Null Functions Research Articles

ANALISIS LAGRANGIAN NULL NONSTANDAR DAN FUNGSI GAUGE UNTUK HUKUM INERSIA NEWTON : SEBUAH REVIEW

This paper describes a review of a journal entitled 'Nonstandard Null Lagrangians and Gauge Functions for Newtonian Law of Inertia' which discusses Nonstandard Lagrangian Null solutions for Newton's Law of Inertia. The purpose of this study is to present in detail the Lagrangian Formalism method for generating Nonstandard Lagrangian Null and its Gauge function for Newton's Law of Inertia, as well as the role of action invariant in generating Lagrangian Null and Exact Gauge functions, by deriving a one-dimensional oscillator arm using the basic Lagrangian equations. The Nonstandard Null Lagrangian is derived from the Nonstandard Lagrangian, then the two Lagrangian Null are entered into the action invariant to make it Exact, after the Nonstandard Lagrangian Null is declared Exact, it is substituted into which represents Newton's Law of Inertia. The results of this research show that an Exact Nonstandard Lagrangian Null can be generated by making the Gauge Function Invariant, so that the first Nonstandard Lagrangian Null and Gauge Function for Newton's Law of Inertia are obtained.

Non-Standard and Null Lagrangians for Nonlinear Dynamical Systems and Their Role in Population Dynamics

Non-standard Lagrangians do not display any discernible energy-like terms, yet they give the same equations of motion as standard Lagrangians, which have easily identifiable energy-like terms. A new method to derive non-standard Lagrangians for second-order nonlinear differential equations with damping is developed and the limitations of this method are explored. It is shown that the limitations do not exist only for those nonlinear dynamical systems that can be converted into linear ones. The obtained results are applied to selected population dynamics models for which non-standard Lagrangians and their corresponding null Lagrangians and gauge functions are derived, and their roles in the population dynamics are discussed.

Hadamard states on spherically symmetric characteristic surfaces, the semi-classical Einstein equations and the Hawking effect

We investigate quasi-free Hadamard states defined via characteristic initial data on null cones centred at the axis of symmetry in spherically symmetric space-times. We characterize the necessary singular behaviour of null boundary two-point functions such that one can define non-linear observables at this null boundary and give formulas for the calculation of these observables. These results extend earlier characterizations of null boundary states defining Hadamard states in the bulk of the null cone. As an application of our derived formulas, we consider their implications for the semi-classical Einstein equations and calculate the vacuum polarization associated with Hawking radiation near a collapsing body.

Nonstandard null Lagrangians and gauge functions and dissipative forces in dynamics

Standard and non-standard Lagrangians that give the same equation of motion are significantly different in their forms, as the latter do not have terms that clearly discernable energy-like expressions. A special family of these Lagrangians are null Lagrangians and their gauge functions. It is shown that non-standard null Lagrangians and their gauge functions can be used to introduce dissipative forces to dynamical systems. With standard null Lagrangians being known for introducing non-dissipative forces, the presented results allow for a complete picture of novel roles played by null Lagrangians in introducing forces to dynamics. Applications of the results to Newton's laws are presented and their Galilean invariance is discussed.

THx2 Programmable Logic Block Architecture for Clockless Asynchronous FPGAs

To address some of the challenges of asynchronous design, we propose a new, decomposable asynchronous logic block architecture based on our TH<inline-formula> <tex-math notation="LaTeX">$x2$ </tex-math></inline-formula> programmable threshold cell, and we use it to implement common threshold functions found in asynchronous, null convention logic circuits. At a minimum, programmable gate arrays require a programmable logic cell that can implement a complete set of logic. It is well known that a NAND function forms a complete set of logic, and in null convention logic, the TH12 and TH22 threshold cells are used to form a basic two-input NAND function. The TH<inline-formula> <tex-math notation="LaTeX">$x2$ </tex-math></inline-formula> threshold cell is capable of performing both TH12 and TH22 operations, so it too forms a complete set of logic. In this paper, we present our eight-transistor mask-programmable gate array logic cell, 16-transistor field-programmable gate array logic cell, and new decomposable field-programmable gate array logic block architecture, all based on the TH<inline-formula> <tex-math notation="LaTeX">$x2$ </tex-math></inline-formula> threshold cell and suitable for implementing null convention logic asynchronous functions. To minimize the TH<inline-formula> <tex-math notation="LaTeX">$x2$ </tex-math></inline-formula> threshold cell area for both TH12 and TH22 modes, we designed a layout with common Euler paths and no diffusion breaks for both modes. The highly compact nature of the TH<inline-formula> <tex-math notation="LaTeX">$x2$ </tex-math></inline-formula> threshold cell&#x2013;along with the symmetry of the mask- and field-programmable gate array logic cells&#x2013;made it an ideal candidate for an asynchronous field-programmable logic block structure. This paper is part of an ongoing project, and it only addresses the programmable logic block architecture, not a complete FPGA fabric.

Effect on null spaces of list-mode imaging systems due to increasing the number of attributes.

There are two types of uncertainty in image reconstructions from list-mode data: statistical and deterministic. One source of statistical uncertainty is the finite number of attributes of the detected particles, which are sampled from a probability distribution on the attribute space. A deterministic source of uncertainty is the effect that null functions of the imaging operator have on reconstructed pixel or voxel values. Quantifying the reduction in this deterministic source of uncertainty when more attributes are measured for each detected particle is the subject of this work. Specifically, upper bounds on an error metric are derived to quantify the error introduced in the reconstruction by the presence of null functions, and these upper bounds are shown to be reduced when the number of attributes is increased. These bounds are illustrated with an example of a two-dimensional single photon emission computed tomography (SPECT) system where the depth of interaction in the scintillation crystal is added to the attribute vector.

Nonstandard Null Lagrangians and Gauge Functions and Dissipative Forces in Dynamics

Nonstandard Null Lagrangians and Gauge Functions and Dissipative Forces in Dynamics

Nonstandard Null Lagrangians and Gauge Functions for Newtonian Law of Inertia

New null Lagrangians and gauge functions are derived and they are called nonstandard because their forms are different than those previously found. The invariance of the action is used to make the Lagrangians and gauge functions exact. The first exact nonstandard null Lagrangian and its gauge function for the law of inertia are obtained, and their physical implications are discussed.

Commutative L-algebras and measure theory

Abstract Measure and integration theory for finitely additive measures, including vector-valued measures, is shown to be essentially covered by a class of commutative L-algebras, called measurable algebras. The domain and range of any measure is a commutative L-algebra. Each measurable algebra embeds into its structure group, an abelian group with a compatible lattice order, and each (general) measure extends uniquely to a monotone group homomorphism between the structure groups. On the other hand, any measurable algebra X is shown to be the range of an essentially unique measure on a measurable space, which plays the role of a universal covering. Accordingly, we exhibit a fundamental group of X, with stably closed subgroups corresponding to a special class of measures with X as target. All structure groups of measurable algebras arising in a classical context are archimedean. Therefore, they admit a natural embedding into a group of extended real-valued continuous functions on an extremally disconnected compact space, the Stone space of the measurable algebra. Extending Loomis’ integration theory for finitely additive measures, it is proved that, modulo null functions, each integrable function can be represented by a unique continuous function on the Stone space.

Bateman Oscillators: Caldirola-Kanai and Null Lagrangians and Gauge Functions

The Lagrange formalism is developed for Bateman oscillators, which includes both damped and amplified systems, and a novel method to derive the Caldirola-Kanai and null Lagrangians is presented. For the null Lagrangians, the corresponding gauge functions are obtained. It is shown that the gauge functions can be used to convert the undriven Bateman oscillators into the driven ones. Applications of the obtained results to quantizatation of the Bateman oscillators are briefly discussed.

Effect on null spaces of list-mode imaging systems due to increasing the size of attribute space.

An upper bound is derived for a figure of merit that quantifies the error in reconstructed pixel or voxel values induced by the presence of null functions for any list-mode system. It is shown that this upper bound decreases as the region in attribute space occupied by the allowable attribute vectors expands. This upper bound allows quantification of the reduction in this error when this type of expansion is implemented. Of course, reconstruction error is also caused by system noise in the data, which has to be treated statistically, but we will not be addressing that problem here. This method is not restricted to pixelized or voxelized reconstructions and can in fact be applied to any region of interest. The upper bound for pixelized reconstructions is demonstrated on a list-mode 2D Radon transform example. The expansion in the attribute space is implemented by doubling the number of views. The results show how the pixel size and number of views both affect the upper bound on reconstruction error from null functions. This reconstruction error can be averaged over all pixels to give a single number or can be plotted as a function on the pixel grid. Both approaches are demonstrated for the example system. In conclusion, this method can be applied to any list-mode system for which the system operator is known and could be used in the design of the systems and reconstruction algorithms.

Divergence zero quaternionic vector fields and Hamming graphs

We give a possible extension of the definition of quaternionic power series, partial derivatives and vector fields in the case of two (and then several) non commutative (quaternionic) variables. In this setting we also investigate the problem of describing zero functions which are not null functions in the formal sense. A connection between an analytic condition and a graph theoretic property of a subgraph of a Hamming graph is shown, namely the condition that polynomial vector field has formal divergence zero is equivalent to connectedness of subgraphs of Hamming graphs H ( d , 2) . We prove that monomials in variables z and w are always linearly independent as functions only in bidegrees ( p , 0), ( p , 1), (0, q ), (1, q ) and (2, 2).

Controlling the false discovery exceedance for heterogeneous tests

Several classical methods exist for controlling the false discovery exceedance (FDX) for large-scale multiple testing problems, among them the Lehmann-Romano procedure (Lehmann and Romano 2005) ($[\mathrm{LR}]$ below) and the Guo-Romano procedure (Guo and Romano 2007) ($[\mathrm{GR}]$ below). While these two procedures are the most prominent, they were originally designed for homogeneous test statistics, that is, when the null distribution functions of the $p$-values $F_{i}$, $1\leq i\leq m$, are all equal. In many applications, however, the data are heterogeneous which leads to heterogeneous null distribution functions. Ignoring this heterogeneity induces a lack of power. In this paper, we develop three new procedures that incorporate the $F_{i}$’s, while maintaining rigorous FDX control. The heterogeneous version of $[\mathrm{LR}]$, denoted $[\mathrm{HLR}]$, is based on the arithmetic average of the $F_{i}$’s, while the heterogeneous version of $[\mathrm{GR}]$, denoted $[\mathrm{HGR}]$, is based on the geometric average of the $F_{i}$’s. We also introduce a procedure $[\mathrm{PB}]$, that is based on the Poisson-binomial distribution and that uniformly improves $[\mathrm{HLR}]$ and $[\mathrm{HGR}]$, at the price of a higher computational complexity. Perhaps surprisingly, this shows that, contrary to the known theory of false discovery rate (FDR) control under heterogeneity, the way to incorporate the $F_{i}$’s can be particularly simple in the case of FDX control, and does not require any further correction term. The performances of the new proposed procedures are illustrated by real and simulated data in two important heterogeneous settings: first, when the test statistics are continuous but the $p$-values are weighted by some known independent weight vector, e.g., coming from co-data sets; second, when the test statistics are discretely distributed, as is the case for data representing frequencies or counts. Our new procedures are implemented in the R package $\mathtt{FDX}$, see Junge and Döhler (2020).

Existence and Uniqueness of Entropy Solution for an Elliptic Problem with Nonlinear Boundary Conditions and Measure-data

In this paper, we study a class of elliptic equations on a bounded domain with nonlinear boundary conditions of type graph and measure - data. First of all, give some spaces and basic assumptions. Next, we apply the classical variational approach. So, we need an essentially bounded estimate on the solution, which is not evident to obtain directly in our problem. The obstacles which we encounter is that we cannot get rid of the non-linear term evaluated as a zero gradient and it appear at the boundary, for the part of the measure-data, a term which cannot vanish, when one uses the integration by parts formula. To overcome this difficulties, we first redefine and extend the function which appears in the third Leray-Lions-type conditions and we add a penalization term on the boundary. Secondly, we consider a smooth domain in order to work with the Sobolev spaces that are the closure of indefinitely differentiable and null functions on the bounary, and to going back later to the classical Sobolev space. Then, we assume that the domain is extensible. Next, we obtain a priori estimates and convergence results in the approach problem, which allow us to delete the penalization term. To finish, we introduce a notion of entropy solution for our main problem and prove that it is the limit of the solution obtained in the variational case.

Null function classification using NFR matrix and decision tree

This research work is based on the classification of the functions which are required and which is not required for the efficient development of the software. The motivation of this work is to identify non required functions to reduce development cost and efforts. The classification is the technique which can classify data into certain number of classes. The NFR matrix is the existing technique which can classify the Null functions. To classify the null functions NFR use clustering method which can modify to increase accuracy of classification. In the existing method NFR matrix is used for the null function classification, it use the clustering for the classification. In this work, decision tree will based with the clustering. It take input result of clustering and generate classified result.

Input-Output pairs detection in a distillation column operating in closed-loop

The detection of no-model input-output (IO) pairs is important because it can speed up the multivariable system identification process, since all the pairs with null transfer functions are previously discarded and it can also improve the identified model quality and so improve the performance of model based controllers. In the available literature, the methods focus just on the open-loop case, since in this case there is not the effect of the controller forcing the main diagonal in the transfer matrix to one and all other terms to zero. In this paper, a modification of a previous method that is able to detect no-model IO pairs in closed-loop systems is presented. Tests are performed using the traditional methods and the proposed one to show its effectiveness.

Null functions in three-dimensional imaging of alpha and beta particles

Null functions of an imaging system are functions in the object space that give exactly zero data. Hence, they represent the intrinsic limitations of the imaging system. Null functions exist in all digital imaging systems, because these systems map continuous objects to discrete data. However, the emergence of detectors that measure continuous data, e.g. particle-processing (PP) detectors, has the potential to eliminate null functions. PP detectors process signals produced by each particle and estimate particle attributes, which include two position coordinates and three components of momentum, as continuous variables. We consider Charged-Particle Emission Tomography (CPET), which relies on data collected by a PP detector to reconstruct the 3D distribution of a radioisotope that emits alpha or beta particles, and show empirically that the null functions are significantly reduced for alpha particles if ≥3 attributes are measured or for beta particles with five attributes measured.

Detection of no-model input-output pairs in closed-loop systems

The detection of no-model input-output (IO) pairs is important because it can speed up the multivariable system identification process, since all the pairs with null transfer functions are previously discarded and it can also improve the identified model quality, thus improving the performance of model based controllers. In the available literature, the methods focus just on the open-loop case, since in this case there is not the effect of the controller forcing the main diagonal in the transfer matrix to one and all the other terms to zero. In this paper, a modification of a previous method able to detect no-model IO pairs in open-loop systems is presented, but adapted to perform this duty in closed-loop systems. Tests are performed by using the traditional methods and the proposed one to show its effectiveness.

Nonparametric Transition-Based Tests for Jump-Diffusions

We develop a specification test for discretely-sampled jump-diffusions, based on a comparison of a nonparametric estimate of the transition density or distribution function to their corresponding parametric counterparts. As a special case, our method applies to pure diffusions. We propose three different discrepancy measures between the null and alternative transition density and distribution functions. We establish the asymptotic null distributions of proposed test statistics and compute their power functions. The finite sample properties are investigated via simulation studies and are compared with those of alternative tests.

Singular value decomposition for photon-processing nuclear imaging systems and applications for reconstruction and computing null functions

Recent advances in technology are enabling a new class of nuclear imaging systems consisting of detectors that use real-time maximum-likelihood (ML) methods to estimate the interaction position, deposited energy, and other attributes of each photon-interaction event and store these attributes in a list format. This class of systems, which we refer to as photon-processing (PP) nuclear imaging systems, can be described by a fundamentally different mathematical imaging operator that allows processing of the continuous-valued photon attributes on a per-photon basis. Unlike conventional photon-counting (PC) systems that bin the data into images, PP systems do not have any binning-related information loss. Mathematically, while PC systems have an infinite-dimensional null space due to dimensionality considerations, PP systems do not necessarily suffer from this issue. Therefore, PP systems have the potential to provide improved performance in comparison to PC systems. To study these advantages, we propose a framework to perform the singular-value decomposition (SVD) of the PP imaging operator. We use this framework to perform the SVD of operators that describe a general two-dimensional (2D) planar linear shift-invariant (LSIV) PP system and a hypothetical continuously rotating 2D single-photon emission computed tomography (SPECT) PP system. We then discuss two applications of the SVD framework. The first application is to decompose the object being imaged by the PP imaging system into measurement and null components. We compare these components to the measurement and null components obtained with PC systems. In the process, we also present a procedure to compute the null functions for a PC system. The second application is designing analytical reconstruction algorithms for PP systems. The proposed analytical approach exploits the fact that PP systems acquire data in a continuous domain to estimate a continuous object function. The approach is parallelizable and implemented for graphics processing units (GPUs). Further, this approach leverages another important advantage of PP systems, namely the possibility to perform photon-by-photon real-time reconstruction. We demonstrate the application of the approach to perform reconstruction in a simulated 2D SPECT system. The results help to validate and demonstrate the utility of the proposed method and show that PP systems can help overcome the aliasing artifacts that are otherwise intrinsically present in PC systems.