In this paper we discuss fractional integrals and fractional derivatives of a function with respect to another function. We present some fundamental properties for both types of fractional operators, such as Taylor's theorem, Leibniz and semigroup rules. We also provide a numerical tool to deal with these operators, by approximating them with a sum involving integer-order derivatives.

This paper provides sufficient conditions to guarantee the existence and multiplicity of positive solutions for a singular semipositone fractional equation, subject to integral boundary value condition and parametric dependence $\mu $. Our approach relies on Krasnoselskii's fixed point theorem. Some examples are presented to illustrate our results.

Recently, there has been a variety of intriguing discoveries regarding the symbolic computation of series containing central binomial coefficients and harmonic-type numbers. In this article, we present a vast generalization of the recently discovered harmonic summation formula $$\sum _{n=1}^{\infty } \binom {2n}{n}^{2} \frac {H_{n}}{32^{n}} = \frac {\Gamma^2\bigl(\frac {1}{4} )}{4 \sqrt {\pi }} ( 1 - \frac {4 \ln 2}{\pi } ) $$ through creative applications of an integration method that we had previously introduced and applied to prove new Ramanujan-like formulas for $\frac {1}{\pi }$. We provide explicit closed-form expressions for natural variants of the above series that cannot be evaluated by state-of-the-art computer algebra systems, such as the elegant symbolic evaluation $$ \sum _{n=1}^{\infty } \frac { \binom {2 n}{n}^2 H_n}{32^n (n + 1)} = 8-\frac {2 \Gamma^2\bigl(\frac {1} {4} )}{\pi ^{3/2}}-\frac {4 \pi ^{3/2}+16 \sqrt {\pi } \ln 2}{\Gamma^2\bigl(\frac {1}{4})} $$ introduced in our present paper. We also discuss some related problems concerning binomial series containing alternating harmonic numbers. We also introduce a new class of harmonic summations for Catalan's constant $G$ and $\frac {1}{\pi }$ such as the series $$ \sum _{n=1}^{\infty } \frac { \binom {2 n}{n}^2 H_n}{ 16^{n} (n+1)^2} = 16+\frac {32 G-64 \ln 2}{\pi }-16 \ln 2 $$ which we prove through a variation of our previous integration method for constructing $\frac {1}{\pi }$ series.

A conjecture of Mordell states that if $p$ is a prime and $p$ is congruent to $3$ modulo $4$, then $p$ does not divide $y$ where $(x,y)$ is the fundamental solution to $x^{2}-py^{2}=1$. The conjecture has been verified for primes not exceeding $10^{7}$. In this article, we show that Mordell's conjecture holds for four conjecturally infinite families of primes.

Triple product sums on the generalized Fibonomial and Lucanomial coefficients are evaluated in closed forms by means of Bailey's summation formulae for two terminating well-poised $_3\phi _2$-series.

We prove that a class of convexity-type results for sequential fractional delta differences is uniformly sharp. More precisely, we consider the sequential difference $\Delta _{1-\mu +a}^{\nu }\Delta _{a}^{\mu }f(t)$, for $t\in \mathbb {N}_{3+a-\mu -\nu }$, and demonstrate that there is a strong connection between the sign of this function and the convexity or concavity of $f$ if and only if the pair $(\mu ,\nu )$ lives in a particular subregion of the parameter space $(0,1)\times (1,2)$.

A \textit {convex polynomial} is a convex combination of the monomials $\{1, x, x^2, \ldots \}$. This paper establishes that the convex polynomials on $\mathbb R$ are dense in $L^p(\mu )$ and weak$^*$ dense in $L^\infty (\mu )$ whenever $\mu $ is a compactly supported regular Borel measure on $\mathbb {R}$ and $\mu ([-1,\infty )) = 0$. It is also shown that the convex polynomials are norm dense in $C(K)$ precisely when $K \cap [-1, \infty ) = \emptyset $, where $K$ is a compact subset of the real line. Moreover, the closure of the convex polynomials on $[-1,b]$ is shown to be the functions that have a convex power series representation.

A continuous linear operator $T$ on a locally convex space $X$ is \textit {convex-cyclic} if there is a vector $x \in X$ such that the convex hull of the orbit of $x$ is dense in $X$. The previous results are used to characterize which multiplication operators on various real Banach spaces are convex-cyclic. Also, it is shown for certain multiplication operators that every nonempty closed invariant convex set is a closed invariant subspace.

In a 1990 paper, W. May studied the question of when isomorphisms of the endomorphism rings of mixed modules are necessarily induced by isomorphisms of the underlying modules. In so doing he introduced a class of mixed modules over a complete discrete valuation domain; we later renamed these modules after their inventor. The class of May modules of countable torsion-free rank is particularly important. A decomposition theorem is established for such modules. The modules in this class are characterized in several ways. Finally, an example is constructed showing that several of these ideas do not extend to May modules of uncountable torsion-free rank.

We study domains in which the set of boundary orbit accumulation points contains a relatively open subset of the boundary. The main result, in complex dimension 2, is that a domain with such a property must be either the ball or the bidisc.

The main motive here is to establish the controllability results for a Volterra integro dynamic inclusion with the impulsive condition on time scales. Fixed point theorem for multivalued maps due to Dhage has been used to establish the main results. Moreover, we give the controllability results for the taken problem with nonlocal effect. Also, an example for two different time scales is given to validate the utilization of these analytical outcomes.

We investigate Condition E and the basic regularity conditions for the two-direction multiscaling functions $\boldsymbol {\phi }$, and then investigate approximation order of $\boldsymbol {\phi }$. By investigating a structure of approximation vectors, we are able to find simple and efficient criteria for Condition E, the basic regularity conditions, and the approximation order of $\boldsymbol {\phi }$. Examples for illustrating the general theory are given.

The paper is a continuation of a recent paper of ours, which considers the normality for a family of holomorphic functions concerning the total derivative in several complex variables. Here we extend the famous Zalcman's lemma concerning the total derivative from one complex variable to several complex variables, and obtain some normality criteria where complex values are shared by every function from the family and its $k$-th total derivative which may be seen as some generalizations of normality for holomorphic functions sharing values in one complex variable. In addition, the case of sharing analytic function by every function from the family and its $k$-th total derivative is also verified.

In this paper we investigate some colored notions of graph homomorphisms. We compare three different notions of colored homomorphisms and determine the number of such homomorphisms between several classes of graphs. More specifically, over all possible colorings of paths, we consider the colorings that yields the largest and smallest number of colored homomorphisms.

In earlier work, we described a relation between the parameters associated with multiplier systems of complex weight on the discrete Hecke groups $G_{\lambda }$ when $1 \leq \lambda \lt 2$, and consequently showed that parabolic Poincare series of nonreal weight on the modular group are not absolutely convergent anywhere. In the current paper we establish an analogous divergence result for all Hecke groups with $\lambda > 2$.

We provide examples to shed some light on the regularity-type properties of the boundary spectrum of a Banach algebra element. In particular, we show that the boundary spectrum is generated by a set that is neither an upper nor a lower semiregularity.

Given $ p, q \in \mathbb {R} $, we generalize the classical Dunkl--Williams inequality for $p$-angular and $q$-angular distances in inner product spaces. We extend the Hile inequality for arbitrary $p$-angular and $q$-angular distances and study some geometric aspects of a generalization of Dunkl--Williams inequality. Introducing power refinements, we show significant power refinements of the generalized Dunkl--Williams inequality under some mild conditions. Among other things, we give new characterizations of inner product spaces with regard to $p$-angular and $q$-angular distances. In particular, we prove that if $ p, q, r \in \mathbb {R} $, $ q \neq 0$ and $ 0\leq p/q\lt 1 $, then $ X $ is an inner product space if and only if for every $ x, y \in X \setminus \{0\} $, $$ \bigl \lVert \lVert x \rVert ^{p-1} x - \lVert y \rVert ^{p-1}y \bigr \rVert \leq \frac {2^{1/r}\bigl \lVert \lVert x \rVert ^{q-1} x - \lVert y \rVert ^{q-1}y \bigr \rVert }{\bigl [\lVert x \rVert ^{r(q-p)} + \lVert y \rVert ^{r(q-p)}\bigr ]^{\frac {1}{r}}}. $$

We investigate the problem of finding pairs of Pythagorean triangles $(a, b, c), (A, B, C)$, with given cathetus ratios $A/a,\, B/b$. In particular, we prove that there are infinitely many essentially different (non-similar) pairs of Pythagorean triangles $(a, b, c), (A, B, C)$ satisfying given proportions, provided that $Aa\neq Bb$.

$\tau $-rigid modules are essential in the $\tau $-tilting theory introduced by Adachi, Iyama and Reiten. In this paper, we give equivalent conditions for Iwanaga-Gorenstein algebras with self-injective dimension at most one in terms of $\tau $-rigid modules. We show that every indecomposable module over iterated tilted algebras of Dynkin type is $\tau $-rigid. Finally, we give a $\tau $-tilting theorem on homological dimension which is an analog to that of classical tilting modules.