Inner Product Research Articles

Geometry of quantum complexity

Computational complexity is a quantum information concept that recently has found applications in the holographic understanding of the black hole interior. We consider quantum computational complexity for $n$ qubits using Nielsen's geometrical approach. In the definition of complexity there is a big amount of arbitrariness due to the choice of the penalty factors, which parametrizes the cost of the elementary computational gates. In order to reproduce desired features in holography, such as ergodicity and exponential maximal complexity for large number of qubits $n$, negative curvatures are required. With the simplest choice of penalties, this is achieved at the price of singular sectional curvatures in the large $n$ limit. We investigate a choice of penalties in which we can obtain negative curvatures in a smooth way. We also analyze the relation between operator and state complexities, framing the discussion with the language of Riemannian submersions. This provides a direct relation between geodesics and curvatures in the unitaries and the states spaces, which we also exploit to give a closed-form expression for the metric on the states in terms of the one for the operators. Finally, we study conjugate points for a large number of qubits in the unitary space and we provide a strong indication that maximal complexity scales exponentially with the number of qubits in a certain regime of the penalties space.

An elementary proof for the representation theorem of analytic isotropic tensor functions of a second-order tensor

Based on the Cayley-Hamilton theorem and fixed-point method, we provide an elementary proof for the representation theorem of analytic isotropic tensor functions of a second-order tensor in a three-dimensional (3D) inner-product space, which avoids introducing the generating function and Taylor series expansion. The proof is also extended to any finite-dimensional inner-product space.

Hadamard's optimized inequality

The aim of the paper is demonstration of equalitymaxσ∈Snσ≠id{1,…,n}{∏i=1n|〈xi,xσ(i)〉|}=maxτ∈Snτ is a transposition⁡{∏i=1n|〈xi,xτ(i)〉|} for each sequence of vectors x1,…,xn in a unitary space. The obtained equality has an important application in optimizing of certain generalization of the Hadamard's inequality proved by Xiaodong Zhang and Shangjun Yang and by the authors of the paper independently. Some illustrating examples are presented in the article. Also a problem connected with the mentioned equality is formulated.

A note on discrete wave packet frames in ℂN

We show that for every nonzero vector [Formula: see text] in [Formula: see text], the discrete wave packet system [Formula: see text] constitutes a frame for the unitary space [Formula: see text]. An application of this result is given, where frame conditions cannot be derived from discrete wavelet systems in [Formula: see text]. The canonical dual of discrete wave packet frame is also discussed.

Weak n-inner product spaces

In this article, we study a generalization of the n-inner product which we name weak n-inner product. As particular case, we consider the n-iterated 2-inner product and we give its representation in terms of the standard k-inner products, $$k\le n$$ , using the Dodgson’s identity for determinants. Finally, we present several applications, including a brief characterization of a linear regression model for the random variables in discrete case and a generalization of the Chebyshev functional using the n-iterated 2-inner product.

English

English

Constructions of Orbit Codes Based on Unitary Spaces Over Finite Fields

Orbit codes, as special constant dimension codes, have attracted much attention due to their applications for error correction in random network coding. This paper is devoted to constructing large orbit codes by making full use of unitary space. Firstly, we construct a cyclic unitary group of order $q^{2n}-1$ by means of the companion matrix of a primitive polynomial over finite fields $\mathbb {F}_{q^{2}}$ , and so the corresponding code is unitary cyclic orbit code. As a special application, a new quaternary orbit code $(6,63,4,3)$ is given. Secondly, we obtain orbit codes with large size using the external direct product of unitary groups acting on the direct sum of subspaces. Finally, a table is given for illustrating our codes improve upon those constructed by Trautmann et al. and Poroch et al.

On 2-inner product spaces and reproducing property

This paper is devoted to study the reproducing property on 2-inner product Hilbert spaces. We focus on a new structure to produce reproducing kernel Hilbert and Banach spaces. According to multi variable computing, this structures play the key role in probability, mathematical finance and machine learning.

Simulation Study on Fire Visibility of Typical Floor Planes of Modern Super High-Rise Office Buildings in China

With the development of office forms, the space form of super high-rise office buildings changed from the unitary efficient office space to a complex space that integrated office, communication, and experience, which also diversified the design of typical floors in the office zone. However, from the perspective of fire prevention, the placement of shared space changed the form of the plane in typical floors in the office zone, affecting the smoke spreading of fire and paths of personnel evacuation. Hence, the subject on the planar relationship among high-rise office buildings based on fire prevention analysis, which optimizes space design, is worthy of discussion. After collecting many cases of super high-rise office buildings in China, this study categorized them into six typical planes and adopted the software PyroSim for comparative simulation of the smoke spreading of fire. By comparing the visibility of different zones in fires, this study analyzed the effects of the area of the office zone, the location of the atrium, and the form of the plane on the fire visibility on the typical floor and put forth the key factors that influence fire visibility, thus optimizing the plane design of the typical floor of super high-rise buildings. The findings show that in the six fire scenes, the area of the office zone is between 1136m2 and 1736m2. The peak of duration for visibility decline at 1.5 m appeared in Scene 3. Its office zone is 1536m2, and the duration of visibility decline is greater than 1000 s. In other fire scenes, the duration of visibility decline is less than 300 s. By comparing the plane arrangement features of the fire scenes, the paper concludes that in a given fire scene, establishing an atrium in the office zone, expanding the length of the adjacent edge between the atrium and the office zone, and building an evacuation corridor between the atrium and the core tube can greatly extend the effective evacuation time. This can serve as reference for the plane design of super high-rise office buildings.

Inner Product of RGB Unit Vectors for Simple and Versatile Detection of Color Transition.

We propose a novel concept for detecting color transition by the inner product (IP) of RGB unit vectors. A digital microscope-based detector and a Visual Basic program were developed in-house. The concept is applied to indicator-based flow titration. The IP is 1 or < 1 if the vector's direction is the same or different, respectively. The IP's change can be used as a criterion for the indicator's color transition. The present IP-based approach is simple, economical, and versatile because it is applicable to any color transition without selecting an analytical wavelength.

Variational Inequalities on 2-Inner Product Spaces

Variational Inequalities on 2-Inner Product Spaces

On Uniqueness of New Orthogonality via 2-HH Norm in Normed Linear Space

This paper generalizes the special case of the Carlsson orthogonality in terms of the 2-HH norm in real normed linear space. Dragomir and Kikianty (2010) proved in their paper that the Pythagorean orthogonality is unique in any normed linear space, and isosceles orthogonality is unique if and only if the space is strictly convex. This paper deals with the complete proof of the uniqueness of the new orthogonality through the medium of the 2-HH norm. We also proved that the Birkhoff and Robert orthogonality via the 2-HH norm are equivalent, whenever the underlying space is a real inner-product space.

Describing topology on the set of persistence diagrams

In Topological Data Analysis, persistence diagrams are a useful tool for visualizing persistent homology. It is proved that the set of persistence diagrams endowed with the bottleneck distance is homeomorphic to the pre-Hilbert space of finitary sequences.We also consider a bitopological space of persistence diagrams and describe its topology.

On the isomorphism of certain primitive Q-polynomial not P-polynomial association schemes

In 2011, Penttila and Williford constructed an infinite new family of primitive Q-polynomial 3-class association schemes, not arising from distance regular graphs, by exploring the geometry of the lines of the unitary polar space H(3,q2), q even, with respect to a symplectic polar space W(3,q) embedded in it.In a private communication to Penttila and Williford, H. Tanaka pointed out that these schemes have the same parameters as the 3-class schemes found by Hollmann and Xiang in 2006 by considering the action of PGL(2,q2), q even, on a non-degenerate conic of PG(2,q2) extended in PG(2,q4). Therefore, the question arises whether the above association schemes are isomorphic. In this paper we provide the positive answer. As by product, we get an isomorphism of strongly regular graphs.

Ulam Stability in Real Inner-Product Spaces

Roughly speaking an equation is called Ulam stable if near each approximate solution of the equation there exists an exact solution. In this paper we prove that Cauchy-Schwarz equation, Ortogonality equation and Gram equation are Ulam stable.This paper is concerned with the Ulam stability of some classical equations arising in thecontext of inner-product spaces. For the general notion of Ulam stability see, e.q., [1]. Roughlyspeaking an equation is called Ulam stable if near every approximate solution there exists anexact solution; the precise meaning in each case presented in this paper is described in threetheorems. Related results can be found in [2, 3, 4]. See also [5] for some inequalities in innerproduct spaces.

Types of Statistical Indicators Characterized by 2-Pre-Hilbert Spaces

In this article, we established new results related to a 2-pre-Hilbert space. Among these results we will mention the Cauchy-Schwarz inequality. We show several applications related to some statistical indicators as average, variance and standard deviation and correlation coefficient, using the standard 2-inner product and some of its properties. We also present a brief characterization of a linear regression model for the random variables in discrete case.

Numerical Radius Inequalities in 2-Inner Product Spaces

In this paper, we have obtained the analogue results on numerical radius inequalities from the classical inner product spaces to 2-inner product spaces. We have established several related reverse inequalities and some well known results in 2-inner product spaces

On Parseval’s Identity and Plancherel Relation in 2-Inner Product Spaces

This paper investigates the foundational concepts of linear 2-normed spaces and 2-inner product spaces, presenting essential definitions and theorems that elucidate their structure. We establish Parseval’s identity for finite orthonormal sets within these spaces and examine its relationship with the Plancherel relation. Our findings include theorems related to orthonormal sets and Bessel's inequality, highlighting their implications for countably infinite orthonormal sets. By contributing to the theoretical framework of 2-inner product spaces, we enhance the understanding of orthogonality and summability in this context. Introduced by Diminnie et al. (1973), the 2-inner product extends traditional inner product notions into a two-dimensional framework, offering valuable insights into unique mathematical structures. Recent contributions, such as the formalization of orthonormal sets by Elumalai and Patricia (2000) and the rigorous proofs of the Riesz theorems by Harikrishnan et al. (2011), further deepen our understanding of these spaces. This paper aims to develop important inequalities based on orthonormal sets and present equivalent statements related to them, fostering continued research in various mathematical contexts.

In-Block Lateralization as a New Technique for Mobilization of the Inferior Alveolar Nerve: A Technique Case Series.

The edentulous atrophic posterior mandible is often a great challenge for implant rehabilitation. Although a number of treatment options have been proposed, including the use of short implants and surgical grafting techniques, in cases of severe bone atrophy, techniques for mobilization of the inferior alveolar nerve (IAN) have been shown to be efficient, with good results. Four female patients underwent IAN lateralization for prosthetic rehabilitation of the posterior mandible from 2013 to 2019, with 3 years to 5 years and 3 months of follow-up. This case series describes a new technique for mobilization of the IAN, named in-block lateralization, to facilitate access to the IAN and to reduce nerve manipulation. The implant is immediately installed (allowing nerve lateralization in unitary spaces), and the original mandibular anatomy is restored with autogenous bone from the original bed during the same surgical procedure. When well indicated and well performed, this new approach provides better and easier visualization of the IAN and safer manipulation aiming to achieve good results for implant stability and minimal risk of neurosensory disturbances, allowing rehabilitation even in unitary spaces.

Deconstructing effective non-Hermitian dynamics in quadratic bosonic Hamiltonians

Unlike their fermionic counterparts, the dynamics of Hermitian quadratic bosonic Hamiltonians are governed by a generally non-Hermitian Bogoliubov-de Gennes effective Hamiltonian. This underlying non-Hermiticity gives rise to a dynamically stable regime, whereby all observables undergo bounded evolution in time, and a dynamically unstable one, whereby evolution is unbounded for at least some observables. We show that stability-to-instability transitions may be classified in terms of a suitably generalized symmetry, which can be broken when diagonalizability is lost at exceptional points in parameter space, but also when degenerate real eigenvalues split off the real axis while the system remains diagonalizable. By leveraging tools from Krein stability theory in indefinite inner-product spaces, we introduce an indicator of stability phase transitions, which naturally extends the notion of phase rigidity from non-Hermitian quantum mechanics to the bosonic setting. As a paradigmatic example, we fully characterize the stability phase diagram of a bosonic analogue to the Kitaev–Majorana chain under a wide class of boundary conditions. In particular, we establish a connection between phase-dependent transport properties and the onset of instability, and argue that stable regions in parameter space become of measure zero in the thermodynamic limit. Our analysis also reveals that boundary conditions that support Majorana zero modes in the fermionic Kitaev chain are precisely the same that support stability in the bosonic chain.