Finite Space Research Articles

Valuation of a financial claim contingent on the outcome of a quantum measurement

We consider a rational agent who at time 0 enters into a financial contract for which the payout is determined by a quantum measurement at some time T > 0. The state of the quantum system is given in the Heisenberg representation by a known density matrix p^ . How much will the agent be willing to pay at time 0 to enter into such a contract? In the case of a finite dimensional Hilbert space H , each such claim is represented by an observable X^T where the eigenvalues of X^T determine the amount paid if the corresponding outcome is obtained in the measurement. We prove, under reasonable axioms, that there exists a pricing state q^ which is equivalent to the physical state p^ such that the pricing function Π0T takes the linear form Π0T(X^T)=P0Ttr(q^X^T) for any claim X^T , where P0T is the one-period discount factor. By ‘equivalent’ we mean that p^ and q^ share the same null space: that is, for any |ξ⟩∈H one has p^|ξ⟩=0 if and only if q^|ξ⟩=0 . We introduce a class of optimization problems and solve for the optimal contract payout structure for a claim based on a given measurement. Then we consider the implications of the Kochen–Specker theorem in this setting and we look at the problem of forming portfolios of such contracts. Finally, we consider multi-period contracts.

The magnitude vector of images

The magnitude of a finite metric space has recently emerged as a novel invariant quantity, allowing to measure the effective size of a metric space. Despite encouraging first results demonstrating the descriptive abilities of the magnitude, such as being able to detect the boundary of a metric space, the potential use cases of magnitude remain under-explored. In this work, we investigate the properties of the magnitude on images, an important data modality in many machine learning applications. By endowing each individual image with its own metric space, we are able to define the concept of magnitude on images and analyse the individual contribution of each pixel with the magnitude vector. In particular, we theoretically show that the previously known properties of boundary detection translate to edge detection abilities in images. Furthermore, we demonstrate practical use cases of magnitude for machine learning applications and propose a novel magnitude model that consists of a computationally efficient magnitude computation and a learnable metric. By doing so, we address one computational hurdle that used to make magnitude impractical for many applications and open the way for the adoption of magnitude in machine learning research.

Interactions and pattern formation in a macroscopic magnetocapillary SALR system of mermaid cereal

When particles are deposited at a fluid interface they tend to aggregate by capillary attraction to minimize the overall potential energy of the system. In this work, we embed floating millimetric disks with permanent magnets to introduce a competing repulsion effect and study their pattern formation in equilibrium. The pairwise energy landscape of two disks is described by a short-range attraction and long-range repulsion (SALR) interaction potential, previously documented in a number of microscopic condensed matter systems. Such competing interactions enable a variety of pairwise equilibrium states, including the possibility of a local minimum energy corresponding to a finite disk spacing. Two-dimensional (2D) experiments and simulations in confined geometries demonstrate that as the areal packing fraction is increased, the dilute repulsion-dominated lattice state becomes unstable to the spontaneous formation of localized clusters, which eventually merge into a system-spanning striped pattern. Finally, we demonstrate that the equilibrium pattern can be externally manipulated by the application of a supplemental vertical magnetic force that remotely enhances the effective capillary attraction.

Semiclassical perspective on Landau levels and Hall conductivity in an anisotropic cubic Dirac semimetal and the peculiar case of star-shaped classical orbits

We study an anisotropic cubic Dirac semimetal subjected to a constant magnetic field. In the case of an isotropic dispersion in the x−y plane, with parameters vx=vy, it is possible to find exact Landau levels, indexed by the quantum number n, using the typical ladder operator approach. Interestingly, we find that the lowest energy level (the zero-energy state in the case of kz=0) has a degeneracy that is 3 times that of other states. This degeneracy manifests in the Hall conductivity as a step at a zero chemical potential 3/2 the size of other steps. Moreover, as n→∞, we find energies En∝n3/2, which means the nth step as a function of the chemical potential roughly occurs at a value μ∝n3/2. We propose that these exciting features could be used to experimentally identify cubic Dirac semimetals. Subsequently, we analyze the anisotropic case vy=λvx, with λ≠1. First, we consider a perturbative treatment around λ≈1 and find that energies En∝n3/2 still hold as n→∞. To gain further insight into the Landau level structure for a maximum anisotropy, we turn to a semiclassical treatment that reveals interesting star-shaped orbits in phase space that close at infinity. This property is a manifestation of weakly localized states. Despite being infinite in length, these orbits enclose a finite phase space volume and permit finding a simple semiclassical formula for the energy, which has the same form as above. Our findings suggest that both isotropic and anisotropic cubic Dirac semimetals should leave similar experimental imprints. Published by the American Physical Society 2024

Immersions into Sasakian space forms

We study immersions of Sasakian manifolds into finite and infinite dimensional Sasakian space forms. After proving Calabi’s rigidity results in the Sasakian setting, we characterise all homogeneous Sasakian manifolds which admit a (local) Sasakian immersion into a nonelliptic Sasakian space form. Moreover, we give a characterisation of homogeneous Sasakian manifolds which can be embedded into the standard sphere both in the compact and noncompact case.

On positively divisible non-Markovian processes

There are some positively divisible non-Markovian processes whose transition matrices satisfy the Chapman–Kolmogorov equation. These processes should also satisfy the Kolmogorov consistency conditions, an essential requirement for a process to be classified as a stochastic process. Combining the Kolmogorov consistency conditions with the Chapman–Kolmogorov equation, we derive a necessary condition for positively divisible stochastic processes on a finite sample space. This necessary condition enables a systematic approach to the manipulation of certain Markov processes in order to obtain a positively divisible non-Markovian process. We illustrate this idea by an example and, in addition, analyze a classic example given by Feller in the light of our approach.

Multimodal transient topology optimization design of heat dissipation structure in electric aircraft power cabin

High power density electronic devices are widely installed in electric aircraft power cabin with finite space, which causes the heat to be dissipated difficultly. In this work, a multimodal transient topology optimization model based on the weight coefficient method is proposed to obtain a lightweight and efficient heat dissipation structure in electric aircraft power cabin. The proposed topology optimization model can consider all flight envelope including takeoff, cruise and landing processes. The effect of the topology weight coefficient on the heat dissipation capacity in all flight envelope is investigated. Results show that the heat dissipation efficiency of the topology optimization structure can be improved by 11.25%, 9.18%, and 11.27% in the takeoff, cruise and landing processes compared with that of the conventional heat dissipation structure, respectively. Increasing the topology weight coefficient by 6.67% can result in an 3.47% improvement in the heat dissipation efficiency during the takeoff and landing processes. The above findings can be used as a reference for the design of the lightweight and efficient heat dissipation structure in electric aircraft.

Technical note: Determining chemical composition of atmospheric single particles by a standard-free mass calibration algorithm

Abstract. The chemical composition of individual particles can be revealed by single-particle mass spectrometers (SPMSs). With higher accuracy in the ratio of mass to charge (m/z), more detailed chemical information could be obtained. In SPMSs, the conventional standard-based calibration methods (internal/external) are constrained by the inhomogeneity of ionization lasers and the finite focusing ability of the inlet system, etc.; therefore, the mass accuracy is restricted. In this study, we obtained the detailed and reliable chemical composition of single particles utilizing a standard-free mass calibration algorithm. In the algorithm, the characteristic distributions of hundreds of ions were concluded and collected in a database denoted as prototype. Each single-particle mass spectrum was initially calibrated by a function with specific coefficients. The range of coefficients was constrained by the magnitude of mass deviation to a finite vector space. To find the optimal coefficient vector, the conformity of each initially calibrated spectrum to the prototype dataset was assessed. The optimum calibrated spectrum was obtained with maximum conformity. For more than 98 % ambient particles, a 20-fold improvement in mass accuracy, from ∼ 10 000 ppm (integer) to ∼ 500 ppm (two decimal places), was achieved. The improved mass accuracy validated the determination of adjacent ions with a m/z difference ∼ 0.05 Th. Furthermore, atmospheric particulate trace elements that were poorly studied before are specified. The obtained detailed single-particle-level chemical information could help explain the source apportionment, reaction mechanism, and mixing state of atmospheric particles.

Weakly compact sets in Orlicz–Bochner sequence spaces

AbstractIn this work, we give three kinds of criteria for weak sets in Orlicz–Bochner sequence spaces without constraints, conditions posited in each criterion are necessary and sufficient. As an application, we give criteria for weak sets in Orlicz sequence spaces. Well‐known conclusions are exhibited once more, such as Schur's theorem, Banach–Alaoglu's theorem, and the boundedly compact principle of finite dimension space. The results obtained show that the weak compactness may not be extrapolated straightforwardly from to , for example, .

Ultra-Generalized Continuous Class F Power Amplifier with Finite Third-Harmonic Load Impedance

This paper proposes the ultra−generalized continuous class F (UCCF) mode, which combines the influences of the drain current conduction angle and overdriven transistor on the drain current waveform to achieve a broader finite third-harmonic load impedance space. The UCCF mode uses α to describe the magnitude of the drain current conduction angle and pcr to describe the drain current peak-clipped ratio. An analysis of the effects of pcr and α on the linearity and efficiency of the UCCF model is presented, establishing a robust theoretical foundation for achieving a balance between these two characteristics. Additionally, an examination of how pcr and α influence the load impedance design space is conducted, demonstrating that the UCCF mode not only offers a broader finite third-harmonic load impedance space but also expands the fundamental and second-harmonic load impedance design space. For practical validation, a PA with frequency of 2.05–2.65 GHz is designed based on CGH40010F. The test results show that S11 is less than −15 dB, the drain efficiency is 67.0–73.2%, and the output power is 40.1–41.0 dBm. The linearity is tested using a 5G NR (New Radio) signal with a bandwidth of 100 MHz and a peak-to-average power ratio of 8 dB at 2.35 GHz. The worst adjacent channel power ratio (ACPR) is −34.8 dBc without digital predistortion (DPD), and −57.8 dBc with DPD. An average output power (Pave) of around 32.4 dBm and an average DE (DEave) of 34.39% were obtained.

Randomized complexity of mean computation and the adaption problem

Recently the adaption problem of Information-Based Complexity (IBC) for linear problems in the randomized setting was solved in Heinrich (2024) [8]. Several papers treating further aspects of this problem followed. However, all examples obtained so far were vector-valued. In this paper we settle the scalar-valued case. We study the complexity of mean computation in finite dimensional sequence spaces with mixed LpN norms. We determine the n-th minimal errors in the randomized adaptive and non-adaptive settings. It turns out that among the problems considered there are examples where adaptive and non-adaptive n-th minimal errors deviate by a power of n. The gap can be (up to log factors) of the order n1/4. We also show how to turn such results into infinite dimensional examples with suitable deviation for all n simultaneously.

An Algorithm for Detecting Sentence Validity

In this paper, we introduce an algorithm for determining the grammatical validity of a sentence. We take a similar approach as in (Preller, 2007) and (Lambek, Type Grammar Revisited, 1999) (Lambek, From word to sentence: a computational algebraic approach to grammar, 2008) by encoding the English words based on word type which we call components. A sentence can be described both algebraically and geometrically. Our algorithm generates the geometric portion called underlinks from the generalized reductions of the algebraic portion. Underlinks uniquely determine the reduction of the components leading to the empty string. This is the mathematical basis for determining if a sentence is valid. We also provide a proof for the algorithm’s time complexity of O(n2) along with a Python implementation. This paper is part of a bigger project based on (Coecke, Mehrnoosh, Clarky, 2010) where we explore the combination of a sentence’s grammar and meaning. This is done by combining two compact closed categories; pregroups represent the grammar of a sentence, and finite dimensional vector spaces describe the meaning of a sentence. Together one compact closed category is created, representing both aspects of the sentence.

Strong blocking sets and minimal codes from expander graphs

A strong blocking set in a finite projective space is a set of points that intersects each hyperplane in a spanning set. We provide a new graph theoretic construction of such sets: combining constant-degree expanders with asymptotically good codes, we explicitly construct strong blocking sets in the ( k − 1 ) (k-1) -dimensional projective space over F q \mathbb {F}_q that have size at most c q k c q k for some universal constant c c . Since strong blocking sets have recently been shown to be equivalent to minimal linear codes, our construction gives the first explicit construction of F q \mathbb {F}_q -linear minimal codes of length n n and dimension k k , for every prime power q q , for which n ≤ c q k n \leq c q k . This solves one of the main open problems on minimal codes.

An inf-sup approach to C0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_0$$\\end{document}-semigroup generation for an interactive composite structure-Stokes PDE dynamics

In this work, we investigate the existence and uniqueness properties of a composite structure (multilayered)–fluid interaction PDE system which arises in multi-physics problems, and particularly in biofluidic applications related to the mammalian blood transportation process. The PDE system under consideration consists of the interactive coupling of 3D Stokes flow and 3D elastic dynamics which gives rise to an additional 2D elastic equation on the boundary interface between these 3D PDE systems. By means of a nonstandard mixed variational formulation, we show that the PDE system generates a C0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_0$$\\end{document}-semigroup on the associated finite energy space of data. In this work, the presence of the pressure term in the 3D Stokes equation adds a great challenge to our analysis. To overcome this difficulty, we follow a methodology which is based on the necessarily non-Leray-based elimination of the associated pressure term, via appropriate nonlocal operators. Moreover, while we express the fluid solution variable via decoupling of the Stokes equation, we construct the elastic solution variables by solving a mixed variational formulation via a Babuska–Brezzi approach.

Quantum diffusion induced by small quantum chaos.

It is demonstrated that quantum systems classically exhibiting strong and homogeneous chaos in a bounded region of the phase space can induce a global quantum diffusion. As an ideal model system, a small quantum chaos with finite Hilbert space dimension N weakly coupled with M additional degrees of freedom which is approximated by linear systems is proposed. By twinning the system the diffusion process in the additional modes can be numerically investigated without taking the unbounded diffusion space into account explicitly. Even though N is not very large, diffusion occurs in the additional modes as the coupling strength increases if M≥3. If N is large enough, a definite quantum transition to diffusion takes place through a critical subdiffusion characterized by an anomalous diffusion exponent.

On quantum computing for artificial superintelligence

Artificial intelligence algorithms, fueled by continuous technological development and increased computing power, have proven effective across a variety of tasks. Concurrently, quantum computers have shown promise in solving problems beyond the reach of classical computers. These advancements have contributed to a misconception that quantum computers enable hypercomputation, sparking speculation about quantum supremacy leading to an intelligence explosion and the creation of superintelligent agents. We challenge this notion, arguing that current evidence does not support the idea that quantum technologies enable hypercomputation. Fundamental limitations on information storage within finite spaces and the accessibility of information from quantum states constrain quantum computers from surpassing the Turing computing barrier. While quantum technologies may offer exponential speed-ups in specific computing cases, there is insufficient evidence to suggest that focusing solely on quantum-related problems will lead to technological singularity and the emergence of superintelligence. Subsequently, there is no premise suggesting that general intelligence depends on quantum effects or that accelerating existing algorithms through quantum means will replicate true intelligence. We propose that if superintelligence is to be achieved, it will not be solely through quantum technologies. Instead, the attainment of superintelligence remains a conceptual challenge that humanity has yet to overcome, with quantum technologies showing no clear path toward its resolution.

Reconstruction of Chitosan Network Orders Using the Meniscus Splitting Method for Designing pH-Responsive Materials.

Chitosan is a product of deacetylated chitin and a natural polymer that is attractive as a functional and biocompatible material in the pursuit of alternative materials to synthetic plastics for a sustainable society. Although hierarchical architectures, from precise molecular structures to nanofibers and twisted structures, have been clarified, the expansion of the anisotropic microstructures of chitosan into millimeter-scale materials is in the process of development. In this study, a chitosan network was reconstructed from an aqueous solution by using the meniscus splitting method to form a three-dimensionally ordered microstructure. A chitosan membrane deposited on the millimeter scale formed a useful anisotropically pH-responsive hydrogel. During the evaporation of the aqueous solution from a finite space, chitosan underwent ordered deposition by capillary force to form a membrane with oriented microstructures and microlayers. Unlike the cast films formed between solid-liquid and air-liquid interfaces, this membrane formed between two air-liquid interfaces. As a result, the membranes with ordered microstructures were capable of signifying directional swelling in aqueous environments and reversible/irreversible swelling-deswelling changes by controlling the pH range. We envision that the anisotropic pH response of the chitosan network can be utilized under physiological conditions as a next-generation material.

Computing a common prior

Morris (1994) and Feinberg (2000) showed that a finite type space admits a common prior if and only if there is no agreeable bet in it.We also consider finite type spaces and observe that the problem of computing a common prior is equivalent to considering the intersection of affine spaces spanned by the types of a player. Therefore, we can apply the Fredholm alternative and conclude that the computational complexity of computing a common prior is strongly polynomial.

Exact time-evolving scattering states in open quantum-dot systems with an interaction: discovery of time-evolving resonant states

We study exact time-evolving many-electron states of an open double quantum-dot system with an interdot Coulomb interaction. A systematic construction of the time-evolving states for arbitrary initial conditions is proposed. For any initial states of one- and two-electron plane waves on the electrical leads, we obtain exact solutions of the time-evolving scattering states, which converge to known stationary scattering eigenstates in the long-time limit. For any initial states of localized electrons on the quantum dots, we find exact time-evolving states of a new type, which we refer to as time-evolving resonant states. In contrast to stationary resonant states, whose wave functions spatially diverge and not normalizable, the time-evolving resonant states are normalizable since their wave functions are restricted to a finite space interval due to causality. The exact time-evolving resonant states enable us to calculate the time-dependence of the survival probability of electrons on the quantum dots for the system with the linearized dispersions. It decays exponentially in time on one side of an exponential point of resonance energies while, on the other side, it oscillates during the decay as a result of the interference of the two resonance energies.

Limits and colimits in the category of Banach $L^{0}$-modules

We prove that the category of Banach L^{0} -modules over a given \sigma -finite measure space is both complete and cocomplete, which means that it admits all small limits and colimits.