Representation Of Integers Research Articles

Representations of integers as sums of four polygonal numbers and partial theta functions

AbstractIn this paper, we consider representations of integers as sums of at most four distinct polygonal numbers with a prescribed number of repeats of each distinct polygonal number. We compare such representations with classical polygonal numbers, and those representations with generalized polygonal numbers. Our main result is that representations with classical polygonal numbers are equidistributed in the sense that the number of representations in the nonnegative quadrant in four-dimensional space is asymptotically $$\frac{1}{16}$$ 1 16 of the representations in the entire space.

Algorithms for representing positive odd integers as the sum of arithmetic progressions

This paper delves into the historical and recent developments in this area of mathematical inquiry, tracing the evolution from Wheatstone’s representation of powers of an integer as sums of arithmetic progressions to extensions of Sylvester’s Theorem (Sylvester and Franklin, [14]). Sylvester’s Theorem, a result that determines the representability of positive integers as sums of consecutive integers, has been the foundation for numerous extensions, including the representation of integers as sums of specific arithmetic progressions and powers of such progressions. The recent works of Ho et al. [3] and Ho et al. [4] have further expanded on Sylvester’s Theorem, offering a procedural approach to compute the representability of positive integers in the context of arithmetic progressions. In this paper, efficient algorithms to compute the number of ways to represent an odd positive integer as sums of powers of arithmetic progressions are presented.

Powers as Fibonacci sums

We examine the equation for positive integers y, a ≥ 2 and k ≥ 3. This equation can be expressed as a problem in terms of the Zeckendorf representations of integers. Using bounds on linear forms in logarithms and Baker-Davenport reduction methods, we are able to completely solve the equation for y ≤ 25000 when k = 3, for y ≤ 1000 when k = 4, for y ≤ 40 when k = 5, and for y ≤ 3 when k = 6.

Integer representations of classical Weyl groups

Integer representations of classical Weyl groups

Explicit evaluation of triple convolution sums of the divisor functions

In this paper, we use the theory of modular forms and give a general method to obtain the convolution sums [Formula: see text] for odd integers [Formula: see text] and [Formula: see text], where [Formula: see text] is the sum of the [Formula: see text]th powers of the positive divisors of [Formula: see text]. We consider four cases, namely (i) [Formula: see text], (ii) [Formula: see text]; [Formula: see text] (iii) [Formula: see text]; [Formula: see text] and (iv) [Formula: see text], and give explicit expressions for the respective convolution sums. We provide several examples of these convolution sums in each case and further use these formulas to obtain explicit formulas for the number of representations of a positive integer [Formula: see text] by certain positive definite quadratic forms. The existing formulas for [Formula: see text] (in [20]), [Formula: see text] (in [7]), [Formula: see text] (in [35]), [Formula: see text], [Formula: see text] (in [30]) and [Formula: see text] (in [31]), which were all obtained by using the theory of quasimodular forms, follow from our method.

Challenges in data representation for efficient execution of encryption operation

Big number operation has always been a bottleneck to computer system as it imposes high demand on computing power. With a limited power available, operations such as exponentiation and multiplication involving large integer belonging to encryption process requires grave scrutiny. One way to address this issue is by replacing an original complex computation into a sequence of small computations that in the end produces the same results. This paper takes an evolutionary approach to survey numerous articles that have contributed to the advancement of integer representation. Numerous representations were proposed, those that come into play concentrated on reducing non-zero digits and limiting non-zero spacing other than allowing subtraction operation. A comparison was made to distinguish the properties of each method from the others. This detailed outlook can be a guide for identifying the correct representation to be chosen for implementation within specific application.

Testability of positive integers

Given a set [Formula: see text] of some positive integers, a positive integer [Formula: see text] is [Formula: see text]-testable if there are two disjoint subsets [Formula: see text] and [Formula: see text] of [Formula: see text] such that [Formula: see text]. The testability of the set [Formula: see text] of positive integers not exceeding [Formula: see text], denoted by [Formula: see text], is the minimum integer [Formula: see text] such that there is a set [Formula: see text] with [Formula: see text] positive integers satisfying that each [Formula: see text] is [Formula: see text]-testable. In this short paper, we prove that [Formula: see text] and propose an algorithm to output the representation of an arbitrary positive integer by the form [Formula: see text], where [Formula: see text] and [Formula: see text] are two disjoint sets of some non-negative integers not exceeding [Formula: see text].

Algebraic Representation of Primes by Hybrid Factorization

The representation of integers by prime factorization, proved by Euclid in the Fundamental Theorem of Arithmetic −also referred to as the Prime Factorization Theorem− although universal in scope, does not provide insight into the algebraic structure of primes themselves. No such insight is gained by summative prime factorization either, where a number can be represented as a sum of up to three primes, assuming Goldbach’s conjecture is true. In this paper, a third type of factorization is introduced, called hybrid prime factorization, defined as the representation of a number as sum −or difference− of two products of primes with no common factors between them. By using hybrid factorization, primes are expressed as algebraic functions of other primes, and primality is established by a single algebraic condition. Following a hybrid factorization approach, sufficient conditions for the existence of Goldbach pairs are derived, and their values are algebraically evaluated, based on the symmetry exhibited by Goldbach primes around their midpoint. Hybrid prime factorization is an effective way to represent, predict, compute, and analyze primes, expressed as algebraic functions. It is shown that the sequence of primes can be generated through an algebraic process with evolutionary properties. Since prime numbers do not follow any predetermined pattern, proving that they can be represented, computed and analyzed algebraically has important practical and theoretical ramifications.

COVID-ECG-RSNet: COVID-19 classification from ECG images using swish-based improved ResNet model

Millions of humansworldwide have been impacted by consecutive COVID-19waves during the past threeyears. The correct treatment for COVID-19 has not yet been identified because vaccine receivers have also contracted the illness. People’s lives can be saved, and patients can be shielded from laborious treatments with the quick and accurate diagnosis of coronavirus. Investigators have utilized various clinical imaging modalities i.e.,CT-Scan and X-ray for the identification of coronavirus, while, less emphasis is spent on the utilization ofECG images in recognizing the COVID-19-affected patients. In contrast to CT-Scan and X-ray images, ECG samples arereadily accessible, hence we utilize them to diagnose coronavirus. Since scientists generally transform the ECG data into integer representation before using any approach, which indicates higher computing overhead. Accurateand quickcoronavirus identification from the ECG modalityis a difficult and cumbersome operation. In this study, we attempted to address these issues by leveraging the ECG filesdirectly in an improved deep learning (DL)methodology called the COVID-ECG-RSNet. Descriptively, a novel DL model is presented by altering the ResNet-50 model by introducing an optimized activation approach called the Swish method during the feature engineering process. Moreover, we introduced multiple dense layers at the end of the proposed CNN structure to ensure more robust sample features for classification purposes. The introduced COVID-ECG-RSNet model is proficient in categorizing the ECG samples as being normal, coronavirus, myocardial infarction (MI), abnormal heartbeats (AHB), and victims with Previous history of Myocardial Infarction (PMI) groups. A vast evaluation with the help of a complicated and publically accessible data sample of ECG images is accomplished to show the effectiveness of our approach in recognizing the coronavirus images in contrast to other categories of heart diseases and healthy samples via using the ECG modality. We have attained an accuracy score of 98.8%, along with precision, and recall scores of 98.9%, and 98.7% respectively indicating the efficacy of suggested approach.

On The Sum of Three Square Formula

Let n,x,y,z be any given integers. The study of n for which n = x2 +y2 + z2 is a very long-standing problem. Recent survey of sizeable literature shows that many researchers have made some progress to come up with algorithms of decomposing integers into sums of three squares. On the other hand, available results on integer representation as sums of three square is still very minimal. If a,b,c,d,k,m,n,u,v and w are any non-negative integers, this study determines the sum of three-square formula of the form abcd+ka2+ma+n = u2 +v2 +w2 and establishes its applications to various cases.

Representations of integers by powers of two primes

Representations of integers by powers of two primes

On the Polynomial Structure of 𝑟𝑘(𝑛)

If 𝒓𝒌(𝒏) is the number of representations of a positive integer n as the sum of k squares, then 𝒓𝒌(𝒏) is a polynomial in k of degree n; here we exhibit expressions for certain coefficients through this polynomial.

Features of digital signal processing algorithms using Galois fields GF(2n+1).

An alternating representation of integers in binary form is proposed, in which the numbers -1 and +1 are used instead of zeros and ones. It is shown that such a representation creates considerable convenience for multiplication numbers modulo p = 2n+1. For such numbers, it is possible to implement a multiplication algorithm modulo p, similar to the multiplication algorithm modulo the Mersenne number. It is shown that for such numbers a simple algorithm for digital logarithm calculations may be proposed. This algorithm allows, among other things, to reduce the multiplication operation modulo a prime number p = 2n+1 to an addition operation.

A Novel 3D Reversible Data Hiding Scheme Based on Integer-Reversible Krawtchouk Transform for IoMT.

To avoid rounding errors associated with the limited representation of significant digits when applying the floating-point Krawtchouk transform in image processing, we present an integer and reversible version of the Krawtchouk transform (IRKT). This proposed IRKT generates integer-valued coefficients within the Krawtchouk domain, seamlessly aligning with the integer representation commonly utilized in lossless image applications. Building upon the IRKT, we introduce a novel 3D reversible data hiding (RDH) algorithm designed for the secure storage and transmission of extensive medical data within the IoMT (Internet of Medical Things) sector. Through the utilization of the IRKT-based 3D RDH method, a substantial amount of additional data can be embedded into 3D carrier medical images without augmenting their original size or compromising information integrity upon data extraction. Extensive experimental evaluations substantiate the effectiveness of the proposed algorithm, particularly regarding its high embedding capacity, imperceptibility, and resilience against statistical attacks. The integration of this proposed algorithm into the IoMT sector furnishes enhanced security measures for the safeguarded storage and transmission of massive medical data, thereby addressing the limitations of conventional 2D RDH algorithms for medical images.

ON MONOTONE INCREASING REPRESENTATION FUNCTIONS

AbstractLet $k\ge 2$ be an integer and let A be a set of nonnegative integers. The representation function $R_{A,k}(n)$ for the set A is the number of representations of a nonnegative integer n as the sum of k terms from A. Let $A(n)$ denote the counting function of A. Bell and Shallit [‘Counterexamples to a conjecture of Dombi in additive number theory’, Acta Math. Hung., to appear] recently gave a counterexample for a conjecture of Dombi and proved that if $A(n)=o(n^{{(k-2)}/{k}-\epsilon })$ for some $\epsilon>0$ , then $R_{\mathbb {N}\setminus A,k}(n)$ is eventually strictly increasing. We improve this result to $A(n)=O(n^{{(k-2)}/{(k-1)}})$ . We also give an example to show that this bound is best possible.

Some Generalized Formula For Sums of Cube

The study of integer representations as a sum of powers is still a very long standing problem. In this work, the study of integer representation as a sum of cube is introduced and investigated for non-zero distinct integer solution. Let \(a_1\), \(a_2\), \(a_3\), ... , \(a_n\) and d be any positive integers such that \(a_n\) - \(a_n\)-1= \(a_n\)-1 - \(a_n\)-2 = ... = \(a_2\) - \(a_1\) = d. This study formulates some general results for sums of n cube. In particular, this research introduces and develops the diophantine equation I =(\(a_1\)+\(a_2\)+\(a_3\)+...+\(a_n\)) L = \(a_1^3\)+\(a_2^3\)+\(a_3^3\)+...+\(a_n^3\) for some integer L. The method involves decomposing integer I into sums of n cube and determination of general representation of integer L using case by case basis.

Highly accurate and fast YOLOv4-based polyp detection

More than 900000 deaths were caused by Colorectal Cancer (CRC) in 2020. Colonoscopy is the gold standard for colorectal cancer screening, with studies concluding that colonoscopies significantly reduce mortality from CRC. It has been shown in the literature that computer-aided detection (CAD) systems can improve adenoma detection. In particular, deep learning models have shown promising results in helping physicians reduce the number of missed lesions during real-time colonoscopies. To keep up with the increase in resolution of the colonoscopies and to perform inference in real-time in smaller medical devices, faster models (i.e., models that can process images at high frame rates) are required.In this work, we use YOLOv4 to detect polyps, which are known to be CRC precursors. To further increase the inference speed of the model, achieve real-time performance, and make the model smaller, we deployed it on NVIDIA TensorRT, which optimizes and quantizes the model using different floating-point and integer representation levels down to 8-bit.Different methods of regularization, data pre-processing, and data augmentation were tested. An analysis of various data augmentation methods was conducted, where we individually tested these techniques by stacking them on top of previously used ones and assessed the improvements. The effects of transfer learning, using custom anchor boxes, the usage of different optimizers, Cross-Iteration Batch Normalization (CBN), and Distance-Intersection Over Union Non-Maximum Suppression (DIOU-NMS) were equally analyzed.We used publicly available datasets to train, test, and validate our model to facilitate comparison to other studies. To evaluate the inference speed, a publicly available video dataset was used. We achieved 82.93% for mAP, 81.44% for precision, 75.96% for recall in the Etis-Larib dataset, and 90.96% for mAP, 88.65% for precision, 87.62% for recall, 88.23% for F1, and 87.86% for F2 using the CVC-ClinicDB dataset. Using an NVIDIA RTX 2080TI graphics processing unit (GPU), a speed of approximately 98 frames per second (FPS) was obtained on three videos from the Colonoscopic video dataset. For the FP16 version of the implementation, the inference speed increased to 163 FPS. For the INT8 version of the model, the inference speed was further increased to approximately 172 FPS.An important conclusion was that for FP16 and INT8 levels of quantization, the accuracy metrics for the model did not suffer much degradation and, in some cases, even increased performance, thus suggesting that a regularization effect can occur when using quantization. This implies that the network may become smaller, faster, and more precise (which could prove to be essential for resource-constrained devices) and that quantization might also be used as a regularization technique. Also, data augmentation techniques proved crucial to overcoming the lack of labeled data, which often happens in the medical field due to privacy reasons.

A Note on the Number of Representations of a Positive Integer as a Sum of Generalized Polygonal Numbers

A Note on the Number of Representations of a Positive Integer as a Sum of Generalized Polygonal Numbers

New Kronecker–Weyl type equidistribution results and Diophantine approximation

An interesting result of Veech more than 50 years ago is a parity, or mod 2, version of the Kronecker–Weyl equidistribution theorem concerning the irrational rotation sequence {qalpha }, q=0,1,2,3,ldots If alpha is badly approximable and bin (0,1) satisfies bne {malpha } for any min {mathbb {Z}}, then the parity of cardinalities of the sets {1leqslant qleqslant N,{:},{qalpha }in [0,b)} as Nrightarrow infty is evenly distributed. We first answer a question of Veech and establish a stronger form of the mod n analog of his result. Furthermore, for irrational alpha and b={malpha } for some min {mathbb {N}}, we give a simple yet precise characterization of those cases that give rise to even distribution. We also obtain time-quantitative description of some very striking violations of uniformity—this part is particularly number theoretic in nature, and involves Ostrowski representations of positive integers and alpha -expansions of real numbers. The Veech discrete 2-circle problem can also be visualized as a problem that concerns 1-direction geodesic flow on a surface obtained by modifying the surface comprising two side-by-side squares by the inclusion of symmetric barriers and gates on the vertical edges, with appropriate modification of the vertical edge identifications. We establish a far-reaching generalization of this case to ones that concern 1-direction geodesic flow on surfaces obtained by modifying a finite square tiled translation surface in analogous but not necessarily symmetric ways.

Arbitrary-size permutation networks using arbitrary-radix switches

Permutation networks accepting an arbitrary number of inputs using arbitrary-radix switches find their applications in a wide range of areas such as datacenter network, network on chip, and network switches. We extend the Benes network to accepting an arbitrary-size input and using arbitrary-radix switches. The resulting network, namely A2-Benes network, has a regular topology as the Benes or Clos network while using fewer switches. For arbitrary n≥1 and w≥2, the number of switches of the n×nA2-Benes network is the total number of 1's and 0's in the radix-w representations of integers 0,…,n−1, not counting the leading 0's; the number of stages is 2⌈logw⁡n⌉−1. The regular topology and fewer switches of the A2-Benes network lead to lower wiring complexity and costs.