Two-variable Function Research Articles

Problems on a Semiaxis for an Integro-Differential Equation with Quadratic Nonlinearity

A functional equation is considered in which a linear combination of a two-variable function and its time derivative is set equal to the double integral of a quadratic expression of the same function with respect to space variables. For the resulting integro-differential equation with quadratic nonlinearity, the Cauchy problem with initial data continuous and bounded on the positive semiaxis is investigated. The convergence of the classical method of successive approximations is proved. The accuracy of the approximation is estimated depending on the index of the iterative solution. It is proved that the problem has a solution in associated function spaces, and the uniqueness of this solution is established. An a priori estimate for solutions from the associated well-posedness class is derived. A guaranteed time interval of solution existence is found.

Adjustable Spectral Characteristics to Design FIR Filter Using Two-Variable Window Function

Adjustable Spectral Characteristics to Design FIR Filter Using Two-Variable Window Function

A reverse Sidorenko inequality

Let H be a graph allowing loops as well as vertex and edge weights. We prove that, for every triangle-free graph G without isolated vertices, the weighted number of graph homomorphisms \(\hom (G, H)\) satisfies the inequality $$\begin{aligned} \hom (G, H ) \le \prod _{uv \in E(G)} \hom (K_{d_u,d_v}, H )^{1/(d_ud_v)}, \end{aligned}$$where \(d_u\) denotes the degree of vertex u in G. In particular, one has $$\begin{aligned} \hom (G, H )^{1/|E(G)|} \le \hom (K_{d,d}, H )^{1/d^2} \end{aligned}$$for every d-regular triangle-free G. The triangle-free hypothesis on G is best possible. More generally, we prove a graphical Brascamp–Lieb type inequality, where every edge of G is assigned some two-variable function. These inequalities imply tight upper bounds on the partition function of various statistical models such as the Ising and Potts models, which includes independent sets and graph colorings. For graph colorings, corresponding to \(H = K_q\), we show that the triangle-free hypothesis on G may be dropped; this is also valid if some of the vertices of \(K_q\) are looped. A corollary is that among d-regular graphs, \(G = K_{d,d}\) maximizes the quantity \(c_q(G)^{1/|V(G)|}\) for every q and d, where \(c_q(G)\) counts proper q-colorings of G.Finally, we show that if the edge-weight matrix of H is positive semidefinite, then $$\begin{aligned} \hom (G, H) \le \prod _{v \in V(G)} \hom (K_{d_v+1}, H )^{1/(d_v+1)}. \end{aligned}$$This implies that among d-regular graphs, \(G = K_{d+1}\) maximizes \(\hom (G, H)^{1/|V(G)|}\). For 2-spin Ising models, our results give a complete characterization of extremal graphs: complete bipartite graphs maximize the partition function of 2-spin antiferromagnetic models and cliques maximize the partition function of ferromagnetic models. These results settle a number of conjectures by Galvin–Tetali, Galvin, and Cohen–Csikvári–Perkins–Tetali, and provide an alternate proof to a conjecture by Kahn.

Determining effective crack lengths from electrical measurements in polymer-supported thin films

Although it is evident that the formation of multiple through-thickness cracks in polymer-supported thin films leads to an increase of the electrical resistance, the attempts to quantify the dependence of resistance growth only based on the induced crack density have not yet been successful. In this paper a recently developed relationship, representing the resistance growth as a two-variable polynomial function of crack density and crack length, is utilized to analyze the crack patterns induced by monotonic and cyclic tensile loading of 250 nm thick Cu films with a 10 nm Cr adhesion layer on polyimide. It is demonstrated that by knowing the in-situ resistance during deformation and post-mortem linear density of induced cracks, it is possible to extract the effective crack lengths, a parameter which is often ignored during experimental characterization of the damage induced through mechanical loading. The described algorithm is not only much more cost-effective in comparison to time-consuming in-situ microscopy methods, it also reflects the reliability of the whole sample rather than of only selected surface areas.

Multiple criteria group decision making based on group satisfaction

To generate solutions to multiple criteria group decision-making (MCGDM) problems that are satisfactory to the decision makers, this paper proposes a new method. To examine whether a group solution is satisfactory to the decision makers, group satisfaction is constructed from alternative assessment and ranking differences between the decision makers and the group. The difference between a decision maker's assessment and a group's assessment is designed based on differences in assessment grades, whose normalization is theoretically proven to construct alternative assessment differences. Inspired by Spearman's rank correlation coefficient, the expected utilities of decision makers’ and the group's assessments are used to construct alternative ranking differences. An abstract two-variable function with specific properties is designed to relate alternative assessment difference to alternative ranking difference to form group satisfaction. From the constructed group satisfaction, the process of generating group-satisfactory solutions to MCGDM problems is presented. The problem of selecting engineering project management software is analyzed by using the proposed method to demonstrate its applicability. To highlight the importance of group satisfaction in MCGDM, relationships and differences between group satisfaction and group consensus are analyzed through the problem and simulation experiments.

Some classes of special functions using Fourier transforms of some two-Variable orthogonal polynomials

ABSTRACTIn this paper some new classes of two-variable orthogonal functions by using Fourier transforms of two-variable orthogonal polynomials are introduced. Orthogonality relations are obtained by using the Parseval identity. Recurrence relations for new families of orthogonal functions are also presented.

Squeezed Hermite polynomial state: nonclassical features and decoherence behavior

We theoretically investigate the nonclassicality of the squeezed Hermite polynomial state (SHPS) by examining the squeezing feature, the oscillation of photon-number distribution (PND) and the negativity of the Wigner function (WF). Our results indicate that the squeezing operation leads to a large squeezing degree and negative volume of the WF, but the Hermite polynomial creation operation has less impact on them, and the two operations have opposite modulations of the PND. In addition, we study the evolution law of the SHPS in the amplitude decay channel (ADC). It is found that the density-operator evolution is related to two conjugated operator Hermite polynomials within normal ordering, which brings about a new two-variable special function and its generating function. We also find that the analytical WF evolution is represented as the compact form of this new special function, and that the large squeezing is more robust against the decoherence of the SHPS in the ADC.

Hole enlarging anchorage technology of surrounding rock of roadway in weakly consolidated formation

Aiming at the problem of surrounding rock deformation control of the roadway in weakly consolidated formation and combined with the calculation method of anchoring force of anchor bolt, this paper analyzes the feasibility of hole enlarging and anchoring and proposes a calculation method for the anchoring force of hole enlarging and anchoring; this paper establishes a simplified mechanics model of cylinder table enlarging, standing circular table enlarging and circular table enlarging with the method of theoretical mechanics and material mechanics and finds after analysis that the anchoring effect is best when the angle between the busbar and the axis of the circular table is θ < 45°. It draws a two-variable function relation curve for the surrounding rock stress σb and the hole enlarging parameters at the supporting region of the enlarged part of the standing circular table and obtains the optimal anchor length L is 130 mm, the optimal hole enlarging angle θ is 15°, and the optimal hole diameter D is 100 mm. It verifies the theoretical analysis results with numerical calculation, and the results show that the theoretical derivation is consistent with the simulation results. The research results make up the inadequacies of the accurate calculation method for the optimal hole enlarging parameters of the surrounding rock hole of roadway in the weakly consolidated formation, and provide a theoretical basis for the popularization and application of the anchoring technology in the roadway support of weakly consolidated formation.

Wigner function for squeezed negative binomial state and evolution of density operator for amplitude decay**Project supported by the National Natural Science Foundation of China (Grant No. 11347026) and the Natural Science Foundation of Shandong Province, China (Grant Nos. ZR2016AM03 and ZR2017MA011).

Using the thermal-entangled state representation and the operator-ordering method, we investigate Wigner function (WF) for the squeezed negative binomial state (SNBS) and the analytical evolution law of density operator in the amplitude decay channel. The results show that the analytical WF is related to the square of the module of single-variable Hermite polynomials, which leads to a new two-variable special function and its generating function, and the parameters s and γ play opposite roles in the WF distributions. Besides, after undergoing this channel, the initial pure SNBS evolves into a new mixed state related to two operator Hermite polynomials within normal ordering, and fully loses its nonclassicality and decays to vacuum at long decay time.

Extracting Information from an Urban Network by Combining a Visibility Index and a City Data Set

Cities can be represented by spatial networks, and the mathematical structure that defines a spatial network is a graph. Taking into account this premise, this paper is focused on analysing information on an urban scale by combining a new ray-casting visibility index with a data set of the urban street network. The visibility index provides information about the most visible buildings or areas. We relate this index with other data extracted from the city, with the aim of generating and analysing information about urban elements. To corroborate this idea, real data are analysed. The dataset is related to the heritage conservation of the buildings of the Villaflora suburb, located in the city of Quito (Ecuador). This information is processed, together with the visibility index, with the aim of determining the conservation degree of the urban areas most visually exposed to pedestrians or visitors. The combination of both values—heritage conservation and visibility index—is carried out by means of two new indices, I P and I N , which are defined using two-variable exponential functions.

Analytical investigation of self-oscillating condition for resonant terahertz extended interaction oscillators

Extended Interaction Oscillators (EIOs) are high-frequency sources of millimeter wave and terahertz radiations. The dimensions of the EIO resonant structure should be smaller to generate higher frequency signals. However, due to the skin effect and decreasing the available beam surface area, the maximum obtainable oscillating frequency of any RF geometry of EIO is limited. Analytical relations for maximum attainable frequencies of resonant EIO geometries in the sense of beam-wave coupling are derived and presented in this paper. They are directly expressed in terms of the standing-wave profile of the RF resonant circuit in this work. The maximum frequencies are expressed in terms of a defined two-variable analytical function U for single-section resonant EIOs. Besides, an algebraic formula is derived for a critical unloaded frequency of an EIO geometry (fcr) in terms of the resonant circuit field distribution. A 4/5 power-law dependency is derived for a maximum oscillating frequency vs a number of periods of the RF circuit. For two completely different EIO geometries, the results of the obtained relations are compared with the outcomes of a particle-in-cell solver, showing acceptable agreement for both the structures. In addition to computational rapidity, the obtained relations present an analytical viewpoint to estimate and improve the high-frequency generation capability of terahertz extended interaction oscillators.

The analysis of the understanding of the three-dimensional (Euclidian) space and the two-variable function concept by university students

Conceptual understanding is being emphasized in mathematics education. Students often have difficulty understanding the multi-variable function, a key concept. Based on the APOS theory, which analyzes the cognitive structures formed by individuals in learning a mathematical concept and produces components related to that learning, this study analyzes the conceptual understanding of three-dimensional spaces and two-variable functions by university students. The genetic decomposition of these concepts proposed by Trigueros and Martinez-Planell is also considered. The analyzes results revealed that only one student constructed the concept of three-dimensional space as an object within the framework of genetic decomposition. Some students could not relate the concepts of two-variable function and three-dimensional space. Students who could perform algebraic operations had problems related to geometric representation. This study suggests the refinement of genetic decomposition to include, e.g., mental construction steps for writing algebraic equations of special surfaces whose graphs are given in R3.

Generalized bivariate Hermite–Gauss sampling

Recently, Asharabi and Al-Haddad in (Turk J Math 41: 387–403, 2017) have established a generalized bivariate sampling series involving samples from the function and its mixed and non-mixed partial derivatives for an entire function of two-variables which satisfies certain growth conditions. This series converges slowly unless the sample values, $$f^{(i,j)}\left( nh,mh\right) $$ , $$i,j\in \mathbb {N}_{\circ }$$ , decay rapidly as $$n,m\longrightarrow \mp \infty $$ . In this paper, we derive a modification of this sampling using a bivariate Gaussian multiplier. This modification highly improved the convergence rate of the bivariate sampling series which will be of an exponential type. Moreover, it is valid for wider classes, the class of entire functions including unbounded functions on $$\mathbb {R}^{2}$$ and the class of analytic functions in a bivariate strip. We show that many known results included in Asharabi and Prestin (IMA J Numer Anal 36: 851–871, 2016) are special cases of our results. Various illustrative examples are presented and they show a rightly good agreement with our theoretical analysis.

Using cycles of research in APOS: The case of functions of two variables

This article reports on an Action-Process-Object-Schema Theory (APOS) based study consisting of three research cycles on student learning of the basic idea of a two-variable functions and its graphical representation. Each of the three research cycles used semi-structured interviews with students to test a conjecture about mental constructions (genetic decomposition) students may use to understand functions of two variables, develop supporting classroom activities based on interview results, and successively improve the conjecture. The article brings together for the first time findings already reported in the literature from the first two research cycles, and the results of the third and final cycle. The final results show that students who were assigned special activities based on the research findings of the first two cycles were more likely to exhibit behavior consistent with a Process conception of function of two variables. An important contribution of the article is that it shows how different APOS research cycles may be used to successively improve students’ understanding of a mathematical notion. Also, the description of findings from the three research cycles, provides a potentially useful guide to improve student learning of function of two variables.

사다리꼴 주름판의 최적형상 및 엄밀진동에 관한 연구

In this study, the refined rigidities, optimal shapes, and free vibration of trapezoidal corrugated plates were analyzed. The rigidities of trapezoidal corrugated plates by material mechanical analysis have large errors as compared with practical behavior. Thus, the refined rigidities of the plates are proposed by calculating two-variable functions by surface fitting for the correction of errors. These rigidities were applied in analyzing the optimal shapes and free vibration of the plates. In addition, a Visual Basic program was developed for the vibration analysis.

On the Lengths of Level Curves

This article establishes several remarkably simple identities concerning the lengths of level curves of a two-variable function, with the most notable one relating length to curvature. Some geometric and analytic applications of the results are considered.

An experimental study on the thermal conductivity of cerium oxide/ethylene glycol nanofluid: developing a new correlation

The main aim of this experimental study is to examine the effect of cerium oxide nanoparticles on the thermal conductivity of ethylene glycol. In this way, cerium oxide nanoparticles with the particle diameter of 10–30 nm have been used for making nanofluid samples. The samples were made in volume concentration range of 0.25–2.5% using a two-step method. Visual observation of nanofluid samples showed that they have acceptable stability. Transient hot wire method was used for measuring the thermal conductivity of the samples. Measurements were done for all samples at temperatures ranging from 25to 50°C. Measurements showed that the thermal conductivity of nanofluid enhanced with increasing temperature and solid volume fraction. The results also showed that the thermal conductivity of ethylene glycol could enhance by about 22% when the nanoparticles volume fraction reaches 2.5%. This enhancement occurred at 50°C. Finally, a mathematical correlation was presented to predict the thermal conductivity ratio of CeO2/EG using curve-fitting. This correlation, a two-variable function of temperature and volume fraction, showed a linear relationship between thermal conductivity ratio and these variables.

Discrete cosine and sine transforms generalized to honeycomb lattice

The discrete cosine and sine transforms are generalized to a triangular fragment of the honeycomb lattice. The honeycomb point sets are constructed by subtracting the root lattice from the weight lattice points of the crystallographic root system A2. The two-variable orbit functions of the Weyl group of A2, discretized simultaneously on the weight and root lattices, induce a novel parametric family of extended Weyl orbit functions. The periodicity and von Neumann and Dirichlet boundary properties of the extended Weyl orbit functions are detailed. Three types of discrete complex Fourier-Weyl transforms and real-valued Hartley-Weyl transforms are described. Unitary transform matrices and interpolating behavior of the discrete transforms are exemplified. Consequences of the developed discrete transforms for transversal eigenvibrations of the mechanical graphene model are discussed.

A note on two-variable Lommel matrix functions

The aim of the present work is to develop a pair of Lommel matrix functions of two variables suggested by the Bessel matrix functions and some of their properties are studied to be special cases of our results.

Ranks of Cross-Commutators and Unitary Module Maps

Let M be an invariant subspace of $$H^2$$ over the bi-disk and $$N=H^2\ominus M$$ . Let $$S_{z,N},S_{w,N}$$ be the compression of the multiplication operators $$T_z,T_w$$ on $$H^2$$ onto N. For a two-variable inner function $$\theta $$ , let $$M_\theta = \theta M$$ and $$N_\theta =H^2\ominus M_\theta $$ . We shall study the relationship of the ranks of the cross-commutators $$[S_{z,N},S^*_{w,N}]$$ and $$[S_{z,N_\theta },S^*_{w,N_\theta }]$$ . We also characterize M such that rank $$[S_{z,N},S^*_{w,N}]$$ $$\not =$$ rank $$[S_{z,N_\theta },S^*_{w,N_\theta }]$$ for any non-constant inner function $$\theta $$ .