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A more flexible neural network (NN) method using the fundamental invariants (FIs) as the input vector is proposed in the construction of potential energy surfaces for molecular systems involving identical atoms. Mathematically, FIs finitely generate the permutation invariant polynomial (PIP) ring. In combination with NN, fundamental invariant neural network (FI-NN) can approximate any function to arbitrary accuracy. Because FI-NN minimizes the size of input permutation invariant polynomials, it can efficiently reduce the evaluation time of potential energy, in particular for polyatomic systems. In this work, we provide the FIs for all possible molecular systems up to five atoms. Potential energy surfaces for OH3 and CH4 were constructed with FI-NN, with the accuracy confirmed by full-dimensional quantum dynamic scattering and bound state calculations. [ABSTRACT FROM AUTHOR]

We examine the phase space structures that govern reaction dynamics in the absence of critical points on the potential energy surface. We show that in the vicinity of hyperbolic invariant tori, it is possible to define phase space dividing surfaces that are analogous to the dividing surfaces governing transition from reactants to products near a critical point of the potential energy surface. We investigate the problem of capture of an atom by a diatomic molecule and show that a normally hyperbolic invariant manifold exists at large atom-diatom distances, away from any critical points on the potential. This normally hyperbolic invariant manifold is the anchor for the construction of a dividing surface in phase space, which defines the outer or loose transition state governing capture dynamics. We present an algorithm for sampling an approximate capture dividing surface, and apply our methods to the recombination of the ozone molecule. We treat both 2 and 3 degrees of freedom models with zero total angular momentum. We have located the normally hyperbolic invariant manifold from which the orbiting (outer) transition state is constructed. This forms the basis for our analysis of trajectories for ozone in general, but with particular emphasis on the roaming trajectories. [ABSTRACT FROM AUTHOR]

Let X be a complex K 3 surface with an effective action of a group G which preserves the holomorphic symplectic form. Let ... be the generating function for the Euler characteristics of the Hilbert schemes of G-invariant length n subschemes. We show that its reciprocal, Zx,g(q)-1 is the Fourier expansion of a modular cusp form of weight 1/2 e(X/G) for the congruence subgroup ⌈0(|G|). We give an explicit formula for ZX,G in terms of the Dedekind eta function for all 82 possible (X, G). We extend our results to various refinements of the Euler characteristic, namely the Elliptic genus, the Xy genus, and the motivic class. As a byproduct of our method, we prove a result which is of independent interest: it establishes an eta product identity for a certain shifted theta function of the root lattice of a simply laced root system. [ABSTRACT FROM AUTHOR]

We show that the reduced SL2(C)-twisted Burau representation can be obtained from the quantum group Uq(sl2) for q = i a fourth root of unity and that representations of Uq.sl2/satisfy a type of Schur-Weyl duality with the Burau representation. As a consequence, the SL2.C/-twisted Reidemeister torsion of links can be obtained as a quantum invariant. Our construction is closely related to the quantum holonomy invariant of Blanchet-Geer-Patureau-Mirand-Reshetikhin, and we interpret their invariant as a twisted Conway potential. [ABSTRACT FROM AUTHOR]

In 2010, B. Balsam and A. Kirillov, Jr. showed that the Turaev-Viro invariants defined for a spherical fusion category A extends to invariants of 3-manifolds with corners. In 2021, A. Kirillov, Jr. described an equivalent formulation for the 2-1 part of the theory (2-manifolds with boundary) using the space of "stringnets with boundary conditions" as the vector spaces associated to 2-manifolds with boundary. Here we construct a similar theory for the 3-2 part of the 4-3-2 theory by L. Crane and D. Yetter (1993). [ABSTRACT FROM AUTHOR]

In 2013, Monod introduced a family of Thompson-like groups which provides natural counterexamples to the von Neumann-Day conjecture. We construct a characterization of conjugacy and an invariant and use them to compute centralizers in one group of this family. [ABSTRACT FROM AUTHOR]

In recent years, great success has been achieved on the classification of symmetry-protected topological (SPT) phases for interacting fermion systems by using generalized cohomology theory. However, the explicit calculation of generalized cohomology theory is extremely hard due to the difficulty of computing obstruction functions. Based on the physical picture of topological invariants and mathematical techniques in homotopy algebra, we develop an algorithm to resolve this hard problem. It is well known that cochains in the cohomology of the symmetry group, which are used to enumerate the SPT phases, can be expressed equivalently in different linear bases, known as the resolutions. By expressing the cochains in a reduced resolution containing much fewer basis than the choice commonly used in previous studies, the computational cost is drastically reduced. In particular, it reduces the computational cost for infinite discrete symmetry groups, like the wallpaper groups and space groups, from infinity to finity. As examples, we compute the classification of two-dimensional interacting fermionic SPT phases, for all 17 wallpaper symmetry groups. [ABSTRACT FROM AUTHOR]

We prove the existence of regular optimal G -invariant partitions, with an arbitrary number ℓ ≥ 2 of components, for the Yamabe equation on a closed Riemannian manifold (M , g) when G is a compact group of isometries of M with infinite orbits. To this aim, we study a weakly coupled competitive elliptic system of ℓ equations, related to the Yamabe equation. We show that this system has a least energy G -invariant solution with nontrivial components and we show that the limit profiles of its components separate spatially as the competition parameter goes to − ∞ , giving rise to an optimal partition. For ℓ = 2 the optimal partition obtained yields a least energy sign-changing G -invariant solution to the Yamabe equation with precisely two nodal domains. [ABSTRACT FROM AUTHOR]

In this paper, we study the boundedness and persistence of positive solution, existence of invariant rectangle, local and global behavior, and rate of convergence of positive solutions of the following systems of exponential difference equations x n + 1 = α 1 + β 1 e − x n − 1 γ 1 + y n , y n + 1 = α 2 + β 2 e − y n − 1 γ 2 + x n , x n + 1 = α 1 + β 1 e − y n − 1 γ 1 + x n , y n + 1 = α 2 + β 2 e − x n − 1 γ 2 + y n , where the parameters α i , β i , γ i for i ∈ { 1 , 2 } and the initial conditions x − 1 , x 0 , y − 1 , y 0 are positive real numbers. Some numerical example are given to illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

We present some inequalities related to partial determinants for positive semidefinite block matrices. Firstly, we introduce the definition of partial matrix functions corresponding to partial traces and partial determinants, and then we provide a unified extension of a recent result of Lin [10], Chang-Paksoy-Zhang [4] and Lin-Sra [12]. Secondly, we give a new generalization of a result of Paksoy-Turkmen-Zhang [15]. Finally, we conclude with an interesting conjecture involving partial determinants. [ABSTRACT FROM AUTHOR]

In this short paper, we establish a reverse of the derived inequalities for sector matrices by Tan and Xie, with Kantorovich constant. Then, as application of our main theorem, some inequalities for determinant and unitarily invariant norm are presented. [ABSTRACT FROM AUTHOR]

We consider a steady submerged laminar jet of viscous incompressible fluid flowing out of a tube and propagating along a solid plane surface. The numerical solution of Navier–Stokes equations is obtained in the stationary three-dimensional formulation. The hypothesis that at large distances from the tube exit the flowfield is described by the self-similar solution of the parabolized Navier–Stokes equations is confirmed. The asymptotic expansions of the self-similar solution are obtained for small and large values of the coordinate in the jet cross-section. Using the numerical solution the self-similarity exponent is determined. An explicit dependence of the self-similar solution on the Reynolds number and the conditions in the jet source is determined. [ABSTRACT FROM AUTHOR]

In this work we study the change of the structure of a regular pencil when we perform small perturbations over some of its rows and the other rows remain unaltered. We provide necessary conditions when several rows are perturbed, and prove them to be sufficient to prescribe the homogenous invariant factors or the Weyr characteristic of the resulting pencil when one row is perturbed. [ABSTRACT FROM AUTHOR]

Following the concept of statistical convergence, in this paper we introduce two more subtle notions, viz. statistical ω -limit set and statistical ω -cluster set than the general ω -limit set in a discrete dynamical system of a continuous function and study some properties related to these two points. [ABSTRACT FROM AUTHOR]

We present a permutation-invariant distance between atomic configurations, defined through a functional representation of atomic positions. This distance enables us to directly compare different atomic environments with an arbitrary number of particles, without going through a space of reduced dimensionality (i.e., fingerprints) as an intermediate step. Moreover, this distance is naturally invariant through permutations of atoms, avoiding the time consuming associated minimization required by other common criteria (like the root mean square distance). Finally, the invariance through global rotations is accounted for by a minimization procedure in the space of rotations solved by Monte Carlo simulated annealing. A formal framework is also introduced, showing that the distance we propose verifies the property of a metric on the space of atomic configurations. Two examples of applications are proposed. The first one consists in evaluating faithfulness of some fingerprints (or descriptors), i.e., their capacity to represent the structural information of a configuration. The second application concerns structural analysis, where our distance proves to be efficient in discriminating different local structures and even classifying their degree of similarity. [ABSTRACT FROM AUTHOR]

A theoretical investigation on the origin dependence of the vibronic polarizabilities, isotropic and anisotropic rotational invariants, and scattering cross sections in Resonance Raman Optical Activity (RROA) spectroscopy is presented. Expressions showing the origin dependence of these polarizabilities were written in the resonance regime using the Franck-Condon (FC) and Herzberg-Teller (HT) approximations for the electronic transition moments. Differently from the far-from-resonance scattering regime, where the origin dependent terms cancel out when the rotational invariants are calculated, RROA spectrum can exhibit some origin dependence even for eigenfunctions of the electronic Hamiltonian. At the FC level, the RROA spectrum is completely origin invariant if the polarizabilities are calculated using a single excited state or for a set of degenerate states. Otherwise, some origin effects can be observed in the spectrum. At the HT level, RROA spectrum is origin dependent even when the polarizabilities are evaluated from a single excited state but the origin effect is expected to be small in this case. Numerical calculations performed for (S)-methyloxirane, (2R,3R)-dimethyloxirane, and (R)-4-F-2-azetidinone at both FC and HT levels using the velocity representation of the electric dipole and quadrupole transition moments confirm the predictions of the theory and show the extent of origin effects and the effectiveness of suggested ways to remove them. [ABSTRACT FROM AUTHOR]

A fourth-power law underlying the steady shock-wave structure and solid viscosity of condensed material has been observed for a wide range of metals and non-metals. The fourth-power law relates the steady-wave Hugoniot pressure to the fourth power of the strain rate during passage of the material through the structured shock wave. Preceding the fourth-power law was the observation in a shock transition that the product of the shock dissipation energy and the shock transition time is a constant independent of the shock pressure amplitude. Invariance of this energy-time product implies the fourth-power law. This property of the shock transition in solids was initially identified as a shock invariant. More recently, it has been referred to as the dissipative action, although no relationship to the accepted definitions of action in mechanics has been demonstrated. This same invariant property has application to a wider range of transient failure phenomena in solids. Invariance of this dissipation action has application to spall fracture, failure through adiabatic shear, shock compaction of granular media, and perhaps others. Through models of the failure processes, a clearer picture of the physics underlying the observed invariance is emerging. These insights in turn are leading to a better understanding of the shock deformation processes underlying the fourth-power law. Experimental result and material models encompassing the dynamic failure of solids are explored for the purpose of demonstrating commonalities leading to invariance of the dissipation action. Calculations are extended to aluminum and uranium metals with the intent of predicting micro-scale dynamics and spatial structure in the steady shock wave. [ABSTRACT FROM AUTHOR]

Double triangle expansion is an operation on 4-regular graphs with at least one triangle which replaces a triangle with two triangles in a particular way. We study the class of graphs which can be obtained by repeated double triangle expansion beginning with the complete graph K5. These are called double triangle descendants of K5. We enumerate, with explicit rational generating functions, those double triangle descendants of K5 with at most four more vertices than triangles. We also prove that the minimum number of triangles in any K5 descendant is four. Double triangle descendants are an important class of graphs because of conjectured properties of their Feynman periods when they are viewed as scalar Feynman diagrams, and also because of conjectured properties of their c2 invariants, an arithmetic graph invariant with quantum field theoretical applications. [ABSTRACT FROM AUTHOR]

We consider the symmetric traveling salesman problem (TSP) with instances represented by complete graphs G with distances between cities as edge weights. A complexity index is an invariant of an instance I by which we predict the execution time of an exact TSP algorithm for I. In the paper [5] we have considered some short edge subgraphs of G and defined several new invariants related to their connected components. Extensive computational experiments with instances on 50 vertices with the uniform distribution of integer edge weights in the interval [1,100] show that there exists correlation between the sequences of selected invariants and the sequence of execution times of the well-known TSP Solver Concorde. In this paper we extend these considerations for instances up to 100 vertices. [ABSTRACT FROM AUTHOR]

Using an extension of the Kontsevich integral to tangles in handlebodies similar to a construction given by Andersen, Mattes and Reshetikhin, we construct a functorZWB ! yA, where B is the category of bottom tangles in handlebodies and yA is the degree-completion of the category A of Jacobi diagrams in handlebodies. As a symmetric monoidal linear category, Ais the linear PROP governing "Casimir Hopf algebras", which are cocommutative Hopf algebras equipped with a primitive invariant symmetric 2-tensor. The functor Z induces a canonical isomorphism grB Š A, where grB is the associated graded of the Vassiliev-Goussarov filtration on B. To each Drinfeld associator ' we associate a ribbon quasi-Hopf algebra H' in yA, and we prove that the braided Hopf algebra resulting from H' by "transmutation" is precisely the image by Z of a canonical Hopf algebra in the braided category B. Finally, we explain how Z refines the LMO functor, which is a TQFT-like functor extending the Le-Murakami-Ohtsuki invariant. [ABSTRACT FROM AUTHOR]

The mini-workshop focused on very recent developments of analytic and algebraic techniques for studying almost complex structures which are not necessarily integrable. It provided a forum to discuss and compare techniques from PDE's, elliptic theory, deep algebraic structures, as well as geometric flows, and new topological invariants, towards attacking several longstanding open problems in the field of complex and almost complex geometry. [ABSTRACT FROM AUTHOR]

We consider a non-linear Hartree energy for bosonic particles in a symmetric double-well potential. In the limit where the wells are far apart and the potential barrier is high, we prove that the ground state and first excited state are given to leading order by an even, respectively odd, superposition of ground states in single wells. The corresponding energies are separated by a small tunneling term that we evaluate precisely. [ABSTRACT FROM AUTHOR]

Conformally Stäckelmanifolds can be characterized as the class of n-dimensional pseudo-Riemannian manifolds (M, G) on which the Hamilton--Jacobi equation G(Vu, Vu) = 0 for null geodesics and the Laplace equation -- ΔG φ = 0 are solvable by R-separation of variables. In the particular case in which the metric has Riemannian signature, they provide explicit examples ofmetrics admitting a set of n-1 commuting conformal symmetry operators for the Laplace-Beltrami operator ΔG. In this paper, we solve the anisotropic Calderón problem on compact 3-dimensional Riemannian manifolds with boundary which are conformally Stäckel, that is we show that the metric of such manifolds is uniquely determined by the Dirichlet-to-Neumann map measured on the boundary of the manifold, up to diffeomorphims that preserve the boundary. [ABSTRACT FROM AUTHOR]

We obtain a complete asymptotic expansion for the eigenvalues of the Dirichlet-to-Neumann maps associated with Schrödinger operators on compact Riemannian surfaces with boundary. For the zero potential, we recover the well-known spectral asymptotics for the Steklov problem. For non-zero potentials, we obtain new geometric invariants determined by the spectrum of what we call the parametric Steklov problem. In particular, for constant potentials parametric Steklov problem, the total geodesic curvature on each connected component of the boundary is a spectral invariant. Under the constant curvature assumption, this allows us to obtain some interior information from the spectrum of these boundary operators. [ABSTRACT FROM AUTHOR]

A group € is said to be "finitely non-co-Hopfian," or "renormalizable," if there exists a self-embedding 'W € ! € whose image is a proper subgroup of finite index. Such a proper self-embedding is called a "renormalization for €." In this work, we associate a dynamical system to a renormalization' of €. The discriminant invariant D' of the associated Cantor dynamical system is a profinite group which is a measure of the asymmetries of the dynamical system. If D' is a finite group for some renormalization, we show that €=C' is virtually nilpotent, where C' is the kernel of the action map. We introduce the notion of a (virtually) renormalizable Cantor action, and show that the action associated to a renormalizable group is virtually renormalizable. We study the properties of virtually renormalizable Cantor actions, and show that virtual renormalizability is an invariant of continuous orbit equivalence. Moreover, the discriminant invariant of a renormalizable Cantor action is an invariant of continuous orbit equivalence. Finally, the notion of a renormalizable Cantor action is related to the notion of a self-replicating group of automorphisms of a rooted tree. [ABSTRACT FROM AUTHOR]

This paper is devoted to a new perspective to classical visual groups and their fundamental invariants. It proposes a new approach to determine visual intrinsic invariants, visual Frenet-Serret frames and the associated formulas, visual angles and drawing visual normals. Moreover, it introduces Visual Structural Theorem (VST) indicating that each regular-smooth curve in a visual geometry satisfies a unique third order differential equation, and offers its inverse as a conjecture. [ABSTRACT FROM AUTHOR]

The class IV2 of 2-nondegenerate constant Levi rank 1 hypersurfaces M5 ⊂ C3 is governed by Pocchiola's two primary invariants W0 and J0. Their vanishing characterizes equivalence of such a hypersurface M5 to the tube M5 LC over the real light cone in R3. When either W0 ≠0 or J0 ≠ 0, by normalization of certain two group parameters c and e, an invariant coframe can be built on M5, showing that the dimension of the CR automorphism group drops from 10 to 5. This paper constructs an explicit feg-structure in case W0 and J0 do not necessarily vanish. Furthermore, Pocchiola's calculations hidden on a computer now appear in details, especially the determination of a secondary invariant R, expressed in terms of the first jet of W0. All other secondary invariants of the feg-structure are also expressed explicitly in terms of W0 and J0. [ABSTRACT FROM AUTHOR]

Copyright of Iraqi Journal of Science is the property of Republic of Iraq Ministry of Higher Education & Scientific Research (MOHESR) and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Recently, the first named author defined a 2-parametric family of groups Gnk [V. O. Manturov, Non-reidemeister knot theory and its applications in dynamical systems, geometry and topology, preprint (2015), arXiv:1501.05208]. Those groups may be regarded as analogues of braid groups. Study of the connection between the groups Gnk and dynamical systems led to the discovery of the following fundamental principle: "If dynamical systems describing the motion of n particles possess a nice codimension one property governed by exactly kk particles, then these dynamical systems admit a topological invariant valued in Gnk". The Gnk groups have connections to different algebraic structures, Coxeter groups, Kirillov-Fomin algebras, and cluster algebras, to name three. Study of the Gnk groups led to, in particular, the construction of invariants, valued in free products of cyclic groups. All generators of the Gnk groups are reflections which make them similar to Coxeter groups and not to braid groups. Nevertheless, there are many ways to enhance Gnk groups to get rid of this 22-torsion. Later the first and the fourth named authors introduced and studied the second family of groups, denoted by Γnk, which are closely related to triangulations of manifolds. The spaces of triangulations of a given manifolds have been widely studied. The celebrated theorem of Pachner [P.L. homeomorphic manifolds are equivalent by elementary shellings, Europ. J. Combin.12(2) (1991) 129-145] says that any two triangulations of a given manifold can be connected by a sequence of bistellar moves or Pachner moves. See also [I. M. Gelfand, M. M. Kapranov and A. V. Zelevinsky, Discriminants, Resultants, and Multidimensional Determinants (Birkhäuser, Boston, 1994); A. Nabutovsky, Fundamental group and contractible closed geodesics, Comm. Pure Appl. Math.49(12) (1996) 1257-1270]; the Γnk naturally appear when considering the set of triangulations with the fixed number of points. There are two ways of introducing the groups Γnk: the geometrical one, which depends on the metric, and the topological one. The second one can be thought of as a "braid group" of the manifold and, by definition, is an invariant of the topological type of manifold; in a similar way, one can construct the smooth version. In this paper, we give a survey of the ideas lying in the foundation of the Gnk and Γnk theories and give an overview of recent results in the study of those groups, manifolds, dynamical systems, knot and braid theories. [ABSTRACT FROM AUTHOR]

Counting the number of Galois orbits of newforms in Sk(Γ0(N)) and giving some arithmetic sense to this number is an interesting open problem. The case N = 1 corresponds to Maeda's conjecture (still an open problem) and the expected number of orbits in this case is 1, for any k ≥ 16. In this article we give local invariants of Galois orbits of newforms for general N and count their number. Using an existence result of newforms with prescribed local invariants we prove a lower bound for the number of non-CM Galois orbits of newforms for Γ0(N) for large enough weight k (under some technical assumptions on N). Numerical evidence suggests that in most cases this lower bound is indeed an equality, thus we leave as a question the possibility that a generalization of Maeda's conjecture could follow from our work. We finish the paper with some natural generalizations of the problem and show some of the implications that a generalization of Maeda's conjecture has. [ABSTRACT FROM AUTHOR]

The Euler characteristic i.e., the difference between the number of vertices |V| and edges |E| is the most important topological characteristic of a graph. However, to describe spectral properties of differential equations with mixed Dirichlet and Neumann vertex conditions it is necessary to introduce a new spectral invariant, the generalized Euler characteristic χ G : = | V | - | V D | - | E | , with | V D | denoting the number of Dirichlet vertices. We demonstrate theoretically and experimentally that the generalized Euler characteristic χ G of quantum graphs and microwave networks can be determined from small sets of lowest eigenfrequencies. If the topology of the graph is known, the generalized Euler characteristic χ G can be used to determine the number of Dirichlet vertices. That makes the generalized Euler characteristic χ G a new powerful tool for studying of physical systems modeled by differential equations on metric graphs including isoscattering and neural networks where both Neumann and Dirichlet boundary conditions occur. [ABSTRACT FROM AUTHOR]

Determinantal point processes are point processes whose correlation functions are given by determinants of matrices. The entries of these matrices are given by one fixed function of two variables, which is called the kernel of the point process. It is well known that there are different kernels that induce the same correlation functions. We classify all the possible transformations of a kernel that leave the induced correlation functions invariant, restricting to the case of symmetric kernels. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the periodicity problems in the finitely generated algebraic structures and exhibit their natural sources in the theory of invariants of finite groups and it forms an interesting and relatively self-contained nook in the imposing edifice of group theory. One of the deepest and important results of the related theory of finite groups is a complete classification of all periodic groups, that is, the finite groups with periodic properties. For every integer k ≥ 2 and a k-generated non-associative algebraic structure S =< A >, where A = fa1; a2;: ::; akg, the sequence...is called the k-nacci sequence of S with respect to the generating set A, denoted kA(S). If kA(S) is periodic, we call the length of the period of the sequence the periodicity length of S with respect to the generating set A, written LENA(S) and the minimum of the positive integers of LENA(S) will be mentioned as periodicity invariant of S, denoted by k(S). However, this invariant has been studied for groups and semi-groups during the years as well as the associative property of S where above sequence was reduced to xi = xi-kxi-k+1: :: xi-3xi-2xi-1, for every i ≥ k+1. Thus, we attempt to give explicit upper bounds for the periodicity invariant of two infinite classes of finite non-associative 3-generated algebraic structures. Moreover, two classes of non-isomorphic Moufang loops of the same periodicity length were obtained in the study. [ABSTRACT FROM AUTHOR]

Collinear search impairment (CSI) is a phenomenon where a task-irrelevant collinear structure impairs a target search in a visual display. It has been suggested that CSI is monocular, occurs without the participants' access to consciousness and is possibly processed at an early visual site (e.g. V1). This effect has frequently been compared with a well-documented opposite effect called attentional capture (AC), in which salient and task-irrelevant basic features (e.g. color, orientation) enhance target detection. However, whether this phenomenon can be attributed to non-attentional factors such as collinear facilitation (CF) has not yet been formally tested. Here we used one well-established property of CF, i.e. that target contrast modulates its effect direction (facilitation vs suppression), to examine whether CSI shared similar signature profiles along different contrast levels. In other words, we tested whether CSI previously observed at the supra-threshold level was reduced or reversed at near-threshold contrast levels. Our results showed that, regardless of the luminance contrast levels, participants spent a longer time searching for targets displayed on the salient singleton collinear structure than those displayed off the structure. Contrast invariance suggests that it is unlikely that CSI is exclusively sub-served by an early vision mechanism (e.g. CF). [ABSTRACT FROM AUTHOR]

We show the existence of semiorthogonal decompositions (SOD) of Pandharipande-Thomas (PT) stable pair moduli spaces on Calabi-Yau 3-folds with irreducible curve classes, assuming relevant moduli spaces are non-singular. The above result is motivated by categorifications of the wall-crossing formula for PT invariants in the derived category, and also by a d-critical analogue of Bondal-Orlov's and Kawamata's D/K equivalence conjecture. We also give SOD of stable pair moduli spaces on K3 surfaces, which categorifies Kawai-Yoshioka's formula proving Katz-Klemm-Vafa's formula for PT invariants on K3 surfaces with irreducible curve classes. [ABSTRACT FROM AUTHOR]

Computing the autotopism group of a partial Latin rectangle can be performed in a variety of ways. This pilot study has two aims: (a) to compare these methods experimentally and (b) to identify the design goals one should have in mind for developing practical software. To this end, we compare six families of algorithms (two backtracking methods and four graph automorphism methods), with and without the use of entry invariants, on two test suites. We consider two entry invariants: one determined by the frequencies of row, column, and symbol representatives, and one determined by 2×2 submatrices. We find: (a) with very few entries, many symmetries often exist, and these should be identified mathematically rather than computationally, (b) with an intermediate number of entries, a quick‐to‐compute entry invariant was effective at reducing the need for computation, (c) with an almost‐full partial Latin rectangle, more sophisticated entry invariants are needed, and (d) the performances for (full) Latin squares and almost‐full partial Latin squares are significantly poorer than other partial Latin rectangles of comparable size, obstructed by the existence of (partial) Latin squares with large (possibly transitive) autotopism groups. [ABSTRACT FROM AUTHOR]

Copyright of Iraqi Journal of Science is the property of Republic of Iraq Ministry of Higher Education & Scientific Research (MOHESR) and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Action Units activation is a set of local individual facial muscle parts that occur in time constituting a natural facial expression event. AUs occurrence activation detection can be inferred as temporally consecutive evolving movements of these parts. Detecting AUs automatically can provide explicit benefits since it considers both static and dynamic facial features. Our work is divided into three contributions: first, we extracted the features from Local Binary Patterns, Local Phase Quantisation, and dynamic texture descriptor LPQTOP with two distinct leveraged network models from different CNN architectures for local deep visual learning for AU image analysis. Second, cascading the LPQTOP feature vector with Long Short‐Term Memory is used for coding longer term temporal information. Next, we discovered the importance of stacking LSTM on top of CNN for learning temporal information in combining the spatially and temporally schemes simultaneously. Also, we hypothesised that using an unsupervised Slow Feature Analysis method is able to leach invariant information from dynamic textures. Third, we compared continuous scoring predictions between LPQTOP and SVM, LPQTOP with LSTM, and AlexNet. A competitive substantial performance evaluation was carried out on the Enhanced CK dataset. Overall, the results indicate that CNN is very promising and surpassed all other methods [ABSTRACT FROM AUTHOR]

We consider some string invariants at genus two that appear in the analysis of the D8ℛ4 and D6ℛ5 interactions in type II string theory. We conjecture a Poisson equation involving them and the Kawazumi-Zhang invariant based on their asymptotic expansions around the non-separating node in the moduli space of genus two Riemann surfaces. [ABSTRACT FROM AUTHOR]

In the paper, lifted left invariant (α, β)-metrics of Douglas type on tangent Lie groups are studied. Suppose that g is a left invariant Riemannian metric on a Lie group G, and F is a left invariant (α, β)-metric of Douglas type induced by g. Using vertical and complete lifts, we construct the vertical and complete lifted (α, β)-metrics Fv and Fc on the tangent bundle TG and give necessary and sufficient conditions for them to be of Douglas type. Then the flag curvature of these metrics are studied. Finally, as some special cases, the flag curvatures of Fv and Fc are given for Randers metrics of Douglas type and Kropina and Matsumoto metrics of Berwald type. [ABSTRACT FROM AUTHOR]

We prove that finite categorically equivalent p-rings have isomorphic additive groups (in particular, they have the same cardinality) and that the number of generators is a categorical invariant for finite rings. We also classify rings of size p3 up to categorical equivalence. [ABSTRACT FROM AUTHOR]

In this article, we consider a generalized equal width wave (GEW) equation which is a significant nonlinear wave equation as it can be used to model many problems occurring in applied sciences. Here we study a Petrov–Galerkin method for the model problem, in which element shape functions are quadratic and weight functions are linear B-splines. We investigate the existence and uniqueness of solutions of the weak form of the equation. Then, we establish the theoretical bound of the error in the semi-discrete spatial scheme as well as of a full discrete scheme at t = tn. Furthermore, a powerful Fourier analysis has been applied to show that the proposed scheme is unconditionally stable. Finally, propagation of solitary waves and evolution of solitons are analyzed to demonstrate the efficiency and applicability of the proposed scheme. The three invariants (I1, I2 and I3) of motion have been commented to verify the conservation features of the proposed algorithms. Our proposed numerical scheme has been compared with other published schemes and demonstrated to be valid, effective and it outperforms the others. [ABSTRACT FROM AUTHOR]

The Lewis solutions describe the exterior gravitational field produced by infinitely long rotating cylinders, and are useful models for global gravitational effects. When the metric parameters are real (Weyl class), the exterior metrics of rotating and static cylinders are locally indistinguishable, but known to globally differ. The significance of this difference, both in terms of physical effects (gravitomagnetism) and of the mathematical invariants that detect the rotation, remain open problems in the literature. In this work we show that, by a rigid coordinate rotation, the Weyl class metric can be put into a 'canonical' form where the Killing vector field ∂t is time-like everywhere, and which depends explicitly only on three parameters with a clear physical significance: the Komar mass and angular momentum per unit length, plus the angle deficit. This new form of the metric reveals that the two settings differ only at the level of the gravitomagnetic vector potential which, for a rotating cylinder, cannot be eliminated by any global coordinate transformation. It manifests itself in the Sagnac and gravitomagnetic clock effects. The situation is seen to mirror the electromagnetic field of a rotating charged cylinder, which likewise differs from the static case only in the vector potential, responsible for the Aharonov–Bohm effect, formally analogous to the Sagnac effect. The geometrical distinction between the two solutions is also discussed, and the notions of local and global staticity revisited. The matching in canonical form to the van Stockum interior cylinder is also addressed. [ABSTRACT FROM AUTHOR]

We prove the existence of automorphisms of ℂk, k ≥ 2, having an invariant, non-recurrent Fatou component biholomorphic to ℂ x (ℂ*)k-1 which is attracting, in the sense that all the orbits converge to a fixed point on the boundary of the component. As a corollary, we obtain a Runge copy of ℂ x (ℂ*) k-1 in ℂk. The constructed Fatou component also avoids k analytic discs intersecting transversally at the fixed point. [ABSTRACT FROM AUTHOR]

A permutationally invariant global potential energy surface for the HOCO system is reported by fitting a larger number of high-level ab initio points using the newly proposed permutation invariant polynomial-neural network method. The small fitting error (∼5 meV) indicates a faithful representation of the potential energy surface over a large configuration space. Full-dimensional quantum and quasi-classical trajectory studies of the title reaction were performed on this potential energy surface. While the results suggest that the differences between this and an earlier neural network fits are small, discrepancies with state-to-state experimental data remain significant. [ABSTRACT FROM AUTHOR]

For discrete classical Molecular dynamics obtained by the "Verlet" algorithm (VA) with the time increment h there exists a shadow Hamiltonian ˜H with energy ˜E(h), for which the discrete particle positions lie on the analytic trajectories for ˜H . Here, we proof that there, independent of such an analytic analogy, exists an exact hidden energy invariance E* for VA dynamics. The fact that the discrete VA dynamics has the same invariances as Newtonian dynamics raises the question, which of the formulations that are correct, or alternatively, the most appropriate formulation of classical dynamics. In this context the relation between the discrete VA dynamics and the (general) discrete dynamics investigated by Lee [Phys. Lett. B 122, 217 (1983)] is presented and discussed. [ABSTRACT FROM AUTHOR]