Well‐posedness of fractional integro‐differential equations in vector‐valued functional spaces

We study the well‐posedness of the fractional differential equations with infinite delay on Lebesgue–Bochner spaces and Besov spaces , where A and B are closed linear operators on a Banach space X satisfying ,  and . Under suitable assumptions on the kernels a and b, we completely characterize the well‐posedness of in the above vector‐valued function spaces on by using known operator‐valued Fourier multiplier theorems. We also give concrete examples where our abstract results may be applied.     

A new first-principles approach for the catenary

This paper reports on findings relating to catenaries since the publication in Expositiones Mathematicae of Denzler and Hinz’s pioneering 1999 paper, Catenaria Vera – the True Catenary. New governing differential equations and explicit solutions are derived for the catenary in positive and negative radial potentials with physical constants incorporated in the derivations. In keeping with precedent by Denzler and Hinz, a measure of historical perspective is offered as homage to Gottfried Wilhelm Leibniz, Christiaan Huygens and Johann Bernoulli, the original first-solvers of the catenary.     

A calibrated model of local food system of Hawaii: What are the economic implications of the state's food goals and policies?

We develop a fully calibrated positive mathematical programming model for Hawaii's local food systems—which captures the production and the consumer sides of the market. Then we use the model to assess two proposed policies—a general excise tax (GET) exemption on locally produced foods, and an investment in agricultural infrastructure. For the GET exemption case, our results indicate an economic gain of $118 per $100 cost. On the other hand, an investment in 1,200 acres of land injected to support local production may generate an economic gain of up to $357 per $100 annual cost of the investment. However, these estimates should be considered preliminary, and thus viewed with caution. Although the model is used to capture Hawaii's local food systems, we believe that our model is generalizable and can be adopted by other economies to assess their respective food localization policies. Recommendations for Resource Managers

• Local food policies need to be based on quantitative terms instead of mere armchair speculation because often their potential outcomes may vary significantly.

• The current modeling framework demonstrates the potential of using positive mathematical programming (PMP) in capturing the intricacies of local food systems. However, this exploratory exercise should be viewed as preliminary in nature and the ensuing results were taken with caution because many important factors such as labor availability may have been left out.

• Thus, further model refinements are necessary to better capture the complexities of local food systems such as farm heterogeneity, availability of farm labor, water availability, and interisland transportation of farm products in the case of Hawaii.

     

Calcite Deposits Differentiate Cave from House‐Farmed Edible Bird's Nest as shown by SEM‐EDX,ATR‐FTIR and Raman Microspectroscopy

The difference between the swiftlet white edible bird's nest from limestone caves versus house‐farmed ones, especially in response to high temperature and stewing time in water where the latter type would disintegrate readily, has been a puzzle for a long time. We show that edible bird's nests from the limestone caves have calcite deposits on the surface of the nest cement as compared to the house‐farmed nests which are built by swiftlets on timber planks. The micron and sub‐micron calcite particles are seen in SEM‐EDX and further characterized by ATR‐FTIR and Raman microspectroscopy. The calcite deposits make it possible for the cave nest to retain a gelatinous texture under the harsh retort conditions at 121 °C for 20 mins in commercial bottling. We show that house‐farmed nests can be soaked in CaCl₂(aq) followed by rinsing with Na₂CO₃(aq) to grow the same calcite deposits on the nest cement with the same characteristic as cave nests. Therefore, there should no longer be a need to harvest cave nests, and we can better conserve the dwindling population and natural habitats of cave swiftlets.     

Gaps in the number of generators of monomial Togliatti systems

Let $I_{d,n}?k[x_0,?,x_n]$ be a minimal monomial Togliatti system of forms of degree d. In [4], Mezzetti and Miró-Roig proved that the minimal number of generators $\mu (I_{d,n})$ of $I_{d,n}$ lies in the interval $[2n+1,\left(\begin{array}{c}n+d?1\\ n?1\end{array}\right)]$. In this paper, we prove that for $n\ge 4$ and $d\ge 3$, the integer values in $[2n+3,3n?1]$ cannot be realized as the number of minimal generators of a minimal monomial Togliatti system. We classify minimal monomial Togliatti systems $I_{d,n}?k[x_0,?,x_n]$ of forms of degree d with $\mu (I_{d,n})=2n+2$ or 3n (i.e. with the minimal number of generators reaching the border of the non-existence interval). Finally, we prove that for $n=4$, $d\ge 3$ and $\mu \in [9,\left(\begin{array}{c}d+3\\ 3\end{array}\right)]?\{11\}$ there exists a minimal monomial Togliatti system $I_{d,n}?k[x_0,?,x_n]$ of forms of degree d with $\mu (I_{n,d})=\mu$.     

On actions of Drinfel'd doubles on finite dimensional algebras

Let q be an nth root of unity for $n>2$ and let $T_n(q)$ be the Taft (Hopf) algebra of dimension $n^2$. In 2001, Susan Montgomery and Hans-Jürgen Schneider classified all non-trivial $T_n(q)$-module algebra structures on an n-dimensional associative algebra A. They further showed that each such module structure extends uniquely to make A a module algebra over the Drinfel'd double of $T_n(q)$. We explore what it is about the Taft algebras that leads to this uniqueness, by examining actions of (the Drinfel'd double of) Hopf algebras H “close” to the Taft algebras on finite-dimensional algebras analogous to A above. Such Hopf algebras H include the Sweedler (Hopf) algebra of dimension 4, bosonizations of quantum linear spaces, and the Frobenius–Lusztig kernel $u_q({\mathfrak{sl}}_2)$.     

On the Lipschitz equivalence of self‐affine sets

Recently Lipschitz equivalence of self‐similar sets on has been studied extensively in the literature. However for self‐affine sets the problem is more awkward and there are very few results. In this paper, we introduce a w‐Lipschitz equivalence by repacing the Euclidean norm with a pseudo‐norm w. Under the open set condition, we prove that any two totally disconnected integral self‐affine sets with a common matrix are w‐Lipschitz equivalent if and only if their digit sets have equal cardinality. The main methods used are the technique of pseudo‐norm and Gromov hyperbolic graph theory on iterated function systems.     

Bounded sets structure of CpX and quasi‐(DF)‐spaces

For wide classes of locally convex spaces, in particular, for the space $C_p(X)$ of continuous real‐valued functions on a Tychonoff space X equipped with the pointwise topology, we characterize the existence of a fundamental bounded resolution (i.e., an increasing family of bounded sets indexed by the irrationals which swallows the bounded sets). These facts together with some results from Grothendieck's theory of $(DF)$‐spaces have led us to introduce quasi‐ $(DF)$‐spaces, a class of locally convex spaces containing $(DF)$‐spaces that preserves subspaces, countable direct sums and countable products. Regular $(LM)$‐spaces as well as their strong duals are quasi‐ $(DF)$‐spaces. Hence the space of distributions $D^{\prime }(\mathrm{\Omega })$ provides a concrete example of a quasi‐ $(DF)$‐space not being a $(DF)$‐space. We show that $C_p(X)$ has a fundamental bounded resolution if and only if $C_p(X)$ is a quasi‐ $(DF)$‐space if and only if the strong dual of $C_p(X)$ is a quasi‐ $(DF)$‐space if and only if X is countable. If X is metrizable, then $C_k(X)$ is a quasi‐ $(DF)$‐space if and only if X is a σ‐compact Polish space.     

Mixing time of an unaligned Gibbs sampler on the square

The paper concerns a particular example of the Gibbs sampler and its mixing efficiency. Coordinates of a point are rerandomized in the unit square ${[0,1]}^2$ to approach a stationary distribution with density proportional to $exp(?A^2{(u?v)}^2)$ for $(u,v)\in {[0,1]}^2$ with some large parameter $A$.Diaconis conjectured the mixing time of this process to be $O\left(A^2\right)$ which we confirm in this paper. This improves on the currently known $O(exp\left(A^2\right))$ estimate.