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Masking of quantum information means that information is hidden from a subsystem and spread over a composite system. Modi et al. proved in [Phys. Rev. Lett. 120, 230501 (2018)] that this is true for some restricted sets of nonorthogonal quantum states and it is not possible for arbitrary quantum states. In this paper, we discuss the problem of masking quantum information encoded in pure and mixed states, respectively. Based on an established necessary and sufficient condition for a set of pure states to be masked by an operator, we find that there exists a set of four states that can not be masked, which implies that to mask unknown pure states is impossible. We construct a masker $S^\sharp$ and obtain its maximal maskable set, leading to an affirmative answer to a conjecture proposed in Modi's paper mentioned above. We also prove that an orthogonal (resp. linearly independent) subset of pure states can be masked by an isometry (resp. injection). Generalizing the case of pure states, we introduce the maskability of a set of mixed states and prove that a commuting subset of mixed states can be masked by an isometry $S^{\diamond}$ while it is impossible to mask all of mixed states by any operator. We also find the maximal maskable sets of mixed states of the isometries ${S^{\sharp}}$ and ${S^{\diamond}}$, respectively.

Given an infinite family of extended real-valued functions fi, i∈I, and a family H of nonempty finite subsets of I, the H-partial robust sum of fi, i∈I, is the supremum, for J∈H, of the finite sums ∑j∈Jfj. These infinite sums arise in a natural way in location problems as well as in functional approximation problems, and include as particular cases the well-known sup function and the so-called robust sum function, corresponding to the set H of all nonempty finite subsets of I, whose unconstrained minimization was analyzed in previous papers of three of the authors (https://doi.org/10.1007/s11228-019-00515-2 and https://doi.org/10.1007/s00245-019-09596-9). In this paper, we provide ordinary and stable zero duality gap and strong duality theorems for the minimization of a given H-partial robust sum under constraints, as well as closedness and convex criteria for the formulas on the subdifferential of the sup-function. This research was supported by the National Foundation for Science & Technology Development (NAFOSTED), Vietnam, Project 101.01-2018.310, and by Ministerio de Ciencia, Innovación y Universidades (MCIU), Agencia Estatal de Investigación (AEI), and European Regional Development Fund (ERDF), Project PGC2018-097960-B-C22.

In the following paper, one studies, given a bounded, connected open set $\Omega$ $\subseteq$ R n , $\kappa$ > 0, a positive Radon measure $\mu$ 0 in $\Omega$ and a (signed) Radon measure $\mu$ on $\Omega$ satisfying $\mu$($\Omega$) = 0 and |$\mu$| $\kappa$$\mu$ 0 , the possibility of solving the equation div u = $\mu$ by a vector field u satisfying |u| $\kappa$w on $\Omega$ (where w is an integrable weight only related to the geometry of $\Omega$ and to $\mu$ 0), together with a mild boundary condition. This extends results obtained in [4] for the equation div u = f , improving them on two aspects: one works here with the divergence equation with measure data, and also construct a weight w that relies in a softer way on the geometry of $\Omega$, improving its behavior (and hence the a priori behavior of the solution we construct) substantially in some instances. The method used in this paper follows a constructive approach of Bogovskii type.

The main purpose of this paper is to use the Hardy&ndash, Littlewood method to study the solvability of mixed powers of primes. To be specific, we consider the even integers represented as the sum of one prime, one square of prime, one cube of prime, and one biquadrate of prime. However, this representation can not be realized for all even integers. In this paper, we establish the exceptional set of this kind of representation and give an upper bound estimate.

We use hypergeometric sheaves on $G_m/F_q$, which are particular sorts of rigid local systems, to construct explicit local systems whose arithmetic and geometric monodromy groups are the finite general linear groups $GL_n(q)$ for any $n \ge 2$ and and any prime power $q$, so long as $q > 3$ when $n=2$. This paper continues a program of finding simple (in the sense of simple to remember) families of exponential sums whose monodromy groups are certain finite groups of Lie type, cf. [Gr], [KT1], [KT2], [KT3] for (certain) finite symplectic and unitary groups, or certain sporadic groups, cf. [KRL], [KRLT1], [KRLT2], [KRLT3]. The novelty of this paper is obtaining $GL_n(q)$ in this hypergeometric way. A pullback construction then yields local systems on $A^1/F_q$ whose geometric monodromy groups are $SL_n(q)$. These turn out to recover a construction of Abhyankar., 26 pages

This paper concerns both norm estimation and pointwise approximation for the Bochner–Riesz means of an arbitrary Morrey function on $${{{{\mathbb {R}}}}^n}$$—Theorems 1.1 and 1.2 for $$L^{p,\lambda }({{{{\mathbb {R}}}}^n})$$—thereby generalizing the corresponding results for $$L^p({{{{\mathbb {R}}}}^n})$$ in Stein (Acta Math 100:93–147, 1958) and Carbery et al. (J Lond Math Soc 38:513–524, 1988). As a side note, this paper also establishes Lemma 4.1 of Tomas–Stein type—if $$f\in L^{p,\lambda }({{{{\mathbb {R}}}}^n})$$ under $$ 2^{-1}(n+1)

We have studied the k-rainbow domination number of C n □ C m for k &ge, 4 (Gao et al. 2019), in which we present the 3-rainbow domination number of C n □ C m , which should be bounded above by the four-rainbow domination number of C n □ C m . Therefore, we give a rough bound on the 3-rainbow domination number of C n □ C m . In this paper, we focus on the 3-rainbow domination number of the Cartesian product of cycles, C n □ C m . A 3-rainbow dominating function (3RDF) f on a given graph G is a mapping from the vertex set to the power set of three colors { 1 , 2 , 3 } in such a way that every vertex that is assigned to the empty set has all three colors in its neighborhood. The weight of a 3RDF on G is the value &omega, ( f ) = &sum, v &isin, V ( G ) | f ( v ) | . The 3-rainbow domination number, &gamma, r 3 ( G ) , is the minimum weight among all weights of 3RDFs on G. In this paper, we determine exact values of the 3-rainbow domination number of C 3 □ C m and C 4 □ C m and present a tighter bound on the 3-rainbow domination number of C n □ C m for n &ge, 5 .

In our Novi Sad conference paper (1999) we described Dehn type surgeries of the famous Gieseking (1912) hyperbolic ideal simplex manifold $\mathcal{S}$, leading to compact fundamental domain $\mathcal{S}(k)$, $k = 2, 3, \dots$ with singularity geodesics of rotation order $k$, but as later turned out with cone angle $2(k-1)/k$. We computed also the volume of $S(k)$, tending to zero if $k$ goes to infinity. That time we naively thought that we obtained orbifolds with the above surprising property. As the reviewer of Math. Rev., Kevin P. Scannell (MR1770996 (2001g:57030)) rightly remarked, "this is in conflict with the well-known theorem of D. A. Kazhdan and G. A. Margulis (1968) and with the work of Thurston, describing the geometric convergence of orbifolds under large Dehn fillings". In this paper we refresh our previous publication. Correctly, we obtained cone manifolds (for $k > 2$), as A. D. Mednykh and V. S. Petrov (2006) kindly pointed out. We complete our discussion and derive the above cone manifold series (Gies.1 and Gies.2) in two geometrically equivalent form, by the half turn symmetry of any ideal simplex. Moreover we obtain a second orbifold series (Gies.3 and 4), tending to the regular ideal simplex as the original Gieseking manifold., 19 pages, 9 figures

In 1900, at the international congress of mathematicians, Hilbert claimed that the Riemann zeta function $\zeta(s)$ is not the solution of any algebraic ordinary differential equations on its region of analyticity. In 2015, Van Gorder considered the question of whether $\zeta(s)$ satisfies a non-algebraic differential equation and showed that it formally satisfies an infinite order linear differential equation. Recently, Prado and Klinger-Logan extended Van Gorder's result to show that the Hurwitz zeta function $\zeta(s,a)$ is also formally satisfies a similar differential equation \begin{equation*}\label{HurDE} T\left[\zeta (s,a) - \frac{1}{a^s}\right] = \frac{1}{(s-1)a^{s-1}}. \end{equation*} But unfortunately in the same paper they proved that the operator $T$ applied to Hurwitz zeta function $\zeta(s,a)$ does not converge at any point in the complex plane $\mathbb{C}$. In this paper, by defining $T_{p}^{a}$, a $p$-adic analogue of Van Gorder's operator $T,$ we establish an analogue of Prado and Klinger-Logan's differential equation satisfied by $\zeta_{p,E}(s,a)$ which is the $p$-adic analogue of the Hurwitz-type Euler zeta functions \begin{equation*}\label{HEZ} \zeta_E(s,a)=\sum_{n=0}^\infty\frac{(-1)^n}{(n+a)^s}. \end{equation*} In contrast with the complex case, due to the non-archimedean property, the operator $T_{p}^{a}$ applied to the $p$-adic Hurwitz-type Euler zeta function $\zeta_{p,E}(s,a)$ is convergent $p$-adically in the area of $s\in\mathbb{Z}_{p}$ with $s\neq 1$ and $a\in K$ with $|a|_{p}>1,$ where $K$ is any finite extension of $\mathbb{Q}_{p}$ with ramification index over $\mathbb{Q}_{p}$ less than $p-1.$, Comment: 18 pages. Final version. Dedicated to the memory of Prof. David Goss (1952-2017)

The first named author is supported in part by NSF grant DMS-1855464, ISF grant 281/17, and the Simons Foundation. The second named author is supported in part by an OTKA NK 104183 grant. The third named author is supported in part by a NSERC and an OTKA NK 104183 grant. In their seminal paper Erdös and Szemerédi formulated conjectures on the size of sumset and product set of integers. The strongest form of their conjecture is about sums and products along the edges of a graph. In this paper we show that this strong form of the Erdös-Szemerédi conjecture does not hold. We give upper and lower bounds on the cardinalities of sumsets, product sets, and ratio sets along the edges of graphs.

This paper considers an edge minimization problem in saturated bipartite graphs. An $n$ by $n$ bipartite graph $G$ is $H$-saturated if $G$ does not contain a subgraph isomorphic to $H$ but adding any missing edge to $G$ creates a copy of $H$. More than half a century ago, Wessel and Bollob\'as independently solved the problem of minimizing the number of edges in $K_{(s,t)}$-saturated graphs, where $K_{(s,t)}$ is the `ordered' complete bipartite graph with $s$ vertices from the first color class and $t$ from the second. However, the very natural `unordered' analogue of this problem was considered only half a decade ago by Moshkovitz and Shapira. When $s=t$, it can be easily checked that the unordered variant is exactly the same as the ordered case. Later, Gan, Kor\'andi, and Sudakov gave an asymptotically tight bound on the minimum number of edges in $K_{s,t}$-saturated $n$ by $n$ bipartite graphs, which is only smaller than the conjecture of Moshkovitz and Shapira by an additive constant. In this paper, we confirm their conjecture for $s=t-1$ with the classification of the extremal graphs. We also improve the estimates of Gan, Kor\'andi, and Sudakov for general $s$ and $t$, and for all sufficiently large $n$., Comment: Reflected minor suggestions from reviewers

In this paper, we consider the problem of computing the rank of a block-structured symbolic matrix (a generic partitioned matrix) $A = (A_{\alpha \beta} x_{\alpha \beta})$, where $A_{\alpha \beta}$ is a $2 \times 2$ matrix over a field $\mathbf{F}$ and $x_{\alpha \beta}$ is an indeterminate for $\alpha = 1,2,\dots, \mu$ and $\beta = 1,2, \dots, \nu$. This problem can be viewed as an algebraic generalization of the bipartite matching problem and was considered by Iwata and Murota (1995). Recent interests in this problem lie in the connection with non-commutative Edmonds' problem by Ivanyos, Qiao, and Subrahamanyam (2018) and Garg, Gurvits, Oliveiva, and Wigderson (2019), where a result by Iwata and Murota implicitly states that the rank and non-commutative rank (nc-rank) are the same for this class of symbolic matrices. The main result of this paper is a simple and combinatorial $O((\mu \nu)^2 \min \{ \mu, \nu \})$-time algorithm for computing the symbolic rank of a $(2 \times 2)$-type generic partitioned matrix of size $2\mu \times 2\nu$. Our algorithm is inspired by the Wong sequence algorithm by Ivanyos, Qiao, and Subrahamanyam for the nc-rank of a general symbolic matrix, and requires no blow-up operation, no field extension, and no additional care for bounding the bit-size. Moreover it naturally provides a maximum rank completion of $A$ for an arbitrary field $\mathbf{F}$., Comment: This is a post-peer-review, pre-copyedit version of an article published in Mathematical Programming. The final authenticated version is available online at: https://doi.org/10.1007/s10107-021-01676-5

In this paper we introduce a general framework for the study of limits of relational structures in general and graphs in particular, which is based on a combination of model theory and (functional) analysis. We show how the various approaches to graph limits fit to this framework and that they naturally appear as "tractable cases" of a general theory. As an outcome of this, we provide extensions of known results. We believe that this put these into next context and perspective. For example, we prove that the sparse--dense dichotomy exactly corresponds to random free graphons. The second part of the paper is devoted to the study of sparse structures. First, we consider limits of structures with bounded diameter connected components and we prove that in this case the convergence can be "almost" studied component-wise. We also propose the structure of limits objects for convergent sequences of sparse structures. Eventually, we consider the specific case of limits of colored rooted trees with bounded height and of graphs with bounded tree-depth, motivated by their role of elementary brick these graphs play in decompositions of sparse graphs, and give an explicit construction of a limit object in this case. This limit object is a graph built on a standard probability space with the property that every first-order definable set of tuples is measurable. This is an example of the general concept of {\em modeling} we introduce here. Our example is also the first "intermediate class" with explicitly defined limit structures., Comment: added journal reference

Amdeberhan's conjectures on the enumeration, the average size, and the largest size of $(n,n+1)$-core partitions with distinct parts have motivated many research on this topic. Recently, Straub and Nath-Sellers obtained formulas for the numbers of $(n, dn-1)$ and $(n, dn+1)$-core partitions with distinct parts, respectively. Let $X_{s,t}$ be the size of a uniform random $(s,t)$-core partition with distinct parts when $s$ and $t$ are coprime to each other. Some explicit formulas for the $k$-th moments $\mathbb{E} [X_{n,n+1}^k]$ and $\mathbb{E} [X_{2n+1,2n+3}^k]$ were given by Zaleski and Zeilberger when $k$ is small. Zaleski also studied the expectation and higher moments of $X_{n,dn-1}$ and conjectured some polynomiality properties concerning them in arXiv:1702.05634. Motivated by the above works, we derive several polynomiality results and asymptotic formulas for the $k$-th moments of $X_{n,dn+1}$ and $X_{n,dn-1}$ in this paper, by studying the beta sets of core partitions. In particular, we show that these $k$-th moments are asymptotically some polynomials of n with degrees at most $2k$, when $d$ is given and $n$ tends to infinity. Moreover, when $d=1$, we derive that the $k$-th moment $\mathbb{E} [X_{n,n+1}^k]$ of $X_{n,n+1}$ is asymptotically equal to $\left(n^2/10\right)^k$ when $n$ tends to infinity. The explicit formulas for the expectations $\mathbb{E} [X_{n,dn+1}]$ and $\mathbb{E} [X_{n,dn-1}]$ are also given. The $(n,dn-1)$-core case in our results proves several conjectures of Zaleski on the polynomiality of the expectation and higher moments of $X_{n,dn-1}$., Comment: This paper has been accepted for publication in SCIENCE CHINA Mathematics

A topological invariant is a numerical parameter associated with molecular graph and plays an imperative role in the study and analysis of quantitative structure activity/property relationships (QSAR/QSPR). The correlation between the entire π-electron energy and the structure of a molecular graph was explored and understood by the first Zagreb index. Recently, Liu et al. (2019) calculated the first general Zagreb index of the F-sum graphs. In the same paper, they also proposed the open problem to compute the general Randić index RαΓ=∑uv∈EΓdΓu×dΓvα of the F-sum graphs, where α∈R and dΓu denote the valency of the vertex u in the molecular graph Γ. Aim of this paper is to compute the lower and upper bounds of the general Randić index for the F-sum graphs when α∈N. We present numerous examples to support and check the reliability as well as validity of our bounds. Furthermore, the results acquired are the generalization of the results offered by Deng et al. (2016), who studied the general Randić index for exactly α=1.

In this paper, using computations done through the LiE software, we compare the tensor product of irreducible selfdual representations of the special linear group with those of classical groups to formulate some conjectures relating the two. In the process a few other phenomenon present themselves which we record as questions. More precisely, under the natural correspondence of irreducible finite dimensional selfdual representations of ${\rm SL}_{2n}({\mathbb C})$ with those of ${\rm Spin}_{2n+1}({\mathbb C})$, it is easy to see that if the tensor product of three irreducible representations of ${\rm Spin}_{2n+1}({\rm C})$ contains the trivial representation, then so does the tensor product of the corresponding representations of ${\rm SL}_{2n}({\rm C})$. The paper formulates a conjecture in the reverse direction. We also deal with the pair $({\rm SL}_{2n+1}({\rm C}), {\rm Sp}_{2n}({\rm C}))$., Comment: Revised version!

This paper considers random processes of the form $$X_{n+1}=a_nX_n+b_n\pmod p$$ where p is odd, $$X_0=0$$ , $$(a_0,b_0), (a_1,b_1), (a_2,b_2),\ldots $$ are i.i.d., and $$a_n$$ and $$b_n$$ are independent with $$P(a_n=2)=P(a_n=(p+1)/2)=1/2$$ and $$P(b_n=1)=P(b_n=0)=P(b_n=-1)=1/3$$ . This can be viewed as a multiplicatively symmetrized version of a random process of Chung, Diaconis, and Graham. This paper shows that order $$(\log p)^2$$ steps suffice for $$X_n$$ to be close to uniformly distributed on the integers mod p for all odd p while order $$(\log p)^2$$ steps are necessary for $$X_n$$ to be close to uniformly distributed on the integers mod p.

This paper studies zeta functions of the form $\sum_{n=1}^{\infty} \chi(n) n^{-s}$, with $\chi$ a completely multiplicative function taking only unimodular values. We denote by $\sigma(\chi)$ the infimum of those $\alpha$ such that the Dirichlet series $\sum_{n=1}^{\infty} \chi(n) n^{-s}$ can be continued meromorphically to the half-plane $\operatorname{Re} s>\alpha$, and denote by $\zeta_{\chi}(s)$ the corresponding meromorphic function in $\operatorname{Re} s>\sigma(\chi)$. We construct $\zeta_{\chi}(s)$ that have $\sigma(\chi)\le 1/2$ and are universal for zero-free analytic functions on the half-critical strip $1/21$. Finally, we show that there exists $\zeta_{\chi}(s)$ with $\sigma(\chi) \le 1/2$ and zeros at any discrete multiset in the strip $1/21/2$; on the Riemann hypothesis, this strip may be replaced by the half-critical strip $1/2 < \operatorname{Re} s < 1$., Comment: This is the final version of the paper which has been accepted for publication in Journal d'Analyse Math\'{e}matique

Given a graph $G$, a probability measure $\mu$ on the subsets of the edge set of $G$ is said to be $1$-independent if events determined by edge sets that are at graph distance at least $1$ apart in $G$ are independent. Call such a probability measure a $1$-ipm on $G$, and denote by $\mathbf{G}_{\mu}$ the associated random spanning subgraph of $G$. Let $\mathcal{M}_{1,\geqslant p}(G)$ (resp. $\mathcal{M}_{1,\leqslant p}(G)$) denote the collection of $1$-ipms $\mu$ on $G$ for which each edge is included in $\mathbf{G}_{\mu}$ with probability at least $p$ (resp. at most $p$). Let $\mathbb{Z}^2$ denote the square integer lattice. Balister and Bollob\'as raised the question of determining the critical value $p_{\star}=p_{1,c}(\mathbb{Z}^2)$ such that for all $p>p_{\star}$ and all $\mu \in \mathcal{M}_{1,\geqslant p}(\mathbb{Z}^2)$, $\left(\mathbf{\mathbb{Z}^2}\right)_{\mu}$ almost surely contains an infinite component. This can be thought of as asking for a $1$-independent analogue of the celebrated Harris--Kesten theorem. In this paper we investigate both this problem and connectivity problems for $1$-ipms more generally. We give two lower bounds on $p_{\star}$ that significantly improve on the previous bounds. Furthermore, motivated by the Russo--Seymour--Welsh lemmas, we define a $1$-independent critical probability for long paths and determine its value for the line and ladder lattices. Finally, for finite graphs $G$ we study $f_{1,G}(p)$ (respectively $F_{1,G}(p)$), the infimum (resp. supremum) over all $\mu\in \mathcal{M}_{1,\geqslant p}(G)$ (resp. all $\mu \in \mathcal{M}_{1,\leqslant p}(G)$) of the probability that $\mathbf{G}_{\mu}$ is connected. We determine $f_{1,G}(p)$ and $F_{1,G}(p)$ exactly when $G$ is a path, a complete graph and a cycle of length at most $5$. Many new problems arise from our work, which are discussed in the final section of the paper., Comment: 43 pages, 3 figures

Given a Khalimsky (for short, K-) topological space X, the present paper examines if there are some relationships between the contractibility of X and the existence of the fixed point property of X. Based on a K-homotopy for K-topological spaces, we firstly prove that a K-homeomorphism preserves a K-homotopy between two K-continuous maps. Thus, we obtain that a K-homeomorphism preserves K-contractibility. Besides, the present paper proves that every simple closed K-curve in the n-dimensional K-topological space, S C K n , l , n &ge, 2 , l &ge, 4 , is not K-contractible. This feature plays an important role in fixed point theory for K-topological spaces. In addition, given a K-topological space X, after developing the notion of K-contractibility relative to each singleton { x } ( &sub, X ) , we firstly compare it with the concept of K-contractibility of X. Finally, we prove that the K-contractibility does not imply the K-contractibility relative to each singleton { x 0 } ( &sub, X ) . Furthermore, we deal with certain conjectures involving the (almost) fixed point property in the categories KTC and KAC, where KTC (see Section 3) (resp. KAC (see Section 5)) denotes the category of K-topological (resp. KA-) spaces, KA-) spaces are subgraphs of the connectedness graphs of the K-topology on Z n .

A cut $\varepsilon$-sparsifier of a weighted graph $G$ is a re-weighted subgraph of $G$ of (quasi)linear size that preserves the size of all cuts up to a multiplicative factor of $\varepsilon$. Since their introduction by Bencz\'ur and Karger [STOC'96], cut sparsifiers have proved extremely influential and found various applications. Going beyond cut sparsifiers, Filtser and Krauthgamer [SIDMA'17] gave a precise classification of which binary Boolean CSPs are sparsifiable. In this paper, we extend their result to binary CSPs on arbitrary finite domains., Comment: Full version of a STACS'19 paper

The circular descent of a permutation $\sigma$ is a set $\{ \sigma(i) \mid \sigma(i) \gt \sigma(i+1) \}$. In this paper, we focus on the enumerations of permutations by the circular descent set. Let $\operatorname{cdes}_n(S)$ be the number of permutations of length $n$ which have the circular descent set $S$. We derive the explicit formula for $\operatorname{cdes}_n(S)$. We describe a class of generating binary trees $T_k$ with weights. We find that the number of permutations in the set $\operatorname{CDES}_n(S)$ corresponds to the weights of $T_k$. As a application of the main results in this paper, we also give the enumeration of permutation tableaux according to their shape.

In this paper, we study the limit distribution of the record values under nonlinear normalization of the form $${\cal T}_n(x)=\exp\{u_{n}(| \log |x||)^{v_{n}}\mbox{sign}(\log |x|)\}\mbox{sign}(x),$$ which is called exponential norming ($e-$norming). The corresponding limit laws of the upper extremes are called $e$-max stable laws (denoted by $U(.)$). In this paper, we show that the limit distributions of the record values under exponential norming are of the form $ {\cal N}(-\log (-\log U(x))),$ where ${\cal N}(.)$ is the standard normal distribution. Moreover, we study the domains of attraction for these types of limit laws. Finally, some illustrative examples are given.

In this paper, we are interested in the multiple solutions of the following fractional critical problem \[ \begin{cases} (-\Delta)^s u = |u|^{2_s^*-2} u + \lambda u &\textrm{in $\Omega$}, \\ u = 0 &\textrm{on $\mathbb{R}^N \setminus \Omega$}, \end{cases} \] where $s \in (0,1)$, $N \gt 4s$, $2^*_s = 2N/(N-2s)$, $\Omega$ is a smooth bounded domain in $\mathbb{R}^N$ and $(-\Delta)^s$ is the fractional Laplace operator. Because the nonlocal property of fractional Laplacian makes the variational functional of the fractional critical problem different from the one of local operator $-\Delta$. To the best of our knowledge, it is still unknown whether multiple solutions of the fractional critical problem exist for all $\lambda \gt 0$. In this paper, we give a partial answer. Precisely, by introducing some new ideas and careful estimates, we prove that for any $s \in (0,1)$, the fractional critical problem has at least $[(N+1)/2]$ pairs of nontrivial solutions if $0 \lt \lambda \neq \lambda_n$, and has $[(N+1-l)/2]$ pairs if $\lambda = \lambda_n$ with multiplicity number $0 \lt l \lt \min \{n,N+2\}$, via constraint method and Krasnoselskii genus. Here $\lambda_n$ denotes the $n$-th eigenvalue of $(-\Delta)^s$ with zero Dirichlet boundary data in $\Omega$ and $[a]$ denotes the least positive integer $k$ such that $k \geq a$.

The weighted T-free 2-matching problem is the following problem: given an undirected graph G, a weight function on its edge set, and a set T of triangles in G, find a maximum weight 2-matching containing no triangle in T. When T is the set of all triangles in G, this problem is known as the weighted triangle-free 2-matching problem, which is a long-standing open problem. A main contribution of this paper is to give the first polynomial-time algorithm for the weighted T-free 2-matching problem under the assumption that T is a set of edge-disjoint triangles. In our algorithm, a key ingredient is to give an extended formulation representing the solution set, that is, we introduce new variables and represent the convex hull of the feasible solutions as a projection of another polytope in a higher dimensional space. Although our extended formulation has exponentially many inequalities, we show that the separation problem can be solved in polynomial time, which leads to a polynomial-time algorithm for the weighted T-free 2-matching problem., A preliminary version of this paper appears in [Kobayashi, Y.: Weighted triangle-free 2-matching problem with edge-disjoint forbidden triangles. In: Integer Programming and Combinatorial Optimization, pp. 280–293. Springer, New York (2020).].

In the paper, the author gives a solution to a conjecture on a double inequality for a function involving the tri- and tetra-gamma functions, which was first posed in Remark 6 of the paper &ldquo, Complete monotonicity of a function involving the tri- and tetragamma functions&rdquo, (2015) and repeated in the seventh open problem of the paper &ldquo, On complete monotonicity for several classes of functions related to ratios of gamma functions&rdquo, (2019).

In this paper we introduce a new class of rings whose elements are a sum of a central and a unit element, namely a ring $R$ is called $CU$ if each element $a\in R$ has a decomposition $a = c + u$ where $c$ is central and $u$ is unit. One of the main results in this paper is that if $F$ is a field which is not isomorphic to $\Bbb Z_2$, then $M_2(F)$ is a $CU$ ring. This implies, in particular, that any square matrix over a field which is not isomorphic to $\Bbb Z_2$ is the sum of a central matrix and a unit matrix.

For a group $G$ generated by $S\doteq \{g_1,\ldots ,g_n\}$, one can construct the Cayley graph $\mathrm {Cayley}({G},{S})$. Given a distance set $D\subset \mathbb Z _{\geq 0}$ and $\mathrm{Cayley}{G}{S}$, one can construct a $D$-neighborhood complex. This neighborhood complex is a simplicial complex to which we can associate a chain complex. Group $G$ acts on this chain complex, and this leads to an action on the homology of the chain complex. These group actions decompose into several representations of $G$. This paper uses tools from group theory, representation theory and homological algebra to further our understanding of the interplay between generated groups, corresponding representations on their associated $D$-neighborhood complexes and the homology of the $D$-neighborhood complexes. This paper is an exposition of the results in my dissertation focusing on the case of two generators.

The (un)weighted stability for some matrices on a graph is one of essential hypotheses in time-frequency analysis and applied harmonic analysis. In the first part of this paper, we show that for a localized matrix in a Beurling algebra, its weighted stabilities for different exponents and Muckenhoupt weights are equivalent to each other, and reciprocal of its optimal lower stability bound for one exponent and weight is controlled by a polynomial of reciprocal of its optimal lower stability bound for another exponent and weight. Banach algebras of matrices with certain off-diagonal decay is of great importance in many mathematical and engineering fields, and its inverse-closed property can be informally interpreted as localization preservation. Let $${{\mathcal {B}}}(\ell ^p_w)$$ be the Banach algebra of bounded linear operators on the weighted sequence space $$\ell ^p_w$$ on a graph. In the second part of this paper, we prove that Beurling algebras of localized matrices on a connected simple graph are inverse-closed in $${{\mathcal {B}}}(\ell ^p_w)$$ for all $$1\le p

We study properties of sets having the minimum length (one-dimensional Hausdorff measure) in the class of closed connected sets Σ ⊂ ℝ2 satisfying the inequality max yϵM dist (y, Σ) ≤ r for a given compact set M ⊂ ℝ2 and given r > 0. Such sets play the role of the shortest possible pipelines arriving at a distance at most r to every point of M where M is the set of customers of the pipeline. In this paper, it is announced that every maximum distance minimizer is a union of finitely many curves having one-sided tangent lines at every point. This shows that a maximum distance minimizer is isotopic to a finite Steiner tree even for a “bad” compact set M, which distinguishes it from a solution of the Steiner problem (an example of a Steiner tree with infinitely many branching points can be found in a paper by Paolini, Stepanov, and Teplitskaya). Moreover, the angle between these lines at each point of a maximum distance minimizer is at least 2π/3. Also, we classify the behavior of a minimizer Σ in a neighborhood of any point of Σ. In fact, all the results are proved for a more general class of local minimizers.

The purpose of this paper is to investigate the invariance of the arithmetic mean with respect to two weighted Bajraktarevic means, i.e., to solve the functional equation $$\begin{aligned} \left( \frac{f}{g}\right) ^{\!\!-1}\!\!\left( \frac{tf(x)+sf(y)}{tg(x)+sg(y)}\right) +\left( \frac{h}{k}\right) ^{\!\!-1}\!\!\left( \frac{sh(x)+th(y)}{sk(x)+tk(y)}\right) =x+y \qquad (x,y\in I), \end{aligned}$$ where $$f,g,h,k:I\rightarrow \mathbb {R}$$ are unknown continuous functions such that g, k are nowhere zero on I, the ratio functions f / g, h / k are strictly monotone on I, and $$t,s\in \mathbb {R}_+$$ are constants different from each other. By the main result of this paper, the solutions of the above invariance equation can be expressed either in terms of hyperbolic functions or in terms of trigonometric functions and an additional weight function. For the necessity part of this result, we will assume that $$f,g,h,k:I\rightarrow \mathbb {R}$$ are four times continuously differentiable.

A Shilla graph is defined as a distance-regular graph of diameter 3 with second eigen-value θ1 equal to a3. For a Shilla graph, let us put a = a3 and b = k/a. It is proved in this paper that a Shilla graph with b2 = c2 and noninteger eigenvalues has the following intersection array: $$\left\{ {\frac{{{b^2}\left( {b - 1} \right)}}{2},\frac{{\left( {b - 1} \right)\left( {{b^2} - b + 2} \right)}}{2},\frac{{b\left( {b - 1} \right)}}{4};1,\frac{{b\left( {b - 1} \right)}}{4},\frac{{b{{\left( {b - 1} \right)}^2}}}{2}} \right\}$$ If Γ is a Q-polynomial Shilla graph with b2 = c2 and b = 2r, then the graph Γ has intersection array $$\left\{ {2tr\left( {2r + 1} \right),\left( {2r + 1} \right)\left( {2rt + t + 1} \right),r\left( {r + t} \right);1,r\left( {r + t} \right),t\left( {4{r^2} - 1} \right)} \right\}$$ and, for any vertex u in Γ, the subgraph Γ3(u) is an antipodal distance-regular graph with intersection array $$\left\{ {t\left( {2r + 1} \right),\left( {2r - 1} \right)\left( {t + 1} \right),1;1,t + 1,t\left( {2r + 1} \right)} \right\}$$ The Shilla graphs with b2 = c2 and b = 4 are also classified in the paper.

The speed of a class of graphs counts the number of graphs on the vertex set $\lbrace 1,\dots, n\rbrace$ inside the class as a function of $n$. In this paper, we investigate this function for many classes of graphs that naturally arise in discrete geometry, for example intersection graphs of segments or disks in the plane. While upper bounds follow from Warren's theorem (a variant of a theorem of Milnor and Thom), all the previously known lower bounds were obtained from ad hoc constructions for very specific classes. We prove a general theorem giving an essentially tight lower bound for the number of graphs on $\lbrace 1,\dots, n\rbrace$ whose edges are defined using the signs of a given finite list of polynomials, assuming these polynomials satisfy some reasonable conditions. This in particular implies lower bounds for the speed of many different classes of intersection graphs, which essentially match the known upper bounds. Our general result also gives essentially tight lower bounds for counting containment orders of various families of geometric objects, including circle orders and angle orders. Some of the applications presented in this paper are new, whereas others recover results of Alon-Scheinerman, Fox, McDiarmid-M\"{u}ller and Shi. For the proof of our result we use some tools from algebraic geometry and differential topology., Comment: 38 pages, improved exposition

The Fibonacci sequence [math] , when reduced modulo [math] is periodic. For example, [math] . The period of [math] is denoted by [math] , so [math] . In this paper we present an algorithm that, given a period [math] , produces all [math] such that [math] . For efficiency, the algorithm employs key ideas from a 1963 paper by John Vinson on the period of the Fibonacci sequence. We present output from the algorithm and discuss the results.

Let M m be a closed smooth manifold equipped with a smooth involution having fixed point set of the form F n ∪ F 3 , where F n and F 3 are submanifolds with dimensions n and 3, respectively, where 3 n m and with the normal bundles over F n and F 3 being nonbounding. The authors of this paper, together with Patricia E. Desideri, previously showed that, when n is even, then m ≤ n + 4 , which we call a small codimension phenomenon. Further, they showed that this small bound is best possible. In this paper we study this problem for n odd, which is much more complicated, requiring more sophisticated techniques involving characteristic numbers. We show in this case that m ≤ M ( n − 3 ) + 6 , where M ( n ) is the Stong–Pergher number (see the definition of M ( n ) in Section 1). Further, we show that this bound is almost best possible, in the sense that there exists an example with m = M ( n − 3 ) + 5 , which means that for n odd the small codimension phenomenon does not occur and the bound in question is meaningful. The existence of these bounds is guaranteed by the famous Five Halves Theorem of J. Boardman, which establishes that, under the above hypotheses, m ≤ 5 2 n .

In this paper we clarify the quadratic irrationalities that can be admitted by an odd-degree complex irreducible character χ of an arbitrary finite group. Write Q ( χ ) to denote the field generated over the rational numbers by the values of χ, and let d > 1 be a square-free integer. We prove that if Q ( χ ) = Q ( d ) then d ≡ 1 (mod 4) and if Q ( χ ) = Q ( − d ) , then d ≡ 3 (mod 4). This follows from the main result of this paper: either i ∈ Q ( χ ) or Q ( χ ) ⊆ Q ( exp ⁡ ( 2 π i / m ) ) for some odd integer m ≥ 1 .

Recently a lot of papers have been devoted to partial infinite reciprocal sums of a higher-order linear recursive sequence. In this paper, we continue this program by finding a sequence which is asymptotically equivalent to partial infinite sums, including a reciprocal of polynomial applied to linear higher order recurrences.

We denote the local "little" and "big" Lipschitz functions of a function $f: {{\mathbb R}}\to {{\mathbb R}}$ by $ {\mathrm {lip}}f$ and $ {\mathrm {Lip}}f$. In this paper we continue our research concerning the following question. Given a set $E {\subset} {{\mathbb R}}$ is it possible to find a continuous function $f$ such that $ {\mathrm {lip}}f=\mathbf{1}_E$ or $ {\mathrm {Lip}}f=\mathbf{1}_E$? In giving some partial answers to this question uniform density type (UDT) and strong uniform density type (SUDT) sets play an important role. In this paper we show that modulo sets of zero Lebesgue measure any measurable set coincides with a ${\mathrm {Lip}} 1$ set. On the other hand, we prove that there exists a measurable SUDT set $E$ such that for any $G_\delta$ set $\widetilde{E}$ satisfying $|E\Delta\widetilde{E}|=0$ the set $\widetilde{E}$ does not have UDT. Combining these two results we obtain that there exists ${\mathrm {Lip}} 1$ sets not having UDT, that is, the converse of one of our earlier results does not hold., Comment: This is the version accepted to appear in Mathematica Slovaca

The symbol 𝓢 c (X) denotes the hyperspace of all nontrivial convergent sequences in a Hausdorff space X. This hyperspace is endowed with the Vietoris topology. In the current paper, we compare the cellularity, the tightness, the extent, the dispersion character, the net weight, the i-weight, the π-weight, the π-character, the pseudocharacter and the Lindelöf number of 𝓢 c (X) with the corresponding cardinal function of X. We also answer a question posed by the authors in a previous paper.

The motivating question of this paper is roughly the following: given a group scheme $G$ over $\mathbb{Z}_p$, $p$ prime, with semisimple generic fiber $G_{\mathbb{Q}_p}$, how far are open subgroups of $G(\mathbb{Z}_p)$ from subgroups of the form $X(\mathbb{Z}_p)\mathbf{K}_p(p^n)$, where $X$ is a subgroup scheme of $G$ and $\mathbf{K}_p(p^n)$ is the principal congruence subgroup $\operatorname{Ker} (G(\mathbb{Z}_p)\rightarrow G(\mathbb{Z}/p^n\mathbb{Z}))$? More precisely, we will show that for $G_{\mathbb{Q}_p}$ simply connected there exist constants $J\ge1$ and $\varepsilon>0$, depending only on $G$, such that any open subgroup of $G (\mathbb{Z}_p)$ of level $p^n$ admits an open subgroup of index $\le J$ which is contained in $X(\mathbb{Z}_p)\mathbf{K}_p(p^{\lceil \varepsilon n\rceil})$ for some proper connected algebraic subgroup $X$ of $G$ defined over $\mathbb{Q}_p$. Moreover, if $G$ is defined over $\mathbb{Z}$, then $\varepsilon$ and $J$ can be taken independently of $p$. We also give a correspondence between natural classes of $\mathbb{Z}_p$-Lie subalgebras of $\mathfrak{g}_{\mathbb{Z}_p}$ and of closed subgroups of $G(\mathbb{Z}_p)$ that can be regarded as a variant over $\mathbb{Z}_p$ of Nori's results on the structure of finite subgroups of $\operatorname{GL}(N_0,\mathbb{F}_p)$ for large $p$. As an application we give a bound for the volume of the intersection of a conjugacy class in the group $G (\hat{\mathbb{Z}}) = \prod_p G (\mathbb{Z}_p)$, for $G$ defined over $\mathbb{Z}$, with an arbitrary open subgroup. In a future paper, this result will be applied to the limit multiplicity problem for arbitrary congruence subgroups of the arithmetic lattice $G (\mathbb{Z})$., fixed a few inaccuracies and made some stylistic changes to appear in JEMS

In 1933, Lehmer exhibited the polynomial $$\begin{aligned} L(z)=z^{10} + z^9 - z^7 - z^6 - z^5 - z^4 - z^3 + z + 1 \end{aligned}$$ with Mahler measure $$\mu _0>1$$ . Then he asked if $$\mu _0$$ is the smallest Mahler measure, not 1. This question became known as the Lehmer conjecture and it was apparently solved in the positive, while this paper was in preparation [19]. In this paper we consider those polynomials of the form $$\chi _A$$ , that is, Coxeter polynomials of a finite dimensional algebra A (for instance $$L(z)=\chi _{\mathbb {E}_{10}}$$ ). A polynomial in $$\mathbb {Z}[T]$$ which is either cyclotomic or with Mahler measure $$\ge \mu _0$$ is called a Lehmer polynomial. We give some necessary conditions for a polynomial to be Lehmer. We show that A being a tree algebra is a sufficient condition for $$\chi _A$$ to be Lehmer.

In the paper [1], V. O. Manturov introduced the groups G k n depending on two natural parameters n > k and naturally related to topology and to the theory of dynamical systems. The group G2 n , which is the simplest part of G k n , is isomorphic to the group of pure free braids on n strands. In the present paper, we study the groups G2 n supplied with additional structures–parity and points; these groups are denoted by G2n,p and G2n,d. First,we define the groups G2n,p and G2n,d, then study the relationship between the groups G2 n , G2n,p, and G2n,d. Finally, we give an example of a braid on n + 1 strands, which is not the trivial braid on n + 1 strands, by using a braid on n strands with parity. After that, the author discusses links in S g × S1 that can determine diagrams with points; these points correspond to the factor S1 in the product S g × S1.

This thesis consists of the following papers.I J. Larsson, The Minimum Matching in Pseudo-dimension 0 0.The second paper lives in a more general setting. There we search for any cover of the ground set V, for general families F. Each set f ∈ F receives weight w(f) uniformly at random from [0,1]. The cost of a cover f_1,f_2,...f_m is then taken to be max_i w(f_i). This is equivalent (after a rescaling) to drawing sets from F at Poisson times, and the cost of a cover is the first time V is covered. This problem is known under a number of names, perhaps most famously the coupon collector problem. In the classical formulation, single elements of V are drawn, not sets. The classical coupon collector thus corresponds to the family F consisting of singleton sets, and we call the version allowing larger sets structured coupon collector problems. The main concern of this paper is to identify relevant properties of F that affect the covering time (i.e. minimal cost of a cover), and to provide (easily checkable) sufficient conditions for concentration of the covering time.For the third paper we narrow the scopes once more, and study the biased random k-SAT problem. The random k-SAT problem can be seen as a special case of the structured coupon collector, but a special case that has far richer structure than the generic case. The ground set is the hypercube Σ_n = {0, 1}^n, and the coupons are all the k-codimensional subcubes of Σ_n. We study a slight variation on this problem: subcubes are drawn with a constant bias towards 0, so that vertices in Σ_n with fewer 1's and more 0's are easier to cover.

In this paper, we improve two known invariant subspace theorems. More specifically, we show that a closed linear subspace M in the Hardy space H p ( D ) ( 1 &le, p &lt, &infin, ) is invariant under the shift operator M z on H p ( D ) if and only if it is hyperinvariant under M z , and that a closed linear subspace M in the Lebesgue space L 2 ( &part, D ) is reducing under the shift operator M e i &theta, on L 2 ( &part, D ) if and only if it is hyperinvariant under M e i &theta, At the same time, we show that there are two large classes of invariant subspaces for M e i &theta, that are not hyperinvariant subspaces for M e i &theta, and are also not reducing subspaces for M e i &theta, Moreover,we still show that there is a large class of hyperinvariant subspaces for M z that are not reducing subspaces for M z . Furthermore, we gave two new versions of the formula of the reproducing function in the Hardy space H 2 ( D ) , which are the analogue of the formula of the reproducing function in the Bergman space A 2 ( D ) . In addition, the conclusions in this paper are interesting now, or later if they are written into the literature of invariant subspaces and function spaces.

This paper studies Menon-type identities involving both multiplicative characters and additive characters. In the paper, we shall give the explicit formula of the following sum \[ \sum_{\substack{a \in \mathbb{Z}_n^{\ast} \\ b_1, \ldots, b_k \in \mathbb{Z}_n}} \gcd(a-1, b_1, \ldots, b_k, n) \chi(a) \lambda_1(b_1) \cdots \lambda_k(b_k), \] where for a positive integer $n$, $\mathbb{Z}_n^{\ast}$ is the group of units of the ring $\mathbb{Z}_n = \mathbb{Z}/n\mathbb{Z}$, $\gcd$ represents the greatest common divisor, $\chi$ is a Dirichlet character modulo $n$, and for a nonnegative integer $k$, $\lambda_1, \ldots, \lambda_k$ are additive characters of $\mathbb{Z}_n$. Our formula further extends the previous results by Sury [13], Zhao-Cao [17] and Li-Hu-Kim [4].

This paper develops an abstract theory for subdifferential operators to give existence and uniqueness of solutions to the initial-boundary problem (P) for the nonlinear diffusion equation in an unbounded domain $\Omega\subset\mathbb{R}^N$ ($N\in{\mathbb N}$), written as \[ \frac{\partial u}{\partial t} + (-\Delta+1)\beta(u) = g \quad \mbox{in}\ \Omega\times(0, T), \] which represents the porous media, the fast diffusion equations, etc., where $\beta$ is a single-valued maximal monotone function on $\mathbb{R}$, and $T>0$. Existence and uniqueness for (P) were directly proved under a growth condition for $\beta$ even though the Stefan problem was excluded from examples of (P). This paper completely removes the growth condition for $\beta$ by confirming Cauchy's criterion for solutions of the following approximate problem (P)$_{\varepsilon}$ with approximate parameter $\varepsilon>0$: \[ \frac{\partial u_{\varepsilon}}{\partial t} + (-\Delta+1)(\varepsilon(-\Delta+1)u_{\varepsilon} + \beta(u_{\varepsilon}) + \pi_{\varepsilon}(u_{\varepsilon})) = g \quad \mbox{in}\ \Omega\times(0, T), \] which is called the Cahn--Hilliard system, even if $\Omega \subset \mathbb{R}^N$ ($N \in \mathbb{N}$) is an unbounded domain. Moreover, it can be seen that the Stefan problem is covered in the framework of this paper.

We extend our result on the convergence of double recurrence Wiener-Wintner averages to the case where we have a polynomial exponent. We will show that there exists a single set of full measure for which the averages \[ \frac{1}{N} \sum_{n=1}^N f_1(T^{an}x)f_2(T^{bn}x)\phi(p(n)) \] converge for any polynomial $p$ with real coefficients, and any continuous function $\phi$ from the torus to the set of complex numbers . We also show that if either function belongs to an orthogonal complement of an appropriate Host-Kra-Ziegler factor that depends on the degree of the polynomial $p$, then the averages converge to zero uniformly for all polynomials. This paper combines the authors' previously announced work., Comment: This is the final version to appear in Journal d'Analyse mathematique. This latest version combines the previously posted papers on the arxiv website as arXiv:1408.3064 and arXiv:1409.0463

In this paper, we show that the number of monic integer polynomials of degree \(d \ge 1\) and height at most H which have no real roots is between \(c_1H^{d-1/2}\) and \(c_2 H^{d-1/2}\), where the constants \(c_2>c_1>0\) depend only on d. (Of course, this situation may only occur for d even.) Furthermore, for each integer s satisfying \(0 \le s < d/2\) we show that the number of monic integer polynomials of degree d and height at most H which have precisely 2s non-real roots is asymptotic to \(\lambda (d,s)H^{d}\) as \(H \rightarrow \infty \). The constants \(\lambda (d,s)\) are all positive and come from a recent paper of Bertok, Hajdu, and Pethő. They considered a similar question for general (not necessarily monic) integer polynomials and posed this as an open question.

In a recent paper, Das introduced the graph $$\mathcal {I}n(\mathbb {V})$$ , called subspace inclusion graph on a finite dimensional vector space $$\mathbb {V}$$ , where the vertex set is the collection of nontrivial proper subspaces of $$\mathbb {V}$$ and two vertices are adjacent if one is properly contained in another. Das studied the diameter, girth, clique number and chromatic number of $$\mathcal {I}n(\mathbb {V})$$ when the base field is arbitrary, and he also studied some other properties of $$\mathcal {I}n(\mathbb {V})$$ when the base field is finite. At the end of the above paper, the author posed the open problem of determining the automorphisms of $$\mathcal {I}n(\mathbb {V})$$ . In this paper, we give the answer to the open problem.

The first aim of this paper is to give a description of the (not necessarily linear) order isomorphisms \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C(X)\rightarrow C(Y)$$\end{document}C(X)→C(Y) where X, Y are compact Hausdorff spaces. For a simple case, suppose X is metrizable and T is such an order isomorphism. By a theorem of Kaplansky, T induces a homeomorphism \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau :X\rightarrow Y$$\end{document}τ:X→Y. We prove the existence of a homeomorphism \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X\times \mathbb {R}\rightarrow Y\times \mathbb {R}$$\end{document}X×R→Y×R that maps the graph of any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f\in C(X)$$\end{document}f∈C(X) onto the graph of Tf. For nonmetrizable spaces the result is similar, although slightly more complicated. Secondly, we let X and Y be compact and extremally disconnected. The theory of the first part extends directly to order isomorphisms \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^{\infty }(X)\rightarrow C^{\infty }(Y)$$\end{document}C∞(X)→C∞(Y). (Here \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^{\infty }(X)$$\end{document}C∞(X) is the space of all continuous functions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X\rightarrow [-\infty ,\infty ]$$\end{document}X→[-∞,∞] that are finite on a dense set.) The third part of the paper considers order isomorphisms T between arbitrary Archimedean Riesz spaces E and F. We prove that such a T extends uniquely to an order isomorphism between their universal completions. (In the absence of linearity this is not obvious.) It follows, that there exist an extremally disconnected compact Hausdorff space X, Riesz isomorphisms \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{}$$\end{document}^ of E and F onto order dense Riesz subspaces of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^{\infty }(X)$$\end{document}C∞(X) and an order isomorphism \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S:C^{\infty }(X)\rightarrow C^{\infty }(X)$$\end{document}S:C∞(X)→C∞(X) such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{Tf}=S\hat{f}$$\end{document}Tf^=Sf^ (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f\in E$$\end{document}f∈E).