Similarity analytical solutions for the Schrӧdinger equation with the Riesz fractional derivative in quantum mechanics

The present article deals with the similarity method to tackle the fractional Schrӧdinger equation where the derivative is defined in the Riesz sense. Moreover, the procedure of reducing a fractional partial differential equation (FPDE) into an ordinary differential equation (ODE) has been efficiently displayed by means of suitable scaled transform to the proposed fractional equation. Furthermore, the ODEs are treated effectively via the Fourier transform. The graphical solutions are also depicted for different fractional derivatives α .     

Fractional diffusion-type equations with exponential and logarithmic differential operators

We deal with some extensions of the space-fractional diffusion equation, which is satisfied by the density of a stable process (see Mainardi et al. (2001)): the first equation considered here is obtained by adding an exponential differential (or shift) operator expressed in terms of the Riesz–Feller derivative. We prove that this produces a random component in the time-argument of the corresponding stable process, which is represented by the so-called Poisson process with drift. Analogously, if we add, to the space-fractional diffusion equation, a logarithmic differential operator involving the Riesz-derivative, we obtain, as a solution, the transition semigroup of a stable process subordinated by an independent gamma subordinator with drift. Finally, we show that an extension of the space-fractional diffusion equation, containing both the fractional shift operator and the Feller integral, is satisfied by the transition density of the process obtained by time-changing the stable process with an independent linear birth process with drift.     

A note on the Hamilton–Waterloo problem with C8-factors and Cm-factors

In this paper, we almost completely solve the Hamilton–Waterloo problem with $C_8$-factors and $C_m$-factors where the number of vertices is a multiple of $8m$.     

Dyck paths with a first return decomposition constrained by height

We study the enumeration of Dyck paths having a first return decomposition with special properties based on a height constraint. We exhibit new restricted sets of Dyck paths counted by the Motzkin numbers, and we give a constructive bijection between these objects and Motzkin paths. As a byproduct, we provide a generating function for the number of Motzkin paths of height $k$ with a flat (resp. with no flats) at the maximal height.     

Riesz potential estimates for a class of double phase problems

A double phase problem with a bounded Borel measure on the right hand side is studied. We prove an optimal pointwise gradient estimate for such a measure data problem via Riesz potentials of the measure under log-Dini continuity assumption on the modulating coefficients. As a consequence, we find an optimized $C^1$ regularity criterion.     

Depth and Stanley depth of the edge ideals of square paths and square cycles

In this paper, we compute depth and Stanley depth for the quotient ring of the edge ideal associated to a square path on n vertices. We also compute depth and Stanley depth for the quotient ring of the edge ideal associated to a square cycle on n vertices, when n≡0,3,4( mod 5), and give tight bounds when n≡1,2( mod 5). We also prove a conjecture of Herzog presented in [5 Herzog, J. (2013). A survey on Stanley depth. In: Bigatti, M. A., Gimenez, P., Sáenz-de-Cabezón, E., eds. Monomial Ideals, Computations and Applications. Lecture Notes in Mathematics, Vol. 2083. Heidelberg: Springer, pp. 3–45. https://arxiv.org/pdf/1702.00781.pdf.[Crossref] , [Google Scholar]], for the edge ideals of square paths and square cycles.     

A generalization of inverse distance weighting and an equivalence relationship to noise-free Gaussian process interpolation via Riesz representation theorem

In this paper, we show the relationship between two seemingly unrelated approximation techniques. On the one hand, a certain class of Gaussian process-based interpolation methods, and on the other hand inverse distance weighting, which has been developed in the context of spatial analysis where there is often a need for interpolating from irregularly spaced data to produce a continuous surface. We develop a generalization of inverse distance weighting and show that it is equivalent to the approximation provided by the class of Gaussian process-based interpolation methods. The equivalence is established via an elegant application of Riesz representation theorem concerning the dual of a Hilbert space. It is thus demonstrated how a classical theorem in linear algebra connects two disparate domains.     

Extremal graphs for the k‐flower

The Turán number of a graph H, , is the maximum number of edges in any graph of order n that does not contain an H as a subgraph. A graph on vertices consisting of k triangles that intersect in exactly one common vertex is called a k‐fan, and a graph consisting of k cycles that intersect in exactly one common vertex is called a k‐flower. In this article, we determine the Turán number of any k‐flower containing at least one odd cycle and characterize all extremal graphs provided n is sufficiently large. Erdős, Füredi, Gould, and Gunderson determined the Turán number for the k‐fan. Our result is a generalization of their result. The addition aim of this article is to draw attention to a powerful tool, the so‐called progressive induction lemma of Simonovits.     

On Hamilton decompositions of infinite circulant graphs

The natural infinite analog of a (finite) Hamilton cycle is a two‐way‐infinite Hamilton path (connected spanning 2‐valent subgraph). Although it is known that every connected 2k‐valent infinite circulant graph has a two‐way‐infinite Hamilton path, there exist many such graphs that do not have a decomposition into k edge‐disjoint two‐way‐infinite Hamilton paths. This contrasts with the finite case where it is conjectured that every 2k‐valent connected circulant graph has a decomposition into k edge‐disjoint Hamilton cycles. We settle the problem of decomposing 2k‐valent infinite circulant graphs into k edge‐disjoint two‐way‐infinite Hamilton paths for , in many cases when , and in many other cases including where the connection set is or .     

Decomposing planar cubic graphs

The 3‐Decomposition Conjecture states that every connected cubic graph can be decomposed into a spanning tree, a 2‐regular subgraph and a matching. We show that this conjecture holds for the class of connected plane cubic graphs.