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Mathematics and Statistics

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We show that every positroid of rank ๐‘Ÿ โ‰ฅ 2 has a good coline. Using the definition of the chromatic number of oriented matroid introduced by J. Neลกetล™il, R. Nickel, and W. Hochstรคttler, this shows that every orientation of a positroid of rank at least 2 is 3-colorable.

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Let ๐‘‡ be a tree, a vertex of degree one is called a leaf. The set of all leaves of ๐‘‡ is denoted by Leaf(๐‘‡). The subtree ๐‘‡ โˆ’ Leaf(๐‘‡) of ๐‘‡ is called the stem of ๐‘‡ and denoted by Stem(๐‘‡). A tree ๐‘‡ is called a caterpillar if Stem(๐‘‡) is a path. In this paper, we give two sufficient conditions for a connected graph to have a spanning tree whose stem is a caterpillar. We also give some examples to show that these conditions are sharp.

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We revisit the problem of property testing for convex position for point sets in โ„๐‘‘. Our results draw from previous ideas of Czumaj, Sohler, and Ziegler (2000). First, their testing algorithm is redesigned and its analysis is revised for correctness. Second, its functionality is expanded by (i) exhibiting both negative and positive certificates along with the convexity determination, and (ii) significantly extending the input range for moderate and higher dimensions.

The behavior of the randomized tester on input set ๐‘ƒ โŠ‚ โ„๐‘‘ is as follows: (i) if ๐‘ƒ is in convex position, it accepts; (ii) if ๐‘ƒ is far from convex position, with probability at least 2/3, it rejects and outputs a (๐‘‘ +2)-point witness of non-convexity as a negative certificate; (iii) if ๐‘ƒ is close to convex position, with probability at least 2/3, it accepts and outputs a subset in convex position that is a suitable approximation of the largest subset in convex position. The algorithm examines a sublinear number of points and runs in subquadratic time for every fixed dimension ๐‘‘.

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We prove zero density theorems for Dedekind zeta functions in the vicinity of the line Re s = 1, improving an earlier result of W. Staล›.

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A positive integer d = โˆ i = 1 r p i d i is said to be an exponential divisor or an e-divisor of n = โˆ i = 1 r p i n i > 1 if ๐‘‘๐‘– โˆฃ ๐‘›๐‘– for all prime divisors ๐‘๐‘– of ๐‘›. In addition, 1 is an e-divisor of 1. It is easy to see that โ„ค+ is a poset under the e-divisibility relation. Utilizing this observation we show that e-convolution of arithmetical functions is an example of the convolution of incidence functions of posets. We also note that the identity, units and the Mรถbius function are preserved in this process.

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Let (๐‘ƒ๐‘›)๐‘›โ‰ฅ0 and (๐‘„๐‘›)๐‘›โ‰ฅ0 be the Pell and Pellโ€“Lucas sequences. Let ๐‘ be a positive integer such that ๐‘ โ‰ฅ 2. In this paper, we prove that the following two Diophantine equations ๐‘ƒ๐‘› = ๐‘๐‘‘๐‘ƒ๐‘š + ๐‘„๐‘˜ and ๐‘ƒ๐‘› = ๐‘๐‘‘๐‘„๐‘š + ๐‘ƒ๐‘˜ with ๐‘‘, the number of digits of ๐‘ƒ๐‘˜ or ๐‘„๐‘˜ in base ๐‘, have only finitely many solutions in nonnegative integers (๐‘š, ๐‘›, ๐‘˜, ๐‘, ๐‘‘). Also, we explicitly determine these solutions in cases 2 โ‰ค ๐‘ โ‰ค 10.

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Grรคtzer and Lakser asked in the 1971 Transactions of the American Mathematical Society if the pseudocomplemented distributive lattices in the amalgamation class of the subvariety generated by ๐Ÿ๐‘› โŠ• ๐Ÿ can be characterized by the property of not having a *-homomorphism onto ๐Ÿ๐‘– โŠ• ๐Ÿ for 1 < ๐‘– < ๐‘›.

In this article, their question from 1971 is answered.

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Mathematica Pannonica
Authors:
Muhammad T. Tajuddin
,
Usama A. Aburawash
, and
Muhammad Saad

This paper introduces and examines the concept of a *-Rickart *-ring, and proves that every Rickart *-ring is also a *-Rickart *-ring. A necessary and sufficient condition for a *-Rickart *-ring to be a Rickart *-ring is also provided. The relationship between *-Rickart *-rings and *-Baer *-rings is investigated, and several properties of *-Rickart *-rings are presented. The paper demonstrates that the property of *-Rickart extends to both the center and *-corners of a *-ring, and investigates the extension of a *-Rickart *-ring to its polynomial *-ring. Additionally, *-Rickart *-rings with descending chain condition on *-biideals are studied, and all *-Rickart (*-Baer) *-rings with finitely many elements are classified.

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Very recently, the authors in [5] proposed the exponential-type operator connected with x 4 3 and studied its convergence estimates. In the present research, we extend the study and obtain the general form of its ๐‘-th order moment; ๐‘ โˆˆ โ„• โˆช {0}. Further, we establish the simultaneous approximation for the operator under consideration.

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A bi-cyclic 4-polytope in โ„4 was introduced by Z. Smilansky as the convex hull of evenly spaced points on a generalized trigonometric moment curve in โ„4. We present combinatorial geometric conditions that yield the face lattices of a class of such 4-polytopes.

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A classical result of Dowker (Bull. Amer. Math. Soc. 50: 120-122, 1944) states that for any plane convex body ๐พ, the areas of the maximum (resp. minimum) area convex ๐‘›-gons inscribed (resp. circumscribed) in ๐พ is a concave (resp. convex) sequence. It is known that this theorem remains true if we replace area by perimeter, or convex ๐‘›-gons by disk-๐‘›-gons, obtained as the intersection of ๐‘› closed Euclidean unit disks. It has been proved recently that if ๐ถ is the unit disk of a normed plane, then the same properties hold for the area of ๐ถ-๐‘›-gons circumscribed about a ๐ถ-convex disk ๐พ and for the perimeters of ๐ถ-๐‘›-gons inscribed or circumscribed about a ๐ถ-convex disk ๐พ, but for a typical origin-symmetric convex disk ๐ถ with respect to Hausdorff distance, there is a ๐ถ-convex disk ๐พ such that the sequence of the areas of the maximum area ๐ถ-๐‘›-gons inscribed in ๐พ is not concave. The aim of this paper is to investigate this question if we replace the topology induced by Hausdorff distance with a topology induced by the surface area measure of the boundary of ๐ถ.

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Studia Scientiarum Mathematicarum Hungarica
Authors:
Koppรกny Encz
,
Mรกrton Marits
,
Benedek Vรกli
, and
Mรกtรฉ Weisz

A ๐‘ž-graph with ๐‘’ edges and ๐‘› vertices is defined as an ๐‘’ ร— ๐‘› matrix with entries from {0, โ€ฆ , ๐‘ž}, such that each row of the matrix (called a ๐‘ž-edge) contains exactly two nonzero entries. If ๐ป is a ๐‘ž-graph, then ๐ป is said to contain an ๐‘ -copy of the ordinary graph ๐น, if a set ๐‘† of ๐‘ž-edges can be selected from ๐ป such that their intersection graph is isomorphic to ๐น, and for any vertex ๐‘ฃ of ๐‘† and any two incident edges ๐‘’, ๐‘“ โˆˆ ๐‘† the sum of the entries of ๐‘’ and ๐‘“ is at least ๐‘ . The extremal number ex(๐‘›, ๐น, ๐‘ž, ๐‘ ) is defined as the maximal number of edges in an ๐‘›-vertex ๐‘ž-graph such that it does not contain contain an ๐‘ -copy of the forbidden graph ๐น.

In the present paper, we reduce the problem of finding ex(๐‘›, ๐น, ๐‘ž, ๐‘ž + 1) for even ๐‘ž to the case ๐‘ž = 2, and determine the asymptotics of ex(๐‘›, ๐ถ2๐‘˜+1, ๐‘ž, ๐‘ž + 1).

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Studia Scientiarum Mathematicarum Hungarica
Authors:
David Conlon
,
Jacob Fox
,
Xiaoyu He
,
Dhruv Mubayi
,
Andrew Suk
, and
Jacques Verstraรซte

For positive integers ๐‘›, ๐‘Ÿ, ๐‘  with ๐‘Ÿ > ๐‘ , the set-coloring Ramsey number ๐‘…(๐‘›; ๐‘Ÿ, ๐‘ ) is the minimum ๐‘ such that if every edge of the complete graph ๐พ๐‘ receives a set of ๐‘  colors from a palette of ๐‘Ÿ colors, then there is guaranteed to be a monochromatic clique on ๐‘› vertices, that is, a subset of ๐‘› vertices where all of the edges between them receive a common color. In particular, the case ๐‘  = 1 corresponds to the classical multicolor Ramsey number. We prove general upper and lower bounds on ๐‘…(๐‘›; ๐‘Ÿ, ๐‘ ) which imply that ๐‘…(๐‘›; ๐‘Ÿ, ๐‘ ) = 2ฮ˜(๐‘›๐‘Ÿ) if ๐‘ /๐‘Ÿ is bounded away from 0 and 1. The upper bound extends an old result of Erdล‘s and Szemerรฉdi, who treated the case ๐‘  = ๐‘Ÿ โˆ’ 1, while the lower bound exploits a connection to error-correcting codes. We also study the analogous problem for hypergraphs.

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We consider a function from the Euclidean three space whose zero set is the image of the standard cuspidal edge. The composition of a parametrized singular surface in the three space with this function provides an approximation of the surface by the standard cuspidal edge. Taking a look at singularities of this composition, we study various approximations of singular surfaces like the cross cap, the generalized cuspidal edge and the swallowtail by standard cuspidal edges.

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Based on Peterโ€™s work from 2003, quadrilaterals can be characterized in the following way: โ€œamong all quadrilaterals with given side lengths ๐‘Ž, ๐‘, ๐‘ and ๐‘‘, those of the largest possible area are exactly the cyclic onesโ€. In this paper, we will give the corresponding characterization for every polygon, by means of quasicyclic polygons properties.

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In this paper, some basic characterizations of a weighted Bloch space with the differentiable strictly positive weight ๐œ” on the unit disc are given, including the growth, the higher order or free derivative descriptions, and integral characterizations of functions in the space.

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We present examples of multiplicative semigroups of positive reals (Beurlingโ€™s generalized integers) with gaps bounded from below.

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In this paper, we propose some new positive linear approximation operators, which are obtained from a composition of certain integral type operators with certain discrete operators. It turns out that the new operators can be expressed in discrete form. We provide representations for their coefficients. Furthermore, we study their approximation properties and determine their moment generating functions, which may be useful in finding several other convergence results in different settings.

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Let ๐‘“ be a normalized primitive cusp form of even integral weight for the full modular group ฮ“ = ๐‘†๐ฟ(2, โ„ค). In this paper, we investigate upper bounds for the error terms related to the average behavior of Fourier coefficients ๐œ†๐‘“ โŠ—๐‘“ โŠ—โ‹ฏโŠ—๐‘™๐‘“ (๐‘›) of ๐‘™-fold product ๐ฟ-functions, where ๐‘“ โŠ— ๐‘“ โŠ— โ‹ฏ โŠ—๐‘™ ๐‘“ denotes the ๐‘™-fold product of ๐‘“. These results improves and generalizes the recent developments of Venkatasubbareddy and Sankaranarayanan [41]. We also provide some other similar results related to the error terms of general product ๐ฟ-functions.

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The paper is devoted to some extremal problems for convex polygons on the Euclidean plane, related to the concept of self Chebyshev radius for the polygon boundary. We consider a general problem of minimization of the perimeter among all ๐‘›-gons with a fixed self Chebyshev radius of the boundary. The main result of the paper is the complete solution of the mentioned problem for ๐‘› = 4: We proved that the quadrilateral of minimum perimeter is a so called magic kite, that verified the corresponding conjecture by Rolf Walter.

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Let ๐‘› โ‰ฅ 2 be an integer. The graph G n ยฏ is obtained by letting all the elements of {0, โ€ฆ , ๐‘› โˆ’ 1} to be the vertices and defining distinct vertices ๐‘ฅ and ๐‘ฆ to be adjacent if and only if gcd(๐‘ฅ + ๐‘ฆ, ๐‘›) โ‰  1. In this paper, we give some bounds for the Castelnuovoโ€“Mumford regularity of the edge ideals and their powers for G n ยฏ .

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Motivated by the examples of Heppes and Wegner, we present several other examples of the following kind: a bounded convex region ๐ท and a convex disk ๐พ in the plane are described, such that every thinnest covering of ๐ท with congruent copies of ๐พ contains crossing pairs.

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In this paper we show that the spherical cap discrepancy of the point set given by centers of pixels in the HEALPix tessellation (short for Hierarchical, Equal Area and iso-Latitude Pixelation) of the unit 2-sphere is lower and upper bounded by order square root of the number of points, and compute explicit constants. This adds to the currently known (short) collection of explicitly constructed sets whose discrepancy converges with order ๐‘ โˆ’1/2, matching the asymptotic order for i.i.d. random point sets. We describe the HEALPix framework in more detail and give explicit formulas for the boundaries and pixel centers. We then introduce the notion of an ๐‘›-convex curve and prove an upper bound on how many fundamental domains are intersected by such curves, and in particular we show that boundaries of spherical caps have this property. Lastly, we mention briefly that a jittered sampling technique works in the HEALPix framework as well.

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We apply a recent general zero density theorem of us (valid for a large class of complex functions) to improve earlier density theorems of Heath-Brown and Paulโ€“Sankaranarayanan for Dedekind zeta functions attached to a number field ๐พ of degree ๐‘› with ๐‘› > 2.

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Asymptotic uniform upper density, shortened as a.u.u.d., or simply upper density, is a classical notion which was first introduced by Kahane for sequences in the real line.

Syndetic sets were defined by Gottschalk and Hendlund. For a locally compact group ๐บ, a set ๐‘† โŠ‚ ๐บ is syndetic, if there exists a compact subset ๐ถ โ‹ ๐บ such that ๐‘†๐ถ = ๐บ. Syndetic sets play an important role in various fields of applications of topological groups and semigroups, ergodic theory and number theory. A lemma in the book of Fรผrstenberg says that once a subset ๐ด โŠ‚ โ„ค has positive a.u.u.d., then its difference set ๐ด โˆ’ ๐ด is syndetic.

The construction of a reasonable notion of a.u.u.d. in general locally compact Abelian groups (LCA groups for short) was not known for long, but in the late 2000โ€™s several constructions were worked out to generalize it from the base cases of โ„ค๐‘‘ and โ„๐‘‘. With the notion available, several classical results of the Euclidean setting became accessible even in general LCA groups.

Here we work out various versions in a general locally compact Abelian group ๐บ of the classical statement that if a set ๐‘† โŠ‚ ๐บ has positive asymptotic uniform upper density, then the difference set ๐‘† โˆ’ ๐‘† is syndetic.

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Recently [3] we proved a general zero density theorem for a large class of functions which included among others the Riemann zeta function, Dedekind zeta functions, Dirichlet ๐ฟ-functions. The goal of the present work is a (slight) improvement of this general theorem which might lead to stronger results in some applications.

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This paper serves as a kick-off to address the question of how to define and investigate the stability of bi-continuous semigroups. We will see that the mixed topology is the key concept in this framework.

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An endo-commutative algebra is a nonassociative algebra in which the square mapping preserves multiplication. In this paper, we give a complete classification of 2-dimensional endo-commutative straight algebras of rank one over an arbitrary non-trivial field, where a straight algebra of dimension 2 satisfies the condition that there exists an element x such that x and x 2 are linearly independent. We list all multiplication tables of the algebras up to isomorphism.

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In this paper, we consider the simultaneous sign changes of coefficients of Rankinโ€“Selberg L-functions associated to two distinct Hecke eigenforms supported at positive integers represented by some certain primitive reduced integral binary quadratic form with negative discriminant D. We provide a quantitative result for the number of sign changes of such sequence in the interval (x, 2x] for sufficiently large x.

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In this paper, we derive several asymptotic formulas for the sum of d(gcd(m,n)), where d(n) is the divisor function and m,n are in Piatetski-Shapiro and Beatty sequences.

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Let ๐‘› โˆˆ โ„•. An element (x 1, โ€ฆ , x ๐‘›) โˆˆ En is called a norming point of T โˆˆ L ( nE) if โ€–x 1โ€– = โ‹ฏ = โ€–xn โ€– = 1 and |T (x 1, โ€ฆ , xn )| = โ€–Tโ€–, where L ( nE) denotes the space of all continuous n-linear forms on E. For T โˆˆ L ( nE), we define

Norm(T) = {(x 1, โ€ฆ , x n) โˆˆ En โˆถ (x 1, โ€ฆ , x n) is a norming point of T}.

Norm(T) is called the norming set of T. We classify Norm(T) for every T โˆˆ L (2 ๐‘‘ โˆ—(1, w)2), where ๐‘‘ โˆ—(1, w)2 = โ„2 with the octagonal norm of weight 0 < w < 1 endowed with x , y d * 1 , w = max x , y , x + y 1 + w .

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In this paper, we introduce and study the class of k-strictly quasi-Fredholm linear relations on Banach spaces for nonnegative integer k. Then we investigate its robustness through perturbation by finite rank operators.

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We construct an algebra of dimension 2โ„ต0 consisting only of functions which in no point possess a finite one-sided derivative. We further show that some well known nowhere differentiable functions generate algebras, which contain functions which are differentiable at some points, but where for all functions in the algebra the set of points of differentiability is quite small.

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Studia Scientiarum Mathematicarum Hungarica
Authors:
Jesรบs A. De Loera
,
Christopher Oโ€™Neill
, and
Chengyang Wang

In this paper, we explore affine semigroup versions of the convex geometry theorems of Helly, Tverberg, and Carathรฉodory. Additionally, we develop a new theory of colored affine semigroups, where the semigroup generators each receive a color and the elements of the semigroup take into account the colors used (the classical theory of affine semigroups coincides with the case in which all generators have the same color). We prove an analog of Tverbergโ€™s theorem and colorful Hellyโ€™s theorem for semigroups, as well as a version of colorful Carathรฉodoryโ€™s theorem for cones. We also demonstrate that colored numerical semigroups are particularly rich by introducing a colored version of the Frobenius number.

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We prove the endomorphism conjecture for graded posets with largest Whitney number at most 4.

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Lovejoy introduced the partition function A l ยฏ n as the number of ๐‘™-regular overpartitions of ๐‘›. Andrews defined (๐‘˜, ๐‘–)-singular overpartitions counted by the partition function C ยฏ k , i n , and pointed out that C ยฏ 3 , 1 n = A 3 ยฏ n . Meanwhile, Andrews derived an interesting divisibility property that C ยฏ 3 , 1 9 n + 3 โ‰ก C ยฏ 3 , 1 9 n + 6 โ‰ก 0 (mod 3). Recently, we constructed the partition statistic ๐‘Ÿ๐‘™-crank of ๐‘™-regular overpartitions and give combinatorial interpretations for some congruences of A l ยฏ n as well as the congruences of Andrews. In this paper, we aim to prove some equalities for the ๐‘Ÿ3-crank of 3-regular overpartitions.

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The Hilbert metric between two points ๐‘ฅ, ๐‘ฆ in a bounded convex domain ๐บ is defined as the logarithm of the cross-ratio ๐‘ฅ, ๐‘ฆ and the intersection points of the Euclidean line passing through the points ๐‘ฅ, ๐‘ฆ and the boundary of the domain. Here, we study this metric in the case of the unit ball ๐”น๐‘›. We present an identity between the Hilbert metric and the hyperbolic metric, give several inequalities for the Hilbert metric, and results related to the inclusion properties of the balls defined in the Hilbert metric. Furthermore, we study the distortion of the Hilbert metric under conformal and quasiregular mappings.

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This article studies a new class of monomial ideals associated with a simple graph ๐บ, called generalized edge ideal, denoted by ๐ผ๐‘”(๐บ). Assuming that all the vertices ๐‘ฅ have an exponent greater than 1 in ๐ผ๐‘”(๐บ), we completely characterize the graph ๐บ for which ๐ผ๐‘”(๐บ) is integrally closed, and show that this is equivalent to ๐ผ๐‘”(๐บ) being normal i.e., all integral powers of ๐ผ๐‘”(๐บ) are integrally clased. We also give a necessary and sufficient condition for I g G = I g G ยฏ when ๐บ is the star-shaped graph. Finally, we give a necessary and sufficient condition when the generalized edge ideal of a complete graph is integrally closed.

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A proper edge coloring of a graph ๐บ is strong if the union of any two color classes does not contain a path with three edges (i.e. the color classes are induced matchings). The strong chromatic index ๐‘ž(๐บ) is the smallest number of colors needed for a strong coloring of ๐บ. One form of the famous (6, 3)-theorem of Ruzsa and Szemerรฉdi (solving the (6, 3)-conjecture of Brownโ€“Erdล‘sโ€“Sรณs) states that ๐‘ž(๐บ) cannot be linear in ๐‘› for a graph ๐บ with ๐‘› vertices and ๐‘๐‘›2 edges. Here we study two refinements of ๐‘ž(๐บ) arising from the analogous (7, 4)-conjecture. The first is ๐‘ž๐ด(๐บ), the smallest number of colors needed for a proper edge coloring of ๐บ such that the union of any two color classes does not contain a path or cycle with four edges, we call it an A-coloring. The second is ๐‘ž๐ต(๐บ), the smallest number of colors needed for a proper edge coloring of ๐บ such that all four-cycles are colored with four different colors, we call it a B-coloring. These notions lead to two stronger and one equivalent form of the (7, 4)-conjecture in terms of ๐‘ž๐ด(๐บ), ๐‘ž๐ต(๐บ) where ๐บ is a balanced bipartite graph. Since these are questions about graphs, perhaps they will be easier to handle than the original special(7, 4)-conjecture. In order to understand the behavior of ๐‘ž๐ด(๐บ) and ๐‘ž๐ต(๐บ), we study these parameters for some graphs.

We note that ๐‘ž๐ด(๐บ) has already been extensively studied from various motivations. However, as far as we know the behavior of ๐‘ž๐ต(๐บ) is studied here for the first time.

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We study a natural set of refinements of the Ehrhart series of a closed polytope, first considered by Chapoton. We compute the refined series in full generality for a simplex of dimension ๐‘‘, a cross-polytope of dimension ๐‘‘, respectively a hypercube of dimension ๐‘‘ โ‰ค 3, using commutative algebra. We deduce summation formulae for products of ๐‘ž-integers with different arguments, generalizing a classical identity due to MacMahon and Carlitz. We also present a characterisation of a certain refined Eulerian polynomial in algebraic terms.

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Over integral domains of characteristics different from 2, we determine all the matrices a b c d which are similar to c a d b .

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We present generalizations of the Pinelis extension of Stolarskyโ€™s inequality and its reverse. In particular, a new Stolarsky-type inequality is obtained. We study the properties of the linear functional related to the new Stolarsky-type inequality, and finally apply these new results in the theory of fractional integrals.

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In this paper, we consider the Feuerbach point and the Feuerbach line of a triangle in the isotropic plane, and investigate some properties of these concepts and their relationships with other elements of a triangle in the isotropic plane. We also compare these relationships in Euclidean and isotropic cases.

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We define the order of the double hypergeometric series, investigate the properties of the new confluent Kampรฉ de Fรฉriet series, and build systems of partial differential equations that satisfy the new Kampรฉ de Fรฉriet series. We solve the Cauchy problem for a degenerate hyperbolic equation of the second kind with a spectral parameter using the high-order Kampรฉ de Fรฉriet series. Thanks to the properties of the introduced Kampรฉ de Fรฉriet series, it is possible to obtain a solution to the problem in explicit forms.

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Let ๐”ผ ๐‘‘ denote the ๐‘‘-dimensional Euclidean space. The ๐‘Ÿ-ball body generated by a given set in ๐”ผ ๐‘‘ is the intersection of balls of radius ๐‘Ÿ centered at the points of the given set. The author [Discrete Optimization 44/1 (2022), Paper No. 100539] proved the following Blaschkeโ€“Santalรณ-type inequalities for ๐‘Ÿ-ball bodies: for all 0 < ๐‘˜ < ๐‘‘ and for any set of given ๐‘‘-dimensional volume in ๐”ผ ๐‘‘ the ๐‘˜-th intrinsic volume of the ๐‘Ÿ-ball body generated by the set becomes maximal if the set is a ball. In this note we give a new proof showing also the uniqueness of the maximizer. Some applications and related questions are mentioned as well.

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We discuss the outline of the shapes of graphs of ฯ‡ 2 statistics for distributions of leading digits of irrational rotations under some conditions on mth convergent. We give some estimates of important coefficients Lk โ€™s, which determine the graphical shapes of ฯ‡2 statistics. This means that the denominator qm of mth convergent and the large partial quotient am +1 determine the outline of shapes of graphs, when we observe values of ฯ‡ 2 statistics with step qm .

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In this note, we introduce the concept of semi-*-IFP, the involutive version of semi-IFP, which is a generalization of quasi-*-IFP and *-reducedness of *-rings. We study the basic structure and properties of *-rings having semi-*-IFP and give results for IFPs in rings with involution. Several results and counterexamples are stated to connect the involutive versions of IFP. We discuss the conditions for the involutive IFPs to be extended into *-subrings of the ring of upper triangular matrices. In *-rings with quasi-*-IFP, it is shown that Kรถtheโ€™s conjecture has a strong affirmative solution. We investigate its related properties and the relationship between *-rings with quasi-*-IFP and *-Armendariz properties.

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In the present paper, we establish the convergence rates of the single logarithm and the iterated logarithm for martingale differences which give some further results for the open question in Stoica [6].

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We give a full, correct proof of the following result, earlier claimed in [1]. If the Continuum Hypothesis holds then there is a coloring of the plane with countably many colors, with no monocolored right triangle.

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