Algebraic Formula Research Articles

Some Results on Oblique Dual and Approximate Oblique Dual Hilbert-Schmidt Frames

The concept of Hilbert-Schmidt frames (HS-frames) is more general than that of g-frames. In this paper, we investigate the oblique dual (OD) and ε -approximate oblique dual ( ε -AOD) HS-frames on closed subspaces W and V of a separable Hilbert space H . We first introduce the concepts of OD and ε -AOD HS-frames. Then we present some conditions for a pair of OD and ε -AOD HS-frames to be symmetric. We also obtain the algebraic formula for all OD and ε -AOD HS-frames on V for a given HS-frame in W . As application, we get the canonical OD HS-frame has a minimum norm among all the representations of elements in W . We also discuss the perturbations of ε -AOD HS-frames. Finally, applying our results, we not only recover some known results but also derive a new result in the classical Hilbert space frame setting.

Accounting for Twins and Other Multiple Births in Perinatal Studies of Live Births Conducted Using Healthcare Administration Data

The analysis of perinatal studies is complicated by twins and other multiple births even when multiples are not the exposure, outcome, or a confounder of interest. In analyses of infant outcomes restricted to live births, common approaches to handling multiples include restriction to singletons, counting outcomes at the pregnancy-level (i.e., by counting if at least one twin experienced a binary outcome), or infant-level analysis including all infants and accounting for clustering of outcomes, such as by using generalized estimating equations or mixed effects models. Several healthcare administration databases only support restriction to singletons or pregnancy-level approaches. For example, in MarketScan insurance claims data, diagnoses in twins are often assigned to a single infant identifier, thereby preventing ascertainment of infant-level outcomes among multiples. Different approaches correspond to different questions, produce different estimands, and often rely on different assumptions. We demonstrate the differences that can arise from these different approaches using Monte Carlo simulations, algebraic formulas, and an applied example.

About Filtration in a Geophysical Bridge with a Seepage Site

Based on the representation of a modular elliptic function in the form of a combination of simple algebraic formulas that conformally map the area of the velocity hodograph (curved triangle) onto a half-plane, a direct definition of filtration rates in a geophysical bridge is given. For the first time, a family of isotope lines of equal filtration rates was constructed for the inner area of the bridge in the absence of water in the downstream. For special cases, the values of the proposed formulas almost completely coincide with the hydromechanical solution of Masket M. (for 4 cases) and the numerical calculations of Khairullin Z.E. (for 2 cases). The well-known Nelson-Skornyakov F.B. formula for the output filtration rates through the lower face of the jumper, adopted by analogy with the flow outlet from the base of the flatbed into the horizontal drainage, gives an underestimation of the results by up to 45%, and therefore cannot be recommended for practical use.

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.