Groups of infinite unitriangular matrices over associative unitary rings are considered. These groups naturally act on infinite dimensional free modules over underlying rings. They are profinite in case underlying rings are finite. Inspired by their connection with groups defined by finite automata the problem to construct faithful representations of free products of groups by banded infinite unitriangular matrices is considered.For arbitrary prime p a sufficient conditions on a finite set of banded infinite unitriangular matrices over unitary associative rings of characteristic p under which they generate the free product of cyclic p-groups is given. The conditions are based on certain properties of the actions on finite dimensional free modules over underlying rings.It is shown that these conditions are satisfied. For arbitrary free product of finite number of cyclic p-groups constructive examples of the sets of infinite unitriangular matrices over unitar associative rings of characteristic p that generate given free product are presented. These infinite matrices are constructed from finite dimensional ones that are nilpotent Jordan blocks.A few open questions concerning properties of presented examples and other types of faithful representations are formulated.
{"title":"Free products of cyclic groups in groups of infinite unitriangular matrices","authors":"A. Oliynyk","doi":"10.30970/ms.60.1.28-33","DOIUrl":"https://doi.org/10.30970/ms.60.1.28-33","url":null,"abstract":"Groups of infinite unitriangular matrices over associative unitary rings are considered. These groups naturally act on infinite dimensional free modules over underlying rings. They are profinite in case underlying rings are finite. Inspired by their connection with groups defined by finite automata the problem to construct faithful representations of free products of groups by banded infinite unitriangular matrices is considered.For arbitrary prime p a sufficient conditions on a finite set of banded infinite unitriangular matrices over unitary associative rings of characteristic p under which they generate the free product of cyclic p-groups is given. The conditions are based on certain properties of the actions on finite dimensional free modules over underlying rings.It is shown that these conditions are satisfied. For arbitrary free product of finite number of cyclic p-groups constructive examples of the sets of infinite unitriangular matrices over unitar associative rings of characteristic p that generate given free product are presented. These infinite matrices are constructed from finite dimensional ones that are nilpotent Jordan blocks.A few open questions concerning properties of presented examples and other types of faithful representations are formulated.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136099708","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the paper we study equiaffine immersions $fcolon (M^n,nabla) rightarrow {mathbb{R}}^{n+2}$ with flat connection $nabla$ and one-dimensional Weingarten mapping. For such immersions there are two types of the transversal distribution equiaffine frame.We give a parametrization of a submanifold with the given properties for both types of equiaffine frame. The main result of the paper is contained in Theorems 1, 2 and Corollary 1: Let $fcolon ({M}^n,nabla)rightarrow({mathbb{R}}^{n+2},D)$ be an affine immersion with pointwise codimension 2, equiaffine structure, flat connection $nabla$, one-dimensional Weingarten mapping then there exists three types of its parametrization:$(i)$ $vec{r}=g(u^1,ldots,u^n) vec{a}_1+intvec{varphi}(u^1)du^1+sumlimits_{i=2}^n u^ivec{a}_i;$$(ii)$ $vec{r}=(g(u^2,ldots,u^n)+u^1)vec{a}+int v(u^1) vec{eta}(u^1)du^1+sumlimits_{i=2}^n u^iintlambda_i(u^1)vec{eta}(u^1)du^1;$$(iii)$ $vec{r}=(g(u^2,ldots,u^n)+u^1)vec{rho}(u^1)+int (v(u^1) - u^1)dfrac{d vec{rho}(u^1)}{d u^1}du^1+sumlimits_{i=2}^n u^iintlambda_i(u^1)dfrac{d vec{rho}(u^1)}{d u^1}du^1.$
{"title":"Equiaffine immersions of codimension two with flat connection and one-dimensional Weingarten mapping","authors":"O. O. Shugailo","doi":"10.30970/ms.60.1.99-112","DOIUrl":"https://doi.org/10.30970/ms.60.1.99-112","url":null,"abstract":"In the paper we study equiaffine immersions $fcolon (M^n,nabla) rightarrow {mathbb{R}}^{n+2}$ with flat connection $nabla$ and one-dimensional Weingarten mapping. For such immersions there are two types of the transversal distribution equiaffine frame.We give a parametrization of a submanifold with the given properties for both types of equiaffine frame. The main result of the paper is contained in Theorems 1, 2 and Corollary 1: Let $fcolon ({M}^n,nabla)rightarrow({mathbb{R}}^{n+2},D)$ be an affine immersion with pointwise codimension 2, equiaffine structure, flat connection $nabla$, one-dimensional Weingarten mapping then there exists three types of its parametrization:$(i)$ $vec{r}=g(u^1,ldots,u^n) vec{a}_1+intvec{varphi}(u^1)du^1+sumlimits_{i=2}^n u^ivec{a}_i;$$(ii)$ $vec{r}=(g(u^2,ldots,u^n)+u^1)vec{a}+int v(u^1) vec{eta}(u^1)du^1+sumlimits_{i=2}^n u^iintlambda_i(u^1)vec{eta}(u^1)du^1;$$(iii)$ $vec{r}=(g(u^2,ldots,u^n)+u^1)vec{rho}(u^1)+int (v(u^1) - u^1)dfrac{d vec{rho}(u^1)}{d u^1}du^1+sumlimits_{i=2}^n u^iintlambda_i(u^1)dfrac{d vec{rho}(u^1)}{d u^1}du^1.$","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136099714","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $mathscr{A}$ be a ring with its center $mathscr{Z}(mathscr{A}).$ An additive mapping $xicolon mathscr{A}to mathscr{A}$ is called a homoderivation on $mathscr{A}$ if
$forall a,bin mathscr{A}colonquad xi(ab)=xi(a)xi(b)+xi(a)b+axi(b).$
An additive map $psicolon mathscr{A}to mathscr{A}$ is called a generalized homoderivation with associated homoderivation $xi$ on $mathscr{A}$ if
$forall a,bin mathscr{A}colonquadpsi(ab)=psi(a)psi(b)+psi(a)b+axi(b).$
This study examines whether a prime ring $mathscr{A}$ with a generalized homoderivation $psi$ that fulfils specific algebraic identities is commutative. Precisely, we discuss the following identities:
$psi(a)psi(b)+abin mathscr{Z}(mathscr{A}),quadpsi(a)psi(b)-abin mathscr{Z}(mathscr{A}),quadpsi(a)psi(b)+abin mathscr{Z}(mathscr{A}),$
$psi(a)psi(b)-abin mathscr{Z}(mathscr{A}),quadpsi(ab)+abin mathscr{Z}(mathscr{A}),quadpsi(ab)-abin mathscr{Z}(mathscr{A}),$
$psi(ab)+bain mathscr{Z}(mathscr{A}),quadpsi(ab)-bain mathscr{Z}(mathscr{A})quad (forall a, bin mathscr{A}).$
Furthermore, examples are given to prove that the restrictions imposed on the hypothesis of the various theorems were not superfluous.
{"title":"On generalized homoderivations of prime rings","authors":"N. Rehman, E. K. Sogutcu, H. M. Alnoghashi","doi":"10.30970/ms.60.1.12-27","DOIUrl":"https://doi.org/10.30970/ms.60.1.12-27","url":null,"abstract":"Let $mathscr{A}$ be a ring with its center $mathscr{Z}(mathscr{A}).$ An additive mapping $xicolon mathscr{A}to mathscr{A}$ is called a homoderivation on $mathscr{A}$ if
 $forall a,bin mathscr{A}colonquad xi(ab)=xi(a)xi(b)+xi(a)b+axi(b).$
 An additive map $psicolon mathscr{A}to mathscr{A}$ is called a generalized homoderivation with associated homoderivation $xi$ on $mathscr{A}$ if
 $forall a,bin mathscr{A}colonquadpsi(ab)=psi(a)psi(b)+psi(a)b+axi(b).$
 This study examines whether a prime ring $mathscr{A}$ with a generalized homoderivation $psi$ that fulfils specific algebraic identities is commutative. Precisely, we discuss the following identities:
 $psi(a)psi(b)+abin mathscr{Z}(mathscr{A}),quadpsi(a)psi(b)-abin mathscr{Z}(mathscr{A}),quadpsi(a)psi(b)+abin mathscr{Z}(mathscr{A}),$
 $psi(a)psi(b)-abin mathscr{Z}(mathscr{A}),quadpsi(ab)+abin mathscr{Z}(mathscr{A}),quadpsi(ab)-abin mathscr{Z}(mathscr{A}),$
 $psi(ab)+bain mathscr{Z}(mathscr{A}),quadpsi(ab)-bain mathscr{Z}(mathscr{A})quad (forall a, bin mathscr{A}).$
 Furthermore, examples are given to prove that the restrictions imposed on the hypothesis of the various theorems were not superfluous.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136099711","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Invesigation of bounds for Hankel determinat of analytic univalent functions is prominent intrest of many researcher from early twenth century to study geometric properties. Many authors obtained non sharp upper bound of third Hankel determinat for different subclasses of analytic univalent functions until Kwon et al. obtained exact estimation of the fourth coefficeient of Caratheodory class. Recently authors made use of an exact estimation of the fourth coefficient, well known second and third coefficient of Caratheodory class obtained sharp bound for the third Hankel determinant associated with subclasses of analytic univalent functions. Let $w=f(z)=z+a_{2}z^{2}+cdots$ be analytic in the unit disk $mathbb{D}={zinmathbb{C}:|z|<1}$, and $mathcal{S}$ be the subclass of normalized univalent functions with $f(0)=0$, and $f'(0)=1$. Let $z=f^{-1}$ be the inverse function of $f$, given by $f^{-1}(w)=w+t_2w^2+cdots$ for some $|w|
{"title":"An exact estimate of the third Hankel determinants for functions inverse to convex functions","authors":"B. Rath, K. S. Kumar, D. V. Krishna","doi":"10.30970/ms.60.1.34-39","DOIUrl":"https://doi.org/10.30970/ms.60.1.34-39","url":null,"abstract":"Invesigation of bounds for Hankel determinat of analytic univalent functions is prominent intrest of many researcher from early twenth century to study geometric properties. Many authors obtained non sharp upper bound of third Hankel determinat for different subclasses of analytic univalent functions until Kwon et al. obtained exact estimation of the fourth coefficeient of Caratheodory class. Recently authors made use of an exact estimation of the fourth coefficient, well known second and third coefficient of Caratheodory class obtained sharp bound for the third Hankel determinant associated with subclasses of analytic univalent functions. Let $w=f(z)=z+a_{2}z^{2}+cdots$ be analytic in the unit disk $mathbb{D}={zinmathbb{C}:|z|<1}$, and $mathcal{S}$ be the subclass of normalized univalent functions with $f(0)=0$, and $f'(0)=1$. Let $z=f^{-1}$ be the inverse function of $f$, given by $f^{-1}(w)=w+t_2w^2+cdots$ for some $|w|<r_o(f)$. Let $mathcal{S}^csubsetmathcal{S}$ be the subset of convex functions in $mathbb{D}$. In this paper, we estimate the best possible upper bound for the third Hankel determinant for the inverse function $z=f^{-1}$ when $fin mathcal{S}^c$.Let $mathcal{S}^c$ be the class of convex functions. We prove the following statements (Theorem):If $fin$ $mathcal{S}^c$, thenbegin{equation*}big|H_{3,1}(f^{-1})big| leq frac{1}{36}end{equation*} and the inequality is attained for $p_0(z)=(1+z^3)/(1-z^3).$","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136099707","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The concept of bounded $L$-index in a direction $mathbf{b}=(b_1,ldots,b_n)inmathbb{C}^nsetminus{mathbf{0}}$ is generalized for a class of analytic functions in the unit polydisc, where $L$ is some continuous function such that for every $z=(z_1,ldots,z_n)inmathbb{D}^n$ one has $L(z)>betamax_{1le jle n}frac{|b_j|}{1-|z_j|},$ $beta=mathrm{const}>1,$ $mathbb{D}^n$ is the unit polydisc, i.e. $mathbb{D}^n={zinmathbb{C}^n: |z_j|le 1, jin{1,ldots,n}}.$ For functions from this class we obtain sufficient and necessary conditions providing boundedness of $L$-index in the direction. They describe local behavior of maximum modulus of derivatives for the analytic function $F$ on every slice circle ${z+tmathbf{b}: |t|=r/L(z)}$ by their values at the center of the circle, where $tinmathbb{C}.$ Other criterion describes similar local behavior of the minimum modulus via the maximum modulus for these functions. We proved an analog of the logarithmic criterion desribing estimate of logarithmic derivative outside some exceptional set by the function $L$. The set is generated by the union of all slice discs ${z^0+tmathbf{b}: |t|le r/L(z^0)}$, where $z^0$ is a zero point of the function $F$. The analog also indicates the zero distribution of the function $F$ is uniform over all slice discs. In one-dimensional case, the assertion has many applications to analytic theory of differential equations and infinite products, i.e. the Blaschke product, Naftalevich-Tsuji product. Analog of Hayman's Theorem is also deduced for the analytic functions in the unit polydisc. It indicates that in the definition of bounded $L$-index in direction it is possible to remove the factorials in the denominators. This allows to investigate properties of analytic solutions of directional differential equations.
{"title":"Analytic in a unit polydisc functions of bounded $L$-index in direction","authors":"A. Bandura, T. Salo","doi":"10.30970/ms.60.1.55-78","DOIUrl":"https://doi.org/10.30970/ms.60.1.55-78","url":null,"abstract":"The concept of bounded $L$-index in a direction $mathbf{b}=(b_1,ldots,b_n)inmathbb{C}^nsetminus{mathbf{0}}$ is generalized for a class of analytic functions in the unit polydisc, where $L$ is some continuous function such that for every $z=(z_1,ldots,z_n)inmathbb{D}^n$ one has $L(z)>betamax_{1le jle n}frac{|b_j|}{1-|z_j|},$ $beta=mathrm{const}>1,$ $mathbb{D}^n$ is the unit polydisc, i.e. $mathbb{D}^n={zinmathbb{C}^n: |z_j|le 1, jin{1,ldots,n}}.$ For functions from this class we obtain sufficient and necessary conditions providing boundedness of $L$-index in the direction. They describe local behavior of maximum modulus of derivatives for the analytic function $F$ on every slice circle ${z+tmathbf{b}: |t|=r/L(z)}$ by their values at the center of the circle, where $tinmathbb{C}.$ Other criterion describes similar local behavior of the minimum modulus via the maximum modulus for these functions. We proved an analog of the logarithmic criterion desribing estimate of logarithmic derivative outside some exceptional set by the function $L$. The set is generated by the union of all slice discs ${z^0+tmathbf{b}: |t|le r/L(z^0)}$, where $z^0$ is a zero point of the function $F$. The analog also indicates the zero distribution of the function $F$ is uniform over all slice discs. In one-dimensional case, the assertion has many applications to analytic theory of differential equations and infinite products, i.e. the Blaschke product, Naftalevich-Tsuji product. Analog of Hayman's Theorem is also deduced for the analytic functions in the unit polydisc. It indicates that in the definition of bounded $L$-index in direction it is possible to remove the factorials in the denominators. This allows to investigate properties of analytic solutions of directional differential equations.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136099713","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the paper, we discussed the distribution of unique range sets and its elements over the extended complex plane from a different point of view and obtained some new results regarding the structure and position of unique range sets. These new results have immense applications like classifying different subsets of C to be or not to be a unique range set, exploring the fact that every bi-linear transformation preserves unique range sets for meromorphic functions, providing simpler and shorter proofs of existence of some unique range sets, unfolding the fact that zeros or poles of any meromorphic function lie in a unique range set, in particular,identifying the Fundamental Theorem of Algebra to a more specific region and many more applications. We have also posed some open questions to unveil the mysterious arrangement of the elements of unique range sets.
{"title":"On the distribution of unique range sets and its elements over the extended complex plane","authors":"S. Mallick","doi":"10.30970/ms.60.1.40-54","DOIUrl":"https://doi.org/10.30970/ms.60.1.40-54","url":null,"abstract":"In the paper, we discussed the distribution of unique range sets and its elements over the extended complex plane from a different point of view and obtained some new results regarding the structure and position of unique range sets. These new results have immense applications like classifying different subsets of C to be or not to be a unique range set, exploring the fact that every bi-linear transformation preserves unique range sets for meromorphic functions, providing simpler and shorter proofs of existence of some unique range sets, unfolding the fact that zeros or poles of any meromorphic function lie in a unique range set, in particular,identifying the Fundamental Theorem of Algebra to a more specific region and many more applications. We have also posed some open questions to unveil the mysterious arrangement of the elements of unique range sets.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136099709","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the Hilbert space $H:=L_2(mathbb{R})$, we consider the impedance Sturm--Liouville operator $T:Hto H$ generated by the differential expression $ -pfrac{d}{dx}{frac1{p^2}}frac{d}{dx}p$, where the function $p:mathbb{R}tomathbb{R}_+$ is of bounded variation on $mathbb{R}$ and $inf_{xinmathbb{R}} p(x)>0$. Existence of the transformation operator for the operator $T$ and its properties are studied.
In the paper, we suggest an efficient parametrization of the impedance function p in term of a real-valued bounded measure $muin boldsymbol M$ via$p_mu(x):= e^{mu([x,infty))}, xinmathbb{R}.$For a measure $muin boldsymbol M$, we establish existence of the transformation operator for the Sturm--Liouville operator $T_mu$, which is constructed with the function $p_mu$. Continuous dependence of the operator $T_mu$ on $mu$ is also proved. As a consequence, we deduce that the operator $T_mu$ is unitarily equivalent to the operator $T_0:=-d^2/dx^2$.
{"title":"Transformation operators for impedance Sturm–Liouville operators on the line","authors":"M. Kazanivskiy, Ya. Mykytyuk, N. Sushchyk","doi":"10.30970/ms.60.1.79-98","DOIUrl":"https://doi.org/10.30970/ms.60.1.79-98","url":null,"abstract":"In the Hilbert space $H:=L_2(mathbb{R})$, we consider the impedance Sturm--Liouville operator $T:Hto H$ generated by the differential expression $ -pfrac{d}{dx}{frac1{p^2}}frac{d}{dx}p$, where the function $p:mathbb{R}tomathbb{R}_+$ is of bounded variation on $mathbb{R}$ and $inf_{xinmathbb{R}} p(x)>0$. Existence of the transformation operator for the operator $T$ and its properties are studied.
 In the paper, we suggest an efficient parametrization of the impedance function p in term of a real-valued bounded measure $muin boldsymbol M$ via$p_mu(x):= e^{mu([x,infty))}, xinmathbb{R}.$For a measure $muin boldsymbol M$, we establish existence of the transformation operator for the Sturm--Liouville operator $T_mu$, which is constructed with the function $p_mu$. Continuous dependence of the operator $T_mu$ on $mu$ is also proved. As a consequence, we deduce that the operator $T_mu$ is unitarily equivalent to the operator $T_0:=-d^2/dx^2$.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136099712","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-06-24DOI: 10.30970/ms.59.2.215-224
Kh.O. Sukhorukova
We consider non-additive measures on the compact Hausdorff spaces, which are generalizations of the idempotent measures and max-min measures. These measures are related to the continuous triangular norms and they are defined as functionals on the spaces of continuous functions from a compact Hausdorff space into the unit segment.The obtained space of measures (called ∗-measures, where ∗ is a triangular norm) are endowed with the weak* topology. This construction determines a functor in the category of compact Hausdorff spaces. It is proved, in particular, that the ∗-measures of finite support are dense in the spaces of ∗-measures. One of the main results of the paper provides an alternative description of ∗-measures on a compact Hausdorff space X, namely as hyperspaces of certain subsets in X × [0, 1]. This is an analog of a theorem for max-min measures proved by Brydun and Zarichnyi.
{"title":"Spaces of non-additive measures generated by triangular norms","authors":"Kh.O. Sukhorukova","doi":"10.30970/ms.59.2.215-224","DOIUrl":"https://doi.org/10.30970/ms.59.2.215-224","url":null,"abstract":"We consider non-additive measures on the compact Hausdorff spaces, which are generalizations of the idempotent measures and max-min measures. These measures are related to the continuous triangular norms and they are defined as functionals on the spaces of continuous functions from a compact Hausdorff space into the unit segment.The obtained space of measures (called ∗-measures, where ∗ is a triangular norm) are endowed with the weak* topology. This construction determines a functor in the category of compact Hausdorff spaces. It is proved, in particular, that the ∗-measures of finite support are dense in the spaces of ∗-measures. One of the main results of the paper provides an alternative description of ∗-measures on a compact Hausdorff space X, namely as hyperspaces of certain subsets in X × [0, 1]. This is an analog of a theorem for max-min measures proved by Brydun and Zarichnyi.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41808743","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-06-24DOI: 10.30970/ms.59.2.123-131
M. Pratsovytyi, O. Baranovskyi, O. Bondarenko, S. Ratushniak
In the paper, we introduce and study a massive class of continuous functions defined on the interval $(0;1)$ using a special encoding (representation) of the argument with an alphabet $ mathbb{Z}={0,pm 1, pm 2,...}$ and base $tau=frac{sqrt{5}-1}{2}$: $displaystyle x=b_{alpha_1}+sumlimits_{k=2}^{m}(b_{alpha_k}prodlimits_{i=1}^{k-1}Theta_{alpha_i})equivDelta^{Phi}_{alpha_1alpha_2...alpha_m(emptyset)},quadx=b_{alpha_1}+sumlimits_{k=2}^{infty}(b_{alpha_k}prodlimits_{i=1}^{k-1}Theta_{alpha_i})equivDelta^{Phi}_{alpha_1alpha_2...alpha_n...},$ where $alpha_nin mathbb{Z}$, $Theta_n=Theta_{-n}=tau^{3+|n|}$,$b_n=sumlimits_{i=-infty}^{n-1}Theta_i=begin{cases}tau^{2-n}, & mbox{if } nleq0, 1-tau^{n+1}, & mbox{if } ngeq 0.end{cases}$ The function $f$, which is the main object of the study, is defined by equalities$displaystylebegin{cases}f(x=Delta^{Phi}_{i_1...i_k...})=sigma_{i_11}+sumlimits_{k=2}^{infty}sigma_{i_kk}prodlimits_{j=1}^{k-1}p_{i_jj}equivDelta_{i_1...i_k...},f(x=Delta^{Phi}_{i_1...i_m(emptyset)})=sigma_{i_11}+sumlimits_{k=2}^{m}sigma_{i_kk}prodlimits_{j=1}^{k-1}p_{i_jj}equivDelta_{i_1...i_m(emptyset)},end{cases}$ where an infinite matrix $||p_{ik}||$ ($iin mathbb{Z}$, $kin mathbb N$) satisfies the conditions 1) $|p_{ik}|<1$ $forall iin mathbb{Z}$, $forall kin mathbb N;quad$2) $sumlimits_{iin mathbb{Z}}p_{ik}=1$ $forall kinmathbb N$; 3) $0
在本文中，我们引入并研究了一类在区间$（0；1）$上定义的连续函数，使用字母表$mathbb｛Z｝=｛0，pm1，pm2，…｝$和基$tau=frac｛sqrt的自变量的特殊编码（表示）{5}-1}{2} $：$displaystyle x=b_｛k-1｝Theta_｛alpha_i｝）equivDelta^｛Phi｝_｛alpha_1alpha_2…alpha_n…｝，$其中$alpha_ninmathbb｛Z｝$，$Theta_n=Theta_{-n｝=tau^｛3+|n|｝$，$b_n=sumlimits_{i=-infty｝^｛n-1｝Theta_i=begin｛cases｝tau^｛2-n｝，&&mbox｛if｝nleq0，1-tau ^｛n+1｝，mbox{if｝ngeq 0.end｛casses｝$函数$f$是研究的主要对象，由等式$displaystyleboot定义{cases}f（x=Δ^｛Phi｝_｛i_1…i_k…｝^{k-1}p_｛i_jj｝equivDelta_｛i_1…i_k…｝，f（x=Delta^｛Phi｝_｛i1…i_m（pemptyset）｝^{k-1}p_｛i_jj｝equivDelta_｛i_1…i_m（emptyset）｝，end｛cases｝$其中一个无限矩阵$|p_｛ik｝|$（$iinmathbb｛Z｝$，$kinmath bb N$）满足条件1）$|p_{ik｝|<1$ for all iinathbb{Z｝$，$ for ll kinath bb N；quad$2）$sumlimits_{iinmathbb｛Z｝｝p_｛ik｝=1$$对于所有kinmath bb N$；3） $0^{k-1}p_｛i_jj｝
{"title":"One class of continuous locally complicated functions related to infinite-symbol $Phi$-representation of numbers","authors":"M. Pratsovytyi, O. Baranovskyi, O. Bondarenko, S. Ratushniak","doi":"10.30970/ms.59.2.123-131","DOIUrl":"https://doi.org/10.30970/ms.59.2.123-131","url":null,"abstract":"In the paper, we introduce and study a massive class of continuous functions defined on the interval $(0;1)$ using a special encoding (representation) of the argument with an alphabet $ mathbb{Z}={0,pm 1, pm 2,...}$ and base $tau=frac{sqrt{5}-1}{2}$: $displaystyle x=b_{alpha_1}+sumlimits_{k=2}^{m}(b_{alpha_k}prodlimits_{i=1}^{k-1}Theta_{alpha_i})equivDelta^{Phi}_{alpha_1alpha_2...alpha_m(emptyset)},quadx=b_{alpha_1}+sumlimits_{k=2}^{infty}(b_{alpha_k}prodlimits_{i=1}^{k-1}Theta_{alpha_i})equivDelta^{Phi}_{alpha_1alpha_2...alpha_n...},$ \u0000where $alpha_nin mathbb{Z}$, $Theta_n=Theta_{-n}=tau^{3+|n|}$,$b_n=sumlimits_{i=-infty}^{n-1}Theta_i=begin{cases}tau^{2-n}, & mbox{if } nleq0, 1-tau^{n+1}, & mbox{if } ngeq 0.end{cases}$ \u0000The function $f$, which is the main object of the study, is defined by equalities$displaystylebegin{cases}f(x=Delta^{Phi}_{i_1...i_k...})=sigma_{i_11}+sumlimits_{k=2}^{infty}sigma_{i_kk}prodlimits_{j=1}^{k-1}p_{i_jj}equivDelta_{i_1...i_k...},f(x=Delta^{Phi}_{i_1...i_m(emptyset)})=sigma_{i_11}+sumlimits_{k=2}^{m}sigma_{i_kk}prodlimits_{j=1}^{k-1}p_{i_jj}equivDelta_{i_1...i_m(emptyset)},end{cases}$ where an infinite matrix $||p_{ik}||$ ($iin mathbb{Z}$, $kin mathbb N$) satisfies the conditions \u00001) $|p_{ik}|<1$ $forall iin mathbb{Z}$, $forall kin mathbb N;quad$2) $sumlimits_{iin mathbb{Z}}p_{ik}=1$ $forall kinmathbb N$; \u00003) $0<sumlimits_{k=2}^{infty}prodlimits_{j=1}^{k-1}p_{i_jj}<infty~~forall (i_j)in L;quad$4) $0<sigma_{ik}equivsumlimits_{j=-infty}^{i-1}p_{jk}<1$ $forall iin mathbb Z, forall kin mathbb N.$ \u0000This class of functions contains monotonic, non-monotonic, nowhere monotonic functions and functionswithout monotonicity intervals except for constancy intervals, Cantor-type andquasi-Cantor-type functions as well as functions of bounded and unbounded variation. The criteria for the function $f$ to be monotonic and to be a function of the Cantor type as well as the criterion of nowhere monotonicity are proved. Expressions for the Lebesgue measure of the set of non-constancy of the function and for the variation of the function are found. Necessary and sufficient conditions for thefunction to be of unbounded variation are established.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41805080","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-06-24DOI: 10.30970/ms.59.2.178-186
A. Kuryliak
It is proved analogues of the classical Wiman's inequality} for the class $mathcal{D}$ of absolutely convergents in the whole complex plane $mathbb{C}^p$ (entire) Dirichlet series of the form $displaystyle F(z)=sumlimits_{|n|=0}^{+infty} a_ne^{(z,lambda_n)}$ with such a sequence of exponents $(lambda_n)$ that ${lambda_ncolon ninmathbb{Z}^p}subset mathbb{C}^p$ and $lambda_nnot=lambda_m$ for all $nnot= m$. For $Finmathcal{D}$ and $zinmathbb{C}^psetminus{0}$ we denote $mathfrak{M}(z,F):=sumlimits_{|n|=0}^{+infty}|a_n|e^{Re(z,lambda_n)},quadmu(z,F):=sup{|a_n|e^{mathop{rm Re}(z,lambda_n)}colon ninmathbb{Z}^ p_+},$ $(m_k)_{kgeq 0}$ is $(mu_{k})_{kgeq 0}$ the sequence $(-ln|a_{n}|)_{ninmathbb{Z}^p_+}$ arranged by non-decreasing. The main result of the paper: Let $Fin mathcal{D}.$ If $(exists alpha > 0)colon$ $intnolimits_{t_0}^{+infty}t^{-2}{(n_1(t))^{alpha}}dt<+infty,$ $n_1(t)overset{def}=sumnolimits_{mu_nleq t} 1,quad t_0>0,$ then there exists a set $Esubsetgamma_{+}(F),$ such that $tau_{2p}(Ecapgamma_{+}(F))=int_{Ecapgamma_{+}(F)}|z|^{-2p}dxdyleq C_p, z=x+iyinmathbb{C}^p,$ and relation $mathfrak{M}(z,F)= o(mu(z,F)ln^{1/alpha} mu(z,F))$ holds as $zto infty$ $(zin gamma_Rsetminus E)$ for each $R>0$, where $gamma_R=Big{zinmathbb{C}^psetminus{0}colon K_F(z)leq R Big},quad K_F(z)=supBig{frac1{Phi_z( t)}int^{ t}_0 frac {{Phi_z}(u)}{u} ducolon t geq t_0Big},$ $gamma(F)={zinmathbb{C}colon limlimits_{tto +infty}Phi_z(t)=+infty},quad gamma_+(F)=mathop{cup}_{R>0}gamma_R$, $Phi_z(t)=frac1{t}lnmu(tz,F)$. In general, under the specified conditions, the obtained inequality is exact.
{"title":"Wiman’s type inequality for entire multiple Dirichlet series with arbitrary complex exponents","authors":"A. Kuryliak","doi":"10.30970/ms.59.2.178-186","DOIUrl":"https://doi.org/10.30970/ms.59.2.178-186","url":null,"abstract":"It is proved analogues of the classical Wiman's inequality} for the class $mathcal{D}$ of absolutely convergents in the whole complex plane $mathbb{C}^p$ (entire) Dirichlet series of the form $displaystyle F(z)=sumlimits_{|n|=0}^{+infty} a_ne^{(z,lambda_n)}$ with such a sequence of exponents $(lambda_n)$ that ${lambda_ncolon ninmathbb{Z}^p}subset mathbb{C}^p$ and $lambda_nnot=lambda_m$ for all $nnot= m$. For $Finmathcal{D}$ and $zinmathbb{C}^psetminus{0}$ we denote \u0000$mathfrak{M}(z,F):=sumlimits_{|n|=0}^{+infty}|a_n|e^{Re(z,lambda_n)},quadmu(z,F):=sup{|a_n|e^{mathop{rm Re}(z,lambda_n)}colon ninmathbb{Z}^ p_+},$ \u0000$(m_k)_{kgeq 0}$ is $(mu_{k})_{kgeq 0}$ the sequence $(-ln|a_{n}|)_{ninmathbb{Z}^p_+}$ arranged by non-decreasing. \u0000The main result of the paper: Let $Fin mathcal{D}.$ If $(exists alpha > 0)colon$ $intnolimits_{t_0}^{+infty}t^{-2}{(n_1(t))^{alpha}}dt<+infty,$ \u0000$n_1(t)overset{def}=sumnolimits_{mu_nleq t} 1,quad t_0>0,$ then there exists a set $Esubsetgamma_{+}(F),$ such that \u0000$tau_{2p}(Ecapgamma_{+}(F))=int_{Ecapgamma_{+}(F)}|z|^{-2p}dxdyleq C_p, z=x+iyinmathbb{C}^p,$ \u0000and relation $mathfrak{M}(z,F)= o(mu(z,F)ln^{1/alpha} mu(z,F))$ holds as $zto infty$ $(zin gamma_Rsetminus E)$ for each $R>0$, where \u0000$gamma_R=Big{zinmathbb{C}^psetminus{0}colon K_F(z)leq R Big},quad K_F(z)=supBig{frac1{Phi_z( t)}int^{ t}_0 frac {{Phi_z}(u)}{u} ducolon t geq t_0Big},$ $gamma(F)={zinmathbb{C}colon limlimits_{tto +infty}Phi_z(t)=+infty},quad gamma_+(F)=mathop{cup}_{R>0}gamma_R$, $Phi_z(t)=frac1{t}lnmu(tz,F)$. In general, under the specified conditions, the obtained inequality is exact.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41942300","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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