Sufficient Conditions For Equality Research Articles

Conditions for equality in Anderson’s theorem

The classical Anderson theorem is a well-known result relevant to multivariate statistics and probability. It establishes an integral inequality for symmetric quasiconcave functions over a shifted symmetric convex set. The aim of this note is twofold. First, we provide necessary and sufficient conditions for equality in Anderson’s theorem, extending the result of Soms (1991). Second, we propose a set of tractable requirements that guarantee strict inequality in Anderson’s theorem. Our results are used to characterize the independence of Gaussian-distributed random variables.

On equality of certain subcentral automorphisms in finite p-groups

Let [Formula: see text] be a group and [Formula: see text] be the group of all automorphisms of [Formula: see text]. In this paper, we study the structure of certain subgroups of the automorphisms group [Formula: see text] and give necessary and sufficient conditions for equality between certain subcentral automorphisms. As consequence of these results, we state a number of interesting corollaries about central automorphisms, autocentral automorphisms and derival automorphisms for finite [Formula: see text]-groups.

Observations on the Lovász θ-Function, Graph Capacity, Eigenvalues, and Strong Products †.

This paper provides new observations on the Lovász θ-function of graphs. These include a simple closed-form expression of that function for all strongly regular graphs, together with upper and lower bounds on that function for all regular graphs. These bounds are expressed in terms of the second-largest and smallest eigenvalues of the adjacency matrix of the regular graph, together with sufficient conditions for equalities (the upper bound is due to Lovász, followed by a new sufficient condition for its tightness). These results are shown to be useful in many ways, leading to the determination of the exact value of the Shannon capacity of various graphs, eigenvalue inequalities, and bounds on the clique and chromatic numbers of graphs. Since the Lovász θ-function factorizes for the strong product of graphs, the results are also particularly useful for parameters of strong products or strong powers of graphs. Bounds on the smallest and second-largest eigenvalues of strong products of regular graphs are consequently derived, expressed as functions of the Lovász θ-function (or the smallest eigenvalue) of each factor. The resulting lower bound on the second-largest eigenvalue of a k-fold strong power of a regular graph is compared to the Alon-Boppana bound; under a certain condition, the new bound is superior in its exponential growth rate (in k). Lower bounds on the chromatic number of strong products of graphs are expressed in terms of the order and the Lovász θ-function of each factor. The utility of these bounds is exemplified, leading in some cases to an exact determination of the chromatic numbers of strong products or strong powers of graphs. The present research paper is aimed to have tutorial value as well.

On Idempotent Units in Commutative Group Rings

Special elements as units, which are defined utilizing idempotent elements, have a very crucial place in a commutative group ring. As a remark, we note that an element is said to be idempotent if r^2=r in a ring. For a group ring RG, idempotent units are defined as finite linear combinations of elements of G over the idempotent elements in R or formally, idempotent units can be stated as of the form id(RG)={∑_(r_g∈id(R))▒〖r_g g〗: ∑_(r_g∈id(R))▒r_g =1 and r_g r_h=0 when g≠h} where id(R) is the set of all idempotent elements [3], [4], [5], [6]. Danchev [3] introduced some necessary and sufficient conditions for all the normalized units are to be idempotent units for groups of orders 2 and 3. In this study, by considering some restrictions, we investigate necessary and sufficient conditions for equalities:i.V(R(G×H))=id(R(G×H)),ii.V(R(G×H))=G×id(RH),iii.V(R(G×H))=id(RG)×Hwhere G×H is the direct product of groups G and H. Therefore, the study can be seen as a generalization of [3], [4]. Notations mostly follow [12], [13].

On Topologized Fundamental Groups with Small Loop Transfer Viewpoints

In this paper, by introducing some kind of small loop transfer spaces at a point, we study the behavior of topologized fundamental groups with the compact-open topology and the whisker topology, $\pi _{1}^{qtop}(X,x_{0})$ and $\pi _{1}^{wh}(X,x_{0})$ , respectively. In particular, we give necessary or sufficient conditions for equality and being topological group of these two topologized fundamental groups. Finally, we give some examples to show that the converse of some of these implications do not hold, in general.

Preservation of a quantum Rényi relative entropy implies existence of a recovery map

It is known that a necessary and sufficient condition for equality in the data processing inequality (DPI) for the quantum relative entropy is the existence of a recovery map. We show that equality in DPI for a sandwiched Rényi relative α-entropy with is also equivalent to this property. For the proof, we use an interpolating family of Lp-norms with respect to a state.

On relations between BLUEs under two transformed linear models

For a given general linear model ℳ={y,Xβ,Σ}, we investigate relationships between the best linear unbiased estimations (BLUEs) under its two transformed models ℳ1={Ay,AXβ,AΣA′} and ℳ2={By,BXβ,BΣB′}. We first establish some expansion formulas for calculating the ranks and inertias of the covariance matrices of BLUEs and their operations under ℳ1 and ℳ2. We then derive from the rank and inertia formulas necessary and sufficient conditions for equalities and inequalities of BLUEs’ covariance matrices to hold. We also give applications of the rank and inertia formulas to two sub-sample models of ℳ.

Theorem of Necessity and Sufficiency of Stable Equilibrium for Generalized Potential Equality between System and Reservoir

The literature reports that equality of temperature, equality of potential and equality of pressure between a system and a reservoir are necessary conditions for the stable equilibrium of the system-reservoir composite or, in the opposite and equivalent logical inference, that stable equilibrium is a sufficient condition for equality. The aim and the first novelty of the present study is to prove that equality of temperature, potential and pressure is also a sufficient condition for stable equilibrium, in addition to necessity, implying that stable equilibrium is a condition also necessary, in addition to sufficiency, for equality. The second novelty is that the proof of the sufficiency of equality (or the necessity of stable equilibrium) is attained by means of the generalization of the entropy property, derived from the generalization of exergy property, which is used to demonstrate that stable equilibrium is a logical consequence of equality of generalized potential. This proof is underpinned by the Second Law statement and the Maximum-Entropy Principle based on generalized entropy which depends on temperature, potential and pressure of the reservoir. The conclusion, based on these two novel concepts, consists of the theorem of necessity and sufficiency of stable equilibrium for equality of generalized potentials within a composite constituted by a system and a reservoir.

Uncertainty and certainty relations for complementary qubit observables in terms of Tsallis’ entropies

Uncertainty relations for more than two observables have found use in quantum information, though commonly known relations pertain to a pair of observables. We present novel uncertainty and certainty relations of state-independent form for the three Pauli observables with use of the Tsallis $$\alpha $$ -entropies. For all real $$\alpha \in (0;1]$$ and integer $$\alpha \ge 2$$ , lower bounds on the sum of three $$\alpha $$ -entropies are obtained. These bounds are tight in the sense that they are always reached with certain pure states. The necessary and sufficient condition for equality is that the qubit state is an eigenstate of one of the Pauli observables. Using concavity with respect to the parameter $$\alpha $$ , we derive approximate lower bounds for non-integer $$\alpha \in (1;+\infty )$$ . In the case of pure states, the developed method also allows to obtain upper bounds on the entropic sum for real $$\alpha \in (0;1]$$ and integer $$\alpha \ge 2$$ . For applied purposes, entropic bounds are often used with averaging over the individual entropies. Combining the obtained bounds leads to a band, in which the rescaled average $$\alpha $$ -entropy ranges in the pure-state case. A width of this band is essentially dependent on $$\alpha $$ . It can be interpreted as an evidence for sensitivity in quantifying the complementarity.

Interactions in the Analysis of Variance

The standard model for the analysis of variance is over-parameterized. The resulting identifiability problem is typically solved by placing linear constraints on the parameters. In the case of the interactions, these require that the marginal sums be zero. Although seemingly neutral, these conditions have unintended consequences: the interactions are of necessity connected whether or not this is justified, the minimum number of nonzero interactions is four, and, in particular, it is not possible to have a single interaction in one cell. There is no reason why nature should conform to these constraints. The approach taken in this article is one of sparsity: the linear factor effects are chosen so as to minimize the number of nonzero interactions subject to consistency with the data. The resulting interactions are attached to individual cells making their interpretation easier irrespective of whether they are isolated or form clusters. In general, the calculation of a sparse solution is a difficult combinatorial problem but the special nature of the analysis of variance simplifies matters considerably. In many cases, the sparse L 0 solution coincides with the L 1 solution obtained by minimizing the sum of the absolute residuals and that can be calculated quickly. The identity of the two solutions can be checked either algorithmically or by applying known sufficient conditions for equality.

Eigenvalues and expansion of bipartite graphs

We prove lower bounds on the largest and second largest eigenvalue of the adjacency matrix of connected bipartite graphs and give necessary and sufficient conditions for equality. We give several examples of classes of graphs that are optimal with respect to the bounds. We prove that BIBD-graphs are characterized by their eigenvalues. Finally we present a new bound on the expansion coefficient of (c, d)-regular bipartite graphs and compare that with with a classical bound.

Hölder type matrix inequalities of Pate, Blakley, and Roy extended to the inner product of Frobenius

Suppose m, n, and k are positive integers, and let 〈·,·〉 be the standard inner product on the spaces Rp, p>0. Recently Pate has shown that if D is an m×n non-negative real matrix, and u and v are non-negative unit vectors in Rn and Rm, respectively, then〈(DDt)kDu,v〉⩾〈Du,v〉2k+1,with equality if and only if 〈(DDt)kDu,v〉=0, or there exists α>0 such that Du=αv and Dtv=αu. This extends to non-symmetric non-square matrices a 1965 result of Blakley and Roy, and resolves a special case of a graph theoretic inequality conjectured by Sidorenko. We generalize the above, obtaining pure matrix inequalities involving the Frobenius inner product, 〈·,·〉f. In particular, we show that if k is a positive integer, and D, X, and Y are non-negative matrices that are m×n,n×p, and m×p, respectively, then∑i=1p‖xi‖‖yi‖2k〈D(DtD)kX,Y〉f⩾(〈DX,Y〉f)2k+1,where X has columns x1,x2,…,xp, Y has columns y1,y2,…,yp, and ‖·‖ is the 2-norm. Necessary and sufficient conditions for equality are also given.

On composition principles for reduced moduli

We obtain the necessary and sufficient conditions for equality holding in the composition principles for generalized reduced moduli. As an application, we give some descriptions of extremal configurations in known and new inequalities for products of the Robin radii of nonoverlapping domains.

On the Asymptotic Efficiency of Distributed Estimation Systems With Constant Modulus Signals Over Multiple-Access Channels

A distributed estimation problem is considered with multiple-access channels between sensors and a fusion center. The sensors phase-modulate their noisy observations before transmitting them to the fusion center, where a signal parameter is estimated. The asymptotic efficiency of this estimator is then determined by using two inequalities that relate the Fisher information and the characteristic function. A necessary and sufficient condition for equality is found for the first time in the literature. The loss in efficiency of the distributed estimation scheme relative to the centralized approach is quantified for different sensing noise distributions. It is shown that this distributed estimation system does not incur an efficiency loss if and only if the sensing noise distribution is Gaussian.

Elementary proofs of two theorems involving arguments of eigenvalues of a product of two unitary matrices

We give elementary proofs of two theorems concerning bounds on the maximum argument of the eigenvalues of a product of two unitary matrices--one by Childs et al. [J. Mod. Phys. 47, 155 (2000)] and the other one by Chau [Quant. Inf. Comp. 11, 721 (2011)]. Our proofs have the advantages that the necessary and sufficient conditions for equalities are apparent and that they can be readily generalized to the case of infinite-dimensional unitary operators.

A comparative analysis of OTF, NPS, and DQE in energy integrating and photon counting digital x‐ray detectors

One of the benefits of photon counting (PC) detectors over energy integrating (EI) detectors is the absence of many additive noise sources, such as electronic noise and secondary quantum noise. The purpose of this work is to demonstrate that thresholding voltage gains to detect individual x rays actually generates an unexpected source of white noise in photon counters. To distinguish the two detector types, their point spread function (PSF) is interpreted differently. The PSF of the energy integrating detector is treated as a weighting function for counting x rays, while the PSF of the photon counting detector is interpreted as a probability. Although this model ignores some subtleties of real imaging systems, such as scatter and the energy-dependent amplification of secondary quanta in indirect-converting detectors, it is useful for demonstrating fundamental differences between the two detector types. From first principles, the optical transfer function (OTF) is calculated as the continuous Fourier transform of the PSF, the noise power spectra (NPS) is determined by the discrete space Fourier transform (DSFT) of the autocovariance of signal intensity, and the detective quantum efficiency (DQE) is found from combined knowledge of the OTF and NPS. To illustrate the calculation of the transfer functions, the PSF is modeled as the convolution of a Gaussian with the product of rect functions. The Gaussian reflects the blurring of the x-ray converter, while the rect functions model the sampling of the detector. The transfer functions are first calculated assuming outside noise sources such as electronic noise and secondary quantum noise are negligible. It is demonstrated that while OTF is the same for two detector types possessing an equivalent PSF, a frequency-independent (i.e., "white") difference in their NPS exists such that NPS(PC) > or = NPS(EI) and hence DQE(PC) < or = DQE(EI). The necessary and sufficient condition for equality is that the PSF is a binary function given as zero or unity everywhere. In analyzing the model detector with Gaussian blurring, the difference in NPS and DQE between the two detector types is found to increase with the blurring of the x-ray converter. Ultimately, the expression for the additive white noise of the photon counter is compared against the expression for electronic noise and secondary quantum noise in an energy integrator. Thus, a method is provided to determine the average secondary quanta that the energy integrator must produce for each x ray to have superior DQE to a photon counter with the same PSF. This article develops analytical models of OTF, NPS, and DQE for energy integrating and photon counting digital x-ray detectors. While many subtleties of real imaging systems have not been modeled, this work is illustrative in demonstrating an additive source of white noise in photon counting detectors which has not yet been described in the literature. One benefit of this analysis is a framework for determining the average secondary quanta that an energy integrating detector must produce for each x ray to have superior DQE to competing photon counting technology.

On a Sufficient Condition for Equality of Two Maximal Monotone Operators

We establish minimal conditions under which two maximal monotone operators coincide. Our first result is inspired by an analogous result for subdifferentials of convex functions. In particular, we prove that two maximal monotone operators T,S which share the same convex-like domain D coincide whenever \(T(x)\cap S(x)\not=\emptyset \) for every x ∈ D. We extend our result to the setting of enlargements of maximal monotone operators. More precisely, we prove that two operators coincide as long as the enlargements have nonempty intersection at each point of their common domain, assumed to be open. We then use this to obtain new facts for convex functions: we show that the difference of two proper lower semicontinuous and convex functions whose subdifferentials have a common open domain is constant if and only if their e-subdifferentials intersect at every point of that domain.

An Analytical Model of NPS and DQE Comparing Photon Counting and Energy Integrating Detectors.

In this work, analytical models of the optical transfer function (OTF), noise power spectra (NPS), and detective quantum efficiency (DQE) are developed for two types of digital x-ray detectors. The two detector types are (1) energy integrating (EI), for which the point spread function (PSF) is interpreted as a weighting function for counting x-rays, and (2) photon counting (PC), for which the PSF is treated as a probability. The OTF is the Fourier transform of the PSF. The two detector types, having the same PSF, possess an equivalent OTF. NPS is the discrete space Fourier transform (DSFT) of the autocovariance of signal intensity. From first principles, it is shown that while covariance is equivalent for both detector types, variance is not. As a consequence, provided the two detector types have equivalent PSFs, a difference in NPS exists such that NPSPC ≥ NPSEI and hence DQEPC ≤ DQEEI. The necessary and sufficient condition for equality is that the PSF is either zero or unity everywhere. A PSF modeled as the convolution of a Lorentzian with a rect function is analyzed in order to illustrate the differences in NPS and DQE. The Lorentzian models the blurring of the x-ray converter, while the rect function reflects the sampling of the detector. The NPS difference between the two detector types is shown to increase with increasing PSF width. In conclusion, this work develops analytical models of OTF, NPS, and DQE for energy integrating and photon counting digital x-ray detectors.

Some topological properties of rough sets and their applications

Properties of approximations of sets establish that lower approximation of union of sets is not equal to the union of their lower approximations in general and a similar result holds for upper approximation of intersection of sets. This confirms to the observation that there is a loss of information in a distributed knowledge base than in an integrated one. We obtain necessary and sufficient conditions for equalities to hold in the above two properties. We find that there are ambiguities in types of union and intersection of rough sets and establish theorems to reduce these ambiguities. Novotny and Pawlak introduced three (bottom, top and total) types of rough equalities, which show that comparison of domains depends upon our knowledge about the universe. We obtain suitable conditions under which the top and bottom rough equalities can be interchanged in these properties. Example of a distributed database of employees is used throughout.

The equality cases for the inequalities of Oppenheim and Schur for positive semi-definite matrices

In this paper, necessary and sufficient conditions for equality in the inequalities of Oppenheim and Schur for positive semidefinite matrices are investigated.