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Abstract: Approximation of convex disks by inscribed and circumscribed polygons is a classical geometric problem whose study is motivated by various applications in robotics and computer aided design. We consider the following optimization problem: given integers \(3\le n\le m-1\) , find the value or an estimate of $$\begin{aligned} r(n,m)=\max _{P\in {\mathcal {P}}_m}\,\, \min _{Q\in {\mathcal {P}}_n,\,Q \supseteq P} \frac{ Q }{ P } \end{aligned}$$ where P varies in the set \({\mathcal {P}}_m\) of all convex m-gons, and, for a fixed m-gon P, the minimum is taken over all n-gons Q containing P; here \( \cdot \) denotes area. It is easy to prove that \(r(3,4)=2\) , and from a result of Gronchi and Longinetti it follows that \(r(n-1, n)= 1+\frac{1}{n}\tan \left( \pi /{n}\right) \tan \left( {2\pi }/{n}\right) \) for all \(n\ge 6\) . In this paper we show that every unit area convex pentagon is contained in a convex quadrilateral of area no greater than \(3/\sqrt{5}\) thus determining the value of r(4, 5). In all cases, the equality is reached only for affine regular polygons. PubDate: 2023-06-05

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Abstract: In 1997 we proved that if n is of the form $$\begin{aligned} 4k, \quad 8k-1\quad {\textrm{or}} \quad 2^{2m+1}(2k-1)+3, \end{aligned}$$ where \(k,m\in {\mathbb {N}} \) , then there are no positive rational numbers x, y, z satisfying $$\begin{aligned} xyz = 1, \quad x+y+z = n. \end{aligned}$$ Recently, N. X. Tho proved the following statement: let \(a\in \mathbb N\) be odd and let either \(n\equiv 0\pmod 4\) or \(n\equiv 7\pmod 8\) . Then the system of equations $$\begin{aligned} xyz = a, \quad x+y+z = an. \end{aligned}$$ has no solutions in positive rational numbers x, y, z. A representative example of our result is the following statement: assume that \(a,n\in {\mathbb {N}}\) are such that at least one of the following conditions holds: \(n\equiv 0\pmod 4\) \(n\equiv 7\pmod 8 \) \(a\equiv 0\pmod 4\) \(a\equiv 0\pmod 2\) and \(n\equiv 3\pmod 4\) \(a^2n^3=2^{2m+1}(2k-1)+27\) for some \(k,m\in {\mathbb {N}}.\) Then the system of equations $$\begin{aligned} xyz = a, \quad x+y+z = an. \end{aligned}$$ has no solutions in positive rational numbers x, y, z. PubDate: 2023-06-05

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Abstract: In this paper we study ideals of points lying on rational normal curves defined in projective plane and projective 3-space. We give an explicit formula for the value of Castelnuovo–Mumford regularity for their ordinary powers. Moreover, we compare the m-th symbolic and ordinary powers for such ideals in order to show whenever the m-th symbolic defect is non-zero. PubDate: 2023-06-05

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Abstract: In this paper, we consider the following fractional Choquard–Kirchhoff equation with magnetic fields and critical exponents $$\begin{aligned} M([u]_{s,A}^{2})(-\Delta )_{A}^{s}u+V(x)u=[ x ^{-\alpha }* u ^{2^{*}_{\alpha ,s}}] u ^{2^{*}_{\alpha ,s}-2}u+\lambda f(x,u) \quad \text {in } {\mathbb {R}}^{N}, \end{aligned}$$ where \(N>2s\) with \(0<s<1\) , \(\lambda >0\) , \(A=(A_{1},A_{2},\ldots ,A_{n})\in ({\mathbb {R}}^{N},{\mathbb {R}}^{N})\) is a magnetic potential, \(2^{*}_{\alpha ,s}=(2N-\alpha )/(N-2s)\) is the fractional Hardy—Littlewood—Sobolev critical exponent with \(0<\alpha <2s\) , \(M([u]_{s,A}^{2})=a+b[u]_{s,A}^{2}\) with \(a,b>0\) , \(u\in ({\mathbb {R}}^{N}, {\mathbb {C}})\) is a complex valued function, \(V\in L^{\infty }({\mathbb {R}}^{N})\) and \(f\in ({\mathbb {R}}^{N}\times {\mathbb {R}},{\mathbb {R}})\) are continuous functions, \((-\Delta )^{s}_{A}\) is a fractional magnetic Laplacian operator. Under some suitable assumptions, by applying the Nehari method and the concentration-compactness principle, we obtain the existence of ground state solutions. PubDate: 2023-06-02

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Abstract: The theory of cylindric algebras was introduced by Tarski in the fifties of the twentieth century, and its intensive study was further pursued by pioneers such as Henkin and Monk and, by the Hungarian mathematicians Andréka, Németi and Sain, and many of their students; to name only a few: Madarász, Marx, Kurucz, Simon, Mikulás, and Sági and many others outside Hungary including the author of this paper. Here we introduce and investigate new notions of representability for cylindric algebras and investigate various connections between such notions. Let \(2<n\le l<m\le \omega \) . Let \(\textsf {CA}_n\) denote the variety of cylindric algebras of dimension n and let \(\textsf {RCA}_n\) denote the variety of representable \(\textsf {CA}_n\) s. We say that an atomic algebra \({{\mathfrak {A}}}\in \textsf {CA}_n\) has the complex neat embedding property up to l and m if \({{\mathfrak {A}}}\in \textsf {RCA}_n\cap \textsf {Nr}_n\textsf {CA}_l\) and \({{{\mathfrak {C}}}{{\mathfrak {m}}}}\textsf {At}{{\mathfrak {A}}}\in \mathbf {S}\textsf {Nr}_n\textsf {CA}_m\) . Fixing the prarameters l at the value n, this is a measure of how much the algebra is representable. The yardstick is how far can its Dedekind–MacNeille completion be dilated, that is to say, counting the number of more extra dimensions its Dedekind–MacNeille completion neatly embeds into. If \({{\mathfrak {A}}}, {{\mathfrak {B}}}\in \textsf {RCA}_n\) are atomic, \({{{\mathfrak {C}}}{{\mathfrak {m}}}}\textsf {At}{{\mathfrak {B}}}\in S\textsf {Nr}_n\textsf {CA}_l\) and \({{{\mathfrak {C}}}{{\mathfrak {m}}}}\textsf {At}{{\mathfrak {A}}}\in S\textsf {Nr}_n\textsf {CA}_m\) , then we say that \({{\mathfrak {A}}}\) is more representable than \({{\mathfrak {B}}}\) . When \(m=\omega \) , we say that \({{\mathfrak {A}}}\) is strongly representable; this is the maximum degree of representability; the algebra in question cannot be ‘more representable’ than that. In this case the atom structure of \({{\mathfrak {A}}}\) , namely \(\textsf {At}{{\mathfrak {A}}}\) , is strongly representable in the sense of Hirsch and Hodkinson. This notion gives an infinite potential spectrum of ‘degrees’ of representability. In this connection, we exhibit various atomic algebras in \(\textsf {RCA}_n\cap \textsf {Nr}_n\textsf {CA}_l\) that do no not have the complex neat embedding property for infinitely many values of l and m. It is known that the class of Kripke frames \(\textsf {Str}(\textsf {RCA}_n)=\{{{\mathfrak {F}}}: {{{\mathfrak {C}}}{{\mathfrak {m}}}}{{\mathfrak {F}}}\in \textsf {RCA}_n\}\) is not elementary. From this it follows that there is some \(n<m<\omega \) such that \(\textsf {Str}(\mathbf {S}\textsf {Nr}_n\textsf {CA}_m)=\{{{\mathfrak {F}}}: {{{\mathfrak {C}}}{{\mathfrak {m}}}}{{\mathfrak {F}}}\in \mathbf {S}\textsf {Nr... PubDate: 2023-06-01

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Abstract: Noetherian quivers were studied and classified in [4], but nothing is known about the coherence of quivers. The purpose of this paper is to prove that any acyclic quiver Q is coherent, or, in other words, that for any coherent ring R the category of representations of Q by modules is locally coherent. PubDate: 2023-06-01

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Abstract: In this paper we prove the following result: Let R be a prime ring with \({\text {char}}(R)\ne 2,3,5\) and let \( T :R \rightarrow R\) be an additive mapping satisfying the relation \( 3T(x^{4})=T(x)x^{3}+xT(x^2)x+x^{3}T(x)\) for all \(x\in R\) . In this case T is of the form \(T(x)=\lambda x\) for all \(x\in R\) and some fixed element \(\lambda \in C\) , where C is the extended centroid of R. PubDate: 2023-06-01

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Abstract: Given any positive integer n, it is well known that there always exists a triangle with rational sides a, b and c such that the area of the triangle is n. For any pair of primes (p, q) such that \(p \not \equiv 1\) (mod 8) and \(p^{2}+1=2q\) , we look into the possibility of the existence of triangles having rational sides with p as the area and \(p^{-1}\) as \(\tan \frac{\theta }{2}\) for one of the angles \(\theta \) . We also discuss the relation of such triangles with the solutions of certain Diophantine equations. PubDate: 2023-06-01

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Abstract: Let d be a square-free positive integer. Let a, b, c and \(x_{0}\) be integers, with \(a>0\) . Also let \(\Omega \) be the set of all primes \(p\in {\mathbb {N}}\) satisfying \(p\mid a \) . Suppose that \(b^2-4ac=t^2d\) , for some integer \(t\ge 1\) . Also suppose that the equation \(4p= x^2-dy^2 \) is solvable in integers x, y, for every \(p\in \Omega \) . We prove that if \( an^2+bn+c \) is 1 or prime for every integer \(n\in [x_{0}, x_{0}+\frac{2^{\sigma }\sqrt{ d}}{3}] \) with \(\sigma \in \{0,1\}\) , then \({\mathbb {Z}}[\alpha ]\) is a unique factorization domain, where \(\alpha =\frac{-1+{\sqrt{d}}}{2}\) if \(d\equiv 1\pmod 4\) and \(\alpha =\sqrt{d}\) otherwise. We also apply this criterion to give an improvement of Mollin–Williams’s result that provides sufficient conditions so that \({\mathbb {Q}}(\sqrt{d})\) has class number one. PubDate: 2023-06-01

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Abstract: In this work, we study the following problem: given a regular symmetric form (linear functional) v, find all the regular symmetric forms u which satisfy the equation \(xu = \lambda x^3v, \lambda \in {\mathbb {C}} - \{0\}\) . We give the second-order recurrence relation of the orthogonal polynomial sequence with respect to u. An example is highlighted in the second degree case. PubDate: 2023-06-01

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Abstract: Let (a, b, c) be a primitive Pythagorean triple satisfying \(a^2+b^2=c^2\) . In 1956, Jeśmanowicz conjectured that the exponential Diophantine equation \((na)^x+(nb)^y=(nc)^z\) has no positive integer solutions other than \((x,y,z)=(2,2,2)\) for any positive integer n. In this paper, we obtain an effective sufficient condition for the conjecture for \((a,b,c)=(4k^2-1,4k,4k^2+1)\) with \(k=p^{\alpha }\) where \(p=4m+3\) is a prime number and \(\alpha \) is a positive integer. In addition, numerical results are included, which say that, for example, if \(\alpha =1\) , then the conjecture is true for \(0\le m\le 10^7\) . If we assume a certain conjecture related to abc conjecture, we can prove completely Jeśmanowicz’ conjecture of the above form for any \(\alpha \) and m. PubDate: 2023-06-01

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Abstract: Let \(h\ge 3\) and i be integers with \(1\le i\le h-1\) . In this paper, we give linear independence results for the values of the functions $$\begin{aligned} g_{h,i}(z):=\sum _{n=1}^{\infty }\frac{z^{in}-z^{(h-i)n}}{1-z^{hn}} , \quad z <1, \end{aligned}$$ at suitable algebraic points. As an application, we deduce arithmetical properties of certain sums of reciprocals of linear recurrence sequences. For example, the six numbers $$\begin{aligned} 1,\quad \sum _{n=1}^{\infty }\frac{1}{L_{2n}},\quad \sum _{n=1}^{\infty }\frac{1}{L_{2n}+1},\quad \sum _{n=1}^{\infty }\frac{1}{L_{2n}-1},\quad \sum _{n=1}^{\infty }\frac{1}{L_{4n}},\quad \sum _{n=1}^{\infty }\frac{1}{L_{4n}-1} \end{aligned}$$ are linearly independent over the field \({\mathbb {Q}}\left( \sqrt{5}\right) \) , where \(\{L_{n}\}_{n\ge 0}\) is the classical Lucas sequence. PubDate: 2023-06-01

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Abstract: Let \(\Lambda \) be an algebra whose quiver is In this paper, we classify the \(\tau \) -tilting modules over \(\Lambda \) when \(l(P_1)\leqslant n-2\) . Moreover, the following recurrence formula for the number of \(\tau \) -tilting \(\Lambda \) -modules holds: $$\begin{aligned} {\tau {-}\mathrm{tilt}\, }\Lambda =\sum ^{l(P_1)}_{i=1}C_{i-1}\cdot {\tau {-}\mathrm{tilt}\, }\Lambda /\langle e_{\leqslant i}\rangle , \end{aligned}$$ where \(e_{\leqslant i}:=e_1+e_2+\cdots +e_i\) and \(C_i=\frac{1}{i+1}\left( {\begin{array}{c}2i\\ i\end{array}}\right) \) is the ith Catalan number. PubDate: 2023-06-01

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Abstract: Let \({\mathcal {A}}\) be a factor von Neumann algebra with dim \(({\mathcal {A}})\ge 2\) . For any \(A, B\in {\mathcal {A}}\) , a product \( A\mathbin {\triangle }B=A^{*}B+B^{*}A\) is called a bi-skew Jordan product. In this paper, it is proved that every bijective map preserving bi-skew Jordan triple product on \({\mathcal {A}}\) is a linear \(*\) -isomorphism, or a conjugate linear \(*\) -isomorphism, or the negative of a linear \(*\) -isomorphism, or the negative of a conjugate linear \(*\) -isomorphism. PubDate: 2023-06-01

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Abstract: Let \(A_{\varphi }\) denote the matrix of rotation with angle \(\varphi \) of the Euclidean plane, FLOOR the function which rounds a real point to the nearest lattice point down on the left and ROUND the function for rounding off a vector to the nearest node of the lattice. We prove under the natural assumption \(\varphi \not = k\frac{\pi }{2}\) that the functions \({{\,\mathrm{FLOOR}\,}}\circ A_{\varphi }\) and \({{\,\mathrm{ROUND}\,}}\circ A_{\varphi }\) are neither surjective nor injective. More precisely we prove lower and upper estimates for the size of the sets of lattice points, which are the image of two lattice points as well as of lattice points, which have no preimages. It turns out that the densities of those sets are positive. PubDate: 2023-06-01

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Abstract: In this article, we consider an infinite family of normal surface singularities with an integral homology sphere link which is related to the family of space monomial curves with a plane semigroup. These monomial curves appear as the special fibers of equisingular families of curves whose generic fibers are a complex plane branch, and the related surface singularities appear in a proof of the monodromy conjecture for these curves. To investigate whether the link of a normal surface singularity is an integral homology sphere, one can use a characterization that depends on the determinant of the intersection matrix of a (partial) resolution. To study our family, we apply this characterization with a partial toric resolution of our singularities constructed as a sequence of weighted blow-ups. PubDate: 2023-06-01

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Abstract: Let \(\mathbb {K}\) be an uncountable field of characteristic zero and let f be a function from \(\mathbb {K}^n\) to \(\mathbb {K}\) . We show that if the restriction of f to every affine plane \(L\subset \mathbb {K}^n\) is regular, then f is a regular function. PubDate: 2023-06-01

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Abstract: Let \({\mathcal {S}}_4^m\) be a class of singular hypersurfaces defined by $$\begin{aligned} x^{m+2}= (y_1^2+y_2^2+y_3^2+y_4^2)z^m, \end{aligned}$$ where \(m\geqslant 2\) is an integer. We establish a more precise asymptotic formula for the number of rational points of bounded height on \({\mathcal {S}}_4^m\) with a power saving error term. PubDate: 2023-06-01

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Abstract: Let G be a group and \(G_0 \subseteq G\) be a subset. A sequence over \(G_0\) means a finite sequence of terms from \(G_0\) , where the order of elements is disregarded and the repetition of elements is allowed. A product-one sequence is a sequence whose elements can be ordered such that their product equals the identity element of the group. We study algebraic and arithmetic properties of monoids of product-one sequences over finite subsets of G and over the whole group G, with a special emphasis on the infinite dihedral group. PubDate: 2023-06-01

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Abstract: In this paper we establish sufficient conditions for the oscillation of all solutions of equation $$\begin{aligned} \varDelta ^4x(n)+p(n)\varDelta x(n+1)+q(n)x(n-\tau )=0 \end{aligned}$$ via comparison with some first order delay difference equations whose oscillatory characters are known. The presented criterion is easily verifiable. Examples are also given to illustrate the main result. PubDate: 2023-06-01