Loading [MathJax]/extensions/TeX/mathchoice.js
Research article

2-SAT discrete Hopfield neural networks optimization via Crow search and fuzzy dynamical clustering approach

  • Received: 27 November 2023 Revised: 14 February 2024 Accepted: 27 February 2024 Published: 06 March 2024
  • MSC : 03B52, 68T27, 68N17, 68W99

  • Within the swiftly evolving domain of neural networks, the discrete Hopfield-SAT model, endowed with logical rules and the ability to achieve global minima of SAT problems, has emerged as a novel prototype for SAT solvers, capturing significant scientific interest. However, this model shows substantial sensitivity to network size and logical complexity. As the number of neurons and logical complexity increase, the solution space rapidly contracts, leading to a marked decline in the model's problem-solving performance. This paper introduces a novel discrete Hopfield-SAT model, enhanced by Crow search-guided fuzzy clustering hybrid optimization, effectively addressing this challenge and significantly boosting solving speed. The proposed model unveils a significant insight: its uniquely designed cost function for initial assignments introduces a quantification mechanism that measures the degree of inconsistency within its logical rules. Utilizing this for clustering, the model utilizes a Crow search-guided fuzzy clustering hybrid optimization to filter potential solutions from initial assignments, substantially narrowing the search space and enhancing retrieval efficiency. Experiments were conducted with both simulated and real datasets for 2SAT problems. The results indicate that the proposed model significantly surpasses traditional discrete Hopfield-SAT models and those enhanced by genetic-guided fuzzy clustering optimization across key performance metrics: Global minima ratio, Hamming distance, CPU time, retrieval rate of stable state, and retrieval rate of global minima, particularly showing statistically significant improvements in solving speed. These advantages play a pivotal role in advancing the discrete Hopfield-SAT model towards becoming an exemplary SAT solver. Additionally, the model features exceptional parallel computing capabilities and possesses the potential to integrate with other logical rules. In the future, this optimized model holds promise as an effective tool for solving more complex SAT problems.

    Citation: Caicai Feng, Saratha Sathasivam, Nurshazneem Roslan, Muraly Velavan. 2-SAT discrete Hopfield neural networks optimization via Crow search and fuzzy dynamical clustering approach[J]. AIMS Mathematics, 2024, 9(4): 9232-9266. doi: 10.3934/math.2024450

    Related Papers:

    [1] Guangwei Hu, Huixue Lao, Huimin Pan . High power sums of Fourier coefficients of holomorphic cusp forms and their applications. AIMS Mathematics, 2024, 9(9): 25166-25183. doi: 10.3934/math.20241227
    [2] M. D. M. Bibiloni-Femenias, O. Valero . Modular relaxed indistinguishability and the aggregation problem. AIMS Mathematics, 2024, 9(8): 21557-21579. doi: 10.3934/math.20241047
    [3] Sufyan Asif, Muhammad Khalid Mahmood, Amal S. Alali, Abdullah A. Zaagan . Structures and applications of graphs arising from congruences over moduli. AIMS Mathematics, 2024, 9(8): 21786-21798. doi: 10.3934/math.20241059
    [4] Afrah. A. N. Abdou . Fixed points of Kannan maps in modular metric spaces. AIMS Mathematics, 2020, 5(6): 6395-6403. doi: 10.3934/math.2020411
    [5] Huafeng Liu, Rui Liu . The sum of a hybrid arithmetic function over a sparse sequence. AIMS Mathematics, 2024, 9(2): 4830-4843. doi: 10.3934/math.2024234
    [6] Fatemeh Lael, Naeem Saleem, Işık Hüseyin, Manuel de la Sen . ˊCiriˊc-Reich-Rus type weakly contractive mappings and related fixed point results in modular-like spaces with application. AIMS Mathematics, 2022, 7(9): 16422-16439. doi: 10.3934/math.2022898
    [7] Martin Bača, Muhammad Imran, Zuzana Kimáková, Andrea Semaničová-Feňovčíková . A new generalization of edge-irregular evaluations. AIMS Mathematics, 2023, 8(10): 25249-25261. doi: 10.3934/math.20231287
    [8] Ali N. A. Koam, Ali Ahmad, Martin Bača, Andrea Semaničová-Feňovčíková . Modular edge irregularity strength of graphs. AIMS Mathematics, 2023, 8(1): 1475-1487. doi: 10.3934/math.2023074
    [9] Bai-Ni Guo, Dongkyu Lim, Feng Qi . Series expansions of powers of arcsine, closed forms for special values of Bell polynomials, and series representations of generalized logsine functions. AIMS Mathematics, 2021, 6(7): 7494-7517. doi: 10.3934/math.2021438
    [10] Abdolsattar Gholidahneh, Shaban Sedghi, Ozgur Ege, Zoran D. Mitrovic, Manuel de la Sen . The Meir-Keeler type contractions in extended modular b-metric spaces with an application. AIMS Mathematics, 2021, 6(2): 1781-1799. doi: 10.3934/math.2021107
  • Within the swiftly evolving domain of neural networks, the discrete Hopfield-SAT model, endowed with logical rules and the ability to achieve global minima of SAT problems, has emerged as a novel prototype for SAT solvers, capturing significant scientific interest. However, this model shows substantial sensitivity to network size and logical complexity. As the number of neurons and logical complexity increase, the solution space rapidly contracts, leading to a marked decline in the model's problem-solving performance. This paper introduces a novel discrete Hopfield-SAT model, enhanced by Crow search-guided fuzzy clustering hybrid optimization, effectively addressing this challenge and significantly boosting solving speed. The proposed model unveils a significant insight: its uniquely designed cost function for initial assignments introduces a quantification mechanism that measures the degree of inconsistency within its logical rules. Utilizing this for clustering, the model utilizes a Crow search-guided fuzzy clustering hybrid optimization to filter potential solutions from initial assignments, substantially narrowing the search space and enhancing retrieval efficiency. Experiments were conducted with both simulated and real datasets for 2SAT problems. The results indicate that the proposed model significantly surpasses traditional discrete Hopfield-SAT models and those enhanced by genetic-guided fuzzy clustering optimization across key performance metrics: Global minima ratio, Hamming distance, CPU time, retrieval rate of stable state, and retrieval rate of global minima, particularly showing statistically significant improvements in solving speed. These advantages play a pivotal role in advancing the discrete Hopfield-SAT model towards becoming an exemplary SAT solver. Additionally, the model features exceptional parallel computing capabilities and possesses the potential to integrate with other logical rules. In the future, this optimized model holds promise as an effective tool for solving more complex SAT problems.



    Let be a prime number and fSk(Γ1(N)) be a cusp form of weight k and level N. Let ρf:Gal(¯Q/Q)GL2(¯F) be a mod Galois representation associated to f. Let L be the fixed field of the kernel of ρf. Then the representation ρf factors through as:

    where π is the canonical restriction map and ϕ is the isomorphism between Gal(L/Q) and the image im(ρf) of ρf. Thus to compute ρf, it suffices to give the Galois extension field L over Q and the isomorphism ϕ.

    In their book [1], Edixhoven et al. propose a polynomial time algorithm to compute ρf associated to level one modular forms. They prove that ρf can be described by a certain polynomial PfQ[x] of degree 21 whose splitting field is the fixed field L of ker(ρf). One can obtain L by adjoining the roots of Pf to Q, and the isomorphism ϕ is induced by a bijection between the roots of Pf and a 2-dimensional subspace of the -torsion of the Jacobian variety J1() of the modular curve X1() associated to Γ1(). This algorithm has been generalized to modular forms of arbitrary levels by Bruin [2]. Likewise, the associated projective representation ˜ρf:Gal(¯Q/Q)PGL2(¯F) can be described by a suitable polynomial ˜PfQ[x] of degree +1.

    The computations depend heavily on and the genus of the modular curve X1(), which is equal to (5)(7)/24. In practice, the most time-consuming part of the algorithm is to approximate the points of J1(), and the precision significantly increases when grows. Consequently, the explicit computations have been done only for the primes no bigger than 43.

    Let Δk be the unique cusp form of level 1 and weight k with k=12,16,18,20,22,26. In practice, this algorithm has been first implemented by Bosman [1,Chapter 7] to evaluate the projective polynomial ˜PΔk for 23 and k+1. In [3,4] and the unpublished paper [5], this algorithm has been improved and more polynomials ˜PΔk have been explicitly computed for 43.

    However, as far as we know, no one has implemented the algorithm to calculate the polynomials for the cases with <k1. In this paper, we shall discuss the algorithms for computing mod Galois representations associated to modular forms of weight k when <k1. We will propose an algorithm of this case and then do explicit computations of the mod projective Galois representations ˜ρΔk for k=16,20,22,26 and all the unexceptional primes for which <k1.

    In the book [1], the authors deal with the case with <k1 by twisting the representations and then reduce the computations to the cases with k+1. In fact, for a form of level one and weight k with <k1, in [1,Proposition 2.5.18] they show a method to obtain a form of weight k+1, such that the two Galois representations associated to the two forms are isomorphic. In this paper, we will prove this result also holds for modular forms of levels greater than 1.

    First, in Section 2, we show a generalization of Sturm bound theorem [6,Theorem 2] to mod modular forms, which gives an explicit method to identify two forms by observing a few coefficients of the q-expansions. Then in Section 3, we use the generalized result to give an explicit method, for a given modular form of type (N,k,ε), to obtain a twist form of type (N,k,ε) with k+1, such that the two Galois representations associated to the two forms are isomorphic up to twist. In fact this is a generalization of [1,Proposition 2.5.18] to modular forms of arbitrary levels. Consequently, the computations of the cases with <k1 boil down to the cases with k+1.

    In the end of Section 3, we prove the corresponding results for the projective representations and then present the algorithm for the projective case.

    In Section 4, we apply the algorithm in Subsection 3.3 to do explicit computations of the mod projective Galois representations ˜ρΔk for k=16,20,22,26 and all the unexceptional primes for which <k1. Here Δk is the normalized cusp form of level one and weight k. The computed projective polynomials ˜PΔk(x) associated to the representations ˜ρΔk are shown in Table 4.

    Lehmer [7] conjectures that Ramanujan's tau function τ(n) is non-vanishing for all n and shows that τ(n)0 for all n<3316799. Serre [8] sums up the congruences of τ(p) modulo exceptional primes of τ(p) and obtains a bound of 15 digits for Lehmer's conjecture with respect to τ(n). Bosman [1] first used the results of modular Galois representations to discuss the non-vanishing coefficients of τ(n) and then this method was developed by others. So far the bound for Lehmer's conjecture with respect to τ(n) is up to 24 digits [5].

    In [9], the authors discuss non-vanishing Fourier coefficients an(Δk) of Δk and achieve the bounds Bk of n such that an(Δk)0 for all n<Bk in the cases with k=16,18,20,22,26.

    In this paper, as an application, we shall discuss the non-vanishing Fourier coefficients of Δk using our results. In fact, for k=16,20,22,26, we obtain higher bounds Bk of n such that an(Δk)0 for all n<Bk. We demonstrate how much the bounds Bk have been improved for k=16,20,22,26 in Table 1. Note that the last column of Table 1 is the approximate quotients of the new and old bounds.

    Table 1.  Comparison between the new and old bounds.
    k new bound old bound new bound
    old bound
    16 169424346446440054199 12604744061516618549 13
    20 1222095705994609939349 74201833676082662549 16
    22 567829713758553825538049 28265095927027650599999 20
    26 3442219356673306598399 818406791865712833299 4

     | Show Table
    DownLoad: CSV

    The method in this paper is different with that in the previous papers. In [4], we compute modular Galois representations only when k1, in which case we say the prime is "large enough". However, in this paper, we discuss the cases when <k1, that is, the prime is small. In this case, we don't have the weight 2 forms by which we can carry out all the computations. Instead, in this paper, we use the twists of the forms by the θ operator. In [9], to obtain the new bounds Bk, we discuss the exceptional primes and observe the congruence formulas, and there is none of new modular Galois representation being computed. In this paper, we do computations in the cases with unexceptional primes by using the new modular Galois representations, which are computed in Subsection 4.2.

    Throughout this paper, we suppose 5 to be a prime and denote ¯F the algebraic closure of the finite field F. All the explicit computations of this paper have been done in the open source software SAGE [10].

    The mod modular forms were first developed by Serre [11] and Swinnerton-Dyer [12], and generalized by Katz [13]. In this subsection we give a brief review of the theory of mod modular forms. For the details, we refer to [14] and [15,Section 2].

    Let be a prime and N1 be prime to . The congruence subgroup Γ1(N) of level N is

    Γ1(N)={(abcd)SL(2,Z)  c0 mod N,   ad1 mod N}. (2.1)

    Let X1() be the modular curve associated to Γ1(). Let k>0 be an even integer. Let E be a generalized elliptic curve over a scheme S and α:(Z/NZ)SE be an embdedding of group schemes. Denote the relative differentials by Ω1E/S and zero section by 0. Let ωE/S:=0Ω1E/S. Then a modular form f of type (N,k) over ¯F is a law, that assigns to each pair (E/S,α) a section of ωkE/S.

    The q-expansions of mod modular forms f at cusp of Γ1(N) have been given by evaluating f on (Eq,α), where q=e2πiz and Eq is the Tate curve over ¯F[[q]](q1). More precisely, the q-expansions of f at are the the power series f(Eq,α)/(dt/t)k¯F[[q]], where dt/t is the standard differential on Eq. This in fact coincides with the usual q-expansions of modular forms, since (Eq,α) corresponds to a neighbourhood of the cusp in the completed up half plane H=HQ, where H is the up half plane. As usual, we denote the n-th coefficient of the q-expansion by an(f).

    Let ε:(Z/(N)Z)¯F be a Dirichlet character. Define an action of Z/(N)Z) on mod form f by

    (a)(E/S,α)=f(E/S,aα),   aZ/(N)Z). (2.2)

    A modular form f of type (N,k) is called a form of type (N,k,ε) if it satisfies

    (a)(E/S,α)=ε(a)f. (2.3)

    One can also define Hecke operators Tp that are coincide with the usual Hecke operators. For instance, we have that all the Tp commute with each other and the eigenvalues determine the q-expansions of f up to a constant factor.

    A modular form f is called cusp form if a0(f)=0. A modular form f of type (N,k,ε) is said to be an eigenform if it is an eigenvector for all the Hecke operators Tp with pN. An eigenform f is said to be normalized if a1(f)=1.

    Let θ=qddq be the classical differential operator n>0an(f)qnn>0nan(f)qn. If f is an eigenform of type (N,k,ε), in [16,Section 2.1], it is shown that θf is an eigenform of type (N,k++1,ε).

    Let A be the Hasse invariant of the Tate curve Eq over ¯F[[q]](q1), then we have:

    Lemma 2.1. The Hasse invariant A is given by A=(dt/t)1. Hence, A is a mod modular form of type (1,1,1).

    Proof. This is Proposition 1.9 c) of [14].

    From this lemma, we know the q-expansion of A is 1. For two forms of types (N,k1,ε) and (N,k2,ε) with k_{1}\equiv k_{2} \mod \ell-1 , we can view the two forms as forms of the same type by multiplying one form by suitable powers of A . This can be used to prove the following proposition, which is a generalization of Sturm bound theorem to modular forms of different weights:

    Proposition 2.2. Let f_{1} and f_{2} be two normalized eigenforms of type (N, k_{1}, \varepsilon) and (N, k_{2}, \varepsilon) , respectively. Let k = max\{k_{1}, k_{2}\} . Suppose that k_{1}\equiv k_{2} \mod \ell-1 and a_{m}(f_{1}) = a_{m}(f_{2}) in \overline{\mathbb{F}}_{\ell} for all m with m \le \frac{k[SL_{2}(\mathbb{Z}):\varGamma_{1}(N)]}{12} . Then f_{1} = f_{2} .

    Proof. Let A be the Hasse invariant. Without loss of generality, we suppose k_{1}\le k_{2} . Then by Lemma 2.1, the form A^{(k_{2}-k_{1})/(\ell-1)} f_{1} is an eigenform of type (N, k_{2}, \varepsilon) . We know A = 1 , and this implies that the form f_{1} is also a form of type (N, k_{2}, \varepsilon) . Since we have a_{m}(f_{1}) = a_{m}(f_{2}) in \overline{\mathbb{F}}_{\ell} for all m with m \le \frac{k[SL_{2}(\mathbb{Z}):\varGamma_{1}(N)]}{12} , it follows from Sturm's theorem that f_{1} = f_{2} .

    A proof of this result for classical modular forms can be found in [17].

    The following well-known theorem takes an important role for our computations:

    Theorem 2.3. Let f be a normalized eigenform of type (N, k, \varepsilon) , then there exist i and k' with 0\le i\le \ell-1, \ k'\le \ell+1 , and a normalized eigenform of type (N, k', \varepsilon) , such that f = \theta^{i}g .

    Proof. See [15,Theorem 3.4].

    In this section, we shall describe the algorithm for computing mod \ell Galois representations associated to modular forms of weight k when \ell < k-1 . We also prove the corresponding results for the projective representations and then present the algorithm for the projective case.

    Deligne [18] proves the following well known theorem:

    Theorem 3.1 (Deligne). Let f be an eigenform of type (N, k, \varepsilon) . Then there exists a continuous semi-simple representation

    \begin{equation} \rho_{f}: Gal(\overline{\mathbb{Q}}/\mathbb{Q}) \rightarrow GL_{2}(\overline{\mathbb{F}}_{\ell}), \end{equation} (3.1)

    which is unramified outside N\ell , and for all primes p\nmid N\ell the characteristic polynomial of \rho_{f} (Frob_{p}) satisfies in \overline{\mathbb{F}}_{\ell}

    \begin{equation} charpol( \rho_{f} (Frob_{p})) = x^2-a_{p}(f)x+ \varepsilon(p)p^{k-1}. \end{equation} (3.2)

    Moreover, \rho_{f} is unique up to isomorphism.

    Let f = \sum_{n > 0} a_{n} (f) q^{n} be an eigenform. Then by definition, the eigenform \theta f has q -expansion \sum_{n > 0} na_{n} (f) q^{n} . It follows from the above theorem that

    \rho_{\theta f} = \rho_{f} \otimes \chi_{\ell},

    where \chi is the mod \ell cyclotomic character. Then for an eigenform f of type (N, k, \varepsilon) with \ell < k-1 , it follows from Theorem 2.3 that there exist an integer i and an eigenform g of type (N, k', \varepsilon) with k' \le \ell+1 , such that \rho_{f} is a twist of \rho_{g} by \chi^{i}_{\ell} , i. e.,

    \begin{equation} \rho_{ f} \cong \rho_{g} \otimes \chi^{i}_{\ell}. \end{equation} (3.3)

    Moreover, we have the following theorem to determine such i and k' :

    Theorem 3.2. Let f_{1} and f_{2} be two normalized eigenforms of type (N, k_{1}, \varepsilon) and (N, k_{2}, \varepsilon) , respectively. Let i be an integer with 0\le i \le \ell-1 . Then f_1 = \theta^{i}f_2 if and only if k_{1}\equiv k_{2}+2i \mod \ell-1 and a_{p}(f_{1}) = p^{i}a_{p}(f_{2}) in \overline{\mathbb{F}}_{\ell} for all primes p with p \le \frac{\ell(\ell+1)[SL_{2}(\mathbb{Z}):\varGamma_{1}(N)]}{12} .

    Proof. We first assume that f_1 = \theta^{i}f_2 . By the argument above, it follows that \rho_{ f_1} and \rho_{f_2} \otimes \chi^{i}_{\ell} are isomorphic. Then by (3.2), we have

    \varepsilon \chi_{\ell}^{k_{1}-1} = \varepsilon \chi_{\ell}^{k_{2}-1+2i}.

    Hence we have k_{1}\equiv k_{2}+2i \mod \ell-1 . Since a_{p}(\theta^i f_{2}) = p^{i}a_{p}(f_{2}) for all primes p , in \overline{\mathbb{F}}_{\ell} we have

    a_{p}(f_{1}) = p^{i}a_{p}(f_{2})

    for all primes p with p \le \frac{\ell(\ell+1)[SL_{2}(\mathbb{Z}):\varGamma_{1}(N)]}{12} .

    For the other direction, we assume that

    k_{1}\equiv k_{2}+2i \mod \ell-1,

    and a_{p}(f_{1}) = p^{i}a_{p}(f_{2}) for all primes p with p \le \frac{\ell(\ell+1)[SL_{2}(\mathbb{Z}):\varGamma_{1}(N)]}{12} .

    It follows from Theorem 2.3 that there exist an integer j with 0\le j \le \ell-1 and a normalized eigenform g_{1} of type (N, k_{g_{1}}, \varepsilon) with k_{g_{1}}\le\ell+1 such that f_{1} = \theta^{j}g_{1} in \overline{\mathbb{F}}_{\ell} . This implies that f_{1} = \theta^{j}g_{1} is a form of type (N, k_{1}, \varepsilon) with k_{1}\le \ell(\ell+1) .

    We set f'_{2} = \theta^i f_{2} . Then for the same reason as above, the form f'_{2} is of type (N, k_{2}', \varepsilon) with k_{2}'\le \ell(\ell+1) . Moreover, we have that \rho_{f_2'} is isomorphic to \rho_{f_2} \otimes \chi^{i}_{\ell} . By the argument in the first paragraph of the proof, we have

    \begin{equation} k_{2}'\equiv k_{2}+2i \mod \ell-1. \end{equation} (3.4)

    Then by the assumption and (3.4), we have

    k_{1} \equiv k_{2}+2i \equiv k_{2}' \mod \ell-1,

    and

    a_{p}(f_{1}) = p^{i}a_{p}(f_{2}) = a_{p}(f_{2}')

    for all primes p with p \le \frac{\ell(\ell+1)[SL_{2}(\mathbb{Z}):\varGamma_{1}(N)]}{12} .

    Since f_1 and f_{2}' are normalized eigenforms, this implies that a_{m}(f_{1}) = a_{m}(f_{2}') for all m\in \mathbb{Z} with m \le \frac{\ell(\ell+1)[SL_{2}(\mathbb{Z}):\varGamma_{1}(N)]}{12} . By Proposition 2.2, we then have that f_{1} = f_{2}' and therefore f_1 = \theta^{i}f_2 . This completes the proof.

    This theorem also provides a method to calculate the values of i and k' for which (3.3) is satisfied. From this point of view, this theorem is a generalization of [1,Proposition 2.5.18] from level one to arbitrary levels. In [2,Theorem 3.5], the author gives an elaborate result of the generalization of [1,Proposition 2.5.18], which is used to theoretically prove that the algorithm described in [2] is in polynomial time. However, it is quite convenient to apply Theorem 3.2 when we do explicit computations.

    In this subsection, we shall describe the algorithm for computing the mod \ell Galois representations associated to modular forms. This algorithm was first proposed by Edixhoven and Couveignes [1] for modular forms of level one and then generalized to forms of arbitrary levels by Bruin [2]. The algorithm that we shall present is slightly different. In fact, we shall apply Theorem 3.2 instead when we reduce the computations to the cases with 2 \le k \leq \ell+1 .

    Now let f be a cuspidal normalized eigenform of type (N, k, \varepsilon) with \ell < k-1 . Theorem 2.3 and 3.2 allow us to explicitly obtain a normalized eigenform f' of type (N, k', \varepsilon) with 2\le k' \le \ell+1 such that \rho_{f} and \rho_{f'} \otimes \chi^{i}_{\ell} are isomorphic. Thus it suffices to compute \rho_{f'} and the question boils down to the case with 2 \le k' \leq \ell+1 .

    In [19,Theorem 2.2], the author shows that if 2 < k \leq \ell+1 and \rho_{f, \lambda} is ireducible, then there is a cuspidal normalized eigenform f_{2} of type (N\ell, 2, \varepsilon_2) such that \rho_{f} is isomorphic to \rho_{f_{2}} . Therefore, for any p \nmid N\ell , this reduces the questions to the case with k = 2 .

    Now suppose that \rho_{f} is a mod \ell Galois representation associated to a cuspidal normalized eigenform of type (N, 2, \varepsilon) . Let X_{1}(\ell) be the modular curve associated to \varGamma_{1}(\ell) and let J_{1}(\ell) denote its Jacobian. Denote \mathbb T the subring of End (J_{1}(\ell)) generated by the Hecke operators T_p over \mathbb Z . Then

    \mathbb T = \mathbb{Z}[ T_{n}, \langle n\rangle : n \in \mathbb Z_{+} \ \mathrm{and} \ (n, \ell) = 1].

    Define a ring homomorphism

    \theta: \mathbb{T} \rightarrow \mathbb F_{\lambda} ,

    given by

    \langle d \rangle \mapsto \varepsilon(d) \ \ and \ \ T_{n} \mapsto a_{n}(f).

    Let \mathfrak m denote the maximal ideal ker \theta and then \mathbb{T}/\mathfrak{m}\subset \overline{\mathbb{F}}_{\ell} . Moreover, we let

    V = J_{1}(\ell)(\overline{\mathbb{Q}})[\mathfrak{m}] = \{x \in J_{1}(\ell)(\overline{\mathbb{Q}}) \ | \ tx = 0 \ \mathrm{for} \ \mathrm{all} \ t \ \mathrm{in} \ \mathfrak m \}.

    Then we have:

    Theorem 3.3. The set V is a 2-dimensional \mathbb{T}/\mathfrak{m} -linear subspace of J_{1}(\ell)(\overline{\mathbb{Q}})[\ell] . Moreover, the representation

    \rho: Gal (\mathbb{\overline{Q}}/\mathbb{Q}) \rightarrow \mathrm{Aut}(V)

    is isomorphic to the modular Galois representation \rho_{f} .

    Proof. See [20,Section 3.2 and 3.3]).

    Let L be the fixed field of {\rm ker}(\rho_{f}) of the Galois representation \rho_{f} . Then the representation \rho_{f} can factor through as:

    where \pi is the canonical restriction map and \phi is the isomorphism between Gal (L/\mathbb{Q}) and the image {\rm im}(\rho_{f}) of \rho_{f} . It can be shown that, to compute \rho_{f} , it suffices to compute a suitable polynomial P_{f}(x)\in \mathbb Q[x] of degree \ell^2-1 with

    P_{f}(x) = \prod\limits_{P\in V-\{0\}}(x-h(P))

    for some suitable function h in the function field of X_1(\ell) . Here h(P) = \sum_{i = 1}^g h(P_i) where g is the genus of X_1(\ell) , and P_i are the points on X_1(\ell) such that each divisor P\in V-\{0\} can be written as \sum_{i = 1}^g (P_i) -gO .

    In fact, it can be shown that the fixed field of \rho_{f} is actually the splitting field of P_{f}\in \mathbb Q[x] . Then one can obtain L by adjoining the roots of P_{f} to \mathbb{Q} , and the isomorphism \phi is induced by the bijection between the roots of P_{f} and the points of the 2-dimensional \mathbb{T}/\mathfrak{m} -linear subspace V of J_{1}(\ell)(\overline{\mathbb{Q}})[\ell] .

    Composed with the canonical projection map GL_{2}(\mathbb{F}_{\lambda})\rightarrow PGL_{2}(\mathbb{F}_{\lambda}) , the representation \rho_{f} in (3.1) gives a projective representation

    \tilde{\rho}_{f}: Gal(\overline{\mathbb{Q}}/\mathbb{Q}) \rightarrow PGL_{2}(\overline{\mathbb{F}}_{\ell}).

    Now we apply Theorem 3.2 to the projective representation cases and then we have:

    Theorem 3.4. Let f_{1} and f_{2} be two normalized eigenforms of type (N, k_{1}, \varepsilon) and (N, k_{2}, \varepsilon) , respectively. Let i be an integer with 0\le i \le \ell-1 . Suppose that k_{1}\equiv k_{2}+2i \mod \ell-1 and a_{p}(f_{1}) = p^{i}a_{p}(f_{2}) in \overline{\mathbb{F}}_{\ell} for all primes p with p \le \frac{\ell(\ell+1)[SL_{2}(\mathbb{Z}):\varGamma_{1}(N)]}{12} . Then \tilde{\rho}_{ f_1} and \tilde{\rho}_{f_2} are isomorphic.

    Proof. It follows from Theorem 3.2 that \tilde{\rho}_{ f_1} and \tilde{\rho}_{f_2} \otimes \chi^{i}_{\ell} are isomorphic. For any \sigma \in Gal (\overline{\mathbb{Q}}/\mathbb{Q}) , we have

    \rho_{f_2} \otimes \chi^{i}_{\ell}(\sigma) = \rho_{f_2}(\sigma) \cdot\chi^{i}_{\ell}(\sigma).

    In PGL_{2}(\mathbb{F}_{\lambda}) , we have

    \overline{\rho_{f_2}(\sigma)} = \overline{\rho_{f_2}(\sigma) \cdot\chi^{i}_{\ell}(\sigma)},

    where, as usual, the bar denotes the quotient by the subgroup of GL_{2}(\overline{\mathbb{F}}_{\ell}) consisting of scalar matrices. Hence we have \tilde{\rho}_{f_2} \otimes \chi^{i}_{\ell} = \tilde{\rho}_{f_2} and this implies \tilde{\rho}_{ f_1} and \tilde{\rho}_{f_2} are isomorphic.

    Now we can describe the algorithm for computing the projective Galois representation \tilde{\rho}_{f} associated to an normalized eigenform of type (N, k, \varepsilon) with \ell < k-1 .

    First, by Theorems 2.3 and 3.4, we can explicitly obtain a normalized eigenform f' of type (N, k', \varepsilon) with 2\le k' \le \ell+1 such that \tilde{\rho}_{ f} and \tilde{\rho}_{f'} are isomorphic. Thus our computations boil down to the case with 2 \le k' \leq \ell+1 . Then again we can reduce the question to the weight 2 case using the same arguments in the previous subsection. Finally, we can compute a suitable polynomial instead for the following reason:

    Let K be the fixed field of {\rm ker}(\tilde{\rho}_{f}) , then the representation \tilde{\rho}_{f} can factor through as:

    where \pi is the canonical restriction map and \varphi is the isomorphism between Gal (L/\mathbb{Q}) and {\rm im}(\tilde{\rho}_{f}) .

    Let V = J_{1}(\ell)(\overline{\mathbb{Q}})[\mathfrak{m}] be the 2-dimensional \mathbb{T}/\mathfrak{m} -linear subspace of J_{1}(\ell)(\overline{\mathbb{Q}})[\ell] as in Theorem 3.3. Then the projective line \mathbb P(V) has \ell+1 points, and it follows that the fixed field of \tilde{\rho}_{f} is in fact the splitting field K of a certain polynomial \tilde P_{f}\in Q[x] of degree \ell+1 , which is given by

    \begin{equation} \tilde P_{f}(x) = \prod\limits_{A\subset \mathbb P(V)}(x-\sum\limits_{P\in A-\{0\}}h(P)). \end{equation} (3.5)

    Moreover, one can obtain K by adjoining the roots of \tilde P_{f, \ell} to \mathbb{Q} and, the isomorphism \varphi is induced by the bijection between the roots of \tilde P_{f} and the points of the projective line \mathbb P(V) . This implies that the projective representation \tilde{\rho}_{f} can be described by the polynomial \tilde P_{f} .

    For k = 16, 18, 20, 22 and 26 , let \Delta_k = \sum_{n > 0}^{\infty}a_{n}q^{n} denote the unique cusp form of level 1 and weight k . A prime \ell is said to be exceptional if the image of \rho_{\Delta_k, \ell} does not contain SL_{2}(\mathbb{F}_{\ell}) . Otherwise, a prime \ell is called unexceptional.

    Bosman [1] first does practical computations and obtains \tilde P_{\Delta_{k}} for modular forms \Delta_{k} of level 1 and of weight k\le 22 , with \ell \leq 23 . Others improve the algorithm and computed the polynomials for more cases. See [4] and [5] for instance. As far as we know, all the polynomials \tilde P_{\Delta_{k}} that have been computed in this method are shown in [1,Section 7.5] and [5,Table 4].

    Note that all the computed polynomials are of the cases with k\le \ell+1 . In this section, we shall apply the algorithm described in Subsection 3.3 to explicitly compute the polynomials \tilde P_{\Delta_{k}} associated to the mod \ell projective Galois representations \tilde{\rho}_{ \Delta_{k}} for k = 16, 20, 22, 26 and all the unexceptional primes \ell , with \ell < k-1 .

    As an application, we shall discuss the non-vanishing Fourier coefficients of \Delta_{k} using our results.

    For a prime \ell , we let \tilde{\Delta}_k = \sum_{n > 0}^{\infty}\tilde{a}_{n}q^{n} , where \tilde{a}_{n} means the reduction of a_{n} mod \ell . Then \tilde{\Delta}_k is a normalized cuspidal eigenform of type (1, k, 1) . We denote by \tilde P_{\Delta_{k}, \ell} the polynomial \tilde P_{\Delta_k}(x) defined in (3.5) which describes the mod \ell projective Galois representation \tilde{\rho}_{\Delta_k} associated to \tilde{\Delta}_k .

    A prime \ell is said to be exceptional if the image of \rho_{f} does not contain SL_{2}(\mathbb{F}_{\ell}) . In Table 2 we list the all unexceptional primes for \Delta_k with \ell < k-1 . Then for the (k, \ell) in Table 2, we shall compute the polynomials \tilde P_{\Delta_{k}, \ell} .

    Table 2.  Small unexceptional primes for \Delta_{k} .
    k \ell
    16 13
    20 17
    22 11
    19
    26 13
    23

     | Show Table
    DownLoad: CSV

    For \Delta_{k} and unexceptional prime \ell , with (k, \ell) in Table 2, we apply Theorem 3.4 to find normalized eigenforms f of type (1, k', 1) with k' < \ell-1 such that \tilde{\rho}_{\Delta_{k}} and \tilde{\rho}_{f} are isomorphic. More precisely, we first obtain all pairs (i, k') such that

    k\equiv k'+2i \mod \ell-1.

    Then we take a pair (i, k') such that a_{p}(f_{1}) \equiv p^{i}a_{p}(f_{2}) \mod \ell for all primes p with p \le \frac{\ell(\ell+1)}{12} . This condition can be verified quickly in SAGE. In fact, by Theorem 2.3, such k' and i do exist and after doing some simple calculations we explicitly obtain the forms that satisfy the conditions in Theorem 3.2. Then by Theorem 3.4, we finally reduce the computations to the cases with k\le \ell+1 . Thus it gives:

    Proposition 4.1. We take values of k, \ell, i, k' in each row of Table 3. Then we have the congruences

    Table 3.  The values of k, \ell, i, k' .
    k \ell i k'
    16 13 2 12
    20 17 2 16
    22 11 1 12
    19 2 18
    26 13 1 12
    23 2 22

     | Show Table
    DownLoad: CSV
    \label{twistforms} \Delta_{k} \equiv \theta^{i}\Delta_{k'} \mod \ell,

    and moreover, we have \tilde{\rho}_{ \Delta_{k}}\cong\tilde{\rho}_{\Delta_{k'}} .

    We take values of k, \ell, i, k' in each row of Table 3. Since \tilde{\rho}_{ \Delta_{k}} and \tilde{\rho}_{\Delta_{k'}} are isomorphic, we have

    \tilde P_{\Delta_{k}, \ell}(x) = \tilde P_{\Delta_{k'}, \ell}(x).

    Fortunately, all the corresponding polynomials \tilde P_{\Delta_{k'}, \ell}(x) have been computed and shown in [1,Section 7.5]. As a result, the polynomials \tilde P_{\Delta_{k}, \ell}(x) associated to the mod \ell projective Galois representation \tilde{\rho}_{ \Delta_{k}} are shown in Table 4.

    Table 4.  The polynomials \tilde{P}_{\varDelta_{k}, \ell} associated to \tilde{\rho}_{ \Delta_{k}} .
    ( k , \ell ) \tilde{P}_{\varDelta_{k}, \ell}
    (16, 13) x^{14}+7x^{13}+26x^{12}+78x^{11}+169x^{10}+52x^{9}-702x^{8}-1248x^{7}+494x^{6}+2561x^{5}+312x^{4}-2223x^{3}+169x^{2}+506x-215
    (20, 17) x^{18}-2x^{17}-17x^{15}+204x^{14}-1904x^{13}+3655x^{12}+5950x^{11}-3672x^{10}-38794x^{9}+19465x^{8}+95982x^{7}-280041x^{6}-206074x^{5}+455804x^{4}+946288x^{3}-1315239x^{2}+606768x-378241
    (22, 11) x^{12}-4x^{11}+55x^{9}-165x^{8}+264x^{7}-341x^{6}+330x^{5}-165x^{4}-55x^{3}+99x^{2}-41x-111
    (22, 19) x^{20}+10x^{19}+57x^{18}+228x^{17}-361x^{16}-3420x^{15}+23446x^{14}+88749x^{13}-333526x^{12}-1138233x^{11}+1629212x^{10}+13416014x^{9}+7667184x^{8}-208954438x^{7}+95548948x^{6}+593881632x^{5}-1508120801x^{4}-1823516526x^{3}+2205335301x^{2}+1251488657x-8632629109
    (26, 13) x^{14}+7x^{13}+26x^{12}+78x^{11}+169x^{10}+52x^{9}-702x^{8}-1248x^{7}+494x^{6}+2561x^{5}+312x^{4}-2223x^{3}+169x^{2}+506x-215
    (26, 23) x^{24}-11x^{23}+46x^{22}-1127x^{20}+6555x^{19}-7222x^{18}-140737x^{17}+1170700x^{16}-2490371x^{15}-16380692x^{14}+99341324x^{13}+109304533x^{12}-2612466661x^{11}+4265317961x^{10}+48774919226x^{9}-244688866763x^{8}-88695572727x^{7}+4199550444457x^{6}-10606348053144x^{5}-25203414653024x^{4}+185843346182048x^{3}-228822955123883x^{2}-1021047515459130x+2786655204876088

     | Show Table
    DownLoad: CSV

    In [9], the authors discuss the non-vanishing Fourier coefficients of \Delta_{k} with k = 16, 18, 20, 22, 26 and achieve the explicit bounds B_k of n such that the Fourier coefficients a_n(\Delta_k)\ne0 for all n < B_k . They first prove that the smallest n for which a_n(\Delta_k) = 0 must be a prime. Then, for each prime p with a_p(\Delta_k) = 0 , they obtain the formulations that such p must satisfy. In addition, the congruence

    a_{p}(\Delta_k)\equiv 0 \mod \ell

    can be verified by the polynomials \tilde P_{\Delta_k, \ell} associated to the projective Galois representations. Precisely, when the polynomial \tilde P_{\Delta_k, \ell}\in \mathbb{Z}[x] , it can be shown that a_{p}(\Delta_k)\equiv 0 \mod \ell is equivalent to \tilde P_{\Delta_k, \ell} \mod{p} having an irreducible factor of degree 2 in \mathbb{F}_{p}[x] . Consequently, one can systematically search for the smallest prime p satisfying the formulations, as well as a_{p}(\Delta_k)\equiv 0 \mod \ell .

    Now we can add the polynomials \tilde P_{\Delta_k, \ell} in Table 4 to the searching computations. That is, for k = 16, 20, 22, 26 and all the small unexceptional primes \ell in Table 2, we can efficiently verify the additional searching conditions

    a_{p}(\Delta_k)\equiv 0 \mod \ell.

    As a result, for k = 16, 20, 22, 26 , we are able to obtain the new bounds B_k of n such that a_n(\Delta_k)\ne0 for all n < B_k .

    Proposition 4.2. Let the pair ( k, B_k ) take the values as in Table 5. Then the coefficients a_n(\Delta_{k}) are non-vanishing for all n with n < B_k in Table 5.

    Table 5.  The bounds B_k .
    k B_k
    16 169424346446440054199
    20 1222095705994609939349
    22 567829713758553825538049
    26 3442219356673306598399

     | Show Table
    DownLoad: CSV

    The computational results of modular Galois representations can be applied to compute the Fourier coefficients of modular forms f according to (3.2). More precisely, if we can calculate mod \ell Galois representations for enough primes \ell whose product exceeds 4p^{(k-1)/2} , the coefficient a_p(f) can be easily computed by Chinese Remainder Theorem. Our results in this paper add the small primes to the list and can be applied to the computations of the Fourier coefficients of modular forms. Besides, for many groups \mathrm{SL}_2(\mathbb{F}_{\ell ^k}) and \mathrm{GL}_2(\mathbb{F}_{\ell ^k}) , it is still unknown whether they are Galois groups of number fields over rational field \mathbb{Q} . Our results are expected to answer some of these questions, since our computations also provide number fields and their Galois groups, namely \mathrm{SL}_2(\mathbb{F}_{\ell ^k}) and \mathrm{GL}_2(\mathbb{F}_{\ell ^k}) .

    In this paper, we give an explicit method, for a given modular form of type (N, k, \varepsilon) , to obtain a twist form of type (N, k', \varepsilon) with k'\le \ell+1 , such that the two Galois representations associated to the two forms are isomorphic up to twist. Then we prove the corresponding results for the projective representations and present the algorithm for the projective case. Moreover, we apply the algorithm in Subsection 3.3 to do explicit computations of the mod \ell projective Galois representations \tilde{\rho}_{ \Delta_{k}} for k = 16, 20, 22, 26 and all the unexceptional primes \ell for which \ell < k-1 . The computed projective polynomials \tilde P_{\Delta_{k}}(x) associated to the representations \tilde{\rho}_{ \Delta_{k}} are shown in Table 4.

    In the end, as an application, we discuss the non-vanishing Fourier coefficients of \Delta_{k} using our results. In fact, for k = 16, 20, 22, 26 , we obtain new higher bounds B_k of n such that a_n(\Delta_k)\ne0 for all n < B_k , which are shown in Table 5.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    Peng Tian is supported by NSFC (NOs: 11601153) and Ha T. N. Tran was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) (funding RGPIN-2019-04209 and DGECR-2019-00428).

    The authors declare that they have no conflicts of interest.



    [1] S. A. Cook, The complexity of theorem-proving procedures, In: Logic, automata, and computational complexity: The works of Stephen A. Cook, 2023,143–152. https://doi.org/10.1145/3588287.3588297
    [2] J. J. Hopfield, Neural networks and physical systems with emergent collective computational abilities, P. Natl. Acad. Sci., 79 (1982), 2554–2558. https://doi.org/10.1073/pnas.79.8.2554 doi: 10.1073/pnas.79.8.2554
    [3] W. A. T. W. Abdullah, Logic programming on a neural network, Int. J. Intell. Syst., 7 (1992), 513–519. https://doi.org/10.1002/int.4550070604 doi: 10.1002/int.4550070604
    [4] S. Sathasivam, W. A. T. W. Abdullah, Logic mining in neural network: reverse analysis method, Computing, 91 (2011), 119–133. https://doi.org/10.1007/s00607-010-0117-9 doi: 10.1007/s00607-010-0117-9
    [5] M. S. M. Kasihmuddin, M. A. Mansor, S. Sathasivam, Hybrid genetic algorithm in the Hopfield network for logic satisfiability problem, Pertanika J. Sci. Technol., 25 (2017), 139–152.
    [6] S. Sathasivam, N. P. Fen, M. Velavan, Reverse analysis in higher order Hopfield network for higher order horn clauses, Appl. Math. Sci., 8 (2014), 601–612. https://doi.org/10.12988/ams.2014.310565 doi: 10.12988/ams.2014.310565
    [7] M. A. Mansor, M. S. M. Kasihmuddin, S. Sathasivam, Artificial immune system paradigm in the Hopfield network for 3-satisfiability problem, Pertanika J. Sci. Technol., 25 (2017), 1173–1188.
    [8] M. S. M. Kasihmuddin, M. A. Mansor, S. Sathasivam, Discrete Hopfield neural network in restricted maximum k-satisfiability logic programming, Sains Malays., 47 (2018), 1327–1335. https://dx.doi.org/10.17576/jsm-2018-4706-30
    [9] S. Sathasivam, M. A. Mansor, A. I. M. Ismail, S. Z. M. Jamaludin, M. S. M. Kasihmuddin, M. Mamat, Novel random k satisfiability for k≤ 2 in Hopfield neural network, Sains Malays., 49 (2020), 2847–2857. https://doi.org/10.17576/jsm-2020-4911-23 doi: 10.17576/jsm-2020-4911-23
    [10] F. L. Azizan, S. Sathasivam, Randomised alpha-cut fuzzy logic hybrid model in Solving 3-satisfiability Hopfield neural network, Malays. J. Fundam. Appl. Sci., 19 (2023), 43–55. https://doi.org/10.11113/mjfas.v19n1.2697 doi: 10.11113/mjfas.v19n1.2697
    [11] S. A. Alzaeemi, S. Sathasivam, Examining the forecasting movement of palm oil price using RBFNN-2SATRA metaheuristic algorithms for logic mining, IEEE Access, 9 (2021), 22542–22557. https://doi.org/10.1109/ACCESS.2021.3054816 doi: 10.1109/ACCESS.2021.3054816
    [12] M. A. Mansor, M. S. M. Kasihmuddin, S. Sathasivam, VLSI circuit configuration using satisfiability logic in Hopfield network, Int. J. Intell. Syst. Appl., 8 (2016), 22. https://doi.org/10.5815/ijisa.2016.09.03 doi: 10.5815/ijisa.2016.09.03
    [13] M. A. Mansor, M. S. M. Kasihmuddin, S. Sathasivam, Enhanced Hopfield network for pattern satisfiability optimization, Int. J. Intell. Syst. Appl., 8 (2016), 27. https://doi.org/10.5815/ijisa.2016.11.04 doi: 10.5815/ijisa.2016.11.04
    [14] M. S. M. Kasihmuddin, M. A. Mansor, S. Sathasivam, Bezier curves satisfiability model in enhanced Hopfield network, Int. J. Intell. Syst. Appl., 8 (2016), 9. https://doi.org/10.5815/ijisa.2016.12.02 doi: 10.5815/ijisa.2016.12.02
    [15] M. S. M. Kasihmuddin, M. A. Mansor, S. Sathasivam, Students' performance via satisfiability reverse analysis method with Hopfield neural network, AIP Conf Proc., 2184 (2019), 060035. https://doi.org/10.1063/1.5136467 doi: 10.1063/1.5136467
    [16] L. C. Kho, M. S. M. Kasihmuddin, M. A. Mansor, S. Sathasivam, 2 Satisfiability logical rule by using ant colony optimization in Hopfield neural network, AIP Conf. Proc., 2184 (2019), 060009. https://doi.org/10.1063/1.5136441 doi: 10.1063/1.5136441
    [17] M. A. Mansor, M. S. M. Kasihmuddin, S. Z. M. Jamaluddin, S. Sathasivam, Pattern 2 satisfiability snalysis via hybrid artificial bee colony algorithm as a learning algorithm, Commun. Comput. Appl. Math., 2 (2020), 12–18.
    [18] S. Sathasivam, M. A. Mansor, M. S. M. Kasihmuddin, H. Abubakar, Election algorithm for random k satisfiability in the Hopfield neural network, Processes, 8 (2020), 568. https://doi.org/10.3390/pr8050568 doi: 10.3390/pr8050568
    [19] M. A. Mansor, M. S. M. Kasihmuddin, S. Sathasivam, Grey wolf optimization algorithm with discrete Hopfield neural network for 3 Satisfiability analysis, J. Phys.: Conf. Ser., 1821 (2021), 012038. https://doi.org/10.1088/1742-6596/1821/1/012038 doi: 10.1088/1742-6596/1821/1/012038
    [20] F. L. Azizan, S. Sathasivam, M. K. M Ali, N. Roslan, C. Feng, Hybridised network of fuzzy logic and a genetic algorithm in solving 3-satisfiability Hopfield neural networks, Axioms, 12 (2023), 250. https://doi.org/10.3390/axioms12030250 doi: 10.3390/axioms12030250
    [21] M. S. Mohd Kasihmuddin, N. S. Abdul Halim, S. Z. Mohd Jamaludin, M. Mansor, A. Alway, N. E. Zamri, et al., Logic mining approach: Shoppers' purchasing data extraction via evolutionary algorithm, J. Inf. Commun. Technol., 22 (2023), 309–335. https://doi.org/10.32890/jict2023.22.3.1 doi: 10.32890/jict2023.22.3.1
    [22] S. Sathasivam, Applying fuzzy logic in neuro symbolic integration, World Appl. Sci. J., 17 (2012), 79–86.
    [23] B. Bollobás, C. Borgs, J. T. Chayes, J. H. Kim, D. B. Wilson, The scaling window of the 2-SAT transition, Random Struct. Algor., 18 (2001), 201–256. https://doi.org/10.1002/rsa.1006 doi: 10.1002/rsa.1006
    [24] M. Fürer, S. P. Kasiviswanathan, Algorithms for counting 2-SAT solutions and colorings with applications, In: Algorithmic aspects in information and management, 2007. https://doi.org/10.1007/978-3-540-72870-2_5
    [25] M. Formann, F. Wagner, A packing problem with applications to lettering of maps, In: Proceedings of the seventh annual symposium on computational geometry, 1991.
    [26] S. Ramnath, Dynamic digraph connectivity hastens minimum sum-of-diameters clustering, SIAM J. Discrete Math., 18 (2004), 272–286. https://doi.org/10.1137/S0895480102396099 doi: 10.1137/S0895480102396099
    [27] R. Miyashiro, T. Matsui, A polynomial-time algorithm to find an equitable home-away assignment, Oper. Res. Lett., 33 (2005), 235–241. https://doi.org/10.1016/j.orl.2004.06.004 doi: 10.1016/j.orl.2004.06.004
    [28] K. J. Batenburg, W. A. Kosters, Solving Nonograms by combining relaxations, Pattern Recogn., 42 (2009), 1672–1683. https://doi.org/10.1016/j.patcog.2008.12.003 doi: 10.1016/j.patcog.2008.12.003
    [29] S. Sathasivam, Clauses representation comparison in neuro-symbolic integration, Proceedings of the World Congress on Engineering, 2010.
    [30] J. C. Dunn, A fuzzy relative of the ISODATA process and its use in detecting compact well-separated clusters, J. Cyb., 3 (1973), 32–57. https://doi.org/10.1080/01969727308546046 doi: 10.1080/01969727308546046
    [31] J. C. Bezdek, R. Ehrlich, W. Full, FCM: The fuzzy c-means clustering algorithm, Comput. Geosci., 10 (1984), 191–203. https://doi.org/10.1016/0098-3004(84)90020-7 doi: 10.1016/0098-3004(84)90020-7
    [32] A. Askarzadeh, A novel metaheuristic method for solving constrained engineering optimization problems: Crow search algorithm, Comput. Struct., 169 (2016), 1–12. https://doi.org/10.1016/j.compstruc.2016.03.001 doi: 10.1016/j.compstruc.2016.03.001
    [33] A. Saha, A. Bhattacharya, P. Das, A. K. Chakraborty, Crow search algorithm for solving optimal power flow problem, In: 2017 Second International Conference on Electrical, Computer and Communication Technologies (ICECCT), 2017. https://doi.org/10.1109/ICECCT.2017.8118028
    [34] A. Lenin Fred, S. N. Kumar, P. Padmanaban, B. Gulyas, H. Ajay Kumar, Fuzzy-Crow search optimization for medical image segmentation, In: Applications of hybrid metaheuristic algorithms for image processing, 2020. https://doi.org/10.1007/978-3-030-40977-7_18
    [35] P. Upadhyay, J. K. Chhabra, Kapur's entropy based optimal multilevel image segmentation using Crow search algorithm, Appl. Soft Comput., 97 (2020), 105522. https://doi.org/10.1016/j.asoc.2019.105522 doi: 10.1016/j.asoc.2019.105522
    [36] G. Y. Abdallh, Z. Y. Algamal, A QSAR classification model of skin sensitization potential based on improving binary Crow search algorithm, Electron. J. Appl. Stat., 13 (2020), 86–95. https://doi.org/10.1285/i20705948v13n1p86 doi: 10.1285/i20705948v13n1p86
    [37] S. Arora, H. Singh, M. Sharma, S. Sharma, P. Anand, A new hybrid algorithm based on grey wolf optimization and Crow search algorithm for unconstrained function optimization and feature selection, IEEE Access, 7 (2019), 26343–26361. https://doi.org/10.1109/ACCESS.2019.2897325 doi: 10.1109/ACCESS.2019.2897325
    [38] F. Davoodkhani, S. Arabi Nowdeh, A. Y. Abdelaziz, S. Mansoori, S. Nasri, M. Alijani, A new hybrid method based on gray wolf optimizer-Crow search algorithm for maximum power point tracking of photovoltaic energy system, In: Modern maximum power point tracking techniques for photovoltaic energy systems, 2020. https://doi.org/10.1007/978-3-030-05578-3_16
    [39] A. B. Pratiwi, A hybrid cat swarm optimization-Crow search algorithm for vehicle routing problem with time windows, In: 2017 2nd international conferences on information technology, information systems and electrical engineering (ICITISEE), 2017. https://doi.org/10.1109/ICITISEE.2017.8285529
    [40] H. M. Farh, A. M. Al-Shaalan, A. M. Eltamaly, A. A. Al-Shamma'A, A novel Crow search algorithm auto-drive PSO for optimal allocation and sizing of renewable distributed generation, IEEE Access, 8 (2020), 27807–27820. https://doi.org/10.1109/ACCESS.2020.2968462 doi: 10.1109/ACCESS.2020.2968462
    [41] K. Gaddala, P. S. Raju, Merging lion with Crow search algorithm for optimal location and sizing of UPQC in distribution network, J. Control Autom. Electr. Syst., 31 (2020), 377–392. https://doi.org/10.1007/s40313-020-00564-1 doi: 10.1007/s40313-020-00564-1
    [42] R. Ganeshan, P. Rodrigues, Crow-AFL: Crow based adaptive fractional lion optimization approach for the intrusion detection, Wireless Pers. Commun., 111 (2020), 2065–2089. https://doi.org/10.1007/s11277-019-06972-0 doi: 10.1007/s11277-019-06972-0
    [43] D. K. Shende, S. S. Sonavane, Crow Whale-ETR: Crow Whale optimization algorithm for energy and trust aware multicast routing in WSN for IoT applications, Wireless Netw., 26 (2020), 4011–4029. https://doi.org/10.1007/s11276-020-02299-y doi: 10.1007/s11276-020-02299-y
    [44] A. M. Anter, A. E. Hassenian, D. Oliva, An improved fast fuzzy c-means using Crow search optimization algorithm for crop identification in agricultural, Expert Syst. Appl., 118 (2019), 340–354. https://doi.org/10.1016/j.eswa.2018.10.009 doi: 10.1016/j.eswa.2018.10.009
    [45] A. M. Anter, M. Ali, Feature selection strategy based on hybrid Crow search optimization algorithm integrated with chaos theory and fuzzy c-means algorithm for medical diagnosis problems, Soft Comput., 24 (2020), 1565–1584. https://doi.org/10.1007/s00500-019-03988-3 doi: 10.1007/s00500-019-03988-3
    [46] L. O. Hall, J. C. Bezdek, S. Boggavarpu, A. Bensaid, Genetic fuzzy clustering, In: Proceedings of the first international joint conference of the north American fuzzy information processing society biannual conference, 1994,411–415. https://doi.org/10.1109/IJCF.1994.375077
    [47] N. Siswanto, A. N. Adianto, H. A. Prawira, A. Rusdiansyah, A Crow search algorithm for aircraft maintenance check problem and continuous airworthiness maintenance program, JSMI, 3 (2019), 10115–10123. https://doi.org/10.30656/jsmi.v3i2.1794 doi: 10.30656/jsmi.v3i2.1794
    [48] D. Gupta, S. Sundaram, J. J. Rodrigues, A. Khanna, An improved fault detection Crow search algorithm for wireless sensor network, Int. J. Commun. Syst., 36 (2023), e4136. https://doi.org/10.1002/dac.4136 doi: 10.1002/dac.4136
    [49] J. Mandala, M. C. Rao, Privacy preservation of data using Crow search with adaptive awareness probability, J. Inf. Secur. Appl., 44 (2019), 157–169. https://doi.org/10.1016/j.jisa.2018.12.005 doi: 10.1016/j.jisa.2018.12.005
    [50] R. Dash, S. Samal, R. Dash, R. Rautray, An integrated TOPSIS Crow search based classifier ensemble: In application to stock index price movement prediction, Appl. Soft Comput., 85 (2019), 105784. https://doi.org/10.1016/j.asoc.2019.105784 doi: 10.1016/j.asoc.2019.105784
    [51] A. G. Hussien, M. Amin, M. Wang, G. Liang, A. Alsanad, A.Gumaei, et al., Crow search algorithm: Theory, recent advances, and applications, IEEE Access, 8 (2020), 173548–173565. https://doi.org/10.1109/ACCESS.2020.3024108 doi: 10.1109/ACCESS.2020.3024108
    [52] J. H. Holland, Adaptation in natural and artificial systems: An introductory analysis with applications to biology, control, and artificial intelligence, MIT Press, 1992.
    [53] M. Alata, M. Molhim, A. Ramini, Optimizing of fuzzy c-means clustering algorithm using GA, Int. J. Comput. Inf. Eng., 2 (2008), 670–675. https://doi.org/10.5281/zenodo.1081049 doi: 10.5281/zenodo.1081049
    [54] Y. Ding, X. Fu, Kernel-based fuzzy c-means clustering algorithm based on genetic algorithm, Neurocomputing, 188 (2016), 233–238. https://doi.org/10.1016/j.neucom.2015.01.106 doi: 10.1016/j.neucom.2015.01.106
    [55] X. Wang, H. Wang, Driving behavior clustering for hazardous material transportation based on genetic fuzzy C-means algorithm, IEEE Access, 8 (2020), 11289–11296. https://doi.org/10.1109/ACCESS.2020.2964648 doi: 10.1109/ACCESS.2020.2964648
    [56] X. Cui, E. C. Yan, Fuzzy c-means cluster analysis based on variable length string genetic algorithm for the grouping of rock discontinuity sets, KSCE J. Civ. Eng., 24 (2020), 3237–3246. https://doi.org/10.1007/s12205-020-2188-2 doi: 10.1007/s12205-020-2188-2
    [57] S. Sathasivam, Upgrading logic programming in Hopfield network, Sains Malays., 39 (2010), 115–128.
    [58] M. Velavan, Z. R. bin Yahya, M. N. bin Abdul Halif, S. Sathasivam, Mean field theory in doing logic programming using Hopfield network, Mod. Appl. Sci., 10 (2016), 154. https://doi.org/10.5539/mas.v10n1p154 doi: 10.5539/mas.v10n1p154
  • Reader Comments
  • © 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(1223) PDF downloads(47) Cited by(1)

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog