Search Results (85)

Let X be a complex projective variety embedded in a complex projective space. The dimensions of the secant varieties of X have an expected value, and it is important to know if they are equal or at least near to this expected value. Blomenhofer and Casarotti proved important results on the embeddings of G-varieties, G being an algebraic group, embedded in the projectivations of an irreducible G-representation, proving that no proper secant variety is a cone. In this paper, we give other conditions which assure that no proper secant varieties of X are a cone, e.g., that X is G-homogeneous. We consider the Segre product of two varieties with the product action and the case of toric varieties. We present conceptual tests for it, and discuss the information we obtained from certain linear projections of X. For the Segre–Veronese embeddings of ${\mathbb{P}}^n\times {\mathbb{P}}^n$ with respect to forms of bidegree $(1,d)$, our results are related to the simultaneous rank of degree d forms in $n+1$ variables. Full article

It has been shown that at the present stage of the evolution of the universe, cosmological acceleration is an inevitable kinematical consequence of quantum theory in semiclassical approximation. Quantum theory does not involve such classical concepts as Minkowski or de Sitter spaces. In classical theory, when choosing Minkowski space, a vacuum catastrophe occurs, while when choosing de Sitter space, the value of the cosmological constant can be arbitrary. On the contrary, in quantum theory, there is no uncertainties in view of the following: (1) the de Sitter algebra is the most general ten-dimensional Lie algebra; (2) the Poincare algebra is a special degenerate case of the de Sitter algebra in the limit $R\to \infty$ where R is the contraction parameter for the transition from the de Sitter to the Poincare algebra and R has nothing to do with the radius of de Sitter space; (3) R is fundamental to the same extent as c and ℏ: c is the contraction parameter for the transition from the Poincare to the Galilean algebra and ℏ is the contraction parameter for the transition from quantum to classical theory; (4) as a consequence, the question (why the quantities (c, ℏ, R) have the values which they actually have) does not arise. The solution to the problem of cosmological acceleration follows on from the results of irreducible representations of the de Sitter algebra. This solution is free of uncertainties and does not involve dark energy, quintessence, and other exotic mechanisms, the physical meaning of which is a mystery. Full article

This work addresses closed-form expressions for the distributions $P\left(M\right)$ of the magnetic quantum numbers M and $Q\left(J\right)$ of total angular momentum J for non-equivalent fermions in single-j orbits. Such quantities play an important role in both nuclear and atomic physics, through the shell models. Using irreducible representations of the rotation group, different kinds of formulas are presented, involving multinomial coefficients, generalized Pascal triangle coefficients, or hypergeometric functions. Special cases are discussed, and the connections between $P\left(M\right)$ (and therefore $Q\left(J\right)$) and mathematical functions such as elementary symmetric, cyclotomic, and Jacobi polynomials are outlined. Full article

While either a spin or point-group adaptation is straightforward when considered independently, the standard technique for factoring isotropic spin Hamiltonians by the total spin S and the irreducible representation $\mathsf{\Gamma }$ of the point group is limited by the complexity of the transformations between different coupling schemes that are related in terms of their site permutations. To overcome these challenges, we apply projection operators directly to uncoupled basis states, enabling the simultaneous treatment of spin and point-group symmetry without the need for recoupling transformations. This provides a simple and efficient approach for the exact diagonalization of isotropic spin models, which we illustrate, with applications in Heisenberg spin rings and polyhedra, including systems that are computationally inaccessible with conventional coupling techniques. Full article

As shown by Dyson in his famous paper “Missed Opportunities”, it follows, even from purely mathematical considerations, that quantum Poincare symmetry is a special degenerate case of quantum de Sitter symmetries. Thus, the usual explanation of why, in particle physics, Poincare symmetry works with a very high accuracy is as follows. A theory in de Sitter space becomes a theory in Minkowski space when the radius of de Sitter space is very high. However, the answer to this question must be given only in terms of quantum concepts, while de Sitter and Minkowski spaces are purely classical concepts. Quantum Poincare symmetry is a good approximate symmetry if the eigenvalues of the representation operators $M_{4\mu }$ of the anti-de Sitter algebra are much greater than the eigenvalues of the operators $M_{\mu \nu }$ ($\mu ,\nu =0,1,2,3$). We explicitly show that this is the case in the Flato–Fronsdal approach, where elementary particles in standard theory are bound states of two Dirac singletons. Full article

Layered orthorhombic single crystals of EuYCuTe₃ are synthesized using the ampoule method from the elemental precursors taken in the ratio of 1 Eu:1 Y:1 Cu:3 Te by heating up to 1120 K with an excess of CsI as flux. The orthorhombic structure of EuYCuTe₃ is established, and structural parameters are obtained using X-ray diffraction. At ambient conditions, the sample crystallizes in the space group Pnma with the unit cell parameters a = 11.2730(7) Å, b = 4.3214(3) Å, c = 14.3271(9) Å. The structure is composed of vertex-connected [CuTe₄]⁷⁻ tetrahedra, which form chains along the [010] direction, and of edge-connected [YTe₆]⁹⁻ octahedra, which form layers parallel to the (010) plane. The Eu²⁺ cations are found in a capped trigonal prismatic coordination of Te²⁻ anions. The structural phase transition from the α to the β phase is discovered upon heating the sample to 323 K, which comes accompanied with a decrease of [CuTe₄]⁷⁻ tetrahedral distortion. The symmetry of the high-temperature phase is established as ordered in the space group Cmcm (a = 4.3231(3) Å, b = 14.3328(9) Å, c = 11.2843(7) Å). The nature and microscopic mechanism of the phase transition is discussed. By cooling it down below 3 K, the soft ferromagnetic properties of EuYCuTe₃ are discovered. The correlation of the ferromagnetic transition temperature in the series of chalcogenides EuYCuCh₃ (Ch = S, Se, Te) with the ionic radius of the chalcogenide anion is established. The structural dynamical elastic properties of α- and β-EuYCuTe₃ were calculated within the ab initio approach. The vibrational mode frequencies and decomposition on irreducible representations, as well as the degree of ion involvement in each mode, were determined. The calculations reveal an imaginary mode in the Y-point of the Brillouin zone in the high symmetry β-EuYCuTe₃ phase. This finding explains the nature of structural reconstruction in EuYCuTe₃ crystal as a second-order phase transition induced by soft mode condensation at the edge of the Brillouin zone. The exfoliation of a single layer is simulated theoretically. The exfoliation energy is estimated, and the dynamical properties of EuYCuTe₃ single layers are studied. Full article

It is shown that it is reasonable to use Galois fields, including those obtained by algebraic extensions, to describe the position of a point in a discrete Cartesian coordinate system in many cases. This approach is applicable to any problem in which the number of elements (e.g., pixels) into which the considered fragment of the plane is dissected is finite. In particular, it is obviously applicable to the processing of the vast majority of digital images actually encountered in practice. The representation of coordinates using Galois fields of the form GF(p²) is a discrete analog of the representation of coordinates in the plane through a complex variable. It is shown that two different types of algebraic extensions can be used simultaneously to represent transformations of discrete Cartesian coordinates described through Galois fields. One corresponds to the classical scheme, which uses irreducible algebraic equations. The second type proposed in this report involves the use of a formal additional solution of some equation, which has a usual solution. The correctness of this approach is justified through the representation of the elements obtained by the algebraic expansion of the second type by matrices defined over the basic Galois field. It is shown that the proposed approach is the basis for the development of new methods of information protection, designed to control groups of UAVs in the zone of direct radio visibility. The algebraic basis of such methods is the solution of systems of equations written in terms of finite algebraic structures. Full article

The proxy-SU(3) symmetry predicts, in a parameter-free way, the collective deformation variables $\beta$ and $\gamma$ in even–even atomic nuclei away from closed shells based on the highest weight irreducible representations (irreps) of SU(3) in the relevant proton and neutron shells, which are the most symmetric irreps allowed by the Pauli principle and the short-range nature of the nucleon–nucleon interactions. The special cases in which the use of the next-highest-weight irrep of SU(3) becomes necessary are pointed out, and numerical results are given for several regions of the nuclear chart, which can be used as input for irrep-mixing calculations. Full article

As shown in our publications, quantum theory based on a finite ring of characteristic p (FQT) is more general than standard quantum theory (SQT) because the latter is a degenerate case of the former in the formal limit $p\to \infty$. One of the main differences between SQT and FQT is the following. In SQT, elementary objects are described by irreducible representations (IRs) of a symmetry algebra in which energies are either only positive or only negative and there are no IRs where there are states with different signs of energy. In the first case, objects are called particles, and in the second antiparticles. As a consequence, in SQT it is possible to introduce conserved quantum numbers (electric charge, baryon number, etc.) so that particles and antiparticles differ in the signs of these numbers. However, in FQT, all IRs necessarily contain states with both signs of energy. The symmetry in FQT is higher than the symmetry in SQT because one IR in FQT splits into two IRs in SQT with positive and negative energies at $p\to \infty$. Consequently, most fundamental quantum theory will not contain the concepts of particle–antiparticle and additive quantum numbers. These concepts are only good approximations at present since at this stage of the universe the value p is very large but it was not so large at earlier stages. The above properties of IRs in SQT and FQT have been discussed in our publications with detailed technical proofs. The purpose of this paper is to consider models where these properties can be derived in a much simpler way. Full article

The argument of this article is threefold. First, the article argues that from its rise in the sixteenth century to our own time, the advancement of modern physics as mathematical-experimental science has been defined by the invention of new mathematical structures. Second, the article argues that quantum theory, especially following quantum mechanics, gives this thesis a radically new meaning by virtue of the following two features: on the one hand, quantum phenomena are defined as essentially different from those found in all previous physics by purely physical features; and on the other, quantum mechanics and quantum field theory are defined by purely mathematical postulates, which connect them to quantum phenomena strictly in terms of probabilities, without, as in all previous physics, representing or otherwise relating to how these phenomena physically come about. While these two features may appear discordant, if not inconsistent, I argue that they are in accord with each other, at least in certain interpretations (including the one adopted here), designated as “reality without realism”, RWR, interpretations. This argument also allows this article to offer a new perspective on a thorny problem of the relationships between continuity and discontinuity in quantum physics. In particular, rather than being concerned only with the discreteness and continuity of quantum objects or phenomena, quantum mechanics and quantum field theory relate their continuous mathematics to the irreducibly discrete quantum phenomena in terms of probabilistic predictions while, at least in RWR interpretations, precluding a representation or even conception of how these phenomena come about. This subject is rarely, if ever, discussed apart from previous work by the present author. It is, however, given a new dimension in this article which introduces, as one of its main contributions, a new principle: the mathematical complexity principle. Full article

We propose a new approach to extend the notion of commutator and Lie algebra to algebras with ternary multiplication laws. Our approach is based on the ternary associativity of the first and second kinds. We propose a ternary commutator, which is a linear combination of six triple products (all permutations of three elements). The coefficients of this linear combination are the cube roots of unity. We find an identity for the ternary commutator that holds due to the ternary associativity of either the first or second kind. The form of this identity is determined by the permutations of the general affine group $GA(1,5)\subset S_5$. We consider this identity as a ternary analog of the Jacobi identity. Based on the results obtained, we introduce the concept of a ternary Lie algebra at cube roots of unity and provide examples of such algebras constructed using ternary multiplications of rectangular and three-dimensional matrices. We also highlight the connection between the structure constants of a ternary Lie algebra with three generators and an irreducible representation of the rotation group. The classification of two-dimensional ternary Lie algebras at cube roots of unity is proposed. Full article

We study point-line configurations through the lens of projective geometry and matroid theory. Our focus is on their realization spaces, where we introduce the concepts of liftable and quasi-liftable configurations, exploring cases in which an n-tuple of collinear points can be lifted to a nondegenerate realization of a point-line configuration. We show that forest configurations are liftable and characterize the realization space of liftable configurations as the solution set of certain linear systems of equations. Moreover, we study the Zariski closure of the realization spaces of liftable and quasi-liftable configurations, known as matroid varieties, and establish their irreducibility. Additionally, we compute an irreducible decomposition for their corresponding circuit varieties. Applying these liftability properties, we present a procedure to generate some of the defining equations of the associated matroid varieties. As corollaries, we provide a geometric representation for the defining equations of two specific examples: the quadrilateral set and the $3\times 4$ grid. While the polynomials for the latter were previously computed using specialized algorithms tailored for this configuration, the geometric interpretation of these generators was missing. We compute a minimal generating set for the corresponding ideals. Full article

The whimsical Las Vegas/Monte Carlo cubic dice are generalized to construct the combinatorial problem of enumerating all n-dimensional hypercube dice and dice of other shapes that exhibit cubic, icosahedral, and higher symmetries. By utilizing powerful generating function techniques for various irreducible representations, we derive the combinatorial enumerations of all possible dice in n-dimensional space with hyperoctahedral symmetries. Likewise, a number of shapes that exhibit icosahedral symmetries such as a truncated dodecahedron and a truncated icosahedron are considered for the combinatorial problem of dice enumerations with the corresponding shapes. We consider several dice with cubic symmetries such as the truncated octahedron, dodecahedron, and Rubik’s cube shapes. It is shown that all enumerated dice are chiral, and we provide the counts of chiral pairs of dice in the n-dimensional space. During the combinatorial enumeration, it was discovered that two different shapes of dice exist with the same chiral pair count culminating to the novel concept of isochiral polyhedra. The combinatorial problem of dice enumeration is generalized to multi-coloring partitions. Applications to chirality in n-dimension, molecular clusters, zeolites, mesoporous materials, cryptography, and biology are also pointed out. Applications to the nonlinear n-dimensional hypercube and other dicey encryptions are exemplified with romantic, clandestine messages: “I love U” and “V Elope at 2”. Full article

The Standard Model is an up-to-date theory that best summarizes current knowledge in particle physics. Although some problems still remain open, it represents the leading model which all physicists refer to. One of the pillars which underpin the Standard Model is represented by the Lorentz invariance of the equations that form its backbone. These equations made it possible to predict the existence of particles and phenomena that experimental physics had not yet been able to detect. The first hint of formulating a fundamental theory of particles can be found in the 1932 Majorana equation, formulated when electrons and protons were the only known particles. Today we know that parts of the hypotheses set by Majorana were not correct, but his equation hid concepts that are found in the Standard Model. In this study, the Majorana equation is revisited and solved for free particles. The time-like, light-like and space-like solutions, represented by infinite-component wave functions, are discussed. Full article

This paper presents a study of fork–join systems. The fork–join system breaks down each customer into numerous tasks and processes them on separate servers. Once all tasks are finished, the customer is considered completed. This design enables the efficient handling of customers. The customers enter the system in a MAP flow. This helps create a more realistic and flexible representation of how customers arrive. It is important for modeling various real-life scenarios. Customers are divided into $K\ge 2$ tasks and assigned to different subsystems. The number of tasks matches the number of subsystems. Each subsystem has a server that processes tasks, and a buffer that temporarily stores tasks waiting to be processed. The service time of a task by the k-th server follows a PH (phase-type) distribution with an irreducible representation (${\beta }_k$, $S_k$), $1\le k\le K$. An analytical solution was derived for the case of $K=2$ when the input MAP flow and service time follow a PH distribution. We have efficient algorithms to calculate the stationary distribution and performance characteristics of the fork–join system for this case. In general cases, this paper suggests using a combination of Monte Carlo and machine learning methods to study the performance of fork–join systems. In this paper, we present the results of our numerical experiments. Full article