Integer Function Research Articles

Robust Stability and Control of Fractional Polynomials Including Integer and Fractional Natural Exponential Functions

Robust Stability and Control of Fractional Polynomials Including Integer and Fractional Natural Exponential Functions

Efficient Homomorphic Evaluation of Arbitrary Uni/Bivariate Integer Functions and Their Applications

Efficient Homomorphic Evaluation of Arbitrary Uni/Bivariate Integer Functions and Their Applications

Fast data-independent KLT approximations based on integer functions

Fast data-independent KLT approximations based on integer functions

On some sums involving the integral part function

Denote by [Formula: see text], [Formula: see text] and [Formula: see text] the number of representations of [Formula: see text] as a product of [Formula: see text] natural numbers, the number of distinct prime factors of [Formula: see text] and the characteristic function of the square-free integers, respectively. Let [Formula: see text] be the integral part of real number [Formula: see text]. For [Formula: see text], we prove that [Formula: see text] for [Formula: see text], where [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] is an arbitrarily small positive number. These improve the corresponding results of Bordellès.

Comparative analysis of the USA's Washington Ferries and road transport carbon emissions using the Trozzi and Vaccaro and Greatest Integer functions.

Countries' sectors are currently under great scrutiny for their response to the greenhouse gas (GHG) emission profile and the general effect of the sectoral activities on the environment. As in the agenda of all sectors, environmental concerns and investigations are of high importance in shipping and maritime transport. Amidst the rising forms of globalization, the need for sustainable transportation is constantly increasing. However, the machines that are the cornerstone of transportation largely depend on fossil fuels, thus resulting in environmental degradation. Notably, environmental-related degradation has continued to account for global warming, climate change, and ocean acidification. Shipping is considered the most environmentally friendly mode of transportation in terms of carbon dioxide (CO2) emissions per ton per mile of transported unit load when compared against road transportation. In this study, six ferry lines (FLs) of Washington State Ferries were calculated to compare ship-generated carbon dioxide (CO2) emissions with those from road transportation as if the carried vehicles had used the highway instead of transport by FL. While making these calculations, the Greatest Integer function (GIF) and Trozzi and Vaccaro function (TVF) were utilized. From the examined three scenarios, i.e., all passengers travel by car instead of ferry as scenario 1, all ferries carry both cars and passengers as scenario 2, and all car-free passengers travel by bus instead of ferry as scenario 3, the outlined results are as follows: (i) none of the cars were carried by the ferry, and car-free passengers preferred traveling by their own cars as observed in scenario 1; (ii) hypothetical scenarios (1 to 3) in which the road vehicles carried on FLs had instead used the highway, and the total potential CO2 emissions of these road vehicles were calculated as 2,638,858.138, 704,958.2998, and 1,394,148.577 tonnes per year, respectively. Policy-wise, this study revealed the management strategies for CO2 emissions reduction for two transport modes, shipping and road transportation, under current conditions.

Behaviour of the Normalized Depth Function

Let $I\subset S=K[x_1,\dots,x_n]$ be a squarefree monomial ideal, $K$ a field. The $k$th squarefree power $I^{[k]}$ of $I$ is the monomial ideal of $S$ generated by all squarefree monomials belonging to $I^k$. The biggest integer $k$ such that $I^{[k]}\ne(0)$ is called the monomial grade of $I$ and it is denoted by $\nu(I)$. Let $d_k$ be the minimum degree of the monomials belonging to $I^{[k]}$. Then, $\text{depth}(S/I^{[k]})\ge d_k-1$ for all $1\le k\le\nu(I)$. The normalized depth function of $I$ is defined as $g_{I}(k)=\text{depth}(S/I^{[k]})-(d_k-1)$, $1\le k\le\nu(I)$. It is expected that $g_I(k)$ is a non-increasing function for any $I$. In this article we study the behaviour of $g_{I}(k)$ under various operations on monomial ideals. Our main result characterizes all cochordal graphs $G$ such that for the edge ideal $I(G)$ of $G$ we have $g_{I(G)}(1)=0$. They are precisely all cochordal graphs $G$ whose complementary graph $G^c$ is connected and has a cut vertex. As a far-reaching application, for given integers $1\le s<m$ we construct a graph $G$ such that $\nu(I(G))=m$ and $g_{I(G)}(k)=0$ if and only if $k=s+1,\dots,m$. Finally, we show that any non-increasing function of non-negative integers is the normalized depth function of some squarefree monomial ideal.

Observer-based quantised control of networked systems with round-robin protocol and transmission delay

This paper addresses the ultimate boundedness control problem for a class of networked nonlinear systems with the round-robin (RR) protocol and uniform quantisation. The communication between sensor nodes and the controller is implemented via a constrained communication channel. The quantised output of the system is transmitted to the remote controller through a communication channel subject to a transmission delay. For the purpose of alleviating possible data collision, the well-known RR communication protocol is deployed to schedule the data transmissions. On the other hand, the uniform quantisation effects of the network are characterised by a round function (i.e. the nearest integer function). The purpose of the addressed problem is to design an observer-based controller for the networked nonlinear systems such that, in the presence of RR protocol and uniform quantisation effects, the closed-loop system is ultimately bounded. The controller is designed based on mean square stability analysis and Lyapunov-like method. A set of sufficient conditions for the ultimate boundedness of the closed-loop system are established and, on the basis of which, the desired controller gains are obtained by solving a set of linear matrix inequalities. The effectiveness of the proposed method is verified by numerical examples.

The Optimized Algorithm of Finding the Shortest Path in a Multiple Graph

In this paper, we study undirected multiple graphs of any natural multiplicity $k>1$. There are edges of three types: ordinary edges, multiple edges and multi-edges. Each edge of the last two types is a union of $k$ linked edges, which connect 2 or $(k+1)$ vertices, correspondingly. The linked edges should be used simultaneously. If a vertex is incident to a multiple edge, it can be also incident to other multiple edges and it can be the common end of $k$ linked edges of some multi-edge. If a vertex is the common end of some multi-edge, it cannot be the common end of another multi-edge. As for an ordinary graph, we can define the integer function of the length of an edge for a multiple graph and set the problem of the shortest path joining two vertices. Any multiple path is a union of $k$ ordinary paths, which are adjusted on the linked edges of all multiple and multi-edges. In the article, we optimize the algorithm of finding the shortest path in an arbitrary multiple graph, which was obtained earlier. We show that the optimized algorithm is polynomial. Thus, the problem of the shortest path is polynomial for any multiple graph.

A modification of Petri nets with anticipation on a position

We propose a modification of Petri nets with strong anticipation on a position. The extension modifies a transition rule by adding a new term that contains an integer function of the new marking in the position. The differences from classic Petri nets are found; for example, the set of markings that are reachable from a current marking by firing the enabled transition can either be empty or contain more than one marking. We consider the construction of a reachability graph and a coverability tree. We give the conditions for the existence of the coverability tree and propose the algorithm for constructing the coverability tree that generalizes the well-known classic algorithm. The main ideas and constructions are illustrated in the examples.

The Polynomial Algorithm of Finding the Shortest Path in a Divisible Multiple Graph

In this paper, we study undirected multiple graphs of any natural multiplicity к > 1. There are edges of three types: ordinary edges, multiple edges and multi-edges. Each edge of the last two types is a union of к linked edges, which connect 2 or (к + 1) vertices, correspondingly. The linked edges should be used simultaneously. If a vertex is incident to a multiple edge, it can be also incident to other multiple edges and it can be the common end of к linked edges of some multi-edge. If a vertex is the common end of some multi-edge, it cannot be the common end of another multi-edge. Divisible multiple graphs are characterized by a possibility to divide the graph into к parts, which are adjusted on the linked edges and which have no common edges. Each part is an ordinary graph. As for an ordinary graph, we can define the integer function of the length of an edge for a multiple graph and set the problem of the shortest path joining two vertices. Any multiple path is a union of к ordinary paths, which are adjusted on the linked edges of all multiple and multi-edges. In the article, we show that the problem of the shortest path is polynomial for a divisible multiple graph. The corresponding polynomial algorithm is formulated. Also we suggest the modification of the algorithm for the case of an arbitrary multiple graph. This modification has an exponential complexity in the parameter к.

Mathematical Model of a Local Grid with Small Modular Reactor NPPs

The technical and economic processes of the Local Grid, regional electricity system comprising small modular reactor nuclear power plants, renewable electricity generators, and electricity storage systems are investigated. Smart grid technologies are used in the management of such a system. In relation to the electricity system, the Local Grid acts as a consumer or producer of electricity depending on the price situation in the wholesale market and its own ability to balance the volume of electricity generation-consumption.
 The Local Grid market is considered to be perfect, otherwise the wholesale electricity market is characterized by imperfect competition. The mathematical model of the Local Grid is offered. The presented model adequately reproduces the characteristics of the interaction between the local market and the wholesale market. The mathematical model is generated in the form of the commitment problem for small modular reactors and electricity storage systems. The model reflects the technological limitations for the loading modes of small modular reactors and units of the energy storage system, which operate in the modes of direct and reverse energy conversion. In particular, the model reflects the shunting modes and start-shutdown modes of small modular reactors and charge-discharge modes of energy storage system units, which together determine flexibility of the operation modes for the entire power system. The commitment problem of small modular reactors of a nuclear power plant, units of the electricity storage system, and transmission lines connecting the Local Grid with the power system is a mixed integer programming problem with an objective function of minimized system costs.
 Cluster integer functions are used to reduce the dimensionality of the problems of mathematical modeling the loading modes of the Local Grid in the descriptions for the sets of the same type small modular reactors, as well as the same type energy storage units. The Local Grid model reproduces the modes of its disconnection from the system operator's network to increase nuclear safety in the event of a natural disaster or hostilities. The results of computational experiments on modeling the commitment modes of the local electrical grid are presented.
 The experiments used standard data on the volume of electricity consumption and its production from renewable energy sources. The results of the computational experiment confirm the adequacy of the suggested mathematical model for the Local Grid. The presented model is suitable for the use both independently – to analyze Local Grid functioning - and in combination with the models of electric power industry - to determine the impact of the Local Grids on the electricity system.
 The article is developed based on the results of the research carried out under target program “Support of State-Priority Research and Scientific and Technical (Experimental) Solutions of the Department of Physical and Technical Problems of Energy of the NAS of Ukraine for 2022 – 2023” with program expenditure classification code 6541230 (applied research).

The Number of Limit Cycles Bifurcating from an Elementary Centre of Hamiltonian Differential Systems

This paper studies the number of small limit cycles produced around an elementary center for Hamiltonian differential systems with the elliptic Hamiltonian function H=12y2+12x2−23x3+a4x4(a≠0) under two types of polynomial perturbations of degree m, respectively. It is proved that the Hamiltonian system perturbed in Liénard systems can have at least [3m−14] small limit cycles near the center, where m≤101, and that the related near-Hamiltonian system with general polynomial perturbations can have at least m+[m+12]−2 small-amplitude limit cycles, where m≤16. Furthermore, in any of the cases, the bounds for limit cycles can be reached by studying the isolated zeros of the corresponding first order Melnikov functions and with the help of Maple programs. Here, [·] represents the integer function.

Prediction of crude oil prices in COVID-19 outbreak using real data

The world has been undergoing a global economic recession for almost two years because of the health crisis stemming from the outbreak and its effects have still continued so far. Especially, COVID-19 reduced consumer spending due to social isolation, lockdown and travel restrictions in 2020. As a result of this, with social and economic life coming to a standstill, oil prices plummeted. With the ongoing uncertainty concerning the COVID-19 pandemic, it has been of great importance for all economic agents to predict crude oil prices. The objective of this paper is to improve a model in order to make more accurate predictions for crude oil price movements. The performance of this model is assessed in terms of some significant criteria comparing our model with its counterparts as well as artificial neural networks (ANNs) and support vector machine (SVM) methods. As for these criteria, root mean square error (RMSE) and mean absolute error (MAE) results show that this model outperforms other models in forecasting crude oil prices. Further, the simulation results for 2021 show that the daily crude oil price forecasts are almost close to the real oil prices.Oil price forecasting has become more and more important for economic agents in COVID-19 period. A consistent model is required to cope with the movements in crude oil prices. A novel method combining fuzzy time series and the greatest integer function is developed. The results show that our model outperforms other counterparts or ANN and SVM methods. We capture non-linearity and volatility in crude oil prices.

Theoretical estimation of entropy of solids by using integer and non-integer [formula omitted]-dimensional Debye functions

A new full analytically formulation of standard entropies of solid metals, for the first time, has been constructed by the use of the integer and non-integer n-dimensional Debye functions. Using n-dimensional Debye functions is one of the most used approach that provides more accurate and reliable results for determining thermophysical properties of solids. Our approximation is valid for arbitrary values of low to melting temperature. We have compared our calculation results with available literature data for some solid metals. As can be seen from comparisons our approximation is satisfactory.

The fractional sum of small arithmetic functions

Motivated by recent results, we study sums of the formSf(x)=∑n≤xf(⌊xn⌋), where f is an arithmetic function and ⌊⋅⌋ denotes the greatest integer function. We improve the error term in the asymptotic formula for Sf(x) in some specific cases.

Formal verification of high-level synthesis

High-level synthesis (HLS), which refers to the automatic compilation of software into hardware, is rapidly gaining popularity. In a world increasingly reliant on application-specific hardware accelerators, HLS promises hardware designs of comparable performance and energy efficiency to those coded by hand in a hardware description language such as Verilog, while maintaining the convenience and the rich ecosystem of software development. However, current HLS tools cannot always guarantee that the hardware designs they produce are equivalent to the software they were given, thus undermining any reasoning conducted at the software level. Furthermore, there is mounting evidence that existing HLS tools are quite unreliable, sometimes generating wrong hardware or crashing when given valid inputs. To address this problem, we present the first HLS tool that is mechanically verified to preserve the behaviour of its input software. Our tool, called Vericert, extends the CompCert verified C compiler with a new hardware-oriented intermediate language and a Verilog back end, and has been proven correct in Coq. Vericert supports most C constructs, including all integer operations, function calls, local arrays, structs, unions, and general control-flow statements. An evaluation on the PolyBench/C benchmark suite indicates that Vericert generates hardware that is around an order of magnitude slower (only around 2× slower in the absence of division) and about the same size as hardware generated by an existing, optimising (but unverified) HLS tool.

EXACT PERIODIC SOLUTIONS OF SECOND-ORDER DIFFERENTIAL EQUATIONS WITH PIECEWISE CONSTANT ARGUMENTS

In the paper is given a method of finding periodical solutions of the differential equation of the form $x''(t)+p(t)x''(t-1)=q(t)x([t])+f(t),$ where $[\cdot]$ denotes the greatest integer function, $p(t)$,$q(t)$ and $f(t)$ are continuous periodic functions of $t$. This reduces $n$-periodic soluble problem to a system of $n+1$ linear equations, where $n=2,3$. Furthermore, by using the well known properties of linear system in the algebra, all existence conditions for $2$ and $3$-periodical solutions are described, and the explicit formula for these solutions are obtained.

Anisotropy as a diagnostic test for distinct tensor-network wave functions of integer- and half-integer-spin Kitaev quantum spin liquids

Contrasting ground states of quantum magnets with the integer- and half-integer-spin moments are the manifestation of many-body quantum interference effects. In this work, we investigate the distinct nature of the integer- and half-integer-spin quantum spin liquids in the framework of the Kitaev's model on the honeycomb lattice. The models with arbitrary spin quantum numbers are not exactly solvable in contrast to the well-known quantum spin liquid solution of the spin-1/2 system. We use the tensor-network wave functions for the integer-and half-integer-spin quantum spin liquid states to unveil the important difference between these states. We find that the distinct sign structures of the tensor-network wave function for the integer- and half-integer-spin quantum spin liquids are responsible for completely different ground states in the spatially anisotropic limit. Hence the spatial anisotropy would be a useful diagnostic test for distinguishing these quantum spin liquid states, both in the numerical computations and experiments on real materials. We support this discovery via extensive numerics including the tensor-network, DMRG, and exact diagonalization computations.

Minimum [formula omitted]-critical bipartite graphs

We study the problem of Minimum k-Critical Bipartite Graph of order (n,m) - MkCBG-(n,m): to find a bipartite G=(U,V;E), with |U|=n, |V|=m, and n>m>1, which is k-critical bipartite, and the tuple (|E|,ΔU,ΔV), where ΔU and ΔV denote the maximum degree in U and V, respectively, is lexicographically minimum over all such graphs. G is k-critical bipartite if deleting at most k=n−m vertices from U creates G′ that has a complete matching, i.e., a matching of size m. We show that, if m(n−m+1)∕n is an integer, then a solution of the MkCBG-(n,m) problem can be found efficiently among (a,b)-regular bipartite graphs of order (n,m), with a=m(n−m+1)∕n, and b=n−m+1. If a=m−1, then all (a,b)-regular bipartite graphs of order (n,m) are k-critical bipartite. For a<m−1, it is not the case. We characterize the values of n, m, a, and b that admit an (a,b)-regular bipartite graph of order (n,m), with b=n−m+1, and give a simple construction that creates such a k-critical bipartite graph whenever possible. Our techniques are based on Hall’s marriage theorem, elementary number theory, linear Diophantine equations, properties of integer functions and congruences, and equations involving them.

On the alternating sums of reciprocal generalized Fibonacci numbers

In this paper, we consider finite alternating sums derived from the generalized Fibonacci numbers [Formula: see text] [Formula: see text] and [Formula: see text], where [Formula: see text] and [Formula: see text] are positive integers with [Formula: see text], [Formula: see text]. Applying the greatest integer function to these sums, we obtain some equalities involving the generalized Fibonacci numbers.