Analysis of lettuce evapotranspiration across soil water

In this study, we evaluated the lettuce accumulated evapotranspiration (ET) across four levels of irrigation, using a nonlinear mixed‐effects model. The plants were grown in protected environments and monitored over 23 consecutive days. When the moisture of the substrate in the vessels reached 50% of their maximum retention capacity, the water levels were elevated according to four treatments: , , , and . The model appeared to provide a good fit to the data and showed that the estimates of the maximum amount of accumulated ET were similar for the three treatments with soil water deficit and lower for . The results of the study supported the idea that optimization of the ET of lettuce plants could be achieved through irrigation with deficit, also indicating that the economical use of water was the most efficient way to boost agricultural production. Recommendations for resource managers

• The continued growth of the world population will result in a decrease of quality and availability of water and also an increase in demand for food. Therefore, sustainability will depend on high agricultural productivity with rational use of water.
• Considered the most efficient technology for boosting agricultural productivity, irrigation is also the largest water consumer in the world. With any kind of irrigation in the vegetable production area, the water intake for the plants must be treated with great caution.
• Both a lack or excess in water can decrease plant productivity. The amount of water available in the soil should be enough to maximize production. It is shown that high levels of irrigation are not necessary for this.
• It is important to balance the amount of irrigated water for an optimal level, in order that the production has its the production is maximized and valuable water resources are not wasted.

     

Catch‐to‐stock dependence: The case of small pelagic fishery with bounded harvesting effort

Biologic characteristics of schooling fish species explain why the rates of harvesting in pelagic fisheries are not proportional to the existent stock size and may exhibit no variation between the periods of fish abundance and scarcity. Therefore, the stock‐dependent nonlinearities in catchability must be reflected in the design of flexible fishing policies, which target the sustainable exploitation of this important natural resource. In this study, such nonlinearities are expressed through eventual variability of the “catch‐to‐stock” parameter that measures the sensitivity of an additional catch yield to marginal changes in the fish‐stock level. Using the optimal control modeling framework, we establish that each value of the “catch‐to‐stock” parameter generates a unique steady‐state size of the fish stock and the latter engenders an optimal fishing policy that can be sustained as long as the “catch‐to‐stock” parameter remains unchanged. We also prove the continuous dependence of the steady‐state stock and underlying fishing policy upon the mentioned “catch‐to‐stock” parameter and then focus on the analysis of the equilibrium responses to changes in this parameter induced by external perturbations. Recommendations for Resource Managers

• Marginal catches of pelagic fish stocks do not react in a linear way to changes in existing stock level, and the latter is captured in our model by the “catch‐to‐stock” parameter . Each observable value of engenders a unique steady‐state stock size that defines an optimal fishing policy, which can be sustained as long as remains unchanged.
• The ability of fishery managers to detect variations in the levels of hyperstability expressed by the “catch‐to‐stock” parameter may help them to anticipate new equilibrium responses in stock evolution and to make timely adjustments in the fishing policy.
• Plausible estimations of the “catch‐to‐stock” parameter , as well as detection of its possible alterations, can be carried out within the framework of Management Strategy Evaluation (MSE) approach where different data collected inside and outside the fishery are contrasted via the validation of a relatively simple decision‐making model (presented in this paper) coupled with other “operation models” of higher complexity.
• If the “catch‐to‐stock” parameter cannot be reasonably assessed (), the fishery managers may rely upon the lower bound of stationary stock size, which depends on economic and biological factors (such as the present and future economic values of the exploited fish stock, its marginal productivity, and underlying dynamics of biological growth).

     

Dynamics of coral reef models in the presence of parrotfish

The decline of coral reefs characterized by macroalgae increase has been a global threat. We consider a slightly modified version of an ordinary differential equation (ODE) model proposed in Blackwood, Hastings, and Mumby [Theor. Ecol. 5 (2012), pp. 105–114] that explicitly considers the role of parrotfish grazing on coral reef dynamics. We perform complete stability, bifurcation, and persistence analysis for this model. If the fishing effort (f) is in between two critical values and , then the system has a unique interior equilibrium, which is stable if and unstable if . If is less (more) than these critical values, then the system has up to two (zero) interior equilibria. Also, we develop a more realistic delay differential equation (DDE) model to incorporate the time delay and treating it as the bifurcation parameter, and we prove that Hopf bifurcation about the interior equilibria could occur at critical time delays, which illustrate the potential importance of the inherent time delay in a coral reef ecosystem. Recommendations for Resource Managers

• One serious threat to coral reefs is overfishing of grazing species, including high level of algal abundance. Fishing alters the entire dynamics of a reef (Hughes, Baird, & Bellwood, 2003), for which the coral cover was predicted to decline rapidly (Mumby, 2006). One major issue is to reverse and develop appropriate management to increase or maintain coral resilience.
• We have provided a detailed local and global analysis of model (Blackwood, Hastings, & Mumby, 2012) and obtained an ecologically meaningful attracting region, for which there is a chance of stable coexistence of coral–algal–fish state.
• The healthy reefs switch to unhealthy state, and the macroalgae–parrotfish state becomes stable as the fishing effort increases through some critical values. Also, for some critical time delays, a switch between healthy and unhealthy reef states occurs through a Hopf bifurcation, which can only appear in the delay differential equation (DDE) model. Eventually, for large enough time delay, oscillations appear and an unhealthy state occurs.

     

Reductions on equivalence relations generated by universal sets

Let X, Y be Polish spaces, , . We say A is universal for Γ provided that each x‐section of A is in Γ and each element of Γ occurs as an x‐section of A. An equivalence relation generated by a set is denoted by , where . The following results are shown:

• (1) If A is a set universal for all nonempty closed subsets of Y, then is a equivalence relation and .
• (2) If A is a set universal for all countable subsets of Y, then is a equivalence relation, and
  • (i) and ;
  • (ii) if , then ;
  • (iii) if every set is Lebesgue measurable or has the Baire property, then .
  • (iv) for , if every set has the Baire property, and E is any equivalence relation, then .

     

Fractal graphs

The lexicographic sum of graphs is defined as follows. Let be a graph. With each associate a graph . The lexicographic sum of the graphs over is obtained from by substituting each by . Given distinct , we have all the possible edges in the lexicographic sum between and if , and none otherwise. When all the graphs are isomorphic to some graph , the lexicographic sum of the graphs over is called the lexicographic product of by and is denoted by . We say that a graph is fractal if there exists a graph , with at least two vertices, such that . There is a simple way to construct fractal graphs. Let be a graph with at least two vertices. The graph is defined on the set of functions from to as follows. Given distinct is an edge of if is an edge of , where is the smallest integer such that . The graph is fractal because . We prove that a fractal graph is isomorphic to a lexicographic sum over an induced subgraph of , which is itself fractal.     

Distinct partial sums in cyclic groups: polynomial method and constructive approaches

Let be an abelian group and consider a subset with . Given an ordering of the elements of , define its partial sums by and for . We consider the following conjecture of Alspach: for any cyclic group and any subset with , it is possible to find an ordering of the elements of such that no two of its partial sums and are equal for . We show that Alspach’s Conjecture holds for prime when and when . The former result is by direct construction, the latter is nonconstructive and uses the polynomial method. We also use the polynomial method to show that for prime a sequence of length having distinct partial sums exists in any subset of of size at least in all but at most a bounded number of cases.     

Variable Hardy–Lorentz spaces

Let be a measurable function on with . We introduce the variable Hardy–Lorentz space for via the radial grand maximal function. Under the assumption that satisfies the log‐Hölder condition, we establish a version of Fefferman–Stein vector‐valued inequality in variable Lorentz space by interpolation. We also construct atomic decompositions for , and develop a theory of real interpolation and formulate the dual space of the variable Hardy–Lorentz space with and . As a byproduct, we obtain a new John–Nirenberg theorem. Furthermore, we get equivalent characterizations of the variable Hardy–Lorentz space by means of the Lusin area function, the Littlewood–Paley g‐function and the Littlewood–Paley ‐function. Finally, we investigate the boundedness of singular operators on for and .     

On polynomial approximations to AC

Classical approximation results show that any circuit of size and depth has an ‐error probabilistic polynomial over the reals of degree . We improve this upper bound to , which is much better for small values of . We then use this result to show that ‐wise independence fools circuits of size and depth up to error at most , improving on Tal's strengthening of Braverman's result that ‐wise independence suffices. To our knowledge, this is the first PRG construction for that achieves optimal dependence on the error . We also prove lower bounds on the best polynomial approximations to . We show that any polynomial approximating the function on bits to a small constant error must have degree at least . This result improves exponentially on a result of Meka, Nguyen, and Vu (Theory Comput. 2016).     

Bipartite spanning sub(di)graphs induced by 2-partitions

For a given -partition of the vertices of a (di)graph , we study properties of the spanning bipartite subdigraph of induced by those arcs/edges that have one end in each . We determine, for all pairs of nonnegative integers , the complexity of deciding whether has a 2-partition such that each vertex in (for ) has at least (out-)neighbours in . We prove that it is -complete to decide whether a digraph has a 2-partition such that each vertex in has an out-neighbour in and each vertex in has an in-neighbour in . The problem becomes polynomially solvable if we require to be strongly connected. We give a characterisation of the structure of -complete instances in terms of their strong component digraph. When we want higher in-degree or out-degree to/from the other set, the problem becomes -complete even for strong digraphs. A further result is that it is -complete to decide whether a given digraph has a -partition such that is strongly connected. This holds even if we require the input to be a highly connected eulerian digraph.     

The Ramsey number of loose cycles versus cliques

Let be the Ramsey number of an -uniform loose cycle of length versus an -uniform clique of order . Kostochka et al. showed that for each fixed , the order of magnitude of is up to a polylogarithmic factor in . They conjectured that for each we have . We prove that , and more generally for every that . We also prove that for every and , if is odd, which improves upon the result of Collier-Cartaino et al. who proved that for every and we have .     

Non‐autonomous forms and invariance

We generalize the Beurling–Deny–Ouhabaz criterion for parabolic evolution equations governed by forms to the non‐autonomous, non‐homogeneous and semilinear case. Let be Hilbert spaces such that V is continuously and densely embedded in H and let be the operator associated with a bounded H‐elliptic form for all . Suppose is closed and convex and the orthogonal projection onto . Given and , we investigate when the solution of the non‐autonomous evolutionary problem remains in and show that this is the case if for a.e. . Moreover, we examine necessity of this condition and apply this result to a semilinear problem.     

The maximum,supremum, and spectrum for critical set sizes in (0,1)‐matrices

If is a partially filled‐in (0,1)‐matrix with a unique completion to a (0,1)‐matrix (with prescribed row and column sums), then we say that is a defining set for . A critical set is a minimal defining set (the deletion of any entry results in more than one completion). We give a new equivalent definition of critical sets in (0,1)‐matrices and apply this theory to , the set of (0,1)‐matrices of dimensions with uniform row and column sum . The smallest possible size for a defining set of a matrix in is [N. Cavenagh, J. Combin. Des. 21 (2013), pp. 253–266], and the infimum (the largest‐smallest defining set size for members of ) is known asymptotically [N. Cavenagh and R. Ramadurai, Linear Algebra Appl. 537 (2018), pp. 38–47]. We show that no critical set of size larger than exists in an element of and that there exists a critical set of size in an element of for each such that . We also bound the supremum (the smallest‐largest critical set size for members of ) between and .     

On Turán problems for Cartesian products of graphs

Let be disjoint sets of sizes and . Let be a family of quadruples, having elements from and from , such that any subset with and contains one of the quadruples. We prove that the smallest size of is as . We also solve asymptotically a more general two‐partite Turán problem for quadruples.     

On the exact decomposition threshold for even cycles

A graph has a -decomposition if its edge set can be partitioned into cycles of length . We show that if , then has a -decomposition, and if , then has a -decomposition, where and (we assume is large and satisfies necessary divisibility conditions). These minimum degree bounds are best possible and provide exact versions of asymptotic results obtained by Barber, Kühn, Lo and Osthus. In the process, we obtain asymptotic versions of these results when is bipartite or satisfies certain expansion properties.     

Distance matching extension and local structure of graphs

A matching in a graph is said to be extendable if there exists a perfect matching of containing . Also, is said to be a distance matching if the shortest distance between a pair of edges in is at least . A graph is distance matchable if every distance matching is extendable in , regardless of its size. In this paper, we study the class of distance matchable graphs. In particular, we prove that for every integer with , there exists a positive integer such that every connected, locally -connected -free graph of even order is distance matchable. We also prove that every connected, locally -connected -free graph of even order is distance matchable. Furthermore, we make more detailed analysis of -free graphs and study their distance matching extension properties.     

The maximum number of columns in supersaturated designs with

Cheng and Tang [Biometrika, 88 (2001), pp. 1169–1174] derived an upper bound on the maximum number of columns that can be accommodated in a two‐symbol supersaturated design (SSD) for a given number of rows () and a maximum in absolute value correlation between any two columns (). In particular, they proved that for (mod ) and . However, the only known SSD satisfying this upper bound is when . By utilizing a computer search, we prove that for , and . These results are obtained by proving the nonexistence of certain resolvable incomplete blocks designs. The combinatorial properties of the RIBDs are used to reduce the search space. Our results improve the lower bound for SSDs with rows and columns, for , and . Finally, we show that a skew‐type Hadamard matrix of order can be used to construct an SSD with rows and columns that proves . Hence, we establish for and for all (mod ) such that . Our result also implies that when is a prime power and (mod ). We conjecture that for all and (mod ), where is the maximum number of equiangular lines in with pairwise angle .     

A complete solution to existence of H designs

An H design is a triple , where is a set of points, a partition of into disjoint sets of size , and a set of ‐element transverses of , such that each ‐element transverse of is contained in exactly one of them. In 1990, Mills determined the existence of an H design with . In this paper, an efficient construction shows that an H exists for any integer with . Consequently, the necessary and sufficient conditions for the existence of an H design are , , and , with a definite exception .     

There are eight‐element orthogonal exponentials on the spatial Sierpinski gasket

The self‐affine measure corresponding to an expanding matrix and the digit set in the space is supported on the spatial Sierpinski gasket, where are the standard basis of unit column vectors in and . In the case and , it is conjectured that the cardinality of orthogonal exponentials in the Hilbert space is at most “4”, where the number 4 is the best upper bound. That is, all the four‐element sets of orthogonal exponentials are maximal. This conjecture has been proved to be false by giving a class of the five‐element orthogonal exponentials in . In the present paper, we construct a class of the eight‐element orthogonal exponentials in the corresponding Hilbert space to disprove the conjecture. We also illustrate that the constructed sets of orthogonal exponentials are maximal.     

On -connected graphs

A graph is called -connected if is -edge-connected and is -edge-connected for every vertex . The study of -connected graphs is motivated by a theorem of Thomassen [J. Combin. Theory Ser. A 110 (2015), pp. 67–78] (that was a conjecture of Frank [SIAM J. Discrete Math. 5 (1992), no. 1, pp. 25–53]), which states that a graph has a -vertex-connected orientation if and only if it is (2,2)-connected. In this paper, we provide a construction of the family of -connected graphs for even, which generalizes the construction given by Jordán [J. Graph Theory 52 (2006), pp. 217–229] for (2,2)-connected graphs. We also solve the corresponding connectivity augmentation problem: given a graph and an integer , what is the minimum number of edges to be added to make -connected. Both these results are based on a new splitting-off theorem for -connected graphs.     

On the relative strengths of fragments of collection

Let be the basic set theory that consists of the axioms of extensionality, emptyset, pair, union, powerset, infinity, transitive containment, Δ₀‐separation and set foundation. This paper studies the relative strength of set theories obtained by adding fragments of the set‐theoretic collection scheme to . We focus on two common parameterisations of the collection: ‐collection, which is the usual collection scheme restricted to ‐formulae, and strong ‐collection, which is equivalent to ‐collection plus ‐separation. The main result of this paper shows that for all ,

1. proves that there exists a transitive model of Zermelo Set Theory plus ‐collection,
2. the theory is ‐conservative over the theory .

It is also shown that (2) holds for when the Axiom of Choice is included in the base theory. The final section indicates how the proofs of (1) and (2) can be modified to obtain analogues of these results for theories obtained by adding fragments of collection to a base theory (Kripke‐Platek Set Theory with Infinity plus ) that does not include the powerset axiom.