Narrow Results

The transformation behavior of the vector-valued theta function of a positive-definite even lattice under the metaplectic group ${Mp}_2(\mathbb{Z})$ is described by the Weil representation. We show that the invariants of this representation are induced from five fundamental invariants. As an application we give simple generating sets for Jacobi forms of singular weight.

We continue the analysis of the modular isomorphism problem for $2$-generated $p$-groups with cyclic derived subgroup, $p>2$, started in [D. García-Lucas, Á. del Río and M. Stanojkowski, On group invariants determined by modular group algebras: Even versus odd characteristic, Algebra Represent. Theory26 (2022) 2683–2707, doi:10.1007/s10468-022-10182-x]. We show that if $G$ belongs to this class of groups, then the isomorphism type of the quotients $G/{(G^{\prime })}^{p^3}$ and $G/{\gamma }_3{(G)}^p$ are determined by its modular group algebra. In fact, we obtain a more general but technical result, expressed in terms of the classification [O. Broche, D. García-Lucas and Á. del Río, A classification of the finite 2-generator cyclic-by-abelian groups of prime-power order, Int. J. Algebra Comput.33(4) (2023) 641–686]. We also show that for groups in this class of order at most $p^{11}$, the modular isomorphism problem has positive answer. Finally, we describe some families of groups of order $p^{12}$ whose group algebras over the field with $p$ elements cannot be distinguished with the techniques available to us.

We introduce some new models of infinite-state transition systems. The basic model, called a (reversal-bounded) counter machine (CM), is a nondeterministic finite automaton augmented with finitely many reversal-bounded counters (i.e. each counter can be incremented or decremented by 1 and tested for zero, but the number of times it can change mode from nondecreasing to nonincreasing and vice-versa is bounded by a constant, independent of the computation). We extend a CM by augmenting it with some familiar data structures: (i) A pushdown counter machine (PCM) is a CM augmented with an unrestricted pushdown stack. (ii) A tape counter machine (TCM) is a CM augmented with a two-way read/write worktape that is restricted in that the number of times the head crosses the boundary between any two adjacent cells of the worktape is bounded by a constant, independent of the computation (thus, the worktape is finite-crossing). There is no bound on how long the head can remain on a cell. (iii) A queue counter machine (QCM) is a CM augmented with a queue that is restricted in that the number of alternations between non-deletion phase and non-insertion phase on the queue is bounded by a constant. A non-deletion (non-insertion) phase is a period consisting of insertions (deletions) and no-changes, i.e., the queue is idle. We show that emptiness, (binary, forward, and backward) reachability, nonsafety, and invariance for these machines are solvable. We also look at extensions of the models that allow the use of linear-relation tests among the counters and parameterized constants as "primitive" predicates. We investigate the conditions under which these problems are still solvable.

There are two main goals in this article, one of them is to minimize the number of invariants needed to obtain Whitney equisingular one parameter families of finitely determined holomorphic map germs f_t:(ℂⁿ,0) → (ℂ^p,0), with n < p. The other is to show how to compute the local Euler obstruction of the stable types which appear in a finitely determined map germ in these dimensions.

The polar multiplicities of all stable types and the 0-stable singularities are the invariants that guarantee the Whitney equisingularity of such families and the polar multiplicities are the numbers that also allow us to compute the local Euler obstruction. Therefore our first step is to describe all stable types which appear when n < p and show the relationship between the polar multiplicities in each stable type. Using the fact that these polar multiplicities are upper semi-continuous we minimize the number of invariants that guarantee Whitney equisingularity of such a family. We also apply the relationship between the polar multiplicities in each stable type and a result of Lê and Teissier to show how to compute the local Euler obstruction of the stable types which appear in these dimensions.

In this paper, a meshless spectral radial point interpolation (MSRPI) method using weighted $\theta$-scheme is formulated for the numerical solutions of a class of nonlinear Kawahara-type evolutionary equations. The formulated method is applied for simulation of single and double solitary waves motion, wave generation and oscillatory shock waves propagation. Quality of approximation is measured via discrete $L_{\infty }$, $L_2$ and $L_{\mathrm{\text{rms}}}$ error norms. Three invariant quantities corresponding to mass, momentum and energy are also computed for the method validation. Stability analysis of the proposed method is briefly discussed and verified computationally. Comparison of the obtained results are made with other existing results in the literature revealing the method superiority.

The covariant free fields of any spin on anti-de Sitter (AdS) spacetimes are studied, pointing out that these transform under isometries according to covariant representations (CRs) of the AdS isometry group, induced by those of the Lorentz group. Applying the method of ladder operators, it is shown that the CRs with unique spin are equivalent with discrete unitary irreducible representations (UIRs) of positive energy of the universal covering group of the isometry one. The action of the Casimir operators is studied finding how the weights of these representations (reps.) may depend on the mass and spin of the covariant field. The conclusion is that on AdS spacetime, one cannot formulate a universal mass condition as in special relativity.

In this paper, we perform a detailed group classification for a generalized convection-reaction-diffusion equation with three unknown functions. Specifically, we determine all the functional forms for the unknown functions where the given equation admits nontrivial Lie point symmetries. The classification problem provides us with eight families of equations summarized in four categories. The admitted Lie symmetries form the four Lie algebras $2A_1$, $A_{4,4}$, $A_{2,1}\oplus A_1$ and $A_{2,1}\oplus A_{2,1}$. For the four families of the classification problem we calculate the one-dimensional optimal system and we derive all the similarity transformations which reduce the partial differential equation into an ordinary differential equation. Applications of the similarity transformations are presented while exact solutions are derived.

The aim of the current paper is to study the multiscalar-tensor theories of gravity without derivative couplings. We construct a few basic objects that are invariant under a Weyl rescaling of the metric and transform covariantly when the scalar fields are redefined. We introduce rules to construct further such objects and put forward a scheme that allows to express the results obtained either in the Einstein frame or in the Jordan frame as general ones. These so-called “translation” rules are used to show that the parametrized post-Newtonian approximation results obtained in the aforementioned two frames indeed are the same if expressed in a general frame.

An experimental analysis of shape classification methods based on moment and autoregressive (AR) invariants is presented. Various types of translation, scale and rotation invariants are used to construct feature vectors for classification. The performance is evaluated using five different objects picked up from real scenes with a TV camera. Silhouettes and contours are extracted from nonoccluded two-dimensional (2D) objects rotated, scaled and translated in 3D space. The feature extraction methods are implemented and systematically tested using several parametric and nonparametric classifiers. The results clearly show the advantage of the method based on the moment invariants.

A central task of computer vision is to automatically recognize objects in real-world scenes. The parameters defining image and object spaces can vary due to lighting conditions, camera calibration and viewing positions. It is therefore desirable to look for geometric properties of the object which remain invariant under such changes. In this paper we present geometric algebra as a complete framework for the theory and computation of projective invariants formed from points and lines in computer vision. We will look at the formation of 3D projective invariants from multiple images, show how they can be formed from image coordinates and estimated tensors (F, fundamental matrix and T, trilinear tensor) and give results on simulated and real data.

The classification of images using regular or geometric moment functions suffers from two major problems. First, odd orders of central moments give zero value for images with symmetry in the x and/or y directions and symmetry at centroid. Secondly, these moments are very sensitive to noise especially for higher order moments. In this paper, a single solution is proposed to solve both these problems. The solution involves the computation of the moments from a reference point other than the image centroid. The new reference centre is selected such that the invariant properties like translation, scaling and rotation are still maintained. In this paper, it is shown that the new proposed moments can solve the symmetrical problem. Next, we show that the new proposed moments are less sensitive to Gaussian and random noise as compared to two different types of regular moments derived by Hu.⁶ Extensive experimental study using a neural network classification scheme with these moments as inputs are conducted to verify the proposed method.

In component-based systems, there are several obstacles to using Design by Contract (DbC), particularly with respect to third-party components. Contracts are particularly valuable when debugging or testing composite software structures that include third-party components. However, existing approaches have critical weaknesses. First, existing approaches typically require a component's source code to be available if you wish to strip (or re-insert) checks. Second, documentation of the contract is either distributed separately from the component or embedded in the component's source code. Third, enabling and disabling specific kinds of checks on separate components from independent vendors can be a significant challenge. This paper describes an approach to representing contracts for .NET components using attributes. This contract information can be retrieved from the compiled component's metadata and used for many purposes. The paper also describes nContract, a tool that automatically generates run-time checks from embedded contracts. Such run-time checks can be generated and added to a system without requiring source code access or recompilation. Further, when checks for a given component are excluded, they impose no run-time overhead. Finally, a highly expressive, fine-grained mechanism for controlling user preferences about which specific checks are enabled or disabled is presented.

Given a finitely generated (fg) group G, the set R(G) of homomorphisms from G to SL₂ℂ inherits the structure of an algebraic variety known as the representation variety of G in SL₂ℂ. This algebraic variety is an invariant of fg presentations of G. Call a group G parafree of rank n if it shares the lower central sequence with a free group of rank n, and if it is residually nilpotent. The deviation of a fg parafree group is the difference between the minimum possible number of generators of G and the rank of G. So parafree groups of deviation zero are actually just free groups. Parafree groups that are not free share a host of properties with free groups. In this paper algebraic geometric invariants involving the number of maximal irreducible components (mirc) of R(G), and the dimension of R(G) for certain classes of parafree groups are computed. It is shown that in an infinite number of cases these invariants successfully discriminate between ismorphism types within the class of parafree groups of the same rank. This is quite surprising, since an n generated group G is free of rank n if and only if Dim(R(G)) = 3n. In fact, a trivial consequence of Theorem 1.8 in this paper is that given an arbitrary positive integer k, and any integer r ≥ 2, there exist infinitely many non-isomorphic fg parafree groups of rank r and deviation 1 with representation varieties of dimension 3r, having more than k mirc of dimension 3r. This paper also introduces many novel and surprising dimension formulas for the representation varieties of certain one-relator groups.

We present a new and efficient method to detect whether or not two given rational plane curves are similar. If both curves are the same, the method finds the symmetries of the curve. The method relies on the introduction of a complex differential invariant that has a nice behavior with respect to Möbius transformations, which are the mappings lying behind the similarities in the parameter space. From a computational point of view, our algorithm only requires bivariate gcds and factoring. The algorithm is implemented in Maple^™ [Maplesoft, a division of Waterloo Maple Inc. Waterloo, Ontario (2021)]. An extensive study of its practical performance is also provided.

Generalized Reidemeister moves provide an extended set of moves to work with virtual knots and links. We introduce virtual tangle moves, generalization of classical rational tangle moves and show that such generalizations are essential to develop new invariants of virtual knots and links. We show that every 2-algebraic virtual link is a virtual 4-move equivalent to a trivial link or Hopf link. The properties of virtual tangle move are analyzed on few existing invariants associated with virtual knots and links.

Virtual knot theory is a generalization of knot theory which is based on Gauss chord diagrams and link diagrams on closed oriented surfaces. A twisted knot is a generalization of a virtual knot, which corresponds to a link diagram on a possibly non-orientable surface. In this paper, we discuss an invariant of twisted links which is obtained from the JKSS invariant of virtual links by use of double coverings. We also discuss some properties of double covering diagrams.

We extend the concepts of trivializing and knotting numbers for knots to spatial graphs and 2-bouquet graphs, in particular. Furthermore, we calculate the trivializing and knotting numbers for projections and pseudodiagrams of 2-bouquet spatial graphs based on the number of precrossings and the placement of the precrossings in the pseudodiagram of the spatial graph.

We create an invariant of virtual $Y$-oriented trivalent spatial graphs using colorings by virtual Niebrzydowski algebras. This paper generalizes the color invariants using virtual tribrackets and Niebrzydowski algebras by Nelson–Pico, and Graves-Nelson-T. We computed all tribrackets, Niebrzydowski algebras and virtual Niebrzydowski algebras of orders 3 and 4, and provide generative code for all data sets.

A rack, which is the algebraic distillation of two of the Reidemeister moves, is a set with a binary operation such that right multiplication is an automorphism. Any codimension two link has a fundamental rack which contains more information than the fundamental group. Racks provide an elegant and complete algebraic framework in which to study links and knots in 3–manifolds, and also for the 3–manifolds themselves. Racks have been studied by several previous authors and have been called a variety of names. In this first paper of a series we consolidate the algebra of racks and show that the fundamental rack is a complete invariant for irreducible framed links in a 3–manifold and for the 3–manifold itself. We give some examples of computable link invariants derived from the fundamental rack and explain the connection of the theory of racks with that of braids.

A v_J/c correction to the Shapiro time delay seems verified by a 2002 Jovian observation by VLBI. In this Essay, this correction is interpreted as an effect of the aberration of light in an optically refractive medium which supplies an analog of Jupiter's gravity field rather than as a measurement of the speed of gravity, as it was first proposed by other authors. The variation of the index of refraction is induced by the Lorentz invariance of the weak gravitational field equations for Jupiter in a uniform translational slow motion with velocity v_J=13.5 km/s. The correction on time delay and deflection is due not to the Kerr (or Lense-Thirring) stationary gravitomagnetic field of Jupiter, but to its Schwarzschild gravitostatic field when measured from the barycenter of the solar system.