Symmetry and Experimental Data

We extend the periodicity of birational rowmotion for rectangular posets to the case when the base field is replaced by a noncommutative ring (under appropriate conditions). This resolves a conjecture from 2014. The proof uses a novel approach and is fully self-contained. Consider labelings of a finite poset \(P\) by \(\left\vert P\right\vert + 2\) elements of a ring \(\mathbb{K}\): one label associated with each poset element and two constant labels for the added top and bottom elements in \(\widehat{P}\). Birational rowmotion is a partial map on such labelings. It was originally defined by Einstein and Propp for \(\mathbb{K}=\mathbb{R}\) as a lifting (via detropicalization) of piecewise-linear rowmotion, a map on the order polytope \(\mathcal{O}(P) := \{\text{order-preserving } f: P \to[0,1]\}\). The latter, in turn, extends the well-studied rowmotion map on the set of order ideals (or more properly, the set of order filters) of \(P\), which correspond to the vertices of \(\mathcal{O}(P)\). Dynamical properties of these combinatorial maps sometimes (but not always) extend to the birational level, while results proven at the birational level always imply their combinatorial counterparts. Allowing \(\mathbb{K}\) to be noncommutative, we generalize the birational level even further, and some properties are in fact lost at this step.In 2014, the authors gave the first proof of periodicity for birational rowmotion on rectangular posets (when \(P\) is a product of two chains) for \(\mathbb{K}\) a field, and conjectured that it survives (in an appropriately twisted form) in the noncommutative case. In this paper, we prove this noncommutative periodicity and a concomitant antipodal reciprocity formula. We end with some conjectures about periodicity for other posets, and the question of whether our results can be extended to (noncommutative) semirings.It has been observed by Glick and Grinberg that, in the commutative case, periodicity of birational rowmotion can be used to derive Zamolodchikov periodicity in the type \(AA\) case, and vice-versa. However, for noncommutative \(\mathbb{K}\), Zamolodchikov periodicity fails even in small examples (no matter what order the factors are multiplied), while noncommutative birational rowmotion continues to exhibit periodicity. Thus, our result can be viewed as a lateral generalization of Zamolodchikov periodicity to the noncommutative setting.Mathematics Subject Classifications: 06A07, 05E99Keywords: Rowmotion, posets, noncommutative rings, semirings, Zamolodchikov periodicity, root systems, promotion, trees, graded posets, Grassmannian, tropicalization

The spin-one Duffin-Kemmer-Petiau oscillator in uniform magnetic field is studied in noncommutative formalism. The corresponding energy is obtained and thereby the corresponding thermal properties are obtained for both commutative and noncommutative cases.

Since the late sixties, various Auslander regularity conditions have been widely investigated in both commutative and noncommutative cases, e.g., [Ba2], [FGR], [Roos], [Lev], etc .... In particular, the dimension and multiplicity theory for modules, the pure (holonomic) module theory over noncommutative Auslander regular rings have been developed in the study of rings of differential operators and enveloping algebras, e.g., [Ka], [Be], [B-B], [K-T], etc ... A typical result concerning purity is the so-called Gabber-Kashiwara theorem (see [Gi] Theorem V8) which has been used to: (1). determine the existence of holonomic submodules in D-modules and the existence of a b-function; (2). determine the equidimensionality of the characteristic variety of a simple module over the enveloping algebra of a Lie algebra. In this chapter we first present a general study of Auslander regularity of filtered rings and graded rings; Then we generalize the important Gabber-Kashiwara theorem and Roos theorem to Zariski rings with commutative associated graded rings including most of the examples given in Ch.II. §2; and consequently the pure module theory over Zariski ring is developed as well.KeywordsExact SequenceSpectral SequenceNoetherian RingRegular RingProjective ResolutionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We develop the superfield approach to the effective potential in three dimensions and calculate the one-loop and two-loop Kählerian effective potential in commutative and noncommutative cases.

Let B be an associative ring with identity, A a subring of B containing the identity of B. If B is commutative then it is customary to define an element b of B to be integral over A if it satisties an equation of the form for some a1, a2, …, an A. This definition does not generalize readily to the case when B is non-commutative. Van der Waerden ([11], p. 75) defines b ∈ B to be integral over A if all powers of b belong to a finite A-module. This definition is quite satisfactory when A satisfies the ascending chain condition for left ideals, but in the general case this type of integrity is not necessarily transitive, even when B is commutative. Krull [6] calls an element b ∈ B which satisfies the above condition almost integral over A (but he only considers the commutative case). The subset Ā of B consisting of all almost integral elements over A is called the complete integral closure of A in B. If Ā = A, A is said to be completely integrally closed in B. More recently (in [3]), Gilmer and Heinzer (see also Bourbaki, [1]) have discussed these properties in the commutative case and have shown that the complete integral closure of A in B need not be completely integrally closed in B. If B is not commutative, the set A of elements of B almost integral over A, may not even form a ring. In [5] p. 122, Jacobson uses a definition equivalent to Van der Waerden's for the non-commutative case but the definition applies only for a very restricted class of rings.

Weak amenability of a weighted group algebra, or a Beurling algebra, is a long-standing open problem. The commutative case has been extensively investigated and fully characterized. We study the non-commutative case. Given a weight function $\omega$ on a locally compact group $G$, we characterize derivations from $L^1(G,\omega)$ into its dual in terms of certain functions. Then we show that for a locally compact IN group $G$, if there is a non-zero continuous group homomorphism $\varphi$: $G\to \mathbb{C}$ such that $\varphi(x)/\omega(x)\omega(x^{-1})$ is bounded on $G$, then $L^1(G,\omega)$ is not weakly amenable. Some useful criteria that rule out weak amenability of $L^1(G,\omega)$ are established. Using them we show that for many polynomial type weights the weighted Heisenberg group algebra is not weakly amenable, neither is the weighted $\boldsymbol{ax+b}$ group algebra. We further study weighted quotient group algebra $L^1(G/H,\hat\omega)$, where $\hat\omega$ is the canonical weight on $G/H$ induced by $\omega$. We reveal that the kernel of the canonical homomorphism from $L^1(G,\omega)$ to $L^1(G/H,\hat\omega)$ is complemented. This allows us to obtain some sufficient conditions under which $L^1(G/H,\hat\omega)$ inherits weak amenability of $L^1(G,\omega)$. We study further weak amenability of Beurling algebras of subgroups. In general, weak amenability of a Beurling algebra does not pass to the Beurling algebra of a subgroup. However, in some circumstances this inheritance can happen. We also give an example to show that weak amenability of both $L^1(H,\omega|_H)$ and $L^1(G/H,\hat\omega)$ does not ensure weak amenability of $L^1(G,\omega)$.

In 1969 Lindenstrauss and Rosenthal showed that if a Banach space is isomorphic to a complemented subspace of an Lp-space, then it is either an Lp-space or isomorphic to a Hilbert space. This is the motivation of this paper where we study non- Hilbertian complemented operator subspaces of non-commutative Lp-spaces and show that this class is much richer than in the commutative case. We investigate the local properties of some new classes of operator spaces for every 2 < p < 1 which can be considered as operator space analogues of the Rosenthal sequence spaces from Banach space theory, constructed in 1970. Under the usual conditions on the defining sequence we prove that most of these spaces are operator Lp-spaces, not completely isomorphic to previously known such spaces. However, it turns out that some column and row versions of our spaces are not operator Lp-spaces and have a rather complicated local structure which implies that the Lindenstrauss-Rosenthal alternative does not carry over to the non-commutative case.

1. This paper contains a generalization in the commutative case of the structure theorem of W. Ambrose for H*-algebras. An immediate result of Ambrose's axioms is that the orthogonal complement of an ideal is an ideal of the same nature. This fact suggests thinking in terms of a Banach algebra whose conjugate space is a Banach algebra such that the annihilator of an ideal in one algebra is an ideal in the other algebra. This is the essential idea in the definition of a commutative, semi-simple, reflexive Banach algebra given in ?2. This section shows that such an algebra is essentially an 1,-space in which a product has been defined by means of co-ordinate multiplication. ?3 contains a few remarks and examples concerning the author's conjecture that a similar structure theorem must exist in the noncommutative case. The term annihilator will be constantly needed in both its ring and vector space sense. The phrase ring annihilator will be used to distinguish the use of the term in the ring sense from its use in the vector space sense. B' will be used to denote the conjugate space of the given algebra B. Greek letters e, 8, and p will be used to denote complex numbers. Elements in B will be denoted by letters x, y, and z, while elements in B' will be denoted by f, g, and h. The value of the linear functionalf at the point x will be denoted by (x, f). Projections in B will be denoted by means of the letter E, while those in B' will be denoted by means of E'. The following theorems from the general theory will be used.

Equivariant matrices, commuting with a group of permutation matrices, are considered. Such matrices typically arise from PDEs and other computational problems where the computational domain exhibits discrete geometrical symmetries. In these cases, group representation theory provides a powerful tool for block diagonalizing the matrix via the Generalized Fourier Transform (GFT). This technique yields substantial computational savings in problems such as solving linear systems, computing eigenvalues and computing analytic matrix functions such as the matrix exponential. The paper is presenting a comprehensive self contained introduction to this field. Building upon the familiar special (finite commutative) case of circulant matrices being diagonalized with the Discrete Fourier Transform, we generalize the classical convolution theorem and diagonalization results to the noncommutative case of block diagonalizing equivariant matrices. Applications of the GFT in problems with domain symmetries have been developed by several authors in a series of papers. In this paper we elaborate upon the results in these papers by emphasizing the connection between equivariant matrices, block group algebras and noncommutative convolutions. Furthermore, we describe the algebraic structure of projections related to non-free group actions. This approach highlights the role of the underlying mathematical structures, and provides insight useful both for software construction and numerical analysis. The theory is illustrated with a selection of numerical examples.

This paper is concerned with the evaluation of the escape system in the ship. The escape system is modeled by a combination of several basic elements and represented with a network graph. Mathematical models of the crowd flows in and between the basic elements are constructed using fluid flow analogy, and their parameters are determined from the experimental data. A method is presented to quantitatively analyze the escape of passengers and to design an optimum escape system of the ship. The proposed method is applied to the evaluation and optimization of the escape system of the actual car ferry and its applicability is demonstrated.

The paper presents examples of the selection of basic statistical objects for modeling certain physicochemical systems largely related to electrochemistry and metal corrosion. Analytic components (sets of molecules and ions of each electrolyte) were selected as basic elements for a statistical description of the thermodynamic properties of “symmetrical” electrolyte-solvents, where equilibria are maintained by protolysis. The model provides close agreement with the experimental data on the concentration dependences of saturated vapor pressures over aqueous solutions of “strong” electrolytes such as solutions of nitric and sulfuric acids. In the problem of the solution of a passive alloy, the passivating film is treated as a solid solution of constituting oxides, and just these oxides are selected as the basic elements of the statistical model. The calculated dependence of the rate of alloy solution on alloy composition is in agreement with experiment. The statistical model of the solution of a crystal formulated in terms of atom positions on the surface provides a correct description of self-consistency between the topography of the atomic relief and the rate of solution. The model is used to explain effects observed in the solution (corrosion) of metals and related to changes in topography caused by nonstationary conditions or the presence of impurities.

Concurrency in Biological Modeling: Behavior, Execution and Visualization

The work concerns the problem of switching processes in VLSI technology, reprogrammable analog arrays technology, and nanotechnology. Informatics is a discipline of science exploring the laws of processes of coding, processing, storing and transferring information or data. The movement and processing of Information is indissolubly connected with the switching of states of Structures carrying the data. In general, the changes of structure state can be forced through the changes of carried data parameters with unchanged structure configuration, through the switching of structure configuration or switching of the structure conformation. Switching is exhibited in different technologies by different basic elements (basic Structures or building blocks) and specified operations performed on them in order to obtain the object or product with desirable properties including also the dynamic changes of those properties. Dealing in turn with so different cases as VLSI Structures, FPAA (Field Programmable Analog Array) Structures and nanostructures, it is possible to notice, that the common feature takes a stand in solving the tasks related with them, rely on choice of certain basic elements critical for structure behavior as e.g. multicon-ductor transmission lines (MTL), configurations of FPAA substructures or whole FPAA systems, molecule conformations, and it is operated on them analyzing the internal switching processes. The design of interconnects in a high-speed integrated circuits and systems requires analysis and computer simulation based on interconnects models in the form of coupled, multiconductor transmission lines. The fundamental parameters for matrix description (or modeling, simulation and design) of MTL include the capacitance matrices. Precisely, these are the matrices per-unit of length, the capacitance coefficients matrix called Maxwell matrix of MTL and the two-terminal capacitance matrix. If the two-terminal matrix is known, the Maxwell matrix used for MTL switching characteristics estimation, can be computed from the two-terminal matrix. In the work, two methods of determination of the Maxwell matrix from measurements are presented: passive, indirect measurements and measurements with active separation. The measurements were performed basing on fabricated VLSI test structures named ST2 and AF4. The measurement results arę complemented with the accuracy analysis. Scaling of structures is conformed to fast switching, high-speed integrated YLSI/ULSI integrated circuits design. In case when the geometrical dimensions decrease, multiconductor interconnections between parts of a system become strongly coupled MTL, which in switching moments have to be recharged with smali geometry devices. In the work the method and numerical computations of scaled (in geometrical sense) MTL for its energetical characterization is presented. This can be used for proper design of MTL geometrical parameters in a YLSI/ULSI structures working with a high-speed switching. The results of measurements of the switching delay in the scaled in dimensions VLSI gates, are also included. Poor matching of the switched MTL reveals with reflections extending the time of switching and can be source of switching noises. The method of so called diagonal matching is presented in the work. The time of switching in the high-speed planar VLSI structures (driver and receiver gates connected via the long microstrip line) depends on dielectric properties of substrate under the signal and ground microstrip lines. Most important for fast switching is recharging of dielectric in the presence of the dielectric absorption phenomenon. In the work, the model of dielectric absorption and its identification algorithm based on the experimental data is presented. This model can be used for simulation of switching transients in the interconnections of different length for given technology, taking into account the dielectric absorption. Nowadays, it is apparent that many designs are evolving towards mixed-signal systems. In these kind of circuits in CMOS technology, the analog part of a system and an interface are composed frequently basing on switched capacitor (SC) technology and in the newest solution basing on field programmable analog arrays. There are two main weak points in fast switching of reconfigurable FPAA: firstly, the time of download of the configuration from supporting program or EEPROM memory with the time of transients in a FPAA chip when the download is finished and the chip undertakes its function, and secondly, for recently accessible FPAAs, there is no possibility of switching its parameters values and a part of active structure on-the-fly (during FPAA operation). In the work, the solution of fast switching of the part of working FPAA for adaptive operation is presented. The solution is based on external control of one or few CAB (Configur-able Analog Błock) in the working FPAA. The solution of the download time problem in FPAA chip operating in continuous time fashion is based on parallelization of the FPAAs and "hiding" of the download time and time of transients after download. Finally, the applications of the modified, fast switched FPAAs systems for adaptive and predictive control with possible application in the microreactors control have been presented. In the nanonetworks, an information is transferred through the state of moving molecules. In a short-distance communication (inside the closed area determined by the borders of nano-process) the state of the signaling molecules is forced through the binding via hydrogen or covalent bonds of another molecul or through the interaction of the environment's parameters.These processes in biological nanonetworks are performed in the signal transduction cascades and the same procedures are applied in the nanotechnical system of informatics. We considred these process with emphasis on the fusion of cascades. In general, these interactions lead to the switching of conformations of the molecule. In the work, we focused on biocompatible nanostructure introducing the standard, formal description of molecular structures , methodology of modeling and simulation of the conformation changes, and hierarchical description of nanonetworks useful in the switching conformation studies. The simulation methodologies based on the dynamic programming, Monte Carlo and Nascent Protein Folding algorithms are investigated. Moreover, nanotechnological, two-stage production processes based on the "gluey matrix" concept are discussed.

A three-dimensional passage flow analysis method aimed at centrifugal impellers

Understanding working fluids flow and heat transfer in Enhanced Geothermal System (EGS) is critically important for assessing production capabilities and heat extraction rates from hot dry rock (HDR) effectively. Water is a conventional working fluid of EGS; therefore, it is necessary to study water–rock coupling heat transfer within HDR fractures. A single fracture is a basic element of complex fracture systems of EGS, so there has been many researches investigating water flow and heat transfer through a single fracture within a cylindrical HDR specimen for preliminary study, and heat transfer coefficient (HTC) is often adopted to evaluate heat transfer performance between water and rock, but there is a lack of knowledge about the inner surface temperature (Ti), which is a key parameter to calculate HTCs, due to limitations of current experimental methods and analytical solutions. Numerical modeling can be an alternative way to help us have a more comprehensive understanding of the distributions of Ti, but the 2D numerical models only reveal the distribution of Ti along the flow direction without its distribution perpendicular to the flow direction, namely, the distribution along radius. Therefore, it is essential to conduct the corresponding 3D numerical modeling to obtain the radial distribution of Ti. In this paper, a 3D numerical model accounting for surface morphology is proposed to simulate water flow and heat transfer through a single fracture within a cylindrical HDR specimen and verified by experimental data. The results reveal that the radial distribution of Ti can be described by a quadratic function mostly, and neither the surface morphology nor the phase state has influence on the tendency of Ti. Finally, four relatively stable formulas to calculate HTCs so far are compared and discussed.