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Research article Special Issues

A continuum model for the tensegrity Maxwell chain

  • A recent study has presented a Maxwell mass–spring model for a chain formed by two different types of tensegrity prisms alternating with lumped masses. Such a model shows tensegrity theta prisms arranged in parallel with minimal regular prisms acting as resonant substructures. It features a tunable frequency bandgap response, due to the possibility of adjusting the width of the bandgap regions by playing with internal resonance effects in addition to mass and spring contrasts. This paper expands such research by presenting a continuum modeling of the tensegrity Maxwell chain, which is useful to conduct analytic studies and to develop finite element models of the plane wave dynamics of the investigated system. In correspondence to the high wave-length limit, i.e., in the low wave number regime, it is shown that the dispersion relations of the discrete and continuum models provide similar results. Analytic solutions to the wave dynamics of physical systems are presented, which validate the predictions of the bandgap response offered by the dispersion relation of the continuum model.

    Citation: Luca Placidi, Julia de Castro Motta, Rana Nazifi Charandabi, Fernando Fraternali. A continuum model for the tensegrity Maxwell chain[J]. Networks and Heterogeneous Media, 2024, 19(2): 597-610. doi: 10.3934/nhm.2024026

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  • A recent study has presented a Maxwell mass–spring model for a chain formed by two different types of tensegrity prisms alternating with lumped masses. Such a model shows tensegrity theta prisms arranged in parallel with minimal regular prisms acting as resonant substructures. It features a tunable frequency bandgap response, due to the possibility of adjusting the width of the bandgap regions by playing with internal resonance effects in addition to mass and spring contrasts. This paper expands such research by presenting a continuum modeling of the tensegrity Maxwell chain, which is useful to conduct analytic studies and to develop finite element models of the plane wave dynamics of the investigated system. In correspondence to the high wave-length limit, i.e., in the low wave number regime, it is shown that the dispersion relations of the discrete and continuum models provide similar results. Analytic solutions to the wave dynamics of physical systems are presented, which validate the predictions of the bandgap response offered by the dispersion relation of the continuum model.



    Let Fq be the finite field of q elements with characteristic p, where q=pr, p is a prime number. Let Fq=Fq{0} and Z+ denote the set of positive integers. Let sZ+ and bFq. Let f(x1,,xs) be a diagonal polynomial over Fq of the following form

    f(x1,,xs)=a1xm11+a2xm22++asxmss,

    where aiFq, miZ+, i=1,,s. Denote by Nq(f=b) the number of Fq-rational points on the affine hypersurface f=b, namely,

    Nq(f=b)=#{(x1,,xs)As(Fq)f(x1,,xs)=b}.

    In 1949, Hua and Vandiver [1] and Weil [2] independently obtained the formula of Nq(f=b) in terms of character sum as follows

    Nq(f=b)=qs1+ψ1(a11)ψs(ass)J0q(ψ1,,ψs), (1.1)

    where the sum is taken over all s multiplicative characters of Fq that satisfy ψmii=ε, ψiε, i=1,,s and ψ1ψs=ε. Here ε is the trivial multiplicative character of Fq, and J0q(ψ1,,ψs) is the Jacobi sum over Fq defined by

    J0q(ψ1,,ψs)=c1++cs=0,ciFqψ1(c1)ψs(cs).

    Though the explicit formula for Nq(f=b) are difficult to obtain in general, it has been studied extensively because of their theoretical importance as well as their applications in cryptology and coding theory; see[3,4,5,6,7,8,9]. In this paper, we use the Jacobi sums, Gauss sums and the results of quadratic form to deduce the formula of the number of Fq2-rational points on a class of hypersurfaces over Fq2 under certain conditions. The main result of this paper can be stated as

    Theorem 1.1. Let q=2r with rZ+ and Fq2 be the finite field of q2 elements. Let f(X)=a1xm11+a2xm22++asxmss, g(Y)=y1y2+y3y4++yn1yn+y2n2t1+ +y2n3+y2n1+bty2n2t++b1y2n2+b0y2n, and l(X,Y)=f(X)+g(Y), where ai,bjFq2, mi1, (mi,mk)=1, ik, mi|(q+1), miZ+, 2|n, n>2, 0tn22, TrFq2/F2(bj)=1 for i,k=1,,s and j=0,1,,t. For hFq2, we have

    (1) If h=0, then

    Nq2(l(X,Y)=0)=q2(s+n1)+γFq2(si=1((γai)mimi1)(qs+2n3+(1)tqs+n3)).

    (2) If hFq2, then

    Nq2(l(X,Y)=h)=q2(s+n1)+(qs+2n3+(1)t+1(q21)qs+n3)si=1((hai)mimi1)+γFq2{h}[si=1((γai)mimi1)(q2n+s3+(1)tqn+s3)].

    Here,

    (γai)mi={1,ifγaiisaresidueofordermi,0,otherwise.

    To prove Theorem 1.1, we need the lemmas and theorems below which are related to the Jacobi sums and Gauss sums.

    Definition 2.1. Let χ be an additive character and ψ a multiplicative character of Fq. The Gauss sum Gq(ψ,χ) in Fq is defined by

    Gq(ψ,χ)=xFqψ(x)χ(x).

    In particular, if χ is the canonical additive character, i.e., χ(x)=e2πiTrFq/Fp(x)/p where TrFq/Fp(y)=y+yp++ypr1 is the absolute trace of y from Fq to Fp, we simply write Gq(ψ):=Gq(ψ,χ).

    Let ψ be a multiplicative character of Fq which is defined for all nonzero elements of Fq. We extend the definition of ψ by setting ψ(0)=0 if ψε and ε(0)=1.

    Definition 2.2. Let ψ1,,ψs be s multiplicative characters of Fq. Then, Jq(ψ1,,ψs) is the Jacobi sum over Fq defined by

    Jq(ψ1,,ψs)=c1++cs=1,ciFqψ1(c1)ψs(cs).

    The Jacobi sums Jq(ψ1,,ψs) as well as the sums J0q(ψ1,,ψs) can be evaluated easily in case some of the multiplicative characters ψi are trivial.

    Lemma 2.3. ([10,Theorem 5.19,p. 206]) If the multiplicative characters ψ1,,ψs of Fq are trivial, then

    Jq(ψ1,,ψs)=J0q(ψ1,,ψs)=qs1.

    If some, but not all, of the ψi are trivial, then

    Jq(ψ1,,ψs)=J0q(ψ1,,ψs)=0.

    Lemma 2.4. ([10,Theorem 5.20,p. 206]) If ψ1,,ψs are multiplicative characters of Fq with ψs nontrivial, then

    J0q(ψ1,,ψs)=0

    if ψ1ψs is nontrivial and

    J0q(ψ1,,ψs)=ψs(1)(q1)Jq(ψ1,,ψs1)

    if ψ1ψs is trivial.

    If all ψi are nontrivial, there exists an important connection between Jacobi sums and Gauss sums.

    Lemma 2.5. ([10,Theorem 5.21,p. 207]) If ψ1,,ψs are nontrivial multiplicative characters of Fq and χ is a nontrivial additive character of Fq, then

    Jq(ψ1,,ψs)=Gq(ψ1,χ)Gq(ψs,χ)Gq(ψ1ψs,χ)

    if ψ1ψs is nontrivial and

    Jq(ψ1,,ψs)=ψs(1)Jq(ψ1,,ψs1)=1qGq(ψ1,χ)Gq(ψs,χ)

    if ψ1ψs is trivial.

    We turn to another special formula for Gauss sums which applies to a wider range of multiplicative characters but needs a restriction on the underlying field.

    Lemma 2.6. ([10,Theorem 5.16,p. 202]) Let q be a prime power, let ψ be a nontrivial multiplicative character of Fq2 of order m dividing q+1. Then

    Gq2(ψ)={q,ifmoddorq+1meven,q,ifmevenandq+1modd.

    For hFq2, define v(h)=1 if hFq2 and v(0)=q21. The property of the function v(h) will be used in the later proofs.

    Lemma 2.7. ([10,Lemma 6.23,p. 281]) For any finite field Fq, we have

    cFqv(c)=0,

    for any bFq,

    c1++cm=bv(c1)v(ck)={0,1k<m,v(b)qm1,k=m,

    where the sum is over all c1,,cmFq with c1++cm=b.

    The quadratic forms have been studied intensively. A quadratic form f in n indeterminates is called nondegenerate if f is not equivalent to a quadratic form in fewer than n indeterminates. For any finite field Fq, two quadratic forms f and g over Fq are called equivalent if f can be transformed into g by means of a nonsingular linear substitution of indeterminates.

    Lemma 2.8. ([10,Theorem 6.30,p. 287]) Let fFq[x1,,xn], q even, be a nondegenerate quadratic form. If n is even, then f is either equivalent to

    x1x2+x3x4++xn1xn

    or to a quadratic form of the type

    x1x2+x3x4++xn1xn+x2n1+ax2n,

    where aFq satisfies TrFq/Fp(a)=1.

    Lemma 2.9. ([10,Corollary 3.79,p. 127]) Let aFq and let p be the characteristic of Fq, the trinomial xpxa is irreducible in Fq if and only if TrFq/Fp(a)0.

    Lemma 2.10. ([10,Lemma 6.31,p. 288]) For even q, let aFq with TrFq/Fp(a)=1 and bFq. Then

    Nq(x21+x1x2+ax22=b)=qv(b).

    Lemma 2.11. ([10,Theorem 6.32,p. 288]) Let Fq be a finite field with q even and let bFq. Then for even n, the number of solutions of the equation

    x1x2+x3x4++xn1xn=b

    in Fnq is qn1+v(b)q(n2)/2. For even n and aFq with TrFq/Fp(a)=1, the number of solutions of the equation

    x1x2+x3x4++xn1xn+x2n1+ax2n=b

    in Fnq is qn1v(b)q(n2)/2.

    Lemma 2.12. Let q=2r and hFq2. Let g(Y)Fq2[y1,y2,,yn] be a polynomial of the form

    g(Y)=y1y2+y3y4++yn1yn+y2n2t1++y2n3+y2n1+bty2n2t++b1y2n2+b0y2n,

    where bjFq2, 2|n, n>2, 0tn22, TrFq2/F2(bj)=1, j=0,1,,t. Then

    Nq2(g(Y)=h)=q2(n1)+(1)t+1qn2v(h). (2.1)

    Proof. We provide two proofs here. The first proof is as follows. Let q1=q2. Then by Lemmas 2.7 and 2.10, the number of solutions of g(Y)=h in Fq2 can be deduced as

    Nq2(g(Y)=h)=c1+c2++ct+2=hNq2(y1y2+y3y4++yn2t3yn2t2=c1)Nq2(yn2t1yn2t+y2n2t1+bty2n2t=c2)Nq2(yn1yn+y2n1+b0y2n=ct+2)=c1+c2++ct+2=h(qn2t31+v(c1)q(n2t4)/21)(q1v(c2))(q1v(ct+2))=c1+c2++ct+2=h(qn2t21+v(c1)q(n2t2)/21v(c2)qn2t31v(c1)v(c2)q(n2t4)/21)(q1v(c3))(q1v(ct+2))=c1+c2++ct+2=h(qnt21+v(c1)q(n2)/21v(c2)qnt31++(1)t+1v(c1)v(c2)v(ct+2)q(n2t4)/21)=qn11+q(n2)/21c1Fq2v(c1)++(1)t+1c1+c2++ct+2=hv(c1)v(c2)v(ct+2)q(n2t4)/21. (2.2)

    By Lamma 2.7 and (2.2), we have

    Nq2(g(Y)=h)=qn11+(1)t+1v(h)q(n2)/21=q2(n1)+(1)t+1v(h)qn2.

    Next we give the second proof. Note that if f and g are equivalent, then for any bFq2 the equation f(x1,,xn)=b and g(x1,,xn)=b have the same number of solutions in Fq2. So we can get the number of solutions of g(Y)=h for hFq2 by means of a nonsingular linear substitution of indeterminates.

    Let k(X)Fq2[x1,x2,x3,x4] and k(X)=x1x2+x21+Ax22+x3x4+x23+Bx24, where TrFq2/F2(A)=TrFq2/F2(B)=1. We first show that k(x) is equivalent to x1x2+x3x4.

    Let x3=y1+y3 and xi=yi for i3, then k(X) is equivalent to y1y2+y1y4+y3y4+Ay22+y23+By24.

    Let y2=z2+z4 and yi=zi for i2, then k(X) is equivalent to z1z2+z3z4+Az22+z23+Az24+Bz24.

    Let z1=α1+Aα2 and zi=αi for i1, then k(X) is equivalent to α1α2+α23+α3α4+(A+B)α24.

    Since TrFq2/F2(A+B)=0, we have α23+α3α4+(A+B)α24 is reducible by Lemma 2.9. Then k(X) is equivalent to x1x2+x3x4. It follows that if t is odd, then g(Y) is equivalent to x1x2+x3x4++xn1xn, and if t is even, then g(Y) is equivalent to x1x2+x3x4++xn1xn+x2n1+ax2n with TrFq2/F2(a)=1. By Lemma 2.11, we get the desired result.

    From (1.1), we know that the formula for the number of solutions of f(X)=0 over Fq2 is

    Nq2(f(X)=0)=q2(s1)+d11j1=1ds1js=1¯ψj11(a1)¯ψjss(as)J0q2(ψj11,,ψjss),

    where di=(mi,q21) and ψi is a multiplicative character of Fq2 of order di. Since mi|q+1, we have di=mi. Let H={(j1,,js)1ji<mi, 1is}. It follows that ψj11ψjss is nontrivial for any (j1,,js)H as (mi,mj)=1. By Lemma 2, we have J0q2(ψj11,,ψjss)=0 and hence Nq2(f(X)=0)=q2(s1).

    Let Nq2(f(X)=c) denote the number of solutions of the equation f(X)=c over Fq2 with cFq2. Let V={(j1,,js)|0ji<mi,1is}. Then

    Nq2(f(X)=c)=γ1++γs=cNq2(a1xm11=γ1)Nq2(asxmss=γs)=γ1++γs=cm11j1=0ψj11(γ1a1)ms1js=0ψjss(γsas).

    Since ψi is a multiplicative character of Fq2 of order mi, we have

    Nq2(f(X)=c)=γ1c++γsc=1(j1,,js)Vψj11(γ1c)ψj11(ca1)ψjss(γsc)ψjss(cas)=(j1,,js)Vψj11(ca1)ψjss(cas)γ1c++γsc=1ψj11(γ1c)ψjss(γsc)=(j1,,js)Vψj11(ca1)ψjss(cas)Jq2(ψj11,,ψjss).

    By Lemma 2.3,

    Nq2(f(X)=c)=q2(s1)+(j1,,js)Hψj11(ca1)ψjss(cas)Jq2(ψj11,,ψjss).

    By Lemma 2.5,

    Jq2(ψj11,,ψjss)=Gq2(ψj11)Gq2(ψjss)Gq2(ψj11ψjss).

    Since mi|q+1 and 2mi, by Lemma 2.6, we have

    Gq2(ψj11)==Gq2(ψjss)=Gq2(ψj11ψjss)=q.

    Then

    Nq2(f(X)=c)=q2(s1)+qs1m11j1=1ψj11(ca1)ms1js=1ψjss(cas)=q2(s1)+qs1(m11j1=0ψj11(ca1)1)(ms1js=0ψjss(cas)1).

    It follows that

    Nq2(f(X)=c)=q2(s1)+qs1si=1((cai)mimi1), (3.1)

    where

    (cai)mi={1,ifcai is a residue of ordermi,0,otherwise.

    For a given hFq2. We discuss the two cases according to whether h is zero or not.

    Case 1: h=0. If f(X)=0, then g(Y)=0; if f(X)0, then g(Y)0. Then

    Nq2(l(X,Y)=0)=c1+c2=0Nq2(f(X)=c1)Nq2(g(Y)=c2)=q2(s1)(q2(n1)+(1)t+1(q21)qn2)+c1+c2=0c1,c2Fq2Nq2(f(X)=c1)Nq2(g(Y)=c2). (3.2)

    By Lemma 2.12, (3.1) and (3.2), we have

    Nq2(l(X,Y)=0)=q2(s+n2)+(1)t+1q2(s1)+hn(1)t+1q2(s2)+n+c1Fq2[q2(s+n2)(1)t+1q2(s2)+n+si=1((c1ai)mimi1)(q2n+s3(1)t+1qn+s3)]=q2(s+n2)+(1)t+1q2(s1)+n(1)t+1q2(s2)+n+q2(s+n1)(1)t+1q2(s1)+nq2(s+n2)+(1)t+1q2(s2)+n+c1Fq2[si=1((c1ai)mimi1)(q2n+s3(1)t+1qn+s3)]=q2(s+n1)+c1Fq2[si=1((c1ai)mimi1)(q2n+s3(1)t+1qn+s3)]. (3.3)

    Case 2: hFq2. If f(X)=h, then g(Y)=0; if f(X)=0, then g(Y)=h; if f(X){0,h}, then g(Y){0,h}. So we have

    Nq2(l(X,Y))=h)=c1+c2=hNq2(f(X)=c1)Nq2(g(Y)=c2)=Nq2(f(X)=0)Nq2(g(Y)=h)+Nq2(f(X)=h)Nq2(g(Y)=0)+c1+c2=hc1,c2Fq2{h}Nq2(f(X)=c1)Nq2(g(Y)=c2). (3.4)

    By Lemma 2.12, (3.1) and (3.4),

    Nq2(l(X,Y)=h)=2q2(s+n2)+(1)t+1q2s+n2(1)t+12q2s+n4+(qs+2n3+(1)t+1(q21)qs+n3)si=1((hai)mimi1)+c1Fq2{h}[q2(s+n2)(1)t+1q2s+n4+si=1((c1ai)mimi1)(q2n+s3(1)t+1qn+s3)].

    It follows that

    Nq2(l(X,Y)=h)=2q2(s+n2)+(1)t+1q2s+n2(1)t+12q2s+n4+(qs+2n3+(1)t+1(q21)qs+n3)si=1((hai)mimi1)+c1Fq2{h}[q2(s+n2)(1)t+1q2s+n4+si=1((c1ai)mimi1)(q2n+s3(1)t+1qn+s3)]=q2(s+n1)+(qs+2n3+(1)t+1(q21)qs+n3)si=1((hai)mimi1)+c1Fq2{h}[si=1((c1ai)mimi1)(q2n+s3+(1)tqn+s3)]. (3.5)

    By (3.3) and (3.5), we get the desired result. The proof of Theorem 1.1 is complete.

    There is a direct corollary of Theorem 1.1 and we omit its proof.

    Corollary 4.1. Under the conditions of Theorem 1.1, if a1==as=hFq2, then we have

    Nq2(l(X,Y)=h)=q2(s+n1)+(qs+2n3+(1)t+1(q21)qs+n3)si=1(mi1)+γFq2{h}[si=1((γh)mimi1)(q2n+s3+(1)tqn+s3)],

    where

    (γh)mi={1,ifγhisaresidueofordermi,0,otherwise.

    Finally, we give two examples to conclude the paper.

    Example 4.2. Let F210=α=F2[x]/(x10+x3+1) where α is a root of x10+x3+1. Suppose l(X,Y)=α33x31+x112+y23+α10y24+y1y2+y3y4. Clearly, TrF210/F2(α10)=1, m1=3, m2=11, s=2, n=4, t=0, a2=1. By Theorem 1.1, we have

    N210(l(X,Y)=0)=10245+(327+323)×20=1126587102265344.

    Example 4.3. Let F212=β=F2[x]/(x12+x6+x4+x+1) where β is a root of x12+x6+x4+x+1. Suppose l(X,Y)=x51+x132+y23+β10y24+y1y2+y3y4. Clearly, TrF212/F2(β10)=1, m1=5, m2=13, s=2, n=4, t=0, a1=a2=1. By Corollary 1, we have

    N212(l(X,Y)=1)=25×12+(647643×4095)×48=1153132559312355328.

    This work was jointly supported by the Natural Science Foundation of Fujian Province, China under Grant No. 2022J02046, Fujian Key Laboratory of Granular Computing and Applications (Minnan Normal University), Institute of Meteorological Big Data-Digital Fujian and Fujian Key Laboratory of Data Science and Statistics.

    The authors declare there is no conflicts of interest.



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