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The present paper gives an explicit classification of the isomorphism classes of non-hyperelliptic genus 4 curves over an algebraically closed field of characteristic 0. A non-hyperelliptic genus 4 curve lies on a quadric in P 3 of rank 3 or 4. In the case of rank 3, we give a set of 60 invariants which classify the isomorphism classes, and in the case of rank 4, we find 65 invariants. These invariants are defined by transvectants and can be efficiently computed on a given example. [ABSTRACT FROM AUTHOR]

There is a growing trend toward the use of natural fibers as reinforcing materials, with flax being a significant part of this market. The mechanical properties of these polymer composites, like those of synthetic fibers, are governed by parameters and material invariants. The challenge is to minimize these parameters, and to reveal these invariants to make stiffness and strength easily comparable with each other and with other composites, while avoiding excessive complexity. To this end, a simple methodology has been developed using the following parameters: Tsai's modulus or the trace of the stiffness tensor and the area of the Omni Failure Envelope in stress space. Based on the analysis of significant published experimental data on flax composites, new insights were found. The trace-normalized longitudinal Young modulus is a material property that were found to be 0.77 for tension and 0.67 compression with a coefficient of variation of 5.6% and 15%, respectively. The area of the Omni Failure Envelopes and the strength are linearly related. The use of the proposed parameters and some invariants has been discussed and they are used to compare and rank them with each other and with other composites, including carbon, aramid, and glass fiber-reinforced polymer composites. [ABSTRACT FROM AUTHOR]

Computational modeling of neurodynamical systems often deploys neural networks and symbolic dynamics. One particular way for combining these approaches within a framework called vector symbolic architectures leads to neural automata. Specifically, neural automata result from the assignment of symbols and symbol strings to numbers, known as Gödel encoding. Under this assignment, symbolic computation becomes represented by trajectories of state vectors in a real phase space, that allows for statistical correlation analyses with real-world measurements and experimental data. However, these assignments are usually completely arbitrary. Hence, it makes sense to address the problem which aspects of the dynamics observed under a Gödel representation is intrinsic to the dynamics and which are not. In this study, we develop a formally rigorous mathematical framework for the investigation of symmetries and invariants of neural automata under different encodings. As a central concept we define patterns of equality for such systems. We consider different macroscopic observables, such as the mean activation level of the neural network, and ask for their invariance properties. Our main result shows that only step functions that are defined over those patterns of equality are invariant under symbolic recodings, while the mean activation, e.g., is not. Our work could be of substantial importance for related regression studies of real-world measurements with neurosymbolic processors for avoiding confounding results that are dependant on a particular encoding and not intrinsic to the dynamics. [ABSTRACT FROM AUTHOR]

In this paper, we give a description of differential operators on tensor products A ⊗ 핂 B {A\otimes_{\mathbb{K}}B} , where A and B are finitely generated 핂 {\mathbb{K}} -algebras. We prove that any differential operator on A ⊗ 핂 B {A\otimes_{\mathbb{K}}B} can be written as a finite sum of D 1 ⊗ D 2 {D_{1}\otimes D_{2}} , where D 1 {D_{1}} and D 2 {D_{2}} are differential operators on A and B, respectively. Moreover, we introduce a series of new invariants, the ( k , s ) {(k,s)} -th Tjurina algebra A ( k , s ) ⁢ ( V ) {A_{(k,s)}(V)} for an isolated hypersurface singularity ( V , ퟎ ) = ( V ⁢ ( f ) , ퟎ ) ⊆ ( ℂ r , ퟎ ) {(V,\boldsymbol{0})=(V(f),\boldsymbol{0})\subseteq(\mathbb{C}^{r},\boldsymbol{% 0})} . We formulate a sharp upper estimate for the dimension of the ℂ {\mathbb{C}} -vector space of differential operators on A ( k , s ) ⁢ ( V ) {A_{(k,s)}(V)} of order at most 1, and we give lower and upper bounds for the dimension of the ℂ {\mathbb{C}} -vector space of differential operators on A ( k , s ) ⁢ ( V ) {A_{(k,s)}(V)} of order at most n. [ABSTRACT FROM AUTHOR]

We give a general procedure to determine the unramified Brauer group of quotients of rational varieties by finite groups. [ABSTRACT FROM AUTHOR]

Let k be an algebraically closed field of characteristic p > 0 , let G = GL n be the general linear group over k , let g = gl n be its Lie algebra and let D s be subalgebra of the divided power algebra of g ⁎ spanned by the divided power monomials with exponents < p s. We give a basis for the G -invariants in D s up to degree n and show that these are also the g -invariants. We define a certain natural restriction property and show that it doesn't hold when s > 1. If s = 1 , then D 1 is isomorphic to the truncated coordinate ring of g of dimension p dim ⁡ (g) and we conjecture that the restriction property holds and show that this leads to a conjectural spanning set for the invariants (in all degrees). We give similar results for the divided power algebras of several matrices and of vectors and covectors, and show that in the second case the restriction property doesn't hold. We also give the dimensions of the filtration subspaces of degree ≤ n of the centre of the hyperalgebra of the Frobenius kernel G s. [ABSTRACT FROM AUTHOR]

We classify generalized Liénard differential equations, sometimes called Liénard type equations, that have integrating factors of a special form. We present first integrals of the related equations. Our families of equations contain almost all known integrable generalized Liénard equations as partial cases. We find the necessary and sufficient conditions of the Liouvillian integrability for the polynomial generalized Liénard equations provided that they are nondegenerate near infinity. The degenerate near infinity polynomial generalized Liénard equations exist only under some restrictions on the degrees and leading coefficients of the polynomials parameterizing the equations. We provide a number of novel integrable polynomial and rational generalized Liénard equations. These equations arise in a variety of applications. [ABSTRACT FROM AUTHOR]

Classical invariant theory is mainly concerned with form invariance. This involves finding some function of the coefficients of a homogeneous polynomial which, under a linear transformation of the variables in the polynomial, has its value changed by a factor which only depends upon the transformation. In this article a wider class of invariants is considered. The polynomials now involves only a single variable. If a function of the coefficients can be found which is unchanged by a simple change of origin then it is said to be polynomial invariant. It is shown that there is an intimate connection between such an invariant and pencils of lines associated with an algebraic curve and this is extended to cover a collection of polynomials. [ABSTRACT FROM AUTHOR]

We introduce the notion of porous invariants for multipath affine loops over the integers. These are invariants definable in (fragments of) Presburger arithmetic and, as such, lack certain tame geometrical properties, such a convexity and connectedness. Nevertheless, we show that in many cases such invariants can be automatically synthesised, and moreover can be used to settle reachability questions for various non-trivial classes of affine loops and target sets. For the class of Z -linear invariants (those defined as conjunctions of linear equations with integer coefficients), we show that a strongest such invariant can be computed in polynomial time. For the more general class of N -semi-linear invariants (those defined as Boolean combinations of linear inequalities with integer coefficients), such a strongest invariant need not exist. Here we show that for point targets the existence of a separating invariant is undecidable in general. However we show that such separating invariants can be computed either by restricting the number of program variables or by restricting from multipath to single-path loops. Additionally, we consider porous targets, represented as Z -semi-linear sets (those defined as Boolean combinations of equations with integer coefficients). We show that an invariant can be computed providing the target spans the whole space. We present our tool porous, which computes porous invariants. [ABSTRACT FROM AUTHOR]

Soft materials exhibit significant nonlinear geometric deformations and stress–strain relationships under external forces. This paper explores weakly nonlinear elasticity theories, including Landau's and Murnaghan's formulations, advancing understanding beyond linear elasticity. We establish connections between these methods and extend strain-energy functions to the third and fourth orders in power of ε , where ε = H · H and 0 < ε ≤ 1 , and H is the perturbation to the deformation gradient tensor F = I + H. Furthermore, we address simplified strain-energy functions applicable to incompressible materials. Through this work, we contribute to a comprehensive understanding of nonlinear elasticity and its relationship to weakly nonlinear elasticity, facilitating the study of moderate deformations in soft material behavior and its practical applications. [ABSTRACT FROM AUTHOR]

Copyright of Ideas y Valores is the property of Universidad Nacional de Colombia 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.)

In principle, the local classification of spacetimes is always possible using the Cartan-Karlhede algorithm. However, in practice, the process of determining equivalence of two spacetimes relies on determining if a set of equations has a solution. Depending on the form of the equations this may be undecideable. Furthemore, if a solution does exist the equations may still be unsolvable in some way. In the case that spacetimes admit Killing vector fields with non-trivial orbits, we propose a new set of invariant quantities, called Killing invariants. These invariants will allow for the sub-classification of spacetimes admitting the same group of symmetries and will, in principle, be substantially less complicated than other sets of invariants. We apply this approach to the class of static spherically symmetric geometries as an illustrative example. [ABSTRACT FROM AUTHOR]

This paper presents the results concerning a space of invariants for second type almost geodesic mappings. After discussing the general formulas of invariants for mappings of symmetric affine connection spaces, based on these formulas, invariants for second type almost geodesic mappings of symmetric affine connection spaces and Riemannian spaces are obtained, as well as their mutual connection. Also, one invariant of Thomas type and two invariants of Weyl type for almost geodesic mappings of the second type were attained. [ABSTRACT FROM AUTHOR]

For a monic polynomial f over a commutative, unitary ring A the splitting algebra A f is the universal A -algebra such that f splits in A f . The symmetric group acts on the splitting algebra by permuting the roots of f. It is known that if the intersection of the annihilators of the elements 2 and D f (where D f depends on f) in A is zero, then the invariants under the group action are exactly equal to A. We show that the converse holds. [ABSTRACT FROM AUTHOR]

The current article discusses Sierpinski networks, which are fractal networks with certain applications in computer science, physics, and chemistry. These networks are typically used in complicated frameworks, fractals, and recursive assemblages. The results derived in this study are in mathematical and graphical format for particular classes of these networks of two distinct sorts with two invariants, K-Banhatti Sombor (KBSO) and Dharwad, along with their reduced forms. These results can facilitate the formation, scalability, and introduction of novel interconnection network topologies, chemical compounds, and VLSI processor circuits. The mathematical expressions employed in this research offer modeling insights and design guidelines to computer engineers. The derived simulation results demonstrate the optimal ranges for a certain network. The optimization assistant tool deployed in this work provides a single maximized value representing the maximum optimized network. These ranges can be put into service to dynamically establish a network according to the requirements of this paper. [ABSTRACT FROM AUTHOR]

We obtain a complete classification of scalar n$$ n $$th‐order ordinary differential equations for all subalgebras of vector fields in the real plane. While softwares like Maple can compute invariants of a given order, our results are for a general n$$ n $$. The n=1,2,3$$ n&#x0003D;1,2,3 $$ cases are well‐known in the literature. Further, it is known that there are three types of n$$ n $$th‐order equations depending upon the point symmetry algebra they possess, namely, first‐order equations which admit an infinite dimensional Lie algebra of point symmetries, second‐order equations possessing the maximum 8‐point symmetries, and higher‐order, n≥3$$ n\ge 3 $$, admitting the maximum n+4$$ n&#x0002B;4 $$ dimensional point symmetry algebra. We show that scalar n$$ n $$th‐order equations for n>5$$ n>5 $$ do not admit maximally an n+3$$ n&#x0002B;3 $$ dimensional real Lie algebra of point symmetries. Moreover, we prove that for n>4$$ n>4 $$, equations can admit two types of n+2$$ n&#x0002B;2 $$ dimensional real Lie algebra of point symmetries: one type resulting in nonlinear equations which are not linearizable via a point transformation and the second type yielding linearizable (via point transformation) equations. Furthermore, we present the types of maximal real n$$ n $$ dimensional and higher than n$$ n $$‐dimensional point symmetry algebras admissible for equations of order n≥4$$ n\ge 4 $$ and their canonical forms. The types of lower‐dimensional point symmetry algebras which can be admitted are shown, and the equations are constructible as well. We state the relevant results in tabular form and in theorems. [ABSTRACT FROM AUTHOR]

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. Theory 26 (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′)p3 and G/γ3(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 p11, the modular isomorphism problem has positive answer. Finally, we describe some families of groups of order p12 whose group algebras over the field with p elements cannot be distinguished with the techniques available to us. [ABSTRACT FROM AUTHOR]

There has been recent interest in novel Clifford geometric invariants of linear transformations. This motivates the investigation of such invariants for a certain type of geometric transformation of interest in the context of root systems, reflection groups, Lie groups and Lie algebras: the Coxeter transformations. We perform exhaustive calculations of all Coxeter transformations for A 8 , D 8 and E 8 for a choice of basis of simple roots and compute their invariants, using high-performance computing. This computational algebra paradigm generates a dataset that can then be mined using techniques from data science such as supervised and unsupervised machine learning. In this paper we focus on neural network classification and principal component analysis. Since the output—the invariants—is fully determined by the choice of simple roots and the permutation order of the corresponding reflections in the Coxeter element, we expect huge degeneracy in the mapping. This provides the perfect setup for machine learning, and indeed we see that the datasets can be machine learned to very high accuracy. This paper is a pump-priming study in experimental mathematics using Clifford algebras, showing that such Clifford algebraic datasets are amenable to machine learning, and shedding light on relationships between these novel and other well-known geometric invariants and also giving rise to analytic results. [ABSTRACT FROM AUTHOR]

The limits of change in stress/strain invariants in the phases of polycrystalline materials with cubic lattices are investigated. The relationship between the local and macroscopic parameters is established on the basis of the following principles: averaged connections, orthogonality of fluctuations of the stress and strain tensors, extremum of discrepancy between the macroscopic measures and suitable average values of microscopic analogues. General expressions for extreme values of stress/strain deviator invariants for the polycrystal phases are obtained. The non-monotonic nature of changes in the extreme values of the invariants of stress/strain deviators and volumetric stresses/strains depending on the phase concentration is revealed. In case of a two-phase polycrystal, as the harder phase increases, the invariants first increase, reaching their maximum value at a concentration of less than 5%, and then, monotonically decrease. Volumetric macrostress has a nonlinear effect on the patterns of changes in volumetric stresses in the grains of a polycrystalline material. [ABSTRACT FROM AUTHOR]

The present paper shows that local similarity theories, proposed for the strongly-stratified boundary layers, can be derived as invariant solutions defined under the Lie-group theory. A system truncated to the mean momentum and buoyancy equations is considered for this purpose. The study further suggests how similarity functions for the mean profiles are determined from the vertical fluxes, with a potential dependence on a measure of the anisotropy of the system. A time scale that is likely to characterize the transiency of a system is also identified as a non-dimensionalization factor. [ABSTRACT FROM AUTHOR]

Local-first software manages and processes private data locally while still enabling collaboration between multiple parties connected via partially unreliable networks. Such software typically involves interactions with users and the execution environment (the outside world). The unpredictability of such interactions paired with their decentralized nature make reasoning about the correctness of local-first software a challenging endeavor. Yet, existing solutions to develop local-first software do not provide support for automated safety guarantees and instead expect developers to reason about concurrent interactions in an environment with unreliable network conditions. We propose LoRe, a programming model and compiler that automatically verifies developer-supplied safety properties for local-first applications. LoRe combines the declarative data flow of reactive programming with static analysis and verification techniques to precisely determine concurrent interactions that violate safety invariants and to selectively employ strong consistency through coordination where required. We propose a formalized proof principle and demonstrate how to automate the process in a prototype implementation that outputs verified executable code. Our evaluation shows that LoRe simplifies the development of safe local-first software when compared to state-of-the-art approaches and that verification times are acceptable. [ABSTRACT FROM AUTHOR]

A rotation-two-component Camassa–Holm (R2CH) system was proposed recently to describe the motion of shallow water waves under the influence of gravity. This is a highly nonlinear and strongly coupled system of partial differential equations. A crucial issue in designing numerical schemes is to preserve invariants as many as possible at the discrete level. In this paper, we present a provable implicit nonlinear difference scheme which preserves at least three discrete conservation invariants: energy, mass, and momentum, and prove the existence of the difference solution via the Browder theorem. The error analysis is based on novel and refined estimates of the bilinear operator in the difference scheme. By skillfully using the energy method, we prove that the difference scheme not only converges unconditionally when the rotational parameter diminishes, but also converges without any step-ratio restriction for the small energy case when the rotational parameter is nonzero. The convergence orders in both settings (zero or nonzero rotation parameter) are O (τ 2 + h 2) for the velocity in the L ∞ -norm and the surface elevation in the L 2 -norm, where τ denotes the temporal stepsize and h the spatial stepsize, respectively. The theoretical predictions are confirmed by a properly designed two-level iteration scheme. Compared with existing numerical methods in the literature, the proposed method demonstrates its effectiveness for long-time simulation over larger domains and superior resolution for both smooth and non-smooth initial values. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This work is a continuation of [Y. Fittouhi and A. Joseph, Parabolic adjoint action, Weierstrass Sections and components of the nilfibre in type A]. Let P be a parabolic subgroup of an irreducible simple algebraic group G. Let P ′ be the derived group of P, and let m be the Lie algebra of the nilradical of P. A theorem of Richardson implies that the subalgebra C [ m ] P ′ , spanned by the P semi-invariants in C [ m ] , is polynomial. A linear subvariety e + V of m is called a Weierstrass section for the action of P ′ on m , if the restriction map induces an isomorphism of C [ m ] P ′ onto C [ e + V ] . Thus, a Weierstrass section can exist only if the latter is polynomial, but even when this holds its existence is far from assured. Let N be zero locus of the augmentation C [ m ] + P ′ . It is called the nilfibre relative to this action. Suppose G = SL (n , C) , and let P be a parabolic subgroup. In [Y. Fittouhi and A. Joseph, loc. cit.], the existence of a Weierstrass section e + V in m was established by a general combinatorial construction. Notably, e ∈ N and is a sum of root vectors with linearly independent roots. The Weierstrass section e + V looks very different for different choices of parabolics but nevertheless has a uniform construction and exists in all cases. It is called the "canonical Weierstrass section". Through [Y. Fittouhi and A. Joseph, loc. cit. Prop. 6.9.2, Cor. 6.9.8], there is always a "canonical" component N e of N containing e. It was announced in [Y. Fittouhi and A. Joseph, loc. cit., Prop. 6.10.4] that one may augment e to an element e VS by adjoining root vectors. Then the linear span E VS of these root vectors lies in N e and its closure is just N e . Yet, this same result shows that N e need not admit a dense P orbit [Y. Fittouhi and A. Joseph, loc. cit., Lemma 6.10.7]. For the above [Y. Fittouhi and A. Joseph, loc. cit., Theorem 6.10.3] was needed. However, this theorem was only verified in the special case needed to obtain the example showing that N e may fail to admit a dense P orbit. Here a general proof is given (Theorem 4.4.5). Finally, a map from compositions to the set of distinct non-negative integers is defined. Its image is shown to determine the canonical Weierstrass section. One may anticipate that the remaining components of N can be similarly described. However, this is a long story and will be postponed for a subsequent paper. These results should form a template for general type. [ABSTRACT FROM AUTHOR]

We obtain a complete topological classification of $k$ -folding map-germs on generic surfaces in $\mathbb {R}^3$ , discover new robust features of surfaces and recover, in a unified way, many of the robust features that were obtained previously by considering the contact of a surface with lines, planes or spheres. [ABSTRACT FROM AUTHOR]

BackgroundMicrobiomes consist of bacteria, viruses, and other microorganisms, and are responsible for many different functions in both organisms and the environment. Past analyses of microbiomes focused on using correlation to determine linear relationships between microbes and diseases. Weak correlations due to nonlinearity between microbe pairs may cause researchers to overlook critical components of the data. With the abundance of available microbiome, we need a method that comprehensively studies microbiomes and how they are related to each other.ResultsWe collected publicly available datasets from human, environment, and animal samples to determine both symmetric and asymmetric Boolean implication relationships between a pair of microbes. We then found relationships that are potentially invariants, meaning they will hold in any microbe community. In other words, if we determine there is a relationship between two microbes, we expect the relationship to hold in almost all contexts. We discovered that around 330,000 pairs of microbes universally exhibit the same relationship in almost all the datasets we studied, thus making them good candidates for invariants. Our results also confirm known biological properties and seem promising in terms of disease diagnosis.ConclusionsSince the relationships are likely universal, we expect them to hold in clinical settings, as well as general populations. If these strong invariants are present in disease settings, it may provide insight into prognostic, predictive, or therapeutic properties of clinically relevant diseases. For example, our results indicate that there is a difference in the microbe distributions between patients who have or do not have IBD, eczema and psoriasis. These new analyses may improve disease diagnosis and drug development in terms of accuracy and efficiency.

The symmetry of space and time properties allows us to use the principle of least action to determine the invariants in the processes occurring on the interface of metals. The invariants established by the author and the criterion allowing to estimate the state of the interface are proposed. [ABSTRACT FROM AUTHOR]

Poisson structures related to affine Courant-type algebroids are analyzed, including those related with cotangent bundles on Lie-group manifolds. Special attention is paid to Courant-type algebroids and their related R structures generated by suitably defined tensor mappings. Lie–Poisson brackets that are invariant with respect to the coadjoint action of the loop diffeomorphism group are created, and the related Courant-type algebroids are described. The corresponding integrable Hamiltonian flows generated by Casimir functionals and generalizing so-called heavenly-type differential systems describing diverse geometric structures of conformal type in finite dimensional Riemannian manifolds are described. [ABSTRACT FROM AUTHOR]

Copyright of Eurasian Music Science Journal is the property of State Conservatory of Uzbekistan 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.)

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. [ABSTRACT FROM AUTHOR]

The Riccati equation is widely used in physics, mechanics and other disciplines, but due to the non integrability of Riccati equation, it is very difficult to solve it. In this paper, the analytic solution of a kind of Riccati equation is given by using the invariant relationship of Riccati equation, which makes the invariant solution of Riccati equation an important path to solve Riccati equation. [ABSTRACT FROM AUTHOR]

Lie similarity transformations for unsteady flow in a thin film under the influence of internal heating and thermal radiation have been deduced earlier. These transformations have been employed to reduce the partial differential equations of the fluid flow and heat transfer to ordinary differential equations. Fluid velocity and temperature profiles were presented by constructing homotopic solutions for the obtained system of equations. Here we apply a general linear combination of all the Lie point symmetries associated with the equations describing the flow and heat transfer with internal heating and thermal radiation, to construct the general Lie similarity transformations. By applying these transformations to the flow, we map them to ordinary differential equations. The reduced system of differential equations contains the arbitrary constants used to construct the generalised Lie similarity transformations through the linear combination of all the Lie symmetries of the flow equations. Further, we determine analytic solutions using the homotopy analysis method for the reduced system of equations. We show that the presence of arbitrary constants in the reduced system of differential equations enables control of the deduced homotopic analytic solutions, i.e., these constants serve as control parameters. We illustrate the effects of these control parameters on the temperature profiles graphically. [ABSTRACT FROM AUTHOR]

Let p be a an odd prime and let G be a finite p-group with cyclic commutator subgroup G ′ . We prove that the exponent and the abelianization of the centralizer of G ′ in G are determined by the group algebra of G over any field of characteristic p. If, additionally, G is 2-generated then almost all the numerical invariants determining G up to isomorphism are determined by the same group algebras; as a consequence the isomorphism type of the centralizer of G ′ is determined. These claims are known to be false for p = 2. [ABSTRACT FROM AUTHOR]

This paper presents the results concerning a space of invariants for second type almost geodesic mappings. After discussing the general formulas of invariants for mappings of symmetric affine connection spaces, based on these formulas, invariants for second type almost geodesic mappings of symmetric affine connection spaces and Riemannian spaces are obtained, as well as their mutual connection. Also, one invariant of Thomas type and two invariants of Weyl type for almost geodesic mappings of the second type were attained.

A "Composition map" is constructed, leaning heavily on earlier work Fittouhi and Joseph (2023) [5,6]). It defines a composition tableau which recovers the "canonical" Weierstrass section e + V described in the first paper above. Moreover without reference to this earlier work , it is then shown that e + V is indeed a Weierstrass section. This results in a huge simplification. Moreover one may read off from the composition tableau the "VS quadruplets" of the second of the above papers, thereby describing the "canonical component" of the nil-fibre in which e lies but does not of itself determine. [ABSTRACT FROM AUTHOR]

Counting polynomials find their way into chemical graph theory through quantum chemistry in two ways: as approximate solutions to the Schrödinger equation or by storing information in a mathematical form and trying to find a pattern in the roots of these expressions. Coefficients count how many times a property occurs, and exponents express the extent of the property. They help understand the origin of regularities in the chemistry of specific classes of compounds. Our objective is to accelerate the research of newcomers into chemical graph theory. One problem in understanding these concepts is in the different approaches and notations of each research study; some researchers provide online tools for computing these mathematical concepts, but these need to be maintained for functionality. We take advantage of similar mathematical aspects of 14 such polynomials that merge theoretical chemistry and pure mathematics; give examples, differences, and similarities; and relate them to recent research. [ABSTRACT FROM AUTHOR]

We characterize the existence of first integrals and invariants (first integrals depending on the time) for the polynomial vector fields which are invariant under an involution. [ABSTRACT FROM AUTHOR]

In this paper, septic B-spline approach with linearization and without linearization technique with the help of Butcher's fifth-order Runge–Kutta scheme is implemented to obtain solitary wave solutions of the modified regularized long wave equation. The error norms L 2 and L ∞ and the three invariants I 1 , I 2 and I 3 are computed on three tests to confirm about the efficiency and accuracy of the proposed methods. The stability analysis is performed using Von-Neumann technique on the linearized septic B-spline approach. The results of the present work are compared with the analytical and existing solutions. [ABSTRACT FROM AUTHOR]