One-parameter Family Research Articles

Collisions of weakly-bound kinks in the Christ-Lee model

We investigate soliton collisions in a one-parameter family of scalar field theories in 1+1 dimensions which was first discussed by Christ and Lee [1]. The models have a sextic potential with three local minima, and for suitably small values of the parameter their kinks have an internal structure in the form of two weakly-bound subkinks. We show that for these values of the parameter kink collisions are best understood as an independent sequence of collisions of these subkinks, and that a static mode analysis is not enough to explain resonant structures emerging in this model. We also emphasise the role of radiation and oscillon formation in the collision process.

Fermi equation of state with finite temperature corrections in quantum space-times approach: Snyder model vs GUP case

We investigate the impact of the deformed phase space associated with the quantum Snyder space on microphysical systems. The general Fermi–Dirac equation of state and specific corrections to it are derived. We put emphasis on non-relativistic degenerate Fermi gas as well as on the temperature-finite corrections to it. Considering the most general one-parameter family of deformed phase spaces associated with the Snyder model allows us to study whether the modifications arising in physical effects depend on the choice of realization. It turns out that we can distinguish three different cases with radically different physical consequences.

Basins of attraction of a one-parameter family of root-finding techniques

Abstract Initial conditions can have a substantial impact on the behavior of iterative root-finding techniques for nonlinear equations. By allowing complex starting points and complex roots, it is possible to examine the basins of attraction in the complex plane in order to compare the performance of various iterative techniques. In this paper, a one-parameter family of third-order root-finding methods is studied by varying its parameter A within −2.0 and 2.4 and applying it to a polynomial equation of high degree (degree 25). This family includes the Euler–Chebyshev’s (A = 0), Halley’s (A = 1) and BSC (A = 2) techniques. According to the results, the one-parameter family provides the best performance for values near A = 1, which equals to the Halley’s method.

The effect of the spatial variation of the evaporative flux on the deposition from a thin sessile droplet

A mathematical model for the effect of the spatial variation of the local evaporative flux on the evaporation of and deposition from a thin pinned particle-laden sessile droplet is formulated and solved. We then analyse the behaviour for a one-parameter family of local evaporative fluxes with the free parameter $n \, (>-1)$ that exhibits qualitatively different behaviours mimicking those that can be obtained by, for example, surrounding the droplet with a bath of fluid or using a mask with one or more holes in it to achieve a desired pattern of evaporation enhancement and/or suppression. We show that when $-1< n<1$ (including the special cases $n=-1/2$ of diffusion-limited evaporation into an unbounded atmosphere and $n=0$ of spatially uniform evaporation), all of the particles are eventually advected to the contact line, and so the final deposit is a ring deposit at the contact line, whereas when $n>1$ all of the particles are eventually advected to the centre of the droplet, and so the final deposit is at the centre of the droplet. In particular, the present work demonstrates that a singular (or even a non-zero) evaporative flux at the contact line is not an essential requirement for the formation of a ring deposit. In addition, we calculate the paths of the particles when diffusion is slower than both axial and radial advection, and show that in this regime all of the particles are captured by the descending free surface before eventually being deposited onto the substrate.

Bismut Ricci flat generalized metrics on compact homogeneous spaces

A generalized metric on a manifold M M , i.e., a pair ( g , H ) (g,H) , where g g is a Riemannian metric and H H a closed 3 3 -form, is a fixed point of the generalized Ricci flow if and only if ( g , H ) (g,H) is Bismut Ricci flat: H H is g g -harmonic and R c ( g ) = 1 4 H g 2 Rc(g)=\tfrac {1}{4}H_g^2 . On any homogeneous space M = G / K M=G/K , where G = G 1 × G 2 G=G_1\times G_2 is a compact semisimple Lie group with two simple factors, under some mild assumptions, we exhibit a Bismut Ricci flat G G -invariant generalized metric, which is proved to be unique among a 4 4 -parameter space of metrics in many cases, including when K K is neither abelian nor semisimple. On the other hand, if K K is simple and the standard metric is Einstein on both G 1 / π 1 ( K ) G_1/\pi _1(K) and G 2 / π 2 ( K ) G_2/\pi _2(K) , we give a one-parameter family of Bismut Ricci flat G G -invariant generalized metrics on G / K G/K and show that it is most likely pairwise non-homothetic by computing the ratio of Ricci eigenvalues. This is proved to be the case for every space of the form M = G × G / Δ K M=G\times G/\Delta K and for M 35 = S O ( 8 ) × S O ( 7 ) / G 2 M^{35}=SO(8)\times SO(7)/G_2 .

Empirically Driven multiwavelength K-corrections at low redshift

ABSTRACT K-corrections – a necessary ingredient for converting between flux in observed bands to flux in rest-frame bands – are critical for comparing galaxies at differing redshifts. These corrections often rely on fits to empirical or theoretical spectral energy distribution (SED) templates of galaxies. However, templates can only produce reliable K-corrections in regimes where SED models are robust. For instance, the templates utilized in some popular software packages are not well-constrained in some bands (e.g. WISE W4 in Kcorrect), which results in ill-behaved K-corrections. We address this shortcoming by developing an empirically driven approach to K-corrections that limits the dependence on SED templates. We perform a polynomial fit for the K-correction as a function of a galaxy’s rest-frame colour determined in a pair of well-constrained bands (e.g. 0(g − r)) and redshift, exploiting the fact that galaxy SEDs can be approximated as a one-parameter family at low redshift. For bands well-constrained by SED templates, our empirically driven K-corrections yield results comparable to the SED fitting methods used by Kcorrect and the GSWLC-M2 catalogue (the updated medium-deep GALEX–SDSS–WISE Legacy Catalogue). However, our method dramatically outperforms Kcorrect derived K-corrections for WISE W4. Our method is also robust to incorrect template assumptions outside of the optical bands and enforces that the K-correction must be zero at z = 0. Our K-corrected photometry and code are publicly available.

Staticity of asymptotically hyperbolic minimal mass extensions

In this paper, we define the Bartnik mass of a domain whose boundary is connected and compact, has scalar curvature bounded below −n(n − 1), and whose extensions are asymptotically hyperbolic manifolds. With this definition, we show that asymptotically hyperbolic admissible extensions of a domain that achieve the Bartnik mass must admit a static potential. Given a non-static admissible extension of a domain, we are able to construct a one-parameter family of metrics that are close to the original metric, have smaller mass, share the same bound on the scalar curvature, and contain the domain isometrically.

Uphill in Reaction-Diffusion Multi-species Interacting Particles Systems

We study reaction-diffusion processes with multi-species particles and hard-core interaction. We add boundary driving to the system by means of external reservoirs which inject and remove particles, thus creating stationary currents. We consider the condition that the time evolution of the average occupation evolves as the discretized version of a system of coupled diffusive equations with linear reactions. In particular, we identify a specific one-parameter family of such linear reaction-diffusion systems where the hydrodynamic limit behaviour can obtained by means of a dual process. We show that partial uphill diffusion is possible for the discrete particle systems on the lattice, whereas it is lost in the hydrodynamic limit.

A bulk-edge correspondence between the second Chern number and the spectral flow

A one-parameter family of semi-quantum Hamiltonians admitting the time-reversal symmetry is defined on the product S2(r1)×S2(r2) of two-spheres of respective radii r1 and r2, and a corresponding one-parameter family of quantum Hamiltonians is defined in terms of the angular momentum operators associated with respective spheres. The second Chern number of the eigen-vector bundle associated with a positive eigenvalue of the semi-quantum Hamiltonian is a piecewise constant function of the parameter with jumps at singular parameter values leading to the emergence of total degeneracy points for the eigenvalues. Correspondingly, the quantum Hamiltonian has edge-state eigenvalues which, as functions of the parameter, pass the zero energy level at the singular parameter values and the bulk-state eigenvalues prove to correspond to eigenvalues of the semi-quantum Hamiltonian. It is shown that the second-Chern number c2(T) at a non-singular parameter value T is in one-to-one correspondence with the spectral flow as a function of the end point of the interval (−∞,T). The bulk-edge correspondence is also shown to hold between the linearized semi-quantum and quantum systems, where the semi-quantum Hamiltonian is linearized at each of the degeneracy points of the eigenvalues and quantized to define linear quantum Hamiltonians. On this level, the spectral flow of each linear quantum Hamiltonian with a parameter μ proves to coincide with the index of the linear quantum Hamiltonian with μ=0.

The Calogero–Bogoyavlenskii–Schiff breaking soliton equation: Recursion operators and higher symmetries

We find two one-parametric families of recursion operators and use them to construct higher symmetries for the Calogero–Bogoyavlenskii–Schiff breaking soliton equation. Then we prove that the recursion operators from the first family pair-wise commute with respect to the Nijenhuis bracket (are compatible).

Diophantine equations with three monomials

TextWe present a general algorithm for solving all two-variable polynomial Diophantine equations consisting of three monomials. Before this work, even the existence of an algorithm for solving the one-parameter family of equations x4+axy+y3=0 has been an open question. We also present an elementary method that reduces the task of finding all integer solutions to a general three-monomial equation to the task of finding primitive solutions to equations with three monomials in disjoint variables. We identify a large class of three-monomial equations for which this method leads to a complete solution. Empirical data suggests that this class contains 100% of three-monomial equations as the number of variables goes to infinity. VideoFor a video summary of this paper, please visit https://youtu.be/ANsJvFTmDCE.

Methods for Applying Matrices when Creating Models of Group Pursuit

Introduction. It is obvious that in the near future, the issues of equipping moving robotic systems with autonomous control elements will remain relevant. This requires the development of models of group pursuit. Note that optimization in pursuit tasks is reduced to the construction of optimal trajectories (shortest trajectories, trajectories with differential constraints, fuel consumption indicators). At the same time, the aspects of automated distribution by goals in group pursuit were not considered. To fill this gap, the presented piece of research has been carried out. Its result should be the construction of a model of automated distribution of pursuers by goals in group pursuit.Materials and Methods. A matrix was formed to study the multiple goal group pursuit. The control parameters for the movement of the pursuers were modified according to the minimum curvature of the trajectory. The methods of pursuit and approach were considered in detail. The possibilities of modifying the method of parallel approach were shown. Matrix simulation was used to build a scheme of multiple goal group pursuit. The listed processes were illustrated by functions in the given coordinate systems and animation. Block diagrams of the phase coordinates of the pursuer at the next step, the time and distance of the pursuer reaching the goal were constructed as a base of functions. In some cases, the location of targets and pursuers was defined as points on the circle of Apollonius. The matrix was formed by samples corresponding to the distribution of pursuers by goals.Results. Nine variants of the pursuit, parallel, proportional and three-point approach on the plane and in space were considered. The maximum value of the goal achievement time was calculated. There were cases when the speed vector of the pursuer was directed arbitrarily and to a point on the Apollonius circle. It was noted that the three-point approach method was convenient if the target was moving along a ballistic trajectory. To modify the method of parallel approach, a network of parallel lines was built on the plane. Here, the length of the arc of the line (which can be of any shape) and the array of reference points of the target trajectory were taken into account. An equation was compiled and solved with these elements. On an array of samples with corresponding time values, the minimum time was found, i.e., the optimal time for simultaneous group achievement of multiple goals was determined. For unified access to the library, the control vector was expressed through a one-parameter family of parallel planes. A library of calculations of control vectors was formed. An example of applying matrix simulation to group pursuit was shown. A scheme of group pursuit of multiple goals was presented. For two goals and three pursuers, six samples corresponding to the distribution of pursuers by goals were considered. The data was presented in the form of a matrix. Based on the research results, the computer program was created and registered – “Parallel Approach on Plane of Group of Pursuers with Simultaneous Achievement of the Goal”.Discussions and Conclusion. The methods of using matrices in modeling group pursuit were investigated. The possibility of modifying the method of parallel approach was shown. Matrix simulation of group pursuit enabled to build its scheme for a set of purposes. The matrix of the distribution of pursuers by goals would be generated at each moment of time. Methods of forming matrices of the distribution of pursuers and targets are of interest in the design of virtual reality systems, for tasks with simulating the process of group pursuit, escape, evasion. The dynamic programming method opens up the possibility of automating the distribution with optimization according to the specified parameters under the formation of the matrix of the distribution of pursuers by goals.

Quantum correlations on the no-signaling boundary: self-testing and more

In device-independent quantum information, correlations between local measurement outcomes observed by spatially separated parties in a Bell test play a fundamental role. Even though it is long-known that the set of correlations allowed in quantum theory lies strictly between the Bell-local set and the no-signaling set, many questions concerning the geometry of the quantum set remain unanswered. Here, we revisit the problem of when the boundary of the quantum set coincides with the no-signaling set in the simplest Bell scenario. In particular, for each Class of these common boundaries containing k zero probabilities, we provide a (5−k)-parameter family of quantum strategies realizing these (extremal) correlations. We further prove that self-testing is possible in all nontrivial Classes beyond the known examples of Hardy-type correlations, and provide numerical evidence supporting the robustness of these self-testing results. Candidates of one-parameter families of self-testing correlations from some of these Classes are identified. As a byproduct of our investigation, if the qubit strategies leading to an extremal nonlocal correlation are local-unitarily equivalent, a self-testing statement provably follows. Interestingly, all these self-testing correlations found on the no-signaling boundary are provably non-exposed. An analogous characterization for the set M of quantum correlations arising from finite-dimensional maximally entangled states is also provided. En route to establishing this last result, we show that all correlations of M in the simplest Bell scenario are attainable as convex combinations of those achievable using a Bell pair and projective measurements. In turn, we obtain the maximal Clauser-Horne-Shimony-Holt Bell inequality violation by any maximally entangled two-qudit state and a no-go theorem regarding the self-testing of such states.

Asymptotics of the deformed higher order Airy-kernel determinants and applications

We study the one-parameter family of Fredholm determinants , , where stands for the integral operator acting on with the higher order Airy kernel. This family of determinants represents a new universal class of distributions which is a higher order analogue of the classical Tracy–Widom distribution. Each of the determinants admits an integral representation in terms of a special real solution to the nth member of the Painlevé II hierarchy. Using the Riemann–Hilbert approach, we establish asymptotics of the determinants and the associated higher order Painlevé II transcendents as for and , respectively. In the case of , we are able to calculate the constant term in the asymptotic expansion of the determinants, while for , the relevant asymptotics exhibit singular behaviours. Applications of our results are also discussed, which particularly include asymptotic statistical properties of the counting function for the random point process defined by the higher order Airy kernel.

A nonsmooth Newton method for solving the generalized complementarity problem

In this paper, we present a new nonsmooth Newton-type algorithm for solving the generalized complementarity problem based on its reformulation as a system of nonlinear equations using a one-parametric family of complementarity functions. We demonstrate, under suitable hypotheses, that this algorithm converges locally and q-quadratically. In addition, we show numerical experiments that allow us to see the good performance of the proposed algorithm.

On Isospectral Composite Beams

We consider a composite system consisting of two identical straight elastic beams under longitudinal vibration connected by an elastic interface capable of counteracting the relative vibration of the two beams with its shearing stiffness. We construct examples of isospectral composite beams, i.e., countable one-parameter families of beams having different shearing stiffness but exactly the same eigenvalues under a given set of boundary conditions. The construction is explicit and is based on the reduction to a one-dimensional Sturm–Liouville eigenvalue problem and the application of a Darboux’s lemma.

Point Pair-Based Expression of Cutter Swept Envelopes in Five-Axis Milling

This paper proposes an efficient method of formulating cutter swept envelopes in five-axis milling based on conformal geometric algebra. The surface of the rotary cutter is represented as a canal surface, as the envelope of a one-parameter family of spheres. The swept envelopes generated by the general rigid motion of the cutter can be regarded as the envelopes of a two-parameter family of spheres. With the help of the dual representations of spheres and planes in conformal space, a point pair-based expression of the swept envelopes is formulated analytically, which represents two surfaces uniformly. By using the present method, the swept surfaces of a conical cutter, a drum-shaped cutter, and a toroidal cutter are illustrated. As an application, two types of collision detection methods for 5-axis milling are presented based on the point pair-based envelope expression. The benchmark examples demonstrate that the use of the proposed envelope expression significantly improves computational efficiency.

Level crossings, attractor points and complex multiplication

We study the complex structure moduli dependence of the scalar Laplacian eigenmodes for one-parameter families of Calabi-Yau n-folds in ℙn+1. It was previously observed that some eigenmodes get lighter while others get heavier as a function of these moduli, which leads to eigenvalue crossing. We identify the cause for this behavior for the torus. We then show that at points in a sublocus of complex structure moduli space where Laplacian eigenmodes cross, the torus has complex multiplication. We speculate that the generalization to arbitrary Calabi-Yau manifolds could be that level crossing is related to rank one attractor points. To test this, we compute the eigenmodes numerically for the quartic K3 and the quintic threefold, and match crossings to CM and attractor points in these varieties. To quantify the error of our numerical methods, we also study the dependence of the numerical spectrum on the quality of the Calabi-Yau metric approximation, the number of points sampled from the Calabi-Yau variety, the truncation of the eigenbasis, and the distance from degeneration points in complex structure moduli space.

Mathcal{N} = 2 CFT3’s from mathcal{N} = 4 gauged supergravity

We use holography and four-dimensional mathcal{N} = 4 gauged supergravity to collect evidence for a large class of interconnected three-dimensional mathcal{N} = 2 conformal field theories. On the gravity side, we construct a one-parameter family of ISO(3) × ISO(3) gaugings of half-maximal supergravity containing a rich structure of mathcal{N} = 2 AdS4 solutions at fixed radius. By looking at excitations around these AdS4 solutions, the spectrum of low lying operators in the dual mathcal{N} = 2 CFT3’s is computed and further arranged into \U0001d52c\U0001d530(2|4) supermultiplets. Upon suitable removal of gauge redundancies, we identify the Zamolodchikov metric on the conformal manifold dual to the AdS4 moduli space, and recover previous results in the S-fold literature. Two special points of mathcal{N} = 4 supersymmetry enhancement occur. While one describes an S-fold CFT3 dual to a non-geometric type IIB twisted compactification, the string-theoretic realisation of the other, if any, is still lacking.

Poisson orders on large quantum groups

We develop a Poisson geometric framework for studying the representation theory of all contragredient quantum super groups at roots of unity. This is done in a uniform fashion by treating the larger class of quantum doubles of bozonizations of all distinguished pre-Nichols algebras [9] belonging to a one-parameter family; we call these algebras large quantum groups. We prove that each of these quantum algebras has a central Hopf subalgebra giving rise to a Poisson order in the sense of [13]. We describe explicitly the underlying Poisson algebraic groups and Poisson homogeneous spaces in terms of Borel subgroups of complex semisimple algebraic groups of adjoint type. The geometry of the Poisson algebraic groups and Poisson homogeneous spaces that are involved and its applications to the irreducible representations of the algebras Uq⊃Uq⩾⊃Uq+ are also described. Besides all (multiparameter) big quantum groups of De Concini–Kac–Procesi and big quantum super groups at roots of unity, our framework also contains the quantizations in characteristic 0 of the 34-dimensional Kac-Weisfeiler Lie algebras in characteristic 2 and the 10-dimensional Brown Lie algebras in characteristic 3. The previous approaches to the above problems relied on reductions to rank two cases and direct calculations of Poisson brackets, which is not possible in the super case since there are 13 kinds of additional Serre relations on up to 4 generators. We use a new approach that relies on perfect pairings between restricted and non-restricted integral forms.