Ideal Theorem Research Articles

A unified approach to the Pythagorean identity and projection theorem for a class of divergences based on M-estimations

Divergences that satisfy Pythagorean-type identity are found to be quite useful in statistical inference. Projection theorem of these divergences explores the associated inference problems through closed-form estimators and a fixed number of sufficient statistics. This work attempts to unify the Pythagorean property and projection theorem of such divergences, including the well-known KL, Basu et al., and Jones et al. divergences, minimization of which result in an M-estimation. This unification characterizes a general form of power-law distributions that includes the well-known Student-t and Student-r families. Rao-Blackwell-type best estimators for these families are derived using sufficient statistics based on the proposed unified class of divergences. Finally, it comments on the efficacy of these estimators through numerical studies.

A Sampling Theorem for Exact Identification of Continuous-Time Nonlinear Dynamical Systems

A Sampling Theorem for Exact Identification of Continuous-Time Nonlinear Dynamical Systems

A General Theorem and Proof for the Identification of Composed CFA Models

In this article, we present a general theorem and proof for the global identification of composed CFA models. They consist of identified submodels that are related only through covariances between their respective latent factors. Composed CFA models are frequently used in the analysis of multimethod data, longitudinal data, or multidimensional psychometric data. Firstly, our theorem enables researchers to reduce the problem of identifying the composed model to the problem of identifying the submodels and verifying the conditions given by our theorem. Secondly, we show that composed CFA models are globally identified if the primary models are reduced models such as the CT-C(M-1) model or similar types of models. In contrast, composed CFA models that include non-reduced primary models can be globally underidentified for certain types of cross-model covariance assumptions. We discuss necessary and sufficient conditions for the global identification of arbitrary composed CFA models and provide a Python code to check the identification status for an illustrative example. The code we provide can be easily adapted to more complex models.

Hyper-power series and generalized real analytic functions

This article is a natural continuation of the paper Tiwari, D., Giordano, P., Hyperseries in the non-Archimedean ring of Colombeau generalized numbers in this journal. We study one variable hyper-power series by analyzing the notion of radius of convergence and proving classical results such as algebraic operations, composition and reciprocal of hyper-power series. We then define and study one variable generalized real analytic functions, considering their derivation, integration, a suitable formulation of the identity theorem and the characterization by local uniform upper bounds of derivatives. On the contrary with respect to the classical use of series in the theory of Colombeau real analytic functions, we can recover several classical examples in a non-infinitesimal set of convergence. The notion of generalized real analytic function reveals to be less rigid both with respect to the classical one and to Colombeau theory, e.g. including classical non-analytic smooth functions with flat points and several distributions such as the Dirac delta. On the other hand, each Colombeau real analytic function is also a generalized real analytic function.

An Identity Theorem for the Fourier–Laplace Transform of Polytopes on Nonzero Complex Multiples of Rationally Parameterizable Hypersurfaces

A set mathcal {S} of points in mathbb {R}^n is called a rationally parameterizable hypersurface if there is vector function varvec{sigma }:mathbb {R}^{n-1}rightarrow mathbb {R}^n having as components rational functions defined on some common domain D such that mathcal {S}={varvec{sigma }(textbf{t}):textbf{t}in D}. A generalized n-dimensional polytope in mathbb {R}^n is a union of a finite number of convex n-dimensional polytopes in mathbb {R}^n. The Fourier–Laplace transform of such a generalized polytope mathcal {P} in mathbb {R}^n is defined by F_{mathcal {P}}(textbf{z})=int _{mathcal {P}}e^{textbf{z}cdot textbf{x}},textbf{dx}. Let gamma be a fixed nonzero complex number. We prove that F_{mathcal {P}_1}(gamma varvec{sigma }(textbf{t}))=F_{mathcal {P}_2}(gamma varvec{sigma }(textbf{t})) for all textbf{t} in O implies mathcal {P}_1=mathcal {P}_2 if O is an open subset of D satisfying some well-defined conditions and we present similar results for the null set of the Fourier–Laplace transform of mathcal {P}. Moreover we show that this theorem can be applied to quadric hypersurfaces that do not contain a line, but at least two points, i.e., in particular to spheres.

Logarithmic Vertex Algebras

We introduce and study the notion of a logarithmic vertex algebra, which is a vertex algebra with logarithmic singularities in the operator product expansion of quantum fields; thus providing a rigorous formulation of the algebraic properties of quantum fields in logarithmic conformal field theory. We develop a framework that allows many results about vertex algebras to be extended to logarithmic vertex algebras, including in particular the Borcherds identity and Kac Existence Theorem. Several examples are investigated in detail, and they exhibit some unexpected new features that are peculiar to the logarithmic case.

A Lyndon’s identity theorem for one-relator monoids

For every one-relator monoid M = langle A mid u=v rangle with u, v in A^* we construct a contractible M-CW complex and use it to build a projective resolution of the trivial module which is finitely generated in all dimensions. This proves that all one-relator monoids are of type mathrm{FP}_{infty }, answering positively a problem posed by Kobayashi in 2000. We also apply our results to classify the one-relator monoids of cohomological dimension at most 2, and to describe the relation module, in the sense of Ivanov, of a torsion-free one-relator monoid presentation as an explicitly given principal left ideal of the monoid ring. In addition, we prove the topological analogues of these results by showing that all one-relator monoids satisfy the topological finiteness property mathrm{F}_infty , and classifying the one-relator monoids with geometric dimension at most 2. These results give a natural monoid analogue of Lyndon’s Identity Theorem for one-relator groups.

Pullback Coherent States, Squeezed States and Quantization

In this semi-expository paper, we define certain Rawnsley-type coherent and squeezed states on an integral K\"ahler manifold (after possibly removing a set of measure zero) and show that they satisfy some properties which are akin to maximal likelihood property, reproducing kernel property, generalised resolution of identity property and overcompleteness. This is a generalization of a result by Spera. Next we define the Rawnsley-type pullback coherent and squeezed states on a smooth compact manifold (after possibly removing a set of measure zero) and show that they satisfy similar properties. Finally we show a Berezin-type quantization involving certain operators acting on a Hilbert space on a compact smooth totally real embedded submanifold of $U$ of real dimension $n$, where $U$ is an open set in ${\mathbb C}{\rm P}^n$. Any other submanifold for which the criterion of the identity theorem holds exhibit this type of Berezin quantization. Also this type of quantization holds for totally real submanifolds of real dimension $n$ of a general homogeneous K\"ahler manifold of real dimension $2n$ for which Berezin quantization exists. In the appendix we review the Rawnsley and generalized Perelomov coherent states on ${\mathbb C}{\rm P}^n$ (which is a coadjoint orbit) and the fact that these two types of coherent states coincide.

"On the maximum modulus principle and the identity theorem in arbitrary dimension"

"We prove an identity theorem for Gˆateaux holomorphic functions on polygonally connected 2- open sets, which yields a very general maximum norm principle and a sublinear “max-min” principle. All results apply in particular to vector-valued functions which are holomorphic (in any sense that implies Gˆateaux holomorphy) on domains in Hausdorff locally convex spaces."

Fuzzy Bayesian Estimation of Linear (Circular) Consecutive k-out-of-n: F System Reliability

The consecutive k-out-of-n: F structure is broadly applicable in industrial and military frameworks. Without lifetime information about the entire design, it is appealing to use fuzzy earlier beliefs on its segments. In this paper the fuzzy Bayesian reliability assessment of the linear (circular) consecutive k-out-of-n: F system is proposed under the squared error loss function. The parameters are known as fuzzy random variables in the process of obtaining fuzzy Bayesian reliability. The conventional strategy for estimating Bayesian reliability will be utilized to construct the fuzzy Bayesian point estimator of the proposed system by applying the Resolution Identity theorem. A comparative study of the derived result with the literature is presented.
 HIGHLIGHTS
 
 Reliability Estimate of Consecutive k-out-of-n: F system
 Parameters are known as fuzzy random variables
 Applied Resolution Identity Theorem to estimate fuzzy Bayesian reliability
 Fuzzy Bayesian system reliability is calculated
 Fuzzy mean time to failure is calculated
 
 GRAPHICAL ABSTRACT

Generalization of Wolff's ideal theorem on H_K(B)^∞(ᵓb)

Generalization of Wolff's ideal theorem on H_K(B)^∞(ᵓb)

Some Properties of Cone Inner Product Spaces over Banach Algebra

The aim of this paper is to introduce a concept of a cone inner product space over Banach algebras. This is done by replacing the co-domain of the classical inner product space by an ordered Banach algebra. Some properties such as Cauchy-Schwarz inequality, parallelogram identity and Pythagoras theorem are established in this setting. Similarly, the notion of cone normed algebra was introduced. Some illustrative examples are given to support our findings.

Using Cauchy theorem to compute complicated real integrals

As the central object in the theory of complex analysis, Holomorphic functions have many elegant mathematical properties. Holomorphic functions are extremely valuable because, on the one hand, they are unexpectedly common, and on the other hand, they may be used to establish extremely powerful theorems. For example, To establish theorems like the prime number theorem, analytic number theorists commonly create holomorphic or meromorphic functions that hold number-theoretic information, such as the Riemann zeta function. Given knowledge about a holomorphic function in a relatively small part of its domain, one may extract information about the function’s behavior in other a priori unrelated sections of its domain, according to Cauchy’s integral formula and the identity theorem (and this is what allows things like contour integration to work). Due to difficulties in obtaining primitives of some real function, the fundamental theorem of Calculus does not work in most cases. In this article, we review the Cauchy theorem and use it as a tool to compute several real integrals.

TUKEY ORDER AMONG IDEALS

AbstractWe investigate the Tukey order in the class of $F_{\sigma }$ ideals of subsets of $\omega $ . We show that no nontrivial $F_{\sigma }$ ideal is Tukey below a $G_{\delta }$ ideal of compact sets. We introduce the notions of flat ideals and gradually flat ideals. We prove a dichotomy theorem for flat ideals isolating gradual flatness as the side of the dichotomy that is structurally good. We give diverse characterizations of gradual flatness among flat ideals using Tukey reductions and games. For example, we show that gradually flat ideals are precisely those flat ideals that are Tukey below the ideal of density zero sets.

Positive solution to Schrödinger equation with singular potential and double critical exponents

In this paper, we consider the following Schrödinger equation with singular potential and double critical exponents: -\Delta u + \frac{A}{|x|^{\alpha}}u = |u|^{2^{*}-2}u + |u|^{p-2}u + \lambda |u|^{2_{\alpha}^{*}-2}u,\quad x\in \mathbb{R}^{N}, where N\geq 3 , \alpha\in(0,2) , p\in(\frac{2N-4+2\alpha}{N-2},2^{*}) , and A,\lambda>0 are two real constants, and 2^{*}=\frac{2N}{N-2} is the Sobolev critical exponent, and 2_{\alpha}^{*}=2+\frac{4\alpha}{2N-2-\alpha} is the critical exponent with respect to the parameter \alpha . First, using the refined Sobolev inequality, we establish the Lions type theorem. Second, we prove that any nonnegative weak solutions of above equation satisfy Pohozaev type identity. Finally, by using perturbation method, Pohozaev type identity and Lions type theorem, we show the existence of positive solution to above equation. We point out that the double critical exponents is an new phenomenon, and we are the first to consider it. Our result partial extends the results in Badiale and Rolando [Rend. Lincei Mat. Appl. 17 (2006)], and Su, Wang and Willem [Commun. Contemp. Math. 9 (2007)].

Identity Theorem in Complex Analysis

Identity Theorem in Complex Analysis

Vector-valued holomorphic functions in several variables

In the present paper we give some explicit proofs for folklore theorems on holomorphic functions in several variables with values in a locally complete locally convex Hausdorff space $E$ over $\mathbb{C}$. Most of the literature on vector-valued holomorphic functions is either devoted to the case of one variable or to infinitely many variables whereas the case of (finitely many) several variables is only touched or is subject to stronger restrictions on the completeness of $E$ like sequential completeness. The main tool we use is Cauchy's integral formula for derivatives for an $E$-valued holomorphic function in several variables which we derive via Pettis-integration. This allows us to generalise the known integral formula, where usually a Riemann-integral is used, from sequentially complete $E$ to locally complete $E$. Among the classical theorems for holomorphic functions in several variables with values in a locally complete space $E$ we prove are the identity theorem, Liouville's theorem, Riemann's removable singularities theorem and the density of the polynomials in the $E$-valued polydisc algebra.

Valiron\u2019s Interpolation Formula and a Derivative Sampling Formula in the Mellin Setting Acquired via Polar-Analytic Functions

In this paper, we first recall some recent results on polar-analytic functions. Then we establish Mellin analogues of a classical interpolation of Valiron and of a derivative sampling formula. As consequences a new differentiation formula and an identity theorem in Mellin–Bernstein spaces are obtained. The main tool in the proofs is a residue theorem for polar-analytic functions.

Geometry and Identity Theorems for Bicomplex Functions and Functions of a Hyperbolic Variable

Let $$\mathbb{D}$$ be the two-dimensional real algebra generated by 1 and by a hyperbolic unit k such that $$k^{2} = 1$$. This algebra is often referred to as the algebra of hyperbolic numbers. A function $$f : \mathbb{D} \rightarrow \mathbb{D}$$ is called $$\mathbb{D}$$-holomorphic in a domain $$\Omega \subset \mathbb{D}$$ if it admits derivative in the sense that $${\rm lim}_{h\rightarrow{0}}\frac{f({\mathfrak{z}_{0}+h)} -f{(\mathfrak{z}_{0})}} {h}$$ exists for every point $$\mathfrak{z}_0$$ in $$\Omega$$, and when h is only allowed to be an invertible hyperbolic number. In this paper we prove that $$\mathbb{D}$$-holomorphic functions satisfy an unexpected limited version of the identity theorem. We will offer two distinct proofs that shed some light on the geometry of $$\mathbb{D}$$. Since hyperbolic numbers are naturally embedded in the four-dimensional algebra of bicomplex numbers, we use our approach to state and prove an identity theorem for the bicomplex case as well.

The geometry of involutions in ranked groups with ati‐subgroup

We revisit the geometry of involutions in groups of finite Morley rank. Our approach unifies and generalises numerous results, both old and recent, that have exploited this geometry; though in fact, we prove much more. We also conjecture that this path leads to a new identification theorem for $\operatorname{PGL}_2(\mathbb{K})$.