Bijective Transformation Research Articles

Picture fuzzy sub-hyperspace of a hyper vector space and its application in decision making problem

<abstract><p>In this paper, the notion of picture fuzzy sub-hyperspace of a hyper vector space is introduced and some related results are investigated on the basis of some basic operations (intersection, union, Cartesian product etc.) on picture fuzzy sets. The concept of picture fuzzy linear transformation with respect to some picture fuzzy sub-hyperspace is initiated here and some important results are studied in this regard. It is shown that with respect to some pre-assumed picture fuzzy sub-hyperspace, linear combination of two picture fuzzy linear transformations is a picture fuzzy linear transformation, composition of two picture fuzzy linear transformations is a picture fuzzy linear transformation and inverse of a bijective picture fuzzy linear transformation is a picture fuzzy linear transformation. The effect of good linear transformation on picture fuzzy sub-hyperspaces is discussed here. It is shown that the image of a picture fuzzy sub-hyperspace is a picture fuzzy sub-hyperspace under bijective good linear transformation and the inverse image of a picture fuzzy sub-hyperspace is a picture fuzzy sub-hyperspace under good linear transformation. Some important results on picture fuzzy sub-hyperspaces in the light of $ (\theta, \phi, \psi) $-cut of picture fuzzy set are studied here. Finally, an application of picture fuzzy sub-hyperspace conditions in decision making problem is presented here.</p></abstract>

Theory of perturbation of electric potential by a 3D object made of an anisotropic dielectric material

The extended boundary condition method (EBCM) was formulated for the perturbation of a source electric potential by a 3D object composed of a homogeneous anisotropic dielectric medium whose relative permittivity dyadic is positive definite. The formulation required the application of Green’s second identity to the exterior region to deduce the electrostatic counterpart of the Ewald–Oseen extinction theorem. The electric potential inside the object was represented using a basis obtained by implementing an affine bijective transformation of space to the Gauss equation for the electric field. The EBCM yields a transition matrix that depends on the geometry and the composition of the 3D object, but not on the source potential.

Theory of Perturbation of Electrostatic Field by an Anisotropic Dielectric Sphere

Summary The boundary-value problem for the perturbation of an electric potential by a homogeneous anisotropic dielectric sphere in vacuum was formulated. The total potential in the exterior region was expanded in series of radial polynomials and tesseral harmonics, as is standard for the Laplace equation. A bijective transformation of space was carried out to formulate a series representation of the potential in the interior region. Boundary conditions on the spherical surface were enforced to derive a transition matrix that relates the expansion coefficients of the perturbation potential in the exterior region to those of the source potential. Far from the sphere, the perturbation potential decays as the inverse of the distance squared from the center of the sphere, as confirmed numerically when the source potential is due to either a point charge or a point dipole.

Preservers of unitary group or equivalence by unitaries

We give the complete description of bijective real-linear transformations on B(H), the algebra of all bounded linear operators on an infinite-dimensional complex Hilbert space H, where every unitary U∈B(H) is mapped to a unitary ϕ(U). In turn, bijective real-linear maps on B(H) preserving equivalence by unitaries will be determined.

Noncovariance of “covariant polymerization” in models of loop quantum gravity

It is possible to implement a certain form of modified gravity inspired by loop quantization through non-bijective canonical transformations. The canonical nature might suggest that such modifications are guaranteed to preserve general covariance. Here, however, we show that a dedicated space-time analysis is still required, even in the case of a bijective canonical transformation. In addition, a complete global analysis is presented for a recent proposal of a non-bijective transformation, showing that it does not preserve general covariance and that the only novel physical effect introduced by the modification is the presence of certain time-reversal hypersurfaces between classical space-time regions. These results provide further insights into the physical interpretation of modified dynamics in models of loop quantum gravity.

On Invariant Subspaces in the Lai\u2013Massey Scheme and a Primitivity Reduction

In symmetric cryptography, the round functions used as building blocks for iterated block ciphers are obtained as the composition of different layers acting as a sequence of bijective transformations providing global increasing complexity. The study of the conditions on such layers which make the group generated by the round functions of a block cipher a primitive group has been addressed in the past years, both in the case of Substitution Permutation Networks and Feistel Networks, giving to block cipher designers the recipe to avoid the imprimitivity attack, which exploits the invariance of some subspaces during the encryption. In the case of Lai–Massey schemes, where both Substitution Permutation Network and Feistel Network features are combined, the resistance against imprimitivity attacks has been a long-standing open problem. In this paper we consider a generalization of such a scheme and we prove its resistance against the imprimitivity attack. Our solution is obtained as a consequence of a more general result in which the problem of proving the primitivity of a generalized Lai–Massey scheme is reduced to the simpler one of proving the primitivity of the group generated by the round functions of a strictly related Substitution Permutation Network. We prove how this implies a reduction in the computational cost of invariant-subspace search.

Boundary-free kernel-smoothed goodness-of-fit tests for data on general interval

We propose kernel-type smoothed Kolmogorov-Smirnov and Cramér-von Mises tests for data on general interval, using bijective transformations. Though not as severe as in the kernel density estimation, utilizing naive kernel method directly to those particular tests will result in boundary problem as well. This happens mostly because the value of the naive kernel distribution function estimator is still larger than 0 (or less than 1) when it is evaluated at the boundary points. This situation can increase the errors of the tests especially the second-type error. In this article, we use bijective transformations to eliminate the boundary problem. Some numerical studies illustrating the estimator and the tests’ performances will be presented in the last part of this article.

On computations with double Schubert automaton and stable maps of multivariate cryptography

The families of bijective transformations Gn of affine space Kn over general commutative ring K of increasing order with the property of stability will be constructed. Stability means that maximal degree of elements of cyclic subgroup generated by the transformation of degreed is bounded by d. In the case K = Fq these transformations of Kncan be of an exponential order. We introduce large groups formed by quadratic transformations and numerical encryption algorithm protected by secure protocol of Noncommutative Cryptography. The construction of transformations is presented in terms of walks on Double Schubert Graphs.

The exact equivalence of distance and kernel methods in hypothesis testing

Distance-based tests, also called "energy statistics", are leading methods for two-sample and independence tests from the statistics community. Kernel-based tests, developed from "kernel mean embeddings", are leading methods for two-sample and independence tests from the machine learning community. A fixed-point transformation was previously proposed to connect the distance methods and kernel methods for the population statistics. In this paper, we propose a new bijective transformation between metrics and kernels. It simplifies the fixed-point transformation, inherits similar theoretical properties, allows distance methods to be exactly the same as kernel methods for sample statistics and p-value, and better preserves the data structure upon transformation. Our results further advance the understanding in distance and kernel-based tests, streamline the code base for implementing these tests, and enable a rich literature of distance-based and kernel-based methodologies to directly communicate with each other.

On the Algebraic Properties of Quasi-affine Bijective Transformations

A quasi-affine transformation, being the whole part of a rational affine transformation, is the discretized form of an affine transformation. Introduced by Marie-André Jacob-Da Col, it has been the subject of numerous studies. This article is devoted to the study of the algebraic structures of some quasi-affine bijective transformations, in particular the discrete translations of isolated points and Pythagorean rotations.

МЕТОД ЗАХИСТУ ІНФОРМАЦІЇ СИСТЕМИ «РОЗУМНИЙ ДІМ» НА БАЗІ НОВОГО ПРОТОКОЛУ ОБМІНУ ДАНИХ

The analysis of security problems of information exchange between the client and the "Smart Home" system is carried out. The priorities of information protection at the stage of information transfer are determined. The analysis of the proposed new wireless network standard Thread is carried out. Thread uses IPv6 and is built on the IEEE 802.15.4 standard, and its main advantage is security. At the same time, there can be up to 250 devices in the network, which are protected by encryption of the banking system level. But this is not enough to ensure the integrity of information. The use of a new encryption algorithm based on the current encryption algorithm based on the blockchain algorithm is proposed, thanks to the use of dynamically changing nonlinear bijective transformation (S-blocks) allows to avoid significant information security problems. This algorithm is used to ensure confidentiality (security during transmission), integrity (security during storage and modification only for authorized users) and authenticity (authenticity of the message source). And provides reliable information protection.

Frequency Fitness Assignment: Making Optimization Algorithms Invariant Under Bijective Transformations of the Objective Function Value

Under frequency fitness assignment (FFA), the fitness corresponding to an objective value is its encounter frequency in fitness assignment steps and is subject to minimization. FFA renders optimization processes invariant under bijective transformations of the objective function value. On TwoMax, Jump, and Trap functions of dimension s, the classical (1 + 1)-EA with standard mutation at rate 1/s can have expected runtimes exponential in s. In our experiments, a (1 + 1)-FEA, the same algorithm but using FFA, exhibits mean runtimes that seem to scale as s <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>2</i></sup> ln s. Since Jump and Trap are bijective transformations of OneMax, it behaves identical on all three. On OneMax, LeadingOnes, and Plateau problems, it seems to be slower than the (1 + 1)-EA by a factor linear in s. The (1 + 1)-FEA performs much better than the (1 + 1)-EA on W-Model and MaxSat instances. We further verify the bijection invariance by applying the Md5 checksum computation as transformation to some of the above problems and yield the same behaviors. Finally, we show that FFA can improve the performance of a memetic algorithm for job shop scheduling.

Flow-Based Independent Vector Analysis for Blind Source Separation

This letter describes a time-varying extension of independent vector analysis (IVA) based on the normalizing flow (NF), called NF-IVA, for determined blind source separation of multichannel audio signals. As in IVA, NF-IVA estimates demixing matrices that transform mixture spectra to source spectra in the complex-valued spatial domain such that the likelihood of those matrices for the mixture spectra is maximized under some non-Gaussian source model. While IVA performs a time-invariant bijective linear transformation, NF-IVA performs a series of time-varying bijective linear transformations (flow blocks) adaptively predicted by neural networks. To regularize such transformations, we introduce a soft volume-preserving (VP) constraint. Given mixture spectra, the parameters of NF-IVA are optimized by gradient descent with backpropagation in an unsupervised manner. Experimental results show that NF-IVA successfully performs speech separation in reverberant environments with different numbers of speakers and microphones and that NF-IVA with the VP constraint outperforms NF-IVA without it, standard IVA with iterative projection, and improved IVA with gradient descent.

Identification of Noncanonical Transcripts Produced by Systematic Nucleotide Exchanges in HIV-Associated Centroblastic Lymphoma.

Noncanonical transcriptions include transcriptions that systematically exchange nucleotides, also called bijective transformations or swinger transformations. Swinger transformation A↔T+C↔G recovers identities of 8 among 9 unknown RNAs differentially expressed in centroblastic lymphoma, a human immunodeficiency virus (HIV)-associated non-Hodgkin's lymphoma. The identified RNAs align with human genes with known anti-HIV1 or oncogenic activities. Function disruption through swinger-transformed transcription potentially enables avoiding antiviral responses and contributes to cancer induction.

Maps on positive cones in operator algebras preserving power means

In this paper we consider power means of positive Hilbert space operators both in the conventional and in the Kubo–Ando senses. We describe the corresponding isomorphisms (bijective transformations respecting those means as binary operations) on positive definite cones and on positive semidefinite cones in operator algebras. We also investigate the question when those two sorts of power means can be transformed into each other.

Commutativity preserving transformations on conjugacy classes of finite rank self-adjoint operators

Let H be a complex Hilbert space and let C be a conjugacy class of rank k self-adjoint operators on H with respect to the action of the group of unitary operators. Under the assumption that dim⁡H≥4k we describe all bijective transformations of C preserving the commutativity in both directions. In particular, it follows from this description that every such transformation is induced by a unitary or anti-unitary operator only in the case when for every operator from C the dimensions of eigenspaces are mutually distinct.

Maps Between Hilbert Space Effect Algebras Preserving Unitary Invariant Norms of the Sequential Product

In this paper, we aim to characterize those bijective transformations on Hilbert space effect algebras that preserve any unitary invariant norm of the so-called sequential product ( A , B ) ↦ A B A . The corresponding characterization provides a generalization of a former result of KovAcs and MolnAr.

Additive Maps Preserving Nilpotent Perturbation of Scalars

Let X be a Banach space over \(\mathbb{F} (=\mathbb{R} \rm{or} \mathbb{C})\) with dimension greater than 2. Let \(\mathcal{N}(X)\) be the set of all nilpotent operators and \(\mathcal{B}_0(X)\) the set spanned by \(\mathcal{N}(X)\). We give a structure result to the additive maps on \(\mathbb{F}I+\mathcal{B}_0(X)\) that preserve rank-1 perturbation of scalars in both directions. Based on it, a characterization of surjective additive maps on \(\mathbb{F}I+\mathcal{B}_0(X)\) that preserve nilpotent perturbation of scalars in both directions are obtained. Such a map Φ has the form either Φ(T) = cAT A−1 +ϕ(T)I for all \(T\in\mathbb{F}I+\mathcal{B}_0(X)\) or Φ(T) = cAT* A−1 + ϕ(T)I for all \(T\in\mathbb{F}I+\mathcal{B}_0(X)\), where c is a nonzero scalar, A is a τ-linear bijective transformation for some automorphism τ of F and ϕ is an additive functional. In addition, if dim X = ∞, then A is in fact a linear or conjugate linear invertible bounded operator.

A Characterization of Unitary-Antiunitary Similarity Transformations via Kubo–Ando Means

In this note, we characterize all those bijective transformations on the set of positive semidefinite matrices which preserve an arbitrary unitarily invariant norm of any given Kubo–Ando mean corresponding to an n-monotone strictly concave generating function. This result strengthens a former theorem of Molnar and Szokol.

Symmetries of Projective Spaces and Spheres

AbstractLet H be either a complex inner product space of dimension at least two or a real inner product space of dimension at least three, and let us fix an $\alpha \in (0,\tfrac{\pi }{2} )$. The purpose of this paper is to characterise all bijective transformations on the projective space P(H) which preserve the quantum angle $\alpha$ (or Fubini–Study distance $\alpha$) between lines in both directions. (Let us emphasise that we do not assume anything about the preservation of other quantum angles). For real inner product spaces and when $H=\mathbb{C}^2$ we do this for every $\alpha$, and when H is a complex inner product space of dimension at least three we describe the structure of such transformations for $\alpha \leq \tfrac{\pi }{4}$. Our result immediately gives an Uhlhorn-type generalisation of Wigner’s theorem on quantum mechanical symmetry transformations, that is considered to be a cornerstone of the mathematical foundations of quantum mechanics. Namely, under the above assumptions, every bijective map on the set of pure states of a quantum mechanical system that preserves the transition probability $\cos ^2\alpha$ in both directions is a Wigner symmetry (thus automatically preserves all transition probabilities), except for the case when $H=\mathbb{C}^2$ and $\alpha = \tfrac{\pi }{4}$ where an additional possibility occurs. (Note that the classical theorem of Uhlhorn is the solution for the $\alpha = \tfrac{\pi }{2}$ case). Usually in the literature, results which are connected to Wigner’s theorem are discussed under the assumption of completeness of H; however, here we shall remove this unnecessary hypothesis in our investigation. Our main tool is a characterisation of bijective maps on unit spheres of real inner product spaces which preserve one spherical angle in both directions.