Bounded in the mean of order $p$ solutions of a difference equation with a jump of the operator coefficient

2021 ◽  
Vol 101 ◽  
pp. 103-108
Author(s):  
M. F. Gorodnii ◽  
I. V. Gonchar
Keyword(s):  
Difference Equation ◽  
Operator Coefficient ◽  
The Mean

scholarly journals Bounded solutions of a difference equation with finite number of jumps of operator coefficient

2020 ◽  
Vol 12 (1) ◽  
pp. 165-172
Author(s):  
A. Chaikovs'kyi ◽  
O. Lagoda
Keyword(s):  
Difference Equation ◽  
Finite Number ◽  
Existence And Uniqueness ◽  
Bounded Solution ◽  
Sufficient Conditions ◽  
Operator Coefficient ◽  
Bounded Solutions ◽  
Piecewise Constant ◽  
Variable Operator ◽  
Necessary And Sufficient

We study the problem of existence of a unique bounded solution of a difference equation with variable operator coefficient in a Banach space. There is well known theory of such equations with constant coefficient. In that case the problem is solved in terms of spectrum of the operator coefficient. For the case of variable operator coefficient correspondent conditions are known too. But it is too hard to check the conditions for particular equations. So, it is very important to give an answer for the problem for those particular cases of variable coefficient, when correspondent conditions are easy to check. One of such cases is the case of piecewise constant operator coefficient. There are well known sufficient conditions of existence and uniqueness of bounded solution for the case of one jump. In this work, we generalize these results for the case of finite number of jumps of operator coefficient. Moreover, under additional assumption we obtained necessary and sufficient conditions of existence and uniqueness of bounded solution.

scholarly journals On the Polynomial Solution of Divided-Difference Equations of the Hypergeometric Type on Nonuniform Lattices

Axioms ◽  
2019 ◽  
Vol 8 (2) ◽  
pp. 47 ◽  
Author(s):  
Mama Foupouagnigni ◽  
Salifou Mboutngam
Keyword(s):  
Difference Equation ◽  
Divided Difference ◽  
Polynomial Solution ◽  
Formal Proof ◽  
Second Order ◽  
Fixed Degree ◽  
Hypergeometric Type ◽  
The Mean ◽  
The Right

In this paper, we provide a formal proof of the existence of a polynomial solution of fixed degree for a second-order divided-difference equation of the hypergeometric type on non-uniform lattices, generalizing therefore previous work proving existence of the polynomial solution for second-order differential, difference or q-difference equation of hypergeometric type. This is achieved by studying the properties of the mean operator and the divided-difference operator as well as by defining explicitly, the right and the “left” inverse for the second operator. The method constructed to provide this formal proof is likely to play an important role in the characterization of orthogonal polynomials on non-uniform lattices and might also be used to provide hypergeometric representation (when it does exist) of the second solution—non polynomial solution—of a second-order divided-difference equation of hypergeometric type.

scholarly journals Posterior sloping angle of the capital femoral physis in slipped capital femoral epiphysis has poor clinical utility and should not guide treatment on prophylactic fixation

2020 ◽  
Vol 28 (2) ◽  
pp. 230949902093782
Author(s):  
Youheng Ou Yang ◽  
Chloe Xiaoyun Chan ◽  
Gloria Hui Min Cheng ◽  
Sumanth Kumar Gera ◽  
Arjandas Mahadev ◽  
...  
Keyword(s):  
Age Of Onset ◽  
Slipped Capital Femoral Epiphysis ◽  
Operator Coefficient ◽  
Number Needed To Treat ◽  
Level Of Evidence ◽  
Treatment Threshold ◽  
Treatment Thresholds ◽  
The Mean ◽  
Femoral Physis

Background: Prophylactic pinning of the uninvolved side after unilateral slipped capital femoral epiphysis (SCFE) is controversial as it balances increased surgical risks against the possibility of protecting a normal hip from initial slip and deformity. A posterior sloping angle (PSA) of greater than 12–14.5° has been proposed by various authors as a treatment threshold to predict for contralateral hip progression and prophylactic pinning. Methods: A retrospective review of a 10-year series of patients with the diagnosis of SCFE and follow-up of 18 months was conducted. Patients were divided into two groups, those with Isolated Unilateral Slips and those who subsequently underwent Subsequent Contralateral Progression. PSA measurements were performed by two clinicians and assessed for inter-observer reliability. Data collected included age, sex, ethnicity, Loder class, endocrinopathy, renal impairment, radiation exposure, and PSA. Results: There were no significant differences between the distribution of gender, site of slip, age of onset, Loder class, and presence of medical comorbidities between the Isolated Unilateral Slip and Subsequent Contralateral Progression groups ( p > 0.05). The mean PSA value was not significantly higher in the Subsequent Contralateral Progression group (17.9 ± 4.32 (10.5–23.5)) compared to the Isolated Unilateral Slip group (15.8 ± 5.31 (6–26)) ( p = 0.32). The receiver operator coefficient-derived ideal treatment threshold of 16.5° gave a sensitivity of 0.71, specificity of 0.64, and number needed to treat of 3. Conclusion: PSA differences between the Subsequent Contralateral Progression and Isolated Unilateral Slip groups were not statistically significant in this series. All proposed treatment thresholds had poor specificity. Prophylactic pinning should not be based on isolated PSA values. Level of evidence: III.

scholarly journals Exponential stability of a kind of stochastic delay difference equations

2006 ◽  
Vol 2006 ◽  
pp. 1-9 ◽  
Author(s):  
Xiaohua Ding
Keyword(s):  
Difference Equation ◽  
Exponential Stability ◽  
Type Theorem ◽  
Stochastic Difference Equation ◽  
Delay Difference Equation ◽  
Stochastic Delay ◽  
Mean Square Exponential Stability ◽  
The Mean ◽  
Delay Difference Equations ◽  
Delay Difference

We present a Razumilchin-type theorem for stochastic delay difference equation, and use it to investigate the mean square exponential stability of a kind of nonautonomous stochastic difference equation which may also be viewed as an approximation of a nonautonomous stochastic delay integrodifferential equations (SDIDEs), and of a difference equation arises from some of the earliest mathematical models of the macroeconomic “trade cycle” with the environmental noise.

AN OVERVIEW OF THE STRETCHED INTERMEDIATE MODEL OF B–Z DNA TRANSITION

2006 ◽  
Vol 01 (03) ◽  
pp. 279-291 ◽  
Author(s):  
WILBER LIM ◽  
YUAN PING FENG
Keyword(s):  
Difference Equation ◽  
Deoxyribonucleic Acid ◽  
Mean Field ◽  
Simulation Method ◽  
Stochastic Difference Equation ◽  
Intermediate Model ◽  
Field Approach ◽  
Dna Strands ◽  
The Mean ◽  
Simulation Results

Recently, the stretched intermediate model was proposed for the B–Z deoxyribonucleic acid (DNA) transition based on simulation results carried out using the Stochastic Difference Equation (SDE) that showed unwinding and elongation of the oligomer during the transition. This model has proven to be successful in resolving the steric dilemma in short oligomers. However, extending the simulation method to larger DNA strands may prove to be computationally challenging. Such difficulty has led us to adopt a mean field approach using phenomenological interaction potentials to simulate the transition. Like the atomistic approach, the SDE simulations based on the mean field approach, also suggest the presence of a stretched intermediate during the transition.

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