Simple Theorem Research Articles

Algebraic Nexus of Fibonacci Forms and Two-Simplex Topology in Multicellular Morphogenesis

Background: Fibonacci patterns and tubular forms both arose early in the phylogeny of multicellular organisms. Tubular forms offer the advantage of a regulated internal milieu, and Fibonacci forms may offer packing efficiencies. The underlying mechanisms behind the cellular genesis of Fibonacci and tubular forms remain unknown. Methods: In a multicellular organism, cells adhere to form a macrostructure and to coordinate further replication. We propose and prove simple theorems connecting cell replication and adhesion to Fibonacci forms and simplicial topology. Results: We identify some cellular and molecular properties whereby the contact inhibition of replication by adhered cells may approximate Fibonacci growth patterns. We further identify how a component 2→3 cellular multiplication step may generate a multicellular structure with some properties of a two-simplex. Tracking the homotopy of a two-simplex to a circle and to a tube, we identify some molecular and cellular growth properties consistent with the morphogenesis of tubes. We further find that circular and tubular cellular aggregates may be combinatorially favored in multicellular adhesion over flat shapes. Conclusions: We propose a correspondence between the cellular and molecular mechanisms that generate Fibonacci cell counts and those that enable tubular forms. This implies molecular and cellular arrangements that are candidates for experimental testing and may provide guidance for the synthetic biology of hollow morphologies.

Fluctuation-response relation as a probe of long-range correlations in nonequilibrium quantum and classical fluids.

The absence of a simple fluctuation-dissipation theorem is a major obstacle for studying systems that are not in thermodynamic equilibrium. We show that for a fluid in a nonequilibrium steady state characterized by a constant temperature gradient the commutator correlation functions are still related to response functions; however, the relation is to the bilinear response of products of two observables, rather than to a single linear response function as is the case in equilibrium. This modified fluctuation-response relation holds for both quantum and classical systems. It is both motivated and informed by the long-range correlations that exist in such a steady state and allows for probing them via response experiments. This is of particular interest in quantum fluids, where the direct observation of fluctuations by light scattering would be difficult. In classical fluids it is known that the coupling of the temperature gradient to the diffusive shear velocity leads to correlations of various observables, in particular temperature fluctuations, that do not decay as a function of distance, but rather extend over the entire system. We investigate the nature of these correlations in a fermionic quantum fluid and show that the crucial coupling between the temperature gradient and velocity fluctuations is the same as in the classical case. Accordingly, the nature of the long-ranged correlations in the hydrodynamic regime also is the same. However, as one enters the collisionless regime in the low-temperature limit the nature of the velocity fluctuations changes: they become ballistic rather than diffusive. As a result, correlations of the temperature and other observables are still singular in the long-wavelength limit, but the singularity is weaker than in the hydrodynamic regime.

Reduced probability densities of long-lived metastable states as those of distributed thermal systems: Possible experimental implications for supercooled fluids

When liquids are cooled sufficiently rapidly below their melting temperature, they may bypass crystalization and, instead, enter a long-lived metastable supercooled state that has long been the focus of intense research. Although they exhibit strikingly different properties, both the (i) long-lived supercooled liquid state and (ii) truly equilibrated (i.e., conventional equilibrium fluid or crystalline) phases of the same material share an identical Hamiltonian. This suggests a mapping between dynamical and other observables in these two different arenas. We formalize these notions via a simple theorem applicable to general systems and illustrate that given a Hamiltonian defining the dynamics: (1) the reduced probability densities of all possible stationary states are linear combinations of reduced probability densities associated with thermal equilibria at different temperatures, chemical potentials, etc. (2) Excusing special cases, amongst all of these stationary states, a clustering of correlations is only consistent with conventional thermal equilibrium states (associated with a sharp distribution of the above state variables). (3) Other stationary states may be modified so as to have local correlations. These deformations concomitantly lead to metastable (yet possibly very long-lived) states. Since the lifetime of the supercooled state is exceptionally long relative to the natural microscopic time scales, their reduced probability densities may be close to those that we find for exact stationary states (i.e., a weighted average of equilibrium probability densities at different state variables). This form suggests several new predictions such as the existence of dynamical heterogeneity stronger than probed for thus far and a relation between the specific heat peak and viscosities. Our theorem may further place constraints on the putative “ideal glass” phase.

Fourier series and its property

Fourier analysis appeared to solve some partial differential equations at first. In 1747, D’ Alembert solved the vibrating string equation using the method of traveling waves. In 1753, D. Bernoulli proposed the solution which for all intents and purposes is the Fourier series, but Euler did not convince of its full generality entirely since he could not make sure whether any function could be expanded in the Fourier seriesFourier analysis is an important theorem of modern mathematics. It deeply influences the development of partial differential equations and information science. This paper will talk about some basic propositions of the Fourier series and some simple theorems of the Fourier transform. It will use an algebraic view to talk about convolution and good kernels. It will refer to some methods of Hilbert space and complex analysis.

A Genelization of Subgroups

Group theory is an important branch of mathematics. It has important applications in many disciplines, such as in physics, chemistry and quantum mechanics. The aim of this document is to provide readers with a greater understanding of the basic knowledge of group theory. Specifically, firstly, the concepts and properties of different types of groups, subgroups and the definition of coset are introduced, which will be used in the next section. Besides, several examples including groups, finite groups, infinite groups, subgroups and abelian groups are listed. Then, a discussion of the two simple theorems and some lemmas will be showed so that readers can be familiar with the properties of groups. This discussion will deepen the understanding of maximal subgroups. This paper investigates the basic definitions and theorems of group theory. In the future, more researchers can pay attention to group theory and the application of group theory in more areas.

Differential Attack With Constants On μ2 Block Cipher

Abstract Differential attack is one of the most important methods in cryptanalysis. When finding a high-probability differential trail, the effect of constant has long been ignored. In this paper, we focus on the effect of constants on the differential attack against $\mu ^2$. $\mu ^2$ is a newly proposed block cipher based on a Type-II generalized Feistel structure. Its 16-bit F function (denoted as F-box) is an ultra-lightweight permutation equipped with different constants. The designer applied the minimum number of active S-boxes to determine $\mu ^2$’ security margin in the design document. However, the F-boxes use different round constants in different rounds; the constants may lead to incompatibility of differential trails of F-boxes. Therefore, to provide a more precise differential attack on $\mu ^2$, we construct an model based on STP (Simple Theorem Prover) constraint solver to search for the valid differential trails with a more precise probability of $\mu ^2$ for different starting rounds. Finally, the related-key differential trail covers one more round than the existing methods. Analyzing the effect of constants on the validity and the probability of the differential trail reminds the designers and the attackers to have a more comprehensive analysis of specific ciphers.

Symplectic Flatness and Twisted Primitive Cohomology

We introduce the notion of symplectic flatness for connections and fibre bundles over symplectic manifolds. Given an A_infty -algebra, we present a flatness condition that enables the twisting of the differential complex associated with the A_infty -algebra. The symplectic flatness condition arises from twisting the A_infty -algebra of differential forms constructed by Tsai, Tseng and Yau. When the symplectic manifold is equipped with a compatible metric, the symplectic flat connections represent a special subclass of Yang–Mills connections. We further study the cohomologies of the twisted differential complex and give a simple vanishing theorem for them.

A Systematic Control Design Method with Active Damping Control in Voltage Source Converters

This paper proposes a systematic control design method for active damping control of grid-connected voltage source converters (VSCs). The proposed control method considers the conventional cascaded control loops and improves them by including additional states feedback-based active damping. In such a way, all control gains are lumped into one control gain matrix based on the proposed formulation. The lumping of all control gains into one matrix leads to a linear optimization problem, so different techniques can be used to calculate control gains. This work calculates them by using a simple but effective optimal control theorem as a noteworthy feature. The proposed control method can overcome the challenges of designing multiple control loops, evaluating wide time scale dynamics, and tuning required control parameters. Moreover, direct relationships between the proposed tuning parameters and system well-known stability and performance indicators such as maximum damping factor, minimum damping ratio, and the control efforts are identified, providing good physical insight. Finally, the proposed control structure and optimal gain calculations ensure power converter robustness against uncertainties in the grid’s short-circuit ratio (SCR) and different operating-point conditions. When the grid’s SCR changes from 10 (strong grid condition) to 1 (ultra-weak grid condition), the system under the proposed control method maintains good stability margins and simultaneously provides a fast dynamic response by facilitating the implementation of a high-bandwidth phase-locked loop (PLL). The performance of the proposed control strategy was investigated analytically and practically by conducting eigenvalue analysis, simulations, and experiments.

A short note on Manin–Mumford

We give a short proof of Manin–Mumford in the multiplicative group based on the pigeon-hole principle and a simple structure theorem for tori contained in subvarieties of [Formula: see text] due to Bombieri and Gubler. The arguments appear to be new and are applicable in other situations. Our proof is quite elementary and we compare it to previous proofs that rely on machinery such as intersection theory or [Formula: see text]-minimality.

Electromagnetic energy transport by tearing fluctuations in a self-organized reversed-field pinch plasma

Fluctuation measurements reveal the outward electromagnetic energy flux needed to drive the dynamo electromotive force supporting magnetic self-organization in a reversed-field pinch plasma. The radial Poynting flux due to tearing mode fluctuations is measured with an insertable probe during magnetic relaxation. This flux corresponds to transient power levels much larger than the input power and comparable to the global equilibrium magnetic energy transient loss rate. The probe measurements of this flux are roughly as predicted by a simple Poynting's theorem model upon substitution of equilibrium measurement data.

A degenerate bifurcation from simple eigenvalue theorem

<abstract><p>A new bifurcation from simple eigenvalue theorem is proved for general nonlinear functional equations. It is shown that in this bifurcation scenario, the bifurcating solutions are on a curve which is tangent to the line of trivial solutions, while in typical bifurcations the curve of bifurcating solutions is transversal to the line of trivial ones. The stability of bifurcating solutions can be determined, and examples from partial differential equations are shown to demonstrate such bifurcations.</p></abstract>

Rigidity of Euler products

We report simple rigidity theorems for Euler products under deformations of Euler factors. Certain products of the Riemann zeta function are rigid in the sense that there exist no deformations which preserve the meromorphy on $\mathbf{C}$.

The lattice and semigroup structure of multipermutations

We study the algebraic properties of binary relations whose underlying digraph is smooth, that is, has no source or sink. Such objects have been studied as surjective hyper-operations (shops) on the corresponding vertex set, and as binary relations that are defined everywhere and whose inverse is also defined everywhere. In the latter formulation, they have been called multipermutations. We study the lattice structure of sets (monoids) of multipermutations over an [Formula: see text]-element domain. Through a Galois connection, these monoids form the algebraic counterparts to sets of relations closed under definability in positive first-order logic without equality. We show one side of this Galois connection, and give a simple dichotomy theorem for the evaluation problem of positive first-order logic without equality on the class of structures whose preserving multipermutations form a monoid closed under inverse. These problems turn out either to be in [Formula: see text] or to be [Formula: see text]-complete. We go on to study the monoid of all multipermutations on an [Formula: see text]-element domain, under usual composition of relations. We characterize its Green relations, regular elements and show that it does not admit a generating set that is polynomial on [Formula: see text].

Spin squeezing from bilinear spin-spin interactions: Two simple theorems

We demonstrate two simple theorems about squeezing induced by bilinear spin-spin interactions that conserve spin parity -- including a vast majority of quantum spin models implemented by state-of-the-art quantum simulators. In particular we show that squeezing captures the first form of quantum correlations which are produced: 1) at equilibrium, by adiabatically turning on the spin-spin interactions starting from a factorized state aligned with an external, arbitrary field; 2) away from equilibrium, by evolving unitarily the same state with the interacting Hamiltonian.

Normal ordered exponential approach to thermal properties and time-correlation functions: general theory and simple examples

A normal ordered exponential parametrisation is used to obtain equations for thermal one-and two-particle reduced density matrices, as well as free energies, partition functions and entropy for both Fermionic (electronic) and Bosonic (vibrational) Hamiltonians. A first principles derivation of the equations, relying only on a simple Wick's theorem and starting from the differential equation , is presented that yields a differential equation for the amplitudes representing density cumulants, as well as the grand potential. In contrast to other approaches reported in the literature, the theory does not use the interaction picture and an integral formulation as a starting point but rather requires a propagation of the resulting differential equation for the amplitudes in the Schrödinger representation. While the theory is applicable to general classes of many-body problems in principle, here, the theory is illustrated using simple model systems. As an application of the related time-dependent formulation numerically exact time-autocorrelation functions and absorption spectra are obtained for harmonic Franck–Condon problems. These examples illustrate the basic soundness of the scheme and are used for pedagogical purposes. Other approaches in the literature are only discussed briefly and no detailed comparative discussion is attempted.

SOME PROPERTIES ON DERIVATIONS OF LATTICES

In this paper we consider some properties of derivations of lattices and show that (i) for a derivation $d$ of a lattice $L$ with the maximum element $1$, it is monotone if and only if $d(x) le d(1)$ for all $xin L$ (ii) a monotone derivation $d$ is characterized by $d(x) = xwedge d(1)$ and (iii) simple characterization theorems of modular lattices and of distributive lattices are given by derivations. We also show that, for a distributive lattice $L$ and a monotone derivation $d$ of it, the set ${rm Fix}_d(L)$ of all fixed points of $d$ is isomorphic to the lattice $L/ker (d)$. We provide a counter example to the result (Theorem 4) proved in [3].

Confined hydrogenlike ions in plasma environments

The behavior of H-like ions embedded in astrophysical plasmas in the form of \emph{dense, strongly and weakly coupled} plasmas are investigated. In these, the increase and decrease in temperature is impacted with a change in confinement radius $(r_{c})$. Two independent and generalized scaling ideas have been applied to modulate the effect of plasma screening constant ($\lambda$) and charge of ion ($Z$) on such systems. Several new relations are derived to interconnect the original Hamiltonian and two scaled Hamiltonians. In exponential cosine screened Coulomb potential (ECSCP) (dense) and weakly coupled plasma (WCP) these scaling relations have provided a linear equation connecting the critical screening constant $(\lambda^{(c)})$ and $Z$. Their ratio offers a state-dependent constant, beyond which, a particular state vanishes. Shannon entropy has been employed to understand the plasma effect on the ion. With increase in $\lambda$, the accumulation of opposite charge surrounding the ion increases leading to a reduction in number of bound states. However, with rise in ionic charge $Z$, this effect can be delayed. The competing effect of plasma charge density ($n_e$) and temperature in WCP and ECSCP is investigated. A recently proposed simple virial-like theorem has been established for these systems. Multipole ($k=1-4$) oscillator strength (OS) and polarizabilities for these are studied considering $1s, 2s$ states. As a bonus, analytical closed-form expressions are derived for $f^{(k)}$ and $\alpha^{(k)} (k=1-4)$ involving $1s$ and $2s$ state, for \emph{free H-like ion}.

Subsequences of Real Valued Sequences and Convergence

Convergence of real valued sequences is a classical subject of study for many mathematicians and it continues to be studied in recent years. Different types of convergence including ordinary, statistical, almost and ideal convergence, and related properties have been researched. In some studies, the relationship of a sequence and its subsequences regarding some type of convergence is investigated, using Lebesgue measure and Baire category. In this paper we revisit ordinary convergence and study the properties of subsequences of a real valued sequence using measure and category. We state and prove some simple theorems offering fresh insights into a classical subject.

Lower Bounds for Non-Elitist Evolutionary Algorithms via Negative Multiplicative Drift.

A decent number of lower bounds for non-elitist population-based evolutionary algorithms has been shown by now. Most of them are technically demanding due to the (hard to avoid) use of negative drift theorems-general results which translate an expected movement away from the target into a high hitting time. We propose a simple negative drift theorem for multiplicative drift scenarios and show that it can simplify existing analyses. We discuss in more detail Lehre's (2010) negative drift in populations method, one of the most general tools to prove lower bounds on the runtime of non-elitist mutation-based evolutionary algorithms for discrete search spaces. Together with other arguments, we obtain an alternative and simpler proof of this result, which also strengthens and simplifies this method. In particular, now only three of the five technical conditions of the previous result have to be verified. The lower bounds we obtain are explicit instead of only asymptotic. This allows us to compute concrete lower bounds for concrete algorithms, but also enables us to show that super-polynomial runtimes appear already when the reproduction rate is only a (1-ω(n-1/2)) factor below the threshold. For the special case of algorithms using standard bit mutation with a random mutation rate (called uniform mixing in the language of hyper-heuristics), we prove the result stated by Dang and Lehre (2016b) and extend it to mutation rates other than Θ(1/n), which includes the heavy-tailed mutation operator proposed by Doerr et al. (2017). We finally use our method and a novel domination argument to show an exponential lower bound for the runtime of the mutation-only simple genetic algorithm on OneMax for arbitrary population size.

Simple homotopy theory and nerve theorem for categories

We study a combinatorial homotopy theory for small categories without a loop (loopfree categories) which is closely related to Whitehead's simple homotopy theory for regular CW-complexes with triangular cells. Quillen's theorem A and the nerve theorem for loopfree categories are considered from the viewpoint of the simple homotopy theory. Moreover, we extend the classical nerve theorem and discuss an application to the topological data analysis for data-sets lying in a non-convex field.