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We introduce a new class of quiver varieties, called saw (quiver) varieties. These can be seen as a generalization of the three existing theories: the quiver gauge theory of monopoles, Kac's representation theory of Dynkin types, and Donaldson's theory on the based maps of P 1. E.g., in our terminology, the handsaw (quiver) varieties and chainsaw varieties are finite and affine A type saw varieties respectively with the maximal symmetry breaking. These varieties are the monopole spaces and the caloron spaces in 3d gauge theory, respectively. We expand the theory over to the DE type quiver gauge theory. The saw varieties are constructed via a Hamiltonian formalism, so that the moment maps defining these varieties naturally arise. The main assertion of the paper is that the moment map μ defining a saw variety is a flat morphism for arbitrary dimension vector, if and only if the underlying graph is of finite ADE types. Furthermore under the assumption of maximal symmetry breaking or all the higher frame dimension vectors, the affine ADE types also attain the flatness property of μ. We give an algebro-geometric description of the momentum-zero scheme μ − 1 (0) and the saw variety defined as the Hamiltonian reduction. Various partition functions given rise to by the above mentioned theories are the byproducts of the geometric structure of the corresponding saw varieties. [ABSTRACT FROM AUTHOR]

Asymmetric cell divisions, where cells divide with respect to a polarized axis and give rise to daughter cells with different fates, are critically important for development. In many such divisions, the conserved PAR polarity proteins accumulate in distinct cortical domains in response to a symmetry breaking cue. The one-cell C. elegans embryo is a paradigm for understanding mechanisms of PAR polarization, but much less is known about polarity in subsequent divisions. Here, we investigate the polarization of the P 1 cell of the two-cell embryo. A posterior PAR-2 domain forms in the first 4 ​min, and polarization becomes stronger over time. Initial polarization depends on the PAR-1 and PKC-3 kinases, and the downstream polarity regulators MEX-5 and PLK-1. However, par-1 and plk-1 mutants exhibit delayed polarization. This late polarization correlates with the time of centrosome maturation and actomyosin flow, and loss of centrosome maturation or myosin in par-1 mutant embryos causes an even stronger polarity phenotype. Based on these and other results, we propose that PAR polarity in the P 1 cell is generated by at least two redundant mechanisms: There is a novel early pathway dependent on PAR-1, PKC-3 and cytoplasmic polarity, and a late pathway that resembles symmetry breaking in the one-cell embryo and requires PKC-3, centrosome associated AIR-1 and myosin flow. [Display omitted] • There are two redundant pathways for polarizing the C.elegans P 1 cell. • Posterior PARs polarize as early as 2 ​min after P 0 cell cytokinesis. • PAR-1, PLK-1, and MEX-5 are all required for early P 1 cell polarization. • Centrosome maturation and myosin flow are required for late polarization in par-1 mutant embryos. • PKC-3 is required for both early and late P 1 cell polarization. [ABSTRACT FROM AUTHOR]

We consider the nonlinear Schrödinger equation Δ u = (1 + ε V 1 (| y |)) u − | u | p − 1 u in R N , N ≥ 3 , p ∈ (1 , N + 2 N − 2). The phenomenon of pattern formation has been a central theme in the study of nonlinear Schrödinger equations. However, the following nonexistence of O (N) symmetry breaking solution is well-known: if the potential function is radial and nondecreasing, any positive solution must be radial. Therefore, solutions of interesting patterns can only exist after violating the assumptions. O (N) symmetry breaking solutions have been presented by Wei and Yan [20]. Symmetry groups of regular polygons describe their solution patterns. Ever since work of Wei and Yan, there have been substantial generalizations but solutions with higher dimensional symmetry have not been constructed. In this study, the existence of nonradial solutions whose symmetry group is a discrete subgroup of O (3) , more precisely, the orientation-preserving regular tetrahedral group is shown. [ABSTRACT FROM AUTHOR]

An elegant strategy for proving impossibility results in distributed computing was introduced in the celebrated FLP consensus impossibility proof. This strategy is local in nature as at each stage, one configuration of a hypothetical protocol for consensus is considered, together with future decisions of possible extensions. This proof strategy has been used in numerous situations related to consensus, leading one to wonder why it has not been used in impossibility results of two other well-known tasks: set agreement and renaming. This paper provides an explanation of why impossibility proofs of these tasks have been of a global nature. It shows that a protocol can always solve such tasks locally, in the following sense. Given a configuration and all its future decisions, if a single successor configuration is selected, then the protocol can reveal all decisions in this branch of executions, satisfying the task specification. This result is shown for both set agreement and renaming, providing evidence that there are no local impossibility proofs for these tasks. • Studies impossibility proofs for sub-consensus tasks in wait-free shared-memory. • Locally-solvable valency tasks capture the FLP strategy for proving impossibility results. • Presents locally-solvable valency tasks for set consensus, weak symmetry breaking and renaming. • This explains why there are no FLP-style proofs for the impossibility of solving these tasks. [ABSTRACT FROM AUTHOR]

Brown algae are a group of multicellular, heterokont algae that have convergently evolved developmental complexity that rivals that of embryophytes, animals or fungi. Early in development, brown algal zygotes establish a basal and an apical pole, which will become respectively the basal system (holdfast) and the apical system (thallus) of the adult alga. Brown algae are interesting models for understanding the establishment of cell polarity in a broad evolutionary context, because they exhibit a large diversity of life cycles, reproductive strategies and, importantly, their zygotes are produced in large quantities free of parental tissue, with symmetry breaking and asymmetric division taking place in a highly synchronous manner. This review describes the current knowledge about the establishment of the apical-basal axis in the model brown seaweeds Ectocarpus , Dictyota , Fucus and Saccharina , highlighting the advantages and specific interests of each system. Ectocarpus is a genetic model system that allows access to the molecular basis of early development and life-cycle control over apical-basal polarity. The oogamous brown alga Fucus , together with emerging comparative models Dictyota and Saccharina, emphasize the diversity of strategies of symmetry breaking in determining a cell polarity vector in brown algae. A comparison with symmetry-breaking mechanisms in land plants, animals and fungi, reveals that the one-step zygote polarisation of Fucus compares well to Saccharomyces budding and Arabidopsis stomata development, while the two-phased symmetry breaking in the Dictyota zygote compares to Schizosaccharomyces fission, the Caenorhabditis anterior-posterior zygote polarisation and Arabidopsis prolate pollen polarisation. The apical-basal patterning in Saccharina zygotes on the other hand, may be seen as analogous to that of land plants. Overall, brown algae have the potential to bring exciting new information on how a single cell gives rise to an entire complex body plan. [ABSTRACT FROM AUTHOR]

[Display omitted] We report a series of experiments carried out with a dichloromethane drop deposited on the surface of an aqueous solution containing a surfactant, cetyltrimethylammonium bromide. After an induction stage during which the drop stays axisymmetric, oscillations occur along the contact line. These oscillations are succeeded by a spectacular spontaneous spinning of the drop. The latter quickly takes the form of a two-tip 'rotor' and the spinning rate stabilizes at a constant value, no longer varying despite the gradual changes of the drop shape and size. The drop eventually disappears due to the continual dissolution and evaporation of dichloromethane. Schlieren visualizations and particle image velocimetry are used to establish a consistent scenario capable of explaining the evolution of the system. The Marangoni effect induced by the dissolution of dichloromethane in the drop vicinity is shown to be responsible for the observed dynamics. Arguments borrowed from dynamical systems theory and from an existing low-order model allow us to explain qualitatively why the system selects the spinning configuration. The geometry of the immersed part of the drop is shown to play a crucial role in this selection process, as well as in the regulation of the spinning rate. [ABSTRACT FROM AUTHOR]

We have conducted an experimental study on the use of rear flexible vertical flaps as adaptive solutions to reduce the drag of a squareback Ahmed body, and on the fluid–structure interaction mechanisms at the turbulent wake. To that aim, wind tunnel experiments were conducted to compare the performance of various configurations including the baseline body, the body with rigid flaps and with flexible flaps. These configurations were tested under different aligned and cross-flow conditions. The results reveal that the flexible adaptive devices effectively reduce the drag within for low values of the dimensionless stiffness quantified through the Cauchy number, C a. Thus, the two-dimensional deformation of the flexible flaps, which undergo progressive inwards reconfiguration (with an averaged tip deflection angle of Θ ≃ 4 °), reduces the bluffness of the flow separation at the body base, thus shrinking the recirculation region. This reconfiguration leads to increased base pressure, resulting into a 8.3% decrease in the global drag, C D , under aligned conditions. Similar drag reductions are observed under yawed conditions. Two regimes are identified in terms of the coupled fluid–structure dynamics. For low C a , the passive reconfiguration of the flaps include small amplitude, periodic oscillations corresponding to the first free deformation mode of a cantilevered beam. Alongside these weak oscillations, the flaps are deformed guided by the changes in the value of the horizontal base pressure gradient, depicting bi-stable behavior which is caused by the synchronization between the Reflectional Symmetry Breaking (RSB) mode, typically present in the wake of three-dimensional bluff bodies, and the flaps deformation. For higher values of C a , the flexible flaps deflect inwardly by about Θ ≃ 20 ° on average, but exhibit vigorous oscillations combining the first and second free deformation modes of a cantilevered beam. These large amplitude oscillations excite the flow separation at the model's trailing edges, leading to significant fluctuations in the separated shear layers and a consequent 31% increase in the global drag. Under yawed conditions, the flaps responses for large values of C a are different due to the asymmetry of the corresponding recirculation region. [ABSTRACT FROM AUTHOR]

• Bending waves can be modulated spatial-temporally by longitudinal waves in an Euler-Bernoulli Beam. • Theoretical and simulated results show the existence of asymmetric wavenumber bandgaps. • Waves in asymmetric wavenumber bandgaps propagate nonreciprocally and grow exponentially with time. Breaking time or/and spatial symmetry can lead to unprecedent mechanical responses of a structure, the design of which is usually very intricate. This paper proposes a simple design of a beam structure with spatiotemporal modulation. In fact, a time-varying axial load applied to the ends of an elastic beam will induce space-time dependent axial force propagating in the form of longitudinal wave in the beam, which can modulate spatiotemporally the propagation of transverse bending waves. The governing equation is derived, of which the Floquet solution leads to a quadratic eigenvalue problem that can be solved for the frequency as a function of the wavenumber. Theoretical results show the existence of asymmetric wavenumber bandgaps and the unique dispersion diagram which is not periodic with respect to time or space. The breaking of temporal symmetry enables nonreciprocal propagation of transverse bending waves, which is confirmed by the finite element simulations. This modulation method offers promise for applications in the fields of elastic wave filter, frequency converter and magnitude amplifier. [ABSTRACT FROM AUTHOR]

We explore the conformal 5D braneworld, where warping emerges through conformal symmetry breaking. Our scenario seamlessly aligns with conventional brane approaches if conformal symmetry remains unbroken. It is shown that a model with a single conformal breaking parameter effectively localizes gravity on the brane, but it falls short in trapping gauge bosons. However, in scenarios with two parameters, gravity is localized, and the model also achieves the localization of zero modes for both gauge and Dirac fields. [ABSTRACT FROM AUTHOR]

Asymmetric cell division is an essential feature of normal development and certain pathologies. The process and its regulation have been studied extensively in the Caenorhabditis elegans embryo, particularly how symmetry of the actomyosin cortical cytoskeleton is broken by a sperm-derived signal at fertilization, upstream of polarity establishment. Diploscapter pachys is the closest parthenogenetic relative to C. elegans , and D. pachys one-cell embryos also divide asymmetrically. However how polarity is triggered in the absence of sperm remains unknown. In post-meiotic embryos, we find that the nucleus inhabits principally one embryo hemisphere, the future posterior pole. When forced to one pole by centrifugation, the nucleus returns to its preferred pole, although poles appear identical as concerns cortical ruffling and actin cytoskeleton. The location of the meiotic spindle also correlates with the future posterior pole and slight actin enrichment is observed at that pole in some early embryos along with microtubule structures emanating from the meiotic spindle. Polarized location of the nucleus is not observed in pre-meiotic D. pachys oocytes. All together our results are consistent with the idea that polarity of the D. pachys embryo is attained during meiosis, seemingly based on the location of the meiotic spindle, by a mechanism that may be present but suppressed in C. elegans. [Display omitted] • In parthenogenetic D. pachys , the first embryonic cell division is asymmetric. • In D. pachys embryos, the nucleus inhabits principally the future posterior pole. • Embryo centrifugation revealed that the posterior pole is defined during meiosis. • Pre-meiotic D. pachys oocytes do not show signs of asymmetry. • The meiotic spindle appears to trigger the location of the future posterior pole. [ABSTRACT FROM AUTHOR]

In this paper, we consider the self-dual O (3) Maxwell-Chern-Simons-Higgs equations on a two-dimensional flat torus arising from the O (3) sigma gauge field model. Our main goal is to obtain the existence and uniqueness of topological solutions with vortices and antivortices. In order to achieve this goal, we show that the nondegeneracy of linearized operator for entire solution holds even when the symmetry is broken. Moreover, we also obtain the uniform bound of L 1 norm of nonlinearities with respect to large charge of electron. We expect that these results would play an significant role to get the asymptotic behavior of all possible solutions and count the total number of solutions. [ABSTRACT FROM AUTHOR]

Living systems at all scales are compartmentalized into interacting subsystems. This paper reviews a mechanism that drives compartmentalization in generic systems at any scale. It first discusses three symmetries of generic physical interactions in a quantum-theoretic description. It then shows that if one of these, a permutation symmetry on the inter-system boundary, is spontaneously broken, the symmetry breaking is amplified by the Free Energy Principle (FEP). It thus shows how compartmentalization generically results from permutation symmetry breaking under the FEP. It finally notes that the FEP asymptotically restores the broken symmetry, showing that the FEP can be regarded as a theory of fluctuations away from a permutation-symmetric boundary, and hence from an entangled joint state of the interacting systems. [ABSTRACT FROM AUTHOR]

We present a nonperturbative theory of bound states in the continuum (BICs) based on the analytical theory of infinite-grating eigenmodes in the case of a planar grating waveguide, that is, 1D photonic crystal slab. We describe a mixed BIC mainly formed by a coherent mixture of two photonic-crystal normal modes in a vicinity of their avoided crossing where both of them have a nonzero coupling with a radiation-loss channel. We reveal the universal properties of the mixed BIC associated with the spatial parity symmetry breaking and self-similar spectral behavior of the bulk normal modes near Γ -point. We show that the mechanism of the mixed-BIC formation constitutes a special kind of the generic Friedrich–Wintgen mechanism. A cutoff from the radiation-loss channel and extremely high-Q narrow resonance are achieved due to the destructive interference of the two crossing normal modes. On the contrary, a conventional symmetry-protected BIC is formed mostly by a single photonic-crystal normal mode which has vanishing coupling Fourier coefficient with every available radiation-loss channel. Implementation of the mixed BICs in various photonic-crystals structures could lead to interesting applications. • Genesis, field constituents, and mechanisms of bound states in the continuum. • Self-contained analytical theory: from the first principles to full mode structure. • Universal path to disclosing content of bound states in the continuum, high-Q modes. • Great method for designing new bound states in the continuum in photonic crystals. • Intuitively appealing picture, interpretation of related high-Q waveguide eigenmodes. [ABSTRACT FROM AUTHOR]

This study aims to analyze the dynamics of a thin Newtonian liquid film on a uniformly heated compliant substrate. We consider the violation of time-reversal symmetry in the liquid, resulting in an additional non-zero term in the liquid stress tensor. Using the long-wave expansion technique, we derive a set of coupled equations governing the film thickness and substrate deformation, accounting for inertia, surface tension, thermocapillarity, and odd viscosity. Through linear stability analysis and spatiotemporal simulations, we observe that the compliant substrate enhances instability, while wall heating exacerbates it. However, the introduction of odd viscosity effectively suppresses these instabilities, as confirmed by the agreement between simulation and theoretical predictions. • Analyzed dynamics of thin film flow along a heated compliant substrate. • Liquid's time-reversal symmetry is broken and this leads to odd viscosity inclusion. • Compliant substrates and wall heating amplify surface wave instability. • Odd viscosity reduces instability. [ABSTRACT FROM AUTHOR]

An effective field theory framework is used to investigate some Lorentz-violating effects on the generation of electromagnetic and gravitational waves, complementing previous work on propagation. Specifically we find solutions to a modified, anisotropic wave equation, sourced by charge or fluid matter. We derive the radiation fields for scalars, classical electromagnetic radiation, and partial results for gravitational radiation. For gravitational waves, the results show longitudinal and breathing polarizations proportional to coefficients for spacetime-symmetry breaking. [ABSTRACT FROM AUTHOR]

Ladders of recollements of abelian categories are introduced, and used to address three general problems. Ladders of a certain height allow to construct recollements of triangulated categories, involving derived categories and singularity categories, from abelian ones. Ladders also allow to tilt abelian recollements, and ladders guarantee that properties like Gorenstein projective or injective are preserved by some functors in abelian recollements. Breaking symmetry is crucial in developing this theory. [ABSTRACT FROM AUTHOR]

During development, a single cell is transformed into a highly complex organism through progressive cell division, specification and rearrangement. An important prerequisite for the emergence of patterns within the developing organism is to establish asymmetries at various scales, ranging from individual cells to the entire embryo, eventually giving rise to the different body structures. This becomes especially apparent during gastrulation, when the earliest major lineage restriction events lead to the formation of the different germ layers. Traditionally, the unfolding of the developmental program from symmetry breaking to germ layer formation has been studied by dissecting the contributions of different signaling pathways and cellular rearrangements in the in vivo context of intact embryos. Recent efforts, using the intrinsic capacity of embryonic stem cells to self-assemble and generate embryo-like structures de novo, have opened new avenues for understanding the many ways by which an embryo can be built and the influence of extrinsic factors therein. Here, we discuss and compare divergent and conserved strategies leading to germ layer formation in embryos as compared to in vitro systems, their upstream molecular cascades and the role of extrinsic factors in this process. • Gastrulation leads to stereotypical germ layer progenitor specification and organization. • Gastrulation relies on tight interactions between signaling pathways and embryonic morphology. • In culture, embryonic substructures can recapitulate certain aspects of gastrulation. • In vitro stem cell models show a high capacity to trigger germ layer formation also de novo. • Such models allow to dissect the specific contributions of molecular, cellular and biophysical mechanisms. [ABSTRACT FROM AUTHOR]

Breaking embryonic symmetry is an essential prerequisite to shape the initially symmetric embryo into a highly organized body plan that serves as the blueprint of the adult organism. This critical process is driven by morphogen signaling gradients that instruct anteroposterior axis specification. Despite its fundamental importance, what triggers symmetry breaking and how the signaling gradients are established in time and space in the mammalian embryo remain largely unknown. Stem cell-based in vitro models of embryogenesis offer an unprecedented opportunity to quantitatively dissect the multiple physical and molecular processes that shape the mammalian embryo. Here we review biochemical mechanisms governing early mammalian patterning in vivo and highlight recent advances to recreate this in vitro using stem cells. We discuss how the novel insights from these model systems extend previously proposed concepts to illuminate the extent to which embryonic cells have the intrinsic capability to generate specific, reproducible patterns during embryogenesis. • We review decades of experimentation in the mouse for biochemical and biophysical mechanisms underlying embryo patterning. • We discuss advances in stem-cell based embryo models with a particular focus on symmetry breaking. • The role for embryonic geometry and size in governing mouse gastrulation is discussed. • We question yet unknown contributing factors to symmetry breaking events in mammalian embryos. [ABSTRACT FROM AUTHOR]

We consider the following eigenvalue optimization in the composite membrane problem with fractional Laplacian: given a bounded domain Ω ⊂ R n , α > 0 and 0 < A < | Ω | , find a subset D ⊂ Ω of area A such that the first Dirichlet eigenvalue of the operator (− Δ) s + α χ D is as small as possible. The solution D is called as an optimal configuration for the data (Ω , α , A). Looking at the well-known extension definition for the fractional Laplacian, in the case s = 1 / 2 this is essentially the composite membrane problem for which the mass is concentrated at the boundary as one is trying to minimize the Steklov eigenvalue. We prove existence of solutions and study properties of optimal configuration D. This is a free boundary problem which could be formulated as a two-sided unstable obstacle problem. Moreover, we show that for some rotationally symmetric domains (thin annuli), the optimal configuration is not rotational symmetric, which implies the non-uniqueness of the optimal configuration D. On the other hand, we prove that for a convex domain Ω having reflection symmetries, the optimal configuration possesses the same symmetries, which implies uniqueness of the optimal configuration D in the ball case. [ABSTRACT FROM AUTHOR]

The initial breaking of left-right (L–R) symmetry in the embryo is controlled by a motile-cilia-driven leftward fluid flow in the left-right organiser (LRO), resulting in L–R asymmetric gene expression flanking the LRO. Ultimately this results in left- but not right-sided activation of the Nodal - Pitx2 pathway in more lateral tissues. While aspects of the initial breaking event clearly vary between vertebrates, events in the Lateral Plate Mesoderm (LPM) are conserved through the vertebrate lineage. Evidence from model systems and humans highlights the role of cilia both in the initial symmetry breaking and in the ability of more lateral tissues to exhibit asymmetric gene expression. In this review we concentrate on the process of L–R determination in mouse and humans. [ABSTRACT FROM AUTHOR]

How an animal establishes its body axis is a fundamental question in developmental biology. The freshwater cnidarian Hydra is an attractive model for studying axis formation because it is radially symmetric, with a single oral-aboral axis. It was recently proposed that the orientation of the new body axis in a regenerating Hydra polyp is determined by the oral-aboral orientation of the actin-myosin contractile processes (myonemes) in the animal's outer epithelial layer. However, it remained unclear how the oral-aboral polarity of the body axis would be defined. As Wnt signaling is known to control axis polarity in Hydra and bilaterians, we hypothesized that it plays a role in axis formation during regeneration of Hydra tissue pieces. We tested this hypothesis using pharmacological perturbations and novel grafting experiments to set Wnt signaling and myoneme orientation perpendicular to each other to determine which controls axis formation. Our results demonstrate that Wnt signaling is the dominant encoder of axis orientation and polarity, in line with its conserved role in axial patterning. • Grafting and alsterpaullone (ALP) used to probe axis specification in Hydra. • Bilateral grafting is used to compete Wnt signaling and myoneme orientation. • New axis of regenerating Hydra tissue is determined by parental Wnt signaling. • Wnt signaling is the dominant encoder of axis information in Hydra. [ABSTRACT FROM AUTHOR]

This paper presents a technique for symmetry reduction that adaptively assigns a prefix of variables in a system of constraints so that the generated prefix-assignments are pairwise nonisomorphic under the action of the symmetry group of the system. The technique is based on McKay's canonical extension framework (McKay, 1998). Among key features of the technique are (i) adaptability—the prefix sequence can be user-prescribed and truncated for compatibility with the group of symmetries; (ii) parallelizability—prefix-assignments can be processed in parallel independently of each other; (iii) versatility—the method is applicable whenever the group of symmetries can be concisely represented as the automorphism group of a vertex-colored graph; and (iv) implementability—the method can be implemented relying on a canonical labeling map for vertex-colored graphs as the only nontrivial subroutine. To demonstrate the practical applicability of our technique, we have prepared an experimental open-source implementation of the technique and carry out a set of experiments that demonstrate ability to reduce symmetry on hard instances. Furthermore, we demonstrate that the implementation effectively parallelizes to compute clusters with multiple nodes via a message-passing interface. [ABSTRACT FROM AUTHOR]

In this paper, we show the existence and non-existence of minimizers of the following minimization problems which include an open problem mentioned by Horiuchi and Kumlin [20] : G a : = inf u ∈ W 0 1 , N (Ω) ∖ { 0 } ⁡ ∫ Ω | ∇ u | N d x (∫ Ω | u | q f a , β (x) d x) N q , where f a , β (x) : = | x | − N (log ⁡ a R | x |) − β. First, we give an answer to the open problem when Ω = B R (0). Next, we investigate the minimization problems on general bounded domains. In this case, the results depend on the shape of the domain Ω. Finally, symmetry breaking property of the minimizers is proved for sufficiently large β. [ABSTRACT FROM AUTHOR]

We show that the topological Kitaev model can be placed in the context of phenomena associated with strongly correlated electron systems. This can be achieved implementing the Gutzwiller projection of the conventional s -wave BCS superconductor. Within that approach, neither spin–orbit coupling nor time-reversal symmetry breaking are needed to produce such an effect. • Emergent topological Kitaev model in a strongly correlated electronic system. • Gutzwwiller projected s-wave BCS superconductor and the infinitely Kondo interaction regime. • Role of classical spin textures in producing non-trivial topological states for correlated conduction electrons. [ABSTRACT FROM AUTHOR]

• Topological interface state (TIS) is achieved at an ultra-low frequency by designing a deep-subwavelength phononic beam. • The phononic beam is engineered with unit cells having acoustic black hole (ABH) configurations to produce the first band gap in a low-frequency regime. • Topological phase of the phononic beam is controlled by modifying the ABH configuration and breaking the inversion symmetry of unit-cell, leading to the formation of low-frequency TIS in a combined structure of two phononic beams with different topological phases. • The designed phononic beam is fabricated, and its TIS is verified experimentally. We propose deep-subwavelength phononic beams capable of supporting topological interface states (TISs) at ultra-low frequencies. The topological phononic beams are designed using unit cells with acoustic black hole (ABH) configurations to obtain the first band gap in a low-frequency range, for which the unit cell size is much smaller than the wavelength of the wave to be controlled. We control the topological phases of the phononic beams by modifying the ABH configuration and breaking the inversion symmetry of unit-cell. Two topologically distinct beams are then connected to produce the TIS in their low band gaps. Numerical and experimental results show that TISs are achieved at low frequencies of 4.3–8.5 Hz using the designed phononic beams whose lattice constants (a) are less than λ /20. The value of a / λ is reduced further by adjusting unit-cell arrays of the phononic beams. Given that topological phononic crystals reported in previous studies usually had lattice constants comparable to or several times smaller than wavelengths, the present work offers the possibility of designing a more compact system for topological states at low frequencies. [ABSTRACT FROM AUTHOR]

Kasner cosmology is a vacuum and anisotropically expanding spacetime in the general relativity context. In this work, such a cosmological model is studied in another context, the bumblebee model, where the Lorentz symmetry is spontaneously broken. By using the bumblebee context it is possible to justify the anisotropic feature of the Kasner cosmology. Thus, the origin of the anisotropy in this cosmological model could be in the Lorentz symmetry breaking. Lastly, an application in the pre-inflationary cosmology is suggested. • Lorentz violation in a cosmological model. • Kasner cosmology is an anisotropic and vacuum model. • The anisotropies in the Kasner model could have an origin in the Lorenz violation. [ABSTRACT FROM AUTHOR]

Abstract We study ground states of two-component Bose–Einstein condensates (BEC) with trapping potentials in R 2 , where the intraspecies interaction (− a 1 , − a 2) and the interspecies interaction − β are both attractive, i. e , a 1 , a 2 and β are all positive. The existence and non-existence of ground states are classified completely by investigating equivalently the associated L 2 -critical constraint variational problem. The uniqueness and symmetry-breaking of ground states are also analyzed under different types of trapping potentials as β ↗ β ⁎ = a ⁎ + (a ⁎ − a 1) (a ⁎ − a 2) , where 0 < a i < a ⁎ : = ‖ w ‖ 2 2 (i = 1 , 2) is fixed and w is the unique positive solution of Δ w − w + w 3 = 0 in R 2. The semi-trivial limit behavior of ground states is tackled in the companion paper [12]. [ABSTRACT FROM AUTHOR]

Abstract We study existence, regularity, and qualitative properties of solutions to the system − Δ u = | v | q − 1 v in Ω , − Δ v = | u | p − 1 u in Ω , ∂ ν u = ∂ ν v = 0 on ∂ Ω , with Ω ⊂ R N bounded; in this setting, all nontrivial solutions are sign changing. Our proofs use a variational formulation in dual spaces, considering sublinear p q < 1 and superlinear p q > 1 problems in the subcritical regime. In balls and annuli we show that least energy solutions (l.e.s.) are foliated Schwarz symmetric and, due to a symmetry-breaking phenomenon, l.e.s. are not radial functions; a key element in the proof is a new L t -norm-preserving transformation, which combines a suitable flipping with a decreasing rearrangement. This combination allows us to treat annular domains, sign-changing functions, and Neumann problems, which are non-standard settings to use rearrangements and symmetrizations. In particular, we show that our transformation diminishes the (dual) energy and, as a consequence, radial l.e.s. are strictly monotone. We also study unique continuation properties and simplicity of zeros. Our theorems also apply to the scalar associated model, where our approach provides new results as well as alternative proofs of known facts. [ABSTRACT FROM AUTHOR]

Abstract The conformal method for the initial value formulation of GR is one of the most powerful and widely usedtools in both analytical studies and numerical simulations of solutions. As is well-known, it exploits a hidden spatial conformal symmetry that can be extracted from the equations. But why conformal symmetry? Motivated by these questions and by well-known obstacles to quantum gravity, I look for the most general geometrodynamical symmetries compatible with a reduced physical configuration space for metric gravity. I argue that they are indeed spatial conformal diffeomorphisms. The next question is: what sort of theories most naturally incorporate this symmetry? Demanding locality for an action for metric gravity compatible with these principles determines the allowed operators, both for the purely gravitational part as well as matter couplings. The symmetries guarantee that there are two gravitational propagating physical degrees of freedom, but no explicit refoliation-invariance. The simplest such system has a geometric interpretation as a geodesic model in infinite-dimensional conformal superspace. One example of solution to the equations of motion corresponds to a static Bianchi IX spatial ansatz. The unique coupling to electromagnetism forces the electromagnetic equations to be hyperbolic, enabling us to "build" a standard space–time causal structure. There are, however, deviations from the standard Maxwell equations when space–time anisotropies become too large. Regarding quantization, with the geometric interpretation and the lack of refoliation invariance, the path integral treatment of the symmetries becomes much less involved than the similar approaches to GR. The symmetries form an (infinite-dimensional) Lie algebra, and no BFV treatment is necessary. Moreover, one can use gauge-invariant variational principles for selecting the boundary conditions of the path integral. We find that the propagator around the homogeneous solution has up to 6th order spatial derivatives, giving it plausible regularization properties as in Horava–Lifschitz. [ABSTRACT FROM AUTHOR]

We propose an exponential spectral process (ESP) method for time discretization of spatial-temporal equations. The proposed ESP method uses explicit iterations at each time step, which allows us to use simple initializations at each iteration. This method has the capacity to obtain high accuracy (up to machine precision) with reasonably large time step sizes. Theoretically, the ESP method is proved to be unconditionally energy stable for arbitrary iteration steps for the case of using two spectral points. To demonstrate the advantages of the ESP method, we consider two applications which have stability difficulties in large time simulations. One of them is the Allen-Cahn equation with the symmetry breaking problem faced by most of the existing time discretizations, and the second one is about the complex Ginzburg-Landau equation which also suffers from large time instability. • The ESP method is able to have high accuracy and large time step sizes at same time. • A simple constant initialization derives the convergence of the ESP method. • The ESP method overcomes the symmetry breaking problem for Allen-Cahn equations. • The ESP method has comparable improvement in accuracy and efficiency for the CGLE. [ABSTRACT FROM AUTHOR]

Hazardous azimuthal thermoacoustic modes often occur in lean-premixed annular combustors. One practical control method is the application of perforated liners with bias flow. However, circumferentially uniform perforated liners are not competent to control them, because the perforated plate covered by a single azimuthally uniform backing cavity cannot provide sufficient acoustic pressure difference across the perforated plate to bring in adequate acoustic damping. In this work, we developed a three-dimensional thermoacoustic model to consider the effects of a novel passive control damper of the circumferentially segmented perforated liner. Emphasis is placed on the effects of the segmented liner on controlling the azimuthal modes. It is found that the azimuthal mode in the premixed annular combustor can be either plenum-dominant or combustor-dominant, with the frequency of the latter one larger than the former. The stabilities of the two kinds of modes behave differently with the change of time delay. The results of the lined combustor cases indicate that the circumferentially segmented perforated liner generally improves the control effectiveness compared with the circumferentially uniform perforated liner thanks to the rigid plates inserted in the backing cavity. At the locally inserted position, the rigid plate enforces a rigid boundary condition that changes the azimuthal pressure mode shape in the backing cavity. As a result, more acoustic damping through the pressure difference across the plate is produced. Symmetry-breaking effects are found if the segmentation is asymmetric or two-fold symmetric. The behavior of the splitting frequencies caused by the segmented liner is also studied, and the symmetric segmented liner turns out to be a better choice because the resulting azimuthal mode is degenerate with two equivalent suppressed growth rates. In order to further enhance the control effect, one significant effective measure is to increase the depth of the backing cavity. • A thermoacoustic model is presented for an azimuthally non-uniform perforated liner. • The impact of asymmetric liners on thermoacoustic instabilities has been studied. • A segmented perforated liner in an annular combustor facilitates symmetry breaking. • Split modes with different mode shapes have different stability behaviors. • A segmented perforated liner helps to control azimuthal modes in annular combustors. [ABSTRACT FROM AUTHOR]

Cognition most singularly involves choice that reduces uncertainty. Reduction of uncertainty implies the existence of an information source ‘dual’ to the cognitive process under study. However, information source uncertainty for path-dependent nonergodic systems cannot be described as a conventional Shannon ‘entropy’ since time averages are not ensemble averages. Nonetheless, the essential nature of information as a form of free energy allows study of nonergodic cognitive systems having complex dynamic topologies whose algebraic expression is in terms of directed homotopy groupoids rather than groups. This permits a significant extension of the Data Rate Theorem linking control and information theories via an analog to the spontaneous symmetry breaking arguments fundamental to modern physics. In addition, the identification of information as a form of free energy enables construction of dynamic empirical Onsager models in the gradient of a classic entropy that can be built from the Legendre transform of even path-dependent information source uncertainties. The methodology provides new analytic tools that should prove useful in understanding failure modes and their dynamics across a broad spectrum of cognitive phenomena, ranging from physiological processes at different scales and levels of organization to critical system automata and institutional economics. [ABSTRACT FROM AUTHOR]

This paper studies the Neumann problem of a nonlocal Allen–Cahn equation in an interval. A main result finds a symmetry breaking (secondary) bifurcation point on the bifurcation curve of solutions with odd-symmetry. Our proof is based on a level set analysis for the associated integral map. A method using the complete elliptic integrals proves the uniqueness of secondary bifurcation point. We also show some numerical simulations concerning the global bifurcation structure. [ABSTRACT FROM AUTHOR]

The Fock–Darwin system is analyzed from the point of view of its symmetry properties in the quantum and classical frameworks. The quantum Fock–Darwin system is known to have two sets of ladder operators, a fact which guarantees its solvability. We show that for rational values of the quotient of two relevant frequencies, this system is superintegrable, the quantum symmetries being responsible for the degeneracy of the energy levels. These symmetries are of higher order and close a polynomial algebra. In the classical case, the ladder operators are replaced by ladder functions and the symmetries by constants of motion. We also prove that the rational classical system is superintegrable and its trajectories are closed. The constants of motion are also generators of symmetry transformations in the phase space that have been integrated for some special cases. These transformations connect different trajectories with the same energy. The coherent states of the quantum superintegrable system are found and they reproduce the closed trajectories of the classical one. [ABSTRACT FROM AUTHOR]

We study the noncommutative corrections on the time-dependent Aharonov–Bohm effect when both the coordinate–coordinate and momentum–momentum noncommutativities are considered. On the noncommutative phase space, while the ordinary gauge symmetry can be kept by the Seiberg–Witten map, but the Lorentz symmetry is broken. Therefore nontrivial noncommutative corrections are expected. We find there are three kinds of noncommutative corrections in general: (1) ξ -dependent correction which comes from the noncommutativity among momentum operators; (2) momentum-dependent correction which is rooted in the nonlocal interactions in the noncommutative extended model; (3) momentum-independent correction which emerges become of the gauge invariant condition on the nonlocal interactions in the noncommutative model. We proposed two dimensionless quantities, which are based on the distributions of the measured phase shift with respect to the external magnetic field and to the cross section enclosed by the particle trajectory, to extract the noncommutative parameters. We find that stronger (weaker) magnetic field strength can give better bounds on the coordinate–coordinate (momentum–momentum) noncommutative parameter, and large parameter space region can be explored by the time-dependent AB effect. [ABSTRACT FROM AUTHOR]

We construct and study perturbative unitarity (i.e., ghost and tachyon analysis) of a 3 + 1 -dimensional noncompact Weyl–Einstein–Yang–Mills model. The model describes a local noncompact Weyl’s scale plus S U ( N ) phase invariant Higgs-like field,conformally coupled to a generic Weyl-invariant dynamical background. Here, the Higgs-like sector generates the Weyl’s conformal invariance of system. The action does not admit any dimensionful parameter and genuine presence of de Sitter vacuum spontaneously breaks the noncompact gauge symmetry in an analogous manner to the Standard Model Higgs mechanism. As to flat spacetime, the dimensionful parameter is generated within the dimensional transmutation in quantum field theories, and thus the symmetry is radiatively broken through the one-loop Effective Coleman–Weinberg potential. We show that the mere expectation of reducing to Einstein’s gravity in the broken phases forbids anti-de Sitter space to be its stable vacua. The model is unitary in de Sitter and flat vacua around which a massless graviton, N 2 − 1 massless scalar bosons, N massless Dirac fermions, N 2 − 1 Proca-type massive Abelian and non-Abelian vector bosons are generically propagated. [ABSTRACT FROM AUTHOR]

In this study, we investigate experimentally the linear and nonlinear thermoacoustic response of an annular combustor model with electroacoustic feedback to study the effects that various types of asymmetric modifications have on the system's eigenvalue degeneracy, and on its self-sustained oscillations. The baseline configuration has a nominal rotational symmetry of degree N = 12 and a mirror symmetry. First, we break the symmetry by modifying the downstream boundary condition, using lids with multiple holes. Some lids reduce the degree of rotational symmetry of the entire system, and may retain or break the mirror symmetry, depending on their orientation; other lids completely break the rotational symmetry but preserve the mirror symmetry. Second, for an axisymmetric downstream boundary condition, we break only the mirror symmetry by modelling for the first time flames for which the centre of heat release is not aligned with the burner axis. This leads the clockwise and counterclockwise spinning modes to have distinct frequencies and growth rates. The effect of all above-mentioned types of asymmetries is investigated over a broad range of flame gain values, which allows assessing how the sensitivity to asymmetries depends on the flame gain. The experimental results are then interpreted with the aid of low-order models and a 3D Helmholtz solver. This experimental study complements recent theoretical and numerical analyses on the effect of asymmetries on the nature of the thermoacoustic spectrum in annular combustors. • Thermoacoustic experiment in an annular combustor model with electroacoustic feedback. • Effect of asymmetric perturbations on the thermoacoustics of annular combustors. • Criteria to predict the (non)degeneracy of thermoacoustic modes tested experimentally. • Dependence of the sensitivity to asymmetries on the distance to the Hopf point. [ABSTRACT FROM AUTHOR]

We introduce a mathematical framework based on simple combinatorial arguments (Kernel Representation) that allows to deal successfully with spin glass problems, among others. Let Ω N be the space of configurations of an N − spins system, each spin having a finite set Ω of inner states, and let μ : Ω N → 0 , 1 be some probability measure. Here we give an argument to encode μ into a kernel function M : 0 , 1 2 → Ω , and use this notion to reinterpret the assumptions of the Replica Symmetry Breaking ansatz (RSB) of Parisi (1980) and Parisi et al. (1987) without using replicas, nor averaging on the disorder. • Mathematical formalism to handle spin-glass problems in constructive way. • Derivation of full-RSB ansatz without using replicas nor averaging on the disorder. • Constructive Cavity method with ROSt. • Alternative derivation for the Parisi ansatz of the SK model. [ABSTRACT FROM AUTHOR]

Cardy–Verlinde formula links the entropy of conformal symmetry field to the total energy and its Casimir energy in a D-dimensional space. To correct black hole thermodynamics, modified dispersion relation can be used which is proposed as a general feature of quantum gravity approaches. In this paper, the thermodynamics of Schwarzschild four-dimensional black hole is corrected using the modified dispersion relation for Fermions and Bosons. Finally, using modified thermodynamics of Schwarzschild four-dimensional black hole, generalization for Cardy–Verlinde formula is obtained. [ABSTRACT FROM AUTHOR]

We solve a 40-year-old open problem on depth optimality of sorting networks. In 1973, Donald E. Knuth detailed sorting networks of the smallest depth known for n ≤ 16 inputs, quoting optimality for n ≤ 8 (Volume 3 of “ The Art of Computer Programming ”). In 1989, Parberry proved optimality of networks with 9 ≤ n ≤ 10 inputs. We present a general technique for obtaining such results, proving optimality of the remaining open cases of 11 ≤ n ≤ 16 inputs. Exploiting symmetry, we construct a small set R n of two-layer networks such that: if there is a depth- k sorting network on n inputs, then there is one whose first layers are in R n . For each network in R n , we construct a propositional formula whose satisfiability is necessary for the existence of a depth- k sorting network. Using an off-the-shelf SAT solver we prove optimality of the sorting networks listed by Knuth. For n ≤ 10 inputs, our algorithm is orders of magnitude faster than prior ones. [ABSTRACT FROM AUTHOR]

Heteroclinic cycles involving two saddle-foci, where the saddle-foci share both invariant manifolds, occur persistently in some symmetric differential equations on the 3-dimensional sphere. We analyse the dynamics around this type of cycle in the case when trajectories near the two equilibria turn in the same direction around a 1-dimensional connection — the saddle-foci have the same chirality. When part of the symmetry is broken, the 2-dimensional invariant manifolds intersect transversely creating a heteroclinic network of Bykov cycles. We show that the proximity of symmetry creates heteroclinic tangencies that coexist with hyperbolic dynamics. There are n -pulse heteroclinic tangencies — trajectories that follow the original cycle n times around before they arrive at the other node. Each n -pulse heteroclinic tangency is accumulated by a sequence of ( n + 1 ) -pulse ones. This coexists with the suspension of horseshoes defined on an infinite set of disjoint strips, where the first return map is hyperbolic. We also show how, as the system approaches full symmetry, the suspended horseshoes are destroyed, creating regions with infinitely many attracting periodic solutions. [ABSTRACT FROM AUTHOR]

From the modified Maxwell theory coupled to gravity, we establish a possible scenario of the violation of the Lorentz symmetry and write an effective metric for the cosmic string spacetime. Then, we investigate the arising of an analogue of the Anandan quantum phase for a relativistic Dirac neutral particle with a permanent magnetic dipole moment in the cosmic string spacetime under Lorentz symmetry breaking effects. Besides, we analyse the influence of the effects of the Lorentz symmetry violation and the topology of the defect on the Aharonov–Casher geometric quantum phase in the nonrelativistic limit. [ABSTRACT FROM AUTHOR]

It is quite relevant the extension of decoration transformation for quantum spin models since most of the real materials could be well described by Heisenberg type models. Here we propose an exact quantum decoration transformation and also showing interesting properties such as the persistence of symmetry and the symmetry breaking during this transformation. Although the proposed transformation, in principle, cannot be used to map exactly a quantum spin lattice model into another quantum spin lattice model, since the operators are non-commutative. However, it is possible the mapping in the “classical” limit, establishing an equivalence between both quantum spin lattice models. To study the validity of this approach for quantum spin lattice model, we use the Zassenhaus formula, and we verify how the correction could influence the decoration transformation. But this correction could be useless to improve the quantum decoration transformation because it involves the second-nearest-neighbor and further nearest neighbor couplings, which leads into a cumbersome task to establish the equivalence between both lattice models. This correction also gives us valuable information about its contribution, for most of the Heisenberg type models, this correction could be irrelevant at least up to the third order term of Zassenhaus formula. This transformation is applied to a finite size Heisenberg chain, comparing with the exact numerical results, our result is consistent for weak xy-anisotropy coupling. We also apply to bond-alternating Ising–Heisenberg chain model, obtaining an accurate result in the limit of the quasi-Ising chain. [ABSTRACT FROM AUTHOR]

In this paper we study a diffuse interface generalized antiferromagnetic model. The functional describing the model contains a Modica–Mortola type local term and a nonlocal generalized antiferromagnetic term in competition. The competition between the two terms results in a frustrated system which is believed to lead to the emergence of a wide variety of patterns. The sharp interface limit of our model is considered in [23] and in [10]. In the discrete setting it has been previously studied in [16,17,19]. The model contains two parameters: τ and ε. The parameter τ represents the relative strength of the local term with respect to the nonlocal one, while the parameter ε describes the transition scale in the Modica–Mortola type term. If τ < 0 one has that the only minimizers of the functional are constant functions with values in { 0 , 1 }. In any dimension d ≥ 1 for small but positive τ and ε , it is conjectured that the minimizers are non-constant one-dimensional periodic functions. In this paper we are able to prove such a characterization of the minimizers, thus showing also the symmetry breaking in any dimension d > 1. [ABSTRACT FROM AUTHOR]

A review is given on the recent developments in the studies of nonlinear optical responses in quantum materials. Especially the broken inversion symmetry, and consequent second order processes in the electric field of light are focused. The geometric nature of the electronic states in solids plays crucial roles, and photo-current has distinct nature from the conventional transport current accordingly. [ABSTRACT FROM AUTHOR]

A review is given on the recent developments in the studies of nonlinear optical responses in quantum materials. Especially the broken inversion symmetry, and consequent second order processes in the electric field of light are focused. The geometric nature of the electronic states in solids plays crucial roles, and photo-current has distinct nature from the conventional transport current accordingly. [ABSTRACT FROM AUTHOR]

• A new complex structure-preserving method is proposed to investigate the dynamic response of the Space Manipulator Systems. • The effects of the parameters of the step-variable cross-section for the manipulator on the stable vibration amplitudes of which are reproduced. • The symmetry breaking of the dynamic model for the Space Manipulator Systems is presented. Decreasing the vibration amplitude of the manipulator in the Space Manipulator Systems (SMS) with a given mass limitation is an effective way to reduce the control costs for the SMS. In this paper, the dynamic response of the SMS modeled by a flexible hub-beam with a step-variable cross-section excited by an external torque is investigated employing a new complex structure-preserving method. The dynamic model of the flexible hub-beam with a step-variable cross-section is deduced firstly. Then, a new structure-preserving method connecting the symplectic precise integration method and the generalized multi-symplectic method is proposed, in which, the time-consuming iteration is avoided and the simulation time is saved. In the numerical simulations, several simple cases for the step-variable cross-section are considered and the stable vibration amplitudes of the flexible beam at the free end are reported. The main contribution of this work is proposing a structure-preserving method for the solving the coupling dynamic problems of the hub-beam system to provide some suggestions on the structure design for the SMS directly. [ABSTRACT FROM AUTHOR]

Since Padmanabhan and Teotonio-Sobrinho (2015) show the emergence of non-Abelian fusion rules in some examples of a class of Abelian models, but does not prove whether these rules also exist in other cases, the purpose of this paper is to present such proof emphasizing the importance of the existence of these rules. By the way, as the ground state of these models can be degenerated as a function of their algebra and, hence, they can support some symmetry-protected topological (SPT) phases, we prove that these non-Abelian fusion rules are always necessary for these SPT phase transitions to occur via a condensation mechanism or/and some global symmetry breaking. • There are a multitude of Abelian systems that support non-Abelian fusion rules. • The ground state degeneracy of these Abelian systems depends on the choice we make for their "gauge" group actions. • Non-Abelian fusion rules are necessary for (phase) transitions between some vacuum states when these "gauge" group actions define k -cycles of different degrees. [ABSTRACT FROM AUTHOR]

Hyperbolic reaction–diffusion equations have recently attracted attention both for their application to a variety of biological and chemical phenomena, and for their distinct features in terms of propagation speed and novel instabilities not present in classical two-species reaction–diffusion systems. We explore the onset of diffusive instabilities and resulting pattern formation for such systems. Starting with a rather general formulation of the problem, we obtain necessary and sufficient conditions for the Turing and wave instabilities in such systems, thereby classifying parameter spaces for which these diffusive instabilities occur. We find that the additional temporal terms do not strongly modify the Turing patterns which form or parameters which admit them, but only their regions of existence. This is in contrast to the case of additional space derivatives, where past work has shown that resulting patterned structures are sensitive to second-order cross-diffusion and first-order advection. We also show that additional temporal terms are necessary for the emergence of spatiotemporal patterns under the wave instability. We find that such wave instabilities exist for parameters which are mutually exclusive to those parameters leading to stationary Turing patterns. This implies that wave instabilities may occur in cases where the activator diffuses faster than the inhibitor, leading to routes to spatial symmetry breaking in reaction–diffusion systems which are distinct from the well studied Turing case. • Turing and wave instabilities are classified in hyperbolic reaction–diffusion systems. • The Turing instability mechanism is unchanged due to second-order time derivatives. • Cross-diffusion modifies the Turing instability and resulting Turing patterns. • Wave instabilities are shown to be possible in systems of two equations. • Second-order time derivatives strongly influence the wave instability. [ABSTRACT FROM AUTHOR]