Fractional domination in prisms

Matthew Walsh

Displaying similar documents to “Fractional domination in prisms”

Fractional Langevin equation with α-stable noise. A link to fractional ARIMA time series

M. Magdziarz, A. Weron (2007)

Studia Mathematica

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We introduce a fractional Langevin equation with α-stable noise and show that its solution $Y_{κ} (t), t \geq 0$ is the stationary α-stable Ornstein-Uhlenbeck-type process recently studied by Taqqu and Wolpert. We examine the asymptotic dependence structure of $Y_{κ} (t)$ via the measure of its codependence r(θ₁,θ₂,t). We prove that $Y_{κ} (t)$ is not a long-memory process in the sense of r(θ₁,θ₂,t). However, we find two natural continuous-time analogues of fractional ARIMA time series with long memory in the framework of...

Multiplicity results for a class of fractional boundary value problems

Nemat Nyamoradi (2013)

Annales Polonici Mathematici

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We prove the existence of at least three solutions to the following fractional boundary value problem: ⎧ $- d / d t (1 / 2_{0} D_{t}^{- σ} (u^{'} (t)) + 1 / 2_{t} D_{T}^{- σ} (u^{'} (t))) - λ β (t) f (u (t)) - μ γ (t) g (u (t)) = 0$ , a.e. t ∈ [0, T], ⎨ ⎩ u (0) = u (T) = 0, where $_{0} D_{t}^{- σ}$ and $_{t} D_{T}^{- σ}$ are the left and right Riemann-Liouville fractional integrals of order 0 ≤ σ < 1 respectively. The approach is based on a recent three critical points theorem of Ricceri [B. Ricceri, A further refinement of a three critical points theorem, Nonlinear Anal. 74 (2011), 7446-7454].

Generalized $H_{m, m}^{m, 0}$ function fractional integration operators in some classes of analytic functions

V. Kiryakova (1988)

Matematički Vesnik

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Initial value problems for fractional functional differential inclusions with Hadamard type derivative

Nassim Guerraiche, Samira Hamani, Johnny Henderson (2016)

Archivum Mathematicum

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We establish sufficient conditions for the existence of solutions of a class of fractional functional differential inclusions involving the Hadamard fractional derivative with order $α \in (0, 1]$ . Both cases of convex and nonconvex valued right hand side are considered.

Set-valued fractional order differential equations in the space of summable functions

Hussein A.H. Salem (2008)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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In this paper, we study the existence of integrable solutions for the set-valued differential equation of fractional type $(D^{α ₙ} - \sum_{i = 1}^{n - 1} a_{i} D^{α_{i}}) x (t) \in F (t, x (φ (t)))$ , a.e. on (0,1), $I^{1 - α ₙ} x (0) = c$ , αₙ ∈ (0,1), where F(t,·) is lower semicontinuous from ℝ into ℝ and F(·,·) is measurable. The corresponding single-valued problem will be considered first.

Functions of finite fractional variation and their applications to fractional impulsive equations

Dariusz Idczak (2017)

Czechoslovak Mathematical Journal

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We introduce a notion of a function of finite fractional variation and characterize such functions together with their weak $σ$ -additive fractional derivatives. Next, we use these functions to study differential equations of fractional order, containing a $σ$ -additive term—we prove existence and uniqueness of a solution as well as derive a Cauchy formula for the solution. We apply these results to impulsive equations, i.e. equations containing the Dirac measures.

Barenblatt solutions and asymptotic behaviour for a nonlinear fractional heat equation of porous medium type

Juan Luis Vázquez (2014)

Journal of the European Mathematical Society

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We establish the existence, uniqueness and main properties of the fundamental solutions for the fractional porous medium equation introduced in [51]. They are self-similar functions of the form $u (x, t) = t^{- α} f (| x | t^{- β})$ with suitable and $β$ . As a main application of this construction, we prove that the asymptotic behaviour of general solutions is represented by such special solutions. Very singular solutions are also constructed. Among other interesting qualitative properties of the equation we prove an Aleksandrov...

Pointwise inequalities and approximation in fractional Sobolev spaces

David Swanson (2002)

Studia Mathematica

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We prove that a function belonging to a fractional Sobolev space $L^{α, p} (ℝ ⁿ)$ may be approximated in capacity and norm by smooth functions belonging to $C^{m, λ} (ℝ ⁿ)$ , 0 < m + λ < α. Our results generalize and extend those of [12], [4], [14], and [11].

Fractional global domination in graphs

Subramanian Arumugam, Kalimuthu Karuppasamy, Ismail Sahul Hamid (2010)

Discussiones Mathematicae Graph Theory

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Let G = (V,E) be a graph. A function g:V → [0,1] is called a global dominating function (GDF) of G, if for every v ∈ V, $g (N [v]) = \sum_{u \in N [v]} g (u) \geq 1$ and $g (\bar{N (v)}) = \sum_{u \notin N (v)} g (u) \geq 1$ . A GDF g of a graph G is called minimal (MGDF) if for all functions f:V → [0,1] such that f ≤ g and f(v) ≠ g(v) for at least one v ∈ V, f is not a GDF. The fractional global domination number $γ_{f g} (G)$ is defined as follows: $γ_{f g} (G)$ = min|g|:g is an MGDF of G where $| g | = \sum_{v \in V} g (v)$ . In this paper we initiate a study of this parameter.

Density of some sequences modulo 1

Artūras Dubickas (2012)

Colloquium Mathematicae

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Recently, Cilleruelo, Kumchev, Luca, Rué and Shparlinski proved that for each integer a ≥ 2 the sequence of fractional parts ${a ⁿ / n}_{n = 1}^{\infty}$ is everywhere dense in the interval [0,1]. We prove a similar result for all Pisot numbers and Salem numbers α and show that for each c > 0 and each sufficiently large N, every subinterval of [0,1] of length $c N^{- 0.475}$ contains at least one fractional part Q(αⁿ)/n, where Q is a nonconstant polynomial in ℤ[z] and n is an integer satisfying 1 ≤ n ≤ N.

Trace inequalities for fractional integrals in grand Lebesgue spaces

Vakhtang Kokilashvili, Alexander Meskhi (2012)

Studia Mathematica

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rning the boundedness for fractional maximal and potential operators defined on quasi-metric measure spaces from $L^{p), θ} (X, μ)$ to $L^{q), q θ / p} (X, ν)$ (trace inequality), where 1 < p < q < ∞, θ > 0 and μ satisfies the doubling condition in X. The results are new even for Euclidean spaces. For example, from our general results D. Adams-type necessary and sufficient conditions guaranteeing the trace inequality for fractional maximal functions and potentials defined on so-called s-sets in ℝⁿ follow. Trace...

Fractional multilinear integrals with rough kernels on generalized weighted Morrey spaces

Ali Akbulut, Amil Hasanov (2016)

Open Mathematics

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In this paper, we study the boundedness of fractional multilinear integral operators with rough kernels [...] TΩ,αA1,A2,…,Ak, $T_{Ω, α}^{A_{1}, A_{2}, ..., A_{k}},$ which is a generalization of the higher-order commutator of the rough fractional integral on the generalized weighted Morrey spaces Mp,ϕ (w). We find the sufficient conditions on the pair (ϕ1, ϕ2) with w ∈ Ap,q which ensures the boundedness of the operators [...] TΩ,αA1,A2,…,Ak, $T_{Ω, α}^{A_{1}, A_{2}, ..., A_{k}},$ from [...] Mp,φ1wptoMp,φ2wq $M_{p, ϕ_{1}} (w^{p}) to M_{p, ϕ_{2}} (w^{q})$ for 1 < p < q < ∞. In all cases the conditions...

On the minus domination number of graphs

Hailong Liu, Liang Sun (2004)

Czechoslovak Mathematical Journal

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Let $G = (V, E)$ be a simple graph. A $3$ -valued function $f V (G) \to {- 1, 0, 1}$ is said to be a minus dominating function if for every vertex $v \in V$ , $f (N [v]) = \sum_{u \in N [v]} f (u) \geq 1$ , where $N [v]$ is the closed neighborhood of $v$ . The weight of a minus dominating function $f$ on $G$ is $f (V) = \sum_{v \in V} f (v)$ . The minus domination number of a graph $G$ , denoted by $γ^{-} (G)$ , equals the minimum weight of a minus dominating function on $G$ . In this paper, the following two results are obtained. (1) If $G$ is a bipartite graph of order $n$ , then $γ^{-} (G) \geq 4 (\sqrt{n + 1} - 1) - n .$ (2) For any negative integer $k$ and any positive integer...

A new characteristic property of Mittag-Leffler functions and fractional cosine functions

Zhan-Dong Mei, Ji-Gen Peng, Jun-Xiong Jia (2014)

Studia Mathematica

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A new characteristic property of the Mittag-Leffler function $E_{α} (a t^{α})$ with 1 < α < 2 is deduced. Motivated by this property, a new notion, named α-order cosine function, is developed. It is proved that an α-order cosine function is associated with a solution operator of an α-order abstract Cauchy problem. Consequently, an α-order abstract Cauchy problem is well-posed if and only if its coefficient operator generates a unique α-order cosine function.

Commutators with fractional integral operators

Irina Holmes, Robert Rahm, Scott Spencer (2016)

Studia Mathematica

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We investigate weighted norm inequalities for the commutator of a fractional integral operator and multiplication by a function. In particular, we show that, for $μ, λ \in A_{p, q}$ and α/n + 1/q = 1/p, the norm $| | [b, I_{α}] : L^{p} (μ^{p}) \to L^{q} (λ^{q}) | |$ is equivalent to the norm of b in the weighted BMO space BMO(ν), where $ν = μ λ^{- 1}$ . This work extends some of the results on this topic existing in the literature, and continues a line of investigation which was initiated by Bloom in 1985 and was recently developed further by the first author, Lacey,...

Regularity of solutions of the fractional porous medium flow

Luis Caffarelli, Fernando Soria, Juan Luis Vázquez (2013)

Journal of the European Mathematical Society

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We study a porous medium equation with nonlocal diffusion effects given by an inverse fractional Laplacian operator. The precise model is $u_{t} = \nabla \cdot (u \nabla {(- Δ)}^{- s} u), 0 < s < 1$ . The problem is posed in ${x \in ℝ^{n}, t \in ℝ}$ with nonnegative initial data $u (x, 0)$ that are integrable and decay at infinity. A previous paper has established the existence of mass-preserving, nonnegative weak solutions satisfying energy estimates and finite propagation. As main results we establish the boundedness and $C^{α}$ regularity of such weak solutions. Finally, we extend...

Suitable domains to define fractional integrals of Weyl via fractional powers of operators

Celso Martínez, Antonia Redondo, Miguel Sanz (2011)

Studia Mathematica

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We present a new method to study the classical fractional integrals of Weyl. This new approach basically consists in considering these operators in the largest space where they make sense. In particular, we construct a theory of fractional integrals of Weyl by studying these operators in an appropriate Fréchet space. This is a function space which contains the $L^{p} (ℝ)$ -spaces, and it appears in a natural way if we wish to identify these fractional operators with fractional powers of a suitable...

Counterexample to a conjecture on the structure of bipartite partitionable graphs

Richard G. Gibson, Christina M. Mynhardt (2007)

Discussiones Mathematicae Graph Theory

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A graph G is called a prism fixer if γ(G×K₂) = γ(G), where γ(G) denotes the domination number of G. A symmetric γ-set of G is a minimum dominating set D which admits a partition D = D₁∪ D₂ such that $V (G) - N [D_{i}] = D_{j}$ , i,j = 1,2, i ≠ j. It is known that G is a prism fixer if and only if G has a symmetric γ-set. Hartnell and Rall [On dominating the Cartesian product of a graph and K₂, Discuss. Math. Graph Theory 24 (2004), 389-402] conjectured that if G is a connected, bipartite graph such that V(G) can...

Remarks on partially square graphs, hamiltonicity and circumference

Hamamache Kheddouci (2001)

Discussiones Mathematicae Graph Theory

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Given a graph G, its partially square graph G* is a graph obtained by adding an edge (u,v) for each pair u, v of vertices of G at distance 2 whenever the vertices u and v have a common neighbor x satisfying the condition $N_{G} (x) \subseteq N_{G} [u] \cup N_{G} [v]$ , where $N_{G} [x] = N_{G} (x) \cup x$ . In the case where G is a claw-free graph, G* is equal to G². We define $σ ° ₜ = m i n \sum_{x \in S} d_{G} (x) : S i s a n i n d e p e n d e n t s e t i n G * a n d | S | = t$ . We give for hamiltonicity and circumference new sufficient conditions depending on σ° and we improve some known results.

Pseudo almost periodicity of fractional integro-differential equations with impulsive effects in Banach spaces

Zhinan Xia (2017)

Czechoslovak Mathematical Journal

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In this paper, for the impulsive fractional integro-differential equations involving Caputo fractional derivative in Banach space, we investigate the existence and uniqueness of a pseudo almost periodic $P C$ -mild solution. The working tools are based on the fixed point theorems, the fractional powers of operators and fractional calculus. Some known results are improved and generalized. Finally, existence and uniqueness of a pseudo almost periodic $P C$ -mild solution of a two-dimensional impulsive...

Iterated neighborhood graphs

Martin Sonntag, Hanns-Martin Teichert (2012)

Discussiones Mathematicae Graph Theory

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The neighborhood graph N(G) of a simple undirected graph G = (V,E) is the graph $(V, E_{N})$ where $E_{N}$ = a,b | a ≠ b, x,a ∈ E and x,b ∈ E for some x ∈ V. It is well-known that the neighborhood graph N(G) is connected if and only if the graph G is connected and non-bipartite. We present some results concerning the k-iterated neighborhood graph $N^{k} (G) : = N (N (. . . N (G)))$ of G. In particular we investigate conditions for G and k such that $N^{k} (G)$ becomes a complete graph.

Left general fractional monotone approximation theory

George A. Anastassiou (2016)

Applicationes Mathematicae

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We introduce left general fractional Caputo style derivatives with respect to an absolutely continuous strictly increasing function g. We give various examples of such fractional derivatives for different g. Let f be a p-times continuously differentiable function on [a,b], and let L be a linear left general fractional differential operator such that L(f) is non-negative over a closed subinterval I of [a,b]. We find a sequence of polynomials Qₙ of degree ≤n such that L(Qₙ) is non-negative...