contractions of interest (any contraction with a cost that is superlinear in input/output size). State-of-the-art sequential and parallel libraries for handling cyclic **group** **symmetry**, both in specific physical applications and in domain-agnostic settings, typically iterate over appro- priate blocks within a block-sparse tensor format [1, 4, 13, 16, 18–21, 28,32, 34, 39]. The use of explicit looping (over possibly small blocks) makes it difficult to reach theoretical peak compute performance. Parallelization of block-wise contractions can be done manually or via specialized software [13,16,19–21,28,32, 34]. However, such parallelization is challeng- ing in the distributed-memory setting, where block-wise multiplication might (depending on contraction and initial tensor data distribution) require communication/redistribution of tensor data. We introduce a general transformation of cyclic **group** symmetric tensors, irreducible representation alignment, which allows all contractions between such tensors to be trans- formed into a single large dense tensor contraction with optimal cost, in which the two input reduced forms as well as the output are indexed by a new auxiliary index. This transformation provides three advantages:

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are smooth maps between manifolds. An r-parameter Lie **group** carries the structure of an r-dimensional manifold. A Lie **group** can also be considered as a topological **group** (i.e., a **group** endowed with a topology with respect to which the **group** operations are continuous) that is also a manifold.

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A common application o f **group** theory is in the classification o f quantum states in terms o f irreducible representations. In particular, it is customary to classify vibrational states [14, 15] in terms o f the underlying point or space **group**. A complication arises for overtones o f degenerate harmonic vibrations which are classifiable in terms o f a much higher order **symmetry** **group**. The **symmetry** adaptation o f the corresponding functions to the underlying point **group** **symmetry** which is restored by addition o f anharmonic terms is solved using the methods developed here. The treatment is particularly interesting because it involves the harmonic oscillator [16, 17] and angular momentum [18, 19, 2 0 , 2 1 ], which are the two basic models o f quantum mechanics, as

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In this paper we shall show that the **symmetry** of a DAE sample of ferroics is the same as that of the **symmetry** of a composite. As a consequence, the concept of latent **symmetry**, which has been shown to explain unexpected symmetries in the point **group** **symmetry** of composites of geometrically shaped objects [11, 12], could also explain unexpected symmetries found in DAE samples of crystalline ferroics. Finally, two theorems by Vlachavas on the **symmetry** of composites are shown to be invalid.

Different lattice structures of the crystal showed the different properties, and crystal geometry can be classi- fied by crystal **symmetry** **group** [13]. With reference to the research methods of **symmetry** **group**, researches on unit cell geometry of braided materials can be summa- rized. According to the space **group** and the space **group**, **symmetry** operations can be deduced from a large num- ber of new three-dimensional yarn crossover methods [12]. A new geometry structure of 3D braided material was deduced by using space point S6 corresponding to **symmetry** operations and the **symmetry** of space **group** P3, and author researched its processing and properties [14]. Under the condition of satisfying all space point **group** C 3 3 [13] **symmetry** operations, a new unit-cell

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Graph theory is a branch of discrete mathematics concerned with relation, between objects. From the point of the graph theory, all organic molecular structures can be drawn as graphs in which atoms and bonds are represented by vertices and edges, respectively. By **symmetry** we mean the automorphism **group** **symmetry** of a graph which is a subgroup of its vertex permutation **group**. The **symmetry** of a graph, also called topological **symmetry**, which need not be the isomorphic to the molecular point **group** **symmetry**. However, it dose represent the maximal **symmetry** of its topological structure.

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If the theory is developed in the globally symmetric space-time, the number and form of the conservation laws are determined by the first Nöether theorem. The number of con- servation laws coincide with the number of Lie **group** **symmetry** invariants. All the known di ff erential conservation laws (energy, momentum, impulse ets.) are given by the first Nöther theorem. The invariant values destined for the experimental testing are obtained by integra- tion of the di ff erential conservation laws arising from the above-mentioned first theorem and its generalization on the Lie’s groups of the global internal **symmetry**. The transformations of such groups depend on the number parameters. This generalization was obtained by me in 1967 [10, 11].

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A crystal is not solely a geometric entity specified by a space **group** **symmetry**. It is made of atoms, ions or molecules that mutually interact in a number of ways or react to surrounding fields, local and external. It accordingly exhibits a variety of static and dynamical physical properties that find transcriptions into a wide range of physical quantities, namely thermodynamic variables featuring a macrostate, wavefunctions or fields describing quantal or classical microscopic states or excitations, response functions to experimental probes, . . .. A macroscopic strain, an electric polarization, . . ., a wavefunction of an individual atom, ion or molecule in its crystal environment, an exciton wavefunction, . . ., a magnetic moment configuration field, an electric quadrupole tensor field, . . ., a magneto-electric susceptibility tensor, a dynamical neutron scattering function, . . . are a few among typical examples. It is explicit from this enumeration that a physical quantity may be of diverse nature, tensorial or spinorial possibly function of time and space, so prone to change under time or space transforms. When these belong to a **symmetry** **group** it is of crucial interest to establish under which conditions a particular quantity might be invariant, or to specify the orbit of all its transforms in the instance where it inherently would be of lower **symmetry**, 8 or else to gauge the fully or partially symmetric

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The highest occupied and lowest unoccupied sets of MOs in transition metal complexes have very predictable forms and occur in doubly and triply degenerate sets, because of the high point **group** **symmetry** (Oh). In this exercise we look at the **symmetry** of the orbitals that result from the interaction of the Cr p and d shell orbitals with the CO ligand s and p orbitals. In each example studied, the objective is to determine the **symmetry** of the orbitals and to relate this to the bonding interactions involved (e.g. which s vs p orbitals of the ligand match the central atom symmetries).

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For modulated structures, the concept of point **symmetry** is more difficult to grasp because one must determine the macroscopic invariance of physical quantities that are related to objects spatially modulated at the atomic scale. A very good discussion on the subject has been given by Dvorák and co-workers [7]. Essentially, the definition given earlier in eq. (3.2) is still valid but needs to be applied on the different components of the standing waves that describe the magnetically ordered state: The rotational part of a **symmetry** operator is a valid operator of the point **group** if and only if the only effect of applying the latter results in de-phasing the initial waves by k t 0 where t 0 is a direct-lattice vector.

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(flavour) groups. In section 3 we specialise to 2 + 1 flavours and give quark mass expansions to second order for the pseudoscalar and vector meson octets and baryon octet and decuplet. (The relation of this expansion to chiral perturbation theory is discussed later in section 5.) In section 4 we extend the formalism to the partially quenched case (when the valence quarks of a hadron do not have to have the same mass as the sea quarks). This is potentially useful as the same expansion coefficients occur, which could allow a cheaper determination of them. We then turn to more specific lattice considerations in sections 6, 7 with emphasis on clover fermions (i.e. non-chiral fermions) used here. This is followed by section 8, which first gives numerical results for the constant singlet quark mass results used here. Flavour singlet quantities prove to be a good way of defining the scale and the consistency of some choices is discussed. We also investigate possible finite size effects. Finally in section 9 the numerical results for the hadron mass spectrum are presented in the form of a series of ‘fan’ plots where the various masses fan out from their common value at the symmetric point. Our conclusions are given in section 10. Several Appendices provide some **group** theory background for this article, discuss the action used here and give tables of the hadron masses found. Mostly we restrict ourselves to the constant surface. However, in a few sec- tions, we also consider variations in m R

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Since there is an approximate **symmetry** between matter and antimatter, one might expect to have approximately the same quantity of matter and antimatter in the world. Moreover, the violation of the **symmetry** under the charge conjugation operator C is not sufficient to explain the absence of antimatter. As shown by Sakharov in 1967, CP **symmetry** should also be violated. During years, astronomers have searched the heaven in their quest for an anti-Universe. The quest stopped in 1964 when an experiment on kaons revealed CP **symmetry** violation. Matter and antimatter are not formed in the same proportion. During the Big Bang, matter and antimatter annihilated, some matter remained, but no antimatter. Then began the quest for the reason of this **symmetry** violation. In 1972, Makoto Kobayashi and Toshihide Maskawa exploited an idea of Nicola Cabibbo to obtain the answer. Their theory implied increasing the number of quarks, which had already raised to 4 thanks to Glashow, Iliopoulos, & Maiani [32]. Completing the miracle of multiplication of quarks, Kobayashi and Maskawa added two other quarks (bottom and top) whose existence was experimentally confirmed a few years later.

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From theoretical DFT calculations Perez-Mato et al. (2008) concluded that a direct transition from the parent phase I4/ mmm to the monoclinic phase with **symmetry** B1a1, as generally assumed, was very unlikely, as the energy landscape did not favour the necessary simultaneous condensation of the three primary distortion modes associated with different irreps that are present in the monoclinic phase. Such ‘avalanche’ first-order transitions are known for two order parameters, but not for three, and in the present case there is no indication in the mode couplings that would promote such an exceptional scenario. The existence of an intermediate phase where one of the order parameters previously condenses solves this puzzle, and the second transition between the P4/mbm and B1a1 symmetries is now reduced to the simultaneous activation of two additional primary distortion modes, together with a change of direction (change from a 2k to a 1k distortion) of the order parameter associated with the intermediate phase. This second phase transition is then analogous to the one in Aurivillius compounds with perovskite blocks with only two layers (Perez-Mato et al., 2004), the difference being the additional previous condensation of the X 2

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The topological quantum number Q of a superconducting or chiral insulating wire counts the number of stable bound states at the end points. We determine Q from the matrix r of reflection amplitudes from one of the ends, generalizing the known result in the absence of time-reversal and chiral **symmetry** to all five topologically nontrivial **symmetry** classes. The formula takes the form of the determinant, Pfaffian, or matrix signature of r, depending on whether r is a real matrix, a real antisymmetric matrix, or a Hermitian matrix. We apply this formula to calculate the topological quantum number of N coupled dimerized polymer chains, including the effects of disorder in the hopping constants. The scattering theory relates a topological phase transition to a conductance peak, of quantized height and with a universal (**symmetry** class independent) line shape. Two peaks which merge are annihilated in the superconducting **symmetry** classes, while they reinforce each other in the chiral **symmetry** classes.

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Point **group** is a type of **group** in chemistry, which is a collection of **symmetry** elements possessed by a shape or form which all pass through one point in space. The stereographic projection is used to visualize the **symmetry** operations of point groups. On the other hand, **group** theory is the study about an algebraic structure known as a **group** in mathematics. This research relates point groups of order at most eight with groups in **group** theory. In this research, isomorphisms, matrix representations, conjugacy classes and conjugacy class graph of point groups of order at most eight are found. The isomorphism between point groups and groups in **group** theory are obtained by mapping the elements of the groups and by showing that the isomorphism properties are fulfilled. Then, matrix representations of point groups are found based on the multiplication table. The conjugacy classes and conjugacy class graph of point groups of order at most eight are then obtained. From this research, it is shown that point groups C1, C3, C 5 and C 7 are isomorphic to the groups Z1, Z3, Z 5 and Z 7 respectively.

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, in which the Au III ion is in a slightly distorted square-planar coordination environment. In the cation, the nitro **group** is essentially coplanar with the imidazole ring, as indicated by an O. . .N—C C torsion angle of 0.2 (4) , while the hydroxyethyl

We consider the time dependent neutron diffusion equation for one energy **group** in cylinder coordinates, assuming translational **symmetry** along the cylinder axis. This problem for a specific energy **group** is solved analytically applying the Hankel transform in the radial coordinate r. Our special interest rests in the build-up factor for a time dependent linear neutron source aligned with the cylinder axis, which in the limit of zero decay constant reproduces also the static case. The new approach to solve the diffusion equation by integral transform technique is presented and results for sev- eral parameter sets and truncation in the solution for the flux and build-up factor are shown and found to be compatible to those of literature [1,2].

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Abstract. We present a new framework unifying interactions in nature by introduc- ing mirror fermions, explaining the hierarchy between the weak scale and the coupling uniﬁcation scale, which is found to lie close to Planck energies. A novel process lead- ing to the emergence of **symmetry** is proposed, which not only reduces the arbitrariness of the scenario proposed but is also followed by signiﬁcant cosmological implications. Phenomenology includes the probability of detection of mirror fermions via the corre- sponding composite bosonic states and the relevant quantum corrections at the LHC.

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However, growth is a mechanical process, and whereas the role of morphogens is indisputable, they cannot be expected to act alone [15–18]. Simply put, genes and GRNs are not everything. Such a reductionist view ne- glects the important fact that living organisms, too, function in an environment where the laws of physics are as valid as in the non-living world, so they are under the influence of the same basic architectural principles (described by the fundamental laws of physics) that shape the non-living natural world [19]. Thus, tracing everything back to molecules while searching for the ul- timate causes of biological processes can be misleading because this kind of approach omits other factors without which the molecular systems could not work properly. Genes constitute the plan for building the body, but molecules can only act in an appropriate set of physical circumstances. Since morphogens act in a physical en- tity – the developing tissue –, tissue morphogenesis should be regarded as a process which is under genetic control but which also occurs by the action of mechan- ical forces [15–26]. Mechanical forces, in contrast to local effects, may also act globally, which can be im- portant while organs develop to achieve their correct sizes and shapes [16]. Since cells are interconnected, cell proliferation and shape changes potentially affect the whole tissue or organ, inducing mechanical stress, even when they are local phenomena [16]. Moreover, the physical environment may not only function as the matrix in which the biological processes occur, but can also be the guiding factor which drives the molecules and cells to act both during the formation of a given tissue and during the functioning of the anatomical structure (see also [15, 20, 23, 27–29]). I suggest that in the case of most symmetrical biological structures this is exactly what happens. **Symmetry** is a response in the geometry of the “ living matter ” to physical forces.

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Over the past decade or so, there has been huge progress in the study of topological phases, particularly for non-interacting systems. Without inter-particle interactions, bosons simply form Bose-Einstein condensate, hence there is no nontrivial bosonic topological phases in this regime. On the other hand, free electrons form band structures. The band strctures can have nontrivial topology, leading to nontrivial topological phases of free fermions. Examples include the famous topological insulators and topological superconductors in two and three spatial dimensions [24– 32]. Topological insulators have a gapped bulk, and gapless boundary states of Dirac fermions. They have the interesting property that the nontrivial gapless boundary states are preserved as long as the charge conservation and time-reversal symmetries are preserved. In other words, the topological insulators are protected by these two symmetries. If either **symmetry** is broken – either spontaneously or explicitly – the boundary states will be gapped, and the topological insulators become ordinary band insulators. This makes them perfect examples of SPT phases. Analogously, topological superconductors can also be understood as SPT phases, with their nontrivial gapless boundary states of Majorana fermions protected by time-reversal **symmetry**. Building on these examples, people gradually gained more understanding of the free fermionic phases, which ultimately led to an exhaustive classification of topological phases of free fermions in all spatial dimensions, known as the ten-fold way [33, 34].

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