Symmetry of Syzygies of a System of Functional Equations Defining a Ring Homomorphism

I deal with an alienation problem for the system of two fundamental Cauchy functional equations with an unknown function f mapping a ring X into an integral domain Y and preserving binary operations of addition and multiplication, respectively. The resulting syzygies obtained by adding (resp. multiplying) these two equations side by side are discussed. The first of these two syzygies was first examined by Jean Dhombres in 1988 who proved that under some additional conditions concering the domain and range rings it forces f to be a ring homomorphism (alienation phenomenon). The novelty of the present paper is to look for sufficient conditions upon f solving the other syzygy to be alien.

Download Full-text

ON A CLASS OF DISTRIBUTIVE FUZZY IMPLICATIONS

International Journal of Uncertainty Fuzziness and Knowledge-Based Systems ◽

10.1142/s0218488501000764 ◽

2001 ◽

Vol 09 (02) ◽

pp. 229-238 ◽

Cited By ~ 34

Author(s):

MICHAŁ BACZYŃSKI

Keyword(s):

Unknown Function ◽

Functional Equations ◽

System Of Functional Equations ◽

Fuzzy Implications ◽

Fuzzy Implication

We deal with the system of functional equations I(x,T(y,z))=T(I(x,y), I(x,z)), I(x,I(y,z))=I(T(x,y),z), where function T:[0,1]2→ [0,1] is a strict t-norm and I:[0,1]2→ [0,1] is an unknown function. Under some assumptions imposed on function I we obtain that I is almost conjugate with the Yager fuzzy implication.

Download Full-text

The interface dynamics of a surfactant drop on a thin viscous film

European Journal of Applied Mathematics ◽

10.1017/s0956792516000474 ◽

2016 ◽

Vol 28 (4) ◽

pp. 656-686 ◽

Cited By ~ 2

Author(s):

MARINA CHUGUNOVA ◽

JOHN R. KING ◽

ROMAN M. TARANETS

Keyword(s):

Diffusion Coefficient ◽

Parabolic Equations ◽

Functional Equations ◽

Surfactant Concentration ◽

Thin Liquid Film ◽

Sufficient Conditions ◽

System Of Functional Equations ◽

Surface Diffusion Coefficient ◽

Finite Speed ◽

Insoluble Surfactant

We study a system of two coupled parabolic equations that models the spreading of a drop of an insoluble surfactant on a thin liquid film. Unlike the previously known results, the surface diffusion coefficient is not assumed constant and depends on the surfactant concentration. We obtain sufficient conditions for finite speed of support propagation and for waiting-time phenomenon by application of an extension of Stampacchia's lemma for a system of functional equations.

Download Full-text

Pembentukan Lapangan Faktor dari Suatu Daerah Integral

Jurnal Natural ◽

10.30862/jn.v8i2.340 ◽

2012 ◽

Vol 8 (2) ◽

Author(s):

Tri Widjajanti ◽

Dahlia Ramlan ◽

Rium Hilum

Keyword(s):

Polynomial Ring ◽

Integral Domain ◽

Rational Numbers ◽

Ring Of Integers ◽

Binary Operations

Ring of integers under the addition and multiplication as integral domain can be imbedded to the field of rational numbers. In this paper we make  a construction such that any integral domain can be  a field of quotient. The construction contains three steps. First, we define element of field F from elements of integral domain D. Secondly, we show that the binary operations in fare well-defined. Finally, we prove that  f : D ® F is an isomorphisma. In this case, the polynomial ring F[x] as the integral domain can be imbedded to the field of quotient.

Download Full-text

Jordan products of quantum channels and their compatibility

Nature Communications ◽

10.1038/s41467-021-22275-0 ◽

2021 ◽

Vol 12 (1) ◽

Author(s):

Mark Girard ◽

Martin Plávala ◽

Jamie Sikora

Keyword(s):

Quantum State ◽

Sufficient Conditions ◽

The Other ◽

Independent Interest ◽

Quantum Channels ◽

Semidefinite Programs ◽

Jordan Product ◽

Marginal Problem ◽

Jordan Products ◽

Product Of Matrices

AbstractGiven two quantum channels, we examine the task of determining whether they are compatible—meaning that one can perform both channels simultaneously but, in the future, choose exactly one channel whose output is desired (while forfeiting the output of the other channel). Here, we present several results concerning this task. First, we show it is equivalent to the quantum state marginal problem, i.e., every quantum state marginal problem can be recast as the compatibility of two channels, and vice versa. Second, we show that compatible measure-and-prepare channels (i.e., entanglement-breaking channels) do not necessarily have a measure-and-prepare compatibilizing channel. Third, we extend the notion of the Jordan product of matrices to quantum channels and present sufficient conditions for channel compatibility. These Jordan products and their generalizations might be of independent interest. Last, we formulate the different notions of compatibility as semidefinite programs and numerically test when families of partially dephasing-depolarizing channels are compatible.

Download Full-text

On a New Geometric Constant Related to the Euler-Lagrange Type Identity in Banach Spaces

Mathematics ◽

10.3390/math9020116 ◽

2021 ◽

Vol 9 (2) ◽

pp. 116

Author(s):

Qi Liu ◽

Yongjin Li

Keyword(s):

Banach Spaces ◽

Product Space ◽

Sufficient Conditions ◽

Normal Structure ◽

The Other ◽

Inner Product ◽

Equivalent Characterization ◽

Geometric Constant ◽

Geometric Constants

In this paper, we will introduce a new geometric constant LYJ(λ,μ,X) based on an equivalent characterization of inner product space, which was proposed by Moslehian and Rassias. We first discuss some equivalent forms of the proposed constant. Next, a characterization of uniformly non-square is given. Moreover, some sufficient conditions which imply weak normal structure are presented. Finally, we obtain some relationship between the other well-known geometric constants and LYJ(λ,μ,X). Also, this new coefficient is computed for X being concrete space.

Download Full-text

Using binary operations to construct a transitive set of block transformations

Discrete Mathematics and Applications ◽

10.1515/dma-2020-0035 ◽

2020 ◽

Vol 30 (6) ◽

pp. 375-389

Author(s):

Igor V. Cherednik

Keyword(s):

Binary Operation ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Binary Operations ◽

The Family ◽

Transitive Set ◽

Necessary And Sufficient

AbstractWe study the set of transformations {ΣF : F∈ 𝓑∗(Ω)} implemented by a network Σ with a single binary operation F, where 𝓑∗(Ω) is the set of all binary operations on Ω that are invertible as function of the second variable. We state a criterion of bijectivity of all transformations from the family {ΣF : F∈ 𝓑∗(Ω)} in terms of the structure of the network Σ, identify necessary and sufficient conditions of transitivity of the set of transformations {ΣF : F∈ 𝓑∗(Ω)}, and propose an efficient way of verifying these conditions. We also describe an algorithm for construction of networks Σ with transitive sets of transformations {ΣF : F∈ 𝓑∗(Ω)}.

Download Full-text

Equipartitioning and balancing points of polygons

Pythagoras ◽

10.4102/pythagoras.v0i71.2 ◽

2010 ◽

Vol 0 (71) ◽

Author(s):

Shunmugam Pillay ◽

Poobhalan Pillay

Keyword(s):

General Theorem ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

The Other ◽

Sufficient Condition ◽

Centre Of Mass ◽

Necessary And Sufficient Condition ◽

Necessary And Sufficient

The centre of mass G of a triangle has the property that the rays to the vertices from G sweep out triangles having equal areas. We show that such points, termed equipartitioning points in this paper, need not exist in other polygons. A necessary and sufficient condition for a quadrilateral to have an equipartitioning point is that one of its diagonals bisects the other. The general theorem, namely, necessary and sufficient conditions for equipartitioning points for arbitrary polygons to exist, is also stated and proved. When this happens, they are in general, distinct from the centre of mass. In parallelograms, and only in them, do the two points coincide.

Download Full-text

MEASURES OF LOCAL ENTROPIES FOR COMPOSITIVE FUZZY PARTITIONS

International Journal of Uncertainty Fuzziness and Knowledge-Based Systems ◽

10.1142/s0218488511007192 ◽

2011 ◽

Vol 19 (04) ◽

pp. 717-728 ◽

Cited By ~ 2

Author(s):

DORETTA VIVONA ◽

MARIA DIVARI

Keyword(s):

Functional Equations ◽

Fuzzy Measure ◽

System Of Functional Equations ◽

Locality Property ◽

Fuzzy Measures ◽

Fuzzy Partitions

The aim of this paper is to characterize of the measures of entropies without probability or fuzzy measure for compositive fuzzy partitions, taking into account the so-called locality property. We propose a system of functional equations, whose solutions give some forms of entropies without probability or fuzzy measures.

Download Full-text