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Substitution tilings with transcendental inflation factor, Discrete Analysis 2024:11, 24 pp. Symbolic dynamics is the study of topological dynamical systems $(X,T)$, called *subshifts*, where $T\colon X\to X$ is the shift transformation acting on a shift-invariant subspace $X$ of the space $K^{\mathbb Z}$ of doubly infinite sequences in some set $K$, which is given the product topology. The set $K$ is called the *alphabet* of the shift. *Substitutions* are special types of subshifts that exhibit a variety of phenomena and have been studied extensively when the alphabet $K$ is finite. The basic idea is that for each $a\in K$ there is a substitution rule of the form $a\mapsto a_1\cdots a_k$, and $X$ consists of the closure of the set of all sequences that are built up from elements of $K$ when we repeatedly apply these substitution rules. Such symbolic substitutions can be made geometric, as so-called *stone inflations* of $\mathbb{R}$, in the following way. An *interval tile* is a finite subinterval of $\mathbb R$ (closed on the left and open on the right, say). Given a set of substitution rules, we assign to each letter $a$ a length $\ell(a)$. A *tile inflation* first inflates the length of each tile by a fixed inflation factor $\lambda>1$, resulting in a tile of length $\lambda \ell(a)$, and then decomposes the inflated tile in the obvious way into tiles of lengths $\ell(a_1),\dots,\ell(a_k)$, where $a\mapsto a_1\cdots a_k$ is the rule of the substitution. Repeating this process results in a tiling of $\mathbb{R}$, which is called a *substitution tiling with inflation factor* $\lambda$. Substitutions on infinite alphabets and the corresponding tilings of $\mathbb{R}$, are natural objects of study in various settings, and their study has been steadily gaining attention. The goal of the current article is to better understand how the landscape of infinite alphabets differs from the more familiar world of finite alphabets. Previous works by N. Manibo, D. Rust, and J. Walton made a first foray into the spectral theory for infinite compact alphabets, and the current paper answers some of the questions raised in those works. In particular, the following question was raised: what inflation factors $\lambda$ can arise for a compact-alphabet substitution tiling of $\mathbb{R}$ with inflation symmetry (which means that if $x$ is an endpoint of one of the intervals, then so is $\lambda x$). The inflation factor is the expansion factor of the substitution rule of the tiling. It is not too hard to show that for finite alphabets such expansion factors must be algebraic integers, but when the alphabet is infinite there are no obvious restrictions. The main results of the paper establish criteria under which substitutions on infinite compact alphabets have certain statistical properties such as well-defined inflation factors, letter frequencies, and natural tile-length functions. Using the methodology they develop, the authors show that any real number greater than $2$ can be the expansion factor of a substitution tiling on the real line, which furthermore produces a uniformly discrete and relatively dense point set (a Delone set). Finally, an example of a substitution with an explicit transcendental inflation factor is constructed, making nice use of a classical transcendence result of Mahler.

Around Furstenberg's times $p$, times $q$ conjecture: times $p$-invariant measures with some large Fourier coefficients, Discrete Analysis 2024:10, 31 pp. In the 1960s, Furstenberg proved several results on the following theme. Let $p$ and $q$ be integers that are multiplicatively independent, which means that $\log p/\log q$ is irrational, or equivalently that no power of $p$ is equal to a power of $q$. Then if $x$ is an irrational number, we expect its base-$p$ representation and its base-$q$ representation to have no common structure. An example of a rigorous result in this direction is the following landmark theorem of Furstenberg. Let $\mathbb T$ denote the circle $\mathbb R/\mathbb Z$ and let $T_m:\mathbb T\to\mathbb T$ denote multiplication by $m$ mod 1. **Theorem.** *If $p$ and $q$ are multiplicatively independent, then the only infinite closed subset of $\mathbb T$ that is invariant under both $T_p$ and $T_q$ is $\mathbb T$.* To see why the hypotheses make sense, note that if $X$ consists of all multiples of $1/m$ for some $m$, then it is closed and invariant under $T_p$ and $T_q$, but finite, and if $X$ consists of all multiples of some irrational number $x$, then it is infinite and invariant under $T_p$ and $T_q$ but not closed. And finally, if $p^m=q^n$ and $d=\mathop{\text{gcd}}(m,n)$, then $p^{m/d}=q^{n/d}$, so we may assume that $m$ and $n$ are coprime. But $p^{m/n}$ is an integer, so $p^m$ must be a perfect $n$th power, from which it follows that $p$ is a perfect $n$th power, and similarly $q$ is a perfect $m$th power. We thus find $r$ such that $p=r^n$ and $q=r^m$. If we then let $a_10$, if $n$ is sufficiently large then every point in $\mathbb T$ is within $\epsilon$ of a point in $T_q^nX$. It is easy to see that this implies the theorem above. The second was a measure-theoretic version of the theorem, which asserts that if $\mu$ is a probability measure on $\mathbb T$ that is invariant under $T_p$ and $T_q$, then $\mu$ is a convex combination of an atomic measure and Lebesgue measure. (As a moderately interesting example of such a measure, let $r$ is a prime and let $G$ be the multiplicative subgroup of $\mathbb F_r^*$ generated by $p$ and $q$. Then the uniform measure on $G$ is invariant under $T_p$ and $T_q$, as is any convex combination of that measure with Lebesgue measure.) The third relates to the second in a similar way to how the first relates to the theorem. It states that if $\mu$ is an non-atomic $T_p$-invariant probability measure on $\mathbb T$, then the measures $T_q^n\mu$ converge to Lebesgue measure in the weak-star topology. This means that for every continuous function $f$ defined on $\mathbb T$, the integral of $f$ with respect to $T_q^n\mu$ converges to the integral of $f$ with respect to Lebesgue measure. The main aim of this paper is to *disprove* this last conjecture, thus shedding interesting light on what a proof of the $\times p,\times q$ conjecture could be like. In fact, the authors disprove it in a very strong way, by showing not just that there is a counterexample, but by showing that a *typical* non-atomic $T_p$-invariant probability measure is a counterexample, in the Baire-category sense. That is, they show that the set of non-atomic $T_p$-invariant probability measures $\mu$ for which $T_q^n\mu$ converges to Lebesgue measure in the weak-star topology is meagre (meaning that it is contained in a countable union of nowhere dense sets). The authors prove this by showing that $$\limsup_{n\to\infty}|\hat\mu(q^n)|>0$$ for all but a meagre set of non-atomic $T_p$-invariant measures $\mu$. Note that if $\mu$ has this property, then $\int\exp(2\pi i x/q)\mathrm{d}T_q^n\mu$ does not tend to zero, whereas $\int\exp(2\pi ix/q)\mathrm{d}\lambda=0$. A natural class of examples of $T_p$-invariant measures is given by letting $A$ be a subset of $\{0,1,\dots,p-1\}$ and taking the obvious measure on the set of all numbers whose base-$p$ representation have all their digits in $A$, where the digits are independent and uniformly distributed in $A$. (More generally one could take any non-uniform distribution on $\{0,1,\dots,p-1\}$.) It is not hard to see that such sets will have large Fourier coefficients at $p^n$. Bearing this kind of example in mind, it might seem rather strange that one can use the $T_p$-invariance of $\mu$ to say something about the Fourier coefficients $\hat\mu(q^n)$ for some $q$ that is multiplicatively independent of $p$. The key turns out to be that there is an arithmetic progression of integers $N$ such that the powers of $q$ are eventually periodic mod $p^N-1$. (A clue as to why it is natural to reduce mod $p^N-1$ is that rational numbers with denominator $p^N-1$ are fixed points of $T_p^N$.) In fact, weaker conditions turn out to suffice: the authors identify a condition they call assumption (H), and show that if $(c_n)$ is any sequence that satisfies (H), then all but a meagre set of non-atomic $T_p$-invariant measures satisfy $$\limsup_{n\to\infty}|\hat\mu(c_n)|>0.$$ It is not clear whether assumption (H) is necessary: the authors ask the very nice question of whether the same conclusion holds for every integer sequence $(c_n)$ that tends to infinity. The paper also contains multidimensional generalizations of the main results, as well as other interesting open problems.

Random restrictions of high-rank tensors and polynomial maps, Discrete Analysis 2024:9, 24 pp. Let $A$ be an $n\times n$ matrix of rank $r$. The following argument shows that if we randomly restrict $A$ by picking each row and each column independently with positive probability $\sigma$, then the resulting submatrix will be likely to have rank at least $\tau r$ for some $\tau>0$ that depends only on $\sigma$. We note first that $A$ has $r$ independent columns. By standard probabilistic estimates, there is an exponentially small (in $r$) probability that the number $s$ of columns chosen in our random selection is smaller than, say, $\sigma r/2$. The restriction of the matrix to those columns has row-rank $s$, so we can find $s$ linearly independent rows, and by standard arguments again there is an exponentially small probability that fewer than $\sigma s/2$ of those rows are selected. So the probability that we do not end up with a matrix of rank at least $\sigma^2 r/4$ is exponentially small. The aim of this paper is to obtain a similar conclusion for tensors. (A tensor can be thought of as a higher-dimensional matrix. For example, a 3-tensor over a field $\mathbb F$ is a 3-dimensional array of elements of $\mathbb F$.) However, for tensors there are many different notions of rank, each one typically taking a property of matrix rank and generalizing it: it turns out that several properties that are equivalent for matrices have natural generalizations that are not equivalent for tensors. There is another problem that turns out to be more serious. Suppose that we have fixed on a notion of tensor rank. The natural thing to try to do in order to generalize the result about matrices would be to prove a result similar to the result that the rank of a matrix equals its row rank and its column rank. However, this is false: for instance, there exists a 3-tensor with slice-rank 4 with no $4\times 4\times 4$ subtensor of slice-rank 4. (Slice rank is a notion introduced by Tao that played a central role in a reformulation of the solution to the cap-set problem by Ellenberg and Gijswijt, building on work of Croot, Lev and Pach.) Let $T$ be a $d$-tensor over a field $\mathbb F$, defined as a function from $[n_1]\times\dots\times [n_d]$ to $\mathbb F$, where $[n]$ is shorthand for $\{1,2,\dots,n\}$. Given a subset $E\subset[n_i]$ define $T_{i,E}$ to be the restriction of $T$ to $[n_1]\times\dots\times[n_{i-1}]\times E\times[n_{i+1}]\times\dots\times[n_d]$. To obtain a generalization of the matrix statement, it would be sufficient to prove that if $T$ is a $d$-tensor of rank $r$, then for any $i\leq d$ we can find a set $E_i$ of size linear in $r$ such that the restriction $T_{i,E_i}$ has rank $|E_i|$. In fact, a somewhat weaker property would also be sufficient, which the authors call the linear core property. This states that one can find a $t\times\dots\times t$ subtensor of $T$ of rank at least $s$, where $t\leq C\mathop{\text{rk}} T$ and $s\geq c\mathop{\text{rk}} T$ for some constants $C,c>0$. Unfortunately, for many kinds of tensor rank, while it is conjectured that they have the linear core property, this has not been proved. The best result we have in that direction, due to Thomas Karam, is that we can at least obtain such a subtensor for values of $t$ and $s$ that depend only on the rank of $T$ (for many different definitions of rank), but the bounds he obtains are exponential rather than linear. For this reason, the authors of the paper choose a different approach. To deal with the multiplicity of definitions of tensor ranks, they prove a more abstract result that applies to all "natural" definitions, which are ones that satisfy three obvious properties. The first is subadditivity -- that $\mathop{\text{rk}}(S+T)\leq\mathop{\text{rk}}S+\mathop{\text{rk}}T$. The second is monotonicity -- that the rank of any restriction $T_{i,E_i}$ is at most the rank of $T$. And the third is a Lipschitz condition: that if $E\subset F$, then $\mathop{\text{rk}}T_{i,F}-\mathop{\text{rk}}T_{i,E}\leq|F\setminus E|$. The problem is then reduced to one of proving a suitable concentration result for subadditive monotone Lipschitz functions on the collection of subsets of $[n]$, or equivalently on $\{0,1\}^n$ with respect to the biased measure where each element is chosen independently with probability $\sigma$. If such a function has maximum value $r$, then the concentration result the authors prove gives an exponential upper bound for the probability that the value of such a function is smaller than $\sigma r$ multiplied by a suitable constant. There is a close connection between $d$-tensors and multivariable polynomials of degree $d$ -- indeed, associated with a $d$-tensor is a $d$-linear map of which the tensor provides the coefficients, and one can also obtain a (symmetric) $d$-linear map from a polynomial by differencing $d$ times. It is thus natural to ask how well randomly restricting a multivariable polynomial to a dense set of variables preserves its rank (which can be defined as the rank of the associated $d$-linear map or $d$-tensor). Sometimes it is possible to deduce results about polynomials directly from corresponding results about tensors and multilinear maps, but for this result it is more difficult. An issue is that there is no canonical way to represent even a homogeneous polynomial as a multilinear map. (Such a canonical representation does exist when the field has high characteristic, but the result proved here is field-independent.) One might instead consider trying to apply the abstract concentration inequality directly by looking at the function that takes a set $E$ to the rank of the restriction of a given polynomial to $E$, but this function turns out not to be subadditive. Instead, the authors prove another abstract result, making use of ideas from the analysis of Boolean functions, this time concerning monotone Lipschitz functions that are not necessarily subadditive. The original motivation for these results was a problem in complexity theory. Very roughly, the results here can be used to show that bounded-depth and bounded fanin circuits supplemented by mod-2 addition gates of arbitrary fanin are insufficiently powerful to decode corrupted error-correcting codes.

This paper aims to analyze one’s understanding of limit of a function based on Concept Image and Concept Definition schemes, which can be used as powerful tools in research that have mathematical comprehension as part of their object of study. For that, we developed and applied a set of tasks involving limits of functions to students who had already been in a Calculus course. Vinner’s study about concept image and concept definition was the theoretical framework of this paper. The results obtained let us observe the plurality of interpretations about limits and other adjacent concepts, which allowed us to trace, in this paper, a reflection upon the cognitive conflicts concerning the (lack of) understanding of these concepts.

Action recognition in videos is one of the essential, challenging and active area of research in the field of computer vision that adopted in various applications including automated surveillance systems, security systems and human computer interaction. In this paper, we present an in-depth comparative analysis of five CNN-RNN models based on pre-trained networks such as InceptionV3, VGG16, MobileNetV2, ResNet152V2 and InceptionResNetV2 with recurrent LSTM units for action recognition on Anomaly-5 dataset. The performance of these models is analyzed and compared in terms of accuracy, precision, recall & F1-scores and computational efficiency. The CNN-RNN architectures we considered for analysis in this paper, the ResNet152V2 based CNN-RNN model exhibits better performance and achieved highest accuracy, precision, recall and F1-score equal to 92.20% due to its ability to capture more complex spatial features. This comparative analysis may guide the researchers in selecting appropriate models for real-world applications for action recognition. In addition of this, a new dataset is developed called Anomaly-5 that can helps as a valuable resource for training and evaluating action recognition algorithms.

Handwritten Character recognition falls under the domain of image classification that has been under research for years. The idea is to make the machine recognize handwritten human characters. The language focused in this research paper is English while using offline handwritten character recognition for identifying English characters. There are many publically available datasets from which EMNIST is the most challenging one. The main idea of this research paper is to propose a deep learning CNN method to help recognize English characters. This research paper proposes a deep learning convolutional neural network that is tested and compared with renowned pre-trained models using transfer learning. These parametric settings address multiple issues and are finalized after experimentation. The same hyper-parametric settings were used for all the models under test and E-Character with the same data augmentation settings. The proposed model named the E-Character recognizer was able to produce 87.31% accuracy. It was better than most of the tested pre-trained models and other proposed methods by other researchers. This research paper further highlighted some of the problems like misclassification due to the similar structure of characters.

Six permutation patterns force quasirandomness, Discrete Analysis 2024:8, 26 pp. Quasirandomness is a major theme in combinatorics, with several important applications. The broad idea is to say that a combinatorial structure is quasirandom if it has important properties that a random structure of the same type will have with high probability. And what turns this idea from a source of possible definitions into a theory with applications is that it often turns out that quite different-looking properties are equivalent. For example, the following three properties of a graph with $n$ vertices and density $\alpha$ are equivalent (up to adjustments to the error terms). 1. The number of quadruples $(x,y,z,w)$ such that $xy, yz, zw$ and $wx$ are edges of $G$ is approximately $\alpha^4n^4$. 1. For every small graph $H$ with $k$ vertices, the number of functions $f:V(H)\to V(G)$ such that $f(x)f(y)$ is an edge of $G$ for every edge $xy$ of $H$ is approximately $\alpha^{|E(H)|}n^k$. 1. For every set $X$ of vertices of $G$, the number of edges $xy$ with $x,y\in X$ differs from $\alpha\binom{|X|}2$ by at most a small multiple of $n^2$. The fact that the first property implies the third is particularly useful because often it is easier to verify the first but also easier to deduce consequences from the third. The theory of quasirandom graphs was developed by Thomason and by Chung, Graham and Wilson. Later, similar equivalences were discovered for other combinatorial structures, such as hypergraphs and subsets of finite Abelian groups. This paper is about a notion of quasirandomness for permutations, originally introduced by Joshua Cooper. As with graphs, it can be defined in several equivalent ways. One central property of a permutation $\pi$ of $\{1,2,\dots,n\}$ is known as _balance_: we say that $\pi$ is balanced if for every pair of intervals $I$ and $J$, $$|I\cap\pi(J)|-|I||J|/n|$$ is at most a small multiple of $n$. That is, $\pi$ spreads every interval out so that it is evenly distributed in $\{1,2,\dots,n\}$. Another, more relevant to this paper, concerns permutation patterns. Given some fixed permutation $\sigma$ of $\{1,2,\dots,k\}$, we say that $\sigma$ _occurs in_ $\pi$ _at_ $(x_1,\dots,x_k)$ if $x_1

New bounds in the discrete analogue of Minkowski's second theorem, Discrete Analysis 2024:7, 6 pp. Minkowski's first theorem in the geometry of numbers states that if a closed centrally symmetric body $K$ in $\mathbb R^n$ has volume at least $2^n$, then it contains a non-zero point in $\mathbb Z^n$. The proof is simple: if $K$ does not contain such a point, then all translates of $K/2$ by a vector with integer coordinates are disjoint (since if $x+K/2$ intersects with $y+K/2$, then $x-y\in K$). But the volume of $K/2$ is $2^{-n}$ times the volume of $K$, and also equal to the density of the union of all the integer translates of $K/2$, and therefore at most 1. More generally (but equivalently), if $\Lambda$ is any lattice in $\mathbb R^n$ and $K$ is a closed centrally symmetric body of volume at least $2^n\det\Lambda$, then $K$ contains a non-zero point of $\Lambda$. (The determinant of a lattice is the volume of a fundamental parallelepiped.) It is easy to see that the bound given by Minkowski's first theorem can be very far from sharp. For instance, if $K$ is a very thin cylinder about the first coordinate axis that does not quite touch the points $(\pm 1,0,0,\dots,0)$, then the theorem implies that $K$ has volume at most $2^n$, when in fact it can be arbitrarily small. On the other hand, if $K$ is the cube $[-t,t]^n$ for $t$ very close to 1, then $K$ has volume very close to $2^n$, so the result cannot be improved without more refined assumptions about $K$. Minkowski's second theorem relates the volume of $K$ to quantities known as _successive minima_. If $\lambda$ increases from zero, then the convex body $\lambda K$ will at first contain no non-zero points of $\Lambda$. The first successive minimum $\lambda_1$ is the minimal $\lambda$ such that $\lambda$ contains a non-zero point of $\Lambda$, which exists by Minkowski's first theorem, and in general the $k$th successive minimum is the minimal $\lambda$ such that $\lambda K$ contains at least $k$ linearly independent points of $\Lambda$. If $\Lambda=\mathbb Z^n$ and $K$ is the cuboid $\prod_{i=1}^n[-t_i,t_i]$ with $t_1\geq t_2\geq\dots\geq t_n$, then $\lambda_i=1/t_i$, so the volume of $K$ is $2^n/\prod_i\lambda_i$. Minkowski's second theorem states that this is an upper bound for a general symmetric convex body and a general lattice, except that one must of course again introduce the normalizing factor $\det\Lambda$. That is, $\mathop{\text{vol}} K\leq 2^n\det\Lambda/\prod_i\lambda_i$. This paper, as is clear from its title, is about a discrete version of Minkowski's second theorem. What makes it discrete is that the quantity being bounded above is no longer the volume of $K$ but the number of lattice points that $K$ contains. Note that in this instance we no longer expect $\det\Lambda$ to appear in the bound. It was conjectured by Betke, Henk and Wills that one should have the upper bound $$|K\cap\Lambda|\leq\prod_{i=1}^n\lfloor 2/\lambda_i+1\rfloor.$$ Later Malikiosis conjectured that the same bound should apply even if one drops the assumption that $K$ is centrally symmetric, where one defines the successive minima of a non-symmetric convex body $K$ to be those of the symmetric convex body $(K-K)/2$. Malikiosis proved that the conjecture is correct up to an explicit factor that grows exponentially with $n$ (which is quite reasonable given that scaling an $n$-dimensional convex body up by $\lambda$ increases its volume by a factor $\lambda^n$), and Freyer and Lucas proved an upper bound of $\prod_{i=1}^n(2/\lambda_i+d)$, which is correct up to a factor $1+o(1)$ when all the $\lambda_i$ converge to zero, reflecting the fact that the discrete problem is a better and better approximation to the continuous one when the successive minima get smaller and smaller. This paper gives a new bound that improves the bounds of Malikiosis, and in some cases improves the bound of Freyer and Lucas as well. If the successive minima are $\lambda_1\leq\dots\leq\lambda_n$, then the upper bound obtained in the paper is $$2^k(1+\lambda_k/2)^k/\lambda_1\dots\lambda_k,$$ where $k$ is maximal such that $\lambda_k\leq 1$. (In fact, the paper proves a slightly better bound than this that is more complicated to define.) Note that when the successive minima are all small, then $k=n$, and the bound is close to $2^k/\lambda_1\dots\lambda_n$, so like the result of Freyer and Lucas this argument approximates the continuous one for bodies that are small in every direction. In fact, both bounds straightforwardly imply Minkowski's second theorem, as one can see by applying them to the convex body $rK$ and letting $r$ tend to infinity. The proof is closely modelled on a proof by Tao and Vu of Minkowski's second theorem, with a twist at the end.

Boolean functions with small approximate spectral norm, Discrete Analysis 2024:6, 22 pp. Let $G$ be a finite Abelian group and let $f:G\to\mathbb C$. A quantity of considerable interest in many contexts is the _spectral norm_ of $f$, that is, the $\ell_1$ norm of the Fourier transform $\hat f$. One way to interpret this is as a measure of "how much randomness" $f$ exhibits, since by the duality of $\ell_1$ and $\ell_\infty$ we can rewrite it as $$\max\{|\langle f,g\rangle|:\|\hat g\|_\infty\leq 1\}.$$ The quantity $\|\hat g\|_\infty$ is small if and only if every Fourier coefficient of $g$ is small, which can be interpreted as saying that $g$ is "arithmetically random" in a certain sense. Therefore, if there is some small $\delta>0$ such that $|\langle f,g\rangle|\leq\delta\|\hat g\|_\infty$ for every $g$, then we learn that not only is $f$ not random in this sense, but it does not even correlate with any random function. One expects functions that are "everywhere non-random" in a sense like this to have some kind of interesting structure, and there are some interesting results that show that various precise versions of this intuition are indeed true. Of particular relevance to this paper is results about functions that take values in $\{0,1\}$, and what it says about such a function if its spectral norm is bounded. If $f$ is the characteristic function of a subgroup $H$ of $G$ and $\chi:G\to\mathbb C$ is a character, then $\hat f(\chi)=0$ unless $\chi$ vanishes identically on $H$, in which case it is $|H|/|G|$. Using the structure theorem for finite Abelian groups, one can show quite easily that the number of characters that do not vanish identically on $H$ is $|G|/|H|$, from which we deduce that $\|\hat f\|_1=1$ for such functions. From that we can deduce that any 01-valued function that can be expressed as a linear combination $\sum_i\lambda_i\mathbf 1_{H_i}$ has spectral norm at most $\sum_i|\lambda_i|$. Also, since we have the simple inequality $\|\hat f*\hat g\|_1\leq\|\hat f\|_1\|\hat g\|_1$, the convolution law implies that if $f$ and $g$ have bounded spectral norm, then so does their pointwise product $fg$. A theorem known as Cohen's idempotent theorem gives the following converse to the above observations. If $G$ is a finite Abelian group (in fact, the theorem is more general and applies to locally compact Abelian groups), and $f:G\to\{0,1\}$ is a function with bounded spectral norm, then $f$ is the characteristic function of a set built out of a bounded number of characteristic functions of cosets of subgroups of $G$ using Boolean operations, where the second bound depends only on the first. This theorem was given a second proof by Green and Sanders, who obtained a reasonable quantitative estimate for the dependence -- Cohen's proof was qualitative. The number of cosets needed is referred to as the _coset complexity_ of $f$. This paper addresses the following question: what if instead of asking for $f$ to have a small spectral norm, we ask only that it should be _approximated_ (uniformly) by a function with small spectral norm? It is not immediately obvious that this gives us any new examples, but the authors show in the paper (see Lemma 3) that if $f:\mathbb F_2^k\to\{0,1\}$ is the function that takes the value 1 at $x$ if and only if $x$ has at most one non-zero coordinate, then the spectral norm of $f$ is at least $\sqrt k/2$, but for every $\epsilon\in(0,1/2)$ there is a function $g$ with spectral norm at most $5\log(1/\epsilon)$ such that $\|g-f\|_\infty\leq\epsilon$. Thus, for any fixed $\epsilon$ in this range we can find a sequence of functions $f_k$ such that $f_k$ has unbounded spectral norm but can be uniformly approximated to within $\epsilon$ by a sequence of functions $g_k$ with bounded spectral norm. In the light of this example, what can one hope to prove about Boolean functions defined on $\mathbb F_2^n$ with small approximate spectral norm? The best one could hope for is to show that the above example is essentially the only obstacle, and that is exactly the type of result that the authors show. More precisely, they define a set $A$ to be $k$-_affine connected_ if there does not exist a $k$-dimensional coset of a subspace that intersects $A$ in a set of $k+1$ affinely independent vectors. It is straightforward to show that if either $A$ or $A^c$ is not $k$-affine connected, then the spectral norm of the characteristic function of $A$ is at least $\sqrt k$. The authors prove a much less obvious converse to this (Theorem 2 in the paper), which implies that if $A$ and $A^c$ are both $k$-affine connected and the characteristic function of $A$ has bounded approximate spectral norm, then $A$ has bounded coset complexity (where the bound depends on $k$, the $\epsilon$ that comes from the approximation, and the size of the $\epsilon$-approximate spectral norm -- note that the result becomes stronger as $\epsilon$ approaches 1/2). Some of the proof follows similar lines to that of Green and Sanders, but at a crucial point a new approach is needed, which takes the form of a sophisticated inductive argument. Thus, among other things this paper provides a new proof of the result of Green and Sanders. Unfortunately, where Green and Sanders obtain exponential bounds, the bounds in this paper are of tower type: it is an interesting open question to determine whether those are necessary. The concluding section of the paper discusses this and other interesting questions that arise naturally out of the paper, as well as placing the results in a wider complexity-theory context.

To further improve the spectral efficiency of the cooperative non-orthogonal multiple access (NOMA) network with half-duplex (HD) relaying and avoid the zero diversity gain due to the imperfect self interference estimation at high signa-to-noise ratio (SNR) value with full-duplex (FD) relaying, this paper developed a novel two-stage relay selection for a two-user multi-relay cooperative NOMA systems with adaptive-duplex (AD) relaying which can switch opportunistically between HD and FD relaying modes. An adaptive duplex based on fixed power allocation factor relay selection (AD-FPA-RS) scheme for cooperative NOMA networks is proposed in this paper, the relay can switch the work mode to the FD or HD state adaptively according to the current channel state information and the result of decode at the relay. Futhermore, the exact outage probability for the proposed AD-FPA-RS scheme is obtained in closed-form expression. The simulation and numerical results show that the proposed AD-FPA-RS scheme not only outperforms the existing HD-FPARS and FD-FPA-RS schemes in terms of outage behavior significantly, but also can obtain full diversity gain without the floor outage performance at high SNR region.

In [1], Craig and Goodman develop the geometrical optics solution of the linearized Korteweg-deVries equation away from caustic, or turning, points. Here we develop an analogous solution valid at caustic points.

Neutrosophic set is a generalization of classical set, fuzzy set, and intuitionistic fuzzy set by employing a degree of truth (T), a degree of indeterminacy (I), and a degree of falsehood (F) associated with an element of the dataset. One of the most essential problems is studying set-theoretic operators in order to be applied to practical applications. Developing Matlab toolboxes for computing the operational matrices in neutrosophic environments is essential to gain more widely-used of neutrosophic algorithms. In this paper, we propose some computing procedures in Matlab for neutrosophic operational matrices, especially i) computing the single-valued neutrosophic matrix; ii) determining complement of a single-valued neutrosophic matrix; iii) computing max-min-min and min-max-max of two single-valued neutrosophic matrices; v) computing power of a single-valued neutrosophic matrix; vi) computing additional operation and subtraction of two single-valued neutrosophic matrices; and ix) computing transpose of a single-valued neutrosophic matrix. Numerical examples are given to illustrate their applicability.

The mathematical model of wood drying based on a detailed transport phenomena considering both heat and moisture transfer have been offered in the article [5]. We apply this model to quickly moving paper sheet. The range of the moisture content correspond to the period of drying and only vapor movement in the web is possible. By averaging we have obtained the desired model as a system of two nonlinear first order ordinary differential equations. Popieriaus džiovinimo matematinis modelis Santrauka Straipsnyje remiamasi [5] darbu, kuriame buvo pasiūlytas detalus šilumos ir dregmes perdavimo medyje matematinis modelis. Autorius pritaiko šiuos rezultatus popieriaus gamybos technologinio ciklo modeliavimui. Nagrinejamas greitai judančios popieriaus mases džiovinimo procesas. Aptarti matematiniai modeliai, aprašantys temperatūros ir dregmes pokyčius ivairiose džiovinimo stadijose. Gautoji diferencialiniu lygčiu sistema vidurkinimo metodu pertvarkoma i seka dvieju paprastuju diferencialiniu lygčiu sistemu, kur kiekviena sistema atitinka viena džiovinimo cilindra. First Published Online: 14 Oct 2010

The class of [Formula: see text]-free graphs has been well studied in the past. In the paper “On the chromatic number of [Formula: see text]-free graphs, Discrete Applied Mathematics, 253 (2019), 14–24”, it was shown that the class of [Formula: see text]-free graphs and [Formula: see text]-free graphs admit a linear χ-binding function. In this paper, we study some subclasses of [Formula: see text]-free graphs which is a superclass of [Formula: see text]-free graphs. We show that [Formula: see text]-free graphs and [Formula: see text]-free graphs also admit linear χ-binding functions. In addition, we give a tight χ-binding function for [Formula: see text]-free graphs and improve the χ-bound for [Formula: see text]-free graphs with ω = 4.

Let G be a finite group, let Irr(G) be the set of all complex irreducible characters of G and let cd(G) be the set of all degrees of characters in [Formula: see text] Let [Formula: see text] be the set of all primes that divide some degrees in [Formula: see text] The character degree graph [Formula: see text] of G is the simple undirected graph with vertex set [Formula: see text] and in which two distinct vertices p and q are adjacent if there exists a character degree [Formula: see text] such that r is divisible by the product pq. In this paper we obtain a necessary condition for the character degree graph [Formula: see text] of a solvable group G to be eulerian. We also prove that every n – 2 regular graph on n vertices where n is even and [Formula: see text] is the character degree graph of some solvable group. In the last part of the paper we give a bound for the number of Eulerian character degree graphs in terms of number of vertices.

In this paper, we consider a general tensor complementarity problem with interval parameters, and study the conditions under which, the existence and uniqueness of the solution of the problem are guaranteed. Furthermore, we proved that the solution set of the interval tensor complementarity problem is not necessarily convex. 1. IntroductionInterval analysis is a branch of numerical analysis that was born in the 1960's. It consists of computing with intervals of reals instead of reals, providing a framework for handling uncertainties and verified computations. The result of an interval computation is an interval, a pair of numbers, an upper and a lower bound, and this pair of numbers guarantees to enclose the exact answer. Maybe we still don’t know the truth, but at least we know how much we don’t know! [4]. How is it possible for interval analysis to guarantee that a computational result is true? The answer is very simple. Using the interval analysis we estimate at each calculation step all kinds of errors: inputs errors, rounding errors and truncation errors. One of the most famous references on IA is probably Moore’s Interval Analysis book. Throughout the paper, vectors are written as $\left\{x, y, \ldots \right\}$, matrices are shown by $ \left\{A, B,\ldots \right\}$ and tensors are written as $ \left\{ \mathcal{A}, \mathcal{B},\ldots \right\}.$ Let $[n]$, $\mathbb{R} (\mathbb{C}) $, and $\mathbb{R}^n (\mathbb{C}^n)$ denote the set $\{ 1, 2,\ldots,n\}$, the set of all real (complex) numbers, and the set of all $n$-dimensional real (complex) vectors; respectively. $x \geq 0 \ \ \ (x > 0)$ means $x_i \geq 0 \ \ \ (x_i > 0)$ for all $i \in \left[ n \right]$. Let $\mathbb{R} _{+}^n = \lbrace x \in \mathbb{R}^n \mid x \geq 0\rbrace$ be the positive cone in $\mathbb{R}^n.$ An order $m$ dimension $n$ real tensor $\mathcal{A} = (a_{i_1 i_2\cdots i_m}),$ denoted by $ \mathcal{A} \in \mathbb{R}^{n_1 \times \cdots \times n_m } ,$ consists of $n^m$ entries: \[a_{i_1 i_2 \cdots i_m} \in \mathbb{R}, \quad \; \forall \; i_j = 1,\cdots,n ,\quad j = 1,\cdots,m.\] If $n_1=\cdots=n_m = n$, then it is said $\mathcal{A}$ is an $m$-order $n$-dimensional cubical tensor or for simplicity just $m$-order $n$-dimensional tensor. A vector is a tensor of order $1$ and a matrix is a tensor of order $2$. A tensor $\mathcal{A} = (a_{i_1 i_2 \cdots i_m}) \in \mathbb{R}^{n_1 \times \cdots \times n_m } $ is called nonnegative (positive) if \[a_{i_1 i_2 \cdots i_m} \ge 0 \; (a_{i_1 i_2 \cdots i_m}>0 ), \quad \; \forall \; i_j = 1,\cdots,n ,\quad j = 1,\cdots,m.\] A tensor $ \mathcal{A}$ is said to be symmetric if its entries $ a_{i_1 i_2 \cdots i_m}$ are invariant under any permutation of $ m $ indices $ ( a_{i_1 i_2 \cdots i_m}). $ All the tensors discussed in this paper are real.\\For any two tensors, $\mathcal{A} = (a_{i_1 \cdots i_m } ),$ and $ \mathcal{B} = (b_{i_1 \cdots i_m } ) \in \mathbb{R}^{n_1 \times \cdots \times n_m } $ of identical orders and dimensions, their inner product is defined as $$\left\langle {\mathcal{A},\mathcal{B}} \right\rangle = \sum\limits_{i_1 \cdots i_m } {a_{i_1 \cdots i_m } b_{i_1 \cdots i_m}}. $$ Definition 1.1. If $\mathcal{A} \in \mathbb{R}^{n_1 \times \cdots \times n_m } $ is an $m$-order tensor and $ B \in \mathbb{R}^{J \times n_k }$ is a matrix, then $\mathcal{A} \times_k B $ denotes the mode-$k$ product of $\mathcal{A}$ with $B$, which is of size $ n_1 \times \cdots \times n_{k-1} \times J \times n_{k+1} \times \cdots \times n_m $, and each element of it is defined as follows \[(\mathcal{A} \times_k B)_{i_1,\ldots,i_{k-1}, j, i_{k+1},\ldots,i_m} = \sum\limits_{i_k = 1}^{n_k } a_{i_1 \cdots i_m } b_{j,i_k}.\]If we do the mode-$k$ product of $ \mathcal{A} $ and $B $ for all possible $k \in [m]$ as \[ \mathcal{A} \times_1 B \times_2 \cdots \times_m B,\] and $B $ is reduced to some row vector, say $x^T=\left( x_1,\ldots,x_n \right),$ the following notations are frequently used in this paper:\begin{align*}&\mathcal{A} x^m \equiv \mathcal{A} \times_1 x^T \times_2 \cdots \times_m x^T = \sum\limits_{i_1 \cdots i_m =1}^n {a_{i_1 \cdots i_m} x_{i_1} \cdots x_{i_m}} \in \mathbb{R}, \\&\mathcal{A} x^{m-1 } \equiv \mathcal{A} \times_2 x^T \times_3 \cdots \times_m x^T = \sum\limits_{i_2 \cdots i_m =1}^n {a_{i, i_2 \cdots i_m} x_{i_2} \cdots x_{i_m}} \in \mathbb{R}^n.\end{align*}We call a number $\lambda \in \mathbb{C}$ an eigenvalue of $\mathcal{A}$ if it and a nonzero vector $x \in \mathbb{C}^n$ are solutions of the following homogeneous polynomial equations:(1.1)\begin{equation}\label{e3}\left( \mathcal{A} x^{m-1} \right)_i = \lambda x_i^{m - 1}, \ \ \ \forall i=1,\ldots,n,\end{equation}and call the solution $x$ an eigenvector of $\mathcal{A}$ associated with the eigenvalue $\lambda.$ If we denote $x^{[m-1]}$ as a vector in $\mathbb{C}^n$ such that its $i$th component is $x_i^{m - 1},$ then (1.1) can be simply expressed as $$\mathcal{A} x^{m-1} = \lambda x^{[m-1]}.$$ The set of all the eigenvalues of $\mathcal{A}$ is called the spectrum of $\mathcal{A}.$ The largest modulus of the elements in the spectrum of $\mathcal{A}$ is called the spectral radius of $\mathcal{A},$ denoted as $\rho(\mathcal{A}).$ Definition 1.2. Let $\mathcal{A}$ be an $m$-order and $n$-dimensional cubical tensor, then$(1)$ $\mathcal{A}$ is called an $\mathcal{Z}$-tensor if all of its non-diagonal elements are non-positive. This definition is equivalent to having $\mathcal{A}=s\mathcal{I}-\mathcal{B}$, where $s > 0$, $\mathcal{B}$ is a non-negative tensor and $\mathcal{I}=(I_{i_1 \cdots i_m })$, is the identity tensor with entries$$I_{i_1 \cdots i_m}=\left\{ \begin{gathered}1, \ \ \ i_1=\cdots= i_m,\hfill \\0, \ \ \ otherwise. \hfill \\\end{gathered} \right.$$$(2)$ $\mathcal{A}$ is called an $\mathcal{M}$-tensor if $\mathcal{A}$ is an $\mathcal{Z}$-tensor and $\mathcal{A} = s\mathcal{I} - \mathcal{B}$, $s \geq \rho(\mathcal{B})$. If $s > \rho(B)$, then $\mathcal{A}$ is called a strong(nonsingular) $\mathcal{M}$-tensor. Definition 1.3. An $m$-order $n$-dimensional tensor $\mathcal{A}$ is said to be P-tensor, if for each nonzero $x \in \mathbb{R}^n $, there exists some index $i$such that(1.2)\begin{equation}\label{e4}x_i \left( {\mathcal{A}x^{m - 1}} \right)_i >0.\end{equation}The tensor $\mathcal{M}(\mathcal{A} )=(m_{i_1 \cdots i_m})$ is called the comparison tensor of $\mathcal{A}$ if$$ m_{i_1 \cdots i_m}= \begin{cases}-\left|a_{i_1 \cdots i_m}\right|, & \text { if } (i_2,\cdots,i_m) \neq (i_1,\cdots,i_1) \\ \left|a_{i_1 \cdots i_m}\right|, & \text { if } (i_2,\cdots,i_m)=(i_1,\cdots,i_1)\end{cases} $$ Definition 1.4. A tensor $\mathcal{A}$ is called an $\mathcal{H}$-tensor, if its comparison tensor is an $\mathcal{M}$-tensor, and it is called a nonsingular $\mathcal{H}$ -tensor, if its comparison tensor is nonsingular. The tensor complementarity problem denoted by the $\operatorname{TCP}(\mathbf{q}, \mathcal{A})$, is to find a vector $\mathbf{x}$ such that:$$ \mathbf{x} \geq 0, ~\mathcal{A} \mathbf{x}^{m-1}+\mathbf{q} \geq 0,~\left\langle\mathbf{x}, \mathcal{A} \mathbf{x}^{m-1}+\mathbf{q}\right\rangle=0, $$ where $\left\langle , \right\rangle $ denotes the inner product. Theorem 1.5. [6] Tensor $\mathcal{A}$ is $P$-tensor if and only if TCP(q, $\mathcal{A}$) has a unique solution for every $q > 0.$ 2. Main ResultsInterval linear algebra is a mathematical field developed from classical linear algebra. The only difference is that we do not work with real numbers but we deal with the real closed intervals $x^I : = \left[ {\underline{x}, \overline{x}} \right]$, where $\underline{x} \leq \overline{x}.$ An interval tensor is a tensor which every of its elements is interval. An $m$-order $n$-dimensional cubical interval tensor is denoted by $\mathcal{A}^I : = \left[{\underline{\mathcal{A}}, \overline{\mathcal{A}} } \right]$, where${\underline{ \mathcal{A}}}$ and ${\overline {\mathcal{A}}}$ are real tensors, and \[\mathcal{A}^I (i_1,\cdots,i_m ) = \left[ {\underline {\mathcal{A}} (i_1,\cdots,i_m), \overline {\mathcal{A}} (i_1,\cdots,i_m )} \right].\] The set of all interval tensors of size $ n_1 \times \cdots \times n_m$, is denoted by $\mathbb{I}\mathbb{R}^{n_1 \times \cdots \times n_m}. $ The parametric form of interval, interval vector and interval tensor can be expressed as follows, respectively. $${a^I=\left\{a(t) \in \mathbb{R}: a(t)=\underline{a}+t\left(\overline{a}-\underline{a}\right), \underline{a} \in \mathbb{R}, \overline{a} \in \mathbb{R}, t \in[0,1]\right\},}$$ $${x^I=\left\{x(t) \in \mathbb{R}^{n}: x(t)=\underline{x}+t\left(\overline{x}-\underline{x}\right), \underline{x} \in \mathbb{R}^{n}, \overline{x} \in \mathbb{R}^{n}, t \in[0,1]\right\},} $$ $${\mathcal{A}^I=\left\{\mathcal{A}(t) \in \mathbb{R}^{n \times n \cdots \times n}: \mathcal{A}(t)=\underline{\mathcal{A}}+t\left(\overline{\mathcal{A}}-\underline{\mathcal{A}}\right), \underline{\mathcal{A}} \in \mathbb{R}^{n \times n \cdots \times n}, \overline{\mathcal{A}} \in \mathbb{R}^{n \times n \cdots \times n}, t \in[0,1]\right\}.}$$ Definition 2.1. For $\mathcal{A}^I=[a_{i_1 \cdots i_m}] \in \mathbb{I R}^{n \times n \cdots \times n}$, we define the comparison tensor of $\mathcal{A}^I$ and it is represented as $\mathcal{M}(\mathcal{A}^I )=(m_{i_1 \cdots i_m}) \in \mathbb{R}^{n \times n \cdots \times n}$ by setting$$ m_{i_1 \cdots i_m}= \begin{cases}-\left|[a_{i_1 \cdots i_m}]\right|, & \text { if } (i_2,\cdots,i_m) \neq (i_1,\cdots,i_1) \\ \left|[a_{i_1 \cdots i_m}]\right|, & \text { if } (i_2,\cdots,i_m)=(i_1,\cdots,i_1)\end{cases}$$ Definition 2.2. An interval tensor $\mathcal{A}^I \in \mathbb{I R}^{n \times n \cdots \times n}$ is called an interval $p$-tensor, $\mathcal{Z}$-tensor, $\mathcal{M}$-tensor and $\mathcal{H}$-tensor, if all $\mathcal{A} \in\mathcal{A}^I$ are $p$-tensor, $\mathcal{Z}$-tensor, $\mathcal{M}$-tensor and $\mathcal{H}$-tensor, respectively. Let $\mathcal{A}^I $ be an $m$-order and $n$-dimensional interval tensor and $q^I \in \mathbb{I R}^{n}$ be an $n$ dimensional interval vector. Then we consider the family of $T CP(\mathcal{A}, q)$'s$$q+\mathcal{A} x^{m - 1} \geq 0, \quad x \geq 0, x^ T ( q+\mathcal{A} x^{m - 1} ) = 0, ~\text {where}~ \mathcal{A} \in\mathcal{A}^I, q^I$$This is equivalent to the following family of $T C P(\mathcal{A}, q)$'s$$w-\mathcal{A} x^{m - 1}=q, \quad x \geq 0, w \geq 0, x^ T w=0, \text { where } \mathcal{A} \in\mathcal{A}^I, q^I.$$The family of $T C P(\mathcal{A}, q)$'s is represented as interval tensor complementarity problem and it is denoted by $\operatorname{ITCP}(\mathcal{A}^I, q^I) \cdot \sum_{x}(\mathcal{A}^I,q^I)$ is denoted as solutions set of $\operatorname{ITCP}(\mathcal{A}^I, q^I)$ and it is defined as $$\left\{x \in \mathbb{R}^{n}: q+ \mathcal{A} x^{m - 1} \geq 0, x \geq 0, x^ T (q+ \mathcal{A} x^{m-1}) =0, \mathcal{A} \in\mathcal{A}^I, q^I\right\}$$ The parametric form of $\operatorname{ITCP}(\mathcal{A}^I,q^I)$ is represented as $T C P(\mathcal{A}(t), q(t)), t \in[0,1]$ and its solution set is defined as for some fixed $t,$$$\sum_{x}(\mathcal{A}(t), q(t))=\left\{x \in \mathbb{R}^{n}: q(t)+\mathcal{A}(t) x^{m - 1} \geq 0, x \geq 0,x^{T} (q(t)+\mathcal{A}(t) x^{m - 1}) =0\right\}$$ Lemma 2.3. For any $\operatorname{ITCP}(\mathcal{A}^I,q^I), \sum_{x}\left(\underline{\mathcal{A}}, \underline{q}\right)$ and $\sum_{x}\left(\overline{\mathcal{A}}, \overline{q}\right)$ are subsets of $\sum_{x}(\mathcal{A}^I,q^I)$. Theorem 2.4. Suppose that $T C P\left(\underline{\mathcal{A}}, \underline{q}\right)$ and $T C P\left(\overline{\mathcal{A}}, \overline{q}\right)$ have unique solutions, then $\sum_{x}(\mathcal{A}(t), q(t))$ is a singleton set, for every $t \in[0,1]$. Proof. Suppose that $T C P\left(\underline{\mathcal{A}}, \underline{q}\right)$ and $T C P\left(\overline{\mathcal{A}}, \overline{q}\right)$ have unique solutions, then from Theorem \ref{pp} we get that $\underline{\mathcal{A}}$ and $\overline{\mathcal{A}}$ are $P$-tensors. Since the positive convex combination of $P$-tensors is also a $P$-tensor, then $\mathcal{A}(t)=t \underline{\mathcal{A}}+(1-t) \overline{\mathcal{A}}$, $t \in[0,1]$, is a $P$-tensor. Hence $\operatorname{TCP}(\mathcal{A}(t), q(t))$ has unique solution for each $t$. Therefore, $\sum_{x}(\mathcal{A}(t), q(t))$ is a singleton set, for every $t \in[0,1]$.Theorem 2.5. The set $\sum_{(w, x)}(\mathcal{A}^I,q^I)$ is not convex.Proof. Let $\left(w^{1}, x^{1}\right)^{T},\left(w^{2}, x^{2}\right)^{T} \in \sum_{(w, x)}(\mathcal{A}^I,q^I)$. Then,(2.1)$$\begin{gathered}& w_{i}^{j}-\sum_{i_2 \cdots i_m=1}^{n} \overline{a}_{i i_2 \cdots i_m} x_{i_2}^{j} \cdots x_{i_m}^{j} \leq \overline{q}_{i}, w_{i}^{j}-\sum_{i_2 \cdots i_m=1}^{n} \underline{a}_{i i_2 \cdots i_m} x_{i_2}^{j} \cdots x_{i_m}^{j} \geq \underline{q}_{i},\\ & x_{i_2}^{j} \cdots x_{i_m}^{j} \geq 0, w_{i}^{j} \geq 0, w_{i}^{j} x_{i}^{j}=0, i_1, \cdots, i_m=1, 2,\ldots,n, ~ j=1,2.\end{gathered}$$Let $0 \leq \lambda \leq 1$. Then multiplying (2.1) by $\lambda$ for $j=1$, and multiplying (2.1) by $(1-\lambda)$ for $j=2$, and adding up these inequalities, we get$$\begin{gathered}\left(\lambda w_{i}^{1}+(1-\lambda) w_{i}^{2}\right)-\sum_{i_2 \cdots i_m=1}^{n} \overline{a}_{i i_2 \cdots i_m}\left(\lambda x_{i_2}^{1} \cdots x_{i_m}^{1}+(1-\lambda) x_{i_2}^{2} \cdots x_{i_m}^{2}\right) \leq \overline{q}_{i}, i=1,2, \ldots, n, \\\left(\lambda w_{i}^{1}+(1-\lambda) w_{i}^{2}\right)-\sum_{i_2 \cdots i_m=1}^{n} \underline{a}_{i i_2 \cdots i_m}\left(\lambda x_{i_2}^{1} \cdots x_{i_m}^{1}+(1-\lambda) x_{i_2}^{2} \cdots x_{i_m}^{2}\right) \geq \underline{q}_{i}, i=1,2, \ldots, n, \\\left(\lambda x_{i_2}^{1} \cdots x_{i_m}^{1}+(1-\lambda) x_{i_2}^{2} \cdots x_{i_m}^{2}\right) \geq 0,\left(\lambda w_{i}^{1}+(1-\lambda) w_{i}^{2}\right) \geq 0, i_1,\cdots,i_m=1, 2,\ldots,n.\end{gathered}$$On the other hand, $\left(\lambda w_{i}^{1}+(1-\lambda) w_{i}^{2}\right)\left(\lambda x_{i_2}^{1} \cdots x_{i_m}^{1}+(1-\lambda) x_{i_2}^{2} \cdots x_{i_m}^{2}\right)=0$ only for $t=0$ and $t=1$. Since convex combination of complementarity variables are not complementarity variables. This gives that $\left(\lambda w^{1}+(1-\lambda) w^{2}, \lambda x^{1}+(1-\lambda) x^{2}\right)^{T} \notin \sum_{(w,x)}(\mathcal{A}^I,q^I)$, and so $\sum_{(w, x)}(\mathcal{A}^I,q^I)$ is not a convex set. Theorem 2.6. Let $\mathcal{A}^I $ be an $m$-order and $n$-dimensional interval $H$-tensor. Then, $\operatorname{ITCP}(\mathcal{A}^I,q^I)$ has a solution for every $q^I \in \mathbb{I R}^{n}$, that is, $\sum_{x}(\mathcal{A}^I,q^I)$ is a non-empty set. Proof. Let $\mathcal{A}^I $ be an $m$-order and $n$-dimensional interval $H$-tensor. Then each $\mathcal{A} \in\mathcal{A}^I$ is an $H$-tensor, and so $T C P(\mathcal{A}, q)$ has a solution. Also, $\sum_{x}(\mathcal{A}, q) \subseteq \sum_{x}(\mathcal{A}^I,q^I)$, which yields that $\sum_{x}(\mathcal{A}^I,q^I)$ is non-empty. 3. ConclusionA methodology is developed to discuss the existence of the solution of tensor complementarity problem, where the parameters are closed intervals. We also proved that the solution set of the interval tensor complementarity problem is not necessarily convex.

In this paper, we study the definitions of ordered pair from the point of view of Kuratowski, Wiener, Hausdorff and Morse. We first present the possible definitions of an ordered pair, and then examine the structure of the Cartesian product of sets, relations, and functions. This paper can be useful for mathematical researchers in creating new insight and better understanding of the concept of ordered pair and its related mathematical concepts, such as ordered and unordered Cartesian product, relation and function, etc.

In this paper, which is the first paper from a trio on important developments of low dimensional topology in the past 100 years, we review the history and major developments in knot theory. This historic account includes the initial attempts at formulating some of the main questions about knots in a mathematical language, putting the definitions and arguments in a rigorous mathematical framework, and employing tools from other fields of mathematics to extract interesting and intriguing results in knot theory. The study starts from Tait's work in nineteenth century and reviews the important steps taken before the introduction of gauge theory. In particular, we will review the prime decomposition of knots and various polynomial invariants constructed for knots and links. We finish the paper by discussing some of the important conjectures in knot theory which have surprisingly simple statement. We also review some of the recent developments around the aforementioned conjecture, including some theorems including contributions from the author.

The main purpose of the present paper is to prove the existence of periodic solutions of the three-body problem in the 3D Kepler formulation. We have solved the same problem in the case when the three particles are considered in an external inertial system. We start with the three-body equations of motion, which are a subset of the equations of motion (previously derived by us) for any number of bodies. In the Minkowski space, there are 12 equations of motion. It is proved that three of them are consequences of the other nine, so their number becomes nine, as much as the unknown trajectories are. The Kepler formulation assumes that one particle (the nucleus) is placed at the coordinate origin. The motion of the other two particles is described by a neutral system with respect to the unknown velocities. The state-dependent delays arise as a consequence of the finite vacuum speed of light. We obtain the equations of motion in spherical coordinates and split them into two groups. In the first group all arguments of the unknown functions are delays. We take their solutions as initial functions. Then, the equations of motion for the remaining two particles must be solved to the right of the initial point. To prove the existence–uniqueness of a periodic solution, we choose a space consisting of periodic infinitely smooth functions satisfying some supplementary conditions. Then, we use a suitable operator which acts on these spaces and whose fixed points are periodic solutions. We apply the fixed point theorem for the operators acting on the spaces of periodic functions. In this manner, we show the stability of the He atom in the frame of classical electrodynamics. In a previous paper of ours, we proved the existence of spin functions for plane motion. Thus, we confirm the Bohr and Sommerfeld’s hypothesis for the He atom.

This paper presents a novel deep-learning network designed to detect intervals of jump discontinuities in single-variable piecewise smooth functions from their noisy samples. Enhancing the accuracy of jump discontinuity estimations can be used to find a more precise overall approximation of the function, as traditional approximation methods often produce significant errors near discontinuities. Detecting intervals of discontinuities is relatively straightforward when working with exact function data, as finite differences in the data can serve as indicators of smoothness. However, these smoothness indicators become unreliable when dealing with highly noisy data. In this paper, we propose a deep-learning network to pinpoint the location of a jump discontinuity even in the presence of substantial noise.

The basic reproduction, or reproductive number, is a useful index that indicates whether or not there will be an epidemic. However, it is also very important to determine whether an epidemic will eventually decrease and disappear or persist as an endemic. Different infectious diseases have different behaviors and mathematical models used to simulated them should capture the most important processes; however, the models also involve simplifications. Influenza epidemics are usually short-lived and can be modeled with ordinary differential equations without considering demographics. Delays such as the infection time can change the behavior of the solutions. The same is true if there is permanent or temporary immunity, or complete or partial immunity. Vaccination, isolation and the use of antivirals can also change the outcome. In this paper, we introduce several new models and use them to find the effects of all the above factors paying special attention to whether the model can represent an infectious process that eventually disappears. We determine the equilibrium solutions and establish the stability of the disease-free equilibrium using various methods. We also show that many models of influenza or other epidemics with a short duration do not have solutions with a disappearing epidemic. The main objective of the paper is to introduce different ways of modeling immunity in epidemic models. Several scenarios with different immunities are studied since a person may not be re-infected because he/she has total or partial immunity or because there were no close contacts. We show that some relatively small changes, such as in the vaccination rate, can significantly change the dynamics; for example, the existence and number of the disease-free equilibria. We also illustrate that while introducing delays makes the models more realistic, the dynamics have the same qualitative behavior.

Deep learning has recently demonstrated the ability to predict long-term patient risk and its stratification when trained on imaging data such as chest radiographs. However, existing methods formulate estimating patient risk as a binary classification, typically ignoring or limiting the use of temporal information, and not accounting for the loss of patient follow-up, which reduces the fidelity of estimation and limits the prediction to a certain time horizon. In this paper, we demonstrate that deep survival and time-to-event prediction models can outperform binary classifiers at predicting mortality and risk of adverse health events. In our study, deep survival models were trained to predict risk scores from chest radiographs and patient demographic information in the Prostate, Lung, Colorectal, and Ovarian (PLCO) cancer screening trial (25,433 patient data points used in this paper) for 2-, 5-, and 10-year time horizons. Binary classification models that predict mortality at these time horizons were built as baselines. Compared to the considered alternative, deep survival models improve the Brier score (5-year: 0.0455 [95% CI, 0.0427–0.0482] vs. 0.0555 [95% CI, (0.0535–0.0575)], p < 0.05) and expected calibration error (ECE) (5-year: 0.0110 [95% CI, 0.0080–0.0141] vs. 0.0747 [95% CI, 0.0718–0.0776], p < 0.05) for those fixed time horizons and are able to generate predictions for any time horizon, without the need to retrain the models. Our study suggests that deep survival analysis tools can outperform binary classification in terms of both discriminative performance and calibration, offering a potentially plausible solution for forecasting risk in clinical practice.

Let $ X $ be an infinite Tychonoff space, and $ Y $ be a topological subspace of $ X $. In this paper, we study some covering properties of the subspace $ C_{p}\left(Y|X\right) $ of $ C_{p}\left(Y\right) $ consisting of those functions $ f\in C\left(Y\right) $ which admit a continuous extension to $ X $ equipped with the relative topology of $ C_{p}\left(Y\right) $. Among other results, we show that $ \left(i\right) $ $ C_{p}(Y|X) $ has a fundamental bounded resolution if and only if $ Y $ is countable; when $ X $ is realcompact and $ Y $ is closed in $ X $, we have $ \left(ii\right) $ if $ C_{p}(Y|X) $ admits a resolution of convex compact sets that swallows the local null sequences in $ C_{p}(Y|X) $, then $ Y $ is countable and discrete; $ \left(iii\right) $ if $ C_{p}(Y|X) $ admits a compact resolution that swallows the compact sets, then $ Y $ is also countable and discrete, and, as a corollary, we deduce that $ C_{p}(Y|X) $ admits a compact resolution that swallows the compact sets if and only if $ C_{p}(Y|X) $ is a Polish space. We also prove that $ \left(iv\right) $ for a metrizable space $ X $, $ C_{p}\left(X\right) $ is a quasi-$ \left(LB\right) $-space if and only if $ X $ is $ \sigma $-compact, and hence for a subspace $ Y $ of $ X $, the space $ C_{p}\left(Y|X\right) $ is a quasi-$ \left(LB\right) $-space. We include some examples and observations that answer natural questions raised in this paper.

In this paper, we studied the stability of the Cauchy problem for a class of first-order linear quasi-periodically forced PDEs on the $ m $-dimensional torus: $ \begin{eqnarray*} \left\{ \begin{array}{l} \partial_t u+(\xi+f(x, \omega t, \xi))\cdot \partial_x u = 0, \\ u(x, 0) = u_0(x), \ \end{array} \right. \end{eqnarray*} $ where $ \xi\in \mathbb{R}^m, x\in \mathbb{T}^m, \omega\in\mathbb{R}^d, $ in the case of multidimensional Liouvillean forced frequency. We proved that for each compact set $ \mathcal{O}\in\mathbb{R}^m $ there exists a Cantor subset $ \mathcal{O}_\gamma $ of $ \mathcal{O} $ with positive Lebesgue measure such that if $ \xi\in\mathcal{O}_\gamma, $ then for a perturbation $ f $ being small in some analytic Sobolev norm, there exists a bounded and invertible quasi-periodic family of linear operator $ \Psi(\omega t) $, such that the above PDEs are reduced by the transformation $ v: = \Psi(\omega t)^{-1}[u] $ into the following PDE: $ \begin{eqnarray*} \partial_t v+ (\xi+ m_\infty(\omega t))\cdot\partial_x v = 0, \end{eqnarray*} $ provided that the forced frequency $ \omega\in \mathbb{R}^d $ possesses finite uniform Diophantine exponent, which allows Liouvillean frequency. The reducibility can immediately cause the stability of the above Cauchy problem, that is, the analytic Sobolev norms of the Cauchy problem are controlled uniformly in time. The proof is based on a finite dimensional Kolmogorov-Arnold-Moser (KAM) theory for quasi-periodically forced linear vector fields with multidimensional Liouvillean forced frequency. As we know, the results on Liouvillean frequency existing in the literature deal with two-dimensional frequency and exploit the theory of continued fractions to control the small divisor problem. The results in this paper partially extend the analysis to higher-dimensional frequency and impose a weak nonresonance condition, i.e., the forced frequency $ \omega $ possesses finite uniform Diophantine exponent. Our result can be regarded as a generalization of analytic cases in the work [R. Feola, F. Giuliani, R. Montalto and M, Procesi, Reducibility of first order linear operators on tori via Moser's theorem, J. Funct. Anal., 2019] from Diophantine frequency to Liouvillean frequency.

This paper is concerned with the attraction-repulsion chemotaxis system (1.1) define on a bounded domain $ \Omega \subset \mathbb{R}^N(N\geq 1) $ with no-flux boundary conditions. The source function $ f $ in this system is a smooth function $ f $ that satisfies $ f(u)\leq a-bu^\eta $ for $ u\geq0 $. It is proven that $ \eta\geq1 $ is sufficient to ensure the boundedness of the solution when $ r < \frac{4(N+1)}{N(N+2)} $ is in the balance case $ \chi\alpha = \xi\gamma $, which improve the relevant results presented in papers such as Li and Xiang (2016), Xu and Zheng (2018), Xie and Zheng (2021), and Tang, Zheng and Li (2023).

The purpose of this paper is to establish a numerical method for solving time-fractional diffusion equations. To obtain the numerical solution, a binary reproducing kernel space is defined, and the orthonormal basis is constructed based on Legendre polynomials in this space. In order to find the $ {\varepsilon} $-approximation solution of time-fractional diffusion equations, which is defined in this paper, the algorithm is designed using the constructed orthonormal basis. Some numerical examples are analyzed to illustrate the procedure and confirm the performance of the proposed method. The results faithfully reveal that the presented method is considerably accurate and effective, as expected.

In this paper, we investigated $ n $-dimensional fractional hybrid differential equations (FHDEs) with nonlinear boundary conditions in a nonlinear coupled system. For this purpose, we used Dhage's fixed point theory, and applied the Krasnoselskii-type coupled fixed point theorem to construct existence conditions of the solution of the FHDEs. To illustrated this idea, suitable examples are presented in $ 3 $-dimensional space at the end of the paper.

In this paper, we investigate the asymptotic stability of the trajectories governed by some delay differential inequalities on time scales. Based on time scale theory and the fixed-point theorem, some sufficient conditions are obtained for guaranteeing asymptotic stability. It is interesting that the inequalities studied in this paper include the generalized Halanay inequalities. Due to the fact that dynamic systems on a time scale unify discrete and continuous systems, the results of this paper have wider application value. Furthermore, some numerical examples verify the main results.

This paper introduced an efficient method to obtain the solution of linear and nonlinear weakly singular kernel fractional integro-differential equations (WSKFIDEs). It used Riemann-Liouville fractional integration (R-LFI) to remove singularities and approximated the regularized problem with a combined approach using the generalized fractional step-Mittag-Leffler function (GFSMLF) and operational integral fractional Mittag matrix (OIFMM) method. The resulting algebraic equations were turned into an optimization problem. We also proved the method's accuracy in approximating any function, as well as its fractional differentiation and integration within WSKFIDEs. The proposed method was performed on some attractive examples in order to show how their solutions behave at various values of the fractional order $ \digamma $. The paper provided a valuable contribution to the field of fractional calculus (FC) by presenting a novel method for solving WSKFIDEs. Additionally, the accuracy of this method was verified by comparing its results with those obtained using other methods.

The aim of this paper is to introduce and study the concept of a hyperideal-based zero-divisor graph associated with a general hyperring. This is a generalized version of the zero-divisor graph associated with a commutative ring. For any general hyperring $ R $ having a hyperideal $ I $, the $ I $-based zero-divisor graph $ \Gamma^{(I)}(R) $ associated with $ R $ is the simple graph whose vertices are the elements of $ R\setminus I $ having their hyperproduct in $ I $, and two distinct vertices are joined by an edge when their hyperproduct has a non-empty intersection with $ I $. In the first part of the paper, we concentrate on some general properties of this graph related to absorbing elements, while the second part is dedicated to the study of the $ I $-based zero-divisor graph associated to the general hyperring $ \mathbb{Z}_n $ of the integers modulo $ n $, when $ n = 2p^mq $, with $ p $ and $ q $ two different odd primes, and fixing the hyperideal $ I $.

Abstract In a recent paper the authors proved a nonuniform local limit theorem concerning normal approximation of the point probabilities P ( S = k ) $P(S=k)$ when S = ∑ i = 1 n X i $S=\sum_{i=1}^{n}X_{i}$ and X 1 , X 2 , … , X n $X_{1},X_{2},\ldots ,X_{n}$ are independent Bernoulli random variables that may have different success probabilities. However, their main result contained an undetermined constant, somewhat limiting its applicability. In this paper we give a nonuniform bound in the same setting but with explicit constants. Our proof uses Stein’s method and, in particular, the K-function and concentration inequality approaches. We also prove a new uniform local limit theorem for Poisson binomial random variables that is used to help simplify the proof in the nonuniform case.

The aim of this work is the numerical study of the Rabinovich-Fabrikant system and its generalized model, which shows the occurrence of very rich dynamic behaviors with the interaction of three parameters of the generalized system. In particular, we observe the period-doubling bifurcation phenomenon leading to chaos, which has rarely been reported in previous works in the Rabinovich-Fabricant system. The complex dynamic behaviors of the system are investigated by using the Lyapunov spectrum, the parameters dependent bifurcation diagram and different sections of the phase space. This study is based on the numerical solution of differential equations and their numerical bifurcation analysis using Matlab software. The obtained results are new, because the generalization of the Rabinovich-Fabrikant system of the current study was proposed and studied for the first time. The generalized model describes the three-mode interaction. It can be used to simulate systems in radio and electronics engineering in which there is a three-mode interaction and which include cubic nonlinear terms. In addition, although the Rabinovich-Fabricant systems simulate systems of a physical nature, and in this regard, the coefficients embedded in them must be positive, its highly nonlinear and chaotic nature, due to the presence of third-order sentences, gives them a unique quality to be applied in secure communication. Resultantly, its artificial generalization with the use of negative parameters which adds to the complexity of its rich dynamics, is of particular importance. 1. IntroductionA dynamic system is classified into two types, discrete and continuous, in terms of time evolution. In this paper, we will look at a particular continuous dynamical system known as the "Rabinovitch-Fabrikant system", which was invented in 1979 by two theoretical physicists, Mikhail Rabinovitch and Anatoly Fabrikant [1]. This system is actually a simplification of a complex nonlinear parabolic equation that models different physical systems such as Tollmien–Schlichting waves in hydrodynamic flows, wind waves on water, Langmuir waves in plasma. Having five equilibrium points, it is not topologically equivalent to many classical systems such as the Lorenz and Chen systems (with three equilibrium points), Rössler system (with two equilibrium points) and the like. In the following, we will call this system the RF system, for short. There are several reasons why this system has been noticed.One is the fact that it models a physical system and, therefore, is not a synthetic model and can be used to model numerous physical phenomena. Another reason is that due to its strong nonlinearity (due to the presence of third-order terms in its mathematical model), a rigorous mathematical analysis cannot be performed on it, hence, the system may show new and interesting properties that have not yet been reported. At the same time, it creates serious challenges for the numerical methods of solving ordinary differential equations. The mathematical model of the Rabinovich-Fabrikant (RF) system is described by the following equations:(1.1)\begin{equation}\label{RF1}\begin{aligned}& \frac{dx_1}{dt}=x_2\left(x_3-1+x_1^2\right)+a x_1 \\& \frac{dx_2}{dt}=x_1\left(3 x_3+1-x_1^2\right)+a x_2 \\& \frac{dx_3}{dt}=-2 x_3\left(b+x_1 x_2\right)\end{aligned}\end{equation}In the Rabinovitch-Fabricant (RF) system modeled by the system of ordinary differential equations (1.1), $a>0$ and $b \in \mathbb{R}$ is the bifurcation parameter. Since it is currently impossible to fully analyze this system mathematically, most of the research is based on numerical and computer analysis. Following this common practice, we will also, in this paper, have an approach based on numerical analysis. 2. Main ResultsDifferent dedicated numerical methods for ODEs, implemented in different software packages, might give different results for the same parameters values and initial conditions. system (1.1) by Danca et al. [2, 3, 4] has been studied. By numerical analysis, they showed that this system exhibits unusual and very rich dynamics, including multistability. Here, we obtain their results with a different numerical approach. They used the 3-step predictor–corrector Local Iterative Linearization (LIL) method, which is an implicit 3-step method [5]. In this paper, to calculate the Lyapunov exponents, we have used the MATDS software package applicable in Matlab software by adding codes to get the desired output, and to solve $\mathbf{ODE}$, the $ode45 $ solver implemented in this software with relative error$RelTol=10^{-7}$ and absolute error $AbsTol=10^{-7}$ are used. The step length used in numerical methods is also considered to be $h=10^{-6}$. 2.1. Generalized Rabinovitch-Fabricant system. Recently, a generalization of the system (1.1) has been presented and numerically investigated as follows [6,8]:\begin{equation*}\label{RF3}\begin{aligned}& \frac{dx_1}{dt}=\left[p\left(x_1^2+x_3\right)+q\left(-x_2^2+3 x_3\right)-1\right], x_2+a x_1 \\& \frac{dx_2}{dt}=\left[p\left(-x_1^2+3 x_3\right)+q\left(x_2^2+x_3\right)+1\right], x_1+a x_2 \\&\frac{dx_3}{dt}=-2 x_3(b+(p+q) x_1 x_2) .\end{aligned}\end{equation*}This system becomes the system (1.1) with $p=1$ and $q=0$. In this section, we introduce the following system as a generalization of the system (1.1):(2.1)\begin{equation}\label{RF2}\begin{aligned}& \frac{dx_1}{dt}=x_2\left(x_3-1+x_1^2\right)+a x_1 \\& \frac{dx_2}{dt}=x_1\left(3 x_3+1-x_1^2\right)+c x_2 \\& \frac{dx_3}{dt}=-2 x_3\left(b+x_1 x_2\right)\end{aligned}\end{equation}This system becomes system (1.1) with $c=a$. This generalization is simpler than generalization (2.1), but it includes the same results in the detection of chaos and especially the process of converting the bifurcation of periodic period to chaos. According to the relationship\[\sum_{i=1}^3 \frac{\partial f_i}{\partial x_i} =(2x_1x_2+a)+(c)+(-2x_1x_2-2b)=a+c-2b,\]The system (2.1) is dissipative whenever $a+c-2b

Skeleton-based action recognition is an important but challenging task in the study of video understanding and human-computer interaction. However, existing methods suffer from two deficiencies. On the one hand, most methods usually involve manually designed convolution kernel which cannot capture spatial-temporal joint dependencies of complex regions. On the other hand, some methods just use the self-attention mechanism, ignoring its theoretical explanation. In this paper, we proposed a unified spatio-temporal graph convolutional network with a self-attention mechanism (SA-GCN) for low-quality motion video data with fixed viewing angle. SA-GCN can extract features efficiently by learning weights between joint points of different scales. Specifically, the proposed self-attention mechanism is end-to-end with mapping strategy for different nodes, which not only characterizes the multi-scale dependencies of joints, but also integrates the structural features of the graph and an ability of self-learning fusion features. Moreover, the attention mechanism proposed in this paper can be theoretically explained by GCN to some extent, which is usually not considered in most existing models. Extensive experiments on two widely used datasets, NTU-60 RGB+D and NTU-120 RGB+D, demonstrated that SA-GCN significantly outperforms a series of existing mainstream approaches in terms of accuracy.

Digital image segmentation plays an important role in noise reduction and pixel clustering for pre-processing of deep learning or feature extraction. The classic Self-Organizing Map (SOM) algorithm is a well-known unsupervised clustering neural network model. This classic method works on continuous data instead of discrete data sets with a widely scattered distribution. The novel SOM(SOM2) modelling solved this problem for the classic, simple tabular discrete data set but not for the digital image data. As the essence of digital image pixels data are different from tabular datasets, we have to look at them differently. This paper proposes exploiting the novel SOM method with a hybrid combination of the fuzzy C-Means and K-means convolution filter as image segmentation and noise reduction with soft and hard segmentation as entropy reduction for natural digital images. The main approach of this paper is the segmentation of image contents for the reduction of noises and saturation pixels by entropy criteria. Based on the resulting paper, the combination of SOM2 with FCM for soft segmentation 47%-and the combination of SOM2 with k-means convolution for hard segmentation 33% can reduce the entropy of the original image on average.

This research paper presented a novel investigation into an implicit coupled system of fractional variable order, which has not been previously studied in the existing literature. The study focused on establishing and developing sufficient conditions for the existence and uniqueness of solutions, as well as the Ulam-Hyers stability, for the proposed coupled system without using semigroup property. By extending the existing conclusions examined for the Atangana-Baleanu-Caputo (ABC) operator, we contributed to advancing the understanding of variable-order fractional differential equations. The paper provided a solid theoretical foundation for further analysis, numerical simulations, and practical applications. The obtained results have implications for designing and controlling systems modeled using fractional variable order equations and serve as a basis for addressing a wide range of dynamical problems. The transformation techniques, qualitative analysis, and illustrative examples presented in this work highlight its unique contributions and potential to serve as a foundation for future research.

This paper delves into three research directions, leveraging the Lovász $ \vartheta $-function of a graph. First, it focuses on the Shannon capacity of graphs, providing new results that determine the capacity for two infinite subclasses of strongly regular graphs, and extending prior results. The second part explores cospectral and nonisomorphic graphs, drawing on a work by Berman and Hamud (2024), and it derives related properties of two types of joins of graphs. For every even integer such that $ n \geq 14 $, it is constructively proven that there exist connected, irregular, cospectral, and nonisomorphic graphs on $ n $ vertices, being jointly cospectral with respect to their adjacency, Laplacian, signless Laplacian, and normalized Laplacian matrices, while also sharing identical independence, clique, and chromatic numbers, but being distinguished by their Lovász $ \vartheta $-functions. The third part focuses on establishing bounds on graph invariants, particularly emphasizing strongly regular graphs and triangle-free graphs, and compares the tightness of these bounds to existing ones. The paper derives spectral upper and lower bounds on the vector and strict vector chromatic numbers of regular graphs, providing sufficient conditions for the attainability of these bounds. Exact closed-form expressions for the vector and strict vector chromatic numbers are derived for all strongly regular graphs and for all graphs that are vertex- and edge-transitive, demonstrating that these two types of chromatic numbers coincide for every such graph. This work resolves a query regarding the variant of the $ \vartheta $-function by Schrijver and the identical function by McEliece et al. (1978). It shows, by a counterexample, that the $ \vartheta $-function variant by Schrijver does not possess the property of the Lovász $ \vartheta $-function of forming an upper bound on the Shannon capacity of a graph. This research paper also serves as a tutorial of mutual interest in zero-error information theory and algebraic graph theory.

The paper presents a comprehensive analysis of child labor in Ecuador, employing advanced statistical tools like logistic regression, neutrosophic Likert scales, and similarity measures to deepen the understanding of this social issue. The integration of these methodologies allows for a nuanced assessment of the various socio-economic factors contributing to child labor. By capturing the uncertainty in human responses, the research highlights the complex interplay between poverty, household income, education levels, and labor types on the incidence of child labor.Key findings suggest that rural location, the age of the child, and the informal nature of the head of the household'swork are the most significant predictors of child labor. Notably, parental education appears to have a less direct influence. Despite various efforts, including government monetary transfers through programs like the BDH, child labor persists, indicating the need for more targeted interventions.The paper proposes future research to extend these models to a broader demographic and geographic data set, emphasizing the potential for these methods to be applied to a variety of social issues. The development ofcomputational tools to automate neutrosophic analysis could greatly benefit large-scale studies, potentially aiding policymakers in designing more effective interventions for vulnerable populations.

In recent years, there has been a notable shift in focus towards the analysis of non-stationary time series, driven largely by the complexities associated with delineating significant asymptotic behaviors inherent to such processes. The genesis of the theory of locally stationary processes arises from the quest for asymptotic inference grounded in nonparametric statistics. This paper endeavors to formulate a comprehensive framework for conducting inference within the realm of locally stationary functional time series by harnessing the conditional $ U $-statistics methodology as propounded by W. Stute in 1991. The proposed methodology extends the Nadaraya-Watson regression function estimations. Within this context, a novel estimator was introduced for the single index conditional $ U $-statistics operator, adept at accommodating the non-stationary attributes inherent to the data-generating process. The primary objective of this paper was to establish the weak convergence of conditional $ U $-processes within the domain of locally stationary functional mixing data. Specifically, the investigation delved into scenarios of weak convergence involving functional explanatory variables, considering both bounded and unbounded sets of functions while adhering to specific moment requirements. The derived findings emanate from broad structural specifications applicable to the class of functions and models under scrutiny. The theoretical insights expounded in this study constitute pivotal tools for advancing the domain of functional data analysis.

In this paper, we proposed a higher-order uniform accuracy scheme for nonlinear $ \psi $-Volterra integral equations in two dimension with weakly singular kernel by using the modified block-by-block method. First, we constructed a high order uniform accuracy scheme method in this paper by dividing the entire domain into some small sub-domains and approximating the integration function with biquadratic interpolation in each sub-domain. Second, we rigorously proved that the convergence order of the higher order uniform accuracy scheme was $ O(h_{s}^{3+\sigma_{1} }+h_{t}^{3+\sigma_{2} }) $ with $ 0 < \sigma_{1}, \sigma_{2} < 1 $ by using the discrete Gronwall inequality. Finally, two numerical examples were used to illustrate experimental results with different values of $ \psi $ to support the theoretical results.

This paper addresses the challenge of adaptive control for nonstrict-feedback nonlinear systems that involve input delay, actuator faults, and external disturbance. To deal with the complexities arising from input delay and unknown functions, we have incorporated Pade approximation and radial basis function neural networks, respectively. An adaptive controller has been developed by utilizing the Lyapunov stability theorem and the backstepping approach. The suggested method guarantees that the tracking error converges to a compact neighborhood that contains the origin and that every signal in the closed-loop system is semi-globally uniformly ultimately bounded. To demonstrate the efficacy of the proposed method, an electromechanical system application example, and a numerical example are provided. Additionally, comparative analysis was conducted between the Pade approximation proposed in this paper and the auxiliary systems in the existing method. Furthermore, error assessment criteria have been employed to substantiate the effectiveness of the proposed method by comparing it with existing results.

The asymptotic synchronization problem of chaotic Lur'e systems in the master-slave framework was explored in this paper. A time-varying delay feedback controller with quantization considerations and a delay-product-type Lyapunov-Krasovskii functional technique were employed to tackle this problem. Consider an error system based on master and slave systems, for which sufficient asymptotic stability requirements are developed to assure that the addressed system achieves proper synchronization. Following that, the desired control gain was determined by finding a feasible solution to these stability requirements. The results of this paper were validated using a numerical example with simulations, which revealed that they were superior to previously published ones.

Accuracy is an important factor to consider when evaluating the performance of a manipulator. The accuracy of a manipulator is determined by its ability to accurately move and position objects in a precise manner. This research paper aims to evaluate the performance of different methods for the kinematic analysis of manipulators. The study employs four distinct techniques, namely mathematical modeling using the closed form solutions method, roboanalyzer, Peter Corke toolbox, and particle swarm optimization, to perform kinematic analysis for manipulators. The KUKA industrial manipulator is used as an illustrative case study in this research due to its widespread use in various industrial applications in addition to its high precision and stability. Its wide usage in the industry makes the results of this research highly relevant and allows for a thorough evaluation of the performance of the different methods being studied. Furthermore, understanding the kinematic analysis of the manipulator can also help in improving the performance and increasing the efficiency of the robot in different tasks. This paper conducts a comparison of the accuracy of the four methods, and the results indicate that particle swarm optimization is the most accurate method. The RoboAnalyzer approach achieved the fastest execution time.

Addressing the boundary value problems of fractional-order differential equations hold significant importance due to their applications in various fields. The aim of this paper was to approximate solutions for a class of boundary value problems involving Caputo fractional-order differential equations employing the AA-iterative scheme. Moreover, the stability and data dependence results of the iterative scheme were given for a certain class of mappings. Finally, a numerical experiment was illustrated to support the results presented herein. The results presented in this paper extend and unify some well-known comparable results in the existing literature.

The iterated function system (IFS) is important in different fields like image compression. An important feature of such systems is that they can be used to generate fractals. Yet, for the obtained fractals, it is difficult to locally control them to generate new ones with desired structures at specific places. In this paper, we gave an attempt to solve this problem based on a nonuniform multiple function system. For this, we first analyzed the multiple function systems needed in the generation of the final desired fractals. Based on such analysis, the final fractals with desired structures at specific places can be generated using the nonuniform multiple function system. Moreover, these two procedures were summarized into two algorithms for convenience. Examples were also given to illustrate the performance of the nonuniform multiple function system and the two algorithms in this paper.

We have studied initial value problems for Caputo fractional differential equations with singular nonlinearities involving the p-Laplacian operator. We have given a precise mathematical analysis of the equivalence of the fractional differential equations and Volterra integral equations studied in this paper. A theorem for the global existence of the solution was proven. In addition, an example was given at the end of the article as an application of the results found in this paper.

A new class of nonlocal boundary value problems consisting of multi-term delay fractional differential equations and multipoint-integral boundary conditions is studied in this paper. We derive a more general form of the solution for the given problem by applying a fractional integral operator of an arbitrary order $ \beta_{\xi} $ instead of $ \beta_{1} $; for details, see Lemma 2. The given problem is converted into an equivalent fixed-point problem to apply the tools of fixed-point theory. The existence of solutions for the given problem is established through the use of a nonlinear alternative of the Leray-Schauder theorem, while the uniqueness of its solutions is shown with the aid of Banach's fixed-point theorem. We also discuss the stability criteria, icluding Ulam-Hyers, generalized Ulam-Hyers, Ulam-Hyers-Rassias, and generalized Ulam-Hyers-Rassias stability, for solutions of the problem at hand. For illustration of the abstract results, we present examples. Our results are new and useful for the discipline of multi-term fractional differential equations related to hydrodynamics. The paper concludes with some interesting observations.

In this paper, we dealt with the tracking control problem of a class of fractional-order uncertain systems with time delays. In order to handle the effects brought by the uncertainties, external disturbances, time-delay terms, and to overcome the obstacles caused by inputs saturation, the tracking controller, which consisted of linear control law, nonlinear law, and robust control law proposed in this paper, was designed by combining the composite nonlinear feedback control method and the properties of fractional order operators. Furthermore, the validation of this tracking controller was proved.

Abstract In continuation of Zayed and Bulboacă work in (J. Inequal. Appl. 2022:158, 2022), this paper discusses the geometric characterization of the normalized form of the generalized Bessel function defined by V ρ , r ( z ) : = z + ∑ k = 1 ∞ ( − r ) k 4 k ( 1 ) k ( ρ ) k z k + 1 , z ∈ U , $$\begin{aligned} \mathrm{V}_{\rho,r}(z):=z+\sum_{k=1}^{\infty} \frac{(-r)^{k}}{4^{k}(1)_{k}(\rho )_{k}}z^{k+1}, \quad z\in \mathbb{U}, \end{aligned}$$ for ρ , r ∈ C ∗ : = C ∖ { 0 } $\rho, r\in \mathbb{C}^{\ast}:=\mathbb{C}\setminus \{0\}$ . Precisely, we will use a sharp estimate for the Pochhammer symbol, that is, Γ ( a + n ) / Γ ( a + 1 ) > ( a + α ) n − 1 $\Gamma (a+n)/\Gamma (a+1)>(a+\alpha )^{n-1}$ , or equivalently ( a ) n > a ( a + α ) n − 1 $(a)_{n}>a(a+\alpha )^{n-1}$ , that was firstly proved by Baricz and Ponnusamy for n ∈ N ∖ { 1 , 2 } $n\in \mathbb{N}\setminus \{1,2\}$ , a > 0 $a>0$ and α ∈ [ 0 , 1.302775637 … ] $\alpha \in [0,1.302775637\ldots ]$ in (Integral Transforms Spec. Funct. 21(9):641–653, 2010), and then proved in our paper by another method to improve it using the partial derivatives and the two-variable functions’ extremum technique for n ∈ N ∖ { 1 , 2 } $n\in \mathbb{N}\setminus \{1,2\}$ , a > 0 $a>0$ and 0 ≤ α ≤ 2 $0\leq \alpha \leq \sqrt{2}$ , and used to investigate the orders of starlikeness and convexity. We provide the reader with some examples to illustrate the efficiency of our theory. Our results improve, complement, and generalize some well-known (nonsharp) estimates, as seen in the Concluding Remarks and Outlook section.

The aim of this paper is to study the sufficient conditions for the existence of attractor of a generalized cyclic iterated function system composed of a complete metric space and a finite collection of generalized cyclic F-contraction mappings. Some examples are presented to support our main results and concepts defined herein. The results proved in the paper extend and generalize various well known results in the existing literature.