Division By Zero Research Articles

Computer-assisted verification of four interval arithmetic operators

Interval arithmetic libraries provide the four elementary arithmetic operators for operand intervals bounded by floating-point numbers. Actual implementations need to make a large case analysis that considers, e.g., magnitude relations between all pairs of argument bounds, positional relations between the arguments and zero, and handling of the special values, ±∞ and NaN. Their correctness is not obvious as they are implemented by human hands, which comes to be critical for the reliability. This work provides a mechanically-verified interval arithmetic library. For this purpose, we utilize the Why3 platform equipped with a specification language for annotated programs and back-end theorem provers. We conduct several proof tasks for each of three properties of the target code: validity, soundness, and tightness; zero division exception handling is also verified for the division code. To accomplish the proof, we propose several techniques for specification/verification. First, we specify additional lemmas that support deductions made by back-end SMT solvers, which enable to discharge proof obligations in floating-point arithmetic containing nonlinear terms. Second, we examine the annotation of tightness, which requires to assume that a computation may result in NaN; we propose specific extremum operators for this purpose. In the experiments, applying the techniques in conjunction with the Alt-Ergo SMT solver and the Coq proof assistant proved the entire code.

Division by Zero

The idea that, as opposed to the conventional viewpoint, division by zero may produce a meaningful result, is long standing and has attracted interest from many sides. We provide a survey of some options for the defining an outcome for the application of division in case the second argument equals zero. The survey is limited by a combination of simplifying assumptions which are grouped together in the idea of a premeadow, which generalises the notion of a transfield.

A Time Domain Passivity Control Scheme for Bilateral Teleoperation

Conventional time domain passivity control inevitably embodies division. Zero division can occur under a tiny force or velocity, which may be inevitable, and will be the cause of control crash. To avoid the zero division problem and control crash, we propose a switching dissipation controller for guaranteed stability. The parametric design of the proposed approach is discussed. The switching time domain passivity control is then applied to teleoperation and safe operation is achieved. Simulation and experimental results are demonstrated to validate the effectiveness of the proposed control scheme.

A novel pulse swallow based frequency divider circuit for a phase-locked loops

A new and unique frequency divider circuit has been proposed for this work. In this paper, we have implemented a $$\varDelta \varSigma$$ΔΣ fractional-N PLL using a proposed pulse swallow based frequency divider and a programmable prescaler divider circuit. Advantages of the proposed pulse swallow based frequency divider circuit have been illustrated and its usefulness are described in details. This novel pulse swallow based frequency divider technique does not include any reset or reload signal for the swallow counter as it is normally triggered by the SR latch output signal in conventional pulse swallow based frequency divider circuit. Insertion of a variable delay element at the output of the program counter can be eliminated which is normally used to settle the problem of arrival of the falling edge of the swallow counter pulse. In addition, preset enable signal can be generated without any frequency dependent delay generation block and in few cases without selection of any frequency dependent RC network block. The residual phase noise output of the divider at 1 MHz offset frequency is $$-174.5$$-174.5 dBc/Hz for a carrier signal frequency of 4.7 GHz and power consumption is 9 mW from a 1.2 V power supply. The design of the fractional-N PLL has been carried out in 130 nm standard CMOS process. There is no zero division in the proposed frequency divider's swallow counter for any counting state due to the novel mathematical calculating algorithm for the pulse swallow divider circuit.

ZD, ZG and IOL Controllers and Comparisons for Nonlinear System Output Tracking with DBZ Problem Conquered in Different Relative‐Degree Cases

AbstractThis paper considers the output tracking control of general‐form single‐input single‐output (SISO) nonlinear system, which may encounter the problem of division by zero (DBZ). First, via the Zhang dynamics (ZD) method, a ZD controller is proposed. Then, based on the ZD controller with the aid of gradient dynamics (GD) method, a Zhang‐gradient (ZG) controller is proposed. For comparison, the conventional input‐output linearization (IOL) controller is presented. The ZD, ZG and IOL controllers are compared in different relative‐degree cases (i.e., the standard relative‐degree case, the loose relative‐degree case and the DBZ relative‐degree case). Note that the ZG controller is valid in three relative‐degree cases, while the ZD and IOL controllers are valid only in the standard relative‐degree case and the loose relative‐degree case. In addition, performances of ZD and ZG controllers are guaranteed via theoretical analyses and computer simulations for the output tracking of general‐form nonlinear system with the DBZ problem conquered.

Division by zero

For any sufficiently strong theory of arithmetic, the set of Diophantine equations provably unsolvable in the theory is algorithmically undecidable, as a consequence of the MRDP theorem. In contrast, we show decidability of Diophantine equations provably unsolvable in Robinson’s arithmetic Q. The argument hinges on an analysis of a particular class of equations, hitherto unexplored in Diophantine literature. We also axiomatize the universal fragment of Q in the process.

Division by zero, pseudo-division by zero, Zhang dynamics method and Zhang-gradient method about control singularity conquering

ABSTRACTIn this paper, the division-by-zero (DBO) problem in the field of nonlinear control, which is traditionally termed the control singularity problem (or specifically, controller singularity problem), is investigated by the Zhang dynamics (ZD) method and the Zhang-gradient (ZG) method. According to the impact of the DBO problem on the state variables of the controlled nonlinear system, the concepts of the pseudo-DBO problem and the true-DBO problem are proposed in this paper, which provide a new perspective for the researchers on the DBO problems as well as nonlinear control systems. Besides, the two classes of DBO problems are solved under the framework of the ZG method. Specific examples are shown and investigated in this paper to illustrate the two proposed concepts and the efficacy of the ZG method in conquering pseudo-DBO and true-DBO problems. The application of the ZG method to the tracking control of a two-wheeled mobile robot further substantiates the effectiveness of the ZG method. In addition, the ZG method is successfully applied to the tracking control of a pure-feedback nonlinear system.

Reality of the Division by Zero z/0 = 0

Reality of the Division by Zero z/0 = 0

A Statistical Analysis of the NLMS and NLMS Algorithms for Correlated Gaussian Signals DOI: 10.14209/jcis.2005.8

The Nonnalized LMS adaptive algorithm is widely used in real life adaptive filtering and control applications mainly due to its simplicity and robustness to input signal power variations. This last characteristic is a consequence of its intrinsic nonnalization procedure. However, in practical applications it is usual the addition of a small positive constant to the nonnalization factor in order to avoid divisions by zero or overflow. In this case, the adaptive algorithm is named e-NLMS. This work presents a statistical analysis of the e-NLMS for Gaussian input signals. Deterministic recursive equations are obtained for the mean weight behavior and mean square error for a large number of adaptive coefficients. In addition, a closed expression is provided for the steady state misadjustment. Monte Carlo simulations show an excellent agreement between theoretical predictions and the algorithm's behavior in steady state. During transient, the new model is conservative and more accurate than the existing models. The developed equations can be lIsed with e =0 to predict the behavior of the NLMS algorithm. Simulations illustrate the better quality of the new NLMS model when compared to others already available in the literature.

χα Absolute Chaos: The Early Romantic Poetics of Complex Form

DIVIDED BY ZERO AND WITH AN INDETERMINATE COUNTERVALUE, THE numerator chaos itself remains indeterminate--indeterminate enough, in early romantic poetics, to cover a broad conceptual field. is taken up into a calculus of speculative concepts. It is equated with poetry; determines modern and interesting; explains romantic; is the foundational of mythology; becomes the organizing principle of the novel; defines enthusiasm and irony; and serves as a designation of epoch and style characterizing the works of Dante, Petrarch, Boccaccio, Shakespeare, Cervantes, Calderon, and Goethe. As ein von Systemen [a chaos of systems], (1) even absolute is determined by a concept whose popularized semantics indicates indeterminacy, disorder, and confusion. Mathematics, too, is proclaimed by Friedrich Schlegel Princip des Chaotischen. Die [mu] [alpha] [theta] [mathematische] Form entsteht durch das Irrationale, Potenzirte, Combinatorische, Progressive pp. [Principle of the chaotic. Mathematical form arises from the irrational, exponential, combinatory, progressive pp. S 16: 336]. Whereas Schlegel inclines to algebraic formalizations and simple equations for chaos, Novalis draws on infinitesimal calculus to develop a poetics of involution when giving vernunftige[s] [rational chaos] the notation Chaos (2) oder [infinity] ([chaos.sup.2] or [infinity]). (2) The early romantic tendency to present an aesthetic concept of chaos in mathematical terms should not be mistaken as an anticipation of the mathematics of fractals. (3) Certainly neither chaos theory nor early romantic thought describe the beautiful by appealing to the traditional idea of bel ordre, understanding it instead as the result of an oscillation between order and chaos. These are the terms in which Friedrich Cramer and Wolfgang Kaempfer conceive the beauty of fractal patterns, in der offenen (irrationalen) Ordnung des Uberganges [the open (irrational) order of transition]. (4) Indeed Novalis' mysterious Chiffernschrift [cipher script] of nature found in Wolken ... im Schnee, in Krystallen und in Steinbildungen ... in den Feilspanen um den Magnet her, und sonderbaren Conjuncturen des Zufalls [clouds ... in the snow, in crystals, and in rock formations ... in the filings surrounding a magnet, and the unusual conjunctures of chance H I: 201] might be understood as an intuitive anticipation of the self-similar structures described by Benoit Mandelbrot and already intimated by Leibniz. (5) For romanticism, chaos is a metaphor for the transition from an old to a new order, connected to the ideas of self-similarity, recursion, self-organization, and complexity; however, romantic thought also transcendentalizes this figure of transition through a poetological calculus with the infinite that relies on a self-generated indeterminacy. Early romanticism did not yet have a mathematics for the description of non-linear, chaotic trajectories as they came to be described in the course of the nineteenth century: in 1824 Sadi Carnot almost discovered the second law of thermodynamics that in 1850 Rudolf Clausius formulated as the principle of entropy. Thermal energy, running chaotically along molecular turbulence, becomes distinguished from the orderly course of mechanical energy along visible lines of force. In 1892 Henri Poincare demonstrated that a system of only three components becomes unpredictable and no longer calculable with linear differentials. Earlier in the Newtonian tradition and following Laplace, it was assumed that the parameters of a system could be so precisely calculated that its development would be predictable. The swinging of a pendulum generally stands as the paradigm of such deterministic behavior. The path of a double pendulum, however, though subjected to Newton's laws of motion like the single pendulum, already becomes unpredictable. With the double pendulum, one pendulum is coupled to and simultaneously independent of the movement of the other, so that minimal effects loop back and thereby introduce unpredictable trajectories of motion. …

And nothing to watch: The divide-by-zero future

And nothing to watch: The divide-by-zero future

Models to Show the Impossibility of Division by Zero

Models to Show the Impossibility of Division by Zero

Intuitive Definitions for Division With Zero

Five different approaches show there really is something to division by 0.

An Early Reference to Division by Zero

An Early Reference to Division by Zero

Absurdities Due to Division by Zero

Absurdities Due to Division by Zero

Dividing by Zero

Dividing by Zero

Dividing by Zero

Dividing by Zero