Algebraic Expansion Research Articles

Towards a theory of area in homogeneous groups

A general approach to compute the spherical measure of submanifolds in homogeneous groups is provided. We focus our attention on the homogeneous tangent space, that is a suitable weighted algebraic expansion of the submanifold. This space plays a central role for the existence of blow-ups. Main applications are area-type formulae for new classes of $$C^1$$ smooth submanifolds and the equality between spherical measure and Hausdorff measure on all horizontal submanifolds.

On the algebraic approach to solvable lattice models

We treat here interaction round the face (IRF) solvable lattice models. We study the algebraic structures underlining such models. For the three block case, we show that the Yang Baxter equation is obeyed, if and only if, the Birman-Murakami-Wenzl (BMW) algebra is obeyed. We prove this by an algebraic expansion of the Yang Baxter equation (YBE). For four blocks IRF models, we show that the BMW algebra is also obeyed, apart from the skein relation, which is different. This indicates that the BMW algebra is a sub-algebra for all models with three or more blocks. We find additional relations for the four block algebra using the expansion of the YBE. The four blocks result, that is the BMW algebra and the four blocks skein relation, is enough to define new knot invariant, which depends on three arbitrary parameters, important in knot theory.

A thrust equation treats propellers and rotors as aerodynamic cycles and calculates their thrust without resorting to the blade element method

The lift generated by a translating wing of known translational speed, lift coefficient and area is calculated by a simple equation. A propeller or rotor generating thrust share the same aerodynamic principles but their different kinematics cause the calculation of their thrust to be laborious. This paper derives a thrust equation from an algebraic expansion of the Prandtl’s dynamic pressure term q∞by adding the rotational kinetic energy of a propeller or rotor to the existing translational kinetic energy term. This thrust equation can be applied directly to propellers and rotors and assumes these to operate as cycles with their available kinetic energy as input and work as output. The thrust equation is a function of the normalized thrust ηT, a nondimensional figure of merit that quantifies the ability to generate thrust and allows for a meaningful comparison with other aerodynamic systems, regardless of their kinematics.

Perturbation approach for computing frequency- and time-resolved photon correlation functions

We propose an alternative formulation of the sensor method presented in [Phys. Rev. Lett 109, 183601 (2012)] for the calculation of frequency-filtered and time-resolved photon correlations. Our approach is based on an algebraic expansion of the joint steady state of quantum emitter and sensors with respect to the emitter-sensor coupling parameter \epsilon. This allows us to express photon correlations in terms of the open quantum dynamics of the emitting system only and ensures that computation of correlations are independent on the choice of a small value of \epsilon. Moreover, using time-dependent perturbation theory, we are able to express the frequency- and time- resolved second-order photon correlation as the addition of three components, each of which gives insight into the physical processes dominating the correlation at different time scales. We consider a bio-inspired vibronic dimer model to illustrate the agreement between the original formulation and our approach.

Microkinetic Rate Theory:Formalization, Current Limitations, Prospects As Basis for Continuum Rate Theory

Microkinetic Rate Theory (MRT) has been an important part of chemical kinetics, being taught in undergraduate physics and chemistry courses, and forming the basis of modern descriptor theory used widely in research. In spite of MRT’s ubiquitous importance, it has remain unformalized and ungeneralized. In this talk, MRT is formalized and generalized. MRT is cast as an algebraic expansion and those matrix elements forming the coefficients of the generalization defined in terms of reaction rate constants, which are in turn connect to statistical mechanics. A transport formulation of MRT is given and it shown that Fickian diffusion is a subset of MRT. MRT is shown to deviate from direct kMC simulations in a certain rate regime which is attributed to phenomenologically distinct stochastic and deterministic rate regimes, similar to diffusion and continuum fluid mechanics, but relevant to chemical and complex kinetics.

Eigenspace-Based Minimum Variance Combined With Delay Multiply and Sum Beamformer: Application to Linear-Array Photoacoustic Imaging

In Photoacoustic imaging, Delay-and-Sum (DAS) algorithm is the most commonly used beamformer. However, it leads to a low resolution and high level of sidelobes. Delay-Multiply-and-Sum (DMAS) was introduced to provide lower sidelobes compared to DAS. In this paper, to improve the resolution and sidelobes of DMAS, a novel beamformer is introduced using Eigenspace-Based Minimum Variance (EIBMV) method combined with DMAS, namely EIBMV-DMAS. It is shown that expanding the DMAS algebra leads to several terms which can be interpreted as DAS. Using the EIBMV adaptive beamforming instead of the existing DAS (inside the DMAS algebra expansion) is proposed to improve the image quality. EIBMV-DMAS is evaluated numerically and experimentally. It is shown that EIBMV-DMAS outperforms DAS, DMAS and EIBMV in terms of resolution and sidelobes. In particular, at the depth of 11 mm of the experimental images, EIBMV-DMAS results in about 113 dB and 50 dB sidelobe reduction, compared to DMAS and EIBMV, respectively. At the depth of 7 mm, for the experimental images, the quantitative results indicate that EIBMV-DMAS leads to improvement in Signal-to-Noise Ratio (SNR) of about 75% and 34%, compared to DMAS and EIBMV, respectively.

Lepton flavour violation in the MSSM: exact diagonalization vs mass expansion

The forthcoming precision data on lepton flavour violating (LFV) decays require precise and efficient calculations in New Physics models. In this article lepton flavour violating processes within the Minimal Supersymmetric Standard Model (MSSM) are calculated using the method based on the Flavour Expansion Theorem, a recently developed technique performing a purely algebraic mass-insertion expansion of the amplitudes. The expansion in both flavour-violating and flavour-conserving off-diagonal terms of sfermion and supersymmetric fermion mass matrices is considered. In this way the relevant processes are expressed directly in terms of the parameters of the MSSM Lagrangian. We also study the decoupling properties of the amplitudes. The results are compared to the corresponding calculations in the mass eigenbasis (i.e. using the exact diagonalization of the mass matrices). Using these methods, we consider the following processes: ℓ → ℓ′γ, ℓ→3ℓ′, ℓ→2ℓ′ℓ′′, h→ℓℓ′ as well as μ→e conversion in nuclei. In the numerical analysis we update the bounds on the flavour changing parameters of the MSSM and examine the sensitivity to the forthcoming experimental results. We find that flavour violating muon decays provide the most stringent bounds on supersymmetric effects and will continue to do so in the future. Radiative ℓ → ℓ′γ decays and leptonic three-body decays ℓ → 3ℓ′ show an interesting complementarity in eliminating “blind spots” in the parameter space. In our analysis we also include the effects of non-holomorphic A-terms which are important for the study of LFV Higgs decays.

Generalizing the mathfrak {bms}_{3} and 2D-conformal algebras by expanding the Virasoro algebra

By means of the Lie algebra expansion method, the centrally extended conformal algebra in two dimensions and the mathfrak {bms}_{3} algebra are obtained from the Virasoro algebra. We extend this result to construct new families of expanded Virasoro algebras that turn out to be infinite-dimensional lifts of the so-called mathfrak {B}_{k}, mathfrak {C}_{k} and mathfrak {D}_{k} algebras recently introduced in the literature in the context of (super)gravity. We also show how some of these new infinite-dimensional symmetries can be obtained from expanded Kač–Moody algebras using modified Sugawara constructions. Applications in the context of three-dimensional gravity are briefly discussed.

B-C^* algebra Metric Space and Some Results Fixed point Theorems

On the aim and properties of - algebra, in this paper we introduce the two concepts algebra and algebra metric space as well as introduce concept convergent and Cauchy sequence in space and to study the existence of fixed point theorems with contraction condition and algebra expansion on this space.

How Einstein’s Theory of Relativity Gives us and the Atomic Bomb

AbstractThe LIGO direct-detection of gravitational waves arriving from cosmic sources—now, happily including (as of this meeting!) the merging of two neutron stars—opens a new chapter in our understanding of physics itself: for General Relativity, conceptually so extremely simple, has robustly produced predictions that have invariably been found to be correct when tested. My poster (page 3 of this paper) is intended for high school students who have just learned simple algebra. My derivation of the famous E = mc2 from the Pythagorean Theorem necessarily requires an algebraic expansion that is due to Newton, but apart from that it is simplicity itself: a transparent introduction to what all of physics is today: the construction of mathematics that, miraculously, reproduces our observations of the world—and which also successfully predicts the results of future observations—as so magnificently demonstrated at this glorious Symposium!

An algebraic expansion of the potential theory for predicting dynamic stability limit of in-line cylinder arrangement under single-phase fluid cross-flow

Flow-induced vibration in square cylinder arrangement under viscous fluid incompressible cross-flow is investigated in the present work. The purpose is to contribute to better modeling and understanding external fluid loads exerted on long thin cylinders inducing flow perturbations. Due to high flow confinement, thin cylinders may be subjected to strong vibrations, which may lead to dynamic instability development. A theoretical approach is developed to determine a stability criterion of the dynamical system. The influence of geometric, mechanical and flow parameters such as reduced velocity and pitch ratio is investigated.The proposed model is derived from the potential flow theory and enhanced through an algebraic phase lag model in order to predict the critical limit of the reduced velocity for a square cylinder arrangement submitted to an external in-line cross flow. A theoretical formulation of the total damping, including added damping in still fluid, the damping due to fluid flow and the damping derived from the phase shift between the fluid load and the tube displacement, is expressed. A function depending on fluid and structure parameters, such as reduced velocity, pitch ratio and Scruton number is thus obtained. It is shown that this function provides a prediction of the dynamic stability limit of the system for several ranges of the major parameters to be considered. The results are compared to experimental reference solutions and to those provided by other theoretical models.This work proposes a consistent original model based on a potential flow theory enriched by using an algebraic formulation based on standard physical assumptions from literature. The major advantage of this model is due to the fact that it is in the same time robust and very user-friendly from a computational point of view thanks to the potential framework. In order to describe fluid and solid dynamics in the domain, terms coming from the potential flow theory are estimated by using a finite element method and complementary terms acting on damping are obtained through an algebraic formulation. Therefore this is a convenient way to propose a hybrid numerical / algebraic model for predicting dynamic instability limit in cylinder arrangements.

A note on modified generalized Riccati equation method combined with new algebra expansion

AbstractIn this present work, we have proposed modified generalized Riccati equation method for finding the exact traveling wave solutions of nonlinear evolution equations (NLEEs) via the mKdV equation. It has been shown that the proposed method is effective and can be used for many other NLEEs in mathematical physics. The obtained results show that the proposed method is very powerful and convenient mathematical tool for NLEEs in science and engineering.

SOME LINEAR AND NONLINEAR EQUATIONS OF HIGHER ORDER

In this article we consider boundary value problems for some linear and nonlinear differential equations with partial derivatives of the sixth, fifth, fourth and third orders. We write out conditions on equation coefficients for which existence and uniqueness of solutions from Sobolev's space occur. If these conditions on equation coefficients are not valid, then there are given examples when solution is not unique, or is not unstable, or does not belong to Sobolev's space from existence and uniqueness theorem even for analytical coefficients and analytical right side of differential equation. After S.P. Novikov’s fundamental study in 1974 the interest to the nonlinear Korteweg-de Vries equation, Kadomtsev-Petviashvili equation and other nonlinear equations significantly grew. In this study of such equations we used methods of algebraic geometry integration and expansion method. In these studies exact solutions of special nonlinear equations series in partial derivatives play a big role. Solvability of similar equations was also studied in articles of A.I. Kozhanov, N.A. Larkin and other authors. The aim of this article is to find some exact solutions for special series partial differential equations. Solution graphs of such problems for linear equations and for the Korteweg-de Vries, Burgers-Korteweg-deVries, and Kadomtsev-Petviashvili equations are constructed.

Soliton solutions for the Kundu–Eckhaus equation with the aid of unified algebraic and auxiliary equation expansion methods

The modified unified algebraic (MUA) and auxiliary equation expansion (AEE) methods are applied to reach the different type soliton solutions of Kundu–Eckhaus (K–E) equation. It is illustrated that MUA and AEE are very effective methods to reach the various types of the solutions of nonlinear evolution equations.

Large-maturity regimes of the Heston forward smile

We provide a full characterisation of the large-maturity forward implied volatility smile in the Heston model. Although the leading decay is provided by a fairly classical large deviations behaviour, the algebraic expansion providing the higher-order terms highly depends on the parameters, and different powers of the maturity come into play. As a by-product of the analysis we provide new implied volatility asymptotics, both in the forward case and in the spot case, as well as extended SVI-type formulae. The proofs are based on extensions and refinements of sharp large deviations theory, in particular in cases where standard convexity arguments fail.

A Cosmological Model Describing the Early Inflation, the Intermediate Decelerating Expansion, and the Late Accelerating Expansion of the Universe by a Quadratic Equation of State

We develop a cosmological model based on a quadratic equation of state \(p/c^2=-(\alpha+1){\rho^2}/{\rho_P}+\alpha\rho-(\alpha+1)\rho_ {\Lambda}\), where \(\rho_P\) is the Planck density and \(\rho_{\Lambda}\) the cosmological density, ``unifying'' vacuum energy and dark energy in the spirit of a generalized Chaplygin gas model. For \(\rho\rightarrow \rho_P\), it reduces to \(p=-\rho_P c^2\) leading to a phase of early accelerating expansion (early inflation) with a constant density equal to the Planck density \(\rho_P=5.16 \times 10^{99}\, {\rm g}/{\rm m}^3\) (vacuum energy). For \(\rho_{\Lambda}\ll\rho\ll \rho_P\), we recover the standard linear equation of state \(p=\alpha \rho c^2\) describing radiation (\(\alpha=1/3\)) or pressureless matter (\(\alpha=0\)) and leading to an intermediate phase of decelerating expansion. For \(\rho\rightarrow \rho_{\Lambda}\), we get \(p=-\rho_{\Lambda} c^2\) leading to a phase of late accelerating expansion (late inflation) with a constant density equal to the cosmological density \(\rho_{\Lambda}=7.02\times 10^{-24}\, {\rm g}/{\rm m}^3\) (dark energy). The pressure is successively negative (vacuum energy), positive (radiation and matter), and negative again (dark energy). We show a nice ``symmetry'' between the early universe (vacuum energy \(+\) \(\alpha\)-fluid) and the late universe (\(\alpha\)-fluid \(+\) dark energy). In our model, they are described by two polytropic equations of state with index \(n=+1\) and \(n=-1\) respectively. Furthermore, the Planck density \(\rho_P\) in the early universe plays a role similar to the cosmological density \(\rho_{\Lambda}\) in the late universe. They represent fundamental upper and lower density bounds differing by \(122\) orders of magnitude. The cosmological constant ``problem'' may be a false problem. We study the evolution of the scale factor, density, and pressure. Interestingly, our quadratic equation of state leads to a fully analytical model describing the evolution of the universe from the early inflation (Planck era) to the late accelerating expansion (de Sitter era). These two phases are bridged by a decelerating algebraic expansion (\(\alpha\)-era). Our model does not present any singularity at \(t=0\) and exists eternally in the past (although it may be incorrect to extrapolate the solution to the infinite past). On the other hand, it admits a scalar field interpretation based on an inflaton, quintessence, or tachyonic field. Our model generalizes the standard \(\Lambda\)CDM model by incorporating naturally a phase of early inflation that avoids the primordial singularity. Furthermore, it describes the early inflation, the intermediate decelerating expansion, and the late accelerating expansion of the universe simultaneously in terms of a single equation of state. We determine the corresponding scalar field potential that unifies the inflaton and quintessence potentials.

<inline-formula> <tex-math notation="LaTeX">$H$ </tex-math></inline-formula>-Transforms for Wireless Communication

The $H$ -transforms are integral transforms that involve Fox’s $H$ -functions as kernels. A large variety of integral transforms can be put into particular forms of the $H$ -transform since $H$ -functions subsume most of the known special functions including Meijer’s $G$ -functions. In this paper, we embody the $H$ -transform theory into a unifying framework for modeling and analysis in wireless communication. First, we systematize the use of elementary identities and properties of the $H$ -transform by introducing operations on parameter sequences of $H$ -functions. We then put forth $H$ -fading and degree-2 irregular $H$ -fading to model radio propagation under composite, specular, and/or inhomogeneous conditions. The $H$ -fading describes composite effects of multipath fading and shadowing as a single $H$ -variate, including most of typical models such as Rayleigh, Nakagami- $m$ , Weibull, $ \alpha - \mu $ , $N\ast $ Nakagami- $m$ , (generalized) $K$ -fading, and Weibull/gamma fading as its special cases. As a new class of $H$ -variates (called the degree- $ \zeta $ irregular $H$ -variate), the degree-2 irregular $H$ -fading characterizes specular and/or inhomogeneous radio propagation in which the multipath component consists of a strong specularly reflected or line-of-sight (LOS) wave as well as unequal-power or correlated in-phase and quadrature scattered waves. This fading includes a variety of typical models such as Rician, Nakagami- $q$ , $ \kappa - \mu $ , $ \eta - \mu $ , Rician/LOS gamma, and $ \kappa - \mu $ /LOS gamma fading as its special cases. Finally, we develop a unifying $H$ -transform analysis for the amount of fading, error probability, channel capacity, and error exponent in wireless communication using the new systematic language of transcendental $H$ -functions. By virtue of two essential operations—called Mellin and convolution operations—involved in the Mellin transform and Mellin convolution of two $H$ -functions, the $H$ -transforms for these performance measures culminate in $H$ -functions. Using the algebraic asymptotic expansions of the $H$ -transform, we further analyze the error probability and capacity at high and low signal-to-noise ratios in a unified fashion.

On the Dual Graphs of Cohen–Macaulay Algebras

Given an algebraic set X P n , its dual graph G(X) is the graph whose vertices are the irreducible components of X and whose edges connect components that intersect in codimension one. Hartshorne’s connectedness theorem says that if (the coordinate ring of) X is Cohen-Macaulay, then G(X) is connected. We present two quantitative variants of Hartshorne’s result: (1) If X is a Gorenstein subspace arrangement, then G(X) is r-connected, where r is the Castelnuovo{Mumford regularity of X. (The bound is best possible; for coordinate arrangements, it yields an algebraic extension of Balinski’s theorem for simplicial polytopes.) (2) If X is an arrangement of lines no three of which meet in the same point, and X is canonically embedded in P n , then the diameter of the graph G(X) is not larger than codimPn X. (The bound is sharp; for coordinate arrangements, it yields an algebraic expansion on the recent combinatorial result that the Hirsch conjecture holds for ag normal simplicial complexes.)

The representation of square root quasi-pseudo-MV algebras

$$\sqrt{'}$$? quasi-MV algebras arising from quantum computation are term expansions of quasi-MV algebras. In this paper, we introduce a generalization of $$\sqrt{'}$$? quasi-MV algebras, called square root quasi-pseudo-MV algebras ($$\sqrt{\hbox {quasi-pMV}}$$quasi-pMV algebras, for short). First, we investigate the related properties of $$\sqrt{\hbox {quasi-pMV}}$$quasi-pMV algebras and characterize two special types: Cartesian and flat $$\sqrt{\hbox {quasi-pMV}}$$quasi-pMV algebras. Second, we present two representations of $$\sqrt{\hbox {quasi-pMV}}$$quasi-pMV algebras. Furthermore, we generalize the concepts of PR-groups to non-commutative case and prove that the interval of a non-commutative PR-group with strong order unit is a Cartesian $$\sqrt{\hbox {quasi-pMV}}$$quasi-pMV algebra. Finally, we introduce non-commutative PR-groupoids which extend abelian PR-groupoids and show that the category of negation groupoids with operators and the category of non-commutative PR-groupoids are equivalent.

A study of the bending motion in tetratomic molecules by the algebraic operator expansion method

We study the bending motion in the tetratomic molecules C2H2 (X̃ (1)Σg (+)), C2H2 (Ã (1)Au) trans-S1, C2H2 (Ã (1)A2) cis-S1, and X̃ (1)A1 H2CO. We show that the algebraic operator expansion method with only linear terms comprised of the basic operators is able to describe the main features of the level energies in these molecules in terms of two (linear) or three (trans-bent, cis-bent, and branched) parameters. By including quadratic terms, the rms deviation in comparison with experiment goes down to typically ∼10 cm(-1) over the entire range of energy 0-6000 cm(-1). We determine the parameters by fitting the available data, and from these parameters we construct the algebraic potential functions. Our results are of particular interest in high-energy regions where spectra are very congested and conventional methods, force-field expansions or Dunham-expansions plus perturbations, are difficult to apply.