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The crossing number of a graph is a significant measure that indicates the complexity of the graph and the difficulty of visualizing it. In this paper, we examine the crossing numbers of join products involving the complete graph K 5 with discrete graphs, paths, and cycles. We analyze optimal drawings of K 5 , identify all five non-isomorphic drawings, and address previously hypothesized crossing numbers for K 5 + P n , and K 5 + C n . Through a simplified approach, we first establish cr (K 5 + D n) and then extend our method to prove the crossing numbers cr (K 5 + P n) and cr (K 5 + C n) . These results also lead to new hypotheses for cr (W m + S n) and cr (W m + W n) involving wheels and stars. Our findings correct previous inaccuracies in the literature and provide a foundation for future research. [ABSTRACT FROM AUTHOR]

In growth-cycle models, consumption smoothing motivates procyclical household savings and monopolistic competition engenders countercyclical horizontal R&D at fixed costs of capital. However, data in the US and in 15 OECD countries indicate countercyclical household saving, procyclical R&D, and high volatility. This paper explains the observed cyclical properties by realistic income taxes, R&D subsidies, R&D labor, and depreciation. R&D subsidies promote R&D to increase productivity by trading the levels for the varieties of intermediates. R&D-subsidy payments increase consumption and decrease household saving rates by raising taxable income beyond output, thus hindering future R&D and growth. Anticipated R&D-subsidy payments increase household-saving rates by raising returns to saving, thus promoting future R&D and growth. The US taxes yield unstable fluctuations with consecutive periods of R&D, low saving rates, and strong growth as in the technology boom. R&D labor and partial depreciation strengthen the roles of saving and R&D in attaining high volatility. [ABSTRACT FROM AUTHOR]

A vertex-edge perfect Roman dominating function on a graph G = (V; E) (denoted by ve-PRDF) is a function f: V (G) f0; 1; 2g such that for every edge uv 2 E, maxf f (u); f (v)g, 0, or u is adjacent to exactly one neighbor w such that f (w) = 2, or v is adjacent to exactly one neighbor w such that f (w) = 2. The weight of a ve-PRDF on G is the sum w(f) = P v2V f (v). The vertex-edge perfect Roman domination number of G (denoted by p veR(G)) is the minimum weight of a ve-PRDF on G. In this paper, we first show that vertex-edge perfect Roman dominating is NP-complete for bipartite graphs. Also, for a tree T, we give upper and lower bounds for p veR(T) in terms of the order n, l leaves and s support vertices. Lastly, we determine p veR(G) for Petersen, cycle and Flower snark graphs. [ABSTRACT FROM AUTHOR]

Considerable studies have examined the relationship between commodity markets and stock markets. This paper studies the cyclical relationship between commodity markets and stock markets with implications for investing based on index relationships. Stock markets are represented by the U.S. S&P 500 index and aggregate commodity markets by the U.S. producer price index (PPI). Tradeable market indexes readily available to investors, namely the S&P GSCI Index and the Bloomberg Commodity Index (BCOM), are also studied. An optimal bandpass filter is used to estimate the cyclical component using a pricing-performance measure of the S&P 500 relative to the PPI, based on annual data from 1871 to 2022. The S&P GSCI and the BCOM indexes are also used to test the robustness of the findings. The impacts of the financial crisis of 2008 and the coronavirus pandemic are also assessed. The overriding conclusion of the study is that a cyclical relationship exists between stock markets and commodity markets for both aggregate and tradeable indexes which can last, from peak to peak, approximately 31 years. Measuring returns and risks in a manner consistent with these cycles can shed new light on the usefulness of commodity investing via tradeable indexes in seeking efficient portfolios. [ABSTRACT FROM AUTHOR]

Let K = K(a, p; λ1, λ2) be the multigraph with: the number of parts equal to p; the number of vertices in each part equal to a; the number of edges joining any two vertices of the same part equal to λ1; and the number of edges joining any two vertices of different parts equal to λ2. The existence of C4-factorizations of K has been settled when a is even; when a ≡ 1 (mod 4) with one exception; and for very few cases when a ≡ 3 (mod 4). The existence of Cz-factorizations of K has been settled when a ≡ 1 (mod z) and λ1 is even, and when a ≡ 0 (mod z). In this paper, we give a construction for Cz-factorizations of K for z = 2a when a is even. [ABSTRACT FROM AUTHOR]

Circuits in a graph are interesting structures and identifying them is of an important relevance for many applications. However, enumerating circuits is known to be a difficult problem, since their number can grow exponentially. In this paper, we propose fast parallel approaches for enumerating elementary circuits of directed graphs based on graphics processing unit (GPU). Our algorithms are based on a massive exploration of the graph in a breadth-first search strategy. Algorithm V-FEC explores the graph starting from different vertices simultaneously. To further reduce the search space, we present T-FEC, another algorithm that uses triplets as an initial set to start exploring. To the best of our knowledge, those are the first parallel GPU-based algorithms for finding all circuits of a given graph. In addition, they find circuits of a given length and circuits with a specific vertex or edge. The evaluation results show that the proposed approaches achieve up to 190x speed-up over Johnson's algorithm, one of the most efficient sequential algorithms for finding circuits. [ABSTRACT FROM AUTHOR]

Let R be a ring with unity. In this paper, we introduce a new graph associated with R called the simple-intersection graph of R, denoted by GS (R). The vertices of GS (R) are the nonzero ideals of R, and two vertices are adjacent if and only if their intersection is a nonzero simple ideal. We study the interplay between the algebraic properties of R, and the graph properties of GS (R) such as connectedness, bipartiteness, girth, dominating sets, etc. Moreover, we determine the precise values of the girth and diameter of GS (R), as well as give a formula to compute the clique and domination numbers of GS (R). [ABSTRACT FROM AUTHOR]

Let G be a connected graph and g be a non-negative integer. A vertex set S of graph G is called a g-extra cut if G − S is disconnected and each component of G − S has at least g + 1 vertices. The g-extra connectivity of G is the minimum cardinality of a g-extra cut of G if G has at least one g-extra cut. For two graphs G 1 = (V 1 , E 1) and G 2 = (V 2 , E 2) , the strong product G 1 ⊠ G 2 is defined as follows: its vertex set is V 1 × V 2 and its edge set is { (x 1 , x 2) (y 1 , y 2) | x 1 = x 2 and y 1 y 2 ∈ E 2 ; or y 1 = y 2 and x 1 x 2 ∈ E 1 ; or x 1 x 2 ∈ E 1 and y 1 y 2 ∈ E 2 } , where (x 1 , x 2) , (y 1 , y 2) ∈ V 1 × V 2 . In this paper, we obtain the g-extra connectivity of the strong product of two paths, the strong product of a path and a cycle, and the strong product of two cycles. [ABSTRACT FROM AUTHOR]

This paper proposes two heuristic algorithms for finding fixed-length circuits and cycles in undirected edge-weighted graphs. It focusses particularly on a largely unresearched practical application where we are seeking attractive round trips for pedestrians and joggers in urban street networks. Our first method is based on identifying suitable pairs of paths that are combined to form a solution; our second is based on local search techniques. Both algorithms display high levels of accuracy, producing solutions within just a few meters of the target. Run times for the local search algorithm are also short, with solutions in large cities often being found in less than one second. [ABSTRACT FROM AUTHOR]

This paper proposes a new modeling framework capturing both the long‐run and the cyclical components of a time series. As an illustration, we apply it to four US macro series, namely, annual and quarterly real gross domestic product (GDP) and GDP per capita. The results indicate that the behavior of US GDP can be captured accurately by a model incorporating both stochastic trends and stochastic cycles that allows for some degree of persistence in the data. Both appear to be mean reverting, although the stochastic trend is nonstationary, while the cyclical component is stationary, with cycles repeating themselves every 6–10 years. [ABSTRACT FROM AUTHOR]

Alspach et al. (J. Austral. Math. Soc. 56(3) (1994) 391–402) constructed an infinite family of tetravalent graphs M(a; m, n) and proved that if n ≥ 9 be odd and a 3 ≡ 1 (mod n) , then M(a; 3, n) is half-arc-transitive. In this paper, we show that if a 3 ≡ 1 (mod n) , then M(a; 3, n) is an infinite family of tetravalent half-arc-transitive Cayley graphs for all integers n except 7 and 14. [ABSTRACT FROM AUTHOR]

The paper promotes the notion of computational experiment supported by a multi-tool digital environment as a means of the development of new mathematical knowledge in the context of education. The main study of the paper deals with the issues of teaching this knowledge to secondary teacher candidates within a graduate capstone mathematics education course. The interplay of mathematics and education is considered through the lens of using technology to enhance one's mathematical background by advancing ideas from mostly known to genuinely unknown. In this paper, the knowns consist of Fibonacci numbers, Pascal's triangle, and continued fractions; among the unknowns are Fibonacci-like polynomials and generalized golden ratios in the form of cycles of various lengths. The paper discusses the interplay of pragmatic and epistemic uses of digital tools by the learners of mathematics. The data for the study were collected over the years through solicited comments by teacher candidates enrolled in the capstone course. The main results indicate the candidates' appreciation of the need for deep mathematical knowledge as an instrument of the modern-day pedagogy aimed at making high schoolers interested in the subject matter. [ABSTRACT FROM AUTHOR]

Research has found strong links between past credit booms and adverse outcomes for macroeconomic aggregates like output and investment. However, are price impacts also seen more widely in broad asset classes such as equity and fixed-income markets? We document such a robust and significant connection using a large sample of historical data for many advanced countries since 1870. Credit boom periods with a high "leverage factor" tend to be predictably followed by unusually low returns to risky equities, in absolute terms and relative to a safe fixed-income portfolio. Fixed income is a safe haven at these times and has slightly higher than normal returns. We show these properties hold in-sample and out-of-sample. Return predictability because of the leverage factor is distinct from that because of momentum (lagged return) and value (cashflow relative to price). Trading strategies built on the leverage factor accrue meaningful excess profits out-of-sample. This paper was accepted by Victoria Ivashina, finance. [ABSTRACT FROM AUTHOR]

Epilepsy is characterized by recurrent, unprovoked seizures. Accurate prediction of seizure occurrence has long been a clinical goal since this would allow to optimize patient treatment, prevent injuries due to seizures, and alleviate the patient burden of unpredictability. Advances in implantable electroencephalographic (EEG) devices, allowing for long-term interictal EEG recordings, have facilitated major progress in this field. Recently, it has been discovered that interictal brain activity demonstrates circadian and multi-dien cycles that are strongly aligned, or phase locked, with seizure risk. Thus, cyclical brain activity patterns have been used to forecast seizures. However, in the effort to develop a clinically useful EEG based seizure forecasting system, challenges remain. Firstly, multiple EEG features demonstrate cyclical patterns, but it remains unclear which feature is best suited for predicting seizures. Secondly, the technology for long-term EEG recording is currently limited in both spatial and temporal sampling resolution. In this study, we compare five established EEG metrics:synchrony, spatial correlation, temporal correlation, signal variance which have been motivated from critical dynamics theory, and interictal epileptiform discharge (IED) which are a traditional marker of seizure propensity. We assess their effectiveness in detecting 24-h and seizure cycles as well as their robustness under spatial and temporal subsampling. Analyzing intracranial EEG data from 23 patients, we report that all examined features exhibit 24-h cycles. Spatial correlation, signal variance, and synchrony showed the highest phase locking with seizures, while IED rates were the lowest. Notably, spatial and temporal correlation were also found to be highly correlated to each other, as were signal variance and IED--suggesting some features may reflect similar aspects of cortical dynamics, whereas others provide complementary information. All features proved robust under subsampling, indicating that the dynamic properties of interictal activity evolve slowly and are not confined to specific brain regions. Our results may aid future translational research by assisting in design and testing of EEG based seizure forecasting systems. [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate eventual periodicity of the following max-type system of difference equations of higher order with four variables ... [ABSTRACT FROM AUTHOR]

A set S of vertices in a graph G is a dominating set of G if every vertex not in S is adjacent to some vertex in S. The domination number, (G), of G is the minimum cardinality of a dominating set of G. The authors proved in [A new lower bound on the domination number of a graph, J. Comb. Optim. 38 (2019) 721-738] that if G is a connected graph of order n = 2 with k = 0 cycles and l leaves, then (G) = (n - l + 2 - 2k)/3. As a consequence of the above bound, (G) = (n - l + 2(1 - k) + m)/3 for some integer m = 0. In this paper, we characterize the class of cactus graphs achieving equality here, thereby providing a classification of all cactus graphs according to their domination number. [ABSTRACT FROM AUTHOR]

In 2013, Duchene, Kheddouci, Nowakowski and Tahraoui introduced a labeled version of the graph packing problem. It led to the introduction of a new graph parameter, the k-packing label-span k. This parameter corresponds, given a graph H on n vertices, to the maximum number of labels we can assign to the vertices of the graph, such that there exists a packing of k copies of H into the complete graph Kn, coherent with the labels of the vertices. The authors intensively studied the labeled packing of cycles, and, among other results, they conjectured that for every cycle Cn of order n = 2k + x, with k = 2 and 1 = x = 2k 1, the value of k(Cn) was 2 if x is 1 and k is even, and x + 2 otherwise. In this paper, we disprove this conjecture by giving a counterexample. We however prove that it gives a valid lower bound, and we give sufficient conditions for the upper bound to hold. We then give some similar results for the labeled packing of circuits. [ABSTRACT FROM AUTHOR]

The topology of an interconnection network can be modeled by a graph G=(V(G),E(G)). The connectivity of graph G is a parameter used to measure the reliability of a corresponding network. The direct product is an important graph product. This paper mainly focuses on the super connectedness of the direct product of graphs and cycles. The connectivity of G, denoted by κ(G), is the size of a minimum vertex set S⊆V(G) such that G−S is not connected or has only one vertex. The graph G is said to be super connected, simply super-κ, if every minimum vertex cut is the neighborhood of a vertex with minimum degree. The direct product of two graphs G and H, denoted by G×H, is the graph with vertex set V(G×H)=V(G)×V(H) and edge set E(G×H)={(u1,v1)(u2,v2)|u1u2∈E(G),v1v2∈E(H)}. In this paper, we give some sufficient conditions for the direct product G×Cn to be super connected, where Cn is the cycle on n vertices. Furthermore, those sufficient conditions are the best possible.

Recently Alexandersson and Getachew proved some multivariate generalizations of a formula for enumerating signed excedances in derangements. In this paper we first relate their work to a recent continued fraction for permutations and confirm some of their observations. Our second main result is two refinements of their multivariate identities, which clearly explain the meaning of each term in their main formulas. We also explore some similar formulas for permutations of type B. [ABSTRACT FROM AUTHOR]

We show that a sequence of graphs with uniformly bounded vertex degrees, number of vertices going to infinity, and with geometric property (T) has many small cycles. We also show that when a small part of such a sequence of graphs with geometric property (T) is changed, it still has geometric property (T), provided that it is still an expander. We use this to give an example of a sequence of graphs with geometric property (T) that has large cycle-free balls. [ABSTRACT FROM AUTHOR]

In order to assess the longevity of foldable electronics, folding or bending tests on model systems need to be performed. However, not all bending tests are created equal in that different configurations lead to different amounts of mechanical damage and thus different electrical responses. Two 90° bending instruments were compared using two model metallic thin film systems on polyimide to establish if the two seemingly similar bending tests yield the same results. The two film systems, namely 300 nm Mo and 130 nm Al on 50 nm Mo, were magnetron-sputtered on polyimide substrates and tested in the custom-built FLEX-E-TEST and the commercially available YUASA test that is capable of in situ resistance measurements. For statistics, 10–12 samples were tested of each film system on each folding device using the same applied bending strain and number of cycles. Samples were intermittently characterized with confocal laser scanning microscopy and electrical resistance to correlate the amount of mechanical damage (crack density) with the electrical normalized resistance ratio of the damaged area. The results show that even with the same bending radius, a similar but not identical amount of mechanical damage forms for both bending devices. Additionally, the resistance as a function of cycles also differs after 10,000 cycles. A closer examination of the damage, especially in the Al/Mo film system, indicates that the speed of the bending, and if the samples experience spring back, can alter the received mechanical damage. The in situ resistance data of the YUASA test were further examined and a suggestion of standardizing how folding or bending test results are reported is provided. [ABSTRACT FROM AUTHOR]

Explaining the complex structure and dynamics of sleep, which consist of alternating and physiologically distinct nonREM and REM sleep episodes, has posed a significant challenge. In this study, we demonstrate that a single wave model concept captures the distinctly different overnight dynamics of the four primary sleep measures—the duration and intensity of nonREM and REM sleep episodes—with high quantitative precision for both regular and extended sleep. The model also accurately predicts how these polysomnographic measures respond to sleep deprivation or abundance. Furthermore, the model passes the ultimate test, as its prediction leads to a novel experimental finding—an invariant relationship between the duration of nonREM episodes and the intensity of REM episodes, the product of which remains constant over consecutive sleep cycles. These results suggest a functional unity between nonREM and REM sleep, establishing a comprehensive and quantitative framework for understanding normal sleep and sleep disorders. [ABSTRACT FROM AUTHOR]

The crossing number of a graph is a significant measure that indicates the complexity of the graph and the difficulty of visualizing it. In this paper, we examine the crossing numbers of join products involving the complete graph K5 with discrete graphs, paths, and cycles. We analyze optimal drawings of K5, identify all five non-isomorphic drawings, and address previously hypothesized crossing numbers for K5+Pn, and K5+Cn. Through a simplified approach, we first establish cr(K5+Dn) and then extend our method to prove the crossing numbers cr(K5+Pn) and cr(K5+Cn). These results also lead to new hypotheses for cr(Wm+Sn) and cr(Wm+Wn) involving wheels and stars. Our findings correct previous inaccuracies in the literature and provide a foundation for future research.

Akin to the Erdős-Rademacher problem, Linial and Morgenstern made the following conjecture in tournaments: for any \(d\in (0,1]\), among all \(n\)-vertex tournaments with \(d\binom{n}{3}\) many 3-cycles, the number of 4-cycles is asymptotically minimized by a special random blow-up of a transitive tournament. Recently, Chan, Grzesik, Král' and Noel introduced spectrum analysis of adjacency matrices of tournaments in this study, and confirmed this for \(d\geq 1/36\). In this paper, we investigate the analogous problem of minimizing the number of cycles of a given length. We prove that for integers \(\ell\not\equiv 2\mod 4\), there exists some constant \(c_\ell>0\) such that if \(d\geq 1-c_\ell\), then the number of \(\ell\)-cycles is also asymptotically minimized by the same extremal examples. In doing so, we answer a question of Linial and Morgenstern about minimizing the \(q\)-norm of a probabilistic vector with given \(p\)-norm for integers \(q>p>1\). For integers \(\ell\equiv 2\mod 4\), however the same phenomena do not hold for \(\ell\)-cycles, for which we can construct an explicit family of tournaments containing fewer \(\ell\)-cycles for any given number of \(3\)-cycles. We propose two conjectures concerning the minimization problem for general cycles.Mathematics Subject Classifications: 05C20, 05C35, 05C38Keywords: Tournaments, cycles, spectrum

A vertex-edge perfect Roman dominating function on a graph $ G = (V, E) $ (denoted by ve-PRDF) is a function $ f:V\left(G\right)\longrightarrow\{0, 1, 2\} $ such that for every edge $ uv\in E $, $ \max\{f(u), f(v)\}\neq0 $, or $ u $ is adjacent to exactly one neighbor $ w $ such that $ f(w) = 2 $, or $ v $ is adjacent to exactly one neighbor $ w $ such that $ f(w) = 2 $. The weight of a ve-PRDF on $ G $ is the sum $ w(f) = \sum_{v\in V}f(v) $. The vertex-edge perfect Roman domination number of $ G $ (denoted by $ \gamma_{veR}^{p}(G) $) is the minimum weight of a ve-PRDF on $ G $. In this paper, we first show that vertex-edge perfect Roman dominating is NP-complete for bipartite graphs. Also, for a tree $ T $, we give upper and lower bounds for $ \gamma_{veR}^{p}(T) $ in terms of the order $ n $, $ l $ leaves and $ s $ support vertices. Lastly, we determine $ \gamma_{veR}^{p}(G) $ for Petersen, cycle and Flower snark graphs.

Let R be a ring with unity. In this paper, we introduce a new graph associated with R called the simple-intersection graph of R, denoted by GS(R). The vertices of GS(R) are the nonzero ideals of R, and two vertices are adjacent if and only if their intersection is a nonzero simple ideal. We study the interplay between the algebraic properties of R, and the graph properties of GS(R) such as connectedness, bipartiteness, girth, dominating sets, etc. Moreover, we determine the precise values of the girth and diameter of GS(R), as well as give a formula to compute the clique and domination numbers of GS(R).

We consider one system of partial derivative equations of the parabolic type as a model of a simple 3D gene network in the presence of diffusion of its three components. Using discretization of the phase portrait of this system, comparison theorems, and other methods of the qualitative theory of differential equations, we show uniqueness of the equilibrium solution to this system and find conditions of instability of this equilibrium. Then, we obtain sufficient conditions of existence of at least one oscillating functioning regime of this gene network. An estimate of lower and upper bounds for periods of these oscillations is given as well. In quite a similar way, these results on the existence of cycles in 3D gene networks can be extended to higher-dimensional systems of parabolic or other evolution equations in order to construct mathematical models of more complicated molecular–genetic systems. [ABSTRACT FROM AUTHOR]

The modified Bessel function (MBF) of the first kind is a fundamental special function in mathematics with applications in a large number of areas. When the order of this function is integer, it has an integral representation which includes the exponential of the cosine function. Here, we generalize this MBF to include a fractional parameter, such that the exponential in the previously mentioned integral is replaced by a Mittag–Leffler function. The necessity for this generalization arises from a problem of communication in networks. We find the power series representation of the fractional MBF of the first kind as well as some differential properties. We give some examples of its utility in graph/networks analysis and mention some fundamental open problems for further investigation. [ABSTRACT FROM AUTHOR]

Many social animals display collective activity cycles based on synchronous behavioural oscillations across group members. A classic example is the colony cycle of army ants, where thousands of individuals undergo stereotypical biphasic behavioural cycles of about one month. Cycle phases coincide with brood developmental stages, but the regulation of this cycle is otherwise poorly understood. Here, we probe the regulation of cycle duration through interactions between brood and workers in an experimentally amenable army ant relative, the clonal raider ant. We first establish that cycle length varies across clonal lineages using long-term monitoring data. We then investigate the putative sources and impacts of this variation in a cross-fostering experiment with four lineages combining developmental, morphological and automated behavioural tracking analyses. We show that cycle length variation stems from variation in the duration of the larval developmental stage, and that this stage can be prolonged not only by the clonal lineage of brood (direct genetic effects), but also of the workers (indirect genetic effects). We find similar indirect effects of worker line on brood adult size and, conversely (but more surprisingly), indirect genetic effects of the brood on worker behaviour (walking speed and time spent in the nest). [ABSTRACT FROM AUTHOR]

Origins-of-life chemical experiments usually aim to produce specific chemical end-products such as amino acids, nucleic acids or sugars. The resulting chemical systems do not evolve or adapt because they lack natural selection processes. We have modified Miller origins-of-life apparatuses to incorporate several natural, prebiotic physicochemical selection factors that can be tested individually or in tandem: freezing-thawing cycles; drying-wetting cycles; ultraviolet light-dark cycles; and catalytic surfaces such as clays or minerals. Each process is already known to drive important origins-of-life chemical reactions such as the production of peptides and synthesis of nucleic acid bases and each can also destroy various reactants and products, resulting selection within the chemical system. No previous apparatus has permitted all of these selection processes to work together. Continuous synthesis and selection of products can be carried out over many months because the apparatuses can be re-gassed. Thus, long-term chemical evolution of chemical ecosystems under various combinations of natural selection may be explored for the first time. We argue that it is time to begin experimenting with the long-term effects of such prebiotic natural selection processes because they may have aided biotic life to emerge by taming the combinatorial chemical explosion that results from unbounded chemical syntheses. [ABSTRACT FROM AUTHOR]

We consider C r ( r ⩾ 1 ) diffeomorphisms f defined on manifolds of dimension ⩾ 3 with homoclinic tangencies associated to saddles. Under generic properties, we show that if the saddle is homoclinically related to a blender then the diffeomorphism f can be C r approximated by diffeomorphisms with C 1 robust heterodimensional cycles. As an application, we show that the classic Simon–Asaoka's examples of diffeomorphisms with C 1 robust homoclinic tangencies also display C 1 robust heterodimensional cycles. In a second application, we consider homoclinic tangencies associated to hyperbolic sets. When the entropy of these sets is large enough we obtain C 1 robust cycles after C 1 perturbations. [ABSTRACT FROM AUTHOR]

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Hydrogen is increasingly being viewed as a significant fuel for future industrial processes as it offers pathways to zero emission. The UK sees hydrogen as one of a handful of low-carbon solutions for transition to net zero. Currently, most hydrogen production is from steam reforming of natural gas or coal gasification, both of which involve the release of carbon dioxide. Hydrogen production from mini decentralised grids via a thermochemical process, coupled with electricity production, could offer favourable economics for small modular reactors (SMRs), whereby demand or grid management as a solution would include redirecting the power for hydrogen production when electricity demand is low. It also offers a clean-energy alternative to the aforementioned means. SMRs could offer favourable economics due to their flexible power system as part of the dual-output function. This study objective is to investigate the critical performance parameters associated with the nuclear power plant (NPP), the cycle working fluids, and control-system design for switching between electricity and hydrogen demand to support delivery as part of a mini grid system for a reactor power delivering up to approximately 600 MWth power. The novelty of the work is in the holistic parametric analysis undertaken using a novel in-house tool, which analyses the NPP using different working fluids, with a control function bolt-on at the offtake for hydrogen production. The results indicate that the flow conditions at the offtake can be maintained. The choice of working fluids affects the pressure component. However, the recuperator and heat-exchanger effectiveness are considered as efficiency-limiting factors for hydrogen production and electricity generation. As such, the benefit of high-technology heat exchangers cannot be underestimated. This is also true when deciding on the thermochemical process to bolt onto the plant. The temperature of the gas at the end of the pipeline should also be considered to ensure that the minimum temperature-requirement status for hydrogen production is met. [ABSTRACT FROM AUTHOR]

Let G be a connected graph and g be a non-negative integer. A vertex set S of graph G is called a g-extra cut if G−S is disconnected and each component of G−S has at least g+1 vertices. The g-extra connectivity of G is the minimum cardinality of a g-extra cut of G if G has at least one g-extra cut. For two graphs G1=(V1,E1) and G2=(V2,E2), the strong product G1⊠G2 is defined as follows: its vertex set is V1×V2 and its edge set is {(x1,x2)(y1,y2)|x1=x2 and y1y2∈E2; or y1=y2 and x1x2∈E1; or x1x2∈E1 and y1y2∈E2}, where (x1,x2),(y1,y2)∈V1×V2. In this paper, we obtain the g-extra connectivity of the strong product of two paths, the strong product of a path and a cycle, and the strong product of two cycles.

The Oberwolfach problem, posed by Ringel in 1967, asks for a decomposition of K2n+1 into edge-disjoint copies of a given 2-factor. We show that this can be achieved for all large n. We actually prove a significantly more general result, which allows for decompositions into more general types of factors. In particular, this also resolves the Hamilton-Waterloo problem for large n. [ABSTRACT FROM AUTHOR]

Fibonacci-like polynomials, the roots of which are responsible for a cyclic behavior of orbits of a second-order two-parametric difference equation, are considered. Using Maple and Wolfram Alpha, the location of the largest and the smallest roots responsible for the cycles of period p among the roots responsible for the cycles of periods 2kp (period-doubling) and kp (period-multiplying) has been determined. These purely computational results of experimental mathematics, made possible by the use of modern digital tools, can be used as a motivation for confirmation through not-yet-developed methods of formal mathematics. [ABSTRACT FROM AUTHOR]