A Computer Program for Orthomodular Lattices

Introduction. Every orthomodular lattice (abbreviated : OML) is the union of its maximal Boolean subalgebras (blocks). The question thus arises how conversely Boolean algebras can be amalgamated in order to obtain an OML of which the given Boolean algebras are the blocks. This question we deal with in the present paper.The problem was first investigated by Greechie [6, 7, 8, 9]. His technique of pasting [6] will also play an important role in this paper. A case solved completely by Greechie [9] is the case that any two blocks intersect either in the bounds only or have the bounds, an atom and its complement in common. This is, of course, a very special situation. The more surprising it is that Greechie's methods, if skillfully applied, yield considerable insight into the structure of OMLs and provide a seemingly unexhaustible source for counter-examples.

Let us denote by ℒ ℱ $[\mathcal{L}\mathcal{F}$ the class of all orthomodular lattices (OMLs) that are locally finite (i.e., L ∈ ℒ ℱ $[L\in \mathcal{L}\mathcal{F}$ provided each finite subset of L generates in L a finite subOML). In this note, we first show how one can obtain new locally finite OMLs from the initial ones and enlarge thus the class ℒ ℱ $[\mathcal{L}\mathcal{F}$ . We find ℒ ℱ $[\mathcal{L}\mathcal{F}$ considerably large though, obviously, not all OMLs belong to ℒ ℱ $[\mathcal{L}\mathcal{F}$ . Then we study states on the OMLs of ℒ ℱ $[\mathcal{L}\mathcal{F}$ . We show that local finiteness may to a certain extent make up for distributivity. For instance, we show that if L ∈ ℒ ℱ $[L\in \mathcal{L}\mathcal{F}$ and if for any finite subOML K there is a state s: K → [0, 1] on K, then there is a state on the entire L. We also consider further algebraic and state properties of ℒ ℱ $[\mathcal{L}\mathcal{F}$ relevant to the quantum logic theory.

We show that the variety of ortholattices has the strong amalgamation property and that the variety of orthomodular lattices has the strong Boolean amalgamation property, i.e. that two orthomodular lattices can be strongly amalgamated over a common Boolean subalgebra. We give examples to show that the variety orthomodular lattices does not have the amalgamation property and that the variety of modular ortholattices does not even have the Boolean amalgamation property. We further show that no non-Boolean variety of orthomodular lattices which is generated by orthomodular lattices of bounded height can have the Boolean amalgamation property.

The present of Speckle noise affects the exploitation of useful information in images, especially, in the ultrasound and SAR image. It is desirable to remove speckle noise in real time processing without damaging the image features. In this paper, we propose a new method to suppress the speckle noise in real- time by using Cellular Neural Network (CNN) technology. Section 2, CNN and Cellular Neural Network Universal Machine (CNN-UM) will be briefly introduced, which are the optimized technology for real time image processing applications. Section 3, a few mathematical models of anisotropic diffusion methods for reducing speckle noise on the SAR images will be surveyed and analyzed. In section 4, is also the main part, presented how to design a speckle denoising filter to eliminate the speckle noise and enhance the image features by converting the conventional denoising algorithm to templates, that make CNN chip understandable and processes as a real-time speckle reducing filter. The experiment's results are given in section 5 and some conclusions and future works are summarized in the last section.

Despite all efforts of network designers to predict the behavior of sources in broadband networks, only network traffic controlling, characterization and modeling. Network traffic measurement gives the designer and opportunity to adjust and control the quality of service mechanisms of modern telecommunication networks. In this paper, we present IP traffic measurements. The traffic measurement systems are divided into several groups in according to several properties. Most significant systems are described. In the main part we present the usage of a personal computer as a tool for IP network traffic measurement in the laboratory for telecommunications. The proposed, designed and developed system is outlined and some future work is declared.

We investigate the QCD phase diagram in the strong coupling limit by using a newly developed auxiliary field Monte-Carlo (AFMC) method. Starting from an effective action in the leading order of the 1/g2 and 1/d expansion with one species of unrooted staggered fermion, we solve the many-body problem exactly by introducing the auxiliary fields and integrating out the temporal links and quark fields. We have a sign problem in AFMC, which is different from the original one in finite density lattice QCD. For low momentum auxiliary field modes, a complex phase cancellation mechanism exists, and the sign problem is not serious on a small lattice. Compared with the mean field results, the transition temperature is found to be reduced by around 10 % and the hadron phase is found to be extended in the larger chemical potential direction by around 20 %, as observed in the monomer-dimer-polymer (MDP) simulations.

A new approach for the determination of the critical percolation concentration, percolation probabilities, and cluster size distributions is presented for the site percolation problem. The novel "cluster multiple labeling technique" is described for both two- and three-dimensional crystal structures. Its distinctive feature is the assignment of alternate labels to sites belonging to the same cluster. These sites are members of a simulated finite random lattice. An algorithm useful for the determination of the critical percolation concentration of a finite lattice is also presented. This algorithm is especially useful when applied in conjunction with the cluster multiple labeling technique. The basic features of this technique are illustrated by applying it to a small planar square lattice. Numerical results are given for a triangular subcrystal containing up to 9 000 000 sites. These results compare favorably with the exact value of the infinite lattice critical percolation concentration.

The aim of this paper is to propose a computationally efficient method at combining a direct dynamics approach and musculoskeletal system in order to generate muscle forces. The estimation method is essentially decomposed into three main parts. The first part is the development of a biomechanical model of upper limb allows to determine the musculotendon lengths. The second part is the processing of electromyography signals (EMG). The last part consists of estimating the musculotendon forces based on a Hill rheological model allowing to represent the elastic behavior of muscles. This study selects the motion of elbow flexion as the research object. The obtained results have confirmed the feasibility of forward approach for estimating muscle forces during dynamic contraction. They have shown a good overall correlation between the estimated muscle forces and the measured EMG data. These estimated muscle forces can be exploited in future experimental work as effective information to design and control exoskeletons.

We exploit a prescription to observe directly the physical properties of the thermodynamic limit under continuously applied field in one-dimensional quantum finite lattice systems. By systematically scaling down the energy of the Hamiltonian of the open system from center toward both ends, one could adopt the edge sites with negligibly small energy scale as a "grand canonical" small particle bath, and an equilibrium states with non-integer arbitrary conserved numbers, e.g., electron numbers or sz, are realized in the main part of the system. This will enable the evaluation of the response functions under continuously varying external field in a small lattice without any fine tuning or scaling of parameters while keeping the standard numerical accuracy. Demonstrations are given on quantum spin systems as well as on a Hubbard model by the density matrix renormalization group calculation.

For a finite poset (partially ordered set) \(U\) and a natural number \(n\), let \(S(U,n)\) denote the largest number of pairwise unrelated copies of \(U\) in the powerset lattice (AKA subset lattice) of an \(n\)-element set. If \(U\) is the singleton poset, then \(S(U,n)\) was determined by E. Sperner in 1928; this result is well known in extremal combinatorics. Later, exactly or asymptotically, Sperner's theorem was extended to other posets by A.P. Dove, J.R. Griggs, G.O.H. Katona, D.J. Stahl, and W.T.Jr. Trotter. We determine \(S(U,n)\) for all finite posets with 0 and 1, and we give reasonable estimates for the "\(V\)-shaped" 3-element poset and, mainly, for the 4-element poset with 0 and three maximal elements. For a lattice \(L\), let \(G_{\min}L\) denote the minimum size of generating sets of \(L\). We prove that if \(U\) is the poset of the join-irreducible elements of a finite distributive lattice \(D\), then the function \(k\mapsto G_{\min}{D^k}\) is the left adjoint of the function \(n\mapsto S(U,n)\). This allows us to determine \(G_{\min}{D^k}\) in many cases. E.g., for a 5-element distributive lattice \(D\), \(G_{\min}{D^{2023}}=18\) if \(D\) is a chain and \(G_{\min}{D^{2023}}=15\) otherwise. The present paper, another recent paper, and a 2021 one indicate that large direct powers of small distributive lattices could be of interest in cryptography.

Periodic boundary conditions (PBCs) are a pivotal concept in the treatment of ideal lattices of infinite extent as a finite lattice. Most undergraduate texts that delve into the analytical treatment of lattice dynamics solve the problem in the reciprocal space with an implicit assumption of PBCs. While the traditional approach centered in the reciprocal or [Formula: see text]-space is elegant for simple one-dimensional lattice structures, the method becomes mathematically cumbersome for more complex lattices with real-world applications, such as a honeycomb lattice involving a [Formula: see text] secular determinant. In this work, we have explored the solution of lattice dynamics in direct space numerically, by constructing unit cells through an explicit implementation of Born–von Karman PBCs and solving the lattice dynamical equations in the displacement–time domain using the fourth-order Runge–Kutta algorithm. The Fast Fourier Transform technique is then used to obtain the phonon frequency spectrum corresponding to the computed instantaneous displacements. The fidelity of the model is validated by the agreement of the computational results obtained for the phonon spectrum at high symmetry points in the Brillouin zone with the analytical values for the given problem. Initially, the dynamics and the phonon spectrum are explored for monatomic and diatomic linear lattices which serve to act as introductory cases for the approach before investigating the 2D monatomic square and honeycomb lattices using the nearest and next-nearest neighbor approximations. A short-range force constant model employing the central forces and angular forces is used for the monatomic square lattice to account for the interatomic interactions. Our computational approach is expected to be physically more intuitive for a student for visualizing the periodicity of a given lattice and understanding its related dynamics. The approach demonstrates the usage of Born–von Karman PBCs in constructing a unit cell that effectively captures the dynamics of a lattice system, complementing the traditional secular determinant formalism. The work also illustrates an instance of how the computer programming can be used as a tool to empower the students to explore abstract concepts in physics. A honing of programming and computational skills of students enables them to interact with dynamic models in physics and gain deeper insight into the real-world applications of physics principles. Target group of this work is the students and educators at undergraduate level who seek a thorough and didactic comprehension of lattice dynamics.

This paper elaborates an alternative energy concept involving a solar power plant with a short diffuser. The main physical part of the proposed alternative energy source is a gravitational vortex (convective vortices) that can be thought of as a heat engine. A solar power plant with a short diffuser could produce electricity without carbon dioxide emissions by transforming the available solar energy into useful mechanical shaft work. The proposed concept is similar to the solar chimney power plant concept, but without a chimney of limited height; in that sense, the design is closer to the atmospheric vortex engine concept. Specific details about the proposed alternative solution, previous research outcomes and planned future research work will be discussed in this paper.

Propagation of low frequency acoustic waves in periodic 2D-lattices of tubes

Using the monomer--dimer representation of the lattice Schwinger model, with $N_f =1$ Wilson fermions in the strong--coupling regime ($\beta=0$), we evaluate its partition function, $Z$, exactly on finite lattices. By studying the zeroes of $Z(k)$ in the complex plane $(Re(k),Im(k))$ for a large number of small lattices, we find the zeroes closest to the real axis for infinite stripes in temporal direction and spatial extent $S=2$ and 3. We find evidence for the existence of a critical value for the hopping parameter in the thermodynamic limit $S\rightarrow \infty$ on the real axis at about $k_c \simeq 0.39$. By looking at the behaviour of quantities, such as the chiral condensate, the chiral susceptibility and the third derivative of $Z$ with respect to $1/2k$, close to the critical point $k_c$, we find some indications for a continuous phase transition.

Nowadays, carbon nano (CN) structures and specifically carbon nanotubes (CNTs), because of the nanotube’s nanoscale shape, are widely used in carrier and separation applications. The conjugation of CNTs with polysaccharide, proteins, drugs, and magnetic nanoparticles provides a chance for smart targeting and trajectory manipulation, which are used in the crucial field of life science applications, including for cancer disease diagnostics and treatments. Providing an optimal procedure for delivering a drug to a specific area based on mathematical criteria is key in systemic delivery design. Trajectory guidance and applied force control are the main parameters affected by systemic delivery. Moreover, a better understanding of the tissue parameters and cell membrane molecular behaviour are other factors that can be indirectly affected by the targeted delivery. Both sides are an essential part of successful targeting. The lung is one of the challenging organs for drug delivery inside the human body. It has a large surface area with a thin epithelium layer. A few severe diseases directly involve human lung cells, and optimal and successful drug delivery to the lung for the treatment procedure is vital. In this paper, we studied functionalized CNTs’ targeted delivery via crossing through the lung cell membrane. Molecular dynamics (MD) software simulated all the interaction forces. Mathematical modelling of the cell membrane and proposed delivery system based on the relation of velocity and force has been considered. Dynamics equations for CNTs were defined in the time and frequency domain using control theory methods. The proposed delivery system consists of two main parts: crossing through the cell membrane and targeting inside the cell. For both steps, a mathematical model and a proper magnetic field profile have been proposed. The designed system provides criteria for crossing through the cell membrane within 30 s to 5 min and a translocation profile of 1 to 100 Å.