Distinct Real Numbers Research Articles

Hereditary completeness of exponential systems [formula omitted] in their closed span in L2(a,b) and spectral synthesis

Suppose that {λn}n=1∞ is a sequence of distinct positive real numbers satisfying the conditions inf{λn+1−λn}>0, and ∑n=1∞λn−1<∞. We prove that the exponential system {eλnt}n=1∞ is hereditarily complete in the closure of the subspace spanned by {eλnt}n=1∞ in the space L2(a,b). We also give an example of a class of compact non-normal operators defined on this closure which admit spectral synthesis.

On eigenvalues of certain special matrices

Let g : R → [ 0 , ∞ ) be a conditionally negative definite function and f : [ 0 , ∞ ) → [ 0 , ∞ ) be a Bernstein function. We prove that the function h = f ∘ g is conditionally negative definite and that for distinct real numbers p 1 , p 2 , … , p n the inertia of the matrix [ h ( p i − p j ) ] is ( 1 , 0 , n − 1 ) if f is non-linear and g ( x ) = 0 only for x = 0. We also present a new and simple proof to show that the matrix [ log ⁡ ( 1 − p i p j ) ] is negative definite for n distinct positive real numbers p i < 1 , ∀ i . These results supplement and unify earlier results proved by a number of authors including Dyn, Goodman and Michelli, Bhatia and Jain, Garg and Aujla, Garg and Agarwal for operator monotone functions.

A New Architecture of a Complex-Valued Convolutional Neural Network for PolSAR Image Classification

Polarimetric synthetic aperture radar (PolSAR) image classification has been an important area of research due to its wide range of applications. Traditional machine learning methods were insufficient in achieving satisfactory results before the advent of deep learning. Results have significantly improved with the widespread use of deep learning in PolSAR image classification. However, the challenge of reconciling the complex-valued inputs of PolSAR images with the real-valued models of deep learning remains unsolved. Current complex-valued deep learning models treat complex numbers as two distinct real numbers, providing limited assistance in PolSAR image classification results. This paper proposes a novel, complex-valued deep learning approach for PolSAR image classification to address this issue. The approach includes amplitude-based max pooling, complex-valued nonlinear activation, and a cross-entropy loss function based on complex-valued probability. Amplitude-based max pooling reduces computational effort while preserving the most valuable complex-valued features. Complex-valued nonlinear activation maps feature into a high-dimensional complex-domain space, producing the most discriminative features. The complex-valued cross-entropy loss function computes the classification loss using the complex-valued model output and dataset labels, resulting in more accurate and robust classification results. The proposed method was applied to a shallow CNN, deep CNN, FCN, and SegNet, and its effectiveness was verified on three public datasets. The results showed that the method achieved optimal classification results on any model and dataset.

Determine matrices receiving given pairwise distinct real numbers as eigenvalues

The paper shows a method to give matrices that receive given pairwise distinct real numbers as eigenvalues. Thereby, multiple examples and exercises are provided with given answers. The method applied does not involve handheld calculator use.

A divide-and-conquer method for constructing a pseudo-Jacobi matrix from mixed given data

For the given signature operator H=Ir⊕−In−r, a pseudo-Jacobi matrix is a self-adjoint matrix relatively to a symmetric bilinear form 〈⋅,⋅〉H, and it is the counterpart of a classical Jacobi matrix to the indefinite scalar product space setting. In this article, we consider an inverse eigenvalue problem for this class of matrices. The main concern is to construct an n×n pseudo-Jacobi matrix from a prescribed n-tuple of distinct real numbers and a Jacobi matrix of order not less than ⌊n2⌋, such that its spectrum is this tuple and the given Jacobi matrix is its trailing principal submatrix. A divide-and-conquer scheme is used to solve this problem, and a necessary and sufficient condition under which the problem is solvable is presented. A numerical algorithm is designed to solve this pseudo-Jacobi matrix inverse eigenvalue problem according to the obtained results. Some illustrative numerical examples are also given to test the reconstructive algorithm.

Joint ergodicity of piecewise monotone interval maps

For , let µ i be a Borel probability measure on which is equivalent to the Lebesgue measure λ and let be µ i -preserving ergodic transformations. We say that transformations are uniformly jointly ergodic with respect to if for any , limN−M→∞1N−M∑n=MN−1f0(T0 nx)⋅f1(T1 nx)⋯fk(Tk nx)=∏i=0k∫fidμi in L2(λ). We establish convenient criteria for uniform joint ergodicity and obtain numerous applications, most of which deal with interval maps. Here is a description of one such application. Let T G denote the Gauss map, , and, for β > 1, let T β denote the β-transformation defined by . Let T 0 be an ergodic interval exchange transformation. Let be distinct real numbers with and assume that for all . Then for any , limN−M→∞1N−M∑n=MN−1f0(T0nx)⋅f1(Tβ1nx)⋯fk(Tβknx)⋅fk+1(TGnx)=∫f0dλ⋅∏i=1k∫fidμβi⋅∫fk+1dμGin L2(λ). We also study the phenomenon of joint mixing. Among other things we establish joint mixing for skew tent maps and for restrictions of finite Blaschke products to the unit circle.

Erdős–Szekeres theorem for multidimensional arrays

The classical Erdős–Szekeres theorem dating back almost a hundred years states that any sequence of (n-1)^2 + 1 distinct real numbers contains a monotone subsequence of length n . This theorem has been generalised to higher dimensions in a variety of ways but perhaps the most natural one was proposed by Fishburn and Graham more than 25 years ago. They defined the concept of a monotone and a lex-monotone array and asked how large an array one needs in order to be able to find a monotone or a lex-monotone subarray of size n \times \cdots \times n . Fishburn and Graham obtained Ackerman-type bounds in both cases. We significantly improve these results. Regardless of the dimension we obtain at most a triple exponential bound in n in the monotone case and a quadruple exponential one in the lex-monotone case.

Finding structure in sequences of real numbers via graph theory: a problem list

We investigate a method of generating a graph $G=(V,E)$ out of an ordered list of $n$ distinct real numbers $a_1, \dots, a_n$. These graphs can be used to test for the presence of interesting structure in the sequence. We describe sequences exhibiting intricate hidden structure that was discovered this way. Our list includes sequences of Deutsch, Erd\H{o}s, Freud & Hegyvari, Recaman, Quet, Zabolotskiy and Zizka. Since our observations are mostly empirical, each sequence in the list is an open problem.

Maximum independent sets in a proper monograph determined through a signature

Let [Formula: see text] be a non-empty, finite graph. If the vertices of G can be bijectively labeled by a set S of positive distinct real numbers with two vertices being adjacent if and only if the positive difference of the corresponding labels is in S, then G is called a proper monograph. The set S is called the signature of G and denoted as G(S). Not all proper monographs have the property that a set of idle vertices can be bijectively mapped to the maximum independent sets. As a result, in this paper, we present the proper monograph labelings of several classes of graphs that satisfy the property mentioned above. We present the proper monograph labelings of graphs such as cycles, [Formula: see text], cycles with paths attached to one or more vertices, and Cycles with an irreducible tree attached to one or more vertices.

Fast and Longest Rollercoasters

For kge 3, a k-rollercoaster is a sequence of numbers whose every maximal contiguous subsequence, that is increasing or decreasing, has length at least k; 3-rollercoasters are called simply rollercoasters. Given a sequence of distinct real numbers, we are interested in computing its maximum-length (not necessarily contiguous) subsequence that is a k-rollercoaster. Biedl et al. (in: ICALP, volume 107 of LIPIcs. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, pp 18:1–18:15, 2018) have shown that each sequence of n distinct real numbers contains a rollercoaster of length at least lceil n/2rceil for n>7, and that a longest rollercoaster contained in such a sequence can be computed in O(nlog n)-time (or faster, in O(n log log n) time, when the input sequence is a permutation of {1,ldots ,n}). They have also shown that every sequence of ngeqslant (k-1)^2+1 distinct real numbers contains a k-rollercoaster of length at least frac{n}{2(k-1)}-frac{3k}{2}, and gave an O(nklog n)-time (respectively, O(n klog log n)-time) algorithm computing a longest k-rollercoaster in a sequence of length n (respectively, a permutation of {1,ldots ,n}). In this paper, we give an O(nk^2)-time algorithm computing the length of a longest k-rollercoaster contained in a sequence of n distinct real numbers; hence, for constant k, our algorithm computes the length of a longest k-rollercoaster in optimal linear time. The algorithm can be easily adapted to output the respective k-rollercoaster. In particular, this improves the results of Biedl et al. (2018), by showing that a longest rollercoaster can be computed in optimal linear time. We also present an algorithm computing the length of a longest k-rollercoaster in O(n log ^2 n)-time, that is, subquadratic even for large values of kle n. Again, the rollercoaster can be easily retrieved. Finally, we show an Omega (n log k) lower bound for the number of comparisons in any comparison-based algorithm computing the length of a longest k-rollercoaster.

Multithreshold multipartite graphs

AbstractWe call a graph a ‐threshold graph if there are distinct real numbers and a mapping such that for any two vertices , we have that if and only if there are odd numbers such that . The least integer such that is a ‐threshold graph is called a threshold number of , and denoted by . The well‐known family of threshold graphs is a set of graphs with . Jamison and Sprague introduced the concept of ‐threshold graph, and proved that exists for every graph . They further obtained a number of interesting results on . In addition, they also proposed several unsolved problems and conjectures, including the following two. Problem: Determine the exact threshold numbers of the complete multipartite graphs. Conjecture: For all even , there is a graph with and . This is equivalent to that for all odd , there is a graph with and , where is the complement of .In this short paper, we give a partial solution of the problem and confirm the conjecture.

New Simpson Type Integral Inequalities for s -Convex Functions and Their Applications

First, we consider a new Simpson’s identity. This identity investigates our main results that consist of some integral inequalities of Simpson’s type for the s –convex functions. From our main results, we obtain some special cases which are discussed in detail. Finally, some applications on the Bessel functions, special means of distinct positive real numbers, and error estimation about Simpson quadrature formula are presented to support our theoretical results.

Determinants of some special matrices

ABSTRACT Let be distinct positive real numbers and m be any integer. Every symmetric polynomial induces a symmetric matrix We obtain the determinants of such matrices with an aim to find the determinants of and for (where is the set of natural numbers) in terms of the Schur polynomials. We also discuss and compute determinant of the matrix for any integer m in terms of the Schur and skew-Schur polynomials.

Functions with identical [formula omitted] norms

Suppose P≔(pj)j=1∞ is a sequence of distinct real numbers pj>0. We prove that the equalities ‖f‖p=‖g‖p,p∈P, imply μ({x∈E:|f(x)|<α})=μ({x∈E:|g(x)|<α}),α≥0, whenever 0<μ(E)<∞ and f,g∈L∞(E) if and only if ∑j=1∞pjpj2+1=∞.

Inertia of some conditionally negative definite matrices

Let $$f:[0,\infty )\rightarrow [0,\infty )$$ be a non-linear operator monotone function and $$g:\mathbb {R}\rightarrow [0,\infty )$$ be a cnd function such that $$g(x)=0$$ only at $$x=0.$$ Let $$p_1,p_2,\ldots ,p_n$$ be any distinct real numbers. Then the matrix $$(f(g(p_i-p_j)))$$ is cnd, see (Kapil et al. in Mediterr J Math 15:199, 2018). We aim to obtain the inertia of such matrices and several other cnd matrices which are arising from the functions having the Weierstrass factorization. These generalize and subsume several existing results.

Matrix recursive polynomial interpolation algorithm: An algorithm for computing the interpolation polynomials

Let x0,x1,…,xn, be a set of n+1 distinct real numbers (i.e., xi≠xj, for i≠j) and ym,k, for m=0,1,…,n, and k=0,1,…,nm, with nm∈N, be given of real numbers, we know that there exists a unique polynomial pN−1 of degree N−1 where N=∑i=0n(ni+1), such that pN−1(k)(xm)=ym,k, for m=0,1,…,n and k=0,1,…,nm. pN−1 is the Hermite interpolation polynomial for the set {(xm,ym,k),m=0,1,…,n,k=0,1,…,nm}. The polynomial pN−1 can be computed by using the Lagrange generalized polynomials. Recently Messaoudi et al. (2018) presented a new algorithm for computing the Hermite interpolation polynomials, for a general case, called Generalized Recursive Polynomial Interpolation Algorithm (GRPIA), this algorithm has been developed without using the Matrix Recursive Interpolation Algorithm (Jbilou and Messaoudi, 1999). Messaoudi et al. (2017) presented also a new algorithm called Matrix Recursive Polynomial Interpolation Algorithm (MRPIA), for a particular case where nm=μ=1, for m=0,1,…,n. In this paper we will give the version of the MRPIA for a particular case nm=μ≥0, for m=0,1,…,n. We will recall the result of the existence of the polynomial pN−1 for this case, some of its properties will also be given. Using the MRPIA, a method will be proposed for the general case, where nm, for some m, are different and some examples will also be given.

A Generalization of Bohr’s Equivalence Theorem

Based on a generalization of Bohr’s equivalence relation for general Dirichlet series, in this paper we study the sets of values taken by certain classes of equivalent almost periodic functions in their strips of almost periodicity. In fact, the main result of this paper consists of a result like Bohr’s equivalence theorem extended to the case of these classes of functions that are associated with countable sets of distinct real numbers having an integral basis.

A Generalized Version of a Local Antimagic Labelling Conjecture

An antimagic labelling of a graph G with m edges is a bijection $$f: E(G) \rightarrow \{1,\ldots ,m\}$$ such that for any two distinct vertices u, v we have $$\sum _{e \in E(v)}f(e) \ne \sum _{e \in E(u)}f(e)$$ , where E(v) denotes the set of edges incident v. The well-known Antimagic Labelling Conjecture formulated in 1994 by Hartsfield and Ringel states that any connected graph different from $$K_2$$ admits an antimagic labelling. A weaker local version which we call the Local Antimagic Labelling Conjecture says that if G is a graph distinct from $$K_2$$ , then there exists a bijection $$f: E(G) \rightarrow \{1,\ldots ,|E(G)|\}$$ such that for any two neighbours u, v we have $$\sum _{e \in E(v)}f(e) \ne \sum _{e \in E(u)}f(e)$$ . This paper proves the following more general list version of the local antimagic labelling conjecture: Let G be a connected graph with m edges which is not a star. For any list L of m distinct real numbers, there is a bijection $$f:E(G) \rightarrow L$$ such that for any pair of neighbours u, v we have that $$\sum _{e \in E(v)}f(e) \ne \sum _{e \in E(u)}f(e)$$ .

A characterization of dual quermassintegrals and the roots of dual Steiner polynomials

Let m≥1, (r0=0,r1,…,rm) be a tuple of distinct real numbers and n≥2. We provide a characterization of those tuples (ω0,ω1,…,ωm) of real numbers such that there exist n-dimensional star bodies K,L with W˜rj(K,L)=ωj, j=0,…,m, where W˜r(K,L) denotes the r-th dual (relative) quermassintegral of K and L. This may be regarded as an analogue within the dual Brunn–Minkowski theory of Shephard's classification of quermassintegrals of two convex bodies.It turns out that the characterization of dual quermassintegrals is related to the moment problem, and based on this relation, we also derive new determinantal inequalities among the dual quermassintegrals. Moreover, this characterization will be the key tool in order to investigate structural properties of the set of roots of dual Steiner polynomials of star bodies.

GRPIA: a new algorithm for computing interpolation polynomials

Let x0,x1, ⋯ , xn, be a set of n + 1 distinct real numbers (i.e., xm ≠ xj, for m ≠ j) and let ym,k, for m = 0, 1, ⋯ , n, and k = 0, 1, ⋯ , rm, with rm ∈ IN, be given real numbers. It is known that there exists a unique polynomial pN− 1 of degree N − 1 with $N={\sum }_{m = 0}^{n}(r_{m}+ 1)$ , such that $p_{N-1}^{(k)}(x_{m})=y_{m,k}$ , for m = 0, 1, ⋯ , n and k = 0, ⋯ , rm. pN− 1 is the Hermite interpolation polynomial for the set {(xm, ym,k), m = 0, 1, ⋯ , n, k = 0, 1, ⋯ , rm}. The polynomial pN− 1 can be computed by using the Lagrange generalized polynomials. Recently, Messaoudi et al. (2017) presented a new algorithm for computing the Hermite interpolation polynomial called the Matrix Recursive Polynomial Interpolation Algorithm (MRPIA), for a particular case where rm = μ = 1, for m = 0, 1, ⋯ , n. In this paper, we will give a new formulation of the Hermite polynomial interpolation problem and derive a new algorithm, called the Generalized Recursive Polynomial Interpolation Algorithm (GRPIA), for computing the Hermite polynomial interpolation in the general case. A new result of the existence of the polynomial pN− 1 will also be established, cost and storage of this algorithm will also be studied, and some examples will be given.