scholarly journals Extremum Modified First Zagreb Connection Index of n-Vertex Trees with Fixed Number of Pendent Vertices

2020 ◽  
Vol 2020 ◽  
pp. 1-6 ◽  
Author(s):  
Sadia Noureen ◽  
Akhlaq Ahmad Bhatti ◽  
Akbar Ali
Keyword(s):  
Molecular Modeling ◽  
Fixed Number ◽  
Graph Invariant ◽  
Connection Number ◽  
Long Time ◽  
Pendent Vertices ◽  
Maximum Minimum

The modified first Zagreb connection index ZC1∗ is a graph invariant that appeared about fifty years ago within a study of molecular modeling, and after a long time, it has been revisited in two papers ((Ali and Trinajstić, 2018) and (Naji et al., 2017)) independently. For a graph G, this graph invariant is defined as ZC1∗G=∑v∈VGdvτv, where dv is the degree of the vertex v and τv is the connection number of v (that is, the number of vertices having distance 2 from v). In this paper, the graphs with maximum/minimum ZC1∗ value are characterized from the class of all n-vertex trees with fixed number of pendent vertices (that are the vertices of degree 1).

scholarly journals Stochastic Models for a Chemostat and Long-Time Behavior

2013 ◽  
Vol 45 (03) ◽  
pp. 822-836 ◽  
Author(s):  
Pierre Collet ◽  
Servet Martínez ◽  
Sylvie Méléard ◽  
Jaime San Martín
Keyword(s):  
Population Size ◽  
Nutrient Concentration ◽  
Bacterial Population ◽  
Fixed Number ◽  
Death Process ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Nutrient Distribution ◽  
Long Time ◽  
Number Of Individuals

We introduce two stochastic chemostat models consisting of a coupled population-nutrient process reflecting the interaction between the nutrient and the bacteria in the chemostat with finite volume. The nutrient concentration evolves continuously but depends on the population size, while the population size is a birth-and-death process with coefficients depending on time through the nutrient concentration. The nutrient is shared by the bacteria and creates a regulation of the bacterial population size. The latter and the fluctuations due to the random births and deaths of individuals make the population go almost surely to extinction. Therefore, we are interested in the long-time behavior of the bacterial population conditioned to nonextinction. We prove the global existence of the process and its almost-sure extinction. The existence of quasistationary distributions is obtained based on a general fixed-point argument. Moreover, we prove the absolute continuity of the nutrient distribution when conditioned to a fixed number of individuals and the smoothness of the corresponding densities.

scholarly journals Stochastic Models for a Chemostat and Long-Time Behavior

2013 ◽  
Vol 45 (3) ◽  
pp. 822-836 ◽  
Author(s):  
Pierre Collet ◽  
Servet Martínez ◽  
Sylvie Méléard ◽  
Jaime San Martín
Keyword(s):  
Population Size ◽  
Nutrient Concentration ◽  
Bacterial Population ◽  
Fixed Number ◽  
Death Process ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Nutrient Distribution ◽  
Long Time ◽  
Number Of Individuals

We introduce two stochastic chemostat models consisting of a coupled population-nutrient process reflecting the interaction between the nutrient and the bacteria in the chemostat with finite volume. The nutrient concentration evolves continuously but depends on the population size, while the population size is a birth-and-death process with coefficients depending on time through the nutrient concentration. The nutrient is shared by the bacteria and creates a regulation of the bacterial population size. The latter and the fluctuations due to the random births and deaths of individuals make the population go almost surely to extinction. Therefore, we are interested in the long-time behavior of the bacterial population conditioned to nonextinction. We prove the global existence of the process and its almost-sure extinction. The existence of quasistationary distributions is obtained based on a general fixed-point argument. Moreover, we prove the absolute continuity of the nutrient distribution when conditioned to a fixed number of individuals and the smoothness of the corresponding densities.

scholarly journals Persistent-Idle Load-Distribution

2020 ◽  
Vol 10 (2) ◽  
pp. 152-169
Author(s):  
Rami Atar ◽  
Isaac Keslassy ◽  
Gal Mendelson ◽  
Ariel Orda ◽  
Shay Vargaftik
Keyword(s):  
Load Distribution ◽  
Stability Region ◽  
Fixed Number ◽  
Small Subset ◽  
Time Intervals ◽  
State Information ◽  
Parallel Server ◽  
Long Time ◽  
The Stability ◽  
Lyapunov Condition

A parallel server system is considered in which a dispatcher routes incoming jobs to a fixed number of heterogeneous servers, each with its own queue. Much effort has been previously made to design policies that use limited state information (e.g., the queue lengths in a small subset of the set of servers, or the identity of the idle servers). However, existing policies either do not achieve the stability region or perform poorly in terms of job completion time. We introduce Persistent-Idle (PI), a new, perhaps counterintuitive, load-distribution policy that is designed to work with limited state information. Roughly speaking, PI always routes to the server that has last been idle. Our main result is that this policy achieves the stability region. Because it operates quite differently from existing policies, our proof method differs from standard arguments in the literature. Specifically, large time properties of reflected random walk, along with a careful choice of a Lyapunov function, are combined to obtain a Lyapunov condition over sufficiently long-time intervals. We also provide simulation results that indicate that job completion times under PI are low for different choices of system parameters, compared with several state-of-the-art load-distribution schemes.

scholarly journals On a Conjecture about the Sombor Index of Graphs

Symmetry ◽  
2021 ◽  
Vol 13 (10) ◽  
pp. 1830
Author(s):  
Kinkar Chandra Das ◽  
Ali Ghalavand ◽  
Ali Reza Ashrafi
Keyword(s):  
Graph Invariant ◽  
Graph Invariants ◽  
Zagreb Indices ◽  
Pendent Vertex ◽  
Vertex Set ◽  
Cyclic Graphs ◽  
Pendent Vertices

Let G be a graph with vertex set V(G) and edge set E(G). A graph invariant for G is a number related to the structure of G which is invariant under the symmetry of G. The Sombor and reduced Sombor indices of G are two new graph invariants defined as SO(G)=∑uv∈E(G)dG(u)2+dG(v)2 and SOred(G)=∑uv∈E(G)dG(u)−12+dG(v)−12, respectively, where dG(v) is the degree of the vertex v in G. We denote by Hn,ν the graph constructed from the star Sn by adding ν edge(s), 0≤ν≤n−2, between a fixed pendent vertex and ν other pendent vertices. Réti et al. [T. Réti, T Došlić and A. Ali, On the Sombor index of graphs, Contrib. Math.3 (2021) 11–18] proposed a conjecture that the graph Hn,ν has the maximum Sombor index among all connected ν-cyclic graphs of order n, where 0≤ν≤n−2. In some earlier works, the validity of this conjecture was proved for ν≤5. In this paper, we confirm that this conjecture is true, when ν=6. The Sombor index in the case that the number of pendent vertices is less than or equal to n−ν−2 is investigated, and the same results are obtained for the reduced Sombor index. Some relationships between Sombor, reduced Sombor, and first Zagreb indices of graphs are also obtained.

Molecular Design and Synthesis of Wettability Alteration Chemicals

2014 ◽  
Vol 884-885 ◽  
pp. 96-99
Author(s):  
Ming Hui Zhou ◽  
Wen Jie Sun
Keyword(s):  
Molecular Modeling ◽  
Molecular Design ◽  
Gas Production ◽  
Dew Point ◽  
Wettability Alteration ◽  
Sharp Drop ◽  
Pressure Drops ◽  
Design And Synthesis ◽  
Dramatic Decline ◽  
Long Time

Many gas-condensate reservoirs experience a sharp drop in gas production owing to condensation near the wellbore as pressure drops lower than the dew point. It has been a challenge for a long time to develop cheaper chemical to stop the dramatic decline in gas production. In this study several molecule formulas are designed, properties of two chemicals for wettability alteration are predicted using molecular modeling method and then synthesized. Analysis result shows high conformity between the prediction and experimental properties.

scholarly journals Distance spectral radius of trees with fixed number of pendent vertices

2013 ◽  
Vol 439 (8) ◽  
pp. 2240-2249 ◽  
Author(s):  
Wenjie Ning ◽  
Liangqi Ouyang ◽  
Mei Lu
Keyword(s):  
Spectral Radius ◽  
Fixed Number ◽  
Pendent Vertices

scholarly journals On the zeroth-order general Randić index, variable sum exdeg index and trees having vertices with prescribed degree

2018 ◽  
Vol 10 (02) ◽  
pp. 1850015 ◽  
Author(s):  
Sohaib Khalid ◽  
Akbar Ali
Keyword(s):  
Real Number ◽  
Fixed Number ◽  
Randić Index ◽  
Zeroth Order ◽  
Positive Real ◽  
Randic Index ◽  
General Randić Index ◽  
General Randic Index ◽  
Pendent Vertices ◽  
Number Of Segments

The zeroth-order general Randić index (usually denoted by [Formula: see text]) and variable sum exdeg index (denoted by [Formula: see text]) of a graph [Formula: see text] are defined as [Formula: see text] and [Formula: see text], respectively, where [Formula: see text] is degree of the vertex [Formula: see text], [Formula: see text] is a positive real number different from 1 and [Formula: see text] is a real number other than [Formula: see text] and [Formula: see text]. A segment of a tree is a path [Formula: see text], whose terminal vertices are branching or/and pendent, and all non-terminal vertices (if exist) of [Formula: see text] have degree 2. For [Formula: see text], let [Formula: see text], [Formula: see text], [Formula: see text] be the collections of all [Formula: see text]-vertex trees having [Formula: see text] pendent vertices, [Formula: see text] segments, [Formula: see text] branching vertices, respectively. In this paper, all the trees with extremum (maximum and minimum) zeroth-order general Randić index and variable sum exdeg index are determined from the collections [Formula: see text], [Formula: see text], [Formula: see text]. The obtained extremal trees for the collection [Formula: see text] are also extremal trees for the collection of all [Formula: see text]-vertex trees having fixed number of vertices with degree 2 (because the number of segments of a tree [Formula: see text] can be determined from the number of vertices of [Formula: see text] having degree 2 and vice versa).

scholarly journals INTERFERENCE OF AN ARRAY OF INDEPENDENT BOSE-EINSTEIN CONDENSATES WITH FIXED NUMBER OF ATOMS

2011 ◽  
Vol 09 (supp01) ◽  
pp. 431-443 ◽  
Author(s):  
SATOSHI ANDO ◽  
KAZUYA YUASA ◽  
MAURO IAZZI
Keyword(s):  
Symmetry Breaking ◽  
Time Of Flight ◽  
Fixed Number ◽  
Interference Fringes ◽  
Asymptotic Values ◽  
Long Time ◽  
Bose Einstein ◽  
The Relationship

Interference of an array of independent Bose-Einstein condensates, whose experiment has been performed recently, is theoretically studied in detail. Even if the number of the atoms in each gas is kept finite and the phases of the gases are not well-defined, interference fringes are observed on each snapshot. The statistics of the snapshot interference patterns, i.e. the average fringe amplitudes and their fluctuations (covariance), are computed analytically, and concise formulas for their asymptotic values for long time of flight are derived. Processes contributing to these quantities are clarified and the relationship with the description on the basis of the symmetry-breaking scenario is revealed.

200kv superconducting cryo electron microscope

1987 ◽  
Vol 45 ◽  
pp. 642-643
Author(s):  
M. Iwatsuki ◽  
Y. Kokubo ◽  
Y. Harada ◽  
J. Lehman
Keyword(s):  
Electron Microscope ◽  
Structure Analysis ◽  
Organic Materials ◽  
Strong Interactions ◽  
Specimen Preparation ◽  
Great Effort ◽  
Electron Microscopic ◽  
Crystal Fields ◽  
Special Skill ◽  
Long Time

In recent years, the electron microscope has been significantly improved in resolution and we can obtain routinely atomic-level high resolution images without any special skill. With this improvement, the structure analysis of organic materials has become one of the interesting targets in the biological and polymer crystal fields.Up to now, X-ray structure analysis has been mainly used for such materials. With this method, however, great effort and a long time are required for specimen preparation because of the need for larger crystals. This method can analyze average crystal structure but is insufficient for interpreting it on the atomic or molecular level. The electron microscopic method for organic materials has not only the advantage of specimen preparation but also the capability of providing various information from extremely small specimen regions, using strong interactions between electrons and the substance. On the other hand, however, this strong interaction has a big disadvantage in high radiation damage.

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