Real Integer Research Articles

Non-Homogeneous Binary Cubic Equation a(x-y)<sup>3</sup>=8bxy

Polynomial equations an interesting subject in theory of numbers, occupy a pivotal role in the realm of mathematics and have a wealth of historical significance. The theoretical importance of polynomial equations of third degree in two unknowns having integral coefficients is great as they are closely connected with many problems of number theory. Specifically, the third degree polynomial equations having two unknowns in connection with elliptic curves occupy a pivotal role in the region of mathematics. This paper discusses on finding many solutions in integers to a typical third degree equation having two variables expressed as a(x-y)<sup>3</sup>=8bxy. The substitution strategy is employed in obtaining successfully different choices of solutions in integers. Some of the special fascinating numbers, namely, Pyramidal numbers, Polygonal numbers, Centered pyramidal numbers, Centered polygonal numbers, Thabit ibn Qurra numbers, Star numbers, Mersenne numbers and Nasty numbers (numbers expressed as product of two numbers in two different ways such that the sum of the factors in one set equals to the difference of factors in another set) are discussed in properties. These special numbers are unique. and have attractive characterization that set them apart from other numbers. The process of formulating second order Ramanujan numbers with base numbers as real integers is illustrated through examples.

Schinzel-type bounds for the Mahler measure on lemniscates

We study the generalized Mahler measure on lemniscates, and prove a sharp lower bound for the measure of totally real integer polynomials that includes the classical result of Schinzel expressed in terms of the golden ratio. Moreover, we completely characterize many cases when this lower bound is attained. For example, we explicitly describe all lemniscates and the corresponding quadratic polynomials that achieve our lower bound for the generalized Mahler measure. It turns out that the extremal polynomials attaining the bound must have even degree. The main computational part of this work is related to finding many extremals of degree four and higher, which is a new feature compared to the original Schinzel’s theorem where only quadratic irreducible extremals are possible.

Relational Program Synthesis with Numerical Reasoning

Learning programs with numerical values is fundamental to many AI applications, including bio-informatics and drug design. However, current program synthesis approaches struggle to learn programs with numerical values. An especially difficult problem is learning continuous values from multiple examples, such as intervals. To overcome this limitation, we introduce an inductive logic programming approach which combines relational learning with numerical reasoning. Our approach, which we call NumSynth, uses satisfiability modulo theories solvers to efficiently learn programs with numerical values. Our approach can identify numerical values in linear arithmetic fragments, such as real difference logic, and from infinite domains, such as real numbers or integers. Our experiments on four diverse domains, including game playing and program synthesis, show that our approach can (i) learn programs with numerical values from linear arithmetical reasoning, and (ii) outperform existing approaches in terms of predictive accuracies and learning times.

Identification of the extended standard linear solid material model by means of experimental dynamical measurements

A novel algebraic procedure for the non-parametric identification of the material model by means of dynamical test measurements is proposed. An extended Standard Linear Solid (SLS) material model is taken into account to model the material linear visco-elastic behavior. It consists of the series arrangement of fractional Kelvin model elements adopting real parameters and integer and non-integer order time differential operators, and of hysteretic Kelvin model elements adopting complex parameters and integer order time differential operators. Hysteretic Kelvin model elements are introduced to take account of the material hysteretic behavior. The material E(j⋅ω) complex modulus, is analytically modeled as the ratio of pseudo polynomials (non-integer power terms) in the j⋅ω Fourier variable. A multi-step, iterative, material model identification technique is here proposed to identify the unknown polynomial coefficients and the non-integer exponent values starting from E(j⋅ω) material discrete estimates from input-output dynamical measurements made on a beam specimen at different ω frequency values. Computational, nonphysical SLS elements resulting from the application of the identification procedure can be found and eliminated, so that a low order optimal model result. Some results obtained by applying the proposed identification technique with real experimental measurements are shown and discussed.

The impact of sustainable energy technologies and demand response programs on the hub’s planning by the practical consideration of tidal turbines as a novel option

This paper investigates a multi-objective optimal energy planning strategy for a hub, incorporating renewable and non-renewable resources, like PV, tidal turbine, fuel-cell, CHP, boiler, micro-turbine, reactor, reformer, electrolyzer, and energy storage by utilizing the time of use program (TOU). In this strategy, tidal turbine, fuel-cell, and reformer technologies are considered novel technologies that simultaneously reduce the proposed hub’s cost and pollution. The hub’s total cost and pollution are considered objective functions. To make the results more realistic, characteristics of the tidal turbine are investigated by utilizing the manufactory’s company information. The problem is then modeled as real mixed integer programming (RMIP) and is solved in GAMS software using a CPLEX solver. Epsilon constraints method and fuzzy satisfying approach are used to select the optimal solution based on the proposed model. Finally, a sensitivity analysis is performed to assess the effective parameters that affect the planning’s results. The results show that the overall pollution is reduced by about 9% by assuming the proposed planning, and the total profit is increased by about 30%.

Correlations of Values of Random Diagonal Forms

Abstract We study the value distribution of diagonal forms in $k$ variables and degree $d$ with random real coefficients and positive integer variables, normalized so that the mean spacing is one. We show that the $\ell $-correlation of almost all such forms is Poissonian when $k$ is large enough depending on $\ell $ and $d$.

Sum of Three Squares is Equal to a Single Square

Sum of three squares is equal to a single square is a new mathematical formula which is true for real integers,fractional numbers and for imaginary numbers also. The formula is built with two consecutive numbers and their product.

A Simple Proof of Bieberbach Conjecture

The famous Riemann mapping theorem, which was established in 1851, states that every proper subdomain of the complex plane that is simply connected (without “holes”) is the image of the unit disk under a univalent analytic mapping f(z). This is the most significant characteristic of univalent analytic functions. The domain D, the image point f(0) in D, and the restriction that f’(0) be a positive real integer each independently define the mapping function f(z). One naturally asks how geometric aspects of the image domain f(D) represent analytic properties of f when considering univalent analytic functions as “Riemann mappings”. This is the perspective of geometric function theory, whose primary issue was solved by de Branges’ proof. The Bieberbach conjecture (hereinafter referred to as “BC”) is essentially a claim on the extreme nature of the Koebe function. Therefore, a key subject in this article is to make an explanation of the Koebe function and intepret why it is the best contender to be “largest” in the sense of the conjecture. Also, a detailed but easy understanding noval proof is given for this conjecture.

On existential definitions of c.e. subsets of rings of functions of characteristic 0

We extend results of Denef, Zahidi, Demeyer and the second author to show the following.(1)Every c.e. set of integers has a single-fold Diophantine definition over the ring of integral functions of any function field of characteristic 0.(2)Every c.e. set of integers has a single-fold Diophantine definition over a polynomial ring over an integral domain Z of characteristic 0.(3)All c.e. subsets of polynomial rings over rings of totally real integers have finite-fold Diophantine definitions. (These are the first examples of infinite rings with this property.)(4)Let K be a one-variable function field over a field of constants k, and let p be any prime of K. If k is algebraic over Q and for some odd prime p embeddable into a finite extension of Qp, then the valuation ring of p has a Diophantine definition over K. If k is embeddable into a real field, then valuation rings are existentially definable for “almost all” primes.(5)Let K be a one-variable function field over a number field and let S be a finite non-empty set of its primes. Then all c.e. subsets of OK,S are Diophantine over OK,S. (Here OK,S is the ring of S-integers or a ring of integral functions.)

Building Pareto Frontiers for Ecosystem Services Tradeoff Analysis in Forest Management Planning Integer Programs

Decision making in modern forest management planning is challenged by the need to recognize multiple ecosystem services and to address the preferences and goals of stakeholders. This research presents an innovative a posteriori preference modeling and multi-objective integer optimization (MOIP) approach encompassing integer programming models and a new technique for generation and interactive visualization of the Pareto frontier. Due to the complexity and size of our management problems, a decomposition approach was used to build the Pareto frontier of the general problem using the Pareto frontiers of its sub-problems. The emphasis was on the approximation of convex Edgeworth–Pareto hulls (EPHs) for the sub-problems by systems of linear inequalities; the generation of Edgeworth–Pareto hulls by the convex approximation of the Pareto frontier evinced a very small discrepancy from the real integer programming solutions. The results thus highlight the possibility of generating the Pareto frontiers of large multi-objective integer problems using our approach. This research innovated the generation of Pareto frontier methods using integer programming in order to address multiple objectives, locational specificity requirements and product even-flow constraints in landscape-level management planning problems. This may contribute to enhancing the analysis of tradeoffs between ecosystem services in large-scale problems and help forest managers address effectively the demand for forest products while sustaining the provision of services in participatory management planning processes.

Singularities of bicomplex holomorphic functions

In the theory of bicomplex holomorphic functions, there is not concept of isolated singularities; that is, such functions do not have singularities just at a point like holomorphic functions in one complex variable. However, there are other type of singularities that behave similarly to the isolated singularities in one complex variable. In this work, we describe how they can be classified in such a way that it resembles the classification made for the complex analysis case. It turns out that to singularities there corresponds their orders which are hyperbolic numbers with integer components, not real integers. We give also the Residue Theorem in the bicomplex analysis setting.

Scenario-based assessment for optimal planning of multi-carrier hub-energy system under dual uncertainties and various scheduling by considering CCUS technology

This paper provides a scenario-based assessment strategy for the optimal scheduling of a hub by considering uncertain parameters (electricity price and wind turbine output power) and carbon capture utilization and storage (CCUS) technology simultaneously. CCUS technology helps overcome pollution issues, on the one hand, and earn revenue for the power system, on the other hand. Three different planning horizons which include deterministic, stochastic, and robust models are investigated and compared. Energy hub (EH) aims to minimize the total cost and environmental pollution simultaneously while considering the effect of objective function priority. The problem is then modeled as real mixed integer programming (RMIP) and is solved in GAMS software by using CPLEX; the results are compared with other solvers. Epsilon constrains (EPC) method and fuzzy satisfying approach are used to select the optimal solution based on the presented model. In the end, a sensitivity analysis is performed. Results show that by utilizing CCUS technology, the cost and pollution for all the planning horizons are reduced properly. CCUS’s profits of about $44.6, $42.7 $, and $50.4 for deterministic, stochastic, and robust planning are obtained, respectively. Moreover, considering function priority has a significant impact on the final results.

A Search For Integral Solutions To The Ternary Bi-Quadratic Equation

This paper deals with the problem of obtaining non-zero distinct integer solutions to the ternary bi-quadratic equation. A few interesting relations among the solution are presented. Given on integer solution of the equation under consideration, integer solutions for various choices of hyperbola and parabolas are exhibited. The formulation of second order Ramanujan Numbers with base numbers as real integers and Gaussian integers is illustrated and also the sequence of Diophantine 3-tuples are exhibited.

Dislocated hybrid synchronization via. tracking control & parameter estimation methods with application

ABSTRACT In this article the dislocated hybrid synchronization between chaotic systems of different structures viz. real integer order and complex fractional order have been performed using two control methods viz. tracking control method and parameter estimation adaptive method. Illustrative example of the application of synchronization in secure communication has been performed. Numerical simulations have been performed using MATLAB which verify the efficacy of both the methods and help compare results.

A Novel Sorting Method for Real and Integer Numbers: An Extension of Counting Sort

Sorting is an essential conceptin the study of data structures. There are many sorting algorithms that can sort elements in a given array or list. Counting sort is a sorting algorithm that has the best time complexity. However, the counting sort algorithm only works for positive integers. In this paper, an extension of the counting sort algorithm is proposed that can sort real numbers and integers (both positive and negative).

Optimal design for cryogenic structured packing column using particle swarm optimization algorithm

Large scale cryogenic air separation is the most efficient and cost-effective approach to produce high-purity air products by present. Structured packing columns (SPC) are widely focused and applied due to their characteristics of high efficiency and energy saving in the cryogenic distillation process. The optimal design of the SPC is to reduce the energy consumption and initial investment, while it is a highly nonlinear and multivariable problem. The coexistence of real variables and integer variables, such as the flow rates and the positions of materials at the inlets/outlets, makes the optimization become a typical mixed integer nonlinear programming (MINLP) problem. The purpose of this paper is to study the optimal design method for the cryogenic SPC using the particle swarm optimization (PSO) algorithm. A modified PSO for handling the MINLP problem (MI-PSO) is proposed. A multi-objective optimal design for the SPC in cryogenic air separation plant with the capacity of 17,000 Nm3/h is investigated as a instance. By MI-PSO algorithm, the total exergy loss theoretically reduces 36.3% and the main condenser heat load decreases 5.4% after optimization, which can provide test prediction for the cryogenic distillation experiment.

Design and Implementation of Real Time Integer Order PID Controller for Infrared Heater

A modified technique is introduced to understand the design of Integer order PID controller. Due to simple tuning rules, conventional PID controller has the great success with the automatic tuning feature and tables that simplify their design. The Integer order PID controller show better robustness performance with the comparison of PID control's experimental to Heating control system (HIL). In Integer order control, design of simulation model of system can be designed straight forwardly with design specification based on frequency analysis which can be change continuously. The simulation analysis and real time results shows that Integer order PID controller is more effective way to enhance the system control performance.

Real option-based optimization for financial incentive allocation in infrastructure projects under public-private partnerships

Financial incentives that stimulate energy investments under public-private partnerships are considered scarce public resources, which require deliberate allocation to the most economically justified projects to maximize the social benefits. This study aims to solve the financial incentive allocation problem through a real option-based nonlinear integer programming approach. Real option theory is leveraged to determine the optimal timing and the corresponding option value of providing financial incentives. The ambiguity in the evolution of social benefits, the decision-maker’s attitude toward ambiguity, and the uncertainty in social benefits and incentive costs are all considered. Incentives are offered to the project portfolio that generates the maximum total option value. The project portfolio selection is formulated as a stochastic knapsack problem with random option values in the objective function and random incentive costs in the probabilistic budget constraint. The linear probabilistic budget constraint is subsequently transformed into a nonlinear deterministic one. Finally, the integer non-linear programming problem is solved, and the optimality gap is computed to assess the quality of the optimal solution. A case study is presented to illustrate how the limited financial incentives can be optimally allocated under uncertainty and ambiguity, which demonstrates the efficacy of the proposed method.

Fraction-free computation of the unit-circle resultant with any singularity profile

We show that an algorithm established in a recent publication in this journal to provide a fraction-free version for the Bistritz test can also be used to compute the unit-circle resultant. The consequence is an effective new algorithm to find this resultant for any complex polynomial that is also an efficient integer algorithm for polynomial with real or Gaussian integer coefficients. The algorithm ends always with the resultant expressed as the minimal degree multivariate polynomial of the polynomial's (indeterminate) coefficients. A simpler form of the algorithm, that equally reaches this expression in all circumstances, but is not guaranteed to stay over integers during possible intermediate abnormal gaps, is also suggested along with the simplest form that the algorithm takes in the strongly regular case.

Symmetric tensor categories in characteristic 2

We construct and study a nested sequence of finite symmetric tensor categories Vec=C0⊂C1⊂⋯⊂Cn⊂⋯ over a field of characteristic 2 such that C2n are incompressible, i.e., do not admit tensor functors into tensor categories of smaller Frobenius–Perron dimension. This generalizes the category C1 described by Venkatesh [28] and the category C2 defined by Ostrik. The Grothendieck rings of the categories C2n and C2n+1 are both isomorphic to the ring of real cyclotomic integers defined by a primitive 2n+2-th root of unity, On=Z[2cos⁡(π/2n+1)].