Math. Interdisc. Res. University of Kashan Mathematics Interdisciplinary Research 2538-3639 University of Kashan 106 10.22052/mir.2021.242239.1286 Original Scientific Paper Inverse Nodal Problem for Polynomial Pencil of a Sturm-Liouville Operator from Nodal Parameters Inverse Nodal Problem for Polynomial Pencil of a Sturm-Liouville Operator from Nodal Parameters Goktas Sertac Department of Mathematics, Mersin University, Mersin, Turkey Biten Esengul Department of Mathematics, Mersin University, Mersin, Turkey 01 09 2021 6 3 171 183 08 05 2021 15 10 2021 Copyright © 2021, University of Kashan. 2021 https://mir.kashanu.ac.ir/article_111634.html

A Sturm-Liouville problem with n-potential functions in the second order differential equation and which contains spectral parameter depending on linearly in one boundary condition is considered. The asymptotic formulas for the eigenvalues, nodal parameters (nodal points and nodal lengths) of this problem are calculated by the Prüfer's substitutions. Also, using these asymptotic formulas, an explicit formula for the potential functions are given. Finally, a numerical example is given.

Eigenvalues Eigenfunctions Prüfer's Substitutions Sturm-Liouville problem Inverse Nodal problem
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Levitan, On the determination of a differential equation from its spectral function,Izvestiya Akademii Nauk SSSR Seriya Matematicheskaya 15 (4) (1951) 309 − 360.  S. Goktas, H. Koyunbakan and T. Gulsen, Inverse nodal problem for polynomial pencil of Sturm-Liouville operator, Math. Methods Appl. Sci. 41 (2018) 7576 − 7582.  I. M. Guseinov, A. A. Nabiev and R. T. Pashaev, Transformation operators and asymptotic formulas for the eigenvalues of a polynomial pencil of Sturm-Liouville operators, Sibirskii Matematicheskii Zhurnal 41 (2000) 554 − 566.  I. M. Guseinov and A. A. Nabiev, A class of inverse problems for a quadratic pencil of Sturm-Liouville operators, Differentsial’nye Uravneniya 36 (3) (2000) 418 − 420.  O. L. Hald and J. R. McLaughlin, Solutions of inverse nodal problems, Inverse Problems 5 (1989) 307 − 347.  H. Hoschtadt, The inverse Sturm-Liouville problem, Commun. Pure Appl. Math. 26 (1973) 715 − 729.  N. B. Kerimov and S. Goktas, E. A. 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Panakhov, Half inverse problem for diffusion operators on the finite interval J. Math. Anal. Appl. 326 (2007) 1024 − 1030.  H. Koyunbakan and E. Yilmaz, Reconstruction of the potential function and its derivatives for the diffusion operator Zeitschrift für Naturforschung A 63a (2008) 127 − 130.  E. A. Maris and S. Goktas, On the spectral properties of a Sturm-Liouville problem with eigenparameter in the boundary condition, HJMS 49 (4) (2020) 1373 − 1382.  J. R. McLaughlin, Inverse spectral theory using nodal points as data-A uniqueness result J. Diff. Eqs. 73 (1988) 354 − 362.  A. Neamaty and Y. Khalili, Determination of a differential operator with discontinuity from interior spectral data Inverse Probl. Sci. Eng. 22 (6) (2014) 1002 − 1008.  A. Neamaty and Y. Khalili, The uniqueness theorem for differential pencils with the jump condition in the finite interval, Iranian J. Sci. Tech. (Sci.) 38 (3.1) (2014) 305 − 309.  S. Mosazadeh and A. Akbarfam, On Hochstadt-Lieberman theorem for impulsive Sturm-Liouville problems with boundary conditions polynomially dependent on the spectral parameter Turkish J. Math. 44 (3) (2018) 778−790.  A. A. Nabiev, On a fundamental system of solutions of the matrix schrödinger equation with a polynomial energy-dependent potential, Math. Methods Appl. Sci. 33 (11) (2010) 1372 − 1383.  E. S. Panakhov, H. Koyunbakan and U. Ic, Reconstruction formula for the potential function of Sturm-Liouville problem with eigenparameter boundary condition, Inverse Probl. Sci. Eng. 18 (1) (2010) 173 − 180.  J. P. Pinasco and C. A. Scarola, Nodal inverse problem for second order Sturm-Liouville operators with indefinite weights, Appl. Math. Comput. 256 (2015) 819 − 830.  E. Şen, Computation of trace and nodal points of eigenfunctions for a Sturm-Liouville problem with retarded argument, Cumhuriyet Sci. J. 39 (3) (2018) 597 − 607.  Y. P. Wang, Y. Hu and C. T. Shieh, The partial inverse nodal problem for differential pencils on a finite interval, Tamkang J. Math. 50 (3) (2019) 307− 319.  Y. P. Wang and C. T. Shieh, X. Wei, Partial inverse nodal problems for differential pencils on a star-shaped graph, Math. Methods Appl. Sci. 43 (15) (2020) 8841 − 8855.  C. -F. Yang, Inverse nodal problems for the Sturm-Liouville operator with a constant delay, J. Diff. Eqs. 257 (4) (2014) 1288 − 1306.  E. Yilmaz and H. Koyunbakan, Reconstruction of potential function and its derivatives for Sturm-Liouville problem with eigenvalues in boundary condition, Inverse Prob. Sci. Eng. 18 (7) (2010) 935 − 944.
Math. Interdisc. Res. University of Kashan Mathematics Interdisciplinary Research 2538-3639 University of Kashan 106 10.22052/mir.2021.240213.1227 Original Scientific Paper Fixed Point of Multivalued Mizoguchi-Takahashi's Type Mappings and Answer to the Rouhani-Moradi's Open Problem Fixed Point of Multivalued Mizoguchi-Takahashi's Type Mappings and Answer to the Rouhani-Moradi's Open Problem Moradi Sirous Department of Mathematics, Faculty of Sciences, Lorestan University, Khorramabad 68151-4-4316, Iran Fathi Zahra Department of Mathematics, Faculty of Science, Arak University, Arak 38156-8-8349, Iran 01 09 2021 6 3 185 194 22 07 2020 24 12 2021 Copyright © 2021, University of Kashan. 2021 https://mir.kashanu.ac.ir/article_111628.html

The fixed point theorem of Nadler (1966) was extended by Mizoguchi and Takahashi in 1989 and then for multi-valued contraction mappings, the existence of fixed point was demonstrated by Daffer and Kaneko (1995). Their results generalized the Nadler’s theorem. In 2009 Kamran generalized Mizoguchi-Takahashi’s theorem. His theorem improve Klim and Wadowski results (2007), and extended Hicks and Rhoades (1979) fixed point theorem. Recently Rouhani and Moradi (2010) generalized Daffer and Kaneko’s results for two mappings. The results of the current work, extend the previous results given by Kamram (2009), as well as by Rouhani and Moradi (2010), Nadler (1969), Daffer and Kaneko (1995), and Mizoguchi and Takahashi (1986) for tow multi-valued mappings. We also give a positive answer to the Rouhani-Moradi’s open problem.

fixed point Mizoguchi-Takahashi fixed point theorem multi-valued mapping weak contraction
 M. Abbas and F. Khojasteh, Common f-endpoint for hybrid generalized multi-valued contraction mappings, RACSAM 108 (2) (2014) 369 − 375.  S. Benchabaney and S. Djebaliz, Common fixed point for multi-valued (ψ,θ, G)-contraction type maps in metric spaces with a graph structure, Appl. Math. E-Notes 19 (2019) 515 − 526.  C. Chifu and G. Petrusel, Existence and data dependence of fixed points and strict fixed points for contractive-type multi-valued operators, Fixed Point Theory Appl. 2007 (2007) 34248.  P. Z. Daffer and H. Kaneko, Fixed points of generalized contractive multi-valued mappings, J. Math. Anal. Appl. 192 (1995) 655 − 666.  T. L. Hicks and B. E. Rhoades, A Banach type fixed point theorem, Math. Japonica 24 (1979) 327 − 330.  T. Kamran, Mizoguchi-Takahashi’s type fixed point theorem, Comput. Math. Appl. 57 (2009) 507 − 511.  D. Klim and D. Wardowski, Fixed point theorems for set-valued contractions in complete metric spaces, J. Math. Anal. Appl. 334 (2007) 132 − 139.  Y. Mahendra Singh, G. A. Hirankumar Sharma and M. R. Singh, Common fixed point theorems for (ψ, φ)-weak contractive conditions in metric spaces, Hacet. J. Math. Stat. 48 (5) (2019) 1398 − 1408.  N. Mizoguchi and W. Takahashi, Fixed point theorems for multivalued mappings on complete metric space, J. Math. Anal. Appl. 141 (1989) 177 − 188.  B. Mohammadi, Strictl fixed points of Ciric-generalized weak quasicontractive multi-valued mappings of integral type, Int. J. Nonlinear Anal. Appl. 9 (2) (2018) 117 − 129.  S. Moradi, Endpoints of multi-valued cyclic contraction mappings, Int. J. Nonlinear Anal. Appl. 9 (1) (2018) 203 − 210.  S. B. Nadler, Multi-valued contraction mappings, Pacific J. Math. 30 (1969) 475 − 488.  S. Reich, Fixed point of contractive functions, Boll. Unione Mat. Ital. 4 (1972) 26 − 42.  B. D. Rouhani and S. Moradi, Common Fixed Point of Multi-valued Generalized ϕ-Weak Contractive Mappings, Fixed Point Theory Appl. 2010 (2010) 708984.  N. Shahzad and A. Lone, Fixed points of multimaps which are not necessarily nonexpansive, Fixed Point Theory Appl. 2 (2005) 169 − 176.  Q. Zhang and Y. Song, Fixed point theory for generalized φ-weak contractions, Appl. Math. Lett. 22 (2009) 75 − 78.
Math. Interdisc. Res. University of Kashan Mathematics Interdisciplinary Research 2538-3639 University of Kashan 106 10.22052/mir.2021.190015.1149 Original Scientific Paper Hopf-Zero Bifurcation in Three-Cell Networks with Two Discrete Time Delays Hopf-Zero Bifurcation in Three-Cell Networks with Two Discrete Time Delays Dadi Zohreh Department of Mathematics, University of Bojnord, Bojnord, I. R. Iran Yazdani Zahra Department of Mathematics, University of Bojnord, Bojnord, I. R. Iran 01 09 2021 6 3 195 214 14 06 2019 31 07 2021 Copyright © 2021, University of Kashan. 2021 https://mir.kashanu.ac.ir/article_111627.html

In this paper, we study a delayed three-cell network which is introduced by coupled cell theory and neural network theory. We investigate this model with two different discrete delays. The aim is to obtain necessary conditions for the stability and the existence of Hopf-zero bifurcation in this model. Moreover, we find the normal form of this bifurcation by using linearization and the Multiple Time Scale method. Finally, the theoretical results are verified by numerical simulations.

Coupled cell theory neural network Stability Hopf-zero bifurcation Normal form
 S. A. Campbell and Y. Yuan, Zero singularities of codimension two and three in delay differential equations, Nonlinearity 21 (11) (2008) 2671 − 2691.  J. Cao, W. Yu and Y. Qu, A new complex network model and convergence dynamics for reputation computation in virtual organizations, Phys. Lett. A 356 (6) (2006) 414 − 425.  T. Chen, H. L. He and G. M. Church Modeling gene expression with differential equations, Pac. Symp. Biocomput. (1999) 29 − 40.  Z. Dadi, Dynamics of two-cell systems with discrete delays, Adv. Comput. Math. 43 (3) (2017) 653 − 676.  Z. Dadi, Z. Afsharnezhad and N. Pariz, Stability and bifurcation analysis in the delay-coupled nonlinear oscillators, Nonlinear Dyn. 70 (1) (2012) 155 − 169.  Z. Dadi and F. Ravanbakhsh, Global Asymptotic and Exponential Stability of Tri-Cell Networks with Different Time Delays, J. Control Optim. Appl. Math. 2 (2) (2017) 45 − 60.  L. Deng, Z. Wu and Q. Wu, Pinning synchronization of complex network with non-derivative and derivative coupling, Nonlinear Dyn. 73 (1) (2013) 775 − 782.  Y. Ding, W. Jiang and P. Yu, Double Hopf bifurcation in delayed van der PolDuffing equation, Int. J. Bifurc. Chaos Appl. Sci. Eng. 23 (1) (2013) 1350014.  Y. Ding, W. Jiang and P. Yu, Hopf-zero bifurcation in a generalized Gopalsamy neural network model, Nonlinear Dyn. 70 (2) (2012) 1037 − 1050.  Y. Du, R. Xu and Q. Liu, Stability and bifurcation analysis for a discrete-time bidirectional ring neural network model with delay, Electron. J. Differ. Equ. 198 (2013) 1 − 12.  M. Golubitsky and I. Stewart, Homeostasis, singularities, and networks, J. Math. Biol. 74 (1-2) (2017) 387 − 407.  S. Guo, Y. Chen and J. Wu, Two-parameter bifurcations in a network of two neurons with multiple delays, J. Differ. Equ. 244 (2) (2008) 444 − 486.  D. Heide, M. Schäfer and M. Greiner, Robustness of networks against fluctuation-induced cascading failures, Phys. Rev. E 77 (5) (2008) 056103.  C. Hu, J. Yu, H. J. Jiang and Z. D. Teng, Pinning synchronization of weighted complex networks with variable delays and adaptive coupling weights, Non-linear Dyn. 67 (2) (2012) 1373 − 1385.  E. Javidmanesh, Z. Dadi, Z. Afsharnezhad and S. Effati, Global stability analysis and existence of periodic solutions in an eight-neuron BAM neural network model with delays, J. Intell. Fuzzy Syst. 27 (1) (2014) 391 − 406.  S. M. Lee, O. M. Kwon and J. H. Park, A novel delay-dependent criterion for delayed neural networks of neutral type, Phys. Lett. A 374 (17) (2010) 1843 − 1848.  X. Li and J. Cao, Delay-dependent stability of neural networks of neutral type with time delay in the leakage term, Nonlinearity 23 (7) (2010) 1709−1726.  X. Liao, S. Guo and C. Li, Stability and bifurcation analysis in tri-neuron model with time delay, Nonlinear Dyn. 49 (1) (2007) 319 − 345.  A. H. Nayfeh, Introduction to Perturbation Techniques, John Wiley and Sons, New York, 2011.  A. H. Nayfeh, Order reduction of retarded nonlinear systems-the method of multiple scales versus center manifold reduction, Nonlinear Dyn. 51 (4) (2008) 483 − 500.  B. Rahman, Y. N. Kyrychko, K. B. Blyuss and S. J. Hogan, Dynamics of a subthalamic nucleus-globus pallidus network with three-time delays, IFAC-PapersOnLine 51 (14) (2018) 294 − 299.  B. Rahman, K. B. Blyuss and Y. N. Kyrychko, Aging transition in system of oscillators with global distributed-delay coupling, Phys. Rev. E 96 (2017) 032203.  B. Rahman, Y. N. Kyrychko and K. B. Blyuss, Dynamics of unidirectionally-coupled ring neural network with discrete and distributed delays, J. Math. Biol. 80 (6) (2020) 1617 − 1653.  B. Rahman, Dynamics of Neural Systems with Time Delays, Ph.D. Thesis, University of Sussex, 2017.  X. Shi, L. Han, Z. Wang and K. Tang, Pinning synchronization of unilateral coupling neuron network with stochastic noise, Appl. Math. Comput. 232 (2014) 1242 − 1248.  S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer, New York, 2003.  X. P. Yan, Bifurcation analysis in a simplified tri-neuron BAM network model with multiple delays, Nonlinear Anal. Real World Appl. 9 (3) (2008) 963 − 976.  P. Yu, Symbolic computation of normal forms for resonant double Hopf bifurcations using a perturbation technique, J. Sound Vib. 247 (4) (2001) 615 − 632.  J. Zhou, S. Li and Z. Yang, Global exponential stability of Hopfield neural networks with distributed delays, Appl. Math. Model. 33 (3) (2009) 1513 −1520.
Math. Interdisc. Res. University of Kashan Mathematics Interdisciplinary Research 2538-3639 University of Kashan 106 10.22052/mir.2021.240429.1263 Original Scientific Paper A Note on the Lempel-Ziv Parsing Algorithm under Asymmetric Bernoulli‎ ‎Model A Note on the Lempel-Ziv Parsing Algorithm under Asymmetric Bernoulli‎ ‎Model Naeini Hojjat Department of Statistics, Science and Research Branch, Islamic Azad University, Tehran, I. R. Iran Kazemi Ramin Department of Statistics, Imam Khomeini International University, Qazvin, I. R. Iran Behzadi Mohammad Hasan Department of Statistics, Science and Research Branch, Islamic Azad University, Tehran, I. R. Iran 01 09 2021 6 3 215 223 10 01 2021 21 07 2021 Copyright © 2021, University of Kashan. 2021 https://mir.kashanu.ac.ir/article_111540.html

‎In this paper‎, ‎by applying analytic‎ ‎combinatorics‎, ‎we obtain an asymptotics for the t-th moment‎ ‎of the number of phrases of length l in the Lempel-Ziv parsing algorithms built over a string generated by an asymmetric Bernoulli‎ ‎model‎. We show that the t-th moment is approximated by its Poisson transform‎.

Lempel-Ziv parsing algorithm phrases digital search tree moment
Math. Interdisc. Res. University of Kashan Mathematics Interdisciplinary Research 2538-3639 University of Kashan 106 10.22052/mir.2021.240266.1238 Original Scientific Paper On the Hosoya Index of Some Families of Graph On the Hosoya index of some families of graph Movahedi Fateme Department of Mathematics, Faculty of Sciences, Golestan University, Gorgan, Iran Akhbari Mohammad Hadi Department of Mathematics, Estahban Branch, Islamic Azad University, Estahban, Iran Kamarulhaili Hailiza School of Mathematical Sciences, Universiti Sains Malaysia, 11800 USM Penang, Malaysia 01 09 2021 6 3 225 234 02 09 2020 14 06 2021 Copyright © 2021, University of Kashan. 2021 https://mir.kashanu.ac.ir/article_111532.html

‎We obtain the exact relations of the Hosoya index that is defined as the sum of the number of all the matching sets‎, ‎on some classes of cycle-related graphs‎. ‎Moreover‎, ‎this index of three graph families‎, ‎namely‎, ‎chain triangular cactus‎, ‎Dutch windmill graph‎, ‎and Barbell graph is determined‎.

Hosoya Index Helm graph graph lotus chain triangular cactus Dutch windmill graph
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Math. Interdisc. Res. University of Kashan Mathematics Interdisciplinary Research 2538-3639 University of Kashan 106 10.22052/mir.2021.240348.1254 Original Scientific Paper A Remark on the Factorization of Factorials A Remark on the Factorization of Factorials Hassani Mehdi Department of Mathematics, University of Zanjan, University Blvd., 45371-38791 Zanjan, I. R. Iran Marie Mahmoud Department of Mathematics, University of Zanjan, University Blvd., 45371-38791 Zanjan, I. R. Iran 01 09 2021 6 3 235 242 05 11 2020 22 07 2021 Copyright © 2021, University of Kashan. 2021 https://mir.kashanu.ac.ir/article_111895.html

The subject of this paper is to study distribution of the prime factors p and their exponents, which we denote by vp (n!), in standard factorization of n! into primes. We show that for each θ > 0 the primes p not exceeding nθ eventually assume almost all value of the sum ∑p⩽nθ vp(n!). Also, we introduce the notion of θ-truncated factorial, defined by n!θ =∏p⩽nθ  pvp (n!) and we show that the growth of log n!1/2 is almost half of growth of log n!1.

Factorial ‎Growth of arithmetic functions Prime number
 G. H. Hardy and S. Ramanujan, The normal number of prime factors of a number n, Q. J. Math. 48 (1917) 76 − 92.  E. G. H. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, 3rd edn. Chelsea Publishing Company, New York, 1974.  J. Lee, The second central moment of additive functions, Proc. Amer. Math. Soc. 114 (4) (1992) 887 − 895.  J. B. Rosser and L. Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962) 64 − 94.