scholarly journals How many modes can a mixture of Gaussians with uniformly bounded means have?

2021 ◽  
Author(s):  
Navin Kashyap ◽  
Manjunath Krishnapur
Keyword(s):  
Upper Bound ◽  
Higher Dimensions ◽  
Explicit Construction ◽  
Mixing Distribution ◽  
Mixture Of Gaussians ◽  
Additional Variance ◽  
Uniformly Bounded

Abstract We show, by an explicit construction, that a mixture of univariate Gaussian densities with variance $1$ and means in $[-A,A]$ can have $\varOmega (A^2)$ modes. This disproves a recent conjecture of Dytso et al. (2020, IEEE Trans. Inf. Theory, 66, 2006–2022) who showed that such a mixture can have at most $O(A^{2})$ modes and surmised that the upper bound could be improved to $O(A)$. Our result holds even if an additional variance constraint is imposed on the mixing distribution. Extending the result to higher dimensions, we exhibit a mixture of Gaussians in ${\mathbb{R}}^{d}$, with identity covariances and means inside ${[-A,A]}^{d}$, that has $\varOmega (A^{2d})$ modes.

scholarly journals Multivariate Lagrange Interpolation at Sinc Points Error Estimation and Lebesgue Constant

2016 ◽  
Vol 8 (4) ◽  
pp. 118 ◽  
Author(s):  
Maha Youssef ◽  
Hany A. El-Sharkawy ◽  
Gerd Baumann
Keyword(s):  
Uniform Convergence ◽  
Error Estimation ◽  
Upper Bound ◽  
Lagrange Interpolation ◽  
Lebesgue Constant ◽  
Explicit Construction ◽  
Interpolation Operator ◽  
Lebesgue Constants ◽  
Set Of Points

This paper gives an explicit construction of multivariate Lagrange interpolation at Sinc points. A nested operator formula for Lagrange interpolation over an $m$-dimensional region is introduced. For the nested Lagrange interpolation, a proof of the upper bound of the error is given showing that the error has an exponentially decaying behavior. For the uniform convergence the growth of the associated norms of the interpolation operator, i.e., the Lebesgue constant has to be taken into consideration. It turns out that this growth is of logarithmic nature $O((log n)^m)$. We compare the obtained Lebesgue constant bound with other well known bounds for Lebesgue constants using different set of points.

scholarly journals Explicit Spectral Decimation for a Class of Self-Similar Fractals

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Sergio A. Hernández ◽  
Federico Menéndez-Conde
Keyword(s):  
Laplace Operator ◽  
Sierpinski Gasket ◽  
Higher Dimensions ◽  
Explicit Construction ◽  
Sierpiński Gasket ◽  
Infinite Collection ◽  
Self Similar ◽  
The Laplace Operator

The method of spectral decimation is applied to an infinite collection of self-similar fractals. The sets considered are a generalization of the Sierpinski Gasket to higher dimensions; they belong to the class of nested fractals and are thus very symmetric. An explicit construction is given to obtain formulas for the eigenvalues of the Laplace operator acting on these fractals.

scholarly journals Improved Results on FuzzyH∞Filter Design for T-S Fuzzy Systems

2010 ◽  
Vol 2010 ◽  
pp. 1-21 ◽  
Author(s):  
Jiyao An ◽  
Guilin Wen ◽  
Wei Xu
Keyword(s):  
Upper Bound ◽  
Fuzzy Systems ◽  
Filter Design ◽  
Linear Matrix ◽  
Time Varying ◽  
Bounded Interval ◽  
Time Varying Delay ◽  
Error System ◽  
Varying Delay ◽  
Uniformly Bounded

The fuzzyH∞filter design problem for T-S fuzzy systems with interval time-varying delay is investigated. The delay is considered as the time-varying delay being either differentiable uniformly bounded with delay derivative in bounded interval or fast varying (with no restrictions on the delay derivative). A novel Lyapunov-Krasovskii functional is employed and a tighter upper bound of its derivative is obtained. The resulting criterion thus has advantages over the existing ones since we estimate the upper bound of the derivative of Lyapunov-Krasovskii functional without ignoring some useful terms. A fuzzyH∞filter is designed to ensure that the filter error system is asymptotically stable and has a prescribedH∞performance level. An improved delay-derivative-dependent condition for the existence of such a filter is derived in the form of linear matrix inequalities (LMIs). Finally, numerical examples are given to show the effectiveness of the proposed method.

scholarly journals Set-Systems with Restricted Multiple Intersections

10.37236/1625 ◽  
2002 ◽  
Vol 9 (1) ◽  
Author(s):  
Vince Grolmusz
Keyword(s):  
Upper Bound ◽  
Explicit Construction ◽  
Partial Answer ◽  
Residue Classes ◽  
Set Systems ◽  
Element Set ◽  
Multiple Intersections

We give a generalization for the Deza-Frankl-Singhi Theorem in case of multiple intersections. More exactly, we prove, that if ${\cal H}$ is a set-system, which satisfies that for some $k$, the $k$-wise intersections occupy only $\ell$ residue-classes modulo a $p$ prime, while the sizes of the members of ${\cal H}$ are not in these residue classes, then the size of ${\cal H}$ is at most $$(k-1)\sum_{i=0}^{\ell}{n\choose i}$$ This result considerably strengthens an upper bound of Füredi (1983), and gives partial answer to a question of T. Sós (1976). As an application, we give a direct, explicit construction for coloring the $k$-subsets of an $n$ element set with $t$ colors, such that no monochromatic complete hypergraph on $$\exp{(c(\log m)^{1/t}(\log \log m)^{1/(t-1)})}$$ vertices exists.

scholarly journals On non-Kähler degrees of complex manifolds

2019 ◽  
Vol 19 (1) ◽  
pp. 65-69 ◽  
Author(s):  
Daniele Angella ◽  
Adriano Tomassini ◽  
Misha Verbitsky
Keyword(s):  
Upper Bound ◽  
Betti Numbers ◽  
Higher Dimensions ◽  
Complex Manifolds ◽  
Complex Surfaces ◽  
Compact Complex Surfaces

Abstract We study cohomological properties of complex manifolds. In particular, under suitable metric conditions, we extend to higher dimensions a result by A. Teleman, which provides an upper bound for the Bott– Chern cohomology in terms of Betti numbers for compact complex surfaces according to the dichotomy b1 even or odd.

scholarly journals Completely order bounded maps on non-commutative $${\varvec{L_p}}$$-spaces

2021 ◽  
Vol 6 (3) ◽  
Author(s):  
Erwin Neuhardt
Keyword(s):  
Upper Bound ◽  
Von Neumann Algebra ◽  
Von Neumann ◽  
Linear Map ◽  
Injective Von Neumann Algebra ◽  
Uniformly Bounded

AbstractWe define norms on $$L_p({\mathcal {M}}) \otimes M_n$$ L p ( M ) ⊗ M n where $${\mathcal {M}}$$ M is a von Neumann algebra and $$M_n$$ M n is the space of complex $$n \times n$$ n × n matrices. We show that a linear map $$T: L_p({\mathcal {M}}) \rightarrow L_q({\mathcal {N}})$$ T : L p ( M ) → L q ( N ) is decomposable if $${\mathcal {N}}$$ N is an injective von Neumann algebra, the maps $$T \otimes Id_{M_n}$$ T ⊗ I d M n have a common upper bound with respect to our defined norms, and $$p = \infty $$ p = ∞ or $$q = 1$$ q = 1 . For $$2p< q < \infty $$ 2 p < q < ∞ we give an example of a map $$T$$ T with uniformly bounded maps $$T \otimes Id_{M_n}$$ T ⊗ I d M n which is not decomposable.

scholarly journals Examining the effect of quantum strategies on symmetric conflicting interest games

2017 ◽  
Vol 15 (05) ◽  
pp. 1750033
Author(s):  
Katarzyna Bolonek-Lasoń
Keyword(s):  
Upper Bound ◽  
Bell Inequality ◽  
Explicit Construction ◽  
Player Game ◽  
Quantum Strategies ◽  
Quantum Counterpart

The explicit construction is presented of two-player game satisfying: (i) symmetry with respect to the permutation of the players; (ii) the existence of upper bound on total payoff following from Bell inequality; (iii) the existence of unfair equilibrium with total payoff saturating the above bound. The quantum counterpart of the game is considered which possesses only fair equilibria and strategies outperforming the classical ones.

scholarly journals Serre–Tate theory for Calabi–Yau varieties

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Piotr Achinger ◽  
Maciej Zdanowicz
Keyword(s):  
Weak Sense ◽  
Higher Dimensions ◽  
Explicit Construction ◽  
K3 Surfaces ◽  
Canonical Coordinates ◽  
Canonical Class ◽  
Witt Vectors ◽  
Hodge Filtration ◽  
Relative Version ◽  
Simply Connected

Abstract Classical Serre–Tate theory describes deformations of ordinary abelian varieties. It implies that every such variety has a canonical lift to characteristic zero and equips the base of its universal deformation with a Frobenius lifting and canonical multiplicative coordinates. A variant of this theory has been obtained for ordinary K3 surfaces by Nygaard and Ogus. In this paper, we construct canonical liftings modulo p 2 {p^{2}} of varieties with trivial canonical class which are ordinary in the weak sense that the Frobenius acts bijectively on the top cohomology of the structure sheaf. Consequently, we obtain a Frobenius lifting on the moduli space of such varieties. The quite explicit construction uses Frobenius splittings and a relative version of Witt vectors of length two. If the variety has unobstructed deformations and bijective first higher Hasse–Witt operation, the Frobenius lifting gives rise to canonical coordinates. One of the key features of our liftings is that the crystalline Frobenius preserves the Hodge filtration. We also extend Nygaard’s approach from K3 surfaces to higher dimensions, and show that no non-trivial families of such varieties exist over simply connected bases with no global one-forms.

scholarly journals Edge Forcing in Butterfly Networks

2021 ◽  
Vol 182 (3) ◽  
pp. 285-299
Author(s):  
G. Jessy Sujana ◽  
T.M. Rajalaxmi ◽  
Indra Rajasingh ◽  
R. Sundara Rajan
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Higher Dimensions ◽  
Zero Forcing ◽  
Minimum Cardinality ◽  
Forcing Number ◽  
Np Complete ◽  
Forcing Set ◽  
Zero Forcing Set

A zero forcing set is a set S of vertices of a graph G, called forced vertices of G, which are able to force the entire graph by applying the following process iteratively: At any particular instance of time, if any forced vertex has a unique unforced neighbor, it forces that neighbor. In this paper, we introduce a variant of zero forcing set that induces independent edges and name it as edge-forcing set. The minimum cardinality of an edge-forcing set is called the edge-forcing number. We prove that the edge-forcing problem of determining the edge-forcing number is NP-complete. Further, we study the edge-forcing number of butterfly networks. We obtain a lower bound on the edge-forcing number of butterfly networks and prove that this bound is tight for butterfly networks of dimensions 2, 3, 4 and 5 and obtain an upper bound for the higher dimensions.

scholarly journals Bounded Degree Spanners of the Hypercube

10.37236/9074 ◽  
2020 ◽  
Vol 27 (3) ◽  
Author(s):  
Rajko Nenadov ◽  
Mehtab Sawhney ◽  
Benny Sudakov ◽  
Adam Zsolt Wagner
Keyword(s):  
Upper Bound ◽  
Maximum Degree ◽  
Short Note ◽  
Explicit Construction ◽  
Bounded Degree ◽  
Iterated Logarithm

 In this short note we study two questions about the existence of subgraphs of the hypercube $Q_n$ with certain properties. The first question, due to Erdős–Hamburger–Pippert–Weakley, asks whether there exists a bounded degree subgraph of $Q_n$ which has diameter $n$. We answer this question  by giving an explicit construction of such a subgraph with maximum degree at most 120. The second problem concerns properties of $k$-additive spanners of the hypercube, that is, subgraphs of $Q_n$ in which the distance between any two vertices is at most $k$ larger than in $Q_n$. Denoting by $\Delta_{k,\infty}(n)$ the minimum possible maximum degree of a $k$-additive spanner of $Q_n$, Arizumi–Hamburger–Kostochka showed that  $$\frac{n}{\ln n}e^{-4k}\leq \Delta_{2k,\infty}(n)\leq 20\frac{n}{\ln n}\ln \ln n.$$ We improve their upper bound by showing that     $$\Delta_{2k,\infty}(n)\leq 10^{4k} \frac{n}{\ln n}\ln^{(k+1)}n,$$where the last term denotes a $k+1$-fold iterated logarithm.

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