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The statistical behavior of scalars passively advected by random flows exhibits intermittency in the form of anomalous multiscaling, in many ways similar to the patterns commonly observed in incompressible high-Reynolds fluids. This similarity suggests a generic dynamical mechanism underlying intermittency, though its specific nature remains unclear. Scalar turbulence is framed in a linear setting that points towards a zero-mode scenario connecting anomalous scaling to the presence of statistical conservation laws; the duality is fully substantiated within Kraichnan theory of random flows. However, extending the zero-mode scenario to nonlinear settings faces formidable technical challenges. Here, we revisit the scalar problem in the light of a hidden symmetry scenario introduced in recent deterministic turbulence studies addressing the Sabra shell model and the Navier–Stokes equations. Hidden symmetry uses a rescaling strategy based entirely on symmetry considerations, transforming the original dynamics into a rescaled (hidden) system; It yields the universality of Kolmogorov multipliers and ultimately identifies the scaling exponents as the eigenvalues of Perron-Frobenius operators. Considering a minimal shell model of scalar advection of the Kraichnan type that was previously studied by Biferale & Wirth, the present work extends the hidden symmetry approach to a stochastic setting, in order to explicitly contrast it with the zero-mode scenario. Our study indicates that the zero-mode and the multiplicative scenarios are intrinsically related. While the zero-mode approach solves the eigenvalue problem for p th order correlation functions, Perron-Frobenius (multiplicative) scenario defines a similar eigenvalue problem in terms of p th order measures. For systems of the Kraichnan type, the first approach provides a quantitative chararacterization of intermittency, while the second approach highlights the universal connection between the scalar case and a larger class of hydrodynamic models. [ABSTRACT FROM AUTHOR]

Synthesis of the originally assigned structure of the styryl tetrahydropyranol-dihydropyranone natural product cryptoconcatone H from C 2-symmetric (±)-1,8-nonadiene-4,6-diol is reported. Desymmetrization by Mitsunobu reaction with crotonic acid established the requisite inter-ring stereochemical relationship and was followed by a highly diastereoselective Re2 O7-catalyzed Prins cyclization with cinnamaldehyde to construct the 2,4,6- cis -tetrahydropyranol ring. Ring-closing metathesis resulted in formation of the dihydropyranone ring and completed the synthesis in three steps and 32% overall yield. The brevity of the synthesis is the result of the recognition of hidden, inter-ring symmetry in the target and the ensuing choice of an appropriately symmetric diol as our starting material. [ABSTRACT FROM AUTHOR]

W. D. Neumann and A. W. Reid conjectured that the figure-eight knot and the two dodecahedral knots are the only hyperbolic knots admitting hidden symmetries. We construct an n-component hyperbolic link whose complement admits hidden symmetries for any n ≥ 4. [ABSTRACT FROM AUTHOR]

Resistance ρ of an one-dimensional disordered system of length L has the log-normal distribution in the limit of large L. Parameters of this distribution depend on the Fermi level position, but are independent on the boundary conditions. However, the boundary conditions essentially affect the distribution of phases entering the transfer matrix and generally change the parameters of the evolution equation for the distribution P (ρ). This visible contradiction is resolved by the existence of the hidden symmetry, whose nature is revealed by the derivation of the equation for the stationary phase distribution and by analysis of its transformation properties. [ABSTRACT FROM AUTHOR]

MOLECULAR SYMMETRY AND DESYMMETRIZATION REACTIONS IN ORGANIC SYNTHESIS. Molecular Symmetry has found several applications in Organic Chemistry. The aim of this article is to showcase its relevance as a tool for synthetic planning. Natural Products and synthetic compounds with pharmaceutical applications were selected to highlight desymmetrization reactions that make use of organocatalysis, biocatalysis or by transition metals. The recognition of structural patterns in retrosynthetic analysis is discussed for compounds that exhibit explicit or hidden symmetry. Limitations of the aforementioned strategies are also described, as well as the untapped potential, challenges and frontiers yet to be explored regarding these concepts.

The asymmetric two-mode quantum Rabi model (tmQRM) breaks Z 2 -symmetry and also implies hidden symmetry. In this paper, the energy spectrum are analytically computed with Bogoliubov transformation and SU (1, 1) Lie algebra. The analytical results are in good agreement with those obtained by numerical calculation. We identify the condition for the bias parameter such that level crossings appear in the spectrum which are caused by the hidden symmetry. This condition is related not only to the field frequency, but also to the coupling strength between the field and qubit. The level crossing points could be determined by the pole energy and recurrence coefficients. This is tested and verified in the low energy levels. The generators of the hidden Z 2 -symmetry, commuting with the Hamiltonian, are derived within the Bogoliubov operator approach. The present results may provide more information to the hidden symmetry in the asymmetric tmQRM. [ABSTRACT FROM AUTHOR]

The hidden Z 2 symmetry of the asymmetric quantum Rabi model (AQRM) has recently been revealed via a systematic construction of the underlying symmetry operator. Based on the AQRM result, we propose an ansatz for the general form of the symmetry operators for AQRM-related models. Applying this ansatz we obtain the symmetry operator for three models: the anisotropic AQRM, the asymmetric Rabiâ€"Stark model (ARSM), and the anisotropic ARSM. [ABSTRACT FROM AUTHOR]

Herein we report the first enantioselective total synthesis of 3,5‐dimethylorsellinic acid‐derived meroterpenoids (−)‐berkeleyone A and its five congeners ((−)‐preaustinoids A, A1, B, B1, and B2) in 12–15 steps, starting from commercially available 2,4,6‐trihydroxybenzoic acid hydrate. Based upon the recognition of latent symmetry within D‐ring, our convergent synthesis features two critical reactions: 1) a symmetry‐breaking, diastereoselective dearomative alkylation to assemble the entire carbon core, and 2) a Sc(OTf)3‐mediated sequential Krapcho dealkoxycarbonylation/carbonyl α‐tert‐alkylation to forge the intricate bicyclo[3.3.1]nonane framework. We also conducted our preliminary biomimetic investigations and uncovered a series of rearrangements (α‐ketol, α‐hydroxyl‐β‐diketone, etc.) responsible for the biomimetic diversification of (−)‐berkeleyone A into its five preaustinoid congeners. [ABSTRACT FROM AUTHOR]

Transverse wrinkles are known to appear in thin rectangular elastic sheets when stretched in the long direction. Numerically computed bifurcation diagrams for extremely thin, highly stretched films indicate entire orbits of wrinkling solutions; cf. Healey, Li, and Cheng [J. Nonlinear Sci., 23 (2013), pp. 777-805]. These correspond to arbitrary phase shifts of the wrinkled pattern in the transverse direction. While such behavior is normally associated with problems in the presence of a continuous symmetry group, an unloaded rectangular sheet possesses only a finite symmetry group. In order to understand this phenomenon, we consider a simpler problem more amenable to analysis-a finite-length beam on a nonlinear softening foundation under axial compression. We first obtain asymptotic results via amplitude equations that are valid as a certain nondimensional beam length becomes sufficiently large. We deduce that any two phase shifts of a solution differ from one another by exponentially small terms in that length. We validate this observation with numerical computations, indicating the presence of solution orbits for sufficiently long beams. We refer to this as "hidden asymptotic symmetry". [ABSTRACT FROM AUTHOR]

Symmetries and tunability are of fundamental importance in wave scattering control, but symmetries are often obvious upon visual inspection, which constitutes a significant vulnerability of metamaterial wave devices to reverse-engineering risks. Here, it is theoretically and experimentally shown that a symmetry in the reduced basis of the "primary meta-atoms" that are directly connected to the outside world is sufficient; meanwhile, a suitable topology of non-local interactions between them, mediated by the internal "secondary" meta-atoms, can hide the symmetry from sight in the canonical basis. Covert symmetry-based scattering control in a cable-network metamaterial featuring a hidden parity ( P $\mathcal {P}$ ) symmetry in combination with hidden- P $\mathcal {P}$ -symmetry-preserving and hidden- P $\mathcal {P}$ -symmetry-breaking tuning mechanisms is experimentally demonstrated. Physical-layer security in wired communications is achieved using the domain-wise hidden P $\mathcal {P}$ -symmetry as a shared secret between the sender and the legitimate receiver. Within the approximation of negligible absorption, the first tuning of a complex scattering metamaterial without mirror symmetry to feature exceptional points (EPs) of PT $\mathcal {PT}$ -symmetric reflectionless states, as well as quasi-bound states in the continuum, is reported. These results are reproduced in metamaterials involving non-reciprocal interactions between meta-atoms, including the first observation of reflectionless EPs in a non-reciprocal system., (© 2023 Wiley-VCH GmbH.)

Quantum field theory L1 on spacetime X1 can be coupled to another quantum field theory L2 on a spacetime X2 via the third quantum field theory L12 living on X12=X1∩X2. We explore several such constructions with two- and four-dimensional X1,X2's and zero- and two-dimensional X12's, in the context of N=2 supersymmetry, non-perturbative Dyson-Schwinger equations, and BPS/CFT correspondence. The companion paper (Nekrasov, "BPS/CFT correspondence V: BPZ and KZ equations from qq-characters", 2017. arXiv:1711.11582 [hep-th]) will show that the BPZ and KZ equations of two-dimensional conformal field theory are obeyed by the half-BPS surface defects in quiver N=2 gauge theories. [ABSTRACT FROM AUTHOR]

In quantum mechanics, accidental degeneracy refers to energy degeneracy that occurs coincidentally, without any protection by symmetry. Here, we prove a theorem stating that any two-fold degeneracy (accidental or not) in a quantum system is protected by a novel hidden symmetry, which can be expressed by an antiunitary operator with its square being -1. In this sense, the so-called accidental degeneracy is not really accidental, and this actually implies a hidden antiunitary symmetry. [ABSTRACT FROM AUTHOR]

We investigate the bifurcation structure of the Kuramoto-Sivashinsky equation with homogeneous Dirichlet boundary conditions. Using hidden symmetry principles, based on an extended problem with periodic boundary conditions and symmetry, we show that the zero solution exhibits two kinds of pitchfork bifurcations: one that breaks the reflection symmetry of the system with Dirichlet boundary conditions and one that breaks a shift-reflect symmetry of the extended system. Using Lyapunov-Schmidt reduction, we show both to be supercritical. We extend the primary branches by means of numerical continuation, and show that they lose stability in pitchfork, transcritical or Hopf bifurcations. Tracking the corresponding secondary branches reveals an interval of the viscosity parameter in which up to four stable equilibria and time-periodic solutions coexist. Since the study of the extended problem is indispensable for the explanation of the bifurcation structure, the Kuramoto-Sivashinsky problem with Dirichlet boundary conditions provides an elegant manifestation of hidden symmetry. [ABSTRACT FROM AUTHOR]

Meso compounds represent a particular family of achiral molecules bearing elements of chirality. Their desymmetrization through enantioselective catalytic methods usually leads to elaborate chiral building blocks containing several stereogenic elements, which can be a very useful and elegant approach in the context of total synthesis. In the present review, the power of this strategy is illustrated through the different possibilities of catalytic enantioselective desymmetrization. From the combination of the hidden symmetry detection and the catalytic enantioselective transformations a new type of economy emerges: the economy of chiral reagents. 1 Introduction 1.1 What Is a Meso Compound? 1.2 Why Is the Catalytic Enantioselective Desymmetrization of Meso Compounds a Powerful Strategy in Total Synthesis? 1.3 Toward an Economy of Chiral Reagents 2 Enzymatic Desymmetrization 2.1 (-)-Sceptrin (Baran, 2006) 2.2 cis-Solamin (Stark, 2006) 2.3 Crocacin C (2010, Bressy/Pons) 3 Metallocatalyzed Desymmetrization 3.1 Quadrigemine C (2002, Overman) 3.2 (+)-Homochelidonine (2007, Lautens) 3.3 (-)-Cyanthiwigin F (2008, Stoltz) 3.4 [5]-Ladderanoic Acid (2016, Gonzalez-Martinez/Boxer/Burns) 4 Organocatalyzed Desymmetrization 4.1 (+)-Hirsutene (2008, List) 4.2 Alstoscholarines (2011, Neuville/Zhu) 4.3 (-)-Diospongin A (2015, Chuzel/Bressy) 5 Conclusion. [ABSTRACT FROM AUTHOR]

It is shown that each element s in the normalizer of the automorphism group Aut(G) of a simple graph G with labeled vertex set V is an Aut(G) invariant isomorphism between G and the graph obtained from G by the s permutation of V—i.e., s is a hidden permutation symmetry of G. A simple example illustrates the theory and the applied notion of system robustness for reconfiguration under symmetry constraint (RUSC) is introduced.

In this work, we prove that the integrating factors can be used as a reduction method. Analytical solutions of the Jaulent–Miodek (JM) equation are obtained using integrating factors as an extension of a recent work where, through hidden symmetries, the JM was reduced to ordinary differential equations (ODEs). Some of these ODEs had no quadrature. We here derive several new solutions for these non-solvable ODEs.

Symmetry is one of the most fundamental concepts of all nature. We discuss the general theory of symmetry based on theory of semigroups. This includes and extends the theory of groups, which is by now broadly used as an algebraic tool for studying many types of symmetries observed by people. However, symmetry groups are only a particular case of more general, symmetry semigroups. The latter are in turn an instance of an even more general groupoids of symmetry, which may further be a topic of special consideration. [ABSTRACT FROM AUTHOR]

We study properties of nonlinear supersymmetry algebras realized in the one-dimensional quantum mechanics of matrix systems. Supercharges of these algebras are differential operators of a finite order in derivatives. In special cases, there exist independent supercharges realizing an (extended) supersymmetry of the same super-Hamiltonian. The extended supersymmetry generates hidden symmetries of the super-Hamiltonian. Such symmetries have been found in models with (2×2)-matrix potentials. [ABSTRACT FROM AUTHOR]

We study relations between functions on the cotangent bundle of a spacetime which are constants of motion for geodesics and functions on the odd-dimensional phase space conserved by the Reeb vector fields of geometrical structures generated by the metric and an electromagnetic field. [ABSTRACT FROM AUTHOR]

The phase space of general relativistic test particle is defined as the 1-jet space of motions. A Lorentzian metric defines the canonical contact structure on the odd-dimensional phase space. In the paper we study infinitesimal symmetries of the gravitational contact phase structure which are not generated by spacetime infinitesimal symmetries, i.e. they are hidden symmetries. We prove that Killing multivector fields admit hidden symmetries of the gravitational contact phase structure and we give the explicit description of such hidden symmetries. [ABSTRACT FROM AUTHOR]

Typical heat engines exhibit a kind of homotypy: the heat exchanges between a cyclic heat engine and its two heat reservoirs abide by the same function type; the forward and backward flows of an autonomous heat engine also conform to the same function type. This homotypy mathematically reflects in the existence of hidden symmetries for heat engines. The heat exchanges between the cyclic heat engine and its two reservoirs are dual under the joint transformation of parity inversion and time-reversal operation. Similarly, the forward and backward flows in the autonomous heat engine are also dual under the parity inversion. With the consideration of these hidden symmetries, we derive a generic nonlinear constitutive relation up to the quadratic order for tight-coupling cyclic heat engines and that for tight-coupling autonomous heat engines, respectively.

It is shown that each element σ in the normalizer of the automorphism group Aut(G) of a simple graph G with labeled vertex set V is an Aut(G) invariant isomorphism between G and the graph obtained from G by the σ permutation of V--i.e., σ is a hidden permutation symmetry of G. A simple example illustrates the theory and the applied notion of system robustness for reconfiguration under symmetry constraint (RUSC) is introduced. [ABSTRACT FROM AUTHOR]

A personal view is presented of developments in the non-linear mechanics of structural buckling over more or less the past four decades. The era has been one of unprecedented change and development in the world of science as a whole, and this is acknowledged by describing ways in which the field has interacted strongly with other areas of study, including mathematics, non-linear dynamics and chaos and structural geology. A framework is provided through two key conferences that have emerged as pivotal moments in time when ideas seemed to take shape in a collective sense rather than just with individuals. Throughout most of the paper, the buckling of the axially compressed cylindrical shell is used to illustrate key features, including the breaking of hidden symmetry, localization and snaking solutions leading to a minimum energy periodic state that accords with a revitalized Maxwell stability criterion. The paper closes with some thoughts to the future, including the modelling of layered structures in geology and potential uses of modern composites. [ABSTRACT FROM PUBLISHER]

The higher order symmetries are investigated in a covariant Hamiltonian formulation. The covariant phase-space approach is extended to include the presence of external gauge fields and scalar potentials. The special role of the Killing–Yano tensors is pointed out. Some nontrivial examples involving Runge–Lenz type conserved quantities are explicitly worked out. [ABSTRACT FROM AUTHOR]

Abstract: The Jarlskog Invariant of the neutrino mapping matrix is calculated based on a phenomenological model which relates the smallness of light lepton masses and (of ) with the smallness of T violation. For small T violating phase in the lepton sector, is proportional to , but and are proportional to . This leads to . Assuming , we find , consistent with the present experimental data. [Copyright &y& Elsevier]

Abstract: We propose that the smallness of the light quark masses is related to the smallness of the T (i.e. CP) violation in hadronic weak interactions. Accordingly, for each of the two quark sectors (“upper” and “lower”) we construct a 3×3 mass matrix in a bases of unobserved quark states, such that the “upper” and “lower” basis states correspond exactly via the W ± transitions in the weak interaction. In the zeroth approximation of our formulation, we assume T conservation by making all matrix elements real. In addition, we impose a “hidden symmetry” (invariance under simultaneous translations of all three basis quark states in each sector), which ensures a zero mass eigenstate in each sector. Next, we simultaneously break the hidden symmetry and T invariance by introducing a phase factor eiχ in the interaction for each sector. The Jarlskog invariant J CKM, as well as the light quark masses are evaluated in terms of the parameters of the model. Comparing formulas, we find that most unknown factors drop out, resulting in a simple relation with , to leading order in χ and m s /m b , with A, λ the Wolfenstein parameters. (Because of the large top quark mass, the contribution from upper quark sector can be neglected.) Setting J CKM =3.08×10−5, m b =4.7GeV (1s mass), m s =95MeV, A =0.818, and λ =0.227, we find , consistent with the accepted value m d =3−7MeV. We make a parallel proposal for the lepton sectors. With the hidden symmetry and in the approximation of T invariance, both the masses of e and ν 1 are zero. The neutrino-mapping matrix V ν is shown to be of the same Harrison–Perkins–Scott form which is in agreement with experiments. We also examine the correction due to T violation, and evaluate the corresponding Jarlskog invariant . [Copyright &y& Elsevier]

Abstract: In this article it is shown that, Sommerfeld''s coefficients for excitonic transitions in quantum wells are determined only with the principle quantum number within the framework of two-dimensional Coulomb potential. This is a consequence of hidden symmetry of two-dimensional Coulomb problem, conditioned by the existence of two-dimensional analog of the Runge–Lentz vector. For the narrow gap semiconductor quantum well with the non-parabolic dispersion law of electron and hole in the two-band Kane model it is shown that two-dimensional excitonic states are described in the frames of an analog of Klein–Gordon equation with the two-dimensional Coulomb potential. The non-stability of the ground state of the two-dimensional Kane''s exciton is shown. [Copyright &y& Elsevier]

The influence of hidden symmetry on two-dimensional excitonic states in semiconductor quantum wells is investigated. It is shown that excitonic states in quantum wells, with the parabolic dispersion law for the electron and hole, and Sommerfeld's coefficients for excitonic transitions are determined only with the principle quantum number within the framework of two-dimensional Coulomb potential. This is a result of hidden symmetry of two-dimensional Coulomb problem, conditioned by the existence of two-dimensional analog of the Runge-Lentz vector. For the narrow gap semiconductor quantum well with the non-parabolic dispersion law of electron and hole, in the two-band Kane model, it is shown that two-dimensional excitonic states are described in the frames of analog of the Klein-Gordon equation with the two-dimensional Coulomb potential. Non-stability of the ground state of the two-dimensional Kane's exciton investigated. [ABSTRACT FROM AUTHOR]

The exact solutions of a one-dimensional Schrödinger equation with a harmonic oscillator plus an inverse square potential are obtained. The ladder operators constructed directly from the normalized wavefunctions are found to satisfy a su(1, 1) algebra. Another hidden symmetry is used to explore the relations between the eigenvalues and eigenfunctions by substituting x → -ix. The vibrational partition function Z is calculated exactly to study thermodynamic functions such as the vibrational mean energy U, specific heat C, free energy F, and entropy S. It is both interesting and surprising to find that both vibrational specific heat C and entropy S are independent of the potential strength α. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007 [ABSTRACT FROM AUTHOR]

Optical traps and lattices provide a new opportunity to study strongly correlated high spin systems with cold atoms. In this article, we review the recent progress on the hidden symmetry properties in the simplest high spin fermionic systems with hyperfine spin F=3/2, which may be realized with atoms of 132Cs, 9Be, 135Ba, 137Ba, and 201Hg. A generic SO(5) or isomorphically, Sp(4) symmetry is proved in such systems with the s-wave scattering interactions in optical traps, or with the on-site Hubbard interactions in optical lattices. Various important features from this high symmetry are studied in the Fermi liquid theory, the mean field phase diagram, and the sign problem in quantum Monte-Carlo simulations. In the s-wave quintet Cooper pairing phase, the half-quantum vortex exhibits the global analogue of the Alice string and non-Abelian Cheshire charge properties in gauge theories. The existence of the quartetting phase, a four-fermion counterpart of the Cooper pairing phase, and its competition with other orders are studied in one-dimensional spin-3/2 systems. We also show that counter-intuitively quantum fluctuations in spin-3/2 magnetic systems are even stronger than those in spin-1/2 systems. [ABSTRACT FROM AUTHOR]

On the basis of the generalization of the theorem about K-odd operators (K is the Dirac's operator), certain linear combination is constructed, which appears to commute with the Dirac Hamiltonian for Coulomb field. This operator coincides with the Johnson and Lippmann operator and is closely connected to the familiar Laplace–Runge–Lenz vector. Our approach guarantees not only derivation of Johnson–Lippmann operator, but simultaneously commutativity with the Dirac Hamiltonian. [ABSTRACT FROM AUTHOR]

Abstract: Basic mechanisms for the buckling of a thin cylindrical shell under torsional loading are reviewed from a post-buckling perspective. Deflections are considered so far into the large-deflection range that the shell is allowed to fold to a flat two-dimensional form, in a mechanism reminiscent of a deployable structure. Critical and initial post-buckling effects are explored through concepts of energy minimization and hidden symmetries. For comparisons with the final large-deflection folded shape, a truss element program is employed. It is shown that, as buckling develops, the mode shape must change to accommodate both the symmetry-breaking aspects of the predominately inwards deflection, and the rotation of peak and valley lines of the buckle pattern necessary to accommodate the geometry of the final folded shape. [Copyright &y& Elsevier]

Based on the ansatz to the wave functions, the quasi-exact solutions of the 2D Schrödinger equation with some anharmonic potentials are reviewed and analyzed if admitting restrictions on the parameters of the potential and the angular momentum m. These potentials are taken as the screened Coulomb potential V(r)=a/r+b/(r+λ), the singular one-fraction power one V(r)=ar-1/2+br-3/2 and the singular two-fraction one V(r)=ar2/3+br-2/3+cr-4/3. The latter one is found that the hidden symmetry exists if substituting r→-ir. It will reverse the signs of E and c of quantum system, leaving the remaining parameters invariant. [ABSTRACT FROM AUTHOR]

The magnetoluminescence spectra of symmetric quantum wells containing an electron gas show an abrupt transition from Landau level behavior (i.e. a linear shift of energy with field) to quadratic (exciton-like) behavior at the field at which the lowest Landau level is just filled, so that the electron filling factor νe = 2. Professor Rashba's insight was that this transition is a natural (although surprising, since the high field limit is by no means reached) consequence of the hidden symmetry which is known to hold in two-dimensional systems at high fields. This interpretation also accounts for the appearance ofmagnetoplasmon, “shake-up” and cyclotron satellites when νe > 2. [ABSTRACT FROM AUTHOR]

We study the nonlinear (polynomial, N-fold,…) supersymmetry algebra in one-dimensional QM. Its structure is determined by the type of conjugation operation (Hermitian conjugation or transposition) and described with the help of the Hamiltonian projection on the zero-mode subspace of supercharges. We show that the SUSY algebra with transposition symmetry is always polynomial in the Hamiltonian if supercharges represent differential operators of finite order. The appearance of the extended SUSY with several (complex or real) supercharges is analyzed in details and it is established that no more than two independent supercharges may generate a nonlinear superalgebra which can be appropriately specified as N=2 SUSY. In this case we find a nontrivial hidden symmetry operator and rephrase it as a nonlinear function of the Hamiltonian on the physical state space. The full N=2 nonlinear SUSY algebra includes “central charges” both polynomial and nonpolynomial (due to a symmetry operator) in the Hamiltonian. [Copyright &y& Elsevier]

We set up a singularity-theoretic framework for classifying one-parameter steady-state bifurcations with hidden symmetries. This framework also permits a non-trivial linearization at the bifurcation point. Many problems can be reduced to this situation; for instance, the bifurcation of steady or periodic solutions to certain elliptic partial differential equations with Neumann or Dirichlet boundary conditions. We formulate an appropriate equivalence relation with its associated tangent spaces, so that the usual methods of singularity theory become applicable. We also present an alternative method for computing those matrix-valued germs that appear in the equivalence relations employed in the classification of equivariant bifurcation problems. This result is motivated by hidden symmetries appearing in a class of partial differential equations defined on an $N$-dimensional rectangle under Neumann boundary conditions.1991 Mathematics Subject Classification: 58C27, 58F14. [ABSTRACT FROM AUTHOR]