 finance ,extreme value theory ,General Mathematics ,Risk premium ,lcsh:T57-57.97 ,05 social sciences ,mixing processes ,Estimator ,01 natural sciences ,010104 statistics & probability ,Mixing (mathematics) ,Distortion ,0502 economics and business ,Statistics ,lcsh:Applied mathematics. Quantitative methods ,tail index estimation ,0101 mathematics ,Extreme value theory ,Mathematics - Abstract
In the actuarial literature, many authors have studied estimation of the reinsurance premium for heavy tailed i.i.d. sequences, especially for the Proportional Hazard (PH) due to Wang. The main aim of this paper is to extend this estimation for heavy tailed dependent sequences satisfying some mixing dependence structure. In this study we prove that the new estimator is asymptotically normal. The behavior of the estimator is examined using simulation for MA(1) process.
- Published
- 2018
44. Stochastic differential equations for random matrices processes in the nonlinear framework
- Author
-
Hacène Boutabia, Sara Stihi, and Selma Meradji
- Subjects
Particle system ,Pure mathematics ,General Mathematics ,\(G\)-Brownian motion matrix ,lcsh:T57-57.97 ,010102 general mathematics ,eigenvalues ,Motion (geometry) ,eigenvectors ,01 natural sciences ,random matrices ,010104 statistics & probability ,Stochastic differential equation ,Nonlinear system ,Matrix (mathematics) ,Mathematics::Probability ,\(G\)-stochastic differential equations ,lcsh:Applied mathematics. Quantitative methods ,Symmetric matrix ,0101 mathematics ,Random matrix ,Eigenvalues and eigenvectors ,Mathematics - Abstract
In this paper, we investigate the processes of eigenvalues and eigenvectors of a symmetric matrix valued process \(X_{t}\), where \(X_{t}\) is the solution of a general SDE driven by a \(G\)-Brownian motion matrix. Stochastic differential equations of these processes are given. This extends results obtained by P. Graczyk and J. Malecki in [Multidimensional Yamada-Watanabe theorem and its applications to particle systems, J. Math. Phys. 54 (2013), 021503].
- Published
- 2018
45. Circulant matrices: norm, powers, and positivity
- Author
-
Marko Lindner
- Subjects
circulant matrix ,15A60, 15B05, 15B48 ,General Mathematics ,lcsh:T57-57.97 ,Matrix norm ,Functional Analysis (math.FA) ,Combinatorics ,Mathematics - Functional Analysis ,Norm (mathematics) ,spectral norm ,lcsh:Applied mathematics. Quantitative methods ,FOS: Mathematics ,eventually positive semigroups ,Circulant matrix ,Mathematics - Abstract
In their recent paper "The spectral norm of a Horadam circulant matrix", Merikoski, Haukkanen, Mattila and Tossavainen study under which conditions the spectral norm of a general real circulant matrix \({\bf C}\) equals the modulus of its row/column sum. We improve on their sufficient condition until we have a necessary one. Our results connect the above problem to positivity of sufficiently high powers of the matrix \({\bf C^\top C}\). We then generalize the result to complex circulant matrices.
- Published
- 2018
46. On the uniform perfectness of equivariant diffeomorphism groups for principal G manifolds
- Author
-
Kazuhiko Fukui
- Subjects
Group (mathematics) ,General Mathematics ,lcsh:T57-57.97 ,010102 general mathematics ,Dimension (graph theory) ,Lie group ,01 natural sciences ,Manifold ,010101 applied mathematics ,Combinatorics ,equivariant diffeomorphism ,principal \(G\) manifold ,lcsh:Applied mathematics. Quantitative methods ,Equivariant map ,Identity component ,Diffeomorphism ,0101 mathematics ,Mathematics::Symplectic Geometry ,uniform perfectness ,Mathematics - Abstract
We proved in [K. Abe, K. Fukui, On commutators of equivariant diffeomorphisms, Proc. Japan Acad. 54 (1978), 52-54] that the identity component \(\text{Diff}\,^r_{G,c}(M)_0\) of the group of equivariant \(C^r\)-diffeomorphisms of a principal \(G\) bundle \(M\) over a manifold \(B\) is perfect for a compact connected Lie group \(G\) and \(1 \leq r \leq \infty\) (\(r \neq \dim B + 1\)). In this paper, we study the uniform perfectness of the group of equivariant \(C^r\)-diffeomorphisms for a principal \(G\) bundle \(M\) over a manifold \(B\) by relating it to the uniform perfectness of the group of \(C^r\)-diffeomorphisms of \(B\) and show that under a certain condition, \(\text{Diff}\,^r_{G,c}(M)_0\) is uniformly perfect if \(B\) belongs to a certain wide class of manifolds. We characterize the uniform perfectness of the group of equivariant \(C^r\)-diffeomorphisms for principal \(G\) bundles over closed manifolds of dimension less than or equal to 3, and in particular we prove the uniform perfectness of the group for the 3-dimensional case and \(r\neq 4\).
- Published
- 2017
47. On nonexistence of global in time solution for a mixed problem for a nonlinear evolution equation with memory generalizing the Voigt-Kelvin rheological model
- Author
-
Volodymyr Il'kiv, Myroslava Vovk, Petro Pukach, and Zinovii Nytrebych
- Subjects
General Mathematics ,lcsh:T57-57.97 ,010102 general mathematics ,Mathematical analysis ,beam vibrations ,Voigt-Kelvin model ,01 natural sciences ,nonlinear evolution equation ,blowup ,010101 applied mathematics ,Vibration ,memory ,Nonlinear system ,Rheology ,boundary value problem ,lcsh:Applied mathematics. Quantitative methods ,Boundary value problem ,0101 mathematics ,Nonlinear evolution ,Time variable ,Mathematics - Abstract
The paper deals with investigating of the first mixed problem for a fifth-order nonlinear evolutional equation which generalizes well known equation of the vibrations theory. We obtain sufficient conditions of nonexistence of a global solution in time variable.
- Published
- 2017
48. The metric dimension of circulant graphs and their Cartesian products
- Author
-
Kevin Chau and Shonda Gosselin
- Subjects
Discrete mathematics ,Hypergraph ,General Mathematics ,lcsh:T57-57.97 ,010102 general mathematics ,0102 computer and information sciences ,Cartesian product ,01 natural sciences ,metric dimension ,Metric dimension ,Vertex (geometry) ,Combinatorics ,symbols.namesake ,Circulant graph ,circulant graph ,010201 computation theory & mathematics ,Computer Science::Discrete Mathematics ,lcsh:Applied mathematics. Quantitative methods ,symbols ,Congruence class ,0101 mathematics ,Resolving set ,cartesian product ,Circulant matrix ,Mathematics - Abstract
Let \(G=(V,E)\) be a connected graph (or hypergraph) and let \(d(x,y)\) denote the distance between vertices \(x,y\in V(G)\). A subset \(W\subseteq V(G)\) is called a resolving set for \(G\) if for every pair of distinct vertices \(x,y\in V(G)\), there is \(w\in W\) such that \(d(x,w)\neq d(y,w)\). The minimum cardinality of a resolving set for \(G\) is called the metric dimension of \(G\), denoted by \(\beta(G)\). The circulant graph \(C_n(1,2,\ldots,t)\) has vertex set \(\{v_0,v_1,\ldots,v_{n-1}\}\) and edges \(v_iv_{i+j}\) where \(0\leq i\leq n-1\) and \(1\leq j\leq t\) and the indices are taken modulo \(n\) (\(2\leq t\leq\left\lfloor\frac{n}{2}\right\rfloor\)). In this paper we determine the exact metric dimension of the circulant graphs \(C_n(1,2,\ldots,t)\), extending previous results due to Borchert and Gosselin (2013), Grigorious et al. (2014), and Vetrik (2016). In particular, we show that \(\beta(C_n(1,2,\ldots,t))=\beta(C_{n+2t}(1,2,\ldots,t))\) for large enough \(n\), which implies that the metric dimension of these circulants is completely determined by the congruence class of \(n\) modulo \(2t\). We determine the exact value of \(\beta(C_n(1,2,\ldots,t))\) for \(n\equiv 2\bmod 2t\) and \(n\equiv (t+1)\bmod 2t\) and we give better bounds on the metric dimension of these circulants for \(n\equiv 0\bmod 2t\) and \(n\equiv 1 \bmod 2t\). In addition, we bound the metric dimension of Cartesian products of circulant graphs.
- Published
- 2017
49. Non-factorizable C-valued functions induced by finite connected graphs
- Author
-
Ilwoo Cho
- Subjects
Redei zeta functions ,gluing on graphs ,General Mathematics ,Symmetric graph ,Mathematics::Number Theory ,01 natural sciences ,law.invention ,Combinatorics ,Arithmetic zeta function ,symbols.namesake ,High Energy Physics::Theory ,law ,non-factorizable graphs ,Mathematics::Quantum Algebra ,0103 physical sciences ,Line graph ,Cograph ,graph zeta functions ,0101 mathematics ,Mathematics ,Universal graph ,Block graph ,lcsh:T57-57.97 ,010102 general mathematics ,High Energy Physics::Phenomenology ,directed graphs ,Riemann zeta function ,Vertex-transitive graph ,lcsh:Applied mathematics. Quantitative methods ,symbols ,010307 mathematical physics ,graph groupoids - Abstract
In this paper, we study factorizability of \(\mathbb{C}\)-valued formal series at fixed vertices, called the graph zeta functions, induced by the reduced length on the graph groupoids of given finite connected directed graphs. The construction of such functions is motivated by that of Redei zeta functions. In particular, we are interested in (i) "non-factorizability" of such functions, and (ii) certain factorizable functions induced by non-factorizable functions. By constructing factorizable functions from our non-factorizable functions, we study relations between graph zeta functions and well-known number-theoretic objects, the Riemann zeta function and the Euler totient function.
- Published
- 2017
50. On criteria for algebraic independence of collections of functions satisfying algebraic difference relations
- Author
-
Hiroshi Ogawara
- Subjects
Discrete mathematics ,Function field of an algebraic variety ,General Mathematics ,lcsh:T57-57.97 ,010102 general mathematics ,Algebraic extension ,Dimension of an algebraic variety ,Vignéras' multiple gamma functions ,01 natural sciences ,Addition theorem ,Algebraic cycle ,\(q\)-polylogarithm functions ,systems of algebraic difference equations ,algebraic independence ,0103 physical sciences ,Algebraic surface ,lcsh:Applied mathematics. Quantitative methods ,Real algebraic geometry ,Algebraic function ,010307 mathematical physics ,0101 mathematics ,Mathematics ,difference algebra - Abstract
This paper gives conditions for algebraic independence of a collection of functions satisfying a certain kind of algebraic difference relations. As applications, we show algebraic independence of two collections of special functions: (1) Vigneras' multiple gamma functions and derivatives of the gamma function, (2) the logarithmic function, \(q\)-exponential functions and \(q\)-polylogarithm functions. In a similar way, we give a generalization of Ostrowski's theorem.
- Published
- 2017
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