Fractional Differential Equations

A graph labeling is an assignment of integers to the vertices or edges, or both, subject to certain conditions. Graph labelings were first introduced in the late 1960s. In the intervening years dozens of graph labelings techniques have been studied in over 1000 papers. Finding out what has been done for any particular kind of labeling and keeping up with new discoveries is difficult because of the sheer number of papers and because many of the papers have appeared in journals that are not widely available. In this survey I have collected everything I could find on graph labeling. For the convenience of the reader the survey includes a detailed table of contents and index.

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A Dynamic Survey of Graph Labeling

Formally we have arrived at the middle of the book. So you may need a pause for recovering, a pause which we want to fill up by some fixed point theorems supplementing those which you already met or which you will meet in later chapters. The first group of results centres around Banach’s fixed point theorem. The latter is certainly a nice result since it contains only one simple condition on the map F, since it is so easy to prove and since it nevertheless allows a variety of applications. Therefore it is not astonishing that many mathematicians have been attracted by the question to which extent the conditions on F and the space Ω can be changed so that one still gets the existence of a unique or of at least one fixed point. The number of results produced this way is still finite, but of a statistical magnitude, suggesting at a first glance that only a random sample can be covered by a chapter or even a book of the present size. Fortunately (or unfortunately?) most of the modifications have not found applications up to now, so that there is no reason to write a cookery book about conditions but to write at least a short outline of some ideas indicating that this field can be as interesting as other chapters. A systematic account of more recent ideas and examples in fixed point theory should however be written by one of the true experts. Strange as it is, such a book does not seem to exist though so many people are puzzling out so many results.

Fixed Point Theory

Background
Laparoscopic surgery for acute appendicitis has been proposed to have advantages over conventional surgery.

Objectives
To compare the diagnostic and therapeutic effects of laparoscopic and conventional 'open' surgery.

Search methods
We searched the Cochrane Library, MEDLINE, EMBASE, LILACS, CNKI, SciSearch, study registries, and the congress proceedings of endoscopic surgical societies.

Selection criteria
We included randomized clinical trials comparing laparoscopic (LA) versus open appendectomy (OA) in adults or children. Studies comparing immediate OA versus diagnostic laparoscopy (followed by LA or OA if necessary) were separately identified.

Data collection and analysis
Two reviewers independently assessed trial quality. Missing information or data was requested from the authors. We used odds ratios (OR), relative risks (RR), and 95% confidence intervals (CI) for analysis.

Main results
We included 67 studies, of which 56 compared LA (with or without diagnostic laparoscopy) vs. OA in adults. Wound infections were less likely after LA than after OA (OR 0.43; CI 0.34 to 0.54), but the incidence of intraabdominal abscesses was increased (OR 1.87; CI 1.19 to 2.93). The duration of surgery was 10 minutes (CI 6 to 15) longer for LA. Pain on day 1 after surgery was reduced after LA by 8 mm (CI 5 to 11 mm) on a 100 mm visual analogue scale. Hospital stay was shortened by 1.1 day (CI 0.7 to 1.5). Return to normal activity, work, and sport occurred earlier after LA than after OA. While the operation costs of LA were significantly higher, the costs outside hospital were reduced. Seven studies on children were included, but the results do not seem to be much different when compared to adults. Diagnostic laparoscopy reduced the risk of a negative appendectomy, but this effect was stronger in fertile women (RR 0.20; CI 0.11 to 0.34) as compared to unselected adults (RR 0.37; CI 0.13 to 1.01).

Authors' conclusions
In those clinical settings where surgical expertise and equipment are available and affordable, diagnostic laparoscopy and LA (either in combination or separately) seem to have various advantages over OA. Some of the clinical effects of LA, however, are small and of limited clinical relevance. In spite of the mediocre quality of the available research data, we would generally recommend to use laparoscopy and LA in patients with suspected appendicitis unless laparoscopy itself is contraindicated or not feasible. Especially young female, obese, and employed patients seem to benefit from LA.

Laparoscopic versus open surgery for suspected appendicitis

제6차 가정과 교육과정운영은 어떻게 할 것인가

We prove new Hermite-Hadamard inequalities for conformable fractional integrals by using convex function, -convex, and coordinate convex functions. We prove new Montgomery identity and by using this identity we obtain generalized Hermite-Hadamard type inequalities.

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Conformable Fractional Integrals Versions of Hermite-Hadamard Inequalities and Their Generalizations

The aim of this research work is to investigate the innovative concept of magnetohydrodynamic (MHD) three-dimensional rotational flow of nanoparticles (single-walled carbon nanotubes and multi-walled carbon nanotubes). This flow occurs in the presence of non-linear thermal radiation along with heat generation or absorption based on the Casson fluid model over a stretching sheet. Three common types of liquids (water, engine oil, and kerosene oil) are proposed as a base liquid for these carbon nanotubes (CNTs). The formulation of the problem is based upon the basic equation of the Casson fluid model to describe the non-Newtonian behavior. By implementing the suitable non-dimensional conditions, the model system of equations is altered to provide an appropriate non-dimensional nature. The extremely productive Homotopy Asymptotic Method (HAM) is developed to solve the model equations for velocity and temperature distributions, and a graphical presentation is provided. The influences of conspicuous physical variables on the velocity and temperature distributions are described and discussed using graphs. Moreover, skin fraction coefficient and heat transfer rate (Nusselt number) are tabulated for several values of relevant variables. For ease of comprehension, physical representations of embedded parameters such as radiation parameter ( R d ) , magnetic parameter ( M ) , rotation parameter ( K ), Prandtl number ( P r ), Biot number ( λ ) , and heat generation or absorption parameter ( Q h ) are plotted and deliberated graphically.

The Rotating Flow of Magneto Hydrodynamic Carbon Nanotubes over a Stretching Sheet with the Impact of Non-Linear Thermal Radiation and Heat Generation/Absorption

This research paper investigates entropy generation analysis on two-dimensional nanofluid film flow of Eyring-Powell fluid with heat amd mass transmission over an unsteady porous stretching sheet in the existence of uniform magnetic field (MHD). The flow of liquid films are taken under the impact of thermal radiation. The basic time dependent equations of heat transfer, momentum and mass transfer are modeled and converted to a system of differential equations by employing appropriate similarity transformation with unsteady dimensionless parameters. Entropy analysis is the main focus in this work and the impact of physical parameters on the entropy profile are discussed in detail. The influence of thermophoresis and Brownian motion has been taken in the nanofluids model. An optima approach has been applied to acquire the solution of modeled problem. The convergence of the HAM (Homotopy Analysis Method) has been presented numerically. The disparity of the Nusslet number, Skin friction, Sherwood number and their influence on the velocity, heat and concentration fields has been scrutinized. Moreover, for comprehension, the physical presentation of the embedded parameters are explored analytically for entropy generation and discussed.

Entropy Generation on Nanofluid Thin Film Flow of Eyring–Powell Fluid with Thermal Radiation and MHD Effect on an Unsteady Porous Stretching Sheet

If G is a connected graph, the distance d(u, v) between two vertices u, v is an element of V(G) is the length of a shortest path between them. Let W = {w(1), w(2), ... , w(k)} be an ordered set of vertices of G and let v be a vertex of G. The representation r(v vertical bar W) of v with respect to W is the k-tuple (d(v, w(1)), d(v, w(2)), ... , d(v, w(k))). If distinct vertices of G have distinct representations with respect to W, then W is called a resolving set or locating set for G. A resolving set of minimum cardinality is called a basis for G and this cardinality is the metric dimension of G, denoted by dim(G). A family G of connected graphs is a family with constant metric dimension if dim(G) does not depend upon the choice of G in G. In this paper, we are dealing with the study of metric dimension of Mobius ladders. We prove that Mobius ladder M-n constitute a family of cubic graphs with constant metric dimension and only three vertices suffice to resolve all the vertices of Mobius ladder M-n except when n equivalent to 2(mod 8). It is natural to ask for the characterization of regular graphs with constant metric dimension.

On the metric dimension of Mobius ladders

A New Idea of Fractal-fractional Derivative with Power Law Kernel for Free Convection Heat Transfer in a Channel Flow between Two Static Upright Parallel Plates

Gohar Ali

Papers

Conformable Fractional Integrals Versions of Hermite-Hadamard Inequalities and Their Generalizations

The Rotating Flow of Magneto Hydrodynamic Carbon Nanotubes over a Stretching Sheet with the Impact of Non-Linear Thermal Radiation and Heat Generation/Absorption

Entropy Generation on Nanofluid Thin Film Flow of Eyring–Powell Fluid with Thermal Radiation and MHD Effect on an Unsteady Porous Stretching Sheet

On the metric dimension of Mobius ladders

A New Idea of Fractal-fractional Derivative with Power Law Kernel for Free Convection Heat Transfer in a Channel Flow between Two Static Upright Parallel Plates