Single Valuedness Research Articles

A method for transforming aliased modes to true modes based on density clustering and Riemann sheets selection in acoustic logging dispersion inversion

Dispersion curves obtained from well logs provide crucial formation information around the borehole. An essential step in the inversion of such curves is to eliminate aliased modes, which can potentially interfere with true modes at higher frequencies. The traditional approach involves manually matching the inversion slowness values with the corresponding wave modes at each frequency, which is time consuming and labor intensive. In addition, the traditional method is applicable for the elimination of the aliased modes that are far from true modes, but it is ineffective in resolving aliased modes that approach or even intersect with the true modes. To address these limitations, a novel method is developed to transform aliased modes into true modes. The proposed method involves two key steps. First, the slownesses of individual modes are automatically picked by the density clustering algorithm. Second, based on the Riemann sheets selection of slowness, aliased modes are transformed into true modes by using the single valuedness of the wave-mode spectra and the correspondence between the amplitude spectrum and the slowness. In addition, a theoretical background is provided by introducing the concept of Riemann sheets to explain the origin of aliased modes in the computation. The proposed method can obtain comprehensive and precise dispersion curves, even in cases for which the true modes are interfered with or crossed by aliased modes. The accuracy of the method is validated by comparing the inversion results with the forward model’s dispersion curves. Furthermore, the robustness of the approach is determined by applying it to synthetic waveforms with noise and field waveforms.

Integer and fractionalized vortex lattices and off-diagonal long-range order

We analyze the implication of off-diagonal long-range order (ODLRO) for inhomogeneous periodic field configurations and multi-component order parameters. For single component order parameters we show that the only static, periodic field configuration consistent with ODLRO is a vortex lattice with integer flux in units of the flux quantum in each unit cell. For a superconductor with g degenerate components, fractional vortices are allowed. Depending on the precise order-parameter manifold, they tend to occur in units of 1/g of the flux quantum. These results are well known to emerge from the Ginzburg-Landau or BCS theories of superconductivity. Our results imply that they are valid even if these theories no-longer apply. Integer and fractional vortex lattices are transparently seen to emerge as a consequence of the macroscopic coherence and single valuedness of the condensate.

Monotonicity and Weighted Prenucleoli: A Characterization Without Consistency

A solution on a set of transferable utility (TU) games satisfies strong aggregate monotonicity (SAM) if every player can improve when the grand coalition becomes richer. It satisfies equal surplus division (ESD) if the solution allows the players to improve equally. We show that the set of weight systems generating weighted prenucleoli that satisfy SAM is open, which implies that for weight systems close enough to any regular system, the weighted prenucleolus satisfies SAM. We also provide a necessary condition for SAM for symmetrically weighted nucleoli. Moreover, we show that the per capita nucleolus on balanced games is characterized by single-valuedness (SIVA), translation covariance (TCOV) and scale covariance (SCOV), and equal adjusted surplus division (EASD), a property that is comparable to but stronger than ESD. These properties together with ESD characterize the per capita prenucleolus on larger sets of TU games. EASD and ESD can be transformed to independence of (adjusted) proportional shifting, and these properties may be generalized for arbitrary weight systems p to I(A)Sp. We show that the p-weighted prenucleolus on the set of balanced TU games is characterized by SIVA, TCOV, SCOV, and IASp and on larger sets by additionally requiring ISp.

On the existence of new integrable cases for Euler-Poisson equations in Newtonian fields

In this article, the rotational motion of a non-symmetric rigid body about a fixed point under the action of multi-Newtonian centers located on one fixed axis is considered. The aim of the work is to derive a technique to obtain such new (particular) integrable cases for the rotational motion of a rigid body in newtonian field of forces whose centers located on the vertical fixed axis. This technique reduces the governing equations and their known first integrals describing the motion to the unknown fourth integral and new system of differential equations. The well-known fourth integral cases are tested to verify the validity of the derived system and some new (particular) integrable cases are obtained with their geometrical interpretations of motion. The particular scope from this work is to extend Arkangel’skii’s theorem and the other related theorems to obtain such new integrable cases of single valuedness of solutions for this problem and its related special cases.

Monotonicity and Weighted Prenucleoli: A Characterization Without Consistency

A solution on a set of transferable utility (TU) games satisfies strong aggregate monotonicity (SAM) if every player can improve when the grand coalition becomes richer. It satisfies equal surplus division (ESD) if the solution allows the players to improve equally. We show that the set of weight systems generating weighted prenucleoli that satisfy SAM is open which implies that for weight systems close enough to any regular system the weighted prenucleolus satisfies SAM. We also provide a necessary condition for SAM for symmetrically weighted nucleoli. Moreover, we show that the per capita nucleolus on balanced games is characterized by single-valuedness (SIVA), translation and scale covariance (COV), and equal adjusted surplus division (EASD), a property that is comparable but stronger than ESD. These properties together with ESD characterize the per capita prenucleolus on larger sets of TU games. EASD and ESD can be transformed to independence of (adjusted) proportional shifting and these properties may be generalized for arbitrary weight systems p to I(A)Sp. We show that the p-weighted prenucleolus on the set of balanced TU games is characterized by SIVA, COV, and IASp; and on larger sets by additionally requiring ISp.

Infimal Convolution and Optimal Time Control Problem III: Minimal Time Projection Set

We continue the investigation of a general minimal time problem with a convex constant dynamics and a lower semicontinuous extended real-valued target function defined on a Banach space. In this paper we obtain an explicit description of the minimal time projection set in terms of the Legendre--Fenchel conjugate function and provide some sufficient conditions for this set to be nonempty. The single valuedness and Lipschitz property of the minimal time projection are also investigated.

Immittance Data Validation by Kramers-Kronig Relations - Derivation and Implications.

Explicitly based on causality, linearity (superposition) and stability (time invariance) and implicit on continuity (consistency), finiteness (convergence) and uniqueness (single valuedness) in the time domain, Kramers‐Kronig (KK) integral transform (KKT) relations for immittances are derived as pure mathematical constructs in the complex frequency domain using the two‐sided (bilateral) Laplace integral transform (LT) reduced to the Fourier domain for sufficiently rapid exponential decaying, bounded immittances. Novel anti KK relations are also derived to distinguish LTI (linear, time invariant) systems from non‐linear, unstable and acausal systems. All relations can be used to test KK transformability on the LTI principles of linearity, stability and causality of measured and model data by Fourier transform (FT) in immittance spectroscopy (IS). Also, integral transform relations are provided to estimate (conjugate) immittances at zero and infinite frequency particularly useful to normalise data and compare data. Also, important implications for IS are presented and suggestions for consistent data analysis are made which generally apply likewise to complex valued quantities in many fields of engineering and natural sciences.

Remarks on unboundedness of set-valued Itô stochastic integrals

The paper deals with set-valued stochastic integrals which are defined as a random multifunctions. Integrable boundedness of a such integral is one of the most important features in potential applications. Unfortunately, up to now there were no correct proofs of such property. Surprisingly, also a negative answer to this problem is still not explained correctly. Hence the problem seems to be still undetermined. We shall show that in general the answer is negative. We shall provide several relatively simple examples both in convex and nonconvex-valued case. As a consequence, we will show that under some restrictions on the class of selections of the integrand the integrable boundedness of set-valued Itô's integral is equivalent to its single valuedness.

Morphisms of Colimits: from Paths to Profunctors

A general kind of morphism of diagrams in a category is introduced. It is the most general notion of morphism which induces a morphism between the colimits of the diagrams. The sense in which it is the most general is made precise. It is expressed in terms of total profunctors which generalize everywhere defined relations. Their functorial properties are developed leading to the notion of cohesive family of diagrams. A complementary notion of deterministic profunctor is also introduced generalizing single valuedness.

Hamiltonian solutions of the 3-body problem in (2+1) gravity

We present a full study of the 3-body problem in gravity in a flat (2+1)-dimensional spacetime, and in the nonrelativistic limit of small velocities. We provide an explicit form of the Arnowitt–Deser–Misner Hamiltonian in a regular coordinate system and we set up all the ingredients for canonical quantization. We emphasize the rôle of a U(2) symmetry under which the Hamiltonian is invariant and which should generalize to a U(N − 1) symmetry for N bodies. This symmetry seems to stem from a braid group structure in the operations of looping of particles around each other, and guarantees the single valuedness of the Hamiltonian. Its rôle for the construction of single-valued energy eigenfunctions is also discussed.

Metric projection and compatibly parameterized families of prox-regular sets in Hilbert space

The aim of the paper is to prove, under appropriate conditions, that the metric projection to a parameterized closed set C ( u ) of a Hilbert space is near a point ( u ¯ , x ¯ ) , with x ¯ ∈ C ( u ¯ ) , single valued and jointly continuous in the parameter variable u and the variable x whose projection is taken on C ( u ) . It is also shown how the single valuedness of the metric projection and its above continuity property near the point ( u ¯ , x ¯ ) is related to similar properties of a suitable truncation of a set-valued mapping associated with the normal cone of C ( u ) . The above properties are obtained under the Attouch–Wets continuity of C ( ⋅ ) near u ¯ and the assumption that the set C ( u ) is prox-regular in x at x ¯ with a compatible parameterization by u at u ¯ . The paper thus extends previous results established by Robinson in finite dimensional Euclidean spaces to Hilbert spaces.

Geometric phase in Bohmian mechanics

Using the quantum kinematic approach of Mukunda and Simon, we propose a geometric phase in Bohmian mechanics. A reparametrization and gauge invariant geometric phase is derived along an arbitrary path in configuration space. The single valuedness of the wave function implies that the geometric phase along a path must be equal to an integer multiple of 2 π. The nonzero geometric phase indicates that we go through the branch cut of the action function from one Riemann sheet to another when we locally travel along the path. For stationary states, quantum vortices exhibiting the quantized circulation integral can be regarded as a manifestation of the geometric phase. The bound-state Aharonov–Bohm effect demonstrates that the geometric phase along a closed path contains not only the circulation integral term but also an additional term associated with the magnetic flux. In addition, it is shown that the geometric phase proposed previously from the ensemble theory is not gauge invariant.

Geometric phase within the complex quantum Hamilton–Jacobi formalism

We derive a geometric phase using the quantum kinematic approach within the complex quantum Hamilton–Jacobi formalism. The single valuedness of the wave function implies that the geometric phase along an arbitrary path in the complex plane must be equal to an integer multiple of 2 π. The nonzero geometric phase indicates that we travel along the path through the branch cut of the phase function from one Riemann sheet to another.

Second Order Cones for Maximal Monotone Operators via Representative Functions

It is shown that various first and second order derivatives of the Fitzpatrick and Penot representative functions for a maximal monotone operator T, in a reflexive Banach space, can be used to represent differential information associated with the tangent and normal cones to the Graph T. In particular we obtain formula for the proto-derivative, as well as its polar, the normal cone to the graph of T. First order derivatives are shown to be useful in recognising points of single-valuedness of T. We show that a strong form of proto-differentiability to the graph of T, is often associated with single valuedness of T.

ANALYTIC PATTERNS FOR CHAOTIC EQUATIONS

Chaotic evolution equations sometimes display regular patterns (fronts, kinks, holes), which often correspond to closed form analytic solutions. In the dissipative-dispersive Kuramoto and Sivashinsky travelling wave reduction u(x − ct), νu‴ + bu″ + μu′ + u2/2 + A = 0, ν ≠ 0, with (ν,b,μ,A) constants, such analytic solutions are known for heteroclinic solutions, but one has also observed (Toh, 1987) homoclinic solutions without corresponding analytic solutions yet. We review the search for the most general analytic solution admissible by this chaotic differential equation. Several investigations, both analytic by the Painlevé test (Thual and Frisch, 1986) and numerical by Padé approximants (Yee, Conte, Musette, 2003) indicate its quite probable single valuedness for any (ν,b,μ,A). Moreover, Nevanlinna theory on the growth of solutions near infinity rules out (Eremenko, 2005) the possibility for the singularities of this unknown closed form single valued expression to be only poles.We propose a qualitative closed form involving a deformed elliptic function.

Non-Abelian hydrodynamics and the flow of spin in spin–orbit coupled substances

Motivated by the heavy ion collision experiments there is much activity in studying the hydrodynamical properties of non-Abelian (quark–gluon) plasmas. A major question is how to deal with color currents. Although not widely appreciated, quite similar issues arise in condensed matter physics in the context of the transport of spins in the presence of spin–orbit coupling. The key insight is that the Pauli Hamiltonian governing the leading relativistic corrections in condensed matter systems can be rewritten in a language of SU(2) covariant derivatives where the role of the non-Abelian gauge fields is taken by the physical electromagnetic fields: the Pauli system can be viewed as Yang–Mills quantum-mechanics in a ‘fixed frame’, and it can be viewed as an ‘analogous system’ for non-Abelian transport in the same spirit as Volovik’s identification of the He superfluids as analogies for quantum fields in curved space time. We take a similar perspective as Jackiw and coworkers in their recent study of non-Abelian hydrodynamics, twisting the interpretation into the ‘fixed frame’ context, to find out what this means for spin transport in condensed matter systems. We present an extension of Jackiw’s scheme: non-Abelian hydrodynamical currents can be factored in a ‘non-coherent’ classical part, and a coherent part requiring macroscopic non-Abelian quantum entanglement. Hereby it becomes particularly manifest that non-Abelian fluid flow is a much richer affair than familiar hydrodynamics, and this permits us to classify the various spin transport phenomena in condensed matter physics in an unifying framework. The “particle based hydrodynamics” of Jackiw et al. is recognized as the high temperature spin transport associated with semiconductor spintronics. In this context the absence of faithful hydrodynamics is well known, but in our formulation it is directly associated with the fact that the covariant conservation of non-Abelian currents turns into a disastrous non-conservation of the incoherent spin currents of the high temperature limit. We analyze the quantum-mechanical single particle currents of relevance to mesoscopic transport with as highlight the Ahronov–Casher effect, where we demonstrate that the intricacies of the non-Abelian transport render this effect to be much more fragile than its abelian analog, the Ahronov–Bohm effect. We subsequently focus on spin flows protected by order parameters. At present there is much interest in multiferroics where non-collinear magnetic order triggers macroscopic electric polarization via the spin–orbit coupling. We identify this to be a peculiarity of coherent non-Abelian hydrodynamics: although there is no net particle transport, the spin entanglement is transported in these magnets and the coherent spin ‘super’ current in turn translates into electric fields with the bonus that due to the requirement of single valuedness of the magnetic order parameter a true hydrodynamics is restored. Finally, ‘fixed-frame’ coherent non-Abelian transport comes to its full glory in spin–orbit coupled ‘spin superfluids’, and we demonstrate a new effect: the trapping of electrical line charge being a fixed frame, non-Abelian analog of the familiar magnetic flux trapping by normal superconductors. The only known physical examples of such spin superfluids are the 3He A- and B-phase where unfortunately the spin–orbit coupling is so weak that it appears impossible to observe these effects.

Quantization with maximally degenerate Poisson brackets: the harmonic oscillator!

Nambu's construction of multi-linear brackets for super-integrable systems can be thought of as degenerate Poisson brackets with a maximal set of Casimirs in their kernel. By introducing privileged coordinates in phase space these degenerate Poisson brackets are brought to the form of Heisenberg's equations. We propose a definition for constructing quantum operators for classical functions which enables us to turn the maximally degenerate Poisson brackets into operators. They pose a set of eigenvalue problems for a new state vector. The requirement of the single valuedness of this eigenfunction leads to quantization. The example of the harmonic oscillator is used to illustrate this general procedure for quantizing a class of maximally super-integrable systems.

Choice of dynamical variables for global reconstruction of model equations from time series.

The success of modeling from an experimental time series is determined to a significant extent by the choice of dynamical variables. We propose a method for preliminary investigation of a time series whose purpose is to find out whether a global dynamical model with smooth functions can be constructed for the chosen variables. The method consists in the estimation of single valuedness and continuity of relations between dynamical variables and variables to enter left-hand sides of model equations. The method is explained with numerical examples. Its efficiency is demonstrated by modeling a real nonlinear electric circuit.

The single-conclusion proof logic and inference rules specification

The logic of single-conclusion ( functional) proofs ( FLP ) is introduced. It combines the verification property of proofs with the single valuedness of proof predicate and describes the operations on proofs induced by modus ponens rule and proof checking. It is proved that FLP is decidable, sound and complete with respect to arithmetical proof interpretations based on single-valued proof predicates. The application to arithmetical inference rules specification and PA -admissibility testing is considered. We show that the provability in FLP gives the complete admissibility test for the rules which can be specified by schemes in FLP -language. The test is supplied with the ground proof extraction algorithm which eliminates the admissible rules from a PA -proof by utilizing the information from the corresponding FLP -proofs.

Quantitative atom-resolved force gradient imaging using noncontact atomic force microscopy

Quantitative force gradient images are obtained using a sub-angström amplitude, off-resonance lever oscillation method during scanning tunneling microscopy imaging. We report the direct observation of short-range bonds, and the measured short-range force interaction agrees well in magnitude and length scale with theoretical predictions for single bonds. Atomic resolution is shown to be associated with the presence of a prominent short-range contribution to the total force interaction. It is shown that the background longer-range interaction, whose relative magnitude depends on the tip structure, has a significant effect on the contrast observed at the atomic scale.