Cardinality of height function’s range in case of maximally many rectangular islands — computed by cuts

2013 ◽  
Vol 11 (2) ◽  
Author(s):  
Eszter Horváth ◽  
Branimir Šešelja ◽  
Andreja Tepavčević
Keyword(s):  
Height Function ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Combinatorial Properties ◽  
Square Cells

AbstractWe deal with rectangular m×n boards of square cells, using the cut technics of the height function. We investigate combinatorial properties of this function, and in particular we give lower and upper bounds for the number of essentially different cuts. This number turns out to be the cardinality of the height function’s range, in case the height function has maximally many rectangular islands.

scholarly journals Homotopical complexity of a billiard flow on the 3D flat torus with two cylindrical obstacles

2017 ◽  
Vol 39 (4) ◽  
pp. 1071-1081
Author(s):  
CALEB C. MOXLEY ◽  
NANDOR J. SIMANYI
Keyword(s):  
Natural Habitat ◽  
Hyperbolic Group ◽  
Height Function ◽  
Upper Bounds ◽  
Upper And Lower Bounds ◽  
Lower And Upper Bounds ◽  
Flat Torus ◽  
Average Speed ◽  
Rotation Vectors ◽  
Rotation Set

We study the homotopical rotation vectors and the homotopical rotation sets for the billiard flow on the unit flat torus with two disjoint and orthogonal toroidal (cylindrical) scatterers removed from it. The natural habitat for these objects is the infinite cone erected upon the Cantor set $\text{Ends}(G)$ of all ‘ends’ of the hyperbolic group $G=\unicode[STIX]{x1D70B}_{1}(\mathbf{Q})$. An element of $\text{Ends}(G)$ describes the direction in (the Cayley graph of) the group $G$ in which the considered trajectory escapes to infinity, whereas the height function $s$ ($s\geq 0$) of the cone gives us the average speed at which this escape takes place. The main results of this paper claim that the orbits can only escape to infinity at a speed not exceeding $\sqrt{3}$ and, in any direction $e\in \text{Ends}(\unicode[STIX]{x1D70B}_{1}({\mathcal{Q}}))$, the escape is feasible with any prescribed speed $s$, $0\leq s\leq 1/(\sqrt{6}+2\sqrt{3})$. This means that the radial upper and lower bounds for the rotation set $R$ are actually pretty close to each other. Furthermore, we prove the convexity of the set $\mathit{AR}$ of constructible rotation vectors, and that the set of rotation vectors of periodic orbits is dense in $\mathit{AR}$. We also provide effective lower and upper bounds for the topological entropy of the studied billiard flow.

scholarly journals On some generalizations of Horadam’s numbers

Filomat ◽  
2018 ◽  
Vol 32 (14) ◽  
pp. 5037-5052
Author(s):  
Hacène Belbachir ◽  
Amine Belkhir
Keyword(s):  
Upper Bounds ◽  
Circulant Matrices ◽  
Lower And Upper Bounds ◽  
Combinatorial Properties

In this paper, we introduce the incomplete Horadam numbers Wn(k), and hyper-Horadam numbers W(k)n, which generalize the Horadam?s numbers defined by the recurrence Wn = pWn-1 + qWn-2, with W0 = a and W1 = b. We give some combinatorial properties. As an application, we evaluate a lower and upper bounds for the spectral norms of r-circulant matrices associated with these two generalizations. Moreover, we establish a new bounds for the spectral norms of r-circulant matrices associated with Horadam?s numbers in terms of incomplete Horadam and hyper-Horadam numbers.

Extension of the Spatial PDI Model to Three Dimensions

1997 ◽  
Vol 84 (1) ◽  
pp. 176-178
Author(s):  
Frank O'Brien
Keyword(s):  
Population Density ◽  
Dimensional Space ◽  
Finite Lattice ◽  
Three Dimensional ◽  
Upper Bounds ◽  
Three Dimensions ◽  
Lower And Upper Bounds ◽  
Density Index ◽  
Three Dimensional Space

The author's population density index ( PDI) model is extended to three-dimensional distributions. A derived formula is presented that allows for the calculation of the lower and upper bounds of density in three-dimensional space for any finite lattice.

Energy of spherical fuzzy graphs

2020 ◽  
pp. 321-332
Author(s):  
S. Yahya Mohamed ◽  
A. Mohamed Ali
Keyword(s):  
Adjacency Matrix ◽  
Upper Bounds ◽  
Fuzzy Graph ◽  
Lower And Upper Bounds ◽  
Fuzzy Graphs

In this paper, the notion of energy extended to spherical fuzzy graph. The adjacency matrix of a spherical fuzzy graph is defined and we compute the energy of a spherical fuzzy graph as the sum of absolute values of eigenvalues of the adjacency matrix of the spherical fuzzy graph. Also, the lower and upper bounds for the energy of spherical fuzzy graphs are obtained.

scholarly journals Relationship between Age and Value of Information for a Noisy Ornstein–Uhlenbeck Process

Entropy ◽  
2021 ◽  
Vol 23 (8) ◽  
pp. 940
Author(s):  
Zijing Wang ◽  
Mihai-Alin Badiu ◽  
Justin P. Coon
Keyword(s):  
Value Of Information ◽  
Markov Models ◽  
Upper Bounds ◽  
Distribution Functions ◽  
Lower And Upper Bounds ◽  
Uhlenbeck Process ◽  
Source Data ◽  
Non Linear ◽  
Queue Model ◽  
Selection Of

The age of information (AoI) has been widely used to quantify the information freshness in real-time status update systems. As the AoI is independent of the inherent property of the source data and the context, we introduce a mutual information-based value of information (VoI) framework for hidden Markov models. In this paper, we investigate the VoI and its relationship to the AoI for a noisy Ornstein–Uhlenbeck (OU) process. We explore the effects of correlation and noise on their relationship, and find logarithmic, exponential and linear dependencies between the two in three different regimes. This gives the formal justification for the selection of non-linear AoI functions previously reported in other works. Moreover, we study the statistical properties of the VoI in the example of a queue model, deriving its distribution functions and moments. The lower and upper bounds of the average VoI are also analysed, which can be used for the design and optimisation of freshness-aware networks. Numerical results are presented and further show that, compared with the traditional linear age and some basic non-linear age functions, the proposed VoI framework is more general and suitable for various contexts.

The Lower and Upper Bounds of Turán Number for Odd Wheels

2021 ◽  
Vol 37 (3) ◽  
pp. 919-932
Author(s):  
Byeong Moon Kim ◽  
Byung Chul Song ◽  
Woonjae Hwang
Keyword(s):  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Turán Number

scholarly journals Lower and upper bounds of quantum battery power in multiple central spin systems

2021 ◽  
Vol 103 (5) ◽  
Author(s):  
Li Peng ◽  
Wen-Bin He ◽  
Stefano Chesi ◽  
Hai-Qing Lin ◽  
Xi-Wen Guan
Keyword(s):  
Upper Bounds ◽  
Spin Systems ◽  
Lower And Upper Bounds ◽  
Battery Power
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