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In this article, the author shows that the equilibrium equations of any one-dimensional theory can be obtained by direct manipulation of the indefinite 3D elasticity equations: σ i j , i + p j = 0. It is found that there is no need to refer to Newton-type or Lagrange-type equations to write the equilibrium in terms of stress resultants. Such 'direct manipulation' is illustrated by considering various problems, from the simplest to the most complex. First, the bar case is analyzed and the well-known equilibrium normal stress resultant equation is obtained: N z z , z + q z = 0 ; second, the Timoshenko beam theory is tackled; finally, the case of arbitrary higher order beam theories based on the Carrera Unified Formulation is treated. The equations of a new, unprecedented higher-order beam theory are derived directly from the indefinite equilibrium equations as an example of the capabilities of the proposed method. The manipulation introduced consists of the following points: (i) the 3D elasticity is first written for each degree of freedom (dof) of the beam theories; (ii) the respective stress resultants are used to replace stress components in the latter equations; (iii) the derivatives on the beam axis z do not affect the nature of the indefinite equilibrium equations and these derivatives are applied directly to the respective stress resultants; (iv) the derivatives on the section coordinate x – y should 1 – change the sign of the respective stress resultants, 2 – be applied to the functions of x, y used to define the stress resultants. These points have been proved by referring to the principle of virtual work and splitting the differential operators of the strain-displacement relations into two parts: over the section derivatives along the beam axis derivatives. Some possible further generalizations are discussed in the appendix; in particular, the extension to is briefly discussed. [ABSTRACT FROM AUTHOR]

In this paper, we study the (classical) Frobenius problem, namely the problem of finding the largest integer that cannot be represented as a nonnegative integer combination of given, relatively prime, (strictly) positive integers (known as the Frobenius number). The main contribution of this paper are observations regarding a previously known upper bound on the Frobenius number where, in particular, we observe that a previously presented argument features a subtle error, which alters the value of the upper bound. Despite this, we demonstrate that the subtle error does not impact upon on the validity of the upper bound, although it does impact on the upper bounds tightness. Notably, we formally state the corrected result and additionally compare the relative tightness of the corrected upper bound with the original. In particular, we show that the updated bound is tighter in all but only a relatively "small" number of cases using both formal techniques and via Monte Carlo simulation techniques. [ABSTRACT FROM AUTHOR]

We give an algorithm that produces all solutions of the equation ∑i=1n1/x i = 1 in integers of the form 2akb, where k is a fixed positive integer that is not a power of 2, a is an element of {0, 1, 2} that can vary from term to term, and b is a nonnegative integer that can vary from term to term. We also completely characterize the pairs (k,n) for which this equation has a nontrivial solution in integers of this form. [ABSTRACT FROM AUTHOR]

The article "Heron Triangles and the Hunt for Unicorns" delves into the search for Pythagorean triples and Heron triangles with rational sides and medians. It explores the concept of perfect triangles with integer sides, medians, and area, as well as the mathematical beauty of Somos-5 sequences and their connection to integrable maps. The text discusses the work of Buchholz and Rathbun in finding Heron triangles with two rational medians using specific algorithms, presenting tables and equations to illustrate the relationships between the parameters of these triangles. It also mentions the existence of an infinite family of such triangles and sporadic cases that do not conform to this family, highlighting the intricate mathematical patterns and challenges in studying these triangles. [Extracted from the article]

Equation discovery, also known as symbolic regression, is the field of machine learning that studies algorithms for discovering quantitative laws, expressed as closed-form equations or formulas, in collections of observed data. The latter is expected to come from measurements of physical systems and, therefore, noisy, moving the focus of equation discovery algorithms towards discovering approximate equations. These loosely match the noisy observed data, rendering them inappropriate for applications in mathematics. In this article, we introduce Diofantos, an algorithm for discovering equations in the ring of integers that exactly match the training data. Diofantos is based on a reformulation of the equation discovery task into the task of solving linear Diophantine equations. We empirically evaluate the performance of Diofantos on reconstructing known equations for more than 27,000 sequences from the online encyclopedia of integer sequences, OEIS. Diofantos successfully reconstructs more than 90% of these equations and clearly outperforms SINDy, a state-of-the-art method for discovering approximate equations, that achieves a reconstruction rate of less than 70%. [ABSTRACT FROM AUTHOR]

In this paper, we establish a new linking theorem for local Lipschitz functionals without the τ -upper semi-continuity assumption. As an application, we study the following equation with strongly indefinite structure and sign-changing discontinuous nonlinearity - Δ u + V (x) u = | u | p - 2 u - μ H e (a - | u |) | u | q - 2 u , a.e. in R N , u ∈ H 1 (R N) , N ≥ 2 , where V is 1-periodic and 0 lies in a gap of the spectrum of - Δ + V , 2 < q < p < 2 ∗ and μ , a > 0 are parameters. We prove the existence of a nontrivial solution of the equation above when μ > 0 is small enough. [ABSTRACT FROM AUTHOR]

This paper proposes a fractal modification of tropical algebra for noise removal and optimal control. Fractal addition, fractal multiplication, and fractal dot product are defined and explained in a fractal space. A fractal tropical polynomials function can model a coastline at any scale, providing a new tool for fractal analysis of any irregular curve. An example is given to solve an optimal control subject to a fractal tropical polynomials function. This paper provides a new window for tropical algebra, combining fractal geometry and algebra. [ABSTRACT FROM AUTHOR]

The so-called Frobenius number in the famous linear Diophantine problem of Frobenius is the largest integer such that the linear equation a 1 x 1 + ⋯ + a k x k = n ( a 1 , ... , a k are given positive integers with gcd (a 1 , ... , a k) = 1) does not have a non-negative integer solution (x 1 , ... , x k). The generalized Frobenius number (called the p -Frobenius number) is the largest integer such that this linear equation has at most p solutions. That is, when p = 0 , the 0 -Frobenius number is the original Frobenius number. In this paper, we introduce and discuss p -numerical semigroups by developing a generalization of the theory of numerical semigroups based on this flow of the number of representations. That is, for a certain non-negative integer p , p -gaps, p -symmetric semigroups, p -pseudo-symmetric semigroups, and the like are defined, and their properties are obtained. When p = 0 , they correspond to the original gaps, symmetric semigroups and pseudo-symmetric semigroups, respectively. [ABSTRACT FROM AUTHOR]

The steering tool combined bearing is a mechanical component that can support forces, guide movement, and bear vibrations at the same time. In the process of analyzing the contact and load distribution inside the combined bearing, there are several contact points, and the static equilibrium equation is statically indeterminate. Based on dynamics theory and Hertz elastic contact theory, the concept of a load integral and the displacement estimation formula for roller bearings are introduced, and a numerical solution to the statically indeterminate problem of the steering tool combined bearing is given. According to the statically indeterminate equation matrix of the combined bearing, the relationship between the load and displacement of the combined bearing is studied to improve the statically indeterminate equation matrix. Moreover, the simulation results of the maximum stress and displacement distribution results of the inner and outer ring eccentricity from 1 to 6mm are compared with the algorithm prediction results. The error between the numerical solution of the statically indeterminate problem of the steering tool combined bearing and the simulation result is small, improving the calculation efficiency. This provides theoretical support for the design and reasonable application of the steering tool combined bearing. [ABSTRACT FROM AUTHOR]

We consider stationary Schrödinger equations with indefinite potential and nonlinearity sublinear at u = 0. Using linking theorem and symmetric mountain pass theorem, existence and multiplicity results are obtained. [ABSTRACT FROM AUTHOR]

In this paper we consider some fragments of IOpen (Robinson arithmetic Q with induction for quantifier-free formulas) proposed by Harvey Friedman and answer some questions he asked about these theories. We prove that I (lit) is equivalent to IOpen and is not finitely axiomatizable over Q , establish some inclusion relations between I (=) , I (≠) , I (⩽) and I (⩽̸) . We also prove that the set of diophantine equations solvable in models of I (=) is (algorithmically) decidable. [ABSTRACT FROM AUTHOR]

Let D be a noncommutative division ring. In a recent paper, Lee and Lin proved that if char D ≠ 2 , the only solution of additive maps f , g on D satisfying the identity f (x) = x n g (x − 1) on D ∖ { 0 } with n ≠ 2 a positive integer is the trivial case, that is, f = 0 and g = 0. Applying Hua's identity and the theory of functional and generalized polynomial identities, we give a complete solution of the same identity for any nonnegative integer n if char D = 2. [ABSTRACT FROM AUTHOR]

In this paper, we find all the solutions of the Diophantine equation from the title. [ABSTRACT FROM AUTHOR]

AbstractConsider a pair of right triangles with an equal hypotenuse. This turns out to solve the diophantine system of equations a2 + b2 = c2 + d2 = e2 in integers. To this system we associate a family of elliptic curves with defining equation y2 = (x−a2)(x−b2)(x−c2)+a2b2c2. We show that there exists a subfamily of rank ≥ 3 over ℚ(m, n, k, ℓ) and obtain a subfamily of rank (exactly) four over ℚ(k) and determine a set of its free generators. Besides, we show there exist infinitely many elliptic curves of rank ≥ 5 parameterized by a rank five quartic elliptic curve. We also find a few particular examples with higher ranks. The families we construct have ℤ/2ℤ torsion subgroups in general. The previous work [13] has obtained similar Pythagorean quadruplet elliptic curve families in two parameters with rank ≥ 3. (Recall that by a Pythagorean quadruplet (a, b, c, d), we mean an integer solution to the quadratic equation a2 + b2 = c2 + d2.) [ABSTRACT FROM AUTHOR]

In this paper, based on the Diophantine equation technique, we first prove the circulant graph C (Z n , { k 1 , k 2 }) is isomorphic to the disjoint union of gcd (n , k 1 , k 2) identical graphs, which belong to one of the three types of graphs: (i) C (Z n { 1 , k }) ; (i i) the Cartesian graph bundle C s □ φ C t over cycles; (i i i) the Cartesian product of the complete graph K 2 and a cycle. Naturally, to study the dispersability of C (Z n , { k 1 , k 2 }) can be reduced to study that of the above three types. Next, we solve the dispersability of the circulant graph C (Z n , { 1 , k }). Specifically, C (Z n , { 1 , k }) is dispersable if it is a bipartite; otherwise, it is nearly dispersable. In addition, we prove that bipartite circulant graphs are dispersable. [ABSTRACT FROM AUTHOR]

We say that a simple graph G is Seidel integral if its Seidel spectrum consists entirely of integers. In this work we establish a characterization of Seidel integral graphs which belong to the class αKa ∪ βKb,b, where mG denotes the m-fold union of the graph G. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study balancing hybrinomials, i.e., polynomials being a generalization of balancing hybrid numbers. We provide some properties of the balancing hybrinomials, including Catalan, Cassini, d'Ocagne, and Vajda identities, among others. Moreover, we present a matrix representation of the hybrinomials. [ABSTRACT FROM AUTHOR]