This paper is an extended review of our doctoral thesis “Isomerism as Intrinsic Symmetry of Molec... more This paper is an extended review of our doctoral thesis “Isomerism as Intrinsic Symmetry of Molecules” in which we present, continue, generalize, and trace out Lunn–Senior’s theory of isomerism in organic chemistry.
In [1] we find certain finite homomorphic images of Artin braid group into appropriate symmetric ... more In [1] we find certain finite homomorphic images of Artin braid group into appropriate symmetric groups, which a posteriori are extensions of the symmetric group on n letters by an abelian group. The main theorem of this paper characterizes completely the extensions of this type that are split.
Given two families of continuous functions u = (up)p∈I and v = (vq)q∈J on a topological space X, ... more Given two families of continuous functions u = (up)p∈I and v = (vq)q∈J on a topological space X, we define a preorder R = R(u, v) on X by the condition that any member of u is an R-increasing and any member of v is an R-decreasing function. It turns out that if the topological space X is quasi-compact and sequentially compact, then any element x ∈ X is R-dominated by an R-maximal element m ∈ X: xRm. In particular, since the (n − 1)-dimensional simplex is a compact subset of R n , then considering its members as portfolios consisting of n financial assets, we obtain the classical 1952 result of Harry Markowitz that any portfolio is dominated by an efficient portfolio. Moreover, several other examples of possible application of this general setup are presented.
In this paper we review the famous article "Eine für die Valenztheorie geeignete Basis der binäre... more In this paper we review the famous article "Eine für die Valenztheorie geeignete Basis der binären Vektorinvarianten" of G. Rumer, E. Teller, and H. Weyl on Rumer diagrams and use a powerful old result from algebraic geometry to enumerate these diagrams with given numbers of atoms and valence bonds.
We define degree of dependence of two events A and B in a probability space by using Boltzmann-Sh... more We define degree of dependence of two events A and B in a probability space by using Boltzmann-Shannon entropy function of an appropriate probability distribution produced by these events and depending on one parameter (the probability of intersection of A and B) varying within a closed interval I. The entropy function attains its global maximum when the events A and B are independent. The important particular case of discrete uniform probability space motivates this definition in the following way. The entropy function has a minimum at the left endpoint of I exactly when one of the events and the complement of the other are connected with the relation of inclusion (maximal negative dependence). It has a minimum at the right endpoint of I exactly when one of these events is included in the other (maximal positive dependence). Moreover, the deviation of the entropy from its maximum is equal to average information that carries one of the binary trials defined by A and B with respect t...
Fundamental Journal of Mathematics and Applications, 2018
By Markowitz geometry we mean the intersection theory of ellipsoids and affine subspaces in a rea... more By Markowitz geometry we mean the intersection theory of ellipsoids and affine subspaces in a real finite-dimensional linear space. In the paper we give a meticulous and self-contained treatment of this arch-classical subject, which lays a solid mathematical groundwork of Markowitz mean-variance theory of efficient portfolios in economics.
Comptes rendus de l'Académie bulgare des sciences: sciences mathématiques et naturelles
Given a linearly ordered set I, every surjective map p : A → I endows the set A with a structure ... more Given a linearly ordered set I, every surjective map p : A → I endows the set A with a structure of set of preferences by "replacing" the elements ι ∈ I with their inverse images p −1 (ι) considered as "balloons" (sets endowed with an equivalence relation), lifting the linear order on A, and "agglutinating" this structure with the balloons. Every ballooning A of a structure of linearly ordered set I is a set of preferences A whose preference relation (not necessarily complete) is negatively transitive and every such structure on a given set A can be obtained by ballooning of certain structure of a linearly ordered set I, intrinsically encoded in A. In other words, the difference between linearity and negative transitivity is constituted of balloons. As a consequence of this characterization, under certain natural topological conditions on the set of preferences A furnished with its interval topology, the existence of a continuous generalized utility function on A is proved.
Polya's fundamental enumeration theorem and some results from Williamson's generalized se... more Polya's fundamental enumeration theorem and some results from Williamson's generalized setup of it are proved in terms of Schur- Macdonald's theory (S-MT) of "invariant matrices". Given a permutation group W ≤ Sd and a one-dimensional characterof W, the polynomial functor Fcorresponding via S-MT to the induced monomial representation U� = ind S d W (�) of Sd, is studied. It turns out that the characteristic ch(F�) is the weighted inventory of some set J(�) of W-orbits in the integer-valued hypercube (0, ∞)d. The elements of J(�) can be distinguished among all W-orbits by a maximum property. The identity ch(F�) = ch(U�) of both characteristics is a consequence of S-MT, and is equivalent to a result of Williamson. Polya's theorem can be obtained from the above identity by the specialization � = 1W, where 1W is the unit character of W.
Introduction In our paper [3], it is shown that the natural environment for P'olya's fund... more Introduction In our paper [3], it is shown that the natural environment for P'olya's fundamental enumeration theorem and for one of its possible generalizations, is Schur-Macdonald's theory of invariant matrices (cf. [4, Ch. I, Appendix]). The main result of this theory is the eqivalence between the category of finite-dimensional linear representations of the symmetric group S d and the category of polynomial homogeneous degree d functors on the category of finite-dimensional linear spaces. The above-mentioned generalization is a particular case of the main equality of characteristics of two objects corresponding each other via Schur-Macdonald's equivalence. For the original version of P'olya's theorem, we suppose that this fact was pointed out to G. P'olya by I. Schur himself (cf. [5, Ch. 1, n o 20]). Quite the contrary, it turns out that such a generalization of Redfield's master theorem lives only in the environment of the representation theory of ...
Polya's fundamental enumeration theorem is generalized in terms of Schur-Macdonald's theo... more Polya's fundamental enumeration theorem is generalized in terms of Schur-Macdonald's theory (S-MT) of invariant matrices. Given a permutation group $W\leq S_d$ and a one-dimensional character $\chi$ of $W$, the polynomial functor $F_\chi$ corresponding via S-MT to the induced monomial representation $U_\chi = ind_W^{S_d}(\chi)$ of $S_d$, is studied. It turns out that the characteristic $ch(F_\chi)$ is the weighted inventory of some set $J(\chi)$ of $W$-orbits in the integer-valued hypercube $[0,\infty)^d$. The elements of $J(\chi) can be distinguished among all $W$-orbits by a maximum property. The identity $ch(F_\chi) = ch(U_\chi)$ of both characteristics is a consequence of S-MT. Polya's theorem can be obtained from the above identity by specialization $\chi=1_W$, where $1_W$ is the unit character of $W$.
Given a loset I, every surjective map p: A ---> I endows the set A with a structure of preorde... more Given a loset I, every surjective map p: A ---> I endows the set A with a structure of preordered set by "replacing" the elements of I with their inverse images via p considered as "bubbles" (sets endowed with an equivalence relation), lifting the structure of loset on A, and "agglutinating" this structure with the bubbles. Every bubbling A of a structure of loset I is a structure of preordered set A (not necessarily complete) whose preorder has negatively transitive asymmetric part and every such structure on a given set A can be obtained by bubbling up of certain structure of a loset I, intrinsically encoded in A. In other words, the difference between linearity and negative transitivity is constituted of bubbles. As a consequence of this characterization, under certain natural topological conditions on the preordered set A furnished with its interval topology, the existence of a continuous generalized utility function on A is proved.
Transactions of the American Mathematical Society, 1988
This paper is devoted to finding necessary and sufficient conditions for a graded ring to be the ... more This paper is devoted to finding necessary and sufficient conditions for a graded ring to be the canonical ring of a minimal surface of general type with K 2 = 2 p g − 3 {K^2} = 2{p_g} - 3 , p g ⩾ 3 {p_g} \geqslant 3 , and such that its canonical linear system has one base point.
Transactions of the American Mathematical Society, 1988
This paper is devoted to finding necessary and sufficient conditions for a graded ring to be the ... more This paper is devoted to finding necessary and sufficient conditions for a graded ring to be the canonical ring of a minimal surface of general type with K2 = 2pg-3, pg > 3, and such that its canonical linear system has one base point. 1. Known results. 1.1. Let 5 be a minimal surface of general type with K2 = 2pg-3, pg > 3. We assume that the canonical linear system \K\ of S has one base point P. Following Horikawa [4, Lemma 2] let ir: S'-► 5 be the quadratic transformation with center P and exceptional curve E.
In this paper the author finds explicitly all finite-dimensional irreducible representations of a... more In this paper the author finds explicitly all finite-dimensional irreducible representations of a series of finite permutation groups that are homomorphic images of Artin braid group.
In this paper, a mathematical characterization of Lunn-Senior's groups of univalent substitution ... more In this paper, a mathematical characterization of Lunn-Senior's groups of univalent substitution isomerism of the linear alkanes, under some natural assumptions that reflect their common properties, is given. For each linear alkane, the number of its monosubstitution derivatives, its di-substitution derivatives, and its tri-substitution derivatives as linear, quadratic, and cubic polynomial expressions, respectively, in their number, is obtained. In principle, the number of derivatives of a given linear alkane with any particular composition can be established. The same explicit expressions for the case of k-substitution homogeneous derivatives of the linear alkanes are obtained by Balasubramanian (Theoret. Chim. Acta (Berl.) 51:37, 1979).
This paper is an extended review of our doctoral thesis “Isomerism as Intrinsic Symmetry of Molec... more This paper is an extended review of our doctoral thesis “Isomerism as Intrinsic Symmetry of Molecules” in which we present, continue, generalize, and trace out Lunn–Senior’s theory of isomerism in organic chemistry.
In [1] we find certain finite homomorphic images of Artin braid group into appropriate symmetric ... more In [1] we find certain finite homomorphic images of Artin braid group into appropriate symmetric groups, which a posteriori are extensions of the symmetric group on n letters by an abelian group. The main theorem of this paper characterizes completely the extensions of this type that are split.
Given two families of continuous functions u = (up)p∈I and v = (vq)q∈J on a topological space X, ... more Given two families of continuous functions u = (up)p∈I and v = (vq)q∈J on a topological space X, we define a preorder R = R(u, v) on X by the condition that any member of u is an R-increasing and any member of v is an R-decreasing function. It turns out that if the topological space X is quasi-compact and sequentially compact, then any element x ∈ X is R-dominated by an R-maximal element m ∈ X: xRm. In particular, since the (n − 1)-dimensional simplex is a compact subset of R n , then considering its members as portfolios consisting of n financial assets, we obtain the classical 1952 result of Harry Markowitz that any portfolio is dominated by an efficient portfolio. Moreover, several other examples of possible application of this general setup are presented.
In this paper we review the famous article "Eine für die Valenztheorie geeignete Basis der binäre... more In this paper we review the famous article "Eine für die Valenztheorie geeignete Basis der binären Vektorinvarianten" of G. Rumer, E. Teller, and H. Weyl on Rumer diagrams and use a powerful old result from algebraic geometry to enumerate these diagrams with given numbers of atoms and valence bonds.
We define degree of dependence of two events A and B in a probability space by using Boltzmann-Sh... more We define degree of dependence of two events A and B in a probability space by using Boltzmann-Shannon entropy function of an appropriate probability distribution produced by these events and depending on one parameter (the probability of intersection of A and B) varying within a closed interval I. The entropy function attains its global maximum when the events A and B are independent. The important particular case of discrete uniform probability space motivates this definition in the following way. The entropy function has a minimum at the left endpoint of I exactly when one of the events and the complement of the other are connected with the relation of inclusion (maximal negative dependence). It has a minimum at the right endpoint of I exactly when one of these events is included in the other (maximal positive dependence). Moreover, the deviation of the entropy from its maximum is equal to average information that carries one of the binary trials defined by A and B with respect t...
Fundamental Journal of Mathematics and Applications, 2018
By Markowitz geometry we mean the intersection theory of ellipsoids and affine subspaces in a rea... more By Markowitz geometry we mean the intersection theory of ellipsoids and affine subspaces in a real finite-dimensional linear space. In the paper we give a meticulous and self-contained treatment of this arch-classical subject, which lays a solid mathematical groundwork of Markowitz mean-variance theory of efficient portfolios in economics.
Comptes rendus de l'Académie bulgare des sciences: sciences mathématiques et naturelles
Given a linearly ordered set I, every surjective map p : A → I endows the set A with a structure ... more Given a linearly ordered set I, every surjective map p : A → I endows the set A with a structure of set of preferences by "replacing" the elements ι ∈ I with their inverse images p −1 (ι) considered as "balloons" (sets endowed with an equivalence relation), lifting the linear order on A, and "agglutinating" this structure with the balloons. Every ballooning A of a structure of linearly ordered set I is a set of preferences A whose preference relation (not necessarily complete) is negatively transitive and every such structure on a given set A can be obtained by ballooning of certain structure of a linearly ordered set I, intrinsically encoded in A. In other words, the difference between linearity and negative transitivity is constituted of balloons. As a consequence of this characterization, under certain natural topological conditions on the set of preferences A furnished with its interval topology, the existence of a continuous generalized utility function on A is proved.
Polya's fundamental enumeration theorem and some results from Williamson's generalized se... more Polya's fundamental enumeration theorem and some results from Williamson's generalized setup of it are proved in terms of Schur- Macdonald's theory (S-MT) of "invariant matrices". Given a permutation group W ≤ Sd and a one-dimensional characterof W, the polynomial functor Fcorresponding via S-MT to the induced monomial representation U� = ind S d W (�) of Sd, is studied. It turns out that the characteristic ch(F�) is the weighted inventory of some set J(�) of W-orbits in the integer-valued hypercube (0, ∞)d. The elements of J(�) can be distinguished among all W-orbits by a maximum property. The identity ch(F�) = ch(U�) of both characteristics is a consequence of S-MT, and is equivalent to a result of Williamson. Polya's theorem can be obtained from the above identity by the specialization � = 1W, where 1W is the unit character of W.
Introduction In our paper [3], it is shown that the natural environment for P'olya's fund... more Introduction In our paper [3], it is shown that the natural environment for P'olya's fundamental enumeration theorem and for one of its possible generalizations, is Schur-Macdonald's theory of invariant matrices (cf. [4, Ch. I, Appendix]). The main result of this theory is the eqivalence between the category of finite-dimensional linear representations of the symmetric group S d and the category of polynomial homogeneous degree d functors on the category of finite-dimensional linear spaces. The above-mentioned generalization is a particular case of the main equality of characteristics of two objects corresponding each other via Schur-Macdonald's equivalence. For the original version of P'olya's theorem, we suppose that this fact was pointed out to G. P'olya by I. Schur himself (cf. [5, Ch. 1, n o 20]). Quite the contrary, it turns out that such a generalization of Redfield's master theorem lives only in the environment of the representation theory of ...
Polya's fundamental enumeration theorem is generalized in terms of Schur-Macdonald's theo... more Polya's fundamental enumeration theorem is generalized in terms of Schur-Macdonald's theory (S-MT) of invariant matrices. Given a permutation group $W\leq S_d$ and a one-dimensional character $\chi$ of $W$, the polynomial functor $F_\chi$ corresponding via S-MT to the induced monomial representation $U_\chi = ind_W^{S_d}(\chi)$ of $S_d$, is studied. It turns out that the characteristic $ch(F_\chi)$ is the weighted inventory of some set $J(\chi)$ of $W$-orbits in the integer-valued hypercube $[0,\infty)^d$. The elements of $J(\chi) can be distinguished among all $W$-orbits by a maximum property. The identity $ch(F_\chi) = ch(U_\chi)$ of both characteristics is a consequence of S-MT. Polya's theorem can be obtained from the above identity by specialization $\chi=1_W$, where $1_W$ is the unit character of $W$.
Given a loset I, every surjective map p: A ---> I endows the set A with a structure of preorde... more Given a loset I, every surjective map p: A ---> I endows the set A with a structure of preordered set by "replacing" the elements of I with their inverse images via p considered as "bubbles" (sets endowed with an equivalence relation), lifting the structure of loset on A, and "agglutinating" this structure with the bubbles. Every bubbling A of a structure of loset I is a structure of preordered set A (not necessarily complete) whose preorder has negatively transitive asymmetric part and every such structure on a given set A can be obtained by bubbling up of certain structure of a loset I, intrinsically encoded in A. In other words, the difference between linearity and negative transitivity is constituted of bubbles. As a consequence of this characterization, under certain natural topological conditions on the preordered set A furnished with its interval topology, the existence of a continuous generalized utility function on A is proved.
Transactions of the American Mathematical Society, 1988
This paper is devoted to finding necessary and sufficient conditions for a graded ring to be the ... more This paper is devoted to finding necessary and sufficient conditions for a graded ring to be the canonical ring of a minimal surface of general type with K 2 = 2 p g − 3 {K^2} = 2{p_g} - 3 , p g ⩾ 3 {p_g} \geqslant 3 , and such that its canonical linear system has one base point.
Transactions of the American Mathematical Society, 1988
This paper is devoted to finding necessary and sufficient conditions for a graded ring to be the ... more This paper is devoted to finding necessary and sufficient conditions for a graded ring to be the canonical ring of a minimal surface of general type with K2 = 2pg-3, pg > 3, and such that its canonical linear system has one base point. 1. Known results. 1.1. Let 5 be a minimal surface of general type with K2 = 2pg-3, pg > 3. We assume that the canonical linear system \K\ of S has one base point P. Following Horikawa [4, Lemma 2] let ir: S'-► 5 be the quadratic transformation with center P and exceptional curve E.
In this paper the author finds explicitly all finite-dimensional irreducible representations of a... more In this paper the author finds explicitly all finite-dimensional irreducible representations of a series of finite permutation groups that are homomorphic images of Artin braid group.
In this paper, a mathematical characterization of Lunn-Senior's groups of univalent substitution ... more In this paper, a mathematical characterization of Lunn-Senior's groups of univalent substitution isomerism of the linear alkanes, under some natural assumptions that reflect their common properties, is given. For each linear alkane, the number of its monosubstitution derivatives, its di-substitution derivatives, and its tri-substitution derivatives as linear, quadratic, and cubic polynomial expressions, respectively, in their number, is obtained. In principle, the number of derivatives of a given linear alkane with any particular composition can be established. The same explicit expressions for the case of k-substitution homogeneous derivatives of the linear alkanes are obtained by Balasubramanian (Theoret. Chim. Acta (Berl.) 51:37, 1979).
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