Algebraic Formula Research Articles

Development of a Program for Searching the Optimum Condition using the Design of Experiments

The Design of Experiments (DOE) is a method that is widely used due to its effectiveness in selecting optimum conditions in the design stage of product development. On the other hand, fast, low-cost, labor-saving, and energy-saving innovative development is also required in the industry. In previous research, a program for quickly searching the optimum condition using the design of experiments is developed and evaluated. Relationships between each parameter and the final property are firstly cleared for an algebraic formula by using the design of experiments. Then the optimum conditions for each parameter were decided by using these formulas in the program. However, when each parameter has several errors in the data, the search accuracy becomes very low. In this research, the improvement for the searching accuracy using the law of error propagation was developed and evaluated. Relationships between each parameter error and the final property are firstly cleared for high-accuracy searching by using the law of error propagation and the previous results in the previous research. And each parameter influence for the final property was cleared. It was found that the large parameter effects could be improved for high-precision search by using high-precision instruments, increasing the number of trials N, and taking measurements in an optimal environment. Relationships between each parameter error and the final property were investigated and evaluated for the proposed method by using a mathematical model. It is concluded from the result that (1) the proposed method is effective for clearing the relationships between each parameter error and the final property, and (2) the proposed method is effective for searching the optimum condition.

An Optimal Scalp Rotation Flap Design: Mathematical and Bio-Mechanical Analysis

The design and implementation of successful rotational flaps of the scalp remains a complex process. There are several described techniques, all of which are based on a two-dimension surface, absent consideration of the convexity and thereby three-dimensional nature of the scalp. This has contributed to flaps that are either too small or unnecessarily large in a bid to compensate.Mathematical analysis using trigonometric and algebraic formulae were used to derive the relationship between the defect, triangulation, the contour of the scalp, the radius of the arc of rotation for the flap and scalp skin tension. Polyurethane models were used to test the applicability of the resultant formula, and a consecutive clinical case series is presented.Rotation flap dimensions were calculated using an apex angle of 30 degrees depicted as a two-dimensional construct. The contour of the defect was then measured as the depth of depression with radius of the rotation flap calculated based on the changes in flap surface area across the defect. Finite Element Analysis was utilised to measure redistribution of tension along the flap reconstructing the defect, and the efficacy of the approach confirmed in the consecutive clinical series. The current study provides a reliable flap design based on mathematical analysis, with evidence-based application to the clinical setting.

Young walls and graded dimension formulas for finite quiver Hecke algebras of types A2n−1(2), Dn(1) and Bn(1)

In this paper, we establish the graded dimension formulas of finite quiver Hecke algebras using level-[Formula: see text] Fock space representations of quantum affine algebras of types [Formula: see text], [Formula: see text] and [Formula: see text].

On $F$-Polynomials for Generalized Quantum Cluster Algebras and Gupta's Formula

We show the polynomial property of $F$-polynomials for generalized quantum cluster algebras and obtain the associated separation formulas under a mild condition. Along the way, we obtain Gupta's formulas of $F$-polynomials for generalized quantum cluster algebras. These formulas specialize to Gupta's formulas for quantum cluster algebras and cluster algebras respectively. Finally, a generalization of Gupta's formula has also been discussed in the setting of generalized cluster algebras.

Algebraic formulas and geometric derivation of source identities

Source identities are fundamental identities between multivariable special functions. We give a geometric derivation of rational and trigonometric source identities. We also give a systematic derivation and extension of various determinant representations for source functions which appeared in previous literature as well as introducing the elliptic version of the determinants, and obtain identities between determinants. We also show several symmetrization formulas for the rational version.

EllipsoHyperbola A Common Approach that Joins the Conic Sections in 2D and 3D Space

This work aspires to interactively reveal through the use of the GeoGebra Software the relationship between the Conic Sections in 3D and the 2D symmetric forms of Conic Sections around O in a coordinate system Oxy, that is Circle, Ellipse, axis x'x, and Hyperbola, showing that these Conic Sections arise from the same algebraic formula and therefore have common characteristics. This manuscript includes short research on the four kinds of Conic Sections through a common approach that joins them in the two-but also in three-dimensional space, revealing the role of the slope of the Generator line of the conic surface and the role of the slope of the cutting plane in their equations.

Visualizing the support of Kostant’s weight multiplicity formula for the rank-2 Lie algebras

Visualizing the support of Kostant’s weight multiplicity formula for the rank-2 Lie algebras

Almost Existentially Closed Models in Positive Logic

This paper explores the concept of almost positively closed models in the framework of positive logic. To accomplish this, we initially define various forms of the positive amalgamation property, such as h-amalgamation and symmetric and asymmetric amalgamation properties. Subsequently, we introduce certain structures that enjoy these properties. Following this, we introduce the concepts of Δ-almost positively closed and Δ-weekly almost positively closed. The classes of these structures contain and exhibit properties that closely resemble those of positive existentially closed models. In order to investigate the relationship between positive almost closed and positive strong amalgamation properties, we first introduce the sets of positive algebraic formulas ET and AlgT and the properties of positive strong amalgamation. We then show that if a model A of a theory T is a ET+A-weekly almost positively closed, then A is a positive strong amalgamation basis of T, and if A is a positive strong amalgamation basis of T, then A is AlT+A-weekly almost positively closed.

Classical and local similarity in problems of turbulent convection: Extension of Prandtl semi-empirical theory for horizontal layers of water and air mediums

A variant of the local similarity theory is considered, assuming a priori specification of two basic dimensional parameters: the second moment of vertical velocity and the “spectral” Prandtl mixing length. Such selection of basic parameters lets us interpret the algebraic formulas of Prandtl's semiempirical turbulence theory for the coefficients of turbulent heat transfer and kinetic energy dissipation as relationships of the local similarity theory. A priori approximations of the basic parameters are based on known relationships and consider the upper boundary of the convective layer as a solid wall. Within the framework of local similarity theory, approximations for the turbulent moments of buoyancy and vertical velocity of arbitrary order are also proposed. The correspondence of these approximations to experimental data has been established in both air and water mediums, as well as in the results of numerical modeling. Such correspondence allows us to conclude that a wide class of vertical profiles of turbulent moments in a windless convective layer depends only on two basic parameters: the second moment of vertical velocity and the “spectral” Prandtl mixing length. The obtained result indicates the universal nature of the theory of local similarity and is a significant addition to the semiempirical Prandtl theory.

Numerical cubature on scattered data by adaptive interpolation

We construct cubature methods on scattered data via resampling on the support of known algebraic cubature formulas, by different kinds of adaptive interpolation (polynomial, RBF, PUM). This approach gives a promising alternative to other recent methods, such as direct meshless cubature by RBF or least-squares cubature formulas.

Game Theory Analysis of Algebraic Content Selection in Middle School Mathematics Curricula

The algebra section of middle school math textbooks is indispensable. The characteristics of the textbook and the rationality of its elements need to be algebra content of middle school math textbooks mainly consists of algebraic formulas, equations and inequalities, functions and other elements. This paper focuses on modeling the analysis based on game theory and then exploring the rationality of the algebra textbook study is conducted by reviewing the literature and constructing an analytical model based on cooperative game and Nash equilibrium theory. It is obtained that in the strategy where all three elements are used at the same time, the one that focuses on the orchestration function can maximize the reorganization of unit algebra content design is needed to achieve rational weighting of individual algebra selection of elements needs to fully reflect the articulation between the individual algebra elements. Algebraic elements are designed to spiral in difficulty to match the needs of teachers and students using them.

Toric differential forms and periods of complete intersections

Let n be an even natural number. We compute the periods of any n2-dimensional complete intersection algebraic cycle inside an n-dimensional non-degenerated intersection of a projective simplicial toric variety. Using this information we determine the cycle class of such algebraic cycles. As part of the proof we develop a toric generalization of a classical theorem of Macaulay about complete intersection Artin Gorenstein rings, and we generalize an algebraic cup formula for residue forms due to Carlson and Griffiths to the toric setting.

Remarkable series concerning $ \binom{3n}{n} $ and harmonic numbers in numerators

<abstract><p>Three classes of infinite series containing binomial coefficient $ \binom{3n}{n} $, harmonic-like numbers, and an independent variable "$ y $" are examined. Several algebraic formulae in closed form are established, including, as special cases, three conjectured values for numerical series by Z.-W. Sun. This is fulfilled by integrating Lambert's series and manipulating the cubic transformations for the $ _3{F_2} $-series through the "coefficient extraction" method.</p></abstract>

Algebraic disturbance estimation for a class of fractional order T-S fuzzy systems with noisy output measurements

This paper concerns with the design of algebraic disturbance estimation method for the fractional order Takagi-Sugeno (T-S) fuzzy systems with noisy output measurements. At first, the overall fractional order fuzzy system model is given for the studied system. By constructing a family of fractional order modulating functions, algebraic integral formulas are derived for the disturbance by using the overall fractional order fuzzy system model. Based on the system’s input and output, these formulas can be algebraically computed in both noise-free and noisy cases. Simulation results are shown to validate the efficiency of the proposed algebraic disturbance estimation method against output measurement noises.

An algebraic formula for two loop renormalization of scalar quantum field theory

We find a general formula for the two-loop renormalization counterterms of a scalar quantum field theory with interactions containing up to two derivatives, extending ’t Hooft’s one-loop result. The method can also be used for theories with higher derivative interactions, as long as the terms in the Lagrangian have at most one derivative acting on each field. We show that diagrams with factorizable topologies do not contribute to the renormalization group equations. The results in this paper will be combined with the geometric method in a subsequent paper to obtain the counterterms and renormalization group equations for the scalar sector of effective field theories (EFT) to two-loop order.

Testing the satisfiability of algebraic formulas over the field of two elements

We construct a probabilistic polynomial algorithm for testing the satisfiability of algebraic formulas of depth 3 over the two-element field, with addition as the top operation in the formulas. An algorithm with the same characteristics exists for the problem of testing whether a polynomial given by formulas of this type is identically zero (PIT problem). However, these problems and algorithms for their solution are essentially different. The probabilistic algorithm for the PIT problem is based on the Schwartz-Zippel lemma, whereas the satisfiability testing algorithm proposed in this paper is based on the Valiant-Vazirani lemma.

Microsatellites’ mutation modeling through the analysis of the Y-chromosomal transmission: Results of a GHEP-ISFG collaborative study

The Spanish and Portuguese Speaking Working Group of the International Society for Forensic Genetics (GHEP-ISFG) organized a collaborative study on mutations of Y-chromosomal short tandem repeats (Y-STRs). New data from 2225 father-son duos and data from 44 previously published reports, corresponding to 25,729 duos, were collected and analyzed. Marker-specific mutation rates were estimated for 33 Y-STRs. Although highly dependent on the analyzed marker, mutations compatible with the gain or loss of a single repeat were 23.2 times more likely than those involving a greater number of repeats. Longer alleles (relatively to the modal one) showed to be nearly twice more mutable than the shorter ones. Within the subset of longer alleles, the loss of repeats showed to be nearly twice more likely than the gain. Conversely, shorter alleles showed a symmetrical trend, with repeat gains being twofold more frequent than reductions. A positive correlation between the paternal age and the mutation rate was observed, strengthening previous findings. The results of a machine learning approach, via logistic regression analyses, allowed the establishment of algebraic formulas for estimating the probability of mutation depending on paternal age and allele length for DYS389I, DYS393 and DYS627. Algebraic formulas could also be established considering only the allele length as predictor for DYS19, DYS389I, DYS389II-I, DYS390, DYS391, DYS393, DYS437, DYS439, DYS449, DYS456, DYS458, DYS460, DYS481, DYS518, DYS533, DYS576, DYS626 and DYS627 loci. For the remaining Y-STRs, a lack of statistical significance was observed, probably as a consequence of the small effective size of the subsets available, a common difficulty in the modeling of rare events as is the case of mutations. The amount of data used in the different analyses varied widely, depending on how the data were reported in the publications analyzed. This shows a regrettable waste of produced data, due to inadequate communication of the results, supporting an urgent need of publication guidelines for mutation studies.

Prediction of large deflection of chromium nanobeams using a hybrid meta-heuristic algorithm

In this paper, the prediction of large deflection of the chromium nanobeams has been studied using hybrid meta-heuristic methods. At first, using the full factorial design of experiment method, 100 experiments were designed in which the length and force applied to the nanobeams were considered as input variables at 25 and 4 levels, respectively. Also, the deflection of the nanobeam was considered as the output. The nanobeam’s deflection was estimated through the regression models by first, second, and third-order equations, respectively. The preliminary results demonstrated that the third-order regression model predicts the deflection with high accuracy and its error is less than the other models. In the following, to reduce the number of required experiments, two levels of the applied force (8 and 11 nN) were used to predict the deflection. Accordingly, the new equations were used for interpolation and extrapolation of the deflection in other levels of forces i.e., 9.5 and 12.5 nN. Finally, the coefficients of both third-order equations obtained by 4 and 2 levels of forces were optimized using the meta-heuristic algorithms i.e., Big Bang-Big Crunch (BBBC) and Ray Optimization algorithm (ROA). Comparing the results showed that by using the BBBC and RAO methods, the error value decreases by 10% and 20%, respectively. The results of this paper showed that the deflections of the chromium nanobeam can be estimated by algebraic formulas with high accuracy and fewer experiments, time, and cost.

МОДЕЛИРОВАНИЕ ПОТРЕБНОСТИ В МЕДИЦИНСКОЙ ПОМОЩИ, ОКАЗЫВАЕМОЙ В АМБУЛАТОРНЫХ УСЛОВИЯХ, ПРИ БОЛЕЗНЯХ СИСТЕМЫ КРОВООБРАЩЕНИЯ У НАСЕЛЕНИЯ СТАРШИХ ВОЗРАСТНЫХ ГРУПП МУЖСКОГО ПОЛА

The purpose of the study: to perform mathematical modeling of the need for outpatient medical care for diseases of the circulatory system in the population of older male age groups. Materials and methods of research. As a source of primary information, we used data on cases of medical care provided to the population of older age groups (60 years and older). The probability of treatment in each one-year age range, starting from the age of 60 years, was studied. 41 points were used to construct the dynamic curve. The calculation was made per 1000 population of the corresponding age and gender. The principle of constructing mathematical models was based on a functional approach. As a result of this approach, a system of linear and nonlinear equations was obtained, in each of which the dependence of the volume of medical care on predictor variables was expressed. In this case, age and gender were used as predictor variables. Results. Algebraic formulas were obtained for modeling the level of need for medical care provided in outpatient settings in relation to three blocks of causes (group A of causes). The impact on the level of need of other reasons can be determined using a correction factor. Such coefficients should be applied to the estimated value of the need for three blocks of reasons included in Group A of reasons.

Combinatorial versus algebraic formulae for the Moore–Penrose inverse of a Laplacian matrix of a threshold graph

We present four different approaches to calculate the Moore–Penrose inverse of a Laplacian matrix of a connected threshold graph. The first one is combinatorial and it benefits from the combinatorial structure of a threshold graph, while the second three are of algebraic nature and explore the particular structure of a matrix in question.