Proof of the operator-norm convergent Trotter product formula on a Banach space is unexpectedly elaborate and a few of known results are based on assumption that at least one of the semigroups involved into this formula is holomorphic. Here we present an example of the operator-norm convergent Trotter product formula on a Banach space, where this condition is relaxed to demand that involved semigroups are contractive.
{"title":"Operator-norm Trotter product formula on Banach spaces","authors":"Valentin Anatol'evich Zagrebnov","doi":"10.4213/im9370e","DOIUrl":"https://doi.org/10.4213/im9370e","url":null,"abstract":"Proof of the operator-norm convergent Trotter product formula on a Banach space is unexpectedly elaborate and a few of known results are based on assumption that at least one of the semigroups involved into this formula is holomorphic. Here we present an example of the operator-norm convergent Trotter product formula on a Banach space, where this condition is relaxed to demand that involved semigroups are contractive.","PeriodicalId":54910,"journal":{"name":"Izvestiya Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135661294","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A system of singular integral equations with monotonic and convex non-linearity on the entire real line is considered. System of this form have applications in many areas of natural science. In particular, such systems arise in the theory of $p$-adic open-closed strings, in the mathematical theory of spatial-temporal epidemic spread within the framework of the well known Diekmann-Kaper model, in the kinetic theory of gases, in the radiative transfer theory. An existence theorem for a non-trivial and bounded solution is proved. The asymptotic behaviour of the constructed solution at $pminfty$ is also studied. Specific examples of non-linearities and kernel functions having an applied character are given.
{"title":"On non-trivial solvability of one system of non-linear integral equations on the real axis","authors":"Khachatur Aghavardovich Khachatryan, Haykanush Samvelovna Petrosyan","doi":"10.4213/im9348e","DOIUrl":"https://doi.org/10.4213/im9348e","url":null,"abstract":"A system of singular integral equations with monotonic and convex non-linearity on the entire real line is considered. System of this form have applications in many areas of natural science. In particular, such systems arise in the theory of $p$-adic open-closed strings, in the mathematical theory of spatial-temporal epidemic spread within the framework of the well known Diekmann-Kaper model, in the kinetic theory of gases, in the radiative transfer theory. An existence theorem for a non-trivial and bounded solution is proved. The asymptotic behaviour of the constructed solution at $pminfty$ is also studied. Specific examples of non-linearities and kernel functions having an applied character are given.","PeriodicalId":54910,"journal":{"name":"Izvestiya Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135661675","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $(V, p)$ be a normal surface singularity. Let $picolon (M, A)to (V, p)$ be a minimal good resolution of $V$. The weighted dual graphs $Gamma$ associated with $A$ completely describes the topology and differentiable structure of the embedding of $A$ in $M$. In this paper, we classify all the weighted dual graphs of $A=bigcup_{i=1}^n A_i$ such that one of the curves $A_i$ is a $-3$-curve, and all the remaining ones are $-2$-curves. This is a natural generalization of Artin's classification of rational triple points. Moreover, we compute the fundamental cycles of maximal graphs (see § 5) which can be used to determine whether the singularities are rational, minimally elliptic or weakly elliptic. We also give formulas for computing arithmetic and geometric genera of star-shaped graphs.
{"title":"Classification of weighted dual graphs consisting of $-2$-curves and exactly one $-3$-curve","authors":"Stephen S.-T. Yau, Qiwei Zhu, Huaiqing Zuo","doi":"10.4213/im9337e","DOIUrl":"https://doi.org/10.4213/im9337e","url":null,"abstract":"Let $(V, p)$ be a normal surface singularity. Let $picolon (M, A)to (V, p)$ be a minimal good resolution of $V$. The weighted dual graphs $Gamma$ associated with $A$ completely describes the topology and differentiable structure of the embedding of $A$ in $M$. In this paper, we classify all the weighted dual graphs of $A=bigcup_{i=1}^n A_i$ such that one of the curves $A_i$ is a $-3$-curve, and all the remaining ones are $-2$-curves. This is a natural generalization of Artin's classification of rational triple points. Moreover, we compute the fundamental cycles of maximal graphs (see § 5) which can be used to determine whether the singularities are rational, minimally elliptic or weakly elliptic. We also give formulas for computing arithmetic and geometric genera of star-shaped graphs.","PeriodicalId":54910,"journal":{"name":"Izvestiya Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135661292","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Oleg Vasilevich Morzhin, Alexander Nikolaevich Pechen
This article considers some control problems for closed and open two-level quantum systems. The closed system's dynamics is governed by the Schrödinger equation with coherent control. The open system's dynamics is governed by the Gorini-Kossakowski-Sudarshan-Lindblad master equation whose Hamiltonian depends on coherent control and superoperator of dissipation depends on incoherent control. For the closed system, we consider the problem for generation of the phase shift gate for some values of phases and final times for which we numerically show that zero coherent control, which is a stationary point of the objective functional, is not optimal; it gives an example of subtle point for practical solving quantum control problems. The two-stage method, which was developed in [A. Pechen, Phys. Rev. A., 84, 042106 (2011)] for generic $N$-level open quantum systems for approximate generation of a given target density matrix, is modified here for the case of two-level systems. We modify the first ("incoherent") stage by numerically optimizing piecewise constant incoherent control instead of using constant incoherent control analytically computed using eigenvalues of the target density matrix. Exact analytical formulas are derived for the system's state evolution, the objective functions and their gradients for the modified first stage. These formulas are then applied in the two-step gradient projection method. The numerical simulations show that the modified first stage's duration can be significantly less than the unmodified first stage's duration, but at the cost of performing optimization in the class of piecewise constant controls. Bibliography: 72 titles.
{"title":"On optimization of coherent and incoherent controls for two-level quantum systems","authors":"Oleg Vasilevich Morzhin, Alexander Nikolaevich Pechen","doi":"10.4213/im9372e","DOIUrl":"https://doi.org/10.4213/im9372e","url":null,"abstract":"This article considers some control problems for closed and open two-level quantum systems. The closed system's dynamics is governed by the Schrödinger equation with coherent control. The open system's dynamics is governed by the Gorini-Kossakowski-Sudarshan-Lindblad master equation whose Hamiltonian depends on coherent control and superoperator of dissipation depends on incoherent control. For the closed system, we consider the problem for generation of the phase shift gate for some values of phases and final times for which we numerically show that zero coherent control, which is a stationary point of the objective functional, is not optimal; it gives an example of subtle point for practical solving quantum control problems. The two-stage method, which was developed in [A. Pechen, Phys. Rev. A., 84, 042106 (2011)] for generic $N$-level open quantum systems for approximate generation of a given target density matrix, is modified here for the case of two-level systems. We modify the first (\"incoherent\") stage by numerically optimizing piecewise constant incoherent control instead of using constant incoherent control analytically computed using eigenvalues of the target density matrix. Exact analytical formulas are derived for the system's state evolution, the objective functions and their gradients for the modified first stage. These formulas are then applied in the two-step gradient projection method. The numerical simulations show that the modified first stage's duration can be significantly less than the unmodified first stage's duration, but at the cost of performing optimization in the class of piecewise constant controls.\u0000Bibliography: 72 titles.","PeriodicalId":54910,"journal":{"name":"Izvestiya Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135661300","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For a general optimal control problem with state and regular mixed constraints we propose a proof of the maximum principle based on the so-called $v$-change of time variable $t mapsto tau$, under which the original time becomes an additional state variable subject to the equation $dt/dtau = v(tau)$, while the additional control variable $v(tau)geqslant 0$ is piecewise constant, and its values become arguments of the new problem.
{"title":"Variations of $v$-change of time in an optimal control problem with state and mixed constraints","authors":"Andrei Venediktovich Dmitruk","doi":"10.4213/im9305e","DOIUrl":"https://doi.org/10.4213/im9305e","url":null,"abstract":"For a general optimal control problem with state and regular mixed constraints we propose a proof of the maximum principle based on the so-called $v$-change of time variable $t mapsto tau$, under which the original time becomes an additional state variable subject to the equation $dt/dtau = v(tau)$, while the additional control variable $v(tau)geqslant 0$ is piecewise constant, and its values become arguments of the new problem.","PeriodicalId":54910,"journal":{"name":"Izvestiya Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135009145","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The paper is concerned with problems of existence, uniqueness, and stabilization of weak solutions of one class of semilinear second-order parabolic differential equations on closed manifolds. These equations are inhomogeneous analogues of the Kolmogorov-Petrovskii-Piskunov-Fisher equation, and have significant applied and mathematical value.
{"title":"On stabilization of solutions of second-order semilinear parabolic equations on closed manifolds","authors":"Dmitry Vasilievich Tunitsky","doi":"10.4213/im9354e","DOIUrl":"https://doi.org/10.4213/im9354e","url":null,"abstract":"The paper is concerned with problems of existence, uniqueness, and stabilization of weak solutions of one class of semilinear second-order parabolic differential equations on closed manifolds. These equations are inhomogeneous analogues of the Kolmogorov-Petrovskii-Piskunov-Fisher equation, and have significant applied and mathematical value.","PeriodicalId":54910,"journal":{"name":"Izvestiya Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135009868","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
New one-dimensional Hardy-type inequalities for a weight function of the form $x^alpha(2-x)^beta$ for positive and negative values of the parameters $alpha$ and $beta$ are put forward. In some cases, the constants in the resulting one-dimensional inequalities are sharp. We use one-dimensional inequalities with additional terms to establish multivariate inequalities with weight functions depending on the mean distance function or the distance function from the boundary of a domain. Spatial inequalities are proved in arbitrary domains, in Davies-regular domains, in domains satisfying the cone condition, in $lambda$-close to convex domains, and in convex domains. The constant in the additional term in the spatial inequalities depends on the volume or the diameter of the domain. As a consequence of these multivariate inequalities, estimates for the first eigenvalue of the Laplacian under the Dirichlet boundary conditions in various classes of domains are established. We also use one-dimensional inequalities to obtain new classes of meromorphic univalent functions in simply connected domains. Namely, Nehari-Pokornii type sufficient conditions for univalence are obtained.
{"title":"Hardy type inequalities for one weight function and their applications","authors":"R. Nasibullin","doi":"10.4213/im9291e","DOIUrl":"https://doi.org/10.4213/im9291e","url":null,"abstract":"New one-dimensional Hardy-type inequalities for a weight function of the form $x^alpha(2-x)^beta$ for positive and negative values of the parameters $alpha$ and $beta$ are put forward. In some cases, the constants in the resulting one-dimensional inequalities are sharp. We use one-dimensional inequalities with additional terms to establish multivariate inequalities with weight functions depending on the mean distance function or the distance function from the boundary of a domain. Spatial inequalities are proved in arbitrary domains, in Davies-regular domains, in domains satisfying the cone condition, in $lambda$-close to convex domains, and in convex domains. The constant in the additional term in the spatial inequalities depends on the volume or the diameter of the domain. As a consequence of these multivariate inequalities, estimates for the first eigenvalue of the Laplacian under the Dirichlet boundary conditions in various classes of domains are established. We also use one-dimensional inequalities to obtain new classes of meromorphic univalent functions in simply connected domains. Namely, Nehari-Pokornii type sufficient conditions for univalence are obtained.","PeriodicalId":54910,"journal":{"name":"Izvestiya Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70326983","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper is concerned with some sufficient conditions for the boundedness of Hardy-Littlewood maximal functions, rough Hausdorff and matrix Hausdorff operators on two-weighted Herz spaces on $p$-adic fields through its atomic decomposition.
{"title":"Two-weight estimates for Hardy-Littlewood maximal functions and Hausdorff operators on $p$-adic Herz spaces","authors":"Kieu Huu Dung, Dao Van Duong","doi":"10.4213/im9404e","DOIUrl":"https://doi.org/10.4213/im9404e","url":null,"abstract":"This paper is concerned with some sufficient conditions for the boundedness of Hardy-Littlewood maximal functions, rough Hausdorff and matrix Hausdorff operators on two-weighted Herz spaces on $p$-adic fields through its atomic decomposition.","PeriodicalId":54910,"journal":{"name":"Izvestiya Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135661299","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Alexander Gennad'evich Kuznetsov, Yuri Gennadievich Prokhorov
We study del Pezzo varieties, higher-dimensional analogues of del Pezzo surfaces. In particular, we introduce ADE classification of del Pezzo varieties, show that in type $mathrm A$ the dimension of non-conical del Pezzo varieties is bounded by $12 - d - r$, where $d$ is the degree and $r$ is the rank of the class group, and classify maximal del Pezzo varieties.
我们研究del Pezzo变体，即del Pezzo曲面的高维类似物。特别地，我们引入del Pezzo变种的ADE分类，证明了在类型A$中，非圆锥del Pezzo变种的维数以$12 - d - r$为界，其中$d$为度，$r$为类群的秩，并对极大del Pezzo变种进行了分类。
{"title":"On higher-dimensional del Pezzo varieties","authors":"Alexander Gennad'evich Kuznetsov, Yuri Gennadievich Prokhorov","doi":"10.4213/im9385e","DOIUrl":"https://doi.org/10.4213/im9385e","url":null,"abstract":"We study del Pezzo varieties, higher-dimensional analogues of del Pezzo surfaces. In particular, we introduce ADE classification of del Pezzo varieties, show that in type $mathrm A$ the dimension of non-conical del Pezzo varieties is bounded by $12 - d - r$, where $d$ is the degree and $r$ is the rank of the class group, and classify maximal del Pezzo varieties.","PeriodicalId":54910,"journal":{"name":"Izvestiya Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135440265","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A. Ber, Matthijs J. Borst, Sander Borst, F. Sukochev
We prove that, for a finite-dimensional real normed space $V$, every bounded mean zero function $fin L_infty([0,1];V)$ can be written in the form $f=gcirc T-g$ for some $gin L_infty([0,1];V)$ and some ergodic invertible measure preserving transformation $T$ of $[0,1]$. Our method moreover allows us to choose $g$, for any given $varepsilon>0$, to be such that $|g|_inftyleq (S_V+varepsilon)|f|_infty$, where $S_V$ is the Steinitz constant corresponding to $V$.
{"title":"A solution to the multidimensional additive homological equation","authors":"A. Ber, Matthijs J. Borst, Sander Borst, F. Sukochev","doi":"10.4213/im9319e","DOIUrl":"https://doi.org/10.4213/im9319e","url":null,"abstract":"We prove that, for a finite-dimensional real normed space $V$, every bounded mean zero function $fin L_infty([0,1];V)$ can be written in the form $f=gcirc T-g$ for some $gin L_infty([0,1];V)$ and some ergodic invertible measure preserving transformation $T$ of $[0,1]$. Our method moreover allows us to choose $g$, for any given $varepsilon>0$, to be such that $|g|_inftyleq (S_V+varepsilon)|f|_infty$, where $S_V$ is the Steinitz constant corresponding to $V$.","PeriodicalId":54910,"journal":{"name":"Izvestiya Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70326549","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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