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On a centennial timescale, solar activity was quantified based on records of instrumental sunspot observations. This article briefly discusses several aspects of the recent archival investigations of historical sunspot records in the 17th to 18th centuries. This article also reviews the recent updates for the active day fraction and positions of the reported sunspot groups of the Maunder Minimum to show their significance within the observational history. These archival investigations serve as base datasets for reconstructing solar activity.

Apart from an account of classical preliminaries, this volume contains a systematic introduction to Sobolev spaces and functions of bounded variation with selected applications. This is installment III of a four part discussion of certain aspects of Real Analysis: Functions of a Single Variable, Curves and Length, Functions of Several Variables, and Surfaces and Area.

In this paper we describe a general method to generate superoscillatory functions of several variables starting from a superoscillating sequence of one variable. Our results are based on the study of suitable infinite order differential operators on holomorphic functions with growth conditions of exponential type, where additional constraints are required when dealing with infinite order differential operators whose symbol is a function that is holomorphic in some open set, but not necessarily entire. The results proved for the superoscillating sequence in several variables are extended to sequences of supershifts in several variables.

This work presents a family of fiber bundles where the total spaces are associated with holomorphic functions on several complex variables and the basis spaces extend the notion of quaternionic slice regular functions of several quaternionic variables. This paper also shows how the fiber bundle theory justifies the domain these slice regular function in several variables.

In this paper, we study upper bounds for the topological complexity of the total spaces of some classes of fibre bundles. We calculate a tight upper bound for the topological complexity of an $n$-dimensional Klein bottle. We also compute the exact value of the topological complexity of $3$-dimensional Klein bottle. We describe the cohomology rings of several classes of generalized projective product spaces with $\mathbb{Z}_2$-coefficients. Then we study the LS-category and topological complexity of infinite families of generalized projective product spaces. We reckon the exact value of these invariants in many specific cases. We calculate the equivariant LS-category and equivariant topological complexity of several product spaces equipped with $\mathbb{Z}_2$-action.

We introduce Wirtinger operators for functions of several quaternionic variables. These operators are real linear partial differential operators which behave well on quaternionic polynomials, with properties analogous to the ones satisfied by the Wirtinger derivatives of several complex variables. Due to the non-commutativity of the variables, Wirtinger operators turn out to be of higher order, except the first ones that are of the first order. In spite of that, these operators commute each other and satisfy a Leibniz rule for products. Moreover, they characterize the class of slice-regular polynomials, and more generally of slice-regular quaternionic functions. As a step towards the definition of the Wirtinger operators, we provide Almansi-type decompositions for slice functions and for slice-regular functions of several variables. We also introduce some aspects of local slice analysis, based on the definition of locally slice-regular function in any open subset of the $n$-dimensional quaternionic space.

This paper presents an exhaustive study about the robustness of several parameterizations, in speaker verification and identification tasks. We have studied several mismatch conditions: different recording sessions, microphones, and different languages (it has been obtained from a bilingual set of speakers). This study reveals that the combination of several parameterizations can improve the robustness in all the scenarios for both tasks, identification and verification. In addition, two different methods have been evaluated: vector quantization, and covariance matrices with an arithmetic-harmonic sphericity measure.

The present paper is devoted to a new multidimensional generalization of the Beurling and Malliavin Theorem, which is a classical result in the Uncertainty Principle in Fourier Analysis. In more detail, we establish by an elegant but simple new method a sufficient condition for a radial function to be a Beurling and Malliavin majorant in several dimensions (this means that the function in question can be minorized by the modulus of a square integrable function which is not zero identically and which has the support of the Fourier transform included in an arbitrary small ball). As a corollary of the radial case, we also get a new sharp sufficient condition in the nonradial case. The latter result provides an answer to a question posed by L. Hörmander. Our proof is different in the cases of odd and even dimensions. In the even dimensional case we make use of one classical formula from the theory of Bessel functions due to N. Ya. Sonin.

This paper is devoted to the study of the dynamic optimization of several controlled crowd motion models in the general planar settings, which is an application of a class of optimal control problems involving a general nonconvex sweeping process with perturbations. A set of necessary optimality conditions for such optimal control problems involving the crowd motion models with multiple agents and obstacles is obtained and analyzed. Several effective algorithms based on such necessary optimality conditions are proposed and various nontrivial illustrative examples together with their simulations are also presented. The implementation of all the considered motion models can be found via the link: https://github.com/tancao1128/Optimal_Control_of_Several_Motion_Models with the instruction and demonstration video uploaded at https://www.youtube.com/watch?v=B8DQ0wvCtIQ.

In this paper, several families of irreducible constacyclic codes over finite fields and their duals are studied. The weight distributions of these irreducible constacyclic codes and the parameters of their duals are settled. Several families of irreducible constacyclic codes with a few weights and several families of optimal constacyclic codes are constructed. As by-products, a family of $[2n, (n-1)/2, d \geq 2(\sqrt{n}+1)]$ irreducible cyclic codes over $\gf(q)$ and a family of $[(q-1)n, (n-1)/2, d \geq (q-1)(\sqrt{n}+1)]$ irreducible cyclic codes over $\gf(q)$ are presented, where $n$ is a prime such that $\ord_n(q)=(n-1)/2$. The results in this paper complement earlier works on irreducible constacyclic and cyclic codes over finite fields.

In the present paper we give some explicit proofs for folklore theorems on holomorphic functions in several variables with values in a locally complete locally convex Hausdorff space $E$ over $\mathbb{C}$. Most of the literature on vector-valued holomorphic functions is either devoted to the case of one variable or to infinitely many variables whereas the case of (finitely many) several variables is only touched or is subject to stronger restrictions on the completeness of $E$ like sequential completeness. The main tool we use is Cauchy's integral formula for derivatives for an $E$-valued holomorphic function in several variables which we derive via Pettis-integration. This allows us to generalise the known integral formula, where usually a Riemann-integral is used, from sequentially complete $E$ to locally complete $E$. Among the classical theorems for holomorphic functions in several variables with values in a locally complete space $E$ we prove are the identity theorem, Liouville's theorem, Riemann's removable singularities theorem and the density of the polynomials in the $E$-valued polydisc algebra.

We describe separating G_2-invariants of several copies of the algebra of octonions over an algebraically closed field of characteristic two. We also obtain a minimal separating and a minimal generating set for G_2-invariants of several copies of the algebra of octonions in case of a field of odd characteristic.

In this paper we will give a scheme-theoretic discussion on the unramified extensions of an arithmetic function field in several variables. The notion of unramified discussed here is parallel to that in algebraic number theory and for the case of classical varieties, coincides with that in Lang's theory of unramified class fields of a function field in several variables. It is twofold for us to introduce the notion of unramified. One is for the computation of the étale fundamental group of an arithmetic scheme; the other is for an ideal-theoretic theory of unramified class fields over an arithmetic function field in several variables. Fortunately, in the paper we will also have operations on unramified extensions such as base changes, composites, subfields, transitivity, etc. It will be proved that a purely transcendental extension over the rational field has a trivial unramified extension. As an application, it will be seen that the affine scheme of a ring over the ring of integers in several variables has a trivial étale fundamental group.

Under consideration methods of constructing trigonometric interpolation splines of two variables on rectangular areas. These methods are easily generalized to the case of trigonometric interpolation splines of several variables on such domains. A numerical example illustrating the main theoretical propositions is considered. The given methods of constructing such splines can be widely used in practice.

Octonionic analysis is becoming eminent due to the role of octonions in the theory of G2 manifold. In this article, a new slice theory is introduced as a generalization of the holomorphic theory of several complex variables to the noncommutative or nonassociative realm. The Bochner-Martinelli formula is established for slice functions of several octonionic variables as well as several quaternionic variables. In this setting, we find the Hartogs phenomena for slice regular functions

In this paper we establish a multivariable non-commutative generalization of Löwner's classical theorem from 1934 characterizing operator monotone functions as real functions admitting analytic continuation mapping the upper complex half-plane into itself. The non-commutative several variable theorem proved here characterizes several variable operator monotone functions, not assumed to be free analytic or even continuous, as free functions that admit free analytic continuation mapping the upper operator poly-halfspace into the upper operator halfspace over an arbitrary Hilbert space. We establish a new abstract integral formula for them using non-commutative topology, matrix convexity and LMIs. The formula represents operator monotone and operator concave free functions as a conditional expectation of a Schur complement of a linear matrix pencil on a tensor product operator algebra. This formula is new even in the one variable case. The results can be applied to any of the various multivariable operator means that has been constructed in the last three decades or so, including the Karcher mean. Thus we obtain an explicit, closed formula for these operator means of several positive

We study the classical Köthe's problem, concerning the structure of non-commutative rings with the property that: ``every left module is a direct sum of cyclic modules". In 1934, Köthe showed that left modules over Artinian principal ideal rings are direct sums of cyclic modules. A ring $R$ is called a ${\it left~Köthe~ring}$ if every left $R$-module is a direct sum of cyclic $R$-modules. In 1951, Cohen and Kaplansky proved that all commutative K{ö}the rings are Artinian principal ideal rings. During the years 1962 to 1965, Kawada solved the Köthe's problem for basic fnite-dimensional algebras: Kawada's theorem characterizes completely those finite-dimensional algebras for which any indecomposable module has square-free socle and square-free top, and describes the possible indecomposable modules. But, so far, the Köthe's problem is open in the non-commutative setting. In this paper, we break the class of left K{ö}the rings into three categories of nested: ${\it left~Köthe~rings}$, ${\it strongly~left~K{ö}the~rings}$ and ${\it very~strongly~left~K{ö}the~rings}$, and then, we solve the Köthe's problem by giving several characterizations of these rings in terms of describing the indec

We present an elementary proof of a general version of Montel's theorem in several variables which is based on the use of tensor product polynomial interpolation. We also prove a Montel-Popoviciu's type theorem for functions $f:\mathbb{R}^d\to\mathbb{R}$ for $d>1$. Furthermore, our proof of this result is also valid for the case $d=1$, differing in several points from Popoviciu's original proof. Finally, we demonstrate that our results are optimal.

In cells, organelles and vesicles are usually transported by cooperation of several motor proteins, including plus-end directed motor kinesin and minus-end directed motor dynein. Many biophysical models have been constructed to understand the mechanism of this motion. However, so far, the basic principle about it remains unclosed. In this paper, based on the recent experimental results and existing theoretical models, a spider-like model is provided. In this model, each motor is regarded as a bead-spring system. The bead can bind to or unbind from the track stochastically, and step forward or backward with fixed step size L and force dependent transition rates. The spring connects the bead to cargo tightly. At any time, the position of cargo is determined by force balance condition. The obvious characteristics of our model are that, the motors interact with each other and they do not share the external load equally. Our results indicate, the stall force of cargo, under which the mean velocity of cargo vanishes, usually decreases with the interactions between motors. If the cargo is pulled by several motors from same motor species, the stall force of cargo is bigger than that of the