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Register a new userGeneralized Outer Bounds on the Finite Geometric Sum of Ellipsoids
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Added by
Navid Hashemi
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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General results on convex bodies are reviewed and used to derive an exact closed-form parametric formula for the boundary of the geometric (Minkowski) sum of $k$ ellipsoids in $n$-dimensional Euclidean space. Previously this was done through iterative algorithms in which each new ellipsoid was added to an ellipsoid approximation of the sum of the previous ellipsoids. Here we provide one shot formulas to add $k$ ellipsoids directly with no intermediate approximations required. This allows us to observe a new degree of freedom in the family of ellipsoidal bounds on the geometric sum. We demonstrate an application of these tools to compute the reachable set of a discrete-time dynamical system.
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General results on convex bodies are reviewed and used to derive an exact closed-form parametric formula for the boundary of the geometric (Minkowski) sum of $k$ ellipsoids in $n$-dimensional Euclidean space. Previously this was done through iterative algorithms in which each new ellipsoid was added to an ellipsoid approximation of the sum of the previous ellipsoids. Here we provide one shot formulas to add $k$ ellipsoids directly with no intermediate approximations required. This allows us to observe a new degree of freedom in the family of ellipsoidal bounds on the geometric sum. We demonstrate an application of these tools to compute the reachable set of a discrete-time dynamical system.
In this paper we consider an extremal problem in geometry. Let $lambda$ be a real number and $A$, $B$ and $C$ be arbitrary points on the unit circle $Gamma$. We give full characterization of the extremal behavior of the function $f(M,lambda)=MA^lambda+MB^lambda+MC^lambda$, where $M$ is a point on the unit circle as well. We also investigate the extremal behavior of $sum_{i=1}^nXP_i$, where $P_i, i=1,...,n$ are the vertices of a regular $n$-gon and $X$ is a point on $Gamma$, concentric to the circle circumscribed around $P_1...P_n$. We use elementary analytic and purely geometric methods in the proof.
The complex representation of real-valued instantaneous power may be written as the sum of two complex powers, one Hermitian and the other non-Hermitian, or complementary. A virtue of this representation is that it consists of a power triangle rotating around a fixed phasor, thus clarifying what should be meant by the power triangle. The in-phase and quadrature components of complementary power encode for active and non-active power. When instantaneous power is defined for a Thevenin equivalent circuit, these are time-varying real and reactive power components. These claims hold for sinusoidal voltage and current, and for non-sinusoidal voltage and current. Spectral representations of Hermitian, complementary, and instantaneous power show that, frequency-by-frequency, these powers behave exactly as they behave in the single frequency sinusoidal case. Simple hardware diagrams show how instantaneous active and non-active power may be extracted from metered voltage and current, even in certain non-sinusoidal cases.
Over-the-air computation (AirComp), which leverages the superposition property of wireless multiple-access channel (MAC) and the mathematical tool of function representation, has been considered as a promising technique for effective collection and computation of massive sensor data in wireless Big Data applications. In most of the existing work on AirComp, optimal system-parameter design is commonly considered under the peak-power constraint of each sensor. In this paper, we propose an optimal transmitter-receiver (Tx-Rx) parameter design problem to minimize the computation mean-squared error (MSE) of an AirComp system under the sum-power constraint of the sensors. We solve the non-convex problem and obtain a closed-form solution. Also, we investigate another problem that minimizes the sum power of the sensors under the constraint of computation MSE. Our results show that in both of the problems, the sensors with poor and good channel conditions should use less power than the ones with moderate channel conditions.
Linear programming (polynomial) techniques are used to obtain lower and upper bounds for the potential energy of spherical designs. This approach gives unified bounds that are valid for a large class of potential functions. Our lower bounds are optimal for absolutely monotone potentials in the sense that for the linear programming technique they cannot be improved by using polynomials of the same or lower degree. When additional information about the structure (upper and lower bounds for the inner products) of the designs is known, improvements on the bounds are obtained. Furthermore, we provide `test functions for determining when the linear programming lower bounds for energy can be improved utilizing higher degree polynomials. We also provide some asymptotic results for these energy bounds.
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