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Characterizing emerging features in cell dynamics using topological data analysis methods


  • Filament-motor interactions inside cells play essential roles in many developmental as well as other biological processes. For instance, actin-myosin interactions drive the emergence or closure of ring channel structures during wound healing or dorsal closure. These dynamic protein interactions and the resulting protein organization lead to rich time-series data generated by using fluorescence imaging experiments or by simulating realistic stochastic models. We propose methods based on topological data analysis to track topological features through time in cell biology data consisting of point clouds or binary images. The framework proposed here is based on computing the persistent homology of the data at each time point and on connecting topological features through time using established distance metrics between topological summaries. The methods retain aspects of monomer identity when analyzing significant features in filamentous structure data, and capture the overall closure dynamics when assessing the organization of multiple ring structures through time. Using applications of these techniques to experimental data, we show that the proposed methods can describe features of the emergent dynamics and quantitatively distinguish between control and perturbation experiments.

    Citation: Madeleine Dawson, Carson Dudley, Sasamon Omoma, Hwai-Ray Tung, Maria-Veronica Ciocanel. Characterizing emerging features in cell dynamics using topological data analysis methods[J]. Mathematical Biosciences and Engineering, 2023, 20(2): 3023-3046. doi: 10.3934/mbe.2023143

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  • Filament-motor interactions inside cells play essential roles in many developmental as well as other biological processes. For instance, actin-myosin interactions drive the emergence or closure of ring channel structures during wound healing or dorsal closure. These dynamic protein interactions and the resulting protein organization lead to rich time-series data generated by using fluorescence imaging experiments or by simulating realistic stochastic models. We propose methods based on topological data analysis to track topological features through time in cell biology data consisting of point clouds or binary images. The framework proposed here is based on computing the persistent homology of the data at each time point and on connecting topological features through time using established distance metrics between topological summaries. The methods retain aspects of monomer identity when analyzing significant features in filamentous structure data, and capture the overall closure dynamics when assessing the organization of multiple ring structures through time. Using applications of these techniques to experimental data, we show that the proposed methods can describe features of the emergent dynamics and quantitatively distinguish between control and perturbation experiments.



    In 1889, Hölder [1] proved that

    nk=1xkyk(nk=1xpk)1p(nk=1yqk)1q, (1.1)

    where {xk}nk=1 and {yk}nk=1 are positive sequences, p and q are two positive numbers, such that

    1/p+1/q=1.

    The inequality reverses if pq<0. The integral form of (1.1) is

    ϵηλ(τ)ω(τ)dτ[ϵηλγ(τ)dτ]1γ[ϵηων(τ)dτ]1ν, (1.2)

    where η,ϵR, γ>1, 1/γ+1/ν=1, and λ,ωC([a,b],R). If 0<γ<1, then (1.2) is reversed.

    The subsequent inequality, widely recognized as Minkowski's inequality, asserts that, for p1, if φ,ϖ are nonnegative continuous functions on [η,ϵ] such that

    0<ϵηφp(τ)dτ< and 0<ϵηϖp(τ)dτ<,

    then

    (ϵη(φ(τ)+ϖ(τ))pdτ)1p(ϵηφp(τ)dτ)1p+(ϵηϖp(τ)dτ)1p.

    Sulaiman [2] introduced the following result pertaining to the reverse Minkowski's inequality: for any φ,ϖ>0, if p1 and

    1<mφ(ϑ)ϖ(ϑ)M, ϑ[η,ϵ],

    then

    M+1(M1)(ϵη(φ(ϑ)ϖ(ϑ))pdϑ)1p(ϵηφp(ϑ)dϑ)1p+(ϵηϖp(ϑ)dϑ)1p(m+1m1)(ϵη(φ(ϑ)ϖ(ϑ))pdϑ)1p. (1.3)

    Sroysang [3] proved that if p1 and φ,ϖ>0 with

    0<c<mφ(ϑ)ϖ(ϑ)M, ϑ[η,ϵ],

    then

    M+1(Mc)(ϵη(φ(ϑ)cϖ(ϑ))pdϑ)1p(ϵηφp(ϑ)dϑ)1p+(ϵηϖp(ϑ)dϑ)1p(m+1mc)(ϵη(φ(ϑ)cϖ(ϑ))pdϑ)1p.

    In 2023, Kirmaci [4] proved that for nonnegative sequences {xk}nk=1 and {yk}nk=1, if p>1,s1,

    1sp+1sq=1,

    then

    nk=1(xkyk)1s(nk=1xpk)1sp(nk=1yqk)1sq, (1.4)

    and if s1,sp<0 and

    1sp+1sq=1,

    then (1.4) is reversed. Also, the author [4] generalized Minkowski's inequality and showed that for nonnegative sequences {xk}nk=1 and {yk}nk=1, if mN, p>1,

    1sp+1sq=1,

    then

    (nk=1(xk+yk)p)1mp(nk=1xpk)1mp+(nk=1ypk)1mp. (1.5)

    Hilger was a trailblazer in integrating continuous and discrete analysis, introducing calculus on time scales in his doctoral thesis, later published in his influential work [5]. The exploration of dynamic inequalities on time scales has garnered significant interest in academic literature. Agarwal et al.'s book [6] focuses on essential dynamic inequalities on time scales, including Young's inequality, Jensen's inequality, Hölder's inequality, Minkowski's inequality, and more. For further insights into dynamic inequalities on time scales, refer to the relevant research papers [7,8,9,10,11,12,13,14,15,16,17,18,19,20]. For the solutions of some differential equations, see, for instance, [21,22,23,24,25,26,27] and the references therein.

    Expanding upon this trend, the present paper aims to broaden the scope of the inequalities (1.4) and (1.5) on diamond alpha time scales. We will establish the generalized form of Hölder's inequality and Minkowski's inequality for the forward jump operator and the backward jump operator in the discrete calculus. In addition, we prove these inequalities in diamond α calculus, where for α=1, we can get the inequalities for the forward operator, and when α=0, we get the inequalities for the backward jump operator. Also, we can get the special cases of our results in the discrete and continuous calculus.

    The paper is organized as follows: In Section 2, we present some definitions, theorems, and lemmas on time scales, which are needed to get our main results. In Section 3, we state and prove new dynamic inequalities and present their special cases in different (continuous, discrete) calculi.

    In 2001, Bohner and Peterson [28] defined the time scale T as an arbitrary, nonempty, closed subset of the real numbers R. The forward jump operator and the backward jump operator are defined by

    σ(ξ):=inf{sT:s>ξ}

    and

    ρ(ξ):=sup{sT:s<ξ},

    respectively. The graininess function μ for a time scale T is defined by

    μ(ξ):=σ(ξ)ξ0,

    and for any function φ: TR, the notation φσ(ξ) and φρ(ξ) denote φ(σ(ξ)) and φ(ρ(ξ)), respectively. We define the time scale interval [η,ϵ]T by

    [η,ϵ]T:=[η,ϵ]T.

    Definition 2.1. (The Delta derivative[28]) Assume that φ: TR is a function and let ξT. We define φΔ(ξ) to be the number, provided it exists, as follows: for any ϵ>0, there is a neighborhood U of ξ,

    U=(ξδ,ξ+δ)T

    for some δ>0, such that

    |φ(σ(ξ))φ(s)φΔ(ξ)(σ(ξ)s)|ϵ|σ(ξ)s|,   sU, sσ(ξ).

    In this case, we say φΔ(ξ) is the delta or Hilger derivative of φ at ξ.

    Definition 2.2. (The nabla derivative [29]) A function λ: TR is said to be - differentiable at ξT, if ψ is defined in a neighborhood U of ξ and there exists a unique real number ψ(ξ), called the nabla derivative of ψ at ξ, such that for each ϵ>0, there exists a neighborhood N of ξ with NU and

    |ψ(ρ(ξ))ψ(s)ψ(ξ)[ρ(ξ)s]|ϵ|ρ(ξ)s|,

    for all sN.

    Definition 2.3. ([6]) Let T be a time scale and Ξ(ξ) be differentiable on T in the Δ and sense. For ξT, we define the diamond α derivative Ξα(ξ) by

    Ξα(ξ)=αΞΔ(ξ)+(1α)Ξ(ξ),  0α1.

    Thus, Ξ is diamond α differentiable if, and only if, Ξ is Δ and differentiable.

    For α=1, we get that

    Ξα(ξ)=αΞΔ(ξ),

    and for α=0, we have that

    Ξα(ξ)=Ξ(ξ).

    Theorem 2.1. ([6]) Let Ξ, Ω: TR be diamond-α differentiable at ξT, then

    (1) Ξ+Ω: TR is diamond-α differentiable at ξT, with

    (Ξ+Ω)α(ξ)=Ξα(ξ)+Ωα(ξ).

    (2) Ξ.Ω: TR is diamond-α differentiable at ξT, with

    (Ξ.Ω)α(ξ)=Ξα(ξ)Ω(ξ)+αΞσ(ξ)ΩΔ(ξ)+(1α)Ξρ(ξ)Ω(ξ).

    (3) For Ω(ξ)Ωσ(ξ)Ωρ(ξ)0, Ξ/Ω: TR is diamond-α differentiable at ξT, with

    (ΞΩ)α(ξ)=Ξα(ξ)Ωσ(ξ)Ωρ(ξ)αΞσ(ξ)Ωρ(ξ)ΩΔ(ξ)(1α)Ξρ(ξ)Ωσ(ξ)Ω(ξ)Ω(ξ)Ωσ(ξ)Ωρ(ξ).

    Theorem 2.2. ([6]) Let Ξ, Ω: TR be diamond-α differentiable at ξT, then the following holds

    (1) (Ξ)αΔ(ξ)=αΞΔΔ(ξ)+(1α)ΞΔ(ξ),

    (2) (Ξ)α(ξ)=αΞΔ(ξ)+(1α)Ξ(ξ),

    (3) (Ξ)Δα(ξ)=αΞΔΔ(ξ)+(1α)ΞΔ(ξ)(Ξ)αΔ(ξ),

    (4) (Ξ)α(ξ)=αΞΔ(ξ)+(1α)Ξ(ξ)(Ξ)α(ξ),

    (5) (Ξ)αα(ξ)=α2ΞΔΔ(ξ)+α(1α)[ΞΔ(ξ)+ΞΔ(ξ)].

    Theorem 2.3. ([6]) Let a,ξT and h: TR, then the diamondα integral from a to ξ of h is defined by

    tah(s)αs=αtah(s)Δs+(1α)tah(s)s, 0α1,

    provided that there exist delta and nabla integrals of h on T.

    It is known that

    (ξah(s)Δs)Δ=h(ξ)

    and

    (ξah(s)s)=h(ξ),

    but in general

    (ξah(s)αs)αh(ξ), for ξT.

    Example 2.1. [30] Let T={0, 1, 2},a=0, and h(ξ)=ξ2 for ξT. This gives

    (ξah(s)αs)α|ξ=1=1+2α(1α),

    so that the equality above holds only when α=Δ or α=.

    Theorem 2.4. ([6]) Let a,b,ξT, β,γ,cR, and Ξ and Ω be continuous functions on [a,b]T, then the following properties hold.

    (1) ba[γΞ(ξ)+βΩ(ξ)]αξ=γbaΞ(ξ)αξ+βbaΩ(ξ)αξ,

    (2) bacΞ(ξ)αξ=cbaΞ(ξ)αξ,

    (3) baΞ(ξ)αξ=abΞ(ξ)αξ,

    (4) caΞ(ξ)αξ=baΞ(ξ)αξ+cbΞ(ξ)αξ.

    Theorem 2.5. ([6]) Let T be a time scale a,bT with a<b. Assume that Ξ and Ω are continuous functions on [a,b]T,

    (1) If Ξ(ξ)0 for all ξ[a,b]T, then baΞ(ξ)αξ0;

    (2) If Ξ(ξ)Ω(ξ) for all ξ[a,b]T, then baΞ(ξ)αξbaΩ(ξ)αξ;

    (3) If Ξ(ξ)0 for all ξ[a,b]T, then Ξ(ξ)=0 if, and only if, baΞ(ξ)αξ=0.

    Example 2.2. ([6]) If T=R, then

    baΞ(ξ)αξ=baΞ(ξ)dξ, for a, bR,

    and if T=N and m<n, then we obtain

    nmΞ(ξ)αξ=n1i=m[αΞ(i)+(1α)Ξ(i+1)], for m, nN. (2.1)

    Lemma 2.1. (Hölder's inequality [6]) If η,ϵT, 0α1, and λ,ωC([η,ϵ]T,R+), then

    ϵηλ(τ)ω(τ)ατ[ϵηλγ(τ)ατ]1γ[ϵηων(τ)ατ]1ν, (2.2)

    where γ>1 and 1/γ+1/ν=1. The inequality (2.2) is reversed for 0<γ<1 or γ<0.

    Lemma 2.2. (Minkowski's inequality[6]) Let η,ϵT, ϵ>η,0α1,p>1, and Ξ,ΩC([η,ϵ]T,R+), then

    (ϵη(Ξ(τ)+Ω(τ))pατ)1p(ϵηΞp(τ)ατ)1p+(ϵηΩp(τ)ατ)1p. (2.3)

    Lemma 2.3. (Generalized Young inequality [4]) If a,b0,sp,sq>1 with

    1sp+1sq=1,

    then

    aspsp+bsqsqab. (2.4)

    Lemma 2.4. (See [31]) If x,y>0 and s>1, then

    (xs+ys)1sx+y, (2.5)

    and if 0<s<1, then

    (xs+ys)1sx+y. (2.6)

    In the manuscript, we will operate under the assumption that the considered integrals are presumed to exist. Also, we denote ατ=ατ1,,ατN  and ϖ(τ)=ϖ(τ1,,τN).

    Theorem 3.1. Assume that ηi, ϵiT,ϵi>ηi,i=1,2,,N,0α1, p,s>0,ps>1 such that

    1sp+1sq=1,

    and ϖ,Θ: TNR+ are continuous functions, then

    ϵNηNϵ1η1ϖ1s(τ)Θ1s(τ)ατ(ϵNηNϵ1η1ϖp(τ)ατ)1sp(ϵNηNϵ1η1Θq(τ)ατ)1sq. (3.1)

    Proof. Applying (2.4) with

    a=ϖ1s(ξ)(ϵNηNϵ1η1ϖp(τ)ατ)1sp

    and

    b=Θ1s(ξ)(ϵNηNϵ1η1Θq(τ)ατ)1sq,

    we get

    ϖp(ξ)spϵNηNϵ1η1ϖp(τ)ατ+Θq(ξ)sqϵNηNϵ1η1Θq(τ)ατϖ1s(ξ)Θ1s(ξ)(ϵNηNϵ1η1ϖp(τ)ατ)1sp×(ϵNηNϵ1η1Θq(τ)ατ)1sq. (3.2)

    Integrating (3.2) over ξi from ηi to ϵi,i=1,2,,N, we observe that

    ϵNηNϵ1η1ϖp(ξ)αξspϵNηNϵ1η1ϖp(τ)ατ+ϵNηNϵ1η1Θq(ξ)αξsqϵNηNϵ1η1Θq(τ)ατ(ϵNηNϵ1η1ϖp(τ)ατ)1sp(ϵNηNϵ1η1Θq(τ)ατ)1sq×ϵNηNϵ1η1ϖ1s(ξ)Θ1s(ξ)αξ,

    and then (note that 1sp+1sq=1)

    ϵNηNϵ1η1ϖ1s(τ)Θ1s(τ)ατ(ϵNηNϵ1η1ϖp(τ)ατ)1sp(ϵNηNϵ1η1Θq(τ)ατ)1sq,

    which is (3.1).

    Remark 3.1. If s=1 and N=1, we get the dynamic diamond alpha Hölder's inequality (2.2).

    Remark 3.2. If T=R, ηi, ϵiR,ϵi>ηi,i=1,2,,N,p,s>0,ps>1 such that

    1sp+1sq=1,

    and ϖ,Θ: RNR+ are continuous functions, then

    ϵNηNϵ1η1ϖ1s(τ)Θ1s(τ)dτ(ϵNηNϵ1η1ϖp(τ)dτ)1sp(ϵNηNϵ1η1Θq(τ)dτ)1sq,

    where dτ=dτ1,,dτN.

    Remark 3.3. If T=N, N=1,η, ϵN,ϵ>η,0α1, p,s>0,ps>1 such that

    1sp+1sq=1,

    and ϖ,Θ are positive sequences, then

    ϵ1τ=η[αϖ1s(τ)Θ1s(τ)+(1α)ϖ1s(τ+1)Θ1s(τ+1)](ϵ1τ=ηαϖp(τ)+(1α)ϖp(τ+1))1sp×(ϵ1τ=ηαΘq(τ)+(1α)Θq(τ+1))1sq.

    Example 3.1. If T=R,ϵR, N=1, η=0,sp=3,sq=, ϖ(τ)=τs, and Θ(τ)=τs, then

    ϵηϖ1s(τ)Θ1s(τ)dτ<(ϵηϖp(τ)dτ)1sp(ϵηΘq(τ)dτ)1sq.

    Proof. Note that

    ϵηϖ1s(τ)Θ1s(τ)dτ=ϵ0τ2dτ=τ33|ϵ0=ϵ33. (3.3)

    Using the above assumptions, we observe that

    ϵηϖp(τ)dτ=ϵ0τspdτ=τsp+1sp+1|ϵ0=ϵsp+1sp+1,

    then

    (ϵηϖp(τ)dτ)1sp=(ϵsp+1sp+1)1sp=ϵ1+1sp(sp+1)1sp. (3.4)

    Also, we can get

    ϵηΘq(τ)dτ=ϵ0τsq(τ)dτ=τsq+1sq+1|ϵ0=ϵsq+1sq+1,

    and so

    (ϵηΘq(τ)dτ)1sq=(ϵsq+1sq+1)1sq=ϵ1+1sq(sq+1)1sq. (3.5)

    From (3.4) and (3.5) (note 1sp+1sq=1), we have that

    (ϵηϖp(τ)dτ)1sp(ϵηΘq(τ)dτ)1sq=ϵ1+1sp(sp+1)1spϵ1+1sq(sq+1)1sq=ϵ3(sp+1)1sp(sq+1)1sq. (3.6)

    Since sp=3 and sq=3/2,

    (sp+1)1sp(sq+1)1sq=413(52)23=413(254)13=(25)13=2.9240177<3,

    then

    ϵ3(sp+1)1sp(sq+1)1sq>ϵ33. (3.7)

    From (3.3), (3.6) and (3.7), we see that

    ϵηϖ1s(τ)Θ1s(τ)dτ<(ϵηϖp(τ)dτ)1sp(ϵηΘq(τ)dτ)1sq.

    The proof is complete.

    Corollary 3.1. If ηi, ϵiT,α=1,ϵi>ηi,i=1,2,,N,p,s>0,ps>1 such that

    1sp+1sq=1

    and ϖ,Θ: TNR+, then

    ϵNηNϵ1η1ϖ1s(τ)Θ1s(τ)Δτ1ΔτN(ϵNηNϵ1η1ϖp(τ)Δτ1ΔτN)1sp(ϵNηNϵ1η1Θq(τ)Δτ1ΔτN)1sq. (3.8)

    Corollary 3.2. If ηi, ϵiT,α=0,ϵi>ηi,i=1,2,,N,p,s>0,ps>1 such that

    1sp+1sq=1,

    and ϖ,Θ: TNR+, then

    ϵNηNϵ1η1ϖ1s(τ)Θ1s(τ)τ1τN(ϵNηNϵ1η1ϖp(τ)τ1τN)1sp(ϵNηNϵ1η1Θq(τ)τ1τN)1sq. (3.9)

    Theorem 3.2. Assume that ηi, ϵiT,ϵi>ηi,i=1,2,,N,0α1,p,s>0,ps>1 such that

    1sp+1sq=1,

    and Ξ,Ω: TNR+ are continuous functions. If 0<mΞ/ΩM<, then

    ϵNηNϵ1η1Ξ1ps(τ)Ω1qs(τ)ατM1p2s2m1q2s2ϵNηNϵ1η1Ξ1qs(τ)Ω1ps(τ)ατ. (3.10)

    Proof. Applying (3.1) with ϖ(τ)=Ξ1p(τ) and Θ(τ)=Ω1q(τ), we see that

    ϵNηNϵ1η1Ξ1ps(τ)Ω1qs(τ)ατ(ϵNηNϵ1η1Ξ(τ)ατ)1sp(ϵNηNϵ1η1Ω(τ)ατ)1sq. (3.11)

    Since

    1sp+1sq=1,

    then (3.11) becomes

    ϵNηNϵ1η1Ξ1ps(τ)Ω1qs(τ)ατ(ϵNηNϵ1η1Ξ1ps(τ)Ξ1qs(τ)ατ)1sp×(ϵNηNϵ1η1Ω1ps(τ)Ω1qs(τ)ατ)1sq. (3.12)

    Since

    Ξ1ps(τ)M1psΩ1ps(τ)

    and

    Ω1qs(τ)m1qsΞ1qs(τ),

    we have from (3.12) that

    ϵNηNϵ1η1Ξ1ps(τ)Ω1qs(τ)ατM1p2s2m1q2s2(ϵNηNϵ1η1Ξ1qs(τ)Ω1ps(τ)ατ)1sp×(ϵNηNϵ1η1Ξ1qs(τ)Ω1ps(τ)ατ)1sq=M1p2s2m1q2s2(ϵNηNϵ1η1Ξ1qs(τ)Ω1ps(τ)ατ)1sp+1sq,

    then we have for

    1sp+1sq=1,

    that

    ϵNηNϵ1η1Ξ1ps(τ)Ω1qs(τ)ατM1p2s2m1q2s2ϵNηNϵ1η1Ξ1qs(τ)Ω1ps(τ)ατ,

    which is (3.10).

    Remark 3.4. Take T=R,ηi, ϵiR,ϵi>ηi,i=1,2,,N,p,s>0,ps>1 such that

    1sp+1sq=1

    and Ξ,Ω: RNR+ are continuous functions. If 0<mΞ/ΩM<, then

    ϵNηNϵ1η1Ξ1ps(τ)Ω1qs(τ)dτM1p2s2m1q2s2ϵNηNϵ1η1Ξ1qs(τ)Ω1ps(τ)dτ.

    Remark 3.5. Take T=N,N=1,η, ϵN,ϵ>η,0α1, p,s>0,ps>1 such that

    1sp+1sq=1,

    and Ξ,Ω are positive sequences. If 0<mΞ/ΩM<, then

    ϵ1τ=η[α(Ξ1ps(τ)Ω1qs(τ))+(1α)(Ξ1ps(τ+1)Ω1qs(τ+1))]M1p2s2m1q2s2ϵ1τ=η[α(Ξ1qs(τ)Ω1ps(τ))+(1α)(Ξ1qs(τ+1)Ω1ps(τ+1))].

    Theorem 3.3. Assume that ηi, ϵiT,ϵi>ηi,i=1,2,,N,0α1, sp<0 such that

    1sp+1sq=1,

    and ϕ,λ: TNR+ are continuous functions, then

    ϵNηNϵ1η1ϕ1s(τ)λ1s(τ)ατ(ϵNηNϵ1η1ϕp(τ)ατ)1sp(ϵNηNϵ1η1λq(τ)ατ)1sq. (3.13)

    Proof. Since sp<0, by applying (3.1) with indices

    sP=spsq=1sp>1,sQ=1sq,

    (note that 1sP+1sQ=1), ϖ(τ)=ϕsq(τ) and Θ(τ)=ϕsq(τ)λsq(τ), then

    ϵNηNϵ1η1ϖ1s(τ)Θ1s(τ)ατ(ϵNηNϵ1η1ϖP(τ)ατ)1sP(ϵNηNϵ1η1ΘQ(τ)ατ)1sQ,

    and substituting with values ϖ(τ) and Θ(τ), the last inequality gives us

    ϵNηNϵ1η1ϕq(τ)[ϕq(τ)λq(τ)]ατ(ϵNηNϵ1η1[ϕsq(τ)]psqατ)sqsp(ϵNηNϵ1η1[ϕsq(τ)λsq(τ)]1s2qατ)sq.

    Thus

    ϵNηNϵ1η1λq(τ)ατ(ϵNηNϵ1η1ϕp(τ)ατ)qp(ϵNηNϵ1η1ϕ1s(τ)λ1s(τ)ατ)sq,

    then we have for sp<0 and sq=sp/(sp1)>0 that

    ϵNηNϵ1η1ϕ1s(τ)λ1s(τ)ατ(ϵNηNϵ1η1ϕp(τ)ατ)1sp(ϵNηNϵ1η1λq(τ)ατ)1sq,

    which is (3.13).

    Remark 3.6. If T=R, ηi, ϵiR,ϵi>ηi,i=1,2,,N,0α1, sp<0 such that

    1sp+1sq=1,

    and ϕ,λ: RNR+are continuous functions, then

    ϵNηNϵ1η1ϕ1s(τ)λ1s(τ)dτ(ϵNηNϵ1η1ϕp(τ)dτ)1sp(ϵNηNϵ1η1λq(τ)dτ)1sq.

    Remark 3.7. If T=N, N=1,η, ϵN, ϵ>η,0α1, sp<0, such that

    1sp+1sq=1,

    and ϕ,λ are positive sequences, then

    ϵ1τ=η[α(ϕ1s(τ)λ1s(τ))+(1α)(ϕ1s(τ+1)λ1s(τ+1))](ϵ1τ=η[αϕp(τ)+(1α)ϕp(τ+1)])1sp(ϵ1τ=η[αλq(τ)+(1α)λq(τ+1)])1sq.

    Theorem 3.4. Assume that ηi, ϵiT,ϵi>ηi,i=1,2,,N,0α1, r>u>t>0, and Ξ,Ω: TNR+are continuous functions, then

    (ϵNηNϵ1η1Ξ(τ)Ωu(τ)ατ)rt(ϵNηNϵ1η1Ξ(τ)Ωt(τ)ατ)ru(ϵNηNϵ1η1Ξ(τ)Ωr(τ)ατ)ut. (3.14)

    Proof. Applying (3.1) with

    p=rtru,q=rtut,

    (note that s=1),

    ϖp(τ)=Ξ(τ)Ωt(τ)and Θq(τ)=Ξ(τ)Ωr(τ),

    we see that

    ϵNηNϵ1η1[Ξ(τ)Ωt(τ)]rurt[Ξ(τ)Ωr(τ)]utrtατ(ϵNηNϵ1η1Ξ(τ)Ωt(τ)ατ)rurt(ϵNηNϵ1η1Ξ(τ)Ωr(τ)ατ)utrt,

    and then

    (ϵNηNϵ1η1Ξ(τ)Ωu(τ)ατ)rt(ϵNηNϵ1η1Ξ(τ)Ωt(τ)ατ)ru(ϵNηNϵ1η1Ξ(τ)Ωr(τ)ατ)ut,

    which is (3.14).

    Remark 3.8. If T=R, ηi, ϵiT, ϵi>ηi,i=1,2,,N,r>u>t>0 and Ξ,Ω: RNR+ are continuous functions, then

    (ϵNηNϵ1η1Ξ(τ)Ωu(τ)dτ)rt(ϵNηNϵ1η1Ξ(τ)Ωt(τ)dτ)ru(ϵNηNϵ1η1Ξ(τ)Ωr(τ)dτ)ut.

    Remark 3.9. If T=N, N=1,η, ϵN, ϵ>η,0α1, r > u > t > 0 and \Xi, \; \Omega \ are positive sequences, then

    \begin{align*} &\left( \sum\limits_{\tau = \eta }^{\epsilon -1}\left[ \alpha \left( \Xi \left( \tau \right) \Omega ^{u}\left( \tau \right) \right) +\left( 1-\alpha \right) \left( \Xi \left( \tau +1\right) \Omega ^{u}\left( \tau +1\right) \right) \right] \right) ^{r-t} \\ &\leq \left( \sum\limits_{\tau = \eta }^{\epsilon -1}\left[ \alpha \left( \Xi \left( \tau \right) \Omega ^{t}\left( \tau \right) \right) +\left( 1-\alpha \right) \left( \Xi \left( \tau +1\right) \Omega ^{t}\left( \tau +1\right) \right) \right] \right) ^{r-u} \\ &\times \left( \sum\limits_{\tau = \eta }^{\epsilon -1}\left[ \alpha \left( \Xi \left( \tau \right) \Omega ^{r}\left( \tau \right) \right) +\left( 1-\alpha \right) \left( \Xi \left( \tau +1\right) \Omega ^{r}\left( \tau +1\right) \right) \right] \right) ^{u-t}. \end{align*}

    Theorem 3.5. Assume that \eta _{i}, \ \epsilon _{i}\in \mathbb{T}, \; \epsilon _{i} > \eta _{i}, \; i = 1, 2, \cdots, N, \; 0\leq \alpha \leq 1 , s\geq 1, \; sp > 1 such that

    \frac{1}{sp}+\frac{1}{sq} = 1,

    and \Xi, \; \Omega : \mathbb{T}^{N}\rightarrow \mathbb{R}^{+} are continuous functions, then

    \begin{align} \begin{aligned} \left( \int_{\eta _{N}}^{\epsilon _{N}}\cdots\int_{\eta _{1}}^{\epsilon _{1}}\left( \Xi \left( \mathbf{\tau }\right) +\Omega \left( \mathbf{\tau } \right) \right) ^{p}\mathbf{\lozenge }_{\alpha }\mathbf{\tau }\right) ^{ \frac{1}{sp}} &\leq \left( \int_{\eta _{N}}^{\epsilon _{N}}\cdots\int_{\eta _{1}}^{\epsilon _{1}}\Xi ^{p}\left( \mathbf{\tau }\right) \mathbf{\lozenge }_{\alpha } \mathbf{\tau }\right) ^{\frac{1}{sp}}+\left( \int_{\eta _{N}}^{\epsilon _{N}}\cdots\int_{\eta _{1}}^{\epsilon _{1}}\Omega ^{p}\left( \mathbf{\tau } \right) \mathbf{\lozenge }_{\alpha }\mathbf{\tau }\right) ^{\frac{1}{sp}}. \end{aligned} \end{align} (3.15)

    Proof. Note that

    \begin{eqnarray} \left( \Xi \left( \mathbf{\tau }\right) +\Omega \left( \mathbf{\tau } \right) \right) ^{p} = \left( \Xi \left( \mathbf{\tau }\right) +\Omega \left( \mathbf{\tau } \right) \right) ^{\frac{1}{s}}\left( \Xi \left( \mathbf{\tau }\right) +\Omega \left( \mathbf{\tau }\right) \right) ^{p-\frac{1}{s}}. \end{eqnarray} (3.16)

    Using the inequality (2.5) with replacing x, \ y by \Xi ^{\frac{1}{s }}\left(\mathbf{\tau }\right) and \Omega ^{\frac{1}{s}}\left(\mathbf{ \tau }\right), respectively, we have

    \begin{equation*} \left( \Xi \left( \mathbf{\tau }\right) +\Omega \left( \mathbf{\tau }\right) \right) ^{\frac{1}{s}}\leq \Xi ^{\frac{1}{s}}\left( \mathbf{\tau }\right) +\Omega ^{\frac{1}{s}}\left( \mathbf{\tau }\right) , \end{equation*}

    therefore, (3.16) becomes

    \begin{eqnarray*} \left( \Xi \left( \mathbf{\tau }\right) +\Omega \left( \mathbf{\tau } \right) \right) ^{p} \leq \Xi ^{\frac{1}{s}}\left( \mathbf{\tau }\right) \left( \Xi \left( \mathbf{\tau }\right) +\Omega \left( \mathbf{\tau }\right) \right) ^{\frac{1 }{s}\left( sp-1\right) }+\Omega ^{\frac{1}{s}}\left( \mathbf{\tau }\right) \left( \Xi \left( \mathbf{\tau }\right) +\Omega \left( \mathbf{\tau }\right) \right) ^{\frac{1}{s}\left( sp-1\right) }, \end{eqnarray*}

    then

    \begin{align} \int_{\eta _{N}}^{\epsilon _{N}}\cdots\int_{\eta _{1}}^{\epsilon _{1}}\left( \Xi \left( \mathbf{\tau }\right) +\Omega \left( \mathbf{\tau }\right) \right) ^{p}\mathbf{\lozenge }_{\alpha }\mathbf{\tau } &\leq \int_{\eta _{N}}^{\epsilon _{N}}\cdots\int_{\eta _{1}}^{\epsilon _{1}}\Xi ^{\frac{1}{s}}\left( \mathbf{\tau }\right) \left( \Xi \left( \mathbf{\tau } \right) +\Omega \left( \mathbf{\tau }\right) \right) ^{\frac{1}{s}\left( sp-1\right) }\mathbf{\lozenge }_{\alpha }\mathbf{\tau } \\ & +\int_{\eta _{N}}^{\epsilon _{N}}\cdots\int_{\eta _{1}}^{\epsilon _{1}}\Omega ^{\frac{1}{s}}\left( \mathbf{\tau }\right) \left( \Xi \left( \mathbf{\tau } \right) +\Omega \left( \mathbf{\tau }\right) \right) ^{\frac{1}{s}\left( sp-1\right) }\mathbf{\lozenge }_{\alpha }\mathbf{\tau }. \end{align} (3.17)

    Applying (3.1) on the two terms of the righthand side of (3.17) with indices sp > 1 and \left(sp\right) ^{\ast } = sp/\left(sp-1\right) (note \frac{1}{sp}+\frac{1}{\left(sp\right) ^{\ast }} = 1 ), we obtain

    \begin{align} & \int_{\eta _{N}}^{\epsilon _{N}}\cdots\int_{\eta _{1}}^{\epsilon _{1}}\Xi ^{ \frac{1}{s}}\left( \mathbf{\tau }\right) \left( \Xi \left( \mathbf{\tau } \right) +\Omega \left( \mathbf{\tau }\right) \right) ^{\frac{1}{s}\left( sp-1\right) }\mathbf{\lozenge }_{\alpha }\mathbf{\tau } \\ & \leq \left( \int_{\eta _{N}}^{\epsilon _{N}}\cdots\int_{\eta _{1}}^{\epsilon _{1}}\Xi ^{p}\left( \mathbf{\tau }\right) \mathbf{\lozenge }_{\alpha } \mathbf{\tau }\right) ^{\frac{1}{sp}}\left( \int_{\eta _{N}}^{\epsilon _{N}}\cdots\int_{\eta _{1}}^{\epsilon _{1}}\left( \Xi \left( \mathbf{\tau } \right) +\Omega \left( \mathbf{\tau }\right) \right) ^{p}\mathbf{\lozenge } _{\alpha }\mathbf{\tau }\right) ^{\frac{sp-1}{sp}} \end{align} (3.18)

    and

    \begin{align} \begin{aligned} \int_{\eta _{N}}^{\epsilon _{N}}\cdots\int_{\eta _{1}}^{\epsilon _{1}}\Omega ^{ \frac{1}{s}}\left( \mathbf{\tau }\right) \left( \Xi \left( \mathbf{\tau } \right) +\Omega \left( \mathbf{\tau }\right) \right) ^{\frac{1}{s}\left( sp-1\right) }\mathbf{\lozenge }_{\alpha }\mathbf{\tau } \leq \left( \int_{\eta _{N}}^{\epsilon _{N}}\cdots\int_{\eta _{1}}^{\epsilon _{1}}\Omega ^{p}\left( \mathbf{\tau }\right) \mathbf{\lozenge }_{\alpha } \mathbf{\tau }\right) ^{\frac{1}{sp}}\left( \int_{\eta _{N}}^{\epsilon _{N}}\cdots\int_{\eta _{1}}^{\epsilon _{1}}\left( \Xi \left( \mathbf{\tau } \right) +\Omega \left( \mathbf{\tau }\right) \right) ^{p}\mathbf{\lozenge } _{\alpha }\mathbf{\tau }\right) ^{\frac{sp-1}{sp}}. \end{aligned} \end{align} (3.19)

    Adding (3.18) and (3.19) and substituting into (3.17), we see that

    \begin{align*} \int_{\eta _{N}}^{\epsilon _{N}}\cdots\int_{\eta _{1}}^{\epsilon _{1}}\left( \Xi \left( \mathbf{\tau }\right) +\Omega \left( \mathbf{\tau }\right) \right) ^{p}\mathbf{\lozenge }_{\alpha }\mathbf{\tau } & \leq \left[ \left( \int_{\eta _{N}}^{\epsilon _{N}}\cdots\int_{\eta _{1}}^{\epsilon _{1}}\Xi ^{p}\left( \mathbf{\tau }\right) \mathbf{\lozenge } _{\alpha }\mathbf{\tau }\right) ^{\frac{1}{sp}}+\left( \int_{\eta _{N}}^{\epsilon _{N}}\cdots\int_{\eta _{1}}^{\epsilon _{1}}\Omega ^{p}\left( \mathbf{\tau }\right) \mathbf{\lozenge }_{\alpha }\mathbf{\tau }\right) ^{ \frac{1}{sp}}\right] \\ & \times \left( \int_{\eta _{N}}^{\epsilon _{N}}\cdots\int_{\eta _{1}}^{\epsilon _{1}}\left( \Xi \left( \mathbf{\tau }\right) +\Omega \left( \mathbf{\tau } \right) \right) ^{p}\mathbf{\lozenge }_{\alpha }\mathbf{\tau }\right) ^{ \frac{sp-1}{sp}}, \end{align*}

    thus,

    \begin{align*} \left( \int_{\eta _{N}}^{\epsilon _{N}}\cdots\int_{\eta _{1}}^{\epsilon _{1}}\left( \Xi \left( \mathbf{\tau }\right) +\Omega \left( \mathbf{\tau } \right) \right) ^{p}\mathbf{\lozenge }_{\alpha }\mathbf{\tau }\right) ^{ \frac{1}{sp}} \leq \left( \int_{\eta _{N}}^{\epsilon _{N}}\cdots\int_{\eta _{1}}^{\epsilon _{1}}\Xi ^{p}\left( \mathbf{\tau }\right) \mathbf{\lozenge }_{\alpha } \mathbf{\tau }\right) ^{\frac{1}{sp}}+\left( \int_{\eta _{N}}^{\epsilon _{N}}\cdots\int_{\eta _{1}}^{\epsilon _{1}}\Omega ^{p}\left( \mathbf{\tau } \right) \mathbf{\lozenge }_{\alpha }\mathbf{\tau }\right) ^{\frac{1}{sp}}, \end{align*}

    which is (3.15).

    Remark 3.10. If s = 1 and N = 1, then we get the dynamic Minkowski inequality (2.3).

    In the following, we establish the reversed form of inequality (3.15).

    Theorem 3.6. Assume that \eta _{i}, \ \epsilon _{i}\in \mathbb{T}, \; \epsilon _{i} > \eta _{i}, \; i = 1, 2, \cdots, N, \; 0\leq \alpha \leq 1 , p < 0, \; 0 < s < 1 and \Xi, \; \Omega : \mathbb{T}^{N}\rightarrow \mathbb{R}^{+} are continuous functions, then

    \begin{align} \begin{aligned} \left( \int_{\eta _{N}}^{\epsilon _{N}}\cdots\int_{\eta _{1}}^{\epsilon _{1}}\left( \Xi \left( \mathbf{\tau }\right) +\Omega \left( \mathbf{\tau } \right) \right) ^{p}\mathbf{\lozenge }_{\alpha }\mathbf{\tau }\right) ^{ \frac{1}{sp}} \geq \left( \int_{\eta _{N}}^{\epsilon _{N}}\cdots\int_{\eta _{1}}^{\epsilon _{1}}\Xi ^{p}\left( \mathbf{\tau }\right) \mathbf{\lozenge }_{\alpha } \mathbf{\tau }\right) ^{\frac{1}{sp}}+\left( \int_{\eta _{N}}^{\epsilon _{N}}\cdots\int_{\eta _{1}}^{\epsilon _{1}}\Omega ^{p}\left( \mathbf{\tau } \right) \mathbf{\lozenge }_{\alpha }\mathbf{\tau }\right) ^{\frac{1}{sp}}.\end{aligned} \end{align} (3.20)

    Proof. Applying (2.6) with replacing x, y by \Xi ^{\frac{1}{s}}\left(\mathbf{\tau }\right) and \Omega ^{\frac{1}{s}}\left(\mathbf{\tau } \right) , respectively, we have that

    \begin{equation} \left( \Xi \left( \mathbf{\tau }\right) +\Omega \left( \mathbf{\tau }\right) \right) ^{\frac{1}{s}}\geq \Xi ^{\frac{1}{s}}\left( \mathbf{\tau }\right) +\Omega ^{\frac{1}{s}}\left( \mathbf{\tau }\right) . \end{equation} (3.21)

    Since

    \begin{eqnarray*} \left( \Xi \left( \mathbf{\tau }\right) +\Omega \left( \mathbf{\tau } \right) \right) ^{p} = \left( \Xi \left( \mathbf{\tau }\right) +\Omega \left( \mathbf{\tau } \right) \right) ^{\frac{1}{s}}\left( \Xi \left( \mathbf{\tau }\right) +\Omega \left( \mathbf{\tau }\right) \right) ^{p-\frac{1}{s}}, \end{eqnarray*}

    by using (3.21), we get

    \begin{align*} \left( \Xi \left( \mathbf{\tau }\right) +\Omega \left( \mathbf{\tau } \right) \right) ^{p}\geq \Xi ^{\frac{1}{s}}\left( \mathbf{\tau }\right) \left( \Xi \left( \mathbf{\tau }\right) +\Omega \left( \mathbf{\tau }\right) \right) ^{\frac{1 }{s}\left( sp-1\right) }+\Omega ^{\frac{1}{s}}\left( \mathbf{\tau }\right) \left( \Xi \left( \mathbf{\tau }\right) +\Omega \left( \mathbf{\tau }\right) \right) ^{\frac{1}{s}\left( sp-1\right) }, \end{align*}

    then

    \begin{align} \int_{\eta _{N}}^{\epsilon _{N}}\cdots\int_{\eta _{1}}^{\epsilon _{1}}\left( \Xi \left( \mathbf{\tau }\right) +\Omega \left( \mathbf{\tau }\right) \right) ^{p}\mathbf{\lozenge }_{\alpha }\mathbf{\tau } \geq \int_{\eta _{N}}^{\epsilon _{N}}\cdots\int_{\eta _{1}}^{\epsilon _{1}}\Xi ^{\frac{1}{s}}\left( \mathbf{\tau }\right) \left( \Xi \left( \mathbf{\tau } \right) +\Omega \left( \mathbf{\tau }\right) \right) ^{\frac{1}{s}\left( sp-1\right) }\mathbf{\lozenge }_{\alpha }\mathbf{\tau } \\ +\int_{\eta _{N}}^{\epsilon _{N}}\cdots\int_{\eta _{1}}^{\epsilon _{1}}\Omega ^{\frac{1}{s}}\left( \mathbf{\tau }\right) \left( \Xi \left( \mathbf{\tau } \right) +\Omega \left( \mathbf{\tau }\right) \right) ^{\frac{1}{s}\left( sp-1\right) }\mathbf{\lozenge }_{\alpha }\mathbf{\tau }. \end{align} (3.22)

    Applying (3.13) on \int_{\eta _{N}}^{\epsilon _{N}}\cdots\int_{\eta _{1}}^{\epsilon _{1}}\Xi ^{\frac{1}{s}}\left(\mathbf{\tau }\right) \left(\Xi \left(\mathbf{\tau }\right) +\Omega \left(\mathbf{\tau }\right) \right) ^{\frac{1}{s}\left(sp-1\right) }\mathbf{\lozenge }_{\alpha }\mathbf{ \tau }, with

    \phi \left( \mathbf{\tau }\right) = \Xi \left( \mathbf{\tau } \right) ,\; \; \; \lambda \left( \mathbf{\tau }\right) = \left( \Xi \left( \mathbf{ \tau }\right) +\Omega \left( \mathbf{\tau }\right) \right) ^{sp-1},

    and the indices sp < 0, \ \left(sp\right) ^{\ast } = sp/\left(sp-1\right) (note \frac{1}{sp}+\frac{1}{\left(sp\right) ^{\ast }} = 1), we see that

    \begin{align} & \int_{\eta _{N}}^{\epsilon _{N}}\cdots\int_{\eta _{1}}^{\epsilon _{1}}\Xi ^{ \frac{1}{s}}\left( \mathbf{\tau }\right) \left( \Xi \left( \mathbf{\tau } \right) +\Omega \left( \mathbf{\tau }\right) \right) ^{\frac{1}{s}\left( sp-1\right) }\mathbf{\lozenge }_{\alpha }\mathbf{\tau } \\ & \geq \left( \int_{\eta _{N}}^{\epsilon _{N}}\cdots\int_{\eta _{1}}^{\epsilon _{1}}\Xi ^{p}\left( \mathbf{\tau }\right) \mathbf{\lozenge }_{\alpha } \mathbf{\tau }\right) ^{\frac{1}{sp}}\left( \int_{\eta _{N}}^{\epsilon _{N}}\cdots\int_{\eta _{1}}^{\epsilon _{1}}\left( \Xi \left( \mathbf{\tau } \right) +\Omega \left( \mathbf{\tau }\right) \right) ^{p}\mathbf{\lozenge } _{\alpha }\mathbf{\tau }\right) ^{\frac{sp-1}{sp}}. \end{align} (3.23)

    Again by applying (3.13) on \int_{\eta _{N}}^{\epsilon _{N}}\cdots\int_{\eta _{1}}^{\epsilon _{1}}\Omega ^{\frac{1}{s}}\left(\mathbf{ \tau }\right) \left(\Xi \left(\mathbf{\tau }\right) +\Omega \left(\mathbf{ \tau }\right) \right) ^{\frac{1}{s}\left(sp-1\right) }\mathbf{\lozenge } _{\alpha }\mathbf{\tau } with

    \phi \left( \mathbf{\tau }\right) = \Omega \left( \mathbf{\tau }\right) ,\; \; \; \lambda \left( \mathbf{\tau }\right) = \left( \Xi \left( \mathbf{\tau }\right) +\Omega \left( \mathbf{\tau } \right) \right) ^{sp-1},

    and the indices sp < 0, \ \left(sp\right) ^{\ast } = sp/\left(sp-1\right) (note \frac{1}{sp}+\frac{1}{\left(sp\right) ^{\ast }} = 1), we have that

    \begin{align} \int_{\eta _{N}}^{\epsilon _{N}}\cdots\int_{\eta _{1}}^{\epsilon _{1}}\Omega ^{ \frac{1}{s}}\left( \mathbf{\tau }\right) \left( \Xi \left( \mathbf{\tau } \right) +\Omega \left( \mathbf{\tau }\right) \right) ^{\frac{1}{s}\left( sp-1\right) }\mathbf{\lozenge }_{\alpha }\mathbf{\tau } & \geq \left( \int_{\eta _{N}}^{\epsilon _{N}}\cdots\int_{\eta _{1}}^{\epsilon _{1}}\Omega ^{p}\left( \mathbf{\tau }\right) \mathbf{\lozenge }_{\alpha } \mathbf{\tau }\right) ^{\frac{1}{sp}} \\ & \times \left( \int_{\eta _{N}}^{\epsilon _{N}}\cdots\int_{\eta _{1}}^{\epsilon _{1}}\left( \Xi \left( \mathbf{\tau }\right) +\Omega \left( \mathbf{\tau } \right) \right) ^{p}\mathbf{\lozenge }_{\alpha }\mathbf{\tau }\right) ^{ \frac{sp-1}{sp}}. \end{align} (3.24)

    Substituting (3.23) and (3.24) into (3.22), we see that

    \begin{align*} \int_{\eta _{N}}^{\epsilon _{N}}\cdots\int_{\eta _{1}}^{\epsilon _{1}}\left( \Xi \left( \mathbf{\tau }\right) +\Omega \left( \mathbf{\tau }\right) \right) ^{p}\mathbf{\lozenge }_{\alpha }\mathbf{\tau } & \geq \left[ \left( \int_{\eta _{N}}^{\epsilon _{N}}\cdots\int_{\eta _{1}}^{\epsilon _{1}}\Xi ^{p}\left( \mathbf{\tau }\right) \mathbf{\lozenge } _{\alpha }\mathbf{\tau }\right) ^{\frac{1}{sp}}+\left( \int_{\eta _{N}}^{\epsilon _{N}}\cdots\int_{\eta _{1}}^{\epsilon _{1}}\Omega ^{p}\left( \mathbf{\tau }\right) \mathbf{\lozenge }_{\alpha }\mathbf{\tau }\right) ^{ \frac{1}{sp}}\right] \\ & \times \left( \int_{\eta _{N}}^{\epsilon _{N}}\cdots\int_{\eta _{1}}^{\epsilon _{1}}\left( \Xi \left( \mathbf{\tau }\right) +\Omega \left( \mathbf{\tau } \right) \right) ^{p}\mathbf{\lozenge }_{\alpha }\mathbf{\tau }\right) ^{ \frac{sp-1}{sp}}, \end{align*}

    then

    \begin{align*} \left( \int_{\eta _{N}}^{\epsilon _{N}}\cdots\int_{\eta _{1}}^{\epsilon _{1}}\left( \Xi \left( \mathbf{\tau }\right) +\Omega \left( \mathbf{\tau } \right) \right) ^{p}\mathbf{\lozenge }_{\alpha }\mathbf{\tau }\right) ^{ \frac{1}{sp}} \geq \left( \int_{\eta _{N}}^{\epsilon _{N}}\cdots\int_{\eta _{1}}^{\epsilon _{1}}\Xi ^{p}\left( \mathbf{\tau }\right) \mathbf{\lozenge }_{\alpha } \mathbf{\tau }\right) ^{\frac{1}{sp}}+\left( \int_{\eta _{N}}^{\epsilon _{N}}\cdots\int_{\eta _{1}}^{\epsilon _{1}}\Omega ^{p}\left( \mathbf{\tau } \right) \mathbf{\lozenge }_{\alpha }\mathbf{\tau }\right) ^{\frac{1}{sp}}, \end{align*}

    which is (3.20).

    Remark 3.11. If \mathbb{T = R} , \eta _{i}, \ \epsilon _{i}\in \mathbb{T}, \; \epsilon _{i} > \eta _{i}, \; i = 1, 2, \cdots, N, \; p < 0, \; 0 < s < 1 and \Xi, \; \Omega : \mathbb{ R}^{N}\rightarrow \mathbb{R}^{+}\ are continuous functions, then

    \begin{eqnarray*} \left( \int_{\eta _{N}}^{\epsilon _{N}}\cdots\int_{\eta _{1}}^{\epsilon _{1}}\left( \Xi \left( \mathbf{\tau }\right) +\Omega \left( \mathbf{\tau } \right) \right) ^{p}\mathbf{d\tau }\right) ^{\frac{1}{sp}} \geq \left( \int_{\eta _{N}}^{\epsilon _{N}}\cdots\int_{\eta _{1}}^{\epsilon _{1}}\Xi ^{p}\left( \mathbf{\tau }\right) \mathbf{d\tau }\right) ^{\frac{1}{ sp}}+\left( \int_{\eta _{N}}^{\epsilon _{N}}\cdots\int_{\eta _{1}}^{\epsilon _{1}}\Omega ^{p}\left( \mathbf{\tau }\right) \mathbf{d\tau }\right) ^{\frac{1 }{sp}}. \end{eqnarray*}

    Remark 3.12. If \mathbb{T = N} , N = 1, \eta, \ \epsilon \in \mathbb{N}, \ \epsilon > \eta, \; 0\leq \alpha \leq 1 , p < 0, \; 0 < s < 1 , and \Xi, \; \Omega are positive sequences, then

    \begin{align*} &\left( \sum\limits_{\tau = \eta }^{\epsilon -1}\left[ \alpha \left( \Xi \left( \tau \right) +\Omega \left( \tau \right) \right) ^{p}+\left( 1-\alpha \right) \left( \Xi \left( \tau +1\right) +\Omega \left( \tau +1\right) \right) ^{p}\right] \right) ^{\frac{1}{sp}} \\ &\geq \left( \sum\limits_{\tau = \eta }^{\epsilon -1}\left[ \alpha \Xi ^{p}\left( \tau \right) +\left( 1-\alpha \right) \Xi ^{p}\left( \tau +1\right) \right] \right) ^{\frac{1}{sp}} +\left( \sum\limits_{\tau = \eta }^{\epsilon -1}\alpha \Omega ^{p}\left( \tau \right) +\left( 1-\alpha \right) \Omega ^{p}\left( \tau +1\right) \right) ^{ \frac{1}{sp}}. \end{align*}

    In this paper, we present novel generalizations of Hölder's and Minkowski's dynamic inequalities on diamond alpha time scales. These inequalities give us the inequalities on delta calculus when \alpha = 1 and the inequalities on nabla calculus when \alpha = 0 . Also, we introduced some of the continuous and discrete inequalities as special cases of our results. In addition, we added an example in our results to indicate the work.

    In the future, we will establish some new generalizations of Hölder's and Minkowski's dynamic inequalities on conformable delta fractional time scales. Also, we will prove some new reversed versions of Hölder's and Minkowski's dynamic inequalities on time scales.

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

    The authors extend their appreciation to the Deputyship for Research & Innovation, Ministry of Education in Saudi Arabia for funding this research work through the project number: ISP23-86.

    The authors declare that they have no conflicts of interest.



    [1] H. Edelsbrunner, J. Harer, Persistent homology-a survey, Contemp. Math., 453 (2008), 257–282. https://doi.org/10.1090/conm/453/08802 doi: 10.1090/conm/453/08802
    [2] L. Wasserman, Topological data analysis, Annu. Rev. Stat. Appl., 5 (2018), 501–532. https://doi.org/10.1146/annurev-statistics-031017-100045 doi: 10.1146/annurev-statistics-031017-100045
    [3] G. Carlsson, Topological methods for data modelling, Nat. Rev. Phys., 2 (2020), 697–708. https://doi.org/10.1038/s42254-020-00249-3 doi: 10.1038/s42254-020-00249-3
    [4] E. J. Amézquita, M. Y. Quigley, T. Ophelders, E. Munch, D. H. Chitwood, The shape of things to come: Topological data analysis and biology, from molecules to organisms, Dev. Dyn., 249 (2020), 816–833. https://doi.org/10.1002/dvdy.175 doi: 10.1002/dvdy.175
    [5] A. Bukkuri, N. Andor, I. K. Darcy, Applications of topological data analysis in oncology, Front. Artif. Intell., 4 (2021), 38. https://doi.org/10.3389/frai.2021.659037 doi: 10.3389/frai.2021.659037
    [6] Y. Skaf, R. Laubenbacher, Topological data analysis in biomedicine: A review, J. Biomed. Inform., 130 (2022), 104082. https://doi.org/10.1016/j.jbi.2022.104082 doi: 10.1016/j.jbi.2022.104082
    [7] K. Garside, R. Henderson, I. Makarenko, C. Masoller, Topological data analysis of high resolution diabetic retinopathy images, PLOS ONE, 14 (2019), e0217413. https://doi.org/10.1371/journal.pone.0217413 doi: 10.1371/journal.pone.0217413
    [8] C. Ellis, M. Lesnick, G. Henselman-Petrusek, B. Keller, J. Cohen, Feasibility of topological data analysis for event-related fMRI, Network Neurosci., 3 (2019), 695–706. https://doi.org/10.1162/netn_a_00095 doi: 10.1162/netn_a_00095
    [9] M. McGuirl, A. Volkening, B. Sandstede, Topological data analysis of zebrafish patterns, PNAS, 117 (2020), 5113–5124. https://doi.org/10.1073/pnas.1917763117 doi: 10.1073/pnas.1917763117
    [10] V. Maroulas, C. P. Micucci, F. Nasrin, Bayesian topological learning for classifying the structure of biological networks, preprint, arXiv: 2009.11974.
    [11] D. Cohen-Steiner, H. Edelsbrunner, J. Harer, Stability of persistence diagrams, Discrete Comput. Geom., 37 (2007), 103–120. https://doi.org/10.1007/s00454-006-1276-5 doi: 10.1007/s00454-006-1276-5
    [12] M. J. Jimenez, M. Rucco, P. Vicente-Munuera, P. Gómez-Gálvez, L. M. Escudero, Topological data analysis for self-organization of biological tissues, in International Workshop on Combinatorial Image Analysis, Springer, 2017,229–242.
    [13] L. L. Bonilla, A. Carpio, C. Trenado, Tracking collective cell motion by topological data analysis, PLOS Comput. Biol., 16 (2020), e1008407. https://doi.org/10.1371/journal.pcbi.1008407 doi: 10.1371/journal.pcbi.1008407
    [14] B. Lin, Topological data analysis in time series: Temporal filtration and application to single-cell genomics, preprint, arXiv: 2204.14048.
    [15] D. Cohen-Steiner, H. Edelsbrunner, D. Morozov, Vines and vineyards by updating persistence in linear time, in Proceedings of the twenty-second annual symposium on Computational geometry, ACM, (2006), 119–126.
    [16] A. Hickok, D. Needell, M. A. Porter, Analysis of spatiotemporal anomalies using persistent homology: case studies with COVID-19 data, preprint, arXiv: 2107.09188.
    [17] C. M. Topaz, L. Ziegelmeier, T. Halverson, Topological data analysis of biological aggregation models, PloS ONE, 10 (2015), e0126383. https://doi.org/10.1371/journal.pone.0126383 doi: 10.1371/journal.pone.0126383
    [18] M. Ulmer, L. Ziegelmeier, C. M. Topaz, A topological approach to selecting models of biological experiments, PloS ONE, 14 (2019), e0213679. https://doi.org/10.1371/journal.pone.0213679 doi: 10.1371/journal.pone.0213679
    [19] M. V. Ciocanel, R. Juenemann, A. T. Dawes, S. A. McKinley, Topological data analysis approaches to uncovering the timing of ring structure onset in filamentous networks, Bull. Math. Biol., 83 (2021), 1–25. https://doi.org/10.1007/s11538-020-00847-3 doi: 10.1007/s11538-020-00847-3
    [20] K. Popov, J. Komianos, G. A. Papoian, MEDYAN: mechanochemical simulations of contraction and polarity alignment in actomyosin networks, PLoS Comput. Biol., 12 (2016), e1004877. https://doi.org/10.1371/journal.pcbi.1004877 doi: 10.1371/journal.pcbi.1004877
    [21] C. A. Mandato, W. M. Bement, Contraction and polymerization cooperate to assemble and close actomyosin rings around Xenopus oocyte wounds, J. Cell Biol., 154 (2001), 785–798. https://doi.org/10.1083/jcb.200103105 doi: 10.1083/jcb.200103105
    [22] R. D. Mortensen, R. P. Moore, S. M. Fogerson, H. Y. Chiou, C. V. Obinero, N. K. Prabhu, et al., Identifying genetic players in cell sheet morphogenesis using a Drosophila deficiency screen for genes on chromosome 2R involved in dorsal closure, G3 Genes Genomes Genetics, 8 (2018), 2361–2387. https://doi.org/10.1534/g3.118.200233 doi: 10.1534/g3.118.200233
    [23] M. V. Ciocanel, A. Chandrasekaran, C. Mager, Q. Ni, G. A. Papoian, A. Dawes, Simulated actin reorganization mediated by motor proteins, PLoS Comput. Biol., 18 (2022), e1010026. https://doi.org/10.1371/journal.pcbi.1010026 doi: 10.1371/journal.pcbi.1010026
    [24] H. A. Benink, W. M. Bement, Concentric zones of active Rhoa and Cdc42 around single cell wounds, J. Cell Biol., 168 (2005), 429–439. https://doi.org/10.1083/jcb.200411109 doi: 10.1083/jcb.200411109
    [25] R. D. Mortensen, R. P. Moore, S. M. Fogerson, H. Y. Chiou, C. V. Obinero, N. K. Prabhu, et al., Supplemental material for Mortensen et al., 2018, GSA J., 2018. https://doi.org/10.25387/g3.6207470.v2 doi: 10.25387/g3.6207470.v2
    [26] D. Legland, I. Arganda-Carreras, P. Andrey, MorphoLibJ: integrated library and plugins for mathematical morphology with ImageJ, Bioinformatics, 32 (2016), 3532–3534, https://doi.org/10.1093/bioinformatics/btw413 doi: 10.1093/bioinformatics/btw413
    [27] A. Clark, Pillow (pil fork) documentation, 2015, Available from: https://pillow.readthedocs.io/en/stable/.
    [28] C. Tralie, N. Saul, R. Bar-On, Ripser.py: A lean persistent homology library for Python, J. Open Source Software, 3 (2018), 925, https://doi.org/10.21105/joss.00925 doi: 10.21105/joss.00925
    [29] U. Bauer, Ripser: efficient computation of Vietoris-Rips persistence barcodes, J. Appl. Comput. Topol., 5 (2021), 391–423, https://doi.org/10.1007/s41468-021-00071-5 doi: 10.1007/s41468-021-00071-5
    [30] J. T. Nardini, B. J. Stolz, K. B. Flores, H. A. Harrington, H. M. Byrne, Topological data analysis distinguishes parameter regimes in the Anderson-Chaplain model of angiogenesis, PLOS Comput. Biol., 17 (2021), e1009094. https://doi.org/10.1371/journal.pcbi.1009094 doi: 10.1371/journal.pcbi.1009094
    [31] J. J. Berwald, J. M. Gottlieb, E. Munch, Computing Wasserstein distance for persistence diagrams on a quantum computer, preprint, arXiv: 1809.06433.
    [32] R. Ghrist, Barcodes: the persistent topology of data, Bull. Am. Math. Soc., 45 (2008), 61–75. https://doi.org/10.1090/S0273-0979-07-01191-3 doi: 10.1090/S0273-0979-07-01191-3
    [33] GitHub, Code for "Topological data analysis distinguishes parameter regimes in the Anderson-Chaplain model of angiogenesis", Available from: https://github.com/johnnardini/Angio_TDA.
    [34] D. Cohen-Steiner, H. Edelsbrunner, J. Harer, Y. Mileyko, Lipschitz functions have l_{p}-stable persistence, Found. Comput. Math., 10 (2010), 127–139. https://doi.org/10.1007/s10208-010-9060-6 doi: 10.1007/s10208-010-9060-6
    [35] C. Tralie, Persim Package in Python, 2021. Available from: https://persim.scikit-tda.org/en/latest/reference/index.html.
    [36] GitHub, Sample code for image analysis and construction of significant topological paths corresponding to the time evolution of 1-dimensional holes (actin-myosin ring channels) in point cloud or binary image datasets, 2022. Available from: https://github.com/veronica-ciocanel/TDA_actomyosin/.
    [37] B. Stolz, H. Harrington, M. Porter, Persistent homology of time-dependent functional networks constructed from coupled time series, Chaos, 27 (2017), 047410. https://doi.org/10.1063/1.4978997 doi: 10.1063/1.4978997
    [38] M. Feng, M. A. Porter, Persistent homology of geospatial data: A case study with voting, SIAM Rev., 63 (2021), 67–99. https://doi.org/10.1137/19M1241519 doi: 10.1137/19M1241519
    [39] B. T. Fasy, F. Lecci, A. Rinaldo, L. Wasserman, S. Balakrishnan, A. Singh, Confidence sets for persistence diagrams, Ann. Stat., 42 (2014), 2301–2339. https://doi.org/10.1214/14-AOS1252 doi: 10.1214/14-AOS1252
    [40] F. Chazal, B. T. Fasy, F. Lecci, A. Rinaldo, A. Singh, L. Wasserman, On the bootstrap for persistence diagrams and landscapes, preprint, arXiv: 1311.0376.
    [41] F. Chazal, B. Fasy, F. Lecci, B. Michel, A. Rinaldo, A. Rinaldo, et al., Robust topological inference: Distance to a measure and kernel distance, J. Mach. Learn. Res., 18 (2017), 5845–5884. https://doi.org/10.48550/arXiv.1412.7197 doi: 10.48550/arXiv.1412.7197
    [42] O. Bobrowski, M. Kahle, P. Skraba, Maximally persistent cycles in random geometric complexes, Ann. Appl. Probab., 27 (2017), 2032–2060. https://doi.org/10.1214/16-AAP1232 doi: 10.1214/16-AAP1232
    [43] O. Bobrowski, M. Kahle, Topology of random geometric complexes: a survey, J. Appl. Comput. Topol., 1 (2018), 331–364. https://doi.org/10.1007/s41468-017-0010-0 doi: 10.1007/s41468-017-0010-0
    [44] N. Chenavier, C. Hirsch, Extremal lifetimes of persistent cycles, Extremes, 25 (2022), 299–330. https://doi.org/10.1007/s10687-021-00430-6 doi: 10.1007/s10687-021-00430-6
    [45] C. Schwayer, M. Sikora, J. Slováková, R. Kardos, C. P. Heisenberg, Actin rings of power, Dev. Cell, 37 (2016), 493–506. https://doi.org/10.1016/j.devcel.2016.05.024 doi: 10.1016/j.devcel.2016.05.024
    [46] R. P. Moore, S. M. Fogerson, U. S. Tulu, J. W. Yu, A. H. Cox, M. A. Sican, et al., Super-resolution microscopy reveals actomyosin dynamics in medioapical arrays, Mol. Biol. Cell., 11 (2022), ar94. https://doi.org/10.1091/mbc.E21-11-0537 doi: 10.1091/mbc.E21-11-0537
    [47] Z. Zhang, Y. Nishimura, P. Kanchanawong, Extracting microtubule networks from superresolution single-molecule localization microscopy data, Mol. Biol. Cell., 28 (2017), 333–345. https://doi.org/10.1091/mbc.E16-06-0421 doi: 10.1091/mbc.E16-06-0421
    [48] D. A. Flormann, M. Schu, E. Terriac, D. Thalla, L. Kainka, M. Koch, et al., A novel universal algorithm for filament network tracing and cytoskeleton analysis, FASEB J., 35 (2021), e21582. https://doi.org/10.1096/fj.202100048R doi: 10.1096/fj.202100048R
    [49] D. Haertter, X. Wang, S. M. Fogerson, N. Ramkumar, J. M. Crawford, K. D. Poss, et al., DeepProjection: Rapid and structure-specific projections of tissue sheets embedded in 3D microscopy stacks using deep learning, bioRxiv, 2021. https://doi.org/10.1101/2021.11.17.468809 doi: 10.1101/2021.11.17.468809
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