Additional Symmetry Condition Research Articles

Existence and multiplicity of solutions for general quasi-linear elliptic equations with sub-cubic nonlinearities

We consider the following quasi-linear Schrödinger equation−∑i,j=1NDj(aij(u)Diu)+12∑i,j=1NDsaij(u)DiuDju+V(x)u=f(u),x∈RN,N≥3, which includes the modified nonlinear Schrödinger equations. Combining a p-Laplacian perturbation argument, we obtain the existence of positive solutions to the problem above and multiple solutions with additional symmetry conditions on aij and f. A new perturbation approach is used to treat the sub-cubic nonlinearity. In particular, for f(u)=|u|p−2u, the case p∈(2,4) is also considered.

Walsh’s Conformal Map onto Lemniscatic Domains for Polynomial Pre-images I

We consider Walsh’s conformal map from the exterior of a compact set E subseteq mathbb {C} onto a lemniscatic domain. If E is simply connected, the lemniscatic domain is the exterior of a circle, while if E has several components, the lemniscatic domain is the exterior of a generalized lemniscate and is determined by the logarithmic capacity of E and by the exponents and centers of the generalized lemniscate. For general E, we characterize the exponents in terms of the Green’s function of E^c. Under additional symmetry conditions on E, we also locate the centers of the lemniscatic domain. For polynomial pre-images E = P^{-1}(Omega ) of a simply-connected infinite compact set Omega , we explicitly determine the exponents in the lemniscatic domain and derive a set of equations to determine the centers of the lemniscatic domain. Finally, we present several examples where we explicitly obtain the exponents and centers of the lemniscatic domain, as well as the conformal map.

On blow-up solutions to the nonlinear Schrödinger equation in the exterior of a convex obstacle

In this paper, we consider the Schrodinger equation with a mass-supercritical focusing nonlinearity, in the exterior of a smooth, compact, convex obstacle of $\R^{d}$ with Dirichlet boundary conditions. We prove that solutions with negative energy blow up in finite time. Assuming furthermore that the nonlinearity is energy-subcritical, we also prove (under additional symmetry conditions) blow-up with the same optimal ground-state criterion than in the work of Holmer and Roudenko on $\R^{d}$. The classical proof of Glassey, based on the concavity of the variance, fails in the exterior of an obstacle because of the appearance of boundary terms with an unfavorable sign in the second derivative of the variance. The main idea of our proof is to introduce a new modified variance which is bounded from below and strictly concave for the solutions that we consider.

Mathematical and computer modeling of a new endoprosthesis with a carbon-carbon composite for interbody fusion of a lumbar spine

Endoprostheses for interbody fusion should have not only biocompatibility and high strength characteristics, but also good visualization with additional research methods (CT, MRI). One such material is carbon. Aim: using mathematical modeling (the finite element method) to develop a model of the interbody prosthesis from a carbon-carbon composite and evaluate the stress-strain state of the vertebral body-implant system.Methods: models created in SolidWorks software. 18 design schemes with endoprostheses from a carbon-carbon composite were developed and studied. Bending load was carried out by angular displacements (2°), compressive — by applying a force of 500 N to the upper cubic element. Additional symmetry conditions were also simulated.Results: a parametric model of the biomechanical system of the lumbar spine was constructed. The smallest equivalent stresses were obtained in the design scheme M17, and the largest — in M13. In the bone elements of the systems, they did not exceed the ultimate strength limits for cortical (160 MPa) and cancellous (18–22 MPa) bones. Contact pressure and displacement are determined for all design schemes.Conclusions: as a result of the numerical values and distribution fields of the components of the stress-strain state in the elements of the studied systems, it was found that the use of the proposed endoprosthesis made of a carbon-carbon composite, the geometric parameters of which correspond to the design scheme M17, is effective for achieving additional stabilization in the system «vertebral body – implant».

Estimation of the products of the inner radii of domains with an additional symmetry condition

In geometric function theory of complex variable extremal problems on non-overlapping domains are well-known classic direction. A lot of such problems are reduced to determination of the maximum of product of inner radii on the system of non-overlapping domains satisfying a certain conditions. Based on these elementary estimates a number of new estimates for functions realizing a conformal mapping of a disc onto domains with certain special properties are obtained. Estimates of this type are fundamental to solving some metric problems arising when considering the cor\-res\-pon\-dence of boundaries under a conformal mapping. Also, on the basis of the results concerning various extremal properties of conformal mappings, the problem of the representability of functions of a complex variable by a uniformly convergent series of polynomials is solved. In this paper, we consider the problem on maximum the products of the inner radii of $n$ disjoint domains with an additional symmetry condition that contain points of extended complex plane and the degree $\gamma$ of the inner radius of the domain that contains the zero point. An upper estimate for the maximum of this product is found for all values of $\gamma\in(0,\,n]$. The main result of the paper generalizes and strengthens the results of the predecessors [1-4] for the case of an arbitrary arrangement of points systems on $\overline{\mathbb{C}}$. In proving the main theorem, the arguments of proving of Lemma 1 [5] and the ideas of proving Theorem 1 [3] played a key role. We also established the conditions under which the structure of points and domains is not important. The corresponding results are obtained for the case when the points are placed on the unit circle and in the case of any fixed $n$-radial system of points.

Almost Complete Transmission of Low Frequency Waves in a Locally Damaged Elastic Waveguide

It is established that low frequency waves in a symmetric orthotropic elastic waveguide pass almost without distortion through a local perturbation of the straight cylinder, i.e., the reflection coefficient is small and the transmission coefficient slightly differs from 1. The results are obtained by the asymptotic analysis of low frequency waves and the corresponding scattering matrix. The formal asymptotic analysis is also performed for an anisotropic elastic waveguide with asymmetric cross-section, but since the canonical system of Jordan chains for the corresponding operator pencil is too complicated, the known results of the theory of perturbations of nonselfadjoint operators provide asymptotic formulas only under additional elastic and geometric symmetry conditions. We define the polarization matrix and prove its main properties.

Amplifying the Randomness of Weak Sources Correlated With Devices

The problem of device-independent randomness amplification against no-signaling adversaries has so far been studied under the assumption that the weak source of randomness is uncorrelated with the (quantum) devices used in the amplification procedure. In this work, we relax this assumption, and reconsider the original protocol of Colbeck and Renner using a Santha-Vazirani (SV) source. To do so, we introduce an SV-like condition for devices, namely that any string of SV source bits remains weakly random conditioned upon any other bit string from the same SV source and the outputs obtained when this further string is input into the devices. Assuming this condition, we show that a quantum device using a~singlet state to violate the chained Bell inequalities leads to full randomness in the asymptotic scenario of a large number of settings, for a restricted set of SV sources (with $0 \leq \varepsilon < (2^{(1/12)} - 1)/(2(2^{(1/12)} + 1)) \approx 0.0144$). We also study a device-independent protocol that allows for correlations between the sequence of boxes used in the protocol and the SV source bits used to choose the particular box from whose output the randomness is obtained. Assuming the SV-like condition for devices, we show that the honest parties can achieve amplification of the weak source, for the parameter range $0 \leq \varepsilon<0.0132$, against a class of attacks given as a mixture of product box sequences, made of extremal no-signaling boxes, with additional symmetry conditions. Composable security proof against this class of attacks is provided.

Incompatibility of efficiency and strategyproofness in the random assignment setting with indifferences

A fundamental resource allocation setting is the random assignment problem in which agents express preferences over objects that are then randomly allocated to the agents. In 2001, Bogomolnaia and Moulin presented the probabilistic serial (PS) mechanism that is an anonymous, Pareto optimal, and weak strategyproof mechanism when the preferences are considered with respect to stochastic dominance. The result holds when agents have strict preferences over individual objects. It has been an open problem whether there exists a mechanism that satisfies the same properties when agents may have indifference among the objects. We show that for this more general domain, there exists no extension of PS that is ex post efficient and weak strategyproof. The result is surprising because it does not even require additional symmetry or fairness conditions such as anonymity, neutrality, or equal treatment of equals. Our result further demonstrates that the lack of weak SD-strategyproofness of the extended PS mechanism of Katta and Sethuraman (2006) is not a design flaw but is due to an inherent incompatibility of efficiency and strategyproofness of PS in the full preference domain.

Symmetry-protected topological phases with uniform computational power in one dimension

We investigate the usefulness of ground states of quantum spin chains with symmetry-protected topological order (SPTO) for measurement-based quantum computation. We show that, in spatial dimension one, if an SPTO phase supports quantum wire, then, subject to an additional symmetry condition that is satisfied in all cases so far investigated, it can also be used for quantum computation.

Symmetry conditions in strain gradient elasticity

We study the variational significance of the “order-of-differentiation” symmetry condition of strain gradient elasticity. This symmetry condition stems from the fact that in strain gradient elasticity, one can interchange the order of differentiation in the components of the second displacement gradient tensor. We demonstrate that this symmetry condition is essential for the validity of free variational formulations commonly employed for deriving the field equations of strain gradient elasticity. We show that relying on this additional symmetry condition, one can restrict consideration to strain gradient constitutive equations with a considerably reduced number of independent material coefficients. We explicitly derive a symmetry unified theory of isotropic strain gradient elasticity with only two independent strain gradient material coefficients. The presented theory has simple stability criteria and its factorized displacement form equations of equilibrium allow for expedient identification of the fundamental solutions operative in specific theoretical and application studies.

Gauge-Higgs Unification Models in Six Dimensions withS2/Z2Extra Space and GUT Gauge Symmetry

We review gauge-Higgs unification models based on gauge theories defined on six-dimensional spacetime withS2/Z2topology in the extra spatial dimensions. Nontrivial boundary conditions are imposed on the extraS2/Z2space. This review considers two scenarios for constructing a four-dimensional theory from the six-dimensional model. One scheme utilizes the SO(12) gauge symmetry with a special symmetry condition imposed on the gauge field, whereas the other employs the E6gauge symmetry without requiring the additional symmetry condition. Both models lead to a standard model-like gauge theory with theSU(3)×SU(2)L×U(1)Y(×U(1)2)symmetry and SM fermions in four dimensions. The Higgs sector of the model is also analyzed. The electroweak symmetry breaking can be realized, and the weak gauge boson and Higgs boson masses are obtained.

Symplectic Geometry of Entanglement

We present a description of entanglement in composite quantum systems in terms of symplectic geometry. We provide a symplectic characterization of sets of equally entangled states as orbits of group actions in the space of states. In particular, using Kostant-Sternberg theorem, we show that separable states form a unique Kaehler orbit, whereas orbits of entanglement states are characterized by different degrees of degeneracy of the canonical symplectic form on the complex projective space. The degree of degeneracy may be thus used as a new geometric measure of entanglement and we show how to calculate it for various multiparticle systems providing also simple criteria of separability. The presented method is general and can be applied also under different additional symmetry conditions stemming, eg. from the indistinguishability of particles.

Tight frames generated by finite nonabelian groups

Let \(\cal H\) be a Hilbert space of finite dimension d, such as the finite signals l2(d) or a space of multivariate orthogonal polynomials, and n ≥ d. There is a finite number of tight frames of n vectors for \(\cal H\) which can be obtained as the orbit of a single vector under the unitary action of an abelian group G (of symmetries of the frame). Each of these so called harmonic frames or geometrically uniform frames can be obtained from the character table of G in a simple way. These frames are used in signal processing and information theory. For a nonabelian group G there are in general uncountably many inequivalent tight frames of n vectors for \(\cal H\) which can be obtained as such a G-orbit. However, by adding an additional natural symmetry condition (which automatically holds if G is abelian), we obtain a finite class of such frames which can be constructed from the character table of G in a similar fashion to the harmonic frames. This is done by identifying each G-orbit with an element of the group algebra ℂG (via its Gramian), imposing the condition in the group algebra, and then describing the corresponding class of tight frames.

On the Gaussian Random Matrix Ensembles with Additional Symmetry Conditions

The Gaussian unitary random matrix ensembles satisfying some additional sym- metry conditions are considered. The eect of these conditions on the limiting normalized counting measures and correlation functions is studied.

On Banach spaces with unconditional bases

LetX be a Banach space with a sequence of linear, bounded finite rank operatorsRn:X→X such thatRnRm=Rmin(n,m) ifn≠m and limn→∞Rnx=x for allx∈X. We prove that, ifRn−Rn−1 factors uniformly through somelp and satisfies a certain additional symmetry condition, thenX has an unconditional basis. As an application, we study conditions on Λ ⊂ ℤ such thatLΛ=closed span\(\left\{ {z^k :k \in \Lambda } \right\} \subset L_1 \left( \mathbb{T} \right)\), where\(\mathbb{T} = \left\{ {z \in \mathbb{C}:\left| z \right| = 1} \right\}\), has an unconditional basis. Examples include the Hardy space\(H_1 = L_{\mathbb{Z}_ + } \).

Sliding mode based harmonic oscillator synchronization

This paper deals with the synchronization problem of harmonic oscillatory systems, such as two- and three-phase harmonic oscillators. Synchronization between referent and controlled harmonic oscillators, based on amplitude and phase control, is pursued by employing a variable structure control system. Synchronization is achieved by organizing sliding mode along the intersection of two appropriate nonlinear sliding surfaces, individually providing amplitude and phase identity. In case of three-phase harmonic oscillator synchronization an additional symmetry condition arises, which is handled by introduction of another sliding surface. Parameter perturbations and external disturbances are taken into consideration in control system design. Variable structure controllers for two- and three-phase synchronization are designed, which provide sliding mode along the intersection of sliding surfaces in finite time in spite of present disturbances. Ideal sliding dynamics is proved to be insensitive to internal and external disturbances resulting in an ideal harmonic response. Mathematical analysis as well as experimental and simulation results are presented.

On the Spectrum of a Class of Second Order Periodic Elliptic Differential Operators

Under an additional symmetry condition, we prove that the spectrum of a second order self-adjoint elliptic differential operator with periodic coefficients is purely absolutely continuous.

Efficiency IV Estimation For Autoregressive Models With Conditional Heterogeneity

This paper analyzes autoregressive time series models where the errors are assumed to be martingale difference sequences that satisfy an additional symmetry condition on their fourth order moments. Under these conditions Quasi Maximum Likelihood estimators of the autoregressive parameters are no longer efficient in the GMM sense. The main result of the paper is the construction of efficient semiparametric instrumental variables estimators for the autoregressive parameters. The optimal instruments are linear functions of the innovation sequence. It is shown that a frequency domain approximation of the optimal instruments leads to an estimator which only depends on the data periodogram and an unknown linear filter. Semiparametric methods to estimate the optimal filter are proposed. The procedure is equivalent to GMM estimators where lagged observations are used as instruments. Due to the additional symmetry assumption on the fourth moments the number of instruments is allowed to grow at the same rate as the sample. No lag truncation parameters are needed to implement the estimator which makes it particularly appealing from an applied point of view.

EFFICIENT IV ESTIMATION FOR AUTOREGRESSIVE MODELS WITH CONDITIONAL HETEROSKEDASTICITY

This paper analyzes autoregressive time series models where the errors are assumed to be martingale difference sequences that satisfy an additional symmetry condition on their fourth-order moments. Under these conditions quasi maximum likelihood estimators of the autoregressive parameters are no longer efficient in the generalized method of moments (GMM) sense. The main result of the paper is the construction of efficient semiparametric instrumental variables estimators for the autoregressive parameters. The optimal instruments are linear functions of the innovation sequence.It is shown that a frequency domain approximation of the optimal instruments leads to an estimator that only depends on the data periodogram and an unknown linear filter. Semiparametric methods to estimate the optimal filter are proposed.The procedure is equivalent to GMM estimators where lagged observations are used as instruments. As a result of the additional symmetry assumption on the fourth moments the number of instruments is allowed to grow at the same rate as the sample. No lag truncation parameters are needed to implement the estimator, which makes it particularly appealing from an applied point of view.

Spin waves in the ferrimagnetic phase of NdCu 2

The spin wave dispersion relation in the ferrimagnetic phase F1 of NdCu 2 has been measured by means of inelastic neutron scattering. In a-direction the expected six dispersion branches could be determined. The spin wave dispersion is compared to that of the ferromagnetic phase F3. In F1 the magnetic Brillouin zone is only one-third of that in F3 introducing additional symmetry conditions for the dispersion relation. The experimental results agree well with these conditions.