Nonconstant Function Research Articles

On the norm-preservation of squares in real algebra representation

One of the main results of the article Gelfand theory for real Banach algebras, recently published in [Rev R Acad Cienc Exactas Fís Nat Ser A Mat RACSAM 114(4):163, 2020] is Proposition 4.1, which establishes that the norm inequality Vert a^{2}Vert le Vert a^{2}+b^{2}Vert for a,bin {mathcal {A}} is sufficient for a commutative real Banach algebra {mathcal {A}} with a unit to be isomorphic to the space {mathcal {C}}_{{mathbb {R}}}({mathcal {K}}) of continuous real-valued functions on a compact Hausdorff space {mathcal {K}}. Moreover, in this proposition is also shown that if the above condition (which involves all the operations of the algebra) holds, then the real-algebra isomorphism given by the Gelfand transform preserves the norm of squares. A very natural question springing from the above-mentioned result is whether an isomorphism of {mathcal {A}} onto {mathcal {C}}_{{mathbb {R}}}({mathcal {K}}) is always norm-preserving of squares. This note is devoted to providing a negative answer to this problem. To that end, we construct algebra norms on spaces {mathcal {C}}_{{mathbb {R}}}({mathcal {K}}) which are (1+epsilon )-equivalent to the sup-norm and with the norm of the identity function equal to 1, where the norm of every nonconstant function is different from the standard sup-norm. We also provide examples of two-dimensional normed real algebras {mathcal {A}} where Vert a^2Vert le k Vert a^2+b^2Vert for all a,bin {mathcal {A}}, for some k>1, but the inequality fails for k=1.

Decision trees based on 1-consequences

In this paper, we study arbitrary infinite binary information systems each of which consists of an infinite set of elements and an infinite set of two-valued non-constant functions (attributes) defined on the set of elements. We consider the notion of a problem over information system, which is described by a finite number of attributes: for a given element, we should determine values of these attributes. As algorithms for problem solving, we study decision trees that use arbitrary attributes from the considered infinite set of attributes and solve the problem based on 1-consequences. In such a tree, we take into account consequences each of which follows from one equation of the kind “attribute = value” obtained during the decision tree work and ignore consequences that can be derived only from at least two equations. As time complexity, we study the depth of decision trees. We prove that in the worst case, with the growth of the number of attributes in the problem description, the minimum depth of decision trees based on 1-consequences grows either as a logarithm or linearly.

Predator-Prey Model with Refuge, Fear and Z-Control

In this paper, we consider a predator-prey model incorporating fear and refuge. Our results show that the predator-free equilibrium is globally asymptotically stable if the ratio between the death rate of predators and the conversion rate of prey into predator is greater than the value of prey in refuge at equilibrium. We also show that the co-existence equilibrium points are locally asymptotically stable if the value of the prey outside refuge is greater than half of the carrying capacity. Numerical simulations show that when the intensity of fear increases, the fraction of the prey inside refuge increases; however, it has no effect on the fraction of the prey outside refuge, in the long run. It is shown that the intensity of fear harms predator population size. Numerical simulations show that the application of Z-control will force the system to reach any desired state within a limited time, whether the desired state is a constant state or a periodic state. Our results show that when the refuge size is taken to be a non-constant function of the prey outside refuge, the systems change their dynamics. Namely, when it is a linear function or an exponential function, the system always reaches the predator-free equilibrium. However, when it is taken as a logistic equation, the system reaches the co-existence equilibrium after long term oscillations.

Homogeneous Fourier and Weyl multipliers on Sobolev spaces related to the Heisenberg group

Inspired by the work of A. Bonami and S. Poornima that a non-constant function which is homogeneous of degree 0 cannot be a Fourier multiplier on homogeneous Sobolev spaces, we establish analogous results for Fourier multipliers on the Heisenberg group Hn and Weyl multipliers on Cn acting on Sobolev Spaces.

Wave dispersion and quantitative accuracy analysis of bond-based peridynamic models with different attenuation functions

Peridynamics (PD) reformulates classical continuum mechanics in terms of spatial integral equations rather than partial differential equations. PD has successfully matched fracture patterns observed in experiments. However, issues remain, such as its dispersion characteristics, constant micromodules, and behavior at material boundaries. The introduction of attenuation functions can effectively address these three problems. In this study, we attempted to conduct a systematic investigation of bond-based peridynamics (BPD) with different attenuation functions. This mainly involved an analysis of the wave dispersion, surface effect, and quantitative accuracy of BPD models with different attenuation functions. Initially, a novel quartic polynomial attenuation function was proposed, and various types of functions were investigated. The corresponding expressions for non-constant micromodulus functions were determined, and attenuation bond-based peridynamic (ABPD) models that considered the effect of attenuation were established. Furthermore, the dispersion relationship of ABPD for the one-dimensional case was derived, and the effects of the horizon size, material point size, and attenuation functions on the dispersion properties were investigated. Finally, the surface effect and the effectiveness of different attenuation functions were examined by considering the attenuation functions and energy correction method in ABPD models under plane stress and plane strain conditions. Thus, the accuracy of ABPD was evaluated. The influence function proposed in this work and other several attenuation functions can be used to minimize wave dispersion for a fixed material point size or horizon size and to reduce the surface effect. The energy correction method had a remarkable influence on the computational accuracies of the classical BPD model and ABPD models. Practical recommendations for selecting the attenuation functions for coupling PD with other methods and for enhancing the accuracy of the BPD model were provided.

Some remarks on the stability of the Cauchy equation and completeness

It was proved in Forti and Schwaiger (C R Math Acad Sci Soc R Can 11(6):215–220, 1989), Schwaiger (Aequ Math 35:120–121, 1988) and with different methods in Schwaiger (Developments in functional equations and related topics. Selected papers based on the presentations at the 16th international conference on functional equations and inequalities, ICFEI, Bȩdlewo, Poland, May 17–23, 2015, Springer, Cham, pp 275–295, 2017) that under the assumption that every function defined on suitable abelian semigroups with values in a normed space such that the norm of its Cauchy difference is bounded by a constant (function) is close to some additive function, i.e., the norm of the difference between the given function and that additive function is also bounded by a constant, the normed space must necessarily be complete. By Schwaiger (Ann Math Sil 34:151–163, 2020) this is also true in the non-archimedean case. Here we discuss the situation when the bound is a suitable non-constant function.

A Liouville theorem on asymptotically Calabi spaces

In this paper, we will study harmonic functions on the complete and incomplete spaces with nonnegative Ricci curvature which exhibit inhomogeneous collapsing behaviors at infinity. The main result states that any nonconstant harmonic function on such spaces yields a definite exponential growth rate which depends explicitly on the geometric data at infinity.

Entire functions that share two pairs of small functions

AbstractIn this paper, we study the unicity of entire functions and their derivatives and obtain the following result: letffbe a non-constant entire function, leta1{a}_{1},a2{a}_{2},b1{b}_{1}, andb2{b}_{2}be four small functions offfsuch thata1≢b1{a}_{1}\not\equiv {b}_{1},a2≢b2{a}_{2}\not\equiv {b}_{2}, and none of them is identically equal to∞\infty. Ifffandf(k){f}^{\left(k)}share(a1,a2)\left({a}_{1},{a}_{2})CM and share(b1,b2)\left({b}_{1},{b}_{2})IM, then(a2−b2)f−(a1−b1)f(k)≡a2b1−a1b2\left({a}_{2}-{b}_{2})f-\left({a}_{1}-{b}_{1}){f}^{\left(k)}\equiv {a}_{2}{b}_{1}-{a}_{1}{b}_{2}. This extends the result due to Li and Yang [Value sharing of an entire function and its derivatives, J. Math. Soc. Japan.51(1999), no. 7, 781–799].

Low complexity multipath and Doppler compensation for direct-sequence spread spectrum signals in underwater acoustic communication

A method to achieve low complexity and low error decoding of direct-sequence spread spectrum signals is demonstrated using measured underwater acoustic communications. Unlike terrestrial radio, the ocean has pronounced and variable multipath (echoes) as well as strong Doppler due to the relatively slow sound speed. The severity of the decoding problem, as analyzed on experimental data, is characterized for both stationary and moving sources. Here, a method based on measuring and updating a passive conjugate matched filter is shown. This method produced low bit-error rates, even in cases with channel fading, by mitigating variable multipath effects. Additionally, Doppler estimation is accomplished by correlating each received symbol with a filter bank comprised of the most recent matched filter shifted by various Doppler offsets. Upon finding the best match, the received symbol is shifted, correlated, and decoded. This method is demonstrated on at-sea experimental data and shown to be a computationally efficient and robust method to receive low SNR. Error free results are shown down to −5 dB input signal-to-noise ratio. The method is then further improved by 2 dB by allowing for non-constant Doppler shift functions within a symbol period, thereby improving Doppler estimation.

Fractal Interpolation Using Harmonic Functions on the Koch Curve

The Koch curve was first described by the Swedish mathematician Helge von Koch in 1904 as an example of a continuous but nowhere differentiable curve. Such functions are now characterised as fractal since their graphs are in general fractal sets. Furthermore, it can be obtained as the graph of an appropriately chosen iterated function system. On the other hand, a fractal interpolation function can be seen as a special case of an iterated function system thus maintaining all of its characteristics. Fractal interpolation functions are continuous functions that can be used to model continuous signals. An in-depth discussion on the theory of affine fractal interpolation functions generating the Koch Curve by using fractal analysis as well as its recent development including some of the research made by the authors is provided. We ensure that the graph of fractal interpolation functions on the Koch Curve are attractors of an iterated function system constructed by non-constant harmonic functions.

Relative logarithmic order of an entire function

In this paper, we extend some results related to the growth rates of entire functions by introducing the relative logarithmic order $\rho _{g}^{l}(f)$ of a nonconstant entire function $f$ with respect to another nonconstant entire function $g.$ Next we investigate some theorems related the behavior of $\rho _{g}^{l}(f)$. We also define the relative logarithmic$ $proximate order of $f$ with respect to $g$ and give some theorems on it.

Essential Self-Adjointness and the $$L^2$$-Liouville Property

We discuss connections between the essential self-adjointness of a symmetric operator and the constancy of functions which are in the kernel of the adjoint of the operator. We then illustrate this relationship in the case of Laplacians on both manifolds and graphs. Furthermore, we discuss the Green’s function and when it gives a non-constant harmonic function which is square integrable.

Characterizations of Hankel operators in the essential commutant of quasicontinuous Toeplitz operators

AbstractThis note characterizes, in terms of interpolating Blaschke products, the symbols of Hankel operators essentially commuting with all quasicontinuous Toeplitz operators on the Hardy space of the unit circle. It also shows that such symbols do not contain the complex conjugate of any nonconstant singular inner function.

Unicity of Entire Functions Concerning Their Shifts and Derivatives

In this paper, we study the unicity of entire functions concerning their shifts and derivatives and prove: Let f be a non-constant entire function of hyper-order less than 1, let c be a non-zero finite value, and let a, b be two distinct finite values. If $$f'(z)$$ and $$f(z+c)$$ share a, b IM, then $$f'(z)\equiv f(z+c)$$ . This improves some results due to Qi and Yang (Comput Methods Funct Theory 20:159–178, 2020).

On Killing Vector Fields on Riemannian Manifolds

We study the influence of a unit Killing vector field on geometry of Riemannian manifolds. For given a unit Killing vector field w on a connected Riemannian manifold (M,g) we show that for each non-constant smooth function f∈C∞(M) there exists a non-zero vector field wf associated with f. In particular, we show that for an eigenfunction f of the Laplace operator on an n-dimensional compact Riemannian manifold (M,g) with an appropriate lower bound on the integral of the Ricci curvature S(wf,wf) gives a characterization of the odd-dimensional unit sphere S2m+1. Also, we show on an n-dimensional compact Riemannian manifold (M,g) that if there exists a positive constant c and non-constant smooth function f that is eigenfunction of the Laplace operator with eigenvalue nc and the unit Killing vector field w satisfying ∇w2≤(n−1)c and Ricci curvature in the direction of the vector field ∇f−w is bounded below by n−1c is necessary and sufficient for (M,g) to be isometric to the sphere S2m+1(c). Finally, we show that the presence of a unit Killing vector field w on an n-dimensional Riemannian manifold (M,g) with sectional curvatures of plane sections containing w equal to 1 forces dimension n to be odd and that the Riemannian manifold (M,g) becomes a K-contact manifold. We also show that if in addition (M,g) is complete and the Ricci operator satisfies Codazzi-type equation, then (M,g) is an Einstein Sasakian manifold.

On the Distribution of Meromorphic Functions of Positive Hyper-Order

Let f(z) be a transcendental meromorphic function, whose zeros have multiplicity at least 3. Set α(z): = β(z)exp (γ(z), where β(z) is a nonconstant elliptic function and γ(z) is an entire function. If σ(f(z)) > σ(α(z)), then f′(z) = α(z) has infinitely many solutions in the complex plane.

Moduli space of logarithmic connections singular over a finite subset of a compact Riemann surface

Let $S$ be a finite subset of a compact connected Riemann surface $X$ of genus $g \geq 2$. Let $\cat{M}_{lc}(n,d)$ denote the moduli space of pairs $(E,D)$, where $E$ is a holomorphic vector bundle over $X$ and $D$ is a logarithmic connection on $E$ singular over $S$, with fixed residues in the centre of $\mathfrak{gl}(n,\C)$, where $n$ and $d$ are mutually corpime. Let $L$ denote a fixed line bundle with a logarithmic connection $D_L$ singular over $S$. Let $\cat{M}'_{lc}(n,d)$ and $\cat{M}_{lc}(n,L)$ be the moduli spaces parametrising all pairs $(E,D)$ such that underlying vector bundle $E$ is stable and $(\bigwedge^nE, \tilde{D}) \cong (L,D_L)$ respectively. Let $\cat{M}'_{lc}(n,L) \subset \cat{M}_{lc}(n,L)$ be the Zariski open dense subset such that the underlying vector bundle is stable. We show that there is a natural compactification of $\cat{M}'_{lc}(n,d)$ and $\cat{M}'_{lc}(n,L)$ and compute their Picard groups. We also show that $\cat{M}'_{lc}(n,L)$ and hence $\cat{M}_{lc}(n,L)$ do not have any non-constant algebraic functions but they admit non-constant holomorhic functions. We also study the Picard group and algebraic functions on the moduli space of logarithmic connections singular over $S$, with arbitrary residues.

Dynamic Adjustment of SIR Model with the Social Permissiveness: An Actual Measure of the Infection Rate

The COVID-19 pandemic has globally impacted the behavior of the social patterns affecting the disease’s contagiousness. This effect deviates the classical SIR model from reproducing the data of COVID-19 in most countries. This study incorporates a non-constant permissiveness function to the SIR model. The resulting model is computationally solved to obtain a likely permissiveness time-function. To solve the adjusted model, a technique based on a proportional-integral controller is applied. The resulting models are compared with previous results obtained by a manual iterative adjusting method.

О СХЕМАХ, ДОПУСКАЮЩИХ КОРОТКИЕ ЕДИНИЧНЫЕ ПРОВЕРЯЮЩИЕ ТЕСТЫ ПРИ ПРОИЗВОЛЬНЫХ НЕИСПРАВНОСТЯХ ФУНКЦИОНАЛЬНЫХ ЭЛЕМЕНТОВ

It is proved that one can implement any non-constant Boolean function in n variables by an irredundant logic network in the basis {&, ⊕, ¬}, containing not more than one dummy input variable and allowing a single fault detection test with length not more than 2n + 3 regarding arbitrary faults of logic gates.

Some characterizations of dual curves in dual 3-space $ \mathbb{D}^{3} $

In this work, we prove that the ratio of torsion and curvature of any dual rectifying curve is a non-constant linear function of its dual arc length parameter. Thereafter, a dual differential equation of third order is constructed for every dual curve. Then, several well-known characterizations of dual spherical, normal and rectifying curves are consequences of this differential equation. Finally, we prove a simple new characterization of dual spherical curves in terms of the Darboux vector.