Form Of Symmetry Research Articles

On the classification of scalar evolution equations with non-constant separant

The ‘separant’ of the evolution equation ut = F, where F is some differentiable function of the derivatives of u up to order m, is the partial derivative where . As an integrability test, we use the formal symmetry method of Mikhailov–Shabat–Sokolov, which is based on the existence of a recursion operator as a formal series. The solvability of its coefficients in the class of local functions gives a sequence of conservation laws, called the ‘conserved densities’ . We apply this method to the classification of scalar evolution equations of orders , for which and are non-trivial, i.e. they are not total derivatives and is not linear in its highest order derivative. We obtain the ‘top level’ parts of these equations and their ‘top dependencies’ with respect to the ‘level grading’, that we defined in a previous paper, as a grading on the algebra of polynomials generated by the derivatives ub+i, over the ring of functions of . In this setting b and i are called ‘base’ and ‘level’, respectively. We solve the conserved density conditions to show that if depends on then, these equations are level homogeneous polynomials in , . Furthermore, we prove that if is non-trivial, then , with while if is trivial, then , where and α, β, γ, λ and μ are functions of . We show that the equations that we obtain form commuting flows and we construct their recursion operators that are respectively of orders 2 and 6 for non-trivial and trivial respectively. Omitting lower order dependencies, we show that equations with non-trivial and b = 3 are symmetries of the ‘essentially non-linear third order equation’; for trivial , the equations with b = 5 are symmetries of a non-quasilinear fifth order equation obtained in previous work, while for b = 3, 4 they are symmetries of quasilinear fifth order equations.

Encapsulation of Two Potassium Cations in Preyssler-Type Phosphotungstates: Preparation, Structural Characterization, Thermal Stability, Activity as an Acid Catalyst, and HAADF-STEM Images.

Dipotassium cation (K+)-encapsulated Preyssler-type phosphotungstate, [P5W30O110K2]13-, was prepared by heating monobismuth (Bi3+)-encapsulated Preyssler-type phosphotungstate, [P5W30O110Bi(H2O)]12-, in acetate buffer in the presence of an excess amount of potassium cations. Characterization of the isolated potassium salt, K13[P5W30O110K2] (1a), and its acid form, H13[P5W30O110K2] (1b), by single crystal X-ray structure analysis, 31P and 183W nuclear magnetic resonance (NMR), Fourier transform infrared (FT-IR) spectroscopy, cyclic voltammetry (CV), high-resolution electrospray ionization mass spectroscopy (HR-ESI-MS), and elemental analysis revealed that two potassium cations are encapsulated in the Preyssler-type phosphotungstate molecule with formal D5h symmetry, which is the first example of a Preyssler-type compound with two encapsulated cations. Incorporation of two potassium cations enhances the thermal stability of the potassium salt, and the acid form shows catalytic activity for hydration of ethyl acetate. Packing of the Preyssler-type molecules was observed by high-resolution high-angle annular dark-field scanning transmission electron microscopy (HAADF-STEM).

Voting and vagueness

How to handle vagueness? One way is to introduce the machinery of acceptable sharpenings, and reinterpret truth as truth-in-all-sharpenings (supervaluationism) or truth-in-some-sharpenings (subvaluationism). A major selling point has been the conservativism of the resulting systems with respect to classical theoremhood and inference. Supervaluationism and subvaluationism possess interesting formal symmetries, a fact that has been used to argue for the subvaluationist approach. However, the philosophical motivation behind each is a different matter. Subvaluationism comes with a standard story (due to Stanislaw Jaśkowski) that is difficult to sign up to. In this paper, I make use of a variant of Putnam’s well-known idea of linguistic deference, and some results in voting theory, to answer this criticism of subvaluationism. The acceptability intuitions of each member of a linguistic community amount to their voting for one or more acceptable sharpenings, with truth then characterised as truth-in-a-(contextually-determined)-sufficiency-of-sharpenings. This produces a family of logical systems that are close relations of subvaluationism, share its conservatism results, yet have stronger philosophical foundations in the workings of externalist content.

Integrability test for evolutionary lattice equations of higher order

A generalized summation by parts algorithm is presented for solving of difference equations of the form Tm(y)−ay=b with variable coefficients, where T denotes the shift operator. Solvability of equations of this type with respect to the coefficients of formal symmetry (or formal recursion operator) provides a convenient integrability test for evolutionary differential-difference equations u,t=f(u−m,…,um).

Generalized symmetries of an \U0001d4dd = 1 supersymmetric Boiti–Leon–Manna–Pempinelli system**Project supported by the National Natural Science Foundation of China (Grant Nos. 11275123, 11175092, 11475052, and 11435005), the Shanghai Knowledge Service Platform for Trustworthy Internet of Things, China (Grant No. ZF1213), and the Talent Fund and K CWong Magna Fund in Ningbo University, China.

The formal series symmetry approach (FSSA), a quite powerful and straightforward method to establish infinitely many generalized symmetries of classical integrable systems, has been successfully extended in the supersymmetric framework to explore series of infinitely many generalized symmetries for supersymmetric systems. Taking the \U0001d4a9 = 1 supersymmetric Boiti–Leon–Manna–Pempinelli system as a concrete example, it is shown that the application of the extended FSSA to this supersymmetric system leads to a set of infinitely many generalized symmetries with an arbitrary function f (t). Some interesting special cases of symmetry algebras are presented, including a limit case f (t) = 1 related to the commutativity of higher order generalized symmetries.

Full propagation-vector star antiferromagnetic order in quantum spin trimer system Ca3CuNi2(PO4)4

We show that the antiferromagnetic structure in the quantum spin trimer system Ca3CuNi2(PO4)4 is based on both arms of propagation vector k star of the paramagnetic space group C2/c. The structure is generated by a symmetric direction of the order parameter of two-dimensional irreducible representation of C2/c with one active magnetic mode and corresponds to the Shubnikov magnetic space group Ca2/c. We reveal the relation between representation analysis in the propagation vector formalism and Shubnikov symmetry. These types of multi-k structures are extremely rarely observed experimentally. To further prove the specific magnetic structure we have performed the calculations of the spin expectation values in the isolated Ni2+–Cu2+–Ni2+ trimer with realistic Hamiltonian. The calculated spin values 〈SNi〉 = 0.9 and 〈SCu〉 = 0.3 are within 10% accuracy in agreement with the experiment, providing a strong complementary argument in favor of a multi-arm magnetic structure.

Multi-k magnetic order in Ca3CuNi2(PO4)4: irrep approach and Shubnikov symmetry

The multi-k magnetic structures with propagation vectors k being the arms of the propagation vector star rarely can be justified experimentally. We show that the antiferromagnetic structure in the low dimensional quantum spin trimer system Ca3CuNi2(PO4)4 is based on the full star of propagation vector k=[1/2,1/2,0] of the paramagnetic space group C2/c. The relation between representation analysis in the propagation vector formalism and Shubnikov magnetic space group (MSG) symmetry is examined in details. A symmetry restrictive MSG that excellently fits the experimental data can be constructed only with the use of the full star. The magnetic structure is further supported by the calculations of the spin expectation values of the isolated Ni-Cu-Ni trimer with realistic Hamiltonian.

Intrinsic Hamiltonian symmetry group structure analysis for orthogonal partial symmetry decomposition

In this paper the partial symmetry method in the intrinsic nuclear frame is analyzed to obtain physical partial symmetries from the so-called formal symmetries of intrinsic sub-Hamiltonians. Derivation of the physical symmetries is illustrated by an example that shows in a detailed way the symmetry structure of the corresponding energy spectra of the total Hamiltonian.

Infinitely many generalized symmetries and Painlevé analysis of a (2 + 1)-dimensional Burgers system

Infinitely many generalized symmetries of a coupled (2 + 1)-dimensional Burgers system are obtained by means of the formal series symmetry approach. It is found that the generalized symmetries constitute a closed infinite-dimensional Lie algebra. Three interesting special cases are presented, including a closed infinite-dimensional Lie algebra and a Kac–Moody–Virasoro-type Lie symmetry algebra. From the first one of the positive flow, a new integrable coupled system of the modified Korteweg–de Vries equation and the potential Boiti–Leon–Manna–Pempinelli equation is constructed. In addition, it is demonstrated that the coupled Burgers system can pass the Painlevé test.

ABOUT THE PHYSICAL REALITY OF ``MAXWELL'S DISPLACEMENT CURRENT'' IN CLASSICAL ELECTRODYNAMICS

This work aims to provide a physical interpretation of the \Maxwell's displacement and generalize the use of the derivative function of the electrical ∞ux \d=dt in the magnetic fleld calculation. The innovative contribution of this study is a mathematical model to describe the origin of magnetic fleld as a variation of electric ∞ux. By this approach, it follows that only the function \d=dt can generate a \magnetic tension: this leads to an interesting uniflcation of electrical phenomena. Displacement current and conduction current are interpreted as complementary aspects of the same phenomena. It is shown how the use of the \d=dt allows the recovery of a formal symmetry in Maxwell's relations of electromagnetic induction law and circuit magnetic law. Included also is a generalized expression of the function \d=dt.

Routes towards an abstract thermodynamics in the late nineteenth century

Two different traditions of research emerged from Rudolf Clausius’s version of thermodynamics. While James Clerk Maxwell and Ludwig Boltzmann pursued the integration of thermodynamics with the kinetic theory of gases, others relied on a macroscopic and more abstract approach that set aside specific mechanical models. This second approach blossomed in about two decades in different countries of Europe and in the United States. Francois Massieu, Josiah Willard Gibbs, Hermann Helmholtz, and then Pierre Duhem explored the connections between the contents of thermodynamics and the formal structures of analytical mechanics. Others like the young Max Planck and Arthur von Oettingen pursued a sort of formal symmetry between thermal and mechanical variables. In the British Isles, Joseph John Thomson developed a dynamical approach to physics and chemistry, making use of the tools of abstract mechanics without excluding microscopic motions. Some developments were logically interconnected, as it was for Massieu’s, Gibbs’s, Helmholtz’s, and Duhem’s, even though they occurred in a largely independent manner. Duhem put forward the most original and most systematic reinterpretation of thermodynamics, which involved a bold upgrading of analytical mechanics and a bold mathematical unification of physics and chemistry. A strong commitment to unification was one of the hallmarks of all these theoretical researches.

NONLOCAL SYMMETRIES, COMPACTON EQUATIONS, AND INTEGRABILITY

We review the theory of nonlocal symmetries of nonlinear partial differential equations and, as examples, we present infinite-dimensional Lie algebras of nonlocal symmetries of the Fokas–Qiao and Kaup–Kupershmidt equations. Then, we consider nonlocal symmetries of a family which contains the Korteweg–de Vries (KdV) and (a subclass of) the Rosenau–Hyman compacton-bearing K(m, n) equations. We find that the only member of the family which possesses nonlocal symmetries (of a kind specified in Sec. 3 below) is precisely the KdV equation. We take this fact as an indication that the K(m, n) equations are not integrable in general, and we use the formal symmetry approach of Shabat to check this claim: we prove that the only integrable cases of the full K(m, n) family are the KdV and modified KdV equations.

On a discrete symmetry of the Bremsstrahlung function in $ \mathcal{N} $ = 4 SYM

We consider the quark anti-quark potential on the three sphere in planar N = 4 SYM and the associated vacuum potential in the near BPS limit with L units of R-charge. The associated Bremsstrahlung function BL has been recently computed analytically by means of the Thermodynamical Bethe Ansatz. We discuss it at strong coupling by computing it at large but nite L. We provide strong support to a special symmetry of the Bremsstrahlung function under the formal discrete Z2 symmetry L ! 1 L. In this context, it is the counterpart of the reciprocity invariance discovered in the past in the spectrum of various gauge invariant composite operators. The Z2 symmetry has remarkable consequences in the scaling limit where L is taken to be large with xed ratio to the 't Hooft coupling. This limit organizes in inverse powers of the coupling and resembles the semiclassical expansion of the dual string theory which is indeed known to capture the leading classical term. We show that the various higher-order contributions to the Bremsstrahlung function obey several constraints and, in particular, the next-to-leading term, formally associated with the string one-loop correction, is completely determined by the classical contribution. The large L limit at strong coupling is also discussed.

3,7,11-Tris{4-[(1R,3S,4S)-neomenth-yl-oxy]phen-yl}tri[1,2,4]triazolo[4,3-a:4',3'-c:4'',3''-e][1,3,5]triazine-chloro-form-ethanol (1/1/1).

The title compound, C54H69N9O3·CHCl3·C2H5OH, was prepared by a threefold nucleophilic substitution of p-neomenthyloxyphenyl­tetra­zole on cyanuric chloride followed by threefold cyclo­elimination of nitro­gen and ring closure. The central tris­triazolotriazine is roughly planar with a maximum deviation of 0.089 (7) Å but the adjacent benzene rings are twisted out of this plane. N—C—C—C torsion angles of −80.2 (9), 159.3 (7) and 50.6 (10)° destroy the formal C3 symmetry. Cavities are found between the phen­oxy residues: one is occupied by a chloro­form mol­ecule, another by ethanol forming a hydrogen bond to a triazole ring while two isopropyl groups point into the third void. One methyl group and the chloro­frm mol­ecule are disorderd and were refined using a split model.

Hamilton formalism and noether symmetry for mechanico—electrical systems with fractional derivatives

This paper presents extensions to the traditional calculus of variations for mechanico—electrical systems containing fractional derivatives. The Euler—Lagrange equations and the Hamilton formalism of the mechanico—electrical systems with fractional derivatives are established. The definition and the criteria for the fractional generalized Noether quasi-symmetry are presented. Furthermore, the fractional Noether theorem and conseved quantities of the systems are obtained by virtue of the invariance of the Hamiltonian action under the infinitesimal transformations. An example is presented to illustrate the application of the results.

On the Classification of Fifth Order Quasi-linear Non-constant Separant Scalar Evolution Equations of the KdV-Type

Fifth order, quasi-linear, non-constant separant evolution equations are of the form \(u_{t} = A(\partial^{5}u/\partial x^{5}) + \tilde{B}\), where A and \(\tilde{B}\) are functions of x , t , u and of the derivatives of u with respect to x up to order 4. We use the existence of a “formal symmetry”, hence the existence of “canonical conservation laws” ρ ( i ) , i = -1,...,5 as an integrability test. We define an evolution equation to be of the KdV-Type, if all odd numbered canonical conserved densities are nontrivial. We prove that fifth order, quasi-linear, non-constant separant evolution equations of KdV type are polynomial in the function a = A 1/5 ; a = (α u 3 2 + β u 3 + γ) -1/2 , where α, β, and γ are functions of x , t , u and of the derivatives of u with respect to x up to order 2. We determine the u 2 dependency of a in terms of P = 4αγ- β 2 > 0 and we give an explicit solution, showing that there are integrable fifth order non-polynomial evolution equations.

Synthesis and Characterization of Chiral Guanidines und Guanidinium Salts Derived from 1-Phenylethylamine

The synthesis of two new chiral guanidines 5 and 12 and derived guanidinium salts 6, 11, 13 -15 with one and three N-(1-phenylethyl) substituents is described. In both cases, the well-precedented, reliable route via chloro-formamidines was taken. Since direct attachment of the N-methyl-N-(1- phenethyl)-amino group failed, the two-step protocol - introduction of the primary 1-phenethylamino group first followed by N-methylation - was employed. Crystal structures and NMR data reveal, that the sterically highly congested “tris” salt - with formal C3 symmetry, albeit unsymmetrical in the crystal - constitutes an intriguing structure with two rotamers present in solution.

A new (2+1)-dimensional generalization of the modified Korteweg-deVries equation and its infinitely many symmetries

Starting from the nonlinear version of bilinear negative Kadomtsev-Petviashvili system, a new (2+1)-dimensional generalization of the modified Korteweg-deVries equation is obtained. For this new system, a set of infinitely many generalized symmetries is found by applying a formal series symmetry approach. And these symmetries constitute a closed infinite-dimensional Lie algebra which is a generalization of w∞ type algebra.

Rhythms, Poetic and Political: The Case of Elizabeth Barrett Browning

Rhythms, Poetic and Political:The Case of Elizabeth Barrett Browning Caroline Levine (bio) How should one go about addressing the politics of prosody? In recent years scholars of Victorian poetry have tended to choose among three alternatives. The first is what we might call the "reflective" model. In this school of thought, poetic meter points us to the temporal patterns of social life, mirroring or enacting a historically specific shaping of experiential time. Herbert F. Tucker, for example, reads the meter of Barrett Browning's "Cry of the Children" as revealing the uncomfortable disjuncture between the embodied time of human life and the jolting experience of factory labor. "Nauseatingly ill-proportioned to human measure," Barrett Browning's "stop-and-start versification mimics the strain and clatter of steam-driven machinery."1 Similarly, Ivan Kreilkamp suggests that Victorian poetry could be read in a newly "super-charged realm of electricity, speed, heat, and light," offering up metrical experiences of social "shock … mobility, acceleration, discontinuity, the transitory, the elusive and the ephemeral."2 One of Kreilkamp's most vivid examples comes from Aurora Leigh, a passage when Aurora flees "southward in the roar of steam" (VII.396). Barrett Browning emerges here as ambivalent, torn between excitement at the "velocity and eroticized power" of the train and horror at its "frightening bath of sound in which individual voice and thought are submerged and overwhelmed" (p. 608). The reflective model, then, is a way of reading metrical forms for their mirroring of lived temporalities, allowing us access to the power and pain of contemporary modes of production.3 The second general way of reading the politics of meter, which we might call the "expressive" model, is one that takes prosody as a purposeful expression of political positions or convictions. Jason Rudy, for example, draws attention to the difference between the metrical patterns of Victorian ballads and those found in Spasmodic poetry. Rudy explains that ballads in the 1850s were simple, widely accessible national songs which, by popular demand, kept to strictly predictable patterns, "offer[ing] insistent regularity [End Page 235] as a model for cultural stability."4 "Through their formal regularity, ballads insist on 'law' rather than 'anarchy,'" he writes (pp. 591-592). And yet, Rudy explains, the same period saw the Spasmodic poets opting for violent, abrupt, jerking rhythms, deliberately repellent to conservative readers. Metrical forms thus express positions on a political spectrum that range from an embrace of unity and stability to radically unsettling conflict and lawlessness. Perhaps the most venerable tradition of reading the politics of literary form—and one that still wields considerable persuasive power—is the Marxist model. Here, critics read literary forms as struggling to contain or repress the reality of social conflicts and contradictions. Garrett Stewart writes, for example, that "in contriving to secure its formal symmetries," a literary text may offer "a formal recompense for totality's absence from any other life sphere."5 Thus the resolutions achieved by a text's prosodic patterns appear as vain but compelling attempts to cover over the terrifying impossibility of achieving real social resolutions. In this light, Tennyson's "sad mechanic exercise" (In Memoriam V.7) is indeed numbing; the comforts of metrical regularity dull our access to the real pain of loss, though rather than the private or intimate loss of a beloved friend they numb us to the very possibility of collective social relationship. In practice, these three models are not mutually exclusive. Isobel Armstrong's magisterial Victorian Poetry moves in astonishingly complex ways among the ideological, the reflective, and the expressive. A poet such as Tennyson emerges in Armstrong's study as both highly self-conscious and deeply ideological, fully intending to resolve social contradictions through myth and yet, in his brilliant lyrics, also exposing the material conditions of social life and the temporal patterns of industrial labor. In the process, the attempt to represent both the subjective and objective conditions of alienated labor—carefully reflecting the social—ultimately undermines both ideological and expressive readings. Reading "The Lotos-Eaters," for example, Armstrong argues that Tennyson's "slumberous cadences betray a real anxiety, as they struggle to represent a materialism which actually makes consciousness more and more...

Continuous Symmetry of C60 Fullerene and Its Derivatives

Conventionally, the I(h) symmetry of fullerene C(60) is accepted, which is supported by numerous calculations. However, this conclusion results from the consideration of the molecule electron system, of its odd electrons in particular, in a closed-shell approximation without taking the electron spin into account. Passing to the open-shell approximation has led to both the energy and the symmetry lowering up to C(i). Seemingly contradicting to a high-symmetry pattern of experimental recording, particularly concerning the molecule electronic spectra, the finding is considered in this Article from the continuous symmetry viewpoint. Exploiting continuous symmetry measure and continuous symmetry level approaches, it was shown that formal C(i) symmetry of the molecule is by 99.99% I(h). A similar continuous symmetry analysis of the fullerene monoderivatives gives a reasonable explanation of a large variety of their optical spectra patterns within the framework of the same C(1) formal symmetry exhibiting a strong stability of the C(60) skeleton. TOC color pictures present chemical portrait of C(60) in terms of atomic chemical susceptibility (Sheka, E. Fullerenes: Nanochemistry, Nanomagnetism, Nanomedicine, Nanophotonics; CRC Press: Taylor and Francis Group, Boca Raton, 2011).