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Dettagli della rivista
Formato
Rivista
eISSN
1338-9750
Pubblicato per la prima volta
12 Nov 2012
Periodo di pubblicazione
3 volte all'anno
Lingue
Inglese

Cerca

Volume 71 (2018): Edizione 1 (December 2018)

Dettagli della rivista
Formato
Rivista
eISSN
1338-9750
Pubblicato per la prima volta
12 Nov 2012
Periodo di pubblicazione
3 volte all'anno
Lingue
Inglese

Cerca

16 Articoli
Accesso libero

Recent Trends in Differential and Difference Equations (A Survey)

Pubblicato online: 25 Jan 2019
Pagine: 1 - 7

Astratto

Abstract

This article briefly describes results of recent publications in the field of differential and difference equations and their applications.

Parole chiave

  • differential equations
  • difference equations

MSC 2010

  • 34Cxx
  • 34Dxx
  • 34Fxx
  • 34Hxx
  • 34Kxx
Accesso libero

On The Dense Controllability for The Parabolic Problem with Time-Distributed Functional

Pubblicato online: 25 Jan 2019
Pagine: 9 - 25

Astratto

Abstract

We consider a control problem for one-dimensional heat equation with quadratic cost functional. We prove the existence and uniqueness of a control function from a prescribed set, and study the structure of the set of accessible temperature functions. We also prove the dense controllability of the problem for some set of control functions.

Parole chiave

  • heat equation
  • quadratic cost functional
  • extremal problem
  • exact controllability
  • dense controllability

MSC 2010

  • 35K20
  • 35Q93
  • 35Q79
  • 49J20
Accesso libero

An Optimal Control Problem for A Viscoelastic Plate in a Dynamic Contact with an Obstacle

Pubblicato online: 25 Jan 2019
Pagine: 27 - 37

Astratto

Abstract

We deal with an optimal control problem governed by a nonlinear hyperbolic initial-boundary value problem describing the perpendicular vibrations of a simply supported anisotropic viscoelastic plate against a rigid obstacle. A variable thickness of a plate plays the role of a control variable. We verify the existence of an optimal thickness function.

Parole chiave

  • Vibrating viscoelastic plate
  • rigid obstacle
  • optimal control
  • variable thickness

MSC 2010

  • 35L85
  • 49J20
  • 49J40
  • 74D10
  • 74K20
Accesso libero

Networks Describing Dynamical Systems

Pubblicato online: 25 Jan 2019
Pagine: 39 - 52

Astratto

Abstract

We consider systems of ordinary differential equations that arise in the theory of gene regulatory networks. These systems can be of arbitrary size but of definite structure that depends on the choice of regulatory matrices. Attractors play the decisive role in behaviour of elements of such systems. We study the structure of simple attractors that consist of a number of critical points for several choices of regulatory matrices.

Parole chiave

  • nonlinear systems of ordinary differential equations
  • attractors
  • gene regulation
  • networks

MSC 2010

  • 34B60
  • 34C60
  • 34D45
Accesso libero

Oscillation Tests for Fractional Difference Equations

Pubblicato online: 25 Jan 2019
Pagine: 53 - 64

Astratto

Abstract

In this paper, we study the oscillatory behavior of solutions of the fractional difference equation of the form

Δ(r(t)g(Δαx(t)))+p(t)f(s=t0t1+α(ts1)(α)x(s))=0,t𝕅t0+1α,$$\Delta \left( {r\left( t \right)g\left( {{\Delta ^\alpha }x(t)} \right)} \right) + p(t)f\left( {\sum\limits_{s = {t_0}}^{t - 1 + \alpha } {{{(t - s - 1)}^{( - \alpha )}}x(s)} } \right) = 0, & t \in {_{{t_0} + 1 - \alpha }},$$

where Δα denotes the Riemann-Liouville fractional difference operator of order α, 0 < α ≤ 1, ℕt0+1−α={t0+1−αt0+2−α…}, t0 > 0 and γ > 0 is a quotient of odd positive integers. We establish some oscillatory criteria for the above equation, using the Riccati transformation and Hardy type inequalities. Examples are provided to illustrate the theoretical results.

Parole chiave

  • oscillation
  • difference equations
  • fractional sum

MSC 2010

  • 26A33
  • 39A12
Accesso libero

The Sturm Separation Theorem for Impulsive Delay Differential Equations

Pubblicato online: 25 Jan 2019
Pagine: 65 - 70

Astratto

Abstract

Wronskian is one of the classical objects in the theory of ordinary differential equations. Properties of Wronskian lead to important conclusions on behaviour of solutions of delay equations. For instance, non-vanishing Wronskian ensures validity of the Sturm separation theorem (between two adjacent zeros of any solution there is one and only one zero of every other nontrivial linearly independent solution) for delay equations. We propose the Sturm separation theorem in the case of impulsive delay differential equations and obtain assertions about its validity.

Parole chiave

  • delay differential equations
  • Sturm separation theorem
  • Wronskian

MSC 2010

  • 34K20
Accesso libero

Stability of The Equilibrium of Nonlinear Dynamical Systems

Pubblicato online: 25 Jan 2019
Pagine: 71 - 80

Astratto

Abstract

The algorithm for estimating the stability domain of zero equilibrium to the system of nonlinear differential equations with a quadratic part and a fractional part is proposed in the article. The second Lyapunov method with quadratic Lyapunov functions is used as a method for studying such systems.

Parole chiave

  • nonlinear differential systems
  • asymptotic stability of the trivial solution
  • quadratic Lyapunov functions
  • Lyapunov matrix equation
  • estimation of convergence of solutions

MSC 2010

  • 34K20
  • 34K25
  • 37B25
  • 37C75
Accesso libero

Stability and Square Integrability of Solutions to third order Neutral Delay Differential Equations

Pubblicato online: 25 Jan 2019
Pagine: 81 - 97

Astratto

Abstract

In this paper, sufficient conditions to guarantee the square integrability of all solutions and the asymptotic stability of the zero solution of a non-autonomous third-order neutral delay differential equation are established. An example is given to illustrate the main results.

Parole chiave

  • uniform ultimate boundedness
  • square integrability
  • Lyapunov functional
  • neutral differential equation of third order

MSC 2010

  • 34K12
  • 34K20
  • 34K40
Accesso libero

Stochastic Differential Equations Describing Systems with Coloured Noise

Pubblicato online: 25 Jan 2019
Pagine: 99 - 107

Astratto

Abstract

In this paper we deal with stochastic differential equations, that describe systems effected by coloured noise. In electrical systems this can be the case, when e.g. transmission line is modelled by means of proper higher-order ladder network. We define the mathematical representation of the coloured noise as a solution of the Langevin equation and formulate the corresponding Itô type stochastic differential equation. Applying this theory we derive the stochastic model of the network and find sets of individual stochastic trajectories numerically via a stochastic version of the backward Euler scheme. Afterwards respective confi-dence intervals are computed statistically while utilizing Student’s t distribution. The theoretical results are illustrated by an example of a higher-order ladder network. Numerical simulations in the example are carried out using Matlab.

Parole chiave

  • Stochastic integral equations
  • Wiener process
  • Itô formula
  • white noise
  • coloured noise
  • higher-order ladder network
  • transmission line model

MSC 2010

  • 34F05
  • 60H10
  • 65C30
Accesso libero

Approximation of Solutions to Nonautonomous Difference Equations

Pubblicato online: 25 Jan 2019
Pagine: 109 - 121

Astratto

Abstract

We study the asymptotic properties of solutions to nonautonomous difference equations of the form

Δmxn=anf(n,xσ(n))+bn,f:𝕅×𝕉𝕉,σ:𝕅𝕅$${\Delta ^m}{x_n} = {a_n}f(n,{x_{\sigma (n)}}) + {b_n},\,\,f:N \times {\Bbb R} \to {\Bbb R},\,\,\sigma :{\Bbb N} \to {\Bbb N}$$

Using the iterated remainder operator and asymptotic difference pairs we establish some results concerning approximative solutions and approximations of solutions. Our approach allows us to control the degree of approximation.

Parole chiave

  • nonautonomous equation
  • difference pair
  • prescribed asymptotic behavior
  • asymptotically polynomial solution
  • bounded solution

MSC 2010

  • 39A10
Accesso libero

Existence of Positive Bounded Solutions of System of Three Dynamic Equations with Neutral Term on Time Scales

Pubblicato online: 25 Jan 2019
Pagine: 123 - 137

Astratto

Abstract

In this paper the system of three dynamic equations with neutral term in the following form

Parole chiave

  • system of difference equation
  • three-dimensional
  • neutral type
  • boundedness
  • nonoscillatory solution
  • time scales

MSC 2010

  • 34N05
  • 39A10
  • 39A22
Accesso libero

Neutral Difference System and its Nonoscillatory Solutions

Pubblicato online: 25 Jan 2019
Pagine: 139 - 148

Astratto

Abstract

The paper deals with a system of four nonlinear difference equations where the first equation is of a neutral type. We study nonoscillatory solutions of the system and we present sufficient conditions for the system to have weak property B.

Parole chiave

  • difference system
  • nonoscillatory solutions
  • neutral term
  • property B
  • weak property B

MSC 2010

  • 39A10
Accesso libero

About Distributed Control in Model of Testosterone Regulation

Pubblicato online: 25 Jan 2019
Pagine: 149 - 154

Astratto

Abstract

In the paper, stability of integro-differential equation is studied. The model of testosterone regulation is considered. The model describes an interaction of: the concentration of hormone (GnRH) which will be denoted as x1, with the concentration of the hormone (LH)-x2 and the concentration of testosterone (Te)-x3 and can be written in the form

{x1(t)+b1x1(t)=0,x2(t)+b2x2(t)g1x1(t)=0,x3(t)+b3x3(t)c10teα1(ts)x2(s)ds=0,t0.$$\left\{ {\matrix{ {{{x'}_1}(t) + {b_1}{x_1}(t) = 0,} \hfill \cr {{{x'}_2}(t) + {b_2}{x_2}(t) - {g_1}{x_1}(t) = 0,} \hfill \cr {{{x'}_3}(t) + {b_3}{x_3}(t) - {c_1}\int\limits_0^t {{e^{ - {\alpha _1}(t - s)}}{x_2}(s)ds = 0,\;t \ge 0.} } \hfill \cr } } \right.$$

The values bi, i = 1, 2, 3 correspond to the respective half-life times of GnRH, LH and Te. The aim of the work is to propose a concept to hold the concentration of testosterone above a corresponding level. In order to achieve this, distributed input control in the form of integral term is used.

Parole chiave

  • integro-differential equations
  • stability
  • reduction method
  • testosterone regulation

MSC 2010

  • 34K20
Accesso libero

Control of Conservation Laws – An Application

Pubblicato online: 25 Jan 2019
Pagine: 155 - 174

Astratto

Abstract

We present here three types of controlled boundary value problems for conservation laws arising from energy co-generation, hydraulic flows and water hammer for hydroelectric power plants and control of the open channel flows (shallow water). The novelty of these models, from the mathematical point of view, is that they are described by nonlinear hyperbolic partial differential equations of the conservation laws with (possibly) nonlinear boundary conditions. At their turn these boundary conditions are controlled by some systems of ordinary differential equations. The engineering requirements for such systems are asymptotic stability and disturbance rejection: these properties have to be achieved by feedback control. In our setting the main tool for tackling these problems is a suitable Lyapunov functional arising from the energy identity. The hints for “guessing” this functional are to be found in the linearized version of the aforementioned mathematical objects.

Parole chiave

  • nonlinear systems
  • conservation laws
  • feedback control
  • energy identity

MSC 2010

  • 35L65
  • 35Q93
  • 93C20
  • 93D15
  • 93D30
Accesso libero

Oscillation Criteria for Forced first Order Nonlinear Neutral Impulsive Difference System

Pubblicato online: 25 Jan 2019
Pagine: 175 - 193

Astratto

Abstract

In this work, we have established sufficient conditions for oscillation and nonoscillation of a class of forced first order neutral impulsive difference equations with deviating arguments and fixed moments of impulsive effect.

Parole chiave

  • oscillation
  • nonoscillation
  • impulsive difference equation
  • nonlinear
  • delay

MSC 2010

  • 39A10
  • 39A12
Accesso libero

On Some Discrete Potential Like Operators

Pubblicato online: 25 Jan 2019
Pagine: 195 - 212

Astratto

Abstract

We consider some discrete pseudo-differential equations in discrete Sobolev–Slobodetskii spaces. For a discrete half-space and certain values of an index of periodic factorization for an elliptic symbol we introduce additional potential-like unknowns and prove existence and uniqueness theorem in appropriate discrete Sobolev–Slobodetskii spaces.

Parole chiave

  • digital pseudo-differential operator
  • discrete Sobolev–Slobodetskii space
  • elliptic symbol
  • periodic factorization
  • potential like operator

MSC 2010

  • 47G30
  • 42B37
16 Articoli
Accesso libero

Recent Trends in Differential and Difference Equations (A Survey)

Pubblicato online: 25 Jan 2019
Pagine: 1 - 7

Astratto

Abstract

This article briefly describes results of recent publications in the field of differential and difference equations and their applications.

Parole chiave

  • differential equations
  • difference equations

MSC 2010

  • 34Cxx
  • 34Dxx
  • 34Fxx
  • 34Hxx
  • 34Kxx
Accesso libero

On The Dense Controllability for The Parabolic Problem with Time-Distributed Functional

Pubblicato online: 25 Jan 2019
Pagine: 9 - 25

Astratto

Abstract

We consider a control problem for one-dimensional heat equation with quadratic cost functional. We prove the existence and uniqueness of a control function from a prescribed set, and study the structure of the set of accessible temperature functions. We also prove the dense controllability of the problem for some set of control functions.

Parole chiave

  • heat equation
  • quadratic cost functional
  • extremal problem
  • exact controllability
  • dense controllability

MSC 2010

  • 35K20
  • 35Q93
  • 35Q79
  • 49J20
Accesso libero

An Optimal Control Problem for A Viscoelastic Plate in a Dynamic Contact with an Obstacle

Pubblicato online: 25 Jan 2019
Pagine: 27 - 37

Astratto

Abstract

We deal with an optimal control problem governed by a nonlinear hyperbolic initial-boundary value problem describing the perpendicular vibrations of a simply supported anisotropic viscoelastic plate against a rigid obstacle. A variable thickness of a plate plays the role of a control variable. We verify the existence of an optimal thickness function.

Parole chiave

  • Vibrating viscoelastic plate
  • rigid obstacle
  • optimal control
  • variable thickness

MSC 2010

  • 35L85
  • 49J20
  • 49J40
  • 74D10
  • 74K20
Accesso libero

Networks Describing Dynamical Systems

Pubblicato online: 25 Jan 2019
Pagine: 39 - 52

Astratto

Abstract

We consider systems of ordinary differential equations that arise in the theory of gene regulatory networks. These systems can be of arbitrary size but of definite structure that depends on the choice of regulatory matrices. Attractors play the decisive role in behaviour of elements of such systems. We study the structure of simple attractors that consist of a number of critical points for several choices of regulatory matrices.

Parole chiave

  • nonlinear systems of ordinary differential equations
  • attractors
  • gene regulation
  • networks

MSC 2010

  • 34B60
  • 34C60
  • 34D45
Accesso libero

Oscillation Tests for Fractional Difference Equations

Pubblicato online: 25 Jan 2019
Pagine: 53 - 64

Astratto

Abstract

In this paper, we study the oscillatory behavior of solutions of the fractional difference equation of the form

Δ(r(t)g(Δαx(t)))+p(t)f(s=t0t1+α(ts1)(α)x(s))=0,t𝕅t0+1α,$$\Delta \left( {r\left( t \right)g\left( {{\Delta ^\alpha }x(t)} \right)} \right) + p(t)f\left( {\sum\limits_{s = {t_0}}^{t - 1 + \alpha } {{{(t - s - 1)}^{( - \alpha )}}x(s)} } \right) = 0, & t \in {_{{t_0} + 1 - \alpha }},$$

where Δα denotes the Riemann-Liouville fractional difference operator of order α, 0 < α ≤ 1, ℕt0+1−α={t0+1−αt0+2−α…}, t0 > 0 and γ > 0 is a quotient of odd positive integers. We establish some oscillatory criteria for the above equation, using the Riccati transformation and Hardy type inequalities. Examples are provided to illustrate the theoretical results.

Parole chiave

  • oscillation
  • difference equations
  • fractional sum

MSC 2010

  • 26A33
  • 39A12
Accesso libero

The Sturm Separation Theorem for Impulsive Delay Differential Equations

Pubblicato online: 25 Jan 2019
Pagine: 65 - 70

Astratto

Abstract

Wronskian is one of the classical objects in the theory of ordinary differential equations. Properties of Wronskian lead to important conclusions on behaviour of solutions of delay equations. For instance, non-vanishing Wronskian ensures validity of the Sturm separation theorem (between two adjacent zeros of any solution there is one and only one zero of every other nontrivial linearly independent solution) for delay equations. We propose the Sturm separation theorem in the case of impulsive delay differential equations and obtain assertions about its validity.

Parole chiave

  • delay differential equations
  • Sturm separation theorem
  • Wronskian

MSC 2010

  • 34K20
Accesso libero

Stability of The Equilibrium of Nonlinear Dynamical Systems

Pubblicato online: 25 Jan 2019
Pagine: 71 - 80

Astratto

Abstract

The algorithm for estimating the stability domain of zero equilibrium to the system of nonlinear differential equations with a quadratic part and a fractional part is proposed in the article. The second Lyapunov method with quadratic Lyapunov functions is used as a method for studying such systems.

Parole chiave

  • nonlinear differential systems
  • asymptotic stability of the trivial solution
  • quadratic Lyapunov functions
  • Lyapunov matrix equation
  • estimation of convergence of solutions

MSC 2010

  • 34K20
  • 34K25
  • 37B25
  • 37C75
Accesso libero

Stability and Square Integrability of Solutions to third order Neutral Delay Differential Equations

Pubblicato online: 25 Jan 2019
Pagine: 81 - 97

Astratto

Abstract

In this paper, sufficient conditions to guarantee the square integrability of all solutions and the asymptotic stability of the zero solution of a non-autonomous third-order neutral delay differential equation are established. An example is given to illustrate the main results.

Parole chiave

  • uniform ultimate boundedness
  • square integrability
  • Lyapunov functional
  • neutral differential equation of third order

MSC 2010

  • 34K12
  • 34K20
  • 34K40
Accesso libero

Stochastic Differential Equations Describing Systems with Coloured Noise

Pubblicato online: 25 Jan 2019
Pagine: 99 - 107

Astratto

Abstract

In this paper we deal with stochastic differential equations, that describe systems effected by coloured noise. In electrical systems this can be the case, when e.g. transmission line is modelled by means of proper higher-order ladder network. We define the mathematical representation of the coloured noise as a solution of the Langevin equation and formulate the corresponding Itô type stochastic differential equation. Applying this theory we derive the stochastic model of the network and find sets of individual stochastic trajectories numerically via a stochastic version of the backward Euler scheme. Afterwards respective confi-dence intervals are computed statistically while utilizing Student’s t distribution. The theoretical results are illustrated by an example of a higher-order ladder network. Numerical simulations in the example are carried out using Matlab.

Parole chiave

  • Stochastic integral equations
  • Wiener process
  • Itô formula
  • white noise
  • coloured noise
  • higher-order ladder network
  • transmission line model

MSC 2010

  • 34F05
  • 60H10
  • 65C30
Accesso libero

Approximation of Solutions to Nonautonomous Difference Equations

Pubblicato online: 25 Jan 2019
Pagine: 109 - 121

Astratto

Abstract

We study the asymptotic properties of solutions to nonautonomous difference equations of the form

Δmxn=anf(n,xσ(n))+bn,f:𝕅×𝕉𝕉,σ:𝕅𝕅$${\Delta ^m}{x_n} = {a_n}f(n,{x_{\sigma (n)}}) + {b_n},\,\,f:N \times {\Bbb R} \to {\Bbb R},\,\,\sigma :{\Bbb N} \to {\Bbb N}$$

Using the iterated remainder operator and asymptotic difference pairs we establish some results concerning approximative solutions and approximations of solutions. Our approach allows us to control the degree of approximation.

Parole chiave

  • nonautonomous equation
  • difference pair
  • prescribed asymptotic behavior
  • asymptotically polynomial solution
  • bounded solution

MSC 2010

  • 39A10
Accesso libero

Existence of Positive Bounded Solutions of System of Three Dynamic Equations with Neutral Term on Time Scales

Pubblicato online: 25 Jan 2019
Pagine: 123 - 137

Astratto

Abstract

In this paper the system of three dynamic equations with neutral term in the following form

Parole chiave

  • system of difference equation
  • three-dimensional
  • neutral type
  • boundedness
  • nonoscillatory solution
  • time scales

MSC 2010

  • 34N05
  • 39A10
  • 39A22
Accesso libero

Neutral Difference System and its Nonoscillatory Solutions

Pubblicato online: 25 Jan 2019
Pagine: 139 - 148

Astratto

Abstract

The paper deals with a system of four nonlinear difference equations where the first equation is of a neutral type. We study nonoscillatory solutions of the system and we present sufficient conditions for the system to have weak property B.

Parole chiave

  • difference system
  • nonoscillatory solutions
  • neutral term
  • property B
  • weak property B

MSC 2010

  • 39A10
Accesso libero

About Distributed Control in Model of Testosterone Regulation

Pubblicato online: 25 Jan 2019
Pagine: 149 - 154

Astratto

Abstract

In the paper, stability of integro-differential equation is studied. The model of testosterone regulation is considered. The model describes an interaction of: the concentration of hormone (GnRH) which will be denoted as x1, with the concentration of the hormone (LH)-x2 and the concentration of testosterone (Te)-x3 and can be written in the form

{x1(t)+b1x1(t)=0,x2(t)+b2x2(t)g1x1(t)=0,x3(t)+b3x3(t)c10teα1(ts)x2(s)ds=0,t0.$$\left\{ {\matrix{ {{{x'}_1}(t) + {b_1}{x_1}(t) = 0,} \hfill \cr {{{x'}_2}(t) + {b_2}{x_2}(t) - {g_1}{x_1}(t) = 0,} \hfill \cr {{{x'}_3}(t) + {b_3}{x_3}(t) - {c_1}\int\limits_0^t {{e^{ - {\alpha _1}(t - s)}}{x_2}(s)ds = 0,\;t \ge 0.} } \hfill \cr } } \right.$$

The values bi, i = 1, 2, 3 correspond to the respective half-life times of GnRH, LH and Te. The aim of the work is to propose a concept to hold the concentration of testosterone above a corresponding level. In order to achieve this, distributed input control in the form of integral term is used.

Parole chiave

  • integro-differential equations
  • stability
  • reduction method
  • testosterone regulation

MSC 2010

  • 34K20
Accesso libero

Control of Conservation Laws – An Application

Pubblicato online: 25 Jan 2019
Pagine: 155 - 174

Astratto

Abstract

We present here three types of controlled boundary value problems for conservation laws arising from energy co-generation, hydraulic flows and water hammer for hydroelectric power plants and control of the open channel flows (shallow water). The novelty of these models, from the mathematical point of view, is that they are described by nonlinear hyperbolic partial differential equations of the conservation laws with (possibly) nonlinear boundary conditions. At their turn these boundary conditions are controlled by some systems of ordinary differential equations. The engineering requirements for such systems are asymptotic stability and disturbance rejection: these properties have to be achieved by feedback control. In our setting the main tool for tackling these problems is a suitable Lyapunov functional arising from the energy identity. The hints for “guessing” this functional are to be found in the linearized version of the aforementioned mathematical objects.

Parole chiave

  • nonlinear systems
  • conservation laws
  • feedback control
  • energy identity

MSC 2010

  • 35L65
  • 35Q93
  • 93C20
  • 93D15
  • 93D30
Accesso libero

Oscillation Criteria for Forced first Order Nonlinear Neutral Impulsive Difference System

Pubblicato online: 25 Jan 2019
Pagine: 175 - 193

Astratto

Abstract

In this work, we have established sufficient conditions for oscillation and nonoscillation of a class of forced first order neutral impulsive difference equations with deviating arguments and fixed moments of impulsive effect.

Parole chiave

  • oscillation
  • nonoscillation
  • impulsive difference equation
  • nonlinear
  • delay

MSC 2010

  • 39A10
  • 39A12
Accesso libero

On Some Discrete Potential Like Operators

Pubblicato online: 25 Jan 2019
Pagine: 195 - 212

Astratto

Abstract

We consider some discrete pseudo-differential equations in discrete Sobolev–Slobodetskii spaces. For a discrete half-space and certain values of an index of periodic factorization for an elliptic symbol we introduce additional potential-like unknowns and prove existence and uniqueness theorem in appropriate discrete Sobolev–Slobodetskii spaces.

Parole chiave

  • digital pseudo-differential operator
  • discrete Sobolev–Slobodetskii space
  • elliptic symbol
  • periodic factorization
  • potential like operator

MSC 2010

  • 47G30
  • 42B37

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