Vectors In Hilbert Space Research Articles

A Closed Formalism for Anatomy-Independent Projection and Optimization of Magnetic Stimulation Coils on Arbitrarily Shaped Surfaces.

Transcranial magnetic stimulation (TMS) is a popular method for the noninvasive stimulation of neurons in the brain. It has become a standard instrument in experimental brain research and has been approved for a range of diagnostic and therapeutic applications. These applications require appropriately shaped coils. Various applications have been established or approved for specific coil designs with their corresponding spatial electric field distributions. However, the specific coil implementation may no longer be appropriate from the perspective of available material and manufacturing opportunities or considering the latest understanding of how to achieve induced electric fields in the head most efficiently. Furthermore, in some cases, field measurements of coils with unknown winding or a user-defined field are available and require an actual implementation. Similar applications exist for magnetic resonance imaging coils. This work aims at introducing a complete formalism free from heuristics, iterative optimization, and ad-hoc or manual steps to form practical stimulation coils with individual turns to either equivalently match an existing coil or produce a given field. The target coil can reside on practically any sufficiently large or closed surface adjacent to or around the head. The method derives an equivalent field through vector projection exploiting the well-known Huygens' and Love's equivalence principle. In contrast to other coil design or optimization approaches recently presented, the procedure is an explicit forward Hilbert-space vector projection or basis change. For demonstration, we map a commercial figure-of-eight coil as one of the most widely used devices and a more intricate coil recently approved clinically for addiction treatment (H4) onto a bent surface close to the head for highest efficiency and lowest field energy. The resulting projections are within ≤4% of the target field and reduce the necessary pulse energy by more than 40%.

On the von Bahr–Esseen inequality for pairwise independent random vectors in Hilbert spaces with applications to mean convergence

Abstract In this correspondence, we prove the von Bahr–Esseen moment inequality for pairwise independent random vectors in Hilbert spaces. Our constant in the von Bahr–Esseen moment inequality is better than that obtained for the real-valued random variables by Chen et al. [The von Bahr–Esseen moment inequality for pairwise independent random variables and applications, J. Math. Anal. Appl. 419 (2014), 1290–1302], and Chen and Sung [Generalized Marcinkiewicz–Zygmund type inequalities for random variables and applications, J. Math. Inequal. 10(3) (2016), 837–848]. The result is then applied to obtain mean convergence theorems for triangular arrays of rowwise and pairwise independent random vectors in Hilbert spaces. Some results in the literature are extended.

The theory of dynamical systems of conflict in the framework of functional analysis

In this article, we give an introduction to the mathematical setting of problems related to the phenomenon of conflict in terms of constructions in Hilbert spaces. The struggle (conflict, game) between opponents (adversaries, players) will be represented by operator transformations of vectors in Hilbert spaces and probabilistic distributions on the territory of life resources. The phenomenon of conflict as a contradiction between opponents appears in mathematical terms as an intersection the domains of definition for operators and overlapping of corresponding measures. Conflict interaction between opponents in the physical sense is described by the specific transformation of states in a Hilbert space. In turn, this is a mapping that changes the spectral measurements. Thus, a complex dynamical system arises, which we call a dynamical system of conflict. Then the following main problems arise as fundamental questions. What reasonable law of engagement (game or war) should be adopted to resolve the initial intersections? What is a fair limiting distribution of the resource territory? In a more general formulation, solving conflict problems means the detailed describing of all possible outcomes on opponents states of the type: victories, defeats, states of equilibrium, compromises as fixed points together with their basins of attraction.

A generalization of van der Corput's difference theorem with applications to recurrence and multiple ergodic averages

We prove a generalization of van der Corput's difference theorem for sequences of vectors in a Hilbert space. This generalization is obtained by establishing a connection between sequences of vectors in the first Hilbert space with a vector in a new Hilbert space whose spectral type with respect to a certain unitary operator is absolutely continuous with respect to the Lebesgue measure. We use this generalization to obtain applications regarding recurrence and multiple ergodic averages when we have measure preserving automorphisms T and S that do not necessarily commute, but T has a maximal spectral type that is mutually singular with the Lebesgue measure.

Equidistribution of Continuous Functions Along Monotone Compact Covers

Abstract We give a necessary and sufficient condition for equidistribution of continuous functions along monotone compact covers on locally compact spaces. We show the existence of equidistributed mappings along Bohr nets arising from group actions. Using almost periodic means, we give an analogue of Weyl’s equidistribution criterion for continuous functions with values in arbitrary topological groups. We prove van der Corput’s inequality on the lattice ℕm for vectors in Hilbert spaces, and use this inequality to extend Hlawka’s equidistribution theorem to functions on the lattice ℕm (m ≥ 1) with values in arbitrary topological groups.

Convergence properties for coordinatewise asymptotically negatively associated random vectors in Hilbert space

Abstract In this work, the authors study some convergence results including weak law of large numbers, strong law of large numbers, complete convergence, and complete moment convergence for weighted sums of coordinatewise asymptotically negatively associated random vectors in Hilbert spaces. These results improve or extend some corresponding ones in the literature.

Integral Results Related to Similarly Separable Vectors in Separable Hilbert Spaces

In this work, we use similarly separable vectors in separable Hilbert spaces to provide generalized integral results related to majorization, Niezgoda, and Ćebysév type inequalities. Next, we furnish some refinements of these inequalities. Theorems obtained in this work extend and improve several known results in the literature. An important aspect of our work is that these inequalities are directly related to Arithmetic, Geometric, Harmonic, and Power means. These means have played an important role in many branches of arts and sciences since the last 2600 years.

The complete moment convergence for coordinatewise pairwise negatively quadrant dependent random vectors in Hilbert space

In this article, we investigate the complete moment convergence for a sequence of coordinatewise pairwise negatively quadrant dependent random vectors in Hilbert space, which is coordinatewise weakly upper bounded by a random vector under special case.

Schr\xf6dinger\u2019s Equation as a Consequence of the Central Limit Theorem Without Assuming Prior Physical Laws

The central limit theorem has been found to apply to random vectors in complex Hilbert space. This amounts to sufficient reason to study the complex–valued Gaussian, looking for relevance to quantum mechanics. Here we show that the Gaussian, with all terms fully complex, acting as a propagator, leads to Schrödinger’s non-relativistic equation including scalar and vector potentials, assuming only that the norm is conserved. No physical laws need to be postulated a priori. It thereby presents as a process of irregular motion analogous to the real random walk but executed under the rules of the complex number system. There is a standard view that Schrödinger’s equation is deterministic, whereas wavefunction “collapse” is probabilistic (by Born’s rule)—we have now a demonstrated linkage to the central limit theorem, indicating a stochastic picture at the foundation of Schrödinger’s equation itself. It may be an example of Wheeler’s “It from bit” with “No underlying law”. Reasons for the primary role of mathbf {C} are open to discussion. The present derivation is compared with recent reconstructions of the quantum formalism, which have the aim of rationalizing its obscurities.

On convergence properties for weighted sums of coordinatewise ANA random vectors in Hilbert spaces

In this article, the complete convergence and complete moment convergence for weighted sums of asymptotically negatively associated random vectors are investigated. The results obtained in this article extend the corresponding ones for random variables in Chen et al. to H-valued random vectors. Weighted version of strong law of large numbers and complete integral convergence result are also presented.

On Kolmogorov Complexity of Unitary Transformations in Quantum Computing

We introduce a notion of Kolmogorov complexity of unitary transformation, which can (roughly) be understood as the least possible amount of information required to fully describe and reconstruct a given finite unitary transformation. In the context of quantum computing, it corresponds to the least possible amount of data to define and describe a quantum circuit or quantum computer program. Our Kolmogorov complexity of unitary transformation is built upon Kolmogorov "qubit complexity" of Berthiaume, W. Van Dam and S. Laplante via mapping from unitary transformations to unnormalized density operators, which are subsequently "purified" into unnormalized vectors in Hilbert space. We discuss the optimality of our notion of Kolmogorov complexity in a broad sense.

The 3-Sphere Instead of Hilbert Space

The Geometric Algebra formalism opens the door to developing a theory replacing conventional quantum mechanics. Generalizations, stemming from changing of complex numbers by geometrically feasible objects in three dimensions, followed by unambiguous definition of states, observables, measurements, bring into reality clear explanations of weird quantum mechanical features, for example primitively considering atoms as a kind of solar system. The three-sphere S3 becomes the playground of the torsion kind states eliminating abstract Hilbert space vectors. The S3 points evolve, governed by updated Schrodinger equation, and act in measurements on observable as operators.

On the Strong Laws of Large Numbers for Sequences of Blockwise Pairwise and Coordinatewise Negatively Dependent Random Vectors in Hilbert Spaces

On the Strong Laws of Large Numbers for Sequences of Blockwise Pairwise and Coordinatewise Negatively Dependent Random Vectors in Hilbert Spaces

Some limiting behavior of the maximum of the partial sum for asymptotically negatively associated random vectors in Hilbert space

In this paper, we establish L 2-convergence and complete convergence results for the maximum of the partial sum of asymptotically negatively associated random vectors in Hilbert space. In addition, we consider Hájek-Rényi inequality and prove some strong law of large numbers for H-valued asymptotically negatively associated random vectors.

Convergence Theorems for m -Coordinatewise Negatively Associated Random Vectors in Hilbert Spaces

In this study, some new results on convergence properties for m -coordinatewise negatively associated random vectors in Hilbert space are investigated. The weak law of large numbers, strong law of large numbers, complete convergence, and complete moment convergence for linear process of H-valued m -coordinatewise negatively associated random vectors with random coefficients are established. These results improve and generalise some corresponding ones in the literature.

From linear algebra to quantum information

Anticipating the realization of quantum computers, we propose the most reader-friendly exposition of quantum information and qubits theory. Although the latter lies within framework of linear algebra, it has some fl avor of quantum mechanics and it would be easier to get used to special symbols and terminologies. Quantum mechanics is described in the language of functional analysis: the state space (the totality of all states) of a quantum system is a Hilbert space over the complex numbers and all mechanical quantities are taken as Hermite operators. Hence some basics of functional analysis is necessary. We make a smooth transition from linear algebra to functional analysis by comparing the elements in these theories: Hilbert space vs. fi nite dimensional vector space, Hermite operator vs. linear map given by a Hermite matrix. Then from Newtonian mechanics to quantum mechanics and then to the theory of qubits. We elucidate qubits theory a bit by accommodating it into linear algebra framework under these precursors.

Weak Laws of Large Numbers for Negatively Superadditive Dependent Random Vectors in Hilbert Spaces

Let $\{X_{n}, {n}\in \mathbb{N}\}$ be a sequence of negatively superadditive dependent random vectors taking values in a real separable Hilbert space. In this paper, we present the weak laws of large numbers for weighted sums (with or without random indices) of $\{X_{n}, {n}\in \mathbb{N}\}$.

Central Limit Theorems for Weighted Sums of Dependent Random Vectors in Hilbert Spaces via the Theory of the Regular Variation

In this paper, based on the theory of regularly varying functions we study central limit theorems for the weighted sum $$S_n=\sum _{j=1}^{m_n}c_{nj}X_{nj}$$ , where $$(X_{nj};1\le j \le m_n,n\ge 1)$$ is a Hilbert-space-valued identically distributed martingale difference array and $$(c_{nj};1\le j \le m_n,n\ge 1)$$ is an array of real numbers. As an application, we present a central limit theorem for moving average processes of martingale differences.

Geometric phase and in-phase superpositions: A fresh perspective on interference in phase space

We apply geometric phase ideas to coherent states to shed light on interference phenomenon in the phase space description of continuous variable Cartesian quantum systems. The motivating idea is Pancharatnam's concept of two Hilbert space vectors being ’’in phase”. This leads to special sums and integrals of vectors which are locally in-phase superpositions. Geometric phase considerations naturally lead to the emergence of phase space areas. Applied to the overcomplete family of coherent states, we are led to preferred in-phase integral representations for various states of physical significance, such as the position, momentum and Fock states as well as the squeezed vacuum state. Interestingly, the Husimi-Kano Q function is maximized along the line of such superpositions. We also get a fresh perspective on the Bohr-Sommerfeld quantization condition. Finally, we use our exact integral representations to obtain asymptotic expansions for state overlaps and matrix elements, leading to phase space area considerations similar to the ones noted earlier in the seminal works of Schleich, Wheeler and collaborators, but now from a geometric phase perspective.

Positive operator-valued measures and densely defined operator-valued frames

In the signal-processing literature, a frame is a mechanism for performing analysis and reconstruction in a Hilbert space. By contrast, in quantum theory, a positive operator-valued measure (POVM) decomposes a Hilbert-space vector for the purpose of computing measurement probabilities. Frames and their most common generalizations can be seen to give rise to POVMs, but does every reasonable POVM arise from a type of frame? We answer this question using a Radon–Nikodym-type result.