38 problems

Let $u=u(t,x)$ be defined for $(t,x)in\mathbb{R}_+\times\mathbb{R}$, let $\beta\in(0,1)$ and $\alpha\in(0,2)$, and let $\dot{W}$ denote space-time white noise. Consider the space-t…

Feigenbaum-number conjecture. The Feigenbaum number exists for fractional maps and fractional difference maps, and has the same value as for regular maps.

The parameters are related by $Q=2q-1$, and $\frac{2}{\alpha+1-Q}$ is the scaling rate obtained from the general formula at $k=1$. Fractional nonlinear Fokker–Planck scaling-rate c…

Order-independence conjecture. The asymptotic behavior of $J_{\omega}^{[p]}$ as its argument tends to infinity is independent of $\omega$ and has decay order $-p/2$.

Taylor-centered extension conjecture. The Taylor-centered extension should yield smoother dependence on the order $r$ for monomials.

Conjecture. The use of $q$-fractional integration by parts may offer a viable path for deriving alternative $q$-Erdélyi-type integrals for the function $\Phi_K$.

Let $0<\alpha\leq\frac{1}{2}$ and define … Here $E_{\rho,\beta}$ denotes the Mittag-Leffler function with parameters $\rho$ and $\beta$. Complete-monotonicity conjecture. The funct…

Let $calpha\in(1,2)$, and let $calpsi(t)$ be the function defined in the paper by equation (psidf). Zero-free conjecture. The function $calpsi(t)$ has no zeros on $(0,\infty)$, nam…

Let $calpha ainleft(frac{3}{5}, 1right)$. The Mittag-Leffler function $E{alpha,1}(z)$ is defined by its standard power series, and the imaginary axis is the set…

Deconvolution conjecture. A conjecture on deconvolution is raised that would permit completing the proposed theory.

Smallest-zero conjecture for $E_{\alpha,1}$. For any $\alpha\in(1,2]$, $E_{\alpha,1}(-z^\alpha)>0$ for all $z\in[0,1.559]$. Moreover, as $\alpha\to1+$,

Smallest-zero conjecture for $E_{\alpha,2}$. For each $\alpha\in[\alpha_0,2]$, the equation

Smallest-zero conjecture for $E_{\alpha,\alpha}$. For any $\alpha\in(1,2]$, $E_{\alpha,\alpha}(-z^\alpha)>0$ for all $z\in[0,2.93785]$. Moreover, as $\alpha\to1+$,

Zeros and analyticity conjecture. The functions satisfy

Let $\mathbb{D}_p^\alpha$ denote the proposed fractional discrete exterior calculus (FDEC) operators, and let the mesh size tend to zero as the number of subdivisions increases. As…

The preceding Fourier-series calculations define the half-order fractional derivative $\frac{\mathrm{d}^{+\frac{1}{2}}}{\mathrm{d}x^{+\frac{1}{2}}}f(x)$, whose truncated Fourier se…

Mainardi's conjecture. For any $t>0$ and fixed $\delta$, $0<\delta<1$, we get the conjectured formula from the source.

Let $x^*$ be a fixed point of a nonlinear map $x_{n+1}=f(x_n)$, and let $f'(x^*)$ denote the derivative of $f$ at that fixed point. Let $a_c(\alpha)$ be the critical lower stabilit…

Let $L$ be either $L_1(0,1)$ or $L_2(0,1)$. For $\alpha\geq 0$, let $I^\alpha$ be a family of bounded linear operators on $L$ satisfying the interpolation conditions … the index la…

Let $n$ and $s$ be the dimension and fractional-order parameters, let $\boldsymbol{x},\boldsymbol{y}$ be points in the underlying Euclidean space, and let $\lambda$ be the temperin…

Let $\mathbb{R}$ denote the real line, let $0<\alpha<1$, and let $p\neq2$. Denote by ${^{\pm}}{W}{^{\alpha,p}}(\mathbb{R})$, $\widehat{W}^{\alpha,p}(\mathbb{R})$, and…

Parameter-relation conjecture. There must be a relation between the parameters such that the fractional relaxation function is completely monotone, as occurs when the derivative or…

Weierstrass-type conjecture. The Fourier series above represents a function of the Weierstrass type that is continuous but nowhere differentiable.

Fractional-friction replacement conjecture. In many practical cases and inside a time interval, a fractional operator might be replaced by the frictional additional term on the cla…

Extension conjecture. The method used to prove the displayed inequalities should also yield analogous inequalities for other fractional derivatives.