The problem of tiling rectangles by polyominoes generated large interest. A related one is the problem of tiling parallelograms by twisted polyominoes. Both problems are related with tilings of (skewed) quadrants by p...The problem of tiling rectangles by polyominoes generated large interest. A related one is the problem of tiling parallelograms by twisted polyominoes. Both problems are related with tilings of (skewed) quadrants by polyominoes. Indeed, if all tilings of a (skewed) quadrant by a tile set can be reduced to a tiling by congruent rectangles (parallelograms), this provides information about tilings of rectangles (parallelograms). We consider a class of tile sets in a square lattice appearing from arbitrary dissections of rectangles in two L-shaped polyominoes and from symmetries of these tiles about the first bisector. Only translations of the tiles are allowed in a tiling. If the sides of the dissected rectangle are coprime, we show the existence of tilings of all (skewed) quadrants that do not follow the rectangular (parallelogram) pattern. If one of the sides of the dissected rectangle is 2 and the other is odd, we also show tilings of rectangles by the tile set that do not follow the rectangular pattern. If one of the sides of the dissected rectangle is 2 and the other side is even, we show a new infinite family of tile sets that follows the rectangular pattern when tiling one of the quadrants. For this type of dis-section, we also show a new infinite family that does not follow the rectangular pattern when tiling rectangles. Finally, we investigate more general dissections of rectangles, with. Here we show infinite families of tile sets that follow the rectangular pattern for a quadrant and infinite families that do not follow the rectangular pattern for any quadrant. We also show, for infinite families of tile sets of this type, tilings of rectangles that do not follow the rectangular pattern.展开更多
We show that the least number of cells (the gap number) one needs to take out from a rectangle with integer sides of length at least 2 in order to be tiled by ribbon right trominoes is less than or equal to 4. If the ...We show that the least number of cells (the gap number) one needs to take out from a rectangle with integer sides of length at least 2 in order to be tiled by ribbon right trominoes is less than or equal to 4. If the sides of the rectangle are of length at least 5, then the gap number is less than or equal to 3. We also show that for the family of rectangles that have nontrivial minimal number of gaps, with probability 1, the only obstructions to tiling appear from coloring invariants. This is in contrast to what happens for simply connected regions. For that class of regions Conway and Lagarias found a tiling invariant that does not follow from coloring.展开更多
As a coprocessor, field-programmable gate array （FPGA） is the hardware computing processor accelerating the computing capacity of coraputers. To efficiently manage the hardware free resources for the placing of task...As a coprocessor, field-programmable gate array （FPGA） is the hardware computing processor accelerating the computing capacity of coraputers. To efficiently manage the hardware free resources for the placing of tasks on FPGA and take full advantage of the partially reconfigurable units, good utilization of chip resources is an important and necessary work. In this paper, a new method is proposed to find the complete set of maximal free resource rectangles based on the cross point of edge lines of running tasks on FPGA area, and the prove process is provided to make sure the correctness of this method.展开更多
(1)Inside the rectangle ABCD,there is a smaller rectangle EFGD.Points B,F,and D lie on the same straight line.AE is 5 cm,ED is 30 cm,and CG is 3 cm.Find the area of the shaded region.
Neuroscience (also known as neurobiology) is a science that studies the structure, function, development, pharmacology and pathology of the nervous system. In recent years, C. Cotardo has introduced coding theory into...Neuroscience (also known as neurobiology) is a science that studies the structure, function, development, pharmacology and pathology of the nervous system. In recent years, C. Cotardo has introduced coding theory into neuroscience, proposing the concept of combinatorial neural codes. And it was further studied in depth using algebraic methods by C. Curto. In this paper, we construct a class of combinatorial neural codes with special properties based on classical combinatorial structures such as orthogonal Latin rectangle, disjoint Steiner systems, groupable designs and transversal designs. These neural codes have significant weight distribution properties and large minimum distances, and are thus valuable for potential applications in information representation and neuroscience. This study provides new ideas for the construction method and property analysis of combinatorial neural codes, and enriches the study of algebraic coding theory.展开更多
We study the extensions of the Bogner-Fox-Schmit element to the whole family of Q_(k) continuously differentiable finite elements on rectangular grids,for all k≥3,in 2D and 3D.We show that the newly defined C_(1) spa...We study the extensions of the Bogner-Fox-Schmit element to the whole family of Q_(k) continuously differentiable finite elements on rectangular grids,for all k≥3,in 2D and 3D.We show that the newly defined C_(1) spaces are maximal in the sense that they contain all C_(1)-Q_(k) functions of piecewise polynomials.We give examples of other extensions of C_(1)-Q_(k) elements.The result is consistent with the Strang’s conjecture(restricted to the quadrilateral grids in 2D and 3D).Some numerical results are provided on the family of C_(1) elements solving the biharmonic equation.展开更多
Based on the TRMM dataset, this paper compares the applicability of the improved MCE(minimum circumscribed ellipse), MBR(minimum bounding rectangle), and DIA(direct indexing area) methods for rain cell fitting. These ...Based on the TRMM dataset, this paper compares the applicability of the improved MCE(minimum circumscribed ellipse), MBR(minimum bounding rectangle), and DIA(direct indexing area) methods for rain cell fitting. These three methods can reflect the geometric characteristics of clouds and apply geometric parameters to estimate the real dimensions of rain cells. The MCE method shows a major advantage in identifying the circumference of rain cells. The circumference of rain cells identified by MCE in most samples is smaller than that identified by DIA and MBR, and more similar to the observed rain cells. The area of rain cells identified by MBR is relatively robust. For rain cells composed of many pixels(N> 20), the overall performance is better than that of MCE, but the contribution of MBR to the best identification results,which have the shortest circumference and the smallest area, is less than that of MCE. The DIA method is best suited to small rain cells with a circumference of less than 100 km and an area of less than 120 km^(2), but the overall performance is mediocre. The MCE method tends to achieve the highest success at any angle, whereas there are fewer “best identification”results from DIA or MBR and more of the worst ones in the along-track direction and cross-track direction. Through this comprehensive comparison, we conclude that MCE can obtain the best fitting results with the shortest circumference and the smallest area on behalf of the high filling effect for all sizes of rain cells.展开更多
Cryptography is deemed to be the optimum strategy to secure the data privacy in which the data is encoded ahead of time before sharing it.Visual Secret Sharing(VSS)is an encryption method in which the secret message i...Cryptography is deemed to be the optimum strategy to secure the data privacy in which the data is encoded ahead of time before sharing it.Visual Secret Sharing(VSS)is an encryption method in which the secret message is split into at least two trivial images called’shares’to cover it.However,such message are always targeted by hackers or dishonest members who attempt to decrypt the message.This can be avoided by not uncovering the secret message without the universal share when it is presented and is typically taken care of,by the trusted party.Hence,in this paper,an optimal and secure double-layered secret image sharing scheme is proposed.The proposed share creation process contains two layers such as threshold-based secret sharing in the first layer and universal share based secret sharing in the second layer.In first layer,Genetic Algorithm(GA)is applied to find the optimal threshold value based on the randomness of the created shares.Then,in the second layer,a novel design of universal share-based secret share creation method is proposed.Finally,Opposition Whale Optimization Algorithm(OWOA)-based optimal key was generated for rectange block cipher to secure each share.This helped in producing high quality reconstruction images.The researcher achieved average experimental outcomes in terms of PSNR and MSE values equal to 55.154225 and 0.79365625 respectively.The average PSNRwas less(49.134475)and average MSE was high(1)in case of existing methods.展开更多
inductive fault analysis is a technique for enumerating likely bridges that is limited by the weighted critical area computation. Based on the rectangle model of a real defect and mathematical morphology, an efficient...inductive fault analysis is a technique for enumerating likely bridges that is limited by the weighted critical area computation. Based on the rectangle model of a real defect and mathematical morphology, an efficient algorithm is presented to compute the weighted critical area of a layout. The algorithm avoids the need to determine which rectangles belong to a net and the merging of the critical area corresponding to a net pair. Experimental resuits showing the algorithm＇s performance are presented.展开更多
Let T<sub>n </sub>be the set of ribbon L-shaped n-ominoes for some n≥4 even, and let T<sup>+</sup><sub>n</sub> be T<sub>n</sub> with an extra 2 x 2 square. We investiga...Let T<sub>n </sub>be the set of ribbon L-shaped n-ominoes for some n≥4 even, and let T<sup>+</sup><sub>n</sub> be T<sub>n</sub> with an extra 2 x 2 square. We investigate signed tilings of rectangles by T<sub>n</sub> and T<sup>+</sup><sub>n</sub> . We show that a rectangle has a signed tiling by T<sub>n</sub> if and only if both sides of the rectangle are even and one of them is divisible by n, or if one of the sides is odd and the other side is divisible by . We also show that a rectangle has a signed tiling by T<sup>+</sup><sub>n, </sub> n≥6 even, if and only if both sides of the rectangle are even, or if one of the sides is odd and the other side is divisible by . Our proofs are based on the exhibition of explicit GrÖbner bases for the ideals generated by polynomials associated to the tiling sets. In particular, we show that some of the regular tiling results in Nitica, V. (2015) Every tiling of the first quadrant by ribbon L n-ominoes follows the rectangular pattern. Open Journal of Discrete Mathematics, 5, 11-25, cannot be obtained from coloring invariants.展开更多
The dynamical equations of a thin rectangle plate subjected to the friction support boundary and its plane force are established in this paper. The local bifurcation of this system is investigated by using L S method...The dynamical equations of a thin rectangle plate subjected to the friction support boundary and its plane force are established in this paper. The local bifurcation of this system is investigated by using L S method and the singularity theory. The Z 2 bifurcation in non degenerate case is discussed. The local bifurcation diagrams of the unfolding parameters and the bifurcation response characters referred to the physical parameters of the system are obtained by numerical simulation. The results of the computer simulation are coincident with the theoretical analysis and experimental results.展开更多
Let and let be the set of four ribbon L-shaped n-ominoes. We study tiling problems for regions in a square lattice by . Our main result shows a remarkable property of this set of tiles: any tiling of the first quadran...Let and let be the set of four ribbon L-shaped n-ominoes. We study tiling problems for regions in a square lattice by . Our main result shows a remarkable property of this set of tiles: any tiling of the first quadrant by , n even, reduces to a tiling by and rectangles, each rectangle being covered by two ribbon L-shaped n-ominoes. An application of our result is the characterization of all rectangles that can be tiled by , n even: a rectangle can be tiled by , n even, if and only if both of its sides are even and at least one side is divisible by n. Another application is the existence of the local move property for an infinite family of sets of tiles: , n even, has the local move property for the class of rectangular regions with respect to the local moves that interchange a tiling of an square by n/2 vertical rectangles, with a tiling by n/2 horizontal rectangles, each vertical/horizontal rectangle being covered by two ribbon L-shaped n-ominoes. We show that none of these results are valid for any odd n. The rectangular pattern of a tiling of the first quadrant persists if we add an extra tile to , n even. A rectangle can be tiled by the larger set of tiles if and only if it has both sides even. We also show that our main result implies that a skewed L-shaped n-omino, n even, is not a replicating tile of order k2 for any odd k.展开更多
We consider tilings of deficient rectangles by the set T4 of ribbon L-tetro-minoes. A tiling exists if and only if the rectangle is a square of odd side. The missing cell has to be on the main NW-SE diagonal, in an od...We consider tilings of deficient rectangles by the set T4 of ribbon L-tetro-minoes. A tiling exists if and only if the rectangle is a square of odd side. The missing cell has to be on the main NW-SE diagonal, in an odd position if the square is (4m+1)×(4m+1) and in an even position if the square is (4m+3)×(4m+3). The majority of the tiles in a tiling follow the rectangular pattern, that is, are paired and each pair tiles a 2×4 rectangle. The tiles in an irregular position together with the missing cell form a NW-SE diagonal crack. The crack is located in a thin region symmetric about the diagonal, made out of a sequence of 3×3 squares that overlap over one of the corner cells. The crack divides the square in two parts of equal area. The number of tilings of a (4m+1)×(4m+1) deficient square by T4? is equal to the number of tilings by dominoes of a 2m×2m square. The number of tilings of a (4m+3)×(4m+3) deficient square by T4? is twice the number of tilings by dominoes of a (2m+1)×(2m+1)?deficient square, with the missing cell placed on the main diagonal. In both cases the counting is realized by an explicit function which is a bijection in the first case and a double cover in the second. If an extra 2×2 tile is added to T4 , we call the new tile set?T+<sub style="margin-left:-6px;">4. A tiling of a deficient rectangle by T+4 exists if and only if the rectangle is a square of odd side. The missing cell has to be on the main NW-SE diagonal, in an odd position if the square is (4m+1)×(4m+1) and in an even position if the square is (4m+3)×(4m+3). The majority of the tiles in a tiling follow the rectangular pattern, that is, are either paired tetrominoes and each pair tiles a 2×4 rectangle, or are 2×2 squares. The tiles in an irregular position together with the missing cell form a NW-SE diagonal crack. The crack is located in a thin region symmetric about the diagonal, made out of a sequence of 3×3 squares that overlap over one of the corner cells. The number of tilings of a (4m+1)×(4m+1) deficient square by T+4 is greater than the number of tilings by dominoes and monomers of a 2m×2m square. The number of tilings of a (4m+3)×(4m+3) deficient square by T+4 is greater than twice the number of tilings by dominoes and monomers of a (2m+1)×(2m+1) deficient square, with the missing cell placed on the main diagonal. We also consider tilings by T4? and T+4 of other significant deficient regions. In particular we show that a deficient first quadrant, a deficient half strip, a deficient strip or a deficient bent strip cannot be tiled by T+4. Therefore T4? and T+4 give examples of tile sets that tile deficient rectangles but do not tile any deficient first quadrant, any deficient half strip, any deficient bent strip or any deficient strip.展开更多
文摘The problem of tiling rectangles by polyominoes generated large interest. A related one is the problem of tiling parallelograms by twisted polyominoes. Both problems are related with tilings of (skewed) quadrants by polyominoes. Indeed, if all tilings of a (skewed) quadrant by a tile set can be reduced to a tiling by congruent rectangles (parallelograms), this provides information about tilings of rectangles (parallelograms). We consider a class of tile sets in a square lattice appearing from arbitrary dissections of rectangles in two L-shaped polyominoes and from symmetries of these tiles about the first bisector. Only translations of the tiles are allowed in a tiling. If the sides of the dissected rectangle are coprime, we show the existence of tilings of all (skewed) quadrants that do not follow the rectangular (parallelogram) pattern. If one of the sides of the dissected rectangle is 2 and the other is odd, we also show tilings of rectangles by the tile set that do not follow the rectangular pattern. If one of the sides of the dissected rectangle is 2 and the other side is even, we show a new infinite family of tile sets that follows the rectangular pattern when tiling one of the quadrants. For this type of dis-section, we also show a new infinite family that does not follow the rectangular pattern when tiling rectangles. Finally, we investigate more general dissections of rectangles, with. Here we show infinite families of tile sets that follow the rectangular pattern for a quadrant and infinite families that do not follow the rectangular pattern for any quadrant. We also show, for infinite families of tile sets of this type, tilings of rectangles that do not follow the rectangular pattern.
文摘We show that the least number of cells (the gap number) one needs to take out from a rectangle with integer sides of length at least 2 in order to be tiled by ribbon right trominoes is less than or equal to 4. If the sides of the rectangle are of length at least 5, then the gap number is less than or equal to 3. We also show that for the family of rectangles that have nontrivial minimal number of gaps, with probability 1, the only obstructions to tiling appear from coloring invariants. This is in contrast to what happens for simply connected regions. For that class of regions Conway and Lagarias found a tiling invariant that does not follow from coloring.
基金Project supported by the Shanghai Leading Academic Discipline Project(Grant No.J50103)the Natural Science Foundation of Jiangxi Province(Grant No.2010GZS0031)the Science Technology Project of Jiangxi Province(Grant No.2010BGB00604)
文摘As a coprocessor, field-programmable gate array （FPGA） is the hardware computing processor accelerating the computing capacity of coraputers. To efficiently manage the hardware free resources for the placing of tasks on FPGA and take full advantage of the partially reconfigurable units, good utilization of chip resources is an important and necessary work. In this paper, a new method is proposed to find the complete set of maximal free resource rectangles based on the cross point of edge lines of running tasks on FPGA area, and the prove process is provided to make sure the correctness of this method.
文摘(1)Inside the rectangle ABCD,there is a smaller rectangle EFGD.Points B,F,and D lie on the same straight line.AE is 5 cm,ED is 30 cm,and CG is 3 cm.Find the area of the shaded region.
文摘Neuroscience (also known as neurobiology) is a science that studies the structure, function, development, pharmacology and pathology of the nervous system. In recent years, C. Cotardo has introduced coding theory into neuroscience, proposing the concept of combinatorial neural codes. And it was further studied in depth using algebraic methods by C. Curto. In this paper, we construct a class of combinatorial neural codes with special properties based on classical combinatorial structures such as orthogonal Latin rectangle, disjoint Steiner systems, groupable designs and transversal designs. These neural codes have significant weight distribution properties and large minimum distances, and are thus valuable for potential applications in information representation and neuroscience. This study provides new ideas for the construction method and property analysis of combinatorial neural codes, and enriches the study of algebraic coding theory.
文摘We study the extensions of the Bogner-Fox-Schmit element to the whole family of Q_(k) continuously differentiable finite elements on rectangular grids,for all k≥3,in 2D and 3D.We show that the newly defined C_(1) spaces are maximal in the sense that they contain all C_(1)-Q_(k) functions of piecewise polynomials.We give examples of other extensions of C_(1)-Q_(k) elements.The result is consistent with the Strang’s conjecture(restricted to the quadrilateral grids in 2D and 3D).Some numerical results are provided on the family of C_(1) elements solving the biharmonic equation.
基金supported by the National Natural Science Foundation of China (Grant Nos. U20A2097,42075087, 91837310)the National Key Research and Development Program of China (Grant No. 2021YFC3000902)。
文摘Based on the TRMM dataset, this paper compares the applicability of the improved MCE(minimum circumscribed ellipse), MBR(minimum bounding rectangle), and DIA(direct indexing area) methods for rain cell fitting. These three methods can reflect the geometric characteristics of clouds and apply geometric parameters to estimate the real dimensions of rain cells. The MCE method shows a major advantage in identifying the circumference of rain cells. The circumference of rain cells identified by MCE in most samples is smaller than that identified by DIA and MBR, and more similar to the observed rain cells. The area of rain cells identified by MBR is relatively robust. For rain cells composed of many pixels(N> 20), the overall performance is better than that of MCE, but the contribution of MBR to the best identification results,which have the shortest circumference and the smallest area, is less than that of MCE. The DIA method is best suited to small rain cells with a circumference of less than 100 km and an area of less than 120 km^(2), but the overall performance is mediocre. The MCE method tends to achieve the highest success at any angle, whereas there are fewer “best identification”results from DIA or MBR and more of the worst ones in the along-track direction and cross-track direction. Through this comprehensive comparison, we conclude that MCE can obtain the best fitting results with the shortest circumference and the smallest area on behalf of the high filling effect for all sizes of rain cells.
基金supported by RUSA PHASE 2.0,Alagappa University,Karaikudi,India。
文摘Cryptography is deemed to be the optimum strategy to secure the data privacy in which the data is encoded ahead of time before sharing it.Visual Secret Sharing(VSS)is an encryption method in which the secret message is split into at least two trivial images called’shares’to cover it.However,such message are always targeted by hackers or dishonest members who attempt to decrypt the message.This can be avoided by not uncovering the secret message without the universal share when it is presented and is typically taken care of,by the trusted party.Hence,in this paper,an optimal and secure double-layered secret image sharing scheme is proposed.The proposed share creation process contains two layers such as threshold-based secret sharing in the first layer and universal share based secret sharing in the second layer.In first layer,Genetic Algorithm(GA)is applied to find the optimal threshold value based on the randomness of the created shares.Then,in the second layer,a novel design of universal share-based secret share creation method is proposed.Finally,Opposition Whale Optimization Algorithm(OWOA)-based optimal key was generated for rectange block cipher to secure each share.This helped in producing high quality reconstruction images.The researcher achieved average experimental outcomes in terms of PSNR and MSE values equal to 55.154225 and 0.79365625 respectively.The average PSNRwas less(49.134475)and average MSE was high(1)in case of existing methods.
文摘inductive fault analysis is a technique for enumerating likely bridges that is limited by the weighted critical area computation. Based on the rectangle model of a real defect and mathematical morphology, an efficient algorithm is presented to compute the weighted critical area of a layout. The algorithm avoids the need to determine which rectangles belong to a net and the merging of the critical area corresponding to a net pair. Experimental resuits showing the algorithm＇s performance are presented.
文摘Let T<sub>n </sub>be the set of ribbon L-shaped n-ominoes for some n≥4 even, and let T<sup>+</sup><sub>n</sub> be T<sub>n</sub> with an extra 2 x 2 square. We investigate signed tilings of rectangles by T<sub>n</sub> and T<sup>+</sup><sub>n</sub> . We show that a rectangle has a signed tiling by T<sub>n</sub> if and only if both sides of the rectangle are even and one of them is divisible by n, or if one of the sides is odd and the other side is divisible by . We also show that a rectangle has a signed tiling by T<sup>+</sup><sub>n, </sub> n≥6 even, if and only if both sides of the rectangle are even, or if one of the sides is odd and the other side is divisible by . Our proofs are based on the exhibition of explicit GrÖbner bases for the ideals generated by polynomials associated to the tiling sets. In particular, we show that some of the regular tiling results in Nitica, V. (2015) Every tiling of the first quadrant by ribbon L n-ominoes follows the rectangular pattern. Open Journal of Discrete Mathematics, 5, 11-25, cannot be obtained from coloring invariants.
文摘The dynamical equations of a thin rectangle plate subjected to the friction support boundary and its plane force are established in this paper. The local bifurcation of this system is investigated by using L S method and the singularity theory. The Z 2 bifurcation in non degenerate case is discussed. The local bifurcation diagrams of the unfolding parameters and the bifurcation response characters referred to the physical parameters of the system are obtained by numerical simulation. The results of the computer simulation are coincident with the theoretical analysis and experimental results.
文摘Let and let be the set of four ribbon L-shaped n-ominoes. We study tiling problems for regions in a square lattice by . Our main result shows a remarkable property of this set of tiles: any tiling of the first quadrant by , n even, reduces to a tiling by and rectangles, each rectangle being covered by two ribbon L-shaped n-ominoes. An application of our result is the characterization of all rectangles that can be tiled by , n even: a rectangle can be tiled by , n even, if and only if both of its sides are even and at least one side is divisible by n. Another application is the existence of the local move property for an infinite family of sets of tiles: , n even, has the local move property for the class of rectangular regions with respect to the local moves that interchange a tiling of an square by n/2 vertical rectangles, with a tiling by n/2 horizontal rectangles, each vertical/horizontal rectangle being covered by two ribbon L-shaped n-ominoes. We show that none of these results are valid for any odd n. The rectangular pattern of a tiling of the first quadrant persists if we add an extra tile to , n even. A rectangle can be tiled by the larger set of tiles if and only if it has both sides even. We also show that our main result implies that a skewed L-shaped n-omino, n even, is not a replicating tile of order k2 for any odd k.
文摘We consider tilings of deficient rectangles by the set T4 of ribbon L-tetro-minoes. A tiling exists if and only if the rectangle is a square of odd side. The missing cell has to be on the main NW-SE diagonal, in an odd position if the square is (4m+1)×(4m+1) and in an even position if the square is (4m+3)×(4m+3). The majority of the tiles in a tiling follow the rectangular pattern, that is, are paired and each pair tiles a 2×4 rectangle. The tiles in an irregular position together with the missing cell form a NW-SE diagonal crack. The crack is located in a thin region symmetric about the diagonal, made out of a sequence of 3×3 squares that overlap over one of the corner cells. The crack divides the square in two parts of equal area. The number of tilings of a (4m+1)×(4m+1) deficient square by T4? is equal to the number of tilings by dominoes of a 2m×2m square. The number of tilings of a (4m+3)×(4m+3) deficient square by T4? is twice the number of tilings by dominoes of a (2m+1)×(2m+1)?deficient square, with the missing cell placed on the main diagonal. In both cases the counting is realized by an explicit function which is a bijection in the first case and a double cover in the second. If an extra 2×2 tile is added to T4 , we call the new tile set?T+<sub style="margin-left:-6px;">4. A tiling of a deficient rectangle by T+4 exists if and only if the rectangle is a square of odd side. The missing cell has to be on the main NW-SE diagonal, in an odd position if the square is (4m+1)×(4m+1) and in an even position if the square is (4m+3)×(4m+3). The majority of the tiles in a tiling follow the rectangular pattern, that is, are either paired tetrominoes and each pair tiles a 2×4 rectangle, or are 2×2 squares. The tiles in an irregular position together with the missing cell form a NW-SE diagonal crack. The crack is located in a thin region symmetric about the diagonal, made out of a sequence of 3×3 squares that overlap over one of the corner cells. The number of tilings of a (4m+1)×(4m+1) deficient square by T+4 is greater than the number of tilings by dominoes and monomers of a 2m×2m square. The number of tilings of a (4m+3)×(4m+3) deficient square by T+4 is greater than twice the number of tilings by dominoes and monomers of a (2m+1)×(2m+1) deficient square, with the missing cell placed on the main diagonal. We also consider tilings by T4? and T+4 of other significant deficient regions. In particular we show that a deficient first quadrant, a deficient half strip, a deficient strip or a deficient bent strip cannot be tiled by T+4. Therefore T4? and T+4 give examples of tile sets that tile deficient rectangles but do not tile any deficient first quadrant, any deficient half strip, any deficient bent strip or any deficient strip.
