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Within the framework of natural orbital functional theory, having a convenient representation of the occupation numbers and orbitals becomes critical for the computational performance of the calculations. Recognizing this, we propose an innovative parametrization of the occupation numbers that takes advantage of the electron-pairing approach used in Piris natural orbital functionals through the adoption of the softmax function, a pivotal component in modern deep-learning models. Our approach not only ensures adherence to the N-representability of the first-order reduced density matrix (1RDM) but also significantly enhances the computational efficiency of 1RDM functional theory calculations. The effectiveness of this alternative parameterization approach was assessed using the W4-17-MR molecular set, which demonstrated faster and more robust convergence compared to previous implementations. [ABSTRACT FROM AUTHOR]

For a set A⊂N$A \subset {\mathbb {N}}$, we characterize the existence of an infinite set B⊂N$B \subset {\mathbb {N}}$ and t∈{0,1}$t \in \lbrace 0,1\rbrace$ such that B+B⊂A−t$B+B \subset A-t$, where B+B={b1+b2:b1,b2∈B}$B+B =\lbrace b_1+b_2\colon b_1,b_2 \in B\rbrace$, in terms of the density of the set A$A$. Specifically, when the lower density d̲(A)>1/2$\underline{\mathop {}\!\mathrm{d}}(A) >1/2$ or the upper density d¯(A)>2/3$\overline{\mathop {}\!\mathrm{d}}(A)> 2/3$, the existence of such a set B⊂N$B\subset {\mathbb {N}}$ and t∈{0,1}$t\in \lbrace 0,1\rbrace$ is assured. Furthermore, whenever d̲(A)>3/4$\underline{\mathop {}\!\mathrm{d}}(A) > 3/4$ or d¯(A)>5/6$\overline{\mathop {}\!\mathrm{d}}(A)>5/6$, we show that the shift t$t$ is unnecessary and we also provide examples to show that these bounds are sharp. Finally, we construct a syndetic three‐coloring of the natural numbers that does not contain a monochromatic B+B+t$B+B+t$ for any infinite set B⊂N$B \subset {\mathbb {N}}$ and number t∈N$t \in {\mathbb {N}}$. [ABSTRACT FROM AUTHOR]

Fix a simply-laced semisimple Lie algebra. We study the crystal B (n λ) , were λ is a dominant minuscule weight and n is a natural number. On one hand, B (n λ) can be realized combinatorially by height n reverse plane partitions on a heap associated to λ . On the other hand, we use this heap to define a module over the preprojective algebra of the underlying Dynkin quiver. Using the work of Saito and Savage–Tingley, we realize B (n λ) via irreducible components of the quiver Grassmannian of n copies of this module. In this paper, we describe an explicit bijection between these two models for B (n λ) and prove that our bijection yields an isomorphism of crystals. Our main geometric tool is Nakajima's tensor product quiver varieties. [ABSTRACT FROM AUTHOR]

Many studies have explored the potential of companion plants to manage pests through top–down or bottom–up effects in agricultural fields. Buckwheat is known as a good candidate for companion planting with various crops to enhance pest management, but rarely in soybeans.We investigated the impact of planting buckwheat as a new companion plant on the population dynamics of major soybean pests and their natural enemies in soybean fields in Andong, Republic of Korea. We recorded numbers of pests and natural enemies at 9–10‐day intervals from 20 soybean plants each in eight small experimental fields throughout the cropping season.Significant reduction in pests, such as Riptortus pedestris (F.) (Hemiptera: Alydidae) adults, and a significant increase in natural enemies, such as Orius species (Hemiptera: Anthocoridae) and crab spiders (Araneae: Thomisidae), were observed in the buckwheat treatment plots. However, there were significant increases in other pests, such as leafminers and leafhoppers, in the buckwheat treatment plots. While we observed effects on R. pedestris adult density, we did not find significant effects on other developmental stages of this pest or its parasitoids.Although there were some concerns in the buckwheat treatment plots, the positive effects of buckwheat in soybean fields appear to outweigh the drawbacks. Buckwheat is likely a suitable companion plant for use in soybeans to manage pests. [ABSTRACT FROM AUTHOR]

We prove that for any twist rigid compact p-adic analytic group G, its twist representation zeta function is a finite sum of terms n_{i}^{-s}f_{i}(p^{-s}), where n_{i} are natural numbers and f_{i}(t)\in \mathbb {Q}(t) are rational functions. Meromorphic continuation and rationality of the abscissa of the zeta function follow as corollaries. If G is moreover a pro-p group, we prove that its twist representation zeta function is rational in p^{-s}. To establish these results we develop a Clifford theory for twist isoclasses of representations, including a new cohomological invariant of a twist isoclass. [ABSTRACT FROM AUTHOR]

A family I of subsets of a set X is an ideal on X if it is closed under taking subsets and finite unions of its elements. An ideal I on X is below an ideal J on Y in the Katětov order if there is a function f : Y → X such that f - 1 [ A ] ∈ J for every A ∈ I . We show that the Hindman ideal, the Ramsey ideal and the summable ideal are pairwise incomparable in the Katětov order, where The Ramsey ideal consists of those sets of pairs of natural numbers which do not contain a set of all pairs of any infinite set (equivalently do not contain, in a sense, any infinite complete subgraph), The Hindman ideal consists of those sets of natural numbers which do not contain any infinite set together with all finite sums of its members (equivalently do not contain IP-sets that are considered in Ergodic Ramsey theory), The summable ideal consists of those sets of natural numbers such that the series of the reciprocals of its members is convergent. [ABSTRACT FROM AUTHOR]

Given an odd prime q, a natural number l and non-zero q-free integers a 1 , a 2 , ... , a l , none of which are equal to 1 or –1, we give necessary and sufficient conditions for the polynomial ∏ j = 1 l (x q − a j) to have roots modulo every positive integer. Consequently: (i) if l ≤ q and none of a 1 , a 2 , ... , a l is a perfect qth power, then the polynomial ∏ j = 1 l (x q − a j) fails to have roots modulo some positive integer; (ii) For every l ∈ N , and every (c j) j = 1 l ∈ (F q ∖ { 0 }) l , the polynomial ∏ j = 1 l (x q − a j) has roots modulo every positive integer if and only if ∏ j = 1 l (x q − rad q (a j c j ))) has roots modulo every positive integer. Here rad q (a j) denotes the q-free part of the integer aj. [ABSTRACT FROM AUTHOR]

The purpose of the present paper is to introduce recursive methods for constructing simple t-designs, s-resolvable t-designs, and large sets of t-designs. The results turn out to be very effective for finding these objects. In particular, they reveal a fundamental property of the considered designs. Consequently, many new infinite series of simple t-designs, t-designs with s-resolutions and large sets of t-designs can be derived from the new constructions. For example, by starting with an important result of Teirlinck stating that for every natural number t and for all N > 1 there is a large set L S [ N ] (t , t + 1 , t + N · ℓ (t)) , where ℓ (t) = ∏ i = 1 t λ (i) · λ ∗ (i) , λ (t) = lcm ( t m | m = 1 , 2 , ... , t) and λ ∗ (t) = lcm (1 , 2 , ... , t + 1) , we obtain the following statement. If (t + 2) is composite, then there is a large set L S [ N ] (t , t + 2 , t + 1 + N · ℓ (t)) for all N > 1 . If (t + 2) is prime, then there is an L S [ N ] (t , t + 2 , t + 1 + N · ℓ (t)) for any N with gcd (t + 2 , N) = 1 . [ABSTRACT FROM AUTHOR]

Random numbers are the critical components of most security algorithms. Generating random numbers from a natural source or any deterministic algorithm suffers from shorter periods, inefficient energy consumption, smaller keyspace, etc. This work proposes a technique to generate random numbers with reduced complexity while increasing the keyspace and period of the random numbers. This work presents a novel method of cross-over operation on two chaotic maps to design a new hybrid chaotic map. The Pseudorandom Number Generator (PRNG) is constructed from the newly designed chaotic map, and the bit sequences are analyzed to justify their security strength. It is then applied to encrypt the images, and their properties are investigated. The proposed system is measured to have a 21.9% increase in keyspace and stronger key sensitivity. The correlation coefficient and entropy of ciphered images were shown to have a uniform distribution of pixel values. The average number of pixels change rate (NPCR) in ciphered images is calculated to be 99.4694%. The unified average changing intensity (UACI) is averaged at 33.4211%. The experimental analysis validates the competence of the proposed chaotic system for cryptography. [ABSTRACT FROM AUTHOR]

This study aims to investigate the interplay between tumor-associated endothelial cells (TECs) and immune cells within the tumor microenvironment (TME) and its impact on tumor prognosis. We conducted single-cell RNA sequencing (scRNA-seq) of tumor, normal, and lymph node tissues obtained from intrahepatic cholangiocarcinoma (ICC) patients to reveal the role of TECs in tumor angiogenesis and their significant heterogeneity. Meanwhile, we identified genes highly expressed in TECs and constructed TEC signatures (TEC.Sig). Next, we calculated TEC scores of samples based on TEC.Sig. Patients with higher TEC scores exhibited a higher frequency of KRAS mutations, which was associated with increased infiltration of neutrophils and immature dendritic cells (iDCs), and decreased numbers of natural killer (NK), CD4 + T, and CD8 + T effector memory (Tem) cells, indicating an inflammation-dominated immunosuppressive phenotype. In contrast, BAP1 mutations and CXCL12 overexpression showed a contrasting trend. Spatial transcriptomics analysis and histological experiments further confirmed that TECs interacted with various tumor-killing immune cells through the CXCL12/CXCR4 axis. Multiple tumor immunotherapy datasets confirmed that the TEC.Sig could predict patient responses to immunotherapy. The TEC score is a promising and reliable biomarker for predicting genetic mutations and prognosis in ICC patients. Enhancing the regulation of the CXCL12/CXCR4 signaling pathway may represent a potential novel therapeutic target for ICC treatment. [ABSTRACT FROM AUTHOR]

Bloch oscillations in small Josephson junctions were predicted theoretically as the quantum dual to Josephson oscillations. A significant consequence of this prediction is the emergence of quantized current steps, so-called dual Shapiro steps, when synchronizing Bloch oscillations to an external microwave signal. These steps potentially enable a fundamental standard of current I, defined via the frequency f of the external signal and the elementary charge e, I = ± n × 2ef, where n is a natural number. Here, we realize this fundamental relation by synchronizing the Bloch oscillations in small Al/AlOx/Al Josephson junctions to sinusoidal drives with frequencies from 1 to 6 GHz and observe dual Shapiro steps up to I ≈ 3 nA. Inspired by today's voltage standards and to further confirm the duality relation, we investigate a pulsed drive regime and observe an asymmetric pattern of dual Shapiro steps. This work confirms quantum duality effects in Josephson junctions and paves the way towards a range of applications in quantum metrology based on well-established fabrication techniques and straightforward circuit design. Biasing a Josephson junction embedded in a high impedance environment with a dc-current results in small voltage oscillations. Here, the authors demonstrate that when those oscillations are locked to an external rf-signal, it results in the creation of quantized current steps, the so-called dual Shapiro steps. [ABSTRACT FROM AUTHOR]

The ultra-high-frequency pulse trains in a Yb:LuAG passively Q-switched laser with a Yb ratio of 15 at. % are experimentally investigated. The pulse trains consist of a low-frequency harmonic spectrum pulse train with a fundamental frequency below 500 KHz as the source and an ultra-high-frequency harmonic spectrum pulse train with an exciting fundamental frequency above 104 KHz. The basic frequency of the low spectrum increases approximately linearly with increasing pump power. The change law of the fundamental frequency of the ultra-high frequency spectrum with the increase in pump power conforms to the frequency-doubling principle of the harmonic spectrum changing to a higher frequency, and increases exponentially with A × 2n (n is a natural number, A = 1.25 × 108, 1.552 × 108, or 1.0 × 108 Hz). In this paper, the base frequency range of the ultra-high-frequency spectrum was 0.077 73 GHz–0.66 THz. The maximum average peak power of an ultra-high-frequency pulse was 18.25 mW, and the minimum width of an ultra-high-frequency pulse was 3.2 ps. [ABSTRACT FROM AUTHOR]

In this study, we investigated how secondary students interpret algebraic expressions that contain literal symbols to stand for variables. We hypothesized that the natural number bias (i.e., the tendency to over-rely on knowledge and experiences based on natural numbers) would affect students to think that the literal symbols stand for natural numbers only rather than for any rational or real number (integrity effect); and that the arithmetical values of the algebraic expressions are of the same sign as the expressions' phenomenal sign (phenomenal sign effect). The participants (138 8th and 9th graders) were asked to evaluate 48 statements about numbers that can or cannot be assigned to six algebraic expressions that contained literal symbols (e.g., a, -d-4). The results supported the main hypothesis of the study with respect to the integrity as well as the phenomenal sign effect and also indicated that the former was stronger than the latter. Additionally, the most salient characteristics of the form of each expression, such as its sign, appeared to affect students' responses regarding the arithmetical values they may represent. Theoretical and educational implications are discussed. [ABSTRACT FROM AUTHOR]

Cinnamides are common core structures that exist in a great number of pharmaceuticals and natural products. The development of efficient methods for preparing cinnamides is in great need. We report herein an efficient polyphosphoric acid (PPA)-promoted direct aldol condensation of an amide for the convenient and straightforward preparation of cinnamides. A variety of cinnamides were obtained in moderate-to-excellent yields (65–89%). This strategy features the use of equivalent amides and a short reaction time. [ABSTRACT FROM AUTHOR]

The Kwajalein micrometeorite collection utilized high volume air samplers fitted with polycarbonate membrane filters to capture particles directly from the atmosphere at the Earth's surface. This initial study focused on identifying cosmic spherule‐like particles, conservatively categorizing them into four groups based on bulk compositional data: Group I exhibit a range of compositions designated terrestrial in origin; group II are Fe‐rich and contain only additional O, S, and/or Ni; group III are silicate spherules with Mg‐to‐Si At% ratios less than 0.4; group IV are silicate spherules with Mg‐to‐Si At% ratios greater than 0.4. Spherules in groups I, II, and III have compositions that are also consistent with particles that are produced in great numbers by natural and/or anthropogenic terrestrial activities (e.g., volcanic microspherules, fly ash from coal fired power plants, etc.) and thus are assumed terrestrial in origin. Group IV spherules exhibit compositions closest to those of cosmic spherules identified in other collections and are, therefore, designated cosmic spherule candidates. Detailed analysis of seven group IV spherules found that whilst five exhibited morphology and compositions consistent with S‐type cosmic spherules, two appear unique to this collection and could not be matched to either terrestrial or extraterrestrial spherules studied to date. [ABSTRACT FROM AUTHOR]

We address the following questions in this paper: (1) Which set or number existence axioms are needed to prove the theorems of 'ordinary' mathematics? (2) How should Frege's theory of numbers be adapted so that it works in a modal setting, so that the fact that equivalence classes of equinumerous properties vary from world to world won't give rise to different numbers at different worlds? (3) Can one reconstruct Frege's theory of numbers in a non-modal setting without mathematical primitives such as "the number of Fs" ( # F ) or mathematical axioms such as Hume's Principle? Our answer to question (1) is 'None'. Our answer to question (2) begins by defining 'x numbers G' as: x encodes all and only the properties F such that being-actually-F is equinumerous to G with respect to discernible objects. We answer (3) by showing that the mere existence of discernible objects allows one to reconstruct Frege's derivation of the Dedekind-Peano axioms in a non-modal setting. [ABSTRACT FROM AUTHOR]

Let S be a discrete semigroup and let S S denote the collection of all functions f : S → S . If (P , ∘) is a subsemigroup of S S by composition operation, then P induces a natural topological dynamical system. In fact, (β S , { T f } f ∈ P) is a topological dynamical system, where β S is the Stone–Čech compactification of S, x ↦ T f (x) = f β (x) : β S → β S and f β is a unique continuous22 extension of f. In this paper, we concentrate on the dynamical system (β S , { T f } f ∈ P) , when S is an arbitrary discrete semigroup and P is a subsemigroup of S S and obtain some relations between subsets of S and subsystems of β S with respect to P. As a consequence, we prove that if (S , +) is an infinite commutative discrete semigroup and C is a finite partition of S, then for every finite number of arbitrary homomorphisms g 1 , ⋯ , g l : N → S , there exist an infinite subset B of the natural numbers and C ∈ C such that for every finite summations n 1 , ⋯ , n k of B there exists s ∈ S such that { s + g i (n 1) , s + g i (n 2) , ⋯ , s + g i (n k) } ⊆ C , ∀ i ∈ { 1 , ⋯ , l }. [ABSTRACT FROM AUTHOR]

A fullerene graph is a 3-connected plane cubic graph in which every face is pentagonal or hexagonal. A set of hexagons H of G is called a resonant pattern if there exists a perfect matching M of G such that exactly three edges of H is contained in M for each member H of H . In this paper we prove for any natural number k that almost all of the family of k disjoint hexagons are resonant patterns in sufficiently large fullerene graphs. [ABSTRACT FROM AUTHOR]

Let d$d$ and n$n$ be natural numbers greater or equal to 2. Let ⟨a,νd,n(x)⟩∈Z[x]$\langle \bm{a}, \nu _{d,n}(\bm{x})\rangle \in \mathbb {Z}[\bm{x}]$ be a homogeneous polynomial in n$n$ variables of degree d$d$ with integer coefficients a$\bm{a}$, where ⟨·,·⟩$\langle \cdot,\cdot \rangle$ denotes the inner product, and νd,n:Rn→RN$\nu _{d,n}: \mathbb {R}^n\rightarrow \mathbb {R}^N$ denotes the Veronese embedding with N=n+d−1d$N=\binom{n+d-1}{d}$. Consider a variety Va$V_{\bm{a}}$ in Pn−1$\mathbb {P}^{n-1}$, defined by ⟨a,νd,n(x)⟩=0$\langle \bm{a}, \nu _{d,n}(\bm{x})\rangle =0$. In this paper, we examine a set of integer vectors a∈ZN$\bm{a}\in {\mathbb {Z}}^N$, defined by A(A;P)={a∈ZN:P(a)=0,∥a∥∞⩽A},$$\begin{equation*} \mathfrak {A}(A;P)=\lbrace \bm{a}\in {\mathbb {Z}}^N:\ P(\bm{a})=0,\ \Vert \bm{a}\Vert _{\infty }\leqslant A\rbrace, \end{equation*}$$where P∈Z[x]$P\in \mathbb {Z}[{\bm{x}}]$ is a nonsingular form in N$N$ variables of degree k$k$ with 2⩽k⩽C(n,d)$2 \leqslant k\leqslant C({n,d})$ for some constant C(n,d)$C({n,d})$ depending at most on n$n$ and d$d$. Suppose P(a)=0$P(\bm{a})=0$ has a nontrivial integer solution. We confirm that the proportion of integer vectors a∈ZN$\bm{a}\in {\mathbb {Z}}^N$ in A(A)$\mathfrak {A}(A)$, whose associated equation ⟨a,νd,n(x)⟩=0$\langle \bm{a}, \nu _{d,n}(\bm{x})\rangle =0$ is everywhere locally soluble, converges to a constant cP$c_P$ as A→∞$A\rightarrow \infty$. Moreover, for each place v$v$ of Q${\mathbb {Q}}$, if there exists a nonzero bv∈QvN$\bm{b}_v\in {\mathbb {Q}}_v^N$ such that P(bv)=0$P(\bm{b}_v)=0$ and the variety Vbv$V_{\bm{b}_v}$ in Pn−1$\mathbb {P}^{n-1}$ admits a smooth Qv$\mathbb {Q}_v$‐point, the constant cP$c_P$ is positive. [ABSTRACT FROM AUTHOR]

Monomial ordering is the process of putting single polynomial terms, or monomials, in a certain order depending on predefined principles. It facilitates the systematic grouping and comparison of monomials, as well as the division and factorization of polynomials and other algebraic operations. Let R := 핂[xi,j|i ∈ [c],j ∈ ℕ], be a polynomial ring over arbitrary field 핂, where c is any fixed natural number. In this paper, we have defined different orderings like lexicographical, degree lexicographical and reverse degree lexicographical ordering on polynomial ring. We have discussed the infinitely many Inc-compatible term ordering on polynomial ring R with respect to two different weight vectors Wc and Wc,n′. [ABSTRACT FROM AUTHOR]

Recent advancements in omics and single-cell analysis highlight the necessity of numerical methods for managing the complexity of biological data. This paper introduces a simulation program for biochemical reaction systems based on the natural number simulation (NNS) method. This novel approach ensures the equitable treatment of all molecular entities, such as DNA, proteins, H2O, and hydrogen ions (H+), in biological systems. Central to NNS is its use of stoichiometric formulas, simplifying the modeling process and facilitating efficient and accurate simulations of diverse biochemical reactions. The advantage of this method is its ability to manage all molecules uniformly, ensuring a balanced representation in simulations. Detailed in Python, NNS is adept at simulating various reactions, ranging from water ionization to Michaelis-Menten kinetics and complex gene-based systems, making it an effective tool for scientific and engineering research. [ABSTRACT FROM AUTHOR]

We study the stable dynamics of non-polynomial automorphisms of ℂ2, more precisely we construct a special class of transcendental Hénon maps of the form F(z,w) = (e−zm + δe2π m iw,z) with m ≥ 2 a natural number and ℝ ∋ δ > 2.One of the points of greatest interest is that we get a combinatorial dynamic behavior, indeed there are cycles of escaping Fatou components where the dynamics vary depending on whether m is even or odd. Moreover, we have two distinct limit functions on each cycle, both of which have generic rank 1. Additionally, each Fatou component in each cycle has two disjoint and hyperbolic limit sets on the line at infinity, with the exception of the Fatou components belonging to the short cycle of period m (that occurs if m is odd) which have the same hyperbolic limit set. [ABSTRACT FROM AUTHOR]

We provide, for each natural number n$n$ and each class among Dn(Σ10)$D_n(\mathbf {\Sigma }^0_1)$, Ďn(Σ10)$\check{D}_n(\mathbf {\Sigma }^0_1)$, D2n+1(Σ10)⊕Ď2n+1(Σ10)$D_{2n+1}(\mathbf {\Sigma }^0_1)\oplus \check{D}_{2n+1}(\mathbf {\Sigma }^0_1)$, a regular language whose associated omega‐power is complete for this class. [ABSTRACT FROM AUTHOR]

A 2n-move is a local change for knots and links which changes 2n-half twists to 0-half twists or vice versa for a natural number n. In 1979, Yasutaka Nakanishi conjectured that the 4-move is an unknotting operation. This is still an open problem. It is known that the ᴦ-polynomial is an invariant for oriented links which is the common zeroth coefficient polynomial of the HOMFLYPT and Kauffman polynomials. In this paper, we show that the 4k-move is not an unknotting operation for any integer k(≥ 2) by using the ᴦ-polynomial, and if ᴦ(K; -1) ≡ 9 (mod 16) then the knot K cannot be deformed into the unknot by a single 4-move. Moreover, we give a one-to-one correspondence between the value ᴦ(K; -1) (mod 16) and the pair (α2(K), α4(K)) (mod 2) of the second and fourth coefficients of the Alexander-Conway polynomial for a knot K. [ABSTRACT FROM AUTHOR]

In this manuscript, we will discuss the solutions of Goursat problems with fuzzy boundary conditions involving gH-differentiability. The solutions to these problems face two main challenges. The first challenge is to deal with the two types of fuzzy gH-differentiability: (i) -differentiability and (i i) -differentiability. The sign of coefficients in Goursat problems and gH-differentiability produces sixteen possible cases. The existing literature does not afford a solution method that addresses all the possible cases of this problem. The second challenge is the mixed derivative term in Goursat problems with fuzzy boundary conditions. Therefore, we propose to discuss the solutions of fuzzy Goursat problems with gH-differentiability. We will discuss the solutions of fuzzy Goursat problems in series form with natural transform and Adomian decompositions. To demonstrate the usability of the established solution methods, we will provide some numerical examples. [ABSTRACT FROM AUTHOR]

The article explores the concept of division by a natural number without the axiom of choice in mathematics. It introduces the principles of "shoe division" and "sock division" as alternative versions of division by n. The text discusses the relationship between sock division and multiplication, presenting a theorem that proves their equivalence. It also explores the implications of sock division for bundles and the different interpretations of multiplication without the axiom of choice. The article concludes with a question about the relationship between divisibility and sock division in ZF set theory. [Extracted from the article]

Artificial nest sites can support populations of endangered species when they are correctly installed. Here we analysed the characteristics and conditions that determined whether the northern house martin Delichon urbicum occupied more than 300 artificial nests around the city of Münster, Germany. We found that artificial nest occupation rates were influenced by various environmental and temporal factors. Positive influences included a longer time since installation and, to a lesser extent, the number of artificial nests at the same site. Negative impacts were observed from higher proportions of sealed surface cover in the surrounding area and, to a lesser extent, southward exposure. The distance to the nearest water body and the number of occupied natural nests showed no significant effect. We compared our results with descriptive evidence from the grey literature and published reports, and we give recommendations for installing artificial house martin nests for conservation practitioners. Future studies should also investigate the potential negative effects of 'dirt boards' below the nests and of gaps between the roof and the artificial nests. [ABSTRACT FROM AUTHOR]

Frequency analysis via electroencephalography (EEG) during general anesthesia is used to develop techniques for measuring anesthesia depth. Variational mode decomposition (VMD) enables mathematical optimization methods to decompose EEG signals into natural number intrinsic mode functions with distinct narrow bands. However, the analysis requires the a priori determination of hyperparameters, including the decomposition number (K) and the penalty factor (PF). In the VMD analysis of EEGs derived from a noninterventional and noninvasive retrospective observational study, we adapted the grey wolf optimizer (GWO) to determine the K and PF hyperparameters of the VMD. As a metric for optimization, we calculated the envelope function of the IMF decomposed via the VMD method and used its envelope entropy as the fitness function. The K and PF values varied in each epoch, with one epoch being the analytical unit of EEG; however, the fitness values showed convergence at an early stage in the GWO algorithm. The K value was set to 2 to capture the α wave enhancement observed during the maintenance phase of general anesthesia in intrinsic mode function 2 (IMF-2). This study suggests that using the GWO to optimize VMD hyperparameters enables the construction of a robust analytical model for examining the EEG frequency characteristics involved in the effects of general anesthesia. [ABSTRACT FROM AUTHOR]

We develop a mathematical approach to formally proving that certain neural computations and representations exist based on patterns observed in an organism's behaviour. To illustrate, we provide a simple set of conditions under which an ant's ability to determine how far it is from its nest would logically imply neural structures isomorphic to the natural numbers ℕ. We generalise these results to arbitrary behaviours and representations and show what mathematical characterisation of neural computation and representation is simplest while being maximally predictive of behaviour. We develop this framework in detail using a path integration example, where an organism's ability to search for its nest in the correct location implies representational structures isomorphic to two‐dimensional coordinates under addition. We also study a system for processing anbn strings common in comparative work. Our approach provides an objective way to determine what theory of a physical system is best, addressing a fundamental challenge in neuroscientific inference. These results motivate considering which neurobiological structures have the requisite formal structure and are otherwise physically plausible given relevant physical considerations such as generalisability, information density, thermodynamic stability and energetic cost. [ABSTRACT FROM AUTHOR]

The article "Golden Moments for Latecomers" presents various examples of amateur mathematicians who have achieved remarkable mathematical results. One example is retired electronics engineer Lee Sallows, who discovered a theorem in elementary geometry. Another example is Belgian military doctor Edouard Zeckendorf, who discovered Zeckendorf's theorem on the representation of natural numbers as a sum of Fibonacci numbers. Hobby mathematician Andrew Beal, who proposed Beal's conjecture, is also mentioned. The article also discusses tiling of surfaces with different polygons, with Richard James III, Marjorie Rice, and David Smith making significant contributions to solving the problem. [Extracted from the article]

The explicit formulae for remainder of sum of powers of successive natural numbers (SPSN) divided by any prime factor p of the number N of sum terms is derived. The remainder is zero, if N contains p j , j > 1. It is also 0, if j = 1 and the exponent is not divisible by p - 1. The expression (N - N / p) (mod p), named "eigen-remainder" introduced for N is the remainder of SPSN in the rest of cases. All natural numbers, having all eigen-remainders equal to 1, are found not exceeding 42. It is demonstrated that in all other cases, there exists a prime divisor of N , which dividing SPSN gives the remainder ≠ 1. This proves Bowen's hypothesis. [ABSTRACT FROM AUTHOR]

Graph theory is used in many areas of chemical sciences, especially in molecular chemistry. It is particularly useful in the structural analysis of chemical compounds and in modeling chemical reactions. One of its applications concerns determining the structural formula of a chemical compound. This can be modeled as a variant of the well-known graph realization problem. In the classical version of the problem, a sequence of natural numbers is given, and the question is whether there exists a graph in which the vertices have degrees equal to the given numbers. In the variant considered in this paper, instead of a sequence of natural numbers, a sequence of sets of natural numbers is given, and the question is whether there exists a multigraph such that each of its vertices has a degree equal to a number from one of the sets. This variant of the graph realization problem matches the nature of the problem of determining the structural formula of a chemical compound better than other variants considered in the literature. We propose a polynomial time exact algorithm solving this variant of the problem. [ABSTRACT FROM AUTHOR]

Let \alpha be a non-zero algebraic number. Let K be the Galois closure of \mathbb {Q}(\alpha) with Galois group G and \bar {\mathbb {Q}} be the algebraic closure of \mathbb {Q}. In this article, among the other results, we prove the following. If f\in \bar {\mathbb {Q}}[G] is a non-zero element of the group ring \bar {\mathbb {Q}}[G] and \alpha is a given algebraic number such that f(\alpha ^n) is a non-zero algebraic integer for infinitely many natural numbers n, then \alpha is an algebraic integer. This result generalises the result of Polya [Rend. Circ Mat. Palermo, 40 (1915), pp. 1–16], Corvaja and Zannier [Acta Math. 193 (2004), pp. 175–191] and Philippon and Rath [J. Number Theory 219 (2021), pp. 198–211]. We also prove the analogue of this result for rational functions with algebraic coefficients. Inspired by a result of B. de Smit [J. Number Theory 45 (1993), pp. 112–116], we prove a finite version of the Polya type result for a binary recurrence sequences of non-zero algebraic numbers. In order to prove these results, we apply the techniques of Corvaja and Zannier along with the results of Kulkarni et al. [Trans. Amer. Math. Soc. 371 (2019), pp. 3787–3804], which are applications of the Schmidt subspace theorem. [ABSTRACT FROM AUTHOR]

In a famous article published in 1975, Kramosil and Michálek introduced a notion of fuzzy metric that was the origin of numerous researches and publications in several frameworks and fields. In 2010, Chistyakov introduced and discussed in detail the concept of modular metric. Since then, some authors have investigated the problem of establishing connections between the notions of fuzzy metric and modular metric, obtaining positive partial solutions. In this paper, we are interested in determining the precise relationship between these two concepts. To achieve this goal, we examine a proof, based on the use of uniformities, of the important result that the topology induced by a fuzzy metric is metrizable. As a consequence of that analysis, we introduce the notion of a weak fuzzy metric and show that every weak fuzzy metric, with continuous t-norm the minimum t-norm, generates a modular metric and, conversely, we show that every modular metric generates a weak fuzzy metric, with continuous t-norm the product t-norm. It follows that every modular metric can be generated from a suitable weak fuzzy metric, and that several examples and properties of modular metrics can be directly deduced from those previously obtained in the field of fuzzy metrics. [ABSTRACT FROM AUTHOR]

We present a detailed analysis of random motions moving in higher spaces with a natural number of velocities. In the case of the so-called minimal random dynamics, under some broad assumptions, we give the joint distribution of the position of the motion (for both the inner part and the boundary of the support) and the number of displacements performed with each velocity. Explicit results for cyclic and complete motions are derived. We establish useful relationships between motions moving in different spaces, and we derive the form of the distribution of the movements in arbitrary dimension. Finally, we investigate further properties for stochastic motions governed by non-homogeneous Poisson processes. [ABSTRACT FROM AUTHOR]

We will prove the Brannan conjecture provided that the parameters α and β verify the conditions 34 ≥ α ≥ β ≥ 1. The basic tool of the study is a new integral representation deduced in this paper. [ABSTRACT FROM AUTHOR]

Let k and n be natural numbers. Let ω k (n) denote the number of distinct prime factors of n with multiplicity k as studied by Elma and the third author [5]. We obtain asymptotic estimates for the first and the second moments of ω k (n) when restricted to the set of h -free and h -full numbers. We prove that ω 1 (n) has normal order log ⁡ log ⁡ n over h -free numbers, ω h (n) has normal order log ⁡ log ⁡ n over h -full numbers, and both of them satisfy the Erdős-Kac Theorem. Finally, we prove that the functions ω k (n) with 1 < k < h do not have normal order over h -free numbers and ω k (n) with k > h do not have normal order over h -full numbers. [ABSTRACT FROM AUTHOR]

We show that a unital ring is generated by its commutators as an ideal if and only if there exists a natural number N such that every element is a sum of N products of pairs of commutators. We show that one can take N ≤ 2 for matrix rings, and that one may choose N ≤ 3 for rings that contain a direct sum of matrix rings – this in particular applies to C*-algebras that are properly infinite or have real rank zero. For Jiang-Su-stable C*-algebras, we show that N ≤ 6 can be arranged. For arbitrary rings, we show that every element in the commutator ideal admits a power that is a sum of products of commutators. Using that a C*-algebra cannot be a radical extension over a proper ideal, we deduce that a C*-algebra is generated by its commutators as a not necessarily closed ideal if and only if every element is a finite sum of products of pairs of commutators. [ABSTRACT FROM AUTHOR]

Consider a commutative monoid (M , + , 0) and a biadditive binary operation μ : M × M → M. We will show that under some additional general assumptions, the operation μ is automatically both associative and commutative. The main additional assumption is localizability of μ , which essentially means that a certain canonical order on M is compatible with adjoining some multiplicative inverses of elements of M. As an application we show that a division ring F is commutative provided that for all a ∈ F there exists a natural number k such that a − k is not a sum of products of squares. This generalizes the classical theorem that every archimedean ordered division ring is commutative to a more general class of formally real division rings that do not necessarily allow for an archimedean (total) order. Similar results about automatic associativity and commutativity are well-known for special types of partially ordered extended rings ("extended" in the sense that neither associativity nor commutativity of the multiplication is required by definition), namely in the uniformly bounded and the lattice-ordered cases, i.e. for (extended) f -rings. In these cases the commutative monoid in question is the positive cone of the partially ordered extended ring. We also discuss how these classical results can be obtained from our main theorem. [ABSTRACT FROM AUTHOR]

We answer a question of Bardakov (Kourovka Notebook, Problem 19.8) which asks for the existence of a pair of natural numbers (c , m) with the property that every element in the free group on the two-element set { a , b } can be represented as a concatenation of c , or fewer, m -almost-palindromes in letters a ± 1 , b ± 1. Here, an m -almost-palindrome is a word which can be obtained from a palindrome by changing at most m letters. We show that no such pair (c , m) exists. In fact, we show that the analogous result holds for all non-abelian free groups. [ABSTRACT FROM AUTHOR]

In this paper, we present a smooth version of Landau’s explicit formula for the von Mangoldt arithmetical function. By assuming the validity of the Riemann hypothesis, we show that in order to determine whether a natural number μ is a prime number, it is sufficient to know the location of a number of nontrivial zeros of the Riemann zeta function of order μlog3 2μ. Next we use Heisenberg’s inequality to support the conjecture that this number of zeros cannot be essentially diminished. [ABSTRACT FROM AUTHOR]

Research on fostering learning about percentages within intelligent tutoring systems (ITSs) is limited. Additionally, there is a lack of data‐driven approaches for improving the design of ITS to facilitate learning about percentages. To address these gaps, we first investigated whether students' understanding of basic mathematical skills (eg, arithmetic, measurement units and geometry) and fractions within an ITS predicts their understanding of percentages. We then applied a psychological network analysis to evaluate interdependencies within the data on 44 subtopics of basic mathematical concepts, fractions and percentages. We leveraged a large‐scale dataset consisting of 2798 students using the ITS bettermarks and working on approximately 4.1 million mathematical problems. We found that advanced arithmetic, measurement units, geometry and fraction understanding significantly predicted percentage understanding. Closer inspection indicated that percentage understanding was best predicted by problems sharing similar features, such as fraction word problems and fraction/natural number multiplication/division problems. Our findings suggest that practitioners and software developers may consider revising specific subtopics which share features with percentage problems for students struggling with percentages. More broadly, our study demonstrates how evaluating interdependencies between subtopics covered within an ITS as a data‐driven approach can provide practical insights for improving the design of ITSs. Practitioner notes What is already known about this topic Longitudinal studies showed that basic mathematical skills predict fraction understanding. There is only limited evidence on whether similar predictions can be observed for percentage understanding—in general and within intelligent tutoring systems. Process data from such intelligent tutoring systems can be leveraged to pursue both educational research questions and optimizing digital learning software. Problems involving percentages typically are word problems requiring multiplications and/or divisions. What this paper adds Similar to the case of fractions, students' performance on advanced arithmetic, measurement units and geometry significantly predicted performance with percentages. Students' performance with fractions also predicted performance with percentages significantly. A psychological network analysis was applied to evaluate specific interdependencies between a range of subtopics (eg, Multiplying and dividing fractions, Adding and subtracting fractions and Calculating with percentages). Fraction word problems and fraction problems involving multiplication/division turned out to be the best predictors of understanding percentages. Implications for practice and/or policy When facing difficulties with percentages, revision of previous mathematical concepts sharing similar features (eg, fraction word problems, fraction/natural number multiplication/division problems) may be advised. Software developers may consider implementing such data‐driven revision recommendations for students facing difficulties within intelligent tutor systems. Psychological network analysis can be utilized as a learning analytics method for easy‐to‐access visualizations illustrating relationships between a large range of different subtopics. What is already known about this topic Longitudinal studies showed that basic mathematical skills predict fraction understanding. There is only limited evidence on whether similar predictions can be observed for percentage understanding—in general and within intelligent tutoring systems. Process data from such intelligent tutoring systems can be leveraged to pursue both educational research questions and optimizing digital learning software. Problems involving percentages typically are word problems requiring multiplications and/or divisions. What this paper adds Similar to the case of fractions, students' performance on advanced arithmetic, measurement units and geometry significantly predicted performance with percentages. Students' performance with fractions also predicted performance with percentages significantly. A psychological network analysis was applied to evaluate specific interdependencies between a range of subtopics (eg, Multiplying and dividing fractions, Adding and subtracting fractions and Calculating with percentages). Fraction word problems and fraction problems involving multiplication/division turned out to be the best predictors of understanding percentages. Implications for practice and/or policy When facing difficulties with percentages, revision of previous mathematical concepts sharing similar features (eg, fraction word problems, fraction/natural number multiplication/division problems) may be advised. Software developers may consider implementing such data‐driven revision recommendations for students facing difficulties within intelligent tutor systems. Psychological network analysis can be utilized as a learning analytics method for easy‐to‐access visualizations illustrating relationships between a large range of different subtopics. [ABSTRACT FROM AUTHOR]

In this paper, the inverse spectral problem method is used to find a solution of the Cauchy problem for the Korteweg–de Vries equation in the class of periodic infinite-gap functions. A simple derivation of the Dubrovin system of differential equations is proposed. The solvability of the Cauchy problem for an infinite system of Dubrovin differential equations in the class of four times continuously differentiable periodic infinite-gap functions is proven. The sum of the uniformly convergent functional series constructed by solving the infinite system of Dubrovin equations and the first trace formula are proven to satisfy the nonlinear Korteweg–de Vries equation. In addition, it is proven that if the number π n is the period of the initial function, then the number π n is the period for the solution of the Cauchy problem with respect to the variable x. Here n ≥ 2 is a natural number. [ABSTRACT FROM AUTHOR]