Branch Singularities Research Articles

The Dazzling Array of Basal Branches in the mtDNA Macrohaplogroup M from India as Inferred from Complete Genomes

Many efforts based on complete mitochondrial DNA (mtDNA) genomes have been made to depict the global mtDNA landscape, but the phylogeny of Indian macrohaplogroup M has not yet been resolved in detail. To fill this lacuna, we took the same strategy as in our recent analysis of Indian mtDNA macrohaplogroup N and selected 56 mtDNAs from over 1,200 samples across India for complete sequencing, with the intention to cover all Indian autochthonous M lineages. As a result, the phylogenetic status of previously identified haplogroups based on control-region and/or partial coding-region information, such as M2, M3, M4, M5, M6, M30, and M33, was solidified or redefined here. Moreover, seven novel basal M haplogroups (viz., M34-M40) were identified, and yet another five singular branches of the M phylogeny were discovered in the present study. The comparison of matrilineal components among India, East Asia, Southeast Asia, and Oceania at the deepest level yielded a star-like and nonoverlapping pattern, reflecting a rapid mode of modern human dispersal along the Asian coast after the initial "out-of-Africa" event.

Spacetime structure of static solutions in Gauss-Bonnet gravity: Charged case

We have studied spacetime structures of static solutions in the $n$-dimensional Einstein-Gauss-Bonnet-Maxwell-$\ensuremath{\Lambda}$ system. Especially we focus on effects of the Maxwell charge. We assume that the Gauss-Bonnet coefficient $\ensuremath{\alpha}$ is non-negative and $4\stackrel{\texttildelow{}}{\ensuremath{\alpha}}/{\ensuremath{\ell}}^{2}\ensuremath{\le}1$ in order to define the relevant vacuum state. Solutions have the $(n\ensuremath{-}2)$-dimensional Euclidean submanifold whose curvature is $k=1$, 0, or $\ensuremath{-}1$. In Gauss-Bonnet gravity, solutions are classified into plus and minus branches. In the plus branch all solutions have the same asymptotic structure as those in general relativity with a negative cosmological constant. The charge affects a central region of a spacetime. A branch singularity appears at the finite radius $r={r}_{b}>0$ for any mass parameter. There the Kretschmann invariant behaves as $O((r\ensuremath{-}{r}_{b}{)}^{\ensuremath{-}3})$, which is much milder than the divergent behavior of the central singularity in general relativity $O({r}^{\ensuremath{-}4(n\ensuremath{-}2)})$. In the $k=1$ and 0 cases plus-branch solutions have no horizon. In the $k=\ensuremath{-}1$ case, the radius of a horizon is restricted as ${r}_{h}<\sqrt{2\stackrel{\texttildelow{}}{\ensuremath{\alpha}}}$ (${r}_{h}>\sqrt{2\stackrel{\texttildelow{}}{\ensuremath{\alpha}}}$) in the plus (minus) branch. Some charged black hole solutions have no inner horizon in Gauss-Bonnet gravity. There are topological black hole solutions with zero and negative mass in the plus branch regardless of the sign of the cosmological constant. Although there is a maximum mass for black hole solutions in the plus branch for $k=\ensuremath{-}1$ in the neutral case, no such maximum exists in the charged case. The solutions in the plus branch with $k=\ensuremath{-}1$ and $n\ensuremath{\ge}6$ have an inner black hole and inner and outer black hole horizons. In the $4\stackrel{\texttildelow{}}{\ensuremath{\alpha}}/{\ensuremath{\ell}}^{2}=1$ case, only a positive mass solution is allowed, otherwise the metric function takes a complex value. Considering the evolution of black holes, we briefly discuss a classical discontinuous transition from one black hole spacetime to another.

Spacetime structure of static solutions in Gauss-Bonnet gravity: Neutral case

We study the spacetime structures of the static solutions in the $n$-dimensional Einstein-Gauss-Bonnet-$\ensuremath{\Lambda}$ system systematically. We assume the Gauss-Bonnet coefficient $\ensuremath{\alpha}$ is non-negative and a cosmological constant is either positive, zero, or negative. The solutions have the $(n\ensuremath{-}2)$-dimensional Euclidean submanifold, which is the Einstein manifold with the curvature $k=1$, 0, and $\ensuremath{-}1$. We also assume $4\stackrel{\texttildelow{}}{\ensuremath{\alpha}}/{\ensuremath{\ell}}^{2}\ensuremath{\le}1$, where $\ensuremath{\ell}$ is the curvature radius, in order for the sourceless solution ($M=0$) to be defined. The general solutions are classified into plus and minus branches. The structures of the center, horizons, infinity, and the singular point depend on the parameters $\ensuremath{\alpha}$, ${\ensuremath{\ell}}^{2}$, $k$, $M$, and branches complicatedly so that a variety of global structures for the solutions are found. In our analysis, the $\stackrel{\texttildelow{}}{M}\mathrm{\text{\ensuremath{-}}}r$ diagram is used, which makes our consideration clear and enables easy understanding by visual effects. In the plus branch, all the solutions have the same asymptotic structure at infinity as that in general relativity with a negative cosmological constant. For the negative-mass parameter, a new type of singularity called the branch singularity appears at nonzero finite radius $r={r}_{b}>0$. The divergent behavior around the singularity in Gauss-Bonnet gravity is milder than that around the central singularity in general relativity. There are three types of horizons: inner, black hole, and cosmological. In the $k=1,0$ cases, the plus-branch solutions do not have any horizon. In the $k=\ensuremath{-}1$ case, the radius of the horizon is restricted as ${r}_{h}<\sqrt{2\stackrel{\texttildelow{}}{\ensuremath{\alpha}}}$ (${r}_{h}>\sqrt{2\stackrel{\texttildelow{}}{\ensuremath{\alpha}}}$) in the plus (minus) branch. The black hole solution with zero or negative mass exists in the plus branch even for the zero or positive cosmological constant. There is also the extreme black hole solution with positive mass. We briefly discuss the effect of the Gauss-Bonnet corrections on black hole formation in a collider and the possibility of the violation of the third law of the black hole thermodynamics.

Physical aspects of the shape of electron emission spectra for ferroelectric electrets

A correlation between the specific features of the electron emission spectrum of a ferroelectric electret and the behavior of the potential at the surface of the sample is determined within the theoretical approach proposed. The physical nature of the main features in experimental emission spectra is interpreted in terms of the behavior of the potential relief. It is revealed that the electret charge relaxation affects the spectrum shape due to the bifurcation (branching) of singularities in the spectrum.

Vacuum structure and flavor symmetry breaking in supersymmetric SO(nc) gauge theories

We determine the vacuum structure and phases of N=1 theories obtained via a mass \mu for the adjoint chiral superfield in N=2, SO(n_c) SQCD. For large number of flavors these theories have two groups of vacua. The first exhibits dynamical breaking of flavor symmetry \USp(2n_f) \to U(n_f) and arises as a relevant deformation of a non-trivial superconformal theory. These are in the confined phase. The second group, in an IR-free phase with unbroken flavor symmetry, is produced from a Coulomb branch singularity with Seiberg's dual gauge symmetry. In the large-\mu regime both groups of vacua are well-described by dual quarks and mesons, and dynamical symmetry breaking in the first group occurs via meson condensation. We follow the description of these vacua from weak to strong coupling and demonstrate a nontrivial agreement between the phases and the number of vacua in the two regimes. We construct the semiclassical monopole flavor multiplets and argue that their multiplicity is consistent with the number of N=1 vacua.

Higgs Branch, Hyper-Kähler Quotient and Duality in SUSY N = 2 Yang–Mills Theories

Low-energy limits of N = 2 supersymmetric field theories in the Higgs branch are described in terms of a nonlinear four-dimensional σ-model on a hyper-Kähler target space, classically obtained as a hyper-Kähler quotient of the original flat hypermultiplet space by the gauge group. We review in a pedagogical way this construction, and illustrate it in various examples, with special attention given to the singularities emerging in the low-energy theory. In particular, we thoroughly study the Higgs branch singularity of Seiberg–Witten SU(2) theory with Nf flavors, interpreted by Witten as a small instanton singularity in the moduli space of one instanton on ℝ4. By explicitly evaluating the metric, we show that this Higgs branch coincides with the Higgs branch of a U(1) N = 2 SUSY theory with the number of flavors predicted by the singularity structure of Seiberg–Witten's theory in the Coulomb phase. We find another example of Higgs phase duality, namely between the Higgs phases of U(Nc)Nf flavors and U(Nf-Nc)Nf flavors theories, by using a geometric interpretation due to Biquard et al. This duality may be relevant for understanding Seiberg's conjectured duality Nc ↔ Nf-Nc in N = 1 SUSY SU(Nc) gauge theories.

Computing critical points and secondary paths in nonlinear structural stability analysis by the finite element method

Stability analysis is a crucial aspect of structural design, particularly for beam, plate and shell type structures. The stability of a structure is often governed by singular points in its equilibrium path. These points usually pose serious problems to ordinary finite element solution methods and require special treatment. In this paper, the state-of-the-art of finite element path-following techniques is critically reviewed with special attention to bracketing procedures for the singular points and branch switching algorithms. Some practical aspects involved will be explained in detail and demonstrated through numerical examples at the end.

On the branching of singularities in complex domains

On the branching of singularities in complex domains

G-signature, G-degree, and symmetries of the branches of curve singularities

G-signature, G-degree, and symmetries of the branches of curve singularities

MULTIPLICITY AND STABILITY OF STEADY SOLUTIONS FOR MARANGONI CONVECTION IN A TWO-DIMENSIONAL RECTANGULAR CONTAINER WITH RIGID SIDEWALLS

A numerical study of steady Marangoni convection (due to surface tension forces) is presented for a liquid layer, confined to a two-dimensional rectangular container with rigid side waits. With a branch-tracing algorithm, branches of steady states and singularities along these branches are calculated. The results are compared with those for which the container has “slippery” sidewalk. Typical new features for rigid sidewalls are the appearance of subcritical instabilities and (for a square container) a limitation on the range of the existence of one-cell patterns.

A unified approach to Painlevé expansions

The Painlevé test for partial differential equations developed by Weiss, Tabor and Carnevale (WTC) is examined in detail and shown to provide a unified approach to the integrable properties of both ordinary and partial differential equations. A simple modification of the WTC procedure used for partial differential equations enables us to determine the Lax pairs, Hirota equations and auto-Bäcklund transformations for ordinary differential equations, including a new Lax pair for an integrable case of the Henon-Heiles system. A detailed study of the KdV hierarchy is made and a complete picture of the pattern of resonances for all solution branches is obtained. The role of the singular branches is examined in detail and important new insights obtained. In particular we find that each singular branch is simply a re-expansion of the principal branch about a point on the pole manifold at which several isolated poles coalesce. A parallel analysis is carried out for the AKNS hierarchy.

Branching of singularities for degenerate hyperbolic operator and Stokes phenomena, IV

Branching of singularities for degenerate hyperbolic operator and Stokes phenomena, IV

A canonical form of a system of microdifferential equations with non-involutory characteristics and branching of singularities

We study a system ℳ of microdifferential (=pseudodifferential) equations. We assume that the characteristic varietyV of ℳ takes the formV=V 1∪V2; hereV 1andV 2are regular involutory submanifolds and intersect normally, andV 1∩V2is non-involutory and 1-codimensional both inV 1and inV 2. We also assume that ℳ has regular singularities alongV. Then we give a canonical form of ℳ in the complex domain. This enables us to investigate the branching of supports of microfunction solutions of ℳ when ℳ is hyperbolic.

Branching of singularities for degenerate hyperbolic operator and Stokes phenomena, III

Branching of singularities for degenerate hyperbolic operator and Stokes phenomena, III

Branching of singularities for a degenerate hyperbolic system

Branching of singularities for a degenerate hyperbolic system

A Fault-Collapsing Analysis in Sequential Logic Networks

Although a sequential circuit M reduces to a combinational network C M after all feedback paths have been cut, an application of Bossen and Hong's checkpoint labeling procedure to C M does not necessarily yield a minimal solution. The set of checkpoints so obtained will include all feedback lines. In this paper, it is shown that these feedback lines are not necessary checkpoints under a “delay equivalence” relation. In addition to this, we also show that not every fanout branch is a necessary checkpoint. Any “singular fanout branches” can be removed from consideration. The results of our analysis lead to a minimal checkpoint labeling procedure for sequential logic networks.

A canonical form of a system of microdifferential equations with non-involutory characteristics and branching of singularities

We study a system ℳ of microdifferential (=pseudodifferential) equations. We assume that the characteristic varietyV of ℳ takes the formV=V1∪V2; hereV1andV2are regular involutory submanifolds and intersect normally, andV1∩V2is non-involutory and 1-codimensional both inV1and inV2. We also assume that ℳ has regular singularities alongV. Then we give a canonical form of ℳ in the complex domain. This enables us to investigate the branching of supports of microfunction solutions of ℳ when ℳ is hyperbolic.

Branching of singularities for degenerate hyperbolic operators and Stokes phenomena, II

Branching of singularities for degenerate hyperbolic operators and Stokes phenomena, II

Branching of singularities for degenerate hyperbolic operators and Stokes phenomena

Branching of singularities for degenerate hyperbolic operators and Stokes phenomena

Impurity-induced first-order Raman spectra of KI and KBr in the fcc and sc phases

The pressure dependences of the impurity-induced Raman spectra of KI and KBr have been measured for both fcc and sc structures, up to 30 kbar. Both materials have been observed to undergo phase transition at about 18 kbar. Four phonon bands have been observed for KI and three distinct peaks for KBr in the sc phase. From the measurement of the temperature dependence of the band intensities for KI between 300 and ∼80 K, these bands have been identified with first-order processes. The mode Grüneisen parameters for the two acoustic Eg modes in fcc phase and two acoustic and one optic mode in sc phase have been measured for both KI and KBr. The impurity-induced phonon bands of KI and KBr in both phases have also been correlated to the corresponding phonon branch singularity or singularities for most peaks.