6 results
Search Results
2. The VC-dimension of axis-parallel boxes on the Torus
- Author
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Clemens Müllner, Thomas Lachmann, and Pierre Gillibert
- Subjects
FOS: Computer and information sciences ,Statistics and Probability ,Computer Science - Machine Learning ,Numerical Analysis ,Control and Optimization ,Algebra and Number Theory ,Discrete Mathematics (cs.DM) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Torus ,0102 computer and information sciences ,01 natural sciences ,Machine Learning (cs.LG) ,Combinatorics ,VC dimension ,010201 computation theory & mathematics ,FOS: Mathematics ,Mathematics - Combinatorics ,Combinatorics (math.CO) ,0101 mathematics ,Computer Science - Discrete Mathematics ,Mathematics - Abstract
We show in this paper that the VC-dimension of the family of d-dimensional axis-parallel boxes and cubes on the d-dimensional torus are both asymptotically d log 2 d . This is especially surprising as in most other examples the VC-dimension usually grows linearly with d in similar settings.
- Published
- 2022
3. On the co-complex-type k-Fibonacci numbers
- Author
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Sakine Hulku, Anthony G. Shannon, and Ömür Deveci
- Subjects
Combinatorics ,Fibonacci number ,Group (mathematics) ,General Mathematics ,Applied Mathematics ,Modulo ,General Physics and Astronomy ,Statistical and Nonlinear Physics ,Complex type ,Mathematics - Abstract
In this paper, we define the co-complex-type k -Fibonacci numbers and then give the relationships between the k -step Fibonacci numbers and the co-complex-type k -Fibonacci numbers. Also, we produce various properties of the co-complex-type k -Fibonacci numbers such as the generating matrices, the Binet formulas, the combinatorial, permanental and determinantal representations, and the finite sums by matrix methods. In addition, we study the co-complex-type k -Fibonacci sequence modulo m and then we give some results concerning the periods and the ranks of the co-complex-type k -Fibonacci sequences for any k and m . Furthermore, we extend the co-complex-type k -Fibonacci sequences to groups. Finally, we obtain the periods of the co-complex-type 2-Fibonacci sequences in the semidihedral group S D 2 m , ( m ≥ 4 ) with respect to the generating pair ( x , y ) .
- Published
- 2021
4. Statistical properties of mutualistic-competitive random networks
- Author
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J. A. Méndez-Bermúdez, Thomas K. Dm. Peron, Yamir Moreno, and C. T. Martinez-Martinez
- Subjects
Physics - Physics and Society ,Current (mathematics) ,Statistical Mechanics (cond-mat.stat-mech) ,General Mathematics ,Applied Mathematics ,MATRIZES ,Structure (category theory) ,FOS: Physical sciences ,General Physics and Astronomy ,Statistical and Nonlinear Physics ,Disordered Systems and Neural Networks (cond-mat.dis-nn) ,Physics and Society (physics.soc-ph) ,Interval (mathematics) ,Condensed Matter - Disordered Systems and Neural Networks ,Vertex (geometry) ,Combinatorics ,Adjacency matrix ,Focus (optics) ,Random matrix ,Condensed Matter - Statistical Mechanics ,Eigenvalues and eigenvectors ,Mathematics - Abstract
Mutualistic networks are used to study the structure and processes inherent to mutualistic relationships. In this paper, we introduce a random matrix ensemble (RME) representing the adjacency matrices of mutualistic networks composed by two vertex sets of sizes n and m − n . Our RME depends on three parameters: the network size n , the size of the smaller set m , and the connectivity between the two sets α , where α is the ratio of current adjacent pairs over the total number of possible adjacent pairs between the sets. We focus on the spectral, eigenvector and topological properties of the RME by computing, respectively, the ratio of consecutive eigenvalue spacings r , the Shannon entropy of the eigenvectors S , and the Randic index R . First, within a random matrix theory approach (i.e. a statistical approach), we identify a parameter ξ ≡ ξ ( n , m , α ) that scales the average normalized measures X ¯ > (with X representing r , S and R ). Specifically, we show that (i) ξ ∝ α n with a weak dependence on m , and (ii) for ξ 1 / 10 most vertices in the mutualistic network are isolated, while for ξ > 10 the network acquires the properties of a complete network, i.e., the transition from isolated vertices to a complete-like behavior occurs in the interval 1 / 10 ξ 10 . Then, we demonstrate that our statistical approach predicts reasonably well the properties of real-world mutualistic networks; that is, the universal curves X ¯ > vs. ξ show good correspondence with the properties of real-world networks.
- Published
- 2021
5. A note on (s,t)-weak tractability of the weighted star discrepancy of regular grids
- Author
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Jie Zhang
- Subjects
Statistics and Probability ,Combinatorics ,Numerical Analysis ,Control and Optimization ,Algebra and Number Theory ,Matching (graph theory) ,Applied Mathematics ,General Mathematics ,Star (graph theory) ,Mathematics - Abstract
In this paper we study ( s , t ) -weak tractability of the weighted star discrepancy with general coefficients of centered regular grids with different mesh-sizes. We give matching necessary and sufficient conditions on ( s , t ) -weak tractability with s > 0 and t ∈ ( 0 , 1 ] in terms of the weight sequences. We also prove that the weighted star discrepancy has ( s , t ) -weak tractability for all s > 0 and t > 1 .
- Published
- 2021
6. On the contraction ratio of iterated function systems whose attractors are Sierpinski n-gons
- Author
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Judy Said and Abdulrahman Ali Abdulaziz
- Subjects
Yield (engineering) ,General Mathematics ,Applied Mathematics ,Gasket ,Regular polygon ,General Physics and Astronomy ,Chaos game ,Statistical and Nonlinear Physics ,Computer Science::Computational Geometry ,Sierpinski triangle ,Combinatorics ,Iterated function system ,Fractal ,Attractor ,Condensed Matter::Statistical Mechanics ,Mathematics::Metric Geometry ,Mathematics - Abstract
In this paper we apply the chaos game to n -sided regular polygons to generate fractals that are similar to the Sierpinski gasket. We show that for each n -gon, there is an exact ratio that will yield a perfect gasket. We then find a formula for this ratio that depends only on the angle π / n .
- Published
- 2021
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